[    {        "title": "1712.05678v2.Large_spin_relaxation_anisotropy_and_valley_Zeeman_spin_orbit_coupling_in_WSe2_Gr_hBN_heterostructures.pdf",        "content": "Large spin relaxation anisotropy and valley-Zeeman spin-orbit coupling in\nWSe2/Gr/hBN heterostructures\nSimon Zihlmann,1,∗Aron W. Cummings,2Jose H. Garcia,2M´ at´ e Kedves,3Kenji\nWatanabe,4Takashi Taniguchi,4Christian Sch¨ onenberger,1and P´ eter Makk1, 3,†\n1Department of Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland\n2Catalan Institute of Nanoscience and Nanotechnology (ICN2),\nCSIC and BIST, Campus UAB, Bellaterra, 08193 Barcelona, Spain\n3Department of Physics, Budapest University of Technology and Economics and Nanoelectronics ’Momentum’\nResearch Group of the Hungarian Academy of Sciences, Budafoki ut 8, 1111 Budapest, Hungary\n4National Institute for Material Science, 1-1 Namiki, Tsukuba, 305-0044, Japan\n(Dated: December 19, 2017)\nLarge spin-orbital proximity effects have been predicted in graphene interfaced with a transition\nmetal dichalcogenide layer. Whereas clear evidence for an enhanced spin-orbit coupling has been\nfound at large carrier densities, the type of spin-orbit coupling and its relaxation mechanism re-\nmained unknown. We show for the first time an increased spin-orbit coupling close to the charge\nneutrality point in graphene, where topological states are expected to appear. Single layer graphene\nencapsulated between the transition metal dichalcogenide WSe 2and hBN is found to exhibit ex-\nceptional quality with mobilities as high as 100 000 cm2V−1s−1. At the same time clear weak\nanti-localization indicates strong spin-orbit coupling and a large spin relaxation anisotropy due to\nthe presence of a dominating symmetric spin-orbit coupling is found. Doping dependent measure-\nments show that the spin relaxation of the in-plane spins is largely dominated by a valley-Zeeman\nspin-orbit coupling and that the intrinsic spin-orbit coupling plays a minor role in spin relaxation.\nThe strong spin-valley coupling opens new possibilities in exploring spin and valley degree of freedom\nin graphene with the realization of new concepts in spin manipulation.\nMOTIVATION/INTRODUCTION\nIn recent years, van der Waals heterostructures (vdW)\nhave gained a huge interest due to their possibility of im-\nplementing new functionalities in devices by assembling\n2D building blocks on demand [1]. It has been shown\nthat the unique band structure of graphene can be en-\ngineered and enriched with new properties by placing it\nin proximity to other materials, including the formation\nof minibands [2–5], magnetic ordering [6, 7], and super-\nconductivity [8, 9]. Special interest has been paid to the\nenhancement of spin-orbit coupling (SOC) in graphene\nsince a topological state, a quantum spin Hall phase, was\ntheoretically shown to emerge [10]. First principles cal-\nculations predicted an intrinsic SOC strength of 12 µeV\n[11], which is currently not observable even in the clean-\nest devices. Therefore, several routes were proposed and\nexplored to enhance the SOC in graphene while preserv-\ning its high electronic quality [12–14]. One of the most\npromising approaches is the combination of a transition\nmetal dichalcogenide (TMDC) layer with graphene in a\nvdW-hetereostructure. TMDCs have very large SOC on\nthe 100 meV–scale in the valence band and large SOC on\nthe order of 10 meV in the conduction band [13].\nThe realization of topological states is not the only\nmotivation to enhance the SOC in graphene. It has been\nshown that graphene is an ideal material for spin trans-\nport [13]. Spin relaxation times on the order of nanosec-\nonds [15, 16] and relaxation lengths of 24 µm [17] have\nbeen observed. However, the presence of only weak SOCin pristine graphene limits the tunability of possible spin-\ntronics devices made from graphene. The presence of\nstrong SOC would enable fast and efficient spin manip-\nulation by electric fields for possible spintronics applica-\ntions, such as spin-filters [18] or spin-orbit valves [19, 20].\nIn addition, enhanced SOC leads to large spin-Hall angles\n[21] that could be used as a source of spin currents or as\na detector of spin currents in graphene-based spintronic\ndevices.\nIt was proposed that graphene in contact to a single\nlayer of a TMDC can inherit a substantial SOC from\nthe underlying substrate [14, 22]. The experimental de-\ntection of clear weak anti-localization (WAL) [23–28] as\nwell as the observation of a beating of Shubnikov de-Haas\n(SdH) oscillations [24] leave no doubt that the SOC is\ngreatly enhanced in graphene/TMDC heterostructures.\nFirst principles calculations of graphene on WSe 2[22]\npredicted large spin-orbit coupling strength and the for-\nmation of inverted bands hosting special edge states. At\nlow energy, the band structure can be described in a\nsimple tight-binding model of graphene containing the\norbital terms and all the symmetry allowed SOC terms\nH=H0+H∆+HI+HVZ+HR[22, 29]:\nH0=/planckover2pi1vF(κkxˆσx+kyˆσy)·ˆs0\nH∆= ∆ˆσz·ˆs0\nHI=λIκˆσz·ˆsz\nHVZ=λVZκˆσ0·ˆsz\nHR=λR(κˆσx·ˆsy−ˆσy·ˆsx).(1)\nHere, ˆσiare the Pauli matrices acting on the pseudospin,arXiv:1712.05678v2  [cond-mat.mes-hall]  18 Dec 20172\nˆsiare the Pauli matrices acting on the real spin and κ\nis either±1 and denotes the valley degree of freedom.\nkxandkyrepresent the k-vector in the graphene plane,\n/planckover2pi1is the reduced Planck constant, vFis the Fermi ve-\nlocity and λi,∆ are constants. The first term H0is\nthe usual graphene Hamiltonian that describes the lin-\near band structure at low energies. H∆represents an\norbital gap that arises from a staggered sublattice poten-\ntial.HIis the intrinsic SOC term that opens a topolog-\nical gap of 2 λI[10].HVZis a valley-Zeeman SOC that\ncouples valley to spin and results from different intrinsic\nSOC on the two sublattices. This term leads to a Zee-\nman splitting of 2 λVZthat has opposite sign in the K\nand K’ valleys and leads to an out of plane spin polar-\nization with opposite polarization in each valley. HRis a\nRashba SOC arising from the structure inversion asym-\nmetry. This term leads to a spin splitting of the bands\nwith a spin expectation value that lies in the plane and is\ncoupled to the momentum via the pseudospin. At higher\nenergies k-dependent terms, called pseudospin inversion\nasymmetric (PIA) SOC come into play, which can be\nneglected at lower doping [29].\nPrevious studies have estimated the SOC strength\nfrom theoretical calculations [23] or extracted only the\nRashba SOC at intermediate [27] or at very high dop-\ning [25] or gave only a total SOC strength [26]. Fur-\nther studies have extracted a combination of Rashba and\nvalley-Zeeman SOC strength form SdH-oscillation beat-\ning measurements [24]. Additionally, a very recent study\nuses the clean limit (precession time) to estimate the SOC\nstrength from diffusive WAL measurements [28].\nHere, we give for the first time a clear and comprehen-\nsive study of SOC at the charge neutrality point (CNP)\nfor WSe 2/Gr/hBN heterostructures. The influence of\nstrong SOC is expected to have the largest impact on\nthe bandstructure close to the CNP. The strength of all\npossible SOC terms is discussed and we find that the re-\nlaxation times are dominated by the valley-Zeeman SOC.\nThe valley-Zeeman SOC leads to a much faster relaxation\nof in-plane spins than out-of plane spins. This asym-\nmetry is unique for systems with strong valley-Zeeman\nSOC and is not present in traditional 2D Rashba sys-\ntems where the anisotropy is 1/2 [18]. Our study is in\ncontrast to previous WAL measurements [25, 27], but is\nin good agreement with recent spin-valve measurements\nreporting a large spin relaxation anisotropy [30, 31].\nMETHODS\nWSe 2/Gr/hBN vdW-heterostructures were assembled\nusing a dry pick-up method [32] and Cr/Au 1D-edge con-\ntacts were fabricated [33]. Obviously a clean interface\nbetween high quality WSe 2and graphene is of utmost\nimportance. A short discussion on the influence of the\nWSe 2quality is given in the Supplemental Material. Af-ter shaping the vdW-heterostructure into a Hall-bar ge-\nometry by a reactive ion etching plasma employing SF 6\nas the main reactive gas, Ti/Au top gates were fabri-\ncated with an MgO dielectric layer to prevent it from\ncontacting the exposed graphene at the edge of the vdW-\nheterostructure. A heavily-doped silicon substrate with\n300 nm SiO 2was used as a global back gate. An optical\nimage of a typical device and a cross section is shown in\nFig. 1 (a). In total, three different samples with a total of\nfour devices were fabricated. Device A, B and C are pre-\nsented in the main text and device D is discussed in the\nSupplemental Material. Standard low frequency lock-in\ntechniques were used to measure two- and four-terminal\nconductance and resistance. Weak anti-localization was\nmeasured at temperatures of 50 mK to 1 .8 K whereas a\nclassical background was measured at sufficiently large\ntemperatures of 30 K to 50 K.\nRESULTS\nDevice Characterization\nThe two-terminal resistance measured from contact 1\nto 2 as a function of applied top and bottom gate is shown\nin Fig. 1 (b). A pronounced resistance maximum, tun-\nable by both gates, indicates the CNP of the bulk of\nthe device whereas a fainter line only changing with V BG\nindicates the CNP from the device areas close to the con-\ntacts, which are not covered by the top gate. From the\nfour-terminal conductivity, shown in Fig. 1 (c), the field\neffect mobility µ/similarequal130 000 cm2V−1s−1and the residual\ndopingn∗= 7×1010cm−2were extracted. The mobil-\nity was extracted from a linear fit of the conductivity\nas a function of density at negative V BG. At positive\nVBGthe mobility is higher as one can easily see from\nFig. 1 (c). At V BG≥25 V, the lever arm of the back\ngate is greatly reduced since the WSe 2layers gets popu-\nlated with charge carriers, i.g. the Fermi level is shifted\ninto some trap states in the WSe 2. Although the WSe 2is\npoorly conducting (low mobility) it can screen potential\nfluctuations due to disorder and this can lead to a larger\nmobility in the graphene layer, as similarly observed in\ngraphene on MoS 2[34].\nFig. 1 (d)shows the longitudinal resistance as a func-\ntion of magnetic field and gate voltage with lines orig-\ninating from the integer quantum Hall effect. At low\nfields, the normal single layer spectrum is obtained with\nplateaus at filling factors ν=±2,±6,±10,±14,...,\nwhereas at larger magnetic fields full degeneracy lift-\ning is observed with plateaus at filling factors ν=\n±2,±3,±4,±5,±6,.... The presence of symmetry bro-\nken states, that are due to electron-electron interactions\n[35], is indicative of a high device quality. In the ab-\nsence of interaction driven symmetry breaking, the spin-\nsplitting of the quantum Hall states could be used to3\ninvestigate the SOC strength [36].\nThe high quality of the devices presented here poses\nsever limitations on the investigation of the SOC strength\nusing WAL theory. Ballistic transport features (trans-\nverse magnetic focusing) are observed at densities larger\nthan 8×1011cm−2. Therefore, a true diffusive regime is\nonly obtained close to the CNP, where the charge carriers\nare quasi-diffusive [37].\n4\n2\n0\n-2\n-4VTG (V)\n20 0 -20\nVBG (V)\n1.5x1041.0 0.5R2T (Ω) (b)\n2.5x103\n2.0\n1.5\n1.0\n0.5\n0.0σ4T (e2/h)\n-4 -2 0 2 4\nVTG (V)n* = 7e10 cm-2\n130'000 cm2/(Vs)(c)\n -30 V\n -15 V\n -0 V\n +15 V\n +30 V\n-20020VBG (V)\n8 6 4 2 0\nBz (T)(d)\n43210log(Rxx(Ω))\n-10-6-22610Gxy (e2/h)\n-20 0 20\nVBG (V)3\n2\n1\n0Rxx (kΩ)(e)\nWSe2hBN\nCr/Au Cr/AuMgOTi/Au\nSiO2\nSi, p++Bz\nB||(a)\n12\n34\n56\nFIG. 1. Device layout and basic characterization of\nWSe 2/Gr/hBN vdW-heterostructures. (a) shows an\noptical image of a device A before the fabrication of the top\ngate, whose outline is indicated by the white dashed rectan-\ngle. On the right, a schematic cross section is shown and the\ndirections of the magnetic fields are indicated. The scale bar\nis 1µm. The data shown in (b)to(e)are from device B. The\ntwo terminal resistance measured from lead 1 to 2 is shown\nas a function of top and back gate voltage. A pronounced re-\nsistance maximum tunable by both gates indicates the charge\nneutrality point (CNP) of the bulk device, whereas a fainter\nline only changing with V BGindicates the CNP from the de-\nvice area close to the contacts that are not covered by the\ntop gate. Cuts in V TGat different V BGof the conductivity\nmeasured in a four-terminal configuration are shown in (c),\nwhich are also used to extract field effect mobility (linear fit\nindicated by black dashed line) and residual doping as indi-\ncated. The fan plot of longitudinal resistance R xxversus V BG\nand B zat V TG=−1.42 V is shown in (d)and a cut at B z\n= 7 T in (e). Clear plateaus are observed at filling factors\nν=±2,±3,±4,... and higher, indicating full lifting of the\nfourfold degeneracy of graphene for magnetic fields >6 T.Magneto conductance\nIn a diffusive conductor, the charge carrier trajectories\ncan form closed loops after several scattering events. The\npresence of time-reversal symmetry leads to a construc-\ntive interference of the electronic wave function along\nthese trajectories and therefore to an enhanced back scat-\ntering probability compared to the classical case. This\nphenomenon is known as weak localization (WL). Con-\nsidering the spin degree of freedom of the electrons, this\ncan change. If strong SOC is present the spin can precess\nbetween scattering events, leading to destructive interfer-\nence and hence to an enhanced forward scattering proba-\nbility compared to the classical case. This phenomenon is\nknown as weak anti-localization [38]. The quantum cor-\nrection to the magneto conductivity can therefore reveal\nthe SOC strength.\nThe two-terminal magneto conductivity ∆ σ=σ(B)−\nσ(B= 0) versus B zand n at T = 0 .25 K and zero perpen-\ndicular electric field is shown in Fig. 2 (a). A clear feature\nat B z= 0 mT is visible, as well as large modulations in B z\nand n due to universal conductance fluctuations (UCFs).\nUCFs are not averaged out since the device size is on the\norder of the dephasing length lφ. Therefore, an ensemble\naverage of the magneto conductivity over several densi-\nties is performed to reduce the amplitude of the UCFs\n[23], and curves as in Fig. 2 (b)result. A clear WAL\npeak is observed at 0 .25 K whereas at 30 K the quan-\ntum correction is fully suppressed due to a very short\nphase coherence time and only a classical background in\nmagneto conductivity remains. This high temperature\nbackground is then subtracted from the low temperature\nmeasurements to extract the real quantum correction to\nthe magneto conductivity [24]. In addition to WL/WAL\nmeasurements the phase coherence time can be extracted\nindependently from the autocorrelation function of UCF\nin magnetic field [39]. UCF as a function of B zwas mea-\nsured in a range where the WAL did not contribute to\nthe magneto conductivity (e.g. 20 mT to 70 mT) and an\naverage over several densities was performed. The in-\nflection point in the autocorrelation, determined by the\nminimum in its derivative, is a robust measure of τφ[40],\nsee Fig. 2 (d).\nFitting\nTo extract the spin-orbit scattering times we use the\ntheoretical formula derived by diagrammatic perturba-\ntion theory [41]. In the case of graphene, the quantum\ncorrection to the magneto conductivity ∆ σin the pres-4\n-10010Bz (mT)\n-2-1012\nn (1011cm-2)(a)\n-1.0-0.50.00.5Δσ (e2/h)-2.5e11 < n < +2.5e11 cm-2, E = 0 V/m\n-0.20-0.15-0.10-0.050.00Δσ (e2/h)\n-10010\nBz (mT)-0.10-0.050.00Δσ - Δσ30 K (e2/h)\n-10010\nBz (mT) 0.25 K\n 30 K 0.25 K\n 1.8 K\n 4 K\n 8 K(b)\n0.8\n0.4\n0.0f(δB) (e4/h2)\n1086420\nδBz (mT)T = 0.25 K\nτφ = 8 ps\nBip = 1 mT(c)\nFIG. 2. Magneto conductivity of device A: (a) Magneto\nconductivity versus B zand n is shown at T = 0 .25 K. A clear\nfeature is observed around B = 0 mT and large modulations\ndue do UCF are observed in B zand n. (b)shows the mag-\nneto conductivity averaged over all traces at different n. The\nWAL peak completely disappears at T = 30 K, leaving the\nclassical magneto conductivity as a background. The 30 K\ntrace is offset vertically for clarity. The quantum correction\nto the magneto conductivity is then obtained by subtracting\nthe high temperature background from the magneto conduc-\ntivity, see (b)on the right for different temperatures. With\nincreasing temperature the phase coherence time shortens and\ntherefore the WAL peak broadens and reduces in height. (c)\nshows the autocorrelation of the magneto conductivity in red\nand its derivative in blue (without scale). The minimum of\nthe derivative indicates the inflection point (B ip) of the auto-\ncorrelation, which is a measure of τφ.\nence of strong SOC is given by:\n∆σ(B) =−e2\n2πh/bracketleftBigg\nF/parenleftBigg\nτ−1\nB\nτ−1\nφ/parenrightBigg\n−F/parenleftBigg\nτ−1\nB\nτ−1\nφ+ 2τ−1asy/parenrightBigg\n−2F/parenleftBigg\nτ−1\nB\nτ−1\nφ+τ−1asy+τ−1sym/parenrightBigg/bracketrightBigg\n,(2)\nwhereF(x) = ln(x) + Ψ(1/2 + 1/x), with Ψ(x) being\nthe digamma function, τ−1\nB= 4eDB/ /planckover2pi1, whereDis the\ndiffusion constant, τφis the phase coherence time, τasy\nis the spin-orbit scattering time due to SOC terms that\nare asymmetric upon z/-z inversion ( HR) andτsymisthe spin-orbit scattering time due to SOC terms that\nare symmetric upon z/-z inversion ( HI,HVZ) [41]. The\ntotal spin-orbit scattering time is given by the sum of the\nasymmetric and symmetric rate τ−1\nSO=τ−1\nasy+τ−1\nsym. In\ngeneral, Eq. 2 is only valid if the intervalley scattering\nrateτ−1\nivis much larger than the dephasing rate τ−1\nφand\nthe rates due to spin-orbit scattering τ−1\nasy,τ−1\nsym.\nIn the limit of very weak asymmetric but strong sym-\nmetric SOC ( τasy/greatermuchτφ/greatermuchτsym), Eq. 2 describes re-\nduced WL since the first two terms cancel and there-\nfore a positive magneto conductivity results. Contrary\nto that, in the limit of very weak symmetric but strong\nasymmetric SOC ( τsym/greatermuchτφ/greatermuchτasy) a clear WAL peak\nis obtained. If both time scales are shorter than τφ, the\nratioτasy/τsymwill determine the quantum correction of\nthe magneto conductivity. In the limit of total weak SOC\n(τasy,τsym/greatermuchτφ) the normal WL in graphene is obtained\n[42], as the first two terms cancel and other terms explic-\nitly involving the inter- and intravalley scattering must\nbe considered (see Supplemental Material).\nSince the second and the third term can produce very\nsimilar dependencies on B zit can be hard to properly\ndistinguish between the influence of τasyandτsymon\n∆σ(B), as also previously reported [24, 28]. It is there-\nfore important to measure and fit the magneto conduc-\ntivity to sufficiently large fields in order to capture the\ninfluence of the second and third term, which only sig-\nnificantly contribute at larger fields (for strong SOC).\nHowever, there is an upper limit of the field scale (the\nso-called transport field Btr) at which the theory of WAL\nbreaks down. The size of the shortest closed loops that\ncan be formed in a diffusive sample is on the order of l2\nmfp,\nwherelmfpis the mean-free path of the charge carriers.\nFields that are larger than Φ 0/l2\nmfp, where Φ 0=h/eis\nthe flux quantum, are not meaningful in the framework\nof diffusive transport.\nIn the most general case there are three different\nregimes in the presence of strong SOC in graphene:\nτasy/lessmuchτsym,τasy∼τsymandτasy/greatermuchτsym. Therefore,\nwe fitted the magneto conductivity with initial fit param-\neters in these three limits. An example is shown in Fig.\n3, where the three different fits are shown as well as the\nextracted parameters. Obviously, the case τasy/greatermuchτsym\n(fit1) andτasy∼τsym(fit2) are indistinguishable and\nfit the data worse than the case τasy/greatermuchτsym(fit3). In\naddition,τφextracted from the UCF matches best for\nfit3. Therefore, we can clearly state that the symmetric\nSOC is stronger than the asymmetric SOC. The flat back-\nground as well as the narrow width of the WAL peak can\nonly be reproduced with the third case. A very similar\nbehaviour was found in device C at the CNP. In device B\n(shown in the Supplemental Material), whose mobility is\nlarger than the one from device A, we cannot clearly dis-\ntinguish the three limits as the transport field is too low\n(≈12 mT) and the flat background at larger field cannot\nbe used to disentangle the different parameters from each5\nother. However, this does not contradict τasy/greatermuchτsym\nand the overall strength of the SOC ( τSO/similarequal0.2 ps) is in\ngood agreement with device A shown here.\nObviously, the extracted time scales should be taken\nwith care as many things can introduce uncertainties in\nthe extracted time scales. First of all, we are looking\nat ensemble-averaged quantities and it is clear that this\nmight influence the precision of the extraction of the time\nscales. In addition, the subtraction of a high temperature\nbackground can lead to higher uncertainty of the quan-\ntum correction. Lastly, the high mobility of the clean\ndevices places severe limitations on the usable range of\nmagnetic field. All these influences lead us to a conser-\nvative estimation of a 50 % uncertainty for the extracted\ntime scales. Nevertheless, the order of magnitude of the\nextracted time scales and trends are still robust.\n-0.14-0.12-0.10-0.08-0.06-0.04-0.020.00Δσ0.25 K - Δσ30 K (e2/h)\n-15-10-5051015\nBz (mT) sample A\n fit1\n fit2\n fit3fit1fit2fit3\nτφ4.14.06.6\nτasy0.660.936.2\nτsym4.11.00.15\nτSO0.570.490.15\nD = 0.12 m2/s\nFIG. 3. Fitting of quantum correction to the magneto\nconductivity of device A The quantum correction to the\nmagneto conductivity is fit using Eq. 2. The results for three\ndifferent limits are shown and their parameters are indicated\n(in units of ps). τφis estimated to be 8 ps from the autocor-\nrelation of UCF in magnetic field, see Fig. 2 (d).\nThe presence of a top and a back gate allows us to tune\nthe carrier density and the transverse electric field inde-\npendently. The spin-orbit scattering rates were found to\nbe electric field independent at the CNP in the range\nof−0.05 V nm−1to 0.08 V nm−1within the precision of\nparameter extraction. Details are given in the Supple-\nmental Material. Within the investigated electric field\nrangeτasywas found to be in the range of 5 ps to 10 ps,\nalways close to τφ.τsymon the other hand was found to\nbe around 0 .1 ps to 0.3 ps whileτpwas around 0 .2 ps to\n0.3 ps, see Supplemental Material for more details. The\nlack of electric field tunability of τasyandτsymin the\ninvestigated electric field range is not so surprising. The\nRashba coupling in this system is expected to change\nconsiderably for electric fields on the order of 1 V nm−1,\nwhich are much larger than the applied fields here. How-\never, such large electric fields are hard to achieve. Inaddition,τsym, which results from λIandλVZis not ex-\npected to change much with electric field as long as the\nFermi energy is not shifted into the conduction or va-\nlence band of the WSe 2[14]. These findings contradict\nanother study [26], which claims an electric field tunabil-\nity of both SOC terms. However, there it is not discussed\nhow accurately those parameters were extracted.\nDensity dependence\nThe momentum relaxation time τpcan be tuned by\nchanging the carrier density in graphene. Fig. 4 shows\nthe dependence of τ−1\nasyandτ−1\nsymonτpin a third device\nC. The lower mobility in device C allowed for WAL mea-\nsurements at higher charge carrier densities not accessi-\nble in devices A and B. At the CNP, τ−1\nasyandτ−1\nsymare\nfound to be consistent across all three devices A, B and\nC. Here,τ−1\nsymincreases with increasing τpwhereasτ−1\nasy\nis roughly constant with increasing τp. The dependence\nof the spin-orbit scattering times on the momentum scat-\ntering time can give useful insights into the dominating\nspin relaxation mechanisms, as will be discussed later.\nIt is important to note that the extracted τasyis always\nvery close to τφ. Therefore, the extracted τasycould be\nshorter than what the actual value would be since τφacts\nas a cutoff.\n2.0x1013\n1.5\n1.0\n0.5τsym-1 (s-1)40\n30\n20\n10σ (e2/h)\n-10x1015-50510\nn (m-2)45'000 cm2/(Vs)34'000 cm2/(Vs)\n120x109\n100\n80\n60\n40τasy-1 (s-1)\n350x10-15 300 250 200\nτp (s)\nFIG. 4. Density dependence of device C: The depen-\ndence of the spin-orbit scattering rates τ−1\nsymandτ−1\nasyas a\nfunction of τpare shown for device C. The error bars on the\nspin-orbit scattering rates are given by a conservative esti-\nmate of 50 %. The two terminal conductivity is shown in the\ninset and the extracted mobilities for the n and p side are\nindicated.6\nIn-plane magnetic field dependence\nAn in-plane magnetic field (B /bardbl) is expected to lift the\ninfluence of SOC on the quantum correction to the mag-\nneto conductivity at sufficiently large fields. This means\nthat a crossover from WAL to WL for z/-z asymmetric\nand a crossover from reduced WL to full WL correction\nfor z/-z symmetric spin-orbit coupling is expected at a\nfield where the Zeeman energy is much larger than the\nSOC strength [41]. The experimental determination of\nthis crossover field allows for an estimate of the SOC\nstrength.\nThe B/bardbldependence of the quantum correction to the\nmagneto conductivity of device A at the CNP and at zero\nperpendicular electric field was investigated, as shown in\nFig. 5. The WAL peak decreases and broadens with\nincreasing B/bardbluntil it completely vanishes at B /bardbl/similarequal3 T.\nNeither a reappearance of the WAL peak, nor a transition\nto WL, is observed at higher B /bardblfields (up to 9 T). A\nqualitatively similar behaviour was observed for device\nD. Fits with equation 2 allow the extraction of τφand\nτSO, which are shown in Fig. 5 (b)for B/bardblfields lower\nthan 3 T. A clear decrease of τφis observed while τSO\nremains constant.\nThe reduction in τφwith increasing B /bardblwas previously\nattributed to enhanced dephasing due to a random vector\npotential created by a corrugated graphene layer in an in-\nplane magnetic field [43]. The clear reduction in τφwith\nconstantτSOand the absence of any appearance of WL at\nlarger B/bardblalso strongly suggests that a similar mechanism\nis at play here. Therefore, the vanishing WAL peak is\ndue to the loss of phase coherence and not due to the\nfact that the Zeeman energy ( Ez) is exceeding the SOC\nstrength. Using the range where WAL is still present, we\ncan define a lower bound of the crossover field when τφ\ndrops below 80 % of its initial value, which corresponds\nto 2 T here. This leads to a lower bound of the SOC\nstrengthλSOC≥Ez∼0.2 meV given a g-factor of 2.\nDISCUSSION\nThe effect of SOC was investigated in high quality\nvdW-heterostructures of WSe 2/Gr/hBN at the CNP, as\nthere the effects of SOC are expected to be most impor-\ntant. The two-terminal conductance measurements are\nnot influenced by contact resistances nor pn-interfaces\nclose to the CNP. At larger doping, the two-terminal con-\nductance would need to be considered with care.\nPhase coherence times around 4 ps to 7 ps were con-\nsistently found from fits to Eq. 2 and from the autocor-\nrelation of UCF. It is commonly known that the phase\ncoherence time is shorter at the CNP than at larger dop-\ning [43, 44]. Moreover, large diffusion coefficients lead to\nlong phase coherence lengths being on the order of the\ndevice size ( lφ=/radicalbig\nDτφ≈1µm), which in turn leads to\n10x10-12\n8\n6\n4\n2\n0τ (s)\n2.5 2.0 1.5 1.0 0.5 0.0\nB|| (T) τφ\n τSO(b)-0.10-0.050.000.05ΔσT=1.8 K - ΔσT=30 K (e2/h)\n-20 -10 0 10 20\nBz (mT) B|| =  0.0 T\n B|| =  0.4 T\n B|| =  1.2 T\n B|| =  2.0 T\n B|| =  2.8 T\n fits(a)\n B|| =  5 T\n B|| =  7 T\n B|| =  9 TFIG. 5. In-plane magnetic field dependence of de-\nvice A: The quantum correction to the magneto conduc-\ntivity at the CNP and at zero perpendicular electric field is\nshown for different in-plane magnetic field strengths B /bardblin\n(a). Here, n was averaged in the range of −1×1011cm−2to\n1×1011cm−2. The WAL peak gradually decreases in height\nand broadens as B /bardblis increased. The traces at B/bardbl= 5, 7, 9 T\nare offset by 0 .03 e2/h for clarity. In (b)the extracted phase\ncoherence time τφand the total spin-orbit scattering time τSO\nare plotted versus B /bardbl.τφclearly reduces, whereas τSOremains\nroughly constant over the full B /bardblrange investigated.\nlarge UCF amplitudes making the analysis harder.\nIn general Eq. 2 is only applicable for short τiv. Since\nτivis unknown in these devices, only an estimate can\nbe given here. WL measurements of graphene on hBN\nfoundτivon the order of picoseconds [45, 46]. Inter-\nvalley scattering is only possible at sharp scattering cen-\ntres as it requires a large momentum change. It is a\nreasonable assumption that the defect density in WSe 2,\nwhich is around 1 ×1012cm−2[47], is larger than in the\nhigh quality hBN [48]. This leads to shorter τivtimes in\ngraphene placed on top of WSe 2and makes Eq. 2 appli-\ncable despite the short spin-orbit scattering times found\nhere. In the case of weaker SOC, Eq. 2 cannot be used.\nInstead, a more complex analysis including τivandτ∗is\nneeded. This was used for device D, and is presented in\nthe Supplemental Material.\nSpin-orbit scattering rates were successfully extracted\nat the CNP and τasywas found to be around 4 ps to\n7 ps whereas τsymwas found to be much shorter, around\n0.1 ps to 0.3 ps. In these systems, if τivis sufficiently\nshort,τasy/2 is predicted to represent the out-of-plane\nspin relaxation time τ⊥andτsymthen represents the in-\nplane spin relaxation time τ/bardbl[18]. For the time scales7\nstated above, a spin relaxation anisotropy τ⊥/τ/bardbl∼20\nis found (see Supplemental Material for detailed calcula-\ntion). This large anisotropy in spin relaxation is unique\nfor systems with a strong valley-Zeeman SOC. Similar\nanisotropies have been found recently in spin valves in\nsimilar systems [30, 31].\nIn order to link spin-orbit scattering time scales to SOC\nstrengths, spin relaxation mechanisms have to be consid-\nered. The simple definition of /planckover2pi1/τSOas the SOC strength\nis only valid in the limit where the precession frequency\nis much larger than the momentum relaxation rate (e.g.\nfull spin precession occurs between scattering events). In\nthe following we concentrate on the parameters from de-\nvice A that were extracted close to the CNP. The de-\npendence on τpin device A can most likely be assumed\nto be very similar to that observed in device C. Within\nthe investigated density range of −2.5×1011cm−2to\n2.5×1011cm−2, including residual doping, an average\nFermi energy of 45 meV was estimated. This is based on\nthe density of states of pristine graphene, which should\nbe an adequate assumption for a Fermi energy larger than\nany SOC strengths.\nThe symmetric spin-orbit scattering time τsymcon-\ntains contributions from the intrinsic SOC and from the\nvalley-Zeeman SOC. Up to now, only the intrinsic SOC\nhas been considered in the analysis of WAL measure-\nments, and the impact of valley-Zeeman SOC has been\nignored. However, as we now explain, it is highly unlikely\nthat intrinsic SOC is responsible for the small values of\nτsym. The intrinsic SOC is expected to relax spin via the\nElliott-Yafet (EY) mechanism [49], which is given as\nτs=/parenleftbigg2EF\nλI/parenrightbigg2\nτp, (3)\nwhereτsis the spin relaxation time, EFis the Fermi\nenergy,λIis the intrinsic SOC strength and τpis the\nmomentum relaxation time [49]. Since the intrinsic\nSOC does not lead to spin-split bands and hence no\nspin-orbit fields exist that could lead to spin preces-\nsion, a relaxation via the Dyakonov-Perel mechanism\ncan be excluded. Therefore, we can estimate λI=\n2EF//radicalBig\nτsymτ−1p∼110 meV using τsym∼0.2 ps, a mean\nFermi energy of 45 meV and a momentum relaxation time\nof 0.3 ps. The extracted value for λIwould correspond\nto the opening of a topological gap of 220 meV. In the\npresence of a small residual doping (here 30 meV), such a\nlarge topological gap should easily be detectable in trans-\nport. However, none of our transport measurements con-\nfirm this. In addition, the increase of τ−1\nsymwithτp, as\nshown in Fig. 4, does not support the EY mechanism.\nOn the other hand, Cummings et al. have shown that\nthe in-plane spins are also relaxed by the valley-Zeeman\nterm via a Dyakonov-Perel mechanism where τivtakesthe role of the momentum relaxation time [18]:\nτ−1\ns=/parenleftbigg2λVZ\n/planckover2pi1/parenrightbigg2\nτiv. (4)\nWhile this equation applies in the motional narrowing\nregime of spin relaxation, our measurement appears to be\nnear the transition where that regime no longer applies.\nTaking this into consideration (see Supplemental Mate-\nrial), we estimate λVZto be in the range of 0 .23 meV to\n2.3 meV for a τsymof 0.2 ps and a τivof 0.1 ps to 1 ps.\nThis agrees well with first principles calculations [22].\nThe large range in λVZcomes from the fact that τivis\nnot exactly known.\nObviously, τsymcould still contain parts that are re-\nlated to the intrinsic SOC ( τ−1\nsym =τ−1\nsym,I +τ−1\nsym,VZ\n). As an upper bound of λI, we can give a scale of\n15 meV, which corresponds to half the energy scale due\nto the residual doping in the system. This would lead\ntoτsym,I∼10 ps. Such a slow relaxation rate ( τ−1\nsym,I )\nis completely masked by the much larger relaxation rate\nτ−1\nsym,VZ coming from the valley-Zeeman term. There-\nfore, the presence of the valley-Zeeman term makes it\nvery hard to give a reasonable estimate of the intrinsic\nSOC strength.\nThe asymmetric spin-orbit scattering time τasycon-\ntains contributions from the Rashba-SOC and from the\nPIA SOC. Since the PIA SOC scales linearly with the mo-\nmentum, it can be neglected at the CNP. Here, τasyrep-\nresents only the spin-orbit scattering time coming from\nRashba SOC. It is known that Rashba SOC can relax the\nspins via the Elliott-Yafet mechanism [49]. In addition,\nthe Rashba SOC leads to a spin splitting of the bands and\ntherefore to a spin-orbit field. This opens a second re-\nlaxation channel via the Dyakonov-Perel mechanism [50].\nIn principle the dependence on the momentum scatter-\ning timeτpallows one to distinguish between these two\nmechanisms. Here, τ−1\nasydoes not monotonically depend\nonτpas one can see in Fig. 4 and therefore we cannot\nunambiguously decide between the two mechanisms.\nAssuming that only the EY mechanism is responsible\nfor spin relaxation, then λR=EF//radicalBig\n4τasyτ−1p∼5.0 meV\ncan be estimated, using τasyof 6 ps, a mean Fermi energy\nof 45 meV and a momentum relaxation time of 0 .3 ps. On\nthe other hand, pure DP-mediated spin relaxation leads\ntoλR=/planckover2pi1//radicalbig2τasyτp∼0.35 meV. The Rashba SOC\nstrength estimated by the EY relaxation mechanism is\nlarge compared to first principles calculations [22], which\nagree much better with the SOC strength estimated by\nthe DP mechanism. This is also in agreement with pre-\nvious findings [25, 27].\nSince there is a finite valley-Zeeman SOC, which is a\nresult of different intrinsic SOC on the A sublattice and\nB sublattice, a staggered sublattice potential can also be\nexpected. The presence of a staggered potential, meaning\nthat the on-site energy of the A atom is different from8\nthe B atom on average, leads to the opening of a trivial\ngap of ∆ at the CNP. Since there is no evidence of an\norbital gap, we take the first principles calculations as an\nestimate of ∆ = 0 .54 meV.\nKnowing all relevant parameters in Eq. 1, a band\nstructure can be calculated, which is shown in Fig. 6.\nThe bands are spin split mainly due to the presence of\nstrong valley-Zeeman SOC but also due to the weaker\nRashba SOC. At very low energies, an inverted band is\nformed due to the interplay of the valley-Zeeman and\nRashba SOC, see Fig. 6 (b). This system was predicted\nto host helical edge states for zigzag graphene nanorib-\nbons, demonstrating the quantum spin Hall effect [22].\nIn the case of stronger intrinsic SOC, which we cannot\nestimate accurately, a band structure as in Fig. 6 (c)is\nexpected with a topological gap appearing at low ener-\ngies. We would like to note here, that this system might\nhost a quantum spin Hall phase. However, its detection\nis still masked by device quality as the minimal Fermi\nenergy is much larger than the topological gap, see also\nFig. 6 (a).\nOur findings are in good agreement with the calcula-\ntions by Gmitra et al. [22]. However, we have to re-\nmark that whereas the calculations were performed for\nsingle-layer TMDCs, we have used multilayer WSe 2as\na substrate. Single-layer TMDCs are direct band-gap\nsemiconductors with the band gap located at the K-point\nwhereas multilayer TMDCs have an indirect band gap.\nSince the SOC results from the mixing of the graphene\norbitals with the WSe 2orbitals, the strength of the in-\nduced SOC depends on the relative band alignment be-\ntween the graphene and WSe 2band, which will be differ-\nent for single- or multilayer TMDCs. This difference was\nrecently shown by Wakamura et al. [28]. Therefore using\nsingle-layer WSe 2to induce SOC might even enhance the\ncoupling found by our studies. Furthermore, the param-\neters taken from Ref. [22] for the orbital gap and for the\nintrinsic SOC therefore have to be taken with care.\nCONCLUSION\nIn conclusion we measured weak anti-localization in\nhigh quality WSe 2/Gr/hBN vdW-heterostructures at the\ncharge neutrality point. The presence of a clear WAL\npeak reveals a strong SOC with a much faster spin relax-\nation of in-plane spins compared to out-of-plane spins.\nWhereas previous studies have also found a clear WAL\nsignal, we present for the first time a complete interpre-\ntation of all involved SOC terms considering their relax-\nation mechanisms. This includes the finding of a very\nlarge spin relaxation anisotropy that is governed by the\npresence of a valley-Zeeman SOC that couples spin to val-\nley. The relaxation mechanism at play here is very special\nsince it relies on intervalley scattering and can only occur\nin materials where a valley degree of freedom is present\n-40-2002040E (meV)\n-100 -50 0 50 100\n|k| (106 m-1)(a)\n∆ = 0.54 meV\nλI = -0.06 meV\nλVZ = 1 meV\nλR = 0.35 meV\n-6-4-20246E (meV)\n-10 -5 0 5 10\n|k| (106 m-1)(b)\n~ 2 λVZ\n8\n4\n0\n-4\n-8E (meV)\n-10 -5 0 5 10\n|k| (106 m-1)(c)\nλI = 5 meVFIG. 6. Possible low energy band structures: (a) and\n(b)show the band structures using the Hamiltonian of Eq. 1\nwith the parameters listed in (a). The unknown parameters\n∆ andλIwere taken from Ref. [22]. In (a), the band struc-\nture is shown in the density range of −2.5×1011cm−2to\n2.5×1011cm−2(CNP), which corresponds the the one inves-\ntigated above. The energy range dominated by charge puddles\nis indicated by the grey shaded region. (b)shows a zoom in\nat low energy. In (c),λIof 5 meV is assumed to show the\nchanges due to the unknown λIat low energy.\nand coupled to spin. This is in excellent good agree-\nment with recent spin-valve measurements that found\nalso very large spin relaxation anisotropies in similar sys-\ntems [30, 31].\nIn addition, we investigated the influence of an in-plane\nmagnetic field on the WAL signature. Due to the loss of\nphase coherence, a lower bound of all SOC strengths of\n0.2 meV can be given, which is in agreement with the\nnumbers presented above. This approach does not de-\npend on accurate fitting of WAL peaks nor on the inter-\npretation of spin-orbit scattering rates.\nThe coupling of spin and valley opens new possibili-\nties in exploring spin and valley degrees of freedom in\ngraphene. In the case of bilayer graphene in proximity to\nWSe 2an enormous gate tunability of the SOC strength\nis predicted since full layer polarization can be achieved\nby an external electric field [19, 20]. This is just one of\nmany possible routes to investigate in the future.\nAcknowledgments\nThe authors gratefully acknowledge fruitful discussions\non the interpretation of the experimental data with Mar-\ntin Gmitra and Vladimir Fal’ko. Clevin Handschin is\nacknowledged for helpful discussions on the sample fab-\nrication. This work has received funding from the Euro-9\npean Unions Horizon 2020 research and innovation pro-\ngramme under grant agreement No 696656 (Graphene\nFlagship), the Swiss National Science Foundation, the\nSwiss Nanoscience Institute, the Swiss NCCR QSIT\nand ISpinText FlagERA network OTKA PD-121052 and\nOTKA FK-123894. P.M. acknowledges support as a\nBolyai Fellow. ICN2 is supported by the Severo Ochoa\nprogram from Spanish MINECO (Grant No. SEV-2013-\n0295) and funded by the CERCA Programme / Gener-\nalitat de Catalunya.\nAuthor contributions S.Z. fabricated and measured\nthe devices with the help of P.M. K.M. contributed to the\nfabrication of device C. S.Z. analysed the data with help\nfrom P.M. and inputs from C.S.. S.Z., P.M., A.W.C.,\nJ.H.G and C.S. were involved in the interpretation of\nthe results. S.Z. wrote the manuscript with inputs from\nP.M., C.S., A.W.C. and J.H.G., K.W. and T.T. provided\nthe hBN crystals used in the devices.\n∗Simon.Zihlmann@unibas.ch\n†peter.makk@mail.bme.hu\n[1] A. K. Geim and I. V. Grigorieva. Van der Waals het-\nerostructures. Nature , 499(7459):419–425, July 2013.\n[2] L. A. Ponomarenko, R. V. Gorbachev, G. L. Yu, D. C.\nElias, R. Jalil, A. A. Patel, A. Mishchenko, A. S.\nMayorov, C. R. Woods, J. R. Wallbank, M. Mucha-\nKruczynski, B. A. Piot, M. Potemski, I. V. Grigorieva,\nK. S. Novoselov, F. Guinea, V. I. Falko, and A. K. Geim.\nCloning of Dirac fermions in graphene superlattices. Na-\nture, 497:594–, May 2013.\n[3] C. R. Dean, L. Wang, P. Maher, C. Forsythe, F. Ghahari,\nY. Gao, J. Katoch, M. Ishigami, P. Moon, M. Koshino,\nT. Taniguchi, K. Watanabe, K. L. Shepard, J. Hone, and\nP. Kim. Hofstadter’s butterfly and the fractal quantum\nHall effect in moir´ e superlattices. Nature , 497:598–, May\n2013.\n[4] B. Hunt, J. D. Sanchez-Yamagishi, A. F. Young,\nM. Yankowitz, B. J. LeRoy, K. Watanabe, T. Taniguchi,\nP. Moon, M. Koshino, P. Jarillo-Herrero, and R. C.\nAshoori. Massive Dirac Fermions and Hofstadter But-\nterfly in a van der Waals Heterostructure. Science ,\n340(6139):1427–1430, 2013.\n[5] Menyoung Lee, John R. Wallbank, Patrick Gal-\nlagher, Kenji Watanabe, Takashi Taniguchi, Vladimir I.\nFal’ko, and David Goldhaber-Gordon. Ballistic mini-\nband conduction in a graphene superlattice. Science ,\n353(6307):1526–1529, 2016.\n[6] Zhiyong Wang, Chi Tang, Raymond Sachs, Yafis Bar-\nlas, and Jing Shi. Proximity-Induced Ferromagnetism in\nGraphene Revealed by the Anomalous Hall Effect. Phys.\nRev. Lett. , 114:016603, Jan 2015.\n[7] Johannes Christian Leutenantsmeyer, Alexey A\nKaverzin, Magdalena Wojtaszek, and Bart J van\nWees. Proximity induced room temperature ferromag-\nnetism in graphene probed with spin currents. 2D\nMaterials , 4(1):014001, 2017.\n[8] D. K. Efetov, L. Wang, C. Handschin, K. B. Efetov,\nJ. Shuang, R. Cava, T. Taniguchi, K. Watanabe, J. Hone,C. R. Dean, and P. Kim. Specular interband Andreev re-\nflections at van der Waals interfaces between graphene\nand NbSe2. Nature Physics , 12:328–, December 2015.\n[9] Landry Bretheau, Joel I-Jan Wang, Riccardo Pisoni,\nKenji Watanabe, Takashi Taniguchi, and Pablo Jarillo-\nHerrero. Tunnelling spectroscopy of Andreev states in\ngraphene. Nature Physics , 13:756–, May 2017.\n[10] C. L. Kane and E. J. Mele. Quantum Spin Hall Effect in\nGraphene. Phys. Rev. Lett. , 95:226801, Nov 2005.\n[11] Sergej Konschuh, Martin Gmitra, and Jaroslav Fabian.\nTight-binding theory of the spin-orbit coupling in\ngraphene. Phys. Rev. B , 82:245412, Dec 2010.\n[12] A. H. Castro Neto and F. Guinea. Impurity-Induced\nSpin-Orbit Coupling in Graphene. Phys. Rev. Lett. ,\n103:026804, Jul 2009.\n[13] Wei Han, Roland K. Kawakami, Martin Gmitra, and\nJaroslav Fabian. Graphene spintronics. Nat Nano ,\n9(10):794–807, October 2014.\n[14] Martin Gmitra and Jaroslav Fabian. Graphene on\ntransition-metal dichalcogenides: A platform for prox-\nimity spin-orbit physics and optospintronics. Phys. Rev.\nB, 92:155403, Oct 2015.\n[15] Marc Dr¨ ogeler, Frank Volmer, Maik Wolter, Bernat\nTerr´ es, Kenji Watanabe, Takashi Taniguchi, Gernot\nG¨ untherodt, Christoph Stampfer, and Bernd Beschoten.\nNanosecond Spin Lifetimes in Single- and Few-Layer\nGraphene-hBN Heterostructures at Room Temperature.\nNano Letters , 14(11):6050–6055, 2014. PMID: 25291305.\n[16] Simranjeet Singh, Jyoti Katoch, Jinsong Xu, Cheng Tan,\nTiancong Zhu, Walid Amamou, James Hone, and Roland\nKawakami. Nanosecond spin relaxation times in single\nlayer graphene spin valves with hexagonal boron nitride\ntunnel barriers. Applied Physics Letters , 109(12):122411,\n2016.\n[17] J. Ingla-Ayn´ es, Marcos H. D. Guimar˜ aes, Rick J. Mei-\njerink, Paul J. Zomer, and Bart J. van Wees. 24 µm\nlength spin relaxation length in boron nitride encapsu-\nlated bilayer graphene. arXiv:1506.00472 , 2015.\n[18] Aron W. Cummings, Jose H. Garcia, Jaroslav Fabian,\nand Stephan Roche. Giant Spin Lifetime Anisotropy in\nGraphene Induced by Proximity Effects. Phys. Rev. Lett. ,\n119:206601, Nov 2017.\n[19] Martin Gmitra and Jaroslav Fabian. Proximity Effects in\nBilayer Graphene on Monolayer wse 2: Field-Effect Spin\nValley Locking, Spin-Orbit Valve, and Spin Transistor.\nPhys. Rev. Lett. , 119:146401, Oct 2017.\n[20] Jun Yong Khoo, Alberto F. Morpurgo, and Leonid Lev-\nitov. On-Demand SpinOrbit Interaction from Which-\nLayer Tunability in Bilayer Graphene. Nano Letters ,\n17(11):7003–7008, 2017. PMID: 29058917.\n[21] Jose H. Garcia, Aron W. Cummings, and Stephan\nRoche. Spin Hall Effect and Weak Antilocalization in\nGraphene/Transition Metal Dichalcogenide Heterostruc-\ntures. Nano Letters , 0(0):null, 2017. PMID: 28715194.\n[22] Martin Gmitra, Denis Kochan, Petra H¨ ogl, and Jaroslav\nFabian. Trivial and inverted Dirac bands and the\nemergence of quantum spin Hall states in graphene\non transition-metal dichalcogenides. Phys. Rev. B ,\n93:155104, Apr 2016.\n[23] Zhe Wang, DongKeun Ki, Hua Chen, Helmuth Berger,\nAllan H. MacDonald, and Alberto F. Morpurgo. Strong\ninterface-induced spin-orbit interaction in graphene on\nWS2. Nature Communications , 6:8339–, September 2015.\n[24] Zhe Wang, Dong-Keun Ki, Jun Yong Khoo, Diego10\nMauro, Helmuth Berger, Leonid S. Levitov, and Al-\nberto F. Morpurgo. Origin and Magnitude of ‘De-\nsigner’ Spin-Orbit Interaction in Graphene on Semicon-\nducting Transition Metal Dichalcogenides. Phys. Rev. X ,\n6:041020, Oct 2016.\n[25] Bowen Yang, Min-Feng Tu, Jeongwoo Kim, Yong Wu,\nHui Wang, Jason Alicea, Ruqian Wu, Marc Bockrath,\nand Jing Shi. Tunable spin-orbit coupling and symmetry-\nprotected edge states in graphene/WS2. 2D Materials ,\n3(3):031012, 2016.\n[26] Tobias V¨ olkl, Tobias Rockinger, Martin Drienovsky,\nKenji Watanabe, Takashi Taniguchi, Dieter Weiss, and\nJonathan Eroms. Magnetotransport in heterostructures\nof transition metal dichalcogenides and graphene. Phys.\nRev. B , 96:125405, Sep 2017.\n[27] Bowen Yang, Mark Lohmann, David Barroso, Ingrid\nLiao, Zhisheng Lin, Yawen Liu, Ludwig Bartels, Kenji\nWatanabe, Takashi Taniguchi, and Jing Shi. Strong\nelectron-hole symmetric Rashba spin-orbit coupling in\ngraphene/monolayer transition metal dichalcogenide het-\nerostructures. Phys. Rev. B , 96:041409, Jul 2017.\n[28] Taro Wakamura, Francesco Reale, Pawel Palczynski, So-\nphie Gu´ eron, Cecilia Mattevi, and H´ el` ene Bouchiat.\nStrong Spin-Orbit Interaction Induced in Graphene by\nMonolayer WS 2.arXiv:1710.07483 , 2017.\n[29] Denis Kochan, Susanne Irmer, and Jaroslav Fabian.\nModel spin-orbit coupling Hamiltonians for graphene sys-\ntems. Phys. Rev. B , 95:165415, Apr 2017.\n[30] Talieh S. Ghiasi, Josep Ingla-Ayn´ es, Alexey A.\nKaverzin, and Bart J. van Wees. Large proximity-\ninduced spin lifetime anisotropy in transition-metal\ndichalcogenide/graphene heterostructures. Nano Letters ,\n17(12):7528–7532, 2017. PMID: 29172543.\n[31] L. A. Ben´ ıtez, J. F. Sierra, W. Savero Torres, A. Ar-\nrighi, F. Bonell, M. V. Costache, and S. O. Valenzuela.\nStrongly anisotropic spin relaxation in graphene/WS 2\nvan der Waals heterostructures. Nature Physics , 2017.\n[32] P. J. Zomer, M. H. D. Guimares, J. C. Brant,\nN. Tombros, and B. J. van Wees. Fast pick up technique\nfor high quality heterostructures of bilayer graphene and\nhexagonal boron nitride. Applied Physics Letters , 105(1),\n2014.\n[33] L. Wang, I. Meric, P. Y. Huang, Q. Gao, Y. Gao, H. Tran,\nT. Taniguchi, K. Watanabe, L. M. Campos, D. A. Muller,\nJ. Guo, P. Kim, J. Hone, K. L. Shepard, and C. R.\nDean. One-Dimensional Electrical Contact to a Two-\nDimensional Material. Science , 342(6158):614–617, 2013.\n[34] Lucas Basnzerus, Kenji Watanabe, Takashi Taniguchi,\nBern Beschoten, and Christoph Stampfer. Dry transfer of\nCVD graphene using MoS2-based stamps. Physics Status\nSolidi RRL , 11:1700136, 2017.\n[35] A. F. Young, C. R. Dean, L. Wang, H. Ren, P. Cadden-\nZimansky, K. Watanabe, T. Taniguchi, J. Hone, K. L.\nShepard, and P. Kim. Spin and valley quantum Hall fer-\nromagnetism in graphene. Nature Physics , 8:550–, May\n2012.\n[36] Tarik P. Cysne, Tatiana G. Rappoport, Jose H. Gar-\ncia, and Alexandre R. Rocha. Quantum Hall Effect in\nGraphene with Interface-Induced Spin-Orbit Coupling.\narXiv:1711.04811 , 2017.\n[37] S. Das Sarma, Shaffique Adam, E. H. Hwang, and Enrico\nRossi. Electronic transport in two-dimensional graphene.\nRev. Mod. Phys. , 83:407–470, May 2011.\n[38] G. Bergmann. Weak anti-localization - An experimentalproof for the destructive interference of rotated spin 1/2.\nSolid State Communications , 42(11):815 – 817, 1982.\n[39] P. A. Lee, A. Douglas Stone, and H. Fukuyama. Univer-\nsal conductance fluctuations in metals: Effects of finite\ntemperature, interactions, and magnetic field. Phys. Rev.\nB, 35:1039–1070, Jan 1987.\n[40] M. B. Lundeberg, J. Renard, and J. A. Folk. Conduc-\ntance fluctuations in quasi-two-dimensional systems: A\npractical view. Phys. Rev. B , 86:205413, Nov 2012.\n[41] Edward McCann and Vladimir I. Fal’ko. z→−zSym-\nmetry of Spin-Orbit Coupling and Weak Localization in\nGraphene. Phys. Rev. Lett. , 108:166606, Apr 2012.\n[42] E. McCann, K. Kechedzhi, Vladimir I. Fal’ko,\nH. Suzuura, T. Ando, and B. L. Altshuler. Weak-\nLocalization Magnetoresistance and Valley Symmetry in\nGraphene. Phys. Rev. Lett. , 97:146805, Oct 2006.\n[43] Mark B. Lundeberg and Joshua A. Folk. Rippled\nGraphene in an In-Plane Magnetic Field: Effects of a\nRandom Vector Potential. Phys. Rev. Lett. , 105:146804,\nSep 2010.\n[44] F. V. Tikhonenko, D. W. Horsell, R. V. Gorbachev, and\nA. K. Savchenko. Weak Localization in Graphene Flakes.\nPhys. Rev. Lett. , 100:056802, Feb 2008.\n[45] WL measuremetns in hBN/Gr/hBN devices revealed in-\ntervalley scattering times on the order of pico seconds.\n[46] Nuno J. G. Couto, Davide Costanzo, Stephan Engels,\nDong-Keun Ki, Kenji Watanabe, Takashi Taniguchi,\nChristoph Stampfer, Francisco Guinea, and Alberto F.\nMorpurgo. Random Strain Fluctuations as Domi-\nnant Disorder Source for High-Quality On-Substrate\nGraphene Devices. Phys. Rev. X , 4:041019, Oct 2014.\n[47] Rafik Addou and Robert M. Wallace. Surface Analysis of\nWSe2 Crystals: Spatial and Electronic Variability. ACS\nApplied Materials & Interfaces , 8(39):26400–26406, 2016.\nPMID: 27599557.\n[48] T. Taniguchi and K. Watanabe. Synthesis of high-purity\nboron nitride single crystals under high pressure by using\nBaBN solvent. Journal of Crystal Growth , 303(2):525 –\n529, 2007.\n[49] H. Ochoa, A. H. Castro Neto, and F. Guinea. Elliot-Yafet\nMechanism in Graphene. Phys. Rev. Lett. , 108:206808,\nMay 2012.\n[50] M. I. Dyakonov and V. I. Perel. Spin Relaxation of Con-\nduction Electrons in Noncentrosymmetric Semiconduc-\ntors. Sov. Phys. Solid State , 13(12):3023–3026, 1972.Supplemental Material: Large spin relaxation anisotropy and valley-Zeeman\nspin-orbit coupling in WSe2/Gr/hBN heterostructures\nSimon Zihlmann,1,∗Aron W. Cummings,2Jose H. Garcia,2M´ at´ e Kedves,3Kenji\nWatanabe,4Takashi Taniguchi,4Christian Sch¨ onenberger,1and P´ eter Makk1, 3,†\n1Department of Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland\n2Catalan Institute of Nanoscience and Nanotechnology (ICN2),\nCSIC and BIST, Campus UAB, Bellaterra, 08193 Barcelona, Spain\n3Department of Physics, Budapest University of Technology and Economics and Nanoelectronics ’Momentum’\nResearch Group of the Hungarian Academy of Sciences, Budafoki ut 8, 1111 Budapest, Hungary\n4National Institute for Material Science, 1-1 Namiki, Tsukuba, 305-0044, Japan\n(Dated: December 19, 2017)\nFABRICATION AND MEASUREMENT DETAILS\nGraphene (obtained from natural graphite, NGS), WSe 2(obtained from hQgraphene) and hBN (grown by Taniguchi\nand Watanabe) were exfoliated with Nitto tape onto Si wafers with 300 nm of SiO 2. The WSe 2/Gr/hBN vdW-\nheterostructures were assembled using a dry pick-up method developed by Zomer et al. [1]. After the assembly of the\nvdW-heterostructures, the stacks were annealed in H 2/N2mixture for 100 min at 200◦C to remove polymer residues\nand to make the stack more homogeneous (merging of bubbles). Higher temperatures were avoided in order not to\ndamage the WSe 2layer. The stacks were shaped into a Hall-bar mesa by standard e-beam lithography and reactive\nion etching using a SF 6, O 2and Ar-based plasma. One-dimensional side contacts were then fabricated with e-beam\nlithography and the evaporation of 10 nm Cr and 50 nm Au. Lift-off was performed in warm acetone. In order to\ninsulate the top gate from the exposed graphene at the edge of the mesa, an insulating MgO layer was evaporated\nbefore the Ti/Au of the top gate.\nStandard low frequency lock-in techniques were used to measure differential conductance and resistance in two-\nand four-terminal configuration. The samples were measured in a3He system at temperatures down to 0 .25 K and\nin a variable temperature insert at temperature of 1 .8 K and higher. The magnetic in-plane field was applied using a\nvector magnet. The small misalignment of the sample plane with the in-plane magnetic field was compensated by a\nfinite offset field in the out-of-plane direction. This offset was found to scale linearly with the applied in-plane field.\nThe back and top gate lever arms ( αBG,αTG) were found from Hall measurements and the charge carrier density\nin the graphene was calculated using a simple capacitance model,\nn=αBG/parenleftbig\nVBG−V0\nBG/parenrightbig\n+αTG/parenleftbig\nVTG−V0\nTG/parenrightbig\n, (1)\nwhereV0\nBGandV0\nBGaccount for some offset doping of the graphene. Similarly, the applied electric field (field direction\nout of plane) was obtained:\nE=1\ndBG/parenleftbig\nVBG−V0\nBG/parenrightbig\n−1\ndTG/parenleftbig\nVTG−V0\nTG/parenrightbig\n, (2)\nwheredBGanddTGdenote the thickness of the back and top gate dielectric. The thicknesses of the bottom WSe 2\nflake and the top hBN flake were determined by atomic force microscopy. To account for the residual doping, the\ndensity was corrected in the following way: ncorr=/radicalbig\nn2+n2∗. It was the corrected density ncorrthat was used for\nthe calculation of the diffusion constant via the Einstein relation and for the estimation of the Fermi energy.\nFITTING OF MAGNETO CONDUCTIVITY DATA FROM DEVICE B\nAs mentioned in the main text, a second device B was investigated as well. A gate-gate map of the resistivity of\ndevice B is shown in Fig. 1 (a). A field effect mobility of ∼25 000 cm2V−1s−1and a residual doping of ∼7×1010cm−2\nwere found. The quantum correction to the magneto conductivity was measured at the charge neutrality point for\ndifferent electric fields. The same analysis was performed as mentioned in the main text. The extracted quantum\ncorrection to the magneto conductivity was also fit using Eq. 1 from the main text considering the three different\ncases as elaborated in the main text. Since the quality of device B is higher than that of device A, the diffusion\nconstant is larger and hence the mean free path lmfpis longer. This leads to a much smaller transport field as thisarXiv:1712.05678v2  [cond-mat.mes-hall]  18 Dec 20172\nscales with l−2\nmfp. Therefore, the fitting range here was limited to 12 mT, which poses serious limits on the quality\nof the fit. It is very difficult to independently extract the different spin-orbit scattering times as obviously seen in\nFig. 1, where basically all three fits overlap. Only at larger fields would the three fits be distinguishable. However,\nthe time scales extracted here do not contradict the results presented in the main text. The strength of the total\nSOC, captured in τSO, is roughly the same for all three fits. As can be seen in Fig. 1, the total spin-orbit scattering\ntimeτSOis more robust with respect to different fitting limits. Therefore, we only consider τSOfor device B in the\nnext section.\n-0.20-0.15-0.10-0.050.00Δσ (e2/h)\n-15-10-5051015\nBz (mT) sample B\n fit1\n fit2\n fit3(b)\nfit1fit2fit3\nτφ3.93.84.2\nτasy0.340.521.2\nτsym4.60.550.18\nτSO0.320.270.16\nD= 0.29 m2/s\nBtr = 12 mT\n-4-2024VTG (V)\n-30-20-10 0102030\nVBG (V) 4e+07 \n 2e+07  0 \n -2e+07  4e+12 \n 3e+12 \n 2e+12 \n 1e+12 \n 0 \n -1e+12 \n -2e+12 \n -3e+12 \n2000 1500 1000 5000ρ (Ω)(a)\nFIG. 1. Data from device B: (a) shows the resistivity as a function of vTGandVBG. Constant density contours are indicated\nwith red solid lines and constant electric field contours is solid black lines. (b) shows the quantum correction to the magneto\nconductivity of device B at zero electric field within a density range of −5×1011cm−2to 5×1011cm−2. The same procedure\nas described in the main text was used. The results for three different limits are shown and their parameters are indicated.\nThe fitting was restricted to the range of the transport field B tr= 12 mT.\nELECTRIC FIELD DEPENDENCE OF THE SPIN-ORBIT SCATTERING RATES\nThe presence of a top and a back gate in our devices allows us to tune the carrier density and the transverse electric\nfield independently in devices A and B. In the case of device A, the SOC strength was found to be electric field\nindependent at the CNP in the range of −5×107V/m to 8 ×107V/m as shown in Fig 2. The electric field range\nwas limited by the fact that at large positive gate voltages the Fermi energy was shifted into the conduction band\nof the WSe 2whereas at large negative gate voltages gate instabilities occurred. Within the investigated electric field\nrangeτasywas found to be in the range of 5 ps to 10 ps, always close to τφ.τsymon the other hand was found to be\naround 0.1 ps to 0.3 ps whileτpwas around 0 .2 ps to 0.3 ps for device A. The total spin-orbit scattering time τSOis\nmostly given by τsym. Device B, where only τφandτSOcould be extracted reliably, shows similar results as device\nA. Therefore, we conclude that the in the electric field range −5×107V/m to 8 ×107V/m no tuning of the SOC\nstrength with electric field is observed. From first principles calculations, the Rashba SOC is expected to change\nby 10 % if the electric field is tuned by 1 ×109V m−1and also the intrinsic and valley-Zeeman SOC parameters are\nexpected to change slightly [2]. However, within the resolution of the extraction of the spin-orbit scattering time\nscales, we cannot establish a clear trend.\nThese findings are in contrast to previous studies that found an electric field tunability of τasyandτSOon a similar\nelectric field scale in graphene/WSe 2devices [3]. However, it is important to note that the changes are small and\nsince no error bars are given, it is hard to tell if the three data points show a clear trend. Another study found a\nlinear tunability of τasyof roughly 10 % on a similar electric field scale in graphene/WS 2devices [4]. There, τsymwas\nneglected with the argument that it cannot lead to spin relaxation. However, it was shown that τsymcan lead to spin\nrelaxation [5] and therefore it cannot be neglected in the analysis. In our case, it is the dominating spin relaxation\nmechanism.3\n6810-13246810-12246810-11τ (s)\n80x106 6040200-20-40-60\nE-field (V/m) τφ, B\n τSO, B\n τp, B τφ, A\n τasy, A\n τsym, A\n τSO, A\n τp, A\nFIG. 2. Electric field dependence of device A and B: The extracted spin-orbit scattering time scales τasy,τsym,τSO\nandτφwere extracted for different perpendicular electric field around the charge neutrality point. In addition, the momentum\nscattering time τpextracted from the diffusion constant is also shown. In the case of device B, only the total spin-orbit scattering\ntimeτSOis given, as a reliable extraction of τasyandτsymwas not possible in this device (see discussion above).\nSPIN RELAXATION ANISOTROPY\nCummings et al. have found a giant spin relaxation anisotropy in systems with strong valley-Zeeman SOC that is\ncommonly found in graphene/TMDC heterostructures [5]. They derived the following equation:\nτ⊥\nτ/bardbl=/parenleftbiggλVZ\nλR/parenrightbigg2τiv\nτp+ 1/2 (3)\nwhereτ⊥is the out-of-plane spin relaxation time, τ/bardblthe in-plane spin relaxation time, λVZis the SOC strength\nof the valley-Zeeman SOC, λRis the SOC strength of the Rashba SOC and τivandτprepresent the intervalley\nand momentum scattering times respectively. If a strong intervalley scattering is assumed, which is a prerequisite\nfor the application of the WAL theory [6], τ⊥is given by τasy/2 andτ/bardblis given by τsym. We therefore get a spin\nrelaxation anisotropy τ⊥/τ/bardbl≈τasy/2τsym≈20, which is much larger than what is expected for usual 2D Rashba\nsystems. Furthermore, assuming a ratio of τiv/τp≈1, which corresponds to very strong intervalley scattering, a ratio\nofλVZ/λR≈6 is expected.\nESTIMATE OF VALLEY-ZEEMAN SOC STRENGTH\nFor a valley-Zeeman SOC strength λVZ, the spin splitting is 2 λVZand the precession frequency is ω= 2λVZ//planckover2pi1.\nIn the D’yakonov-Perel’ (DP) regime of spin relaxation, when ωτiv<1, the in-plane spin relaxation rate is τ−1\ns/bardbl=\n(2λVZ//planckover2pi1)2τiv. However, if ωτiv>1, then the spin can fully precess before scattering randomizes the spin-orbit field,\nand the spin lifetime scales with the intervalley time, τs/bardbl= 2τiv. A plot of these two regimes is shown below, where we\nhave taken our derived limits of λVZ= 0.23 and 2.3 meV (see below) as well as the DFT-derived value of 1 .19 meV.\nConsidering this behavior, the condition τs/bardbl≥2τivshould always be satisfied. Meanwhile, our measurements\nrevealedτsym= 0.2 ps andτiv≈0.1−1 ps, which violates this condition for all except the smallest value of τiv.\nOne way to account for this is to consider the impact of spin-orbit disorder on the in-plane spin lifetime. Assuming\nthat theτs/bardblfrom uniform valley-Zeeman SOC is given by 2 τiv, and the rest comes from spin-orbit disorder, we can\nestimate an upper bound of λVZ=/planckover2pi1//radicalbig\n4(2τiv)τiv= 0.23 meV to 2 .3 meV.\nAnother possibility is that since our measurements are right around the transition point ωτiv= 1, we could be\nextracting the in-plane spin precession frequency; τ−1\nsym=ω. Doing so would give λVZ=/planckover2pi1/2τsym= 1.6 meV, which\nfits in the range derived above. Overall, since the experiments appear to be close to this transition point, all methods\nof deriving the strength of λVZtend to give similar values, from a few tenths up to a few meV depending on the4\nFIG. 3. Dependence of in-plane spin relaxation time τs/bardblon intervalley scattering time τiv. Red and blue lines show the\ndependence in the DP regime of spin relaxation, for the largest and smallest estimated values of λVZ. The black dashed line\nshow the value derived from DFT [2]. The green line shows the dependence in the coherent spin precession regime.\nestimate of τiv. We would like to note that it is not fully understood how the spin precession frequency enters into\nthe WAL correction and how the corresponding SOC strength would then be extracted. Therefore, further theoretical\nwork is needed.\nDATA FROM DEVICE D\nThe third sample with device D is a WSe 2/Gr/hBN stack with a very thin WSe 2(3 nm) as substrate. The gate-gate\nmap of the two terminal resistance is shown in Fig. 4 (a). Due to the very thin bottom WSe 2the mobility in this device\nis around 50 000 cm2V−1s−1and a residual doping of 2 .5×1011cm−2is found. A typical magneto conductivity trace\nof this device is shown in Fig. 4. Mostly, positive magneto conductivity is observed with only a very small feature\nthat shows negative magneto conductivity at 30 mK, which was absent at 1 .8 K. The magneto conductivity of device\nD could not be fitted with the standard WAL formula presented in the main text. However, similar curve shapes\ncould be reproduced by including the influence of τivandτ∗. A complete formula can be derived from equation 9 of\nRef. [6]. If all relaxation gaps are included and if disorder SOC is neglected one arrives at the following form:\n∆σ(B) =−e2\n2πh/bracketleftBigg\nF/parenleftBigg\nτ−1\nB\nτ−1\nφ/parenrightBigg\n−F/parenleftBigg\nτ−1\nB\nτ−1\nφ+ 2τ−1asy/parenrightBigg\n−2F/parenleftBigg\nτ−1\nB\nτ−1\nφ+τ−1asy+τ−1sym/parenrightBigg\n−F/parenleftBigg\nτ−1\nB\nτ−1\nφ+ 2τ−1\niv/parenrightBigg\n−2F/parenleftBigg\nτ−1\nB\nτ−1\nφ+τ−1\n∗/parenrightBigg\n+F/parenleftBigg\nτ−1\nB\nτ−1\nφ+ 2τ−1\niv+ 2τ−1asy/parenrightBigg\n+ 2F/parenleftBigg\nτ−1\nB\nτ−1\nφ+τ−1\n∗+ 2τ−1asy/parenrightBigg\n+2F/parenleftBigg\nτ−1\nB\nτ−1\nφ+ 2τ−1\niv+τ−1asy+τ−1sym/parenrightBigg\n+ 4F/parenleftBigg\nτ−1\nB\nτ−1\nφ+τ−1\n∗+τ−1asy+τ−1sym/parenrightBigg/bracketrightBigg\n.(4)\nHowever, the addition of two more parameters makes it very hard to unambiguously extract all parameters exactly.\nTherefore, we do not extract any spin-orbit time scales from this device. The influence of τivandτ∗are much weaker\nfor the data presented in the main text.\nThe long phase coherence time τφ∼25 ps is attributed to the lower temperature (T= 30 mK) at which the mea-\nsurement was performed. At higher temperature (1 .8 K), the phase coherence is significantly shorter ∼4 ps(broader5\n4\n2\n0\n-2\n-4VTG (V)\n-40 -20 0 20 40\nVBG (V) 3e+08  2e+08  1e+08  0  -1e+08  -2e+08  -3e+08  6e+12 \n 4e+12 \n 2e+12 \n 0 \n -2e+12 \n -4e+12 (a)\n15 10 5R2terminal (kΩ)\n0.5\n0.4\n0.3\n0.2\n0.1\n0.0Δσ (e2/h)\n-20-10 01020\nBz (mT)τφ =  2.5e-11 s\nτasy = 2.8e-11 s\nτsym = 3.0e-11 s\nτiv = 1.5e-12 s\nτ* = 8.0e-13 s\nD = 0.041 m/s2 30 mK\n standard WAL\n complete WAL\n 1.8 K(b)\nFIG. 4. Data from device D: (a) shows a gate-gate map of the two-terminal resistance of device D. Constant density (red\nsolid line, in units of cm−2) and electric field (black solid lines, in units of V m−1) lines are superimposed on top of that. (b)\nshows the quantum quantum correction of the magneto conductivity at zero electric field in the density range of −5×1011cm−2\nto 5×1011cm−2. It shows a WL dip with a tiny feature of WAL around zero Bzat a temperature of 30 mK. A possible fit\n(red) and its parameters, including the influence of τivandτ∗, are indicated. The low magnetic field range can be reasonably\nwell described by the standard WAL formula without τivandτ∗. As a comparison, the magneto conductivity is also shown at\n4 K. This trace is vertically offset by −0.06 e2/h for clarity.\ndip and reduced overall correction) and the influence of the SOC on the magneto conductivity (WAL) is not observed\nany longer.\nBothτasyandτsymseem to be very close to τφin sample D. In particular, τsymis much longer than in the devices\npresented in the main text. We conclude that even though there is some indication of SOC in sample D, its overall\nstrength must be smaller than in the devices presented in the main text. Certainly the SOC relevant for τsymmust\nbe smaller as this time scale is two orders of magnitude longer than in device A and B. This large difference cannot\nbe explained by the shorter τpthat is roughly a factor of 5 shorter in device D than in device A and B.\nINFLUENCE OF WSE 2QUALITY\nIn addition to WSe 2obtained from hQ graphene, we also investigated devices with WSe 2obtained from Nanosurf\nas an alternative source. In general, devices with WSe 2from Nanosurf showed more gate instabilities. Some devices\nshowed mobilities around 20 000 cm2V−1s−1. Magneto conductivity was measured in order to investigate possible\nenhanced SOC, but in none of the devices we did we find a pronounced WAL signature. Some devices showed\nsignatures of WL, whereas some did not show any clear magneto conductivity. For some devices it was impossible to\nmeasure magneto conductivity as the devices were not stable enough.\n∗Simon.Zihlmann@unibas.ch\n†peter.makk@mail.bme.hu\n[1] P. J. Zomer, M. H. D. Guimares, J. C. Brant, N. Tombros, and B. J. van Wees. Fast pick up technique for high quality\nheterostructures of bilayer graphene and hexagonal boron nitride. Applied Physics Letters , 105(1), 2014.\n[2] Martin Gmitra, Denis Kochan, Petra H¨ ogl, and Jaroslav Fabian. Trivial and inverted Dirac bands and the emergence of\nquantum spin Hall states in graphene on transition-metal dichalcogenides. Phys. Rev. B , 93:155104, Apr 2016.\n[3] Tobias V¨ olkl, Tobias Rockinger, Martin Drienovsky, Kenji Watanabe, Takashi Taniguchi, Dieter Weiss, and Jonathan\nEroms. Magnetotransport in heterostructures of transition metal dichalcogenides and graphene. Phys. Rev. B , 96:125405,\nSep 2017.6\n[4] Bowen Yang, Min-Feng Tu, Jeongwoo Kim, Yong Wu, Hui Wang, Jason Alicea, Ruqian Wu, Marc Bockrath, and Jing Shi.\nTunable spin-orbit coupling and symmetry-protected edge states in graphene/WS2. 2D Materials , 3(3):031012, 2016.\n[5] Aron W. Cummings, Jose H. Garcia, Jaroslav Fabian, and Stephan Roche. Giant Spin Lifetime Anisotropy in Graphene\nInduced by Proximity Effects. Phys. Rev. Lett. , 119:206601, Nov 2017.\n[6] Edward McCann and Vladimir I. Fal’ko. z→−zSymmetry of Spin-Orbit Coupling and Weak Localization in Graphene.\nPhys. Rev. Lett. , 108:166606, Apr 2012."    },    {        "title": "1403.6159v1.Optical_spin_injection_in_graphene_with_Rashba_spin_orbit_interaction.pdf",        "content": "arXiv:1403.6159v1  [cond-mat.mes-hall]  24 Mar 2014Optical spin injection in graphene with Rashba spin-orbit i nteraction\nM. Inglot,1V. K. Dugaev,1,2E. Ya. Sherman,3,4and J. Barna´ s5,6\n1Department of Physics, Rzesz´ ow University of Technology,\nal. Powsta´ nc´ ow Warszawy 6, 35-959 Rzesz´ ow, Poland\n2Departamento de F´ ısica and CFIF, Instituto Superior T´ ecn ico,\nUniversidade de Lisboa, av. Rovisco Pais, 1049-001 Lisbon, Portugal\n3Department of Physical Chemistry, Universidad del Pa´ ıs Va sco UPV-EHU, 48080 Bilbao, Spain\n4IKERBASQUE Basque Foundation for Science, Bilbao, Spain\n5Faculty of Physics, Adam Mickiewicz University, ul. Umulto wska 85, 61-614 Pozna´ n, Poland\n6Institute of Molecular Physics, Polish Academy of Sciences ,\nul. M. Smoluchowskiego 17, 60-179 Pozna´ n, Poland\n(Dated: October 8, 2018)\nWe calculate the efficiency of infrared optical spin injectio n in single-layer graphene with Rashba\nspin-orbit coupling and for in-plane magnetic field. The inj ection rate in the photon frequency range\ncorresponding to the Rashba splitting is shown to be proport ional to the ratio of the Zeeman and\nRashbasplittings. Asaresult, large spinpolarization can becontrollably achievedfor experimentally\navailable values of the spin-orbit coupling and in magnetic fields below 10 Tesla.\nPACS numbers: 72.25.Fe, 78.67.Wj, 81.05.ue, 85.75.-d\nI. INTRODUCTION\nGraphene–a two-dimensionalhexagonallattice ofcar-\nbon atoms – was discovered about eight years ago1–3\nand is now one of the most promising materials for fu-\nture nanoelectronics. The high application potential of\nthis novel material is associated with some peculiari-\nties of its electronic and phonon transport properties4\nas well as with its outstanding mechanical5,6and optical\nproperties7. Optoelectronic properties of graphene are\nalso very promising for applications8. Moreover, owing\nto a very long spin relaxation time, which is expected\ndue to a very weak spin-orbit interaction, graphene is\nalso attractive for applications in spin electronics (see,\ne.g., Ref. [9]).\nHowever, to utilize the outstanding properties of\ngraphene for spin-dependent transport, one needs to\nhave a reliable method of controllable spin injection\nand spin manipulation. The possibility of a relatively\nstrong Rashba spin-orbit coupling has been reported in\nRefs. [10,11] for graphene deposited on a Ni (or Ni/Au)\nsubstrate. Such a strong spin-orbit coupling formally en-\nables spin manipulation in graphene. However, even in\nthe absence of a substrate leading to strong spin-orbit\ncoupling, experiments report spin relaxation time on the\ntimescaleoftheorderoforlessthanonenanosecond12–14,\nwhich makes applications of graphene in spin electron-\nics rather difficult. In all the experiments aimed at the\nmeasurements of spin relaxation time, spins are injected\nen masse from a ferromagnetic contact giving rise to\nsome charge/spin density distribution, which influences\nits subsequent dynamics. Several theoretical approaches\n(see for example Refs. [15–19]) have been proposed to\ndescribe spin relaxation. However, most of them demon-\nstrated spin relaxation time much longer than that ob-\nserved experimentally.\nThere are several experimental techniques which canbe used to manipulate and control electron spin in\ngraphene. For example, spin current and spin density\nin graphene nanodisks can be manipulated by varying\nlength of the corresponding zigzag edge20. Quantum\npumping of Dirac fermions and spin current in a mono-\nlayer graphene in perpendicular magnetic field, with the\ngate voltage as a control parameter, has been proposed\nin Ref. [21]. Furthermore, the method of spin current\ngeneration in a monolayer graphene through adiabatic\nquantum pumping by two oscillating in time potentials\nhas been described in Refs. [22,23] and for the bilayer\ngraphene in Ref. [24].\nIt is well-known that spin-orbit coupling can lead to a\ndirect optical spin injection - the technique extensively\nused in the physics of semiconductors25. For graphene,\nthe spin-orbit coupling influences the optical response in\nthe infrared frequency range26. In this paper we con-\nsider the infrared optical spin injection in a single-layer\ngraphene by linearly polarized light. The graphene is as-\nsumed to be deposited on a substrate which leads to the\nRashba spin-orbit coupling. Due to this interaction, the\nelectronic spectrum of graphene near the Dirac points\nsplits into four bands with parabolic dependence on the\nelectron momentum at small wave vectors and linear dis-\npersion at large wave vectors27. Splitting of the sub-\nbands is determined by the spin-orbit coupling strength.\nWe show that optical spin injection becomes allowed in\nthe presence of an external magnetic field, and the injec-\ntion efficiency is of the order of the ratio of the Zeeman\nsplitting and the spin-orbit coupling matrix element. By\nmodifying the infrared light frequency, one can change\nthe absorption region in the momentum space, and thus\ncontrol the spin injection.\nIn Sec. 2 we derive some general formula for optical\nspin injection efficiency in graphene. Numerical results\non spin injection rate are presented and described in\nSec. 3. Summary and final conclusions are in Sec. 4.2\nII. SPIN INJECTION RATE AND EFFICIENCY\nWeassumeanexternalmagneticfield Borientedinthe\ngraphene plane. Hamiltonian describing the low energy\nelectron excitations near the Dirac point Kin graphene\nwith the Rashba spin-orbit interaction takes then the\nform28\nˆH=v(τ·k)+g(B·σ)+λ(τxσy−τyσx),(1)\nwhereg=gLµB/2,λ=α/2 withαbeing the cou-\npling constant of Rashba spin-orbit interaction,27and\ngLis the Land´ e factor. The matrices τandσare the\nPauli matrices in the sublattice and spin space, respec-\ntively. The third term of the above Hamiltonian stands\nfor the Rashba spin-orbit coupling induced by the sub-\nstrate. Note, the first term is diagonal in the spin space\nand forabbreviationthe correspondingunit matrix isnot\nwritten explicitly. Similarly, the second term is diagonal\nin the sublattice spaceand the correspondingunit matrix\nis not written explicitly, too.\nElectronic spectrum corresponding to the Hamiltonian\n(1) consists of four energy bands. In the limit of weak\nmagnetic field, gB/λ→0, this spectrum is described by\nthe formulas\nE(0)\nnk=±λ±(λ2+v2k2)1/2, (2)\nwith all possible combination of the + and −signs, and\nthe index nlabeling the bands in the order of increasing\nenergy (see Fig. 1).\nAs in the usual two-dimensional electron gas with\nRashba spin-orbit interaction, the expectation value of\nthe spin z-component in eigenstates of Hamiltonian (1)\nforB= 0 is equal to zero. However, unlike to the two-\ndimensional electron gas, the expectation value of the in-\nplane spin depends on the wave vector and is relatively\nsmall for low-energy electron states. Indeed, the expec-\ntation value, /angbracketleftΨ(0)\nnk|σ|Ψ(0)\nnk/angbracketright, of electron spin in the state\nΨ(0)\nnkforB= 0 is given by27\ns≡ /angbracketleftΨ(0)\nnk|σ|Ψ(0)\nnk/angbracketright=ξv(k׈z)√\nλ2+v2k2, (3)\nwhereξ=±1 is the band index, ξ= 1 forn= 2,3 and\nξ=−1 forn= 1,4 (cf. Ref. [27]). Thus, when vk≪λ,\nthe expectation value of spin is small, |s| ≪1. Moreover,\nthe spins are perpendicular to the wave vectors, similarly\nas in two-dimensional electron gas. We note that the\nupper index (0) at the eigenfunctions and eigenenergies\nindicates they are for B= 0.\nIn the following, we take the in-plane magnetic field B\nalongthe x-axis and assume it is ratherweak, gB/λ≪1.\nThe former assumptions justifies the absence of Landau\nquantization, while the latter condition assures that the\nband dispersion is only weakly perturbed by the static\nmagnetic field, and the resulting spin injection is linear\nin the applied magnetic field B. The four-band struc-\nture is presented in Fig. 1, which also shows the assumed\nposition of the Fermi level µ.FIG. 1: (Color online). Band structure of graphene with a\nRashba spin-orbit interaction in the gB/λ≪1 limit. Ar-\nrows show the k-dependent orientation of electron spins in\nthe eigenstates of the Hamiltonian. Here we consider only\nthe vicinity of the KDirac point, taking into account that\nthe other valley gives exactly the same result for light abso rp-\ntion and spin injection.\nTaking into account the first term of Eq. (1), Hamilto-\nnian describing interaction of graphene with an external\nperiodic electromagnetic field A(t) =A0e−iωtcan be\nwritten as\nˆHA=−ev\n/planckover2pi1c(τ·A). (4)\nThis periodic perturbation leads to electron transitions\nbetween the bands shown in Fig. 1. The spin states of\nelectrons involved in the transitions are then modified\naccordingly. Without loss of generality, we assume in the\nfollowing that the electromagnetic field is oriented along\nthey-axis,A0= (0,A0,0).\nThe total absorption rate of photons can be calculated\nas the sum of all allowed transitions,\nI(ω) =/summationdisplay\nnn′In→n′(ω), (5)\nwhereIn→n′(ω) correspondstothe absorptionassociated\nwith the transitions of electrons from the subband nto\nthe subband n′, which can be calculated from the Fermi\ngolden rule as\nIn→n′(ω) =2π\n/planckover2pi1/integraldisplayd2k\n(2π)2/vextendsingle/vextendsingle/vextendsingle/angbracketleftΨnk|ˆHA|Ψn′k/angbracketright/vextendsingle/vextendsingle/vextendsingle2\n×δ(Enk+/planckover2pi1ω−En′k)f(Enk)[1−f(En′k)].(6)\nHere, Ψ nkandEnkare the eigenfunctions and eigen-\nvalues of the total Hamiltonian (1), and f(Enk) is the\ncorresponding Fermi distribution function.\nIt is convenient to introduce an independent of the\nsystem parameter I0, defined as\nI0=ω\n4/parenleftBige\n/planckover2pi1c/parenrightBig2\nA2\n0, (7)3\nand rewrite Eq. (5) as\nI(ω) =I0/summationdisplay\nnn′/tildewideIn→n′(ω)≡I0/tildewideI(ω), (8)\nwith the system-dependent functions /tildewideIn→n′(ω). SinceA2\n0\nin Eq. (7) is related to the incident flux qofy-polarized\nphotons by the formula A2\n0= 4π/planckover2pi1cq/ω, Eq. (7) can be\npresented in the form\nI0=πe2\n/planckover2pi1cq. (9)\nThe ratio I0/q=πe2//planckover2pi1ccorresponds to the absorp-\ntion coefficient of graphene without Rashba spin-orbit\ncoupling.3,29,30In the limit of large frequency, /planckover2pi1ω≫λ,\nthe absorption rate (8) is constant and does not de-\npend on frequency, like in the case of graphene with zero\nRashba coupling, I(ω)→I0. Thus,/tildewideI(ω) can be consid-\neredasaratioofabsorptioncoefficientsforgraphenewith\nRashba spin-orbit interaction and of graphene without\nRashba interaction. In other words, /tildewideI(ω) is the absorp-\ntion coefficient normalized to that for graphene without\nRashba interaction.\nNow, let us define the spin injection rate for the i-th\ncomponent of the spin density. Following Eq. (6), we\nwrite\nJn→n′\ni(ω) =2π\n/planckover2pi1/integraldisplayd2k\n(2π)2/vextendsingle/vextendsingle/vextendsingle/angbracketleftΨnk|ˆHA|Ψn′k/angbracketright/vextendsingle/vextendsingle/vextendsingle2\n×(/angbracketleftΨn′k|σi|Ψn′k/angbracketright−/angbracketleftΨnk|σi|Ψnk/angbracketright)\n×δ(Enk+/planckover2pi1ω−En′k)f(Enk)[1−f(En′k)].(10)\nSimilarlytothecaseofabsorption,weintroducethe total\nspin injection rate as Ji(ω) =/summationtext\nn,n′Jn→n′\ni(ω) and write\nJi(ω) =I0/tildewideJi(ω) andJn→n′\ni(ω) =I0/tildewideJn→n′\ni(ω). Thus,\n/tildewideJi(ω) and/tildewideJn→n′\ni(ω) can be considered as normalized to\nI0spin injection rates. Before discussing numerical re-\nsults based on the above formula, let us discuss briefly\nphysical origin of the spin injection.\nIn the absence of magnetic field, symmetry of the ma-\ntrix elements and spin expectation values (as shown in\nFig. 1) as well as the independence of energy Enkon the\nmomentum orientationlead to zerospin injection rate, as\nrequired by the time-reversal symmetry. In an in-plane\nmagnetic field, in turn, each subband is shifted in en-\nergy,Enk−E(0)\nnk=g/parenleftBig\n/angbracketleftΨ(0)\nnk|σ|Ψ(0)\nnk/angbracketright·B/parenrightBig\n.As a result,\nthe lines of energy conservation, Enk+/planckover2pi1ω−En′k= 0,\nfor the transitions changing the electron spin, such as\n1→3 and 2 →4, are not simple circles anymore and\nacquire a distortion of the order of ( gB/λ)cosϕ, where\nϕis the angle between kand the x-axis. In addition,\nthe 4-component wave functions are modified in the first\norder perturbation as\nΨnk−Ψ(0)\nnk=g/summationdisplay\nn′/angbracketleftBig\nΨ(0)\nn′k/vextendsingle/vextendsingle/vextendsingle(σ·B)/vextendsingle/vextendsingle/vextendsingleΨ(0)\nnk/angbracketrightBig\nE(0)\nnk−E(0)\nn′kΨ(0)\nn′.(11)Accordingly, expectation values of the spin components\n/angbracketleftΨnk|σi|Ψnk/angbracketrightand of the interband matrix elements\n/angbracketleftΨnk|ˆHA|Ψn′k/angbracketrightacquirefirst-ordermodificationintheap-\nplied magnetic field, which results in a nonzero spin in-\njection.\nIII. INFRARED ABSORPTION AND SPIN\nINJECTION: NUMERICAL RESULTS\nNow we present some numerical results on the absorp-\ntion of linearly polarized light and the associated spin\ninjection. In our calculations we assumed the tempera-\ntureT= 1 K.\nFIG. 2: (Color online). (a) The normalized absorption coef-\nficients corresponding to indicated interband transitions , cal-\nculated for α= 2λ= 13 meV. (b) The total normalized ab-\nsorption coefficient for twodifferentvalues oftheRashbaspi n-\norbit parameter, as indicated. Both figures are calculated f or\nmagnetic field B= 5 T and for the chemical potential µ= 5\nmeV.\nLet us begin with the normalized absorption coeffi-\ncients presented in Fig. 2 as a function of the frequency\nω. Figure 2a shows the normalized absorption coeffi-\ncients for individual inter-band transitions, while Fig. 2b\nshows the total normalized absorption coefficient for two\ndifferent values of the Rashba parameter α. In the lat-\nter case, the thin solid line corresponds to the absorption4\ncoefficient in the absence of Rashba coupling.\nAs one can see in Fig. 2, the spin-orbit coupling\nstrongly modifies the absorption, in agreement with the\nresultsofRef. [26]. The frequencythresholdforthe inter-\nband transitions is determined by the Pauli blocking and\nalso depends on the chemical potential µand the spin-\norbit splitting 2 λ. In the case considered here, µ <2λ,\nthe transitions of highest frequency occur between the\nn= 1 and n′= 4 bands and start at the K-point with\nzero matrix element. At high frequencies, /planckover2pi1ω≫λ, the\ntotal absorption coefficient approaches that for a pure\nsingle-layer graphene without spin-orbit coupling.\nFIG. 3: (Color online). (a) The total injection rate for the\nspin component σxin a magnetic field of B= 5 T parallel to\nthex-axis and external electromagnetic field polarized along\nthey-axis. The dashed green (solid red) line corresponds to\nthe coupling constant α= 2λ= 13 meV ( α= 2λ= 5 meV).\nAll results are for the chemical potential µ= 5 meV. (b) Spin\ninjection efficiency for spin change per absorbed photon for\nthe same parameters and injection geometry.\nLet us consider now numerical results on spin injection\nshowninFig.3forthe magneticfield B= 5T,whichcor-\nresponds to the Zeeman splitting 2 gBof approximately\n0.6 meV. We show there only the injection rate for the\nspinx-component, and for clarity we simplified there the\nnotation by omitting the index i,˜Jx(ω)≡˜J(ω). Fig-ure 3a shows the total normalized spin injection rate as\na function of frequency for two different values of the\nRashba coupling parameter. The dependence of the spin\ninjectionrateonthe lightfrequencyisrathercomplicated\ndue to several interband transitions involved and com-\nplex dependence of the matrix elements on the transition\nfrequency. For the spin-conserving transitions between\nthe states characterized by the same ξin Eq. (3), such\nas 1→4 and 2→3, the main contribution to the spin\ninjection comes from changes in the expectation values\nof/angbracketleftΨnk|σ|Ψnk/angbracketright, while for the other transitions all the\nchanges in the system make comparable contributions.\nNote, the spin injected is opposite to the magnetic field.\nThe spin injection efficiency can be defined as the av-\nerage spin injected by a single photon. This efficiency\nis given by |/tildewideJω)//tildewideI(ω)|, and is shown in Fig. 3b for the\nsame Rashba parameters as in Fig. 3a. As one can see in\nFig. 3b, the efficiency can reach 0.2 per incident photon.\nIn general, the injection rate is of the order of gB/λ, and\ncan be manipulated by changing the photon frequency\nin the range of the order of λ. Since the transitions are\nrather complicated, the ratio gB/λshould be considered\nas an order-of-magnitude estimate only.\nSimilarspininjection, thoughconsiderablyweaker,can\nbe obtained for the electric field along the magnetic field,\ni.e. along the x-axis.\nIV. SUMMARY AND CONCLUSIONS\nWe have considered theoretically spin injection in\nsingle-layergrapheneinthepresenceofRashbaspin-orbit\ncoupling and in-plane external magnetic field. We have\nfound that the spin injection is efficient at frequencies of\nthe order of spin-orbit band splitting, with the efficiency\nbeing of the order of the ratio of Zeeman and Rashba\nsplittings.\nFor experimentally achievable parameters of the spin-\norbit coupling and magnetic field, the injection efficiency\ncan achieve 0.2 per absorbed photon. This result shows\nthat optical spin injection opens a way for a controllable\nandefficient method ofspindensity andspin currentgen-\neration in graphene, not requiring presence of any ferro-\nmagnetic contact.\nAcknowledgements\nThis work is partly supported by the National Science\nCenter in Poland as a research project in years 2011 –\n2014 and Grants Nos. DEC-2011/01/N/ST3/00394 and\nDEC-2012/06/M/ST3/00042. TheworkofEYSwassup-\nported by the University of Basque Country UPV/EHU\nunder program UFI 11/55, Spanish MEC (FIS2012-\n36673-C03-01), and ”Grupos Consolidados UPV/EHU\ndel Gobierno Vasco” (IT-472-10).5\n1K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M.\nI. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A.\nFirsov, Nature 438, 197 (2005).\n2Y. Zhang, Y. W. Tan, H. L. Stormer, and P. Kim, Nature\n438, 201 (2005).\n3M. I. Katsnelson, Graphene: Carbon in Two Dimensions\n(Cambridge Univ. Press, 2012).\n4D. L. Nika and A. A. Balandin, J. Phys.: Condens. Matter\n24, 233203 (2012).\n5A. R. Ranjbartoreh, B. Wang, X. Shen, and G. Wang, J.\nAppl. Phys. 109, 014306 (2011).\n6H. Chen, M. B. Møller, K. J. Gilmore, G. G. Wallace, and\nD. Li, Adv. Mater. 20, 3557 (2008).\n7L. A. Falkovsky, J. Phys.: Conf. Ser. 129, 012004 (2008).\n8F. Bonaccorso, Z. Sun, T. Hasan, andA.C. Ferrari, Nature\nPhotonics 4, 611 (2010).\n9F. S. M. Guimar˜ aes, A. T. Costa, R. B. Muniz, and M. S.\nFerreira, Phys. Rev. B 81, 233402 (2010)\n10Y. S. Dedkov, M. Fonin, U. R¨ udiger, and C. Laubschat,\nPhys. Rev. Lett. 100, 107602 (2008).\n11A. Varykhalov, J. S´ anchez-Barriga, A. M. Shikin, C.\nBiswas, E. Vescovo, A. Rybkin, D. Marchenko, and O.\nRader, Phys. Rev. Lett. 101, 157601 (2008).\n12N. Tombros, C. Jozsa, M. Popinciuc, H. T. Jonkman, and\nB. J. van Wees, Nature 448, 571 (2007).\n13Wei Han and R. K. Kawakami, Phys. Rev. Lett. 107,\n047207 (2011)\n14T. Y. Yang, J. Balakrishnan, F. Volmer, A. Avsar, M.\nJaiswal, J. Samm, S. R. Ali, A. Pachoud, M. Zeng, M.\nPopinciuc, G. G¨ untherodt, B.Beschoten, andB. ¨Ozyilmaz,\nPhys. Rev. Lett. 107, 047206 (2011).\n15V. K. Dugaev, E. Ya. Sherman, and J. Barna´ s, Phys. Rev.\nB83, 085306 (2011).16L. Wang and M. W. Wu, Phys. Rev. B 87, 205416 (2013);\nP. Zhang, Y. Zhou and M. W. Wu, J. Appl. Phys. 112,\n073709 (2012).\n17D. V. Fedorov, M. Gradhand, S. Ostanin, I. V.\nMaznichenko, A. Ernst, J. Fabian, and I. Mertig, Phys.\nRev. Lett. 110, 156602 (2013), S. Konschuh, M. Gmitra,\nD.Kochan, andJ. Fabian, Phys.Rev.B 85, 115423 (2012).\n18S. Fratini, D. Gos´ albez-Mart´ ınez, P. M. C´ amara, and J.\nFern´ andez-Rossier, Phys. Rev. B 88, 115426 (2013).\n19H. Ochoa, A. H. Castro Neto, and F. Guinea, Phys. Rev.\nLett.108, 206808 (2012).\n20M. Ezawa, Physica E 42, 703 (2010).\n21R. P. Tiwari and M. Blaauboer, Appl. Phys. Lett. 97,\n243112 (2010).\n22Q. Zhang, K. S. Chan, and Z. Lin, Appl. Phys. Lett. 98,\n032106 (2011).\n23D. Greenbaum, S. Das, G. Schwiete, and P. G. Silvestrov,\nPhys. Rev. Lett. 75, 195437 (2007).\n24J. F. Liu and K. S. Chan, Nanotechnology 22, 395201\n(2011).\n25Spin Physics in Semiconductors (Springer Series in Solid-\nState Sciences) by M.I. Dyakonov (Berlin, 2008)\n26P. Ingenhoven, J. Z. Bern´ ad, U. Z¨ ulicke, and R. Egger,\nPhys. Rev. B 81, 035421 (2010).\n27E. I. Rashba, Phys. Rev. B 79, 161409 (2009).\n28C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 226801\n(2005).\n29V. P. Gusynin and S. G. Sharapov, Phys. Rev. B 73,\n245411 (2006).\n30A. B. Kuzmenko, E. van Heumen, F. Carbone, and D. van\nder Marel, Phys. Rev. Lett. 100, 117401 (2008)."    },    {        "title": "1808.08029v1.Residual_spin_susceptibility_in_the_spin_triplet__orbital_singlet_model.pdf",        "content": "Residual spin susceptibility in the spin-triplet, orbital-singlet model\nYue Yu,1Alfred K. C. Cheung,1S. Raghu,1, 2and D. F. Agterberg3\n1Department of Physics, Stanford University, Stanford, CA 94305\n2SLAC National Accelerator Laboratory, Menlo Park, CA 94025\n3Department of Physics, University of Wisconsin, Milwaukee, WI 53201\nNuclear magnetic resonance (NMR) and Knight shift measurements are critical tools in the iden-\nti\fcation of spin-triplet superconductors. We discuss the e\u000bects of spin orbit coupling on the Knight\nshift and susceptibilities for a variety of spin triplet multi-orbital gap functions with orbital-singlet\ncharacter and compare their responses to \"traditional\" single band spin-triplet ( px+ipy) supercon-\nductors. We observe a non-negligible residual spin-susceptibility at low temperature.\nI. INTRODUCTION\nA working de\fnition for an unconventional supercon-\nductor is one whose gap function averaged over the Fermi\nsurface is less than the maximum value of the absolute\nvalue of the gap at any point on the Fermi surface. This\nallows for gaps which do not exhibit isotropic s-wave\npairing. In particular, the possibility for pairing in odd\nparity channels allows for spin-triplet pairing , the most\nnotable purported example being Sr 2RuO 4in which the\nsimplest descriptions involve models with only intra-band\npairing. For Sr 2RuO 4, there is still no consensus on the\nactual form of the superconducting gap with various ex-\nperiments showing con\ricting results1{5. In light of this,\nit is useful to consider what other systems might be can-\ndidates for realizing spin-triplet pairing. One recent work\nproposes that a di\u000berent kind of spin-triplet pairing may\nbe realized in iron-based superconductors that only pos-\nsess hole pockets6. Based on angle-resolved photoemis-\nsion spectroscopy and heat capacity measurements, the\nauthors argue in favor of s-wave gaps in these materials\nand present a new mechanism for its realization.\nThe key ingredients to their proposal are (1) the pres-\nence of spin orbit coupling (SOC), and (2) the multi-\norbital nature of these systems which introduces the pos-\nsibility for inter-band paired gap functions7,8. This re-\nsults in the stabilization of an even parity orbital-singlet,\nspin-triplet pairing state. Indeed, the s-wave iron-based\nsuperconductors are expected to exhibit sizable SOC9\nand seem to have the ingredients necessary for the model\nproposed6,10{12. The same pairing state has also been the\nfocus of study using dynamical mean \feld theory8,13{15.\nThese studies all reveal a pairing instability within the\nstrong coupling limit.\nIn order to evaluate the validity of the proposed model\nfor the relevant iron-based supercondcutors, it is crucial\nto identify the experimental signatures in nuclear mag-\nnetic resonance (NMR) and Knight shift measurements {\nkey experimental testing grounds for unconventional su-\nperconductors2,16{19. To this end, in this article we com-\npare the results in Knight shift between the well-studied\nsingle band spin-triplet state ~d(~k) = (0;0;kx+iky) and\nthe inter-band model of Ref. 6. In the absence of SOC,\nwe \fnd an invariant Knight shift which stays constant\ninto the superconducting phase in every direction for theinter-band model. This distinguishes it from the intra-\nband (kx+iky)^zstate, where there is a drop in the\nKnight shift for a \feld applied along the zdirection10,20.\nWhen including SOC, we observe a substantial decrease\nin spin susceptibilities, which agrees with previous the-\noretical predictions6. However, we observe a non-zero\nresidual spin susceptibility at low temperature, in con-\ntrast with zero residual spin susceptibility for intra-band\nspin-singlet pairing. Our results on spin susceptibility\nand Knight shift reveal that the pairing state driven\nby SOC has both intra-band spin-singlet and inter-band\nspin-triplet properties.\nThis article is organized as follows. In Section II, we\nintroduce our mean \feld theory model involving pairing\nin the orbital singlet, spin triplet channel. In Section III,\nwe explain how observables in NMR and Knight shift\nexperiments can be calculated within our model. Our\nresults are discussed in Section IV. Concluding remarks\nappear in Section V.\nII. THE MODEL\nWe consider a three band model for d-orbital electrons\nwith tetragonal symmetry. The Hamiltonian is given by\nH=H0+HSOC+HBCS, where:\nH0=X\n~k;a;b;\u001bhab\n0(~k)cy\n~ka\u001bc~kb\u001b; (1)\nand:\nhab\n0(~k) =2\n4\u000fyz(~k)V(~k) 0\nV(~k)\u000fxz(~k) 0\n0 0 \u000fxy(~k)3\n5; (2)\nis the kinetic energy part of the tight binding model with\nhybridization between xzandyzorbitals.cy\n~ka\u001b\u0000\nc~ka\u001b\u0001\nis\nthe creation(annihilation) operator of electrons in orbital\na=yz,xz, oryzand spin\u001b=\";#. We consider a quasi-\ntwo dimensional material, where the dispersion in the\nz-direction is neglected. Here, the form and value of the\nunhybridized dispersions and the hybridization potentialarXiv:1808.08029v1  [cond-mat.supr-con]  24 Aug 20182\nare given in Ref. 21:\n\u000fyz(~k) =\u0000\u000f0\u00002tcosky\u00002t?coskx\n\u000fxz(~k) =\u0000\u000f0\u00002tcoskx\u00002t?cosky\n\u000fxy(~k) =\u0000\u000f\u00002t0(coskx+ cosky) + 4t00coskxcosky\nV(~k) =\u00002Vsinkxsinky;\n(3)\nwhich was originally proposed for Sr 2RuO 4. However,\nwe use this model only as a speci\fc example; our analy-sis is not limited to this particular material. We choose\nour unit of energy to be tin the following analysis. With-\nout spin orbit coupling (SOC) and superconductivity, the\nband structure from the diagonalization of H0is given\nin Ref. 21. The minimal band gap at the Fermi sur-\nface \u0001 band\u00190:05tis between the dxyband and one of\nthe hybridized bands. We add the spin orbit coupling\nHSOC =\u0015~L\u0001~S. The form of the BCS interaction we use\nis:\nHBCS=\u0000X\n~k;~k0;a;b;f\u001bigVab\u001b1\u001b2\u001b3\u001b4(~k;~k0)c\u0000~ka\u001b 1c~kb\u001b2cy\n~k0b\u001b3cy\n\u0000~k0a\u001b4;(4)\nUsing a mean \feld decomposition, we can calculate the\ngap function:\n\u0001ab\u001b3\u001b4(~k0) =X\n~k\u001b1\u001b2Vab\u001b1\u001b2\u001b3\u001b4(~k;~k0)hc\u0000~ka\u001b 1c~kb\u001b2i:(5)\nThus we obtain the mean \feld Hamiltonian, which is now\nrewritten into Bogoliubov-de-Gennes (BdG) form:\n\ty\n~k\u0011[cy\n~kX\";cy\n~kX#;cy\n~kY\";cy\n~kY#;cy\n~kZ\";cy\n~kZ#]\nHMF=X\n~kjkx>0[\ty\n~k;\tT\n~\u0000k]hBdG(~k)\u0014\t~k\n\t\u0003\n~\u0000k\u0015\nhBdG(~k) =\u0012^h(~k) ^\u0001(~k)\n^\u0001(~k)y\u0000^hT(~\u0000k)\u0013\n;\n^h(~k) =hab\n0(~k)\n\u001b0\n+\u0015(Lx\n\u001bx+Ly\n\u001by+Lz\n\u001bz); (6)\nas a 12 by 12 matrix. Here \u001bx;y;z; 0are the Pauli matrices\nand identity matrix in spin space. For simplicity, the\norbital angular momentum operators ( La)bc=\u0000i\u000fabcare\nassumed to be the same as for electrons with L= 1.\nFor the case of intra-band pairing, the superconduct-\ning order parameter can be rewritten as \u0001( ~k) = (I3)\ni\u001by~d(~k)\u0001~ \u001b, where~ \u001b= (\u001bx;\u001by;\u001bz) andI3is the 3 by\n3 identity matrix in orbital space20. Additionally, the\nmulti-band nature of the model introduces the possibil-\nity for inter-band coupled gap functions. We consider\na BCS gap \u0001 BCS = 0:01t, which is of the order of the\nband gap near EF. Thus, the inter-band pairing cannot\nbe fully neglected. In this calculation, we consider a local\nmodel (~k-independent model) of orbital-singlet and spin-\ntriplet pairing, which has the following gap function, andthe corresponding Vab\u001b1\u001b2\u001b3\u001b4(~k;~k0):\nVab\u001b1\u001b2\u001b3\u001b4(~k;~k0) =gf(~k)y\nba\u001b2\u001b1f(~k0)ab\u001b3\u001b4\n^\u0001ab\u001b1\u001b2(~k) = \u0001fab\u001b1\u001b2(~k)\nf(~k)\u0011Lz\n(i\u001bz\u001by); (7)\ncouplingyzandxzorbitals. Since the induced intraband\nspin-singlet state does not depend on a speci\fc direc-\ntion, the results on susceptibilities do not show qualita-\ntive di\u000berences in di\u000berent directions, which will be con-\n\frmed in the next section. Thus, other pairings including\n(Lx\n\u001bx+Ly\n\u001by)i\u001bywill give similar results. The gap\nfunction is even parity, with time reversal symmetry and\ninversion symmetry. Note that other pairing states (e.g.\nintra-band pairings or other forms of inter-band pairing)\nare not considered here. The two electrons comprising\nthe Cooper pair form an orbital singlet using the xzand\nyzorbitals. In spin space, they form a triplet pair with\ntotal spin rotating within the x\u0000yplane. The overall\nCooper pair is odd under particle exchange. The ~d-vector\nof this triplet-spin pairing contains only a z-component.\nIf we turn on the SOC, this inter-band pair could de-\nvelop an intra-band spin-singlet component6. The \fnite\nintra-band pairing as induced by SOC helps increase the\nsuperconducting transition temperature, as we will ex-\nplain in Sec. IV.\nIII. NMR AND KNIGHT SHIFT\nWe now consider observables in NMR experiments and\nthe Knight shift, which are key experimental techniques\nin identifying spin triplet superconductors17,19. Under-\nstanding the Knight shift experiment is vital in distin-\nguishing between di\u000berent types of gap functions20,22,23.\nWe will \frst summarize the theoretical background of3\nthis experiment. Then we will show results for the inter-\nband paired state under di\u000berent SOC strengths.\nIn atomic physics, the \feld induced non-zero spin andorbit angular momentum of the electrons generate a hy-\nper\fne \feld experienced by the nuclear spin24:\n~Bhf=\u00002\u00160\u0016Bhr\u00003i(~L+\u0018L(L+ 1)~S\u00003\n2\u0018[~L(~L\u0001~S) + (~L\u0001~S)~L]); (8)\nwhich leads to the Knight shift in the NMR response.\nThe hyper\fne \feld can be decomposed into orbital and\nspin contributions. The orbital angular momentum gen-\nerates a current and hence, a magnetic \feld, contribut-\ning to the \frst term of the hyper\fne \feld in Eq. 8. The\ndipole-dipole interaction between electron spin and nu-\nclear spin leads to the remaining two terms in the hyper-\n\fne \feld, under the approximation known as the Equiva-\nlent Operator Method25. Given the atomic wavefunction\nof an electron with angular momentum l, the strength of\nthe dipole-dipole interaction is \u0018= 2=[(2l\u00001)(2l+ 3)]25.\nIn the following calculations, we choose `= 2 which gives\n\u0018= 2=21. The Fermi contact interaction will also con-\ntribute to the hyper\fne \feld, which is neglected here for\nL6= 0 systems.\nKnight shift tensor !Kis then determined by the hy-\nper\fne \feld through\n~Bhf= !K\u0001~B: (9)\nHere~Bis the external magnetic \feld. The spin and or-\nbital contribution towards the diagonal elements of the\nKnight shift is directly related to the spin and orbital\nsusceptibilities. Under spin orbit coupling, ~Land~Sare\nno longer good quantum numbers, and we will take the\nexpectation value of the hyper\fne \feld operator in the\nsimulation. The orbital contribution, which is propor-\ntional to the orbital magnetic susceptibility, does not\nchange dramatically upon entering the superconducting\nphase. This is because in the Kubo formula for orbital\nsusceptibility, orbital angular momentum couples states\nwith di\u000berent energy, so the energy shift by superconduc-\ntivity will not have a strong e\u000bect. The orbital contri-\nbution can be extracted from measurements within the\nnormal state and then substracted in the superconduct-\ning Knight shift17,19. The spin contribution has a similar\nbehavior as spin susceptibility and acts as a key feature\nfor distinguishing between spin-singlet and spin-triplet\ngap functions. In the following simulations, we numer-\nically diagonalize the BdG Hamiltonian and obtain the\nsusceptibilities and the diagonal elements of the Knight\nshift tensor.IV. RESULTS AND DISCUSSION\nThe gap equation can now be solved numerically, and\nthe parameter \u0001 in Eq. 7 is obtained self-consistently for\na givengand temperature T. After obtaining \u0001, the\nsusceptibilities and Knight shift are calculated from the\nKubo formula. We now compare the Knight shift results\nfor the single band ~d(~k) = (0;0;kx+iky) state and the\ninter-band orbital-singlet, spin-triplet state. In order to\nshed light on the role of SOC in the formation of intra-\nband pairing, we consider three SOC regimes: \u0015= 0\n(i.e. zero SOC) where only inter-band pairing is present,\n\u0015\u0018\u0001BCS when the SOC strength is comparable to the\nsize of the superconducting gap, and \u0015\u001d\u0001BCS. The\nKnight shift and susceptibilities are all normalized to a\ndimensionless number, for which the Knight shift and\nsusceptibilities in the normal state is unity.\nIf the spin orbit coupling is absent, the critical temper-\nature for superconductivity is found to be exponentially\nsmall. We present here an unrealistic scenario with a\nnon-zero order parameter \u0001 BCS under zero/small SOC,\nas shown in Fig. 2 and Fig. 3. This setup can be achieved\nby choosing a relatively large coupling coe\u000ecient gin\nEq. 7, which raises the critical temperature to numer-\nically accessible values. This small SOC regime serves\nonly to illustrate the key results, namely the residual\nspin-susceptibility. Furthermore, to better illustrate the\ne\u000bect of SOC, we choose di\u000berent gparameters for the\ntwo curves \u0015= 0 and\u0015= \u0001 0, such that the supercon-\nductivity gap at the lowest temperature reached is \fxed\nto be \u0001 0= 0:01t.\nFor purely inter-band pairing without SOC, we numer-\nically obtain the susceptibility and Knight shift. Fig. 2\nshows the responses in the presence of an out-of-plane\nmagnetic \feld while Fig. 3 is in the presence of an in-\nplane magnetic \feld. In striking contrast to the intra-\nbandkx+ikymodel2,10,20, the spin and orbital suscepti-\nbilities and the Knight shift show no decrease in the su-\nperconducting phase in any direction. We further check\nthe density of states. By rede\fning the Fermi surface\nto be the states with energy near the chemical potential\njE\u0000EFj<\u0001BCS, we \fnd that there is no dramatic\nchange in the density of states on the Fermi surface.\nHere we provide a qualitative explanation for the above\n\fndings. For intra-band pairing, the Cooper pairs are\ncomposed of electrons with the same energies, and the4\nΔ\"#$%\n~Δ\"#$%Δ\"&'Δ\"&'\nFIG. 1: Schematic representation of the change of band struc-\nture for a two band system with inter-band coupling for (left)\nno superconductivity and (right) with superconductivity. The\ndistance between the BCS gaps and the Fermi surface is of\norder \u0001 Band. If the superconducting order parameter \u0001 BCS\nis much smaller than the energy di\u000berence of the two bands\n\u0001Band, then the states near the Fermi surface are approxi-\nmately unchanged. Thus the density of states at the Fermi\nsurface is una\u000bected by superconductivity.\n0.4 0.5 0.6 0.7\nT/000.51lz\n=00\n=10\n0.4 0.5 0.6 0.7\nT/000.51sz\n=00\n=10\n0.4 0.5 0.6 0.7\nT/000.51KSz\n=00\n=10\n0.4 0.5 0.6 0.7\nT/000.51/0=00\n=10\nFIG. 2: Orbital susceptibility, spin susceptibility and gap\nfunction for triplet pairing \u0001( ~k) =Lz\ni\u001by\u001bzwhen apply-\ning an out-of-plane magnetic \feld for \u0015= 0 and\u0015= \u0001 BCS.\nWithin the range of the gap function, no drop in susceptibili-\nties and Knight shift is observed. Note that the zero temper-\nature gap function should be larger than 0 :01t, but we only\nfocus on the range where \u0001 BCS\u001c\u0001band.\ncontribution to the superconductivity gap is mainly from\nelectrons near the Fermi surface ( jE\u0000EFj\u0019\u0001BCS).\nWhen BCS states are formed, a superconducting gap\nthen opens at the Fermi surface. The density of states\nat the Fermi surface vanishes, and there is a substantial\ndrop in spin susceptibility. In contrast, the inter-band\npairing involves electrons at di\u000berent bands. In a rough\napproximation, let us assume the pairing only happens\nat the band crossing (Fig. 1). When Cooper pairs are\nformed, a superconducting gap will open above and below\nthe Fermi surface. The distance from the Fermi surface\nis of order \u0001 band. If \u0001band\u001d\u0001BCS, the Fermi surface\nis then approximately unchanged. Therefore, the inter-\nband superconductivity does not exhibit any decrease in\n0.4 0.5 0.6 0.7\nT/000.51lx\n=00\n=10\n0.4 0.5 0.6 0.7\nT/000.51sx\n=00\n=10\n0.4 0.5 0.6 0.7\nT/000.51KSx\n=00\n=10\n0.4 0.5 0.6 0.7\nT/000.51/0=00\n=10FIG. 3: Orbital susceptibility, spin susceptibility, and gap\nfunction for triplet pairing \u0001( ~k) =Lz\ni\u001by\u001bzwhen apply-\ning a magnetic \feld in the in-plane x-direction. The same\nparameters are applied as in Fig. 2.\nKnight shift, even though ~dis inz-direction.\nAs we turn on SOC while still keeping it weak ( \u0015\u0018\n\u0001BCS<\u0001Band), the SOC is not yet su\u000ecient to gener-\nate a considerable intra-band spin-singlet pairing. As a\nresult, the decrease spin-susceptibility and Knight shift\nremains small (red curves in Fig. 2and Fig. 3). However,\nwe observe a higher critical temperature, which is due to\na reduction of the minimal band gap by SOC.\n0.1 0.2 0.3 0.4\nT/000.51/0=100\n=200\n=300\n=400\n=500\n0.1 0.2 0.3 0.4\nT/000.51KSz=100\n=200\n=300\n=400\n=500\n0.1 0.2 0.3 0.4\nT/000.51lz=100\n=200\n=300\n=400\n=500\n0.1 0.2 0.3 0.4\nT/000.51sz=100\n=200\n=300\n=400\n=500\nFIG. 4: Orbital susceptibility, spin susceptibility and gap\nfunction for triplet pairing \u0001( ~k) =Lz\ni\u001by\u001bzwhen adding\nz-direction magnetic \feld in the large SOC regime.\nWe now consider the regime with large SOC. The in-\nduced spin-singlet pairing state greatly reduces the spin-\nsusceptibility and the Knight shift in every direction, as\nshown in Fig. 4. The induced intra-band pairing provides\nsu\u000ecient superconducting instability, leading to signif-\nicant enhancement of the critical temperature, as pre-\ndicted in the theoretical work6.\nIn Fig. 5 and Fig. 6, we focus on the residual suscepti-5\n0 20 40\n/000.51lz/lznlz\n0 20 40\n/000.51sz/sznsz\n0 20 40\n/000.51KSz/KSznKSz\nFIG. 5: Residual Knight shift and susceptibilities for an ap-\nplied magnetic \feld in the z-direction, as a function of SOC.\nThe strength of SOC varies from 0 to 50\u0001 BCS, i.e. in a regime\nof strong SOC.\n0 20 40\n/000.51lx/lxnlx\n0 20 40\n/000.51sx/sxnsx\n0 20 40\n/000.51KSx/KSxnKSx\nFIG. 6: Residual Knight shift and susceptibilities under a\nmagnetic \feld applied in the xdirection, as a function of SOC\nstrength. The strength of SOC varies from 0 to 50\u0001 BCS.\nbilities and Knight shift for di\u000berent SOC. We compare\nthe results at two temperature. At the lower tempera-\ntureT= 0:01\u0001BCS, we add a BCS gap \u0001 BCS = 0:01.\nAt the \\higher\" temperature T= \u0001BCS, we consider\nthe normal state (with zero BCS gap). We take the ratio\nof the susceptibility in the \frst state to its value in the\nsecond state as a measure of the residual susceptibility.\nThe aim is to observe the residual susceptibility at low\ntemperature as we tune both the BCS term and SOC\nstrength.\nThe residual spin susceptibility \frst exhibits a con-tinuous drop as SOC is increased from zero, due to the\nformation of intra-band pairing. It reaches a minimum\nat around\u0015= 10\u0001BCS. More importantly, the resid-\nual spin susceptibility never goes to zero, even when the\norbital susceptibility is approximately unchanged. This\nfeature can be explained by returning to the pedagogical\nmodel with \u0015= 0 (see Fig. 2 and Fig. 3 and accompa-\nnying discussion). This is a crucial di\u000berence compared\nwith the well-studied kx+ikymodel, in which the non-\nzero spin susceptibility under SOC is accompanied by a\ndecreasing orbital susceptibility10,20.\nThe non-zero spin susceptibility persists as the SOC is\ntuned to very large values. This may be due to mixing\nbetween an originally vanishing spin susceptibility and\na non-zero orbital susceptibility, similar to the kx+iky\nmodel (in which the z-direction susceptibility exhibits\nsimilar features) or in a simple single-band spin-singlet\nmodel22,23. The system becomes more complex due to\nband crossings. Therefore, for future work, the analysis\nfor very large SOC could be performed for a model with\na large band gap.\nThere is no qualitative di\u000berence between the drops in\ndi\u000berent directions, which con\frms the predicted contri-\nbution from spin-singlet pairing when SOC is present6.\nThus, the Knight shift in di\u000berent directions provides\na tool in identifying the inter-band state and the SOC\nstrength.\nV. CONCLUSION\nIn summary, we have calculated the Knight shift for\nthe even-parity orbital-singlet spin-triplet superconduc-\ntors in a quasi-two dimensional tight binding model, by\nnumerically solving the gap equation. In contrast to the\nkx+ikyunconventional superconductor, the inter-band\nmodel exhibits no decrease in Knight shift in any di-\nrection. After introducing the required large spin orbit\ncoupling, the predicted intra-band spin singlet state is\nobserved, and there is a drop in spin susceptibility and\nKnight shift. However, the residual spin susceptibility is\nnon-zero even under large SOC in contrast to spin-singlet\nintra-band pairing state.\nAcknowledgments\nY.Y., A.C., and S.R. were supported by the DOE\nO\u000ece of Basic Energy Sciences, contract DEAC02-\n76SF00515. D. F. A. was supported by the Gordon\nand Betty Moore Foundation's EPiQS Initiative through\nGrant No. GBMF4302.\n1T. Sca\u000edi, Weak-Coupling Theory of Topological Super-\nconductivity: The Case of Strontium Ruthenate (2017),ISBN 978-3-319-62866-0.\n2K. Ishida, H. Mukuda, Y. Kitaoka, K. Asayama, Z. Mao,6\nY. Mori, and Y. Maeno, Nature 396, 658 (1998).\n3J. Xia, Y. Maeno, P. T. Beyersdorf, M. M. Fejer, and\nA. Kapitulnik, Phys. Rev. Lett. 97, 167002 (2006), URL\nhttps://link.aps.org/doi/10.1103/PhysRevLett.97.\n167002 .\n4C. W. Hicks, J. R. Kirtley, T. M. Lippman, N. C. Koshnick,\nM. E. Huber, Y. Maeno, W. M. Yuhasz, M. B. Maple, and\nK. A. Moler, Phys. Rev. B 81, 214501 (2010), URL https:\n//link.aps.org/doi/10.1103/PhysRevB.81.214501 .\n5A. Steppke, L. Zhao, M. E. Barber, T. Sca\u000edi, F. Jerzem-\nbeck, H. Rosner, A. S. Gibbs, Y. Maeno, S. H. Simon, A. P.\nMackenzie, et al., Science 355, eaaf9398 (2017).\n6O. Vafek and A. V. Chubukov, Phys. Rev. Lett.\n118, 087003 (2017), URL https://link.aps.org/doi/\n10.1103/PhysRevLett.118.087003 .\n7A. Ramires and M. Sigrist, Phys. Rev. B 94,\n104501 (2016), URL https://link.aps.org/doi/10.\n1103/PhysRevB.94.104501 .\n8S. Hoshino and P. Werner, Phys. Rev. Lett. 115,\n247001 (2015), URL https://link.aps.org/doi/10.\n1103/PhysRevLett.115.247001 .\n9S. V. Borisenko, D. Evtushinsky, Z.-H. Liu, I. Morozov,\nR. Kappenberger, S. Wurmehl, B. B uchner, A. Yaresko,\nT. Kim, M. Hoesch, et al., Nature Physics 12, 311 (2016).\n10D. Nisson and N. Curro, New Journal of Physics 18,\n073041 (2016).\n11M. W. Haverkort, I. S. El\fmov, L. H. Tjeng, G. A.\nSawatzky, and A. Damascelli, Phys. Rev. Lett. 101,\n026406 (2008), URL https://link.aps.org/doi/10.\n1103/PhysRevLett.101.026406 .\n12A. Ramires, D. F. Agterberg, and M. Sigrist, arXivpreprint arXiv:1802.00361 (2018).\n13A. Klejnberg and J. Spalek, Journal of Physics: Condensed\nMatter 11, 6553 (1999).\n14J. Spa lek, Phys. Rev. B 63, 104513 (2001), URL https:\n//link.aps.org/doi/10.1103/PhysRevB.63.104513 .\n15J. E. Han, Phys. Rev. B 70, 054513 (2004), URL https:\n//link.aps.org/doi/10.1103/PhysRevB.70.054513 .\n16A. P. Mackenzie and Y. Maeno, Reviews of Modern Physics\n75, 657 (2003).\n17A. Abragam, The principles of nuclear magnetism , 32 (Ox-\nford university press, 1961).\n18N. Curro, Reports on Progress in Physics 72, 026502\n(2009).\n19A. Rigamonti, F. Borsa, and P. Carretta, Reports on\nProgress in Physics 61, 1367 (1998).\n20P. A. Frigeri, D. F. Agterberg, A. Koga, and M. Sigrist,\nPhys. Rev. Lett. 92, 097001 (2004), URL https://link.\naps.org/doi/10.1103/PhysRevLett.92.097001 .\n21A. Akbari and P. Thalmeier, Phys. Rev. B 88,\n134519 (2013), URL https://link.aps.org/doi/10.\n1103/PhysRevB.88.134519 .\n22A. Abrikosov and L. Gorkov, Sov. Phys. JETP 15, 752\n(1962).\n23P. W. Anderson, Phys. Rev. Lett. 3, 325 (1959), URL\nhttps://link.aps.org/doi/10.1103/PhysRevLett.3.\n325.\n24A. Abragam and B. Bleaney, Electron paramagnetic reso-\nnance of transition ions (OUP Oxford, 2012).\n25J. S. Gri\u000eth, The theory of transition-metal ions (Cam-\nbridge University Press, 1971)."    },    {        "title": "1205.6950v1.Effect_of_spin_orbit_coupling_on_magnetic_and_orbital_order_in_MgV__2_O__4_.pdf",        "content": "arXiv:1205.6950v1  [cond-mat.str-el]  31 May 2012Effect of spin-orbit coupling on magnetic and orbital order i n MgV 2O4\nRamandeep Kaur, T. Maitra and T. Nautiyal\nDepartment of Physics, Indian Institute of Technology Roor kee, Roorkee- 247667, Uttarakhand, India\n(Dated: May 30, 2018)\nRecent measurements on MgV 2O4single crystal have reignited the debate on the role of spin- orbit\n(SO) coupling in dictating the orbital order in Vanadium spi nel systems. Density functional the-\nory calculations were performed using the full-potential l inearized augmented-plane-wave method\nwithin the local spin density approximation (LSDA), Coulom b correlated LSDA+U, and with SO\ninteraction (LSDA+U+SO) to study the magnetic and orbital o rdering in low temperature phase\nof MgV 2O4. It is observed that the spin-orbit coupling in the experime ntally observed antiferro-\nmagnetic phase, affects the orbital order differently in alte rnate V-atom chains along c-axis. This\nobservation is found to be consistent with the experimental prediction that the effect of spin-orbit\ncoupling is intermediate between that in case of ZnV 2O4and MnV 2O4.\nPACS numbers: 71.20.-b, 75.25.Dk, 71.70.Ej, 71.27.+a\nI. INTRODUCTION\nVanadium spinels AV 2O4(A=Mg, Zn, Cd) are being studied extensively in recent years1–6as they provide a very\ninteresting playground for the study of competing interactions on a frustrated lattice in 3-dimension. The Vanadium\n(V) ions at the B-sites of the spinel structure form a pyrochlore la ttice, with corner sharing tetrahedra, which is\ngeometrically frustrated. In its 3+ valence state, V ion has two elec trons in the d-shell which, because of a strong\nHund’s coupling, alignparalleltoeachothertherebyimpartingahighs pin state( S= 1) tothe ion. Thusin this family\nof spinels, there is a magnetic ion on a geometrically frustrated lattic e resulting in competing ground states. Things\nget more involvedwhen the partial occupancy of triply degenerate t2gorbitals by the two d-electronsmakes the orbital\ndegree of freedom unfrozen. As both spin and orbital degrees of freedom remain active, there is a high possibility\nof spin-orbit (SO) coupling playing important role in the low energy phy sics of this family of systems. Role of this\ninteraction has been a matter of debate recently1,3,4. The manifestation of the interplay of orbital, spin and lattice\ndegrees of freedom in these systems culminates in experiments as a sequence of phase transitions1,7,8. A structural\ntransition, often followed by magnetic transition as the temperatu re is lowered, signifies competing interactions trying\nto stabilize a particular ground state with gradual lifting of the frus tration.\nMgV2O4, with a normal spinel structure, has been reported to undergo a structural transition at 51 K from cubic to\ntetragonalphaseand a magnetictransitionat 42Kfrom non-magn eticto an antiferromagnetic(AFM)phaseconsisting\nof alternating antiferromagnetic chains of V atoms running parallel toaandbdirections as one goes along c-axis1,17.\nThe high temperature (HT) phase has a cubic spinel structure with F¯43msymmetry where the V ion is surrounded\nby an almost perfect O6octahedron with all the six V-O bonds having same length. This leads t o a sizable ( ∼2.5 eV)\nt2g-egcrystal field splitting of the d levels. There is of course a small trigon al distortion also present in this phase.\nExperimental results further reveal that the structural tran sition to the tetragonal phase at 51 K is accompanied by\na compression along c-axis with c/a=0.9941. This lowers the symmetr y to space group I¯4m2. Hence, in addition to\nthet2g-egsplitting arising from roughly O6octahedral coordination, a further splitting occurs due to the te tragonal\ncompression where the low lying t2gtriplet splits into a lower energy singlet ( dxyorbital) and a higher energy doublet\nofdyzanddzxorbitals. The orbital degeneracy is thereby partially lifted with this s tructural distortion. Now out of\nthe two d electrons, one goes to the lower energy dxyorbital while the other still has a choice as it occupies the doubly\ndegenerate dyzanddzxorbitals. This opens up a possibility of orbital order in this system. St ructural transition also\npartially lifts the frustration of the V−Vbonds in the pyrochlore lattice. This then brings in the second trans ition,\nat lower temperature of 42 K where a long range antiferromagnetic order sets in1. Thus the presence of any orbital\norder and the magnetic order observed at low temperatures in all t he Vanadium spinels are interrelated.\nSeveraltheoreticalmodels havebeen proposedin the last few yea rstoexplain the possible orbitalorderin Vanadium\nspinels so as to be consistent with the observed antiferromagnetic order. Among these, the model proposed by\nTsunetsugu and Motome3is based on strong coupling Kugel-Khomskii Hamiltonian and predicts a n orbital order\nwhere at each V site, d xyorbital is occupied by one electron and the second electron occupie s either d xzor dyz\norbital, alternately, along the c-axis. However, this type of orbita l order was found to be of lower symmetry than\nthat (I41/amd) observed experimentally for ZnV 2O4. In an alternative theoretical model, Tchernychov4considered\na dominant SO interaction which then led to the proposal that the se cond electron would occupy a complex orbital\nof type d xz±idyzat each V site. This orbital order is found to be consistent with the u nderlying crystal symmetry.\nAlso it explains the low magnetic moment per V ion observed in these sys tems as a large negative orbital moment is\nexpected from a strong SO coupling. These findings were also corro borated by electronic structure calculations5for2\nZnV2O4.\nHowever, recent measurements on other members of the Vanadiu m spinel family raise doubts about the presence\nof a strong spin-orbit interaction effect. In fact, there has been a tremendous effort, from both theoreticians and\nexperimentalists working on these systems, to bring out a unified pic ture in terms of the important interactions which\nunderlie the two phase transitions (one structural and the other magnetic). In ZnV 2O4the SO coupling is found to\nbe significant both from theory as well as experiments4,5,10whereas in case of MnV 2O4there seems to be very little\nor no effect of the SO interaction on the orbital order11,12. Recently Wheeler et al.1performed neutron diffraction\nmeasurements on MgV 2O4single crystal and speculated on the basis of their observations th at MgV 2O4might come\nintermediate between ZnV 2O4and MnV 2O4as far as strength of SO coupling is concerned. Hence it is expected that\nin MgV 2O4the occupied orbitals, instead of being completely real (Tsunetsug u and Motome model) or completely\ncomplex (Tchernychov model), could be a mixture of real and comple x orbitals. In the previous theoretical study on\nMgV2O413, the issue of impact of SO coupling on orbital order has not been inve stigated. However, as stated above,\nSO coupling in MgV 2O4is expected to be non-negligible from experimental observations. I n order to investigate\nthoroughly the effect of SO interaction on magnetic and orbital ord er in MgV 2O4, we have carried out first principle\nelectronic structure calculations incorporating SO coupling. Such a calculation is definitely expected to unfurl the\nstrength of SO coupling in this system, the nature of orbital order (if there is any) and correlation of experimentally\nobserved magnetic order with the orbital order, if present.\nII. METHODOLOGY\nWe undertake an electronic structure calculation using full-potent ial linearized augmented-plane-wave method with\nthe basis chosen to be linearized augmented plane waves as implement ed in WIEN2K code14. The calculations have\nbeen carried out with no shape approximation to the potential and c harge density. These calculations were performed\nat three levels of sophistication using local spin density approximatio n(LSDA), Coulomb correlatedLSDA+U approx-\nimation, and with SO interaction i.e. LSDA+U+SO approximation. To rem ove the self-Coulomb and self-exchange-\ncorrelation energy included in LSDA approximation, we use self-inter action corrected scheme (LSDA+U(SIC))15,\nwhich is appropriate for the strongly correlated systems. The cor rected energy functional is written as15\nE=ELSDA−[UN(N−1)/2−JN(N−2)/4]+1/2/summationdisplay\nm,m′,σUmm′nmσnm′−σ+1/2/summationdisplay\nm/negationslash=m′,m′,σ(Umm′−Jmm′)nmσnm′σ\nHere ELSDAis the standard LSDA energy functional, U represents the on-site Coulomb interaction, J is the exchange\nparameter and n mσare the occupations of the localized orbitals. N is the total number o f localized electrons.\nIn the LSDA+U+SO calculations, SO coupling was considered within the scalar relativistic approximation and the\nsecond variational method was employed16. In this method, the eigen value problem is first solved separately fo r spin\nup and spin down states without inclusion of the SO interaction term ( HSO) in the total Hamiltonian. The resulting\neigen values and eigen functions are then used to solve new eigen valu e problem with the H SOterm in the total\nHamiltonian. This method is more efficient and computationally less expe nsive than the calculation in which H SOis\nincluded in the total Hamiltonian by doubling the dimension of the origina l eigenvalue problem in order to calculate\nthe non-zero matrix elements between spin-up and spin-down stat es. In this method, the calculation of H SOmatrix\nelements involves much less number of basis functions than in the orig inal basis set.\nMgV2O4crystallizes in tetragonal structure with symmetry I¯4m2 (space group 119) at low temperatures1. Atomic\npositions and lattice constants were taken from the experimental data1. The atomic sphere radii were chosen to be\n1.96, 1.99, and 1.78 a.u. for Mg, V, and O, respectively. We have used 50kpoints in the irreducible part of the\nBrillouin zone for the self-consistent calculations. In order to mode l the low temperature magnetic order observed in\nthe experiment, we have constructed a supercell (with 8 inequivale nt Vanadium atoms). The lowering of symmetry\nof this unit cell arises due to the experimentally observed antiferro magnetic ordering. The network of corner sharing\nV4O4cubes of low temperature structure is shown in Fig. 1 with the magne tic order. The 8 inequivalent Vanadium\natoms considered in the calculation are also marked in the figure with t he corresponding orientation of spins at that\nparticularsite. Onecanseetheantiferromagneticchainsalong a(...V3−V7−V3−V7...)andb(...V6−V2−V6−V2...)\naxes alternating along c-axis. In each V 4O4cube there are 4 inequivalent V atoms. Due to the presence of coop erative\ntrigonal distortion along c-axis resulting in alternating compression and expansion of cube faces, there is a further\nsymmetry breaking and hence successive cubes along c-axis no long er remain equivalent. Furthermore, the V 4O4\ncube containing V1, V5, V2 and V6 does not have the same spin arran gement as that containing V3, V7, V4 and V8.\nTherefore to model the experimentally observed magnetic order o ne needs to consider 8 inequivalent V atoms in the\nunit cell.3\nFIG. 1: Corner sharing network of V 4O4cubes in the low temperature structure of MgV 2O4showing the experimentally\nobserved magnetic order. The solid and dotted lines joining the V atoms (shown in one cube) represent the shorter V-V FM\nbonds (2.971 ˚A) and longer V-V AFM bonds (2.98 ˚A) respectively.\nFIG. 2: Spin polarized (a) total DOS within LSDA, (b) partial DOS for V d-states around the Fermi level within LSDA+U,\n(c) partial d-DOS within LSDA+U+SO (U-J = 2 eV)in the low temp erature AFM phase.\nIII. RESULTS\nOur LSDA calculations of experimentally observed antiferromagnet ically ordered phase show that total energy of this\nphase is indeed lower than that of the corresponding ferromagnet ic (FM) phase by 0.4 eV per formula unit. The\ndensity of states (DOS) of this antiferromagnetic state (within LS DA) is shown in Fig. 2(a). It is observed that LSDA\ngivesa metallic state whereasthe system is knownto be a Mott insulat or17. Thus AFM interactionalone is not able to\nopen up the gap. Around the Fermi level, mainly V d-states are seen to be present. In an effort to have the insulating\ngap as observed experimentally, we included Coulomb correlation in ou r calculations within LSDA+U approximation.4\nFIG. 3: Real space electron density at each V site within (a) L SDA+U (b) LSDA+U+SO along two different directions.\nIsosurface used for both corresponds to 0.5 e/ ˚A3.\nWe performed calculations with U eff(=U-J) values in the range 1 to 4 eV as found to be relevant from the literature\non Vanadium spinel systems5,11,13. We present the results for U eff= 2 eV in the following, nevertheless it may be\nnoted that our conclusions remain valid in the whole range of U effvalues considered by us. In Fig. 2(b) we show\nthe partial DOS of five d-orbitals as these are the states present around the Fermi level. As expected, the application\nof Coulomb correlation Uis able to open up a small gap of 0.12 eV which increases with the increas e inU. The gap\noriginates because of the the splitting of the t 2glevels in addition to the t 2g-egsplitting due to octahedral field. The\nfurther splitting of t 2gis primarily caused by the antiferromagnetic interactions which get e nhanced in the presence\nof Coulomb correlations.\nAnother observation that can clearly be made from the partial DOS of d-orbitals (Fig. 2(b)) is that among the\noccupied t 2gorbitals, one orbital (i.e. d x2−y2) is more populated while the other two (d xzand dyz) essentially have the\nsame occupancy and seem to be degenerate. The higher occupanc y of dx2−y2orbital is a result of the presence of the\ntetragonal compression along c-axis at low temperatures. Howev er, closer analysis of occupancies of the apparently\ndegenerate d xzand dyzorbitals at each Vanadium site shows that there is a tendency towar ds orbital ordering. Table\nI lists the orbital occupancies of d x2−y2, dxzand dyzfor the 8 inequivalent V atoms in the unit cell considered. One\nclearly observes that the occupancy of d xzand dyzorbitals are different as one moves along the c-axis whereas that\nof dx2−y2remains the same. The orbital polarization increases on increasing t he value of Uand alternates for the\ndxzand dyzorbitals (see for example, V1 and V2) in successive Vanadium layers a long c-axis. This is similar to an\nA-type antiferro-orbital order where the antiferromagnetic V c hains parallel to the ab-plane have ferro-orbital order\n(e.g. similar orbital occupancies of V1 and V5 ions or that of V2 and V6 ions) whereas along c-axis there is an\nantiferro-orbital order between d xzand dyzorbitals (see occupancies of V1 and V2 or that of V5 and V6 in Table\nI). This is consistent with the previous theoretical observation on the same system13. The observed intra-chain ferro-\norbital order is also consistent with experimental antiferromagne tic order as per Goodenough-Kanamori-Anderson\nrules18.The orbital order described above is also revealed in the calculated r eal space electron density at each V site\nshown in Fig. 3(a). This orbital order was predicted by Tsunetsugu and Motome3for Vanadium spinels from their\ncalculations based on Kugel- Khomskii model in strong coupling limit.\nAs mentioned earlier, the influence of spin-orbit coupling on the magn etic and orbital order in these systems\nis continuously debated but no conclusion has been reached yet. In order to investigate the effect of spin-orbit\ninteraction in this particular system, we also performed a calculation with spin-orbit coupling within LSDA+U+SO\napproximation. The solution obtained within LSDA+U+SO has a lower en ergy than that obtained within LSDA+U\nby 0.095 eV per formula unit for Ueff=2eV. The partial DOS (Fig. 2(c)) clearly shows a non-negligible impac t of SO\nin general, with an increased energy gap compared to that with LSDA +U.\nThe analysis of orbital occupancies in this case indeed leads to some im portant and interesting observations. The\napparent degeneracy of d xzand dyzorbitals observed within LSDA+U is no longer present and there is a co mplete\nlifting of degeneracy of all the t 2gorbitals (see Table I). Even though, likewise LSDA+U, the antiferro magnetic V\nchains parallel to the ab-plane are still ferro-orbitally ordered and along c-axis these chains are anti-ferro orbitally\nordered, the orbital polarizations in adjacent chains along c-axes are significantly different in the presence of SO\ninteraction. For example, if we compare the occupancies (Table I) f or V1 and V2 with LSDA+U and LSDA+U+SO,5\nTABLE I: Orbital occupancies and spin magnetic moment withi n LSDA+U and LSDA+U+SO (U eff=2 eV)\norbital occupation spin magnetic\ndx2−y2dxzdyzmoment ( µB)\nWith LSDA+U\nV1 (V3) 0.6650.5530.3861.53 (-1.53)\nV2 (V4) 0.6650.3860.5531.53 (-1.53)\nV5 (V7) 0.6650.5530.386-1.53 (1.53)\nV6 (V8) 0.6650.3860.553-1.53 (1.53)\nWith LSDA+U+SO\nV1 (V3) 0.5950.4510.5911.574 (-1.574)\nV2 (V4) 0.7590.2140.6651.571 (-1.571)\nV5 (V7) 0.5950.4510.591-1.574 (1.574)\nV6 (V8) 0.7590.2140.665-1.571 (1.571)\nTABLE II: Calculated orbital moments, total magnetic momen t (J) (in µB) and angle of J w.r.t. z-axis within LSDA+U+SO\n(Ueff= 2 eV). The spin magnetic moment is along z axis and is listed i n Table I\nµorbital µtotalangle\nxyzJ\nV1-0.3550.000-0.4661.1517.79\nV2-0.015-0.030-0.5101.051.90\nV5-0.3550.0000.466-1.15162.21\nV6-0.015-0.0300.510-1.05178.10\nwe note that orbital occupancies of d x2−y2are no longer same in presence of SO interaction. Furthermore, th e\npolarization of the d xzand dyzorbitals are also very different (i.e. at V2 the polarization of d yzorbital w.r.t. d xzis\nmuch stronger than that at V1). This implies that V chains in success ive layers along c-axis are affected differently by\nthe SO interaction. This is also reflected in the orbital moments of V io ns (listed in Table II and depicted in Fig. 4).\nThe calculated electronic density at each V site is shown in Fig. 3(b) wh ich brings out the impact of SO interaction.\nIn Fig. 4 we show two successive Vanadium chains along c-axis with the calculated electron density at each\nVanadium site in the presence of both Coulomb correlation and SO inte raction. We have also marked the direction\nof orbital and magnetic moments at each site. Effect of SO interact ion is clearly different on the two chains and so is\nthe arrangement of the orbital moments. One chain shows the can ted orbital arrangement and orbital moments are\nmaking an angle of 17.790with the c-axis whereas in other chain orbital moment makes an angle of 1.900(almost\ncollinear orbital arrangement) with the c-axis (Table II). On one ch ain (V1-V5-V1-V5) due to canting of orbital\nFIG. 4: Electron densities for two successive Vanadium chai ns along c-axis within LSDA+U+SO showing the impact of SO\ninteraction on them. The directions of corresponding orbit al and magnetic moments are also shown below each chain.6\nmoment, the effect of SO interaction reduces whereas in the other chain orbital moments align almost opposite to\nthe magnetic moment implying a substantial SO interaction. The obse rvation that the SO interaction appears to\naffect alternate V chains along c-axis differently, is interesting. This also substantiates the speculation of Wheeler\net al.1that SO interaction in MgV 2O4may not be as large as that in ZnV 2O4or as small as that in MnV 2O4as\ndiscussed earlier. The magnitude of orbital moments observed in ou r calculation also corroborates this fact. Thus our\nresults show that a small but non-negligible spin-orbit coupling, along with the significant trigonal distortion present\nin MgV 2O4structure, has a substantial effect on the orbital order of this s ystem. This observation is consistent with\nthe experimental observations by Wheeler et al.1of antiferromagnetic chains with a strongly reduced moment and\nthe one-dimensional behavior and a single band of excitations proje cted by the inelastic neutron scattering.\nIV. CONCLUSIONS\nTo conclude, we have studied the effect of spin-orbit interaction on magnetic and orbital order in the low temperature\ntetragonalphase ofMgV 2O4. We observethat even thoughthe orbitalmoments arerelativelys mall comparedto those\nof ZnV 2O4, the orbital order in successive Vanadium chains is differently affect ed in the presence of SO interaction.\nIn one chain (V1-V5-V1-V5, parallel to crystallographic aaxis) the three t 2gorbitals are nearly equally populated\ngiving rise to a canted (non-collinear) arrangement of orbital mome nts whereas in the other (V2-V6-V2-V6, parallel\ntobaxis), the orbitals are highly polarized leading to a collinear arrangeme nt of orbital moments. These results\nimply that SO interaction in MgV 2O4is non-negligible and has a significant effect on orbital order. Howeve r it is not\nvery strong unlike ZnV 2O4and at the same time not very weak unlike MnV 2O4.\nV. ACKNOWLEDGEMENT\nThis work is supported by the DST (India) fast track project (gra nt no.: SR/FTP/PS-74/2008). RD acknowledges\nCSIR (India) for a research fellowship.\n1E.M. Wheeler, B. Lake, A.T. M. Nazmul Islam, M. Reehuis, P. St effens, T. Guidi and A. H. Hill, Phys. Rev. B 82, 140406(R)\n(2010).\n2Paolo G Radaelli, New J. Phys. 7, 53 (2005).\n3H. Tsunetsugu and Y. Motome, Phys. Rev. B 68060405 (2003); ibid.Prog. Theor. Phys. Suppl. 160, 203 (2005).\n4O. Tchernyshyov, Phys. Rev. Lett. 93157206 (2004).\n5T. Maitra and R. Valent´ ı; Phys. Rev. Lett. 99, 126401, (2007).\n6G. Giovanetti, A. Stroppa,S. Picozzi,D. Baldomir,V. Pardo ,S. Blanco-Canosa,F. Rivadulla,S. Jodlauk,D. Niermann,J .\nRohrkamp,T. Lorenz,S. Streltsov,D. I. Khomskii,and J. Hem berger; Phys. Rev. B 83, 060402(R) (2011).\n7M. Reehuis, A. Krimmel, N. Bottgen , A. Loidl and A. Prokofiev, Eur. Phys. J. B 35, 311 (2003).\n8V. O. Garlea, R. Jin, D. Mandrus, B. Roessli, Q. Huang, M. Mill er, A. J. Schultz, and S. E. Nagler, Phys. Rev. Lett. 100,\n066404 (2008).\n9H. Mamiya, M. Onoda, T. Furubayashi, J. Tang, and I. Nakatani , J. Appl. Phys. 81, 5289 (1997).\n10S. H. Jung, J. Noh1, J. Kim, C. L. Zhang, S. W. Cheong and E. J. Ch oi; J. Phys.: Condens. Matter 20175205 (2008).\n11S. Sarkar, T. Maitra, Roser Valent, and T. Saha-Dasgupta; Ph ys. Rev. Lett. 102, 216405 (2009).\n12S.-H. Baek, N. J. Curro, K.-Y. Choi, A. P. Reyes, P. L. Kuhns, H . D. Zhou, and C. R. Wiebe, Phys. Rev. B 80, 140406(R)\n(2009).\n13S. K. Pandey; Phys. Rev. B. 84, 094407 (2011).\n14P. Blaha, K. Schwartz, G. K. H. Madsen, D. Kvasnicka and J. Lui tz; WIEN2K edited by K. Schwarz (Techn. University\nWien, Austria, 2001), ISBN 3-9501031-1-2.\n15V.I. Anisimov, I.V. Solovyev, M.A. Korotin, M.T. Czyzyk, an d G.A. Sawatzky, Phys. Rev. B 48, 16929 (1993)\n16D.D. Koelling et al., J. Phys. C 10, 3107 (1977); A.H. MacDona ld et al., ibid. 13, 2675 (1980)\n17H. Mamiya, M. Onoda, Solid State Communications 95, 217 (1995),\n18J. Kanamori, Prog. Theor. Phys. 17177 (1957); J. B. Goodenough, J. Phys. Chem. Solids 6, 287 (1958); P. W. Anderson,\nPhys. Rev. 79350 (1950)."    },    {        "title": "1812.09270v1.Influence_of_spin_orbit_and_spin_Hall_effects_on_the_spin_Seebeck_current_beyond_linear_response.pdf",        "content": "arXiv:1812.09270v1  [cond-mat.mes-hall]  21 Dec 2018Influence of spin-orbit and spin-Hall effects on the spin Seeb eck current beyond linear\nresponse: A Fokker-Planck approach\nL. Chotorlishvili1, Z. Toklikishvili2, X.-G. Wang3, V.K. Dugaev4, J. Barna´ s5,6J. Berakdar1\n1Institut f¨ ur Physik, Martin-Luther Universit¨ at Halle-W ittenberg, D-06120 Halle/Saale, Germany\n2Faculty of Exact and Natural Sciences, Tbilisi State Univer sity, Chavchavadze av.3, 0128 Tbilisi, Georgia\n3School of Physics and Electronics, Central South Universit y, Changsha 410083, China\n4Department of Physics and Medical Engineering,\nRzeszow University of Technology, 35-959 Rzeszow, Poland\n5Faculty of Physics, Adam Mickiewicz University, ul. Umulto wska 85, 61-614 Poznan, Poland\n6Institute of Molecular Physics, Polish Academy of Sciences ,\nul. M. Smoluchowskiego 17, 60-179 Pozna´ n, Poland\n(Dated: December 24, 2018)\nWe study the spin transport theoretically in heterostructu res consisting of a ferromagnetic metal-\nlic thin film sandwiched between heavy-metal and oxide layer s. The spin current in the heavy metal\nlayer is generated via the spin Hall effect, while the oxide la yer induces at the interface with the\nferromagnetic layer a spin-orbital coupling of the Rashba t ype. Impact of the spin Hall effect and\nRashba spin-orbit coupling on the spin Seebeck current is ex plored with a particular emphasis on\nnonlinear effects. Technically, we employ the Fokker-Planc k approach and contrast the analytical\nexpressions with full numerical micromagnetic simulation s. We show that when an external mag-\nnetic field H0is aligned parallel (antiparallel) to the Rashba field, the s pin-orbit coupling enhances\n(reduces) the spin pumping current. In turn, the spin Hall eff ect and the Dzyaloshinskii-Moriya\ninteraction are shown to increase the spin pumping current.\nI. INTRODUCTION\nIn a seminal paper [1], Bychkov and Rashba explored\nthe impact of spin-orbit (SO) interaction on the proper-\nties of two-dimensional semiconductor heterostructures.\nSince then, the basic idea of Bychkov and Rashba was\ncarried over to other research areas of physics. It was\nshown, for instance, that the SO interaction plays a sig-\nnificant role in the quantum spin Hall Effect in graphene\n[2], Bose-Einstein condensates [3], and in the orbital-\nbased electron-spin control [4]. Recent experiments [5, 6]\nrevealed the role of SO interaction in the motion of do-\nmain walls, as well. Combining the SO coupling and\nthermal effects bring in new insight and phenomena. A\nthermal bias applied to a ferromagnetic insulator leads\nto the formation of a thermally assisted magnonic spin\ncurrent that is proportional to the temperature gradi-\nent. This phenomenon falls in the class of spin See-\nbeck effects and may be useful for thermal control of\nmagnetic moments [7–23]. The objective of this paper\nis to study the impact of SO interaction on the forma-\ntion and transport of thermally assisted magnonic spin\ncurrent in spin-active multilayers. We investigate two\ndifferent heterostructures which include a layer of fer-\nromagnetic metal sandwiched between heavy metal and\noxide materials, see Fig. 1 and Fig. 2. In both cases,\nan inversion asymmetry is caused by two different inter-\nfaces – heavy-metal/ferromagnet and ferromagnet/oxide\nones. A large SO coupling is present in the heavy metal\n[24–28]. This study is motivated by the experimental\nwork in Ref. [27] with a particular attention to the\nsystems Pt/Co/AlO xand Ta/CoFeB/MgO. Moreover,\nthe torques generated by strong SO coupling are gener-\nally different from the Slonczewski’s spin-transfer torque[24, 30, 31], with the prospect for novel physical effects in\nthe heavy-metal/ferromagnetic-metal/oxide heterostruc-\ntures. An applied an electric voltage (see Fig.1) gener-\nates charge current in the ferromagneticand heavy metal\nlayers. This current in the heavy-metal layer leads to\nspin current due to the spin Hall effect, which is then\ninjected into the thin ferromagnetic layer [32–39] and\nacts as an extra torque on the localized magnetic mo-\nments in the ferromagnet. The induced torque influences\nthe magnetization dynamics, which is the topic of this\nwork. To describe the influence of the spin current on\nthe magnetization dynamics in the ferromagnetic layer\nwe add a relevant term to the Landau-Lifshitz-Gilbert\n(LLG) equation. In turn, the Rashba SO coupling at\nthe ferromagnet/oxide interface in the presence of the\ncharge current results in a spin polarization at the in-\nterface, with the exchange coupling exerting a torque on\nthe ferromagnetic layer, as well. Thus, the SHE and the\nRashba SO coupling influence the magnetization dynam-\nics in the ferromagnetic layer through the Rashba and\nSHE torques, both incorporated into the LLG equation\n(the Rashba and SHE fields). The considered setup al-\nlows to formally investigate the interplay/competition of\nthe torques due to Rashba SO interaction and SH effect.\nThe Rashba SO torque acts field-like, while the torque\ndue to the spin current generated viathe spin Hall effect\nis predominantlyofdamping/antidumpinglike in nature.\nWe utilizethe Fokker-Planckmethod [40] forthe stochas-\ntic LLG equation for studying the magnetic dynamics\nbeyond the linear response regime. The influence of the\nRashba-type SO coupling on the magnonic spin current\nwas studied in the works [41–43].\nIn the system shown in Fig.1, the normal metal with\ntemperature TNis attached to the ferromagnet with2\nTF> TN. We consider the spin current flowing from the\nferromagnetic to a normal metal layer. Magnons from\nthe high-temperature region diffuse to the lower temper-\naturepartgivingrisetoamagnonicspincurrentandthus\nalso to the spin Seebeck effect (SSE) [44–46]. Magnonic\nspin current pumped from the ferromagnet into the nor-\nmal metal, Isp, increases with the temperature differ-\nence,Isp∼TF−TN. However, the spin current in-\njected from the ferromagnetic layer to the normal metal\nis not the only spin current that crosses the normal-\nmetal/ferromagnet interface. The fluctuating spin cur-\nrentIflis generated in the normal metal and flows to-\nwards the ferromagnet, i.e. in the direction opposite to\nthe magnonic spin pumping current. The quantity of in-\nterest is therefore the total spin current, Itot=Isp+Ifl\nthat crossesthe normal-metal/ferromagnetinterface. We\nshow that Itotis drastically influenced by the proximity\nof the heavy metal (due to spin Hall effect) and the ox-\nide (due to Rashba spin-orbit coupling). In the second\nsystem (see Fig.2) an additional normal-metal layer is\nattached to the ferromagnetic one.\nThe paper is organized as follows. In Sec. IIwe in-\ntroduce the model under consideration. In Sec. III\nandIVwe explore the spin current in two different\nheterostructures. For the sake of simplicity, we neglect\nDzyaloshinskii-Moriya interaction (DMI). Effects of the\nDMI term and magnetocrystalline anisotropy are ex-\nplorednumericallybymicromagneticsimulationsandare\ndescribed in Sec. V. Section VIsummarizesthe findings.\nMain technical details are deferred to the appendices.\nII. THEORETICAL MODEL\nFor the heavy-metal/ferromagnetic-metal/oxide sand-\nwich we choose the ferromagnetic metallic layer to be\nin direct contact with a nonmagnetic metallic layer, as\nshown in Fig.1. We also assume that, due to a strong\nelectron-phonon interaction, the local thermal equilib-\nrium between electrons and phonons in both ferromag-\nnetic and normal-metal layers is established, Tp\nF=Te\nF=\nTFandTp\nN=Te\nN=TN. The magnon temperature Tm\nF\nin the ferromagneticlayerdiffers in generalfrom the tem-\nperature of electrons/phonons, Tm\nF/negationslash=TF[46].\nAt nonzero temperatures, the thermally activated\nmagnetization dynamics in the ferromagnet gives rise to\na spin current flowing into the normal metal. This effect\nisknownasspin pumping [44, 47–49]. Thecorresponding\nexpression for the spin current density reads [46, 50]\nIsp(t) =/planckover2pi1\n4π[grm(t)×˙m(t)+gi˙m(t)],(1)\nwheregrandgiare the real and imaginary parts\nof the dimensionless spin mixing conductance of\nthe ferromagnet/normal-metal ( F|N) interface, while\nm(t) =M(t)/Msis the dimensionless unit vector along\nthe magnetization orientation (here Msis the saturation\nmagnetization) and ˙m≡dm/dt. The spin current isa tensor describing the spatial distribution of the cur-\nrent flow and orientation of the flowing spin (magnetic\nmoment). Due to the geometry of the system under con-\nsideration, the spin current flows along the y-axis, see\nFig. 1. In turn, the spin polarization of the current de-\npends on the orientation of the magnetic moment and its\ntime derivative. The averagespin depends on the ground\nstate magnetic order which in our case is collinear with\nthe external magnetic field (applied along the y-axis).\nTherefore, the only nonzero component of the average\nspin current tensor is Iy\nsp.\nThermal noise in the normal-metal layer activates a\nfluctuating spin current flowing from the normal metal\nto the ferromagnet [47],\nIfl(t) =−MsV\nγm(t)×ζ′(t). (2)\nHere,Vis the total volume of the ferromagnet, γis the\ngyromagnetic factor, and ζ′(t) =γh′(t) withh′(t) de-\nnoting the random magnetic field. In the classical limit,\nkBT≫/planckover2pi1ω0, the correlation function /angb∇acketleftζ′\ni(t)ζ′\nj(t′)/angb∇acket∇ightofζ′(t)\nreads\n/angb∇acketleftζ′\ni(t)ζ′\nj(t′)/angb∇acket∇ight=2α′γkBTN\nMsVδijδ(t)≡σ′2δijδ(t),(3)\nwhere/angb∇acketleft.../angb∇acket∇ightdenotes the ensemble average, and i,j=\nx,y,z. Furthermore, ω0is the ferromagnetic resonance\nfrequency and α′is the contribution to the damping con-\nstant due to spin pumping, α′=γ/planckover2pi1gr/4πMsV. We em-\nphasize that the correlator(Eq.(3)) is proportionalto the\ntemperature TN.\nThe total spin current flowing through the\nferromagnet/normal-metal interface is given by the sum\nofpumping and fluctuating spin currents, Itot=Isp+Ifl.\nFor clarity of notation, we omit here (and also in the\nfollowing) the time dependence of spin currents, nor-\nmalized magnetization, random magnetic fields, and\ntheir correlators. This dependence will be restored if\nnecessary. According to Eqs. (1) and (2), the total\naverage spin current flowing across the interface can be\nwritten in the following form [46]:\n/angb∇acketleftItot/angb∇acket∇ight=MsV\nγ[α′/angb∇acketleftm×˙m/angb∇acket∇ight−/angb∇acketleftm×ζ′/angb∇acket∇ight].(4)\nNow, we assume that a spatially uniform current of\ndensityja=jaixis injected along the x-axis. This cur-\nrent gives rise to additional torques owing to the spin\nHall effect and Rashba spin-orbit interaction. Thus, the\nmagnetizationdynamics is then modified and is governed\nby the stochastic LLG equation [27]:\ndm\ndt=−γm×(Heff+h)+αm×˙m+τSO,(5)\nwhereαis the Gilbert damping constant, his the time-\ndependentrandommagneticfieldintheferromagnet,and\nHeffis an effective field. This effective field consists of3\nthreecontributions: theexchangefield, theexternalmag-\nnetic field oriented along the y-axis, and the field corre-\nsponding to the DM interaction:\nHeff=2A\nµ0Ms∇2m+H0y−1\nµ0MsδEDM\nδm,\nEDM=D/bracketleftbig\nmz∇m−/parenleftbig\nm∇/parenrightbig\nmz/bracketrightbig\n. (6)\nFor the sake of simplicity, in the analytical part we take\ninto account only the external magnetic field. In turn,\nthe term τSOin Eq.(5) describes SO torques related to\nthe Rashba SO coupling and the spin Hall effect,\nτSO=−γm×HR+γηξm×(m×HR) (7)\n+γm×(m×HSH),\nwhereξis a non-adiabatic parameter, and η= 1 when\nthe torque has Slonczewski-like form, while η= 0 in the\nopposite case [27]. In the above equation, the DM inter-\naction enters the effective magnetic field, while the effect\nof Rashba SO coupling and spin Hall effect are included\nby means of the extra torqueadded to the LLG equation.\nAs already mentioned above, the charge current flow-\ning in the thin ferromagnetic layer leads to spin polar-\nization at the ferromagnet/oxide interface. The accumu-\nlated spin density in vicinity of the interface interacts\nwith the local magnetization by means of the exchange\ncoupling. This effect may be described by an effective\nRashba field HR=HRiy[5, 24, 32]:\nHR=αRP\nµ0µBMs(iz×ja) =αRPja\nµ0µBMsiy,(8)\nwhereαRis the Rashba parameter and Pis the degree of\nspin polarization of conduction electrons [32]. The first\nterminEq.(7)correspondstotheout-of-planetorqueand\nisrelatedtotheeffectivefield HR. Thistorqueisoriented\nperpendicularly to the ( m,HR) plane. The second term\nin Eq.(7) captures the effects of spin diffusion inside the\nmagnetic layer and the spin current associated with the\nRashba interaction at the interface. For more details, we\nrefer to the work [24].\nThe last term in Eq.(7) corresponds to the spin Hall\ntorque[33,34], expressedbythespinHallfield HSH. The\nspin current is generateddue to the spin Hall effect in the\nheavy metal layer and is injected into the ferromagnetic\nlayer. For moredetails, we referto the references [36–39].\nThe explicit expression for HSHreads:\nHSH=/planckover2pi1θSHja\nµ02eMsLziy, (9)\nwhereLzisthethicknessoftheferromagneticlayer,while\nθSHis the spin Hall angle (defined as the ratio of spin\ncurrent and charge current densities).\nAs already mentioned above, total random magnetic\nfieldh(t) has two contributions from different noise\nsources: the thermal random field h0(t), and the ran-\ndom field h′(t). Since the random fields are statisti-\ncally independent, their correlatorsare additive and fully/s122\n/s121\n/s78/s111/s114/s109/s97/s108\n/s77/s101/s116/s97/s108/s79/s120/s105/s100/s101\n/s70/s101/s114/s114/s111/s109/s97/s103/s110/s101/s116/s105/s99\n/s84\n/s78/s84\n/s70/s73\n/s102/s108\n/s73\n/s83/s80/s72\n/s82/s44/s32/s83/s72\n/s72/s101/s97/s118/s121\n/s77/s101/s116/s97/s108/s106\n/s97/s120\nFIG. 1. Schematic illustration of the system. A ferromagnet ic\nmetallic layer is sandwiched between the oxide and heavy\nmetal layers. The injected current jaflows in the ferromag-\nnetic and heavy metal layers in the xdirection. The Rashba\nfieldHRand the spin Hall field are oriented along the yaxis.\nThe normal metal with the temperature TNis attached to\nthe ferromagnetic layer. The temperature of the ferromag-\nnetic layer TFis different from TN.\ndetermined by the total (enhanced) magnetic damping\nα=α0+α′[46] (with α0being the damping parameter\nof the ferromagnetic material, i.e., without contributions\nfrom pumping currents),\n/angb∇acketleftζi(t)ζj(0)/angb∇acket∇ight=2αγkBTm\nF\nMsVδijδ(t) =σ2δijδ(t),(10)\nwhereζ(t) =γh(t), andαTm\nF=α0TF+α′TN.\nIII. SPIN CURRENT: N/F STRUCTURE\nTheinjected electricalcurrentcreatesatransversespin\ncurrent in the heavy-metal layer via the spin Hall effect\n(or spin accumulation at the boundaries of the sample)\n[44]. In turn, the Rashba SO interaction in the presence\nofchargecurrentgivesrisetoadditionaltorqueasalready\ndescribed above. In the case under consideration, the\nRashba SO field, Eq.(8), and the spin Hall field, Eq.(9),\nare oriented along the yaxis. When temperature of the\nferromagnetic film differs from that of the normal metal,\nTF/negationslash=TN, the spin Seebeck current emerges in the Fe/N\ncontact. Note, this current also exists in the absence of\nspin-orbitinteraction and for ja= 0. Below, we calculate\nthe total spin current in the N/F structure, taking into\naccount the Rashba SO field and the spin Hall effect.\nIn order to calculate the spin pumping current, /angb∇acketleftIsp/angb∇acket∇ight=\nMsV\nγα′/angb∇acketleftm×˙m/angb∇acket∇ight, we use Eq.(A1) (see Appendix A) and\nfind\n/angb∇acketleftIsp/angb∇acket∇ight=α′MsV\nγ(−/angb∇acketleftm×ω2/angb∇acket∇ight−/angb∇acketleftm×m×ω1/angb∇acket∇ight),(11)4\nwhereω1andω2are defined in the Appendix A, see\nEq.(A1). Utilizing Eq.(A2) and Eq.(A15) we find mean\nvalues of the magnetization components (see Appendix\nB):\n/angb∇acketleftmy/angb∇acket∇ight=−L(βω2),/angb∇acketleftm2\ny/angb∇acket∇ight= 1−2L(βω2)\nβω2,\n/angb∇acketleftm2\nx/angb∇acket∇ight=/angb∇acketleftm2\nz/angb∇acket∇ight=L(βω2)\nβω2,(12)\nwhereβ= 2/σ2, andL(x) = coth x−1\nxis the Langevin\nfunction. From Eq.(12) we obtain /angb∇acketleftm×ω2/angb∇acket∇ight= 0, and\n/angb∇acketleft(m×˙m)x/angb∇acket∇ight= 0,/angb∇acketleft(m×˙m)y/angb∇acket∇ight=−ω1(1−/angb∇acketleftm2\ny)/angb∇acket∇ight),/angb∇acketleft(m×\n˙m)z/angb∇acket∇ight= 0. Thus, the only nonzero component of the spin\npumping current is Iy\nsp,\n/angb∇acketleftIy\nsp/angb∇acket∇ight=α′MsV\nγω1(1−/angb∇acketleftm2\ny/angb∇acket∇ight)\n=α′MsV\nγ2ω1\nβω2L(βω2). (13)\nFor the evaluation of the fluctuating spin current\n/angb∇acketleftIfl/angb∇acket∇ight=−MsV\nγ/angb∇acketleftm×ζ′/angb∇acket∇ight, we linearize the LLG equation,\nEq.(A1), near to the equilibrium point: /angb∇acketleftmx/angb∇acket∇ight=/angb∇acketleftmz/angb∇acket∇ight=\n0,/angb∇acketleftmy/angb∇acket∇ight=−L(βω2):\n˙mx=ω1mz+ω2/angb∇acketleftmy/angb∇acket∇ightmx−/angb∇acketleftmy/angb∇acket∇ightζz(t),\n˙mz=−ω1mz+ω2/angb∇acketleftmy/angb∇acket∇ightmz+/angb∇acketleftmy/angb∇acket∇ightζx(t).(14)\nFourier transforming to the frequency domain (˜ g=/integraltext\ngeiωtdtandg=/integraltext\n˜ge−iωtdω/2π), from Eq.(14) we ob-\ntain ˜mi(ω) =/summationtext\njχij(ω)˜ζj(ω), where i,j=x,z, and\nχij(ω) =/angb∇acketleftmy/angb∇acket∇ight\n(ω2/angb∇acketleftmy/angb∇acket∇ight+iω)2+ω2\n1\n×/parenleftbigg\nω1 (ω2/angb∇acketleftmy/angb∇acket∇ight+iω)\n−(ω2/angb∇acketleftmy/angb∇acket∇ight+iω)ω1/parenrightbigg\n,(15)\n/angb∇acketleftmi(t)ζ′\nx(0)/angb∇acket∇ight=σ′2/integraldisplay+∞\n−∞χij(ω)e−iωtdω\n2π.(16)\nEquation(16) has nonzero elements:\n/angb∇acketleftmz(t)ζ′\nx(0)/angb∇acket∇ight=−/angb∇acketleftmx(t)ζ′\nz(0)/angb∇acket∇ight\n=−σ′2/angb∇acketleftmy/angb∇acket∇ight/integraldisplay+∞\n∞ω2/angb∇acketleftmy/angb∇acket∇ight+iω\n(ω2/angb∇acketleftmy/angb∇acket∇ight+iω)2+ω2\n1e−iωtdω\n2π.(17)\nDetails of calculating the integral in Eq.(17) are pre-\nsented in Appendix C. Taking into account Eq.(17) one\nobtains\n/angb∇acketleftm×ζ′/angb∇acket∇ighty=/angb∇acketleftmzζ′\nx−mxζ′\nz/angb∇acket∇ight=−σ′2L(βω2).(18)\nThefluctuatingspincurrenthasonlyonenonzerocompo-\nnent, i.e., the ycomponent – similarly as the spin pump-\ning current does,\n/angb∇acketleftIy\nfl/angb∇acket∇ight=MsV\nγσ′2L(βω2). (19)We emphasize that when calculating the spin pumping\ncurrent,wedidnotemployalinearizationprocedure. Ac-\ncordingly, the expression for the spin pumping current,\nEq.(13), is valid for an arbitrary deviation of the magne-\ntization from the ground state magnetic order, even for\nthermally assistedmagnetization-reversalinstability pro-\ncesses, meaning the transversal components mx, mycan\nbe arbitrarily large. On the other hand, the expression\nfor the fluctuation spin current, Eq.(19), was obtained\nupon a linearization near the equilibrium point, as de-\nscribed at the beginning of this paragraph. Taking into\naccount the above derived formula Eq.(13) and Eq.(19)\nfor spin pumping and fluctuation currents, respectively,\nwe deduce the following expression of the total spin cur-\nrent:\n/angb∇acketleftIy\ntot/angb∇acket∇ight=MsV\nγL(βω2)/bracketleftbig\nα′σ2ω1\nω2+σ′2/bracketrightbig\n.(20)\nWhenHeff= (0,H0,0), where H0is the external mag-\nnetic field orientedalongthe yaxis[5], then using Eq.(3),\nEq.(10) and Eq.(A2) one obtains from Eq.(20),\n/angb∇acketleftIy\ntot/angb∇acket∇ight= 2α′kBL/parenleftbiggMsV(αH0+(α−ηξ)HR−HSH)\nαkBTm\nF/parenrightbigg\n×/parenleftbiggα(H0+HR+αHSH)Tm\nF\nαH0+(α−ηξ)HR−HSH−TN/parenrightbigg\n, (21)\nwhereη= 0,1 and we inspect in the following the η= 0\ncase.\nWe analyze now in more details Eq.(21) for η= 0 and\nfor several asymptotic cases. Let us begin with the case\nof a negligible spin Hall effect. Assuming a small HSH,\nHSH≪αHR, αH0, we derive from Eq.(21) the spin cur-\nrent in the following two regimes: (i) The low temper-\nature regime, MsV(H0+HR)/kBTm\nF≫1, and (ii) the\nhigh temperature regime, MsV(H0+HR)/kBTm\nF≪1.\nThese two regimes can be equivalently referred to as\nthe high and weak magnetic field limits, respectively.\nIn particular, in the low temperature limit, upon tak-\ning into account the property of the Langevin function,\nL/parenleftbig\nx/parenrightbig\n= coth/parenleftbig\nx/parenrightbig\n−1/x,L/parenleftbig\nx≫1/parenrightbig\n≈1, we find that the\nspin current depends neither on the SO coupling nor on\nthe external magnetic field, /angb∇acketleftIy\ntot/angb∇acket∇ight= 2α′kB(Tm\nF−TN),\nand is solely determined by the temperature bias. In\nthe low-temperature regime (strong magnetic field), the\nmagnetic fluctuations are small and the spin current is\nthen linear in the averaged square of these fluctuations.\nThe latter in turn are linear in the relevant temperature.\nAccordingly, the spin current is proportional to the tem-\nperature bias. In the high-temperature limit (or equiva-\nlently a small magnetic field), the magnetic fluctuations\nare relatively large. Taking into account the asymptotic\nlimit of the Langevin function, L/parenleftbig\nx≪1/parenrightbig\n≈x/3, in\nthe high magnon temperature limit, the spin current is\n/angb∇acketleftIy\ntot/angb∇acket∇ight= (2/3)α′MsV(H0+HR)(Tm\nF−TN)/Tm\nF. Thus,\nthe spin current is reduced by the factor ( H0+HR)/Tm\nF,\nwhich decreases with increasing magnon temperature or\ndecreasing magnetic field. Note, the spin current is en-5\nhanced when the Rashba and the external fields are par-\nallel and is reduced in the antiparallel case. Remark-\nably, the saturation of the spin current is observed in\nthe high magnon temperature limit, Tm\nF≫TN, where\n/angb∇acketleftIy\ntot/angb∇acket∇ight ≈(2/3)α′MsV(H0+HR).\nLet us assume now a sizable spin-Hall field that can-\nnot be neglected. The first specific case is when HSH≈\nα/parenleftbig\nH0+HR/parenrightbig\n. Taking into account the asymptotic limit\nof the Langevin function, L/parenleftbig\nx≪1/parenrightbig\n≈x/3 in the\nhigh magnon temperature limit, MsV/parenleftbig\nαH0+αHR−\nHSH/parenrightbig\n/αkBTm\nF≪1, one finds the following expression\nfor the spin current:\n/angb∇acketleftIy\ntot/angb∇acket∇ight=2\n3α′MsV/bracketleftbigg/parenleftbig\nH0+HR+αHSH/parenrightbig\n−TN\nTm\nF/parenleftbig\nαH0+αHR−HSH/parenrightbig\nα/bracketrightbigg\n. (22)\nSinceα/parenleftbig\nH0+HR/parenrightbig\n≈HSH, the second term for any finite\nTN/Tm\nFin Eq.(22) is small and can be neglected. Thus,\nthe saturated spin current is\n/angb∇acketleftIy\ntot/angb∇acket∇ight=2\n3α′MsV(H0+HR+αHSH).(23)\nThe expression for the saturated spin current, Eq.(23),\ndoesnotdependonthetemperature. However,Eq.(23)is\nvalid only if the magnon temperature Tm\nFis high enough.\nThus, by tuning the applied external magnetic field H0,\na nonzero spin pumping current can be achieved at arbi-\ntrary and even at equal temperatures Tm\nF=TN. For the\nopposite external and Rashba fields, H0=−HR, from\nEq.(21) follows\n/angb∇acketleftIy\ntot/angb∇acket∇ight= 2α′kBL/parenleftbiggMsVHSH\nαkBTm\nF/parenrightbigg/parenleftbig\nα2Tm\nF+TN/parenrightbig\n.(24)\nThe obtained result is remarkable as it shows that the\nnet pumping current is finite at arbitrary nonzero tem-\nperatures Tm\nFandTNin the absence of the applied tem-\nperature gradient.\nFinally we explore the case when the fields are com-\nparableHSH≈HR≈H0, and since α <1,HSH≫\nαHR, αHR.\n/angb∇acketleftIy\ntot/angb∇acket∇ight= 2α′kBL/parenleftbiggMsV(αH0+αHR−HSH)\nαkBTm\nF/parenrightbigg\n×/braceleftbiggα/parenleftbig\nH0+HR+αHSH)Tm\nF\nαH0+αHR−HSH)−TN/bracerightbigg\n.\nThe expression of the total spin current in this case reads\n/angb∇acketleftIy\ntot/angb∇acket∇ight= 2α′kBL/parenleftbiggMsVHSH\nαkBTm\nF/parenrightbigg\n×/braceleftbiggα/parenleftbig\nH0+HR/parenrightbig\nTm\nF\nHSH+TN/bracerightbigg\n. (25)\nIn the low magnon temperature limit we deduce\n/angb∇acketleftIy\ntot/angb∇acket∇ight= 2α′kB/braceleftbiggα/parenleftbig\nH0+HR/parenrightbig\nTm\nF\nHSH+TN/bracerightbigg\n,(26)/s122\n/s121\n/s78/s111/s114/s109/s97/s108\n/s77/s101/s116/s97/s108/s79/s120/s105/s100/s101\n/s70/s101/s114/s114/s111/s109/s97/s103/s110/s101/s116/s105/s99\n/s84\n/s78 /s49/s84\n/s70/s73\n/s102/s108\n/s73\n/s83/s80/s72\n/s82/s44/s32/s83/s72\n/s72/s101/s97/s118/s121\n/s77/s101/s116/s97/s108/s106\n/s97/s120\n/s84\n/s78 /s50/s78/s111/s114/s109/s97/s108\n/s77/s101/s116/s97/s108\n/s73\n/s83/s80/s73\n/s102/s108/s78/s111/s114/s109/s97/s108\n/s77/s101/s116/s97/s108\nFIG. 2. Schematic illustration of the N/F/N System. The\nferromagnetic film is attached to two nonmagnetic layers, N1\non the left and N2on the right side. The temperatures of the\nlayersN1andN2are different. Other notation as in Fig.1.\nwhile in the high magnon temperature limit one finds\n/angb∇acketleftIy\ntot/angb∇acket∇ight=2\n3MsV/braceleftbigg/parenleftbig\nH0+HR/parenrightbig\n+HSH\nαTN\nTFm/bracerightbigg\n.(27)\nAs we see from Eq.(26),(27) the role of the field HSHis\ndifferent. In the lowmagnontemperaturelimit it reduces\nthe spin pumping current, while in the high magnontem-\nperature limit it enhances the fluctuating spin current.\nIn the analytical calculation, we assumed that temper-\natures of the magnon subsystem and normal metal are\nfixed during the process. However, this is an approxima-\ntion because the temperatures of the subsystems change\nslightly during the equilibration process. For illustra-\ntion, we consider the case when the external and Rashba\nfields,H0andHR, are parallel and we neglect the spin\nHall term. Then from Eq.(21) we deduce:\n/angb∇acketleftIy\ntot/angb∇acket∇ight= 2α′kBL/parenleftbiggMsV/parenleftbig\nH0+HR/parenrightbig\nkBTm\nF/parenrightbigg/parenleftbig\nTm\nF−TN/parenrightbig\n.(28)\nApparently the total spin current is zero when Tm\nF=TN.\nHowever, the magnon temperature Tm\nFthat we used for\nderivation of the Fokker-Planck equation, is the initial\nmagnon temperature. The electric current due to the\nRashba field modifies the magnon density and magnon\ntemperature, leading to a slight difference in effective\nmagnon temperatures Tm\nF/parenleftbig\nja/parenrightbig\n−Tm\nF/parenleftbig\nja= 0/parenrightbig\n=δTm\nF.\nThis correction is beyond the Fokker-Planck equation.\nTherefore, due to the temperature correction δTm\nF, in\nthe numerical calculations, we expect to obtain a fi-\nnite net current even when the initial magnon temper-\nature is equal to the temperature of the normal metal,\nTm\nF/parenleftbig\nja= 0/parenrightbig\n=TN.\nIV. SPIN CURRENT IN THE N/F/N\nSTRUCTURE6\nThe same method has been utilized to calculate the\nspin current in the N/F/N system shown schematically\nin Fig.2. We calculate the spin current defined as the\ndifference of spin currents flowing through the two inter-\nfaces,\nItot=Itot1−Itot2 (29)\nItot1=Ifl1+Isp1,Itot2=Ifl2+Isp2.\nHereItot1andItot2is the total spin current in the first\nand second interfaces. The total spin current includes\nfour terms. Two terms Isp1andIsp2describe the spin\npumping currents from the ferromagnetic layer to the\nleftN1and to the right N2metallic layers, respectively.\nIn turn, the terms Ifl1andIfl2describe the fluctuat-\ning spin currents flowing from the left and right metallic\nlayers towards the ferromagnetic layer. We assume that\nthe two metals have different temperatures TN1andTN2.\nThespinpumpingcurrentflowingfromtheferromagnetic\nlayer towards metallic layers ( i= 1,2) reads\n/angb∇acketleftIy\nsp1/angb∇acket∇ight=α′(TN1)MsV\nγ2ω1\nβω2L(βω2),(30)\n/angb∇acketleftIy\nsp2/angb∇acket∇ight=−α′(TN2)MsV\nγ2ω1\nβω2L(βω2).\nIn turn, the fluctuating currents have the components\n/angb∇acketleftIy\nfl1/angb∇acket∇ight= 2α′(TN1)kBTN1L(βω2),(31)\n/angb∇acketleftIy\nfl2/angb∇acket∇ight=−2α′(TN2)kBTN2L(βω2).\nAs we can see from Eq.(30) and Eq.(31), the difference\nin the two components of the spin pumping current and\nfluctuating current is related to the temperature depen-\ndence of the damping constant α′(TN). For convenience\nwe denote α′(TN1) =α′andα′(TN2) =α′+∆α. If the\ndifference between the temperatures of the metals TN1\nandTN2is not too large, the variation of the damping\nconstant ∆ α′is very small |∆α′|/α′<<1 [51, 52]. In\nsuch a case\n/angb∇acketleftIy\ntot1/angb∇acket∇ight= 2α′kBL(βω2)/parenleftbigg\nαω1\nω2Tm\nF+TN1/parenrightbigg\n,(32)\n/angb∇acketleftIy\ntot2/angb∇acket∇ight=−2α′kBL(βω2)/parenleftbigg\nαω1\nω2Tm\nF+TN2/parenrightbigg\n,\nand total spin current:\n/angb∇acketleftIy\ntot/angb∇acket∇ight= 2α′kBL(βω2)/parenleftbigg\n2αω1\nω2Tm\nF+TN1+TN2/parenrightbigg\n.(33)\nWhenHeff= (0,H0,0), then using Eq.(3), Eq.(10) and\nEq.(A2) one obtains from Eq.(33):\n/angb∇acketleftIy\ntot/angb∇acket∇ight= 2α′kBL/parenleftbiggMsV(αH0+(α−ηξ)HR−HSH)\nαkBTm\nF/parenrightbigg\n×/parenleftbigg\n2α(H0+HR+αHSH)Tm\nF\nαH0+(α−ηξ)HR−HSH−TN1−TN2/parenrightbigg\n.(34)Whenη= 0 andHSH≪αH0,αHRfrom Eq.(34) we get:\n/angb∇acketleftIy\ntot/angb∇acket∇ight= 2α′kBL/parenleftbiggMsV(H0+HR)\nkBTm\nF/parenrightbigg\n×(35)\n/parenleftbig\n2Tm\nF−TN1−TN2/parenrightbig\n.\nAgain we see that the larger is the difference between\nmagnon and metal temperatures, the larger is the total\nspin current.\nV. EFFECT OF DM INTERACTION\nIn order to explore the role of DM interaction, we per-\nformed micromagnetic simulations for a finite-size N/F\nsystem. To be more specific, we study Pt/Co/AlO where\nthe Co layer is 500 nm ×50 nm large with a thickness of\n10 nm. The Co layer is sandwiched between Pt and AlO\nfilms. The parameters describe the Co layer: the satura-\ntion magnetization of Ms= 106A/m, and the damping\nconstant α= 0.2. The Rashba field, HR=αRP\nµBµ0Msja,\ncan be estimated assuming P= 0.5,αR= 10−10eVm,\nand the spatially uniform current density jaalong the x\naxis of the order of 1012A/m2. Due to the structure of\nthe Rashba field, an increase in the magnitude of current\ndensityjais formally equivalent to the corresponding in-\ncrease in the SO constant αR. Thus, the dependence of\nthe total spin current on the current density jais equiv-\nalent to the dependence of the total spin current on the\nSO constant αR.\nIn Fig. (3), the total spin current is plotted as a func-\ntion of the electric current density ja(assumed negative),\nfor the case when the external magnetic field H0and the\nRashbafield HRareparallel. When ja= 0, the totalspin\ncurrentissolelythespinSeebeckcurrentandisabsentfor\nequal temperatures, Tm\nF/parenleftbig\nja= 0/parenrightbig\n=TN. However, when\n|ja|>0, the total spin current is nonzero, as well. As it\nwasalreadymentioned above,the reasonofanonzeronet\nspin current is a slight shift of the magnon temperature,\nTm\nF/parenleftbig\n|ja|>0/parenrightbig\n−Tm\nF/parenleftbig\nja= 0/parenrightbig\n=δTm\nFand of the magnon\ndensity, that occur due to the charge current ja. Ap-\nparently, in case of antiparallel Rashba HRand external\nH0magnetic fields, the total net spin current decreases\nwith increasing magnitude of the charge current. This\nnumerical result is consistent with the analytical results\nobtained in the previous section. As we see, the effect of\nthe DM interaction is diverse: when δTm\nF>0 and spin\ncurrent is positive/angbracketleftbig\nIy\ntot/angbracketrightbig\n=/angbracketleftbig\nIy\nsp/angbracketrightbig\n+/angbracketleftbig\nIy\nfl/angbracketrightbig\n>0,/angbracketleftbig\nIy\nfl/angbracketrightbig\n<0\n(i.e. ferromagnetic layer is hotterthan the normal metal\nlayer), the DM interaction enhances the current. How-\never, in the case δTm\nF<0, when fluctuating spin current\nis larger than the spin pumping current, and the total\nnet current is negative/angbracketleftbig\nIy\ntot/angbracketrightbig\n<0, the DM interaction re-\nduces the spin current. This means that the Rashba HR\nfield alwayshas a positive contributionto the spin pump-\ning current. The situation is the same when the spin Hall\neffect is included, see Fig. (4). As one can see, the spin\nHall effect has the opposite effect, it always decreases the7\n/s48/s46/s48\n/s45/s51/s46/s48/s120/s49/s48/s49/s49\n/s45/s54/s46/s48/s120/s49/s48/s49/s49\n/s45/s57/s46/s48/s120/s49/s48/s49/s49/s45/s49/s46/s48/s120/s49/s48/s45/s55/s48/s46/s48/s49/s46/s48/s120/s49/s48/s45/s55/s50/s46/s48/s120/s49/s48/s45/s55/s32\n/s32/s73/s121 /s116/s111/s116/s40/s74/s47/s109/s50\n/s41\n/s106\n/s97/s32/s40/s65/s47/s109/s50\n/s41/s32/s84 /s32/s61/s32/s53/s48/s75/s44/s32 /s68 /s32/s61/s32 /s50/s46/s52/s32/s51\n/s32/s74/s47/s109/s50\n/s32/s84 /s32/s61/s32/s49/s48/s48/s75/s44/s32 /s68 /s32/s61/s32 /s50/s46/s52/s32/s51\n/s32/s74/s47/s109/s50\n/s32/s84 /s32/s61/s32/s49/s53/s48/s75/s44/s32 /s68 /s32/s61/s32 /s50/s46/s52/s32/s51\n/s32/s74/s47/s109/s50\n/s32/s84 /s32/s61/s32/s53/s48/s75/s44/s32 /s68 /s32/s61/s32/s48\n/s32/s84 /s32/s61/s32/s49/s48/s48/s75/s44/s32 /s68 /s32/s61/s32/s48\n/s32/s84 /s32/s61/s32/s49/s53/s48/s75/s44/s32 /s68 /s32/s61/s32/s48\nFIG. 3. Total spin current Iy\ntotin the absence of the spin\nHall effect ( θSH= 0), plotted as a function of the electric\ncurrent density ja. The external field H0and the Rashba\nfieldHRare parallel. The magnon temperature is T≡Tm\nF=\n50 K (squares line), 100 K (circles line) and 150 K (triangles\nline). The DMI constant is assumed D= 0 (solid dots) and\nD=−2.4×10−3J/m2(open dots). The external magnetic\nfieldH0= 4×105A/m and the normal metal temperature\nTN= 50 K are assumed.\nspin pumping current. Therefore for δTm\nF>0 the total\nspin current without the spin Hall effect is larger, while\nforδTm\nF<0 it is smaller.\nFinally, we consider the case when the Rashba field\nHRand the external magnetic H0field are parallel, see\nFig. (4). Note that a switching of the direction of the\nmagnetic field alters the ground state magnetic order.\nTherefore, the spin current changes sign. As we see from\nFig. (5), the spin current increases with the electric cur-\nrent density |ja|. This result is also consistent with the\nanalytical result obtained in the previous section.\nIn order to see the effect of magnetocrystalline\nanisotropy, we repeated the calculations with the\nanisotropy term being included. Results of the calcula-\ntions, plotted in Fig.(6), Fig. (7), and Fig. (8) show that\nthe magnetocrystalline anisotropy has no significant in-\nfluence on the spin current, so the effects discussed above\nhold in the presence of the anisotropy, as well.\nVI. CONCLUSIONS\nIn this paper, we have considered two different het-\nerostructuresconsistingofathinferromagneticfilmsand-\nwiched between heavy-metal and oxide layers. Interfac-\ning the ferromagnetic layer to the heavy metal may re-\nsult in spin Hall torque exerted on the magnetic moment,\nwhile at the interface of the oxide material a spin-orbit\ncoupling of Rashba type emerges. Both factors (the spin\nHall effect and the Rashba spin-orbit coupling) have a\nsignificant influence on the magnetic dynamics, and thus\nalso on the spin pumping current. The total spin cur-/s48/s46/s48\n/s45/s51/s46/s48/s120/s49/s48/s49/s49\n/s45/s54/s46/s48/s120/s49/s48/s49/s49\n/s45/s57/s46/s48/s120/s49/s48/s49/s49/s45/s56/s46/s48/s120/s49/s48/s45/s56/s45/s52/s46/s48/s120/s49/s48/s45/s56/s48/s46/s48/s52/s46/s48/s120/s49/s48/s45/s56/s56/s46/s48/s120/s49/s48/s45/s56\n/s32/s32/s73/s121 /s116/s111/s116\n/s40/s74/s47/s109/s50\n/s41\n/s106\n/s97/s32/s40/s65/s47/s109/s50\n/s41/s32\n/s83/s72/s68\n/s32\n/s83/s72/s68 /s32/s61/s32 /s50/s46/s52/s32/s51\n/s32/s74/s47/s109/s50\n/s32\n/s83/s72/s68\n/s32\n/s83/s72/s68 /s32/s61/s32 /s50/s46/s52/s32/s51\n/s32/s74/s47/s109/s50\nFIG. 4. The total spin current Iy\ntotwithout thecontributionof\nthe spin Hall effect ( θSH= 0, squares line) and with the spin\nHalleffect( θSH= 0.08, triangles line), plottedasafunctionof\nthe electric current density ja. The external field H0and the\nRashba field HRare parallel. The DM interaction constant\nis assumed D= 0 (solid dots) and D=−2.4×10−3J/m2\n(open dots). The external magnetic field H0= 4×105A/m,\nthe magnon temperature Tm\nF= 100 K, and the normal metal\ntemperature TN= 50 K are assumed.\n/s48/s46/s48\n/s45/s50/s46/s48/s120/s49/s48/s49/s48\n/s45/s52/s46/s48/s120/s49/s48/s49/s48\n/s45/s54/s46/s48/s120/s49/s48/s49/s48\n/s45/s56/s46/s48/s120/s49/s48/s49/s48/s45/s54/s46/s48/s120/s49/s48/s45/s55/s45/s52/s46/s48/s120/s49/s48/s45/s55/s45/s50/s46/s48/s120/s49/s48/s45/s55/s48/s46/s48\n/s32/s32/s73/s121 /s116/s111/s116\n/s40/s74/s47/s109/s50\n/s41\n/s106\n/s97/s32/s40/s65/s47/s109/s50\n/s41/s32\n/s83/s72/s68 /s32/s61/s32\n/s32\n/s83/s72/s68 /s32/s61/s32\n/s32\n/s83/s72/s68 /s32/s61/s32 /s50/s46/s52/s32/s51\n/s32/s74/s47/s109/s50\nFIG. 5. The total spin current Iy\ntotwith the spin Hall ef-\nfect (θSH= 0.08, circles line) and without spin Hall effect\n(θSH= 0, squares line), plotted as a function of the electric\ncurrent density ja. The external field H0and the Rashba\nfieldHRare antiparallel. The DMI constant is assumed D=\n0 (solid dots) and D=−2.4×10−3J/m2(open dots). The\nexternal magnetic field is H0=−9×105A/m, the magnon\ntemperature is TF= 100 K, and the temperature of normal\nmetal is TN= 50 K.\nrent crossing the ferromagnetic/normal-metal interface\nhas two contributions: the spin current pumped from\nthe ferromagnetic metal to the normal one, and the spin\nfluctuating currentflowingin the opposite direction. The\nspin Hall effect and the Rashba spin-orbit coupling influ-\nenceonlyspin pumpingcurrentandthereforeimpactalso8\n/s48/s46/s48\n/s45/s51/s46/s48/s120/s49/s48/s49/s49\n/s45/s54/s46/s48/s120/s49/s48/s49/s49\n/s45/s57/s46/s48/s120/s49/s48/s49/s49/s45/s49/s46/s48/s120/s49/s48/s45/s55/s48/s46/s48/s49/s46/s48/s120/s49/s48/s45/s55/s50/s46/s48/s120/s49/s48/s45/s55/s32\n/s32/s73/s121 /s116/s111/s116/s40/s74/s47/s109/s50\n/s41\n/s106\n/s97/s32/s40/s65/s47/s109/s50\n/s41/s32/s84 /s32/s61/s32/s53/s48/s75/s44/s32 /s68 /s32/s61/s32 /s50/s46/s52/s32/s51\n/s32/s74/s47/s109/s50\n/s32/s84 /s32/s61/s32/s49/s48/s48/s75/s44/s32 /s68 /s32/s61/s32 /s50/s46/s52/s32/s51\n/s32/s74/s47/s109/s50\n/s32/s84 /s32/s61/s32/s49/s53/s48/s75/s44/s32 /s68 /s32/s61/s32 /s50/s46/s52/s32/s51\n/s32/s74/s47/s109/s50\n/s32/s84 /s32/s61/s32/s53/s48/s75/s44/s32 /s68 /s32/s61/s32/s48\n/s32/s84 /s32/s61/s32/s49/s48/s48/s75/s44/s32 /s68 /s32/s61/s32/s48\n/s32/s84 /s32/s61/s32/s49/s53/s48/s75/s44/s32 /s68 /s32/s61/s32/s48\nFIG. 6. Total spin current Iy\ntotin the absence of spin Hall\ncontribution ( θSH= 0), plotted as a function of the electric\ncurrent density ja. The local magnetization and the Rashba\nfield are parallel. The magnon temperature is T≡Tm\nF=\n50 K (squares line), 100 K (circles line) and 150 K (tri-\nangles line). The DMI constant D= 0 (solid dots) and\nD=−2.4×10−3J/m2(open dots). The magnetocrystalline\nanisotropy constant Ky= 3×105J/m3and the normal metal\ntemperature TN= 50 K. The effective anisotropy field is\nHani= 2Kymyey/(µ0Ms).\nthe total spin current. We explored the spin Seebeck cur-\nrent beyond the linear response regime, and found the\nfollowing interesting features: if the external magnetic\nfieldH0is parallel to the Rashba SO field HR, then the\nSO coupling enhances the spin current, in the case of\nan antiparallel magnetic field H0and a Rashba SO field\nHR, the SO coupling decreases the spin current. The\nspin Hall effect and the DM interaction always increase\nthe spin pumping current. The results are confirmed an-\nalyticallybymeansofthe Fokker-Planckequationand by\ndirectmicromagneticnumericalcalculationsforaspecific\nsample.\nACKNOWLEDGMENTS\nThis work is supported by the DFG through the SFB\n762 and SFB-TRR 227 and by the National Research\nCenter in Poland as a research project No. DEC-\n2017/27/B/ST3/02881.\nAppendix A: Derivation of the Fokker-Plank\nequation\nFor the derivation of the Fokker-Plank equation, we\nfollow Ref.[53] and use the functional integration method\nin order to average the dynamics over all possible real-\nizations of the random noise field. First we rewrite LLG/s48/s46/s48\n/s45/s51/s46/s48/s120/s49/s48/s49/s49\n/s45/s54/s46/s48/s120/s49/s48/s49/s49\n/s45/s57/s46/s48/s120/s49/s48/s49/s49/s45/s56/s46/s48/s120/s49/s48/s45/s56/s45/s52/s46/s48/s120/s49/s48/s45/s56/s48/s46/s48/s52/s46/s48/s120/s49/s48/s45/s56/s56/s46/s48/s120/s49/s48/s45/s56\n/s32/s32/s73/s121 /s116/s111/s116\n/s40/s74/s47/s109/s50\n/s41\n/s106\n/s97/s32/s40/s65/s47/s109/s50\n/s41/s32\n/s83/s72/s68 /s32/s61/s32\n/s32\n/s83/s72/s68 /s32/s61/s32 /s50/s46/s52/s32/s51\n/s32/s74/s47/s109/s50\n/s32\n/s83/s72/s68 /s32/s61/s32\n/s32\n/s83/s72/s68 /s32/s61/s32 /s50/s46/s52/s32/s51\n/s32/s74/s47/s109/s50\nFIG. 7. Total spin current Iy\ntotin the absence of spin Hall\neffect (θSH= 0, squares line) and with the Hall effect( θSH=\n0.08, triangles line), plotted as a function of the electric cu r-\nrent density ja, for the case when the local magnetization and\nthe Rashba field are parallel. The DM interaction constant\nD= 0 (solid dots) and D=−2.4×10−3J/m2(open dots).\nThe magnetocrystalline anisotropy constant Ky= 3×105\nJ/m3, the magnon temperature Tm\nF= 100 K and the normal\nmetal temperature TN= 50 K. The effective anisotropy field\nisHani= 2Kymyey/(µ0Ms).\nequation (Eq.(5)) in the form:\ndm\ndt=−m×(ω1+ζ(t))+m×m×ω2,(A1)\nwhere\nω1=ωeff+ωR+αωSH,\nω2=−αωeff−αωR+ηξωR+ωSH,(A2)\nωeff=γHeff,ωR=γHR,ωSH=γHSH,\nandγ→γ/(1 +α2). Here ζ(t) is a random Langevin\nfield with the following correlation properties:\n/angb∇acketleftζ(t)/angb∇acket∇ight= 0, (A3)\n/angb∇acketleftζi(t)ζj(t′)/angb∇acket∇ight=σ2δijδ(t−t′). (A4)\nWe introduce the probability distribution function of the\nrandom Gaussian noise ζ:\nF[ζ(t)] =1\nZζexp/bracketleftbigg\n−1\nσ2/integraldisplay+∞\n−∞dτζ2(τ)/bracketrightbigg\n,(A5)\nwhereZζ=/integraltext\nDζFis the noise partition function. With\nthe help of Eq.(A5) the average of any noise functional\nAζcan be written as\n/angb∇acketleftA[ζ]/angb∇acket∇ightζ=/integraldisplay\nDζA[ζ]F[ζ]. (A6)\nConsidering the obvious identity:\nδζατ\nδζβ(t)=δαβδ(τ−t), (A7)9\n/s48/s46/s48\n/s45/s50/s46/s48/s120/s49/s48/s49/s48\n/s45/s52/s46/s48/s120/s49/s48/s49/s48\n/s45/s54/s46/s48/s120/s49/s48/s49/s48\n/s45/s56/s46/s48/s120/s49/s48/s49/s48/s45/s54/s46/s48/s120/s49/s48/s45/s55/s45/s52/s46/s48/s120/s49/s48/s45/s55/s45/s50/s46/s48/s120/s49/s48/s45/s55/s48/s46/s48\n/s32/s32/s73/s121 /s116/s111/s116\n/s40/s74/s47/s109/s50\n/s41\n/s106\n/s97/s32/s40/s65/s47/s109/s50\n/s41/s32\n/s83/s72/s68 /s32/s61/s32\n/s32\n/s83/s72/s68 /s32/s61/s32\n/s32\n/s83/s72/s68 /s32/s61/s32 /s50/s46/s52/s32/s51\n/s32/s74/s47/s109/s50\nFIG. 8. The total spin current Iy\ntotwith the spin Hall effect\n(θSH= 0.08, circles line) and without spin Hall effect ( θSH=\n0, squares line), plotted as a function of the electric curre nt\ndensityja. The local magnetization and the Rashba field are\nantiparallel. DMIconstant D=0(soliddots)and D=−2.4×\n10−3J/m2(open dots). The external magnetic field H0=\n−5×105A/m, the magnetocrystalline anisotropy constant\nKy= 3×105J/m3, the magnon temperature Tm\nF= 100 K,\nand the normal metal temperature TN= 50 K. The effective\nanisotropy field is Hani= 2Kymyey/(µ0Ms).\nwe can calculate first and second variations of F[ζ(t)]:\nδF[ζ]\nδζα(t)=−1\nσ2ζα(t)F[ζ], (A8)\nδ2F[ζ]\nδζα(t)δζβ(t′)= [1\nσ4ζα(t)ζβ(t′)−1\nσ2δαβδ(t−t′)]F[ζ].\n(A9)\nFor arbitrary nwe have:\n/integraldisplay\nDζδnF[ζ]\nδζα1(t1)δζα2(t2)...δζαn(tn)= 0.(A10)\nTakingintoaccountEq.(A8)toEq.(A10), weobtain(A3)\nand (A4). Now, we introduce the distribution function:\nf(N,t) =/angb∇acketleftπ([ζ],t)/angb∇acket∇ightζ,π([ζ],t) =δ(N−m(t)),(A11)\non the sphere |N|= 1. Taking into account the relation\n[53]˙π=−∂π\n∂N˙m(t) and the equation ofmotion, Eq.(A1),\nwe deduce the Fokker-Plank equation:\n∂f\n∂t=∂\n∂N[(N×ω1)−(N×N×ω2)\n+N×/angb∇acketleftζ(t)π([ζ],t)/angb∇acket∇ightζ]. (A12)\nTo calculate /angb∇acketleftζ(t)π([ζ],t)/angb∇acket∇ightζwe use the standard proce-\ndure, discussed for example in Ref. [53], and obtain\n/angb∇acketleftζ(t)π([ζ],t)/angb∇acket∇ightζ=−σ2\n2N×∂f\n∂N.(A13)The Fokker-Plank equation in the final form reads\n∂f\n∂t=∂\n∂N[(N×ω1)\n−(N×N×ω2)−σ2\n2N×∂f\n∂N].(A14)\nThe stationary solution of the Fokker-Plank equation\nwhenω1||ω2has the form\nf(N) =e−2\nσ2/integraltext\ndN·ω2\n/integraltext\ndNe−2\nσ2/integraltext\ndN·ω2. (A15)\nAppendix B: Mean values of magnetization\nExploiting the parametrization:\nmx= sinθcosϕ,my= sinθcosϕ,mz= cosθ,\n0≤θ≤π,0≤ϕ≤2π, (B1)\nand taking into account Eq.(A15) and the parametriza-\ntion Eq.(B1), we can write the probability distribution\nform:\ndw(θ,ϕ) =1\nZf(θ,ϕ)dm, (B2)\nf(θ,ϕ) = exp(−βω2sinθsinϕ),\ndm= sinθdθdϕ,β =2\nσ2.\nHereZ=4πsin(βω2)\nβω2is the partition function. From\nEq.(B2) we can calculate the mean values of the mag-\nnetization:\n/angb∇acketleftmx/angb∇acket∇ight=/angb∇acketleftmz/angb∇acket∇ight= 0,/angb∇acketleftmy/angb∇acket∇ight=−L(βω2) (B3)\n/angb∇acketleftm2\nx/angb∇acket∇ight=/angb∇acketleftm2\nz/angb∇acket∇ight=L(βω2)\nβω2,/angb∇acketleftm2\ny/angb∇acket∇ight= 1−2L(βω2)\nβω2\n/angb∇acketleftmxmy/angb∇acket∇ight=/angb∇acketleftmzmy/angb∇acket∇ight= 0.\nAppendix C: Derivation of Eq.(17)\nTo calculate (17) we utilize the Jourdan’s lemma,\n/integraldisplay+∞\n−∞ω2/angb∇acketleftmy/angb∇acket∇ight+iω\n(ω2/angb∇acketleftmy/angb∇acket∇ight+iω)2+ω2\n1e−iωtdω\n2π(C1)\n=/integraldisplay+∞\n−∞ω2/angb∇acketleftmy/angb∇acket∇ight−iω\n(ω2/angb∇acketleftmy/angb∇acket∇ight−iω)2+ω2\n1eiωtdω\n2π\n=/braceleftbigg\n−1\n2(e−i(ω1+iω2/angbracketleftmy/angbracketright)t+ei(ω1−iω2/angbracketleftmy/angbracketright)t) ift >0,\n0 ift <0.\nThisintegralisdiscontinuousat t= 0, thereforethevalue\n/angb∇acketleftm(t)×ζ′(0)/angb∇acket∇ightyatt= 0 is given by the average of the\nvalues at t= 0±. Therefore, from Eq.(C1) we deduce\nEq.(18)\n/angb∇acketleftm(0)×ζ′(0)/angb∇acket∇ighty=σ′2/angb∇acketleftmy/angb∇acket∇ight=−σ′2L(βω2).10\n[1] Y. A. Bychkov and E. I. Rasbha, P. Zh. Eksp. Teor. Fiz.,\n39, 66 (1984).\n[2] C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 226801\n(2005).\n[3] Y.J. Lin, K. Jimenez-Garcia, I. B. Spielman, Nature 471,\n83 (2011).\n[4] E. I. Rashba and Al. L. Efros, Phys. Rev. Lett. 91,\n126405 (2003).\n[5] I. M. Miron, T. Moore, H. Szambolics, L. D. Buda-\nPrejbeanu, S. Auffret, B. Rodmacq, S. Pizzini, J. Vogel,\nM. Bonfim, A. Schuhl, and G. Gaudin, Nature Mater.\n10, 419 (2011).\n[6] P. P. J. Haazen, E. Mure, J. H. Franken, R. Lavrijsen,\nH. J. M. Swagten, and B. Koopmans, Nature Mater. 12,\n299 (2013).\n[7] A. A. Kovalev, V. A. Zyuzin, and B. Li, Phys. Rev. B\n95, 165106 (2017).\n[8] U. Ritzmann, D. Hinzke, and U. Nowak, Phys. Rev. B\n95, 054411 (2017).\n[9] P. Yan, G. E. W. Bauer, and H. Zhang, Phys. Rev. B 95,\n024417 (2017).\n[10] J. Barker and G. E. W. Bauer, Phys. Rev. Lett. 117,\n217201 (2016).\n[11] K. I. Uchida, H. Adachi, T. Kikkawa, A. Kirihara, M.\nIshida, S. Yorozu, S. Maekawa E. Saitoh Proceedings of\nthe IEEE 104, 1946 (2016).\n[12] E. J. Guo, J. Cramer, A. Kehlberger, C. A. Ferguson, D.\nA. MacLaren, G. Jakob, and M. Kl¨ aui, Phys. Rev. X 6,\n031012 (2016).\n[13] M. Schreier, F. Kramer, H. Huebl, S. Gepr¨ ags, R. Gross,\nS. T. B. Goennenwein, T. Noack, T. Langner, A. A.\nSerga, B. Hillebrands, and V. I. Vasyuchka, Phys. Rev.\nB93, 224430 (2016).\n[14] S. R. Boona, S. J. Watzman, and J. P. Heremans, APL\nMater.4, 104502 (2016).\n[15] V. Basso, E. Ferraro, and M. Piazzi, Phys. Rev. B 94,\n144422 (2016).\n[16] G. Lefkidis and S. A. Reyes, Phys. Rev. B 94, 144433\n(2016).\n[17] S. M. Rezende, A. Azevedo, and R. L. Rodriguez-Suarez,\nJ. Phys. D: Appl. Phys. 51, 174004 (2018).\n[18] M. Bialek, S. Brechet and J. P. Ansermet, J. Phys. D\nAppl. Phys. 51, 164001 (2018).\n[19] J. M. Bartell, C. L. Jermain, S. V. Aradhya, J. T. Brang-\nham, F. Yang, D. C. Ralph, and G. D. Fuchs, Phys. Rev.\nApplied 7, 044004 (2017).\n[20] H. Yua, S. D. Brechetb, J. P. Ansermet, Phys. Lett. A\n381, 825 (2017); L. Karwacki, J. Magn. Magn. Mater.\n441, 764 (2017).\n[21] S. R. Etesami, L. Chotorlishvili, A. Sukhov, and J. Be-\nrakdar Phys. Rev. B 90, 014410 (2014); S. R. Etesami,\nL. Chotorlishvili, J. Berakdar, Appl. Phys. Lett. 107,\n132402 (2015).\n[22] L. Chotorlishvili, X.-G. Wang, Z. Toklikishvili, and J .\nBerakdar, Phys. Rev. B 97, 144409 (2018).\n[23] X.-G. Wang, L. Chotorlishvili, G.-H. Guo, A. Sukhov,\nV. Dugaev, J. Barna´ s, and J. Berakdar, Phys. Rev. B\n94, 104410 (2016); X.-G Wang, L. Chotorlishvili, G.-H\nGuo, J. Berakdar, J. Phys. D: Appl. Phys. 50, 495005\n(2017); X.-G. Wang, L. Chotorlishvili, and J. Berakdar,\nFront. Mater. 4, 19 (2017); X.-G.Wang, Z.-X. Li, Z.-W.Zhou, Y.-Z. Nie, Q.-L Xia, Z.-M Zeng, L. Chotorlishvili,\nJ. Berakdar, and G.-H. Guo, Phys. Rev. B 95, 020414(R)\n(2017).\n[24] X. Wang and A. Manchon, Phys. Rev. Lett. 108, 117201\n(2012); A. Manchon and S. Zhang, Phys. Rev. B 79,\n094422 (2009).\n[25] B. Gu, I. Sugai, T. Ziman, G. Y. Guo, N. Nagaosa, T.\nSeki, K. Takanashi, and S. Maekawa, Phys. Rev. Lett.\n105, 216401 (2010).\n[26] L. Liu, T. Moriyama, D. C. Ralph, and R. A. Buhrman,\nPhys. Rev. Lett. 106, 036601 (2011).\n[27] E. Martinez, S. Emori, and G. S. D. Beach, Appl. Phys.\nLett.103, 072406 (2013).\n[28] K.-W. Kim, S.-M. Seo, J. Ryu, K.-J. Lee, and H.-W. Lee,\nPhys. Rev. B 85, 180404(R) (2012).\n[29] S. Emori, U. Bauer, S.-M. Ahn, E. Martinez, and G. S.\nD. Beach, Nature Mater. 12, 611 (2013).\n[30] R. Lavrijsen, P. P. Haazen, E. Mure, J. H. Franken, J. T.\nKohlhepp, H. J. M. Swagten, and B. Koopmans, Appl.\nPhys. Lett. 100, 262408 (2012).\n[31] Yu. A. Bychkov and E. I. Rashba, JETP Lett. 39, 78\n(1984).\n[32] P. Gambardella and I. M. Miron, Philos. Trans. R. Soc.\nLondon, Ser. A 369, 3175 (2011); I. M. Miron, G.\nGaudin, S. Auffret, B. Rodmacq, A. Schuhl, S. Pizzini,\nJ. Vogel, P. Gambardella, Nature Mater. 9, 230 (2010).\n[33] M. Dyakonov and V. Perel, JETP Lett. 13, 467 (1971).\n[34] J. E. Hirsch, Phys. Rev. Lett. 83, 1834 (1999).\n[35] B. Gu, I. Sugai, T. Ziman, G. Y. Guo, N. Nagaosa, T.\nSeki, K. Takanashi, and S. Maekawa, Phys. Rev. Lett.\n105, 216401 (2010).\n[36] L. Liu, T. Moriyama, D. C. Ralph, and R. A. Buhrman,\nPhys. Rev. Lett. 106, 036601 (2011).\n[37] L. Liu, C.-F. Pai, Y. Li, H. W. Tseng, D. C. Ralph, and\nR. A. Buhrman, Science 336, 555 (2012).\n[38] K. Kondou, H. Sukegawa, S. Mitani, K. Tsukagoshi, and\nS. Kasai, Appl. Phys. Express 5, 073002 (2012).\n[39] S.-M. Seo, K.-W. Kim, J. Ryu, H.-W. Lee, and K.-J. Lee,\nAppl. Phys. Lett. 101, 022405 (2012).\n[40] Z. Li and S. Zhang, Phys. Rev. B 69, 134416 (2004).\n[41] N. Okuma and K. Nomura Phys. Rev. B 95, 115403\n(2017).\n[42] F. Mahfouzi and N. Kioussis Phys. Rev. B 95, 184417\n(2017)\n[43] B. K. Nikolic, K. Dolui, Marko Petrovi´ c, P. Plechaˇ c, T .\nMarkussen, K. Stokbro arXiv:1801.05793.\n[44] E. Saitoh and K. I. Uchida, Spin Seebeck effect, Spin\nCurrent, EditedbySadamichiMaekawa, Sergio O.Valen-\nzuela, Eiji Saitoh, Takashi Kimura, Oxford University\npress, (2012).\n[45] K. I. Uchida, S. Takahashi, K. Harii, J. leda, W.\nKoshibee, K. Ando, S. Maekawa, E. Saitoh, Nature Let-\nters,455, 778 (2008).\n[46] J. Xiao, G. E. W. Bauer, K. Uchida, E. Saitoh, and S.\nMaekawa, Phys. Rev B 81, 214418 (2010).\n[47] J. Foros, A. Brataas, Y. Tserkovnyak and G. E. Bauer,\nPhys. Rev. Lett. 95, 016601 (2005).\n[48] Y. Tserkovniak, A. Brataas and G. E. W. Bauer, Phys.\nRev. Lett. 88, 117601 (2002).\n[49] H. Adachi, K. I. Uchida, E. Saitoh and S. Maekawa, Rep.\nProg. Phys. 76, 036501 (2013).11\n[50] L. Chotorlishvili, Z. Toklikishvili, V. K. Dugaev, J.\nBarna´ s, S. Trimper and J. Berakdar, Phys. Rev B 88,\n144429, (2013).\n[51] X. Joyeux, T. Devolder, Joo-von kim, Y. Gomez de la\nTorre, S. Eimer, C. Chappert, J. Appl. Phys. 110, 063915\n(2011).[52] L. Chotorlishvili, Z. Toklikishvili, S. R. Etesami, V. K.\nDugaev, J. Barna´ s, J. Berakdar, J. Magn. Magn. Mater.\n396, 254 (2015).\n[53] D. A. Garanin, Phys. Rev. B 55, 3050 (1997)."    },    {        "title": "1801.08349v2.Spin_relaxation_anisotropy_in_a_nanowire_quantum_dot_with_strong_spin_orbit_coupling.pdf",        "content": "Spin-relaxation anisotropy in a nanowire quantum dot with strong spin-orbit coupling\nZhi-Hai Liu ( 刘志海)1and Rui Li ( 李睿)2,\u0003\n1Quantum Physics and Quantum Information Division,\nBeijing Computational Science Research Center, Beijing 100193, China\n2Key Laboratory for Microstructural Material Physics of Hebei Province,\nSchool of Science, Yanshan University, Qinhuangdao 066004, China\n(Dated: June 30, 2021)\nWe study the impacts of the magnetic \feld direction on the spin-manipulation and the spin-\nrelaxation in a one-dimensional quantum dot with strong spin-orbit coupling. The energy spectrum\nand the corresponding eigenfunctions in the quantum dot are obtained exactly. We \fnd that no\nmatter how large the spin-orbit coupling is, the electric-dipole spin transition rate as a function of\nthe magnetic \feld direction always has a \u0019periodicity. However, the phonon-induced spin relaxation\nrate as a function of the magnetic \feld direction has a \u0019periodicity only in the weak spin-orbit\ncoupling regime, and the periodicity is prolonged to 2 \u0019in the strong spin-orbit coupling regime.\nI. INTRODUCTION\nIn recent decades, the spin-orbit couling (SOC) in III-\nV semiconductor materials has promoted great advances\nin the studies of spintronics. For instance, the pseudospin\nqubit in a spin-orbit coupled quantum dot is control-\nlable by an external ac electric-\feld via electric-dipole\nspin resonant (EDSR) [1{14]. Furthermore, a spin-orbit\ncoupled nanowire epitaxially covered by superconductors\nhas been proved to be a promising system for search-\ning the Majorana quasiparticles [15{18]. Thus, from the\nviewpoint of both the fundamental science and the prac-\ntical applications, an accurate understanding of the SOC\ne\u000bect in quantum system becomes important.\nThere are two kinds of SOCs in III-V semiconductor\nmaterials: the Dresselhaus SOC generated by the bulk in-\nversion asymmetry and the Rashba SOC induced by the\nstructure inversion asymmetry [19{21]. Moreover, by ex-\nploiting the electric-\feld dependence of the Rashba SOC\nin semiconductor nanostructures [22, 23], it provides a\npromising method for investigating the strong SOC ef-\nfect in quantum system.\nIn semiconductor quantum dot, in order to observe\nthe nontrivial SOC e\u000bect one should \frst break the\ntime-reversal symmetry by applying an external mag-\nnetic \feld [10, 18]. In the presence of both the mag-\nnetic \feld and the SOC, only a few models are exact\nsolvable. For example, an analytic solution for a two-\ndimensional (2D) quantum dot with hard-wall con\fning\npotential was given in Ref. [24{26]. The exact energy\nspectrum and wavefunctions of a 1D square well quan-\ntum dot were given in Ref. [27]. In all of the above mod-\nels, the magnetic \feld direction is \fxed. However, from\nboth the theoretical and the experimental viewpoints, the\nmagnetic \feld direction plays an important role for the\nobservable SOC e\u000bect in quantum dot [8{11, 28{33]. It\nis desirable to clarify the in\ruences of the magnetic \feld\ndirection on the spin properties when SOC is strong.\n\u0003ruili@ysu.edu.cn\n(b)A nanowire quantum dot \nx 0 -a aV(x)\n(a)\nxz\noB\nϕFIG. 1. (a) The schematic diagram of a nanowire quantum\ndot with large SOC. (b) The con\fning potential of the quan-\ntum dot along the wire axis (x-axis), and an external magnetic\n\feldBapplied on the x\u0000zplane.\nIn this paper, we obtain exactly the eigen-energies and\n-functions of an electron con\fned in an 1D quantum dot\nwith large SOC. Our special interest is focused on the\ninterplay between the SOC and the magnetic \feld di-\nrection. When the magnetic \feld direction is rotated\non a plane, we study both the electric-dipole transition\nrate and the phonon-induced relaxation rate between the\nlowest Zeeman sublevels. The anisotropy of the e\u000bective\nLand\u0013 e g-factor is revealed [34]. Here, in order to facilitate\nthe study of the in\ruence of the magnetic-\feld direction\non the SOC e\u000bects, the original g-factor is assumed to be\na constant, i.e., the corresponding bulk value. We \fnd\nthat no matter how large the SOC is, the Rabi frequency\nas a function of the magnetic \feld direction always has a\n\u0019periodicity. While for the phonon-induced spin relax-\nation rate [35], with the increase of the SOC, the period-\nicity of the relaxation rate changes from \u0019in the weak\nSOC regime to 2 \u0019in the strong SOC regime.arXiv:1801.08349v2  [cond-mat.mes-hall]  18 Jul 20182\nII. THE MODEL\nWe consider a spin-orbit coupled quasi-1D quantum\ndot, where an electron con\fned in an in\fnite square well\nand subject to an external Zeeman \feld [11]. The model\nunder consideration is shown schematically in Fig. 1. The\nnanowire material can be chosen as those with strong\nSOC, e.g., InAs and InSb. Note that our approach is\nalso applicable to materials with weak SOC.\nAs illustrated in Fig. 1, the con\fning potential along\nthe axial direction is modeled by an in\fnite square well\nV(x) =(\n1;jxj>a;\n0;jxj\u0014a;(1)\nwhereais the half-width of the potential well. In the\npresence of an external magnetic \feld applied on the x\u0000z\nplaneB=B(cos';sin'), the Hamiltonian describing\nthe nanowire quantum dot reads [28]\nH=p2\n2me+\u000bp\u001bz+\u0001\n2(\u001bxcos'+\u001bzsin') +V(x);\n(2)\nwheremeis the electron e\u000bective mass, p=\u0000i~@=@x is\nthe canonical momentum along the wire, \u000bis the Rashba\nSOC strength, \u0001 = g\u0016BBcorresponds to the Zeeman\nsplitting (with gand\u0016Bbeing the Land\u0013 e factor and the\nBohr magneton, respectively [27, 36]), and 'is the mag-\nnetic \feld direction. It should be noted that, in the pres-\nence of the magnetic \feld, there is a vector potential term\nAx=\u0000(y=2)Bsin'. However, for a quasi-1D quantum\ndot, we can set y= 0 because the motion of the electron\nis only allowed in the axial direction [28, 37].\nWe \frst give the boundary condition of our model. Be-\ncause the con\fning potential is in\fnite outside the well.\nThus, the electron is strictly con\fned inside the well and\nthe wave function is zero at the boundary sites\n\t(\u0006a) = 0; (3)\nwhere \t(x) = [\t\"(x) \t#(x)]Tis the eigenfunction of the\nquantum dot, with \t \";#(x) being its two components.\nIn experiments, the quantum-dot SOC depends mostly\non both the material parameters and the external electric\n\feld [19, 22, 23]. As an explicit example, in our following\ncalculations, we have chosen the InSb as our nanowire\nmaterial [38, 39]. Unless otherwise speci\fed, the model\nparameters are listed in Table. I.\nIII. THE ENERGY SPECTRUM AND THE\nWAVE FUNCTIONS\nInside the well, the Hamiltonian can be reduced to the\nfollowing bulk Hamiltonian [ V(x) = 0 in Eq. (2)]\nHb=p2\n2me+\u000bp\u001bz+\u0001\n2(\u001bxcos'+\u001bzsin'):(4)TABLE I. The relevant parameters of the InSb nanowire\nquantum dot we are considering. Most of the parameter val-\nues are taken from Refs. 38 and 39.\nme=m0ag a (nm) B(T)xso(nm)'\n0:0136\u000050:6 50 0 :05 40\u0018200 0\u00182\u0019\nD(eV)cl(m/s)\u001a(kg=m3)\u001al(kg=m)b\n6.6 3690 5774.7 1.8142 \u000210\u000012\nam0is the electron mass.\nb\u001al=\u001a\u0002\u0019r2\n0, withr0= 10nm being the radius of the nanowire.\nThe eigenstates of the bulk Hamiltonian can be obtained\nby solving the bulk Schr odinger equation Hb (x) =\nE (x). Speci\fcally, there are several kinds of bulk wave\nfunctions with respect to the energy region [27]. In\nour following calculations, we focus on the energy region\nwhere only bulk plane-wave solutions are allowed.\nBy solving the bulk Schr odinger equation, we \fnd there\nare four independent plane-wave solutions\n 1;2(x) =eik1;2x\u0012\ncos\u001e1;2\nsin\u001e1;2\u0013\n; 3;4(x) =eik3;4x\u0012\nsin\u001e3;4\n\u0000cos\u001e3;4\u0013\n;\n(5)\nwherek1;2;3;4is a function of the energy E(the detailed\nexpressions are given in Appendix A) and\n\u001ei=1\n2arctan\u0012\u0001 cos'\n2\u000b~ki+ \u0001 sin'\u0013\n: (6)\nEach independent solution does not satisfy the hard-wall\nboundary conditions in Eq.(3), i.e.,  i(\u0006a)6= 0. How-\never, a linear combination of all the degenerate bulk\nwave functions can ful\fll the boundary condition [24{\n27]. Therefore, the eigenstate of Hamiltonian (2) can be\nwritten as\n\t(x) =4X\ni=1ci i(x); (7)\nwhereciare the coe\u000ecients to be determined. Imposing\nthe hard-wall boundary conditions on \t( x), we obtain\nan equation array for the coe\u000ecients ci:M\u0001C= 0,\nwhereC= [c1c2c3c4]Tand the detailed expression of\nMis given in Appendix A. The matrix Mnow is only a\nfunction of E. The condition that there exists nontrivial\nsolution reads\nDet[M] = 0: (8)\nIndeed, Eq. (8) indicates an implicit transcendental equa-\ntion forE, and the roots of this equation give us the\nenergy spectrum of the quantum dot. Once the energy\nspectrum is obtained, we can obtain the coe\u000ecients ci\nby solving M\u0001C= 0, such that the corresponding eigen-\nfunctions can be obtained.\nLet \t 0(x) and \t 1(x) be the two lowest eigenstates in\nthe quantum dot, and the corresponding energies are E03\n0 � 2�\nϕ1.64\n1.60\n1.56\n1.52\n1.48()    meV E−36\n−40\n−44\n−48\n−52\neffecttive g-factor(a)\n(b)E0E1\nMagnetic field direction\n0.000.060.12()2\n0| |xΨ ()2\n1| |xΨ↑\n↓\n↑\n↓\n−1.0−0.50.00.51.00.000.060.12\n/xa\nFIG. 2. (a) The two lowest energy levels E0andE1as\na function of the magnetic \feld direction 'for the SOC\nlengthxso= 50 nm. The e\u000bective g-factor is de\fned as\nge\u0011(E0\u0000E1)=(B\u0016B). (b) The probability density distri-\nbutions of the lowest two eigenstates j\t0(x)j2andj\t1(x)j2\nfor magnetic direction '=\u0019=6. The solid lines represent the\nspin-up components and the dashed lines correspond to the\nspin-down components.\nandE1, respectively ( E0< E 1). When the SOC length\nxso\u0011~=(me\u000b) is chosen as xso= 50 nm, the lowest\ntwo energy levels as a function of the magnetic \feld di-\nrection'are shown in Fig. 2(a). The e\u000bective g-factor\nge\u0011\u0000(E1\u0000E0)=(B\u0016B) as a function of the angle 'is\nalso given. When the Zeeman \feld is perpendicular to\nthe spin-orbit \feld, i.e., '= 0,\u0019, and 2\u0019, the e\u000bective\nZeeman splitting reaches its minimum and gebecomes\nmaximal [28]. When the Zeeman \feld is parallel to the\nspin-orbital \feld, i.e., '=\u0019=2, 3\u0019=2, the e\u000bective Zee-\nman splitting reaches its maximum and geequals to the\nbulk value ( ge=\u000050:6). We also show the probabil-\nity density distribution in the quantum dot for the two\nlowest eigenstates \t 0(x) and \t 1(x) [see Fig. 2(b)]. As\ncan be seen from the \fgure, for a general magnetic \feld\ndirection'=\u0019=6, the eigenfunction contains both the\nspin-up component and the spin-down component. The\nspin-up component is dominant in the ground state and\nthe spin-down component is dominant in the \frst excited\nstate.\n0 � 2�0.01.2\n0.6\n0.0\n2.24.4(b)xso= 200 nm\nϕ Magnetic field direction(a)\n310 ( )−\n↓↑×Ω310 ( )−\n↓↑×Ωxso= 40 nmmaxima\nminimaFIG. 3. The electric-dipole spin transition rate \n #\", in unit of\neEa=h, as a function of the magnetic \feld direction ', under\ndi\u000berent SOC strengths. Panel (a) show the result for SOC\nlengthxso= 200 nm, and panel (b) show the result for SOC\nlengthxso= 40 nm.\nIV. ELECTRIC-DIPOLE SPIN RESONANCE\nIn the presence of an external magnetic \feld, the res-\nonant electric-dipole spin transition rate in the quan-\ntum dot was usually calculated using approximated wave\nfunctions, either the SOC or the Zeeman \feld was treated\nperturbatively [7, 9{12]. Here, in our exactly solvable\nmodel, the dependence of the Rabi frequency on the mag-\nnetic \fled direction is investigated.\nWhen an alternating electric \feld is applied along the\nx-axis, the electric-driving Hamiltonian reads\nHe\u0000d=p2\n2me+\u000bp\u001bz+g\u0016B\n2B\u0001\u001b+V(x) +eExcos!t;\n(9)\nwhereEand!are the amplitude and frequency of the\nalternating \feld, respectively. Generally, under a small\nac electric \feld the electric-dipole interaction can be\nregarded as a perturbation [9, 10], and the resonant\nelectric-dipole transition rate, i.e., the Rabi frequency,\ncan be calculated:\n\nij=eE\nhZa\n\u0000a\ty\ni(x)x\tj(x)dx; (10)\nwithhbeing the Plank constant. In the rest of this paper,\nwe only consider the electric-dipole transition between\nthe lowest Zeeman sublevels [40].\nThe spin-\rip transition rate \n #\", in unit of eEa=h, as\na function of the magnetic direction is shown in Fig. 3.\nFigure 3(a) shows the result in the weak SOC regime\nxso>a, and Fig. 3(b) shows the result in the strong SOC\nregimexso< a[41]. When the Zeeman \feld is perpen-\ndicular to the spin-orbital \feld, the spin and the orbital\ndegrees of freedom are hybridized to maximal, such that4\nwhen'= 0,\u0019, and 2\u0019, the Rabi frequency reaches its\nmaximum. When the Zeeman \feld is parallel to the spin-\norbit \feld, there is no mixing of the spin and the orbital\ndegrees of freedom, i.e., the operator \u001bzis a conserved\nquantity, such that the Rabi frequency becomes zero at\nthe sites'=\u0019=2 and 3\u0019=2. No matter how large the\nSOC is, we \fnd that the Rabi frequency as a function\nof the magnetic \feld direction always has a \u0019periodicity\n[see Fig. 3].\nV. THE PHONON-INDUCED SPIN\nRELAXATION\nOn the one hand, the presence of SOC facilitates the\nmanipulation of the electron spin, on the other hand, the\nSOC also mediates an spin-phonon interaction, which is\nharmful to the spin lifetime [42{50]. Here we study the\ndependence of the phonon-induced relaxation rate on the\nmagnetic \feld direction in the quantum dot.\nDue to the high excitation energy of the optical\nphonons, the phonon-induced spin relaxation in a semi-\nconductor quantum dot is almost always caused by the\nacoustic phonons [51{55]. Moreover, for the energetically\nclose two levels, i.e., the lowest Zeeman sublevels, the\nmulti-phonon transition induced by anharmonic phonon\nterms can also be ignored [56, 57]. Generally, there are\ntwo kinds of acoustic electron-phonon (e-ph) interactions:\nthe piezoelectric interaction and the deformation poten-\ntial interaction [58{62]. For narrow-gap semiconductor\nmaterials with strong Rashba SOC and large g-factor,\nthe phonon-induced relaxation is dominated by the de-\nformation potential phonons [63, 64]. The Hamiltonian\ndescribing the e-ph deformational interaction reads [65]\nHe\u0000ph=X\nq\u0012~\n2\u001al!qL\u00131=2\neiqxDjqj(bq+by\n\u0000q);(11)\nwhere\u001alis the mass density of the nanowire, Lis the\nlength of the nanowire, Dis the deformation potential\ncoupling strength, b(by) denotes the phonon annihilation\n(creation) operator, qand!qcorrespond to the wave vec-\ntor and angular frequency of the acoustic wave. Thus, the\ntotal Hamiltonian describing the quantum-dot-phonon\nsystem reads\nHt=H+He\u0000ph+X\nq~!qby\nqbq: (12)\nThe phonon-induced relaxation rate between the energy\nlevelsiandjcan be calculated by using the Fermi golden\nrule [52, 53]\n\u0000ij=D2q2\n2\u001al~!qcljWij(q)j2[n(T) + 1]\u000e(~!q\u0000\u0001ij):(13)\nHere \u0001 ij=jEi\u0000Eijis the energy di\u000berence between\nthe relevant levels, clis the wave velocity, n(T) =\n0.500\n0.250\n0.0000.070\n0.035\n0.000\n0 � 2�\nϕ Magnetic field direction() MHz↓↑Γ () MHz↓↑Γ(a)\n(b)xso= 200 nm\nxso= 40 nmmaxima\nminimaFIG. 4. The phonon-induced spin relaxation rate \u0000 #\"as a\nfunction of the magnetic \feld direction ', under di\u000berent SOC\nstrengths. Panel (a) shows the result in the weak SOC regime\nwithxso= 200 nm; while panel (b) shows the result in the\nstrong SOC regime with xso= 40 nm.\n[exp(\u0001 ij=kBT)\u00001]\u00001is the average phonon number,\nand the electron-phonon matrix element Wij(q) is given\nby\nWij(q) =Za\n\u0000a\ty\nj(x)eiqx\ti(x)dx: (14)\nIn the case of low temperature, kBT\u001c\u0001ij, the aver-\nage phonon number n(T)\u00190. Because the exact wave\nfunctions \t i(x) and \t j(x) are already obtained, the tran-\nsition element Wij(q) can be calculated accurately, and\nhence the relaxation rate \u0000 ij.\nMore speci\fcally, the spin relaxation rate \u0000 #\"between\nthe two lowest energy levels as a function of the angle '\nis shown in Fig. 4. When the magnetic \feld is parallel\nto SOC \feld, i.e., '=\u0019=2 and 3\u0019=2, there is no spin\nrelaxation \u0000#\"= 0 due to the fact that \u001bzis a good\nquantum number. For a relatively weak SOC xso= 200\nnm, at the sites '= 0,\u0019, and 2\u0019, the relaxation rate\nreaches its maximal value [see Fig. 4(a)]. The magnetic\n\feld dependence in this case is very similar to that of\nthe Rabi frequency shown in Fig. 3. However, when the\nSOC is strong, i.e., xso= 40 nm, we \fnd that the sites\nfor the maximal relaxation rate are a little bit deviation\nfrom'=\u0019and 2\u0019[see Fig. 4(b)]. This is actually a\nstrong SOC e\u000bect in the quantum dot. Specially, we can\nexpand the electron-phonon operator as follows\neiqx= 1 +iqx+1\n2(iqx)2+\u0001\u0001\u0001: (15)\nWhen the SOC is weak, the contributions from the high-\norder terms (/x2or higher orders) to the transition ele-\nment (14) are negligible, such that the relaxation rate and\nthe Rabi frequency share the same periodicity. When the\nSOC becomes strong, the contributions from the high-\norder terms become important, such that the sites for5\nthe maximal relaxation rate deviate from the sites of the\nweak SOC. Therefore, the relaxation rate as a function\nof the angle 'shows a 2\u0019period, in stark contrast to a\n\u0019period in the weak SOC regime [31, 32, 55].\nVI. CONCLUSION\nIn this paper, we analytically solve the 1D hard-wall\nquantum dot problem in the presence of both the strong\nSOC and the magnetic \feld. The EDSR and the phonon-\ninduced spin relaxation are studied in details with speci\fc\ninterest focused on the interplay between the SOC and\nthe magnetic \feld direction. In di\u000berent SOC regimes, we\n\fnd that the phonon-induced spin relaxation rate showsdi\u000berent periodic oscillation over the magnetic direction.\nThe 2\u0019periodicity can be served as a signature of the\nstrong SOC e\u000bect in quantum dot.\nThe results of our calculations will help clarify\nthe in\ruence of magnetic \feld direction on the spin-\nmanipulation and the spin-relaxation in quantum dot un-\nder the e\u000bect of strong SOC.\nACKNOWLEDGEMENTS\nThis work was supported by National Natural Sci-\nence Foundation of China (grant No. 11404020) and\nPostdoctoral Science Foundation of China (grant No.\n2014M560039).\nAppendix A: The detailed expressions of kiand M\nIn this Appendix, the detailed expressions of the wave vectors ki(i= 1;2;3;4) as a function of the energy Eare\npresented and the detailed form of the matrix Mis also given.\nExpand the bulk Hamiltonian Hbin Eq. (4) in the spin space\b\nj\"i;j#i\t\n, the bulk Schr odinger equation Hb (x) =\nE (x) can be rewritten as\n \np2\n2me+\u0001\n2sin'+\u000bp\u0001\n2cos'\n\u0001\n2cos'p2\n2me\u0000\u0001\n2sin'\u0000\u000bp!\n (x) =E (x); (A1)\nwhere we have used the identities: \u001bzj\"i=j\"i,\u001bzj#i=\u0000j#i ,\u001bxj\"i=j#i, and\u001bxj#i=j\"i. The eigenstate  (x)\nis assumed to have the form of\n (x) =eikx\u0012\n\u001f1\n\u001f2\u0013\n: (A2)\nSubstituting Eq. (A2) into Eq. (A1), we have\n \n~2k2\n2me+\u0001\n2sin'+~\u000bk\u0000E\u0001\n2cos'\n\u0001\n2cos'~2k2\n2me\u0000\u0001\n2sin'\u0000~\u000bk\u0000E!\neikx\u0012\n\u001f1\n\u001f2\u0013\n= 0: (A3)\nMathematically, the condition that there exists a nontrivial solution to Eq. (A3) reads\nDet \n~2k2\n2me+\u0001\n2sin'+~\u000bk\u0000E\u0001\n2cos'\n\u0001\n2cos'~2k2\n2me\u0000\u0001\n2sin'\u0000~\u000bk\u0000E!\n= 0: (A4)\nEssentially, Eq. (A4) implies a quartic equation of k\n~4k4\n4m2e\u0000\u0012~2E\nme+\u000b2~2\u0013\nk2\u0000\u000b~\u0001 sin'k+E2\u0000\u00012\n4= 0: (A5)\nAfter some tedious algebra, Eq. (A5) can be rewritten as a product of two factor\n(k2+k0k+\u0011)(k2\u0000k0k+\u0010) = 0; (A6)\nwhere\n\u0011= (k3\n0+fk0\u0000j)=(2k0);\n\u0010= (k3\n0+fk0+j)=(2k0): (A7)6\nHerek0is a root of the following equation\nk6\n0+ 2fk4\n0+ (f2\u00004r)k2\n0\u0000j2= 0; (A8)\nwhere the parameters\nj=\u00004\u000bm2\ne\u0001 sin'=~3;\nr=\u0000\n4E2\u0000\u00012\u0001\nm2\ne=~4;\nf=\u00004\u0000\nEme+\u000b2m2\ne\u0001\n=~2: (A9)\nOne solution of Eq. (A8) can be written as\nk0=s\n\u00002f\u00002p\nf2+ 12rcos\r\n3; (A10)\nwhere the angle\n\r=1\n3arccos\"\n\u00002f3+ 72fr\u000027j2\n2p\n(f2+ 12r)3#\n: (A11)\nThen, we can obtain four independent solutions to Eq. (A6)\nk1=\u0000k0+p\nk2\n0\u00004\u0011\n2; k 2=\u0000k0\u0000p\nk2\n0\u00004\u0011\n2;\nk3=k0+p\nk2\n0\u00004\u0010\n2; k 4=k0\u0000p\nk2\n0\u00004\u0010\n2; (A12)\nwhere the complicated dependences of the wave vectors kionEcan be re\rected by Eq. (A9).\nIn the following, the detailed expression for the matrix Mis given. Substituting the eigenfunction [given in Eq. (7)]\ninto the boundary condition [see Eq. (3)], we obtain\nc1eik1acos\u001e1+c2eik2acos\u001e2+c3eik3asin\u001e3+c4eik4asin\u001e4= 0;\nc1eik1asin\u001e1+c2eik2asin\u001e2\u0000c3eik3acos\u001e3\u0000c4eik4acos\u001e4= 0;\nc1e\u0000ik1acos\u001e1+c2e\u0000ik2acos\u001e2+c3e\u0000ik3asin\u001e3+c4e\u0000ik4asin\u001e4= 0;\nc1e\u0000ik1asin\u001e1+c2e\u0000ik2asin\u001e2\u0000c3e\u0000ik3acos\u001e3\u0000c4e\u0000ik4acos\u001e4= 0: (A13)\nThe above equation array can be written as matrix equation M\u0001C= 0, where the matrix Mreads\nM=0\nBB@eik1acos\u001e1eik2acos\u001e2eik3asin\u001e3eik4asin\u001e4\neik1asin\u001e1eik2asin\u001e2\u0000eik3acos\u001e3\u0000eik4acos\u001e4\ne\u0000ik1acos\u001e1e\u0000ik2acos\u001e2e\u0000ik3asin\u001e3e\u0000ik4asin\u001e4\ne\u0000ik1asin\u001e1e\u0000ik2asin\u001e2\u0000e\u0000ik3acos\u001e3\u0000e\u0000ik4acos\u001e41\nCCA: (A14)\nIt should be noted that \u001eialso depends on ki(i= 1\u00004) [see Eq. (6)], such that matrix Monly depends on the\nenergyE[see Eqs. (A9) and (A12)]. The condition there exists nontrivial solution for the coe\u000ecients Creads\nDet[M] = 0: (A15)\nSolving this equation, we can obtain the exact energy spectrum of the quantum dot.\n[1] K. C. Nowack, F. H. L. Koppens, Y. V. Nazarov, and L.\nM. K. Vandersypen, Science 318, 1430 (2007).[2] S. Nadj-Perge, S. M. Frolov, E. P. A. M. Bakkers, and L.\nP. Kouwenhoven, Nature 468, 1084 (2010).7\n[3] K. D. Petersson, L. W. McFaul, M. D. Schroer, M. Jung,\nJ. M. Taylor, A. A. Houck, and J. R. Petta, Nature 490,\n380 (2012).\n[4] X. Hu, Y.-X. Liu, and F. Nori, Phys. Rev. B 86, 035314\n(2012).\n[5] E. I. Rashba, Phys. Rev. B 78, 195302 (2008).\n[6] A. Pfund, I. Shorubalko, K. Ensslin, and R. Leturcq,\nPhys. Rev. B 76, 161308(R)(2007).\n[7] P. Stano and J. Fabian, Phys. Rev. B 77, 045310 (2008).\n[8] S. Nadj-Perge, V. S. Pribiag, J. W. G. van den Berg, K.\nZuo, S. R. Plissard, E. P. A. M. Bakkers, S. M. Frolov,\nand L. P. Kouwenhoven, Phys. Rev. Lett. 108, 166801\n(2012).\n[9] E. I. Rashba and Al. L. Efros, Phys. Rev. Lett. 91,\n126405 (2003).\n[10] V. N. Golovach, M. Borhani, and D. Loss, Phys. Rev. B\n74, 165319 (2006).\n[11] R. Li, J. Q. You, C. P. Sun, and F. Nori, Phys. Rev.\nLett. 111, 086805 (2013); R. Li, Phys. Scr. 91, 055801\n(2016).\n[12] J. Fan, Y. Chen, G. Chen, L. Xiao, S. Jia, and F. Nori,\nSci. Rep. 6, 38851 (2016).\n[13] Y. Tokura, W. G. van der Wiel, T. Obata, and S.\nTarucha, Phys. Rev. Lett. 96, 047202 (2006).\n[14] D. V. Khomitsky, L. V. Gulyaev, and E. Ya. Sherman,\nPhys. Rev. B 85, 125312 (2012).\n[15] M. T. Deng, S. Vaitiekenas, E. B. Hansen, J. Danon, M.\nLeijinse, K. Flensberg, J. Nyg\u0017 ard, P. Krogstrup, and C.\nM. Marcus, Science 354, 1557 (2016).\n[16] S. Li, N. Kang, P. Caro\u000b, and H. Q. Xu, Phys. Rev. B\n95, 014515 (2017).\n[17] Roman M. Lutchyn, Jay D. Sau, and S. Das Sarma,\nPhys. Rev. Lett. 105, 077001 (2010).\n[18] Jay D. Sau and S. Das Sarma, Nature Commun. 3, 964\n(2012).\n[19] R. Winkler, Spin-orbit Coupling E\u000bects in Two-\ndimensional Electron and Hole Systems, (Springer,\nBerlin, 2003).\n[20] G. Dresselhaus, Phys. Rev. 100, 580 (1955).\n[21] Yu. A. Bychkov and E. I. Rashba, J. Phys. C 17, 6039\n(1984).\n[22] D. Liang and X. Gao, Nano Lett. 12(6), 3263-3267\n(2012).\n[23] J. Nitta, T. Akazaki, H. Takayanagi, and T. Enoki, Phys.\nRev. Lett. 78, 1335 (1997).\n[24] E. N. Bulgakov and A. F. Sadreev, JETP Lett. 73, 505\n(2001).\n[25] E. Tsitsishvili, G. S. Lozano, and A. O. Gogolin, Phys.\nRev. B 70, 115316 (2004).\n[26] G. A. Intronati, P. I. Tamborenea, D. Weinmann, and R.\nA. Jalabert, Phys. Rev. B 88, 045303 (2013).\n[27] R. Li, Phys. Rev. B 97, 085430 (2018); R. Li, Z.-H. Liu,\nY. Wu, and C. S. Liu, Sci. Rep. 8, 7400 (2018).\n[28] M. P. Nowak and B. Szafran, Phys. Rev. B 87, 205436\n(2013).\n[29] F. Dolcini and L. Dell'Anna, Phys. Rev. B 78, 024518\n(2008).\n[30] V. I. Falko, B. L. Altshuler, and O. Tsyplyatyev, Phys.\nRev. Lett. 95, 076603 (2005).\n[31] P. Scarlino, E. Kawakami, P. Stano, M. Sha\fei, C. Reichl,\nW. Wegscheider, and L. M. K. Vandersypen, Phys. Rev.\nLett. 113, 256802 (2014).\n[32] A. Hofmann, V. F. Maisi, T. Kr ahenmann, C. Reichl, W.\nWegscheider, K. Ensslin, and T. Ihn, Phys. Rev. Lett.119, 176807 (2017).\n[33] J.-T. Hung, E. Marcellina, B. Wang, A. R. Hamilton,\nand D. Culcer, Phys. Rev. B 95, 195316 (2017).\n[34] Y. Tomohiro and M. Eto, International Symposium on\nAccess Spaces: 36-40, (2011).\n[35] M. Borhani and X. Hu, Phys. Rev. B 85, 125132 (2012).\n[36] Y. Y. Wang and M. W. Wu, Phys. Rev. B 77, 125323\n(2008).\n[37] Z.-H. Liu, R. Li, X. Hu, and J. Q. You, Sci. Rep. 8, 2302\n(2018).\n[38] R. de Sousa and S. Das Sarma, Phys. Rev. B 68, 155330\n(2003).\n[39] C. F. Destefani and S. E. Ulloa, Phys. Rev. B 72, 115326\n(2005).\n[40] J. H. Jiang and M. W. Wu, Phys. Rev. B 75, 035307\n(2007).\n[41] X. Liu, X.-J. Liu, and J. Sinova, Phys. Rev. B 84, 035318\n(2011).\n[42] A. V. Khaetskii and Y. V. Nazarov, Phys. Rev. B 61,\n12639 (2000); Phys. Rev. B 64, 125316 (2001).\n[43] A. Kha, R. Joynt, and D. Culcer, Appl. Phys. Lett.107,\n172101 (2015).\n[44] S. Amasha, K. MacLean, Iuliana P. Radu, D. M.\nZumb uhl, M. A. Kastner, M. P. Hanson, and A. C. Gos-\nsard, Phys. Rev. Lett. 100, 046803 (2008).\n[45] A. Pfund, I. Shorubalko, K. Ensslin, and R. Leturcq,\nPhys. Rev. Lett. 99, 036801 (2007).\n[46] T. Meunier, I. T. Vink, L. H. Willems van Beveren, K.-J.\nTielrooij, R. Hanson, F. H. L. Koppens, H. P. Tranitz,\nW. Wegscheider, L. P. Kouwenhoven, and L. M. K. Van-\ndersypen, Phys. Rev. Lett. 98, 126601 (2007).\n[47] A. Bermeister, D. Keith, and D. Culcer, Appl. Phys.\nLett.105, 192102 (2014).\n[48] J. Jing, P. Huang, and X. Hu, Phys. Rev. A 90, 022118\n(2014).\n[49] L. M. Woods, T. L. Reinecke, and Y. Lyanda-Geller,\nPhys. Rev. B 66, 161318(R) (2002).\n[50] D. Q. Wang, O. Klochan, J.-T. Hung, D. Culcer, I. Far-\nrer, D. A. Ritchie, and A. R. Hamilton, Nano Lett.\n16(12), 7685-7689 (2016).\n[51] M. Trif, V. N. Golovach, and D. Loss, Phys. Rev. B 77,\n045434 (2008).\n[52] P. Stano and J. Fabian, Phys. Rev. Lett. 96, 186602\n(2006); Phys. Rev. B 74, 045320 (2006).\n[53] V. N. Golovach, A. Khaetskii, and D. Loss, Phys. Rev.\nLett. 93, 016601 (2004).\n[54] O. Olendski and T. V. Shahbazyan, Phys. Rev. B 75,\n041306(R) (2007).\n[55] C. Segarra, J. Planelles, J. I. Climente, and F. Rajadell,\nNew. J. Phys 17, 033014 (2015).\n[56] T. Inoshita and H. Sakaki, Solid-State Electronics 37,\n1175-1178 (1994).\n[57] I. A. Dmitriev and R. A. Suris, Phys. Rev. B 90, 155431\n(2014).\n[58] Y. Y. Wang and M. W. Wu, Phys. Rev. B 74, 165312\n(2006).\n[59] J. L. Cheng, M. W. Wu, and C. Lu, Phys. Rev. B 69,\n115318 (2004).\n[60] A. Poudel, L. S. Langsjoen, M. G. Vavilov, and R. Joynt,\nPhys. Rev. B 87, 045301 (2013).\n[61] M. Raith, P. Stano, F. Baru\u000ba, and J. Fabian, Phys.\nRev. Lett. 108, 246602 (2012).\n[62] S. Prabhakar, R. Melnik, and L. L. Bonilla, Phys. Rev.\nB87, 235202 (2013).8\n[63] C. L. Romano, G. E. Marques, L. Sanz and A. M. Al-\ncalde, Phys. Rev. B 77, 033301 (2008).\n[64] A. M. Alcalde, Q. Fanyao, and G. E. Marques, Physica\nE (Amsterdam) 20, 228 (2004).[65] A. N. Cleland, Foundations of Nanomechanics: From\nSolid-State Theory to Device Applications, (Springer,\nBerlin, 2003)."    },    {        "title": "2403.09112v1.Spin_Orbit_Coupled_Insulators_and_Metals_on_the_Verge_of_Kitaev_Spin_Liquids_in_Ilmenite_Heterostructures.pdf",        "content": "Spin-Orbit Coupled Insulators and Metals on the Verge of Kitaev Spin Liquids in\nIlmenite Heterostructures\nYi-Feng Zhao,1,∗Seong-Hoon Jang,2and Yukitoshi Motome1,†\n1Department of Applied Physics, University of Tokyo, Bunkyo, Tokyo 113-8656, Japan\n2Institute for Materials Research, Tohoku University, Aoba, Sendai, 980-8577, Japan\nCompetition and cooperation between electron correlation and relativistic spin-orbit coupling give\nrise to diverse exotic quantum phenomena in solids. An illustrative example is spin-orbit entan-\ngled quantum liquids, which exhibit remarkable features such as topological orders and fractional\nexcitations. The Kitaev honeycomb model realizes such interesting states, called the Kitaev spin\nliquids, but its experimental feasibility is still challenging. Here we theoretically investigate hexag-\nonal heterostructures including a candidate for the Kitaev magnets, an ilmenite oxide MgIrO 3, to\nactively manipulate the electronic and magnetic properties toward the realization of the Kitaev spin\nliquids. For three different structure types of ilmenite bilayers MgIrO 3/ATiO 3with A= Mn, Fe,\nCo, and Ni, we obtain the optimized lattice structures, the electronic band structures, the stable\nmagnetic orders, and the effective magnetic couplings, by combining ab initio calculations and the\neffective model approaches. We find that the spin-orbital coupled bands characterized by the pseu-\ndospin jeff= 1/2, crucially important for the Kitaev-type interactions, are retained in the MgIrO 3\nlayer for all the heterostructures, but the magnetic state and the band gap depend on the types of\nheterostructures as well as the Aatoms. In particular, one type becomes metallic irrespective of\nA, while the other two are mostly insulating. We show that the insulating cases provide spin-orbit\ncoupled Mott insulating states with dominant Kitaev-type interactions, accompanied by different\ncombinations of subdominant interactions depending on the heterostructural type and A, while the\nmetallic cases realize spin-orbit coupled metals with various doping rates. Our results indicate that\nthese hexagonal heterostructures are a good platform for engineering electronic and magnetic prop-\nerties of the spin-orbital coupled correlated materials, including the possibility of Majorana Fermi\nsurfaces and topological superconductivity.\nI. INTRODUCTION\nStrong electron correlations, represented as Coulomb\nrepulsion U, play a pivotal role in 3 dtransition metal\ncompounds and lead to a plethora of intriguing phenom-\nena, such as the Mott transition and high-temperature\nsuperconductivity [1, 2]. The other key concept of quan-\ntum materials, the spin-orbit coupling (SOC), repre-\nsented as λ, is a relativistic effect entangling the spin\ndegree of freedom and the orbital motion of electrons,\nwhich is an essential ingredient in the topological insu-\nlators [3, 4]. Beyond their independent effects, synergy\nbetween Uandλhas attracted increasing attention re-\ncently due to the emergence of new states of matter, such\nas axion insulators [5, 6] and topological semimetals [7–\n9]. In general, it is difficult for the SOC to dramatically\ninfluence the electronic properties in 3 dtransition metal\ncompounds since λis much smaller than U. However,\nwhen proceeding to 4 dand 5 dsystems, the dorbitals\nare spatially more spread out, which reduces U, and at\nthe same time, the relativistic effect becomes larger for\nheavier atoms, enhancing λ. Hence, in these systems,\nthe competition and cooperation between Uandλplay\na decisive role in their electronic states and allow us to\naccess the intriguing regime that yields the exotic corre-\nlated states of matter [10].\n∗zyf@g.ecc.u-tokyo.ac.jp\n†motome@ap.t.u-tokyo.ac.jpOne of the striking examples is the spin-orbit coupled\nMott insulator, typically realized in the iridium oxides\nwith Ir4+valence, e.g., Sr 2IrO4[11, 12]. In each Ir ion\nlocated in the center of the IrO 6octahedron, the crystal\nfield energy, which is significantly larger than Uandλ,\nsplits the dorbital manifold into low-energy t2gand high-\nenergy egones. For Sr 2IrO4, five electrons occupy the t2g\norbitals and make the system yield the t5\n2glow-spin state.\nUsually, the partially-filled orbital causes a metallic state\naccording to the conventional band theory, but the insu-\nlating state was observed in experiments [13]. Consider-\ning the large SOC, the t2gmanifold continues to split into\nhigh-energy doublet characterized with the pesudospin\njeff= 1/2 and low-energy quartet with jeff= 3/2; the\nlatter is fully occupied and the former is half filled. Fi-\nnally, a Mott gap is opened in the half-filled jeff= 1/2\nband by U. This accounts for the insulating nature of\nthe system, and the Mott insulating state realized in the\nspin-orbital coupled bands is called the spin-orbit cou-\npled Mott insulator.\nThe quantum spin liquid (QSL), one of the most exotic\nquantum states in the spin-orbit coupled Mott insulators,\nhas received increasing attention due to the emergence\nof remarkable properties, e.g., fractional excitations [14]\nand topological orders [15]. In the QSL, long-range mag-\nnetic ordering is suppressed down to zero temperature\ndue to strong quantum fluctuations, though the localized\nmagnetic moments are quantum entangled [16–19]. The\npresence of fractional quasiparticles that obey the non-\nabelian statistics is not only of great fundamental phys-arXiv:2403.09112v1  [cond-mat.str-el]  14 Mar 20242\nical research, but also promising toward quantum com-\nputation [20]. In general, one route to realizing the QSL\ndepends on geometrical frustration. Indeed, experiments\nhave evidenced several candidates of QSL in antiferro-\nmagnets with lattice structures including triangular unit,\nwhere magnetic frustration is the common feature [21–\n23]. The other route to the QSL is the so-called exchange\nfrustration caused by the conflicting constraints between\nanisotrpic exchange interactions [24]. The strong spin-\norbital entanglement in the spin-orbit coupled Mott in-\nsulators, in general, gives rise to spin anisotropy, offering\na good playground for the exchange frustration, even on\nthe lattices without geometrical frustration.\nThe Kitaev model is a distinctive quantum spin model\nrealizing the exchange frustration [25]. The model is de-\nfined for S=1\n2local magnetic moments on the two-\ndimensional (2D) honeycomb lattice with the Ising-type\nbond-dependent anisotropic interactions, whose Hamil-\ntonian is given by\nH=X\nαX\n⟨i,j⟩αKαSα\niSα\nj, (1)\nwhere α=x, y, z denote the three bonds of the honey-\ncomb lattice, and Kαis the exchange coupling constant\non the αbond; the sum of ⟨i, j⟩αis taken for nearest-\nneighbor sites of iandjon the αbonds. In this model,\nthe orthogonal anisotropies on the x, y, z bonds provide\nthe exchange frustration. Importantly, the model is ex-\nactly solvable, and the ground state is a QSL with frac-\ntional excitations, itinerant Majorana fermions and local-\nized Z 2fluxes [25]. Moreover, the anyonic excitations in\nthis model hold promise for applications in quantum com-\nputing [25, 26]; especially, non-Abelian anyons, which fol-\nlow braiding rules similar to those of conformal blocks for\nthe Ising model, appear under an external magnetic field.\nThe exchange frustration in the Kitaev-type interac-\ntions can be realized in real materials when two con-\nditions are met [27]. First, at each magnetic ion,\nthe spin-orbit coupled Mott insulating state with pseu-\ndospin jeff= 1/2 should be realized. Second, the pseu-\ndospins need to interact with each other through the lig-\nands shared by neighboring octahedra which forms edge-\nsharing network. Over the past decade, enormous efforts\nhave been devoted to exploring the candidate materials\nfor the Kitaev QSL that meet these conditions [28–31].\nFortunately, dominant Kitaev-type interactions were in-\ndeed discovered in several materials, such as A2IrO3with\nA= Li and Na [32–38] and α-RuCl 3[39–46]. Not only the\n5dand 4 dcandidates, 3 dtransition metal compounds like\nCo oxides have also been investigated [47, 48]. In addi-\ntion to these examples with ferromagnetic (FM) Kitaev\ninteractions, the antiferromagnetic (AFM) Kitaev QSL\ncandidates were also predicted by ab initio calculations,\ne.g., f-electron based magnets [49, 50] and polar spin-\norbit coupled Mott insulators α-RuH 3/2X3/2with X=\nCl and Br in the Janus structure [51].\nAlthough a plethora of Kitaev QSL candidates have\nbeen investigated, those realizing the Kitaev QSL in theground state are still missing, since a long-range magnetic\norder due to parasitic interactions such as the Heisenberg\ninteraction hinders the Kitaev QSL. Considerable efforts\nhave been dedicated to suppressing the parasitic inter-\nactions and/or enhancing the Kitaev-type interaction.\nOne way is to utilize heterostructures that incorporate\nthe Kitaev candidates. For example, the Kitaev interac-\ntion is promoted more than 50% for the heterostructure\ncomposed of 2D monolayers of α-RuCl 3and graphene\ncompared to the pristine α-RuCl 3, predicted by ab initio\ncalculations [52]. The heterostructures between a 2D α-\nRuCl 3and three-dimensional (3D) topological insulator\nBiSbTe 1.5Se1.5evidenced the charge transfer phenomena,\nalbeit the magnetic properties were not reported [53].\nWithin the realm of Kitaev heterostructures, remarkably\nfew studies have been designed for the composite 3D/3D\nsuperlattices due to the fabrication challenge [54]. To\ndate, few attempts [55] have been made to investigate\nthe development of the electronic band structure and the\nmagnetic properties, particularly whether the Kitaev in-\nteraction is still dominant when constructing the 3D/3D\nheterostructures using Kitaev QSL candidates.\nIn this paper, we theoretically study the electronic\nand magnetic properties in bilayer heterostructures as\nan interface of 3D/3D superlattices using a recently-\nsynthesized iridium ilmenite MgIrO 3[56] and other il-\nmenite magnets ATiO 3with A= Mn, Fe, Co, and Ni as\nthe substrate. This material choice is motivated by two\nkey considerations: (i) All of these materials have been\nsuccessfully synthesized in experiments, which is helpful\nfor the epitaxial growth of multilayer heterostructures,\nand (ii) the ilmenite MgIrO 3is identified as a good can-\ndidate for Kitaev magnets [57, 58]. We consider three\nconfigurations of heterostructures, classified by type-I, II,\nand III, which are all chemically allowed due to the char-\nacteristics of the alternative layer stacking in ilmenites,\nas shown in Fig. 1. The electronic band structures, mag-\nnetic ground states, and the effective exchange interac-\ntions are systematically investigated by employing the\ncombinatorial of ab initio calculations, construction of\nthe effective tight-binding model, and perturbation ex-\npansions. We find that (i) the spin-orbit coupled bands\ncharacterized by the effective pseudospin jeff= 1/2, a\nkey demand for Kitaev-type interactions, are still pre-\nserved in the MgIrO 3layer for all types of the het-\nerostructures, (ii) type-I and III heterostructures realize\nspin-orbit coupled Mott insulators excluding Mn type-\nIII, whereas type-II ones are spin-orbit coupled metals\nwith doped jeff= 1/2 bands with various carrier concen-\ntrations, and (iii) in almost all of the insulating cases,\nthe Kitaev-type interactions are predominant, whereas\nthe forms and magnitudes of the other parasitic interac-\ntions depend on the specific types of the heterostructures\nand the Aatoms.\nThe structure of the remaining article is as follows.\nIn Sec. II, we provide a detailed description of the opti-\nmized lattice structures of MgIrO 3/ATiO 3heterostruc-\ntures with A= Mn, Fe, Co, and Ni. In Sec. III, we3\nintroduce the methods employed in this work, including\nthe means for structural optimization and the ab initio\ncalculations with LDA+SOC+ Uscheme (Sec. III A), the\nestimates of the effective transfer integral and construc-\ntion of the multiorbital Hubbard model (Sec. III B),\nand the second-order perturbation that is used in the\nestimation of exchange interactions (Sec. III C). In\nSec. IV, we systematically display the results of the elec-\ntronic band structures for three types of heterostruc-\nture MgIrO 3/ATiO 3. In Sec. IV A, we present the elec-\ntronic band structures for the paramagnetic state ob-\ntained by LDA+SOC, together with the projected den-\nsity of states (PDOS) derived by the maximally-localized\nWannier function (MLWF). In Sec. IV B, we discuss the\nstable magnetic states within LDA+SOC+ Uand show\ntheir band structures and PDOS. In Sec. V, we derive\nthe effective exchange coupling constants for the het-\nerostructures for which the LDA+SOC+ Ucalculations\nsuggest spin-orbit coupled Mott insulating nature, and\nshow their location in the phase diagram for the K-J-Γ\nmodel. In Sec. VI, we discuss the possibility of the real-\nization of Majorana Fermi surfaces (Sec. VI A) and exotic\nsuperconducting phases (Sec. VI B) in the heterostruc-\ntures, and the feasibility of these heterostructures in ex-\nperiments (Sec. VI C). Section VII is devoted to the sum-\nmary and prospects. In Appendix A, we present the de-\ntails of the energy difference of the magnetic orders and\nthe effective exchange couplings between the Aions. We\npresent additional information on orbital projected band\nstructures for different specific types of heterostructures\nofAatoms in Appendix B and the band structures of\nmonolayer MgIrO 3in Appendix C.\nII. HETEROSTRUCTURES\nMgIrO 3andATiO 3both belong to ilmenite oxides\nABO3with R¯3 space group. The lattice structure con-\nsists of alternative stacking of honeycomb layers with\nedge-sharing AO6octahedra and those with BO6octa-\nhedra. The common stacking layer structures reduce the\nlattice mismatch to form the heterostructures and also\nmake them feasible to fabricate in experiments. In this\nstudy, we consider heterostructures composed of mono-\nlayer of MgIrO 3andATiO 3with the balance chemical\nformula, to clarify the interface effect on the electronic\nproperties of 3D/3D superlattices. Specifically, we con-\nstruct three types of heterostructures, distinguished by\nthe intersurface atoms and pertinent octahedra in the\nmiddle layer, labeled as type-I, II, and III and shown\nin Fig. 1. For type-I, the top and bottom layers are\nmade of honeycomb networks of IrO 6and TiO 6octa-\nhedra, respectively, whereas the sandwiching honeycomb\nlayer is formed of alternating MgO 6andAO6octahedra.\nIn the type-II, the bottom layer is replaced by AO6hon-\neycomb layer, resulting in a mixture of MgO 6and TiO 6\nin the middle layer. The type-III has a similar constitu-\ntion of top and bottom layers to type-I, while the middle\ntype-III\nIrMgATi(a)\n(b)\n(c)TiMg/AIr\nAMg/Ti\nIrIr\nATitype-I\ntype-II\nacbabcFIG. 1. Schematic pictures of crystal structures for three\ntypes of the heterostructures MgIrO 3/ATiO 3: (a) type-I, (b)\ntype-II, and (c) type-III with A= Mn, Fe, Co, or Ni. The\nleft and right panels show the side views and the bird’s-\neye views, respectively. The type-I is composed of the top\nhoneycomb layer with edge-sharing IrO 6octahedra and the\nbottom honeycomb layer of TiO 6, sandwiching a honeycomb\nlayer of alternating MgO 6andAO6octahedra. In the type-\nII, the bottom is replaced by the AO6honeycomb layer,\nleaving a mixture of MgO 6and TiO 6in the middle, and\nin the type-III, the middle is replaced by the AO6honey-\ncomb layer. In type-I and II, the chemical formula is com-\nmonly given by Mg 2Ir2O6/A2Ti2O6, but that for type-III is\nMgAIr2O6/A2Ti2O6. The crystal structures are embodied by\nVESTA [59].\nlayer is fully composed of AO6octahedra. We inten-\ntionally design these structures to balance their chemical\nvalences and prevent the presence of redundant charges.\nThis can be derived from the chemical formula for each\ntype of the heterostructure, such as Mg 2Ir2O6/A2Ti2O6\nfor type-I and II, and Mg AIr2O6/A2Ti2O6for type-III,\nrespectively.\nWe optimize the lattice structures of the heterostruc-\ntures by the optimization scheme in Sec. III A. The in-\nformation of the stable lattice structures, including the\nin-plane lattice constant, the bond distance between ad-\njacent Ir atoms and O atoms, and the angle between the\nneighboring Ir, O, and Ir atoms, are listed in Table I\nfor three types of the heterostructures with different A\natoms. For comparison, the experimental structures of\nthe bulk MgIrO 3are also listed. We find all the in-plane\nconstants are close to the bulk value of 5.158 ˚A, in which4\nthe maximum and minimum lattice mismatch is 1.2% and\n0.1% respectively of type-II for the Fe atom and type-III\nfor the Mn atom. See also the discussion in Sec. VI C.\nMeanwhile, not only the in-plane constants but also the\nbond distances of Ir atoms are both enlarged as the in-\ncrease of ionic radii of Aatoms. The heterostructural\ntype can significantly influence the bond distance and an-\ngle as well. For example, the angle between neighboring\nIr atoms and the intermediate O atom θIr−O−Ir= 96.69◦\nof type-II for the Ni case largely increases from that of\n94.03◦of the bulk case. In terms of the Ir-Ir bond length\n(dIr−Ir), the length of 2.986 ˚A for the bulk [56] is sub-\nstantially decreased to 2.930 ˚A of type-II for the Fe case.\nMeanwhile, the Ir-O bond length dIr−Oof all cases are en-\nlarged compared with that of the bulk system of 1.942 ˚A,\nin which type-III with Co atoms is maximally influenced.\nTABLE I. Structural information of optimized heterostruc-\ntures for MgIrO 3/ATiO 3(A= Mn, Fe, Co, and Ni): ade-\nnotes the in-plane lattice constant, and dandθrepresent the\nbond distance and the angle between neighboring ions, respec-\ntively. The experimental information on the bulk MgIrO 3is\nalso shown for comparison.\nA type a(˚A) dIr−Ir(˚A) dIr−O(˚A) θIr−O−Ir(◦)\nMnI 5.167 2.986 1.997 95.92\nII 5.127 2.962 2.009 95.57\nIII 5.152 2.977 1.985 95.96\nFeI 5.104 2.951 2.012 94.32\nII 5.068 2.930 2.004 94.00\nIII 5.083 2.940 2.008 93.98\nCoI 5.116 2.961 1.999 94.76\nII 5.113 2.955 1.990 95.26\nIII 5.115 2.960 2.019 94.31\nNiI 5.148 2.977 2.006 95.03\nII 5.181 2.994 1.997 96.69\nIII 5.173 2.994 1.995 94.18\nbulk [56] 5.158 2.986 1.942 94.03\nIII. METHODS\nA.Ab initio calculations\nIn the ab initio calculations, we use the QUANTUM\nESPRESSO [60] based on the density functional the-\nory [61]. The exchange-correlation potential is treated\nas Perdew-Zunger functional by using the projector-\naugmented-wave method [62, 63]. Under the consider-\nation of the SOC effect, the fully relativistic functional is\nutilized for all the atoms except oxygens [64]. To obtain\nstable structures for heterostructures, we initially con-\nstruct a bilayer MgIrO 3structure using the experimental\nstructure for the bulk material. Subsequently, we replace\nthe lower half with ATiO 3layer to create three different\ntypes of heterostructures in Fig. 1. Then, we perform\nfull optimization for both lattice parameters and the po-\nsition of each ion until the residual force becomes lessthan 0.0001 Ry/Bohr. During the optimization proce-\ndure, the structural symmetry is retained as R¯3 space\ngroup. The 20 ˚A thick vacuum is adopted to eliminate\nthe interaction between adjacent layers. The 6 ×6×1 and\n12×12×1 Monkhorst-Pack k-points meshes are utilized\nfor the structural optimization and self-consistent cal-\nculations, respectively [65]. The self-consistent conver-\ngence is set to 10−8Ry and the kinetic energy is chosen\nto 80 Ry for all the structural configurations, which are\nrespectively small and large enough to guarantee accu-\nrate results. To simulate the electron correlation effects\nfor 3delectrons of Aatoms and 5 delectrons of Ir atom,\nwe adopt the LDA+SOC+ Ucalculations [66] with the\nCoulomb repulsions UA= 5.0 eV, 5.3 eV, 4.5 eV, and\n6.45 eV with A= Mn, Fe, Co, and Ni atoms, respec-\ntively, and UIr= 3.0 eV, accompanying with the Hund’s-\nrule coupling with JH/U= 0.1 according to previous\nworks [67, 68].\nBased on the ab initio results, we also obtain the ML-\nWFs by using the kpoints increased to 18 ×18×1 within\nthe Momkhorst-Pack scheme [65]. We select the t2g, 2p,\nand 3 dorbitals respectively of Ir, O, and Aatoms to\nconstruct the MLWFs by employing the Wannier90 [69].\nHerein, we include O 2 pandA3dorbitals due to their\nsignificant contribution near the Fermi level, as detailed\nin Figs. 2-5. By utilizing the MLWFs, we construct the\ntight-binding models and calculate their band structures\nfor comparison. We also calculate the PDOS of each\natomic orbital, including the effective angular momen-\ntum of Ir atoms jeff, from the tight-binding models. We\nconsider the non-relativistic ab initio calculations and rel-\native MLWFs for the estimation of transfer integrals (see\nSec. III B).\nB. Multiorbital Hubbard model\nTo estimate the effective exchange interactions be-\ntween the magnetic Ir ions, we need the effective transfer\nintegrals between neighboring Ir t2gorbitals with the as-\nsociation of O 2 porbitals by constructing MLWFs with\nLDA calculation in the paramagnetic state. It is notice-\nable that the effects of relativistic SOC and electron cor-\nrelation are not taken into account in this calculation\nto circumvent the doublecounting in constructing the ef-\nfective spin models. Specifically, the effective transfer\nintegral tis estimated as [57]\ntiu,jv+X\nptiu,pt∗\njv,p\n∆p−uv. (2)\nThe first term denotes the direct hopping between two\nadjacent Ir atoms, where tiu,jv represents the transfer\nintegral between orbital uat site iand orbital vat site j.\nThe second term denotes the indirect hopping between\nthe two Ir atoms via the shared O 2 porbitals, where\ntiu,prepresents the transfer integral between Ir atom u\norbital at site iand ligand atom porbital, and ∆ p−uv5\nis the harmonic mean of the energy of uandvorbitals\nmeasured from that of porbitals. Herein, we consider\nonly hopping processes between the nearest-neighbor Ir\natoms.\nUsing the effective transfer integrals, we construct a\nmultiorbital Hubbard model with one hole occupying the\nt2gorbitals, whose Hamiltonian is given by\nH=Hhop+Htri+Hsoc+HU. (3)\nThe first term denotes the kinetic energy of the t2gelec-\ntrons as\nHhop=−X\ni,jc†\ni(ˆTγ\nij⊗σ0)cj, (4)\nwhere the matrix ˆTγ\nijincludes the effective transfer in-\ntegrals estimated by Eq. (2), γis the x,y, and zbond\nconnected by neighboring sites iandjwhich belong to\ndifferent honeycomb sublattices, and σ0denotes the 2 ×2\nidentity matrix; c†\ni= (c†\ni,yz,↑,c†\ni,yz,↓,c†\ni,zx,↑,c†\ni,zx,↓,c†\ni,xy,↑,\nc†\ni,xy,↓) denote the creation of one hole in the t2gorbitals\n(yz,zx, and xy) carrying spin up ( ↑) or down ( ↓) at site\ni. The second term in Eq. (3) denotes the trigonal crystal\nsplitting as\nHtri=−X\nic†\ni(ˆTtri⊗σ0)ci, (5)\nwith ˆTtriin the form of\nˆTtri=\n0 ∆ tri∆tri\n∆tri0 ∆ tri\n∆tri∆tri0\n. (6)\nThe third term denotes the SOC as\nHsoc=−λ\n2X\nic†\ni\n0 iσz−iσy\n−iσz0 iσx\niσy−iσx0\nci,(7)\nwhere σ{x,y,z}are Pauli matrices, and λis the SOC co-\nefficient; for instance, λof Ir atom is estimated at about\n0.4 eV [70, 71]. The last term denotes the onsite Coulomb\ninteractions as [72, 73]\nHU=X\niUniu↑niu↓\n+X\ni,u<v,σ[U′niuσniv¯σ+ (U′−JH)niuσnivσ]\n+X\ni,u̸=vJH(c†\niu↑c†\niv↓ciu↓civ↑+c†\niu↑c†\niu↓civ↓civ↑),(8)\nwith niuσ=c†\niuσciuσ; ¯σ=↓(↑) for σ=↑(↓). In Eq. (8),\nthe first, second, and third summations represent the in-\ntraorbital Coulomb interaction in the same orbital with\nopposite spins, the interorbital Coulomb interactions be-\ntween orbital uand orbital v, and the spin-flip and pair-\nhopping processes, respectively.C. Second-order perturbation\nFor Ir5+ions, the t2gmanifold splits into a doublet and\na quartet under the SOC, which are respectively charac-\nterized by the pseudospin jeff= 1/2 and 3 /2. In the\nground state, the latter is fully occupied and the former\nis half filled, which is described by the Kramers doublet\n|jeff= 1/2,+⟩and|jeff= 1/2,−⟩[27, 74]:\n|jeff= 1/2,+⟩=1√\n3(|dyz↓⟩+i|dzx↓⟩+|dxy↑⟩),(9)\n|jeff= 1/2,−⟩=1√\n3(|dyz↑⟩ −i|dzx↑⟩ − | dxy↓⟩).(10)\nWhen the system is in the spin-orbit coupled Mott\ninsulating state with the low-spin d5configuration, the\nlow-energy physics can be described by the pseudospin\njeff= 1/2 degree of freedom. In this case, the effective\nexchange interactions between the pseudospins can be\nestimated by using the second-order perturbation theory\nin the atomic limit, where the three terms in Eq. (3),\nHtri+Hsoc+HU, are regarded as unperturbed Hamil-\ntonian, and Hhopis treated as perturbation. The en-\nergy correction for a neighboring pseudospin pair in the\nsecond-order perturbation is given by\nE(2)\nσ′\ni,σ′\nj;σi,σj=X\nn⟨σ′\niσ′\nj|Hhop|n⟩⟨n|Hhop|σiσj⟩\nE0−En,(11)\nwhere σiandσ′\nidenote the pseudospin + or −at site\ni,|σiσj⟩and⟨σ′\niσ′\nj|is the initial and final states dur-\ning the perturbation process, respectively, and |n⟩is the\nintermediate state with 5 d4-5d6or 5d6-5d4electron con-\nfiguration; E0is the ground state energy for the 5 d5-\n5d5electron configuration, and Enis the energy eigen-\nvalue for the intermediate state |n⟩. Here, |n⟩andEnare\nobtained by diagonalizing the unperturbed Hamiltonian\nHtri+Hsoc+HU.\nThe effective pseudospin Hamiltonian is written in the\nform of\nH=X\nγ=x,y,zX\n⟨i,j⟩ST\niJγ\nijSj, (12)\nwhere i, jdenote the neighboring sites, and γdenotes the\nthree types of Ir-Ir bonds on the MgIrO 6honeycomb layer\nthat are related by C3rotation. The coupling constant\nJγ\nijis explicitly given, e.g., for the zbond as\nJz\nij=\nJΓ Γ′\nΓJΓ′\nΓ′Γ′K\n, (13)\nwhere J,K, Γ, and Γ′represent the coupling con-\nstants for the isotropic Heisenberg interaction, the bond-\ndependent Ising-like Kitaev interaction, and two types of\nthe symmetric off-diagonal interactions. Using the per-\nturbation energy E(2)\nσ′\ni,σ′\nj,σi,σjobtained by Eq. (11), the6\ncoupling constants are calculated as\nJ= 2E(2)\n+,−;−,+, (14)\nK= 2\u0010\nE(2)\n+,+;+,+−E(2)\n+,−;+,−\u0011\n, (15)\nΓ = 2Imn\nE(2)\n−,−;+,+o\n, (16)\nΓ′= 4Ren\nE(2)\n+,+;+,−o\n. (17)\nIV. ELECTRONIC BAND STRUCTURE\nA. LDA+SOC results for paramagnetic states\nLet us begin with the electronic band structures from\nLDA+SOC calculations. The results for MgIrO 3/ATiO 3\nwith A= Mn, Fe, Co, and Ni are shown in Figs. 2(a),\n3(a), 4(a), and 5(a), respectively. Here we display the\nband structures in the paramagnetic state obtained by\ntheab initio calculations (black solid lines) and the\nMLWF analysis (blue dashed lines), together with the\natomic orbitals PDOS including 5 dof Ir atoms, 3 dofA\natoms, and 2 pof O atoms related to IrO 6andAO6oc-\ntahedra. It is obvious that the systems are metallic for\nall the types of heterostructures, regardless of the choice\nofAatoms. The strong SOC splits the t2gbands of Ir\natoms into jeff= 1/2 and jeff= 3/2 bands, as depicted\nin the PDOS. Specifically, the jeff= 1/2 bands are pre-\ndominated to form the metallic bands in the proximity of\nthe Fermi level, covering the energy region almost from\n−1.0 eV to 0 .2 (0.4) eV for A= Mn with type-II (type-I\nand III), from −0.8 eV to 0 .2 eV for A= Fe with all\ntypes, from −0.6 (−1.0) eV to 0 .2 eV for A= Co with\ntype-II (type-I and III), and from −0.4 eV to 0 .4 eV for\nA= Ni with type-I and II, and −0.8 eV to 0 .2 eV with\ntype-III. On the other hand, the jeff= 3/2 bands pri-\nmarily occupy the energy region below the jeff= 1/2\nbands.\nFrom the PDOS in right panels of Figs. 2(a), 3(a),\n4(a), and 5(a), the 3 dbands of Aatoms and the O 2 p\nbands of AO6octahedra simultaneously across the Fermi\nlevel, with hybridization with the Ir 5 dbands. Notably,\nthe energy range of the A3dbands closely overlaps with\nthat of the Ir 5 d jeff= 1/2 bands for A= Mn and Fe, but\nit overlaps with both jeff= 1/2 and jeff= 3/2 manifold\nforA= Co and Ni. As to the O 2 pbands, the energy\nrange of the PDOS overlaps with that for corresponding\nIr 5dorA3dencapsulated in the octahedra, suggesting\nIr-O and A-O hybridization.\nB. LDA+SOC+ Uresults for magnetic states\n1. Magnetic ground states\nThe bulk counterpart of each constituent of the het-\nerostructures exhibits some magnetic long-range ordersin the ground state. In the bulk MgIrO 3, Ir ions show a\nzigzag-type AFM order with the magnetic moments ly-\ning almost within the honeycomb plane [58]. In the bulk\nATiO 3, the A= Mn and Fe ions show N´ eel-type AFM\norders with out-of-plane magnetic moments [75], while\ntheA= Co and Ni ions support N´ eel orders with in-\nplane magnetic moments [75, 76]. It is intriguing to ex-\namine how these magnetic orders in the bulk are affected\nby making heterostructures. We determine the stable\nmagnetic ground states for each heterostructure through\nab initio calculations by including the effect of electron\ncorrelations based on the LDA+SOC+ Umethod. To\ndetermine the potential magnetic ground state for each\nheterostructure, we compare the energy across a total of\n16 magnetic configurations among all combinations of fol-\nlowing types of the magnetic orders: FM and N´ eel orders\nwith in-plane and out-of-plane magnetic moments for A\nlayer, and N´ eel and zigzag orders with in-plane magnetic\nmoments, as well as FM with in-plane and out-of-plane\nmagnetic moments for Ir layer, within a 2 ×2×1 supercell\nsetup.\nTABLE II. Stable magnetic orders obtained by the\nLDA+SOC+ Ucalculations: FM, N´ eel, and zigzag denotes\nthe ferromagnetic, N´ eel-type antiferromagnetic, and zigzag-\ntype antiferromagnetic orders, respectively. While the direc-\ntions of the magnetic moments are all in-plane for the Ir lay-\ners, those for Acan be in-plane (“in”) or out-of-plane (“out”)\ndepending on Aand type of the heterostructure.\nA layer type-I II III\nMnIr in-zigzag in-zigzag in-zigzag\nMn in-N´ eel out-FM in-N´ eel\nFeIr in-N´ eel in-N´ eel in-FM\nFe out-N´ eel in-FM in-FM\nCoIr out-FM in-FM in-N´ eel\nCo in-N´ eel in-N´ eel in-N´ eel\nNiIr in-N´ eel in-zigzag in-zigzag\nNi out-N´ eel in-FM in-N´ eel\nThe results of the most stable magnetic state are listed\nin Table II. The details of the energy comparison are\nshown in Appendix A. In most cases, the Aions show\nN´ eel orders as in the bulk cases, but the direction of mag-\nnetic moments is changed from the bulk in some cases.\nFor instance, type-I and III with A= Mn and type-III\nwith A= Fe switch the moment direction from out-of-\nplane to in-plane. While all the types with A= Co retain\nthe in-plane N´ eel states, type-I with A= Ni is changed\ninto the out-of-plane N´ eel state. The results indicate that\nthe direction of magnetic moments are sensitively altered\nby making the heterostructures with MgIrO 3. Mean-\nwhile, the other cases, type-II with A= Mn, Fe, and Ni\nas well as type-III with A= Fe, are stabilized in the FM\nstate. These results are in good accordance with the ef-\nfective magnetic couplings between the Aions estimated\nby a similar perturbation theory in Sec. III C [77, 78],\nattesting to the reliability of magnetic properties in het-\nerostructures (see Appendix A).7\n!MK!-2-101Energy (eV)\n!MK!-2-101Energy (eV)!MK!-2-101Energy (eV)\n!MK!-2-101Energy (eV)!MK!-2-101Energy (eV)(a)type-Itype-IItype-III\n!MK!-2-101Energy (eV)(b)jeff =1/2 jeff =3/23d2pjeff =1/2 jeff =3/23d2pjeff =1/2 jeff =3/23d2pIrMnOMn2pOIr2p2pIrMnOMnOIrIrMnOMnOIr\nFIG. 2. The band structures of MgIrO 3/MnTiO 3for type-I, II, and III with (a) the LDA+SOC calculations for the paramagnetic\nstate and (b) the LDA+SOC+ Ucalculations for the stable magnetic orders (see Sec. IV B 2). The black lines represent the\nelectronic structure obtained by the ab initio calculations, and the light-blue dashed curves represent the electronic dispersions\nobtained by tight-binding parameters using the MLWFs. The right panels in each figure denote the PDOS for different orbitals\non specific atoms: The red and blue lines represent the jeffmanifolds of Ir atoms, the cyan and orange lines represent the 2 p\norbitals of O atoms in IrO 6octahedra (O Ir) and MnO 6octahedra (O Mn), respectively, and the green line represents the 3 d\norbitals of Mn atoms. The Fermi energy is set to zero.\nThe magnetic states in the Ir layer are more complex\ndue to the possibility of the zigzag state. For A= Mn, the\nmagnetic ground states of the Ir layer in all three types\nprefer the in-plane zigzag state as in the bulk of MgIrO 3.\nIn contrast, for A= Fe, type-I and II are stable in the\nin-plane N´ eel state, but type-III prefers the in-plane FM\nstate. For A= Co, only type-III stabilizes the in-plane\nN´ eel state, while others exhibit the out-of-plane FM state\nfor type-I and the in-plane FM state for type-II. Lastly,\nforA= Ni, both type-II and type-III prefer the in-plane\nzigzag state, while it changes into the in-plane N´ eel state\nin type-I. These results indicate that the magnetic state\nin the Ir honeycomb layer is susceptible to both Aions\nand the heterostructure type. We will discuss this point\nfrom the viewpoint of the effective magnetic couplings in\nSec. V.\n2. Band structures\nWe present the band structures obtained by the\nLDA+SOC+ Ucalculations in Figs. 2(b), 3(b), 4(b), and\n5(b) for A= Mn, Fe, Co, and Ni, respectively. In these\ncalculations, we adopt the stable magnetic states in Ta-\nble II, except for the cases with in-plane zigzag order inthe Ir layer. For the zigzag cases, for simplicity, we re-\nplace them by the in-plane N´ eel solutions, keeping the\nAlayer the same as the stable one. This reduces signifi-\ncantly the computational cost of the MLWF analysis for\nthe zigzag state with a larger supercell. We confirm that\nthe band structures for the N´ eel state are similar to those\nfor the zigzag state, and the energy differences between\nthe two states are not large as shown in Appendix A.\nWhen we turn on Coulomb repulsions for both Ir and\nAatoms, most of the type-I and III heterostructures be-\ncome insulating, except for Mn type-III. The band gaps,\nobtained by Eg=Ec−Ev, are shown in Fig. 6, where Ec\ndenotes the energy of conduction band minimum and Ev\nis that of valence band maximum. In all cases, except for\ntype-I with A= Mn and Ni and type-III with A= Co,\nthe gap is defined by the jeff= 1/2 bands of Ir ions, that\nis, both conduction and valence bands are jeff= 1/2, and\nthejeff= 1/2 bands is half filled. It is worth highlight-\ning that there are four jeff= 1/2 bands, which originate\nfrom different sites of Ir atoms with opposite magnetic\nmoments; in the bulk and monolayer cases they are de-\ngenerate in pair [57], but the degeneracy is lifted in the\nheterostructures and two out of four are occupied in the\nhalf-filled insulating state. Meanwhile, in the cases of\ntype-I with A= Mn and Ni and type-III with A= Co,8\n!MK!-2-101Energy (eV)\n!MK!-2-101Energy (eV)!MK!-2-101Energy (eV)\n!MK!-2-101Energy (eV)!MK!-2-101Energy (eV)(a)type-Itype-IItype-III\n!MK!-2-101Energy (eV)(b)jeff =1/2 jeff =3/2jeff =1/2 jeff =3/2jeff =1/2 jeff =3/23d2pIrFeOFe2pOIrIrFeOFeOIrIrFeOFeOIr3d2p2p3d2p2p\nFIG. 3. The band structures of MgIrO 3/FeTiO 3for type-I, II, and III obtained by (a) the LDA+SOC calculations for the\nparamagnetic state and (b) the LDA+SOC+ Ucalculations for the stable magnetic orders (see Sec. IV B 2). The notations are\ncommon to Fig. 2.\n!MK!-2-101Energy (eV)\n!MK!-2-101Energy (eV)!MK!-2-101Energy (eV)\n!MK!-2-101Energy (eV)!MK!-2-101Energy (eV)(a)type-Itype-IItype-III\n!MK!-2-101Energy (eV)(b)jeff =1/2 jeff =3/23d2pjeff =1/2 jeff =3/23d2pjeff =1/2 jeff =3/23d2pIrCoOCo2pOIr2p2pIrCoOCoOIrIrCoOCoOIr\nFIG. 4. The band structures of MgIrO 3/CoTiO 3for type-I, II, and III obtained by (a) the LDA+SOC calculations for the\nparamagnetic state and (b) the LDA+SOC+ Ucalculations for the stable magnetic orders (see Sec. IV B 2). The notations are\ncommon to Fig. 2.9\n!MK!-2-101Energy (eV)\n!MK!-2-101Energy (eV)!MK!-2-101Energy (eV)\n!MK!-2-101Energy (eV)!MK!-2-101Energy (eV)(a)type-Itype-IItype-III\n!MK!-2-101Energy (eV)(b)jeff =1/2 jeff =3/23d2pjeff =1/2 jeff =3/23d2pjeff =1/2 jeff =3/23d2pNiONi2pIrOIr2p2pNiONiIrOIrNiONiIrOIr\nFIG. 5. The band structures of MgIrO 3/NiTiO 3for type-I, II, and III obtained by (a) the LDA+SOC calculations for the\nparamagnetic state and (b) the LDA+SOC+ Ucalculations for the stable magnetic orders (see Sec. IV B 2). The notations are\ncommon to Fig. 2.\nthe 3 dbands of Aions hybridized with O 2 porbitals\nintervene near the Fermi level, and the gap is defined\nbetween the jeff= 1/2 and 3 dbands. In these cases,\nhowever, a larger gap is well preserved in the jeff= 1/2\nbands, as shown in Figs. 2(b), 4(b), and 5(b). We note\nthat the Co type-III is a further exception since the gap\nopens between the highest-energy jeff= 1/2 band and\nthe Co 3 dband; the jeff= 1/2 bands are not half filled\nbut 3 /4 filled (see Appendix B). We plot the band gap\ndefined by the jeff= 1/2 bands by red asterisks in Fig. 6,\nincluding the 3 /4-filled case for the Co type-III.\nThese results clearly indicate that the inclusion of both\nSOC and Ueffects results in the opening of a band gap\nin the jeff= 1/2 bands at half filling in type-I and III\nheterostructures excluding Mn type-III and Co type-III.\nThis suggests the formation of spin-orbit coupled Mott\ninsulators in the Ir honeycomb layers, which are corner-\nstone of the Kitaev candidate materials [27], motivating\nus to further investigate the effective exchange interac-\ntion in Sec. V. The Co type-III is in an interesting state\nwith 3 /4 filling of jeff= 1/2 bands, but we exclude it\nfrom the following analysis of the effective exchange in-\nteractions in Sec. V.\nDistinct from the emergence of spin-orbit coupled\nMott insulator, the LDA+SOC+ Uband structures show\nmetallic states for type-II heterostructures. The jeff=\n1/2 bands do not show a clear gap and cross the Fermi\nlevel, resulting in the spin-orbit coupled metals. Notably,\nin all cases, the upper jeff= 1/2 bands are partiallyTABLE III. Electronic states of each heterostructure obtained\nby the LDA+SOC+ Ucalculations. SOCI and SOCM denote\nspin-orbit coupled insulator and metal, respectively. e and h\nin the parentheses represent the carriers in the SOCM doped\nto the mother SOCI. The asterisk for the Co type-III indicates\nthat the system is in the 3 /4-filled insulating state of the\njeff= 1/2 bands.\nA type-I II III\nMn SOCI SOCM(e) SOCM(h)\nFe SOCI SOCM(e) SOCI\nCo SOCI SOCM(e) SOCI*\nNi SOCI SOCM(e) SOCI\ndoped, realizing electron-doped Mott insulators. The\ndoping rate varies with Aatoms. We note that the type-\nIII heterostructure of Mn also exhibits a metallic state,\nbut in this case, holes are doped to the lower jeff= 1/2\nband. See Appendix B for the orbital projected band\nstructures.\nWe summarize the electronic states in Table III. The\ntype-I and III heterosuructures are all spin-orbit cou-\npled Mott insulators (SOCI) except for the type-III Mn\ncase, while the type-II are all spin-orbit coupled met-\nals (SOCM). For the SOCM, we also indicate the nature\nof carriers, electrons or holes; the type-II heterostruc-\ntures are all electron doped, while the type-III Mn is\nhole doped.10\nMn Fe Co Ni\nAtoms00.10.20.30.40.50.60.7Band gap (eV)type-I\ntype-III\ntype-I:  jeff=1/2\ntype-III:  jeff=1/2\nFIG. 6. The band gap in the insulating states for type-I and\nIII obtained by the LDA+SOC+ Ucalculations. The open\ncircles represent the gaps opening in the jeff= 1/2 bands for\nMn and Ni of type-I and Co of type-III. Note that the Co\ntype-III is exceptional since the jeff= 1/2 bands are at 3 /4\nfilling, rather than half filling in the other cases; see the text\nfor details.\nV. EXCHANGE INTERACTIONS\nThe electronic band structure analysis reveals the\nemergence of the spin-orbit coupled Mott insulating state\nin the MgIrO 6layer of type-I and III heterostructures,\nexcept for the Mn and Co type-III. In these cases, the\nlow-energy physics is expected to be described by effec-\ntive pseudospin models with dominant Kitaev-type inter-\nactions [27]. The effective exchange interactions can be\nderived by means of the second-order perturbation for the\nmultiorbital Hubbard model (Secs. III B and III C). We\nsetUIr= 3.0 eV, JH/UIr= 0.1, and λ= 0.4 eV in the\nperturbation calculations. The results are summarized\nin Fig. 7. For comparison, we also plot the estimates for\nmonolayer and bulk MgIrO 3. For the bulk case, its po-\ntential for hosting Kitaev spin liquids was demonstrated\nin the previous study [57]. Regarding the monolayer,\nwe obtain the results from the band structures shown\nin Appendix C, which illustrate the preservation of the\njeff= 1/2 manifold and spin-orbit coupled insulating na-\nture.\nIn type-I heterostructures, the dominant interaction\nis the FM Kitaev interaction K < 0 for almost all A\natoms, except for Mn. Particularly for A= Ni, the abso-\nlute value of Kis significantly larger than the others,\neven considerable when compared with the monolayer\nand bulk MgIrO 3. The subdominant interaction is the\noff-diagonal symmetric interaction Γ >0. The other off-\ndiagonal symmetric interaction Γ′as well as the Heisen-\nberg interaction Jis weaker than them. In the Mn case,\nall the interactions are exceptionally weak, presumably\nbecause of the intervening Mn 3 dband and its hybridiza-tion with the Ir jeff= 1/2 bands. Meanwhile, for the\ntype-III heterostructures, since the Mn and Co cases ex-\nhibit a metallic state and 3 /4 occupation of jeff= 1/2\nbands, respectively, we only calculate the effective mag-\nnetic constants for Fe and Ni. In these cases also, the\ndominant interaction is the FM K, accompanied by the\nsubdominant Γ interaction, as shown in Fig. 7(a).\nThus, in all cases except the Mn type-I heterostructure,\nthe dominant magnetic interaction in the spin-orbit cou-\npled Mott insulating state in the Ir honeycomb layer is\neffectively described by the FM Kitaev interaction. Since\nΓ′is smaller than the other exchange constants, the low-\nenergy magnetic properties can be well described by the\ngeneric K-J-Γ model [79, 80], which has been widely and\nsuccessfully applied to study the Kitaev QSLs. We sum-\nmarize the obtained effective exchange interactions of K,\nJ, and Γ by using the parametrization\n(K, J, Γ) =N(sinθsinϕ,sinθcosϕ,cosθ), (18)\nwhere N= (K2+J2+ Γ2)−1/2is the normalization fac-\ntor. Figure 7(b) presents the results except for Mn type-I.\nOur heterostructures distribute in the region near the FM\nKonly case ( θ=π/2 and ϕ= 3π/2). We find a general\ntrend that larger Aatoms make the systems closer to the\nFMKonly case; the best is found for Ni type-I and III. In\nthe previous studies for the K-J-Γ model [79, 80], a keen\ncompetition between different magnetic phases was found\nin this region, which does not allow one to conclude the\nstable ground state in the thermodynamic limit. Given\nthat this region appears to be connected to the solvable\npoint for the FM Kitaev QSL, our heterostructures pro-\nvide a promising platform for investigating the Kitaev\nQSL physics and related phase competition by finely tun-\ning the magnetic interactions via the proximity effect in\nthe heterostructures.\nVI. DISCUSSION\nOur systematic study of ilmenite heterostructures\nMgIrO 3/ATiO 3with A= Mn, Fe, Co, and Ni reveals\ntheir fascinating electronic and magnetic properties. The\nheterostructures in the paramagnetic state are metallic\nin terms of band structures obtained by LDA+SOC, re-\ngardless of types and Aatoms. When incorporating the\neffect of electron correlation by the LDA+SOC+ Ucal-\nculations, type-II heterostructures remain metallic across\nentire Aatoms, whereas type-I and III heterostrcutures\nturn into insulating states, except for Mn type-III. As\na consequence, the electronic states of heterostructures\nare classified into the spin-orbit coupled insulators and\nmetals, each holding unique properties. The insulating\ncases possess the jeff= 1/2 pseudospin degree of freedom,\nand furnish a fertile playground to investigate the Kitaev\nQSL. In these cases, however, due to the magnetic prox-\nimity effects from the Alayer, we may expect interesting\nmodification of the QSL state, as discussed in Sec. VI A\nbelow. Meanwhile, the metallic cases open avenues for11\n-150-100-50050Coupling const. (meV)(b)\n◆◆◆◆★★▲▲▲▲♡♡♡♡AFM K only\nFM K onlyFM J onlyAFM J onlyFe-ICo-INi-IFe-IIINi-IIIbulkmonolayerΓ only𝜙 = 0𝜙 =𝜋/2𝜙 = 𝜋\n𝜙 = 3𝜋/2(a)\nCoAtomFeNimonolayerbulkJKΓΓ′\nMn𝜃 = 𝜋/2𝜃 = 3𝜋/8𝜃 = 𝜋/4𝜃 = 𝜋/8\nFIG. 7. The effective magnetic constants of heterostructures\nfor different Aatoms ( A= Mn, Fe, Co, and Ni) of (a) type-I\n(solid line with pentagram) and type-III (dashed line with hol-\nlow pentagram). For comparison, we also show the results for\nmonolayer and bulk. In (b), we summarize the results of K,\nJ, and Γ in (a) except for Mn type-I by using the parametriza-\ntion in Eq. (18). The parameters of the intraorbital Coulomb\ninteraction, Hund’s coupling, and spin-orbit coupling are set\ntoUIr= 3.0 eV, JH/UIr= 0.1, and λ= 0.4 eV, respectively,\nin the perturbation calculations.\nexploring spin-orbit coupled metals, relatively scarce in\nstrongly correlated systems [81–85]. In Sec. VI B, we dis-\ncuss the possibility of exotic superconductivity in our self-\ndoped heterostructures. In addition, we discuss the fea-\nsibility of fabrication of these heterostructures and iden-\ntification of the Kitaev QSL nature in experiments in\nSec. VI C.\nA. Majorana Fermi surface by magnetic proximity\neffect\nIn the pure Kitaev model, the spins are fractionalized\ninto itinerant Majorana fermions and localized Z2gauge\nfluxes [25]. The former has gapless excitations at thenodal points of the Dirac-like dispersions at the K and\nK’ points on the Brillouin zone edges, while the latter is\ngapped with no dispersion. When an external magnetic\nfield is applied, the Dirac-like nodes of Majorana fermions\nare gapped out, resulting in the emergence of quasipar-\nticles obeying non-Abelian statistics [25]. Beyond the\nuniform magnetic field, the Majorana dispersions are fur-\nther modulated by an electric field and a staggered mag-\nnetic field [86, 87]. For instance, with the existence of\nthe staggered magnetic field, the Dirac-like nodes at the\nK and K’ points are shifted in the opposite directions\nin energy to each other, leading to the formation of the\nMajorana Fermi surfaces. Moreover, the introduction of\nboth uniform and staggered magnetic fields can lead to\nfurther distinct modulations of the Majorana Fermi sur-\nfaces around the K and K’ points, which are manifested\nby nonreciprocal thermal transport carried by the Majo-\nrana fermions [87].\nIn our heterostructures of type-I and III, the Alayer\nsupports a N´ eel order in most cases (Table II). It can\ngenerate an internal staggered magnetic field applied to\nthe Ir layer through the magnetic proximity effect. This\nmimics the situations discussed above, and hence, it may\nresult in the Majorana Fermi surfaces in the possible Ki-\ntaev QSL in the Ir layer. The combination of the uniform\nand staggered magnetic fields could also be realized by\napplying an external magnetic field to these heterostruc-\ntures. Thus, the ilmenite heterostructures in proximity\nto the Kitaev QSL in the Ir layer hold promise for the\nformation of Majorana Fermi surfaces and resultant ex-\notic thermal transport phenomena, providing a unique\nplatform for identifying the fractional excitations in the\nKitaev QSL.\nB. Exotic superconductivity by carrier doping\nQSLs have long been discussed as mother states of ex-\notic superconductivity [19, 88, 89]. There, the introduc-\ntion of mobile carriers to insulating QSLs possibly in-\nduces superconductivity in which the Cooper pairs are\nmediated by strong spin entanglement in the QSLs. A\nrepresentative example discussed for a long time is high-\nTccuprates; here, the d-wave superconductivity is in-\nduced by carrier doping to the undoped antiferromag-\nnetic state that is close to a QSL of so-called resonat-\ning valence bond (RVB) type [89–91]. A similar ex-\notic superconducting state was also discussed for an irid-\nium oxide Sr 2IrO4with spin-orbital entangled jeff= 1/2\nbands [92, 93]. Carrier doped Kitaev QSLs have also gar-\nnered extensive attention due to its potential accessibil-\nity to unconventional superconductivity that may possess\nmore intricate paring from the unique QSL properties.\nIt was reported that doping into the Kitaev model with\nadditional Heisenberg interactions ( K-Jmodel) led to a\nspin-triplet topological superconducting state [94], where\nthe pairing nature is contingent upon the doping concen-\ntration. Furthermore, the competition between Kand12\nJalso significantly impacts the superconducting state;\nfor example, Kprefers a p-wave superconducting state,\nwhereas Jtends to favor a d-wave one [95, 96]. Even\ntopological superconductivity is observed in an extended\nK-J-Γ model [97].\nIn the present work, we found metallic states in the\nspin-orbital coupled jeff= 1/2 bands in the Ir layer for all\ntype-II heterostructures (electron doping) and the type-\nIII Mn heterostructure (hole doping) (see Table III). Be-\nsides, in the type-III Co heterostructure, electron doping\noccurs in the Ir layer, resulting in the 3 /4-filled insulat-\ning state in the jeff= 1/2 bands. These appealing results\nsuggest that our ilmenite heterostructures offer a plat-\nform for studying exotic metallic and superconducting\n(even topological) properties with great flexibility by var-\nious choices of materials combination, which have been\nscarcely realized in the bulk systems.\nC. Experimental feasibility\nThe bulk compounds of ilmenite ATiO 3with A= Mn,\nFe, Co, and Ni have been successfully synthesized and\ninvestigated for over half a century due to its fruitful\nmagnetic and novel electronic properties [75, 76, 98–100].\nTechnically, the Fe case, however, is more challenging\ncompared to the others, as its synthesis needs very high\npressure and high temperature conditions [101, 102]. Be-\nsides, the iridium ilmenite MgIrO 3has also been syn-\nthesized as a power sample, where a magnetic phase\ntransition was observed at 31 .8 K [56]. The experimen-\ntal lattice parameters are 5 .14˚A for ATiO 3with A=\nMn [75, 103–105], 5 .09˚A for A= Fe [101, 106, 107],\n5.06˚A for A= Co [76, 108, 109], and 5 .03˚A for A=\nNi [75, 107, 110, 111], respectively, as well as that is\n5.16˚A for MgIrO 3. The relatively small lattice mismatch\nbetween these materials also ensure the possibility of\ncombining them to create heterostructure with different\ncompounds. Indeed, we demonstrated this in Sec. II; see\nTable I. More excitingly, the IrO 6honeycomb lattice has\nbeen successfully incorporated into the ilmenite MnTiO 3\nwith the formation of several Mn-Ir-O layers [54]. This\ndevelopment lightens the fabrication of a supercell be-\ntween MgIrO 3andATiO 3.\nThe verification of Kitaev QSL poses a significant chal-\nlenge even though the successful synthesis of aforemen-\ntioned heterostructures. First of all, it is crucial to iden-\ntify the spin-orbital entangled electronic states with the\nformation of the jeff= 1/2 bands in these heterostruc-\ntures, as they are essential for the Kitaev interactions\nbetween the pseudospins. Several detectable spectro-\nscopic techniques are useful for this purpose, applicable\nto both bulk and heterostructures [11, 34, 35, 39, 42,\n43, 46, 112, 113]. Even the Kitaev exchange interaction\ncan be directly uncovered in experiment [36, 37]. How-\never, the key challenge lies in probing the intrinsic prop-\nerties of Kitaev QSL, such as fractional spin excitations.\nThus far, despite cooperative studies between theoriesand experiments on, for instance, dynamical spin struc-\nture factors [44, 114–119] and the thermal Hall effect and\nits half quantization [45, 120–122], have been developed\nto identify the fractional excitations in Kitaev QSL, di-\nrectly applying them on the heterostructures is still a\ngreat challenge. A promising experimental tool would be\nthe Raman spectroscopy, given its successful application\nto not only bulk [41, 123, 124] but also atomically thin\nlayers [125, 126]. The signals might be enhanced by pil-\ning up the heterostructures. Besides, many proposals for\nprobing the Kitaev QSL in thin films and heterostruc-\ntures have been recently made, such as local probes like\nscanning tunneling microscopy (STM) and atomic force\nmicroscopy (AFM) [127–131] as well as the spin Seebeck\neffect [132]. Additionally, as mentioned in Sec. VI A, the\nobservation of the Majorana Fermi surfaces by thermal\ntransport measurements in some particular heterostruc-\ntures is also interesting.\nVII. SUMMARY\nTo summarize, we have conducted a systematic inves-\ntigation of the electronic and magnetic properties of the\nbilayer structures composed by the ilmenites ATiO 3with\nA= Mn, Fe, Co, and Ni, in combination with the can-\ndidate for Kitaev magnets MgIrO 3. We have designed\nand labeled three types of heterostructure, denoted as\ntype-I, II, and III, distinguished by the atomic config-\nurations at the interface. Our analysis of the electronic\nband structures based on the ab initio calculations has re-\nvealed that the spin-orbital coupled bands characterized\nby the pseudospin jeff= 1/2, one of the fundamental\ncomponent for the Kitaev interactions, is retained in the\nMgIrO 3layer for all the types of heterostructures. We\nfound that the MgIrO 3/ATiO 3heterostructures of type-\nI and III are mostly spin-orbit coupled insulators, while\nthose of type-II are spin-orbit coupled metals, irrespec-\ntive of the Aatoms. In the insulating heterostructures of\ntype-I and III, based on the construction of the multior-\nbital Hubbard models and the second-order perturbation\ntheory, we further found that the low-energy magnetic\nproperties can be described by the jeff= 1/2 pseudospin\nmodels in which the estimated exchange interactions are\ndominated by the Kitaev-type interaction. We showed\nthat the parasitic subdominant interactions depend on\nthe type of the heterostructure as well as the Aatoms,\noffering the playground for systematic studies of the Ki-\ntaev spin liquid behaviors. Moreover, the stable N´ eel or-\nder in the ATiO 3layer acts as a staggered magnetic field\nthrough the magnetic proximity effect, leading to the\npotential realization of Majorana Fermi surfaces in the\nMgIrO 3layer. Meanwhile, in the metallic heterostruc-\ntures of type-II as well as type-I Mn, we found that the\nnature of carriers and the doping rates vary depending\non the heterostructures. This provides the possibility of\nsystematically studying the spin-orbit coupled metals, in-\ncluding exploration of unconventional superconductivity13\ndue to the unique spin-orbital entanglement.\nIn recent decades, significant progress has been made\nin the study of QSLs, primarily focusing on the discov-\nery and expansion of new members in bulk materials.\nHowever, there has been limited exploration of creating\nand manipulating the QSLs in heterostructures despite\nthe importance for device applications. Our study has\ndemonstrated that the Kitaev-type QSL could be sur-\nveyed in ilmenite oxide heterostructures, displaying re-\nmarkable properties distinct from the bulk counterpart,\nsuch as flexible tuning of the Kitaev-type interactions\nand other parasitic interactions, and carrier doping to\nthe Kitaev QSL. Besides the van der Waals heterostruc-\ntures such as the combination of α-RuCl 3and graphene,\nour finding would enlighten an additional route to ex-\nplore the Kitaev QSL physics including the utilization of\nMajorana and anyonic excitations for future topological\ncomputing devices.\nACKNOWLEDGMENTS\nWe thank Y. Kato, M. Negishi, S. Okumura, A.\nTsukazaki, and L. Zh. Zhang, for fruitful discussions.\nThis work was supported by JST CREST Grant (No. JP-\nMJCR18T2). Parts of the numerical calculations were\nperformed in the supercomputing systems of the Insti-\ntute for Solid State Physics, the University of Tokyo.\nAppendix A: Detailed ab initio data for energy and\nmagnetic coupling\nIn this Appendix, we present the details of ab initio\nresults for various types of heterostructures. Tables IV-\nVII list the energy differences between different magnetic\nstates for MgIrO 3/ATiO 3heterostructures with A= Mn,\nFe, Co, and Ni. The bold elements in these tables are the\nlowest-energy state in each type, utilized for the calcu-\nlations of band structures in Sec. IV B 1. We also show\nin Table VIII the effective magnetic coupling constants\nbetween Aatoms, in which negative and positive value\nindicates the FM and AFM coupling, respectively. Note\nthat the Aatoms comprise a triangular lattice at the in-\nterface in type I, a honeycomb lattice at the ATiO 3layer,\nand a honeycomb lattice at the interface, as depicted in\nFig. 1.\nAppendix B: Orbital projected band structures\nIn this Appendix, we show the projection of the band\nstructure to the Ir 5 dorbitals for type-II heterostructures\nin Fig. 8 and type-III of Mn and Co in Fig. 9. The green\nshaded bands include high-energy four jeff= 1/2 bands\nand low-energy eight jeff= 3/2 bands. The results in\nFig. 8 indicate that electrons are doped to the half-filled\njeff= 1/2 bands, realizing the spin-orbit coupled metallicTABLE IV. Energy differences between different magnetic or-\ndered states obtained by the LDA+SOC+ Ucalculations for\nMgIrO 3/MnTiO 3: FM, N´ eel, and zigzag denotes the ferro-\nmagnetic, N´ eel-type antiferromagnetic, and zigzag-type anti-\nferromagnetic orders, respectively. While the directions of the\nmagnetic moments are all in-plane for the Ir layers, those for\nAcan be in-plane (“in”) or out-of-plane (“out”). The bold\nnumbers denote the low-energy states used for calculating the\nband structures in Sec. IV B 1.\nmagnetic state energy/Ir (meV)\nIr Mn I II III\nin-FMinFM 323.5 4.868 975.4\nN´ eel 1.492 46.52 1002\noutFM 18.13 4.268 1065\nN´ eel 3.695 44.554 978.2\nout-FMinFM 4.266 7.343 990.8\nN´ eel 0.773 43.68 993.2\noutFM 35.71 1.584 1014\nN´ eel 76.55 40.34 879.6\nN´ eelinFM 27.18 8.127 78.52\nN´ eel 26.98 42.83 48.51\noutFM 20.45 4.474 121.0\nN´ eel 2.953 39.39 22.45\nzigzaginFM 0.307 2.923 78.80\nN´ eel 0.000 21.55 0.000\noutFM 20.05 0.000 120.49\nN´ eel 4.893 39.62 22.95\nTABLE V. Energy differences between different magnetic or-\ndered states for MgIrO 3/FeTiO 3. The notations are common\nto Table IV.\nmagnetic state energy/Ir (meV)\nIr Fe I II III\nin-FMinFM 9.090 1.437 0.000\nN´ eel 17.00 15.75 13.14\noutFM 15.71 21.80 2.631\nN´ eel 1.170 27.02 14.25\nout-FMinFM 10.68 0.973 0.092\nN´ eel 12.12 0.948 13.01\noutFM 55.58 21.13 1.739\nN´ eel 8.115 26.29 8.833\nN´ eelinFM 24.64 0.000 0.056\nN´ eel 11.80 0.721 12.97\noutFM 14.48 20.21 1.244\nN´ eel 0.000 25.37 8.596\nzigzaginFM 15.58 1.233 0.056\nN´ eel 4.376 13.60 4.222\noutFM 14.35 21.42 2.244\nN´ eel 4.743 26.59 9.608\nstates for all Aatoms. The doping rates are large (small)\nforA= Mn and Co (Fe and Ni). Meanwhile, Fig. 9(a)\nshows that the jeff= 1/2 bands are slightly hole doped in\nthe type-III with A= Mn. Figure 9(b) indicates that the\ntype-III with A= Co achieves an insulating state with\n3/4-filled jeff= 1/2 bands.14\nTABLE VI. Energy differences between different magnetic or-\ndered states for MgIrO 3/CoTiO 3. The notations are common\nto Table IV.\nmagnetic state energy/Ir (meV)\nIr Co I II III\nin-FMinFM 188.2 20.02 0.795\nN´ eel 191.3 0.000 0.253\noutFM 40.23 25.63 72.39\nN´ eel 1.201 235.7 87.28\nout-FMinFM 0.174 20.06 0.800\nN´ eel 0.000 0.019 0.071\noutFM 295.3 25.85 260.4\nN´ eel 47.55 0.044 49.64\nN´ eelinFM 189.5 19.56 0.749\nN´ eel 184.3 19.36 0.000\noutFM 17.53 24.82 33.16\nN´ eel 0.617 234.9 31.16\nzigzaginFM 191.8 19.88 0.748\nN´ eel 189.6 19.39 1.969\noutFM 40.81 25.45 34.02\nN´ eel 1.476 130.0 32.03\nTABLE VII. Energy differences between different magnetic\nordered states for MgIrO 3/NiTiO 3. The notations are com-\nmon to Table IV.\nmagnetic state energy/Ir (meV)\nIr Ni I II III\nin-FMinFM 2.990 0.097 37.80\nN´ eel 4.998 70.93 35.99\noutFM 3.536 1.255 26.76\nN´ eel 0.156 69.45 35.74\nout-FMinFM 6.901 2.039 37.48\nN´ eel 7.311 70.74 35.98\noutFM 20.94 2.385 24.79\nN´ eel 6.220 69.61 61.76\nN´ eelinFM 14.06 1.678 48.98\nN´ eel 13.00 18.68 54.35\noutFM 4.941 0.294 26.51\nN´ eel 0.000 18.631 37.36\nzigzaginFM 4.971 0.000 36.97\nN´ eel 6.275 33.08 0.000\noutFM 3.451 0.042 26.20\nN´ eel 3.876 68.36 35.45\nAppendix C: Band structure of monolayer MgIrO 3\nIn this Appendix, we show the electronic band struc-\ntures of monolayer MgIrO 3obtained through ab ini-\ntiocalculations with the LDA+SOC [Fig. 10(a)] and\nLDA+SOC+ Uscheme [Fig. 10(b)]. We set UIr=\n3.0 eV and JH/UIr= 0.1 in the LDA+SOC+ Ucalcu-lations. In the LDA+SOC result, the system behaves\nas an insulating state with a tiny band gap of approx-\nimately ∼0.096 eV. However, the introduction of Uin\nthe LDA+SOC+ Ucalculation results in a larger band\ngap, characteristic of the spin-orbit coupled insulator.\nWe also calculate the PDOS of the jeff= 1/2 and 3 /2\nTABLE VIII. Effective magnetic coupling constants between\ntheAatoms for three types of heterostructures. The unit is\nin meV.\nA type-I II III\nMn 0.667 -0.712 2.417\nFe 0.017 -0.607 -0.149\nCo 0.065 0.262 6.458\nNi 0.088 -0.126 0.405\n(a) Mn\n(b) Fe\n(c) Co(d) Ni1.00.50.0-0.5-1.0-1.5-2.0ΓMKΓ1.00.50.0-0.5-1.0-1.5-2.0ΓMKΓ1.00.50.0-0.5-1.0-1.5-2.0ΓMKΓ1.00.50.0-0.5-1.0-1.5-2.0ΓMKΓEnergy (eV)Energy (eV)\nFIG. 8. Projection to the Ir 5 dorbitals of the band structure\nfor the type-II MgIrO 3/ATiO 3heterostructures with (a) A=\nMn, (b) Fe, (c) Co, and (d) Ni. The gray lines depict the band\nstructures shown in the middle panels of Figs. 2(b)-5(b), and\nthe green shade represents the weight of Ir 5 dorbitals. The\nFermi level is set to zero.\nmanifolds for Ir atoms, as shown in the right panels of\nFigs. 10(a) and 10(b). The PDOS of Ir atoms certifies\nthat the jeff= 1/2 and jeff= 3/2 manifold are retained\nto support the spin-orbit coupled Mott insulating state\nin the monolayer, as in the bulk case [57].\n[1] N. F. Mott, Metal-Insulator Transition , Rev. Mod.\nPhys. 40, 677 (1968).[2] M. Imada, A. Fujimori, and Y. Tokura, Metal-Insulator\nTransitions , Rev. Mod. Phys. 70, 1039 (1998).15\n(a)Mn\njeff = 1/2jeff = 1/2jeff = 1/2jeff = 1/2Co 3d(b) Co1.00.50.0-0.5-1.0-1.5-2.0ΓMKΓEnergy (eV)1.00.50.0-0.5-1.0-1.5-2.0ΓMKΓ\nFIG. 9. Projection to the Ir 5 dorbitals of the band structure\nfor the type-III heterostructures of (a) MgIrO 3/MnTiO 3and\n(b) MgIrO 3/CoTiO 3. The notations are common to Fig. 8.\n[3] X.-L. Qi and S.-C. Zhang, Topological Insulators and\nSuperconductors , Rev. Mod. Phys. 83, 1057 (2011).\n[4] M. Z. Hasan and C. L. Kane, Colloquium: Topological\ninsulators , Rev. Mod. Phys. 82, 3045 (2010).\n[5] X. Wan, A. Vishwanath, and S. Y. Savrasov, Compu-\ntational Design of Axion Insulators Based on 5dSpinel\nCompounds , Phys. Rev. Lett. 108, 146601 (2012).\n[6] D. M. Nenno, C. A. Garcia, J. Gooth, C. Felser, and\nP. Narang, Axion Physics in Condensed-Matter Sys-\ntems, Nat. Rev. Phys. 2, 682 (2020).\n[7] X. Wan, A. M. Turner, A. Vishwanath, and S. Y.\nSavrasov, Topological Semimetal and Fermi-Arc Surface\nStates in the Electronic Structure of Pyrochlore Iridates ,\nPhys. Rev. B 83, 205101 (2011).\n[8] A. A. Burkov and L. Balents, Weyl Semimetal in a\nTopological Insulator Multilayer , Phys. Rev. Lett. 107,\n127205 (2011).\n[9] N. P. Armitage, E. J. Mele, and A. Vishwanath, Weyl\nand Dirac Semimetals in Three-Dimensional Solids ,\nRev. Mod. Phys. 90, 015001 (2018).\n[10] W. Witczak-Krempa, G. Chen, Y. B. Kim, and L. Ba-\nlents, Correlated Quantum Phenomena in the Strong\nSpin-Orbit Regime , Annu. Rev. Condens. Matter Phys.\n5, 57 (2014).\n[11] B. J. Kim, H. Jin, S. J. Moon, J.-Y. Kim, B.-G. Park,\nC. S. Leem, J. Yu, T. W. Noh, C. Kim, S.-J. Oh, J.-H.\nPark, V. Durairaj, G. Cao, and E. Rotenberg, Novel\nJeff= 1/2Mott State Induced by Relativistic Spin-\nOrbit Coupling in Sr2IrO4, Phys. Rev. Lett. 101, 076402\n(2008).\n[12] S. J. Moon, H. Jin, K. W. Kim, W. S. Choi, Y. S. Lee,\nJ. Yu, G. Cao, A. Sumi, H. Funakubo, C. Bernhard,\nand T. W. Noh, Dimensionality-Controlled Insulator-\nMetal Transition and Correlated Metallic State in 5d\nTransition Metal Oxides Srn+1IrnO3n+1(n= 1,2, and\n∞), Phys. Rev. Lett. 101, 226402 (2008).\n[13] M. K. Crawford, M. A. Subramanian, R. L. Harlow,\nJ. A. Fernandez-Baca, Z. R. Wang, and D. C. Johnston,\nStructural and Magnetic Studies of Sr2IrO4, Phys. Rev.\nB49, 9198 (1994).\n[14] N. Read and B. Chakraborty, Statistics of the Excita-\ntions of the Resonating-Valence-Bond State , Phys. Rev.\nB40, 7133 (1989).\n[15] X.-G. Wen, Topological Orders and Chern-Simons The-\nory in Strongly Correlated Quantum Liquid , Int. J. Mod.\nPhys. B 5, 1641 (1991).\n!MK!-2-101Energy (eV)!MK!-2-101Energy (eV)(a)\n(b)FIG. 10. Electronic band structure of monolayer MgIrO 3ob-\ntained by the (a) LDA+SOC and (b) LDA+SOC+ Ucalcula-\ntions for the in-plane N´ eel AFM state. In (b), the parameters\nof the Coulomb interaction and Hund’s coupling are respec-\ntively set to UIr= 3.0 eV and JH/UIr= 0.1. The notions are\ncommon to Fig. 2.\n[16] P. W. Anderson, Resonating Valence Bonds: A New\nKind of Insulator? , Mater. Res. Bull. 8, 153 (1973).\n[17] L. Balents, Spin Liquids in Frustrated Magnets , Nature\n464, 199 (2010).\n[18] Y. Zhou, K. Kanoda, and T.-K. Ng, Quantum Spin Liq-\nuid States , Rev. Mod. Phys. 89, 025003 (2017).\n[19] C. Broholm, R. Cava, S. Kivelson, D. Nocera, M. Nor-\nman, and T. Senthil, Quantum Spin Liquids , Science\n367, eaay0668 (2020).\n[20] C. Nayak, S. H. Simon, A. Stern, M. Freedman, and\nS. Das Sarma, Non-Abelian Anyons and Topological\nAuantum Computation , Rev. Mod. Phys. 80, 1083\n(2008).\n[21] P. W. Anderson, Ordering and Antiferromagnetism in\nFerrites, Phys. Rev. 102, 1008 (1956).\n[22] S. Sachdev, Kagom´ e- and Triangular-Lattice Heisen-\nberg Antiferromagnets: Ordering from Quantum Fluc-\ntuations and Quantum-Disordered Ground States with\nUnconfined Bosonic Spinons , Phys. Rev. B 45, 12377\n(1992).16\n[23] A. Ramirez, Strongly Geometrically Frustrated Magnets ,\nAnnu. Rev. Mater. Sci. 24, 453 (1994).\n[24] Z. Nussinov and J. van den Brink, Compass Models:\nTheory and Physical Motivations , Rev. Mod. Phys. 87,\n1 (2015).\n[25] A. Kitaev, Anyons in an Exactly Solved Model and Be-\nyond, Ann. Phys. 321, 2 (2006).\n[26] A. Y. Kitaev, Fault-Tolerant Auantum Computation by\nAnyons , Ann. Phys. 303, 2 (2003).\n[27] G. Jackeli and G. Khaliullin, Mott Insulators in the\nStrong Spin-Orbit Coupling Limit: From Heisenberg to\na Quantum Compass and Kitaev Models , Phys. Rev.\nLett. 102, 017205 (2009).\n[28] H. Takagi, T. Takayama, G. Jackeli, G. Khaliullin, and\nS. E. Nagler, Concept and Realization of Kitaev Quan-\ntum Spin Liquids , Nat. Rev. Phys. 1, 264 (2019).\n[29] Y. Motome and J. Nasu, Hunting Majorana Fermions in\nKitaev Magnets , J. Phys. Soc. Jpn. 89, 012002 (2020).\n[30] Y. Motome, R. Sano, S. Jang, Y. Sugita, and Y. Kato,\nMaterials Design of Kitaev Spin Liquids Beyond the\nJackeli–Khaliullin Mechanism , J. Phys. Condens. Mat-\nter32, 404001 (2020).\n[31] S. Trebst and C. Hickey, Kitaev Materials , Phys. Rep.\n950, 1 (2022).\n[32] Y. Singh and P. Gegenwart, Antiferromagnetic Mott In-\nsulating State in Single Crystals of the Honeycomb Lat-\ntice Material Na2IrO3, Phys. Rev. B 82, 064412 (2010).\n[33] Y. Singh, S. Manni, J. Reuther, T. Berlijn, R. Thomale,\nW. Ku, S. Trebst, and P. Gegenwart, Relevance of the\nHeisenberg-Kitaev Model for the Honeycomb Lattice Iri-\ndates A2IrO3, Phys. Rev. Lett. 108, 127203 (2012).\n[34] R. Comin, G. Levy, B. Ludbrook, Z.-H. Zhu, C. N. Veen-\nstra, J. A. Rosen, Y. Singh, P. Gegenwart, D. Stricker,\nJ. N. Hancock, D. van der Marel, I. S. Elfimov, and\nA. Damascelli, Na 2IrO3as a Novel Relativistic Mott\nInsulator with a 340 meV Gap, Phys. Rev. Lett. 109,\n266406 (2012).\n[35] C. H. Sohn, H.-S. Kim, T. F. Qi, D. W. Jeong, H. J.\nPark, H. K. Yoo, H. H. Kim, J.-Y. Kim, T. D. Kang, D.-\nY. Cho, G. Cao, J. Yu, S. J. Moon, and T. W. Noh, Mix-\ning between Jeff=1\n2and3\n2Orbitals in Na2IrO3: A Spec-\ntroscopic and Density Functional Calculation Study ,\nPhys. Rev. B 88, 085125 (2013).\n[36] S. Hwan Chun, J.-W. Kim, J. Kim, H. Zheng, C. C.\nStoumpos, C. Malliakas, J. Mitchell, K. Mehlawat,\nY. Singh, Y. Choi, etal.,Direct Evidence for Dominant\nBond-Directional Interactions in a Honeycomb Lattice\nIridate Na2IrO3, Nat. Phys. 11, 462 (2015).\n[37] S. D. Das, S. Kundu, Z. Zhu, E. Mun, R. D. McDon-\nald, G. Li, L. Balicas, A. McCollam, G. Cao, J. G. Rau,\nH.-Y. Kee, V. Tripathi, and S. E. Sebastian, Magnetic\nAnisotropy of the Alkali Iridate Na2IrO3at High Mag-\nnetic Fields: Evidence for Strong Ferromagnetic Kitaev\nCorrelations , Phys. Rev. B 99, 081101 (2019).\n[38] H. Zhao, B. Hu, F. Ye, M. Lee, P. Schlottmann, and\nG. Cao, Ground State in Proximity to a Possible Ki-\ntaev Spin Liquid: The Undistorted Honeycomb Iridate\nNaxIrO3(0.60≤x≤0.80), Phys. Rev. B 104, L041108\n(2021).\n[39] K. W. Plumb, J. P. Clancy, L. J. Sandilands, V. V.\nShankar, Y. F. Hu, K. S. Burch, H.-Y. Kee, and Y.-J.\nKim, α-RuCl 3: A Spin-Orbit Assisted Mott Insulator on\na Honeycomb Lattice , Phys. Rev. B 90, 041112 (2014).[40] Y. Kubota, H. Tanaka, T. Ono, Y. Narumi, and\nK. Kindo, Successive Magnetic Phase Transitions in α-\nRuCl 3: XY-like Frustrated Magnet on the Honeycomb\nLattice , Phys. Rev. B 91, 094422 (2015).\n[41] L. J. Sandilands, Y. Tian, K. W. Plumb, Y.-J. Kim,\nand K. S. Burch, Scattering Continuum and Possible\nFractionalized Excitations in α-RuCl 3, Phys. Rev. Lett.\n114, 147201 (2015).\n[42] A. Koitzsch, C. Habenicht, E. M¨ uller, M. Knupfer,\nB. B¨ uchner, H. C. Kandpal, J. van den Brink, D. Nowak,\nA. Isaeva, and T. Doert, JeffDescription of the Honey-\ncomb Mott Insulator α-RuCl 3, Phys. Rev. Lett. 117,\n126403 (2016).\n[43] S. Sinn, C. H. Kim, B. H. Kim, K. D. Lee, C. J. Won,\nJ. S. Oh, M. Han, Y. J. Chang, N. Hur, H. Sato,\netal.,Electronic Structure of the Kitaev Material α-\nRuCl 3Probed by Photoemission and Inverse Photoemis-\nsion Spectroscopies , Sci. Rep. 6, 39544 (2016).\n[44] S.-H. Do, S.-Y. Park, J. Yoshitake, J. Nasu, Y. Motome,\nY. S. Kwon, D. Adroja, D. Voneshen, K. Kim, T.-H.\nJang, etal.,Majorana Fermions in the Kitaev Quantum\nSpin System α-RuCl 3, Nat. Phys. 13, 1079 (2017).\n[45] Y. Kasahara, T. Ohnishi, Y. Mizukami, O. Tanaka,\nS. Ma, K. Sugii, N. Kurita, H. Tanaka, J. Nasu, Y. Mo-\ntome, etal.,Majorana Quantization and Half-Integer\nThermal Quantum Hall Effect in a Kitaev Spin Liquid ,\nNature 559, 227 (2018).\n[46] H. Suzuki, H. Liu, J. Bertinshaw, K. Ueda, H. Kim,\nS. Laha, D. Weber, Z. Yang, L. Wang, H. Taka-\nhashi, etal.,Proximate Ferromagnetic State in the Ki-\ntaev Model Material α-RuCl 3, Nat. Commun. 12, 4512\n(2021).\n[47] R. Sano, Y. Kato, and Y. Motome, Kitaev-Heisenberg\nHamiltonian for High-Spin d7Mott Insulators , Phys.\nRev. B 97, 014408 (2018).\n[48] H. Liu and G. Khaliullin, Pseudospin Exchange Inter-\nactions in d7Cobalt Compounds: Possible Realization\nof the Kitaev Model , Phys. Rev. B 97, 014407 (2018).\n[49] R. Jang, Seong-Hoon and, Y. Kato, and Y. Motome,\nAntiferromagnetic Kitaev Interaction in f-Electron\nBased Honeycomb Magnets , Phys. Rev. B 99, 241106\n(2019).\n[50] S.-H. Jang, R. Sano, Y. Kato, and Y. Motome, Compu-\ntational Design of f-Electron Kitaev Magnets: Honey-\ncomb and Hyperhoneycomb Compounds A2PrO 3(A=\nAlkali Metals) , Phys. Rev. Mater. 4, 104420 (2020).\n[51] Y. Sugita, Y. Kato, and Y. Motome, Antiferromagnetic\nKitaev Interactions in Polar Spin-Orbit Mott Insulators ,\nPhys. Rev. B 101, 100410 (2020).\n[52] S. Biswas, Y. Li, S. M. Winter, J. Knolle, and R. Va-\nlent´ ı, Electronic Properties of α-RuCl 3in Proximity to\nGraphene , Phys. Rev. Lett. 123, 237201 (2019).\n[53] S. Mandal, D. Mallick, A. Banerjee, R. Gane-\nsan, and P. S. Anil Kumar, Evidence of\nCharge Transfer in Three-Dimensional Topolog-\nical Insulator/Antiferromagnetic Mott Insulator\nBi1Sb1Te1.5Se1.5/α-RuCl 3Heterostructures , Phys.\nRev. B 107, 245418 (2023).\n[54] K. Miura, K. Fujiwara, K. Nakayama, R. Ishikawa,\nN. Shibata, and A. Tsukazaki, Stabilization of A\nHoneycomb Lattice of IrO6Octahedra by Formation\nof Ilmenite-Type Superlattices in MnTiO 3, Commun.\nMater. 1, 55 (2020).17\n[55] B. Kang, M. Park, S. Song, S. Noh, D. Choe, M. Kong,\nM. Kim, C. Seo, E. K. Ko, G. Yi, J.-W. Yoo, S. Park,\nJ. M. Ok, and C. Sohn, Honeycomb Oxide Heterostruc-\nture as a Candidate Host for A Kitaev Quantum Spin\nLiquid , Phys. Rev. B 107, 075103 (2023).\n[56] Y. Haraguchi, C. Michioka, A. Matsuo, K. Kindo,\nH. Ueda, and K. Yoshimura, Magnetic Ordering with\nan XY-like Anisotropy in the Honeycomb Lattice Iri-\ndates ZnIrO 3andMgIrO3Synthesized via a Metathesis\nReaction , Phys. Rev. Mater. 2, 054411 (2018).\n[57] S.-H. Jang and Y. Motome, Electronic and Mag-\nnetic Properties of Iridium Ilmenites AIrO3(A=\nMg,Zn,and Mn), Phys. Rev. Mater. 5, 104409 (2021).\n[58] X. Hao, H. Jiang, R. Cui, X. Zhang, K. Sun, and\nY. Xu, Electronic and Magnetic Properties of Spin–\nOrbit-Entangled Honeycomb Lattice Iridates MIrO 3\n(M = Cd ,Zn,and Mg), Inorg. Chem. 61, 15007 (2022).\n[59] K. Momma and F. Izumi, VESTA 3 for Three-\nDimensional Visualization of Crystal, Volumetric and\nMorphology Data , J. Appl. Crystallogr. 44, 1272 (2011).\n[60] P. Giannozzi, S. Baroni, N. Bonini, M. Calan-\ndra, R. Car, C. Cavazzoni, D. Ceresoli, G. L.\nChiarotti, M. Cococcioni, I. Dabo, etal.,QUAN-\nTUM ESPRESSO: A Modular and Open-Source Soft-\nware Project for Quantum Simulations of Materials , J.\nPhys.: Condens. Matter 21, 395502 (2009).\n[61] P. Hohenberg and W. Kohn, Inhomogeneous Electron\nGas, Phys. Rev. 136, B864 (1964).\n[62] J. P. Perdew and A. Zunger, Self-Interaction Correc-\ntion to Density-Functional Approximations for Many-\nElectron Systems , Phys. Rev. B 23, 5048 (1981).\n[63] P. E. Bl¨ ochl, Projector Augmented-Wave Method , Phys.\nRev. B 50, 17953 (1994).\n[64] A. Dal Corso, Pseudopotentials Periodic Table: From H\ntoPu, Comput. Mater. Sci. 95, 337 (2014).\n[65] H. J. Monkhorst and J. D. Pack, Special Points for\nBrillouin-Zone Integrations , Phys. Rev. B 13, 5188\n(1976).\n[66] A. I. Liechtenstein, V. I. Anisimov, and J. Zaa-\nnen, Density-Functional Theory and Strong Interac-\ntions: Orbital Ordering in Mott-Hubbard Insulators ,\nPhys. Rev. B 52, R5467 (1995).\n[67] B. L. Chittari, Y. Park, D. Lee, M. Han, A. H. MacDon-\nald, E. Hwang, and J. Jung, Electronic and Magnetic\nProperties of Single-Layer MPX3Metal Phosphorous\nTrichalcogenides , Phys. Rev. B 94, 184428 (2016).\n[68] M. Arruabarrena, A. Leonardo, M. Rodriguez-Vega,\nG. A. Fiete, and A. Ayuela, Out-of-Plane Magnetic\nAnisotropy in Bulk Ilmenite CoTiO 3, Phys. Rev. B 105,\n144425 (2022).\n[69] A. A. Mostofi, J. R. Yates, G. Pizzi, Y.-S. Lee, I. Souza,\nD. Vanderbilt, and N. Marzari, An Updated Version of\nWannier90: A Tool for Obtaining Maximally-Localised\nWannier Functions , Comput. Phys. Commun. 185,\n2309 (2014).\n[70] Y. Yamaji, Y. Nomura, M. Kurita, R. Arita, and\nM. Imada, First-Principles Study of the Honeycomb-\nLattice Iridates Na2IrO3in the Presence of Strong Spin-\nOrbit Interaction and Electron Correlations , Phys. Rev.\nLett. 113, 107201 (2014).\n[71] B. H. Kim, G. Khaliullin, and B. I. Min, Electronic Ex-\ncitations in the Edge-Shared Relativistic Mott Insulator:\nNa2IrO3, Phys. Rev. B 89, 081109 (2014).[72] A. M. Ole´ s, Antiferromagnetism and Correlation of\nElectrons in Transition Metals , Phys. Rev. B 28, 327\n(1983).\n[73] L. M. Roth, Simple Narrow-Band Model of Ferromag-\nnetism Due to Intra-Atomic Exchange , Phys. Rev. 149,\n306 (1966).\n[74] G. Khaliullin, Orbital Order and Fluctuations in\nMott Insulators , Prog. Theor. Phys. Suppl. 160,\n155 (2005), https://academic.oup.com/ptps/article-\npdf/doi/10.1143/PTPS.160.155/5162453/160-155.pdf.\n[75] G. Shirane, S. J. Pickart, and Y. Ishikawa, Neutron\nDiffraction Study of Antiferromagnetic MnTiO 3and\nNiTiO 3, J. Phys. Soc. Japan 14, 1352 (1959).\n[76] R. Newnham, J. Fang, and R. Santoro, Crystal Structure\nand Magnetic Properties of CoTiO 3, Acta Crystallogr.\n17, 240 (1964).\n[77] A. I. Liechtenstein, M. Katsnelson, V. Antropov, and\nV. Gubanov, Local Spin Density Functional Approach\nto the Theory of Exchange Interactions in Ferromag-\nnetic Metals and Alloys , J. Magn. Magn. Mater. 67, 65\n(1987).\n[78] X. He, N. Helbig, M. J. Verstraete, and E. Bousquet,\nTB2J: A Python Package for Computing Magnetic In-\nteraction Parameters , Comput. Phys. Commun. 264,\n107938 (2021).\n[79] J. G. Rau, E. K.-H. Lee, and H.-Y. Kee, Generic Spin\nModel for the Honeycomb Iridates beyond the Kitaev\nLimit , Phys. Rev. Lett. 112, 077204 (2014).\n[80] J. Rusnaˇ cko, D. Gotfryd, and J. c. v. Chaloupka, Kitaev-\nlike Honeycomb Magnets: Global Phase Behavior and\nEmergent Effective Models , Phys. Rev. B 99, 064425\n(2019).\n[81] M. Hanawa, Y. Muraoka, T. Tayama, T. Sakakibara,\nJ. Yamaura, and Z. Hiroi, Superconductivity at 1 K in\nCd2Re2O7, Phys. Rev. Lett. 87, 187001 (2001).\n[82] K. Ohgushi, J.-i. Yamaura, M. Ichihara, Y. Ki-\nuchi, T. Tayama, T. Sakakibara, H. Gotou, T. Yagi,\nand Y. Ueda, Structural and Electronic Properties of\nPyrochlore-Type A2Re2O7(A= Ca ,Cd,and Pb ), Phys.\nRev. B 83, 125103 (2011).\n[83] J. Harter, Z. Zhao, J.-Q. Yan, D. Mandrus, and\nD. Hsieh, A Parity-Breaking Electronic Nematic Phase\nTransition in the Spin-Orbit Coupled Metal Cd2Re2O7,\nScience 356, 295 (2017).\n[84] Z. Hiroi, J.-i. Yamaura, T. C. Kobayashi, Y. Matsub-\nayashi, and D. Hirai, Pyrochlore Oxide Superconduc-\ntorCd2Re2O7Revisited , J. Phys. Soc. Jpn. 87, 024702\n(2018).\n[85] S. Tajima, D. Hirai, T. Yajima, D. Nishio-Hamane,\nY. Matsubayashi, and Z. Hiroi, Spin–Orbit-Coupled\nMetal Candidate PbRe 2O6, J. Solid. State. Chem. 288,\n121359 (2020).\n[86] R. Chari, R. Moessner, and J. G. Rau, Magnetoelectric\nGeneration of a Majorana-Fermi Surface in Kitaev’s\nHoneycomb Model , Phys. Rev. B 103, 134444 (2021).\n[87] K. Nakazawa, Y. Kato, and Y. Motome, Asymmetric\nModulation of Majorana Excitation Spectra and Nonre-\nciprocal Thermal Transport in the Kitaev Spin Liquid\nUnder a Staggered Magnetic Field , Phys. Rev. B 105,\n165152 (2022).\n[88] P. W. Anderson, The Resonating Valence Bond State\ninLa2CuO 4and Superconductivity , Science 235, 1196\n(1987).18\n[89] P. A. Lee, N. Nagaosa, and X.-G. Wen, Doping a Mott\nInsulator: Physics of High-Temperature Superconductiv-\nity, Rev. Mod. Phys. 78, 17 (2006).\n[90] J. G. Bednorz and K. A. M¨ uller, Possible High TcSu-\nperconductivity in the Ba-La-Cu-OSystem , Z. Phys. B\n64, 189 (1986).\n[91] P. W. Anderson, G. Baskaran, Z. Zou, and T. Hsu,\nResonating-Valence-Bond Theory of Phase Transitions\nand Superconductivity in La2CuO 4-Based Compounds ,\nPhys. Rev. Lett. 58, 2790 (1987).\n[92] H. Watanabe, T. Shirakawa, and S. Yunoki, Monte\nCarlo Study of an Unconventional Superconducting\nPhase in Iridium Oxide Jeff=1/2Mott Insulators In-\nduced by Carrier Doping , Phys. Rev. Lett. 110, 027002\n(2013).\n[93] Y. J. Yan, M. Q. Ren, H. C. Xu, B. P. Xie, R. Tao,\nH. Y. Choi, N. Lee, Y. J. Choi, T. Zhang, and D. L.\nFeng, Electron-Doped Sr2IrO4: An Analogue of Hole-\nDoped Cuprate Superconductors Demonstrated by Scan-\nning Tunneling Microscopy , Phys. Rev. X 5, 041018\n(2015).\n[94] Y.-Z. You, I. Kimchi, and A. Vishwanath, Doping a\nSpin-orbit Mott Insulator: Topological Superconductiv-\nity from the Kitaev-Heisenberg Model and Possible Ap-\nplication to (Na2/Li2)IrO 3, Phys. Rev. B 86, 085145\n(2012).\n[95] T. Hyart, A. R. Wright, G. Khaliullin, and B. Rosenow,\nCompetition between d-Wave and Topological p-Wave\nSuperconducting Phases in the Doped Kitaev-Heisenberg\nModel , Phys. Rev. B 85, 140510 (2012).\n[96] S. Okamoto, Global Phase Diagram of a Doped Kitaev-\nHeisenberg Model , Phys. Rev. B 87, 064508 (2013).\n[97] J. Schmidt, D. D. Scherer, and A. M. Black-Schaffer,\nTopological Superconductivity in the Extended Kitaev-\nHeisenberg Model , Phys. Rev. B 97, 014504 (2018).\n[98] Y. Ishikawa and S.-i. Akimoto, Magnetic Properties of\ntheFeTiO 3-Fe2O3Solid Solution Series , J. Phys. Soc.\nJapan 12, 1083 (1957).\n[99] H. Kato, M. Yamada, H. Yamauchi, H. Hiroyoshi,\nH. Takei, and H. Watanabe, Metamagnetic Phase Tran-\nsitions in FeTiO 3, J. Phys. Soc. Japan 51, 1769 (1982).\n[100] B. Yuan, I. Khait, G.-J. Shu, F. C. Chou, M. B. Stone,\nJ. P. Clancy, A. Paramekanti, and Y.-J. Kim, Dirac\nMagnons in a Honeycomb Lattice Quantum XY Magnet\nCoTiO 3, Phys. Rev. X 10, 011062 (2020).\n[101] B. A. Wechsler and C. T. Prewitt, Crystal Structure\nof Ilmenite (FeTiO 3)at High Temperature and at High\nPressure , Am. Mineral. 69, 176 (1984).\n[102] A. Raghavender, N. H. Hong, K. J. Lee, M.-H. Jung,\nZ. Skoko, M. Vasilevskiy, M. Cerqueira, and A. Saman-\ntilleke, Nano-Ilmenite FeTiO 3: Synthesis and Charac-\nterization , J. Magn. Magn. Mater. 331, 129 (2013).\n[103] H. Yamauchi, H. Hiroyoshi, M. Yamada, H. Watan-\nabe, and H. Takei, Spin Flopping in MnTiO 3, J. Magn.\nMagn. Mater. 31, 1071 (1983).\n[104] J. Ko and C. T. Prewitt, High-Pressure Phase Transi-\ntion in MnTiO 3from the Ilmenite to the LiNbO 3Struc-\nture, Phys. Chem. Miner. 15, 355 (1988).\n[105] N. Mufti, G. R. Blake, M. Mostovoy, S. Riyadi, A. A.\nNugroho, and T. T. M. Palstra, Magnetoelectric Cou-\npling in MnTiO 3, Phys. Rev. B 83, 104416 (2011).\n[106] Y. Ishikawa and S. Sawada, The Study on Substances\nHaving the Ilmenite Structure I. Physical Properties of\nSynthesized FeTiO 3andNiTiO 3Ceramics , J. Phys. Soc.Jpn.11, 496 (1956).\n[107] Y. Ishikawa and S.-i. Akimoto, Magnetic Property and\nCrystal Chemistry of Ilmenite (MeTiO 3)and Hematite\n(αFe2O3)System I. Crystal Chemistry , J. Phys. Soc.\nJpn.13, 1110 (1958).\n[108] T. Acharya and R. Choudhary, Structural, Dielec-\ntric and Impedance Characteristics of CoTiO 3, Mater.\nChem. Phys. 177, 131 (2016).\n[109] B. Yuan, I. Khait, G.-J. Shu, F. Chou, M. Stone,\nJ. Clancy, A. Paramekanti, and Y.-J. Kim, Dirac\nMagnons in a Honeycomb Lattice Quantum XY Mag-\nnetCoTiO 3, Physical Review X 10, 011062 (2020).\n[110] G. S. Heller, J. J. Stickler, S. Kern, and A. Wold, An-\ntiferromagnetism in NiTiO 3, J. Appl. Phys. 34, 1033\n(1963).\n[111] J. K. Harada, L. Balhorn, J. Hazi, M. C. Kemei, and\nR. Seshadri, Magnetodielectric Coupling in the Ilmenites\nMTiO 3(M= Co ,Ni), Phys. Rev. B 93, 104404 (2016).\n[112] H. Gretarsson, J. P. Clancy, X. Liu, J. P. Hill, E. Bozin,\nY. Singh, S. Manni, P. Gegenwart, J. Kim, A. H. Said,\nD. Casa, T. Gog, M. H. Upton, H.-S. Kim, J. Yu,\nV. M. Katukuri, L. Hozoi, J. van den Brink, and Y.-\nJ. Kim, Crystal-Field Splitting and Correlation Effect\non the Electronic Structure of A2IrO3, Phys. Rev. Lett.\n110, 076402 (2013).\n[113] X. Zhou, H. Li, J. A. Waugh, S. Parham, H.-S. Kim,\nJ. A. Sears, A. Gomes, H.-Y. Kee, Y.-J. Kim, and\nD. S. Dessau, Angle-Resolved Photoemission Study of\nthe Kitaev Candidate α-RuCl 3, Phys. Rev. B 94, 161106\n(2016).\n[114] J. Knolle, D. L. Kovrizhin, J. T. Chalker, and R. Moess-\nner, Dynamics of a Two-Dimensional Quantum Spin\nLiquid: Signatures of Emergent Majorana Fermions and\nFluxes , Phys. Rev. Lett. 112, 207203 (2014).\n[115] J. Knolle, D. L. Kovrizhin, J. T. Chalker, and R. Moess-\nner, Dynamics of Fractionalization in Quantum Spin\nLiquids , Phys. Rev. B 92, 115127 (2015).\n[116] A. Banerjee, C. Bridges, J.-Q. Yan, A. Aczel, L. Li,\nM. Stone, G. Granroth, M. Lumsden, Y. Yiu, J. Knolle,\netal.,Proximate Kitaev Quantum Spin Liquid Be-\nhaviour in a Honeycomb Magnet , Nat. Mater. 15, 733\n(2016).\n[117] J. Yoshitake, J. Nasu, and Y. Motome, Fractional Spin\nFluctuations as a Precursor of Quantum Spin Liquids:\nMajorana Dynamical Mean-Field Study for the Kitaev\nModel , Phys. Rev. Lett. 117, 157203 (2016).\n[118] J. Yoshitake, J. Nasu, Y. Kato, and Y. Motome, Ma-\njorana Dynamical Mean-Field Study of Spin Dynamics\nat Finite Temperatures in the Honeycomb Kitaev Model ,\nPhys. Rev. B 96, 024438 (2017).\n[119] J. Yoshitake, J. Nasu, and Y. Motome, Tempera-\nture Evolution of Spin Dynamics in Two- and Three-\ndimensional Kitaev Models: Influence of Fluctuating Z2\nflux, Phys. Rev. B 96, 064433 (2017).\n[120] J. Nasu, J. Yoshitake, and Y. Motome, Thermal Trans-\nport in the Kitaev Model , Phys. Rev. Lett. 119, 127204\n(2017).\n[121] T. Yokoi, S. Ma, Y. Kasahara, S. Kasahara,\nT. Shibauchi, N. Kurita, H. Tanaka, J. Nasu, Y. Mo-\ntome, C. Hickey, etal.,Half-Integer Quantized Anoma-\nlous Thermal Hall effect in the Kitaev Material Candi-\ndateα-RuCl 3, Science 373, 568 (2021).\n[122] L. E. Chern, E. Z. Zhang, and Y. B. Kim, Sign Structure\nof Thermal Hall Conductivity and Topological Magnons19\nfor In-Plane Field Polarized Kitaev Magnets , Phys. Rev.\nLett. 126, 147201 (2021).\n[123] J. Nasu, J. Knolle, D. L. Kovrizhin, Y. Motome, and\nR. Moessner, Fermionic Response from Fractionaliza-\ntion in an Insulating Two-Dimensional Magnet , Nat.\nPhys. 12, 912 (2016).\n[124] A. Glamazda, P. Lemmens, S.-H. Do, Y. Choi, and K.-\nY. Choi, Raman Spectroscopic Signature of Fractional-\nized Excitations in the Harmonic-Honeycomb Iridates\nβ- and γ-Li2IrO3, Nat. Commun. 7, 12286 (2016).\n[125] B. Zhou, Y. Wang, G. B. Osterhoudt, P. Lampen-\nKelley, D. Mandrus, R. He, K. S. Burch, and E. A.\nHenriksen, Possible Structural Transformation and En-\nhanced Magnetic Fluctuations in Exfoliated α-RuCl 3, J.\nPhys. Chem. Solids 128, 291 (2019).\n[126] J.-H. Lee, Y. Choi, S.-H. Do, B. H. Kim, M.-J. Seong,\nand K.-Y. Choi, Multiple Spin-Orbit Excitons in α-\nRuCl 3from Bulk to Atomically Thin Layers , npj Quan-\ntum Mater. 6, 43 (2021).\n[127] J. Feldmeier, W. Natori, M. Knap, and J. Knolle,\nLocal Probes for Charge-Neutral Edge States in Two-Dimensional Quantum Magnets , Phys. Rev. B 102,\n134423 (2020).\n[128] R. G. Pereira and R. Egger, Electrical Access to Ising\nAnyons in Kitaev Spin Liquids , Phys. Rev. Lett. 125,\n227202 (2020).\n[129] E. J. K¨ onig, M. T. Randeria, and B. J¨ ack, Tunneling\nSpectroscopy of Quantum Spin Liquids , Phys. Rev. Lett.\n125, 267206 (2020).\n[130] M. Udagawa, S. Takayoshi, and T. Oka, Scanning Tun-\nneling Microscopy as a Single Majorana Detector of Ki-\ntaev’s Chiral Spin Liquid , Phys. Rev. Lett. 126, 127201\n(2021).\n[131] T. Bauer, L. R. D. Freitas, R. G. Pereira, and R. Eg-\nger,Scanning Tunneling Spectroscopy of Majorana Zero\nModes in a Kitaev Spin Liquid , Phys. Rev. B 107,\n054432 (2023).\n[132] Y. Kato, J. Nasu, M. Sato, T. Okubo, T. Misawa, and\nY. Motome, Spin Seebeck Effect as a Probe for Majo-\nrana Fermions in Kitaev Spin Liquids , arXiv preprint\narXiv:2401.13175 (2024)."    },    {        "title": "1304.7968v2.Can_electrostatic_field_lift_spin_degeneracy_.pdf",        "content": "arXiv:1304.7968v2  [quant-ph]  7 Aug 2014Can electrostatic field lift spin degeneracy?\nT. G. Tenev1,∗and N. V. Vitanov1\n1Department of Physics, Sofia University, 5 James Bourchier B lvd, Sofia 1164, Bulgaria\n(Dated: October 31, 2021)\nThere are two well known mechanisms which lead to lifting of e nergy spin degeneracy of single\nelectron systems - magnetic field and spin-orbit coupling. W e investigate the possibility for existence\nof a third mechanism in which electrostatic field can lead to l ifting of spin-degeneracy directly\nwithout the mediation of spin-orbit coupling. A novel argum ent is provided for the need of spin-\norbit coupling different from the usual relativistic consid erations. It is shown that due to preserved\ntranslational invariance spin splitting purely by electro static field is not possible for Bloch states. A\npossible lifting of spin degeneracy by electrostatic field c haracterized by broken inversion and broken\ntranslational invariance is considered.\nI. INTRODUCTION\nThe field of spintronics1, a term coined by Wolf2, has\nattracted considerable attention in the past 20 years.\nThis interest was initially triggered by the discovery of\nthe effect of giant magnetoresistance (GMR)3,4which\nnow has established commercial applications. A paral-\nlel interest in semiconductor spintronics has arisen due\nto the proposal for a ballistic spin-transistor5which has\nrecently been realized6. A generalization of the device\noperating in diffusive regime has been proposed7as well\nas other devices1. They depend on the spin splitting\ncaused by the spin-orbit coupling through several differ-\nent mechanisms usually classified in two big groups. His-\ntorically the first group often referred to as Dresselhaus\nspin-orbit coupling8–10, or alternatively, bulk inversion\nasymmetry11–13(BIA), has as a necessary condition the\nbroken space inversion invariance of a bulk crystal. It\nalso appears in a modified form1,11,12,14,15in quasi two-\ndimensional (Q2D) structures grown in crystals lacking\ninversion symmetry. The second group referred to as\neither Rashba spin-orbit coupling16–18or structure in-\nversion asymmetry1,11,12(SIA) leads to spin splitting in\nsystems lacking macroscopic inversion symmetry. In all\nthese cases the inversion asymmetry is merely a neces-\nsary condition and not a sufficient one. The other neces-\nsary condition is taking spin-orbit coupling into account\nin various ways but usually through the folding down\nprocedure11–13,18from 8 band or 14 band models. Here\nwe consider whether the breaking of inversion invariance\nof electrostatic field in several classes of systems can also\nbe a sufficient condition for lifting of spin degeneracy.\nWe examine a symmetry argument11–13,19due to\nKittel19. We provide novel derivation of the symmetry\nargument11–13,19relying only on the commutation prop-\nerties of the time-reversal ˆKand space inversion ˆIop-\nerators with the translation ˆTRnand spin operators ˆSu.\nThis is unlike the usual derivation19relying on explicit\nform of Bloch states. We show a novel argument for the\nneed for introduction of a spin-orbit coupling term simi-\nlar to the argument motivating Maxwell to introduce the\ndisplacement current term in the Maxwell’s equations.\nWe note that the symmetry argument11–13,19involvingtime-reversal and space inversion invariance does not re-\nquire the presence of a spin-orbit coupling term in the\nHamiltonian but merely to take the spin degree of free-\ndom of the electron into account. The application of this\nsymmetry argument, to a nonrelativistic model of elec-\ntron with spin moving in electrostatic field characterized\nby discrete translational invariance, naturally suggests\nthe hypothesis that electrostatic field with broken space\ninversionsymmetry can lead to lifting ofspin degeneracy.\nWe explorethe hypothesisby perturbationmethod treat-\nment and show that if discrete translational invariance is\npreserved electrostatic field can not lift spin-degeneracy.\nIf both translational and space inversion symmetry of a\nperturbing electrostatic field are broken a possible con-\ntribution to spin splitting by electrostatic field is indi-\ncated as far as the perturbation method is applicable.\nPhysically this would be naturally explained since we\nknow21that every non-accidental degeneracy stems from\nsymmetries of the underlying system and in general re-\nduction of symmetries leads to lifting of degeneracies.\nIn Sec. II we introduce the theoretical models. The\nbasic symmetry argument is presented in Sec. III. The\ntransformation properties of Bloch states in a model ne-\nglecting spin-orbit coupling but taking into account the\nspin degree of freedom are given in Sec. IIIA. They are\nused in Sec. IIIB to show the appearance of at least four-\nfold degeneracy in the spectrum as a consequence of the\ncombination of time-reversal and space inversion invari-\nance and in Sec. IIIC to show the lifting of spin degen-\neracy of Bloch states as a consequence of broken space\ninversion invariance. How the presented symmetry ar-\ngument differs from the usual one11–13,19is discussed in\nSec. IIID. In Sec. IV a new argument for the introduc-\ntion of spin-orbit coupling is presented different from the\nusual relativistic arguments12,21,23,24. We then explore\nthe hypothesis formulated in Sec. IIID by perturbation\nmethod treatment in Sec. V. In Sec. VI we discuss cer-\ntain aspects of the utilized model and methods and we\npresent our conclusions in Sec. VII.2\nII. THEORETICAL MODELS\nWe first consider two basic models of a nonrelativistic\nelectron moving in a pure electrostatic field characterized\nby discrete translational invariance. We focus our atten-\ntionon3Dmodelsofthetricliniccrystalsystem. Thetwo\nclasses of models differ from each other by the properties\nof the electrostatic potential with respect to space inver-\nsion symmetry. In both models we take spin degree of\nfreedom into account but neglect the spin-orbit coupling\nterm. The eigenstates of both models are Bloch states19,\nwhich can be characterizedby two quantum numbers: (i)\ncrystal wavevector and (ii) spin index which is the eigen-\nvalue of the u-component ˆSuof the spin vector operator\nˆS. Furthermore we employ the standard Born-von Kar-\nman periodic boundary conditions.\nThe first class of models represents 3D crystals of\nthe triclinic pinacoidal symmetry class. Its Hamiltonian\ntakes the form\nˆH0=ˆp2\n2mˆσ0+V0(r)ˆσ0. (1)\nThis represents a nonrelativistic electron moving in a\npure electrostatic potential φ0(r) with potential energy\nV0=−eφ0(r) characterized by translational invariance,\n[ˆTRn,φ0(r)] = 0, time-reversal invariance [ ˆK,φ0(r)] = 0,\nand space inversion invariance [ ˆI,φ0(r)] = 0.\nThe second class of models represents 3D crystals of\nthe triclinic pedial symmetry class in which the overall\npotential does not possess space inversion invariance. A\ngeneral potential not possessing space inversion invari-\nance can be split into parts symmetric with respect to\nspaceinversion,that isonethatcommuteswith the space\ninversion operator ˆI, and a part antisymmetric with re-\nspect to space inversion, that is one that anticommutes\nwithˆI. We consider the electrostatic field of the second\nclass of models as made of the symmetric part φ0(r) and\nan antisymmetric one φ(r), satisfying ˆIφ(r)ˆI+=−φ(r).\nThe Hamiltonian takes the form\nˆH=ˆp2\n2mˆσ0+V0ˆσ0−eφ(r)ˆσ0, (2)\nWhile the Hamiltonian in such systemspossessesdiscrete\ntranslational invariance [ ˆH,ˆTRn] = 0, and time-reversal\ninvariance, [ ˆH,ˆK] = 0, it is no longer invariant with\nrespect to space inversion, [ ˆH,ˆI]/ne}ationslash= 0.\nInordertoemphasizethatwehavetakenspindegreeof\nfreedom into account we have written the Hamiltonians\n(1) and (2) with the 2 ×2 identity matrix σ0.\nIII. SYMMETRY ARGUMENT\nA. Transformation of Bloch States\nWeconsiderageneralBlochstate |k,su/an}b∇acket∇i}httheproperties\nof which are identical in the two models considered. Itsatisfies the Bloch theorem19,\nˆTRn|k,su/an}b∇acket∇i}ht=e−ik·Rn|k,su/an}b∇acket∇i}ht, (3)\nwhere we use the active convention20,21for the space\ntranslation operator ˆTRn. The time-reversaltransformed\nstate|k′,s′\nu/an}b∇acket∇i}ht=ˆK|k,su/an}b∇acket∇i}htof a Bloch state |k,su/an}b∇acket∇i}htis\nstill a Bloch state because the time-reversal operator\ncommutes21with the spatial translation operators ˆTRn.\nApplying the time-reversal operator ˆKto the Bloch the-\norem (3), taking into account that [ ˆTRn,ˆK] = 0 and that\nˆKas antilinear operator does not commute with com-\nplex scalars cbut satisfies the identity21ˆKc=c∗ˆK,\none obtains the identity ˆTRn|k′,s′\nu/an}b∇acket∇i}ht=eik·Rn|k′,s′\nu/an}b∇acket∇i}ht.\nComparing it with the Bloch theorem, ˆTRn|k′,s′\nu/an}b∇acket∇i}ht=\ne−ik′·Rn|k′,s′\nu/an}b∇acket∇i}ht, shows that k′=−k. Applying ˆKto\nthe relation ˆSu|k,su/an}b∇acket∇i}ht=su|k,su/an}b∇acket∇i}htand using the rela-\ntionˆKˆSu=−ˆSuˆK, which follows directly from the\ndefinition21of the time-reversal operator ˆK, one obtains\nˆSuˆK|k,su/an}b∇acket∇i}ht=−suˆK|k,su/an}b∇acket∇i}ht. Thus the ˆK-transformed\nstate|k′,s′\nu/an}b∇acket∇i}ht=ˆK|k,su/an}b∇acket∇i}htis an eigenstate of ˆSuwith\nan eigenvalue −su, therefore s′\nu=−su. Summarizing,\nthe time-reversal operator ˆKtransforms a Bloch state\n|k,su/an}b∇acket∇i}htrepresenting an electron moving with a wavevec-\ntorkand a spin pointing ”up” the axis uinto the Bloch\nstate|−k,−su/an}b∇acket∇i}ht,\nˆK|k,su/an}b∇acket∇i}ht=|−k,−su/an}b∇acket∇i}ht, (4)\nrepresenting an electron moving in the opposite direction\nwith acrystalwavevector −kanda spinpointing ”down”\nthe axis u.\nThe space translationand space inversionoperatorsdo\nnot commute but satisfy20the identity\nˆIˆTRn=ˆT−RnˆI. (5)\nApplying the spaceinversionoperator ˆIto the Blochthe-\norem, (3), using Eq.(5) and the fact that ˆIdoes not act\non the phase factor e−ik·Rnone obtains\nˆT−RnˆI|k,su/an}b∇acket∇i}ht=e−ikRnˆI|k,su/an}b∇acket∇i}ht. (6)\nSince the ˆI-transformed Bloch state I|k,su/an}b∇acket∇i}htsatisfies the\nBlochtheorem it is still a Blochstate, but in generalwith\ndifferent quantum numbers |k′,s′\nu/an}b∇acket∇i}ht=ˆI|k,su/an}b∇acket∇i}ht. Com-\nparing Eq.(6) with the Bloch theorem ˆT−Rn|k′,s′\nu/an}b∇acket∇i}ht=\neik′Rn|k′,s′\nu/an}b∇acket∇i}htone obtains that k′=−k. By definition21\nthe space inversion operator ˆIcommutes with any com-\nponent of the spin vector operator ˆs. As a consequence,\nusing the usual procedure applied above to ˆSu|k,su/an}b∇acket∇i}ht=\nsu|k,su/an}b∇acket∇i}htgives that s′\nu=su. Therefore, the space in-\nversion operator ˆImaps a Bloch state |k,su/an}b∇acket∇i}ht, describ-\ning electron motion with crystal wavevector kand spin\npointing in the direction of axis u, into the state\nˆI|k,su/an}b∇acket∇i}ht=|−k,su/an}b∇acket∇i}ht, (7)3\nrepresenting an electron motion with the same orienta-\ntion of spin but moving in the opposite direction with a\nwavevector −k. Using the definition of the conjugation\noperator19ˆC=ˆKˆIand Eqs. (4) and (7), one obtains the\naction of the ˆCon Bloch states,\nˆC|k,su/an}b∇acket∇i}ht=|k,−su/an}b∇acket∇i}ht. (8)\nB. Spin Degeneracy\nSupposing the spectrum problem of the Hamiltonian\nˆH0with space-inversion invariant electrostatic potential\nφ0(r) solved its eigenvalue-eigenvectorproblem takes the\nform of the identity\nˆH0|k,su/an}b∇acket∇i}ht ≡E0\nk,su|k,su/an}b∇acket∇i}ht. (9)\nThe eigenvalues E0\nk,suofˆH0are labeled with quantum\nnumbers k,suand the eigenstates of H0and|k,su/an}b∇acket∇i}htsat-\nisfying the Bloch theorem possess all the properties of\nBloch states, in particular Eq. (4) and Eq. (6).\nApplying the time-reversal operator ˆKto Eq.(9) and\nusing the time-reversal invariance of the Hamiltonian\n[ˆH,ˆK] = 0, we obtain\nˆH0ˆK|k,su/an}b∇acket∇i}ht ≡E0\nk,suˆK|k,su/an}b∇acket∇i}ht. (10)\nDue to the time-reversalinvarianceof ˆH0the twolinearly\nindependent states |k,su/an}b∇acket∇i}htand| −k,−su/an}b∇acket∇i}ht=ˆK|k,su/an}b∇acket∇i}ht\ncorrespond to the same eigenvalue of ˆH0, which we\nnow denote as E0\nθ≡E0\nk,su=E0\n−k,−su. The pairs\nof linearly independent states ( |k,su/an}b∇acket∇i}ht,| −k,−su/an}b∇acket∇i}ht) and\n(|−k,su/an}b∇acket∇i}ht,|k,−su/an}b∇acket∇i}ht) span respectively the 2D subspaces\nε0\nθandε0\n−θof the Hilbert space of the single-particle\nsystem. The 2-fold degeneracy of the eigenenergies E0\nθ\nandE0\n−θ≡E0\n−k,su=E0\nk,−su, to which the subspaces\nε0\nθandε0\n−θcorrespond, is a consequence of the time-\nreversalinvarianceof ˆH0andistherealizationofKramers\ndegeneracy21,22in the system described by Eq. (9). By\ntheir construction the subspaces ε0\nθandε0\n−θare invari-\nant with respect to the time reversal symbolically writ-\nten asˆKε0\n±θ=ε0\n±θ. Comparing the spectrum equation\nˆH0|−k,−su/an}b∇acket∇i}ht=E0\n−k,−su|−k,−su/an}b∇acket∇i}htfor astate |−k,−su/an}b∇acket∇i}ht\nwith Eq. (10) allows us to express the Kramers degener-\nacy in the studied system in the form\nE0\nk,su=E0\n−k,−su. (11)\nActing on the left of Eq. (9) with the space-inversion\noperator ˆI,using the hypothesis [ ˆH,ˆI] = 0 and the result\n|−k,su/an}b∇acket∇i}ht ≡ˆI|k,su/an}b∇acket∇i}htfrom Eq.(7), oneobtainsthe identity\nˆH0|−k,su/an}b∇acket∇i}ht ≡E0\nk,su|−k,su/an}b∇acket∇i}ht. (12)\nIt showsthat the space-inversioninvarianceofthe Hamil-\ntonianˆH0requires that the Bloch states |k,su/an}b∇acket∇i}htand\n|−k,su/an}b∇acket∇i}htbelong to the same eigenenergy,\nE0\nk≡E0\nk,su=E0\n−k,su. (13)Expression (12) shows that the states |k,su/an}b∇acket∇i}htand\n|−k,su/an}b∇acket∇i}ht, respectively belonging to the subspaces εθand\nε−θ, must belong to the same degenerate eigenvalue E0\nk\nas a consequence of the space inversion invariance of ˆH0.\nTakingintoaccounttheconsequencesofthetime-reversal\ninvariance of ˆH0given in Eqs. (10) and (11), all eigenen-\nergiesE0\nkofˆH0must be 4-fold degenerate. To every en-\nergy value E0\nkcorresponds a four-dimensional subspace\nε0\nk, which is a direct sum, ε0\nk=ε0\nθ+ε0\n−θ, of the two sub-\nspacesε0\nθandε0\n−θ. The subspace ε0\nkis invariant and re-\nducible with respect to space inversion and time-reversal\nwritten symbolically as ˆKε0\nk=ε0\nkandˆIε0\nk=ε0\nk.\nCombination of the space inversion and time-reversal\ninvarianceofthe Hamiltonian ˆH0ofthe considered trans-\nlational invariant system is equivalent to spin degener-\nacy. Formally this is illustrated using the conjugation\noperator ˆC=ˆKˆIwhich commutes with the Hamiltonian\nˆH0, [ˆH0,ˆC] = 0 if it commutes separately with ˆKand\nˆI. Using the usual procedure of applying the operator\nˆCto Eq.(9) and taking into account Eq.(8) one obtains\nthe identity ˆH0|k,−su/an}b∇acket∇i}ht=E0\nk,su|k,−su/an}b∇acket∇i}ht. Therefore the\nBloch states |k,su/an}b∇acket∇i}htand|k,−su/an}b∇acket∇i}htdescribing an electron\nwith opposite spins belong to the same degenerate en-\nergy value E0\nk. The subspace ε0\nkcorresponding to E0\nkis\ninvariant with respect to ˆC. The spin degeneracy can be\nviewed also as a consequence of the SU(2) invariance of\nthe Hamiltonian ˆH0.\nC. Broken Spin Degeneracy\nThe problem for the spectrum of the Hamiltonian ˆH\nshown in Eq (2) is given by the identity\nˆH|κ,σu/an}b∇acket∇i}ht ≡Eκ,σu|κ,σu/an}b∇acket∇i}ht. (14)\nThe Bloch states |κ,σu/an}b∇acket∇i}htare the common set of eigen-\nstates of the commuting operators ˆHandˆTRnandEκ,σu\nare the corresponding eigenvalues of the Hamiltonian ˆH.\nThey transform among each other according to relations\n(4), (7) and (8) because the Hamiltonian ˆHgiven in\nEq (2) is translational invariant.\nThe spectrum (14) of the Hamiltonian ˆHpossesses the\ntime-reversal induced properties derived from Eq. (10)\nand Eq. (11). It is Kramers degenerate and each of its\neigenenergies are two-fold degenerate, Eκ,σu=E−κ,−σu.\nHowever because the space inversion invariance of the\nelectrostatic potential φ(r), and therefore of ˆHis bro-\nken, the degeneracy due to space inversion invariance is\nlifted,Eκ,σu/ne}ationslash=E−κ,σu. This requires the lifting of spin\ndegeneracy\nEκ,σu/ne}ationslash=Eκ,−σu. (15)\nThe detailed proof follows in the next paragraph.\nThe broken spin degeneracy is proved by applying the\nconjugation operator ˆCto Eq. (14). However, because4\n[ˆH,ˆI]/ne}ationslash= 0 and hence [ ˆH,ˆC]/ne}ationslash= 0 we can not interchange\nthe positions of ˆCandˆH. Instead, using ˆ1 =C−1Cand\n|κ,−σu/an}b∇acket∇i}ht=ˆC|κ,σu/an}b∇acket∇i}htone obtains from Eq. (8) that\nˆCˆHˆC−1|κ,−σu/an}b∇acket∇i}ht=Eκ,σu|κ,−σu/an}b∇acket∇i}ht.(16)\nThe Bloch state |κ,−σu/an}b∇acket∇i}ht, as an eigenfunction of ˆTRn,\nand because [ ˆH,ˆTRn] = 0, is still an eigenstate of ˆH\nwith an eigenvalue Eκ,−σu. However, because of broken\ninversion invariance,\nˆCˆHˆC−1=ˆIˆKˆHˆK−1ˆI−1=ˆIˆHˆI−1/ne}ationslash=ˆH\n|κ,−σu/an}b∇acket∇i}htis not anymore an eigenstate of ˆHwith the\neigenvalue Eκ,σu. Instead the Bloch state |κ,−σu/an}b∇acket∇i}htis\neigenstate of some other operator ˆH′=ˆIˆHˆI−1with\nthe eigenvalue Eκ,σu. Therefore the states |κ,σu/an}b∇acket∇i}htand\n|κ,−σu/an}b∇acket∇i}htdo not correspond to the same energy. Elec-\ntrons in Bloch states characterized by the same wavevec-\ntorκbut having opposite spin orientations do not posses\nthe same energy, the result given in Eq. (15).\nD. Discussion of Symmetry Argument\nInthewellknowntreatment8,11,12,19itissupposedthat\nthe spin-orbit coupling is part of the model Hamiltonian\nand that the spin degeneracy is lifted by the spin-orbit\ncoupling term\nHSO=¯h\n4m2c2ˆσ·(∇V(r)׈p) (17)\nif the electrostatic potential V(r) =V0(r)−eφ(r) does\nnot possess space inversion invariance. However, close\nexamination of the symmetry analysis developed in the\ntext above shows that there is no such requirement. The\nsymmetryanalysisisvalidalsoforanonrelativisticmodel\nthat contains just electrostatic fields φ0(r) andφ(r) and\ndoes not contain spin-orbit coupling or magnetic field. It\nsuggests that electrostatic field alone without the media-\ntion of spin-orbit coupling can lead to lift of spin degen-\neracy given that φ0(r) andφ(r) possess discrete transla-\ntional invariance and are characterized by preserved and\nbroken space inversion symmetry respectively.\nIV. NOVEL ARGUMENT FOR\nINTRODUCTION OF SPIN-ORBIT COUPLING\nTERM\nTaking spin into account suggests examining for SU(2)\nsymmetry. Since we consider models of a nonrelativistic\nelectron in electrostatic field neglecting spin-orbit cou-\npling, the Hamiltonians ˆH0andˆHcommute with ev-\nery component of the spin vector operator ˆSand there-\nfore the Hamiltonians ˆH0andˆHare SU(2) invariant.\nThey commute, [ ˆRs\nu(φ),ˆH] = 0, with every spin rotation\noperator21ˆRs\nu(φ) =e−iφˆSuforarbitraryaxis uandangleof rotation φ. As a consequence of the SU(2) invariance,\nthe spin degeneracy for the model with Hamiltonians ˆH0\nandˆHmust be preserved, in particular for ˆH\nEκ,σu=Eκ,−σu (18)\nThis result, however, contradicts the result (15) stem-\nming from symmetry analysis based solely on time-\nreversalandbrokenspaceinversioninvariancewhenspin-\norbit coupling is neglected.\nApossiblewaytoresolvethisinconsistencyistheintro-\nduction of a term in the model Hamiltonian that breaks\ntheSU(2) invariance and at the same time is consis-\ntent with the different cases of symmetry analysis in-\nvolving just time-reversal and space-inversion operators.\nThe spin-orbit coupling term (17), which does not com-\nmute with any spin-rotation operator21ˆRs\nu(φ), satisfies\nthe above requirements and resolves the noted inconsis-\ntency. We interpret this as a novel argument for intro-\nduction of the spin-orbit coupling term different from the\nusual21,23,24purely relativistic considerations, which give\nan incorrect numerical factor by 1 /2. This difference is\naccounted for by Thomas precession or by taking the\nnonrelativistic limit of the Dirac equation21,23,24. Thus a\nrealistic model of electron dynamics requires taking into\naccount the spin-orbit coupling term as a minimum; oth-\nerwise we would encounter the above mentioned incon-\nsistencies. Therefore in all subsequent models treated\nwithin perturbation method the spin-orbit coupling term\nis part of the considered Hamiltonian. As a consequence\nSU(2) symmetryis alwaysbrokenand there isno require-\nment for preservation of spin-degeneracy.\nV. PERTURBATION METHOD TREATMENT\nA model with spin-orbit coupling does not exclude the\npossibility that electrostatic field by itself leads to lift-\ning of spin degeneracy suggested by the symmetry anal-\nysis based on time-reversal and space-inversion invari-\nance in Sec. IIID. It merely shows that if such splitting\nexists it will lead to additional numerical factor in the\nspin splitting alreadycaused by spin-orbit coupling when\nthe space-inversion invariance of the electrostatic field is\nbroken. Since the crystal wavevector kvaries in discrete\nstepsweinvestigatethisoptionusingstandardstationary\nperturbation method for degenerate levels.\nThe unperturbed Hamiltonian is ˆH0=ˆp2\n2m+\nV0(r), where V0(r) possesses space inversion invariance,\n[V0,ˆI] = 0, while the perturbation δˆVconsists of an\nelectrostatic potential φ(r) which is odd with respect to\nspace inversion, {φ(r),ˆI}= 0, and the spin-orbit cou-\npling terms\nˆU1=¯h\n4m2c2ˆσ·(∇V0(r)׈p), (19a)\nˆU2=−e¯h\n4m2c2ˆσ·(∇φ(r)׈p).(19b)5\nUnlike in the standard k·ˆpmethod where the k= 0\nstationary states are used as unperturbed basis, we use\nthe stationary states |k,su/an}b∇acket∇i}htofˆH0for arbitrary k/ne}ationslash= 0.\nThis choice is naturally suggested by the symmetry anal-\nysis above since for k= 0 we have just two-fold Kramers\ndegeneracy that is not lifted as far as time-reversal in-\nvariance is preserved. We consider 3D model of triclinic\npedial and triclinic pinacoidal systems in which cases the\ndegeneracy of every energy level E0\nkof the unperturbed\nHamiltonian ˆH0is exactly four-fold. The corresponding\nsubspace ε0\nkis spanned by the four Bloch states |k,su/an}b∇acket∇i}ht,\n|k,−su/an}b∇acket∇i}ht,|−k,su/an}b∇acket∇i}htand|−k,−su/an}b∇acket∇i}ht.\nThe first-ordercorrection E(1)\nktothe energyeigenvalue\nE(0)\nkand the zeroth-order states |0/an}b∇acket∇i}htare determined from\nthe eigenvalue equation\nˆP0\nkδˆVˆP0\nk|0k/an}b∇acket∇i}ht=E(1)|0k/an}b∇acket∇i}ht, (20)\nwhereˆP0\nkis the projector to the subspace ε0\nkandδˆV=\n−eφ(r) +ˆU1+ˆU2.The first-order correction E(1)\nkto\nthe energy is determined by the solution of the secular\nequation det/bracketleftBig\nˆP0\nkδVˆP0\nk−E(1)\nk/bracketrightBig\n= 0 corresponding to the\neigenvalue problem (20). The condition for time-reversal\ninvariance simplifies the secular equation by introducing\nrelationships between its matrix elements,\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglea−E(1)\nkc d 0\nc∗b−E(1)\nk0 d\nd∗0b−E(1)\nk−c\n0 d∗−c∗a−E(1)\nk/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle= 0,(21)\nwherethe matrixelements a=a1+a2+α,b=b1+b2+α,\nc=c1+c2andd=d1+d2+βhavecontributionsfromthe\nthree different perturbing terms. The matrix elements\ndue to the spin-orbit coupling term ˆU1are denoted as\na1,b1,c1,d1, the ones due to ˆU2asa2,b2,c2,d2, and\nthe ones due to the electrostatic field φ(r) asαandβ,\na1=/an}b∇acketle{tsu,k|ˆU1|k,su/an}b∇acket∇i}ht, a2=/an}b∇acketle{tsu,k|ˆU2|k,su/an}b∇acket∇i}ht,(22a)\nb1=/an}b∇acketle{t−su,k|ˆU1|k,−su/an}b∇acket∇i}ht,b2=/an}b∇acketle{t−su,k|ˆU2|k,−su/an}b∇acket∇i}ht,(22b)\nc1=/an}b∇acketle{tsu,k|ˆU1|k,−su/an}b∇acket∇i}ht, c2=/an}b∇acketle{tsu,k|ˆU2|k,−su/an}b∇acket∇i}ht,(22c)\nd1=/an}b∇acketle{tsu,k|ˆU1|−k,su/an}b∇acket∇i}ht, d2=/an}b∇acketle{tsu,k|ˆU2|−k,su/an}b∇acket∇i}ht,(22d)\nα=/an}b∇acketle{tk|δV(r)|k/an}b∇acket∇i}ht, β=/an}b∇acketle{tk|δV(r)|−k/an}b∇acket∇i}ht.(22e)\nTaking into account the properties of the electrostatic\npotential φ(r) and the terms ˆU1andˆU2leads to further\nsimplification of the above result. The inversion invari-\nanceˆIˆU1ˆI+=ˆU1requires that a1=b1,c1= 0 and that\nd1is a real number. The condition that the term ˆU2\nmust be odd ˆIˆU2ˆI+=−ˆU2with respect to space inver-\nsion invariance, requires a2=−b2andd2to be purely\nimaginary, while it does not place any restrictions on c2.\nThe fact that the electrostatic field φ(r) is odd with re-\nspect to inversion invariance, ˆIφ(r)ˆI+=−φ(r), requires\nα=−αand thus α= 0, while βneeds to be purelyimaginary. Taking into account these simplifications the\nfirst order corrections to the energy take the form\nE(1)±\nk=a1±/radicalBig\na2\n2+|c2|2+|d1+d2+β|2.(23)\nEq. (23) shows that the spin-orbit coupling term ˆU1\npossessing space inversion invariance leads to a shift\nof the first-order energy levels by the amount a1=\n/an}b∇acketle{tsu,k|ˆU1|k,su/an}b∇acket∇i}ht.\nA. Perturbation with Translational Invariance\nThe expression (23) shows that the four-fold degen-\neracy of E0\nkcan be lifted up to two twofold degenerate\nlevels in first-order perturbation method. The spin split-\nting caused by the spin-orbit coupling term ˆU2, which\nlacks inversion invariance, is embodied in the matrix ele-\nmentsa2,c2andb2withinthisapproach. Thepresenceof\nthe matrix element βin the energy correction (23) leaves\nopen the optionthat the electrostaticfield leadstoan ad-\nditionalcontributiontothespinsplitting. However,since\nthe condition ˆIφ(r)ˆI+=−φ(r) requires that either βis\npurely imaginary or 0, the consideration of time-reversal\ninvariance and broken space inversion invariance within\nperturbation method does not offer conclusive evidence\nthatβ/ne}ationslash= 0.\nWe have supposed that the nonperturbed and per-\nturbed electrostatic potentials possess discrete transla-\ntional invariance, ˆTRnφ(r)ˆT+\nRn=φ(r). Consequently,\nthe terms ˆU1andˆU2also possess discrete translational\ninvariance, ˆTRnˆU1ˆT+\nRn=ˆU1, andˆTRnˆU2ˆT+\nRn=ˆU2\nrespectively. Using this and the Bloch theorem for\nd2=/an}b∇acketle{tsu,k|ˆU2|−k,su/an}b∇acket∇i}htwe find\n/an}b∇acketle{tsu,k|ˆU2|−k,su/an}b∇acket∇i}ht=/an}b∇acketle{tsu,k|ˆT+\nRnˆU2ˆTRn|−k,su/an}b∇acket∇i}ht=\n=e2ik·R/an}b∇acketle{tsu,k|ˆU2|−k,su/an}b∇acket∇i}ht,(24)\nwhich for arbitrary kand translation vector Rcan be\nsatisfied only if d2= 0. Similar reasoning shows that\nd1= 0 and β= 0. Thus the first-order energy correction\n(23) takes the form\nE(1)±\nk=a1±/radicalBig\na2\n2+|c2|2. (25)\nTherefore it is not possible to have lifting of spin degen-\neracy of Bloch states purely by translationally invariant\nelectrostatic field, which is odd with respect to space in-\nversion invariance, but the reason for this is the discrete\ntranslational invariance of the perturbing electrostatic\nfieldφ(r). This is interesting since in the usual sym-\nmetry argument translational invariance is not explicitly\nconsidered, nor any attention is devoted to it. Further-\nmore,allenergycorrectionsofhigherorderdisappeardue\nto translational invariance since all /an}b∇acketle{tsu,k|ˆU|k′,su/an}b∇acket∇i}ht= 0.\nThis makes Eq. (25) exact to all orders in the perturba-\ntion method treatment.6\nB. Perturbation without Translational Invariance\nNatural continuation of the above line of thought is to\nconsider perturbing electrostatic field φ′(r) with corre-\nsponding energy V′(r) =−eφ′(r) not possessing a center\nof inversion, ˆIV′(r)ˆI+/ne}ationslash=V′(r), andlacking translational\ninvariance\nˆTRnV′(r)ˆT+\nRn/ne}ationslash=V′(r), (26)\nto which we will refer as external. Its correspondingfirst-\norder matrix element which is not necessarily zero due\nto time-reversal invariance is β′=/an}b∇acketle{tsu,k|ˆV′| −k,su/an}b∇acket∇i}ht=\n/an}b∇acketle{t−su,k|ˆV′| −k,−su/an}b∇acket∇i}ht. It is defined between zeroth-\norder states with spin, but since the electrostatic field\ndoes not contain operator acting directly on the spin-\ndegree of freedom it can be written shorthanded as\nβ′=/an}b∇acketle{tk|ˆV′|−k/an}b∇acket∇i}ht. Because V′(r) is not invariant with re-\nspect to space inversion, the disappearance of the matrix\nelementβ′is not guaranteed. Examples of electrostatic\nfields possessing these properties are the built-in poten-\ntial in p-n junctions and externally applied electric fields.\nThese are usually orders of magnitude smaller than the\nbulk electrostatic fields.\nBy using the Bloch theorem and the inequality (26) we\nobtain the inequality\n/an}b∇acketle{tsu,k|V′(r)|−k,su/an}b∇acket∇i}ht /ne}ationslash=e2ik·R/an}b∇acketle{tsu,k|V′(r)|−k,su/an}b∇acket∇i}ht,\n(27)\nfor the matrix element β′. Therefore for any given\nkand arbitrary translation vector Rwe must have\nβ′/ne}ationslash=e2ik·Rβ′. This condition excludes the possibility the\nmatrix element to be equal to zero and thus β′/ne}ationslash= 0.\nIn calculating the first-order corrections to the en-\nergy we neglect the matrix elements a′,b′and\nc′stemming from the spin-orbit coupling term\n¯h\n4m2c2ˆσ·(∇V′(r)׈p)sincetheyshouldbemuchsmaller\nthanβ′=/an}b∇acketle{tsu,k|ˆV′|−k,su/an}b∇acket∇i}htbecause of the prefactor\n¯h\n4m2c2. Since β′/ne}ationslash= 0 it will appear in the expressions\nfor the energy splitting. For the triclinic pedial system\nin which the bulk potential does not possess space in-\nversion invariance, the first-order corrections to the en-\nergy,E(1)±\nk=a1±/radicalbig\na2\n2+|c2|2+|β′|2, contain contribu-\ntions from the spin-orbit coupling a2andc2and a contri-\nbution purely from electrostatic field in β′. The original\nfourfold degeneracy is lifted to two distinct two-fold de-\ngenerate levels. The energy splitting between them is\ngiven by\n∆E(1)\nk= 2/radicalBig\na2\n2+|c2|2+|β′|2. (28)\nFor the triclinic pinacoidal system where the bulk elec-\ntrostatic potential possesses space inversion invariance,\na2andc2are identically zero. The expression for the\nfirst-order energy corrections takes the form E(1)±\nk=\na1±|β′|. The energy splitting,\n∆E(1)\nk= 2|β′|, (29)between the two twofold degenerate levels has contribu-\ntions only directly from the electrostatic field through\nβ′.\nC. The lifted degeneracy\nA heuristic argument supporting the interpretation of\nthe energy splittings (28) and (29) as spin splitting is\nthe usual interpretation of the energy splitting due to\nspin-orbit coupling in systems lacking space inversion\ninvariance8,11,12,19as spin splitting. This is embodied\nin the matrix elements a2andc2in Eq. (28). The addi-\ntional effect due to electrostatic field with broken space\ninversion and translational invariance is embodied in β′\nin Eq. (28). Of course because of broken translational in-\nvariance of V′(r) the perturbed states |Ψ/an}b∇acket∇i}htare no longer\nBloch states.\nThe energy splittings (28) and (29) can be interpreted\nas pure spin-splitting for the zero-th order Bloch states.\nThis is easily demonstrated using the invariances of the\nsubspaces ε(0)\nk±as shown in the following paragraphs.\nSince the fourfold degeneracy of E(0)\nkis not completely\nremoved,eachofthefirst-ordercorrectionstoenergy E(1)\nk±\ncorresponds21to a two-dimensional subspace ε(0)\nk±of the\nfour-dimensional subspace ε(0)\nkof the unperturbed prob-\nlem (9). The subspaces ε(0)\nk±are mutually exclusive and\ntheir direct sum ε(0)\nk++ε(0)\nk−=ε(0)\nkis the four-dimensional\nsubspace ε(0)\nk. The zeroth-order state |0k/an}b∇acket∇i}ht, which is the\nprojection of the perturbed state |Ψ/an}b∇acket∇i}htonto the subspace\nε(0)\nk, cannot be determined uniquely, only its belonging\ntooneofthe subspaces ε(0)\nk±canbe inferredfromEq.(20).\nSymmetry arguments suggest that Kramers degener-\nacy is preserved and therefore the remaining twofold de-\ngeneracy in first order is the Kramers degeneracy. In or-\nder to show this we suppose the equation for first-order\nenergy correction to be solved and consider it as identity\nˆP(0)\nkV′(r)ˆP(0)\nk|0±/an}b∇acket∇i}ht ≡E(1)\nk±|0±/an}b∇acket∇i}ht, (30)\nwhere the states |0±/an}b∇acket∇i}htare arbitrarily chosen zeroth-order\nstates belonging to the subspaces ε(0)\nk±, respectively, and\ntherefore to ε(0)\nk. Applying the time reversal opera-\ntor to Eq.(30) and using that [ ˆK,ˆP(0)\nk] = 0, because\nε0\nkis invariant with respect to ˆK, [ˆK,V′(r)] = 0 by\nhypothesis and E(1)\nk±is real, we obtain the identity\nˆP(0)\nkV′(r)ˆP(0)\nkˆK|0±/an}b∇acket∇i}ht ≡E(1)\nk±ˆK|0±/an}b∇acket∇i}ht. TheˆK-transformed\nstateˆK|0±/an}b∇acket∇i}htis orthogonal21to|0±/an}b∇acket∇i}htsince we consider a\nsingle-electronsystemtakingintoaccountthespindegree\nof freedom . As shown above, ˆK|0±/an}b∇acket∇i}htalso identically sat-\nisfies Eq. (30) with the eigenvalues E(1)\nk±and therefore be-\nlong to the subspaces ε(0)\nk±, respectively. Since we choose\n|0±/an}b∇acket∇i}htarbitrarily, and by the above argument every ˆK|0±/an}b∇acket∇i}ht7\nbelongs to ε(0)\nk±, the subspaces ε(0)\nk±are invariant with\nrespect to the time-reversal operator ˆK,ˆKε(0)\nk±=ε(0)\nk±.\nThus the remaining two-fold degeneracy in first order is\nprecisely the Kramers degeneracy.\nThe two-dimensional subspaces ε(0)\nk±are, however, not\ninvariantwith respect to the space-inversionoperatorbe-\ncause the perturbation V′(r) does not commute with it,\n[ˆI,V′(r)]/ne}ationslash= 0. This is provedby againapplying the time-\nreversal operator ˆIto the identity (30) satisfied by arbi-\ntrary state |0±/an}b∇acket∇i}ht ∈ε(0)\nk±. However, since [ ˆI,V′(r)]/ne}ationslash= 0\nand by using, ˆI−1ˆI= 1, Eq.(30) transforms into the\nidentity, ˆP(0)\nkˆIV′(r)ˆI−1ˆP(0)\nk|0±/an}b∇acket∇i}ht=E(1)\nk±ˆI|0±/an}b∇acket∇i}htfor theˆI-\ntransformed state ˆI|0±/an}b∇acket∇i}ht. This shows that ˆI|0±/an}b∇acket∇i}htdoes\nnot satisfy Eq.(30) with eigenvalue E(1)\nk±but a different\nequation with the eigenvalue E(1)\nk±becauseV′/ne}ationslash=ˆIV′ˆI−1.\nTherefore the ˆI-transformed states ˆI|0±/an}b∇acket∇i}htdo not belong\nto the subspaces ε(0)\nk±. The choice of |0±/an}b∇acket∇i}htis arbitrary\napart from the condition |0±/an}b∇acket∇i}ht ∈ε(0)\nk±and therefore for\nevery state |0±/an}b∇acket∇i}htbelonging to ε(0)\nk±theˆI-transformed\nstateˆI|0±/an}b∇acket∇i}htdoes not belong toε(0)\nk±. However ,the four-\ndimensional subspace ε(0)\nk=ε(0)\nk++ε(0)\nk−is invariant with\nrespect to ˆI,ˆIε(0)\nk=ε(0)\nkand therefore ˆI|0±/an}b∇acket∇i}htmust be-\nlong toε(0)\nk. Since ˆI|0±/an}b∇acket∇i}htdoes not belong to ε(0)\nk±the\nonly remaining option is that it belongs to the other\ntwo-dimensional subspace ε(0)\nk∓. Thus the space inver-\nsion operator ˆImaps every state |0+/an}b∇acket∇i}ht ∈ε(0)\nk+to a state\nˆI|0+/an}b∇acket∇i}ht ∈ε(0)\nk−belonging to the other two-dimensional\nsubspace ε(0)\nk−and vice versa, symbolically written as\nˆIε(0)\nk±=ε(0)\nk∓. In other words the zeroth-order states |0±/an}b∇acket∇i}ht\nandˆI|0±/an}b∇acket∇i}htbelong to the two different first-order energy\ncorrections E(1)\nk±separated from each other by the energy\ndifference ∆ E(1)\nk.\nNow consider any arbitrary zeroth-order state with\nsome spin polarization belonging to the subspace ε(0)\nk+\nwhich we denote as |0+/an}b∇acket∇i}htand apply the spin-flip\noperator19ˆCto obtain ˆC|0+/an}b∇acket∇i}ht=ˆIˆK|0+/an}b∇acket∇i}ht=ˆI|0+′/an}b∇acket∇i}ht. From\nthe previous results we know that |0+′/an}b∇acket∇i}ht=ˆK|0+/an}b∇acket∇i}htalso be-\nlongs to the subspace ε(0)\nk+, while the ˆI-transformed state\nˆC|0+/an}b∇acket∇i}ht=ˆI|0+′/an}b∇acket∇i}htbelongstotheothertwo-dimensionalsub-\nspaceε(0)\nk−. Therefore the spin-flip operator ˆC, Eq. (8),\nmaps any zeroth-order state |0k/an}b∇acket∇i}ht ∈ε(0)\nk±to a state be-\nlonging to the other two-dimensional set |0k/an}b∇acket∇i}ht ∈ε(0)\nk∓sim-\nilarly to the space inversion operator ˆI. By definition a\nzeroth-order state corresponds to the subspace ε(0)\nk±if it\nsatisfies Eq.(20) with the corresponding eigenvalue E(1)\nk±.\nSo the zeroth-order states |0k/an}b∇acket∇i}ht ∈ε(0)\nk±andˆC|0k/an}b∇acket∇i}ht ∈ε(0)\nk∓\ncharacterizedby identical quantum numbers but describ-\ning opposite spin orientations belong to the two differentfirst-order energy corrections E(1)\nk±. This constitutes the\nproof.\nA question might arise as to why should a matrix el-\nementβ′which is diagonal in spin indices be thought of\nas a spin splitting even just for the zero-th order states.\nA perspective to understand the issue is that indeed if\nthe secular problem to be solved reduces to 2 ×2 matrix\nthen matrix element diagonal in spin indices can not be\ninterpreted as spin splitting. However we consider a 4 ×4\nmatrix and the matrix element β′while diagonal in spin\nindices is not diagonal in the 4 ×4 matrix. They are on\nthe third diagonal and they lead to lifting of the origi-\nnal four-fold degeneracy into two two-fold degeneracies.\nAbove we have presented detailed argument for the in-\nterpretation of these as pure spin splitting for the zeroth\norder Bloch states.\nHowever the perturbed states |Ψ/an}b∇acket∇i}htare no longer Bloch\nstates because the perturbation V′(r) breaks the space\ntranslation invariance. As a consequence the crystal\nwavevector kis no longer a good quantum number for\ntheperturbedstatesandinhigherordersintheperturba-\ntion method the lifted degeneracy can not be interpreted\npurely as spin degeneracy. Instead it can be interpreted\nas lifting of some sort of orbital-spin degeneracy similar\ntothefinestructuresplittingduetospin-orbitcouplingin\natomic systems. This is within the perturbation method\ntreatment.\nVI. DISCUSSION\nThe primary goal of the study has been to explore the\nhypothesis whether an electrostatic field can lift the spin\ndegeneracyof Bloch states. Integral new contributions of\nthis have been the novel point for introduction of spin-\norbit coupling presented in Sec. IV and the null result\nof Sec. VA, which have naturally shaped the presenta-\ntion. The hypothesis presented in mathematical detail\nin Sec. III and discussed in subsection IIID has been in-\nspired by a symmetry argument which to the best of our\nknowledge is due to Kittel19. The original argument has\nbeen used to predict the lifting of spin degeneracy in the\npresence of spin-orbit coupling. In subsection IIID we\nnoted that it is valid also for a model of electron moving\nin an electrostatic field in which we take spin degree into\naccount but neglect spin-orbit coupling. Such a model is\neasily justified on the ground that in nature there is no\nelectron without spin.\nThe trivial consequence of such a model would be dou-\nble degeneracy of all levels due to the spin-degree of free-\ndom. In such a model, where spin-orbit coupling is ne-\nglected, SU(2) symmetryis preservedasnotedin Sec. IV.\nThus the double spin degeneracy of such a model can be\nviewed also as a consequence of SU(2) symmetry. Of\ncourse when spin-orbit coupling is taken into account,\nSU(2) symmetry is broken21, and it is not possible to\nintroduce spin quantum number as quantum number of\ntype constant of motion. However we treat spin-orbit8\ncoupling terms as perturbations and in the zeroth-order\nmodel the spin can be introduced as a separate quantum\nnumber.\nThe original symmetry argument due to Kittel19does\nnot use the irreducible representations of the point\ngroups. We considered formulating the symmetry argu-\nment using the irreducible representations of the point\ngroups, but for the particular purpose we concluded that\nworking with the symmetry operators themselves is suffi-\ncient. This is so, because we consider a conceptual ques-\ntion, thereforewetestthehypothesisonthesimplestpos-\nsible systems which posses the required characteristics of\nthe problem. These are3Dcrystalsofthe triclinic crystal\nsystem: the triclinic pedial and triclinic pinacoidal crys-\ntal classes. These are the least symmetric classes, which\nhaveincommononlyarotationby2 πandatime-reversal\nsymmetry. In addition the pedial class is characterized\nwith a broken space inversion symmetry, while the only\nother symmetry element of the pinacoidal class is the\nspace inversion.\nThe compatibility relations of the irreducible repre-\nsentations can be used to calculate the effects, including\nlifting of degeneracy, by electric fields which do not break\nthe translational invariance. However, this approach\nwould not offer an answer to the hypothesis whether an\nelectrostatic field alone would break the spin degeneracy.\nThis is so because based solely on the symmetry argu-\nment we cannot say whether the spin splitting is caused\nby the electrostatic field alone or by the spin-orbit cou-\npling. This is the reasonto use the perturbation method.\nIndeed, when we use the perturbation method it turns\noutthat ifthe translationalinvarianceoftheelectrostatic\nfield is preserved no spin-splitting occurs. On the other\nhand the perturbation method treatment indicates a lift-\ning of the spin degeneracy when both the translation and\ninversion invariances are broken. However, if the trans-\nlational invariance is broken we can no longer talk about\nspace symmetry groups and their subgroups - the pointgroups. This is so because the discussion of the space\ngroups requires the translational invariance of the crys-\ntal lattice.\nVII. CONCLUSIONS\nWe have scrutinized the symmetry argument based on\nspace inversion and time-reversal invariance predicting\nthe appearance of spin splitting in case of broken space\ninversion symmetry. A novel argument for the need for\nintroduction of a term breaking SU(2) invariance like the\nspin-orbit coupling term has been presented. This ar-\ngument is different from the usual arguments based on\nspecial relativity used for the introduction of spin-orbit\ncoupling term. We have shown that in systems possess-\ning discrete translational invariance it is not possible to\nhave spin splitting solely by electrostatic field with bro-\nkenspace inversionsymmetrydue to the preservedtrans-\nlational invariance while a spin splitting exists due to the\ncombinationofspin-orbitcoupling and electrostaticfield.\nThe possibility for lifting of spin degeneracy due to elec-\ntrostatic field without the mediation of spin-orbit cou-\npling has been investigated using perturbation method\ntreatment suggesting its possible existence in systems\ncharacterized by both broken space inversion invariance\nand broken translational invariance, as far as the pertur-\nbation method is applicable. There is a possibility that\nthe last result is a quirk of the application of the per-\nturbation method to the case of electrostatic field with\nbroken translational invariance. The problem might be\nfurther clarified and the results tested theoretically by\ntreating the same scenario in the Dirac equation. If they\nare confirmed the possibility needs to be tested experi-\nmentally, most suitable for which maybe systems of the\ntriclinic pinacoidal symmetry system.\n∗tenev@phys.uni-sofia.bg\n1I.ˇZuti´ c, J. Fabian, and S. Das Sarma, Rev. Mod. Phys.\n76, 323 (2004).\n2S. A. Wolf, A. Y. Chtchelkanova, and D. M. Treger, IBM\nJournal of Research and Development 50, 101 (2006).\n3M. N. Baibich, J. M. Broto, A. Fert, F. N. Van Dau,\nF. Petroff, P. Etienne, G. Creuzet, A. Friederich, and\nJ. Chazelas, Phys. Rev. Lett. 61, 2472 (1988).\n4G. Binasch, P. Gr¨ unberg, F. Saurenbach, and W. Zinn,\nPhys. Rev. B 39, 4828 (1989).\n5S. Datta and B. Das, Appl. Phys. Lett. 56, 665 (1990).\n6H. C. Koo, J. H. Kwon, J. Eom, J. Chang, S. H.\nHan, and M. Johnson, Science 325, 1515 (2009),\nhttp://www.sciencemag.org/cgi/reprint/325/5947/1515 .pdf.\n7J. Schliemann, J. C. Egues, and D. Loss, Phys. Rev. Lett.\n90, 146801 (2003).\n8G. Dresselhaus, Physical Review 100, 580 (1955).\n9E. O. Kane, J. Phys. Chem. Solids. 1, 249 (1957).10M. H. Weiler, R. L. Aggarwal, and B. Lax, Phys. Rev. B\n17, 3269 (1978).\n11W. Zawadski and P. Prfeffer, Semicond. Sci. Technol. 19,\nR1 (2004).\n12R. Winkler, Spin-Orbit Coupling Effects in Two Dimen-\nsional Electron and Hole Systems (Springer Verlag, 2002).\n13J. Fabian, A. Matos-Abiague, C. Erther, P. Stane, I. Zutic,\nActa Physica Slovaca 57(2007).\n14R. Eppenga and M. F. H. Schuurmans, Phys. Rev. B 37,\n10923 (1988).\n15E. A. de Andrada e Silva, Phys. Rev. B 46, 1921 (1992).\n16Y. A. Bychkov and E. I. Rashba, Journal of Physics C:\nSolid State Physics 17, 6039 (1984).\n17F. J. Ohkawa and Y. Uemura, Journal of the Physical So-\nciety of Japan 37, 1325 (1974).\n18R. Lassnig, Phys. Rev. B 31, 8076 (1985).\n19C. Kittel, Quantum Theory of Solids , 2nd ed. (John Wiley\n& Sons, 1987) p. 528.9\n20W.-K. Tung, Group Theory in Physics (World Scientific,\n2008).\n21A. Messiah, Quantum Mechanics (Dover Publications,\n1999).\n22J. J. Sakurai, Modern Quantum Mechanics (Addison-\nWesley Publishing Company, 1994).23J. J. Sakurai, Advanced Quantum Mechanics (Addison-\nWesley Publishing Company, 1967).\n24Bernd Thaller, The Dirac Equation (Springer-Verlag,\n1992)."    },    {        "title": "1310.7847v1.Self_Quenching_of_Nuclear_Spin_Dynamics_in_Central_Spin_Problem.pdf",        "content": "arXiv:1310.7847v1  [cond-mat.mes-hall]  29 Oct 2013Self-Quenching of Nuclear Spin Dynamics in Central Spin Pro blem\nArne Brataas1and Emmanuel I. Rashba2\n1Department of Physics, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway\n2Department of Physics, Harvard University, Cambridge, Mas sachusetts 02138, USA\nWe consider, in the framework of the central spin s= 1/2 model, driven dynamics of two electrons\nin a double quantum dot subject to hyperfine interaction with nuclear spins and spin-orbit coupling.\nThe nuclear subsystem dynamically evolves in response toLa ndau-Zener singlet-triplet transitions of\ntheelectronic subsystemcontrolled byexternalgate volta ges. Withoutnoise andspin-orbit coupling,\nsubsequent Landau-Zener transitions die out after about 104sweeps, the system self-quenches, and\nnuclear spins reach one of the numerous glassy dark states. W e present an analytical model that\ncaptures this phenomenon. We also account for the multi-nuc lear-specie content of the dots and\nnumerically determine the evolution of around 107nuclear spins in up to 2 ×105Landau-Zener\ntransitions. Without spin-orbit coupling, self-quenchin g is robust and sets in for arbitrary ratios\nof the nuclear spin precession times and the waiting time bet ween Landau-Zener sweeps as well as\nunder moderate noise. In presence of spin-orbit coupling of a moderate magnitude, and when the\nwaiting time is in resonance with the precession time of one o f the nuclear species, the dynamical\nevolution of nuclear polarization results in stroboscopic screening of spin-orbit coupling. However,\nsmall deviations from the resonance or strong spin-orbit co upling destroy this screening. We suggest\nthat the success of the feedback loop technique for building nuclear gradients is based on the effect\nof spin-orbit coupling.\nPACS numbers:\nI. INTRODUCTION\nElectrical operation of electron spins in semiconduc-\ntor double quantum dots (DQD) is one of the cen-\ntral avenues of semiconductor spintronics1and quantum\ncomputing.2–5There are three basic types of electronic\nspin qubits, (i) the Loss-DiVincenzo2qubits operating\nsingle-electron spins, (ii) singlet-triplet qubits operating\na two-electron system and (iii) three electron qubits.6\nThe second type is the center of our attention. Most\nwidely explored singlet-triplet DQD qubits7are based on\nGaAs8,9andInAs.10–12BothinGaAsandInAs, thereare\nthree species of nuclei possessing non-vanishing angular\nmomenta, and the coupling between electron and nuclear\nspins (mostly through contact interaction) strongly in-\nfluences electron-spin dynamics. Primarily, this coupling\nhas a destructive effect causing electron spin relaxation,\nand many theoretical studies have focused on the chal-\nlenging problem of determining the relaxation rate of an\nelectron spin interacting with about N≈106nuclear\nspins.13–20The problem of an electron spin interacting\nwith a bath of nuclear spins is known as the central spin\nproblem.\nHowever, a controllable nuclear spin polarization, act-\ning as an effective magnetic field, can also become a re-\nsource for manipulating electron spins.21,22In particu-\nlar, the difference (gradient) in the nuclear polarization\nof the left and right dots can be used for σxrotations\nof aS-T0singlet-triplet qubit on the Bloch sphere.8Ef-\nficient control of a vast ensemble of nuclear spins is very\nchallenging, and many analytical and numerical works\nhave been carried on this subject.23–31The principal ex-\nperimental tool for polarizing nuclear spins and building\ngradients is based on driving a two-electron DQD elec-trically through the avoided crossing ( S-T+anticrossing)\nof its singlet level Sand theT+component of its elec-\ntronic triplet T= (T+,T0,T−).T+is the lowest energy\ntriplet component because the electron g-factor is nega-\ntive,g <0, both in GaAs and InAs. The width of the\nanticrossing is controlled by hyperfine and spin-orbit32\ninteractions. When the electron state changes from S\ntoT+orvice versa by passaging through the S-T+an-\nticrossing, and there is no spin-orbit coupling, up to one\nquantum of the angular momentum is transferred to the\nnuclear subsystem, and such transfer facilitates polariz-\ning the nuclear bath by performing multiple passages.\nUnfortunately, experimental data show that the nu-\nclear polarization saturates at a rather low level, typi-\ncally of about 1%.33The origin of this low saturation\nlevel remains unclear and constitutes the critical obsta-\ncle for achieving higher levels of nuclear spin polariza-\ntion. We recently uncovered a mechanism of dynamic\nself-quenching which, in absence of spin-orbit (SO) cou-\npling, results in fast suppression ofthe transversenuclear\npolarization under stationary pumping.30This is caused\nby screening the random field of the initial nuclear spin\nfluctuation by the nuclear polarization produced through\npumping and closing the anticrossing. This conclusion is\nin a qualitative agreement with the data of Refs. 25,31\nin the strong magnetic field limit, and the states with\nvanishing transverse magnetization are known as “dark\nstates”. Meantime, by applying feedback loops, exper-\nimenters managed to achieve considerable and control-\nlable nuclear spin polarizations.34This poses a challeng-\ning question in which way closing the S-T+intersection\ndue to the self-quenching mechanism could be avoided.\nOur data of Ref. 30 indicate that SO coupling changes\nthepatternsofself-quenchingdramatically,whichimplies\nthat it is the spin-orbit coupling that might resolve the2\nproblem. The main goal of the current paper is to shed\nmore light on the mechanisms controlling the transfer of\nangularmomentumfromtheelectronqubittothenuclear\nbath. For this purpose, we solve numerically the equa-\ntions describing coupled electron and nuclear spin dy-\nnamics for DQDs of a realistic size of more than N∼106\nnuclear spins and a shape of two overlapping Gaussian\ndistributions. These simulations follow up to 2 ×105\nsweeps and unveil intimate patterns of transferring spin\npolarizationfromtheelectronictothenuclearsubsystem.\nTo get quantitative insight onto the long time dynam-\nics of spin pumping by multiple passagesacross the S-T+\nanticrossing, we restrict ourselves to the strong magnetic\nfield regime when the Zeeman split-off T0andT−compo-\nnents of the electron spin triplet are well separated from\ntheSandT+states, hence, transitionsto these statesare\ndisregarded. Therefore, with a semiclassical description\nof nuclear spins, electrons form a two-level system, and\npassages across the S-T+anticrossing are described by\nthe Landau-Zenertype theory.35,36The detailed patterns\ndepend on the shape on the pulses on the gates and the\ninstantaneousnuclearconfiguration. Inturn, duringeach\nLZ sweep the nuclear configuration changes due to the\ndirect transfer of the angular momentum and shake-up\nprocesses.29,30Between the LZ sweeps, this configuration\nchanges because of the difference in the Larmor preces-\nsion rates of different nuclear species. To follow the long\nterm evolution, we solve the problem of the coupled dy-\nnamics self-consistently. From the mathematical point of\nview, we arrive to a central spin s= 1/2 problem with a\ndriven dynamics of the electron spin. Hence, beyond the\napplication to the spin pumping problem, our results are\nof general interest for coupled dynamics of many body\nsystems.\nIn this paper we prove, both analytically and numer-\nically, that self-quenching into dark states is a generic\nproperty of the pseudospin s= 1/2 model in absence of\nSO coupling, and that self-quenching sets in after only\nabout 104sweeps. We also demonstrate that this result\nstands under a moderate noise. However, the main fo-\ncus of the paper is on the effect of SO coupling. Because\nthe SO field is static while the hyperfine Overhauserfield\noscillates in time with the Larmor frequencies of nuclei,\nself-quenching cannot set in. Nevertheless, if the wait-\ning time between LZ sweeps coincides with the Larmor\nperiod of one of the species, self-quenching sets in stro-\nboscopically (as was demonstrated in our previous paper\nfor a single-specie model30). More specifically, the Over-\nhauser field of the resonant specie screens the SO field\nduring the LZ sweeps (whose duration is small compared\nwith Larmor periods). Therefore, during the sweeps the\nelectron and nuclear subsystems become decoupled. As\ndistinct from self-quenching in absence of SO coupling,\nthe stroboscopic self-quenching is fragile. Even a small\ndeviationfromtheresonance,about1%, destroysthedel-\nicatecompensationoftheSOandhyperfinecontributions\nduring the LZ sweeps. Moreover, we were able to observe\nthe stroboscopic self-quenching only for moderate valuesof the SO coupling that do not exceed considerably the\nrandom fluctuations of the hyperfine field.\nWe conclude that it is the SO coupling that endows\nthe nuclear subsystem with a long term dynamics under\nthe stationary LZ pumping. Therefore, we suggest that\nSO coupling is critical for efficient operating the feedback\nloops that requireaccumulation of largepolarizationgra-\ndients at the scale of about 106sweeps.34Effect of SO\ncoupling at a shorter time scale has been recently un-\nveiled by Neder et al.37by comparing with experimental\ndata of Ref. 38.\nII. OUTLINE AND BASIC RESULTS\nIn Sec. III, following two introductory sections, we\npresent the basic equations of the driven coupled\nelectron-nuclear spin dynamics of the central spin-1 /2\nproblem that is the basis for all following calculations. In\nSec. IV, we find an analytical solution for a simple model\ndemonstrating the phenomenon of self-quenching which\nreveals basic factors controlling its rate. Our numerical\ntechnique that allows following the coupled dynamics of\nthe electron 1 /2-pseudospin and about 107nuclear spins\nduring up to 2 ×105LZ sweeps is described in Sec. V. It\nalso includes parameters of the double quantum dot and\nLZ pulses used in simulations.\nSec. VI is the central one. It opens with the nuclear\nparameters of InAs and GaAs used in simulations, and\nincludes the results of simulations and their discussion.\nIn this section, we demonstrate that self-quenching is a\ngeneric and robust property of the coupled dynamics in\nabsence of spin-orbit coupling, and analyze its specific\nfeatures in systems consisting of two and three nuclear\nspecies. Next, we introduce SO coupling and demon-\nstrate that it eliminates self-quenching and causes the\nnuclear subsystem to exhibit a persistent, but irregu-\nlar, oscillatory dynamics. We also demonstrate the phe-\nnomenonofstroboscopicself-quenchingthat sets in when\nthe waiting time between LZ sweeps is in resonance with\nthe Larmor period of one of the nuclear species and show\nthat it is very sensitive to deviations from the exact res-\nonance. Finally, we suggest that the SO induced nuclear\ndynamicsiscriticalforthe feedbacklooptechniquedevel-\noped by Bluhm et al.34for building controllable nuclear\npolarization gradients.\nWe summarize our results in Sec. VII and estimate\nthe strength of SO coupling in InAs and GaAs double\nquantum dots in Appendix A.\nIII. BASIC EQUATIONS\nHyperfine electron-nuclear interactions and SO cou-\nplinggovernthecouplingbetweenelectronstatesin A3B5\nquantum dots utilized for quantum computing purposes.\nNuclear spins are dynamic and can be controlled by ma-\nnipulating magnetic fields and electronic states.3\nWe consider electrons in double quantum dots inter-\nacting with nuclear spins via the hyperfine interaction.\nWhen there are two electrons in the dot, the orthogonal\nbasis consists of singlet and triplet spin states. Hyperfine\nand SO interactionscouple these states. By changing the\ngate voltages that confine electrons and determine sin-\nglet and triplet energies, a transition from a singlet Sto\na triplet electron state T+(or vice versa) is accompanied\nby a change in the nuclear spin states. Our focus is on\nwhat happens to the nuclear spins as we repeat Landau-\nZener (LZ) transitions many times, up to 2 ×105, and\nhow the changes in the nuclear spin states in turn affect\nelectrons in the quantum dot.\nWe define a LZ sweep in the following way. We assume\nthat the quantum dot is first set in the singlet state, then\na change in the gate voltages drives a (partial) transition\nto the triplet state, and finally one electron is taken out\nof the system and re-inserted so that the system again\nis in its singlet state. During the sweep, the dynamics\nof the electronic qubit is controlled by the electric field\nproduced by the gates and the nuclear polarization as\ndescribed by Eq. (5) below. In turn, semiclassical dy-\nnamics of nuclear spins is driven by the Knight fields\n∆jλ(t) arising from electron dynamics\n/planckover2pi1dIjλ\ndt=∆jλ×Ijλ, (1)\nwhere the sub index jλdenotes a nuclear specie λat a\nlattice site j. Assuming the time-scale TLZof the LZ\nsweeps is much shorter than the nuclear precession times\ntλin the external magnetic field, the total effect of the\ntime-dependent fields ∆jλ(t) on each nuclear spin can\nbe integrated over the LZ sweep. Then the change of an\nindividual nuclear spin during a sweep is\n△Ijλ=Γjλ×Ijλ, (2)\nwhereΓjλaccounts for the effective magnetic field in-\nduced by the hyperfine interaction during the LZ sweep\nand depends on the configuration of all the nuclear spins\nbefore the sweep.\nLandau-Zener sweeps are repeated many times. Be-\ntween consecutive LZ sweeps, electrons are in the singlet\nstate and do not interact with nuclear spins. During\nthis waiting time Twbetween consecutive sweeps, nu-\nclear spins precess in an external magnetic field Bap-\nplied along the z-direction. The changes of the nuclear\nspins between LZ sweeps are\n△Ix\njλ= cosφjλIx\njλ−sinφjIy\njλ, (3a)\n△Iy\njλ= cosφjλIy\njλ+sinφjλIx\njλ, (3b)\n△Iz\njλ= 0, (3c)\nwhere the superscripts x,y, andzdenote Cartesian com-\nponentsofthenuclearspins,thetransversephasechanges\nareφjλ=−2πTw/tλin terms of the spin precession\ntimestλ= 2π/planckover2pi1/gλµIB, wheregλ=µλ/Iλis theg-factor\nfor a nuclear specie λ,µλis its magnetic moment, and\nµI= 3.15×10−8eV/T is the nuclear magneton.We also model the influence of noise by adding phe-\nnomenologically a random magnetic field along the z-\ndirection for each nuclear spin so that the accumu-\nlated phases in Eq. 3 change to φeff\njλ=φjλ+φnoise\njλ→\n−2πTw(1/tλ+rjλ/τ), whererjλare random numbers\nin the interval from −1 to 1. This procedure simulates\na randomization of the transverse components of nuclear\nspinsafteratime ofthe order τ, andτistermedthe noise\ncorrelation time in what follows. While simulations de-\nscribed below were performed by using random sets of\nrjλ, we mention that averaging over the noise results ef-\nfectivelyinchangingthephase-dependentfactorsinEq.3\nas∝an}bracketle{tcosφeff\njλ∝an}bracketri}htn= cosφjλsin(2πTw/τ)/(2πTw/τ), and sim-\nilarly for ∝an}bracketle{tsinφjλ∝an}bracketri}htn. Therefore, this model of transverse\nnoise leads to a semiclassical dephasing of the transverse\ncomponents of nuclear spins on the time scale τ.\nLet us next review how electronic Landau-Zener\nsweeps influence nuclear spins via Γjλ.29The hyperfine\nelectron-nuclear interaction is\nHhf=Vs/summationdisplay\nλAλ/summationdisplay\nj∈λ/summationdisplay\nm=1,2δ(Rjλ−rm)(Ijλ·s(m)),\n(4)\nwheres(m) =σ(m)/2 are the electron-spin operators\nin terms of the vector of Pauli matrices σ(m) for each\nelectronm= (1,2),Ijλare the nuclear spin operators,\nAλis the electron-nuclear coupling constant for a specie\nλ, andVsis the volume per single nuclear spin. We\nconsider GaAs or InAs quantum dots below; hyperfine\ncoupling parameters for them can be found in Sec. VI.\nAssuming that gate voltages keep the system close to\nthe singlet S- tripletT+transition, the effective Hamil-\ntonian describing the electron qubit is\nH(ST+)=/parenleftbigg\nǫSv+\nv−ǫT+−η/parenrightbigg\n, (5)\nwhereǫSis the singlet energy and ǫT+is the triplet T+\nenergy in the external magnetic field B=Bˆzwhen nu-\nclear spins are unpolarized. By retaining only SandT+\nstates, the problem is reduced to a 1/2 pseudospin prob-\nlem, and we apply the term the central spin problem in\nthis sense.\nThe energies ǫSandǫT+are controlled by the gate\nvoltages. The off-diagonal components v±=v±\nn+v±\nSO,\ncoupling the singlet Sand triplet T+states, contain con-\ntributions from nuclear spins\nv±\nn=Vs/summationdisplay\nλAλ/summationdisplay\nj∈λρjλI±\njλ, I±\njλ= (Ix\njλ±iIy\njλ)/√\n2,(6)\nand SO coupling v±\nSO.26,39When nuclear spins are po-\nlarized, the energy of the triplet state is affected by the\nOverhauser shift\nη=−Vs/summationdisplay\nλAλ/summationdisplay\nj∈λζjλIz\njλ. (7)\nThe singlet-triplet electron-nuclear couplings are\nρjλ=/integraldisplay\ndrψ∗\nS(r,Rjλ)ψT(r,Rjλ), (8)4\nand the electron-nuclear couplings in the T+state are\nζjλ=/integraldisplay\ndr/vextendsingle/vextendsingleψT(r,Rjλ)2/vextendsingle/vextendsingle, (9)\nwhereψS(ψT) is the orbital part of the singlet (triplet)\nwavefunction. Beyondthe2 ×2S-T+model, itispossible\nto define a hyperfine term that determines the singlet S-\ntripletT0coupling(asinEq.4in Ref. 29), but it isnot at\nthe centerof ourattention and will not be discussed here.\nThe effect of the electron spin T0andT−components is\ncritical for the development of nuclear polarization gra-\ndients and has been investigated in Refs. 25,31.\nIn terms of these parameters, the changes of the nu-\nclear spins △Ijλ=Γjλ×Ijλduring a Landau-Zener\nsweep are determined by coefficients\nΓ(x)\njλ=−VsAλρjλ(Pvy+Qvx)/(2v2),(10a)\nΓ(y)\njλ=VsAλρjλ(Pvx−Qvy)/(2v2), (10b)\nΓz\njλ=VsAλζjλR/(2v), (10c)\nwithv2=|v+|2=/parenleftbig\nv2\nx+v2\ny/parenrightbig\n/2. In these expressions,\n0≤P≤1 is theS-T+transition probability, a real\nnumberQis the shake-up parameter defined via29\nP+iQ=−i2v−/integraldisplayTLZ\n−TLZdt\n/planckover2pi1cS(t)c∗\nT+(t) (11)\nin terms of the singlet (triplet) amplitude cS(t) (cT(t)),\nand\nR= 2vTLZ/integraldisplay\n−TLZdt/vextendsingle/vextendsinglecT+(t)/vextendsingle/vextendsingle2//planckover2pi1 (12)\naccounts for the Overhauser shift due to the triplet\nT+component of the electron state during the interval\n(−TLZ,TLZ). In the absence of SO coupling, v±\nSO= 0,\nthe change ∆ Izof the total angular momentum of nuclei\nIz=/summationtext\njIz\njduring a single sweep equals ∆ Iz=−P, as\nfollows from the angular momentum conservation.29\nUsing the amplitudes ( cS(t),cT+(t)) found from solv-\ning the time-dependent Schr¨ odinger equation with the\nHamiltonian H(ST+)of Eq. (5) in combination with the\ndynamical equations for nuclear spins of Eq. (2) makes\nour approach completely self-consistent.\nWe assume that electrons are loaded into the singlet\n(0,2) state with energies far away from the S-T+an-\nticrossing. After loading electrons, gate voltages are\nchanged to bring the system closer to the level anticross-\ning, and this change is performed fast enough to keep\nthe system in the singlet state.40From there on, an LZ\nsweep brings the system through the anticrossing. After\nthe slow sweep, the system is moved back to the recharg-\ning point where it exchanges electrons with the reservoir.\nThis back motion is fast at the scale of the narrow anti-\ncrossingand therefore does not influence the nuclear spin\nsubsystem, but slow at the scale of the electron Zeeman\nsplitting to keep the system inside the S−T+subspace.IV. SIMPLE MODEL WITH ANALYTICAL\nSOLUTION\nLet us first present a simplified model that can be\nsolved analytically and that manifests basic features of\nself-quenching30in the absence of SO coupling, v±\nSO= 0.\nDifferent versions of this “box” or “giant spin” model\nwere applied to various problems, see Refs. 25,26,41,42.\nSubsequently, we will in Section V outline a more com-\nplex and extensive numerical procedure and discuss nu-\nmerical results for realistic models in Section VI. Re-\nstricting ourselves to a single nuclear specie, we sim-\nplify the system by modelling it as a box inside which\nthe electron wave functions are independent on posi-\ntion, all nuclei possess the same Larmor frequency, and\nall hyperfine coupling constants Aλand electron-nuclei\ncouplignsρjλare equal, Aλ=¯Aandρjλ= ¯ρ; typi-\ncally,¯A≈10−4eV. Then Eqs. (6) and (8) simplify to\nvα=A0Iα,α= (+,−,z) withA0=Vs¯A¯ρ∼¯A/N. Here\nNis the number of nuclei in the box, and Iα=/summationtext\njIα\nj\nare components of the “giant” collective angular momen-\ntum of nuclei. In the framework of this model, nuclear\nspin precession in the Zeeman and Overhauserfields does\nnot influence the coupled electron-nuclearspin dynamics,\n∆z= 0. With these assumptions, Eq. (1) becomes\ndI+\ndt=−i\n/planckover2pi1∆+Iz,dI−\ndt=i\n/planckover2pi1∆−Iz,dIz\ndt=i\n/planckover2pi1(∆+I−−∆−I+).\n(13)\nForLZ pulses, the energylevel difference changeslinearly\nwitht,ǫS(t)−ǫT+(t) =β2t//planckover2pi1, for−TLZ≤t≤TLZ, and\nthedynamicsofthequbit iscontrolledbyadimensionless\nparameterγ=v+v−/β2. The equation of motion for γ\nfollowing from (13) is\ndγ\ndt=−A2\n0\n2β2d\ndt(Iz)2. (14)\nDuring each sweep, Izchanges by ∆ Iz=−Pandγ\nchanges by ∆ γ= (A2\n0/β2)PIz. Precession of the collec-\ntive nuclear spin Iin the external magnetic field during\nthe waiting times between sweeps changes neither Iznor\nγand is disregarded. The discrete number of sweeps\nncan be considered as a continous variable since the\nchanges ∆γand ∆Izduring a single sweep are small as\ncompared to γandIz. We then arrive at the differential\nequations that determine the evolution of the γ(n) and\nIz(n):\ndγ\ndn=A2\n0P\nβ2Iz, (15a)\ndIz\ndn=−P. (15b)\nThis central result for the simple model clarifies the dif-\nferent modes of self-quenching. The evolution of the LZ\nparameterγthat controls the LZ probability Pdiffers\nin two scenarios that manifest themselves for opposite\nsigns ofIz. (i) When Izis initially negative, it continues5\nto decrease (becoming more negative) and magnitudes of\nbothγand the LZ probability Pdecrease, hence, the\nprocess slows down. Finally, self-quenching sets in ex-\nponentially, see Eq. 19 below. (ii) When Izis initially\npositive,γfirst increases so that the LZ probability P\nbecomes larger. However, since Izonly can be reduced,\nit eventually becomes negative and self-quenching of sce-\nnario (i) sets in. So, self-quenching ultimately sets in\ngenerically independent on the original sign of Iz.\nWe can get a more detailed insight into the self-\nquenchingdynamicsbyusingthe firstintegralofEq.1443\nγ=−(A2\n0/2β2)(Iz)2+γ0, (16)\nwhereγ0is an integration constant that depends on the\ninitial values of γandIz,γiandIz\ni. Obviously, Eq. 16\ndictates that γ≤γ0, andγ0≥0 becauseγ≥0 by\ndefinition. Therefore,\nIz=±√\n2(β/A0)√γ0−γ, (17)\nand\ndγ/dn=±√\n2(A0/β)√γ0−γP(γ).(18)\nIn scenario (i), when Iz\ni<0, the minus sign should be\nchosen in Eqs. 17 and 18, and γ(n) decreases monoton-\nically. In scenario (ii), when Iz\ni>0, the dynamics first\nfollows the plus branches of Eqs. 17 and 18, and γ(n)\nincreases until it reaches its maximum value γ=γ0. At\nthis point, Iz(n) vanishes, changes sign, and continues\nto decrease as follows from Eq. 16 and Eq. 15b [because\nP(γ0)>0]. At the same point, the signs in Eqs. 17\nand 18 switch from plus to minus, and afterwards γ(n)\ndecreases monotonically as follows from Eq. 15a.\nThe detailed asymptotic behavior of γ(n) forn→ ∞\ndepends on P(γ). For long LZ sweeps with 2 TLZ≫/planckover2pi1/v,\nPLZ(γ) = 1−e−2πγ≈2πγ, and\nγ(n)∝exp[−2π/radicalbig\n2γ0(A0/β)n]. (19)\nEquation(19) describesanexponentialdecaywith anon-\nuniversal exponent. The rate of decay increases with de-\ncreasingβ, when sweeps become more adiabatic. There-\nfore, in absence of SO coupling the large- nbehavior of\nγ(n) is exponential, and self-quenching sets on for arbi-\ntrary initial conditions.\nLet us make a rough estimate of the number of sweeps\nn∞before self-quenching sets in based on Eq. 19. A\ntypical original fluctuation includes N1/2spins, hence,\nv∼A0√\nN. For LZ pulses with an amplitude of about v\nand duration of about /planckover2pi1/v,βis aboutβ∼v. Therefore,\nn∞∼β/A0∼√\nN, i.e., about the number of nuclear\nspins in a typical fluctuation. The dependence of n∞\nonβdemonstrates the effect of the sweep duration TLZ,\nn∞is smaller for longer sweeps. A similar estimate for\nthe length ∆ nof the exponential tail in Eq. (19), with\n2π√2γ0≈10, results in ∆ n∼n∞/10, i.e., it is shorter\nthann∞by a numerical factor.More detailed estimates for both regimes require spe-\ncificassumptionsabouttheshapeanddurationofsweeps.\nFor sufficiently long sweeps, PLZ(γ) can be used for P(γ)\nand Eq. (18) can be integrated. The number of sweeps\nn=n(γi,γf), inunitsof β/(√\n2A0), betweentheinitial γi\nand finalγfvalues ofγis plotted in Fig. 1 for two modes;\nthe value of γ0has been chosen equal to γ0= 2. Fig. 1(a)\nis plotted for Iz\ni<0 and Fig. 1(b) for Iz\ni>0. Front sec-\ntions ofn(γi,γf) surfaces by γf= 0 planes demonstrate\nn∞(γi), the number of sweeps before self-quenching. For\nIz\ni<0, the curve increases fast with γiand reaches its\nmaximum value at γ=γ0. It is achieved at a ridge at\nthen(γi,γf) surface that originates from the square-root\nsingularity in the dn/dγdependence and is well seen in\nFig. 1(a). For Iz\ni>0, then∞(γi) dependence is much\nslower and becomes fast only near γ=γ0. In both cases,\nn∞∼β/A0, in agreement with the previous estimate.\nTherefore,themodelnotonlyprovidesanalyticaljusti-\nfication of the self-quenching phenomenon found numer-\nically in Ref. 30 for systems without SO coupling but\nrelates, for single-specie systems, two modes of behavior\n(monotonicandnonmonotonic)tothedifferenceininitial\nconditions. It is the first analytical solution of the cen-\ntral spin problem (i) describing dynamical evolution of a\npumped system into a “dark state”25,44and (ii) estab-\nlishing a connection between the initial and final states\nof the system.\nV. NUMERICAL PROCEDURE\nDuring a sweep, the difference in the singlet and triplet\nenergiesǫS−ǫT+varies linearly in time within the sweep-\ning interval −TLZ≤t≤TLZ. We impose no restrictions\nontherelativemagnitudeofthesweepduration2 TLZand\nthe inverse S-T+coupling /planckover2pi1/v, but, as stated above, TLZ\nis long as compared to the inverse electron Zeeman en-\nergy. Furthermore, it is assumed that the variationof the\nenergies of both the upper and lower spectrum branches\nis symmetric with respect to the S-T+anti-crossing for\nthefirsttransition when the initial position of the T+\nlevel isη=ηi. We denote the amplitude of the change\nin the energy difference between the singlet and triplet\nenergies as ǫmax. In other words, we use\nǫS(t) =ǫmaxt/2TLZ (20a)\nǫT+(t)−η=−ǫmaxt/2TLZ−(η−ηi) (20b)\nin Eq. (5).45Note that as a result of the dynamical nu-\nclear polarization, LZ sweeps become asymmetric with\nrespect to the anticrossing point because of the changing\nOverhausershift η. There is no longer any traditional LZ\npassagewhenever |η−ηi|>ǫmax, i.e. after the anticross-\ning point passes across one of the ends of the sweeping\ninterval. This naturally implies a slowdown in accumu-\nlating dynamical nuclear polarization. Maintaining the\nLZ passages requires additional feedback mechanisms by\nchanging the energy level difference, which we introduce\nbelow by shifting the edges of the integration interval.6\nFigure 1: Number of sweeps nbetween the initial and final\nvalues of the Landau-Zener parameter γfor two modes; nin\nunitsofβ/(√\n2A0). (a)Initial nuclear polarization is negative,\nIz\ni<0. (b) Initial nuclear polarization is positive, Iz\ni>0. In\ntheplots, thelower boundsof γiandγfwere chosen tobe 0.01\nto cut off logarithmic singularities in n(γ) developing because\nofthePLZ(γ)factor inEq.(18). Curves n∞(γi)infrontpanels\nshow the number of sweeps before the self-quenching sets in.\nSee text for details.\nAssuming |η−ηi| ≤ǫmax, theSandT+states are\ndegenerate at t∗=−TLZ(η−ηi)/ǫmax. To avoid trivial\nquenching due to the shift in ηcaused by the accumu-\nlating polarization far away from the degeneracy point,\nthe electronic energies were renormalized after every 100\nsweeps keeping η−ηi≈0 and ensuring the S-T+anti-\ncrossing be passed during all LZ sweeps, −TLZ< t∗<\nTLZ. As a result, the center of the sweep was perma-\nnently kept close to the anticrossingpoint. Such aregime\ncan be achieved experimentally by applying appropriate\nfeedback loops.\nIn order to relate the properties of the sweeps to\nthe conventional notations of the LZ transition proba-\nbilities in the limit TLZ→ ∞, it is helpful to intro-\nduce the dimensionless initial τi=−TLZ[1+t∗/TLZ]β//planckover2pi1\nand finalτf=TLZ[1−t∗/TLZ]β//planckover2pi1times, where β=\n(ǫmax/planckover2pi1/TLZ)1/2. The Landau-Zener parameter is γ=\n(v/β)2. When −τi≫√γandτf≫√γ, the transition\nprobability converges towards the Landau-Zener result\nPLZ= 1−exp(−2πγ).\nWe consider a simple model for the electron wavefunc-\ntions. The orbital part of the singlet wave function is\nψS(1,2) = cosνψR(1)ψR(2)\n+sinν[ψL(1)ψR(2)+ψL(2)ψR(1)]/√\n2 (21)and the triplet part is\nψT(1,2) = [ψL(1)ψR(2)−ψ(2)ψR(1)]/√\n2,(22)\nwhereψL(ψR) denotes the wave function in the left\n(right) dot. The angle νdepends on the electron Zeeman\nenergy. We assume the electrons are in the lowest orbital\nharmonic oscillator state so that the wave functions are\nψ(x,y,z) =exp/bracketleftbig\n−(x2+y2)/l2−z2/w2/bracketrightbig\n/radicalbig\nwl2(π/2)3/2,(23)\nwherelis the lateral size of each dot and wis its height.\nFor two dots that are separated by a distance dwe form\nan orthonormal basis set based on the functions ψ(x−\nd/2,y,z) andψ(x+d/2,y,z), that defines the above ψL\nandψR. While both dots are chosen of the same size,\nhyperfine couplings in them differ due to the dependence\nofρjλof Eq. 8 on the mixing angle ν.\nWe solve the nuclear dynamics numerically by using\nMathematica 9. First, we include all nuclear spins that\nare in the vicinity of the double quantum dot and sat-\nisfy the condition that the electron-nuclear coupling con-\nstantsζjλ≥κMax{ζjλ}, whereκis a small parameter.\nWe checked, by changing κ, that our results converged\nand have found that reducing κbelowκ= 0.01 does not\nproduce any visible changes in the plots we present. Ini-\ntial configurations of the nuclear spin directions are cho-\nsen by a pseudo-random number generator. The initial\nnuclear spin configuration determines the 2 ×2 electron\nS-T+Hamiltonian. We solve the time-dependent 2 ×2\ndifferentialequationnumericallyforlinearLZsweepsand\ncompute the probability P, the shake-up parameter Q,\nand the time-integrated effect of the Overhauser shift of\nthe triplet state T+described by the parameter R. We\nthen let the nuclear spins precess in the external mag-\nnetic field and a random noise field before the next LZ\nsweep takes place. We record all electron singlet-triplet\ncoupling parameters as a function of the sweep number,\nas well asP,Q, and the change in the total magnetiza-\ntion.\nWe choose realistic parameters for a double quantum\ndot of a height w= 3˚A, sizel= 50˚A, and distance\nd= 100˚A. We consider an external magnetic field of\nB=10 mT. Using a cut-off κ= 0.01 implies that we\nexplicitly include in our calculations around ten millions\nspins. A single such calculation takes about one week\non our state-of-the-art workstation. We have studied the\nevolution of the nuclear spin dynamics durin up to 2 ×\n105LZ sweeps for 107spins and used various pseudo-\nrandom initial configurations of nuclear spins. While the\ndetailed pattern of the dynamics depends on the initial\nconditions, all basic regularities were exactly the same\nin all simulations. Hence, our results are representative\nfor the generic behavior of a pumped electron-nuclear\nsystem.7\nVI. DYNAMICAL NUCLEAR POLARIZATION\nWe are now ready to discuss numerical results for dy-\nnamical polarization of nuclear spins. In all our simu-\nlations we consider double dots of the size w= 3 nm,\nl= 50 nm, and d= 100 nm.\nGaAs (InAs) has 8 nuclear spins per cubic unit cell so\nthat the effective volume per site is Vs=a3/8, where\nthe lattice constant is a= 5.65˚A (a= 6.06˚A). When\nall nuclear spins are fully polarized in GaAs (InAs), the\nOverhauser field seen by the electrons is 5 .3 T (0.86\nT). We accept the following values of electron g-factors,\ngGaAs=−0.44 (gInAs=−8). The other parameters re-\nflecting the abundance, nuclear g-factors, and hyperfine\ncoupling constants are listed in Table I for GaAs and Ta-\nble II for InAs. From these values, it can be understood\nthat in our simulations GaAs behaves as a three-specie\nsystem, whereas InAs behaves as a two-specie system.\nAlthough there are three distinct species in InAs, two of\nthem behave in the same way with respect to the preces-\nsionrateinanexternalmagneticfieldandthecouplingto\nelectrons so that InAs is an effective two-specie system.\n69Ga71Ga75As\np30%20%50%\ng1.31.70.96\nA(µeV)779994\nI3/23/23/2\nTable I: Nuclear abundances p, nuclear g-factors, hyperfine\ncoupling constants A, and nuclear spin in GaAs.46,47\n113In115In75As\np2%48%50%\ng1.21.20.96\nA(µeV)14014076\nI9/29/23/2\nTable II: Nuclear abundances p, nuclear g-factors, hyperfine\ncoupling constants A, and nuclear spin in InAs.48,49\nWe will consider systems with different number of nu-\nclear species to deduce coupled electron-nuclear dynam-\nicsphenomenathat arerobustwith respecttoorstrongly\ninfluenced by the number of species. To this end, we\nchoose InAs and GaAs as model systems. These systems\nhavedifferentmagnitudesoftheSOsplitting; itismodest\nin GaAs but strong in InAs, see Appendix A for details.\nIn Sec. VIA, where calculations for nuclear parameters\nof InAs of Table II are carried out, we use modest val-\nues of SO coupling v±\nSOto illustrate how the dynamics\nbecomes increasinglycomplex and irregularwith increas-\ning strength of the spin-orbit interaction. Nevertheless,\nthis allows making conclusions about the expected nu-\nclear dynamics in InAs for realistic values of v±\nSO, see the\nend of Sec. VIA. The SO coupling constant v±\nSOis a com-\nplex number. Without loss of generality, we will assumein the remainder of the paper that it is real and positive,\nas well as use a simplified notation, v±\nSO=vSO.\nA. Two-specie systems: InAs\nLet us first consider InAs which effectively consists of\ntwo species because the parameters of113In and115In\npractically coincide. Therefore, species113In and115In\nbehave as a single specie and75As as a second specie.\nWe first demonstratethat, in absenceofSO coupling, the\ndynamical evolution of nuclear spins in InAs is similar to\nthe dynamics in GaAs reported earlier.30In all our InAs\nsimulations, we startin the sameinitial (pseudo-random)\nconfiguration of the nuclear spins. We have checked that\nsimilarresultsareobtainedwhenwestartinseveralother\nconfigurations. In all our simulations in this section, the\nwaiting time between LZ sweeps equals the precession\ntime of specie75As in the external magnetic field, Tw=\nt75As.\nWe start by presenting results for a system without\nspin-orbitcoupling, vSO= 0, toprovethatself-quenching\noccurs and investigate its stability with respect to nu-\nclear noise. Fig. 2(a) shows the evolution of the magni-\ntude of the singlet-triplet coupling |v±\nn|with increasing\nnumber of sweeps n. For a sweep duration of TLZ= 40\nns, the initial LZ probability for the first few sweeps is\nP∼0.5, see Fig. 3, and the singlet-triplet coupling is\nself-quenched already after about 20000 sweeps. The\nnumber of sweeps nrequired to reach self-quenching is\nabout the same as for GaAs.30The appearance of sev-\neral peaks of v±\nnin the rangeof 5000-20000sweeps before\nthe self-quenching sets in is typical of multi-specie sys-\ntems. In contrast to single-specie systems, and especially\nthe model of Sec. IV, in multi-specie systems final self-\nquenching is usually preceeded by partial self-quenchings\nfollowed by revivals. We attribute this behavior to com-\npetition between subsystems with the different Zeeman\nprecession times. As seen in Fig. 3, in each peak of |v±\nn|\ntheS-T+transitionprobability Pincreasesstrongly,near\nitIzshows a step-like behavior (not shown), and accom-\npanying peaks of Qindicate massive shakeups which flop\nmany nuclear spins per LZ sweep. The model of Sec. IV\nthat only deals with the total magnetization Izdoes not\ndescribe such events and provides a smoothened picture\nof the nuclear spin evolution.\nTransverse noise transforms the dynamical evolution\ninto a dissipative one. Fig. 2 shows the effect of the in-\ncrease of the level of noise from (a) through (b) to (c).\nIn Fig. 2(a), the noise correlation time is of the order of\nthe self-quenching time τ/t75As= 10000. In this case,\ntransverse noise only modestly perturbs the nuclear spin\nevolutionascomparedtothenon-dissipativeregime(sim-\nulated and analyzed, but not shown). Note the presence\nof a long slightly visible tail with irregular oscillations\nalong it. In contrast, Fig. 2(b) and (c) demonstrate that\nwhen the transverse noise correlation time τis shorter\nthan the typical self-quenching set-in time in un-noisy8\n50000 100000n100200300/VertBar1vn/PlusMinus/VertBar1/LParen1neV/RParen1/LParen1a/RParen1\n50000 100000n100200300/VertBar1vn/PlusMinus/VertBar1/LParen1neV/RParen1/LParen1b/RParen1\n50000 100000n100200300/VertBar1vn/PlusMinus/VertBar1/LParen1neV/RParen1/LParen1c/RParen1\nFigure 2: Hyperfine-induced singlet-triplet coupling |v±\nn|for\na double quantum dot with the nuclear parameters of InAs\nin absence of SO coupling with increasing level of noise as\na function of sweep number n. (a)τ/t75As= 10000, (b)\nτ/t75As= 1000, and (c) τ/t75As= 100. The other param-\neters are Tw=t75As,TLZ= 40 ns.\nsystems, self-quenchingissuppressedandeventuallydoes\nnot happen at all; in particular, Fig. 2(b) demonstrates\na possibility of revivals. We conclude that for high noise\nlevels the chaotic evolution of the nuclear spins persists,\nbut themagnitudesofthe peaksof |v±\nn|seemtogradually\ndecrease in time.\nFig. 2(a) suggests a glassy behavior of the nuclear\nsystem with an extensive manifold of dark states sepa-\nrated by low barriers. In the absence of noise, repeated\nLZ sweeps cause the system to end in one of the dark\nstates (usually after passing through several peaks of\n|v±\nn|). Weak noise produces slow diffusion between adja-\ncentdarkstatesacrosslowsaddle points. Duringthis dif-\nfusion, the magnetization Izchanges only slightly. With\nincreasing noise, the system experiences revivals as seen\ninFig.2(b)asasharppeakin |v±\nn|. Duringsuchevents P50000 100000n0.51.P\n50000 100000n48Q\nFigure 3: (a) Landau-Zener transition probability Pas func-\ntion of the sweep number nfor a double quantum dot with\nthe nuclear parameters of InAs in absence of SO coupling. (b)\nShake-up parameter Qas a function of sweep number n. The\nparameters are as in Fig. 2(a)\nincreasesstrongly, Izshowsstep-likebehavior, and peaks\ninQ(not shown) indicate massive shakeups, similarly to\nthe patterns discussed as applied to Fig. 2(a) above.\nWe demonstrated earlier that SO coupling is screened\nstroboscopicallyin asingle-speciesystem.30Next, wewill\ndemonstrate that SO coupling can be screened strobo-\nscopically also in multi-specie systems, and investigate\nthis phenomenon in more detail. Fig. 4 shows simula-\ntions of the singlet-triplet coupling v±\nnforvSO= 62 neV\nand three LZ sweep durations TLZ. In comparison, the\nstraight black line indicates the value of the spin-orbit\ncouplingvSO= 62 neV (which is independent of the\nsweep number n). We see that in all these simulations,\nthe spin-orbit coupling eventually becomes screened so\nthat all the colored lines approach the black line which\nimpliesthat |v±\nn|=|vSO|. ForlongerLZsweepdurations,\noscillations of v±\nnare more rapid, but screening eventu-\nally occurs faster because nuclear spins are more strongly\naffected during each sweep.\nScreening of the SO coupling even in multi-specie sys-\ntems sounds counter-intuitive at first glance. Indeed, the\nspin-orbitcoupling vSOisstaticwhilethetransversecom-\nponents of the nuclearspins contributing to v±\nnprecessin\ntime. In InAs, twonuclearspecies113Inand115In precess\nat the same frequency and behave effectively as a single\nspin specie whereas the third spin specie,75As, precesses\nat a different frequency. So, while screening indicates\nthat the magnitude of the singlet-triplet coupling v±\nnre-\nmains finite, it must inevitably precess in time. There-9\n50000 100000n100200300/VertBar1vn/PlusMinus/VertBar1/LParen1neV/RParen1\nFigure 4: Transverse nuclear polarization for a double quan -\ntum dot with the nuclear parameters of InAs as a function\nof sweep number nfor the spin-orbit coupling vSO= 62\nneV (black line) and the Landau-Zener sweep durations (red\ncurve)TLZ= 80 ns, (green curve) TLZ= 40 ns, and (blue\ncurve)TLZ= 20 ns. Resonant pumping with Tw=t75As, the\npolarization is plotted at multiples of Tw. See text for details.\nfore, it cannot compensate the spin-orbit coupling vSO\nat all instants of time. The screening we observe is only\npossible because the waiting time Twis exactly equal to\nthe precession time of the75As specie,Tw=t75As. The\ndata used in our plots of |v±\nn|were taken at exact mul-\ntiples of the the waiting time, which was equal to the\nprecession time of the75As specie. Therefore, the self-\nquenching that manifests itself in Fig. 4 is a stroboscopic\nself-quenching.\nStroboscopic self-quenching can be understood in the\nfollowing way. The dynamical evolution of nuclear spins\ncauses self-quenching of the sum of the contributions\nfromthetransversecomponentsofspecies113Inand115In\n(that are out of resonance with the pumping period Tw,\nhence, their contribution to v±\nnvanishes). In contrast,\nthe contribution from the specie75As tov±\nnexactly com-\npensatesthespin-orbitcoupling v±\nSOateverytimeinstant\nwhen a LZ sweep happens. In other words, the matrix\nelementsv±\nn(t) changein time harmonicallywith the am-\nplitudev±\nSOand a period t75As:\nv±\nn(t) =v±\nSOcos(2πt/t75As). (24)\nThis generalizes our previous findings of the screening\nof SO coupling in single-specie systems.30For a single-\nspecie imitation of GaAs, we found that the SO coupling\nwas screened in such a way that that the matrix element\nchanged harmonically with the amplitude vSOand a pe-\nriodtGaAs, wheretGaAsis the average precession time of\nthe three nuclear spin species in GaAs.30\nLet us now demonstrate explicitly that when self-\nquenching sets in, the sum of the contributions from the\ntransversecomponents of113In and115In to|v±\nn|vanishes\nwhile the contribution from75As equalsvSO. We show\nin Fig. 5(a) the contribution from113In and114In to|v±\nn|\nas a function of the number of sweeps n. Clearly, it van-\nishes for large n. On the other hand,75As whose nuclear\nprecession time equals the waiting time Tw, makes a con-\ntribution to |v±\nn|that exactly compensates |v±\nSO|at allintegers ofTw, see Fig. 5(b). Hence, Eq. (24) is satisfied.\n50000 100000n100200300/VertBar1vn/PlusMinus/VertBar1/LParen1eV/RParen1/LParen1a/RParen1Sumof specie1 and2\n50000 100000n100200300/VertBar1vn/PlusMinus/VertBar1/LParen1eV/RParen1/LParen1b/RParen1Specie3\nFigure 5: (a) Sum of contributions from113In and115In\nto hyperfine-induced singlet-triplet coupling v±\nnfor spin-orbit\ncoupling vSO= 62 neV (black line) as a function of sweep\nnumber n. (b) Contribution from75As to hyperfine-induced\nsinglet-triplet coupling v±\nnfor spin-orbit coupling vSO= 62\nneV (black line) as a function of sweep number n. The LZ\nsweep duration is TLZ= 80 ns.\nWe note that the contributions from113In and115In\ntov±\nnvanish not only stroboscopically but identically,\nat each instant of time (not shown). We have also\nchecked that in systems without spin-orbit coupling, self-\nquenching sets in for all species and for an arbitraryratio\nbetweenTwand the precession times of the species (not\nshown). For three-specie systems the last statement is\nproven below, see Sec. VIB.\nNow we will illustrate that stroboscopic screening of\nSO coupling can be practically achieved only for small\nand moderate magnitudes of vSO. Since stroboscopic\nscreening implies that the contribution from75As tov±\nn\ncompensates vSOwhile the combined contribution from\nspecies113In and115In vanishes, we show in Fig. 6 the\nevolution of the contribution of75As tov±\nnas a function\nof sweep number for three values of spin-orbit coupling\nvSO= 31,62 and 91 neV.50While all the results in Fig. 6\nwere found for the same value of TLZand the same initial\nconditions, screening sets in at n≈75000 forvSO= 31\nneV, is delayed to n≈125000 for vSO= 62neV, and is\nfar from complete even at n= 200000 for vSO= 93 neV.\nThese data suggest that stroboscopic self-quenching sets\nin whenvSO/lessorsimilarv0\nn, wherev0\nn≈A/√\nNis a typical fluctu-\nation of the Overhauser field, and cannot be practically\nachieved for vSO/greaterorsimilarv0\nn; see estimates of the magnitude10\nof the spin-orbit coupling in Appendix A. This criterion\nresembles the criterion of the phase transition of Ref. 26.\nOne should keep in mind that with the interval be-\ntween LZ pulses of about 1 µs, a set of n∼106pulses\ntakes about 1 s which is a typical scale of nuclear spin\ndiffusion51, which is not taken into account in the above\nconsiderations. Weexpect, buthavenotchecked, thatin-\nhomogeneity of magnetic field should have a detrimental\neffect on stroboscopic self-quenching. Therefore, we con-\nclude that stroboscopic quenching of SO coupling is less\ngeneric and more fragile than self-quenching in systems\nwithout spin-orbit coupling.\n50000 100000 150000 200000n20406080100/VertBar1vn/PlusMinus/VertBar1/LParen1eV/RParen1\nFigure 6: Contribution of the specie75As to the singlet-\ntriplet coupling v±\nnas a function of sweep number nfor differ-\nent values of vSO: 31 neV (red curve), 62 neV (black curve),\nand 91 neV (blue curve). Duration of LZ pulses TLZ= 40\nns. Waiting time between consecutive LZ pulses equals the\nprecession time of75As,Tw=t75As. Nuclear parameters of\nInAs.\nWe estimatein Appendix Athat SOcouplingisweaker\nor comparable to (stronger than) the typical nuclear\npolarization induced singlet-triplet coupling in GaAs\n(InAs). As a consequence, we expect the SO coupling\nmight be stroboscopically screened in GaAs systems, but\nthat stroboscopic screening is improbable in InAs sys-\ntems.\nB. Three-specie systems: GaAs\nIn this section, we present new results for GaAs\nthat complete the picture of the generic nature of self-\nquenching in multi-specie systems. Furthermore, we\nshow that screening of the SO coupling requires that the\nwaiting time Twis in resonance with the precession time\nof one of the nuclear species. When the resonance con-\ndition is not satisfied, screening of the SO coupling is\npartial and irregular.\nIn Sec. VIA and in Ref. 30, self-quenching in multi-\nspecie systems in absence of SO coupling was demon-\nstarted only under the conditions when the waiting time\nTwwas in resonance with the precession time t75Asof the\n75As specie,Tw=t75As. We demonstrate here that whileself-quenching is generic and independent of the waiting\ntime, the evolution towards the self-quenched states de-\npends on the waiting time.\nTo this end, we plot in Fig. 7 the evolution of the\nsinglet-triplet coupling v±\nnfor two different values of the\nwaiting time Tw. Fig. 7(a) displays results of simulations\nfor the resonant case when Tw=t75As, in which pro-\nnounced oscillations are distinctly seen. For n/greaterorsimilar3000,\nthe plot consists of five branches that reflect coupled dy-\nnamics of three species. In contrast, in the absence of\nthe resonance, Fig. 7(b), the evolution is chaotic. Nev-\nertheless, self-quenching sets-in in both cases and, what\nis most remarkable, at the same time scale of n≈104.\nRemarkably, the processes of Figs. 7(a) and 7(b) ended\nin states with the same Iz(not shown). While the set\nof dark states is vast (as follows from our discussion in\nSection IV), this observation indicates that the number\nof strong attraction centers in which self-quenching ends\nis more scant.\nWe conclude that self-quenching in systems without\nspin-orbit coupling is generic and robust, at least in the\nframework of S-T+scheme.\nWe checked that not only does the total matrix ele-\nmentv±\nnvanish, but also the matrix elements for all of\nthree species contributing to it. Because between the LZ\nsweeps the electron subsystem is in its singlet state, the\nKnight shift vanishes, and according to Eq. (3) all nuclei\nbelonging to some specie precess with the same speed.\nTherefore, the self-organization of the nuclear subsystem\nthat annihilates its coupling to the electron spin persists\nduring the free precession periods.\nFinally, we demonstate that while self-quenching is a\ngenericfeatureintheabsenceofSOcouplingregardlessof\ntheratiobetweenthewaitingtimebetweentheLZsweeps\nTwand the nuclear precession times tλ, in presence of SO\ncoupling the stroboscopic self-quenching is not generic\nandhighlysensitivetothisratio. Onlymodestdeviations\nfromtheresonancedestroysthescreeningofSOcoupling.\nIn Fig. 8, we plot |v±\nn|under the conditions when the\nwaiting time is in exact resonance with the precession\ntime of75As (red curve), and when there is a 1% devi-\nation from the resonance (black curve). While the SO\ncoupling is clearly screened in resonance, only a tiny de-\nviation from resonance destroys screening.\nMore insights into the sensitivity of the screening of\nSO coupling to the deviation from the resonance can be\ngained from Fig. 9 that displays the contributions to v±\nn\nfrom eachofthe species. Using the same intial conditions\nas in Fig. 8, we plot the evolution of the matrix elements\n|v±\nn|for both the resonant and slightly off-resonance\nregimes. Initially they follow each other closely. How-\never, after a couple of thousand sweeps, the deviations\nbecome significant. Ultimately, the contributions from\n69Ga and71Ga do not vanish in the non-resonant case,\nand the contributions from75As does not screen the SO\ncoupling.\nThe critical sensitivity of stroboscopic self-quenching\nto small deviations from resonance looks indicative of11\n2000 4000 6000 8000 10000n10203040/VertBar1vn/PlusMinus/VertBar1/LParen1neV/RParen1\n2000 4000 6000 8000 10000n10203040/VertBar1vn/PlusMinus/VertBar1/LParen1neV/RParen1\nFigure 7: Transverse nuclear polarization as a function of\nsweep number nfor a GaAs double quantum dot in absence\nof spin-orbit coupling and transverse noise. Duration of LZ\npulsesTLZ= 80 ns. (a) The waiting time is in resonance\nwith the75As precession time, Tw=t75As= 13.7µs. (b)\nThe waiting time is incommensurate with the75As precession\ntime,Tw= 1.39t75As= 19.1µs.\n10000 20000 30000n255075/VertBar1vn/PlusMinus/VertBar1/LParen1neV/RParen169Ga,71Ga, and75As\nFigure 8: Transverse nuclear polarization as a function of\nsweep number nfor a GaAs double quantum dot with vSO=8\nneV. The duration of LZ pulses TLZ= 80 ns. The red curve\nshows results for the waiting time in resonance with the pre-\ncession time of75As,Tw=t75Ar, and the black curve for a\n1% deviation from the resonance.\na chaotic behavior of the system.52This is not surpris-\ningbecausethe system ofintegro-differentialequationsof\nEq. (1) is highly nonlinear because the coefficients ∆jλ\ndepend through Eqs. (10) on the electronic amplitudes\ncS,cT+that, in turn, depend on all nuclear angular mo-\nmentaIjλ. In this context, we speculate that a strong\nrevival of all black curves in Fig. 9 near n≈15000 where\nall red curves saturate, and the return of black curves\nclose to their initial values near n≈28000, is reminis-cent of the strange attractor pattern.52These signatures\nofchaoticnucleardynamicinSOcoupledsystemsrequire\na more detailed study.\nWeconcludethatstroboscopicscreeningoftheSOcou-\npling is not a robust phenomenon.\nWhile the above simulations are focused on the large\nnregion, we mention that commensurability oscillations\nin the polarization accumulation per sweep were ob-\nserved experimentally38and described theoretically37in\nthe smallnregion,n/lessorsimilar104.\n10000 20000 30000n1020/VertBar1vn/PlusMinus/VertBar1/LParen1neV/RParen1/LParen1a/RParen169Ga\n10000 20000 30000n1020/VertBar1vn/PlusMinus/VertBar1/LParen1neV/RParen1/LParen1b/RParen171Ga\n10000 20000 30000n1020/VertBar1vn/PlusMinus/VertBar1/LParen1neV/RParen1/LParen1c/RParen175As\nFigure 9: Transverse nuclear polarization as a function of\nsweep number nfor a GaAs double quantum dot with vSO=8\nneV. The duration of LZ pulses TLZ= 80 ns. Red curves\nshow results for the waiting time Twin resonance with the\nprecession time of75As,Tw=t75As, while black curves the\ndata for 1% off-resonance regime.\nIn addition to the regular investigation of the trans-\nverse magnetization, we also followed the time de-\npendence of the longitudinal magnetization vz\nn=\nVs/summationtext\nλAλ/summationtext\nj∈λρjλIz\njλ. In presence of SO coupling, it\nshows an oscillating sign-alternating behavior, and we\nwere unable to detect any signatures of its accumulation.12\nSummarizing the results of Secs. VIA and VIB, we\nconclude that SO coupling eliminates self-quenching and\ncauses the nuclei of a pumped system to exhibit a per-\nsistent irregular dynamics. We speculate that this phe-\nnomenon is closelyrelated to the feedback looptechnique\nfor building controllable nuclear gradients which is inher-\nently based on employingsuch a dynamics.34Indeed, any\nlong-term control of the nuclear ensemble by alternating\nS→T+andT+→Ssweeps is impossible after the self-\nquenching set-in time that is of a millisecond scale in\nabsence of SO coupling. Our data, especially Fig. 8, sug-\ngest that near the resonance between the waiting time\nof LZ pulses and the Larmor frequency of one of the\nnuclear species the quasi-periods of nuclear fluctuations\nbecome longer and are controlled by the deviations from\nthe exact resonance. We also expect that under these\nconditions the nuclear gradients should be dominated by\nthe resonant specie.\nVII. CONCLUSIONS\nAn analytical solution of a simplified model, and ex-\ntensive numerical simulations for a realistic geometry,\nprovethat self-quenchingis a genericpropertyofthe cen-\ntral spin-1/2 problem in absence of spin-orbit coupling.\nAs applied to a double quantum dot of a GaAs type,\nwhere electron and nuclear spins are coupled viahyper-\nfine interaction, pumping nuclear magnetization across a\nS-T+avoided crossing through successive Landau-Zener\nsweeps ceases after about 104sweeps. This is a result\nof the screening of the initial fluctuation of the nuclear\nmagnetization by the injected magnetization and van-\nishing of the S-T+anticrossing width, and this sort of\nself-quenchingisrobust. Under the influence ofmoderate\nnoise, thesystemwandersthroughasetofdarkstatesbe-\nlongingtoawastlandscapeofthesystemincludingabout\n106nuclear spins coupled through inhomogeneous elec-\ntron spin density. With time intervals depending on the\nlevel of the noise, the system experiences revivals when\nadditional magnetization is injected, and afterwards it\nwanders through a new set of dark states.\nDue to the violation of the angular momentum conser-\nvation, spin-orbit coupling changes the situation drasti-\ncally. Self-quenching sets in only stroboscopically under\nthe condition that the waiting time between consecutive\nLandau-Zener sweeps is in resonance with the Larmor\nprecession time of one of the nuclear species. Then the\nprecessing Overhauser field of the resonant specie com-\npensates the spin-orbit field vSOduring the sweep, while\ncontributions of other species vanish. This sort of self-\nquenching is fragile and sensitive even to minor deviation\nfrom the resonance. Generically, injection of nuclear po-\nlarization oscillates in time and changes sign. Therefore,\nspin-orbit coupling causes the nuclear magnetization of a\npumpedS-T+doublequantumdottoexhibit apersistent\ndynamics.\nWe suggest that the feedback loop technique for build-ing controllable nuclear field gradients34is based on the\noscillatory behavior of the nuclear spin magnetization\ncaused by spin-orbit coupling. Spin-orbit coupling is a\nnatural mechanism of overcoming self-quenching. The\ntechnique employs persistent oscillations and selects the\nsign of the pumping response to the changing magneti-\nzation gradient.\nAcknowledgments\nA.B.wouldliketothankB.I.Halperinforhishospital-\nity at Harvard University where this work was initiated.\nWe are grateful to B. I. Halperin, C. M. Marcus, L. S.\nLevitov, H. Bluhm, K. C. Nowack, M. Rudner, and L.\nM. K. Vandersypen for useful discussions. E. I. R. was\nsupported by the Office of the Director of National In-\ntelligence, Intelligence Advanced Research Projects Ac-\ntivity (IARPA), through the Army Research Office grant\nW911NF-12-1-0354and by the NSF through the Materi-\nals Work Network program DMR-0908070.\nAppendix A: Spin-orbit coupling\nThe Rashba spin-orbit Hamiltonian is\nHso=α/summationdisplay\nn=1,2[σx(n)ky(n)−σy(n)kx(n)],(A1)\nwhereαis the strength of SO interaction, and kx(n) and\nky(n) are the in-plane momenta for the electron n. The\nsinglet and triplet states are Ψ S(1,2) =ψS(1,2)χS(1,2)\nand Ψ T+(1,2) =ψT(1,2)χT+(1,2), where the orbital\ncomponents of the singlet and triplet wave functions\nhave been defined in Sec. V and the spin parts of the\nwave functions are χS(1,2) = (| ↑1∝an}bracketri}ht| ↓2∝an}bracketri}ht−| ↓ 1∝an}bracketri}ht ↑2∝an}bracketri}ht)/√\n2\nandχT+(1,2) =| ↑1∝an}bracketri}ht| ↑2∝an}bracketri}ht. In terms of the orbital\nwave functions, the SO induced S-T+coupling is then\nv+\nSO=iαcosν∝an}bracketle{tψR|kx+iky|ψL∝an}bracketri}ht. This matrix element\ncan be estimated similarly to Refs. 26 and 39. It depends\nexponentially on the overlap between the wave functions\nof the dots, ψLandψR. Therefore, the SO coupling v+\nSO\ncan be tuned and strongly decreases with the interdot\ndistanced. In InAs quantum wires the SO coupling pa-\nrameter is around α∼10−11eVm53corresponding to a\nspin precession length of lSO=/planckover2pi12/(2m∗α) of around 100\nnanometers. With typical parameters of d= 100 nm\nandl= 50 nm and cos ν= 1/√\n2, we find that vSOis\naround 4 ×10−5eV. This is about two orders of mag-\nnitude larger than the typical S-T+coupling 10−7eV\ninduced by the hyperfine interaction, but decreases with\ndexponentially. Theoretical estimates of vSOhold only\nwith exponential accuracy. Pre-exponential factors are\nmodel dependent, and a somewhat different estimate was\nproposed in Ref. 54. In GaAs quantum dots,55the SO\ncoupling constant αis two orders of magnitude smaller\nthan in InAs with lSO≈30µm, so that the SO coupling13\nmay be comparable to the hyperfine induced coupling,\nand is usually considered as weaker than it. These esti-\nmates should be treated with caution since the SO cou-\npling is not only a function of the material but is sample\nspecific.\nFor GaAs, an estimate of a typical fluctuation as v0\nn≈\nA/√\nNwithAfrom Table II and N≈106results inv0\nn≈\n100 neV. Our estimates of vSOof Ref. 30 gave vSO≈50\nneV, while the estimate of Ref. 37 is vSO≈15 neV.56Recently Shafiei et al.57managed to resolve the SO and\nhyperfine components of electric dipole spin resonance\n(EDSR)58in the same system, a GaAs double quantum\ndot. Their data suggest that both components were of\na comparable magnitude, with the hyperfine component\nmaybe somewhat stronger.\nKeeping in mind the uncertainty in the values of vSO,\nwe carry out simulations both with SO coupling and\nwithout it.\n1I.ˇZuti´ c, J. Fabian, and S. Das Sarma, Rev. Mod. Phys. 76\n323 (2004)\n2D. Loss and D. P. DiVincenzo, Phys. Rev. A 57, 120\n(1998).\n3L. P. Kouwenhoven and C. Marcus, Physics World 11(6),\n35 (1998).\n4J. Levy, Phys. Rev. Lett. 89, 147902 (2002).\n5Awschalom D. D., D. Loss, and N. Samarth (2002)\nSemiconductor Spintronics and Quantum Computation\n(Springer, Berlin)\n6D. P. DiVincenzo, D. Bacon, J. Kempe, G. Burkard and\nK. B. Whaley, Nature (London) 408, 339 (2000).\n7R. Hanson, L. P. Kouwenhoven, J. R. Petta, S.Tarucha,\nand L. M. K. Vandersypen, Rev. Mod. Phys. 70, 1217\n(2007).\n8J. R. Petta, A. C. Johnson, J. M. Taylor, E. A. Laird, A.\nYacoby, M. D. Lukin, C. M. Marcus, M. P. Hanson, and\nA. C. Gossard, Science 309, 2180 (2005).\n9R. Brunner, Y.-S. Shin, T. Obata, M. Pioro-Ladrire,\nT. Kubo, K. Yoshida, T. Taniyama, Y. Tokura, and S.\nTarucha, Phys. Rev. Lett. 107, 146801 (2011)\n10A. Pfund, I. Shorubalko, K. Ensslin, and R. Leturcq, Phys.\nRev. B76, 161308 (2007).\n11S. Nadj-Perge, S. M. Frolov, E. P. A. M. Bakkers, and L.\nP. Kouwenhoven, Nature 468, 1084 (2010).\n12K. D. Petersson, L. W. McFaul, M. D. Schroer, M. Jung,\nJ. M. Taylor, A. A. Houck, and J. R. Petta, Nature 490,\n380 (2012).\n13I. A. Merkulov, Al. L. Efros, and M. Rosen, Phys.Rev. B\n65, 205309 (2002).\n14A. Khaetskii, D. Loss, and L. Glazman, Phys. Rev.B 67,\n195329 (2003).\n15S. I. Erlingsson and Yu. V. Nazarov, Phys. Rev. B 70,\n205327 (2004).\n16W. A. Coish and D. Loss, Phys. Rev. B 72, 125337 (2005)\n17W. Yao, R.-B. Liu, and L. J. Sham, Phys. Rev. B 74,\n195301 (2006).\n18G. Chen, D. L. Bergman, and L. Balents, Phys. Rev. B 76,\n045312 (2007)\n19L. Cywi´ nski, W. M. Witzel, and S. Das Sarma, Phys. Rev.\nLett.102, 057601 (2009).\n20D. Stanek, C. Raas, and G. S. Uhrig, arXiv:1308.0242.\n21D. Klauser, W. A. Coish, and D. Loss, Phys. Rev. B 78,\n205301 (2008).\n22E. A. Chekhovich, M. N. Makhonin, A. I. Tartakovskii, A.\nYacoby, H. Bluhm, K. C. Nowack, and L. M. K. Vander-\nsypen, Nature Materials 12, 494 (2013).\n23G. Ramon and X. Hu, Phys. Rev. B 75, 161301(R) (2007).\n24M. Stopa, J. J. Krich, and A. Yacoby, Phys. Rev. B 81,041304(R) (2010).\n25M. Gullans, J. J. Krich, J. M. Taylor, H. Bluhm, B. I.\nHalperin, C. M. Marcus, M. Stopa, A. Yacoby, and M. D.\nLukin, Phys. Rev. Lett. 104, 226807 (2010).\n26M. S. Rudner and L. S. Levitov, Phys. Rev. B 82, 155418\n(2010).\n27M. S. Rudner, I. Neder, L. S. Levitov, and B. I. Halperin,\nPhys. Rev. B 82, 041311 (2010).\n28H. Ribeiro and G. Burkard, Phys. Rev. Lett. 102, 216802\n(2009).\n29A. Brataas and E. I. Rashba, Phys. Rev. B 84, 045301\n(2011).\n30A. Brataas and E. I. Rashba, Phys. Rev. Lett. 109, 236803\n(2012).\n31M. Gullans, J. J. Krich, J. M. Taylor, B. I. Halperin, M.\nD. Lukin, Phys. Rev. B 88, 035309 (2013).\n32R. Winkler, Spin-orbit Coupling Effects in Two-\nDimensional Electron and Hole Systems (Springer Berlin\nHeidelberg, 2010).\n33D. J. Reilly, J. M. Taylor, J. R. Petta, C. M. Marcus, M. P.\nHanson, and A. C. Gossard, Phys. Rev. Lett. 104, 236802\n(2010).\n34H. Bluhm, S. Foletti, D. Mahalu, V. Umansky, and A.\nYacoby, Phys. Rev. Lett. 105, 216803 (2010).\n35L. D. Landau and E. M. Lifshitz, Quantum Mechanics\n(Pergamon, New York) 1977.\n36For brevity, we use this traditional notation for the\nLandau-Majorana-St¨ uckelberg-Zener problem.59,60\n37I. Neder, M. S. Rudner, and B. I. Halperin,\narXiv:1309.3027.\n38S. Foletti, H. Bluhn, D. Mahalu, V. Umansky, ans A. Ya-\ncoby, Nature Physics 5, 903 (2009).\n39D. Stepanenko, M. Rudner, B. I. Halperin, and D. Loss,\nPhys. Rev. B 85, 075416 (2012).\n40For thedetails ofthechoice of initial conditions see Ref. 2 9.\n41S. M. Ryabchenko and Y. G. Semenov, Zh. Eksp. Teor.\nFiz.84, 1419 (1983) [Sov. Phys. JETP 57, 825 (1983).\n42A. C. Durst, R. N. Bhatt, and P. A. Wolff, Phys. Rev. B\n65, 235205 (2002).\n43This equationcanbealso derivedbychangingthevariables\nin Eq. 15 so that γdepends on Izinstead of nby using\ndγ/dIz= (dγ/dn)/(dIz/dn) which results in dγ/dIz=\n−A2\n0Iz/β2. Because this equation is independent of the\nsweep number nand the probability P, it can be readily\nintegrated.\n44J. M. Taylor, A. Imamoglu, and M. D. Lukin, Phys. Rev.\nLett91, 246802 (2003).\n45Because our procedures are numerical, they can be applied\nto an arbitrary tdependencies of ǫS−ǫT+. We choose a14\nlinear dependence only to stay closer to the terms of the\nstandard Landau-Zener theory.\n46J. Schliemann, A. Khaetskii, and D. Loss, J. Phys. Con-\ndens. Matter 15, R1809-R1833 (2003).\n47J. M. Taylor, J. R. Petta, A. C. Johnson, A. Yacoby, C.\nM. Marcus, and M. D. Lukin, Phys. Rev. B 76, 035315\n(2007).\n48M. Gueron, Phys. Rev. 135, A200 (1964).\n49M. Syperek, D. R. Yakovlev, I. A. Yugova, J. Misiewicz,\nI. V. Sedova, S. V. Sorokin, A. A. Toropov, S. V. Ivanov,\nand M. Bayer, Phys. Rev. B 84, 085304) (2011).\n50It is seen from Figs. 4 - 6 that SO interaction makes the\ncoupled electron-nuclei dynamics highly volatile. Oscill a-\ntions of|v±\nn|are accompanied with strong shakeups. E.g.,\nthe shakeup factor Qis typically Q∼15 in the peaks of\n|v±\nn|for thevSO= 62 neV curve of Fig. 6.\n51E. A. Chechnovich, M. N. Makhonin, A. I. Tartakovskii,\nA. Yacoby, H. Bluhm, K. C. Nowack, and L. M. K. Van-\ndersypen, Nature Physics 12, 494 (2013).\n52E. N. Lorenz, The Essence of Chaos , (U. WashingtonPress, Seattle) 1993.\n53S. E. Hern´ andez, M. Akabori, K. Sladek, Ch. Volk, S.\nAlagha, H. Hardtdegen, M. G. Pala, N. Demarina, D.\nGr¨ utzmacher, Th. Sch¨ apers, Phys. Rev. B 82, 235303\n(2010).\n54C. Fasth, A. Fuhrer, L. Samuelson, V. N. Golovach, and\nD. Loss, Phys. Rev. Lett. 98, 266801 (2007).\n55K. C. Nowack, F. H. L. Koppens, Yu. V. Nazarov, and L.\nM. K. Vandersypen, Science 318, 1430 (2007).\n56Caption to their Fig. 6.\n57M. Shafiei, K. C. Nowack, C. Reichl, W. Wegscheider,\nand L. M. K. Vandersypen, Phys. Rev. Lett. 110, 107601\n(2013).\n58E. I. RashbaandV. I.Sheka, in: Landau Level Spectroscpy ,\nedited by E. I. Landwehr and G. Rashba (North-Holland,\nAmsterdam, 1991), Vol. 1, p. 131.\n59E. Majorana, Nuovo Cimento 9, 43 (1932).\n60E. C. G. St¨ uckelberg, Helv. Phys. Acta 5, 369 (1932)."    },    {        "title": "1007.5037v1.Current_induced_torques_in_the_presence_of_spin_orbit_coupling.pdf",        "content": "arXiv:1007.5037v1  [cond-mat.mes-hall]  28 Jul 2010Current-induced torques in the presence of spin-orbit coup ling\nPaul M. Haney and M. D. Stiles\nCenter for Nanoscale Science and Technology, National Inst itute of\nStandards and Technology, Gaithersburg, Maryland 20899-6 202, USA\nIn systems with strong spin-orbit coupling, the relationsh ip between spin-transfer torque and\nthe divergence of the spin current is generalized to a relati on between spin transfer torques, total\nangular momentum current, and mechanical torques. In ferro magnetic semiconductors, where the\nspin-orbit coupling is large, these considerations modify the behavior of the spin transfer torques.\nOne example is a persistent spin transfer torque in a spin val ve: the spin transfer torque does not\ndecay away from the interface, but approaches a constant val ue. A second example is a mechanical\ntorque at single ferromagnetic-nonmagnetic interface.\nIntroduction — Since the prediction [1–3] of spin trans-\nfer torques in non-collinear ferromagnetic metal circuits,\nthey have been the subject of extensive research [4, 5].\nThe possibility of using spin transfer torque to im-\nprove the commercial viability of magnetic random ac-\ncess memory (MRAM) [6], and the rich non-equilibrium\nphysics involved establish the topic as one of practical\nand fundamental interest. These torques arise from the\nexchange interaction between non-equilibrium, current-\ncarrying electrons and the spin-polarized electrons that\nmake up the magnetization. In systems where the spin-\norbit coupling is weak, the torque on the magnetization\ncan be computed from the change in the spins flowing\nthrough the region containing the magnetization. This\nrelation is a consequence of conservation of total spin.\nHere, we consider systems in which the spin-orbit cou-\npling cannot be neglected (and hence total spin is no\nlonger conserved).\nIn systems where spin angular momentum is not con-\nserved, the relationship between the spin transfer torque\nand the flow of spins needs to be generalized. Conserva-\ntion oftotalangular momentum implies that mechanical\ntorques on the lattice of the material accompany changes\nin the magnetization [7, 8]. This effect has been used for\ndecades to measure the g-factor of metals. More recent\ntheoretical [9, 10] and experimental [11] work considers\nthe current-inducedmechanicaltorquespresentatthe in-\nterfaceofaferromagnetandnon-magnet, similarin spirit\nto the spin transfer torques on the magnetization present\nin spin valves.\nIn this article we develop a theory for current-induced\ntorques (both spin transfer torques and mechanical\ntorques) in systems with strong spin-orbit coupling, and\napply it to a model of dilute magnetic semiconductors.\nWe find that by accounting for the orbital angular mo-\nmentum of the electrons, we can relate the change in\ntotal angular momentum flow to spin transfer torques\nand mechanical torques. We study two system geome-\ntries where these torques play important roles. The first\nis a spin-valve geometry, which is used to study the fea-\ntures ofspin transfertorques in the presence ofspin-orbit\ncoupling. The second is a single interface between a fer-\nromagnet and non-magnet, which elucidates the physics\nunderlying current-induced mechanical torques.Formalism — We consider a Hamiltonian consisting of\na spin-independent kinetic and potential energy H0=\n−¯h2∇2\n2m+V(r), an exchange splitting ∆, and an atomic-\nlike spin-orbit interaction parameterized by α:\nH=H0+∆\n¯h(M·ˆs)\nMs+α\n¯h2/parenleftBig\nˆL·ˆs/parenrightBig\n,(1)\nwhereˆLandˆsare the electron angular momentum and\nspin operators, respectively [12]. The exchange splitting\narises from a magnetization M, with magnitude Ms. We\ntreat the magnetization within mean field theory.\nWe consider the torque on the magnetization due to\nelectric current flow. The spin transfer torque τSTT\nat position rfrom electronic states with spin density\ns(r) is proportional to the component of spin trans-\nverse to the magnetization [14]: τSTT(r) =dM(r)\ndt=\n−∆\n¯h2(M(r)×s(r)). In the absence of spin-orbit coupling,\nthis torque can be related to the divergence of a spin\ncurrent, which offers conceptual and computational sim-\nplicity [15]. In the following we analyze how spin-orbit\ncoupling changes this simple result. One consequence is\nan expression for the mechanical torque τlat.\nWe develop an expression for τSTTby evaluating\nthe time-dependence of the electron spin and angular\nmomentum densities. To do so, we adopt a Heisen-\nberg picture of time evolution, and evaluatedˆO(r)\ndt=\ni\n¯h/bracketleftBig\nH,ˆψ†(r)ˆOˆψ(r)/bracketrightBig\n,whereˆψ(r) is the position operator,\nfor the operators ˆO=ˆ s,ˆL. This procedure leads to [4]:\ndˆ s\ndt=∇·ˆQs(r)−ˆτSTT+α\n¯h2/parenleftBig\nˆL׈s/parenrightBig\n(2)\nwhereˆQs(r) =ˆψ†(r)ˆv⊗ˆ sˆψ(r), and the velocity operator\nis given by ˆ v=i¯h\n2m/parenleftBig← −∇−− →∇/parenrightBig\n; here the arrowsuperscript\nspecifies the direction in which the gradient acts. In ad-\ndition:\ndˆL\ndt=∇·ˆQL(r)−ˆτlat+α\n¯h2/parenleftBig\nˆs׈L/parenrightBig\n(3)\nwhereˆQL(r) =ˆψ†(r)1\n2/parenleftBig\nˆvˆL+ˆLˆ v/parenrightBig\nˆψ(r) (the product2\nFIG. 1: Left and right panels shows GaMnAs band structure\nwithout and with spin-orbit, respectively (for γ2=γ3= 2.4).\n(arrows indicate spin direction of eigenstates). The inset\nshows the direction of bulk magnetization, and spin, veloc-\nity, and k vectors for a single state (in black, red, blue, and\ngreen). The torque from the misalignment between magne-\ntization and spin equals the torque from the misalignment\nbetween velocity and k vectors.\nof non-commuting operators ˆLandˆ vis symmetrized).\nWe’ve defined ˆτlat(r) =i\n¯hˆψ†(r)/bracketleftBig\nH0,ˆL/bracketrightBig\nˆψ(r), which is\nnonzeroforapotentials V(r)whichbreakrotationalsym-\nmetry [16].\nWe define a total angularmomentum ˆJ=ˆL+ˆs, a total\nangular momentum current ˆQJ=ˆQL+ˆQs, and combine\nEqs. (2) and (3) to obtain:\ndˆJ\ndt−∇·ˆQJ=−ˆτSTT−ˆτlat. (4)\nFinally, we take the expectation value of Eqs. (2-4), re-\nplacing operatorsby densities. Eq. (4) is ourmain formal\nresult. When spin-orbit coupling is important, the total\nangular momentum in the conduction electrons couples\nboth to the magnetization and the lattice. The coupling\nofelectronspintothelatticerequiresbothspin-orbitcou-\npling and crystal field potential. The term τlatchanges\nthe physical picture of spin transfer torque substantially,\nas is illustrated by considering Eq. (4) for a single bulk\neigenstate:dJ\ndtand∇·QJvanish, however τSTTand\nτlatmay both be non-zero, implying a coupling from the\nangular momentum of the lattice to the magnetization.\nThis coupling flows from the lattice to the orbital sub-\nsystem through the crystal field, which then couples to\nthe spin through spin-orbit coupling, and finally to the\nmagnetization through the exchange interaction.\nApplication to DMS — We apply this general formal-\nism to a model of a dilute magnetic semiconductor\n(DMS). DMSs are semiconductor host materials which\nbecome ferromagnetic when doped with magnetic atoms.Ga1−xMnxAs is the archetype for these materials, and\ncan be described as a system of local moments of Mn\nd-electrons, whose interaction is mediated by holes in\nthe semiconductor valence band [17]. The valence states\nare described by the Kohn-Luttinger Hamiltonian HKL\n0,\nwhich represents a small- kexpansion for a periodic H0,\nacting in the ℓ= 1 subspace (describing valence states).\nIt is given by:\nHKL\n0=¯h2\n2m/parenleftbigg\n(γ1+4γ2)k2−6γ2\n¯h2(L·k)2\n−6\n¯h2(γ3−γ2)/summationdisplay\ni/negationslash=jkikjLiLj\n,(5)\nwhereLare the spin-1 matrices for the p-state orbitals,\nγ1, γ2, γ3are Luttinger parameters, and kis the Bloch\nwave-vector. Figure 1 shows how the presence of spin-\norbit coupling affects the band structure.\nFor periodic systems the velocity operator can be writ-\nten as:ˆ v=1\n¯h∂H\n∂k, and spin and angular momentum cur-\nrent densities are again defined as symmetrized products\nofˆvandˆL, andˆvandˆs. The dynamics of the magne-\ntization occur on a much longer time scale than that of\nthe electronic states, so we compute the dynamics from\na sum over scattering states, for whichds\ndt=dL\ndt= 0. For\nthe Luttinger Hamiltonian, the z-component of ˆτlatis:\nˆτz\nlat= (ˆvׯhk)z+6(γ2−γ3)\n¯hm{(kxLy+kyLx),\n(kxLx−kyLy)} (6)\nwhere the brackets on the second term indicate an an-\nticommutator. Other components are given by cyclic\npermutation of indices. The first term of Eq. (6) can\nbe written as ˆvׯhk=d\ndt(ˆrׯhk). This term can be\ninterpreted as a torque on the crystal angular momen-\ntumˆrׯhk, and results from the misalignment between\nwave vector and velocity. It is generically nonzero for\nany material with a non-spherical Fermi surface. In the\nspherical approximation ( γ2=γ3), Eq. (4) implies that\nthe net flux of total angular momentum into a volume\nis equal to the change of magnetization plus the crystal\nangular momentum inside the volume.\nSTT in spin-valves — We firstconsiderasystem tostudy\ntheτSTTterm of Eq. (4). Figure 2(a) shows the geome-\ntry; current flows in the ˆ z-direction, perpendicular to the\nmagnetization of both layers. We focus on the compo-\nnent of torque which is in the plane spanned by the two\nmagnetization directions. This in-plane torque is deter-\nmined by the out-of-plane (or ˆ z-component) spin density\n[14]. For the results presented here, we use the param-\neter values: ( γ1,γ2,γ3) = (6.85,2.1,2.9), ∆ = 0.27 eV,\nα= 0.11 eV,EF= 0.16 eV (EFis measured from the\ntop of the valence band). The tunnel barrier is described\nby Eqs. (1) and (5), with ∆ = 0, and with an energy\noffset so that the top of the valence band is 0.1 eV below\nEF. We calculate the eigenstates numerically and apply3\nFIG. 2: (a)Spin valve geometry: FM layers’ magnetization\npoints in the ˆ xand ˆy(out-of-page) directions. (b) The spin\ntransfer torque versus position away from the left normal\nmetal-FM interface, which decays to zero in the absence of\nspin-orbit coupling, and does not in its presence. (c) Plot o f\nthe total spin transfer torque, the net flux of spin current,\nand net flux of total angular momentum current versus FM\nthickness. The linear dependence for large thickness is due to\na persistent spin transfer torque.\nboundary conditions as described in Ref. [18].\nFigure 2(b) shows the spin transfer torque density as\na function of distance away from the interface. We find\nthat forα= 0 (no spin-orbitcoupling), the torque decays\nto zero away from the interface, as expected [15]. For\nα/ne}ationslash= 0, the torque oscillates around a nonzero value, and\nextends into the bulk. Figure 2(c) shows that the total\nspin transfer torque as a function of ferromagnetic (FM)\nlayer thickness LFMis proportional to thickness for large\nLFM. This is in contrast to the metallic spin valve, where\nthe torque is an interface effect and becomes constant for\nlargeLFM.\nThis persistent spin transfer torque arises because the\nspins of individual eigenstates are not aligned with the\nmagnetization (see Fig. 1) in the presence of spin-orbit\ncoupling. The misalignment gives rise to a torque be-\ntween the lattice and the magnetization. In equilib-\nrium, these torques cancel when summed over all occu-\npied states. However, the presence of a current changes\nthe occupation of the bulk states and can give rise to\na torque [19, 20] in systems without inversion symme-\ntry. Inversion symmetry is only very weakly broken in\nbulk GaMnAs, and is not included in the Kohn Luttinger\nHamiltonian, Eq. (5). Here, interfaces between materials\nbreaks inversion symmetry.\nThe combination of an interface and a current flow\nchanges the occupation of the bulk states near the Fermi\nenergy (depending on the transmission probabilities of\nindividual states across the interface) and induces coher-\nence between these states. The change in the occupation\nprobabilities gives rise to a persistent transverse spin ac-\ncumulation, which only decays through other scattering\nmechanisms not included here ( e.g.defect scattering).0 0.5 1−0.500.511.5\nα/α0STT/I (h/e)\n  \nSTT total\nQJout−QJin\n  \n−τlat\npersistent STT\nFIG. 3: The total spin transfer torque, the net flux of total\nangular momentum, −τlat, and the persistent component of\nthe spin transfer torque on a FM layer with LFM= 30 nm as\nαis increased from 0 to α0= 0.11 eV.\nThis spin accumulation gives rise to the persistent spin\ntransfer torque. The coherence between the states modi-\nfies the spin accumulation and the torque near the inter-\nface but these corrections decay away from the interface\ndue to dephasing.\nFigure 3 shows, as a function of the spin-orbit coupling\nconstantα, the values of total spin transfer torque, the\nangular momentum current flux, - τlat, and the persistent\ncontribution to spin transfer torque (for LFM= 30 nm).\nWe determine the persistent contribution from the slope\nof the integrated total versus FM width LFMat large\nLFM(see Fig. 2c). This procedure neglects the contri-\nbutions from coherence near the interface. In this exam-\nple, the spin transfer torque increases with the addition\nof spin-orbit coupling, largely because of the addition of\nthe persistent term. This qualitative behavior depends\non system parameters: for EF= 0.34 eV, for example,\nthe spin-orbit coupling decreases the total torque.\nNanomechanical torques in wires — We next consider a\nsystem which exemplifies that physics of the τlatterm of\nEq. (4): a single interface between GaMnAs and GaAs,\nwith the direction of the magnetization parallel to the\ncurrent flow (see Fig. 4a). This is similar to the geom-\netry considered in previous theoretical and experimental\nwork [9–11]. The vanishing magnetization in GaAs im-\npliesτSTT= 0, so that τlat=∇·QJ, and its total value\ncan be deduced from Qin\nJ−Qout\nJ. We use the same pa-\nrameters as before, except EF= 0.06 eV, and the top of\nthe valence band of both layers coincide.\nFigure 4c shows τlatin the GaAs layer as a function\nof distance away from the interface (assuming electron\nparticle flow from left to right). The total torque (dark\ncurve) shows oscillatory decay, while the torque from\na particular channel (light curve) shows simple oscilla-\ntion. The behavior of the single channel is illustrated in\nFig. 4b. We assume specular scattering, so that the in-\ncident state chosen (black circle) transmits into the four\nstates of GaAs with equal kx, ky(also shown with black4\nFIG. 4: (a) shows the system geometry. We take electron\nparticle flow from left to right, and consider the mechanical\ntorqueτlatin the z direction. (b) shows slices of the Fermi\nsurface for the different layers, with /angb∇acketleftJ(k)/angb∇acket∇ightsuperimposed, and\nalso shows the J-character of the states specified by the black\ncircle. Also shown is the transmission probability for each\nof the states in the GaAs. (c) shows the total mechanical\ntorque density in the GaAs as a function of distance from the\ninterface (dark curve), and the contribution from the singl e\nincoming state specified in (b) (dashed curve).\ncircles). The character of these states, along with the\ntransmissionprobability, is shown in Fig. 4b. The incom-\ning state couples most strongly to the state with similar\nJcharacter, but also partially transmits into other states\nwith different Jcharacter and wave vector kz. These dif-\nferent scattering channels interfere with each other, lead-\ning to an oscillatory J(z), with an oscillation period in-\nversely proportional to the splitting of kzwave-vectorsof\nthe different sheets of the Fermi surface. This splitting is\nfrom the lattice crystal field and spin-orbit coupling, the\nagents responsible for τlat. Different channels have dif-\nferent oscillation periods, so that their total decays away\nfrom the interface, ashappens for spin transfertorquesin\nferromagnets [15]. For the parameters used here, we findQin\nJz= 1.20¯hI\ne, due to the polarization of the states from\nthe magnetization, while Qout\nJz= 0.46¯hI\ne. Mechanisms\nnot considered here, such as spin-flip scattering, ensure\nthatQout\nJzdecays to zero away from the interface.\nThemechanicaltorqueis Qin\nJz−Qout\nJz= 0.74¯hI\ne. Forap-\npropriate experimental conditions, this torque is greater\nthan the thermal fluctuations and is a measurable ef-\nfect. We refer the reader to Ref. [9–11] for details of\ntreatment of the torsion dynamics and experimental de-\ntails. The formalism developed here generalizes previous\nwork to allow for microscopic evaluation of the electronic\nstructure contribution to the current-induced mechanical\ntorque. For systems with nonzero magnetization, the mi-\ncroscopic form of τlatis necessary to determine the parti-\ntioning of total angular momentum flux between torques\nonthemagnetizationandtorquesonthelattice. Ourthe-\nory neglects other mechanisms of spin relaxation, such as\ndisorder-induced spin-flip scattering, so that full calcula-\ntions will require microscopic calculations like these to\nbe embedded in diffusive transport calculations.\nConclusion — We have shown how atomic-like spin-orbit\ncoupling affects current-induced torques: both the spin\ntransfer torque on the magnetization and the mechanical\ntorque on the lattice. In GaMnAs spin valves, we find\na contribution to the spin transfer torque that persists\nthroughout the bulk. This result may explain experi-\nments which find critical currents which are up to an\norder of magnitude smaller than the value expected from\na simple accounting of the net spin current flux [21, 22].\nFor a single interface between GaMnAs and GaAs, we\nmicroscopically compute the mechanical torque due to\nscattering from the interface. These results highlight im-\nportant, qualitativelydifferentphysicsatplaywhenspin-\norbit coupling is strong.\nTheauthorsacknowledgehelpfulconversationswithA.\nH. MacDonald.\n[1] L. Berger, J. Appl. Phys. 3, 2156 (1978); ibid. 3, 2137\n(1979).\n[2] J. Slonczewski, J. Magn. Magn. Mat. 62, 123, (1996).\n[3] L. Berger, Phys. Rev. B 54, 9353 (1996).\n[4] D. C. Ralph and M. D. Stiles, J. Magn. Magn. Mater.\n320, 1190 (2007).\n[5] M. D. Stiles and J. Miltat, Top. Appl. Phys. 101, 225\n(2006).\n[6] J. A. Katine and E. E. Fullerton, J. Magn. Magn. Mater.\n320, 1217 (2007).\n[7] O. W. Richardson, Phys. Rev. 26, 248 (1908).\n[8] A. Einstein and A. de Hass, Verhandlungen der\nDeutschen Physikalischen Gesellschaft, 17, 152 (1915).\n[9] P. Mohanty et al., Phys. Rev. B 70, 195301 (2004).\n[10] A. A. Kovalev et al., Phys. Rev. B 75, 014430 (2007).\n[11] G. Zolfagharkhani et al., NatureNanotech. 3, 720 (2008).\n[12] In addition to atomic-like angular momentum, there is a\ncontribution to the total orbital angular momentum from\nitinerant motion through the lattice. The distinction be-tween “local” and “itinerant” orbital angular momentum\nis discussed in Ref. [13] . In this work, we consider only\nthe atomic-like contribution.\n[13] T. Thonhauser et al., Phys Rev. Lett. 95, 137205 (2005).\n[14] A. S. N´ u˜ nez and A. H. MacDonald, Solid State. Comm.\n139, 31 (2006).\n[15] M. D. Stiles and A. Zangwill, Phys. Rev. B 66, 014407\n(2002).\n[16] For H0=−¯h2∇2/2m+V(r), our definition of ˆ τlatis\nequivalent to/bracketleftbig\nV(r),ˆL/bracketrightbig\n. We use ˆ τlat=/bracketleftbig\nH0,ˆL/bracketrightbig\nin antici-\npation of other forms of H0, in particular the k·pform\nof the Luttinger Hamiltonian.\n[17] T. Jungwirth et al., Rev. Mod. Phys. 78, 809 (2006).\n[18] A. M. Malik et al., Phys. Rev. B 59, 2861 (1999).\n[19] A. Manchon and S. Zhang, Phys. Rev. B 78, 212405\n(2008).\n[20] Ion Garate and A. H. MacDonald, Phys. Rev. B 80,\n134403 (2009).\n[21] D. Chiba et al., Phys. Rev. Lett. 93, 216602 (2004).5\n[22] M. Elsen et al., Phys. Rev B 73, 035303 (2006)."    },    {        "title": "1106.4349v3.Effective_one_body_Hamiltonian_of_two_spinning_black_holes_with_next_to_next_to_leading_order_spin_orbit_coupling.pdf",        "content": "arXiv:1106.4349v3  [gr-qc]  5 Sep 2013Effective one body Hamiltonian of two spinning black-holes w ith\nnext-to-next-to-leading order spin-orbit coupling\nAlessandro Nagar\nInstitut des Hautes Etudes Scientifiques, 91440 Bures-sur- Yvette, France\n(Dated: May 24, 2018)\nBuilding on the recently computed next-to-next-to-leadin g order (NNLO) post-Newtonian (PN)\nspin-orbitHamiltonian for spinningbinaries [1] we improv e theeffective-one-body(EOB) description\nof the dynamics of two spinning black-holes by including NNL O effects in the spin-orbit interaction.\nThe calculation that is presented extends to NNLO the next-t o-leading order (NLO) spin-orbit\nHamiltonian computed in Ref. [2]. The presentEOB Hamiltoni an reproduces the spin-orbit coupling\nthrough NNLO in the test-particle limit case. In addition, i n the case of spins parallel or antiparallel\nto the orbital angular momentum, when circular orbits exist , we find that the inclusion of NNLO\nspin-orbit terms moderates the effect of the NLO spin-orbit c oupling.\nPACS numbers: 04.25.-g,04.25.Nx\nI. INTRODUCTION\nCoalescing black-hole binaries are among the most\npromising gravitational wave (GW) sources for the cur-\nrently operating network of ground-based interferomet-\nric GW detectors. Since the spin-orbit interaction can\nincrease the binding energy of the last stable orbit, and\nthereby leading to large GW emission, it is reasonable\nto think that the first detections will concern binary sys-\ntems made of spinning binaries. For this reason, there is\na urgent need of template waveforms accurately describ-\ning the GW emission from coalescing spinning black-hole\nbinaries. These template waveforms will be functions of\nat least eight intrinsic real parameters: the two masses\nm1andm2and the two spin-vectors S1andS2. Be-\ncause of the multidimensionality of the parameter space,\nit seems unlikely for state-of-the-art numerical simula-\ntions to densely sample this parameter space. This gives\na boost to develop analytical methods for computing the\nneeded, densely spaced, bank of accurate template wave-\nforms. Among the existing analytical methods for com-\nputing the motion and the dynamics of black hole (and\nneutron star) binaries, the most complete and the most\npromising is the effective-one-body approach (EOB) [3–\n8]. Several recent works have shown the possibility of\ngetting an excellent agreement between the EOB ana-\nlytical waveforms and the outcome of numerical simula-\ntions of coalescing black-hole (and inspiralling neutron-\nstar [9, 10]) binaries. A considerable part of the current\nliterature deals with nonspinning black-holesystems [11–\n18], with different (though not extreme) mass ratios (see\nin particular [19, 20]) or in the (circularized) extreme-\nmass-ratio limit [21–25] (notably including spin [26]).\nThe work at the interface between numerical relativity\nand the analytical EOB description of spinning binaries\nhas been developing fast in recent years. The first EOB\nHamiltonian which included spin effects was conceived in\nRef. [6]. It was shown there that one could map the 3PN\ndynamics, together with the leading-order (LO) spin-\norbit andspin-spin dynamical effects ofa binarysystems,\nonto an effective test-particle movingin a Kerr-typemet-ric, together with an additional spin-orbit interaction.\nIn Ref. [27] the use of the nonspinning EOB Hamilto-\nnian augmented with PN-type spin-orbit and spin-spin\nterms allowed to carryout the first (and up to now, only)\nanalytical exploratory study of the dynamics and wave-\nforms from coalescing spinning binaries with precessing\nspins. Recently, Ref. [2], building upon the PN-expanded\nHamiltonian of [28], extended the EOB approach of [6]\nso to include the next-to-leading-order (NLO) spin-orbit\ncouplings (see also Refs. [29, 30] for a derivation of these\ncouplings in the harmonic-coordinates equations of mo-\ntion and Ref. [31] for a derivation using an effective field\ntheory approach). Using this model (with the addition\nof EOB-resummed radiation reaction force [7, 22, 32]),\nRef. [33] performed the first comparison with numerical-\nrelativity simulations of nonprecessing, spinning, equal-\nmass, black-holes binaries. Then, building on Ref. [2, 6]\nand Ref. [34], Ref. [35] worked out an improved Hamil-\ntonian for spinning black-hole binaries.\nRecently, Hartung and Steinhoff [1] havecomputed the\nPN-expanded spin-orbit Hamiltonian at next-to-next-to-\nleading order (NNLO), pushing one PN order further\nthe previous computation of Damour, Jaranowski and\nSch¨ afer [28]. The result of Ref. [1] completes the knowl-\nedge of the PN Hamiltonian for binary spinning black-\nholes up to and including 3.5PN.\nThis paper belongs to the lineage of Refs. [2, 6] and\nit aims at exploiting the PN-expanded Hamiltonian of\nRef. [1] so as to obtain the NNLO-accurate spin-orbit\ninteraction as it enters the EOB formalism. Note that,\nby contrast to Refs. [35] and [2], we shall not discuss\nhere spin-spin interactions, nor shall we try to propose a\nspecific way to incorporate our NNLO spin-orbit results\ninto some complete, resummed EOB Hamiltonian. Al-\nthough the Hamiltonian that we shall discuss here does\nnot resum all the spin-orbit terms entering the formal\n“spinning test-particle limit”, we shall check that it con-\nsistently reproduces the “spinning test-particle” results\nof Ref. [35].\nThe paper is organized as follows: in Sec. II we re-\ncall the structure of the PN-expanded spin-orbit Hamil-2\ntonian(inArnowitt-Deser-Misner(ADM) coordinates)of\nRef.[1]andthenweexpressitinthecenterofmassframe.\nSection III explicitly performs the canonical transforma-\ntion from ADM coordinates to EOB coordinates and fi-\nnally computes the effective Hamiltonian, and, in partic-\nular, the effective gyro-gravitomagneticratios. In Sec. IV\nwediscuss the caseofcircularequatorialorbits, wederive\nthe test-mass limit and we exploit the gauge freedom to\nsimplify the expression of the final Hamiltonian.\nWe adopt the notation of [2] and we use the letters\na,b= 1,2 as particle labels. Then, ma,xa= (xi\na),pa=\n(pai), andSa= (Sai) denote, respectively, the mass, the\nposition vector, the linear momentum vector, and the\nspin vector of the ath body; for a∝ne}ationslash=bwe also define\nrab≡xa−xb,rab≡ |rab|,nab≡rab/rab,|·|stands here\nfor the Euclidean length of a 3-vector.\nII. PN-EXPANDED HAMILTONIAN IN ADM\nCOORDINATES\nWe closely follow the procedure of Ref. [2]. The start-\ning point of the calculation is the PN-expandend two-\nbody Hamiltonian Hwhich can be decomposed as the\nsum of an orbital part, Ho, a spin-orbit part, Hso(linear\nin the spins) and a spin-spin term Hss(quadratic in the\nspins), that we quote here for completeness but that weare not going to discuss in the paper. It reads\nH(xa,pa,Sa) =Ho(xa,pa)+Hso(xa,pa,Sa)\n+Hss(xa,pa,Sa). (1)\nThe orbital Hamiltonian Hoincludes the rest-mass con-\ntribution and is explicitly known (in ADM-like coordi-\nnates) up to the 3PN order [36, 37]. It has the structure\nHo(xa,pa) =/summationdisplay\namac2+HoN(xa,pa)\n+1\nc2Ho1PN(xa,pa)+1\nc4Ho2PN(xa,pa)\n+1\nc6Ho3PN(xa,pa)+O/parenleftbigg1\nc8/parenrightbigg\n.(2)\nThe spin-orbit Hamiltonian Hsocan be written as\nHso(xa,pa,Sa) =/summationdisplay\naΩa(xb,pb)·Sa.(3)\nHere, the quantity Ωais the sum of three contributions:\nthe LO ( ∝1/c2), the NLO ( ∝1/c4), and the NNLO one\n(∝1/c6),\nΩa(xb,pb) =ΩLO\na(xb,pb)+ΩNLO\na(xb,pb)+ΩNNLO\na(xb,pb).\n(4)\nThe 3-vectors ΩLO\naandΩNLO\nawere explicitly computed\nin Ref. [28], while ΩNNLO\nacan be read off Eq.(5) of\nRef. [1]. We write them here explicitly for completeness.\nFor the particle label a= 1, we have\nΩLO\n1=G\nc2r2\n12/parenleftbigg3m2\n2m1n12×p1−2n12×p2/parenrightbigg\n, (5a)\nΩNLO\n1=G2\nc4r3\n12/parenleftBigg/parenleftbigg\n−11\n2m2−5m2\n2\nm1/parenrightbigg\nn12×p1+/parenleftbigg\n6m1+15\n2m2/parenrightbigg\nn12×p2/parenrightBigg\n+G\nc4r2\n12/parenleftBigg/parenleftbigg\n−5m2p2\n1\n8m3\n1−3(p1·p2)\n4m2\n1+3p2\n2\n4m1m2−3(n12·p1)(n12·p2)\n4m2\n1−3(n12·p2)2\n2m1m2/parenrightbigg\nn12×p1\n+/parenleftbigg(p1·p2)\nm1m2+3(n12·p1)(n12·p2)\nm1m2/parenrightbigg\nn12×p2+/parenleftbigg3(n12·p1)\n4m2\n1−2(n12·p2)\nm1m2/parenrightbigg\np1×p2/parenrightBigg\n, (5b)\nΩNNLO\n1=G\nr2\n12/bracketleftbigg/parenleftbigg7m2(p2\n1)2\n16m5\n1+9(n12·p1)(n12·p2)p2\n1\n16m4\n1+3p2\n1(n12·p2)2\n4m3\n1m2\n+45(n12·p1)(n12·p2)3\n16m2\n1m2\n2+9p2\n1(p1·p2)\n16m4\n1−3(n12·p2)2(p1·p2)\n16m2\n1m2\n2\n−3(p2\n1)(p2\n2)\n16m3\n1m2−15(n12·p1)(n12·p2)p2\n2\n16m2\n1m2\n2+3(n12·p2)2p2\n2\n4m1m3\n2\n−3(p1·p2)p2\n2\n16m2\n1m2\n2−3(p2\n2)2\n16m1m3\n2/parenrightbigg\nn12×p1+/parenleftbigg\n−3(n12·p1)(n12·p2)p2\n1\n2m3\n1m23\n−15(n12·p1)2(n12·p2)2\n4m2\n1m2\n2+3p2\n1(n12·p2)2\n4m2\n1m2\n2−p2\n1(p1·p2)\n2m3\n1m2+(p1·p2)2\n2m2\n1m2\n2\n+3(n12·p1)2p2\n2\n4m2\n1m2\n2−(p2\n1)(p2\n2)\n4m2\n1m2\n2−3(n12·p1)(n12·p2)p2\n2\n2m1m3\n2−(p1·p2)p2\n2\n2m1m3\n2/parenrightbigg\nn12×p2\n+/parenleftbigg\n−9(n12·p1)p2\n1\n16m4\n1+p2\n1(n12·p2)\nm3\n1m2\n+27(n12·p1)(n12·p2)2\n16m2\n1m2\n2−(n12·p2)(p1·p2)\n8m2\n1m2\n2−15(n12·p1)p2\n2\n16m2\n1m2\n2\n+(n12·p2)p2\n2\nm1m3\n2/parenrightbigg\np1×p2/bracketrightbigg\n+G2\nr3\n12/bracketleftbigg/parenleftbigg\n−3m2(n12·p1)2\n2m2\n1+/parenleftbigg\n−3m2\n2m2\n1+27m2\n2\n8m3\n1/parenrightbigg\np2\n1+/parenleftbigg177\n16m1+11\nm2/parenrightbigg\n(n12·p2)2\n+/parenleftbigg11\n2m1+9m2\n2m2\n1/parenrightbigg\n(n12·p1)(n12·p2)+/parenleftbigg23\n4m1+9m2\n2m2\n1/parenrightbigg\n(p1·p2)\n−/parenleftbigg159\n16m1+37\n8m2/parenrightbigg\np2\n2/parenrightbigg\nn12×p1+/parenleftbigg4(n12·p1)2\nm1+13p2\n1\n2m1\n+5(n12·p2)2\nm2+53p2\n2\n8m2−/parenleftbigg211\n8m1+22\nm2/parenrightbigg\n(n12·p1)(n12·p2)\n−/parenleftbigg47\n8m1+5\nm2/parenrightbigg\n(p1·p2)/parenrightbigg\nn12×p2\n+/parenleftbigg\n−/parenleftbigg8\nm1+9m2\n2m2\n1/parenrightbigg\n(n12·p1)+/parenleftbigg59\n4m1+27\n2m2/parenrightbigg\n(n12·p2)/parenrightbigg\np1×p2/bracketrightbigg\n+G3\nr4\n12/bracketleftbigg/parenleftbigg181m1m2\n16+95m2\n2\n4+75m3\n2\n8m1/parenrightbigg\nn12×p1−/parenleftbigg21m2\n1\n2+473m1m2\n16+63m2\n2\n4/parenrightbigg\nn12×p2/bracketrightbigg\n(5c)\nThe expressions for ΩLO\n2,ΩNLO\n2andΩNNLO\n2can be ob-\ntained from the above formulas by exchanging the parti-\ncle labels 1 and 2.\nLet us consider now the dynamics of the relative mo-\ntion of the two body system in the center of mass frame,\nwhich is defined by setting p1+p2= 0. Following [2],\nwe rescale the phase-space variables R≡x1−x2and\nP≡p1=−p2of the relative motion as\nr≡R\nGM,p≡P\nµ≡p1\nµ=−p2\nµ,(6)\nwhereM=m1+m2andµ≡m1m2/M. In addition, we\nrescale the original time variable Tand any part of the\nHamiltonian as\nt≡T\nGM,ˆHNR≡HNR\nµ,(7)\nwhereHNR≡H−Mc2denotes the “nonrelativistic”\nHamiltonian, i.e. the Hamiltonian withouth the rest-\nmass contribution. As in [2] we work with the following\ntwo, basic combinations of the spin vectors:\nS≡S1+S2=m1ca1+m2ca2, (8)\nS∗≡m2\nm1S1+m1\nm2S2=m2ca1+m1ca2,(9)where we have also introduced the Kerr parameters of\nthe individual black-holes, a1≡S1/(m1c) anda2≡\nS2/(m2c). We recall that in the formal1“spinning test\nmass limit” where, for example, m2→0 andS2→0,\nwhile keeping a2=S2/(m2c) fixed, one has a “back-\nground mass” M≃m1, a “background spin” Sbckgd≡\nMcabckgd≃S1=m1ca1, a “test mass” µ≃m2, and\na “test spin” Stest=S2=m2ca2≃µcatest[with\natest≡Stest/(µc)]. Then, in this limit the combina-\ntionS≃S1=m1ca1≃Mcabckgd=Sbckgdmea-\nsures the background spin, while the other combination,\nS∗≃m1ca2≃Mcatest=MStest/µmeasures the (spe-\ncific) test spin atest=Stest/(µc). Finally, since the use\nof the rescaled variables corresponds to a rescaling of the\naction by a factor 1 /(GMµ), it is also natural to work\nwith the corresponding rescaled variables\n¯SX≡SX\nGMµ, (10)\nfor any label X (X= 1 ,2, ,∗).\n1As noted in Ref. [2] this formal limit is not relevant for the p hysi-\ncally mostimportant case of binary black holes, forwhich a2→0\nandm2→0.4\nUsing the definitions (6)-(10), the center-of-mass spin-\norbit Hamiltonian (divided by µ) in terms of the rescaled\nvariables has the structure\nˆHso(r,p,¯S,¯S∗)≡Hso(r,p,¯S,¯S∗)\nµ(11)\n=1\nc2ˆHso\nLO(r,p,¯S,¯S∗)\n+1\nc4ˆHso\nNLO(r,p,¯S,¯S∗)\n+1\nc6ˆHso\nNNLO(r,p,¯S,¯S∗)+O/parenleftbigg1\nc8/parenrightbigg\n,\n(12)\nand it can be written as\nˆHso(r,p,¯S,¯S∗) =ν\nc2r2/parenleftbig\ngADM\ns(¯S,n,p)+gADM\nS∗(¯S∗,n,p)/parenrightbig\n,\n(13)with the following definitions: ν≡µ/Mis the sym-\nmetric mass ratios and ranges from 0 (test-body limit)\nto 1/4 (equal-mass case); the notation ( V1,V2,V3)≡\nV1·(V2×V3) =ǫijkVi\n1Vj\n2Vk\n3stands for the Euclidean\nmixed products of 3-vectors; n≡r/|r|;gADM\nSandgADM\nS∗\nare the two (dimensionless) gyro-gravitomagnetic ratios\nas introduced (up to NLO accuracy) in [2]. These two\ncoefficients parametrize the coupling between the spin\nvectors and the apparent gravito-magnetic field seen in\nthe rest-frame of a moving particle. Their explicit ex-\npressions including the NNLO contribution read\ngADM\nS= 2+1\nc2/parenleftbigg /parenleftbigg/parenleftbigg19\n8νp2+3\n2ν(n·p)2−/parenleftBig\n6+2ν/parenrightBig1\nr/parenrightbigg /parenrightbigg/parenrightbigg\n+1\nc4/braceleftBigg\n−9\n8ν/parenleftBig\n1−22\n9ν/parenrightBig\np4−3\n4ν/parenleftBig\n1−9\n4ν/parenrightBig\np2(n·p)2+15\n16ν2(n·p)4\n+1\nr/bracketleftbigg\n−157\n8ν/parenleftBig\n1+39\n314ν/parenrightBig\np2−16ν/parenleftBig\n1+45\n256ν/parenrightBig\n(n·p)2+1\nr21\n2/parenleftBig\n1+ν/parenrightBig/bracketrightbigg/bracerightBigg\n, (14a)\ngADM\nS∗=3\n2+1\nc2/parenleftbigg /parenleftbigg/parenleftbigg/parenleftBig\n−5\n8+2ν/parenrightBig\np2+3\n4ν(n·p)2−/parenleftBig\n5+2ν/parenrightBig1\nr/parenrightbigg /parenrightbigg/parenrightbigg\n+1\nc4/braceleftBigg\n1\n16/parenleftBig\n7−37ν+39ν2/parenrightBig\np4+9\n16ν(2ν−1)p2(n·p)2\n+1\nr/bracketleftbigg1\n8/parenleftBig\n27−129ν−39\n2ν2/parenrightBig\np2−6ν/parenleftBig\n1+15\n32ν/parenrightBig\n(n·p)2+1\nr/parenleftbigg75\n8+41\n4ν/parenrightbigg/bracketrightbigg/bracerightBigg\n. (14b)\nThe label “ADM” on the gyro-gravitomagnetic ra-\ntios (14) is a reminder that, although the LO values are\ncoordinate independent, both the NLO and NNLO con-\ntributions to these ratios actually depend on the defi-\nnition of the phase-space variables ( r,p). In the next\nSection we shall introduce the two, related, effective\ngyro-gravitomagneticratios that enter the effective EOB\nHamiltonian, written in effective (or EOB) coordinates,\naccording to the prescriptions of [2].\nIII. EFFECTIVE HAMILTONIAN AND\nEFFECTIVE GYRO-GRAVITOMAGNETIC\nRATIOS\nFollowing Ref. [2], two operations have to be per-\nformed on the Hamiltonian written in the center of massframe so to cast it in a form that can be resummed in\na way compatible to previous EOB work. First of all,\none needs to transform the (ADM) phase-space coordi-\nnates (xa,pa,Sa) by a canonical transformation compat-\nible with the one used in previous EOB work. Second,\none needs to compute the effective Hamiltonian corre-\nsponding to the canonically transformed realHamilto-\nnian. Following the same procedure adopted in [2], we\nstart by performing the purely orbital canonical trans-\nformation which was found to be needed to go from the\nADM coordinates used in the PN-expanded Hamiltonian\nto the coordinates used in the EOB dynamics. Since in\nRef. [2] one was concerned only with the NLO spin-orbit\ninteraction, it was enough to consider the 1PN-accurate\ntransformation. In the present study, because one is\nworkingat NNLO in the spin-orbitinteraction, oneneeds\nto take into account the complete 2PN-accurate canon-5\nical transformation introduced in [3]. The transforma-\ntion changes the ADM phase-space variables ( r,p,¯S,¯S∗)\nto (r′,p′,¯S,¯S∗) and it is explicitly given by Eqs. (6.22)-\n(6.23) of [3]. To our purpose, we actually need to use\ntheinverserelations r=r(r′,p′) andp=p(r′,p′), soto replace ( r,p) with (r′,p′) in Eq. (13). The needed\ntransformation is easily found by solving, by iteration,\nEqs. (6.22)-(6.23) of [3], and we explicitly quote it here\nfor future convenience. It reads\nri−r′\ni=1\nc2/bracketleftBigg\n−/parenleftBig\n1+ν\n2/parenrightBigr′i\nr′+ν\n2p′2r′\ni+ν(r′·p′)p′\ni/bracketrightBigg\n+1\nc4/braceleftBigg/bracketleftBigg\n1\n4r′2/parenleftbig\n−ν2+7ν−1/parenrightbig\n+3ν\n4/parenleftBigν\n2−1/parenrightBigp′2\nr′−ν\n8(1+ν)p′4−ν/parenleftbigg\n2+5\n8ν/parenrightbigg(r′·p′)2\nr′3/bracketrightBigg\nr′\ni\n+/bracketleftBigg\nν\n2/parenleftBig\n−5+ν\n2/parenrightBigr′·p′\nr′+ν\n2(ν−1)p′2(r′·p′)/bracketrightBigg\np′\ni/bracerightBigg\n, (15)\npi−p′\ni=1\nc2/bracketleftBigg\n−/parenleftBig\n1+ν\n2/parenrightBigr′·p′\nr′3r′\ni+/parenleftBig\n1+ν\n2/parenrightBigp′\ni\nr′−ν\n2p′2p′\ni/bracketrightBigg\n+1\nc4/braceleftBigg/bracketleftBigg\n1\nr′2/parenleftbigg5\n4−3\n4ν+ν2\n2/parenrightbigg\n+ν\n8(1+3ν)p4−ν\n4/parenleftbigg\n1+7\n2ν/parenrightbiggp2\nr′+ν/parenleftBig\n1+ν\n8/parenrightBig(r′·p′)2\nr′3/bracketrightBigg\np′\ni\n+/bracketleftBigg/parenleftbigg\n−3\n2+5\n2ν−3\n4ν2/parenrightbiggr′·p′\nr′4+3\n4ν/parenleftBigν\n2−1/parenrightBig\np2r′·p′\nr′3+3\n8ν2(r′·p′)3\nr′5/bracketrightBigg\nr′i/bracerightBigg\n. (16)\nAs pointed out in [3], in the test-mass limit ( ν→0)\none hasr′i=/bracketleftbig\n1+1/(2c2r)/bracketrightbig\nri, which is the relation be-\ntween Schwarzschild ( r′) and isotropic ( r) coordinates\nin a Schwarzschild spacetime2. When this transforma-\ntion is applied to to the spin-orbit Hamiltonian in ADMcoordinates, Eq. (13), one gets a transformed Hamilto-\nnian of the form ˆH′(r′,p′,¯S,¯S∗) =ˆH′\no(r′,p′,¯S,¯S∗) +\nˆH′so(r′,p′,¯S,¯S∗), with the NNLO spin-orbit contribu-\ntion that explicitly reads\nˆH′so\nNNLO(r′,p′,¯S,¯S∗) =ν\nr′2/braceleftBigg\n(¯S∗,n′,p′)/bracketleftBigg\nν\nr′2/parenleftBig\n−8+ν\n2/parenrightBig\n+1\nr′/bracketleftbigg/parenleftbigg\n−13\n4ν−3\n4ν2/parenrightbigg\np′2+/parenleftbigg43\n4ν−75\n16ν2/parenrightbigg\n(n′·p′)2/bracketrightbigg\n+/parenleftbigg\n−3\n8ν+9\n16ν2/parenrightbigg\np′4+/parenleftbigg9\n4ν−3\n16ν2/parenrightbigg\np′2(n′·p′)2+135\n16ν2(n′·p′)2/bracketrightBigg\n,\n+(¯S∗,n′,p′)/bracketleftBigg\n−1\nr′2/parenleftbigg1\n2+53\n8ν+5\n8ν2/parenrightbigg\n+1\nr′/bracketleftbigg/parenleftbigg1\n4−53\n16ν+3\n8ν2/parenrightbigg\np′2+/parenleftbigg5\n4+121\n8ν−3ν2/parenrightbigg\n(n′·p′)2/bracketrightbigg\n+/parenleftbigg7\n16−3\n16ν+ν2\n4/parenrightbigg\np′4+/parenleftbigg57\n16ν−3\n4ν2/parenrightbigg\np′2(n′·p′)2+15\n2ν2(n′·p′)2/bracketrightBigg/bracerightBigg\n,(17)\n2As a check of the transformation (15)-(16) one can explicitl y\nverify that it preserves the orbital angular momemntum at 2P Norder, i.e. r′×p′=r×p+O/parenleftbigg1\nc6/parenrightbigg\n.6\nwhere we introduced the radial unit vector n′=r′/|r′|.\nWith this result in hands, we can further perform on\nit a secondary purely spin-dependent , canonical transfor-\nmation that affects both the NLO and NNLO spin orbit\nterms. This transformation can be thought as a gauge\ntransformation related to the arbitrariness in choosing\na spin-supplementary condition and in defining a local\nframe to measure the spin vectors. Such gauge condition\ncan then be conveniently chosen so to simplify the spin-\norbit Hamiltonian. This procedure was pushed forward,\nat NLO accuracy in Ref. [2]. In that case, the canonical\ntransformation was defined by means of a 2PN-accurate\ngenerating function, that was chosen proportional to the\nspins and with two arbitrary ( ν-dependent) dimension-\nless coefficients a(ν) andb(ν). Using rescaled variables,\nthe NLO generating function of [2] reads\n¯Gs2PN=1\nc4ν(n′·p′)\nr′/parenleftbig\na(ν)(¯S,n′,p′)+b(ν)(¯S∗,n′,p′)/parenrightbig\n.\n(18)\nIn Ref. [2] the parameters a(ν) andb(ν) were selected\nso to remove the terms proportional to p2in the final\n(effective) Hamiltonian. Letusrecallthat, atlinearorder\nin the¯Gs2PN, that was enough for the NLO case, the new\nHamiltonian was computed as ˆH′′so(y′′) =ˆH′so(y′′)−\n{ˆH′,¯Gs2PN}(y′′), were we address collectively with y′′=\n(r′′,p′′,¯S′′,S′′∗) the new phase space-variables.\nWe wish now to introduce a more general gauge trans-\nformation such to act also on the NNLO terms of the\nHamiltonian. To do so, in addition to the NLO part\n¯Gs2PNof the spin-dependent generating function men-\ntioned above, one also needs to introduce a NNLO con-\ntribution of the form\n¯Gs3PN=1\nc6ν/braceleftBigg\n(n′·p′)\nr′/bracketleftbiggα(ν)\nr′+β(ν)(n′·p′)2+γ(ν)p′2/bracketrightbigg\n×(¯S,n′,p′)\n+(n′·p′)\nr′/bracketleftbiggδ(ν)\nr′+ζ(ν)(n′·p′)2+η(ν)p′2/bracketrightbigg\n×(¯S∗,n′,p′)/bracerightBigg\n, (19)\nwith six, arbitrary, ν-dependent dimensionless coeffi-\ncients. We shall then consider the effect of a spin-\ndependent generating function of the form ¯Gs=¯Gs2PN+\n¯Gs3PN. Since ¯Gsstarts at 2PN order, it turns out that\npossible quadratic terms in the generating function are\nof order c−8, i.e. at 4PN and thus are of higher order\nthan the NNLO accuracy that we are currently consid-\nering in the spin-orbit Hamiltonian. The consequence is\nthat the purely spin-dependent gauge transformation at\nNNLO will involve onlythe contribution linear in ¯Gs.\nIn other terms, we only need to consider the following\ntransformation on the Hamiltonian\nˆH′′(y′′) =ˆH′(y′′)−{ˆH′,¯Gs}(y′′).(20)Extracting from this equation the spin-dependent terms,\nwe find that the relevant terms in the new spin-orbit\nHamiltonian up to NNLO are then given by\nˆH′′so\nLO(r′′,p′′,¯S′′,¯S′′∗) =H′so\nLO(y′′),\nˆH′′so\nNLO(r′′,p′′,¯S′′,¯S′′∗) =H′so\nNLO(y′′)−{H′\noN,¯Gs2PN}(y′′),\nˆH′′so\nNNLO(r′′,p′′,¯S′′,¯S′′∗) =ˆH′so\nNNLO(y′′)\n−/bracketleftBig\n{ˆH′\noN,¯Gs3PN}\n+{ˆH′\no1PN,¯Gs2PN}\n+{H′so\nLO,¯Gs2PN}/bracketrightBig\n(y′′).(21)\nNote that the single prime in these equations explic-\nitly addresses the various contribution to the spin-orbit\nHamiltonian as computed after the purely orbital canon-\nical transformation mentioned above (note however that\nonly the functional form of ˆH′\no1PNis modified by the ac-\ntion of the orbital canonical transformation).\nFurther simplifications occur in the third Poisson\nbracket of Eq. (21). First of all, since we are inter-\nested in computing only the contribution to the spin-\norbit interaction, the terms quadratic in spins are ne-\nglected. In addition, from the basic relation {Si,Sj}=\nǫijkSkonecanshowbyastraightforwardcalculationthat\n{Hso′\nLO,¯Gs2PN}= 0 (always at linear order in the spin).\nConsequently, the effect of the purely spin-dependent\ncanonical transformation is fully taken into account by\nthe two Poisson brackets involving the generating func-\ntions¯Gs2PNand¯Gs3PN, and the purely orbital contribu-\ntions to the Hamiltonian, ˆH′\noNandˆH′\no1PN.\nFor simplicity of notation, we shall omit hereafter the\ndouble primes from the transformed Hamiltonian. We\nnow need to connect the real Hamiltonian Hto the effec-\ntive oneHeff, which is more closely linked to the descrip-\ntion of the EOB quasigeodesic dynamics. The relation\nbetween the two Hamiltonians is given by [3]\nHeff\nµc2≡H2−m2\n1c4−m2\n2c4\n2m1m2c4(22)\nwheretherealHamiltonian Hcontainstherest-masscon-\ntributions Mc2. In terms of the nonrelativistic Hamilto-\nnianˆHNR, this equation is equivalent to\nˆHeff\nc2= 1+ˆHNR\nc2+ν\n2(ˆHNR)2\nc4, (23)\nwhere it is explicitly\nˆHNR=/parenleftBigg\nˆHoN+ˆHo1PN\nc2+ˆHo2PN\nc4+ˆHo3PN\nc6/parenrightBigg\n+/parenleftBiggˆHso\nLO\nc2+ˆHso\nNLO\nc4+ˆHso\nNNLO\nc6/parenrightBigg\n.(24)\nBy expanding in powers of 1 /c2up to 3PN fractional ac-\ncuracy (and in powers of the spin) the exact effective7\nHamiltonian, one easily finds that the spin-orbit part\nof the effective Hamiltonian ˆHeff(i.e., the part which\nis linear-in-spin) reads\nˆHso\neff=1\nc2ˆHso\nLO+1\nc4/parenleftBig\nˆHso\nNLO+νˆHoNˆHso\nLO/parenrightBig\n+1\nc6/bracketleftBig\nˆHso\nNNLO+ν/parenleftBig\nˆHoNˆHso\nNLO+ˆHo1PNHso\nLO/parenrightBig/bracketrightBig\n.\n(25)\nCombining this result with the effect of the generating\nfunction discussed above, we get the transformed spin-\norbit part of the effective Hamiltonian in the form as\nˆHso\neff=ν\nc2r2/parenleftbig\ngeff\nS(¯S,n,p)+geff\nS∗(¯S∗,n,p)/parenrightbig\n.(26)Theeffectivegyro-gravitomagneticratios geff\nSandgeff\nS∗dif-\nfer from the ADM ones introduced above because of the\neffect of the (orbital+spin) canonical transformation and\nbecauseofthe transformationfrom HtoHeff. They have\nthe structure\ngeff\nS= 2+1\nc2geffNLO\nS(a)+1\nc4geffNNLO\nS(a;α,β,γ) (27)\ngeff\nS∗=3\n2+1\nc2geffNLO\nS(b)+1\nc4geffNNLO\nS∗(b;δ,ζ,η),(28)\nwhere we made it apparent the dependence on the ( ν-\ndependent) NLO and NNLO gauge parameters. Includ-\ning the new NNLO terms, they read\ngeff\nS= 2+1\nc2/bracketleftBigg/parenleftbigg3\n8ν+a/parenrightbigg\np2−/parenleftbigg9\n2ν+3a/parenrightbigg\n(n·p)2/parenrightBigg\n−1\nr(ν+a)/bracketrightBigg\n+1\nc4/bracketleftBigg\n−1\nr2/parenleftbigg\n9ν+3\n2ν2+a+α/parenrightbigg\n+1\nr/bracketleftbigg\n(n·p)2/parenleftbigg35\n4ν−3\n16ν2+6a−4α−3β−2γ/parenrightbigg\n+p2/parenleftbigg\n−17\n4ν+11\n8ν2−3a\n2+α−γ/parenrightbigg/bracketrightbigg\n+/parenleftbigg9\n4ν−39\n16ν2+3a\n2+3β−3γ/parenrightbigg\np2(n·p)2+/parenleftbigg135\n16ν2−5β/parenrightbigg\n(n·p)4\n+/parenleftbigg\n−5\n8ν−a\n2+γ/parenrightbigg\np4/bracketrightBigg\n, (29)\ngeff\nS∗=3\n2+1\nc2/bracketleftBigg/parenleftbigg\n−5\n8+1\n2ν+b/parenrightbigg\np2−/parenleftbigg15\n4ν+3b/parenrightbigg\n(n·p)2−1\nr/parenleftbigg1\n2+5\n4ν+b/parenrightbigg/bracketrightBigg\n+1\nc4/bracketleftBigg\n−1\nr2/parenleftbigg1\n2+55\n8ν+13\n8ν2+b+δ/parenrightbigg\n+1\nr/bracketleftBigg\n(n·p)2/parenleftbigg5\n4+109\n8ν+3\n4ν2+6b−4δ−3ζ−2η/parenrightbigg\n+p2/parenleftbigg1\n4−59\n16ν+3\n2ν2−3b\n2+δ−η/parenrightbigg/bracketrightBigg\n+/parenleftbigg57\n16ν−21\n8ν2+3b\n2+3ζ−3η/parenrightbigg\np2(n·p)2+/parenleftbigg15\n2ν2−5ζ/parenrightbigg\n(n·p)4\n+/parenleftbigg7\n16−11\n16ν−ν2\n16−b\n2+η/parenrightbigg\np4/bracketrightBigg\n. (30)\nThis is the central result of the paper. The NNLO con-\ntribution to the gyro-gravitomagnetic ratios computed\nhere is the crucial, new, information that it is needed to\nimprove to the next PN order the spin-dependent EOB\nHamiltonian (either in the version of Ref. [2] or [35]). Let\nus recall in this respect that in the EOB approach of [2]\nthe relative dynamics can be equivalently represented by\nthe dynamics of a spinning effective particle with effec-\ntive spin σmoving onto a ν-deformed Kerr-type metric.The gyro-gravitomagnetic ratios enter the definition of\nthe test-spin vector σas\nσ=1\n2/parenleftbig\ngeff\nS−2/parenrightbig\nS+1\n2/parenleftbig\ngeff\nS∗−2/parenrightbig\nS∗,(31)\nthat can then be inserted in Eqs. (4.16) of Ref. [2] to get\nthe spin-orbit interaction additional to the leading Kerr-\nmetric part. Together with Eqs. (4.17), (4.18) and (4.19)\nof Ref. [2] this defines the real EOB-improved,resummed8\nHamiltonian for spinning binaries at NNLO in the spin-\norbit interaction.\nIV. LIMITS, CHECKS AND GAUGE FIXING\nA. The extreme-mass-ratio limit\nThe effective spin-orbit Hamiltonian (26) is naturally\nconnected to the test-mass ( ν→0) Hamiltonian explic-\nitly obtained3in [34]. To show this in a concrete case,\nlet us consider the spin-orbit Hamiltonian of a spinning\ntest-particle on Schwarzschild spacetime written explic-\nitly using isotropic coordinates, as given by Eq. (5.12) of\nRef. [34]. By considering the Schwarzschild metric writ-\nten as\nds2=−f(r)dt2+h(r)(dx2+dy2+dz2),(32)\nwhererlabelsheretheisotropicradius4,r2=x2+y2+z2,\n(that is meant to be expressed in rescaled units, where\nnowM≃m1is the background mass and µ≃m2is the\ntest-particle mass), with\nh=/parenleftbigg\n1+1\n2c2r/parenrightbigg4\n, (33)\nand using rescaled variables (and making explicit the\nspeed of light) Eq. (5.12) of Ref. [34] can be written as\nˆHso\nISO=ν\nc2r2gISO\n0/parenleftbig\nn,p,¯S∗\n0/parenrightbig\n. (34)\nIn this equation, ¯S∗\n0is the (rescaled) spin of the\ntest-mass and we have introduced the test-mass gyro-\ngravitomagnetic ratio in isotropic coordinates gISO\n0, that\nis known in closed form [34] and reads\ngISO\n0=h−3/2\n√Q/parenleftbig\n1+√Q/parenrightbig/bracketleftbigg\n1−1\n2c2r+/parenleftbigg\n2−1\n2c2r/parenrightbigg/radicalbig\nQ/bracketrightbigg\n,\n(35)\nwhere\nQ= 1+1\nc2p2\nh. (36)\nBy transforming the Hamiltonian (34) from isotropic\nto Schwarzschild coordinates using the ν→0 limit of\nthe (purely orbital) canonical transformation given by\n3Note in passing that the simple procedure described in Ref. [ 28]\nto obtain the spin-orbit Hamiltonian is totally general and can\nbe applied, in particular, to the test-mass case.\n4Note that we use the same notation for the isotropic radius on\nSchwarzschild spacetime and the ADM radial coordinates. Th ere\nis no ambiguity here since for the Schwarzschild spacetime A DM\ncoordinates do actually coincide with isotropic coordinat esEqs. (15)-(16), expanding in powers of 1 /c2, (and drop-\nping again the primes for simplicity) one obtains\nˆHso\nSchw=ν\nc2r2gSchw\n0/parenleftbig\nn,p,¯S∗/parenrightbig\n. (37)\nwith\ngSchw\n0=3\n2−1\nc2/parenleftbigg1\n2r+5\n8p2/parenrightbigg\n+1\nc4/bracketleftbigg\n−1\n2r2+1\nr/parenleftbigg5\n4(n·p)2+1\n4p2/parenrightbigg\n+7\n16p4/bracketrightbigg\n.\n(38)\nIn theν→0 (Schwarzschild) limit, one has lim ν→0(H−\nconst.)/µ= limν→0ˆHeff(when dropping inessential con-\nstants), ¯S∗=¯S∗\n0and¯S= 0. One then finds that the\nresult (38) agrees in the ν→0 limit with Eq. (30) when\nthe gauge parameters ( b,δ,ζ,η) are simply zero.\nIn addition, in the ν→0 limit where the background\nis a Kerr black hole, i.e. ¯S∝ne}ationslash= 0, Eq. (29) consistently\nexhibits that both the NLO and NNLO contributions\nbecome pure gauge, that can just be set to zero by de-\nmanding ( a,α,β,γ) to vanish.\nB. Circular equatorial orbits\nLet us consider now the situation where both individ-\nualspinsareparallel(orantiparallel)tothe (rescaled)or-\nbital angular momentum vector ℓ=rn×p. [Note that\nin this Section the quantity rdenotes the EOB radial\ncoordinate (further modified by spin-dependent gauge\nterms, see below)]. In this case, circular orbits exists\n(but in the general case, when the spin vectors are not\naligned with ℓ, there are no circular orbits). One can\nthen consistently set everywhere the radial momentum\nto zero, pr≡n·p= 0 and express the total (orbital\nplus spin-orbit part) real, PN-expanded and canonically\ntransformed Hamiltonian, H(y′′)≡H′′\no(y′′) +H′′so(y′′)\n(dropping hereafter the primes for simplicity) as a func-\ntion ofr,ℓ(using the link p2=ℓ2/r2, whereℓ≡ |ℓ|) and\nof the two scalars ˆ aand ˆa∗measuring the projection of\nthe basic spin combinations SandS∗along the direction\nof the orbital angular momentum ℓ. Following the same\nnotation of [2], we introduce here the dimensionless spin\nvariables corresponding to SandS∗\nˆa≡cS\nGM2,ˆa∗≡cS∗\nGM2, (39)\nand we define the projections as\nˆa·ℓ= ˆaℓ,ˆa∗·ℓ= ˆa∗ℓ, (40)\nwith the scalars ˆ aand ˆa∗positive or negative depending\non whether say ˆais parallel or antiparallel to ℓ. The se-\nquence of circular (equatorial) orbits5is then determined\n5To avoid confusion, let us stress that we are here considerin g the\ncircular orbits of the PN-expanded real Hamiltonian and notthe9\nby the constraint\n∂H(r,ℓ,ˆa,ˆa∗)/∂r= 0, (41)\n(or equivalently by ∂Heff/∂r= 0). To start with, let\nus consider first the link between the nonrelativistic en-\nergy (per unit mass µ) and the orbital angular momen-\ntum along circular orbits. The relevance of this quantity\nin the nonspinning case, say Ecirc(ℓ)≡HNR\no(ℓ)/µ, was\npointed out in Ref. [38], since it provides a completely\ngauge-invariant characterization of the dynamics of cir-\ncular orbits. When the black holes are spinning, the\nsame property of gauge-invariance is maintained when\nthe spins are parallel (or antiparallel) to the orbital an-gular momentum, so that it is meaningful to explicitly\ncompute Ecirc(ℓ,ˆa,ˆa∗)≡HNR\no(ℓ)/µ+HNR\nso(ℓ,ˆa,ˆa∗)/µin\nthiscase. Since itis agauge-invariantquantity, theresult\nis independent of the canonical transformations that we\nhave performed on the two-body Hamiltonian in ADM\ncoordinates, so that it gives a reliable check of the pro-\ncedure we followed. As a first operation, we need to\nsolve, iteratively, the constraint (41) so to obtain the\nEOB coordinate radius rin function of ( ℓ,ˆa,ˆa∗). This\nfunction (that is not invariant and depends explicitly on\nthe gauge parameters) reads (putting back the explicit\ndouble primes on ras a reminder that this is the EOB\nradial coordinate)\nr′′(ℓ,ˆa,ˆa∗) =ℓ2/braceleftBigg\n1+1\nc2/bracketleftbigg\n−3\nℓ2+1\nc1\nℓ3/parenleftbigg\n6ˆa+9\n2ˆa∗/parenrightbigg/bracketrightbigg\n+1\nc4/bracketleftBigg\n(−9+3ν)1\nℓ4+1\nc1\nℓ5/parenleftBigg\nˆa/parenleftbigg\n33−17\n8ν+a(ν)/parenrightbigg\n+ˆa∗/parenleftbigg157\n8−5\n2ν+b(ν)/parenrightbigg/parenrightBigg/bracketrightBigg\n+1\nc6/bracketleftBigg/parenleftbigg\n−54+257\n3ν−41\n16π2ν/parenrightbigg1\nℓ6+1\nc1\nℓ7/parenleftBigg\nˆa/parenleftbigg1197\n4−1973\n16ν+3\n4ν2+6a(ν)+α(ν)+γ(ν)/parenrightbigg\n+ˆa∗/parenleftbigg2777\n16−1633\n16ν+7\n16ν2+6b(ν)+δ(ν)+η(ν)/parenrightbigg/parenrightBigg/bracketrightBigg/bracerightBigg\n. (42)\nThe function Ecirc(ℓ,ˆa,ˆa∗) is obtained by inserting this\nrelation in the expression of H(r,ℓ,ˆa,ˆa∗), and it reads\nEcirc(ℓ,ˆa,ˆa∗) =−1\n2ℓ2/braceleftBigg\n1+1\nc2/parenleftbigg1\n4(9+ν)1\nℓ2−1\nc1\nℓ3(4ˆa+3ˆa∗)/parenrightbigg\n+1\nc4/bracketleftBigg\n1\n8/parenleftbig\n81−7ν+ν2/parenrightbig1\nℓ4−1\nc1\nℓ5/parenleftbigg/parenleftbigg\n36+3\n4ν/parenrightbigg\nˆa+99\n4ˆa∗/parenrightbigg/bracketrightBigg\n+1\nc6/bracketleftBigg\n2\nℓ6o1(ν)+1\nc1\nℓ7/parenleftbigg\nˆa/parenleftbigg\n−324+54ν−5\n8ν2/parenrightbigg\n+ˆa∗/parenleftbigg\n−1701\n8+195\n4ν/parenrightbigg/parenrightbigg/bracketrightBigg/bracerightBigg\n(43)\nwhere we defined\n2o1(ν) =3861\n64−8833\n192ν+41\n32π2ν−5\n32ν2+5\n64ν3,(44)\nfor the 3PN-accurate orbital part, with a slight abuse of\nthe notationofRef. [38]. Note that, asit should, Eq. (43)\ncircular orbits of the EOB-resummed real Hamiltonian, as do ne\nin Sec. V of Ref. [2]. This analysis is postponed to future wor k.is totally independent of the eight gauge parameters. We\nhave further verified that that the same result (43) is ob-\ntained starting from the PN-expanded Hamiltonian writ-\nten in ADM coordinates and in the center of mass frame,\nEqs. (13)-(14).\nAs a last remark, let us note that, as it was the case\nat NLO [2], the effective gyro-gravitomagnetic ratios for\ncircular orbits are gauge independent also at NNLO. To\nsee this explicitly, one just imposes in Eqs. (29)-(30) the10\ncondition ( n·p) = 0 and the (approximate) link\np2=1\nr+1\nc23\nr2+O(ˆa,ˆa∗), (45)\nthat is obtained by inverting Eq. (42) at 1PN accuracy\nand neglecting the linear-in-spin terms (that would give\nquadratic-in-spin contributions). At NNLO, one obtains\ngeff\nScirc= 2−1\nc25\n8ν1\nr−1\nc4/parenleftbigg51\n4ν+1\n8ν2/parenrightbigg1\nr2,(46)\ngeff\nS∗\ncirc=3\n2−1\nc2/parenleftbigg9\n8+3\n4ν/parenrightbigg1\nr\n−1\nc4/parenleftbigg27\n16+39\n4ν+3\n16ν2/parenrightbigg1\nr2.(47)\nThese equations indicate that the inclusion of NNLO\nspin-orbit coupling has the effect of reducing the magni-\ntude of the gyro-gravitomagneticratios. The NNLO and\nNLO spin-orbit contributions act then in the same direc-\ntion, so to reduce the repulsive effect ofthe LO spin-orbit\ncoupling which is, by itself, responsible for allowing the\nbinary system to orbit on very close, and very bound, or-\nbits (see also Ref. [6] and the discussion in Sec. VI of [2]).\nWe postpone to future work a detailed quantitative anal-\nysis of the properties of the binding energy entailed by\nEqs. (46)-(47).\nC. Gauge fixing\nWe can finally exploit the flexibility introduced by\nthe spin-dependent gauge transformation so to consid-erably simplify the expression of the effective gyro-\ngravitomagnetic ratios, Eqs. (29)-(30). This is helpful in\nthe study of the dynamics of a binary system with gener-\nically oriented spins. Reference [2] found it convenient to\nfix the NLO gauge parameters ( a(ν),b(ν)) to\na(ν) =−3\n8ν, b(ν) =5\n8−ν\n2(48)\nso to suppress the dependence on p2at NLO. One can\nfollow the same route at NNLO, i.e., by choosing the six\ngauge parameters so to suppress the terms proportional\ntop2,p4andp2(n·p)2. Inthiswaythespin-orbitHamil-\ntonian is expressed in a way that the circular (gauge-\ninvariant) part is immediately recognizable. With ( a,b)\nfixed as per Eq. (48), one easily sees that the aforemen-\ntioned NNLO terms are removed by the following choices\nof the NNLO gauge parameters\nα(ν) =11\n8ν(3−ν), (49)\nβ(ν) =1\n16ν(13ν−2), (50)\nγ(ν) =7\n16ν, (51)\nδ(ν) =1\n16(9+54ν−23ν2), (52)\nη(ν) =1\n16/parenleftbig\n−2+7ν+ν2/parenrightbig\n, (53)\nζ(ν) =1\n16/parenleftbig\n−7−8ν+15ν2/parenrightbig\n. (54)\nThe effective gyro-gravitomagnetic ratios are then sim-\nplified to\ngeff\nS= 2+1\nc2/braceleftbigg\n−1\nr5\n8ν−27\n8ν(n·p)2/bracerightbigg\n+1\nc4/braceleftbigg\n−1\nr2/parenleftbigg51\n4ν+ν2\n8/parenrightbigg\n+1\nr/parenleftbigg\n−21\n2ν+23\n8ν2/parenrightbigg\n(n·p)2+5\n8ν(1+7ν)(n·p)4/bracerightbigg\n, (55)\ngeff\nS∗=3\n2+1\nc2/braceleftbigg\n−1\nr/parenleftbigg9\n8+3\n4ν/parenrightbigg\n−/parenleftbigg9\n4ν+15\n8/parenrightbigg\n(n·p)2/bracerightbigg\n+1\nc4/braceleftbigg\n−1\nr2/parenleftbigg27\n16+39\n4ν+3\n16ν2/parenrightbigg\n+1\nr/parenleftbigg69\n16−9\n4ν+57\n16ν2/parenrightbigg\n(n·p)2+/parenleftbigg35\n16+5\n2ν+45\n16ν2/parenrightbigg\n(n·p)4/bracerightbigg\n.(56)\nThis result extends the information of Eqs. (3.15a) and\n(3.15b) of Ref. [2] at NNLO accuracy. The circular-\norbit result mentioned above is immediately recovered\n“at sight” by imposing ( n·p) = 0. With this result in\nhand, one can proceed similarly to Sec. IV of Ref. [2] (as\noutlined above) to introduce the spin-dependent EOB-resummed real Hamiltonian including NNLO spin-orbit\ncouplings.11\nV. CONCLUSIONS\nBuilding on the recently-computed next-to-next-to-\nleading order PN-expanded spin-orbit Hamiltonian for\ntwo spinning compact objects [1], we computed the ef-\nfective gyro-gravitomagnetic ratios entering the EOB\nHamiltonian at next-to-next to-leading order in the spin-\norbit interaction. This result is obtained by a straightfor-\nward extension of the procedure followed in [2] to derive\nthe NLO spin-orbit EOB Hamiltonian. We discussed in\ndetail the test-particle limit and the case of equatorial\ncircular orbits, when the spins are parallel or antiparallel\nto the orbital angular momentum. In this case, one finds\nthat the NNLO spin-orbit terms moderate the effect ofthe spin-orbit coupling (as the NLO terms was already\ndoing [2]).\nFinally, while this paper was under review process,\nRef. [39] appeared in the archives as a preprint: that\nstudy uses the Lie method to obtain effective gyro-\ngravitomagnetic coefficients that are physically equiva-\nlent to the ones presented here. In addition, it also works\nout two classes of EOB Hamiltonians that are different\nfrom the one considered here.\nACKNOWLEDGMENTS\nI am grateful to Thibault Damour for suggesting this\nproject and for maieutic discussions during its develope-\nment.\n[1] J. Hartung and J. Steinhoff, arXiv:1104.3079 [gr-qc].\n[2] T. Damour, P. Jaranowski, G. Schaefer, Phys. Rev. D78,\n024009 (2008). [arXiv:0803.0915 [gr-qc]].\n[3] A. Buonanno, T. Damour, Phys. Rev. D59, 084006\n(1999). [gr-qc/9811091].\n[4] A. Buonanno, T. Damour, Phys. Rev. D62, 064015\n(2000). [gr-qc/0001013].\n[5] T. Damour, P. Jaranowski, G. Schaefer, Phys. Rev. D62,\n084011 (2000). [gr-qc/0005034].\n[6] T. Damour, Phys. Rev. D 64, 124013 (2001) [arXiv:gr-\nqc/0103018].\n[7] T. Damour, B. R. Iyer, A. Nagar, Phys. Rev. D79,\n064004 (2009). [arXiv:0811.2069 [gr-qc]].\n[8] T. Damour, A. Nagar, Phys. Rev. D81, 084016 (2010).\n[arXiv:0911.5041 [gr-qc]].\n[9] L. Baiotti, T. Damour, B. Giacomazzo, A. Nagar,\nL. Rezzolla, Phys. Rev. Lett. 105, 261101 (2010).\n[arXiv:1009.0521 [gr-qc]].\n[10] L. Baiotti, T. Damour, B. Giacomazzo, A. Nagar and\nL. Rezzolla, arXiv:1103.3874 [gr-qc].\n[11] A. Buonanno, G. B. Cook, F. Pretorius, Phys. Rev. D75,\n124018 (2007). [gr-qc/0610122].\n[12] A. Buonanno, Y. Pan, J. G. Baker, J. Centrella,\nB. J. Kelly, S. T. McWilliams and J. R. van Meter, Phys.\nRev. D76, 104049 (2007) [arXiv:0706.3732 [gr-qc]].\n[13] T. Damour, A. Nagar, Phys. Rev. D77, 024043 (2008).\n[arXiv:0711.2628 [gr-qc]].\n[14] T.Damour, A.Nagar, E. N.Dorband, D.Pollney, L.Rez-\nzolla, Phys. Rev. D77, 084017 (2008). [arXiv:0712.3003\n[gr-qc]].\n[15] M. Boyle, A. Buonanno, L. E. Kidder, A. H. Mroue,\nY. Pan, H. P. Pfeiffer, M. A. Scheel, Phys. Rev. D78,\n104020 (2008). [arXiv:0804.4184 [gr-qc]].\n[16] T. Damour, A. Nagar, M. Hannam, S. Husa and\nB. Bruegmann, Phys. Rev. D 78, 044039 (2008)\n[arXiv:0803.3162 [gr-qc]].\n[17] T. Damour and A. Nagar, Phys. Rev. D 79, 081503\n(2009) [arXiv:0902.0136 [gr-qc]].\n[18] A. Buonanno, Y. Pan, H. P. Pfeiffer, M. A. Scheel,\nL. T. Buchman, L. E. Kidder, Phys. Rev. D79, 124028\n(2009). [arXiv:0902.0790 [gr-qc]].[19] Y. Pan, A. Buonanno, M. Boyle, L. T. Buchman,\nL. E. Kidder, H. P. Pfeiffer and M. A. Scheel,\narXiv:1106.1021 [gr-qc].\n[20] A. L. Tiec, A. H. Mroue, L. Barack, A. Buo-\nnanno, H. P. Pfeiffer, N. Sago and A. Taracchini,\narXiv:1106.3278 [gr-qc].\n[21] A. Nagar, T. Damour and A. Tartaglia, Class. Quant.\nGrav.24, S109 (2007) [arXiv:gr-qc/0612096].\n[22] T. Damour and A. Nagar, Phys. Rev. D 76, 064028\n(2007) [arXiv:0705.2519 [gr-qc]].\n[23] S. Bernuzzi and A. Nagar, Phys. Rev. D 81, 084056\n(2010) [arXiv:1003.0597 [gr-qc]].\n[24] S. Bernuzzi, A. Nagar, A. Zenginoglu, Phys. Rev. D83,\n064010 (2011). [arXiv:1012.2456 [gr-qc]].\n[25] N. Yunes, A. Buonanno, S. A. Hughes, M. Coleman\nMiller, Y. Pan, Phys. Rev. Lett. 104, 091102 (2010).\n[arXiv:0909.4263 [gr-qc]].\n[26] N. Yunes, A. Buonanno, S. A. Hughes, Y. Pan, E. Ba-\nrausse, M. C. Miller and W. Throwe, Phys. Rev. D 83,\n044044 (2011) [arXiv:1009.6013 [gr-qc]].\n[27] A. Buonanno, Y. Chen and T. Damour, Phys. Rev. D\n74, 104005 (2006) [arXiv:gr-qc/0508067].\n[28] T. Damour, P. Jaranowski and G. Schaefer, Phys. Rev.\nD77, 064032 (2008) [arXiv:0711.1048 [gr-qc]].\n[29] G. Faye, L. Blanchet, A. Buonanno, Phys. Rev. D74,\n104033 (2006). [gr-qc/0605139].\n[30] L. Blanchet, A. Buonanno, G. Faye, Phys. Rev. D74,\n104034 (2006). [gr-qc/0605140].\n[31] M. Levi, Phys. Rev. D82, 104004 (2010).\n[arXiv:1006.4139 [gr-qc]].\n[32] Y. Pan, A. Buonanno, R. Fujita, E. Racine, H. Tagoshi,\nPhys. Rev. D83, 064003 (2011). [arXiv:1006.0431 [gr-\nqc]].\n[33] Y.Pan, A.Buonanno, L.T. Buchman, T. Chu, L.E. Kid-\nder, H. P. Pfeiffer and M. A. Scheel, Phys. Rev. D 81,\n084041 (2010) [arXiv:0912.3466 [gr-qc]].\n[34] E. Barausse, E. Racine and A. Buonanno, Phys. Rev. D\n80, 104025 (2009) [arXiv:0907.4745 [gr-qc]].\n[35] E. Barausse and A. Buonanno, Phys. Rev. D 81, 084024\n(2010) [arXiv:0912.3517 [gr-qc]].\n[36] T. Damour, P. Jaranowski and G. Schaefer, Phys. Rev.\nD62, 021501 (2000) [Erratum-ibid. D 63, 029903 (2001)]12\n[arXiv:gr-qc/0003051].\n[37] T. Damour, P. Jaranowski and G. Schaefer, Phys. Lett.\nB513, 147 (2001) [arXiv:gr-qc/0105038].\n[38] T. Damour, P. Jaranowski and G. Schaefer, Phys. Rev.\nD62, 044024 (2000) [arXiv:gr-qc/9912092].[39] E. Barausse, A. Buonanno, [arXiv:1107.2904 [gr-qc]]."    },    {        "title": "2311.14719v1.Thermal_Spin_Orbit_Torque_with_Dresselhaus_Spin_Orbit_Coupling.pdf",        "content": "ThermalSpin-OrbitTorquewithDresselhaus\nSpin-OrbitCoupling\nChun-YiXue,Ya-RuWang,Zheng-ChuanWang*\nSchoolofPhysicalSciences，\nUniversityofChineseAcademyofSciences,Beijing100049,China.\n*wangzc@ucas.ac.cn\nAbstract\nBasedonthespinorBoltzmannequation,weobtainatemperature\ndependentthermalspin-orbittorqueintermsofthelocalequilibrium\ndistributionfunctioninatwo-dimensionalferromagnetwith\nDresselhausspin-orbitcoupling.Wealsoderivethecontinuity\nequationofspinaccumulationandspincurrent—thespindiffusion\nequationinDresselhausferromagnet,whichcontainsthethermal\nspin-orbittorqueunderlocalequilibriumassumption.This\ntemperaturedependentthermalspin-orbittorqueoriginatesfromthe\ntemperaturegradientappliedtothesystem,itisalsosensitiveto\ntemperatureduetothelocalequilibriumdistributionfunctiontherein.\nInthespindiffusionequation,wecansingleouttheusualspin-orbit\ntorqueaswellasthespintransfertorque,whichisconcededtoour\npreviousresults.Finally,weillustratethembyanexampleof\nspin-polarizedtransportthroughaferromagnetwithDresselhaus\nspin-orbitcouplingdrivenbytemperaturegradient,thosetorquesincludingthermalspin-orbittorquearedemonstratednumerically.\nPACS:75.60.Jk,72.15Jf,75.70.Tj,85.70.-w\nI.Introduction\nAsanewbranchofspintronics,spin-orbitronicshasbeen\nexploredwhateverintheoriesorexperiments[1-2],becauseitcan\nprovideanefficientwaytomanipulatethelocalmagneticmomentin\nthedevicesofspintronicsviaspin-orbittorques[3-5].Sincethe\ndiscoveryofgiantmagnetoresistance[6]andspintransfertorque[7],\nnowadaysmagnetoresistancerandomaccessmemory(MRAM)\ndrivenbyspin-polarizedcurrenthasbeendesignedandrealized\nindustrially[8-10].ThefirstgenerationMRAMistoggleMRAM\n[11],whichconsistofatransistorandamagnetictunneljunction\n(MTJ),butthisstructurebringsobviousdisadvantagestothetoggle\nMRAMduetothebigmagneticfield[12].Thesecondgeneration\nMRAMarespintransfertorqueMRAM(STT-MRAM)[13]and\nperpendicularspintransfertorqueMRAM(pSTT-MRAM)[14],in\nwhichmagnetizationreversalinSTT-MRAMandpSTT-MRAMrely\nonthespin-polarizedelectricalcurrentratherthanbigmagneticfield,\nsotheyhavefasterwritingspeed.However,STT-MRAMdepends\nonthermalactivationtostartswitching[15],soithasaninitial\nlatencywhichrestrictsitsmaximumcachespeed.Thus,oneproposedthethirdgenerationMRAMtosolvethisproblem,which\nisdrivenbyspinorbittorque(SOT).SOTcanbeemployedtoswitch\nthemagnetizationinasystemwithabrokeninversionsymmetry.\nSinceSOTneedalowercriticalcurrent,ithasbetterthermal\nstability[16].Recently,ThermalSOThasbeenobservedin\nexperiments[17-18].Tillnow,SOT-MRAMisagoodcandidateof\nmagneticstoragedeviceforbetterperformance.\nIn2007,HatamiproposedSTTwhichcanbedrivenbythermal\nspincurrent[19].Similarly,SOTcanalsobedrivenbythermalspin\ncurrentintheprocessofspin-polarizedelectrontransport,whichis\ncalledthermalSOT(TSOT).TSOTwasfirstlyproposedby\nFreimuthin2014intermsofBerryphase[20-21]whichisexpressed\nbythequantumstatesofelectrons,whilethestatesusuallyshouldbe\ncalculatedbytheDensityFunctionTheory(DFT),itissomewhat\ncumbersome.Thus,inthismanuscriptwewillgiveanother\nexpressionofTSOTbyuseofdistributionfunctioninthespinor\nBoltzmannequation(SBE).\nTheSBEwasfirstlyproposedbyShengetalin1996atsteady\nstate[22],theyderivedthisequationfromKadanoffnonequilibrium\nGreenfunction(NEGF)formalismbasedonthegradient\napproximation[23].ThenSBEwasaccomplishedbyLevyetalin\n2004,whichcouldillustratethetimedependentprocessofspintransporteffectively[24].SBEwasalsoextendedtothecasebeyond\ngradientapproximationin2013[25].In2019,Wangetal.\nsuccessfullyincludedtheRashbaandDresslhausspin-orbitcoupling\nintotheSBE[26],whichishelpfulforustoinvestigatetheSOT\nfromthespindiffusionequation–thecontinuityequationforthe\nspinaccumulationandspincurrent.Inthismanuscript,ourpurpose\nistoinvestigateTSOTina2-dimensionalferromagnetbymeansof\nSBE.WewillfindthatanunusualSOTdrivenbythetemperature\ngradientinthesystemwithDresselhausspinorbitcoupling,itisjust\ntheTSOTwewant.\nII.TheoreticalFormalism\nConsiderthespin-polarizedelectrontransportinatwo-\ndimensionalferromagnetwithDresselhausSOC.It’sHamiltonian\ncanbewrittenas\u0000=−ℏ2\n2\u0000\u00002\u00000+\u0000\u0000\u0000⋅\u0000\u0000\u0000+\u0000,whereis\nelectroniceffectivemass,Jisthes-dexchangecouplingconstant,\n\u0000\u0000\u0000istheunitvectorforthelocalmagneticmomentofferromagnet,\n\u00000istheunitmatrix,\u0000\u0000\u0000isthePaulimatrixvector,\u0000describes\ntheDresselhausSOCintwodimensionalferromagnet.In1955,\nDresselhausgavetheHamiltonianofDresselhausSOCinthree\ndimensionalsystemwithabulkinversionasymmetry(BIA)[29].\nWhensuchathreedimensionalsystemissubjectedtostrainatthe\ninterfaceorathinlayer,theHamiltonianwillreducetothefollowingsimplerform:\u0000=(\u0000−\u0000)[26],whereisthe\ncouplingconstantofDresselhausSOC,andarethexandy\ncomponentsofgradientoperator,andthe\u0000and\u0000arethexand\nycomponentsofthePaulioperator\u0000\u0000\u0000.ByKadanoffnonequilibrium\nGreenfunction(NEGF)formalism,wecanobtaintheSBEforthe\nspinordistributionoftransportelectron[24]:\n\n+\u0000\u0000\u0000⋅\n\u0000\u0000−\u0000\u0000⋅\n\u0000\u0000\u0000+\nℏ\u0000,\u0000=−\u0000\nd#1\nwhere\u0000isthespinordistributionfunctionoftransportelectron,\nwhichcanbeexpandedtoa2×2matrix\u0000=↑↑↑↓\n↓↑↓↓,\u0000isthe\nspinorenergy,it’sdefinedas\u0000()=()\u00000+1\n2()\u0000\u0000\u0000(,).\nIntheSBEframework,wecandecomposethespinordistribution\nfunctionintotwopartsbyusingthecompletebasisofmatrices\u00000\nandthe\u0000\u0000\u0000,\u0000(\u0000\u0000,\u0000\u0000)=(\u0000\u0000,\u0000\u0000)\u00000+\u0000\u0000(\u0000\u0000,\u0000\u0000)\u0000\u0000\u0000,where\u0000\u0000,\u0000\u0000is\nscalardistributionfunctionand\u0000\u0000\u0000\u0000,\u0000\u0000isvectordistribution\nfunction.Underthelocalequilibriumassumption,thespinor\ndistributionfunctioncanalsobewrittenasfollow:\n\u0000\u0000\u0000,\u0000\u0000,=\u00000\u0000\u0000,\u0000\u0000+1\u0000\u0000,\u0000\u0000\u00000+1\u0000\u0000,\u0000\u0000⋅\u0000\u0000\u00002\nwhere\u00000(\u0000\u0000,\u0000\u0000)isthelocalequilibriumdistributionfunction.\n1(\u0000\u0000,\u0000\u0000)and1(\u0000\u0000,\u0000\u0000)arethenonequilibriumpartsofspinor\ndistribution.Intheferromagnet,wetakethelocalequilibrium\ndistributionfunctionasadiagonalmatrix\u00000(\u0000\u0000,\u0000\u0000)=↑↑0\n0↓↓,herethediagonalcomponentsaretakenasFermidistribution\nfunctions↑↑(↓↓)=()±1\n2−()\n()+1−1,whereis\nBoltzmannconstant,()isthechemicalpotentialand≈\n(FermiEnergy)atfinitetemperature.Wecanexpandthelocal\nequilibriumdistributionbasedonthecompletebasis\n\u00000,\u0000,\u0000,\u0000,whichis\u00000(\u0000\u0000,\u0000\u0000)=1\n2(↑↑+↓↓)\u00000+1\n2(↑↑−\n↓↓)\u0000,sothescalardistributionandvectordistributioncanbe\nrewrittenas=1\n2(↑↑+↓↓)+1and=1,1,1+1\n2(↑↑−\n↓↓).\nInspintronics,theSBEwithDresselhausspin-orbitcouplingina\ntwo-dimensionalmagneto-electricsystemunderanexternalelectric\nfield\u0000\u0000hadbeengivenbyChaoYangetal.in2019[26]\n\n+\u0000\u0000\n⋅\u0000\u0000−\u0000\u0000⋅\u0000\u0000\u0000\u0000,\u0000\u0000+\nℏ\n−\n=−\n#3\nand\n\n+\u0000\u0000\n⋅\u0000\u0000−\u0000\u0000⋅\u0000\u0000\u0000\u0000\u0000\u0000,\u0000\u0000−\nℏ\u0000\u0000\u0000×\u0000\u0000\u0000\u0000,\u0000\u0000+\nℏ\u0000\u0000\n−\u0000\u0000\n\n+\nℏ2\u0000\u0000−\u0000\u0000×\u0000\u0000\u0000\u0000,\u0000\u0000=−\u0000\u0000\n#4\nwhere−(\n)and−(\u0000\u0000\n)representthecollisionterms.\nUndertherelaxationtimeapproximationassumption,wecan\nderivetheequationsforthescalardistributionfunctionandthe\nvectordistributionfunctionbasedonEqs.(3)and(4),whichcontainthelocalequilibriumdistributionfunctionas:\n\n+\u0000\u0000\n⋅\u0000\u0000−\u0000\u0000⋅\u0000\u00001\n2↑↑+↓↓+\n+\u0000\u0000\n⋅\u0000\u0000−\u0000\u0000⋅\u0000\u00001\n+\nℏ1\n−1\n=−−\n#5\n\n+\u0000\u0000\n⋅\u0000\u0000−\u0000\u0000⋅\u0000\u0000\u0000\u0000−\u0000\u0000\u0000×\u0000\u0000+\n\nℏ\u0000\u00001\n2↑↑+↓↓\n−\u0000\u00001\n2↑↑+↓↓\n+\n\nℏ\u0000\u00001\n−\u0000\u00001\n\nℏ−\nℏ2\u0000\u0000−\u0000\u0000×\u0000\u0000=−\u0000\u0000−\u0000\u0000\n#6\nwhereandaretherelaxationtimesofelectronandspinflip,\nrespectively.Eq.(5)andEq.(6)arecoupledtogether,weshould\nsolvethemsimultaneously.Thephysicalobservablesinthe\nspin-polarizedtransportcanbeexpressedbythesolutionsofscalar\nandvectordistributionfunctions.Thechargedensityandcharge\ncurrentaredefinedasfollow,\n\u0000\u0000,=1\u0000 \u0000\u0000,\u0000\u0000,\u0000\u0000#7\nand\n\u0000\u0000\u0000,=\u0000\u00001\u0000\u0000,\u0000\u0000, \u0000 \u0000\u0000#8\nwhicharethemomentumintegralsoverthescalardistributions.\nSimilarly,thespinaccumulationandspincurrentdensityaredefine\nas\u0000\u0000\u0000\u0000\u0000,=\u0000\u0000\u0000\u0000\u0000,\u0000\u0000,\u0000\u0000#9\nand\n\u0000\u0000\n\u0000\u0000,=\u0000\u0000\u0000\u0000\u0000 \u0000\u0000,\u0000\u0000,\u0000\u0000#10\nwhicharethemomentumintegralsoverthevectordistributions.It\nshouldbenotedthatthespincurrent\u0000\u0000\n(\u0000\u0000,)isatensor.Moreover,\nwecanalsodefinethethermalcurrentdensityas\u0000\n\u0000\u0000\u0000,=\u0000\u00001\u0000 \u0000\u0000,\u0000\u0000,\u0000\u0000#11\nwhereisthescalarenergyofelectron.Sowhenwegetthe\nsolutionsofscalarandvectordistributionfunctions,wecanobtain\ntheseabovephysicalobservablesaccordingly.\nOntheotherhand,wecanalsoobtainthecontinuityequations\nsatisfiedbythesephysicalobservables.Ifweintegratethe\nmomentumovertheFermisurfaceonthebothsidesofEq.(5)and\nEq.(6),wehave\n\n+\u0000\u0000⋅\u0000=−\n+\u0000\u0000\n⋅\u0000\u0000−\u0000\u0000⋅\u0000\u00001\n2↑↑+↓↓ \u0000 \u0000\u0000\n−\nℏ\n−\n−−\n#12\nand\u0000\u0000\u0000\n+\u0000\u0000⋅\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000=\nℏ\u0000\u0000\u0000×\u0000\u0000\u0000\n−\n\nℏ\u0000\u00001\n2↑↑+↓↓\n−\u0000\u00001\n2↑↑+↓↓\n\u0000\u0000−\nℏ\u0000 \u0000\u0000\n−\u0000\u0000\n\n+\nℏ2\u0000\u0000−\u0000\u0000×\u0000\u0000 \u0000 \u0000\u0000=−\u0000\u0000\u0000−\u0000\u0000\u0000\n#13\nEq.(12)isjustthecontinuityequationforchargedensityandcharge\ncurrent,whileEq.(13)isthecontinuityequationforthespin\naccumulationandspincurrent,thelatterisalsocalledspindiffusion\nequation.Whenthetime≫,theequationwillarriveatasteady\nstate,thenthespindiffusionequationwillreduceto\n\nℏ\u0000\u0000\u0000×\u0000\u0000\u0000=\u0000\u0000⋅\u0000\u0000\n+\nℏ\u0000\u00001\n2↑↑+↓↓\n−\u0000\u00001\n2↑↑+↓↓\n\u0000\u0000 \u0000\n+\nℏ\u0000\u0000\n−\u0000\u0000\n−\nℏ2\u0000\u0000−\u0000\u0000×\u0000\u0000 \u0000 \u0000\u0000#14\nFromtheabovesteadystateequation,wecanreadoutallthetorques\nexistinginthisspin-polarizedtransportprocess.Ontheleftsideof\nthisequation,theterm\nℏ\u0000\u0000\u0000×\u0000\u0000\u0000isjustthespintransfertorque\ngivenbyLevyetal.[27].Ontherighthandsideside,\u0000\u0000⋅\u0000\u0000\nisthe\ndivergenceofspincurrent,whichalsocanmakeacontributiontothe\nusualSTTasshownbyZhangetal[28].Besides,theterm\n\nℏ(\u0000\u0000\n−\u0000\u0000\n)−\nℏ2(\u0000\u0000−\u0000\u0000)×\u0000\u0000 \u0000 \u0000\u0000correspondstothe\nusualspin-orbittorquepresentedbyWangetal.[26],whilethe\ntemperaturedependentterm\nℏ\u0000\u00001\n2↑↑+↓↓\n− \u0000\u0000\u00001\n2↑↑+↓↓\n\u0000\u0000isanewterm,itisinducedbythegradientof\nlocalequilibriumdistributionfunction,werefertothisasthe\nthermalSOT.Whenthegradientoftemperatureisappliedonly\nalongx-direction,itcanbeexpressedas:\n=−\nℏ.\n\u0000\u00001\n2(−+1\n2\n)\n1+(−−1\n2\n)2⋅−+1\n2\n+ \u0000\n(−−1\n2\n)\n1+(−−1\n2\n)2⋅−−1\n2\n1\n2\u0000\u0000\n(15)\nwecanseethatitisproportionaltothegradientoftemperature,\nwhichisconcededtothedefinitionofTSOTgivenbyFreimuthetal\n[20-21],sothistermisjusttheTSOTwesearchfor,itisthecentral\nresultinthismanuscript.Inthenext,wewillevaluatethesetorques\nnumericallyinaferromagnetwithDresselhausSOC.\nIII.NumericalResults\nWeconsideratwo-dimensionalferromagnetwithDresselhaus\nSOC,wherethesystemischosenasarectangularferromagnetwith\nageometryof25×25nm².Thetemperaturedistributionissimply\nchosenas()=0+,whichislinearlydependentonthe\npositionofxcomponent,where0isaconstant,kisthetemperaturegradient.FromEq.(15),wecanseethatthetemperature\ngradientwillinducethermalspin-orbittorque.\nInordertoquantifythesetorquesandcurrents,weneedtosolve\nEq.(3)combiningwith(4)simultaneously,becausethescalar\ndistributionfunctionandvectordistributionfunctionarecoupled\ntogetherintheseequations.Tosimplifycalculation,wechosethe\nunitvectorofmagnetizationasafixedvector\u0000\u0000\u0000=(0,0,1),the\nequilibriumscalardistributionfunctionischosenas=1\n2↑↑+\n↓↓,andtheequilibriumvectordistributionfunctionisadoptedas\n\u0000\u0000=((\n+\n),(\n+\n),(\n+\n\n))[22].Thedifferentialequations(3)and(4)aresolvedby\ndifferencemethod.Thephysicalconstantandparametersarelisted\ninTableⅠ,whereweadoptthematerialsparametersofferromagnet.\nTableⅠ.Thephysicalconstantsandparameters\nPhysicalconstants/parametersSymbolValueUnit\nMomentumrelaxationtime  10−13s\nSpin-fliprelaxationtime  10−12s\nFermienergy  4 eV\nFermiwavevector 1.02×1010−1\ns-dexchangecouplingstrengthJ 0.1 eV\nElectricalfield E −5×104.−1Temperaturegradient  5×109.−1\nInFig.1,weplotthechargecurrentdensityasafunctionof\npositionxandy.Thevariationofchargecurrentwithrespectto\npositionandtimeisgovernedbythecontinuityequationofcharge\ndensityandchargecurrentdensity(12).Forsimplicity,weonly\nstudythechargecurrentatsteadystate.Wecanseethatthecharge\ncurrentdensitydecreasesgraduallyalongboththexandydirections,\nwhichisduetotheresistanceintheferromagnet,inourcalculationit\nisconcernedwiththemomentumrelaxationtimeofelectrons.\nSincetheexternalelectricfieldisappliedonlyalongthex-direction,\nthevariationofchargecurrentalongy-directionismainlycausedby\ntheDresselhausSOC.\nFig.1ThechargecurrentdensityFig.2Thethermalcurrentdensity\nvsposition vsposition\nWealsoshowthecurveofthermalcurrentdensityasafunction\nofpositioninFig.2,whichissimilartothechargecurrentdensity\nbecauseoftheirdefinitions,italsodecreaseswithpositiongradually.\nHereweonlyconsiderthethermalcurrentdensitycarriedbythetransportelectrons.Besidestheexternalelectricfield,thethermal\ncurrentdensitycanalsobedrivenbythetemperaturegradient.Since\ntheelectricfieldandgradientoftemperatureareallalongthex-axis,\nthevariationofthermalcurrentdensityalongy-directionisprimarily\ninducedbytheDresselhausSOC.\nBecausewechoosethemagnetizationofferromagnet\u0000\u0000\u0000=\n(0,0,1),sothezcomponentofSTTis0.InFig.3,weshowthexand\nycomponentsofSTTdensityasafunctionofposition.Theusual\nSTTisthespaceintegralofthisdensityovertherectangular\nferromagnet.ItisshownthatthemagnitudeofSTTdensityis\ndifferentatdifferentpositionforboththexandycomponents.The\nmagnetizationofferromagnetat=10iseasiesttobe\nswitchedbythebiggerSTT,andishardesttofliparoundtheline\n=becauseofthesmallerSTT.Theswitchingofmagnetization\nwillproducethespinwavewithinthetwo-dimensionalferromagnet.\nFig.3(a)Thex-componentofSTTdensity.(b)They-componentofSTTdensity\nThespincurrentdensityvspositionisshowninFig.4.Sincethespincurrentisatensor,weonlydrawthexx-,xy-andxz-\ncomponentsofspincurrentdensityasafunctionofposition,they\nvaryobviouslyaroundtheline=.Thevariationofspincurrent\nwithpositionandtimesatisfiesthecontinuityequation(13)forthe\nspinaccumulationandspincurrent.AccordingtoEq.(14),the\ndivergenceofspincurrentwillmakeacontributiontotheZhang-like\nSTT[28].\nFig.4(a)Thexx-componentofspincurrent(b)Thexy-componentofspin\ncurrent(c)Thexz-componentofspincurrent\nInFig.5,wedrawtheSOTasafunctionofposition.Theusual\nSOTisexpressedas\nℏ(\u0000\u0000\n−\u0000\u0000\n)−\nℏ2(\u0000\u0000−\u0000\u0000)×\u0000\u0000 \u0000 \u0000\u0000,\nsoitisnotsensitivetothetemperature.Itdecreasesalongx-\ndirectionobviously,whilethereissmallvariationalongy-direction,\nbecausetheexternalelectricfieldisappliedalongx-direction.For\ncomparison,wealsoplottheTSOTatdifferenttemperature300K,\n200Kand100KinFig.6,respectively,whichdependonthe\ntemperatureandit’sgradientobviously.Thehigheroftemperature,\nthebiggerofTSOT,becausetherearemorepolarizedelectronFig.5TheSOTvsposition.Fig.6TheTSOTvspositionatdifferent\ntemperatureT=300K,200K,100K.\nparticipatingintransportathighertemperature.ComparedFig.6\nwithFig.5,wecanfindthattheTSOTissmallerthantheusualSOT,\nwhiletheTSOTcanbecomebiggerwhenweincreasethe\ntemperature,sotheTSOTcannotbenegligibleathigher\ntemperature.Certainly,TSOTisalsoproportionaltothetemperature\ngradient,itwillplayanimportantroleathighertemperaturegradient.\nItshouldbepointedoutthattheTSOTiscalculatedbyEq.(15),we\nonlyneedtheexpressionoflocalequilibriumdistributionfunction,it\nisverysimplethanFreimuth’sexpressionofBerryphase[20-21],\nbecausethelatterneedtheelectronicwavefunctionobtainedusually\nbyDFT.BymeansofEq.(15),wecancalculatetheTSOTeasier,\nthisistheadvantageofSBEmethod.\nIV.SummaryandDiscussions\nInthispaper,wehavederivedtheTSOTinatwo-dimensional\nferromagnetwithDresselhausSOCbySBEunderthelocal\nequilibriumassumption.TheusualSOTisinducedbytheexternal\nelectricfieldappliedtothesystem,whiletheTSOTisdrivenbythegradientoftemperature.WealsofindthatTSOTisverysensitiveto\nthetemperature,thehighertemperature,thebiggerTSOT.Our\nresultsshowthattheTSOTissmallerthanSOT,butitcannotbe\nnegligibleathighertemperature.Certainly,TSOTisalso\nproportionaltothetemperaturegradient,accordingtoitsexpression\nEq.(15).BecausethedirectexperimenttoobserveTSOTin\ntwo-dimensionalferromagnetswithDresselhausspin-orbitcoupling\nhaven’tbeencarriedoutnow,weonlypredicttheoreticallythatone\ncanobservetheeffectsofTSOTinthecaseofbiggradientof\ntemperatureandhighertemperature.\nTosimplifyourcalculation,weonlychooseasimpleuniform\nmagnetizationina2-dimensionalferromagnet,whileinrealitythe\nmagnetizationusuallyvarieswithtimeandposition.Thevariationof\nmagnetizationwouldhaveinfluenceonthetransportpropertiesof\nthespin-polarizedelectrons.Ifweconsiderthevariationof\nmagnetization,thecalculationwillbecomemuchmorecomplicated,\nitisleftforfutureexploration.\nAcknowledgments\nThisstudyissupportedbytheNationalKeyR&DProgramof\nChina(GrantNo.2022YFA1402703),theStrategicPriority\nResearchProgramoftheChineseAcademyofSciences(GrantNo.\nXDB28000000).WealsothankProf.GangSu,Zhen-Gang.Zhu,Bo.GuandQing-Bo.Yanfortheirhelpfuldiscussions.\nAUTHORCONTRIBUTIONSTATEMENT\nInthiswork,Zheng-ChuanWangproposedtheidea,Chun-Yi\nXueperformedthecalculation,analyzedthenumericalresults,and\nwrotethemanuscript.Ya-RuWangassistedwiththecalculationand\nanalysis.\nDataAvailabilityStatement\nDatasetsgeneratedduringthecurrentstudyareavailablefromthe\ncorrespondingauthoronreasonablerequest.\nReferences\n[1]A.ManchonandS.Zhang,Phys.Rev.B78,212405(2008).\n[2]T.KuschelandG.Reiss,Nat.Nanotechnol.10,22(2015).\n[3]A.Matos-AbiagueandR.L.Rodrıguez-Suarez,Phys.Rev.B80,\n094424(2009).\n[4]K.M.D.HalsandA.Brataas,Phys.Rev.B88,085423(2013).\n[5]A.Brataas,A.D.KentandH.Ohno,Nat.Mater.11:372-81\n(2012).\n[6]M.N.Baibichetal.,Phys.Rev.Lett.61,2472(1988).\n[7]J.C.Slonczewski,J.Magn.Magn.Mater.159,L1(1996).\n[8]W.J.GallagherandS.S.P.Parkin,IBMJ.Res.Dev.50,5\n(2006).\n[9]X.Hanetal.,Appl.Phys.Lett.118,120502(2021).[10]S.Aggarwaletal.,2019IEEEInt.ElectronDevicesMeet.\n(IEDM)pp.2.1.1–2.1.4(2019).\n[11]L.Savtchenkoetal.,USPatent6546906B1(2003).\n[12]M.Motoyoshietal.,Dig.Tech.Pap.-Symp.VlSITechnol.p.22\n(2004).\n[13]C.Chappert,A.FertandF.N.Van-Dau,Nat.Mater.6:813–23\n(2007).\n[14]T.Kishietal.,Tech.Dig.-Int.ElectronDevicesMeet.309\n(2008).\n[15]R.Patel,E.Ipek,E.G.Friedman,MicroelectronicsJournal,\n45(2),133-143(2014).\n[16]C.Bi,N.SatoandS.X.Wang,AdvancesinNon-Volatile\nMemoryandStorageTechnology,203-235(2019).\n[17]J.M.Kimetal.,NanoLett.20(11):7803-7810(2020).\n[18]D.J.Kimetal.,Nat.Commun.8,1400(2017).\n[19]M.HatamiandG.E.W.Bauer,Phys.Rev.Lett.99,066603\n(2007).\n[20]F.Freimuth,S.BlugelandY.Mokrousov,J.Phys:Condens.\nMatt.26,104202(2014).\n[21]G.Geranton,F.Freimuth,S.BlugelandY.Mokrousov,Phys.\nRev.B91,014417(2015).\n[22]L.Sheng,D.Y.Xing,Z.D.Wang,andJ.Dong,Phys.Rev.B55,5908(1997).\n[23]L.P.Kadanoff,G.Baym,andJ.D.Trimmer,Quantum\nStatisticalMechanics(W.A.Benjamin,1962).\n[24]J.Zhang,P.M.Levy,S.ZhangandV.Antropov,Phys.Rev.Lett.\n93,256602(2004).\n[25]Z.C.Wang,TheEuro.Phys.Jour.B86,206(2013).\n[26]C.Yang,Z.C.Wang,Q.R.ZhengandG.Su,TheEuro.Phys.\nJour.B92,136(2019).\n[27]A.S.Shpiro,P.M.Levy,S.Zhang,Phys.Rev.B67,104430\n(2003).\n[28]S.ZhangandZ.Li,Phys.Rev.Lett.93,27204(2004).\n[29]G.Dresselhaus,A.F.Kip,andC.Kittel,Phys.Rev.\n100,618(1955)."    },    {        "title": "1203.4079v2.Spin_orbit_couplings_between_distant_electrons_trapped_individually_on_liquid_helium.pdf",        "content": "arXiv:1203.4079v2  [quant-ph]  13 Nov 2012Spin-orbit couplings between distant electrons trapped in dividually on\nliquid Helium\nM. Zhang1and L. F. Wei∗1,2\n1Quantum Optoelectronics Laboratory, School of Physics,\nSouthwest Jiaotong University, Chengdu 610031, China\n2State Key Laboratory of Optoelectronic Materials and Techn ologies,\nSchool of Physics and Engineering, Sun Yat-sen University, Guangzhou 510275, China\n(Dated: November 10, 2018)\nAbstract\nWe propose an approach to entangle spins of electrons floatin g on the liquid Helium by coherently ma-\nnipulating their spin-orbit interactions. Theconfigurati on consists ofsingleelectrons, confinedindividually\non liquid Helium by the micro-electrodes, moving along the s urface as the harmonic oscillators. It has\nbeen known that the spin of an electron could be coupled to its orbit (i.e., the vibrational motion) by prop-\nerly applying a magnetic field. Based on this single electron spin-orbit coupling, here we show that a\nJaynes-Cummings (JC) type interaction between the spin of a n electron and the orbit of another electron at\na distance could be realized via the strong Coulomb interact ion between the electrons. Consequently, the\nproposed JC interaction could be utilized to realize a stron g orbit-mediated spin-spin coupling and imple-\nment the desirable quantum information processing between the distant electrons trapped individually on\nliquid Helium.\nPACSnumbers: 73.20.-r, 03.67.Lx, 33.35.+r\n∗weilianfu@gmail.com\n1I. INTRODUCTION\nThe interactions between the microscopic particles, e.g., the ions in Paul trap [1], the neutral\natoms confined in optical lattice [2], and the electrons in Pe nning trap [3], etc., relate usually to\ntheir masses and the inter-particle forces. Due to the small mass and the strong Coulomb interac-\ntion,theinteractingelectronscouldbeused toimplementq uantuminformationprocessing(QIP).\nTheideaofquantumcomputingwithstrongly-interactingel ectronsonliquidHeliumwasfirstpro-\nposedbyPlatzmanandDykmanin1999[4]. Intheirproposal,t hetwolowerhydrogen-likelevels\nof the surface-state electron are encoded as a qubit, and the effectively interbit couplings can be\nrealizedbytheelectricdipole-dipoleinteraction. Whent heliquidheliumiscooledontheorderof\nmK temperature the qubit possesses long coherent time (e.g. , up to the order of ms) [5, 6]. Inter-\nestingly,Lyonsuggested[7]thatthequbitscouldalsobeen coded by thespinsoftheelectrons on\nliquidHelium,andestimatedthatthequbitcoherenttimeco uldreach 100s[7]. Heshowedfurther\nthat the magneticdipole-dipoleinteractions between the s pinscould be used to couplethe qubits,\nif the electrons are confined closed enough. For example, the coupling strength can reach to the\norder of kHz for the distance d= 0.1µm between the electrons [7]. Remarkably, recent experi-\nments[8–10]demonstratedthemanipulationsofelectrons( confining,transporting,anddetecting)\nonliquidHeliuminthesingle-electronregime. Thisprovid esreallytheexperimentalplatformsto\nrealizetherelevantQIPwithelectrons onliquidHelium[11 –14].\nHere, we propose an alternative approach to implement QIP wi th electronic spins on liquid\nHeliumbycoherentlymanipulatingthespin-orbitinteract ionsoftheelectrons. Inourproposal,the\nvirtuesoflong-livedspinstates(toencodethequbit)ands trongCoulombinteraction(forrealizing\ntheexpectably-fastinterbitoperations)arebothutilize d. Theelectronsaretrappedindividuallyon\nthe surface of liquid Helium by the micro-electrodes. In the plane of liquid Helium surface each\nelectron moves as a harmonic oscillator. It has been showed t hat such an external orbit-vibration\ncould be effectively coupled to the internal spin of a single electron by applying a magnetic field\nwith a gradient along the vibrational axis [13]. Interestin gly, we show that the spin of an electron\ncould be coupled to the vibrational motion of another distan t electron [as a Jaynes-Cummings\n(JC) type interaction], by designing a proper virtual excit ation of the electronic vibration. The\npresent JC interaction could be utilized to significantly en hance the spin-spin coupling between\nthe distant electrons, and implement the desirable quantum computation with the spin qubits on\nliquidHelium.\n2Qe\nHLiquid Helium xz\ny\n+I\nBzBsPotential h\nFIG. 1: (Color online) Sketch for a single electron trapped o n the surface of liquid Helium. The liquid\nHelium provides z-directional confinement, and the micro-electrode Q (below the Helium surface at depth\nH) traps the electron in x-yplane. The desirable spin qubit is generated by an applied un iform magnetic\nfieldBs, and the spin-orbit coupling of the trapped electron is obta ined by applying a current to another\nmicro-electrode I (upon the liquid Helium surface at the hei ghth).\nThepaperisorganizedasfollows: InSec. IIwediscusstheme chanismforspin-orbitcoupling\nwithasingleelectrontrappedonliquidHelium[13],andthe nshowhowtoutilizesuchacoupling\nto realize the desirable quantum gate with the single electr on. By using the electron-electron\nCoulomb interaction, in Sec. III, we propose an approach to i mplement the JC coupling between\nthe spin of an electron and the orbital motion of another elec tron. Based on such a distant spin-\norbit interaction, we show that a two-qubit controlled-NOT (CNOT) gate and an orbit-enhanced\ncoupling between the distant spins could be implemented. Fi nally, we give a conclusion in Sec.\nIV.\nII. SPIN-ORBITCOUPLINGWITHA SINGLETRAPPEDELECTRON\nWe consider first a single electron trap shown in Fig. 1 [13], w herein an electron (with mass\nmeand charge e) on liquid Helium is weakly attracted by its dielectric imag e potential V(z) =\n−Λe2/z(withΛ = (ε−1)/4(ε+ 1)andεbeing the dielectric constant of liquid Helium). Due\ntothePauliexclusionprinciple,thereisanbarrier(about 1eV)topreventtheelectronpenetrating\ninto the liquid Helium. As a consequence, z-directional confinement of the electron is realized,\nyielding an one-dimensional (1D) hydrogenlike atom with th e spectrum En=−/planckover2pi1R/n2[15].\nHere,R= Λ2e4me/(2/planckover2pi12)≈170GHz and rb=/planckover2pi12/(mee2Λ)≈7.6nm are theeffectiveRydberg\n3energy and Bohr radius, respectively. In x-yplane, the electron can be confined by the micro-\nelectrode Q located at Hbeneath the liquid Helium surface. Typically, x,y,z≪H, and thus the\npotentialoftheelectron can bedescribed by[5]\nU(x,y,z)≈ −Λe2\nz+E⊥z+me\n2(ν2\nxx2+ν2\nyy2) (1)\nwithE⊥=eQ/H2,νx=νy=/radicalbig\neQ/(meH3), andQbeing the effective charge of the micro-\nelectrode. This potential indicates that the motions of the trapped electron are a 1D Stark-shifted\nhydrogen along the z-direction, and a 2D harmonic oscillator in the plane parall el to the liquid\nHeliumsurface. TheHamiltonianfortheorbitalmotionsoft hetrapped electron can bewrittenas\nˆHo=/summationdisplay\nnEn|na/an}bracketri}ht/an}bracketle{tna|+/summationdisplay\nk=x,y/planckover2pi1νk(ˆa†\nkˆak+1\n2). (2)\nHere,|na/an}bracketri}htis thenth boundstateofthehydrogenlikeatom, ˆa†\nkandˆakare thebosonicoperatorsof\nthevibrationalquantaoftheelectron alongthe k-direction.\nA spin qubit is generated by applying an uniform magnetic fiel dBsalongxdirection, and\nits Hamiltonian reads ˆHq= (gµBBs)ˆσx/2. Here, the Pauli operator is defined as ˆσx=| ↑/an}bracketri}ht/an}bracketle{t↑\n| − | ↓/an}bracketri}ht/an}bracketle{t↓ | with| ↓/an}bracketri}htand| ↑/an}bracketri}htbeing the two spin states. g= 2is the electronic g-factor, and\nµB= 9.3×10−24J/T is the Bohr magneton. The spin-orbit coupling of the trap ped electron can\nbe realized by applying a dc current Ito the electrode I (located upon the liquid Helium surface\nwith a height h) [13]. Typically, x,z≪hand the magnetic field generated by the current I\nreads/vectorB= (Bx,0,Bz)withBx≈µ0I(1−z/h)/(2πh)andBz≈µ0Ix/(2πh2). Here,µ0is\nthe permeability of free space. Therefore, the Hamiltonian describing theinteraction between the\nmagnetic field and spin can be expressed as: ˆHsb=gµB(Bzˆσz+B′\nxˆσx)/2withB′\nx=Bs+Bx,\nˆσz= ˆσ−+ ˆσ+,ˆσ−=| ↓/an}bracketri}ht/an}bracketle{t↑ |andˆσ+=| ↑/an}bracketri}ht/an}bracketle{t↓ |. Consequently, the total Hamiltonian of the\ntrapped electronin theappliedmagneticfields reads\nˆH=/planckover2pi1νs\n2ˆσx+ˆHo+ˆHsx, (3)\nwith\nˆHsx=gµBµ0I\n4πh2/radicalbigg\n/planckover2pi1\n2meνx(ˆax+ˆa†\nx)ˆσz. (4)\nThefirstandsecondtermsintherighthandofEq.(3)describe thefreeHamiltonianofthetrapped\nelectron,with νs= (gµB//planckover2pi1)[Bs+(µ0I/2πh)]beingthetransitionfrequencybetweenitstwospin\nstates, and ˆHsxdescribes the coupling between the spin and the orbital moti on along x-direction.\n4Note that the coupling between the spin and z-directional orbital motion is neglected, due to the\nlarge-detuning. Also, theappliedstrongfield Bs(e.g.,0.06T)does notaffect theinteraction ˆHsx,\nalthoughitwillchangeslightlytheelectron’s motionsint hey-zplane[16].\nObviously,theHamiltonianin Eq.(3)can besimplifiedas\nˆHe=/planckover2pi1Ω/parenleftbig\neiδtˆσ+ˆa+e−iδtˆσ−ˆa†/parenrightbig\n(5)\nintheinteractionpicture. Here, δ=νs−νxisthedetuning,\nΩ =gµBµ0I\n4πh2√2/planckover2pi1meνx(6)\nis the coupling strength, and ˆa= ˆax,ˆa†= ˆa†\nx. Note that, the Hamiltonian in Eq. (5) can also\nbe obtained by applying an ac current I(t) =Icos(ωt)with frequency ω=νx−νs+δto the\nelectrode. Specially, when δ= 0, this Hamiltonian describes a JC-type interaction between the\nspin and orbit of the single electron. In fact, Ref. [13] has a rranged this spin-orbit coupling of\na single electron to increase the interaction between the sp in and a quantized microwave field.\nAlternatively, we will utilize this spin-orbit coupling (t ogether with the electron-electron strong\nCoulomb interaction) to realize a strong interaction betwe en two electronic spins and generate\ncertain typicalquantumgates.\nFor the typical parameters: I= 1mA,h= 0.5µm, andνx= 10GHz [5, 13], we have\nΩ≈5.2MHz. Thisissignificantlylargerthanthedecoherencerate( whichistypicallyontheorder\nof10kHz [5, 13]) of thevibrational states of thetrapped electro n. Thus, the aboveJC interaction\nprovidesapossibleapproachtoimplementQIPbetweenthesp inandorbitstatesofasingletrapped\nelectron. FortheJCinteraction,thestate-evolutionscan belimitedintheinvariant-subspaces {| ↓\n,0/an}bracketri}ht}and{| ↓,1/an}bracketri}ht,| ↑,0/an}bracketri}ht},with|0/an}bracketri}htand|1/an}bracketri}htbeingthegroundandfirstexcitedstatesoftheharmonic\noscillator. Thus,aphasegate ˆP=|0,↓/an}bracketri}ht/an}bracketle{t0,↓ |+|0,↑/an}bracketri}ht/an}bracketle{t0,↑ |+|1,↓/an}bracketri}ht/an}bracketle{t1,↓ |−|1,↑/an}bracketri}ht/an}bracketle{t1,↑ |couldbe\nimplementedbyapplyingacurrentpulsetotheelectrodeI. T herelevantduration tissettosatisfy\ntheconditions: sin(Ωt)≈0andcos(√\n2Ωt)≈ −1(e.g.,Ωt≈37.7numerically). Consequently,a\nCNOTgatewiththesingleelectroncouldberealizedas ˆS=ˆR(π/2,−π/2)ˆPˆR(π/2,π/2),where\nˆR(α,β) = (| ↑/an}bracketri}ht/an}bracketle{t↑ |+| ↓/an}bracketri}ht/an}bracketle{t↓ |)cos(α)−i[exp(iβ)| ↑/an}bracketri}ht/an}bracketle{t↓ |+exp(−iβ)| ↓/an}bracketri}ht/an}bracketle{t↑ |]sin(α)isanarbitrary\nsingle-bit rotation [17]. This CNOT gate operation, betwee n the spin states and the two selected\nvibrational states of a single electron [18], is an intermed iate step for the later CNOT operation\nbetween twodistantspinqubits.\n5Q1 Q2e1 e2\nd\nLiquid Helium \n+I1\n+I2\nH Hh h\nPotential Bsz\nx y\nFIG. 2: (Color online) Two electrons (denoted by e1ande2) are confined individually in two potential\nwells with the distance d, which is sufficiently large (e.g., d= 10µm) such that the magnetic dipole-\ndipole coupling between theelectronic spins isnegligible . Theorbital motionsof thetwoelectrons arealso\ndecoupled from each other, since they are trapped in large-d etuning regime. By applying a current to the\nelectrode I 1the spin of the electron e1could be coupled to the vibrational motions of electron e2, via a\nvirtual excitation of the vibrational motion of electron e1.\nIII. SPIN-ORBITJC COUPLINGBETWEENTHEDISTANTELECTRONS\nWithout loss of generality, we consider here two electrons ( denoted by e1ande2) trapped\nindividually in two potential wells, see Fig. 2. Suppose tha t the distance dbetween the potential\nwells is sufficiently large (e.g., d= 10µm), such that the directly magnetic interaction between\nthe two spins could be neglected. Thus, the interaction betw een the two electrons leaves only\nthe Coulomb one. Specially, the Coulomb interaction along t hex-direction can be approximately\nwrittenas\nV(x)≈e2\n2πǫ0d3x1x2 (7)\nwithxjbeing the displacement of electron ejfrom its potential minima. By controlling the volt-\nages applied on the electrodes Q1andQ2, the vibrational frequencies of the electrons are set as\nthelarge-detuning(and thustheelectrons aredecoupled fr omeach other).\nTo couple the initially-decoupled electrons, we apply a cur rentIto the electrode I 1. As dis-\ncussed previously,such acurrent induces aspin-orbitcoup ling[i.e., ˆHein Eq. (5)]of theelectron\ne1. Therefore, thepresenttwo-electronssystemcan bedescri bed bythefollowingHamiltonian\nˆHee=ˆHe+/planckover2pi1˜Ω/parenleftBig\nei∆tˆaˆb†+e−i∆tˆa†ˆb/parenrightBig\n(8)\nin the interaction picture. Where, ˆbandˆb†are the bosonic operators of the vibrational motion of\nelectrone2alongx-direction, ∆ =ν2x−ν1xisthedetuningbetweenthetwoelectronicvibrations\n6alongx-direction, and\n˜Ω =e2\n4πǫ0med3√ν1xν2x, (9)\nthe coupling strength. Numerically, for d= 10µm andνjx= 10GHz we have ˜Ω≈25MHz.\nAbove,thespinofelectron e2wasdropped,asthedriving(inducedbyelectrode I1)onthisspinis\nnegligible(dueto d≫h).\nThe dynamical evolution ruled by the Hamiltonian in Eq. (8) i s given by the following time-\nevolutionoperator\nˆU(t) = 1+/parenleftbig−i\n/planckover2pi1/parenrightbig/integraltextt\n0ˆHee(t1)dt1\n+/parenleftbig−i\n/planckover2pi1/parenrightbig2/integraltextt\n0ˆHee(t1)/integraltextt1\n0ˆHee(t2)dt2dt1+···.(10)\nWeassume δ= ∆forsimplicity,thentheabovetime-evolutionoperatorcan beapproximatedas\nˆU(t)≈exp/parenleftbigg\n−it\n/planckover2pi1ˆHeff/parenrightbigg\n, (11)\nwiththeeffectiveHamiltonian\nˆHeff=/planckover2pi1Ω2\nδ/bracketleftbig\nˆa†ˆa(ˆσ+ˆσ−−ˆσ−ˆσ+)+ ˆσ+ˆσ−/bracketrightbig\n+/planckover2pi1˜Ω2\nδ/parenleftBig\nˆb†ˆb−ˆa†ˆa/parenrightBig\n+/planckover2pi1Ω˜Ω\nδ/parenleftBig\nˆσ+ˆb+ ˆσ−ˆb†/parenrightBig\n.(12)\nThesecondterm intherighthand ofEq.(10)andthetermsrela tingto thehighorders of Ω/δand\n˜Ω/δwere neglected,since Ω,˜Ω≪δ. Furthermore, at theexperimentaltemperature(e.g., 20mK)\nthe electrons are frozen well into their vibrational ground states (about 40mK for the vibrational\nfrequency ∼10GHz). Thismeansthattheexcitationofthevibrationofelec trone1isvirtual,and\nthus the terms in Eq. (12) related to ˆa†ˆacan be adiabatically eliminated. As a consequence, the\nHamiltonianinEq. (12)reduces to\nˆHeff=/planckover2pi1Ω2\nδˆσ+ˆσ−+/planckover2pi1˜Ω2\nδˆb†ˆb+/planckover2pi1Ω˜Ω\nδ/parenleftBig\nˆσ+ˆb+ ˆσ−ˆb†/parenrightBig\n(13)\nand furtherreads (for Ω =˜Ω)\nˆHJC=/planckover2pi1Ω2\nδ/parenleftBig\nˆσ+ˆb+ ˆσ−ˆb†/parenrightBig\n(14)\nin the interaction picture. Obviously, this Hamiltonian de scribes a JC-type coupling between the\nspinofelectron e1and theorbitalmotionofelectron e2.\n70 2 4 6 810 1200.20.40.60.81\nt (s)Occupancies |↑1,01,02〉 |↓1,01,12〉\n×10−7\nFIG. 3: (Color online) Numerical solutions for the Hamilton ian in Eq. (8): the occupancy evolutions of\nstates| ↑1,01,02/an}bracketri}ht(bluecurve)and | ↓1,01,12/an}bracketri}ht(redcurve),with ˜Ω = Ω = 25 MHzand δ= ∆ = 250 MHz.\nTypically, the effective coupling strength can reach Ω′= Ω2/δ≈2.5MHz for d= 10µm,\nν1x= 10GHz, and δ= 250MHz. With these parameters and the Hamiltonian in E.q (8), Fi g. 3\nshowsnumerically theoccupancy evolutionsof thestates | ↑1,01,02/an}bracketri}htand| ↓1,01,12/an}bracketri}ht. Here,| ↓j/an}bracketri}ht\nand| ↑j/an}bracketri}htare the two spin states of electron ej, and|0j/an}bracketri}htand|1j/an}bracketri}htare the two lower vibrational\nstates of the electron. Obviously, the results are well agre ement with the solutions (i.e., the time-\ndependentoccupanciesof | ↑1,02/an}bracketri}htand| ↓1,12/an}bracketri}ht)fromtheHamiltonian ˆHJC. Thisverifiesthevalid-\nityofˆHJC. Thespin-orbitJCcoupling(14)couldbeusedtoimplementQ IPbetweentheseparately\ntrapped electrons. For example, by applying a current pulse with the duration t=π/(2Ω′)to an\nelectrode, e.g., I 1, a two-qubit operation ˆV1,2(π/2) =| ↓1,02/an}bracketri}ht/an}bracketle{t↓1,02|−i| ↓1,12/an}bracketri}ht/an}bracketle{t↑1,02|between\nthe electrons could be implemented. Consequently, a CNOT ga te between the qubits encoded by\ntheelectronicspinscouldbeimplementedbytheoperationa lsequence ˆC=ˆV1,2(π/2)ˆS2ˆV1,2(π/2),\nwithˆS2being the single-electron CNOT gate operated on the electro ne2. After this two-spin\nCNOT operation, the vibrational motions of the trapped elec trons return to their initial ground\nstates.\nFurthermore, the mechanism used above for the distant spin- orbit coupling can be utilized\nto implement an orbit-mediated spin-spin interaction, whe rein the degrees freedom of the orbits\nof the two electrons are adiabatically eliminated. Indeed, by applying the current pulses to the\nelectrodes simultaneously,theHamiltonianoftheindivid ually-drivenelectrons reads:\nˆH′\nee=/planckover2pi1Ω/parenleftbig\neiδtˆσ+ˆa+e−iδtˆσ−ˆa†/parenrightbig\n+/planckover2pi1˜Ω/parenleftBig\neiδtˆaˆb†+e−iδtˆa†ˆb/parenrightBig\n+/planckover2pi1G/parenleftBig\neiηtˆτ+ˆb+e−iηtˆτ−ˆb†/parenrightBig\n.(15)\n80 20 40 60 80 100 12000.20.40.60.81\nt  (s)Occupancies\n× 10−6|↓1,01,02,↑2〉 |↑1,01,02,↓2〉\nFIG. 4: (Color online) Numerical solutions for the Hamilton ian in Eq. (15): the occupancy evolutions of\nthe states | ↓1,01,02,↑2/an}bracketri}ht(blue curve) and | ↑1,01,02,↓2/an}bracketri}ht(red curve), with ˜Ω = 25MHz,Ω = 2.6MHz,\nδ= 250MHz,and η= Ω2/δ.\nHere, the first and third terms describe respectively the spi n-orbit couplings of the electrons e1\nande2, and the second term describes the Coulomb interaction betw een the electrons. Gandη\nare the coupling strength and the detuning between the spin a nd orbital motions of electron e2,\nrespectively. ˆτ−=| ↓2/an}bracketri}ht/an}bracketle{t↑2|andˆτ+=| ↑2/an}bracketri}ht/an}bracketle{t↓2|are the corresponding spin operators of electron\ne2. The spin-orbit couplings, i.e., the first and third terms in the Hamiltonian, can be realized by\napplying the ac currents I1(t) =I1cos(ω1t)andI2(t) =I2cos(ω2t)to the electrodes I 1and I2\nrespectively,with thefrequencies ω1=ν1x−νs+δandω2=ν2x−νs+η. Here the ac currents\nare appliedtorelatively-easilysatisfytheaboverequire ments forthedetunings.\nWiththehelp ofEq.(13), Eq.(15)can beeffectivelysimplifi edas\nˆH′\nee=ˆHeff+/planckover2pi1G/parenleftBig\neiηtˆτ+ˆb+e−iηtˆτ−ˆb†/parenrightBig\n, (16)\ni.e.,\nˆH′\nee=/planckover2pi1Ω˜Ω\nδ/parenleftBig\neiγtˆσ+ˆb+e−itγˆσ−ˆb†/parenrightBig\n+/planckover2pi1G/parenleftBig\nei(η−˜Ω2/δ)tˆτ+ˆb+e−i(η−˜Ω2/δ)tˆτ−ˆb†/parenrightBig\n(17)\nin the interaction picture, with γ= (Ω2−˜Ω2)/δ. We select G= Ω˜Ω/δandη= Ω2/δfor\nsimplicity,suchthat\nˆH′\nee=/planckover2pi1G/parenleftBig\neiγtˆσ+ˆb+e−itγˆσ−ˆb†/parenrightBig\n+/planckover2pi1G/parenleftBig\neiγtˆτ+ˆb+e−iγtˆτ−ˆb†/parenrightBig\n. (18)\nByrepeatingthesamemethodforderivingtheeffectiveHami ltonianˆHeff,i.e.,neglectingtheterms\nrelatingto thehighorders of G/γinthetime-evolutionoperatorand eliminatingadiabatica llythe\n9termsrelating to ˆb†ˆb, wehave\nˆH′\neff=/planckover2pi1G2\nγ(ˆσ+ˆτ−+ ˆσ−ˆτ+). (19)\nThis is an effectively interaction between the two spins, me diated by their no-excited orbital mo-\ntions[19].\nNumerically, for ˜Ω≈25MHz,Ω≈2.6MHz, and δ≈250MHz, we have |γ| ≈2.5MHz,\nG≈0.26MHz, and Ω′′=|G2/γ| ≈27kHz. With these parameters, Fig. 4 shows numerically\nthetime-dependentoccupanciesof | ↓1,01,02,↑2/an}bracketri}htand| ↑1,01,02,↓2/an}bracketri}htfromtheHamiltonianinEq.\n(15). This provides the validity of the simplified Hamiltoni an in Eq. (19). Obviously,the present\norbit-mediated spin-spin coupling is significantly weaker than the above spin-orbit JC coupling\n(14) between the electrons, but still stronger than the dire ctly magnetic dipole-dipole coupling\n(which is estimated as ∼10−3Hz for the same distance) between the spins. Since the cohere nce\ntime of the spin qubit is very long (e.g., could be up to minute s [7]), the orbit-mediated spin-\nspin coupling demonstrated above could be utilized to gener ate the spins entanglement and thus\nimplementthedesirableQIP.\nFinally, we would like to emphasize that, the considered dou ble-trap configuration shown in\nFig. 2 seems similarly to that of the recent ion-trap experim ents [20, 21]. There, two ions are\nconfinedintwopotentialwellsseparatedby 40µm[20](or 54µm[21]),andtheion-ionvibrational\ncoupling ˆHii=/planckover2pi1˜Ω[exp(i∆t)ˆaˆb†+ exp(−i∆t)ˆa†ˆb]is achieved up to ˜Ω≈10kHz [20] (or ˜Ω≈\n7kHz [21]). The coupling between theions was manipulated tun ablyby controllingthe potential\nwells(viasweepingthevoltagesontherelevantelectrodes )toadiabaticallytunetheoscillatorsinto\nor out of resonance, i.e., ∆ = 0or∆≫˜Ω[22], respectively. Instead, in the present proposal we\nsuggested a JC-type coupling (and consequently an orbit-me diated spin-spin coupling) between\nthe two separated electrons. Therefore, the operational st eps for implementing the QIP should be\nrelativelysimple. Moreinterestingly,heretheelectron- electroncouplingstrength ˜Ωissignificantly\nstronger (about 103times) than that between the trapped ions (e.g,9Be+[20]), since the mass of\nelectron ismuchsmallerthanthatoftheions.\nIV. CONCLUSION\nWe have suggested an approach to implement the QIP with elect ronic spins on liquid helium.\nTwolong-livedspinstatesofthetrappedelectronwereenco dedasaqubit,andthestrongCoulomb\n10interactionbetweentheelectronswasutilizedasthedatab us. Thespin-orbitJCcouplingbetween\nthe spin of an electron and the vibrational motion of another distant electron is generated by\ndesigning a virtual excitation of the electronic vibration . Such a distant spin-orbit interaction\nis further utilized to realize an orbit-mediated spin-spin coupling and implement the desirable\nquantumgates.\nCompared with the ions in the Paul traps, here a feature is tha t the mass of the electron is\nmuch smaller than that of ions, and thus a strong Coulomb coup ling up to 25 MHz between the\nelectrons could reached for a distance of d= 10µm. Finally, the construction suggested here\nfor implementing quantum computation with trapped electro ns on the liquid helium should be\nscalable, andhopefullybefeasiblewithcurrent micro-sca letechnique.\nAcknowledgements : This work was partly supported by the National Natural Scie nce\nFoundationofChinaGrantsNo. 11204249,11147116,1117437 3,and90921010,theMajorState\nBasic Research Development Program of China Grant No. 2010C B923104, and the open project\nofStateKeyLaboratoryofFunctionalMaterials forInforma tics.\n[1] W.Paul,Rev.Mod.Phys. 62,531(1990); D.J.Wineland, C.Monroe, W.M.Itano, D.Leibfr ied, B.E.\nKing,andD.M.Meekhof, J.Res.Natl.Inst.Stand.Technol. 103, 259(1998); J.I.CiracandP.Zoller,\nPhys. Rev. Lett. 74, 4091 (1995).\n[2] O.Morsch and M.Oberthaler, Rev.Mod. Phys. 78, 179 (2006).\n[3] L. S. Brownand G. Gabrielse, Rev. Mod. Phys. 58, 233 (1986); L. Lamata, D. Porras, and J. I. Cirac,\nPhys. Rev. A 81, 022301 (2010).\n[4] P.M.Platzman and M.I. Dykman, Science 284, 1967 (1999).\n[5] M. I.Dykman, P.M.Platzman, and P.Seddighrad, Phys. Rev . B67, 155402 (2003).\n[6] E.Collin,W.Bailey,P.Fozooni,P.G.Frayne,P.Glasson ,K.Harrabi,M.J.Lea,andG.Papageorgiou,\nPhys. Rev. Lett. 89, 245301 (2002).\n[7] S.A.Lyon, Phys. Rev.A 74, 052338 (2006).\n[8] F. R. Bradbury, M. Takita, T. M. Gurrieri, K. J. Wilkel, K. Eng, M. S. Carroll, and S. A. Lyon, Phys.\nRev. Lett. 107, 266803 (2011); K.Kono, Physics, 4, 110 (2011).\n[9] G. Papageorgiou, P. Glasson, K. Harrabi, V. Antonov, E. C ollin, P. Fozooni, P. G. Frayne, M. J. Lea,\nand D. G. Rees, Appl. Phys. Lett. 86, 153106 (2005); G. Sabouret, F. R. Bradbury, S. Shankar, J. A .\n11Bert, and S.A.Lyon, Appl. Phys. Lett. 92, 082104 (2008).\n[10] M. Koch, G.Aub ¨ ock, C.Callegari, and W.E.Ernst, Phys. Rev. Lett. 103, 035302 (2009).\n[11] M. Zhang, H.Y. Jiaand L.F.Wei, Phys. Rev.A 80, 055801 (2009).\n[12] M. Zhang, H.Y. Jiaand L.F.Wei, Opt. Lett. 351686 (2010).\n[13] D. I. Schuster, A. Fragner, M. I. Dykman, S. A. Lyon, and R . J. Schoelkopf, Phys. Rev. Lett. 105,\n040503 (2010).\n[14] S.Mostame and R.Sch ¨utzhold, Phys. Rev. Lett. 101, 220501 (2008).\n[15] C. C. Grimes and T. R. Brown, Phys. Rev. Lett. 32, 280 (1974); D. Konstantinov, M. I. Dykman, M.\nJ. Lea, Y.Monarkha, and K.Kono, Phys.Rev. Lett. 103, 096801 (2009).\n[16] S.S.Sokolov, Phys.Rev. B 51, 2640 (1995).\n[17] Alternatively, the single-bit operations could be imp lemented by applying an ac current I(t)(with\nfrequency ω=νsand phase θon the electrode I2, see, Fig. 2), which generates a magnetic field\nBz≈µ0I(t)d/[2π(d2+h2)]along the z-direction toexcite resonantly the spin of electron e1.\n[18] C. Monroe, D. M. Meekhof, B. E. King, W. M. Itano, and D. J. Wineland, Phys. Rev. Lett. 754714\n(1995).\n[19] K.Mølmer and A.Sørensen, Phys. Rev.Lett. 821835 (1999).\n[20] K. R. Brown, C. Ospelkaus, Y. Colombe, A. C. Wilson, D. Le ibfried, and D. J. Wineland, Nature\n(London) 471, 196 (2011).\n[21] M. Harlander, R.Lechner, M.Brownnutt, R.Blatt, and W. H¨ ansel, Nature (London) 471, 200 (2011).\n[22] M. Zhang and L.F.Wei, Phys. Rev.A, 83064301 (2011).\n12"    },    {        "title": "2012.02810v1.Effects_of_hybridization_and_spin_orbit_coupling_to_induce_odd_frequency_pairing_in_two_band_superconductors.pdf",        "content": "arXiv:2012.02810v1  [cond-mat.supr-con]  4 Dec 2020✐✐\n“paper” — 2020/12/8 — 1:58 — page 1 — #1\n✐✐\n✐\n✐✐\n✐Effects of hybridization and spin-orbit\ncoupling to induce odd frequency pairing in\ntwo-band superconductors\nMoloud Tamadonpour and Heshmatollah Yavari\nThe effects of spin independent hybridization potential and spin-\norbit coupling on two-band superconductor with equal time s -wave\ninterband pairing order parameter is investigated theoret ically. To\nstudy symmetry classes in two-band superconductors the Gor ’kov\nequations are solved analytically. By defining spin singlet and spin\ntriplet s-wave order parameter due to two-band degree of fre e-\ndom the symmetry classes of Cooper pair are studied. For spin\nsinglet case it is shown that spin independent hybridizatio n gen-\nerates Cooper pair belongs to even-frequency spin singlet e ven-\nmomentum even-band parity (ESEE) symmetry class for both in -\ntraband and interband pairing correlations. For spin tripl et order\nparameter, intraband pairing correlation generates odd-f requency\nspin triplet even-momentum even-band parity (OTEE) symme-\ntry class whereas, interband pairing correlation generate s even-\nfrequency spin triplet even-momentum odd-band parity (ETE O)\nclass. For the spin singlet, spin-orbit coupling generates pairing cor-\nrelation that belongs to odd-frequency spin singlet odd-mo mentum\neven-band parity (OSOE) symmetry class and even-frequency spin\nsinglet even-momentum even-band parity (ESEE) for intraba nd\nand interband pairing correlation respectively. In the spi n triplet\ncase for itraband and interband correlation, spin-orbit co upling\ngenerates even-frequency spin triplet odd-momentum even- band\nparity (ETOE) and even-frequency spin triplet even-moment um\nodd-band parity (ETEO) respectively.\n1. Introduction and summary\nSymmetries of order parameter in superconductors affect the ir physical prop-\nerties. The total wave function of a pair of fermions, in acco rdance with the\nPauli principle, should be asymmetric under the permutatio n of orbital, spin\nand time (or equivalently Matsubara frequency) coordinate s [1]. This leads\n1✐✐\n“paper” — 2020/12/8 — 1:58 — page 2 — #2\n✐✐\n✐\n✐✐\n✐2 Moloud Tamadonpour and Heshmatollah Yavari\nto four classes allowed combinations for the symmetries of t he wave function.\nThis would imply that if the pairing is even in time, spin sing let pairs have\neven parity (ESE) and spin triplet pairs have odd parity (ETO ). While if\nthe pairing is odd in time, spin singlet pairs have odd parity (OSO) and\nspin triplet pairs have even parity (OTE). Black-Schaffer and Balatsky [2]\nhave shown that the multiband superconducting order parame ter has an ex-\ntra symmetry classification that originates from the band de gree of freedom,\nso called even-band-parity and odd band-parity. As a conseq uence, Cooper\npairs can be classified into eight symmetry classes [3].\nTransport properties of multi-band superconductor are qua litatively dif-\nferent from those of the one-band superconductor. For insta nce, two-band\nsystem with the non-magnetic impurity violates Anderson th eorem [4]. As\na result, lots of efforts have been devoted to understanding t he properties\nof such systems both theoretically and experimentally. For these materials\nband symmetry plays important role. A main hypothesis of the model is the\nformation of the Cooper pairs inside one energy band and tran sition of this\npair from one band to another which leads to intra and inter ba nd electronic\ninteractions. Multi-band model explained lots of strange p hysical properties\nof superconductive systems and were consistent with experi mental data. Fa-\nmous multiband superconductors are MgB2 [5, 6] and the iron-b ased super-\nconductors [7–9]. The nature of their two bands requires tha t the multiband\napproach be used to describe their properties. On the contra ry for cuprates\ndespite their multiband nature a single-band approach is mo re appropriate.\nFrom a general symmetry analysis of even and odd-frequency p airing\nstates, it was shown that odd-frequency pairing always exis ts in the form of\nodd-interband (orbital) pairing if there is any even-frequ ency even-interband\npairing present consistent with the general symmetry requi rements [10]. The\nappearance of odd-frequency Cooper pairs in two-band super conductors by\nsolving the Gor’kov equation was discussed analytically [1 1]. They considered\nthe equal-time s-wave pair potential and introduced two typ es of hybridiza-\ntion potentials between the two conduction bands. One is a sp in-independent\nhybridization potential and the other is a spin-dependent h ybridization po-\ntential derived from the spin-orbit interaction.\nThe effect of random nonmagnetic impurities on the supercond ucting\ntransition temperature in a two-band superconductor, by as suming the equal-\ntime spin-singlet s-wave pair potential in each conduction band and the hy-\nbridization between the two bands as well as the band asymmet ry was stud-\nied theoretically [11, 12]. The effect of single-quasiparti cle hybridization or\nscattering in a two-band superconductor by performing pert urbation theory✐✐\n“paper” — 2020/12/8 — 1:58 — page 3 — #3\n✐✐\n✐\n✐✐\n✐Effects of hybridization and spin-orbit coupling 3\nto infinite order in the hybridization term, in a multiband su perconductor\nwas investigated [13].\nThe superconducting state of multi-orbital spin-orbit cou pled systems in\nthe presence of an orbitally driven inversion asymmetry, by assuming that\nthe interorbital attraction is the dominant pairing channe l, was studied [14].\nThey have shown that in the absence of the inversion symmetry , supercon-\nducting states that avoid mixing of spin-triplet and spin-s inglet configura-\ntions are allowed, and remarkably, spin-triplet states tha t are topologically\nnontrivial can be stabilized in a large portion of the phase d iagram. The\nimpact of strong spin-orbit coupling (SOC) on the propertie s of new class\nsuperconductors has attracted much attentions. It has been the subject of\ngreat theoretical and experimental interest [15, 16]. The f ormation of unex-\npected multi-component superconductors states allows for superconductors\nwith magnetism and SOC. It was shown that for multi-orbital s ystems such\nas the Fe-pnictides SOC coupling, is much smaller than the or bit Hund’s\ncoupling [17–19], In contrast for multiband systems such as Ir-based oxide\nmaterials it was found that the SOC interaction is comparabl e to the on-site\nCoulomb interaction [20]. The combined effect of Hund’s and S OC coupling\non superconductivity in multi-orbital systems was investi gated and it was\nshown that Hund’s interaction leads to orbital-singlet spi n-triplet supercon-\nductivity, where the Cooper pair wave function is antisymme tric under the\nexchange of two orbitals [21]. Combined effect of the spin-or bit coupling and\nscattering on the nonmagnetic disorder on the formation of t he spin reso-\nnance peak in iron-based superconductors was also studied [ 22].\nIn this paper by using Gor’kov equation the effects of spin-or bit coupling\nand hybridization on the possibility of odd frequency pairi ng of a two-band\nsuperconductor with an equal time s-wave interband pairing order parameter\nare investigated theoretically.\n2. Formalism\n2.1. Two-band model\nThe basic physics of multiband superconductors can be obtai ned by intro-\nducing a two-band model. We start with a normal two-band Hami ltonian as✐✐\n“paper” — 2020/12/8 — 1:58 — page 4 — #4\n✐✐\n✐\n✐✐\n✐4 Moloud Tamadonpour and Heshmatollah Yavari\n[12]\n(2.1)\nˇHN=/integraldisplay\ndr/bracketleftig\nψ†\n1,↑(r), ψ†\n1,↓(r), ψ†\n2,↑(r), ψ†\n2,↓(r)/bracketrightig\nˇHN(r)\nψ1,↑(r)\nψ1,↓(r)\nψ2,↑(r)\nψ2,↓(r)\n,\nwhere\n(2.2) ˇHN=/parenleftbiggξ1kˆσ0/parenleftbig\nυeiθ+V/parenrightbig\nˆσ0 /parenleftbig\nυe−iθ+V∗/parenrightbig\nˆσ0ξ2kˆσ0/parenrightbigg\n.\nHereψα,σ(r)is the annihilation ( ψ†\nα,σ(r))creation) operator of an electron\nwith spin (σ=↑,↓) at theαth conduction band, ξαk=/planckover2pi12k2/2me−µFis the\ndispersion energy of band α,meis the mass of an electron, µFis the chemical\npotential. The spin independent hybridization potential i s a complex number\ncharacterized by a phase θ.υeiθdenotes the hybridization between the two\nbands, which is much smaller than the Fermi energy in the two c onduction\nbands. In the absence of spin flip hybridization the spin-orb it coupling poten-\ntial isV(k) =ηˆz.(σ×/vectork) =η(kyσx−kxσy), whereηis the parameter that\ndescribes the strength of the Rashba spin-orbit coupling an dˆzis the unit\nvector perpendicular to the superconducting surface. This potential is odd-\nmomentum-parity functions satisfying V(k) =−V(−k). Throughout this pa-\nper, Pauli matrices in spin, two-band, particle- hole space s are respectively\ndenoted by ˆσj,ˆρjandˆτjforj= 1−3. Superconducting order parameter in\nbandαis:\n(2.3) ˆ∆αα′(k) =/parenleftbigg∆11(k) ∆12(k)\n∆21(k) ∆22(k)/parenrightbigg\n.\nWe focus only on interband superconducting order parameter (∆11(k) =\n∆22(k) = 0) . The interband s-wave pair potential, is defined by [12]\n(2.4) ∆12,σσ′(r) =g/an}bracketle{tψ1,σ(r)ψ2,σ′(r)/an}bracketri}ht.\nheregis interband attractive interaction between two electrons . By assuming\nthe spatially uniform order parameter the Fourier transfor mation of the pair\npotential becomes\n(2.5) ∆12,↑↓=g\nVvol/summationdisplay\nk/an}bracketle{tψ1,↑(k)ψ2,↓(−k)/an}bracketri}ht.✐✐\n“paper” — 2020/12/8 — 1:58 — page 5 — #5\n✐✐\n✐\n✐✐\n✐Effects of hybridization and spin-orbit coupling 5\nIn the two-band model, for spin singlet the order parameter i s symmetric\n(antisymmetric) under the permutation of band (spin) indic es.\n(2.6) ∆12,↑↓= ∆21;↑↓=−∆1,2;↓↑,\nBut for spin triplet the order parameter is antisymmetric (sy mmetric) under\nthe permutation of band (spin) indices.\n(2.7) ∆12,↑↓=−∆21;↑↓= ∆1,2;↓↑.\nFor simplicity we omit the indices of ∆αα′. The Hamiltonian describing su-\nperconductor in the Nambu space, can be written as [11]\n(2.8)⌣HS(T)=1\n2/summationdisplay\nkψ†\nk,σ/parenleftiggˇHN(k)ˇ∆S(T)\nˇ∆†\nS(T)−ˇH∗\nN(−k)/parenrightigg\nψk,σ,\nwhere the spin-singlet and spin triplet pair potentials ( ˇ∆Sandˇ∆T) are\nrespectively given by\n(2.9) ˇ∆S= ∆ˆρ1iˆσ2,\n(2.10) ˇ∆T= ∆iˆρ2ˆσ1.\nFor a two-band system, the Bogoliubov- de Gennes Hamiltonian can be de-\nscribed by 8×8matrix reflecting spin, particle- hole and two band degrees\nof freedom. In particle-hole space N1, by considering the spin of electron as\n↑and for hole as ↓, while in particle-hole space N2, we consider the spin of\nelectron as ↓and for hole as ↑, we can describe the Hamiltonian ˇHS(T)by a\n4×4matrix [3, 12]\n(2.11)\nˇH0=\nξkυeiθ+V(k) 0 ∆\nυe−iθ+V∗(k)ξk −sspin∆ 0\n0 −sspin∆ −ξk −υe−iθ−V∗(−k)\n∆ 0 −υeiθ−V(−k) −ξk\n\nheresspin=−1for spin singlet and sspin= 1for spin triplet. To discuss\nthe effects of hybridizations and spin-orbit interaction on the properties of\nsuperconductors, we calculate the Green’s functions by sol ving the Gor’kov✐✐\n“paper” — 2020/12/8 — 1:58 — page 6 — #6\n✐✐\n✐\n✐✐\n✐6 Moloud Tamadonpour and Heshmatollah Yavari\nequation [23]\n(2.12)/parenleftbig\niωn−ˇH0/parenrightbigˇG0(k,iωn) =ˇ1,\n(2.13) ˇG0(k,iωn) =/parenleftigg\nˆG0(k,iωn) ˆF0(k,iωn)\n−sspinˆF†\n0(−k,iωn)−ˆG∗\n0(−k,iωn)/parenrightigg\n.\nwhereωn= (2n+1)πkBTis the Matsubara frequency ( kBis the Boltz-\nmann constant), and ˇ1is the identity matrix in spin×band×particle−\nholespace.ˇG0is a4×4matrix where the diagonal components are nor-\nmal Green’s function and non-diagonal components are anoma lous Green’s\nfunction.\n2.2. Spin Singlet Pairing Order\nAccording to Equation (2.11), the Hamiltonian of a two-band superconductor\nwith spin singlet configuration in the presence of spin-orbi t coupling is\n(2.14)\nˇH0=\nξk υeiθ+η(ky+ikx) 0 ∆\nυe−iθ+η(ky−ikx) ξk ∆ 0\n0 ∆ −ξk −υe−iθ+η(ky−ikx)\n∆ 0 −υeiθ+η(ky+ikx) −ξk\n.\nBy using Equation (2.12) and (2.13) , for spin singlet the solu tion of the\nnormal Green’s function within the first order of ∆is calculated as\n(2.15)\nˆG0(k,iωn) =∆\nZ0{[(ξ−iωn)(ν2+η2k2−2νη(kxsinθ+kycosθ)+(ξ+iωn)\n×/parenleftbig\n−ξ2−ω2\nn/parenrightbig\n]ˆρ0+[(−νcosθ−ηky)(−(ξ+iωn)2+ν2+η2k2\n−2νη(kxsinθ+kycosθ))]ˆρ1+[(νsinθ+ηkx)(−(ξ+iωn)2\n+ν2+η2k2−2νη(kxsinθ+kycosθ))]ˆρ2}\nhere\n(2.16)\nZ0=ξ4+2ξ2/parenleftbig\nω2\nn−ν2/parenrightbig\n+/parenleftbig\nω2\nn+ν2/parenrightbig2−8iηνξωn(kxsinθ+kycosθ)\n+2cos2θη2ν2(k2\nx−k2\ny)−4sin2θη2ν2kxky+2η2k2/parenleftbig\nω2\nn−ξ2/parenrightbig\n+η4k4.\nthatkx=kcosφandky=ksinφ, whereφis the angle between momentum\nand thexaxis. The matrix form of the normal Green’s function ( Eq. (2. 15))✐✐\n“paper” — 2020/12/8 — 1:58 — page 7 — #7\n✐✐\n✐\n✐✐\n✐Effects of hybridization and spin-orbit coupling 7\ncan be written as\n(2.17) ˆG0(k,iωn) =/parenleftbiggG11(k,iωn)G12(k,iωn)\nG21(k,iωn)G22(k,iωn)/parenrightbigg\n.\nwhere\n(2.18)\nG11(k,iωn) =∆\nZ0[(ξ−iωn)(ν2+η2(k2\nx+k2\ny)−2νη(kxsinθ+kycosθ)\n−(ξ+iωn)(ξ2+ω2\nn)]\n(2.19)\nG12(k,iωn) =∆\nZ0[{−νeiθ−η(ikx+ky)}{−(ξ+iωn)2+ν2+η2k2\n−2νη(kxsinθ+kycosθ)}]\n(2.20)\nG21(k,iωn) =∆\nZ0[{−νe−iθ−η(−ikx+ky)}{−(ξ+iωn)2+ν2+η2k2\n−2νη(kxsinθ+kycosθ)}]\n(2.21)\nG22(k,iωn) =∆\nZ0[(ξ−iωn)(νe−iθ−η(−ikx+ky))(νeiθ−η(ikx+ky)\n−(ξ+iωn)/parenleftbig\nξ2+ω2\nn/parenrightbig\n]\nBy using Equation (2.12) and (2.13), the anomalous Green’s fu nction can be\nobtained as\n(2.22)\nˆF0(k,iωn) =∆\nZ0[(2νξcosθ+2ηωniky) ˆρ0+/parenleftbig\n−(ν2+ξ2+ω2\nn)+η2k2/parenrightbig\nˆρ1\n+(2νη(kxcosθ−kysinθ)) ˆρ2+(2νξisinθ−2ηωnkx) ˆρ3].\nIn particle- hole space N1, the matrix form of the anomalous Green’s func-\ntion (Eq. (2.22)) is\n(2.23) ˆFN1\n0(k,iωn) =/parenleftbiggF11,↑↓(k,iωn)F12,↑↓(k,iωn)\nF21,↑↓(k,iωn)F22,↑↓(k,iωn)/parenrightbigg\n,\nwhere\n(2.24) F11,↑↓(k,iωn) =∆\nZ0[2νξeiθ−2ηωnkx+2ηωniky],\n(2.25)\nF12,↑↓(k,iωn) =∆\nZ0[−(ν2+ξ2+ω2\nn)+η2k2−2iνη(kxcosθ−kysinθ)],✐✐\n“paper” — 2020/12/8 — 1:58 — page 8 — #8\n✐✐\n✐\n✐✐\n✐8 Moloud Tamadonpour and Heshmatollah Yavari\n(2.26)\nF21,↑↓(k,iωn) =∆\nZ0[−(ν2+ξ2+ω2\nn)+η2k2+2iνη(kxcosθ−kysinθ)],\n(2.27) F22,↑↓(k,iωn) =∆\nZ0[2νξeiθ−2ηωnkx+2ηωniky].\nIn particle- hole space N2, the matrix form of the anomalous Green’s function\nis\n(2.28)ˆFN2\n0(k,iωn) =/parenleftbigg\nF11,↓↑(k,iωn)F12,↓↑(k,iωn)\nF21,↓↑(k,iωn)F22,↓↑(k,iωn)/parenrightbigg\n=−ˆFN1\n0(k,iωn).\nIn the absence of spin-orbit coupling ( η) the anomalous Green’s function\n(Eq.(2.22)) becomes\n(2.29) ˆF0(k,iωn) =∆\nZ0[2νξcosθˆρ0−(ν2+ξ2+ω2\nn)ˆρ1+2νξisinθˆρ3],\nhere\n(2.30) Z0=ξ4+2ξ2/parenleftbig\nω2\nn−ν2/parenrightbig\n+/parenleftbig\nω2\nn+ν2/parenrightbig2.\nThe matrix form of the anomalous Green’s function in Equatio n (2.29) in\nparticle-hole spaces N1andN2, are\n(2.31)\nˆFN1\n0(k,iωn) =/parenleftbigg\nF11,↑↓(k,iωn)F12,↑↓(k,iωn)\nF21,↑↓(k,iωn)F22,↑↓(k,iωn)/parenrightbigg\n=∆\nZ0/parenleftbigg\n2νξcosθ+2iνξsinθ−(ν2+ξ2+ω2\nn)\n−(ν2+ξ2+ω2\nn) 2νξcosθ−2iνξsinθ/parenrightbigg\n=∆\nZ0/parenleftbigg\n2ξνeiθ−(ν2+ξ2+ω2\nn)\n−(ν2+ξ2+ω2\nn) 2ξνe−iθ/parenrightbigg\n,\nand\n(2.32)\nˆFN2\n0(k,iωn) =/parenleftbiggF11,↓↑(k,iωn)F12,↓↑(k,iωn)\nF21,↓↑(k,iωn)F22,↓↑(k,iωn)/parenrightbigg\n=−ˆFN1\n0(k,iωn)\n=∆\nZ0/parenleftbigg\n−2νξcosθ−2iνξsinθ(ν2+ξ2+ω2\nn)\n(ν2+ξ2+ω2\nn)−2νξcosθ+2iνξsinθ/parenrightbigg\n=∆\nZ0/parenleftbigg−2ξνeiθ(ν2+ξ2+ω2\nn)\n(ν2+ξ2+ω2\nn)−2ξνe−iθ/parenrightbigg\n.✐✐\n“paper” — 2020/12/8 — 1:58 — page 9 — #9\n✐✐\n✐\n✐✐\n✐Effects of hybridization and spin-orbit coupling 9\nThe intraband pairing correlations become\n(2.33) F11,↑↓(k,iωn)−F11,↓↑(k,iωn) =4∆\nZ0ξνeiθ,\n(2.34) F22,↑↓(k,iωn)−F22,↓↑(k,iωn) =4∆\nZ0ξνe−iθ.\nHybridization generates ρ0andρ3components which belongs to even fre-\nquency symmetry class. It means that in the presence of inter band cou-\npling, hybridization generates even frequency intra- subl attice pairing in\nthe system. These components belong to even-frequency spin -singlet even-\nmomentum even-band parity (ESEE) symmetry class. This resu lt is in agree-\nment with the equation (20) presented in Ref [12]. Equation ( 2.33) and (2.34)\nare in agreement with the Equation (62) and (63) reported in R ef [3] in the\nfirst order of ∆(|∆|2= 0) and equal energy bands ( ξ−= 0) and both belong\nto the (ESEE) symmetry class. The band symmetry generates in terband\npairing correlation:\n(2.35)[F12,↑↓(k,iωn)−F12,↓↑(k,iωn)]+[F21,↑↓(k,iωn)−F21,↓↑(k,iωn)]\n=−4∆\nZ0(ν2+ξ2+ω2\nn).\nwhich belongs to (ESEE). This result is in agreement with the Equation (65)\npresented in Ref [3].\n(2.36)[F12,↑↓(k,iωn)−F12,↓↑(k,iωn)]−[F21,↑↓(k,iωn)−F21,↓↑(k,iωn)]\n=2∆\nZ3iωnξ−.\nwhich belongs to the symmetry (OSEO) class. We considered a t wo-band\nsuperconductor with an equal dispersion energy in each band (ξ+=ξ−). In\nthis case the interband pairing correlation due to band asym metry is\n(2.37)\n[F12,↑↓(k,iωn)−F12,↓↑(k,iωn)]−[F21,↑↓(k,iωn)−F21,↓↑(k,iωn)] = 0.\nIn the absence of hybridization within the second order of th e spin-orbit\ncoupling constant ( η), we obtain\n(2.38)\nˆG0(k,iωn) =∆\nZ0[(η2k2−(ξ+iωn)2)(ξ−iωn) ˆρ0+η(ky+ikx)(ξ+iωn)2ˆρ1],\nwhere\n(2.39) Z0= (ξ2+ω2\nn)2+2η2k2/parenleftbig\nω2\nn−ξ2/parenrightbig\n.✐✐\n“paper” — 2020/12/8 — 1:58 — page 10 — #10\n✐✐\n✐\n✐✐\n✐10 Moloud Tamadonpour and Heshmatollah Yavari\nEquation (2.22) can be rewritten as\n(2.40) ˆF0(k,iω) =∆\nZ0[2iηωnkyˆρ0−2ηωnkxˆρ3+/parenleftbig\n−ξ2−ω2\nn+η2k2/parenrightbig\nˆρ1].\nIn particle- hole space N1andN2,the matrix form of the anomalous Green’s\nfunction (Eq. (2.40)) is\n(2.41)ˆFN1\n0(k,iωn) =/parenleftbigg\nF11,↑↓(k,iωn)F12,↑↓(k,iωn)\nF21,↑↓(k,iωn)F22,↑↓(k,iωn)/parenrightbigg\n=∆\nZ0/parenleftbigg2iηωnky−2ηωnkx−ξ2−ω2\nn+η2k2\n−ξ2−ω2\nn+η2k22iηωnky+2ηωnkx/parenrightbigg\n,\n(2.42)\nˆFN2\n0(k,iωn) =/parenleftbiggF11,↓↑(k,iωn)F12,↓↑(k,iωn)\nF21,↓↑(k,iωn)F22,↓↑(k,iωn)/parenrightbigg\n=−ˆFN1\n0(k,iωn)\n=∆\nZ0/parenleftbigg−2iηωnky+2ηωnkxξ2+ω2\nn−η2k2\nξ2+ω2\nn−η2k2−2iηωnky−2ηωnkx/parenrightbigg\n.\nThe intraband pairing correlations are\n(2.43) F11,↑↓(k,iωn)−F11,↓↑(k,iωn) =4∆\nZ0iωnη(ky+ikx),\n(2.44) F22,↑↓(k,iωn)−F22,↓↑(k,iωn) =4∆\nZ0iωnη(ky−ikx).\nSpin-orbit coupling generates ρ0andρ3components which belong to odd\nfrequency symmetry class. It means that in the presence of in ter-band cou-\npling, spin-orbit coupling generates odd-frequency intra sublattice pairing in\nthe system. These components belong to odd-frequency spin- singlet odd-\nmomentum even-band parity (OSOE) symmetry class. In Ref [3] the intra-\nband pairing correlation is written as\n(2.45) F11,↑↓(k,iωn)+F11,↓↑(k,iωn) =∆\nZ3(ξ+−ξ−)V3,\n(2.46) F22,↑↓(k,iωn)+F22,↓↑(k,iωn) =∆\nZ3(ξ++ξ−)V3.✐✐\n“paper” — 2020/12/8 — 1:58 — page 11 — #11\n✐✐\n✐\n✐✐\n✐Effects of hybridization and spin-orbit coupling 11\nThe hybridization generates pairing correlations that bel ong to the (ETOE)\nclass. The band asymmetry generates interband pairing corr elation as\n(2.47)\n[F12,↑↓(k,iωn)−F12,↓↑(k,iωn)]−[F21,↑↓(k,iωn)−F21,↓↑(k,iωn)] = 0.\nIn Ref [3] for spin orbit hybridization the band asymmetry ge nerates the\ninterband pair correlation as\n(2.48)\n[F12,↑↓(k,iω)−F12,↓↑(k,iω)]−[F21,↑↓(k,iω)−F21,↓↑(k,iω)] =2∆\nZ3iωnξ−.\nwhich belongs to the odd-frequency spin-singlet even-mome ntum odd-band\nparity symmetry (OSEO). The interband pairing correlation due to band\nsymmetry is\n(2.49)[F12,↑↓(k,iωn)−F12,↓↑(k,iωn)]+[F21,↑↓(k,iωn)−F21,↓↑(k,iωn)]\n=−4∆\nZ0(ξ2+ω2\nn−η2k2).\nThis component belongs to even-frequency spin-singlet eve n-momentum even-\nband parity (ESEE) symmetry class. For spin singlet, hybrid ization potential\ngenerates ESEE symmetry class due to both intra and interban d correlation,\nwhereas the spin dependent hybridization potential genera tes this class only\nfor interband pairing correlation due to band symmetry. In t his case the odd\nfrequency pairing arises only due to intraband pairing corr elations for spin\ndependent hybridization potential.\n2.3. Spin Triplet Pairing Order\nBy considering Equation (2.11), the Hamiltonian of a two-ban d supercon-\nductor with spin triplet configuration in the presence of spi n-orbit coupling\nis\n(2.50)\nˇH0=\nξk υeiθ+η(ky+ikx) 0 ∆\nυe−iθ+η(ky−ikx) ξk −∆ 0\n0 −∆ −ξk −υe−iθ+η(ky−ikx)\n∆ 0 −υeiθ+η(ky+ikx) −ξk\n.\nThe solution of the anomalous Green’s function within the fir st order of ∆\nis calculated as\n(2.51)\nˆF0(k,iωn) =∆\nZ0[(−2iηξkx+2νωnsinθ)ˆρ0+(2iνη(kxcosθ−kysinθ)) ˆρ1\n+(i(ν2−ξ2−ω2\nn)−iη2(k2\nx+k2\ny))ˆρ2+(−2ηξky−2iνωncosθ)ˆρ3].✐✐\n“paper” — 2020/12/8 — 1:58 — page 12 — #12\n✐✐\n✐\n✐✐\n✐12 Moloud Tamadonpour and Heshmatollah Yavari\nThe matrix form of the anomalous Green’s function (Eq. (2.51 ) ) can be\nwritten as\n(2.52) ˆF11,↑↓(k,iωn) =∆\nZ0/parenleftig\n−2iνωneiθ−2iηξkx−2ηξky/parenrightig\n,\n(2.53)\nˆF12,↑↓(k,iωn) =∆\nZ0[(ν2−ξ2−ω2\nn)−η2k2+2iνη(kxcosθ−kysinθ)],\n(2.54)\nˆF21,↑↓(k,iωn) =∆\nZ0[(−ν2+ξ2+ω2\nn)+η2k2+2iνη(kxcosθ−kysinθ)],\n(2.55) ˆF22,↑↓(k,iωn) =∆\nZ0/parenleftig\n2iνωne−iθ−2iηξkx+2ηξky/parenrightig\n.\nIn the absence of spin-orbit coupling ( η= 0) the anomalous Green’s\nfunction Equation (2.51) becomes\n(2.56)ˆF0(k,iωn) =∆\nZ0[2νωnsinθˆρ0+i(ν2−ξ2−ω2\nn)ˆρ2−2iνωncosθˆρ3].\nhere\n(2.57) Z0=ξ4+2ξ2/parenleftbig\nω2\nn−ν2/parenrightbig\n+/parenleftbig\nω2\nn+ν2/parenrightbig2.\nIn particle-hole space N1andN2, the matrix form of the anomalous Green’s\nfunction (Eq. (2.56)) is\n(2.58)\nˆFN1\n0(k,iωn) =/parenleftbiggF11,↑↓(k,iωn)F12,↑↓(k,iωn)\nF21,↑↓(k,iωn)F22,↑↓(k,iωn)/parenrightbigg\n=∆\nZ0/parenleftbigg\n2νωn(sinθ−icosθ) (ν2−ξ2−ω2\nn)\n−(ν2−ξ2−ω2\nn) 2νωn(sinθ+icosθ)/parenrightbigg\n,\n(2.59)ˆFN2\n0(k,iωn) =/parenleftbiggF11,↓↑(k,iωn)F12,↓↑(k,iωn)\nF21,↓↑(k,iωn)F22,↓↑(k,iωn)/parenrightbigg\n=ˆFN1\n0(k,iωn)\n=∆\nZ0/parenleftbigg\n2νωn(sinθ−icosθ) (ν2−ξ2−ω2\nn)\n−(ν2−ξ2−ω2\nn) 2νωn(sinθ+icosθ)/parenrightbigg\n.\nThe intraband pairing correlations becomes\n(2.60) F11,↑↓(k,iωn)+F11,↓↑(k,iωn) =−4∆\nZ0iωnνe−iθ,✐✐\n“paper” — 2020/12/8 — 1:58 — page 13 — #13\n✐✐\n✐\n✐✐\n✐Effects of hybridization and spin-orbit coupling 13\n(2.61) F22,↑↓(k,iωn)+F22,↓↑(k,iωn) =4∆\nZ0iωnνeiθ.\nHybridization generates ρ0andρ3which belongs to odd frequency sym-\nmetry class. These components belong to odd-frequency spin -triplet even-\nmomentum even-band parity (OTEE) symmetry class. This resu lt is in agree-\nment with the Equation (24) presents in Ref [12] In the first or der of∆\n(|∆|2= 0) and equal energy bands ( ξ−= 0) Equation (2.60) and (2.61)\nare coincide with the Equation (83) and (84) presented in Ref [3] and both\nbelong to the (OTEE) symmetry class. The band symmetry gener ates inter-\nband pairing correlation as\n(2.62)[F12,↑↓(k,iωn)+F12,↓↑(k,iωn)]−[F21,↑↓(k,iωn)+F21,↓↑(k,iωn)]\n=4∆\nZ0(ν2−ξ2−ω2\nn).\nwhich belongs to even-frequency spin triplet even-momentu m odd-band par-\nity (ETEO) symmetry class. In Ref [3] the interband pairing c orrelation due\nto band asymmetry is\n(2.63)[F12,↑↓(k,iωn)+F12,↓↑(k,iωn)]+[F21,↑↓(k,iωn)+F21,↓↑(k,iωn)]\n=2∆\nZ5iωnξ−.\nThus the band hybridization generates pairing correlation s that belong to\nthe odd-frequency spin triplet even-momentum even-band pa rity (OTEE)\nclass. Since we considered a two-band superconductor with a n equal energy\nbands, the interband pairing correlation due to band asymme try is\n(2.64)\n[F12,↑↓(k,iωn)+F12,↓↑(k,iωn)]+[F21,↑↓(k,iωn)+F21,↓↑(k,iωn)] = 0.\nIn the absence of hybridization, we obtain\n(2.65) ˆF0(k,iωn) =∆\nZ0[−2iηξkxˆρ0−2ηξkyˆρ3−i/parenleftbig\nξ2+ω2\nn+η2k2/parenrightbig\nˆρ2].\nThe matrix form of the anomalous Green’s function in Equatio n (2.65) in\nparticle- hole spaces N1andN2, are\n(2.66)ˆFN1\n0(k,iωn) =/parenleftbiggF11,↑↓(k,iωn)F12,↑↓(k,iωn)\nF21,↑↓(k,iωn)F22,↑↓(k,iωn)/parenrightbigg\n=∆\nZ0/parenleftbigg−2iηξkx−2ηξky−(ξ2+ω2\nn+η2k2)\n(ξ2+ω2\nn+η2k2)−2iηξkx+2ηξky/parenrightbigg\n,✐✐\n“paper” — 2020/12/8 — 1:58 — page 14 — #14\n✐✐\n✐\n✐✐\n✐14 Moloud Tamadonpour and Heshmatollah Yavari\n(2.67)ˆFN2\n0(k,iωn) =/parenleftbigg\nF11,↓↑(k,iωn)F12,↓↑(k,iωn)\nF21,↓↑(k,iωn)F22,↓↑(k,iωn)/parenrightbigg\n=ˆFN1\n0(k,iωn)\n=∆\nZ0/parenleftbigg\n−2iηξkx−2ηξky−(ξ2+ω2\nn+η2k2)\n(ξ2+ω2\nn+η2k2)−2iηξkx+2ηξky/parenrightbigg\n.\nThe intraband pairing correlations are\n(2.68) F11,↑↓(k,iωn)+F11,↓↑(k,iωn) =−4∆\nZ0ξη(ky+ikx),\n(2.69) F22,↑↓(k,iωn)+F22,↓↑(k,iωn) =4∆\nZ0ξ(ky−ikx).\nSpin-orbit coupling generates ˆρ0andˆρ3which belong to even frequency sym-\nmetry class. These components belong to even-frequency spi n-triplet odd-\nmomentum even-band parity (ETOE) symmetry class. In Ref [3] the intra-\nband pairing correlation is calculated as\n(2.70) F11,↑↓(k,iωn)−F11,↓↑(k,iωn) =−∆\nZ5iωnV3,\n(2.71) F22,↑↓(k,iωn)−F22,↓↑(k,iωn) =∆\nZ0iωnV3.\nThe hybridization generates pairing correlations that bel ong to the odd-\nfrequency spin singlet odd-momentum even-band parity (OSO E) class. As\nmentioned in Ref [3] the interband pair correlation can be wr itten as\n(2.72)[F12,↑↓(k,iωn)+F12,↓↑(k,iωn)]+[F21,↑↓(k,iωn)+F21,↓↑(k,iωn)]\n=2∆\nZ5iωnξ−.\nThus the spin-orbit coupling generates pairing correlatio ns that belong to\nthe odd-frequency spin triplet even-momentum even-band pa rity (OTEE)\nclass. In contrast in our formalism the band asymmetry gener ates interband\npairing correlation as\n(2.73)\n[F12,↑↓(k,iωn)+F12,↓↑(k,iωn)]+[F21,↑↓(k,iωn)+F21,↓↑(k,iωn)] = 0.\nThe interband pairing correlation due to band symmetry is\n(2.74)[F12,↑↓(k,iωn)+F12,↓↑(k,iωn)]−[F21,↑↓(k,iωn)+F21,↓↑(k,iωn)]\n=−4∆\nZ0(ξ2+ω2\nn+η2k2).✐✐\n“paper” — 2020/12/8 — 1:58 — page 15 — #15\n✐✐\n✐\n✐✐\n✐Effects of hybridization and spin-orbit coupling 15\nThese components belong to even-frequency spin-triplet ev en-momentum\nodd-band parity (ETEO) symmetry class. Thus, for spin tripl et, the spin\ndependent and spin independent hybridization both generat e the same sym-\nmetry class ETEO due to interband pairing correlation. The o dd frequency\npairing arises in the presence of spin independent hybridiz ation due to in-\ntraband pairing correlations.\n3. Conclusion\nWithin the theoretical model the existence of odd frequency pairs in two\nband superconductors by incorporating both spin independe nt hybridization\nand spin dependent spin-orbit interaction is investigated . This model also\nincludes both the one-particle hybridization term and all p ossible intraband\nand interband superconducting pairing interaction terms i n a two-band sys-\ntem.\nThe normal and anomalous thermal Green’s functions have bee n calcu-\nlated in the Nambu formalism as elements of the Fourier trans formed4×4\nmatrix Green’s function by taking into account of all possib le intraband\nand interband superconducting interaction terms coupling both bands in\nthe mean field approximation. By assuming that the attractive interaction\nacts on two electrons with different spins in different conduc tion bands dif-\nferent symmetry classes were demonstrated in the presence o f hybridization\nand spin-orbit coupling.\nThe role of intraband and interband pairing correlations to emerge the\nodd frequency in a two-band superconductor was examined. Fo r spin singlet,\nthe odd-frequency is generated by spin dependent hybridiza tion potential\nowing to intraband pairing correlations in agreement with t he odd frequency\ngenerated by the interband pair correlation due to band asym metry in Ref\n[3]. On the other hand, for spin triplet the spin independent hybridization\npotential generates the odd-frequency pairing due to intra band correlations\nin agreement with the result of Ref [12].\nReferences\n[1] M. Sigrist and K. Ueda, Phenomenological theory of unconventional su-\nperconductivity . Rev. Mod. Phys., 63, 239, (1991).\n[2] A. M. Black-Schaffer and A. V. Balasky, Proximity-induced unconven-\ntional superconductivity in topological insulators . Phys. Rev. B, 87,\n220506(R), (2013).✐✐\n“paper” — 2020/12/8 — 1:58 — page 16 — #16\n✐✐\n✐\n✐✐\n✐16 Moloud Tamadonpour and Heshmatollah Yavari\n[3] Y. Asano and A. Sasaki, Odd-frequency Cooper pairs in two-band su-\nperconductors and their magnetic response . Phys. Rev. B, 92, 224508,\n(2015).\n[4] V. A. Moskalenko and M. E. Palistrant, Two-band model determination\nof the critical temperature of a superconductor with an impu rity. Sov.\nPhys. JETP, 22, 526, (1966).\n[5] X. X. Xi, Two-band superconductor magnesium diboride . Rep. Prog.\nPhys., 71, 116501, (2008).\n[6] J. Nagamatsu, N. Nakagawa, T. Muranaka, Y. Zenitani, J. A kimutsu,\nSuperconductivity at 39 K in magnesium diboride . Nature, 410, 63,\n(2001).\n[7] Y. Shun-Li and L. Jian-Xin, Spin fluctuations and unconventional su-\nperconducting pairing in iron-based superconductors . Chin. Phys. B, 22,\n087411, (2013).\n[8] Y. Tanaka, P. M. Shirage and A. Iyo, Disappearance of Meissner Effect\nand Specific Heat Jump in a Multiband Superconductor . J Supercond\nNov Magn, 23, 253–256, (2010).\n[9] P. O. Sprau et al, Discovery of orbital-selective Cooper pairing in FeSe .\nscience, 357, 75-80, (2013).\n[10] A. M. Black-Schaffer and A. V. Balasky, Odd-frequency superconducting\npairing in multiband superconductors . Phys. Rev. B, 88, 104514, (2013).\n[11] Y. Asano and A. A. Golubov, Green’s-function theory of dirty two-band\nsuperconductivity . Phys. Rev. B, 97, 214508, (2018).\n[12] Y. Asano, A Sasaki and A. A. Golubov, Dirty two-band superconductivity\nwith interband pairing order . New J. Phys., 20, 043020, (2018).\n[13] L. Komendov’a, A. V. Balatsky, and A. M. Black-Schaffer, Band hy-\nbridization induced odd-frequency pairing in multiband su perconductors .\nPhys. Rev. B, 92, 094517, (2015).\n[14] Y. Fukaya, S. Tamura, K. Yada K, Y. Tanaka, P. Gentile, an d M. Cuoco,\nInterorbital topological superconductivity in spin-orbi t coupled supercon-\nductors with inversion symmetry breaking . Phys. Rev. B , 97, 174522,\n(2018).\n[15] V. P. Mineev and M. Sigrist, Basic theory of superconductivity in metals\nwithout inversion center . Lect. Notes Phys., 847, 129–154, (2012).✐✐\n“paper” — 2020/12/8 — 1:58 — page 17 — #17\n✐✐\n✐\n✐✐\n✐Effects of hybridization and spin-orbit coupling 17\n[16] M. Smidman, M. B Salamon, H. Q. Yuan and D. F. Agterberg, Spin-\ntriplet p-wave pairing in a three-orbital model for iron pni ctide super-\nconductors . Rep. Prog. Phys, 80, 036501, (2017).\n[17] P. Lee and X. G. Wen, S, Superconductivity and spin-orbit coupling in\nnon-centrosymmetric materials: A review . EPhys. Rev. B, 78, 2, (2008).\n[18] W. L. Yang et al, Evidence for weak electronic correlations in iron pnic-\ntidesPhys. Rev. B, 80, 014508, (2009).\n[19] P. Fazekas, Lecture Notes on Electron Correlation and Magnetism .\nWorld Scientific, Singapore, (1999).\n[20] H. Kuriyama et al, Epitaxially stabilized iridium espinel oxide without\ncations in the tetrahedral site . Appl. Phys. Lett., 96, 27010, (2010).\n[21] M. P. Christoph and H. Y. Kee, Identifying spin-triplet pairing in spin-\norbit coupled multi-band superconductors . EPL, 98, 2, (2011).\n[22] M. M. Korshunov and Y. N. Togushova, Spin-orbit coupling and im-\npurity scattering on the spin resonance peak in three orbita l model for\nFe-based superconductors . Journal of Siberian Federal University Math-\nematics and Physics, 11, 998, (2018).\n[23] L. P. Gor’kov, Theory of superconducting alloys in a strong magnetic\nfield near the critical tepmerature . JETP, 10, 998, (1960).\nDepartment of Physics, University of Isfahan,\nIsfahan 81746, Iran\nE-mail address :h.yavary@sci.ui.ac.ir✐✐\n“paper” — 2020/12/8 — 1:58 — page 18 — #18\n✐✐\n✐\n✐✐\n✐"    },    {        "title": "1205.2162v3.3D_quaternionic_condensations__Hopf_invariants__and_skyrmion_lattices_with_synthetic_spin_orbit_coupling.pdf",        "content": "arXiv:1205.2162v3  [cond-mat.quant-gas]  10 Feb 20163D quaternionic condensations, Hopf invariants, and skyrm ion lattices with synthetic\nspin-orbit coupling\nYi Li,1,2Xiangfa Zhou,3and Congjun Wu1\n1Department of Physics, University of California, San Diego , La Jolla, California 92093, USA\n2Princeton Center for Theoretical Science, Princeton Unive rsity, Princeton, NJ 08544\n3Key Laboratory of Quantum Information, University of Scien ce and Technology of China, CAS, Hefei, Anhui 230026, China\nWe study the topological configurations of the two-componen t condensates of bosons with the\n3D/vector σ·/vector pWeyl-type spin-orbit coupling subject to a harmonic trappi ng potential. The topology\nof the condensate wavefunctions manifests in the quaternio nic representation. In comparison to\ntheU(1) complex phase, the quaternionic phase manifold is S3and the spin orientations form\ntheS2Bloch sphere through the 1st Hopf mapping. The spatial distr ibutions of the quaternionic\nphases exhibit the 3D skyrmion configurations, and the spin d istributions possess non-trivial Hopf\ninvariants. Spin textures evolve from the concentric distr ibutions at the weak spin-orbit coupling\nregime to the rotation symmetry breaking patterns at the int ermediate spin-orbit coupling regime.\nIn the strong spin-orbit coupling regime, the single-parti cle spectra exhibit the Landau-level type\nquantization. In this regime, the three-dimensional skyrm ion lattice structures are formed when\ninteractions are below the energy scale of Landau level mixi ngs. Sufficiently strong interactions\ncan change condensates into spin-polarized plane-wave sta tes, or, superpositions of two plane-waves\nexhibiting helical spin spirals.\nPACS numbers: 03.75.Mn, 03.75.Lm, 03.75.Nt, 67.85.Fg\nI. INTRODUCTION\nQuantum mechanical wavefunctions generally speak-\ning are complex-valued. However, for the single com-\nponent boson systems, their ground state many-body\nwavefunctions are highly constrained, which are usu-\nally positive-definite [1], as a consequence of the Perron-\nFrobeniustheoreminthemathematicalcontextofmatrix\nanalysis [2]. This is a generalization of the “no-node”\ntheorem of the single-particle quantum mechanics, for\nexample, both the ground state wavefunctions of har-\nmonic oscillators and hydrogen atoms are nodeless. Al-\nthough the positive-definiteness does not apply to the\nmany-body fermion wavefunctions because Fermi statis-\ntics necessarilyleads to nodal structures, it remains valid\nfor many-body boson systems. It applies under the fol-\nlowing conditions: the Laplacian type kinetic energy, the\narbitrary single-particle potential, and the coordinate-\ndependent interactions. The positive-definiteness of the\nground state wavefunctions implies that time-reversal\n(TR) symmetry cannot be spontaneously broken in con-\nventional Bose-Einstein condensates (BEC), such as the\nsuperfluid4He and most ground state BECs of ultra-cold\nalkali bosons [3].\nIt would be interesting to seek unconventional BECs\nbeyond the constraint of positive-definite condensate\nwavefunctions [4]. The spin-orbit coupled boson systems\nareanideal platformto studythis classofexoticstatesof\nbosons, which can spontaneously breaking the TR sym-\nmetry. In addition to a simple Laplacian, the kinetic\nenergy contains the spin-orbit coupling term linearly de-\npendent on momentum. If the bare interaction is spin-\nindependent, the condensate wavefunctions are heavily\ndegenerate. An “order-from-disorder” calculation based\non the zero-point energy of the Bogoliubov spectra wasperform to select the condensate configuration[4]. Inside\nthe harmonic trap, it is predicted that the condensates\nspontaneously develop the half-quantum vortex coexist-\ning with 2D skyrmion-type spin textures [5]. Experi-\nmentally, spin-orbit coupled bosons have been realized\nin exciton systems in semi-conducting quantum wells.\nSpin texture configurations similar to those predicted in\nRef. [5] have been observed [6]. On the other hand,\nthe progress of synthetic artificial gauge fields in ultra-\ncold atomic gases greatly stimulates the investigation of\nthe above exotic states of bosons [7, 8]. Extensive stud-\nies have been performed for bosons with the 2D Rashba\nspin-orbit coupling, which exhibit various spin structures\narising from the competitions among the spin-orbit cou-\npling, interaction,andtheconfiningtrapenergy[5,9–16].\nMost studies so far have been on the two-dimensional\nspin-orbit coupled bosons. It would be interesting to fur-\nther consider the unconventional condensates of bosons\nwith the three-dimensional Weyl-type spin-orbit cou-\npling, whose experimental realization has been proposed\nby the authors through atom-light interactions in a com-\nbined tripod and tetrapod level system [20] and also by\nAnderson et al.[21]. As will be shown below, the qua-\nterinon representationprovides a natural and most beau-\ntiful description of the topological condensation configu-\nrations. Quaternions are an extension of complex num-\nbers as the first discovered non-commutative division al-\ngebra, which has provided a new formulation of quantum\nmechanics [17–19]. Similarly to complex numbers whose\nphasesspanaunitcircle S1, thequaternionicphasesspan\na three dimensional unit sphere S3. The spin distribu-\ntions associated with quaternionic wavefunctions are ob-\ntained through the 1st Hopf map S3→S2as will be\nexplained below. It would be interesting to search for\nBECswith non-trivialtopologicaldefects associatedwith2\nthe quaternionic phase structure. It will be a new class\nof unconventional BECs beyond the “no-node” theorem\nbreaking TR symmetry spontaneously.\nIn this article, we consider the unconventional conden-\nsatewavefunctionswiththe3DWeyl-typespin-orbitcou-\npling/vector σ·/vector p. The condensation wavefunctions exhibit topo-\nlogically non-trivial configurations as 3D skyrmions, and\nspin density distributions are also non-trivial with non-\nzero Hopf invariants. These topological configurations\ncan be best represented as defects of quaternion phase\ndistributions. Spatial distributions of the quaternionic\nphase textures and spin textures are concentric at weak\nspin-orbit couplings. As increasing spin-orbit coupling,\nthese textures evolve to lattice structures which are the\n3D quaternionic analogy of the 2D Abrikosov lattice of\nthe usual complex condensate.\nThe rest part of this article is organized as follows.\nIn Sect. II, we define the model Hamiltonian. In Sect.\nIII, the condensate wavefunctions in the weak spin-orbit\nregimearestudied. Topologicalanalysesonthe skyrmion\nconfigurations and Hopf invariants are performed by us-\ning the quaternion representation. In Sect. IV, the\nskyrmion lattice configuration of the spin textures is\nstudied in the intermediate and strong spin-orbit cou-\npling regimes. In Sect. V, superpositions of plane-wave\ncondensate configurations are studied. Conclusions are\nmade in Sect. VI.\nII. THE MODEL HAMILTONIAN\nWe consider a two-component boson system with the\n3D spin-orbit coupling of the /vector σ·/vector p-type confined in a har-\nmonic trap. The free part of the Hamiltonian is defined\nas\nH0=/integraldisplay\nd3/vector r ψ†\nγ(/vector r)/braceleftBig\n−/planckover2pi12/vector∇2\n2m+i/planckover2pi1λ/vector σγδ·(/vector∇)\n+1\n2mω2/vector r2/bracerightBig\nψδ(/vector r), (1)\nwhereγandδequal↑and↓referring to two internal\nstates of bosons; /vector σare Pauli matrices; mis the bo-\nson mass;λis the spin-orbit coupling strength with the\nunit of velocity; ωis the trap frequency. At the single-\nparticle level, Eq. (1) satisfies the Kramer-type time-\nreversal symmetry of T= (−iσ2)Cwith the property of\nT2=−1. However, parity is broken by spin-orbit cou-\npling. In the absence ofthe trap, good quantum numbers\nfor the single-particle states are the eigenvalues ±1 of he-\nlicity/vector σ·/vector p/|p|, wherepis the momentum. This results in\ntwo branches of dispersions\nǫ±(/vectork) =/planckover2pi12\n2m(k∓kso)2, (2)\nwhere/planckover2pi1kso=mλ. The lowest single-particle energy\nstates lie in the sphere with the radius ksodenoted as the\nspin-orbit sphere. It corresponds to a spin-orbit lengthscalelso= 1/ksoin real space. The harmonic trap has a\nnatural length scale lT=/radicalBig\n/planckover2pi1\nmω, and thus the dimension-\nless parameter α=lTksodescribes the relative spin-orbit\ncoupling strength.\nAs for the interaction Hamiltonian, we use the contact\ns-wave scattering interaction defined as\nHint=gγδ\n2/integraldisplay\nd3/vector r ψ†\nγ(/vector r)ψ†\nδ(/vector r)ψδ(/vector r)ψγ(/vector r).(3)\nTwo different interaction parameters are allowed, in-\ncluding the intra and inter-component ones defined as\ng↑↑=g↓↓=g, andg↑↓=cg, wherecis a constant.\nIn the previous study of the 2D Rashba spin-orbit cou-\npling with harmonic potentials [5, 15], the single-particle\neigenstates are intuitively expressed in the momentum\nrepresentation: the low energy state lies around a ring in\nmomentum space, and the harmonic potential becomes\nthe planar rotor operator on this ring subject to a π-\nflux, which quantizes the angular momentum jzto half\nintegers. Similar picture also applies in 3D [5, 22]. The\nlow energy states are around the spin-orbit sphere. In\nthe projected low energy Hilbert space, the eigenvectors\nread\nψ+(/vectork) = (cosθk\n2,sinθk\n2eiφk)T. (4)\nThe harmonic potential is again a rotor Hamiltonian on\nthe spin-orbit sphere subject to the Berry gauge connec-\ntion as\nVtp=1\n2m(i∇k−/vectorAk)2(5)\nwith the moment of inertial I=Mkk2\nsoandMk=\n/planckover2pi12/(mω2)./vectorAk=i/angb∇acketleftψ+(/vectork)|∇k|ψ+(/vectork)/angb∇acket∇ightis the vector po-\ntential of a U(1) magnetic monopole, which quantizes\nthe angular momentum jto half-integers. While the ra-\ndial energy is still quantized in terms of /planckover2pi1ω, the angular\nenergydispersion with respect to jis stronglysuppressed\nat large values of αas\nEnr,j,jz≈/parenleftBig\nnr+j(j+1)\n2α2/parenrightBig\n/planckover2pi1ω+const,(6)\nwherenris the radial quantum number. As further\nshown in Ref. [20], in the case α≫1, all the states\nwith the same nrbut different jandjzare nearly degen-\nerate, thus can be viewed as one 3D Landau level with\nspherical symmetry but the broken parity. If filled with\nfermions, the system belongs to the Z2-class of 3D strong\ntopological insulators.\nNow we load the system with bosons. The interaction\nenergy scale is defined as Eint=gN0/l3\nT, whereN0is\nthe total particle number in the condensate. The corre-\nsponding dimensionless parameter is β=Eint//planckover2pi1ω. At\nthe Hartree-Fock level, the Gross-Pitaevskii energy func-\ntional is defined in terms of the condensate wavefunction3\nΨ = (Ψ ↑,Ψ↓)Tas\nE=/integraldisplay\nd3/vector r(Ψ†\n↑,Ψ†\n↓)/braceleftBig\n−/planckover2pi12∇2\n2m−iλ/planckover2pi1/vector∇·/vector σ+1\n2mω2r2\n+g/parenleftbigg\nn↑+cn↓0\n0cn↑+n↓/parenrightbigg/bracerightBig/parenleftbigg\nΨ↑\nΨ↓/parenrightbigg\n, (7)\nwheren↑,↓(/vector r) =N0|Ψ↑,↓(/vector r)|2are the particle densities of\ntwo components, respectively, and Ψ( /vector r) is normalized as/integraltext\nd3/vector rΨ†(/vector r)Ψ(/vector r) = 1. The condensate wavefunction Ψ( /vector r)\nis solved numerically by using the standard method of\nimaginary time evolution. The dimensionless form of the\nGross-Pitaevskii equation is\nE′=/integraldisplay\nd3/vector r′(˜Ψ†\n↑,˜Ψ†\n↓)/braceleftBig\n−/vector∇′2\n2−iα/vector∇′·/vector σ+r′2\n2\n+β/parenleftbigg\n˜n↑+c˜n↓0\n0c˜n↑+ ˜n↓/parenrightbigg/bracerightBig/parenleftbigg˜Ψ↑\n˜Ψ↓/parenrightbigg\n,(8)\nwhereE′=E/(/planckover2pi1ω),/vector∇′=lT/vector∇;/vector r′=/vector r/lT;˜Ψ↑and˜Ψ↓\narethe renormalizedcondensate wavefunctionssatisfying/integraltext\nd3r′|˜Ψ↑|2+|˜Ψ↓|2= 1; ˜n↑=|˜Ψ↑|2and ˜n↓=|˜Ψ↓|2.\nIII. THE WEAK SPIN-ORBIT COUPLING\nREGIME\nIn this section, we consider the condensate configura-\ntion in the limit of weak spin-orbit coupling, say, α∼1.\nIn this regime, the single-particle spectra still resemble\nthose of the harmonictrap. We study the case that inter-\nactions arenot strongenough to mix stateswith different\nangular momenta.\nA. The spin-orbit coupled condensate\nIn this regime, the condensate wavefunction Ψ remains\nthe same symmetry structure as the single-particle wave-\nfunction over a wide range of interaction parameter β,\ni.e., Ψ remains the eigenstates of j=1\n2as confirmed\nnumerically below. Ψ can be represented as\nΨj=jz=1\n2(r,ˆΩ) =f(r)Y+\nj,jz(ˆΩ)+ig(r)Y−\nj,jz(ˆΩ),(9)\nwheref(r) andg(r) are real radial functions. Y±\nj,jz(ˆΩ)\nare the spin-orbit coupled spherical harmonic functions\nwith even and odd parities, respectively. For example,\nfor the case of j=jz=1\n2, they are\nY+\n1\n2,1\n2(r,ˆΩ) =/parenleftbigg\n1\n0/parenrightbigg\n, Y−\n1\n2,1\n2(r,ˆΩ) =/parenleftbiggcosθ\nsinθeiφ/parenrightbigg\n,(10)\nwhose orbital partial-wavecomponents are sandp-wave,\nrespectively. The TR partner of Eq. (9) is ψjz=−1\n2=\nˆTψj=jz=1\n2=iσ2ψ∗\nj=jz=1\n2. The two terms in Eq. (9)\nare of opposite parity eigenvalues, mixed by the paritybreaking spin-orbit coupling /vector σ·/vector p. The coefficient iof the\nY−\njjzterm is because the matrix element /angb∇acketleftY+\njjz|/vector σ·/vector p|Y−\njjz/angb∇acket∇ight\nis purely imaginary.\nFor the non-interacting case, the radial wavefunctions\nuptoaGaussianfactorcanbeapproximatedbyspherical\nBessel functions as\nf(r)≈j0(ksor)e−r2/2l2\nT, g(r)≈j1(ksor)e−r2/2l2\nT,(11)\nwhich correspond to the sandp-partial waves, respec-\ntively. Both of them oscillate along the radial direction\nand the pitch values are around kso. Atr= 0,f(r)\nreaches the maximum and g(r) is 0. As rincreases,\nroughly speaking, the zero points of f(r) corresponds to\nthe extrema of g(r) and vise versa. Repulsive interac-\ntions expand the spatial distributions of f(r) andg(r),\nbut the above picture still holds qualitatively. In other\nwords, there is aπ\n2-phase shift between the oscillations\noff(r) andg(r).\nB. The quaternion representation\nCan we have unconventional BECs with non-trivial\nquaternionic condensate wavefunctions? Actually, the\ntopological structure of condensate wavefunction Eq. (9)\nmanifests clearly in the quaternion representation as\nshown below.\nWe define the following mapping from the complex\ntwo-component vector Ψ = (Ψ ↑,Ψ↓)Tto a quaternion\nvariable through\nξ=ξ0+ξ1i+ξ2j+ξ3k, (12)\nwhere\nξ0= ReΨ ↑,ξ1= ImΨ ↓,ξ2=−ReΨ↓,ξ3= ImΨ ↑.(13)\ni,j,kare the imaginary units satisfying i2=j2=k2=\n−1, and the anti-commutation relation ij=−ji=k.\nThe TR transformation on ξis just−jξ.\nEq. (9) can be expressed in the quaternionic exponen-\ntial form as\nξj=jz=1\n2(r,ˆΩ) =|ξ(r)|e/vector ω(ˆΩ)γ(r)=|ξ|(cosγ+/vector ωsinγ),(14)\nwhere\n|ξ(r)|= [f2(r)+g2(r)]1\n2,\n/vector ω(ˆΩ) = sinθcosφ i+sinθsinφ j+cosθ k,\ncosγ(r) =f(r)/|ξ(r)|,sinγ(r) =g(r)/|ξ(r)|.(15)\nω(ˆΩ) is the imaginary unit along the direction of ˆΩ sat-\nisfying/vector ω2(ˆΩ) =−1. According to the oscillating proper-\nties off(r) andg(r),γ(r) spirals as rincreases. At the\nn-th zero point of g(r) denotedrn,γ(rn) =nπwhere\nn≥0 and we define r0= 0, while at the n-th zero point\noff(r) denotedr′\nn,γ(r′\nn) = (n−1\n2)πwheren≥1.\nIn 3D, the condensate wavefunctions can be topolog-\nically non-trivial because the homotopy group of the4\nquaternionic phase is π3(S3) =Z[23, 24]. The corre-\nsponding winding number, i.e. the Pontryagin index, of\nthe mapping S3→S3is the 3D skyrmion number. The\nspatial distribution of the quaternionic phase e/vector ω(ˆΩ)γ(r)\ndefined in Eq. 14, which lies on S3, exhibits a topo-\nlogically nontrivial mapping from R3toS3, i.e., a 3D\nmultiple skyrmion configuration. This type of topolog-\nical defects are non-singular which is different from the\nusual vortex in single component BEC. In realistic trap-\nping systems, the coordinate space is the open R3. At\nlarge distance r≫lT,|ξ(r)|decays exponentially, where\nthe quaternionic phase and the mapping are not well-defined. Nevertheless, in each concentric spherical shell\nwithrn<r<r n+1,γ(r) winds from nπto (n+1)π, and\nω(ˆΩ) covers all the directions, thus this shell contributes\n1 to the winding number of e/vector ω(ˆΩ)γ(r)onS3. If the system\nsize is truncated at the order of lT, the skyrmion number\ncan be approximated at the order of lTkso=α.\nThere exists an interesting difference from the previ-\nously studied 2D case: Although the spin density dis-\ntribution exhibit the 2D skyrmion configuration due to\nπ2(S2) [5, 15, 16], the 2D condensation wavefunctions\nhave no well-defined topology due to π2(S3) = 0.\nFIG. 1: The distribution of /vectorS(/vector r) in a) the xz-plane and in the horizontal planes with b)z= 0 andc)z/lT=1\n2.\nThe unit length is set as lT= 1 in all the figures in this article. The color scale shows the magnitude of out-plane\ncomponent Syina) andSzinb) andc). The parameter values are α= 1.5,c= 1, andβ= 30, and the length unit\nin these and all the figures below is lT.\nC. The Hopf mapping and Hopf invariant\nExotic spin textures in spinor condensates have been\nextensively investigated [25–27]. In our case, the 3D spin\ndensity distributions /vectorS(/vector r) exhibit a novel configuration\nwith non-trivial Hopf invariants due to the non-trivial\nhomotopy group π3(S2) =Z[23, 24]./vectorS(/vector r) can be ob-\ntained from ξ(r) through the 1st Hopf map defined as\n/vectorS(/vector r) =1\n2ψ†\nγ/vector σγβψδ, or, in the quaternionicrepresentation,\n1\n2¯ξkξ=Sxi+Syj+Szk, (16)\nwhere¯ξ=ξ0−ξ1i−ξ2j−ξ3kisthequaternionicconjugate\nofξ. The Hopf invariant of the 1st Hopf map is just 1\n[24]. The real space concentric spherical shell rn< r <\nrn+1maps to the quaternionic phase S3, and the latter\nfurthermapstothe S2Blochspherethroughthe1stHopf\nmap. The winding number of the first map is 1, and the\nHopf invariant of the second map is also 1, thus the Hopf\ninvariantoftheshell rn<r<r n+1toS2is1. Rigorously\nspeaking, the magnitude of /vectorS(/vector r) decays exponentially at\nr≫lT, and thus the total Hopf invariant is not well-\ndefined in the open R3space. Again, if we truncate thesystem size at lT, the Hopf invariant is approximately at\nthe order of α.\nFIG. 2: The Hopf fibration of the spin texture configura-\ntion in Fig. 1. Every circle represents a spin orientation,\nand every two circles are linked with the linking number\n1.\nNext we present numeric results for the spin textures\nassociated with the condensation wavefunction Eq. 9 as5\nplotted in Fig. 1. Explicitly, /vectorS(/vector r) is expressed as\n/bracketleftbigg\nSx(/vector r)\nSy(/vector r)/bracketrightbigg\n=g(r)sinθ/bracketleftbigg\ncosφ−sinφ\nsinφcosφ/bracketrightbigg/bracketleftbigg\ng(r)cosθ\nf(r)/bracketrightbigg\n,\nSz(/vector r) =f2(r)+g2(r)cos2θ, (17)\nIn thexz-plane, the in-plane components SxandSzform\na vortex in the half plane of x >0 andSyis prominent\nin the core. The contribution at large distance is ne-\nglected, where /vectorS(/vector r) decays exponentially. Due to the\naxial symmetry of /vectorS(/vector r) in Eq. 17, the 3D distribution is\njust a rotation of that in Fig. 1 a) around the z-axis. In\nthexy-plane, spin distribution exhibits a 2D skyrmion\npattern, whose in-plane components are along the tan-\ngential direction. As the horizontal cross-section shifted\nalong thez-axis,/vectorS(/vector r) remains 2D skyrmion-like, but its\nin-plane components are twisted around the z-axis. The\nspin configuration at z=−z0can be obtained by a com-bined operation of TR and rotation around the y-axis\n180◦, thus its in-plane components are twisted in an op-\nposite way compared to those at z=z0. Combining\nthe configurations on the vertical and horizontal cross\nsections, we complete the 3D distribution of /vectorS(/vector r) with\nnon-zero Hopf invariant.\nThe non-trivial structure of the Hopf invariant of the\nabove spin configuration can be revealed by plotting its\nHopf fibration in terms of the linked non-crossing circles\nin real space, as shown in Fig. 2. For all the points on\neach circle, their normalized spin polarizations /angb∇acketleft/vector σ/angb∇acket∇ight/|/angb∇acketleft/vector σ/angb∇acket∇ight|\nare the same, corresponding to a single point on the S2\nsphere. Inaddition, everytwocirclesarelinkedwith each\nother with the linking number 1, which is the standard\nHopf bundle structure describing a many-to-one map\nfromS3toS2. Ultracoldbosonswithsyntheticspin-orbit\ncoupling providea novel platform to study such beautiful\nmathematical ideas in realistic physics systems.\nFIG. 3: The distribution of /vectorS(/vector r) in horizontal cross-sections with a) z/lT=−0.5, b)z/lT= 0, c)z/lT= 0.5,\nrespectively. The color scale shows the value of Sz, and parameter values are α= 4,β= 2, andc= 1.\nIV. THE INTERMEDIATE AND STRONG\nSPIN-ORBIT COUPLING REGIME\nA. The intermediate spin-orbit coupling strength\nNext we consider the case of the intermediate spin-\norbit coupling strength, i.e., 1 < α <10, at which the\nsingle-particle spectra evolve from the case of the har-\nmonic potential to Landau level-like as shown in Eq. 6.\nInteractions are sufficiently strong to mix a few lowest\nenergy states with different angular momenta j. As a re-\nsult, rotationalsymmetryisbrokenandcomplexpatterns\nappear.\nIn this case, the topology of condensate wavefunctions\nisstill 3Dskyrmion-likemapping from R3toS3, andspin\ntextures with the non-trivial Hopf invariant are obtained\nthrough the 1st Hopf map. Compared to the weak spin-\norbit coupling case, the quaternionic phase skyrmions\nand spin textures are no longer concentric, but split to a\nmulti-centered pattern. The numeric results of /vectorS(/vector r) areplotted in Fig. 3 for different horizontal cross-sections.\nIn thexy-plane,/vectorSexhibits the 2D skyrmion pattern as\nshown in Fig. 3 ( b): The in-plane components form two\nvortices and one anti-vortex, while Sz’s inside the vortex\nand anti-vortex cores are opposite in direction, thus they\ncontribute to the skyrmion number with the same sign.\nThe spin configuration at z=z0>0 is shown in Fig. 3\n(a), which is twisted around the z-axis clock-wise. After\nperforming the combined TR and rotation around the y-\naxis 180◦, we arrive at the configuration at z=−z0in\nFig. 3(b).\nB. The strong spin-orbit coupling regime\nWe next consider the case of strong spin-orbit cou-\npling, i.e., α≫1. The single-particle spectra already\nexhibit the Landau-level type quantization in this regime\nas shown in Eq. 6. The single-particle eigenstates with\nnr= 0 are nearly degenerate i.e., they form the low-6\nest Landau level states. We assume that the interaction\nstrength is enough to mix states inside the lowest Lan-\ndau level but is still relatively weak not to induce inter-\nLandau level mixing.\nIn this regime, the length scale of each skyrmion is\nshortened as enlarging the spin-orbit coupling strength.\nAs we can imagine, more and more skyrmions appear\nand will form a 3D lattice structure, which is the SU(2)\ngeneralization of the 2D Abrikosov lattice of the usual\nU(1) superfluid. We have numerically solved the Gross-\nPitaevskiiequationEq. 7andfound thelattice structure:\nEach lattice site is a single skyrmion of the condensate\nwavefunction ξ(/vector r), whose spin configuration exhibits the\ntexture configuration approximately with a unit Hopf in-\nvariant. The numeric results for the spin texture config-uration are depicted in Fig. 4 a) andb) for two different\nhorizontal cross sections parallel to the xy-plane. In each\ncross section, spin textures form a square lattice, and the\nlattice constant dis estimated approximately the spin-\norbit length scale as\nd≃2πlso= 2πlT/α. (18)\nFor two horizontal cross sections with a distance of\n∆z≃d/2, their square lattice configurations are dis-\nplacedalongthediagonaldirection: Thesitesatonelayer\nsit abovethe plaquette centersofthe adjacentlayer. As a\nresult, the overall three-dimensional configuration of the\ntopological defects is a body-centered cubic ( bcc) lattice,\nand its size is finite confined by the trap.\nFIG. 4: The distribution of /vectorS(/vector r) in horizontal cross-sections with (a) z/lT= 0, (b)z/lT= 0.2, respectively. The color\nscale shows the value of Szand parameter values are α= 22,β= 1, andc= 1. The overall lattice exhibits the bcc\nstructure.\nV. THE EFFECT OF STRONG INTERACTIONS\nIn this section, we present the condensate configura-\ntions in the case that both spin-orbit coupling and inter-\nactions are strong, such that different Landau levels are\nmixed by interactions.\nIn this case, the effect of the harmonic trapping poten-\ntial becomes weak compared with interaction energies,\nthus we can approximate the condensate wavefunctions\nas superpositions of plane-wave states. The plane-wave\ncomponents are located on the spin-orbit sphere and the\ncondensate wavefunctions are no longer topological. At\nc= 1, the interaction is spin-independent, and bosons\nselect a superposition of a pair of states ±/vectorkon the spin-\norbit sphere, say, ±ksoˆz. The condensate wavefunction\nis written as\nψ(/vector r) =/radicalbigg\nNa\nN0eiksoz| ↑/angb∇acket∇ight+/radicalbigg\nNb\nN0e−iksoz| ↓/angb∇acket∇ight,(19)withNa+Nb=N0. The density of Eq. 19 in real\nspace is uniform to minimize the interaction energy at\nthe Hartree-Fock level. However, all the different parti-\ntions ofNa,byield the same Hartree-Fock energy. The\nquantum zero point energy from the Bogoliubov modes\nremoves this accidental degeneracy through the “order-\nfrom-disorder” mechanism, which selects the equal par-\ntitionNa=Nb. The calculation is in parallel to that\nof the 2D Rashba case performed in Ref. [5], thus will\nnot be presented here. In this case, the condensate is a\nspin helix propagates along z-axis and spin spirals in the\nxy-plane.\nAtc/negationslash= 1, the spin-dependent part of the interaction\ncan be written as\nHsp=1−c\n2g/integraldisplay\nd3r(ψ†\n↑ψ↑−ψ†\n↓ψ↓)2.(20)\nAtc >1, the interaction energy at Hartree-Fock level\nis minimized for the condensate wavefunction of a plane\nwave state eiksoz| ↑/angb∇acket∇ight, or, its TR partner.7\nForc<1,/angb∇acketleftHsp/angb∇acket∇ightis minimized if /angb∇acketleftSz/angb∇acket∇ight= 0 in space. At\nthe Hartree-Fock level, the condensate can either be a\nplane-wave state with momentum lying in the equator of\nthe spin-orbit sphere and spin polarizing in the xy-plane,\nor, the spin spiral state described by Eq. 19 with Na=\nNb. An“order-from-disorder”analysisontheBogoliubov\nzero-point energies indicates that the spin spiral stateis selected. We also present the numerical results for\nEq. (4) in the main text with a harmonic trap in Fig.\n5 for the case of c <1. The condensate momenta of\ntwo spin components have opposite signs, thus the trap\ninhomogeneity already prefers the spin spiral state Eq.\n19 at the Hartree-Fock level.\nFIG. 5: The density profile (a) for ↑-component, and that for ↓-component is the same. Phase profiles for (b) ↑and\n(c)↓- components, respectively. Parameter values are a= 10,β= 50, andc= 0.5.\nVI. CONCLUSION\nIn summary, we have investigated the two-component\nunconventional BECs driven by the 3D spin-orbit\ncoupling. In the quaternionic representation, the\nquaternionic phase distributions exhibit non-trivial 3D\nskyrmionconfigurationsfrom R3toS3. Thespinorienta-\ntion distributions exhibit texture configurations charac-\nterizedby non-zeroHopfinvariantsfrom R3toS2. These\ntwo topological structures are connected through the 1st\nHopf map from S3toS2. At large spin-orbit coupling\nstrength, the crystalline order of spin textures, or, wave-\nfunction skyrmions, are formed, which can be viewed as\na generalization of the Abrikosov lattice in 3D.\nNote added.— Near the completion of this manuscript,\nwe became aware of a related work by Kawakami et al.[28], in which the condensate wavefunction in the weak\nspin-orbit coupling case was studied.\nAcknowledgments.— Y.L. thanks the Princeton Cen-\nter for Theoretical Science at Princeton University for\nsupport. X. F. Z. acknowledges the support of NFRP\n(2011CB921204, 2011CBA00200), the Strategic Priority\nResearch Program of the Chinese Academy of Sciences\n(Grant No. XDB01030000),NSFC (11004186,11474266),\nand the Major Research plan of the National Natural\nScience Foundation of China (91536219). C. W. is sup-\nported by the NSF DMR-1410375 and AFOSR FA9550-\n14-1-0168. C. W. acknowledges the support from the\nPresidents Research Catalyst Awards of University of\nCalifornia, and National Natural Science Foundation of\nChina (11328403).\n[1] R. P. Feynman, Statistical Mechanics, A Set of Lectures\n(Berlin: Addison-Wesley, 1972).\n[2] R. B. Bapat and T. Raghavan, Non-Negative Matri-\nces and Applications (Cambridge University Press, Cam-\nbridge, United Kingdom, 1997).\n[3] A. J. Leggett, Rev. Mod. Phys. 73, 307 (2001).\n[4] C. Wu, Mod. Phys. Lett. 23, 1 (2009).\n[5] C. Wu , I. Mondragon-Shem, arXiv:0809.3532; C. Wu ,\nI. Mondragon-Shem, and X. F. Zhou, Chin. Phys. Lett.,\n28, 097102 (2011).\n[6] A.A. High et al., Nature 483, 584 (2012). A.A. High et\nal., arXiv:1103.0321.\n[7] Y.-J. Lin et al., Nature 462, 628 (2009).[8] Y.-J. Lin et al., Nature 471, 83 (2011).\n[9] T. Stanescu et al., Phys. Rev. A 78, 023616 (2008).\n[10] T.-L. Ho et al., Phys. Rev. Lett. 107, 150403 (2011).\n[11] C. Wang et al., Phys. Rev. Lett. 105, 160403 (2010).\n[12] S.-K. Yip, Phys. Rev. A 83, 043616 (2011).\n[13] Y. Zhang et al., Phys. Rev. Lett. 108, 035302 (2012).\n[14] X.-F. Zhou et al., Phys. Rev. A 84, 063624 (2011).\n[15] H. Hu et al., Phys. Rev. Lett. 108, 010402(2012).\n[16] S. Sinha, R. Nath, and L. Santos, arXiv:1109.2045.\n[17] A. V. Balatsky, cond-mat/9205006.\n[18] S. L. Adler, Quaternionic Quantum Mechanics and\nQuantum Fields (OxfordUniversityPress, Oxford, 1995).\n[19] D.Finkelstein et al., J.Math.Phys.(N.Y.) 3, 207(1962).8\n[20] Y. Li et al., Phys. Rev. B 85, 125122 (2012).\n[21] B. M. Anderson et al., arXiv:1112.6022.\n[22] S. K. Ghosh et al., Phys. Rev. A, 84, 053629 (2011).\n[23] F. Wilczek and A. Zee, Phys. Rev. Lett. 51, 2250 (1983).\n[24] M. Nakahara, Geometry, topology, and physics , (Taylor\n& Francis, 2003)\n[25] F. Zhou, Int. J. Mod. Phys. B 17, 2643-2698 (2003); E.Demler and F. Zhou, Phys. Rev. Lett. 88, 163001 (2002).\nG. W.SemenoffandF.Zhou, Phys.Rev.Lett. 98, 100401\n(2007).\n[26] J. Zhang and T. L. Ho, arXiv:0908.1593.\n[27] D. M. Stamper-Kurn, and M. Ueda, arXiv:1205.1888.\n[28] T. Kawakami et al., arXiv:1204.3177."    },    {        "title": "1004.3066v1.Spin_Orbit_Coupling_and_Spin_Waves_in_Ultrathin_Ferromagnets__The_Spin_Wave_Rashba_Effect.pdf",        "content": "arXiv:1004.3066v1  [cond-mat.mes-hall]  18 Apr 2010Spin Orbit Coupling and Spin Waves in Ultrathin Ferromagnet s:\nThe Spin Wave Rashba Effect\nA. T. Costa,1R. B. Muniz,1S. Lounis,2A. B. Klautau,3and D. L. Mills2\n1Instituto de F´ ısica, Universidade Federal Fluminense, 24 210-340 Niter´ oi, RJ, Brasil.\n2Department of Physics and Astronomy,\nUniversity of California Irvine, California, 92697, U. S. A .\n3Departamento de Fisica, Universidade Federal do Par´ a, Bel ´ em, PA, Brazil.\nAbstract\nWe present theoretical studies of the influence of spin orbit coupling on the spin wave excitations\nof the Fe monolayer and bilayer on the W(110) surface. The Dzy aloshinskii-Moriya interaction is\nactive in such films, by virtue of the absence of reflection sym metry in the plane of the film. When\nthe magnetization is in plane, this leads to a linear term in t he spin wave dispersion relation for\npropagation across the magnetization. The dispersion rela tion thus assumes a form similar to\nthat of an energy band of an electron trapped on a semiconduct or surfaces with Rashba coupling\nactive. We also show SPEELS response functions that illustr ate the role of spin orbit coupling in\nsuch measurements. In addition to the modifications of the di spersion relations for spin waves, the\npresence of spin orbit coupling in the W substrate leads to a s ubstantial increase in the linewidth\nof the spin wave modes. The formalism we have developed appli es to a wide range of systems,\nand the particular system explored in the numerical calcula tions provides us with an illustration\nof phenomena which will be present in other ultrathin ferrom agnet/substrate combinations.\n1I. INTRODUCTION\nThestudyofspindynamicsinultrathinferromagnetsisoffundamen talinterest, sincenew\nphysics arises in these materials that has no counterpart in bulk mag netism. Examples are\nprovided by relaxation mechanisms evident in ferromagnetic resona nce and Brillouin light\nscattering studies,1–3and also for the large wave vectors probed by spin polarized electro n\nloss spectroscopy (SPEELS).4Of course, by now the remarkable impact of ultrathin film\nstructures on magnetic data storage is very well known, and othe r applications that exploit\nspin dynamics in such materials are envisioned. Thus these issues are important from a\npractical point of view as well as from that of fundamental physics .\nTheoretical studies of the nature of spin waves in ultrathin films ads orbed on metal sub-\nstrates have been carried out for some years now, along with comp arison with descriptions\nprovided with the Heisenberg model.5In this paper, we extend the earlier theoretical treat-\nments to include the influence of spin orbit coupling on the spin wave sp ectrum of ultrathin\nfilms. This extension is motivated by a most interesting discussion of t he ground state of the\nMn monolayer on the W(110) surface. A nonrelativistic theoretical study of this system pre-\ndicted that the ground state would be antiferromagnetic in charac ter.7This prediction was\nconfirmed by spin polarized scanning tunneling microscope studies of the system.8However,\nrecent experimental STM data with a more sensitive instrument sho wed a more complex\nground state, wherein the ground state is in fact a spin density wav e.9One can construct\nthe new state by beginning with the antiferromagnet, and then sup erimposing on this a\nlong wavelength modulation on the direction of the moments on the lat tice. The authors\nof ref. 9 argued that the lack of reflection symmetry of the syste m in the plane of the film\nactivates theDzyaloshinskii Moriya (DM) interaction, andthe news tate hasitsorigininthis\ninteraction. They also presented relativistic and ab initio calculations that gave an excellent\naccount of the new data. The reflection symmetry is broken simply b y the presence of the\nsubstrate upon which the film is grown. This argument to us is most int riguing, since one\ncan then conclude that the DM interaction must be active in any ultra thin ferromagnet;\nthe substrate is surely always present. The DM interaction has its o rigin in the spin orbit\ninteraction, which of course is generally very weak in magnets that in corporate the 3d tran-\nsition elements as the moment bearing entities. However, in the case of the Mn monolayer\non W(110) hybridization between the Mn 3d and the W 5d orbitals activ ates the very large\n2W spin orbit coupling, with the consequence that the strength of th e DM interaction can\nbe substantial, as illustrated by the calculations presented in ref. 9 . One may expect to see\nsubstantial impact of the DMinteraction in other ultrathin magnets grown on 5d substrates,\nand possibly 4d substrates as well.\nWe have here another example of new physics present in ultrathin ma gnets that is not\nencountered inthebulk formofthematerial fromwhich theultrath instructure isfabricated.\nThe purpose of this paper is to present our theoretical studies of spin orbit effects on spin\nwaves and also on the dynamic susceptibility of a much studied ultrath in film/substrate\ncombination, theFemonolayer andbilayer onW(110). Wefindstriking effects. Forinstance,\nwhen the magnetization is in plane, as we shall see the DM interaction in troduces a term\nlinear in wave vector in the dispersion relation of spin waves. Thus the uniform spin wave\nmode at zero wave vector acquires a finite group velocity. We find th is to be in the range\nof 2×105cm/sec for the Fe monolayer on W(110). Furthermore, left/right asymmetries\nappear in the SPEELS response functions. Thus, we shall see that spin orbit coupling has\nclear effects on the spin excitations of transition metal ultrathin fe rromagnets grown on 5d\nsubstrates.\nWe comment briefly on the philosophy of the approach used here, an d in various earlier\npublications.5Numerous authors proceed as follows. One may generate a descrip tion of the\nmagnetic ground state of the adsorbed films by means of an electro nic structure calculation\nbased on density functional theory. It is then possible to calculate , within the framework of\nanadiabaticapproximation, effectiveHeisenbergexchangeintegra lsJijbetweenthemagnetic\nmoments in unit cell i and unit cell j. These may be entered into a Heise nberg Hamilto-\nnian, and then spin wave dispersion relations may be calculated throu gh use of spin wave\ntheory. It has been known for decades10that in the itinerant 3d magnets, effective exchange\ninteractions calculated in such a manner have very long range in real space. Thus, one must\ninclude a very large number of distant neighbors in order to obtain co nverged results. This\nis very demanding to do with high accuracy for the very numerous dis tant neighbors, since\nthe exchange interactions become very small as one moves out into distant neighbor shells.\nAt a more fundamental level, as noted briefly above, discussions in e arlier publications\nshow that in systems such as we study here, the adiabatic approxim ation breaks down badly,\nwith qualitative consequences.5First, spin wave modes of finite wave vector have very short\nlifetimes, by virtue of decay into the continuum of particle hole pairs ( Stoner excitations)\n3even at the absolute zero of temperature5,11whereas in Heisenberg model descriptions their\nlifetime is infinite. In multi layer films, the earlier calculations show that as a consequence of\nthe short lifetime, the spectrum of spin fluctuations at large wave v ectors contains a single\nbroad feature which disperses with wave vector in a manner similar to that of a spin wave;\nthis is consistent with SPEELS data on an eight layer film of Co on Cu(10 0).4This picture\nstands in contrast to that offered by the Heisenberg model, in which a film of N layers has\nN spin wave modes for each wave vector, and each mode has infinite lif etime.\nThe method developed earlier, and extended here to incorporate s pin orbit coupling,\ntakes due account of the breakdown of the adiabatic approximatio n and also circumvents\nthe need to calculate effective exchange interactions in real space out to distant neighbor\nshells. Weworkdirectlyinwave vectorspacethroughstudyofthew ave vectorandfrequency\ndependent susceptibility discussed below, denotedas χ+,−(/vectorQ/bardbl,Ω;l⊥,l′\n⊥). Theimaginarypart\nof this object, evaluated for l⊥=l′\n⊥and considered as a function of frequency Ω for fixed\nwave vector /vectorQ/bardblprovides us with the frequency spectrum of spin fluctuations on lay erl⊥for\nthe wave vector chosen. Spin waves appear as peaks in this functio n, very much as they do\nin SPEELS data, and in a manner very similar to that used by experimen talist we extract\na dispersion relation for spin waves by following the wave vector depe ndence of the peak\nfrequency. We never need to resort to a real space summation pr ocedure over large number\nof neighbors, coupled by very tiny exchange couplings. The spin wav e exchange stiffness\ncan be extracted either by fitting the small wave vector limit of the d ispersion relation so\ndetermined, or alternatively by utilizing an expression derived earlier5which once againdoes\nnot require a summation in real space.\nWecommentonanotherfeatureofthepresentstudy. Inearlierc alculations,5,11,14asinthe\npresent paper, an empirical tight binding description forms the bas is for our description of\nthe electronic structure. Within this approach, referred to as a m ulti band Hubbard model,\nwe can generate the wave vector and frequency dependent susc eptibility for large systems.\nIn the earlier papers, effective tight binding parameters were extr acted from bulk electronic\nstructure calculations. The present studies are based on tight bin ding parameters obtained\ndirectly from a RS-LMTO-ASA calculation for the Fe/W(110) system . We also obtain tight\nbinding parameters by fitting KKR based electronic structure calcu lations for the ultrathin\nfilm/substrate combinations of interest. We find that spin waves in t he Fe/W(110) system\nare quite sensitive to the empirical tight binding parameters which ar e employed, though as\n4we shall see the various descriptions provide very similar pictures of the one electron local\ndensity of states.\nWe note that Udvardi and Szunyogh12have also discussed the influence of spin orbit\ncoupling on the dispersion relation of spin waves in the Fe monolayer on W(110) within\nthe framework of the adiabatic approach discussed above, where exchange interactions and\nother magnetic parameters are calculated in real space. We shall d iscuss a comparison with\nour results and theirs below. There are differences. Most particula rly, we note that in\nFig. 3, the authors of ref. 12 provide two dispersion curves for pr opagation perpendicular\nto the magnetization, whereas in a film such as this with one spin per un it cell there can\nbe only one magnon branch. Additionally and very recently, Bergman n and coworkers45\ninvestigatedwithinanadiabaticapproachfinitetemperatureeffect sonthemagnonspectrum\nof Fe/W(110).\nInsection II, we comment onourmeans of introducing spin orbit cou pling into the theory.\nThe results of our calculations are summarized in Section III and con cluding remarks are\nfound in section IV.\nII. CALCULATION OF THE DYNAMIC SUSCEPTIBILITY IN THE PRESENCE\nOF SPIN ORBIT COUPLING\nTheformalism forincluding spin orbitcoupling effects in our description of spin dynamics\nis quite involved, so in this section we confine our attention to an outlin e of the key steps,\nand an exposition of the overall structure of the theory. Our sta rting point is the multi band\nHubbard model of the system that was employed in our earlier study of spin dynamics in\nultrathin ferromagnets. The starting Hamiltonian is written as5\nH=/summationdisplay\nij/summationdisplay\nµνσTµν\nijc†\niµσcjνσ+1\n2/summationdisplay\nµνµ′ν′/summationdisplay\niσσ′Ui;µν,µ′ν′c†\niµσc†\niνσ′ciν′σ′ciµ′σ (1)\nwhereiandjaresite indices, σ,σ′refer to spin, and µ,νtothe tight binding orbitals, nine in\nnumberforeachsite, whichareincludedinourtreatment. TheCoulo mbinteractionsoperate\nonly within the 3d orbitals on a given lattice site. The film, within which fer romagnetism\nis driven by the Coulomb interactions, sits on a semi-infinite substrat e within which the\nCoulomb interaction is ignored.\n5In our empirical tight binding picture, the spin orbit interaction adds a term we write as\nHSO=/summationdisplay\ni/summationdisplay\nµνλi\n2/bracketleftBig\nLz\nµν(c†\niµ↑ciν↑−c†\niµciν↓)+L+\nµνc†\niµ↓ciν↑+L−\nµνc†\niµ↑ciν↓/bracketrightBig\n(2)\nwhere/vectorLis the angular momentum operator, λiis the local spin-orbit coupling constant,\nL±=Lx±iLyandLα\nµν=/an}bracketle{tµ|Lα|ν/an}bracketri}ht. We assume that the spin orbit interaction, present both\nwithin the ferromagnetic film and the substrate, operates only with in the 3d atomic orbitals.\nA convenient tabulation of matrix elements of the orbital angular mo mentum operators is\nfound in ref. 13.\nInformation on the spin waves follows from the study of the spectr al density of the trans-\nverse dynamic susceptibility χ+,−(/vectorQ/bardbl,Ω;l⊥,l′\n⊥) as discussed above. From the text around\nEq. (1) of ref. 14, we see that this function describes the amplitud e of the transverse spin\nmotion (the expectation value of the spin operator S+in the layer labeled l⊥) to a fictitious\ntransverse magnetic field of frequency Ω and wave vector /vectorQ/bardblparallel to the film surface that\nis applied to layer l′\n⊥of the sample. The spectral density, given by Im {χ+,−(/vectorQ/bardbl,Ω;l⊥,l′\n⊥)},\nwhen multiplied by the Bose Einstein function n(Ω) = [exp( βΩ)−1]−1is also the amplitude\nof thermal spin fluctuations of wave vector /vectorQ/bardbland frequency Ω in layer l⊥. We obtain in-\nformation regarding the character (frequency, linewidth, and am plitude in layer l⊥) of spin\nwaves from the study of this function, as discussed earlier.5\nOur previous analyses are based on the study of the dynamic susce ptibility just described\nthrough use of the random phase approximation (RPA) of many bod y theory. The Feynman\ndiagrams included in this method are the same as those incorporated into time dependent\ndensity functional theory, though use of our Hubbard model allow s us to solve the result-\ning equation easily once the very large array of irreducible particle ho le propagators are\ngenerated numerically.\nOur task in the present paper is to extend the RPA treatment to inc orporate spin orbit\ncoupling. The extension is non trivial. The quantity of interest, refe rred to in abbreviated\nnotation as χ+,−, may be expressed as a commutator of the spin operators S+andS−whose\nprecise definition is given earlier.5,12With spin orbit coupling ignored, the RPA decoupling\nprocedure leads to a closed equation for χ+,−. When the RPA decoupling is carried out in\nits presence, we are led to a sequence of four coupled equations wh ich include new objects\nwe may refer to as χ−,−,χ↑,−andχ↓,−. The number of irreducible particle hole propagators\nthat must be computed likewise is increased by a factor of four. For a very simple version\n6of a one band Hubbard model, and for a very different purpose, Fuld e and Luther carried\nout an equivalent procedure many years ago15. In what follows, we provide a summary of\nkey steps along with expressions for the final set of equations.\nTo generate the equation of motion, we need the commutator of th e operator S+\nµν(l,l′) =\nc†\nlµ↑cl′ν↓with the Hamiltonian. One finds\n[S+\nµν(l,l′),HSO] =1\n2/summationdisplay\nη{λl′L+\nνηc†\nlµ↑cl′η↑−λlL+\nηµc†\nlη↓cl′ν↓+λl′Lz\nνηc†\nlµ↑cl′η↓−λlLz\nηµc†\nlη↑cl′ν↓}.(3)\nThe last two terms on the right hand side of Eq. 3 lead to terms in the e quation of motion\nwhich involve χ+,−whereas the first two terms couple us to the entities χ↑,−andχ↓,−.\nWhen we write down the commutator of these new correlation funct ions with the spin orbit\nHamiltonian, we are led to terms which couple into the function χ−,−which is formed from\nthe commutator of two S−operators. In the absence of spin orbit coupling, a consequence o f\nspin rotation invariance of the Hamiltonian is that the three new func tions just encountered\nvanish. But they do not in its presence, and they must be incorpora ted into the analysis.\nOnethen introduces theinfluence of theCoulomb interaction into th eequation ofmotion,\nand carries out an RPA decoupling of the resulting terms. The analys is is very lengthy, so\nhere we summarize only the structure that results from this proce dure. Definitions of the\nvarious quantities that enter are given in the Appendix. We express the equations of motion\nin terms of a 4 ×4 matrix structure, where in schematic notation we let χ(1)=χ+,−,\nχ(2)=χ↑,−,χ(3)=χ↓,−andχ(4)=χ−,−. The four coupled equations then have the form\nΩχ(s)=A(s)+/summationdisplay\ns′(Bss′+˜Bss′)χ(s′)(4)\nEach quantity in Eq. 4 has attached to it four orbital indices, and fo ur site indices. To be\nexplicit, χ(2)=χ↑,−which enters Eq. (4) is formed from the commutator of the operat or\nc†\nlµ↑cl′ν↑withc†\nmµ′↓cm′ν′↑and in full we denote this quantity as χ(2)\nµν;µ′ν′(ll′;mm′). The site\nindices label the planes in the film, and we suppress reference to Ω an d/vectorQ/bardbl. The products\non the right hand side of Eq. 4 are matrix multiplications that involve th ese various indices.\nFor instance, the object/summationtext\ns′Bss′χ(s′)is labeled by four orbital and four site indices so\n[Bss′χ(s′)]µν,µ′ν′(ll′;mm′) =/summationdisplay\nγδ/summationdisplay\nnn′Bss′\nµν,γδ(ll′;nn′)χ(s′)\nγδ,µ′ν′(nn′;mm′). (5)\nOne proceeds by writing Eq. 4 in terms of the dynamic susceptibilities t hat characterize\nthe non-interacting system. These, referred to also as the irred ucible particle hole propaga-\ntors, are generated by evaluating the commutators which enter in to the definition of χ(s)in\n7the non interacting ground state. These objects, denoted by χ(0s)obey a structure similar\nto Eq. 4,\nΩχ(0s)=A(s)+/summationdisplay\ns′Bss′χ(s′). (6)\nIt is then possible to relate χ(s)toχ(0s)through the relation, using four vector notation,\n/vector χ(Ω) =/vector χ(0)(Ω)+(Ω −B)−1˜B/vector χ(Ω). (7)\nThe matrix structure Γ ≡(Ω−B)−1may be generated from the definition of B, which may\nbe obtained from the equation of motion of the non-interacting sus ceptibility, Eq. 6. Then\n˜Bfollows from the equation of motion of the full susceptibility, as gene rated in the RPA.\nOne may solve Eq. 7\n/vector χ(Ω) = [I−(Ω−B)−1¯B]−1/vector χ(0)(Ω), (8)\nso our basic task is to compute the non interacting susceptibility mat rix/vector χ(0)and then carry\nout the matrix inversion operation displayed in Eq. 8. For this we requ ire the single particle\nGreens functions (SPGFs) associated with our approach.\nTo generate the SPGFs, we set up an effective single particle Hamilton ianHspby intro-\nducing a mean field approximation for the Coulomb interaction. The ge neral structure of\nthe single particle Hamiltonian is\nHsp=/summationdisplay\nij/summationdisplay\nµνσ˜Tµνσ\nijc†\niµσcjνσ+/summationdisplay\ni/summationdisplay\nµν{α∗\ni;µνc†\niµ↓ciν↑+αi;µνc†\niµ↑ciν↓} (9)\nwhere the effective hopping integral ˜Tµνσ\nijcontains the spin diagonal portion of the spin orbit\ninteraction, along withthemeanfield contributions fromtheCoulomb interaction. The form\nwe use for the latter is stated below. The coefficients in the spin flip te rms are given by\nαi;µν=λiL−\nµν−/summationdisplay\nηγUi;ηµ,νγ/an}bracketle{tc†\niη↓ciγ↑/an}bracketri}ht. (10)\nWe then have the eigenvalue equation that generates the single par ticle eigenvalues and\neigenfunctions in the form Hsp|φs/an}bracketri}ht=Es|φs/an}bracketri}ht; we can write this in the explicit form\n/summationdisplay\nl/summationdisplay\nησ′/bracketleftBig\nδσσ′˜Tµησ′\nil+δil(δσ′↓δσ↑α∗\nl;µη+δσ′↑δσ↓αl;µη)/bracketrightBig\n/an}bracketle{tlησ′|φs/an}bracketri}ht=Es/an}bracketle{tiµσ|φs/an}bracketri}ht.(11)\nThe single particle Greens function may be expressed in terms of the quantities that enter\nEq. 11. We have for this object the definition\nGiµσ;jνσ′(t) =−iθ(t)/an}bracketle{t{ciµσ(t),c†\njνσ′(0)}/an}bracketri}ht (12)\n8and one has the representation\nGiµσ;jνσ′(Ω) =/summationdisplay\ns/an}bracketle{tiµσ|φs/an}bracketri}ht/an}bracketle{tφs|jνσ′/an}bracketri}ht\nΩ−Es+iη. (13)\nThese functions may be constructed directly from their equations of motion, which read,\nafter Fourier transforming with respect to time,\n−/summationdisplay\nl/summationdisplay\nησ′′/bracketleftBig\nδσσ′′˜Tµησ′′\nil+δil(δσ′′↓δσ↑α∗\nl;µη+δσ′′↑δσ↓αl;µη)/bracketrightBig\nGlησ′′;jνσ′+ΩGiµσ;jνσ′=δσσ′δµνδij.\n(14)\nForthecasewherethesubstrateissemi infinite, ourmeansofgen eratinganumerical solution\nto the hierarchy of equations stated in Eq. 14 has been discussed e arlier. What remains is\nto describe how the Coulomb interaction enters the effective hoppin g integrals ˜Tµνσ\nijthat\nappear in Eq. 9, Eq. 11 and Eq. 14.\nThere are, of course, a large number of Coulomb matrix elements in t he original Hamil-\ntonian, even if the Coulomb interactions are confined to within the 3d shell. Through the\nuse of group theory,17the complete set of Coulomb matrix elements may be expressed in\nterms of three parameters. These are given in Table I of the first c ited paper in ref. 5.\nIn subsequent work, we have found that a much simpler structure18nicely reproduces re-\nsults obtained with the full three parameter form. We use the simple r one parameter form\nhere, for which Ui;µν,µ′ν′=Uiδµν′δµ′ν. Then in the mean field approximation, the Coulomb\ncontribution to the single particle Hamiltonian assumes the form\nH(C)\nsp=−/summationdisplay\niUimi\n2/summationdisplay\nµ(c†\niµ↑ciµ↑−c†\niµ↓ciµ↓) (15)\nHeremiis the magnitude of the moment on site i. The Coulomb interactions Uiare non zero\nonly within the ultrathin ferromagnet, and the moments mi, determined self consistently,\nvary from layer to layer when we consider multi layer ferromagnetic films.\nIt should be noted that when the Ansatz just described is employed in Eq. 10, the term\nfrom the Coulomb interaction on the right hand side becomes propor tional to the transverse\ncomponent of the moment located on site iand this vanishes identically. Thus, despite the\ncomplexity introduced by the spin orbit coupling, when the simple one p arameter Ansatz for\nthe Coulomb matrix elements is employed, one needs no parameters b eyond the moment on\neach layer in the self consistent loops that describe the ground sta te. In the present context,\nthis is an extraordinarily large savings in computational labor, and th is will allow us to\n9address very large systems in the future. It is the case that cert ain off diagonal elements\nsuch as/an}bracketle{tc†\nmµ′↓cl′ν↑/an}bracketri}htappear in the quantities defined in the Appendix. Notice, for example ,\nthe expressions in Eqs. A.1. Once the ground state single particle Gr eens functions are\ndetermined, such expectation values are readily computed.\nIII. RESULTS AND DISCUSSION\nIn earlier studies of Fe layers on W(110),19,20as noted above, the electronic structure\nwas generated through use of tight binding parameters obtained f rom bulk electronic struc-\nture calculations. These calculations generate effective exchange interactions comparable\nin magnitude to those found in the bulk transition metals,20with the consequence that for\nboth monolayer Fe and bilayer Fe on W(110) the large wave vector sp in waves generated\nby theory are very much stiffer than found experimentally21,22though it should be noted\nthat for the bilayer, the calculated value of the spin wave exchange stiffness is in excellent\naccord with the data.23Subsequent calculations which construct the spin wave dispersion\nrelation from adiabatic theory based on calculations of effective exc hange integrals also gen-\nerate spin waves for the monolayer substantially stiffer than found experimentally,12though\nthey are softer than in our earlier work by a factor of two or so. We remark that it has\nbeen suggested that the remarkably soft spin waves found exper imentally may have origin\nin carbon contamination of the monolayer and bilayer.20We remark here that this can be\nintroduced during the SPEELS measurement. We note that the mag netic properties of Fe\nmonolayers grown on carbon free W(110)24differ dramatically from those grown on surfaces\nnow known to be contaminated by carbon.25In the former case, the domain walls have a\nthickness of 2.15 nm,24whereas in the latter circumstance very narrow walls with thickness\nbounded from above by 0.6 nm are found.25This suggests that the strength of the effective\nexchange is very different in the two cases, with stiffer exchange in t he carbon free samples.\nThe considerations of the previous paragraph have motivated us t o carry out a series of\nstudies of the effective exchange in the Fe monolayer on W(110) with in the framework of\nthree different electronic structure calculations. We find that alth ough all three give local\ndensity of states that are very similar, along with very similar energy bands when these are\nexamined, theintersiteexchangeinteractionsvarysubstantially. First, wehaveemployed the\nparameter set used earlier that is based on bulk electronic structu res19,20in new calculations\n10we call case A. In case B, we have employed an approach very similar t o that used in ref. 12,\nthough in what follows our calculation of effective exchange integrals is non relativistic.\nThis is the Korringa Kohn Rostoker Greens Function (KKR-GF) meth od,26which employs\nthe atomic sphere approximation and makes use of the Dyson equat ionG=g+gVGas\ngiven in matrix notation. This allows us to calculate the Greens functio nGof an arbitrary\ncomplex system given the perturbing potential Vand the Greens function gof a reference\nunperturbed system. Within the Local Spin Density Approximation ( LSDA),27We consider\na slab of five monolayers of W with the experimental lattice constant on top of which an Fe\nmonolayer is deposited and relaxed by -12.9%12with respect to the W interlayer distance.\nAngular momenta up to lmax= 3 were included in the Greens functions with a kmesh of\n6400 points in the full two dimensional Brillouin zone. The effective exc hange interactions\nwere calculated within the approximation of infinitesmal rotations28that allows one to use\nthe magnetic force theorem. This states that the energy change due to infinitesmal rotations\nin the moment directions can be calculated through the Kohn Sham eig envalues.\nMethod C is the Real Space Linear-Muffin-Tin-Orbital approach as im plemented, also,\nin the atomic sphere approximation (RS-LMTO-ASA).29–33Due to its linear scaling, this\nmethod allows one to address the electronic structure of systems with a large number of\natoms for which the basic eigenvalue problem is solved in real space us ing the Haydock\nrecursion method. The Fe overlayer on the W(110) substrate was simulated by a large\nbcc slab which contained ∼6800 atoms, arranged in 12 atomic planes parallel to the (110)\nsurface, with the experimental lattice parameter of bulk W. One em pty sphere overlayer\nis included, and self consistent potential parameters were obtaine d for the empty sphere\noverlayer, the Fe monolayer, and the three W layers underneath u sing LSDA.34For deeper\nW layers we use bulk potential. Nine orbitals per site (the five 3d and 4 s p complex) were\nused to describe the Fe valence band and the empty sphere overlay er, and for W the fully\noccupied 4f orbitals were also included in the core. To evaluate the or bital moments we use\na scalar relativistic (SR) approach and include a spin orbit coupling ter mλ/vectorL·/vectorSat each\nvariational step.35In the recursion method the continued fraction has been terminat ed after\n30 recursion levels with the Beer Pettifor terminator.36The TB parameters so obtained are\ninserted into our semi-empirical scheme and this allows us to generat e the non interacting\nsusceptibilities which enter our full RPA description of the response of the structure.\nIn order to compare the electronic structures generated by the approaches just described,\n11we turn our attention to the local density of states for the major ity and minority spins in\nthe adsorbed Fe monolayer. These are summarized in Fig. 1.\nThe local densities of states (LDOS) generated by the three sets of TB parameters have\napproximately the same overall features, as we see from Fig. 1. Th e main differences appear\nin the majority spin band, which overlaps the 5d states in the W subst rate over a larger\nenergy range than the minority band. This is also true if we compare t he LDOS generated\nby the tight binding parameters extracted from the KKR electronic structure to the LDOS\nobtaineddirectly fromtheKKR calculations (reddashed linein Fig.1b) . The Fe-Whopping\nparameters are indeed the least accurate portion of our paramet rization scheme. In case A\nwejustusedtheFe-FebulkparameterstodescribetheFe-Whopp ing. IncaseBweextracted\nTB parameters for Fe by fitting a KKR calculation of an unsupported Fe monolayer with a\nlattice parameter matching that of the W substrate. For the Fe-W hopping we used the Fe\nparameters obtained from the fitting, scaled to mimic the Fe-W dista nce relaxation. The\nrelaxation parameter was chosen to give the correct spin magnetic moment for the adsorbed\nFe monolayer. In case C all parameters were directly provided by th e RS-LMTO-ASA code,\nbut in the DFT calculations the Fe-W distance was assumed to be equa l to the distance\nbetween W layers. Thus, the main difference between cases B and C is the treatment of the\nmixing between Fe and W states and this is expected to affect more st rongly the states that\noccupy the same energy range.\nAs noted above, while the local density of states provided by the th ree approaches to\nthe electronic structure are quite similar as we see from Fig. 1 (and t he same is true of the\nelectronic energy bands themselves if these are examined), the ex change interactions differ\nsubstantially for the three cases. For the first, second and third neighbors we have (in meV)\n42.5, 3.72 and 0.46 for model A, 28.7, -7.87 and 0.31, for model B and 1 1.23, -7.31 and 0.22\nfor model C. The authors of ref. 12 find 10.84, -3.34 and 3.64 for th ese exchange integrals.\nWe now turn to our studies of spin excitations in the Fe monolayer and Fe bilayer on\nW(110) within the framework of the electronic structure generat ed through use of the ap-\nproach in case C. We will discuss the influence of spin orbit coupling on b oth the trans-\nverse wave vector dependent susceptibility though study of the s pectral density function\nA(/vectorQ/bardbl,Ω;l⊥) =−Im{χ+,−(/vectorQ/bardbl,Ω;l⊥)}discussed in section I. This function, for fixed wave\nvector/vectorQ/bardbl, when considered as a function of frequency Ω, describes the fre quency spectrum\nof the fluctuations of wave vector /vectorQ/bardblof the transverse magnetic moment in layer l⊥as noted\n12-6 -4 -2 0 2\nE-EF (eV)-2024LDOS (states/eV)-2024LDOS (states/eV)-2024LDOS (states/eV)a)\nb)\nc)\nFIG. 1: (color online) For the Fe monolayer adsorbed on W(110 ), we show the local density of\nstates in the Fe monolayer. The majority spin density of stat es is shown positive and the minority\nspin density of states is negative. The zero of energy is at th e Fermi energy. In (a), bulk electronic\nstructure parameters are used as in the second of the two pape rs cited in 5 (caseA). In (b), we\nhave the density of states generated by method B. The black cu rve is found by fitting the KKR\nelectronic energy band structure to tight binding paramete rs as described in the text, and the red\ncurve is calculated directly from the KKR calculation. In (c ) we have the local density of states\ngenerated by method C.\n13above. In the frequency regime where spin waves are encountere d, this function is closely\nrelated to (but not identical to) the response function probed in a SPEELS measurement.\nIn Fig. 2, for the Fe monolayer on W(110), we show the spectral de nsity function cal-\nculated for three values of |/vectorQ/bardbl|, for propagation across the magnetization. Thus, the wave\nvector is directed along the short axis in the surface. This is the dire ction probedin SPEELS\nstudies ofthe Femonolayer onthissurface.22Ineach figure, we show three curves. The green\ndashed curve is calculated with spin orbit coupling set to zero. We sho w only a single curve\nfor this case, because the spectral density is identical for the tw o directions of propagation\nacross the magnetization, + /vectorQ/bardbland−/vectorQ/bardbl. When spin orbit coupling is switched on, for the\ntwo directions just mentioned the response function is very differe nt, as we see from the red\nandblack curve inthevariouspanels. These spin wave frequencies, deduced fromthepeakin\nthe response functions as discussed in section I, differ for the two directions of propagation,\nand also note that the peak intensities and linewidths differ as well. It is the absence of both\ntime reversal symmetry and reflection symmetry which renders + /vectorQ/bardbland−/vectorQ/bardblinequivalent\nfor this direction of propagation. The system senses this breakdo wn of symmetry through\nthe spin orbit interaction. If one considers propagation parallel to the magnetization, the\nasymmetries displayed in Fig. 2 are absent. The reason is that for th is direction of propaga-\ntion, reflection in the plane that is perpendicular to both the magnet ization and the surface\nis a goodsymmetry operation of the system, but takes + /vectorQ/bardblinto−/vectorQ/bardblthus rendering the two\ndirections equivalent. Recall, of course, that the magnetization is a pseudo vector in regard\nto reflections. Notice how very broad the curves are for large wav e vectors; the lifetime of\nthe spin waves is very short indeed.\nAs discussed in section I, we may construct a spin wave dispersion cu rve by plotting the\nmaxima in spectral density plots such as those illustrated in Fig. 2 as a function of wave\nvector. We show dispersion relations constructed in this manner in F ig. 3, with spin orbit\ncoupling both present and absent. In Fig. 3a, and for propagation perpendicular to the\nmagnetization we show the dispersion curve so obtained for wave ve ctors throughout the\nsurface Brillouin zone, and in Fig. 3b we show its behavior for small wav e vectors.\nLet is first consider Fig, 3(a). Here the dispersion curve extends t hroughout the two\ndimensional Brillouin zone. At the zone boundary, quite clearly the slo pe of the dispersion\ncurve does not vanish. In this direction of propagation, the natur e of the point at the zone\nboundary does not require the slope to vanish. What is most striking , clearly, is the anomaly\n140 10 20 30 40 50\nΩ (meV)05×1031×104-Imχ+,−(a)\n0 50 100 150 200\nΩ (meV)050010001500-Imχ+,-(b)\n0 100 200 300\nΩ (meV)0200400600800-Imχ+,-(c)\nFIG. 2: (color online) Thespectral density functions A(/vectorQ/bardbl,Ω;l⊥) evaluated in theFe monolayer for\nthree values of the wave vector in the direction perpendicul ar to the magnetization. We have (a)\n|/vectorQ/bardbl|= 0.4˚A−1, (b)|/vectorQ/bardbl|= 1.0˚A−1and (c)|/vectorQ/bardbl|= 1.4˚A−1. The green curve (dashed) is calculated\nwith spin orbit coupling set to zero; the spectral density he re is independent of the sign of /vectorQ/bardbl. The\nred and black curves are calculated with spin orbit coupling turned on. Now we see asymmetries\nfor propagation across the magnetization, with the red curv e/vectorQ/bardbldirected from left to right and the\nblack curve from right to left.\nin the vicinity of 1 ˚A−1. This feature is evident in the calculation with spin orbit coupling\nabsent, and for positive values of the wave vector the feature be comes much more dramatic\nwhen spin orbit coupling is switched on. Anomalies rather similar to thos e in the black\ncurve in Fig. 3(a) appear in the green dispersion curve found in Fig. 3 of ref. 12, though\nthese authors did not continue their calculation much beyond the 1 ˚A−1regime. Our spin\nwaves are very much softer than theirs in this spectral region, no tice. In Fig. 3 of ref. 12,\none finds two dispersion curves, one a mirror image of the second. T hus, these authors\n15-1.5 -1-0.5 00.5 11.5\nQ (A-1)050100150200Ω (meV)(a)\n-0.4 -0.2 0 0.2 0.4\nQ (A-1)010203040Ω (meV)(b)\nFIG. 3: (color online) Spin wave dispersion relations const ructed from peaks in the spectral den-\nsity, for the Fe monolayer on W(110). The wave vector is in the direction perpendicular to the\nmagnetization. The red curve is constructed in the absence o f spin orbit coupling, it is included in\nthe black curve.\ndisplay two spin wave frequencies for each wave vector. This surely is not correct. For a\nstructure with one atom per unit cell, there is one and only one spin wa ve mode for each\nwave vector, though as discussed above for the structure explo red here symmetry allow the\nleft/right asymmetry in the dispersion curve illustrated in our Fig. 3.\nIn Fig. 3b, again with spin orbit coupling switched on and off, we show an expanded\nview of the dispersion curve for small wave vectors. With spin orbit in teraction switched off,\nat zero wave vector we see a zero frequency spin wave mode, as re quired by the Goldstone\ntheorem when the underlying Hamiltonian is form invariant under spin r otation. The curve\nis also symmetrical, and is accurately fitted by the form Ω( /vectorQ/bardbl) = 149Q2\n/bardbl(meV), with the\nwave vector in ˚A−1, whereas with spin orbit coupling turned on the dispersion relation is\nfitted by Ω( /vectorQ/bardbl) = 3.4−11.8Q/bardbl+143Q2\n/bardbl(meV). Spin orbit coupling introduces an anisotropy\ngap atQ/bardbl= 0, and most striking is the term linear in wave vector. This has its orig in in the\nDzyaloshinskii Moriya interaction whose presence, as argued by th e authors of ref. 9, has its\norigin in the absence of both time reversal and inversion symmetry, for the adsorbed layer.\nAt long wavelengths, one may describe spin waves by classical long wa velength phe-\nnomenology. The linear term in the dispersion curve has its origin in a te rm in the energy\ndensity of the spin system of the form\nVDM=−Γ/integraldisplay\ndxdzS y(x,z)∂Sx(x,z)\n∂x(16)\n16HereSα(x,z) is a spin density, the xzplane is parallel to the surface, and the magnetization\nis parallel to the zdirection.\nOne interesting feature of the spin wave mode whose dispersion rela tion is illustrated in\nFig. 3b is that at Q/bardbl= 0, the mode has a finite group velocity. The fit to the dispersion\ncurve gives this group velocity to be∂Ω(/vectorQ/bardbl)\n∂Q/bardbl≈2×105cm/s , which is in the range of acoustic\nphonon group velocities.\nWeturnnowtoourcalculationsofspinwavesandtheresponsefunc tionsfortheFebilayer\non W(110). Let us first note that experimentally the orientation of the magnetization in\nthe bilayer appears to be dependent on the surface upon which the bilayer is grown. For\ninstance, when the bilayer is on the stepped W(110) surface, it is ma gnetized perpendicular\nto the surface,25a result inagreement with abinitio calculations of the anisotropy realiz ed in\nthe epitaxial bilayer.38However, in the SPEELS studies of spin excitations in the bilayer22,39\nthe magnetization is in plane. In our calculations, we find for model B t he magnetization is\nperpendicular to the surface, whereas in model C it lies in plane, along the long axis very\nmuch as in the SPEELS experiments. The anisotropy in the bilayer is no t particularly large,\non the order of 0.5 meV/Fe atom, and one sees from these results t hat it is a property quite\nsensitive to the details of the electronic structure. The fact that model B and model C give\nthe two different stable orientation of the magnetization allows us to explore spin excitations\nfor the two different orientations of the magnetization.\nWe first turn our attention to the case where the magnetization lies in plane. The bilayer\nhas two spin wave modes, an acoustic mode for which the magnetizat ion in the two planes\nprecesses in phase, and an optical mode for which they precess 18 0 degrees out of phase.\nIn Fig. 4, we show calculations of the dynamic susceptibility in the freq uency range of the\nacoustic mode for two values of the wave vector, Q/bardbl= +0.5˚A−1 andQ/bardbl=−0.5˚A−1. A\nspin orbit induced left right asymmetry is clearly evident both in the pe ak frequency and\nthe height of the feature. Very recently, beautiful measuremen ts of spin orbit asymmetries\nin the Fe bilayer have appeared,39and the results of our Fig. 4 are to be compared with\nFig. 3 of ref. 39. Theory and experiment are very similar, both in reg ard to the intensity\nasymmetry and also the spin orbit induced frequency shift, though our calculated spin wave\nfrequencies are a little stiffer than those found experimentally.\nAs remarked above, in Fig. 4 we show only the acoustical spin wave mo de frequency\nregime. In Fig. 5, for the spectral densities in the innermost layer ( upper panel) and the\n170 50 100\nΩ (meV)050010001500200025003000-Imχ+,−\nFIG. 4: (color online) The spectral density in the innermost layer, in the acoustic spin wave regime,\nfor wave vectors of Q/bardbl= +0.5˚A−1(black curve) and Q/bardbl=−0.5˚A−1(red curve). Model C has\nbeen used for the calculation. In the ground state, the magne tization lies in plane along the long\naxis.\noutermost layer (lower panel) we show the spectral densities for t he entire spin wave regime,\nincluding the region where the optical spin wave is found. It is clear th at the spin orbit\ninducedfrequency shiftsarelargestfortheopticalmodewhich, u nfortunatelyisnotobserved\nin the experiments.39\nIn Fig. 6 for a sequence of wave vectors, all chosen positive, we sh ow a sequence of spectra\ncalculated for the entire frequency range so both the acoustic an d optical spin wave feature\nare displayed. The black curves show the spectral density of the in nermost Fe layer, and\nthe red curves are for the outer layer. The optical spin wave mode , not evident in the data,\nshows clearly in these figures. Notice that for wave vectors great er than 1 ˚A−1the acoustical\nmode is localized in the outer layer and the optical mode is localized on th e inner layer.\nThe optical mode is very much broader than the acoustical mode at large wave vectors, by\nvirtue of the strong coupling to the electron hole pairs in the W 5d ban ds.\nAninteresting issue is the absence of the optical mode fromthe SPE ELS spectra reported\n18-1500\n-1000\n-500\n0-Imχ+,−\n0 50 100 150 200 250\nΩ (meV)-2000\n-1000\n0-Imχ+,−\nFIG. 5: (color online) For the wave vector Q/bardbl= 0.5˚A−1we show the spectral densities in the\ninnermost Fe layer (upper panel) and in the outermost layer ( lower panel) for the Fe bilayer on\nW(110). Thefigureincludestheoptical spinwave feature. As inFig. 4, theblack curveiscalculated\nforQ/bardblpositive, and the red curves are for Q/bardblnegative. The calculations employ model C.\nin refs. 22 and 39. We note that these spectra are taken with only t wo beam energies, 4 eV\nand 6.75 eV. At such very low energies, the beam electron will sample b oth Fe layers,\nso the SPEELS signal will be a coherent superposition of electron wa ves backscattered\nfrom each layer; the excitation process involves coherent excitat ion of both layers by the\nincident electron. As a consequence of the 180 degree phase differ ence in spin motions\nassociated with the two modes it is quite possible, indeed even probab le, that for energies\nwhere the acoustical mode is strong the intensity of the optical mo de is weak, by virtue\nof quantum interference effects in the excitation scattering amplit ude. In earlier studies\nof surface phonons, it is well documented that on surfaces where two surface phonons of\ndifferent polarization exist for the same wave vector, one can be sile nt and one active in\nelectron loss spectroscopy.40It would require a full multiple scattering analysis of the spin\nwaveexcitationprocesstoexplorethistheoretically. Whileearlier41calculationsthataddress\nSPEELS excitation of spin waves described by the Heisenberg model could be adapted for\n190200040006000Imχ+,−\n010002000Imχ+,−\n05001000Imχ+,−\n050010001500Imχ+,−\n050010001500Imχ+,−\n0 100 200 300 400 500\nΩ (meV)05001000Imχ+,−Qy=0.4 A-1\nQy=0.6 A-1\nQy=0.8 A-1\nQy=1.0 A-1\nQy=1.2 A-1\nQy=1.4 A-1\nFIG. 6: (color online) For the Fe bilayer and for several valu es of the wave vector (all positive), we\nshow the spectral density functions for the innermost layer adjacent to the substrate (black curve)\nand those for the outer layer of the film. The calculations emp loy model C.\n20-1.5 -1-0.5 00.5 11.5\nQ (A-1)0100200300Ω (meV)\nFIG. 7: (color online) For the Fe bilayer with magnetization in plane, we show the spin wave\ndispersion curves calculated with spin orbit coupling (bla ck points) and without spin orbit coupling\n(red curves). Model C has been employed for these calculatio ns.\nthis purpose, in principle, a problem is that at such low beam energies it is necessary to take\ndue account of image potential effects to obtain meaningful result s.42This is very difficult\nto do without considerable information on the electron reflectivity o f the surface.42It would\nbe of great interest to see experimental SPEELS studies of the Fe bilayer with a wider range\nof beam energies to search for the optical mode, if this were possib le.\nIn Fig. 7, we show dispersion curves for the optical and acoustic sp in wave branches for\nthe bilayer. The magnetization lies in plane, and one can see that on th e scale of this figure,\nthe spin orbit effects on the dispersion curve are rather modest co mpared to those in the\nmonolayer. For small wave vectors, with spin orbit coupling present , the dispersion curve of\nthe acoustic spin wave branch is fitted by the form Ω( Q/bardbl) = 0.49−0.85Q/bardbl+243Q2\n/bardbl(meV)\nso at long wavelengths the influence of the Dzyaloshinskii Moriya inte raction is more than\none order of magnitude smaller than it is in the monolayer.\nIf the magnetization is perpendicular to the surface, then symmet ry considerations show\nthat there are no left/right asymmetries in the spin wave propagat ion characteristics. One\nmay see this as follows. Consider a wave vector /vectorQ/bardblin the plane of the surface, which\nalso is perpendicular to the magnetization, and thus perpendicular t o the long axis. The\nreflection Rin the plane perpendicular to the surface and which contains the mag netization\n210 100 200 300 400\nΩ (meV)050010001500200025003000Imχ+,−\nFIG. 8: (color online) For the bilayer and the case where the m agnetization is perpendicular to the\nsurface (model B), and for Q/bardbl= 0.6˚A−1, we show spectral density function calculated for positive\nvalues of Q/bardbl(continuous lines) and negative values of Q/bardbl(symbols). The black curve is the spectral\ndensity for the inner layer, and the red curve is the outermos t Fe layer.\nsimultaneously changes the sign of wave vector and the magnetizat ion. If this is followed\nby the time reversal operation T, then/vectorQ/bardblremains reversed in sign but the magnetization\nchanges back to its original orientation. Thus the product RTleaves the system invariant\nbut transforms /vectorQ/bardblinto−/vectorQ/bardbl. The two propagation directions are then equivalent.\nWe illustrate this in Fig. 8 where, for Q/bardbl= 0.6˚A−1, where it is shown that the spectral\ndensities calculated for the two directions of propagation are ident ical, with spin orbit cou-\npling switched on. Model B, in which the magnetization is perpendicular to the surface, has\nbeen used in these calculations. The spectral densities calculated f or the two signs of Q/bardbl\ncannot be distinguished to within the numerical precision we use.\nIV. CONCLUDING REMARKS\nWe have developed the formalism which allows one to include the influenc e of spin orbit\ncouplingonthespinexcitationsofultrathinferromagnetsonsemiin finitemetallicsubstrates.\nOur approach allows us to calculate the full dynamic susceptibility of t he system, so as\nillustrated by the calculations presented in section III we can examin e the influence of spin\norbit coupling on the linewidth (or lifetime) of spin excitations, along wit h their oscillator\n22strength. As in previous work, we can then construct effective dis persion curves by following\npeaks in the spectral density as a function of wave vector, withou t resort to calculations of\nlarge numbers of very small distant neighbor exchange interaction s. The results presented\nin Fig. 4 are very similar to the experimental data reported in ref. 39 , as discussed above,\nthough we see that in the bilayer the influence of the Dzyaloshinskii M oriya interaction is\nconsiderably more modest than in the monolayer.\nWe will be exploring other issues in the near future. One interest in ou r minds is the\ninfluence of spin orbit coupling on the spin pumping contributions to th e ferromagnetic\nresonance linewidth, as observed in ferromagnetic resonance (FM R) studies of ultrathin\nfilms.43It has been shown earlier44that the methodology employed in the present paper\n(without spin orbit coupling included) can be applied to the description of the spin pumping\ncontribution to the FMR linewidth, and in fact an excellent quantitativ e account of the\ndata on the Fe/Au(100) system was obtained. It is possible that fo r films grown on 4d\nand 5d substrates that spin orbit coupling can influence the spin pum ping relaxation rate\nsubstantially. This willrequirecalculationsdirected towardmuchthic ker filmsthanexplored\nhere. The formalism we have developed and described in the present paper will allow such\nstudies in the future.\nAcknowledgments\nThis research was supported by the U. S. Department of Energy, through grant No.\nDE-FG03-84ER-45083. S. L. wishes to thank the Alexander von Hu mboldt Foundation\nfor a Feodor Lynen Fellowship. A.T.C. and R.B.M. acknowledge support from CNPq and\nFAPERJ and A.B.K. was supported also by the CNPq, Brazil.\nAppendix\nIn this Appendix we provide explicit expressions for the various quan tities which enter\nthe equations displayed in Section II. While these expressions are un fortunately lengthy, it\nwill be useful for them to be given in full.\nA(1)\nµν,µ′ν′(ll′;mm′) =δl′mδνµ′/an}bracketle{tc†\nlµ↑cm′ν′↑/an}bracketri}ht−δlm′δµν′/an}bracketle{tc†\nmµ′↓cl′ν↓/an}bracketri}ht\n23A(2)\nµν,µ′ν′(ll′;mm′) =−δlm′δµν′/an}bracketle{tc†\nmµ↓cl′ν↑/an}bracketri}ht\nA(3)\nµν,µ′ν′(ll′;mm′) =δl′mδνµ′/an}bracketle{tc†\nlµ↓cm′ν′↑/an}bracketri}ht\nA(4)\nµν,µ′ν′(ll′;mm′) = 0 (A.1)\nThe various expectation values in the equations above and those dis played below are\ncalculated from the single particle Greens functions once the self co nsistent ground state\nparameters are determined. Then\n˜B11\nµν,µ′ν′(ll′;mm′) =/summationdisplay\nη/parenleftBig\nUl;µ′η,µν′/an}bracketle{tc†\nlη↓cl′ν↓/an}bracketri}htδlmδlm′−Ul′;µ′ν,ην′/an}bracketle{tc†\nlµ↑cl′η↑/an}bracketri}htδl′mδl′m′/parenrightBig\n˜B12\nµν,µ′ν′(ll′;mm′) =/summationdisplay\nη/parenleftBig\nUl′;µ′ν,ν′η/an}bracketle{tc†\nlµ↑cl′η↓/an}bracketri}htδl′mδl′m′−Ul;ηµ′,µν′/an}bracketle{tc†\nlη↑cl′ν↓/an}bracketri}htδlmδlm′+\n+Ul;µ′η,µν′/an}bracketle{tc†\nlη↑cl′ν↓/an}bracketri}htδlmδlm′/parenrightBig\n˜B13\nµν,µ′ν′(ll′;mm′) =/summationdisplay\nη/parenleftBig\nUl′;µ′ν,ν′η/an}bracketle{tc†\nlµ↑cl′η↓/an}bracketri}htδl′mδl′m′−Ul;ηµ′,µν′/an}bracketle{tc†\nlη↑cl′ν↓/an}bracketri}htδlmδlm′−\n−Ul′;µ′ν,ην′/an}bracketle{tc†\nlµ↑cl′η↓/an}bracketri}htδl′mδl′m′/parenrightBig\n˜B14\nµν,µ′ν′(ll′;mm′) = 0\n(A.2)\n˜B21\nµν,µ′ν′(ll′;mm′) =/summationdisplay\nηUl;µ′η,µν′/an}bracketle{tc†\nlη↓cl′ν↑/an}bracketri}htδlmδlm′\n˜B22\nµν,µ′ν′(ll′;mm′) =/summationdisplay\nη/bracketleftBig\n(Ul′;µ′ν,ν′η−Ul′;µ′ν,ην′)/an}bracketle{tc†\nlµ↑cl′η↑/an}bracketri}htδl′mδl′m′−\n−(Ul;ηµ′,µν′−Ul;µ′η,µν′)/an}bracketle{tc†\nlη↑cl′ν↑/an}bracketri}htδlmδlm′/bracketrightBig\n˜B23\nµν,µ′ν′(ll′;mm′) =/summationdisplay\nη/parenleftBig\nUl;µ′ν,ν′η/an}bracketle{tc†\nlµ↑cl′η↑/an}bracketri}htδl′mδl′m′−Ul;ηµ′,µν′/an}bracketle{tc†\nlη↑cl′ν↑/an}bracketri}htδlmδl′m′/parenrightBig\n˜B24\nµν,µ′ν′(ll′;mm′) =−/summationdisplay\nηUl′;µ′ν,ην′/an}bracketle{tc†\nlµ↑cl′η↓/an}bracketri}htδl′mδl′m′\n(A.3)\n˜B31\nµν,µ′ν′(ll′;mm′) =−/summationdisplay\nηUl′;µ′ν,ην′/an}bracketle{tc†\nlµ↓cl′η↑/an}bracketri}htδl′mδl′m′\n˜B32\nµν,µ′ν′(ll′;mm′) =/summationdisplay\nη/parenleftBig\nUl;µ′ν,ν′η/an}bracketle{tc†\nlµ↓cl′η↓/an}bracketri}htδl′mδl′m′−Ul;ηµ′,µν′/an}bracketle{tc†\nlη↓cl′ν↓/an}bracketri}htδlmδl′m′/parenrightBig\n24˜B33\nµν,µ′ν′(ll′;mm′) =/summationdisplay\nη/bracketleftBig\n(Ul′;µ′ν,ν′η−Ul′;µ′ν,ην′)/an}bracketle{tc†\nlµ↓cl′η↓/an}bracketri}htδl′mδl′m′−\n−(Ul;ηµ′,µν′−Ul;µ′η,µν′)/an}bracketle{tc†\nlη↓cl′ν↓/an}bracketri}htδlmδlm′/bracketrightBig\n˜B34\nµν,µ′ν′(ll′;mm′) =/summationdisplay\nηUl;µ′η,µν′/an}bracketle{tc†\nlη↑cl′ν↓/an}bracketri}htδlmδlm′\n(A.4)\n˜B41\nµν,µ′ν′(ll′;mm′) = 0\n˜B42\nµν,µ′ν′(ll′;mm′) =/summationdisplay\nη/parenleftBig\nUl′;µ′ν,ν′η/an}bracketle{tc†\nlµ↓cl′η↑/an}bracketri}htδl′mδl′m′−Ul;ηµ′,µν′/an}bracketle{tc†\nlη↓cl′ν↑/an}bracketri}htδlmδlm′−\n−Ul′;µ′ν,ην′/an}bracketle{tc†\nlµ↓cl′η↑/an}bracketri}htδl′mδl′m′/parenrightBig\n˜B43\nµν,µ′ν′(ll′;mm′) =/summationdisplay\nη/parenleftBig\nUl′;µ′ν,ν′η/an}bracketle{tc†\nlµ↓cl′η↑/an}bracketri}htδl′mδl′m′−Ul;ηµ′,µν′/an}bracketle{tc†\nlη↓cl′ν↑/an}bracketri}htδlmδlm′+\n+Ul;µ′η,µν′/an}bracketle{tc†\nlη↓cl′ν↑/an}bracketri}htδlmδlm′/parenrightBig\n˜B44\nµν,µ′ν′(ll′;mm′) =/summationdisplay\nη/parenleftBig\nUl;µ′η,µν′/an}bracketle{tc†\nlη↑cl′ν↑/an}bracketri}htδlmδl′m−Ul′;µ′ν,ην′/an}bracketle{tc†\nlµ↓cl′η↓/an}bracketri}htδl′mδl′m′/parenrightBig\n(A.5)\nB11\nµν,µ′ν′(ll′;mm′) =˜Tνν′↓\nl′m′δlmδµµ′−(˜Tµµ′↑\nlm)∗δl′m′δνν′\nB12\nµν,µ′ν′(ll′;mm′) =α∗\nl′;ν′νδlmδl′m′δµµ′\nB13\nµν,µ′ν′(ll′;mm′) =−α∗\nl;µµ′δlmδl′m′δνν′\nB14\nµν,µ′ν′(ll′;mm′) = 0 (A.6)\nB21\nµν,µ′ν′(ll′;mm′) =αl′;νν′δlmδl′m′δµµ′\nB22\nµν,µ′ν′(ll′;mm′) =˜Tνν′↑\nl′m′δlmδµµ′−(˜Tµµ′↑\nlm)∗δl′m′δνν′\nB23\nµν,µ′ν′(ll′;mm′) = 0\nB24\nµν,µ′ν′(ll′;mm′) =−α∗\nl;µµ′δlmδl′m′δνν′ (A.7)\nB31\nµν,µ′ν′(ll′;mm′) =−αl;µ′µδlmδl′m′δνν′\n25B32\nµν,µ′ν′(ll′;mm′) = 0\nB33\nµν,µ′ν′(ll′;mm′) =˜Tνν′↓\nl′m′δlmδµµ′−(˜Tµµ′↓\nlm)∗δl′m′δνν′\nB34\nµν,µ′ν′(ll′;mm′) =α∗\nl′;ν′νδlmδl′m′δµµ′ (A.8)\nB41\nµν,µ′ν′(ll′;mm′) = 0\nB42\nµν,µ′ν′(ll′;mm′) =−αl;µ′µδlmδl′m′δνν′\nB43\nµν,µ′ν′(ll′;mm′) =αl′;νν′δlmδl′m′δµµ′\nB11\nµν,µ′ν′(ll′;mm′) =˜Tνν′↑\nl′m′δlmδµµ′−(˜Tµµ′↓\nlm)∗δl′m′δνν′ (A.9)\n1Rodrigo Arias and D. L. Mills, Phys. Rev. B 60, 7395 (1999); A. Azevedo, A. B. Oliveira, F. M.\nde Aguiar, and S. M. Rezende, Phys. Rev. B 62, 5331 (2000); D. L. Mills and S. M. Rezende\nin Spin Dynamics in Confined Structures II, (Springer Verlag , Heidelberg, 2002); P. Landeros,\nRodrigo E. Arias and D. L. Mills, Phys. Rev. B 77, 214405, (2008); J. Lindner, I. Barsukov, C.\nRaeder, C. Hassel, O. Posth, R. Meckenstock, P. Landeros and D. L. Mills, Phys. Rev. B 80,\n224421 (2009).\n2R.Urban, G. Woltersdorf andG. Heinrich, Phys.Rev. Lett. 87, 217204 (2001), Y.Tserkovnyak,\nA. Brataas and G. E. W. Bauer, Phys. Rev. Lett. 88, 117601 (2002); Phys. Rev. B 66, 224403\n(2002); D. L. Mills, Phys. Rev. B 68, 014419 (2003); M. Zwierzycki, Y. Tserkovnyak, P. J.\nKelly, A. Brataas, and G. E. W. Bauer, Phys. Rev. B 71, 064420 (2005); A. T. Costa, R. B.\nMuniz and D. L. Mills, Phys. Rev. B 73, 054426 (2006).\n3S. M. Rezende, A. Azevedo, M. A. Lucena and F. M. Aguiar, Phys. Rev. B63, 214416 (2001).\n4R. Vollmer, M. Etzkorn, P. S. Anil Kumar, H. Ibach and J. R. Kir schner, Phys. Rev. Lett.\n91, 14720 (2003); for a theoretical discussion of the linewidt h and dispersion observed in the\nexperiment, see A. T. Costa, R. B. Muniz Phys. Rev. B 70, 0544406 (2004).\n5Early papers are H. Tang, M. Plihal and D. L. Mills, Journal of Magnetism and Magnetic\nMaterials 187, 23 (1998); R. B. Muniz and D. L. Mills, Phys. Rev. B 66, 174417 (2002); A. T.\nCosta, R. B. Muniz and D. L. Mills, Phys. Rev. B 68, 224435 (2003).\n266For an explicit comparison between the predictions of the He isenberg model and that of an\nitinerant electron description of the spin excitations, se e Fig. 4 and Fig. 5 of the theoretical\npaper cited in ref. 4.\n7S. Blgel, M. Weinert and P. H. Dederichs, Phys. Rev. Lett. 60, 1077 (1988).\n8S. Heinze, M. Bode, A. Kubetzka, O. Pietzsch, X. Nie, S. Blgel , and R. Wiesendanger, Science\n311, 1805 (2000).\n9M. Bode, M. Heide, K. von Bergmann, P. Ferriani, S. Heinze, G. Bihlmayer, A. Kubetzka, O.\nPietzsch, S. Blgel and R. Wiesendanger, Nature 447, 190 (2007).\n10P. Lederer and A.Blandin, Phil. Mag. 14, 363 (1966).\n11See also A. T. Costa, R. B. Muniz and D. L. Mills, Phys. Rev. B 73, 214403 (2006).\n12L. Udvardi and L. Szunyogh, Phys. Rev. Lett. 102, 207204 (2009).\n13E. Abate and M. Asdente, Phys. Rev. 140, A1303 (1965).\n14A. T. Costa, Roberto Bechara Muniz and D. L. Mills, Phys. Rev. B73, 054426 (2006).\n15Peter Fulde and Alan Luther, Phys. Rev. 175, 337 (1968).\n16See the discussion in the second paper cited in footnote 5.\n17See Chapter 6 and Chapter 7 and Appendix C of Group Theory and Q uantum Mechanics, M.\nTinkham, (McGraw Hill, New York, 1964).\n18See D. M. Edwards, page 114 of Moment Formation in Solids, Vol . 117 of NATO Advanced\nStudy Institute, Series B: Physics (Plenum, New York, 1984) , edited by W. J. L. Buyers.\n19See the second and third papers cited in footnote 5.\n20A. T. Costa, R. B. Muniz, J. X. Cao, R. Q. Wu and D. L. Mills, Phys . Rev. B 78, 054439\n(2008).\n21J. Prokop, W. X. Tang, Y. Zhang, I Tudosa, T. R. F. Peixoto, Kh. Zakeri, and J. Kirschner,\nPhys. Rev. Lett. 102, 177206 (2009).\n22W. X. Tang, Y. Zhang, I. Tudosa, J. Prokop, M. Etzkorn and J. Ki rschner, Phys. Rev. Lett.\n99, 087202 (2007).\n23Seethe discussion in A. T. Costa, R. B. Munizand D. L. Mills, I EEE Transactions on Magnetics\n44, 1974 (2008).\n24S. Krause, G. Herzog, T. Stapelfeldt, L. Berbil-Bautista, M . Bode E. Y. Vedmedenko, and R.\nWiesendanger, Phys. Rev. Lett. 103, 127202 (2009).\n25M. Pratzer, H. J. Elmers, M. Bode, O. Pietzsch, A. Kubetzka an d R. Wiesendanger, Phys. Rev.\n27Lett.88, 057201 (2002).\n26N. Papanikolaou, R. Zeller and P. H. Dederichs, J. Phys. Cond ens. Matter 14, 2799 (2002).\n27S. H. Vosko, L. Wilk and M. Nusair, Can. J. Phys. 58, 1200 (1980).\n28A. I. Liechtenstein, M. I. Katsnelson, V. P. Antropov, and V. A. Gubanov, J. Magnetism and\nMagnetic Materials 67, 65 (1987).\n29P. R. Peduto, S. Frota-Pessa and M. S. Methfessel, Phys. Rev. B44, 13283 (1991).\n30S. Frota-Pessˆ oa, Phys. Rev. B 46, 14570 (1992).\n31R. Haydock, Solid State Physics, edited by H. Ehrenreich, F. Seitz and D. Turnbull, (Academic\nPress, New York, 1980), vol. 35.\n32O. K. Andersen, Phys. Rev. B 12, 3060 (1975).\n33O. K. Andersen, O. Jepsen and D. Gltzel, Highlights of Conden sed Matter Theory, edited by\nF. Bassani, F. Fumi, and M. P. Tosi, (North Holland, Amsterda m, 1985).\n34U. von Barth and L. A. Hedin, J. Phys. C. 5, 1629 (1972).\n35O. Eriksson, B. Johansson, R. C. Albers, A. M. Boring, and M. S . S. Brooks, Phys. Rev. B 42,\n2707 (1990).\n36N. Beer and D. Pettifor, The Electronic Structure of Complex Systems, (Plenum Press, New\nYork, 1984).\n37M. Plihal and D. L. Mills, Phys. Rev. B 58, 14407 (1998).\n38M. Heide, G. Bihlmayer and S. Blgel, Phys. Rev. B 78, 140403 (2008).\n39Kh. Zakeri, Y. Zhang, J. Prokop, T. H. Chuang, N. Sakr, W. X. Ta ng and J. Kirschner, Phys.\nRev. Lett. 104, 137203 (2010).\n40B. M. Hall andD. L. Mills, Phys. Rev. B 34, 8318 (1986); BurlM. Hall, D. L. Mills, Mohammed\nH. Mohammed and L. L. Kesmodel, Phys. Rev. B 38, 5856 (1988).\n41M. P. Gokhale, A. Ormeci and D. L. Mills, Phys. Rev. B 46, 8978 (1992).\n42B. M. Hall, S. Y. Tong and D. L. Mills, Phys. Rev. Lett. 50, 1277 (1983).\n43R. Urban, G. Woltersdorf, and B. Heinrich, Phys. Rev. Lett. 87, 217204 (2001).\n44A. T. Costa, R. B. Muniz and D. L. Mills, Phys. Rev. B 73, 054426 (2006).\n45A. Bergman, A. Taroni, L. Bergqvist, J. Hellsvik, B. Hrvarss on, and O. Eriksson, Phys. Rev. B\n81, 144416 (2010).\n28"    },    {        "title": "1408.1838v3.Spin_Orbital_Order_Modified_by_Orbital_Dilution_in_Transition_Metal_Oxides__From_Spin_Defects_to_Frustrated_Spins_Polarizing_Host_Orbitals.pdf",        "content": "Spin-Orbital Order Modi\fed by Orbital Dilution in Transition Metal Oxides:\nFrom Spin Defects to Frustrated Spins Polarizing Host Orbitals\nWojciech Brzezicki,1, 2Andrzej M. Ole\u0013 s,3, 1and Mario Cuoco2\n1Marian Smoluchowski Institute of Physics, Jagiellonian University,\nprof. S.  Lojasiewicza 11, PL-30348 Krak\u0013 ow, Poland\n2CNR-SPIN, IT-84084 Fisciano (SA), Italy, and\nDipartimento di Fisica \\E. R. Caianiello\", Universit\u0013 a di Salerno, IT-84084 Fisciano (SA), Italy\n3Max-Planck-Institut f ur Festk orperforschung, Heisenbergstrasse 1, D-70569 Stuttgart, Germany\n(Dated: 24 December 2014)\nWe investigate the changes in spin and orbital patterns induced by magnetic transition metal\nions without an orbital degree of freedom doped in a strongly correlated insulator with spin-orbital\norder. In this context we study the 3 dion substitution in 4 dtransition metal oxides in the case of\n3d3doping at either 3 d2or 4d4sites which realizes orbital dilution in a Mott insulator. Although we\nconcentrate on this doping case as it is known experimentally and more challenging than other oxides\ndue to \fnite spin-orbit coupling, the conclusions are more general. We derive the e\u000bective 3 d\u00004d(or\n3d\u00003d) superexchange in a Mott insulator with di\u000berent ionic valencies, underlining the emerging\nstructure of the spin-orbital coupling between the impurity and the host sites and demonstrate\nthat it is qualitatively di\u000berent from that encountered in the host itself. This derivation shows\nthat the interaction between the host and the impurity depends in a crucial way on the type of\ndoubly occupied t2gorbital. One \fnds that in some cases, due to the quench of the orbital degree\nof freedom at the 3 dimpurity, the spin and orbital order within the host is drastically modi\fed by\ndoping. The impurity acts either as a spin defect accompanied by an orbital vacancy in the spin-\norbital structure when the host-impurity coupling is weak, or it favors doubly occupied active orbitals\n(orbital polarons) along the 3 d\u00004dbond leading to antiferromagnetic or ferromagnetic spin coupling.\nThis competition between di\u000berent magnetic couplings leads to quite di\u000berent ground states. In\nparticular, for the case of a \fnite and periodic 3 datom substitution, it leads to striped patterns either\nwith alternating ferromagnetic/antiferromagnetic domains or with islands of saturated ferromagnetic\norder. We \fnd that magnetic frustration and spin degeneracy can be lifted by the quantum orbital\n\rips of the host but they are robust in special regions of the incommensurate phase diagram.\nOrbital quantum \ructuations modify quantitatively spin-orbital order imposed by superexchange. In\ncontrast, the spin-orbit coupling can lead to anisotropic spin and orbital patterns along the symmetry\ndirections and cause a radical modi\fcation of the order imposed by the spin-orbital superexchange.\nOur \fndings are expected to be of importance for future theoretical understanding of experimental\nresults for 4 dtransition metal oxides doped with 3 d3ions. We suggest how the local or global\nchanges of the spin-orbital order induced by such impurities could be detected experimentally.\nPACS numbers: 75.25.Dk, 03.65.Ud, 64.70.Tg, 75.30.Et\nI. INTRODUCTION\nThe studies of strongly correlated electrons in transi-\ntion metal oxides (TMOs) focus traditionally on 3 dma-\nterials [1], mainly because of high-temperature super-\nconductivity discovered in cuprates and more recently\nin iron-pnictides, and because of colossal magnetoresis-\ntance manganites. The competition of di\u000berent and com-\nplex types of order is ubiquitous in strongly correlated\nTMOs mainly due to coupled spin-charge-orbital where\nfrustrated exchange competes with the kinetic energy of\ncharge carriers. The best known example is spin-charge\ncompetition in cuprates, where spin, charge and super-\nconducting orders intertwine [2] and stripe order emerges\nin the normal phase as a compromise between the mag-\nnetic and kinetic energy [3, 4]. Remarkable evolution\nof the stripe order under increasing doping is observed\n[5] and could be reproduced by the theory based on the\nextended Hubbard model [6]. Hole doping in cuprates\ncorresponds to the removal of the spin degree of free-dom. Similarly, hole doping in a simplest system with the\norbital order in d1con\fguration removes locally orbital\ndegrees of freedom and generates stripe phases which in-\nvolve orbital polarons [7]. It was predicted recently that\norbital domain walls in bilayer manganites should be par-\ntially charged as a result of competition between orbital-\ninduced strain and Coulomb repulsion [8], which opens a\nnew route towards charge-orbital physics in TMOs. We\nwill show below that the stripe-like order may also occur\nin doped spin-orbital systems. These systems are very\nchallenging and their doping leads to very complex and\nyet unexplored spin-orbital-charge phenomena [9].\nA prerequisite to the phenomena with spin-orbital-\ncharge coupled degrees of freedom is the understanding\nof undoped systems [10], where the low-energy physics\nand spin-orbital order are dictated by e\u000bective spin-\norbital superexchange [11{13] and compete with spin-\norbital quantum \ructuations [14{16]. Although ordered\nstates occur in many cases, the most intriguing are quan-\ntum phases such as spin [17] or orbital [18] liquids. Re-arXiv:1408.1838v3  [cond-mat.str-el]  30 Jan 20152\ncent experiments on a copper oxide Ba 3CuSb 2O9[19, 20]\nhave triggered renewed e\u000borts in a fundamental search for\na quantum spin-orbital liquid [21{24], where spin-orbital\norder is absent and electron spins are randomly choosing\norbitals which they occupy. A signature of strong quan-\ntum e\u000bects in a spin-orbital system is a disordered state\nwhich persists down to very low temperatures. A good\nexample of such a disordered spin-orbital liquid state is\nas well FeSc 2S4which does not order in spite of \fnite\nCurie-Weiss temperature \u0002 CW=\u000045 K [25], but shows\ninstead signatures of quantum criticality [26, 27].\nSpin-orbital interactions may be even more challenging\n| for instance previous attempts to \fnd a spin-orbital\nliquid in the Kugel-Khomskii model [14] or in LiNiO 2[28]\nturned out to be unsuccessful. In fact, in the former case\ncertain types of exotic spin order arise as a consequence\nof frustrated and entangled spin-orbital interactions [29,\n30], and a spin-orbital entangled resonating valence bond\nstate was recently shown to be a quantum superposition\nof strped spin-singlet covering on a square lattice [31]. In\ncontrast, spin and orbital superexchange have di\u000berent\nenergy scales and orbital interactions in LiNiO 2are much\nstronger and dominated by frustration [32]. Hence the\nreasons behind the absence of magnetic long range order\nare more subtle [33]. In all these cases orbital \ructuations\nplay a prominent role and spin-orbital entanglement [34]\ndetermines the ground state.\nThe role of charge carriers in spin-orbital systems is\nunder very active investigation at present. In doped\nLa1\u0000x(Sr,Ca)xMnO 3manganites several di\u000berent types\nof magnetic order compete with one another and occur\nat increasing hole doping [35{37]. Undoped LaMnO 3\nis an antiferromagnetic (AF) Mott insulator, with large\nS= 2 spins for 3 d4ionic con\fgurations of Mn3+ions\nstabilized by Hund's exchange, coupled via the spin-\norbital superexchange due to egandt2gelectron exci-\ntations [38]. The orbital egdegree of freedom is re-\nmoved by hole doping when Mn3+ions are generated,\nand this requires careful modeling in the theory that\ntakes into account both 3 d4and 3d3electronic con\fg-\nurations of Mn3+and Mn4+ions [39{44]. In fact, the\norbital order changes radically with increasing doping\nin La 1\u0000x(Sr,Ca)xMnO 3systems at the magnetic phase\ntransitions between di\u000berent types of magnetic order [37],\nas weel as at La 0:7Ca0:3MnO 3/BiFeO 3heterostructures,\nwhere it o\u000bers a new route to enhancing multiferroic func-\ntionality [45]. The double exchange mechanism [46] trig-\ngers ferromagnetic (FM) metallic phase at su\u000ecient dop-\ning; in this phase the spin and orbital degrees of freedom\ndecouple and spin excitations are explained by the or-\nbital liquid [47, 48]. Due to distinct magnetic and kinetic\nenergy scales, even low doping may su\u000ece for a drastic\nchange in the magnetic order, as observed in electron-\ndoped manganites [49].\nA rather unique example of a spin-orbital system with\nstrongly \ructuating orbitals, as predicted in the theory\n[50{52] and seen experimentally [53{55], are the per-\novskite vanadates with competing spin-orbital order [56].In theset2gsystemsxyorbitals are \flled by one electron\nand orbital order of active fyz;zxgorbitals is strongly\nin\ruenced by doping with Ca (Sr) ions which replace Y\n(La) ones in YVO 3(LaVO 3). In this case \fnite spin-\norbit coupling modi\fes the spin-orbital phase diagram\n[57]. In addition, the AF order switches easily from the\nG-type AF (G-AF) toC-type AF (C-AF) order in the\npresence of charge defects in Y 1\u0000xCaxVO3. Already at\nlowx'0:02 doping the spin-orbital order changes and\nspectral weight is generated within the Mott-Hubbard\ngap [58]. Although one might imagine that the orbital\ndegree of freedom is thereby removed, a closer inspec-\ntion shows that this is not the case as the orbitals are\npolarized by charge defects [59] and readjust near them\n[60]. Removing the orbital degree of freedom in vana-\ndates would be only possible by electron doping generat-\ning instead d3ionic con\fgurations, but such a doping by\ncharge defects would be very di\u000berent from the doping\nby transition metal ions of the same valence considered\nbelow.\nAlso in 4dmaterials spin-orbital physics plays a role\n[61], as for instance in Ca 2\u0000xSrxRuO 4systems with Ru4+\nions in 4d4con\fguration [62{66]. Recently it has been\nshown that unconventional magnetism is possible for\nRu4+and similar ions where spin-orbit coupling plaus a\nrole [67, 68]. Surprisingly, these systems are not similar\nto manganites but to vanadates where one \fnds as well\nions with active t2gorbitals. In the case of ruthenates\nthet4\n2gRu4+ions have low S= 1 spin as the splitting\nbetween the t2gandeglevels is large. Thus the undoped\nCa2RuO 4is a hole analogue of a vanadate [50, 51], with\nt2gorbital degree of freedom and S= 1 spin per site in\nboth cases. This gives new opportunities to investigate\nspin-orbital entangled states in t2gsystem, observed re-\ncently by angle resolved photoemission [69].\nHere we focus on a novel and very di\u000berent doping\nfrom all those considered above, namely on a substitu-\ntional doping by other magnetic ions in a plane built\nby transition metal and oxygen ions, for instance in the\n(a;b) plane of a monolayer or in perovskite ruthenates\nor vanadates. In this study we are interested primarily\nin doping of a TMO with t2gorbital degrees of freedom,\nwhere doped magnetic ions have no orbital degree of free-\ndom and realize orbital dilution . In addition, we deal with\nthe simpler case of 3 ddoped ions where we can neglect\nspin-orbit interaction which should not be ignored for 4 d\nions. We emphasize that in contrast to manganites where\nholes within egorbitals participate in transport and are\nresponsible for the colossal magnetoresitance, such doped\nhole are immobile due to the ionic potential at 3 dsites\nand form defects in spin-orbital order of a Mott insulator.\nWe encounter here a di\u000berent situation from the dilution\ne\u000bects in the 2D egorbital system considered so far [70]\nas we deel with magnetic ions at doped sites. It is chal-\nlenging to investigate how such impurities modify locally\nor globally spin-orbital order of the host.\nThe doping which realizes this paradigm is by either\nMn4+or Cr3+ions with large S= 3=2 spins stabilized by3\nFIG. 1. (a) Schematic view of the orbital dilution when the\n3d3ion with no orbital degree of freedom and spin S= 3=2\nsubstitutes 4 d4one with spin S= 1 on a bond having speci\fc\nspin and orbital character in the host (gray arrows). Spins\nare shown by red arrows and doubly occupied t2gorbitals\n(doublons) are shown by green symbols for aandcorbitals,\nrespectively. (b) If an inactive orbital along the bond is re-\nmoved by doping, the total spin exchange is AF. (c) On the\ncontrary, active orbitals at the host site can lead to either FM\n(top) or AF (bottom) exchange coupling, depending on the\nenergy levels mismatch and di\u000berence in the Coulomb cou-\nplings between the impurity and the host. We show the case\nwhen the host site is unchanged in the doping process.\nHund's exchange, and orbital dilution occurs either in a\nTMO with d2ionic con\fguration as in the vanadium per-\novskites, or in 4 dMott insulators as in ruthenates. It has\nbeen shown that dilute Cr doping for Ru reduces the tem-\nperature of the orthorhombic distortion and induces FM\nbehavior in Ca 2Ru1\u0000xCrxO4(with 0<x< 0:13) [71]. It\nalso induces surprising negative volume thermal expan-\nsion via spin-orbital order. Such defects, on one hand,\ncan weaken the spin-orbital coupling in the host, but on\nthe other hand, may open a new channel of interaction\nbetween the spin and orbital degree of freedom through\nthe host-impurity exchange, see Fig. 1. Consequences of\nsuch doping are yet unexplored and are expected to open\na new route in the research on strongly correlated oxides.\nThe physical example for the present theory are the in-\nsulating phases of 3 d\u00004dhybrid structures, where doping\nhappens at d4transition metal sites, and the value of the\nspin is locally changed from S= 1 toS= 3=2. As a\ndemonstration of the highly nontrivial physics emerging\nabFIG. 2. Schematic view of C-AF spin order coexisting with\nG-AO orbital order in the ( a;b) plane of an undoped Mott\ninsulator with 4 d4ionic con\fgurations. Spins are shown by\narrows while doubly occupied xyandyzorbitals (canda\ndoublons, see text) form a checkerboard pattern. Equivalent\nspin-orbital order is realized for V3+ions in (b;c) planes of\nLaVO 3[56], with orbitals standing for empty orbitals (holes).\nin 3d\u00004doxides, remarkable e\u000bects have already been\nobserved, for instance, when Ru ions are replaced by Mn,\nTi, Cr or other 3 delements. The role of Mn doping in\nthe SrRuO Ruddlesden-Popper series is strongly linked\nto the dimensionality through the number nof RuO 2\nlayers in the unit cell. The Mn doping of the SrRuO 3\ncubic member drives the system from the itinerant FM\nstate to an insulating AF con\fguration in a continuous\nway via a possible unconventional quantum phase tran-\nsition [72]. Doping by Mn ions in Sr 3Ru2O7leads to\na metal-to-insulator transition and AF long-range order\nfor more than 5% Mn concentration [73]. Subtle orbital\nrearrangement can occur at the Mn site, as for instance\nthe inversion of the crystal \feld in the egsector observed\nvia x-ray absorption spectroscopy [74]. Neutron scatter-\ning studies indicate the occurrence of an unusual E-type\nantiferromagnetism in doped systems (planar order with\nFM zigzag chains with AF order between them) with mo-\nments aligned along the caxis within a single bilayer [75].\nFurthermore, the more extended 4 dorbitals would a\npriori suggest a weaker correlation than in 3 dTMOs due\nto a reduced ratio between the intraatomic Coulomb in-\nteraction and the electron bandwidth. Nevertheless, the\n(e\u000bective)d-bandwidth is reduced by the changes in the\n3d-2p-3dbond angles in distorted structures which typi-\ncally arise in these materials. This brings these systems\non the verge of a metal-insulator transition [76], or even\ninto the Mott insulating state with spin-orbital order, see\nFig. 2. Hence, not only 4 dmaterials share common fea-\ntures with 3 dsystems, but are also richer due to their sen-\nsitivity to the lattice structure and to relativistic e\u000bects\ndue to larger spin-orbit [77] or other magneto-crystalline\ncouplings.\nTo simplify the analysis we assume that onsite\nCoulomb interactions are so strong that charge degrees of\nfreedom are projected out, and only virtual charge trans-\nfer can occur between 3 dand 4dions via the oxygen lig-4\nands. For convenience, we de\fne the orbital degree of\nfreedom as a doublon (double occupancy) in the t4\n2gcon-\n\fguration. The above 3 ddoping leads then e\u000bectively to\nthe removal of a doublon in one of t2gorbitals which we\nlabel asfa;b;cg(this notation is introduced in Ref. [16]\nand explained below) and to replacing it by a t3\n2gion.\nTo our knowledge, this is the only example of remov-\ning the orbital degree of freedom in t2gmanifold realized\nso far and below we investigate possible consequences of\nthis phenomenon. Another possibility of orbital dilution\nwhich awaits experimental realization would occur when\nat2gdegree of freedom is removed by replacing a d2ion\nby ad3one, as for instance by Cr3+doping in a vanadate\n| here a doublon is an empty t2gorbital, i.e., \flled by\ntwo holes.\nBefore presenting the details of the quantitative anal-\nysis, let us concentrate of the main idea of the superex-\nchange modi\fed by doping in a spin-orbital system. The\nd3ions have singly occupied all three t2gorbitals and\nS= 3=2 spins due to Hund's exchange. While a pair\nofd3ions, e.g. in SrMnO 3, is coupled by AF superex-\nchange [48], the superexchange for the d3\u0000d4bond has\na rather rich structure and may also be FM. The spin\nexchange depends then on whether the orbital degree of\nfreedom is active and participates in charge excitations\nalong a considered bond or electrons of the doublon can-\nnot move along this bond due to the symmetry of t2g\norbital, as explained in Fig. 1. This qualitative di\u000ber-\nence to systems without active orbital degrees of freedom\nis investigated in detail in Sec. II.\nThe main outcomes of our analysis are: (i) the de-\ntermination of the e\u000bective spin-orbital exchange Hamil-\ntonian describing the low-energy sector for the 3 d\u00004d\nhybrid structure, (ii) establishing that a 3 d3impurity\nwithout an orbital degree of freedom modi\fes the orbital\norder in the 4 d4host, (iii) providing the detailed way how\nthe microscopic spin-orbital order within the 4 d4host is\nmodi\fed around the 3 d3impurity, and (iv) suggesting\npossible spin-orbital patterns that arise due to periodic\nand \fnite substitution (doping) of 4 datoms in the host\nby 3dones. The emerging physical scenario is that the\n3dimpurity acts as an orbital vacancy when the host-\nimpurity coupling is weak and as an orbital polarizer of\nthe bonds active t2gdoublon con\fgurations when it is\nstrongly coupled to the host. The tendency to polarize\nthe host orbitals around the impurity turns out to be ro-\nbust and independent of spin con\fguration. Otherwise,\nit is the resulting orbital arrangement around the impu-\nrity and the strength of Hund's coupling at the impurity\nthat set the character of the host-impurity magnetic ex-\nchange.\nThe remaining of the paper is organized as follows. In\nSec. II we introduce the e\u000bective model describing the\nspin-orbital superexchange at the 3 d\u00004dbonds which\nserves to investigate the changes of spin and orbital or-\nder around individual impurities and at \fnite doping.\nWe arrive at a rather general formulation which empha-\nsizes the impurity orbital degree of freedom, being a dou-blon, and present some technical details of the deriva-\ntion in Appendix A. The strategy we adopt is to analyze\n\frst the ground state properties of a single 3 d3impu-\nrity surrounded by 4 d4atoms by investigating how the\nspin-orbital pattern in the host may be modi\fed at the\nnearest neighbor (NN) sites to the 3 datom. This study\nis performed for di\u000berent spin-orbital patterns of the 4 d\nhost with special emphasis on the alternating FM chains\n(C-AF order) which coexist with G-type alternating or-\nbital (G-AO) order, see Fig. 2. We address the impurity\nproblem within the classical approximation in Sec. III A.\nAs explained in Sec. III B, there are two nonequivalent\ncases which depend on the precise modi\fcation of the or-\nbital order by the 3 dimpurity, doped either to replace a\ndoublon in aorbital (Sec. III C) or the one in corbital\n(Sec. III D).\nStarting from the single impurity solution we next ad-\ndress periodic arrangements of 3 datoms at di\u000berent con-\ncentrations. We demonstrate that the spin-orbital or-\nder in the host can be radically changed by the presence\nof impurities, leading to striped patterns with alternat-\ning FM/AF domains and islands of fully FM states. In\nSec. IV A we consider the modi\fcations of spin-orbital\norder which arise at periodic doping with macroscopic\nconcentration. Here we limit ourselves to two represen-\ntative cases: (i) commensurate x= 1=8 doping in Sec.\nIV B, and (ii) two doping levels x= 1=5 andx= 1=9 be-\ning incommensurate with underlying two-sublattice order\n(Fig. 2) which implies simultaneous doping at two sub-\nlattices, i.e., at both aandcdoublon sites, as presented\nin Secs. IV C and IV D. Finally, in Sec. V A we inves-\ntigate the modi\fcations of the classical phase diagram\ninduced by quantum \ructuations, and in Secs. V B and\nV C we discuss representative results obtained for \fnite\nspin-orbit coupling (calculation details of the treatment\nof spin-orbit interaction are presented in Appendix B).\nThe paper is concluded by a general discussion of possi-\nble emerging scenarios for the 3 d3impurities in 4 d4host,\na summary of the main results and perspective of future\nexperimental investigations of orbital dilution in Sec. VI.\nII. THE SPIN-ORBITAL MODEL\nIn this Section we consider a 3 dimpurity in a strongly\ncorrelated 4 dTMO and derive the e\u000bective 3 d3\u00004d4\nspin-orbital superexchange. It follows from the coupling\nbetween 3dand 4dorbitals via oxygen 2 porbitals due to\nthep\u0000dhybridization. In a strongly correlated system\nit su\u000eces to concentrate on a pair of atoms forming a\nbondhiji, as the e\u000bective interactions are generated by\ncharge excitations d4\nid4\nj\u000bd5\nid3\njalong a single bond [12].\nIn the reference 4 dhost both atoms on the bond hijiare\nequivalent and one considers,\nH(i;j) =Ht(i;j) +Hint(i) +Hint(j): (1)\nThe Coulomb interaction Hint(i) is local at site iand we\ndescribe it by the degenerate Hubbard model [80], see5\nbelow.\nWe implement a strict rule that the hopping within\nthet2gsector is allowed in a TMO only between two\nneighboring orbitals of the same symmetry which are ac-\ntive along the bond direction [15, 78, 79], and neglect\nthe interorbital processes originating from the octahe-\ndral distortions such as rotation or tilting. Indeed, in\nideal undistorted (perovskite or square lattice) geometry\nthe orbital \ravor is conserved as long as the spin-orbit\ncoupling may be neglected. The interorbital hopping ele-\nments are smaller by at least one order of magnitude and\nmay be treated as corrections in cases where distortions\nplay a role to the overall scenario established below.\nThe kinetic energy for a representative 3 d-2p-4dbond,\ni.e., after projecting out the oxygen degrees of freedom,\nis given by the hopping in the host /thbetween sites i\nandj,\nHt(i;j) =\u0000thX\n\u0016(\r);\u001b\u0010\ndy\ni\u0016\u001bdj\u0016\u001b+dy\nj\u0016\u001bdi\u0016\u001b\u0011\n:(2)\nHeredy\ni\u0016\u001bare the electron creation operators at site iin\nthe spin-orbital state ( \u0016\u001b). The bondhijipoints along\none of the two crystallographic directions, \r=a;b, in\nthe two-dimensional (2D) square lattice. Without distor-\ntions, only two out of three t2gorbitals are active along\neach bondh12iand contribute to Ht(i;j), while the third\norbital lies in the plane perpendicular to the \raxis and\nthus the hopping via oxygen is forbidden by symmetry.\nThis motivates a convenient notation as follows [15],\njai\u0011jyzi;jbi\u0011jxzi;jci\u0011jxyi; (3)\nwith thet2gorbital inactive along a given direction \r2\nfa;b;cglabeled by the index \r. We consider a 2D square\nlattice with transition metal ions connected via oxygen\norbitals as in a RuO 2(a;b) plane of Ca 2RuO 4(SrRuO 3).\nIn this casejai(jbi) orbitals are active along the b(a)\naxis, whilejciorbitals are active along both a;baxes.\nTo derive the superexchange in a Mott insulator, it\nis su\u000ecient to consider a bond which connects nearest\nneighbor sites,hiji\u0011h 12i. Below we consider a bond\nbetween an impurity site i= 1 occupied by a 3 dion and\na neighboring host 4 dion at sitej= 2. The Hamiltonian\nfor this bond can be then expressed in the following form,\nH(1;2) =Ht(1;2) +Hint(1) +Hint(2) +Hion(2):(4)\nThe total Hamiltonian contains the kinetic energy term\nHt(1;2) describing the electron charge transfer via oxy-\ngen orbitals, the onsite interaction terms Hint(m) for the\n3d(4d) ion at site m= 1;2, and the local potential of the\n4datom,Hion(2), which takes into account the mismatch\nof the energy level structure between the two (4 dand 3d)\natomic species and prevents valence \ructuations when\nthe host is doped, even in the absence of local Coulomb\ninteraction.\nThe kinetic energy in Eq. (4) is given by,\nHt(1;2) =\u0000tX\n\u0016(\r);\u001b\u0010\ndy\n1\u0016\u001bd2\u0016\u001b+dy\n2\u0016\u001bd1\u0016\u001b\u0011\n;(5)wheredy\nm\u0016\u001bis the electron creation operator at site m=\n1;2 in the spin-orbital state ( \u0016\u001b). The bondh12ipoints\nalong one of the two crystallographic directions, \r=a;b,\nand again the orbital \ravor is conserved [15, 78, 79].\nThe Coulomb interaction on an atom at site m= 1;2\ndepends on two parameters [80]: (i) intraorbital Coulomb\nrepulsionUm, and (ii) Hund's exchange JH\nm. The label m\nstands for the ion and distinguishes between these terms\nat the 3dand 4dion, respectively. The interaction is\nexpressed in the form,\nHint(m) =UmX\n\u0016nm\u0016\"nm\u0016#\u00002JH\nmX\n\u0016<\u0017~Sm\u0016\u0001~Sm\u0017\n+\u0012\nUm\u00005\n2JH\nm\u0013X\n\u0016<\u0017\n\u001b\u001b 0nm\u0016\u001bnm\u0017\u001b0\n+JH\nmX\n\u00166=\u0017dy\nm\u0016\"dy\nm\u0016#dm\u0017#dm\u0017\": (6)\nThe terms standing in the \frst line of Eq. (6) con-\ntribute to the magnetic instabilities in degenerate Hub-\nbard model [80] and decide about spin order, both in an\nitinerant system and in a Mott insulator. The remaining\nterms contribute to the multiplet structure and are of im-\nportance for the correct derivation of the superexchange\nwhich follows from charge excitations, see below.\nFinally, we include a local potential on the 4 datom\nwhich encodes the energy mismatch between the host\nand the impurity orbitals close to the Fermi level and\nprevents valence \ructuations on the 4 dion due to the 3 d\ndoping. This term has the following general structure,\nHion(2) =Ie\n2 \n4\u0000X\n\u0016;\u001bn2\u0016\u001b!2\n; (7)\nwith\u0016=a;b;c .\nThe e\u000bective Hamiltonian for the low energy processes\nis derived from H(1;2) (4) by a second order expansion\nfor charge excitations generated by Ht(1;2), and treating\nthe remaining part of H(1;2) as an unperturbed Hamil-\ntonian. We are basically interested in virtual charge ex-\ncitations in the manifold of degenerate ground states of\na pair of 3dand 4datoms on a bond, see Fig. 3. These\nquantum states are labeled as\b\nek\n1\t\nwithk= 1;:::; 4 and\nfep\n2gwithp= 1;:::; 9 and their number follows from the\nsolution of the onsite quantum problem for the Hamilto-\nnianHint(i). For the 3 datom the relevant states can be\nclassi\fed according to the four components of the total\nspinS1= 3=2 for the 3dimpurity atom at site m= 1,\nthree components of S2= 1 spin for the 4 dhost atom\nat sitem= 2 and for the three di\u000berent positions of the\ndouble occupied orbital (doublon). Thus, the e\u000bective\nHamiltonian will contain spin products ( ~S1\u0001~S2) between\nspin operators de\fned as,\n~Sm=1\n2X\n\rdy\nm\r\u000b~ \u001b\u000b\fdm\r\f; (8)6\n3d atom  4d atomabc\neI2site 1                site 2 \nFIG. 3. Schematic representation of one con\fguration be-\nlonging to the manifold of 36 degenerate ground states for a\nrepresentative 3 d\u00004dbondh12ias given by the local Coulomb\nHamiltonian Hint(m) (6) withm= 1;2. The dominant ex-\nchange processes considered here are those that move one of\nthe four electrons on the 4 datom to the 3 dneighbor and back.\nThe stability of the 3 d3-4d4charge con\fgurations is provided\nby the local potential energy Ie\n2, see Eq. (7).\nform= 1;2 sites and the operator of the doublon posi-\ntion at site m= 2,\nD(\r)\n2=\u0010\ndy\n2\r\"d2\r\"\u0011\u0010\ndy\n2\r#d2\r#\u0011\n: (9)\nThe doublon operator identi\fes the orbital \rwithin the\nt2gmanifold of the 4 dion with a double occupancy (oc-\ncupied by the doublon) and stands in what follows for\nthe orbital degree of freedom. It is worth noting that the\nhopping (5) does not change the orbital \ravor thus we\nexpect that the resulting Hamiltonian is diagonal in the\norbital degrees of freedom with only D(\r)\n2operators.\nFollowing the standard second order perturbation ex-\npansion for spin-orbital systems [12], we can write the\nmatrix elements of the low energy exchange Hamiltonian,\nH(\r)\nJ(i;j), for a bondh12ik\ralong the\raxis as follows,\n\nek\n1;el\n2\f\fH(\r)\nJ(1;2)\f\fek0\n1;el0\n2\u000b\n=\u0000X\nn1;n21\n\"n1+\"n2\n\u0002\nek\n1;el\n2\f\fHt(1;2)\f\fn1;n2\u000b\n\u0002\nn1;n2\f\fHt(1;2)\f\fek0\n1;el0\n2\u000b\n;(10)\nwith\"nm=En;m\u0000E0;mbeing the excitation energies for\natoms at site m= 1;2 with respect to the unperturbed\nground state. The superexchange Hamiltonian H(\r)\nJ(1;2)\nfor a bond along \rcan be expressed in a matrix form\nby a 36\u000236 matrix, with dependence on Um,JH\nm, and\nIeelements. There are two types of charge excitations:\n(i)d3\n1d4\n2\u000bd4\n1d3\n2one which creates a doublon at the 3 d\nimpurity, and (ii) d3\n1d4\n2\u000bd2\n1d5\n2one which adds another\ndoublon at the 4 dhost site in the intermediate state.\nThe second type of excitations involves more doubly oc-\ncupied orbitals and has much larger excitation energy. It\nis therefore only a small correction to the leading term\n(i), as we discuss in Appendix A.\nSimilar as in the case of doped manganites [48], the\ndominant contribution to the e\u000bective low-energy spin-\norbital Hamiltonian for the 3 d\u00004dbond stems from thed3\n1d4\n2\u000bd4\n1d3\n2charge excitations, as they do not involve\nan extra double occupancy and the Coulomb energy U2.\nThe 3d3\n14d4\n2\u000b3d4\n14d3\n2charge excitations can be analyzed\nin a similar way as the 3 d3\ni3d4\nj\u000b3d4\ni3d3\njones for anhiji\nbond in doped manganites [48]. In both cases the total\nnumber of doubly occupied orbitals does not change, so\nthe main contributions come due to Hund's exchange. In\nthe present case, one more parameter plays a role,\n\u0001 =Ie+ 3(U1\u0000U2)\u00004(JH\n1\u0000JH\n2); (11)\nwhich stands for the mismatch potential energy (7) renor-\nmalized by the onsite Coulomb interactions fUmgand by\nHund's exchange fJH\nmg. On a general ground we expect\n\u0001 to be a positive quantity, since the repulsion Umshould\nbe larger for smaller 3 dshells than for the 4 dones and\nUmis the largest energy scale in the problem.\nLet us have a closer view on this dominant contribu-\ntion of the e\u000bective low-energy spin-orbital Hamiltonian\nfor the 3d\u00004dbond, given by Eq. (A2). For the anal-\nysis performed below and the clarity of our presentation\nit is convenient to introduce some scaled parameters re-\nlated to the interactions within the host and between the\nhost and the impurity. For this purpose we employ the\nexchange couplings JimpandJhost,\nJimp=t2\n4\u0001; (12)\nJhost=4t2\nh\nU2; (13)\nwhich follow from the virtual charge excitations gener-\nated by the kinetic energy, see Eqs. (2) and (5). We use\ntheir ratio to investigate the in\ruence of the impurity on\nthe spin-orbital order in the host. Here this the hopping\namplitude between two t2gorbitals at NN 4 datoms,JH\n2\nandU2refer to the host, and \u0001 (11) is the renormalized\nionization energy of the 3 d\u00004dbonds. The results de-\npend as well on Hund's exchange element for the impurity\nand on the one at host atoms,\n\u0011imp=JH\n1\n\u0001; (14)\n\u0011host=JH\n2\nU2; (15)\nNote that the ratio introduced for the impurity, \u0011imp(14),\nhas here a di\u000berent meaning from Hund's exchange used\nhere for the host, \u0011host(15), which cannot be too large\nby construction, i.e., \u0011host<1=3.\nWith the parametrization introduced above, the dom-\ninant term in the impurity-host Hamiltonian for the im-\npurity spin ~Siinteracting with the neighboring host spins\nf~Sjgatj2N(i), deduced fromH(\r)\n3d\u00004d(1;2) Eq. (A2),\ncan be written in a rather compact form as follows\nH3d\u00004d(i)'X\n\r;j2N(i)n\nJS(D(\r)\nj)(~Si\u0001~Sj) +EDD(\r)\njo\n;\n(16)7\n0.0 0.2 0.4 0.6 0.8 1.0\nηimp-0.20.00.20.40.6JS/Jimp, ED/JimpED\nJS(Dγ= 1)\nJS(Dγ= 0)22\nFIG. 4. Evolution of the spin exchange JS(D(\r)\n2) and the\ndoublon energy ED, both given in Eq. (16) for increasing\nHund's exchange \u0011impat the impurity.\nwhere the orbital (doublon) dependent spin couplings\nJS(D(\r)\nj) and the doublon energy EDdepend on \u0011imp.\nThe evolution of the exchange couplings are shown in\nFig. 4. We note that the dominant energy scale is E\r\nD,\nso for a single 3 d\u00004dbond the doublon will avoid occupy-\ning the inactive ( \r) orbital and the spins will couple with\nJS(D(\r)\nj= 0) which can be either AF if \u0011imp.0:43 or\nFM if\u0011imp>0:43. Thus the spins at \u0011imp=\u0011c\nimp'0:43\nwill decouple according to the H(\r)\nJ(i;j) exchange.\nLet us conclude this Section by writing the complete\nsuperexchange Hamiltonian,\nH=H3d\u00004d+H4d\u00004d+Hso; (17)\nwhereH3d\u00004d\u0011P\niH3d\u00004d(i) includes all the 3 d\u00004d\nbonds around impurities, H4d\u00004dstands for the the ef-\nfective spin-orbital Hamiltonian for the 4 dhost bonds,\nandHsois the spin-orbit interaction in the host. The\nformer term we explain below, while the latter one is\nde\fned in Sec. V B, where we analyze the quantum cor-\nrections and the consequences of spin-orbit interaction.\nThe superexchange in the host for the bonds hijialong\nthe\r=a;baxes [81],\nH4d\u00004d=JhostX\nhijik\rn\nJ(\r)\nij(~Si\u0001~Sj+ 1) +K(\r)\nijo\n;(18)\ndepends on J(\r)\nijandK(\r)\nijoperators acting only in the\norbital space. They are expressed in terms of the pseu-\ndospin operators de\fned in the orbital subspace spanned\nby the two orbital \ravors active along a given direction\n\r, i.e.,\nJ(\r)\nij=1\n2(2r1+ 1)\u0000\n~ \u001ci\u0001~ \u001cj\u0001(\r)\u00001\n2r2\u0000\n\u001cz\ni\u001cz\nj\u0001(\r)\n+1\n8\u0000\nninj\u0001(\r)(2r1\u0000r2+1)\u00001\n4r1\u0000\nni+nj\u0001(\r);(19)K(\r)\nij=r1(~ \u001ci\u0001~ \u001cj)(\r)+r2\u0000\n\u001cz\ni\u001cz\nj\u0001(\r)+1\n4(r1+r2)\u0000\nninj\u0001(\r)\n\u00001\n4(r1+ 1)\u0000\nni+nj\u0001(\r): (20)\nwith\nr1=\u0011host\n1\u00003\u0011host; r 2=\u0011host\n1 + 2\u0011host; (21)\nstanding for the multiplet structure in charge excitations,\nand the orbital operators f~ \u001c(\r)\ni;n(\r)\nigthat for the \r=c\naxis take the form:\n~ \u001c(c)\ni=1\n2\u0000\nay\niby\ni\u0001\n\u0001~ \u001b\u0001\u0000\naibi\u0001|; (22)\nn(c)\ni=ay\niai+by\nibi: (23)\nFor the directions \r=a;bin the considered ( a;b) plane\none \fnds equivalent expressions by cyclic permutation of\nthe axis labelsfa;b;cgin the above formulas. This prob-\nlem is isomorphic with the spin-orbital superexchange in\nthe vanadium perovskites [50, 51], where a hole in the\nfa;bgdoublet plays an equivalent role to the doublon in\nthe present case. The operators fay\ni;by\ni;cy\nigare the dou-\nblon (hard core boson) creation operators in the orbital\n\r=a;b;c , respectively, and they satisfy the local con-\nstraint,\nay\niai+by\nibi+cy\nici= 1; (24)\nmeaning that exactly onedoublon (9) occupies one of\nthe threet2gorbitals at each site i. These bosonic occu-\npation operators coincide with the previously used dou-\nblon occupation operators D(\r)\nj, i.e.,D(\r)\nj=\ry\nj\rjwith\n\r=a;b;c . Below we follow \frst the classical procedure\nto determine the ground states of single impurities in Sec.\nIII, and at macroscopic doping in Sec IV.\nIII. SINGLE 3d IMPURITY IN 4d HOST\nA. Classical treatment of the impurity problem\nIn this Section we describe the methodology that we\napplied for the determination of the phase diagrams for a\nsingle impurity reported below in Secs. III C, and next at\nmacroscopic doping, as presented in Sec. IV. Let us con-\nsider \frst the case of a single 3 dimpurity in the 4 dhost.\nSince the interactions in the model Hamiltonian are only\ne\u000bective ones between NN sites, it is su\u000ecient to study\nthe modi\fcation of the spin-orbital order around the im-\npurity for a given spin-orbital con\fguration of the host\nby investigating a cluster of L= 13 sites shown in Fig. 5.\nWe assume the C-AF spin order (FM chains coupled an-\ntiferromagnetically) accompanied by G-AO order within\nthe host which is the spin-orbital order occurring for the\nrealistic parameters of a RuO 2plane [81], see Fig. 2. Such\na spin-orbital pattern turns out to be the most relevant\none when considering the competition between the host8\nFIG. 5. Schematic top view of the cluster used to obtain\nthe phase diagrams of the 3 dimpurity within the 4 dhost in\nan (a;b) plane. The impurity is at the central site i= 0\nwhich belongs to the corbital host sublattice. For the outer\nsites in this cluster the spin-orbital con\fguration is \fxed and\ndetermined by the undoped 4 dhost (with spins and cor-\nbitals shown here) having C-AF/G-AO order, see Fig. 2. For\nthe central i= 0 site the spin state and for the host sites\ni= 1;:::; 4 the spin-orbital con\fgurations are determined by\nminimizing the energy of the cluster.\nand the impurity as due to the AO order within the ( a;b)\nplane. Other possible con\fgurations with uniform orbital\norder and AF spin pattern, e.g. G-AF order, will also be\nconsidered in the discussion throughout the manuscript.\nThe sitesi= 1;2;3;4 inside the cluster in Fig. 5 have\nactive spin and orbital degrees of freedom while the im-\npurity at site i= 0 has only spin degree of freedom. At\nthe remaining sites the spin-orbital con\fguration is as-\nsigned, following the order in the host, and it does not\nchange along the computation.\nTo determine the ground state we assume that the\nspin-orbital degrees of freedom are treated as classical\nvariables. This implies that for the bonds between atoms\nin the host we use the Hamiltonian (18) and neglect quan-\ntum \ructuations, i.e., in the spin sector we keep only the\nzth (Ising) spin components and in the orbital one only\nthe terms which are proportional to the doublon occupa-\ntion numbers (9) and to the identity operators. Similarly,\nfor the impurity-host bonds we use the Hamiltonian Eq.\n(16) by keeping only the zth projections of spin operators.\nSince we do neglect the \ructuation of the spin amplitude\nit is enough to consider only the maximal and minimal\nvalues ofhSz\niifor spinS= 3=2 at the impurity sites and\nS= 1 at the host atoms. With these assumptions we can\nconstruct all the possible con\fgurations by varying the\nspin and orbital con\fgurations at the sites from i= 1\ntoi= 4 in the cluster shown in Fig. 5. Note that the\nouter ions in the cluster belong all to the same sublattice,\nso two distinct cases have to be considered to probe all\nthe con\fgurations. Since physically it is unlikely that asingle impurity will change the orbital order of the host\nglobally thus we will not compare the energies from these\ntwo cases and analyze two classes of solutions separately,\nsee Sec. III B. Then, the lowest energy con\fguration in\neach class provides the optimal spin-orbital pattern for\nthe NNs around the 3 dimpurity. In the case of degener-\nate classical states, the spin-orbital order is established\nby including quantum \ructuations.\nIn the case of a periodic doping analyzed in Sec. IV,\nwe use a similar strategy in the computation. Taking\nthe most general formulation, we employ larger clusters\nhaving both size and shape that depend on the impurity\ndistribution and on the spin-orbital order in the host.\nFor this purpose, the most natural choice is to search\nfor the minimum energy con\fguration in the elementary\nunit cell that can reproduce the full lattice by a suitable\nchoice of the translation vectors. This is computation-\nally expensive but doable for a periodic distribution of\nthe impurities that is commensurate to the lattice be-\ncause it yields a unit cell of relatively small size for dop-\ning around x= 0:1. Otherwise, for the incommensurate\ndoping the size of the unit cell can lead to a con\fguration\nspace of a dimension that impedes \fnding of the ground\nstate. This problem is computationally more demanding\nand to avoid the comparison of all the energy con\fgu-\nrations, we have employed the Metropolis algorithm at\nlow temperature to achieve the optimal con\fguration it-\neratively along the convergence process. Note that this\napproach is fully classical, meaning that the spins of the\nhost and impurity are treated as Ising variables and the\norbital \ructuations in the host's Hamiltonian Eq. (18)\nare omitted. They will be addressed in Sec. V A.\nB. Two nonequivalent 3ddoping cases\nThe single impurity problem is the key case to start\nwith because it shows how the short-range spin-orbital\ncorrelations are modi\fed around the 3 datom due to\nthe host and host-impurity interactions in Eq. (17).\nThe analysis is performed by \fxing the strength between\nHund's exchange and Coulomb interaction within the\nhost (6) at \u0011host= 0:1, and by allowing for a variation of\nthe ratio between the host-impurity interaction (16) and\nthe Coulomb coupling at the impurity site. The choice\nof\u0011host= 0:1 is made here because this value is within\nthe physically relevant range for the case of the ruthe-\nnium materials. Small variations of \u0011hostdo not a\u000bect\nthe obtained results qualitatively.\nAs we have already discussed in the model derivation,\nthe sign of the magnetic exchange between the impurity\nand the host depends on the orbital occupation of the\n4ddoublon around the 3 dimpurity. The main aspect\nthat controls the resulting magnetic con\fguration is then\ngiven by the character of the doublon orbitals around\nthe impurity, depending on whether they are active or\ninactive along the considered 3 d\u00004dbond. To explore\nsuch a competition quantitatively we investigate G-AO9\norder for the host with alternation of aandcdoublon\ncon\fgurations accompanied by the C-AF spin pattern,\nsee Fig. 2. Note that the aorbitals are active only along\nthebaxis, while the corbitals are active along the both\naxes:aandb[79]. This state has the lowest energy for the\nhost in a wide range of parameters for Hund's exchange,\nCoulomb element and crystal-\feld potential [81].\nDue to the speci\fc orbital pattern of Fig. 2, the 3 dim-\npurity can substitute one of two distinct 4 dsites which\nare considered separately below, either with aor withc\norbital occupied by the doublon. Since the two 4 datoms\nhave nonequivalent surrounding orbitals, not always ac-\ntive along the 3 d\u00004dbond, we expect that the result-\ning ground state will have a modi\fed spin-orbital order.\nIndeed, if the 3 datom replaces the 4 done with the dou-\nblon in the aorbital, then all the 4 dneighboring sites\nhave active doublons along the connecting 3 d\u00004dbonds\nbecause they are in the corbitals. On the contrary, the\nsubstitution at the 4 dsite withcorbital doublon con-\n\fguration leads to an impurity state with its neighbors\nhaving both active and inactive doublons. Therefore, we\ndo expect a more intricate competition for the latter case\nof an impurity occupying the 4 dsite withcorbital con-\n\fguration. Indeed, this leads to frustrated host-impurity\ninteractions, as we show in Sec. III D.\nC. Doping removing a doublon in aorbital\nWe start by considering the physical situation where\nthe 3dimpurity replaces a 4 dion with the doublon within\ntheaorbital. The ground state phase diagram and the\nschematic view of the spin-orbital pattern are reported\nin Fig. 6 in terms of the ratio Jimp=Jhost(14) and the\nstrength of Hund's exchange coupling \u0011imp(12) at the 3 d\nsite. There are three di\u000berent ground states that appear\nin the phase diagram. Taking into account the struc-\nture of the 3 d\u00004dspin-orbital exchange (16) we expect\nthat, in the regime where the host-impurity interaction\nis greater than that in the host, the 4 dneighbors to the\nimpurity tend to favor the spin-orbital con\fguration set\nby the 3d\u00004dexchange. In this case, since the orbitals\nsurrounding the 3 dsite already minimize the 3 d\u00004d\nHamiltonian, we expect that the optimal spin con\fgura-\ntion corresponds to the 4 dspins aligned either antifer-\nromagnetically or ferromagnetically with respect to the\nimpurity 3dspin.\nThe neighbor spins are AF to the 3 dspin impurity\nin the AFaphase, while the FM aphase is just obtained\nfrom AFaby reversing the spin at the impurity, and hav-\ning all the 3 d\u00004dbonds FM. It is interesting to note that\ndue to the host-impurity interaction the C-AF spin pat-\ntern of the host is modi\fed in both the AF aand the\nFMaground states. Another intermediate con\fguration\nwhich emerges when the host-impurity exchange is weak\nin the intermediate FS aphase where the impurity spin\nis undetermined and its con\fguration in the initial C-\nAF phase is degenerate with the one obtained after the\n0.00.20.40.60.81.0ηimp\n0.0 0.5 1.0 1.5 2.0 2.5 3.0\nJimp/JhostAFaFMa\nAFa FSa abFSa\n?FIG. 6. Top | Phase diagram of the 3 dimpurity in the ( a;b)\nplane with the C-AF/G-AO order in 4 dhost for the impurity\ndoped at the sublattice with an a-orbital doublon. Di\u000berent\ncolors refer to local spin order around the impurity, AF and\nFM, while FS indicates the intermediate regime of frustrated\nimpurity spin. Bottom | Schematic view of spin-orbital pat-\nterns for the ground state con\fgurations shown in the top\npanel. The 3 datom is at the central site, the dotted frame\nhighlights the 4 dsites where the impurity induces a a spin\nreversal. In the FS aphase the question mark stands for that\nthe frustrated impurity spin within the classical approach but\nfrustration is released by the quantum \ructuations of the NN\ncorbitals in the adirection resulting in small AF couplings\nalong theaaxis, and spins obey the C-AF order (small ar-\nrow). The labels FM aand AFarefer to the local spin order\naround the 3 dimpurity site with respect to the host | these\nstates di\u000ber by spin inversion at the 3 datom site.\nspin-inversion operation. This is a singular physical sit-\nuation because the impurity does not select a speci\fc\ndirection even if the surrounding host has a given spin-\norbital con\fguration. Such a degeneracy is clearly veri-\n\fed at the critical point \u0011c\nimp'0:43 where the amplitude\nof the 3d\u00004dcoupling vanishes when the doublon occu-\npies the active orbital. Interestingly, such a degenerate\ncon\fguration is also obtained at Jimp=Jhost<1 when the\nhost dominates and the spin con\fguration at the 4 dsites\naround the impurity are basically determined by Jhost. In\nthis case, due to the C-AF spin order, always two bonds\nare FM and other two have AF order, independently of\nthe spin orientation at the 3 dimpurity. This implies that10\norbital\n   fliporbital\n   flip0(a) (b)\nFIG. 7. Schematic view of the two types of orbital bonds\nfound in the 4 dhost: (a) an active bond with respect to\norbital \rips, ( \u001c\r+\ni\u001c\r\u0000\nj+H:c:), and (b) an inactive bond, where\norbital \ructuations are blocked by the orbital symmetry |\nhere the orbitals are static and only Ising terms contribute to\nthe ground state energy.\nboth FM or AF couplings along the 3 d\u00004dbonds per-\nfectly balance each other which results in the degenerate\nFSaphase.\nIt is worth pointing out that there is a quite large re-\ngion of the phase diagram where the FS astate is stabi-\nlized and the spin-orbital order of the host is not a\u000bected\nby doping with the possibility of having large degeneracy\nin the spin con\fguration of the impurities. On the other\nhand, by inspecting the corbitals around the impurity\n(Fig. 6) from the point of view of the full host's Hamilto-\nnian Eq. (18) with orbital \rips included, ( \u001c\r+\ni\u001c\r\u0000\nj+H:c:),\none can easily \fnd out that the frustration of the impu-\nrity spin can be released by quantum orbital \ructuations.\nNote that the corbitals around the impurity in the a\n(b) direction have quite di\u000berent surroundings. The ones\nalong theaaxis are connected by two active bonds along\nthebaxis with orbitals a, as in Fig. 7(a), while the\nones alongbare connected with only one activeaorbital\nalong the same baxis. This means that in the perturba-\ntive expansion the orbital \rips will contribute only along\nthebbonds (for the present G-AO order) and admix the\naorbital character to corbitals along them, while such\nprocesses will be blocked for the bonds along the aaxis,\nas also forborbitals along the baxis, see Fig. 7(b).\nThis fundamental di\u000berence can be easily included in\nthe host-impurity bond in the mean-\feld manner by set-\ntinghDi\u0006b;\ri= 0 for the bonds along the baxis and\n0<hDi\u0006a;\ri<1 for the bonds along the aaxis. Then\none can easily check that for the impurity spin point-\ning downwards we get the energy contribution from the\nspin-spin bonds which is given by E#=\u000b(\u0011host)hDi\u0006a;\ri,\nand for the impurity spin pointing upwards we have\nE\"=\u0000\u000b(\u0011host)hDi\u0006a;\ri, with\u000b(\u0011host)>0. Thus, it is\nclear that any admixture of the virtual orbital \rips in the\nhost's wave function polarize the impurity spin upwards\nso that the C-AF order of the host will be restored.\nD. Doping removing a doublon in corbital\nLet us move to the case of the 3 datom replacing the\ndoublon at corbital. As anticipated above, this con\fgu-\nration is more intricate because the orbitals surroundingthe impurity, as originated by the C-AF/AO order within\nthe host, lead to nonequivalent 3 d\u00004dbonds. There are\ntwo bonds with the doublon occupying an inactive or-\nbital (and has no hybridization with the t2gorbitals at\n3d atom), and two remaining bonds with doublons in\nactivet2gorbitals.\nSince the 3d\u00004dspin-orbital exchange depends on the\norbital polarization of 4 dsites we do expect a competition\nwhich may modify signi\fcantly the spin-orbital correla-\ntions in the host. Indeed, one observes that three con\fg-\nurations compete, denoted as AF1 c, AF2cand FMc, see\nFig. 8. In the regime where the host-impurity exchange\ndominates the system tends to minimize the energy due\nto the 3d\u00004dspin-orbital coupling and, thus, the orbitals\nbecome polarized in the active con\fgurations compatible\nwith theC-AF/G-AO pattern and the host-impurity spin\ncoupling is AF for \u0011imp\u00140:43, while it is FM otherwise.\nThis region resembles orbital polarons in doped mangan-\nites [39, 42]. Also in this case, the orbital polarons arise\nbecause they minimize the double exchange energy [46].\nOn the contrary, for weak spin-orbital coupling be-\ntween the impurity and the host there is an interesting\ncooperation between the 3 dand 4datoms. Since the\nstrength of the impurity-host coupling is not su\u000ecient to\npolarize the orbitals at the 4 dsites, it is preferable to\nhave an orbital rearrangement to the con\fguration with\ninactive orbitals on 3 d\u00004dbonds and spin \rips at 4 d\nsites. In this way the spin-orbital exchange is optimized\nin the host and also on the 3 d\u00004dbonds. The resulting\nstate has an AF coupling between the host and the im-\npurity as it should when all the orbitals surrounding the\n3datoms are inactive with respect to the bond direction.\nThis modi\fcation of the orbital con\fguration induces the\nchange in spin orientation. The double exchange bonds\n(with inactive doublon orbitals) along the baxis are then\nblocked and the total energy is lowered, in spite of the\nfrustrated spin-orbital exchange in the host. As a result,\nthe AF1cstate the spins surrounding the impurity are\naligned and antiparallel to the spin at the 3 dsite.\nConcerning the host C-AF/G-AO order we note that\nit is modi\fed only along the direction where the FM cor-\nrelations develop and spin defects occur within the chain\ndoped by the 3 datom. The FM order is locally disturbed\nby the 3ddefect antiferromagnetically coupled spins sur-\nrounding it. Note that this phase is driven by the or-\nbital vacancy as the host develops more favorable orbital\nbonds to gain the energy in the absence of the orbital\ndegrees of freedom at the impurity. At the same time\nthe impurity-host bonds do not generate too big energy\nlosses as: (i) either \u0011impis so small that the loss due to\nEDis compensated by the gain from the superexchange\n/JS(D(\r)\nj= 1) (all these bonds are AF), see Fig. 4, or\n(ii)Jimp=Jhostis small meaning that the overall energy\nscale of the impurity-host exchange remains small. Inter-\nestingly, if we compare the AF1 cwith the AF2 cground\nstates we observe that the disruption of the C-AF/G-\nAO order is anisotropic and occurs either along the FM\nchains in the AF1 cphase or perpendicular to the FM11\n0.00.20.40.60.81.0ηimp\n0.0 0.5 1.0 1.5 2.0 2.5 3.0\nJimp/JhostAF1cFMc\nAF2c\nAF1c AF2c ab\nFIG. 8. Top | Phase diagram of the 3 dimpurity in the\n4dhost withC-AF/G-AO order and the impurity doped at\nthecdoublon sublattice. Di\u000berent colors refer to local spin\norder around the impurity: AF c1, AFc2, and FM. Bottom |\nSchematic view of spin-orbital patterns for the two AF ground\nstate con\fgurations shown in the top panel; the FM cphase\ndi\u000bers from the AF c2 one only by spin inversion at the 3 d\natom. The 3 datom is at the central site and has no doublon\norbital, the frames highlight the spin-orbital defects caused\nby the impurity. As in Fig. 6, the labels AF and FM refer to\nthe impurity spin orientation with respect to the neighboring\n4dsites.\nchains in the AF2 cphase. No spin frustration is found\nhere, in contrast to the FS aphase in the case of adoublon\ndoping, see Fig. 6.\nFinally, we point out that a very similar phase diagram\ncan be obtained assuming that the host has the FM/ G-\nAO order with aandborbitals alternating from site to\nsite. Such con\fguration can be stabilized by a distortion\nthat favors the out-of-plane orbitals. In this case there\nis no di\u000berence in doping at one or the other sublattice.\nThe main di\u000berence is found in energy scales | for the\nG-AF/C-AO order the diagram is similar to the one of\nFig. 8 if we rescale Jimpby half, which means that the\nG-AF order is softer than the C-AF one. Note also that\nin the peculiar AF1 cphase the impurity does not induce\nany changes in the host for the FM/AO ordered host.\nThus we can safely conclude that the observed change in\nthe orbital order for the C-AF host in the AF1 cphase\nis due to the presence of the corbitals which are notdirectional in the ( a;b) plane.\nSummarizing, we have shown the complexity of local\nspin-orbital order around t3\n2gimpurities in a 4 t4\n2ghost. It\nis remarkable that such impurity spins not only modify\nthe spin-orbital order around them in a broad regime\nof parameters, but also are frequently frustrated. This\nhighlights the importance of quantum e\u000bects beyond the\npresent classical approach which release frustration as we\nshow in Sec. V A.\nIV. PERIODIC 3d DOPING IN 4d HOST\nA. General remarks on \fnite doping\nIn this Section we analyze the spin-orbital patterns due\nto a \fnite concentration xof 3dimpurities within the 4 d\nhost withC-AF/G-AO order, assuming that the 3 dim-\npurities are distributed in a periodic way. The study is\nperformed for three representative doping distributions\n| the \frst one x= 1=8 is commensurate with the un-\nderlying spin-orbital order and the other two are incom-\nmensurate with respect to it, meaning that in such cases\ndoping at both aandcdoublon sites is imposed simulta-\nneously.\nAs the impurities lead to local energy gains due to\n3d\u00004dbonds surrounding them, we expect that the most\nfavorable situation is when they are isolated and have\nmaximal distances between one another. Therefore, we\nselected the largest possible distances for the three dop-\ning levels used in our study: x= 1=8,x= 1=5, and\nx= 1=9. This choice allows us to cover di\u000berent regimes\nof competition between the spin-orbital coupling within\nthe host and the 3 d\u00004dcoupling. While single impu-\nrities may only change spin-orbital order locally, we use\nhere a high enough doping to investigate possible global\nchanges in spin-orbital order, i.e., whether they can oc-\ncur in the respective parameter regime. The analysis is\nperformed as for a single impurity, by assuming the clas-\nsical spin and orbital variables and by determining the\ncon\fguration with the lowest energy. For this analysis\nwe set the spatial distribution of the 3 datoms and we\ndetermine the spin and orbital pro\fle that minimizes the\nenergy.\nB.C-AF phase with x= 1=8doping\nWe begin with the phase diagram obtained at x= 1=8\n3ddoping, see Fig. 9. In the regime of strong impurity-\nhost coupling the 3 d\u00004dspin-orbital exchange deter-\nmines the orbital and spin con\fguration of the 4 datoms\naround the impurity. The most favorable state is when\nthe doublon occupies corbitals at the NN sites to the im-\npurity. The spin correlations between the impurity and\nthe host are AF (FM), if the amplitude of \u0011impis below\n(above)\u0011c\nimp, leading to the AF aand the FM astates,\nsee Fig. 9. The AF astate has a striped-like pro\fle with12\nAF chains alternated by FM domains (consisting of three\nchains) along the diagonal of the square lattice. Even if\nthe coupling between the impurity and the host is AF\nfor all the bonds in the AF astate, the overall con\fgura-\ntion has a residual magnetic moment originating by the\nuncompensated spins and by the cooperation between\nthe spin-orbital exchange in the 4 dhost and that for the\n3d\u00004dbonds. Interestingly, at the point where the domi-\nnant 3d\u00004dexchange tends to zero (i.e., for \u0011imp'\u0011c\nimp),\none \fnds a region of the FS aphase which is analogous\nto the FSaphase found in Sec. III C for a single impu-\nrity, see Fig. 6. Again the impurity spin is frustrated\nin purely classical approach but this frustration is easily\nreleased by the orbital \ructuations in the host so that\ntheC-AF order of the host can be restored. This state\nis stable for the amplitude of \u0011impbeing close to \u0011c\nimp.\nThe regime of small Jimp=Jhostratio is qualitatively\ndi\u000berent | an orbital rearrangement around the impu-\nrity takes place, with a preference to move the doublons\ninto the inactive orbitals along the 3 d\u00004dbonds. Such\norbital con\fgurations favor the AF spin coupling at all\nthe 3d\u00004dbonds which is stabilized by the 4 d\u00004dsu-\nperexchange [38]. This con\fguration is peculiar because\nit generally breaks inversion and does not have any plane\nof mirror symmetry. It is worth pointing out that the\noriginal order in the 4 dhost is completely modi\fed by\nthe small concentration of 3 dions and one \fnds that the\nAF coupling between the 3 dimpurity and the 4 dhost\ngenerally leads to patterns such as the AF cphase where\nFM chains alternate with AF ones in the ( a;b) plane.\nAnother relevant issue is that the cooperation between\nthe host and impurity can lead to a fully polarized FM a\nstate. This implies that doping can release the orbital\nfrustration which was present in the host with the C-\nAF/G-AO order.\nC. Phase diagram for periodic x= 1=5doping\nNext we consider doping x= 1=5 with a given periodic\nspatial pro\fle which concerns both doublon sublattices.\nWe investigate the 3 dspin impurities separated by the\ntranslation vectors ~ u= (i;j) and~ v= (2;\u00001) (one can\nshow that for general periodic doping x,j~ uj2=x\u00001) so\nthere is a mismatch between the impurity periodicity and\nthe two-sublattice G-AO order in the host. One \fnds\nthat the present case, see Fig. 10, has similar general\nstructure of the phase diagram to the case of x= 1=8\n(Fig. 9), with AF correlations dominating for \u0011implower\nthan\u0011c\nimpand FM ones otherwise. Due to the speci\fc\ndoping distribution there are more phases appearing in\nthe ground state phase diagram. For \u0011imp< \u0011c\nimpthe\nmost stable spin con\fguration is with the impurity cou-\npled antiferromagnetically to the host. This happens\nboth in the AF vacancy (AF v) and the AF polaronic\n(AFp) ground states. The di\u000berence between the two\nAF states arises due to the orbital arrangement around\nthe impurity. For weak ratio of the impurity to the host\n0.0 0.5 1.0 1.5 2.0 2.5 3.0\nJimp/Jhost0.00.20.40.60.81.0ηimp\nFMa\nFSa \nAFc AFaAFc \nFSa AFa \nab\n???\n?FIG. 9. Top panel | Ground state diagram obtained for pe-\nriodic 3ddopingx= 1=8. Di\u000berent colors refer to local spin\norder around the impurity: AF a, AFc, FSa, and FMa. Bot-\ntom panel | Schematic view of the ground state con\fgura-\ntions within the four 8-site unit cells (indicated by blue dashed\nlines) for the phases shown in the phase diagram. The ques-\ntion marks in FS aphase indicate frustrated impurity spins\nwithin the classical approach | the spin direction (small ar-\nrows) is \fxed only by quantum \ructuations. The 3 datoms\nare placed at the sites where orbitals are absent.\nspin-orbital exchange, Jimp=Jhost, the orbitals around the\nimpurity are all inactive ones. On the contrary, in the\nstrong impurity-host coupling regime all the orbitals are\npolarized to be in active (polaronic) states around the im-\npurity. Both states have been found as AF1 cand AF2c\nphase in the single impurity problem (Fig. 8).\nMore generally, for all phases the boundary given by\nan approximate hyperbolic relation \u0011imp/J\u00001\nimpsepa-\nrates the phases where the orbitals around impurities in13\n0.0 0.5 1.0 1.5 2.0 2.5 3.0\nJimp/Jhost0.00.20.40.60.81.0ηimp\nAFp FMp \nFMv \nAFv FSvFS1p \nFS2p \n??\n??\nab??\nFIG. 10. Ground state diagram for x= 1=5 periodic concentration of 3 dimpurities (sites where orbitals are absent) with\nschematic views of the ground state con\fgurations obtained for the unit cell of 20 sites. Spin and orbital order are shown by\narrows and orbitals occupied by doublons; magnetic phases (AF, FS, and FM) are highlighted by di\u000berent color. The question\nmarks in FS states (red circles) indicate frustrated impurity spins within the classical approach.\n? ?(a) FSv (b) FS1p, FS2p\nFIG. 11. Isotropic surrounding of the degenerate impurity\nspins in the FS vand FSpphases in the case of x= 1=5 pe-\nriodic doping (Fig. 10). Frames mark the clusters which are\nnot connected with orbitally active bonds.thec-orbital sublattice are all inactive (small \u0011imp) from\nthose where all the orbitals are active (large \u0011imp). The\ninactive orbital around the impurity stabilize always the\nAF coupling between the impurity spin and host spins\nwhereas the active orbitals can give either AF or FM\nexchange depending on \u0011imp(hence\u0011c\nimp, see Fig. 4).\nSince the doping does not match the size of the elemen-\ntary unit cell, the resulting ground states do not exhibit\nspeci\fc symmetries in the spin-orbital pattern. They are\ngenerally FM due to the uncompensated magnetic mo-\nments and the impurity feels screening by the presence\nof the surrounding it host spins being antiparallel to the\nimpurity spin.\nBy increasing Hund's exchange coupling at the 3 dion\nthe system develops fully FM state in a large region of\nthe ground state diagram due to the possibility of suit-\nable orbital polarization around the impurity. On the\nother hand, in the limit where the impurity-host bonds14\nab0.0 0.5 1.0 1.5 2.0 2.5 3.0\nJimp/Jhost0.00.20.40.60.81.0ηimp\nAFp FMp \nFMv \nAFv FSvFS1p \nFS2p ??\n??\n??\nFIG. 12. Ground state diagram for x= 1=9 periodic concentration of 3 dimpurities with schematic views of the ground state\ncon\fgurations obtained for the unit cell of 36 sites. Spin order (AF, FS, and FM) is highlighted by di\u000berent color. The question\nmarks in FS states (red squares) indicate frustrated impurity spins within the classical approach | the spin is \fxed here by\nquantum \ructuations (small arrows). Doped 3 datoms are at the sites where orbitals are absent.\nare weak, so either for \u0011imp'\u0011c\nimpand large enough\nJimp=Jhostso that all orbitals around the impurity are\nactive, or just for small Jimp=Jhostwe get the FS phases\nwhere the impurity spin at the a-orbital sublattice is un-\ndetermined in the present classical approach. This is a\nsimilar situation to the one found in the FS aphase of a\nsingle impurity problem and at x= 1=8 periodic doping,\nsee Figs. 6 and 9, but there it was still possible to iden-\ntify the favored impurity spin polarization by considering\nthe orbital \rips in the host around the impurity.\nHowever, the situation here is di\u000berent as the host's\norder is completely altered by doping and has became\nisotropic, in contrast to the initial C-AF order (Fig. 2)\nwhich breaks the planar symmetry between the aandb\ndirection. It was precisely this symmetry breaking that\nfavored one impurity spin polarization over the other one.\nHere this mechanism is absent | one can easily checkthat the neighborhood of the corbitals surrounding im-\npurities is completely equivalent in both directions (see\nFig. 11 for the view of these surroundings) so that the\norbital \rip argument is no longer applicable. This is a\npeculiar situation in the classical approach and we indi-\ncate frustration in spin direction by question marks in\nFig. 10.\nIn Fig. 11 we can see that both in FS vacancy (FS v)\nand FS polaronic (FS p) phase the orbitals are grouped in\n3\u00023 clusters and 2\u00022 plaquettes, respectively, that en-\ncircle the degenerate impurity spins. For the FS vphase\nwe can distinguish between two kind of plaquettes with\nnon-zero spin polarization di\u000bering by a global spin inver-\nsion. In the case of FS pphases we observe four plaquettes\nwith zero spin polarization arranged in two pairs related\nby a point re\rection with respect to the impurity site.\nIt is worthwhile to realize that these plaquettes are com-15\npletely disconnected in the orbital sector, i.e., there are\nno orbitally active bonds connecting them (see Fig. 7 for\nthe pictorial de\fnition of orbitally active bonds). This\nmeans that quantum e\u000bects of purely orbital nature can\nappear only at the short range, i.e., inside the plaquettes.\nHowever, one can expect that if for some reason the two\ndegenerate spins in a single elementary cell will polarize\nthen they will also polarize in the same way in all the\nother cells to favor long-range quantum \ructuations in\nthe spin sector related to the translational invariance of\nthe system.\nD. Phase diagram for periodic x= 1=9doping\nFinally we investigate low doping x= 1=9 with a given\nperiodic spatial pro\fle, see Fig. 12. Here the impurities\nare separated by the translation vectors ~ u= (0;3) and\n~ v= (3;0). Once again there is a mismatch between the\nperiodic distribution of impurities and the host's two-\nsublattice AO order, so we again call this doping incom-\nmensurate as it also imposes doping at both doublon\nsublattices. The ground state diagram presents gradu-\nally increasing tendency towards FM 3 d\u00004dbonds with\nincreasing\u0011imp, see Fig. 12. These polaronic bonds po-\nlarize as well the 4 d\u00004dbonds and one \fnds an almost\nFM order in the FM pstate. Altogether, we have found\nthe same phases as at the higher doping of x= 1=5,\nsee Fig. 10, i.e., AF vand AFpat low values \u0011imp, FMv\nand FMpin the regime of high \u0011imp, separated by the\nregime of frustrated impurity spins which occur within\nthe phases: FS v, FS1p, and FS2p.\nThe di\u000berence between the two AF (FM) states in Fig.\n12 is due to the orbital arrangement around the impurity.\nAs for the other doping levels considered so far, x= 1=8\nandx= 1=5, we \fnd neutral (inactive) orbitals around\n3dimpurities in the regime of low Jimp=Jhostin AFvand\nFMvphases which lead to spin defects within the 1D FM\nchains in the C-AF spin order. A similar behavior was\nreported for single impurities in the low doping regime in\nSec. III. This changes radically above the orbital tran-\nsition for both types of local magnetic order, where the\norbitals reorient into the active ones. One \fnds that spin\norientations are then the same as those of their neigh-\nboring 4datoms, with some similarities to those found\natx= 1=5, see Fig. 10.\nFrustrated impurity spins occur in the crossover regime\nbetween the AF and FM local order around impurities.\nThis follows from the local con\fgurations around them,\nwhich include two \"-spins and two#-spins accompanied\nbycorbitals at the NN 4 dsites. This frustration is easily\nremoved by quantum \ructuations and we suggest that\nthis happens again in the same way as for x= 1=8 doping,\nas indicated by small arrows in the respective FS phases\nshown in Fig. 12.V. QUANTUM EFFECTS BEYOND THE\nCLASSICAL APPROACH\nA. Spin-orbital quantum \ructuations\nSo far, we analyzed the ground states of 3 dimpurities\nin the (a;b) plane of a 4 dsystem using the classical ap-\nproach. Here we show that this classical picture may be\nused as a guideline and is only quantitatively changed by\nquantum \ructuations if the spin-orbit coupling is weak.\nWe start the analysis by considering the quantum prob-\nlem in the absence of spin-orbit coupling (at \u0015= 0). The\norbital doublon densities,\nN\r\u0011X\ni2hosthni\ri; (25)\nwith\r=a;b;c , and totalSzare conserved quantities and\nthus good quantum numbers for a numerical simulation.\nTo determine the ground state con\fgurations in the pa-\nrameters space and the relevant correlation functions we\ndiagonalize exactly the Hamiltonian matrix (17) for the\ncluster ofL= 8 sites by means of the Lanczos algorithm.\nIn Fig. 13(a) we report the resulting quantum phase\ndiagram for an 8-site cluster having one impurity and\nassuming periodic boundary conditions, see Fig. 13(b).\nThis appears to be an optimal cluster con\fguration be-\ncause it contains a number of sites and connectivities that\nallows us to analyze separately the interplay between the\nhost-host and the host-impurity interactions and to sim-\nulate a physical situation when the interactions within\nthe host dominate over those between the host and the\nimpurity. Such a problem is a quantum analogue of the\nsingle unit cell presented in Fig. 9 for x= 1=8 periodic\ndoping.\nAs a general feature that resembles the classical phase\ndiagram, we observe that there is a prevalent tendency\nto have AF-like (FM-like) spin correlations between the\nimpurity and the host sites in the region of \u0011impbelow\n(above) the critical point at \u0011c\nimp'0:43 which separates\nthese two regimes, with intermediate con\fgurations hav-\ning frustrated magnetic exchange. As we shall discuss\nbelow it is the orbital degree of freedom that turns out\nto be more a\u000bected by the quantum e\u000bects. Following\nthe notation used for the classical case, we distinguish\nvarious quantum AF (QAF) ground states, i.e., QAF cn\n(n= 1;2) and QAF an(n= 1;2), as well as a uni-\nform quantum FM (QFM) con\fguration, i.e., QFM a, and\nquantum frustrated one labeled as QFS a.\nIn order to visualize the main spin-orbital patterns con-\ntributing to the quantum ground state it is convenient to\nadopt a representation with arrows for the spin and el-\nlipsoids for the orbital sector at any given host site. The\narrows stand for the on-site spin projection hSz\nii, with the\nlength being proportional to the amplitude. The length\nscale for the arrows is the same for all the con\fgurations.\nMoreover, in order to describe the orbital character of\nthe ground state we employed a graphical representation\nthat makes use of an ellipsoid whose semi-axes fa;b;cg16\n0.0 0.3 0.6 0.9 1.2 1.5\nJimp/Jhost0.000.250.500.751.00 ηimpQFMa\nQFSa1\nQAFc2 QAFc1QFSa2\nQAFa1\nQAFa2\n12 8 34 5 673 2\n6 41\n71\n71\n7(a)\n(b)\nab\nFIG. 13. (a) Phase diagram for the quantum problem at zero spin-orbit simulated on the 8-site cluster in the presence of\none-impurity. Arrows and ellipsoids indicate the spin-orbital state at a given site i, while the shapes of ellipsoids re\rect the\norbital avarages: hay\niaii,hby\nibiiandhcy\nicii(i.e., a circle in the plane perpendicular to the axis \rimplies 100% occupation of the\norbital\r). (b) The periodic cluster of L= 8 sites used, with the orbital dilution (3 d3impurity) at site i= 8. The dotted lines\nidentify the basic unit cell adopted for the simulation with the same symmetries of the square lattice.\nlength are given by the average amplitude of the squared\nangular momentum components f(Lx\ni)2;(Ly\ni)2;(Lz\ni)2g, or\nequivalently by the doublon occupation Eq. (9). For in-\nstance, for a completely \rat circle (degenerate ellipsoid)\nlying in the plane perpendicular to the \raxis only the\ncorresponding \rorbital is occupied. On the other hand,\nif the ellipsoid develops in all three directions fa;b;cgit\nimplies that more than one orbital is occupied and the\ndistribution can be anisotropic in general. If all the or-\nbitals contribute equally, one \fnds an isotropic spherical\nellipsoid.\nDue to the symmetry of the Hamiltonian, the phases\nshown in the phase diagram of Fig. 13(a) can be\ncharacterized by the quantum numbers for the z-th\nspin projection, Sz, and the doublon orbital occupa-\ntionN\u000b(25), (Sz;Na;Nb;Nc): QAFc1 (\u00003:5;2;2;3),\nQFSa2 (\u00001:5;3;1;3), QAFa1 (\u00005:5;1;3;3), QAFa2 and\nQAFa2 (\u00005:5;2;2;3), QFSa1 (\u00000:5;3;0;4), and QFM a\n(\u00008:5;2;1;4). Despite the irregular shape of the clus-\nter [Fig. 13(b)] there is also symmetry between the\naandbdirections. For this reason, the phases with\nNa6=Nbcan be equivalently described either by the\nset (Sz;Na;Nb;Nc) or (Sz;Nb;Na;Nc).\nThe outcome of the quantum analysis indicates thatthe spin patterns are quite robust as the spin con\fgu-\nrations of the phases QAF a, QAFc, QFSaand QFMa\nare the analogues of the classical ones. The e\u000bects of\nquantum \ructuations are more evident in the orbital sec-\ntor where mixed orbital patterns occur if compared to\nthe classical case. In particular, orbital inactive states\naround the impurity are softened by quantum \ructua-\ntions and on some bonds we \fnd an orbital con\fguration\nwith a superposition of active and inactive states. The\nunique AF states where the classical inactive scenario is\nrecovered corresponds to the QAF c1 and QAF c2 ones in\nthe regime of small \u0011imp. A small hybridization of active\nand inactive orbitals along both the AF and FM bonds is\nalso observed around the impurity for the QFS aphases\nas one can note by the shape of the ellipsoid at host sites.\nMoreover, in the range of large \u0011impwhere the FM state\nis stabilized, the orbital pattern around the impurity is\nagain like in the classical case.\nA signi\fcant orbital rearrangement is also obtained\nwithin the host. We generally obtain an orbital pattern\nthat is slightly modi\fed from the pure AO con\fguration\nassumed in the classical case. The e\u000bect is dramatically\ndi\u000berent in the regime of strong impurity-host coupling\n(i.e., for large Jimp) with AF exchange (QAF a2) with the17\nformation of an orbital liquid around the impurity and\nwithin the host, with doublon occupation represented\nby an almost isotropic shaped ellipsoid. Interestingly,\nthough with a di\u000berent orbital arrangement, the QFS a1\nand the QFS a2 states are the only ones where the C-AF\norder of the host is recovered. For all the other phases\nshown in the diagram of Fig. 13 the coupling between\nthe host and the impurity is generally leading to a uni-\nform spin polarization with FM or AF coupling between\nthe host and the impurity depending on the strength of\nthe host-impurity coupling. Altogether, we conclude that\nthe classical spin patterns are only quantitatively modi-\n\fed and are robust with respect to quantum \ructuations.\nB. Finite spin-orbit coupling\nIn this Section we analyze the quantum e\u000bects in the\nspin and orbital order around the impurity in the pres-\nence of the spin-orbit coupling at the host d4sites. For\nthet4\n2gcon\fguration the strong spin-orbit regime has\nbeen considered recently by performing an expansion\naround the atomic limit where the angular ~Liand spin\n~Simomenta form a spin-orbit singlet for the amplitude\nof the total angular momentum, ~Ji=~Li+~Si(i.e.,J= 0)\n[67]. The instability towards an AF state starting from\ntheJ= 0 liquid has been obtained within the spin-wave\ntheory [68] for the low energy excitations emerging from\nthe spin-orbital exchange.\nIn the analysis presented here we proceed from the\nlimit of zero spin-orbit to investigate how the spin and\norbital order are gradually suppressed when approaching\ntheJ= 0 spin-orbit singlet state. This issue is addressed\nby solving the full quantum Hamiltonian (17) exactly on\na cluster of L= 8 sites including the spin-orbital ex-\nchange for the host and that one derived for the host-\nimpurity coupling (17) as well as the spin-orbit term,\nHso=\u0015X\ni2host~Li\u0001~Si: (26)\nwhere the sum includes the ions of the 4 dhost and we\nuse the spin S= 1 and the angular momentum L= 1, as\nin the ionic 4 d4con\fgurations. Here \u0015is the spin-orbit\ncoupling constant at 4 dhost ions, and the components\nof the orbital momentum ~Li\u0011fLx\ni;Ly\ni;Lz\nigare de\fned\nas follows:\nLx\ni=iX\n\u001b(dy\ni;xy\u001bdi;xz\u001b\u0000dy\ni;xz\u001bdi;xy\u001b);\nLy\ni=iX\n\u001b(dy\ni;xy\u001bdi;yz\u001b\u0000dy\ni;yz\u001bdi;xy\u001b);\nLz\ni=iX\n\u001b(dy\ni;xz\u001bdi;yz\u001b\u0000dy\ni;yz\u001bdi;xz\u001b): (27)\nTo determine the ground state and the relevant corre-\nlation functions we use again the Lanczos algorithm for\nthe cluster of L= 8 sites. Such an approach allows us tostudy the competition between the spin-orbital exchange\nand the spin-orbit coupling on equal footing without any\nsimplifying approximation. Moreover, the cluster calcu-\nlation permits to include the impurity in the host and\ndeal with the numerous degrees of freedom without mak-\ning approximations that would constrain the interplay of\nthe impurity-host versus host-host interactions.\nFinite spin-orbit coupling signi\fcantly modi\fes the\nsymmetry properties of the problem. Instead of the\nSU(2) spin invariance one has to deal with the rotational\ninvariance related to the total angular momentum per\nsite~Ji=~Li+~Si. Though the ~Li\u0001~Siterm in Eq. (26)\ncommutes with both total ~J2andJz, the full Hamilto-\nnian for the host with impurities Eq. (17) has a reduced\nsymmetry because the spin sector is now linearly coupled\nto the orbital which has only the cubic symmetry. Thus\nthe remaining symmetry is a cyclic permutation of the\nfx;y;zgaxes.\nMoreover,Jzis not a conserved quantity due to the or-\nbital anisotropy of the spin-orbital exchange in the host\nand the orbital character of the impurity-host coupling.\nThere one has a Z2symmetry associated with the parity\noperator (-1)Jz. Hence, the ground state can be clas-\nsi\fed as even or odd with respect to the value of Jz.\nThis symmetry aspect can introduce a constraint on the\ncharacter of the ground state and on the impurity-host\ncoupling since the Jzvalue for the impurity is only due\nto the spin projection while in the host it is due to the\ncombination of the orbital and spin projection. A direct\nconsequence is that the parity constraint together with\nthe unbalance between the spin at the host and the im-\npurity sites leads to a nonvanishing total projection of\nthe spin and angular momentum with respect to a sym-\nmetry axis, e.g. the zthaxis. It is worth to note that a\n\fxed parity for the impurity spin means that it prefers\nto point in one direction rather than the other one which\nis not the case for the host's spin and angular momen-\ntum. Thus the presence on the impurity for a \fxed P\nwill give a nonzero polarization along the quantization\naxis for every site of the system. Such a property holds\nfor any single impurity with a half-integer spin.\nAnother important consequence of the spin-orbit cou-\npling is that it introduces local quantum \ructuations in\nthe orbital sector even at the sites close to the impurity\nwhere the orbital pattern is disturbed. The spin-orbit\nterm makes the on-site problem around the impurity ef-\nfectively analogous to the Ising model in a transverse \feld\nfor the orbital sector, with nontrivial spin-orbital entan-\nglement [34] extending over the impurity neighborhood.\nIn Figs. 14 and 15 we report the schematic evolu-\ntion of the ground state con\fgurations for the cluster\nofL= 8 sites, with one-impurity and periodic boundary\nconditions as a function of increasing spin-orbit coupling.\nThese patterns have been determined by taking into ac-\ncount the sign and the amplitude of the relevant spatial\ndependent spin and orbital correlation functions. The\narrows associated to the spin degree of freedom can lie in\nxyplane or out-of-plane (along z, chosen to be parallel18\nQAFc1\nQAFa1 QAFc2 QFMa\na\nba\nba\nba\nb λ1\nλ2\nλ3\nλ4\nλ5\nλ10λ1\nλ2\nλ3\nλ4\nλ7\nλ10λ1\nλ2\nλ4\nλ5\nλ7\nλ10λ1\nλ2\nλ3\nλ4\nλ7\nλ10\nFIG. 14. Evolution of the ground state con\fgurations for the AF phases for selected increasing values of spin-orbit coupling\n\u0015m, see Eq. (28). Arrows and ellipsoids indicate the spin-orbital state at a given site i. Color map indicates the strength of the\naverage spin-orbit, h~Li\u0001~Sii, i.e., red, yellow, green, blue, violet correspond to the growing amplitude of the above correlation\nfunction. Small arrows at \u00155and\u001510indicate quenched magnetization at the impurity by large spin-orbit coupling at the\nneighboring host sites.\nto thecaxis) to indicate the anisotropic spin pattern.\nThe out-of-plane arrow length is given by the on-site ex-\npectation value of hSz\niiwhile the in-plane arrow length\nis obtained by computing the square root of the second\nmoment, i.e.,p\nh(Sx\ni)2iandp\nh(Sy\ni)2iof thexandyspin\ncomponents corresponding to the arrows along aandb,\nrespectively.\nMoreover, the in-plane arrow orientation for a given\ndirection is determined by the sign of the correspond-\ning spin-spin correlation function assuming as a refer-ence the orientation of the impurity spin. The ellipsoid\nis constructed in the same way as for the zero spin-orbit\ncase above, with the addition of a color map that indi-\ncates the strength of the average ~Li\u0001~Si(i.e., red, yel-\nlow, green, blue, violet correspond with a growing ampli-\ntude of the local spin-orbit correlation function). The\nscale for the spin-orbit amplitude is set to be in the\ninterval 0 < \u0015 < J host. The selected values for the\nground state evolution are given by the relation (with19\nm= 1;2;:::; 10),\n\u0015m=\u0014\n0:04 + 0:96(m\u00001)\n9\u0015\nJhost: (28)\nThe scale is set such that \u00151= 0:04Jhostand\u001510=Jhost.\nThis range of values allows us to explore the relevant\nphysical regimes when moving from 3 dto 4dand 5dmate-\nrials with corresponding \u0015being much smaller that Jhost,\n\u0015\u0018Jhost=2 and\u0015 > J host, respectively. For the per-\nformed analysis the selected values of \u0015(28) are also rep-\nresentative of the most interesting regimes of the ground\nstate as induced by the spin-orbit coupling.\nLet us start with the quantum AF phases QAF c1,\nQAFc2, QFSa1, QFSa2, QAFa1, and QAF a2. As one\ncan observe the switching on of the spin-orbit coupling\n(i.e.,\u00151in Fig. 14) leads to anisotropic spin patterns\nwith unequal moments for the in-plane and out-of-plane\ncomponents. From weak to strong spin-orbit coupling,\nthe character of the spin correlations keeps being AF be-\ntween the impurity and the neighboring host sites in all\nthe spin directions. The main change for the spin sector\noccurs for the planar components. For weak spin-orbit\ncoupling the in-plane spin pattern is generally AF for the\nwhole system in all the spatial directions (i.e., G-AF or-\nder). Further increase of the spin-orbit does not modify\nqualitatively the character of the spin pattern for the out-\nof-plane components as long as we do not go to maximal\nvalues of\u0015\u0018Jhostwhere localhSz\niimoments are strongly\nsuppressed. In this limit the dominant tendency of the\nsystem is towards formation of the spin-orbital singlets\nand the spin patterns shown in Fig. 14 are the e\u000bect of\nthe virtual singlet-triplet excitations [67].\nConcerning the orbital sector, only for weak spin-orbit\ncoupling around the impurity one can still observe a rem-\niniscence of inactive orbitals as related to the orbital va-\ncancy role at the impurity site in the AF phase. Such\nan orbital con\fguration is quickly modi\fed by increasing\nthe spin-orbit interaction and it evolves into a uniform\npattern with almost degenerate orbital occupations in all\nthe directions, and with preferential superpositions of c\nand (a;b) states associated with dominating LxandLy\norbital angular components (\rattened ellipsoids along the\ncdirection). An exception is the QAF c2 phase with the\norbital inactive polaron that is stable up to large spin-\norbit coupling of the order of Jhost.\nWhen considering the quantum FM con\fgurations\nQFMa1 in Fig. 14, we observe similar trends in the evo-\nlution of the spin correlation functions as obtained for\nthe AF states. Indeed, the QFM aexhibits a tendency to\nform FM chains with AF coupling for the in-plane compo-\nnents at weak spin-orbit that evolve into more dominant\nAF correlations in all the spatial directions within the\nhost. Interestingly, the spin exchange between the impu-\nrity and the neighboring host sites shows a changeover\nfrom AF to FM for the range of intermediate-to-strong\nspin-orbit amplitudes.\nA peculiar response to the spin-orbit coupling is ob-\ntained for the QFS a1 phase, see Fig. 15, which showed\nQFSa2 QFSa1\nλ1\nλ2\nλ3λ4\nλ8\nλ10λ1\nλ2\nλ3\nλ5\nλ6\nλ10a\nba\nbFIG. 15. Evolution of the ground state con\fgurations for\nthe QFSa1 and QFSa2 phases for selected increasing values\nof spin-orbit coupling \u0015m, see Eq. (28). Arrows and ellip-\nsoids indicate the spin-orbital state at a given site i. Color\nmap indicates the strength of the average spin-orbit, h~Li\u0001~Sii,\ni.e., red, yellow, green, blue, violet correspond to the growing\namplitude of the above local correlation function.\na frustrated spin pattern around the impurity already in\nthe classical regime, with FM and AF bonds. It is re-\nmarkable that due to the close proximity with uniform\nFM and the AF states, the spin-orbit interaction can lead\nto a dramatic rearrangement of the spin and orbital cor-\nrelations for such a con\fguration. At weak spin-orbit\ncoupling (i.e., \u0015'\u00151) the spin-pattern is C-AF and the\nincreased coupling ( \u0015'\u00152)) keeps the C-AF order only20\nfor the in-plane components with the exception of the im-\npurity site. It also modulates the spin moment distribu-\ntion around the impurity along the zdirection. Further\nincrease of \u0015leads to complete spin polarization along\nthezdirection in the host, with antiparallel orientation\nwith respect to the impurity spin. This pattern is guided\nby the proximity to the FM phase. The in-plane compo-\nnents develop a mixed FM-AF pattern with a strong xy\nanisotropy most probably related to the di\u000berent bond\nexchange between the impurity and the host.\nWhen approaching the regime of a spin-orbit coupling\nthat is comparable to Jhost, the out-of-plane spin compo-\nnents dominate and the only out-of-plane spin polariza-\ntion is observed at the impurity site. Such a behavior is\nunique and occurs only in the QFS aphases. The coop-\neration between the strong spin-orbit coupling and the\nfrustrated host-impurity spin-orbital exchange leads to\nan e\u000bective decoupling in the spin sector at the impu-\nrity with a resulting maximal polarization. On the other\nhand, as for the AF states, the most favorable con\fgu-\nration for strong spin-orbit has AF in-plane spin corre-\nlations. The orbital pattern for the QFS astates evolves\nsimilarly to the AF cases with a suppression of the active-\ninactive interplay around the impurity and the setting of\na uniform-like orbital con\fguration with unquenched an-\ngular momentum on site and predominant in-plane com-\nponents. The response of the FM state is di\u000berent in\nthis respect as the orbital active states around the impu-\nrity are hardly a\u000bected by the spin-orbit while the host\nsites far from the impurity the local spin-orbit coupling\nis more pronounced.\nFinally, to understand the peculiar evolution of the\nspin con\fguration it is useful to consider the lowest order\nterms in the spin-orbital exchange that couple directly\nthe orbital angular momentum with the spin. Taking into\naccount the expression of the spin-orbital exchange in the\nhost (26) and the expression of ~Lione can show that the\nlow energy terms on a bond that get more relevant in\nthe Hamiltonian when the spin-orbit coupling makes a\nnon-vanishing local angular momentum. As a result, the\ncorresponding expressions are:\nHa(b)\nhost(i;j)\u0019Jhostn\na1~Si\u0001~Sj+b1Sz\niSz\njLy(x)\niLy(x)\njo\n+\u0015n\n~Li\u0001~Si+~Lj\u0001~Sjo\n; (29)\nwith positive coe\u000ecients a1andb1that depend on r1and\nr2(21). A de\fnite sign for the spin exchange in the limit\nof vanishing spin-orbit coupling is given by the terms\nwhich go beyond Eq. (29). Then, if the ground state\nhas isotropic FM correlations (e.g. QFM a) at\u0015= 0, the\ntermSz\niSz\njLy(x)\niLy(x)\njwould tend to favor AF-like con-\n\fgurations for the in-plane orbital angular components\nwhen the spin-orbit interaction is switched on. This op-\nposite tendency between the zandfx;ygcomponents is\ncounteracted by the local spin-orbit coupling that pre-\nvents to have coexisting FM and AF spin-orbital corre-\nlations. Such patterns would not allow to optimize the~Li\u0001~Siamplitudes. One way out is to reduce the zthspin\nprojection and to get planar AF correlations in the spin\nand in the host. A similar reasoning applies to the AF\nstates where the negative sign of the Sz\niSz\njcorrelations\nfavors FO alignment of the angular momentum compo-\nnents. As for the previous case, the opposite trend of in-\nand out-of-plane spin-orbital components is suppressed\nby the spin-orbit coupling and the in-plane FO correla-\ntions for thefLx;Lygcomponents leads to FM patterns\nfor the in-plane spin part as well.\nSummarizing, by close inspection of Figs. 14 and 15\none \fnds an interesting evolution of the spin patterns in\nthe quantum phases:\n(i) For the QAF states (Fig. 14), a spin canting devel-\nops at the host sites (i.e., the relative angle is between\n0 and\u0019) while the spins on impurity-host bonds are al-\nways AF. The canting in the host evolves, sometime in\nan inhomogeneous way, to become reduced in the strong\nspin-orbit coupling regime where ferro-like correlations\ntend to dominate. In this respect, when the impurity is\ncoupled antiferromagnetically to the host it does not fol-\nlow the tendency to form spin canting.\n(ii) In the QFM states (Fig. 15), at weak spin-orbit\none observes spin-canting in the host and for the host-\nimpurity coupling that persists only in the host whereas\nthe spin-orbit interaction is increasing.\nC. Spin-orbit coupling versus Hund's exchange\nTo probe the phase diagram of the system in presence\nof the spin-orbit coupling ( \u0015 > 0) we solved the same\ncluster ofL= 8 sites as before along three di\u000berent cuts\nin the phase diagram of Fig. 13(a) for three values of \u0015,\ni.e., small\u0015= 0:1Jhost, intermediate \u0015= 0:5Jhost, and\nlarge\u0015=Jhost. Each cut contained ten points, the cuts\nwere parameterized as follows: (i) Jimp= 0:7Jhostand\n0\u0014\u0011imp\u00140:7, (ii)Jimp= 1:3Jhostand 0\u0014\u0011imp\u00140:7,\nand (iii)\u0011imp=\u0011c\nimp'0:43 and 0\u0014Jimp\u00141:5Jhost.\nIn Fig. 16(a) we show the representative spin-orbital\ncon\fgurations obtained for \u0015= 0:5Jhostalong the \frst\ncut shown in Fig. 16(b). Values of \u0011impare chosen as\n\u0011imp=\u0011m\u00110:7(m\u00001)\n9; (30)\nwithm= 1;:::; 10 but not all the points are shown in\nFig. 16(a) | only the ones for which the spin-orbital\ncon\fguration changes substantially.\nThe cut starts in the QAF c2 phase, according to the\nphase diagram of Fig. 16(b), and indeed we \fnd a simi-\nlar con\fguration to the one shown in Fig. 14 for QAF c2\nphase at\u0015=\u00155. Moving up in the phase diagram from\n\u00111to\u00112we see that the con\fguration evolves smoothly\nto the one which we have found in the QAF a1 phase at\n\u0015=\u00155(not shown in Fig. 14). The evolution of spins is\nsuch that the out-of-plane moments are suppressed while\nin-plane ones are slightly enhanced. The orbitals become\nmore spherical and the local spin-orbit average, h~Li\u0001~Sii,21\nFIG. 16. (a) Evolution of the ground state con\fgurations as\nfor increasing \u0011impand for a \fxed value of spin-orbit coupling\n\u0015= 0:5Jhostalong a cut in the phase diagram shown in panel\n(b), i.e., for Jimp= 0:7Jhostand 0\u0014\u0011imp\u00140:7. Arrows\nand ellipsoids indicate the spin-orbital state at a given site\ni. Color map indicates the strength of the average spin-orbit,\nh~Li\u0001~Sii, i.e., red, yellow, green, blue, violet correspond to the\ngrowing amplitude of the above correlation function.\nbecomes larger and more uniform, however for the apical\nsitei= 7 in the cluster [Fig. 13(b)] the trend is oppo-\nsite | initially large value of spin-orbit coupling drops\ntowards the uniform value. The points between \u00113and\n\u00117we skip as the evolution is smooth and the trend is\nclear, however the impurity out-of-plane moment begins\nto grow above \u00115, indicating proximity to the QFS a1\nphase. For this phase at intermediate and high \u0015the\nimpurity moment is much larger than all the others (see\nFig. 15).\nFor\u0011imp=\u00117the orbital pattern clearly shows that\nwe are in the QFS a1 phase at\u0015=\u00155which agrees with\nthe position of the \u00117point in the phase diagram, see\nFig. 16(b). On the other hand, moving to the next \u0011imp\npoint upward along the cut Eq. (30) we already observe\na con\fguration which is very typical for the QFM aphase\nat intermediate \u0015(here\u0015=\u00157shown in Fig. 14 but also\n\u00156, not shown). This indicates that the QFS a1 phase can\nbe still distinguished at \u0015= 0:5Jhostand its position in\nthe phase diagram is similar as in the \u0015= 0 case, i.e., as\nan intermediate phase between the QAF a1(2) and QFM a\none.\nFinally, we have found that also the two other cuts\nwhich were not shown here, i.e., for Jimp= 1:3Jhostand\nincreasing\u0011impand for\u0011imp=\u0011c\nimp'0:43 and increas-\ningJimpcon\frm that the overall character of the phase\ndiagram of Fig. 13(a) is preserved at this value of spin-orbit coupling, however \frstly, the transitions between\nthe phases are smooth and secondly, the subtle di\u000ber-\nences between the two QFS a, QAFaand QAFcphases\nare no longer present. This also refers to the smaller\nvalue of\u0015, i.e.,\u0015= 0:1Jhost, but already for \u0015=Jhost\nthe out-of-plane moments are so strongly suppressed (ex-\ncept for the impurity moment in the QFS a1 phase) and\nthe orbital polarization is so weak (i.e., almost spheri-\ncal ellipsoids) that typically the only distinction between\nthe phases can be made by looking at the in-plane spin\ncorrelations and the average spin-orbit, h~Li\u0001~Sii. In this\nlimit we conclude that the phase diagram is (partially)\nmelted by large spin-orbit coupling but for lower values\nof\u0015it is still valid.\nVI. SUMMARY AND CONCLUSIONS\nWe have derived the spin-orbital superexchange model\nfor 3d3impurities replacing 4 d4(or 3d2) ions in the 4 d\n(3d) host in the regime of Mott insulating phase. Al-\nthough the impurity has no orbital degree of freedom, we\nhave shown that it contributes to the spin-orbital physics\nand in\ruences strongly the orbital order. In fact, it tends\nto project out the inactive orbitals at the impurity-host\nbonds to maximize the energy gain from virtual charge\n\ructuations. In this case the interaction along the su-\nperexchange bond can be either antiferromagnetic or fer-\nromagnetic, depending on the ratio of Hund's exchange\ncoupling at impurity ( JH\n1) and host ( JH\n2) ions and on\nthe mismatch \u0001 between the 3 dand 4datomic energies,\nmodi\fed by the di\u000berence in Hubbard U's and Hund's\nexchangeJH's at both atoms. This ratio, denoted \u0011imp\n(14), replaces here the conventional parameter \u0011=JH=U\noften found in the spin-orbital superexchange models of\nundoped compounds (e.g., in the Kugel-Khomskii model\nfor KCuF 3[14]) where it quanti\fes the proximity to fer-\nromagnetism. On the other hand, if the overall coupling\nbetween the host and impurity is weak in the sense of\nthe total superexchange, Jimp, with respect to the host\nvalue,Jhost, the orbitals being next to the impurity may\nbe forced to stay inactive which modi\fes the magnetic\nproperties | in such cases the impurity-host bond is al-\nways antiferromagnetic.\nAs we have seen in the case of a single impurity, the\nabove two mechanisms can have a nontrivial e\u000bect on\nthe host, especially if the host itself is characterized by\nfrustrated interactions, as it happens in the parameter\nregime where the C-AF phase is stable. For this rea-\nson we have focused mostly on the latter phase of the\nhost and we have presented the phase diagrams of a sin-\ngle impurity con\fguration in the case when the impu-\nrity is doped on the sublattice where the orbitals form\na checkerboard pattern with alternating candaorbitals\noccupied by doublons. The diagram for the c-sublattice\ndoping shows that in some sense the impurity is never\nweak, because even for a very small value of Jimp=Jhost\nit can release the host's frustration around the impurity22\nsite acting as an orbital vacancy. On the other hand, for\nthea-sublattice doping when the impurity-host coupling\nis weak, i.e., either Jimp=Jhostis weak or\u0011impis close to\n\u0011c\nimp, we have identi\fed an interesting quantum mech-\nanism releasing frustration of the impurity spin (that\ncannot be avoided in the purely classical approach). It\nturned out that in such situations the orbital \rips in the\nhost make the impurity spin polarize in such a way that\ntheC-AF order of the host is completely restored.\nThe cases of the periodic doping studied in this pa-\nper show that the host's order can be completely altered\nalready for rather low doping ofx= 1=8, even if the\nJimp=Jhostis small. In this case we can stabilize a ferri-\nmagnetic type of phase with a four-site unit cell having\nmagnetizationhSz\nii= 3=2, reduced further by quantum\n\ructuations. We have established that the only param-\neter range where the host's order remains unchanged is\nwhen\u0011impis close to\u0011c\nimpandJimp=Jhost&1. The latter\nvalue is very surprising as it means that the impurity-host\ncoupling must be large enough to keep the host's order\nunchanged | this is another manifestation of the orbital\nvacancy mechanism that we have already observed for\na single impurity. Also in this case the impurity spins\nare \fxed with the help of orbital \rips in the host that\nlift the degeneracy which arises in the classical approach.\nWe would like to point out that the quantum mechanism\nthat lifts the ground state degeneracy mentioned above\nand the role of quantum \ructuations are of particular\ninterest for the periodically doped checkerboard systems\nwithx= 1=2 doping which is a challenging problem for\nfuture research.\nFrom the point of view of generic, i.e., non-periodic\ndoping, the most representative cases are those of a\ndoping which is incommensurate with the two-sublattice\nspin-orbital pattern. To uncover the generic rules in such\ncases, we have studied periodic x= 1=5 andx= 1=9 dop-\ning. One \fnds that when the period of the impurity posi-\ntions does not match the period of 2 for both the spin and\norbital order of the host, interesting novel types of order\nemerge. In such cases the elementary cell must be dou-\nbled in both lattice directions which clearly gives a chance\nof realizing more phases than in the case of commensu-\nrate doping. Our results show that indeed, the number of\nphases increases from 4 to 7 and the host's order is altered\nin each of them. Quite surprisingly, the overall character\nof the phase diagram remained unchanged with respect\nto the one for x= 1=8 doping and, if we ignore the dif-\nferences in con\fguration, it seems that only some of the\nphases got divided into two versions di\u000bering either by\nthe spin bond's polarizations around impurities (phases\naround\u0011c\nimp), or by the character of the orbitals around\nthe impurities (phases with inactive orbitals in the limit\nof small enough product \u0011impJimp, versus phases with ac-\ntive orbitals in the opposite limit). Orbital polarization\nin this latter region resembles orbital polarons in doped\nmanganites [42, 43] | also here such states are stabilized\nby the double exchange [46].\nA closer inspection of underlying phases reveals how-ever a very interesting degeneracy of the impurity spins\natx= 1=5 that arises again from the classical approach\nbut this time it cannot be released by short-range orbital\n\rips. This happens because the host's order is already\nso strongly altered that it is no longer anisotropic (as it\nwas the case of the C-AF phase) and there is no way\nto restore the orbital anisotropy around the impurities\nthat could lead to spin-bonds imbalance and polarize the\nspin. In the case of lower x= 1=9 doping such an e\u000bect\nis absent and the impurity spins are always polarized, as\nit happens for x= 1=8. It shows that this is rather a\npeculiarity of the x= 1=5 periodic doping.\nIndeed, one can easily notice that for x= 1=5 every\natom of the host is a nearest neighbor of some impurity.\nIn contrast, for x= 1=8 we can \fnd three host's atoms\nper unit cell which do not neighbor any impurity and for\nx= 1=9 there are sixteen of them. For this reason the\nimpurity e\u000bects are ampli\fed for x= 1=5 which is not\nunexpected although one may \fnd somewhat surprising\nthat the ground state diagrams for the lowest and the\nhighest doping considered here are very similar. This\nsuggests that the cooperative e\u000bects of multiple impuri-\nties are indeed not very strong in the low-doping regime,\nso the diagram obtained for x= 1=9 can be regarded as\ngeneric for the dilute doping regime with uniform spatial\npro\fle.\nFor the representative case of x= 1=8 doping, we have\npresented the consequences of quantum e\u000bects beyond\nthe classical approach. Spin \ructuations are rather weak\nfor the considered case of large S= 1 andS= 3=2 spins,\nand we have shown that orbital \ructuations on superex-\nchange bonds are more important. They are strongest in\nthe regime of antiferromagnetic impurity-host coupling\n(which suggests importance of entangled states [34]) and\nenhance the tendency towards frustrated impurity spin\ncon\fgurations but do not destroy other generic trends\nobserved when the parameters \u0011impandJimp=Jhostin-\ncrease.\nIncreasing spin-orbit coupling leads to qualitative\nchanges in the spin-orbital order. When Hund's ex-\nchange is small at the impurity sites, the antiferromag-\nnetic bonds around it have reduced values of spin-orbit\ncoupling term, but the magnetic moments reorient and\nsurvive in the ( a;b) planes, with some similarity to the\nphenomena occurring in the perovskite vanadates [57].\nThis quenches the magnetic moments at 3 dimpurities\nand leads to almost uniform orbital occupancies at the\nhost sites. In contrast, frustration of impurity spins is re-\nmoved and the impurity magnetization along the caxis\nsurvives for large spin-orbit coupling.\nWe would like to emphasize that the orbital dilution\nconsidered here in\ruences directly the orbital degrees of\nfreedom in the host around the impurities. The synthesis\nof hybrid compounds having both 3 dand 4dtransition\nmetal ions will likely open a novel route for unconven-\ntional e\u000bects in complex materials. There are several\nreasons for expecting new scenarios in mixed 3 d\u00004d\nspin-orbital-lattice materials, and we pointed out only23\nsome of them. On the experimental side, the changes\nof local order could be captured using inelastic neutron\nscattering or resonant inelastic x-ray scattering (RIXS).\nIn fact, using RIXS can also bring an additional advan-\ntage: RIXS, besides being a perfect probe of both spin\nand orbital excitations, can also (indirectly) detect the\nnature of orbital ground state (supposedly also including\nthe nature of impurities in the crystal) [82]. Unfortu-\nnately, there are no such experiments yet but we believe\nthat they will be available soon.\nShort range order around impurities could be inves-\ntigated by the excitation spectra at the resonant edges\nof the substituting atoms. Taking them both at \fnite\nenergy and momentum can dive insights into the nature\nof the short range order around the impurity and then\nunveil information of the order within the host as well.\nEven if there are no elastic superlattice extra peaks one\ncan expect that the spin-orbital correlations will emerge\nin the integrated RIXS spectra providing information of\nthe impurity-host coupling and of the short range order\naround the impurity. Even more interesting is the case\nwhere the substituting atom forms a periodic array with\nsmall deviation from the perfect superlattice when one\nexpects the emergence of extra elastic peaks which will\nclearly indicate the spin-orbital reconstruction. In our\ncase an active orbital diluted site cannot participate co-\nherently in the host spin-orbital order but rather may\nto restructure the host ordering [83]. At dilute impurity\nconcentration we may expect broad peaks emerging at\n\fnite momenta in the Brillouin zone, indicating the for-\nmation of coherent islands with short range order around\nimpurities.\nWe also note that local susceptibility can be suit-\nably measured by making use of resonant spectroscopies\n(e.g. nuclear magnetic resonance (NMR), electron spin\nresonance (ESR), nuclear quadrupole resonance (NQR),\nmuon spin resonance ( \u0016SR), etcetera ) that exploit the\ndi\u000berent magnetic or electric character of the atomic nu-\nclei for the impurity and the host in the hybrid system.\nFinally, the random implantation of the muons in the\nsample can provide information of the relaxation time\nin di\u000berent domains with unequal dopant concentration\nwhich may be nonuniform. For the given problem the\ndi\u000berences in the resonant response can give relevant in-\nformation about the distribution of the local \felds, the\noccurrence of local order and provide access to the dy-\nnamical response within doped domains. The use of local\nspectroscopic resonance methods has been widely demon-\nstrated to be successful when probing the nature and the\nevolution of the ground state in the presence of spin va-\ncancies both for ordered and disordered magnetic con\fg-\nurations [84{87].\nIn summary, this study highlights the role of spin de-\nfects which lead to orbital dilution in spin-orbital sys-\ntems. Using an example of 3 d3impurities in a 4 d4(or\n3d2) host we have shown that impurities change radi-\ncally the spin-orbital order around them, independently\nof the parameter regime. As a general feature we havefound that doped 3 d3ions within the host with spin-\norbital order have frustrated spins and polarize the or-\nbitals of the host when the impurity-host exchange as\nwell as Hund's exchange at the impurity are both suf-\n\fciently large. This remarkable trend is independent of\ndoping and is expected to lead to global changes of spin-\norbital order in doped materials. While the latter e\u000bect\nis robust, we argue that the long-range spin \ructuations\nresulting from the translational invariance of the system\nwill likely prevent the ground state from being macro-\nscopically degenerate, so if the impurity spins in one unit\ncell happens to choose its polarization then the others will\nfollow. On the contrary, in the regime of weak Hund's\nexchange 3d3ions act not only as spin defects which or-\nder antiferromagnetically with respect to their neighbors,\nbut also induce doublons in inactive orbitals.\nFinally, we remark that this behavior with switching\nbetween inactive and active orbitals by an orbitally neu-\ntral impurity may lead to multiple interesting phenomena\nat macroscopic doping when global modi\fcations of the\nspin-orbital order are expected to occur. Most of the re-\nsults were obtained in the classical approximation but we\nhave shown that modi\fcations due to spin-orbit coupling\ndo not change the main conclusion. We note that this\ngeneric treatment and the general questions addressed\nhere, such as the release of frustration for competing\nspin structures due to periodic impurities, are relevant\nto double perovskites [88]. While the local orbital polar-\nization should be similar, it is challenging to investigate\ndisordered impurities, both theoretically and in experi-\nment, to \fnd out whether their in\ruence on the global\nspin-orbital order in the host is equally strong.\nACKNOWLEDGMENTS\nWe thank Maria Daghofer and Krzysztof Wohlfeld for\ninsightful discussions. W. B. and A. M. O. kindly ac-\nknowledge support by the Polish National Science Cen-\nter (NCN) under Project No. 2012/04/A/ST3/00331.\nW. B. was also supported by the Foundation for Pol-\nish Science (FNP) within the START program. M. C.\nacknowledges funding from the EU | FP7/2007-2013\nunder Grant Agreement No. 264098 | MAMA.\nAppendix A: Derivation of 3d\u00004dsuperexchange\nHere we present the details of the derivation of the low\nenergy spin-orbital Hamiltonian for the 3 d3\u00004d4bonds\naround the impurity at site i.H3d\u00004d(i), which follows\nfrom the perfurbation theory, as given in Eq. (10). Here\nwe consider a single 3 d3\u00004d4bondhiji. Two contri-\nbutions to the e\u000bective Hamiltonian follow from charge\nexcitations: (i)H(\r)\nJ;43(i;j) due tod3\nid4\nj\u000bd4\nid3\nj, and (ii)\nH(\r)\nJ;25(i;j) due tod3\nid4\nj\u000bd2\nid5\nj. Therefore the low energy24\nHamiltonian is,\nH(\r)\nJ(i;j) =H(\r)\nJ;43(i;j) +H(\r)\nJ;25(i;j): (A1)\nConsider \frst the processes which conserve the num-\nber of doubly occupied orbitals, d3\nid4\nj\u000bd4\nid3\nj. Then by\nmeans of spin and orbital projectors, it is possible to ex-\npressH(\r)\nJ;43(i;j) fori= 1 andj= 2 as\nH(\r)\nJ;43(1;2) =\n\u0000\u0010\n~S1\u0001~S2\u0011t2\n18\u001a4\n\u0001\u00007\n\u0001 + 3JH\n2\u00003\n\u0001 + 5JH\n2\u001b\n+D(\r)\n2\u0010\n~S1\u0001~S2\u0011t2\n18\u001a4\n\u0001\u00001\n\u0001 + 3JH\n2+3\n\u0001 + 5JH\n2\u001b\n+\u0010\nD(\r)\n2\u00001\u0011t2\n12\u001a8\n\u0001+1\n\u0001 + 3JH\n2\u00003\n\u0001 + 5JH\n2\u001b\n;(A2)\nwith the excitation energy \u0001 de\fned in Eq. (11). The\nresulting e\u000bective 3 d\u00004dexchange in Eq. (A2) consists\nof three terms: (i) The \frst one does not depend on the\norbital con\fguration of the 4 datom and it can be FM or\nAF depending on the values \u0001 and the Hund's exchange\non the 3dion. In particular, if \u0001 is the largest or the\nsmallest energy scale, the coupling will be either AF or\nFM, respectively. (ii) The second term has an explicit\ndependence on the occupation of the doublon on the 4 d\natom via the projecting operator D(\r)\n2. This implies that\na magnetic exchange is possible only if the doublon occu-\npies the inactive orbital for a bond along a given direction\n\r. Unlike in the \frst term, the sign of this interaction\nis always positive favoring an AF con\fguration at any\nstrength of \u0001 and JH\n1. (iii) Finally, the last term de-\nscribes the e\u000bective processes which do not depend on\nthe spin states on the 3 dand 4datoms. This contri-\nbution is of pure orbital nature, as it originates from the\nhopping between 3 dand 4datoms without a\u000becting their\nspin con\fguration, and for this reason favors the occupa-\ntion of active t2gorbitals along the bond by the doublon.\nWithin the same scheme, we have derived the ef-\nfective spin-orbital exchange that originates from the\ncharge transfer processes of the type 3 d3\n14d4\n2\u000b3d2\ni4d5\nj,\nH(\r)\nJ;25(1;2). The e\u000bective low-energy contribution to the\nHamiltonian for i= 1 andj= 2 reads\nH(\r)\nJ;25(1;2) =t2\nU1+U2\u0000\u0000\n\u0001 + 3JH\n2\u00002JH\n1\u0001\n\u0002\u001a1\n3D(\r)\n2\u0010\n~S1\u0001~S2\u0011\n+1\n3\u0010\n~S1\u0001~S2\u0011\n\u00001\n2\u0010\nD(\r)\n2+ 1\u0011\u001b\n:(A3)\nBy inspection of the spin structure involved in the ele-\nmental processes that generate H(\r)\nJ;25(1;2), one can note\nthat it is always AF independently of the orbital con-\n\fguration on the 4 datom exhibiting with a larger spin-\nexchange and an orbital energy gain if the doublon is\noccupying the inactive orbital along a given bond. We\nhave veri\fed that the amplitude of the exchange termsinH(\r)\nJ;25(1;2) is much smaller than the ones which enter\ninH(\r)\nJ;43(1;2) which justi\fes that one may simplify Eq.\n(A1) fori= 1 andj= 2 to\nH(\r)\nJ(1;2)'H(\r)\nJ;43(1;2); (A4)\nand neglectH(\r)\nJ;25(1;2) terms altogether. This approxi-\nmation is used in Sec. II.\nAppendix B: Orbital operators in the L-basis\nThe starting point to express the orbital operators ap-\npearing in the spin-orbital superexchange model (17) is\nthe relation between quenched jaii,jbii, andjciiorbitals\nat siteiand the eigenvectors j1ii,j0ii, andj\u00001iiof the\nangular momentum operator Lz\ni. These are known to be\njaii=1p\n2(j1ii+j\u00001ii);\njbii=\u0000ip\n2(j1ii\u0000j\u00001ii);\njcii=j0ii: (B1)\nFrom this we can immediately get the occupation number\noperators for the doublon,\nD(a)\ni=ay\niai=jaiihaji= 1\u0000(Lx\ni)2;\nD(b)\ni=by\nibi=jbiihbji= 1\u0000(Ly\ni)2;\nD(c)\ni=cy\nici=jciihcji= 1\u0000(Lz\ni)2; (B2)\nand the relatedfn(\r)\nigoperators,\nn(a)\ni=by\nibi+cy\nici= (Lx\ni)2;\nn(b)\ni=cy\nici+ay\niai= (Ly\ni)2;\nn(c)\ni=ay\niai+by\nibi= (Lz\ni)2: (B3)\nThe doublon hopping operators have a slightly di\u000berent\nstructure that re\rects their noncommutivity, i.e.,\nay\nibi=jaiihbji=iLy\niLx\ni;\nby\nici=jbiihcji=iLz\niLy\ni;\ncy\niai=jciihaji=iLx\niLz\ni: (B4)\nThese relations are su\u000ecient to write the superexchange\nHamiltonian for the host-host and impurity-host bonds\nin thefLx\ni;Ly\ni;Lz\nigoperator basis for the orbital part.\nHowever, in practice it is more convenient to work with\nreal operators\b\nL+\ni;L\u0000\ni;Lz\ni\t\nrather than with the origi-\nnal ones,fLx\ni;Ly\ni;Lz\nig. Thus we write the \fnal relations\nwhich we used for the numerical calculations in terms of\nthese operators,\nD(a)\ni=\u00001\n4h\u0000\nL+\ni\u00012+\u0000\nL\u0000\ni\u00012i\n+1\n2(Lz\ni)2;\nD(b)\ni=1\n4h\u0000\nL+\ni\u00012+\u0000\nL\u0000\ni\u00012i\n+1\n2(Lz\ni)2;25\nD(c)\ni= 1\u0000(Lz\ni)2; (B5)\nfor the doublon occupation numbers and going directly\nto the orbital ~ \u001cioperators we \fnd that,\n\u001c+(a)\ni =1\n2\u0000\nL\u0000\ni\u0000L+\ni\u0001\nLz\ni;\n\u001c+(b)\ni=\u0000i\n2Lz\ni\u0000\nL+\ni+L\u0000\ni\u0001\n;\n\u001c+(c)\ni=i\n4h\u0000\nL+\ni\u00012\u0000\u0000\nL\u0000\ni\u00012i\n\u0000i\n2Lz\ni; (B6)\nfor the o\u000b-diagonal part and\n\u001cz(a)\ni=1\n8h\u0000\nL+\ni\u00012+\u0000\nL\u0000\ni\u00012i\n+3\n4(Lz\ni)2\u00001\n2;\n\u001cz(b)\ni=1\n8h\u0000\nL+\ni\u00012+\u0000\nL\u0000\ni\u00012i\n\u00003\n4(Lz\ni)2+1\n2;\u001cz(c)\ni=\u00001\n4h\u0000\nL+\ni\u00012+\u0000\nL\u0000\ni\u00012i\n; (B7)\nfor the diagonal one. Note that the complex phase in\n\u001c+(b)\ni and\u001c+(c)\ni is irrelevant and can be omitted here as\n\u001c+(\r)\ni is always accompanied by \u001c\u0000(\r)\nj on a neighboring\nsite. This is a consequence of the cubic symmetry in the\norbital part of the superexchange Hamiltonian and it can\nbe altered by a presence of a distortion, e.g., octahedral\nrotation. For completeness we also give the backward\nrelation between angular momentum components, fL\u000b\nig\nwith\u000b=x;y;z , and the orbital operators f\u001c\u000b(\r)\nig; these\nare:\nLx\ni= 2\u001cx(a)\ni;\nLy\ni= 2\u001cx(b)\ni;\nLz\ni= 2\u001cy(c)\ni: (B8)\n[1] M. Imada, A. Fujimori, and Y. Tokura, Metal-Insulator\nTransitions , Rev. Mod. Phys. 70, 1039 (1998).\n[2] E. Berg, E. Fradkin, S. A. Kivelson, and J. M. Tran-\nquada, Striped Superconductors: how Spin, Charge and\nSuperconducting Orders Intertwine in the Cuprates , New\nJ. Phys. 11, 115004 (2011).\n[3] P. A. Lee, N. Nagaosa, and X.-G. Wen, Doping a Mott\nInsulator: Physics of High-Temperature Superconductiv-\nity, Rev. Mod. Phys. 78, 17 (2006).\n[4] M. Vojta, Lattice Symmetry Breaking in Cuprate Super-\nconductors: Stripes, Nematics, and Superconductivity ,\nAdv. Phys. 58, 699 (2009).\n[5] K. Yamada, C. H. Lee, K. Kurahashi, J. Wada, S. Waki-\nmoto, S. Ueki, H. Kimura, Y. Endoh, S. Hosoya, G. Shi-\nrane, R. J. Birgeneau, M. Greven, M. A. Kastner, and\nY. J. Kim, Doping Dependence of the Spatially Modulated\nDynamical Spin Correlations and the Superconducting-\nTransition Temperature in La2\u0000xSrxCuO 4, Phys. Rev. B\n57, 6165 (1998).\n[6] M. Fleck, A. I. Lichtenstein, and A. M. Ole\u0013 s, Spectral\nProperties and Pseudogap in the Stripe Phases of Cuprate\nSuperconductors , Phys. Rev. B 64, 134528 (2001).\n[7] P. Wr\u0013 obel and A. M. Ole\u0013 s, Ferro-Orbitally Ordered\nStripes in Systems with Alternating Orbital Order , Phys.\nRev. Lett. 104, 206401 (2010).\n[8] Q. Li, K. E. Gray, S. B. Wilkins, M. Garcia Fernan-\ndez, S. Rosenkranz, H. Zheng, and J.F. Mitchell, Predic-\ntion and Experimental Evidence for Thermodynamically\nStable Charged Orbital Domain Walls , Phys. Rev. X 4,\n031028 (2014).\n[9] J. Zaanen and A. M. Ole\u0013 s, Carriers Binding to Excitons:\nCrystal-Field Excitations in Doped Mott-Hubbard Insula-\ntors, Phys. Rev. B 48, 7197 (1993).\n[10] Y. Tokura and N. Nagaosa, Orbital Physics in\nTransition-Metal Oxides , Science 288, 462 (2000).\n[11] K. I. Kugel and D. I. Khomskii, The Jahn-Teller Ef-\nfect and Magnetism: Transition Metal Compounds , Sov.\nPhys. Usp. 25, 231 (1982).[12] A. M. Ole\u0013 s, G. Khaliullin, P. Horsch, and L. F. Feiner,\nFingerprints of Spin-Orbital Physics in Cubic Mott In-\nsulators: Magnetic Exchange Interactions and Optical\nSpectral Weights , Phys. Rev. B 72, 214431 (2005).\n[13] K. Wohlfeld, M. Daghofer, S. Nishimoto, G. Khaliullin,\nand J. van den Brink. Intrinsic Coupling of Orbital Ex-\ncitations to Spin Fluctuations in Mott Insulators , Phys.\nRev. Lett. 107, 147201 (2011).\n[14] L. F. Feiner, A. M. Ole\u0013 s, and J. Zaanen, Quantum Melt-\ning of Magnetic Order due to Orbital Fluctuations , Phys.\nRev. Lett. 78, 2799 (1997).\n[15] G. Khaliullin and S. Maekawa, Orbital Liquid in Three-\nDimensional Mott Insulator: LaTiO 3, Phys. Rev. Lett.\n85, 3950 (2000).\n[16] G. Khaliullin, Orbital Order and Fluctuations in Mott\nInsulators , Prog. Theor. Phys. Suppl. 160, 155 (2005).\n[17] Leon Balents, Spin Liquids in Frustrated Magnets , Na-\nture (London) 464, 199 (2010).\n[18] L. F. Feiner and A. M. Ole\u0013 s, Orbital Liquid in Ferro-\nmagnetic Manganites: The Orbital Hubbard Model for eg\nElectrons , Phys. Rev. B 71, 144422 (2005).\n[19] S. Nakatsuji, K. Kuga, K. Kimura, R. Satake,\nN. Katayama, E. Nishibori, H. Sawa, R. Ishii, M. Hagi-\nwara, F. Bridges, T. U. Ito, W. Higemoto, Y. Karaki, M.\nHalim, A. A. Nugroho, J. A. Rodriguez-Rivera, M. A.\nGreen, and C. Broholm, Spin-Orbital Short-Range Order\non a Honeycomb-Based Lattice , Science 336, 559 (2012).\n[20] J. A. Quilliam, F. Bert, E. Kermarrec, C. Payen,\nC. Guillot-Deudon, P. Bonville, C. Baines, H. Luetkens,\nand P. Mendels, Singlet Ground State of the Quantum\nAntiferromagnet Ba3CuSb 2O9, Phys. Rev. Lett. 109,\n117203 (2012).\n[21] B. Normand and A. M. Ole\u0013 s, Frustration and Entangle-\nment in the t2gSpin-Orbital Model on a Triangular Lat-\ntice: Valence-bond and Generalized Liquid States , Phys.\nRev. B 78, 094427 (2008).\n[22] P. Corboz, M. Lajk\u0013 o, A. M. La uchli, K. Penc, and\nF. Mila, Spin-Orbital Quantum Liquid on the Honeycomb\nLattice , Phys. Rev. X 2, 041013 (2012).26\n[23] J. Nasu and S. Ishihara, Dynamical Jahn-Teller E\u000bect in\na Spin-Orbital Coupled System , Phys. Rev. B 88, 094408\n(2012).\n[24] A. Smerald and F. Mila, Exploring the Spin-Orbital\nGround State of Ba3CuSb 2O9, Phys. Rev. B 90, 094422\n(2014).\n[25] V. Fritsch, J. Hemberger, N. B uttgen, E.-W. Scheidt, H.-\nA. Krug von Nidda, A. Loidl, and V. Tsurkan, Spin and\nOrbital Frustration in MnSc 2S2andFeSc 2S4, Phys. Rev.\nLett. 92, 116401 (2004).\n[26] G. Chen, L. Balents, and A. P. Schnyder, Spin-Orbital\nSinglet and Quantum Critical Point on the Diamond Lat-\ntice: FeSc 2S4, Phys. Rev. Lett. 102, 096406 (2009).\n[27] L. Mittelst adt, M. Schmidt, Z. Wang, F. Mayr,\nV. Tsurkan, P. Lunkenheimer, D. Ish, L. Balents, J.\nDeisenhofer, and A. Loidl, Spin-Orbiton and Quantum\nCriticality in FeSc 2S4, Phys. Rev. B 91, in press (2015).\n[28] F. Vernay, K. Penc, P. Fazekas, and F. Mila, Orbital De-\ngeneracy as a Source of Frustration in LiNiO 2, Phys.\nRev. B 70, 014428 (2004).\n[29] W. Brzezicki, J. Dziarmaga, and A. M. Ole\u0013 s, Non-\ncollinear Magnetic Order Stabilized by Entangled Spin-\nOrbital Fluctuations , Phys. Rev. Lett. 109, 237201\n(2012).\n[30] W. Brzezicki, J. Dziarmaga, and A. M. Ole\u0013 s, Exotic\nSpin Orders Driven by Orbital Fluctuations in the Kugel-\nKhomskii Model , Phys. Rev. B 87, 064407 (2013).\n[31] P. Czarnik and J. Dziarmaga, Striped Critical Spin Liquid\nin a Spin-Orbital Entangled RVB State in a Projected\nEntangled-Pair State Representation , Phys. Rev. B 91,\n045101 (2015).\n[32] F. Reynaud, D. Mertz, F. Celestini, J. Debierre,\nA. M. Ghorayeb, P. Simon, A. Stepanov, J. Voiron, and\nC. Delmas, Orbital Frustration at the Origin of the Mag-\nnetic Behavior in LiNiO 2, Phys. Rev. Lett. 86, 3638\n(2001).\n[33] A. Reitsma, L. F. Feiner, and A. M. Ole\u0013 s, Orbital and\nSpin Physics in LiNiO 2and NaNiO 2, New J. Phys. 7,\n121 (2005).\n[34] A. M. Ole\u0013 s, Fingerprints of Spin-Orbital Entanglement\nin Transition Metal Oxides , J. Phys.: Condens. Matter\n24, 313201 (2012).\n[35] E. Dagotto, T. Hotta, and A. Moreo, Colossal Magne-\ntoresistant Materials: The Key Role of Phase Separation ,\nPhys. Rep. 344, 1 (2001).\n[36] E. Dagotto, Open Questions in CMR Manganites, Rel-\nevance of Clustered States and Analogies with Other\nCompounds Including the Cuprates , New J. Phys. 7, 67\n(2005).\n[37] Y. Tokura, Critical Features of Colossal Magnetoresistive\nManganites , Rep. Prog. Phys. 69, 797 (2006).\n[38] L. F. Feiner and A. M. Ole\u0013 s, Electronic Origin of Mag-\nnetic and Orbital Ordering in Insulating LaMnO 3, Phys.\nRev. B 59, 3295 (1999).\n[39] R. Kilian and G. Khaliullin, Orbital Polarons in the\nMetal-Insulator Transition of Manganites , Phys. Rev. B\n60, 13458 (1999).\n[40] M. Cuoco, C. Noce, and A. M. Ole\u0013 s, Origin of the Optical\nGap in Half-Doped Manganites , Phys. Rev. B 66, 094427\n(2002).\n[41] A. Weisse and H. Fehske, Microscopic Modelling of Doped\nManganites , New J. Phys. 6, 158 (2004).\n[42] M. Daghofer, A. M. Ole\u0013 s, and W. von der Linden, Orbital\nPolarons versus Itinerant egElectrons in Doped Mangan-ites, Phys. Rev. B 70, 184430 (2004).\n[43] J. Geck, P. Wochner, S. Kiele, R. Klingeler, P. Reutler,\nA. Revcolevschi, and B. B uchner, Orbital Polaron Lattice\nFormation in Lightly Doped La1\u0000xSrxMnO 3, Phys. Rev.\nLett. 95, 236401 (2005).\n[44] A. M. Ole\u0013 s and G. Khaliullin, Dimensional Crossover and\nthe Magnetic Transition in Electron Doped Manganites ,\nPhys. Rev. B 84, 214414 (2011).\n[45] Y. M. Sheu, S. A. Trugman, L. Yan, J. Qi, Q. X. Jia,\nA. J. Taylor, and R. P. Prasankumar, Polaronic Trans-\nport Induced by Competing Interfacial Magnetic Order in\naLa0:7Ca0:3MnO 3/BiFeO 3Heterostructure , Phys. Rev.\nX4, 021001 (2014).\n[46] P.-G. de Gennes, E\u000bects of Double Exchange in Magnetic\nCrystals , Phys. Rev. 118, 141 (1960).\n[47] G. Khaliullin and R. Kilian, Theory of Anomalous\nMagnon Softening in Ferromagnetic Manganites , Phys.\nRev. B 61, 3494 (2000).\n[48] A. M. Ole\u0013 s and L. F. Feiner, Why Spin Excitations in\nMetallic Ferromagnetic Manganites are Isotropic , Phys.\nRev. B 65, 052414 (2002).\n[49] H. Sakai, S. Ishiwata, D. Okuyama, A. Nakao, H. Nakao,\nY. Murakami, Y. Taguchi, and Y. Tokura, Electron Dop-\ning in the Cubic Perovskite SrMnO 3:Isotropic Metal ver-\nsus Chainlike Ordering of Jahn-Teller Polarons , Phys.\nRev. B 82, 180409 (2010).\n[50] G. Khaliullin, P. Horsch, and A. M. Ole\u0013 s, Spin Order due\nto Orbital Fluctuations: Cubic Vanadates , Phys. Rev.\nLett. 86, 3879 (2001).\n[51] G. Khaliullin, P. Horsch, and A. M. Ole\u0013 s, Theory of Op-\ntical Spectral Weights in Mott Insulators with Orbital De-\ngrees of Freedom , Phys. Rev. B 70, 195103 (2004).\n[52] P. Horsch, A. M. Ole\u0013 s, L. F. Feiner, and G. Khaliullin,\nEvolution of Spin-Orbital-Lattice Coupling in the RVO3\nPerovskites , Phys. Rev. Lett. 100, 167205 (2008).\n[53] C. Ulrich, G. Khaliullin, J. Sirker, M. Reehuis, M. Ohl,\nS. Miyasaka, Y. Tokura, and B. Keimer, Magnetic Neu-\ntron Scattering Study of YVO 3:Evidence for an Orbital\nPeierls State , Phys. Rev. Lett. 91, 257202 (2003).\n[54] J.-S. Zhou, J. B. Goodenough, J.-Q. Yan, and\nY. Ren, Superexchange Interaction in Orbitally Fluctu-\nating RVO 3, Phys. Rev. Lett. 99, 156401 (2007).\n[55] M. Reehuis, C. Ulrich, K. Proke\u0014 s, S. Mat'a\u0014 s, J. Fu-\njioka, S. Miyasaka, Y. Tokura, and B. Keimer, Structural\nand Magnetic Phase Transitions of the Orthovanadates\nRVO3(R=Dy,Ho,Er) as seen via Neutron Di\u000braction ,\nPhys. Rev. B 83, 064404 (2011).\n[56] J. Fujioka, T. Yasue, S. Miyasaka, Y. Yamasaki,\nT. Arima, H. Sagayama, T. Inami, K. Ishii, and\nY. Tokura, Critical Competition between two Distinct\nOrbital-Spin Ordered States in Perovskite Vanadates ,\nPhys. Rev. B 82, 144425 (2010).\n[57] P. Horsch, G. Khaliullin, and A. M. Ole\u0013 s, Dimeriza-\ntion versus Orbital-Moment Ordering in a Mott Insulator\nYVO 3, Phys. Rev. Lett. 91, 257203 (2003).\n[58] J. Fujioka, S. Miyasaka, and Y. Tokura, Doping Vari-\nation of Anisotropic Charge and Orbital Dynamics in\nY1\u0000xCaxVO3:Comparison with La1\u0000xSrxVO3, Phys.\nRev. B 77, 144402 (2008).\n[59] P. Horsch and A. M. Ole\u0013 s, Defect States and Spin-Orbital\nPhysics in Doped Vanadates Y1\u0000xCaxVO3, Phys. Rev. B\n84, 064429 (2011).\n[60] A. Avella, P. Horsch, and A. M. Ole\u0013 s, Defect States and\nExcitations in a Mott Insulator with Orbital Degrees of27\nFreedom: Mott-Hubbard Gap versus Optical and Trans-\nport Gaps in Doped Systems , Phys. Rev. B 87, 045132\n(2013).\n[61] T. Hotta, Orbital Ordering Phenomena in d- andf-\nElectron Systems , Rep. Prog. Phys. 69, 2061 (2006).\n[62] T. Mizokawa, L. H. Tjeng, G. A. Sawatzky, G. Ghir-\ninghelli, O. Tjernberg, N. B. Brookes, H. Fukazawa,\nS. Nakatsuji, and Y. Maeno, Spin-Orbit Coupling in the\nMott Insulator Ca2RuO 4, Phys. Rev. Lett. 87, 077202\n(2001).\n[63] J. S. Lee, Y. S. Lee, T. W. Noh, S.-J. Oh, J. Yu, S. Nakat-\nsuji, H. Fukazawa, and Y. Maeno, Electron and Orbital\nCorrelations in Ca2\u0000xSrxRuO 4Probed by Optical Spec-\ntroscopy , Phys. Rev. Lett. 89, 257402 (2002).\n[64] A. Koga, N. Kawakami, T. M. Rice, and M. Sigrist,\nOrbital-Selective Mott Transitions in the Degenerate\nHubbard Model , Phys. Rev. Lett. 92, 216402 (2004).\n[65] Z. Fang, K. Terakura, and N. Nagaosa, Orbital Physics\nin Ruthenates: First-Principles Study , New J. Phys. 7,\n66 (2005).\n[66] T. Sugimoto, D. Ootsuki, and T. Mizokawa, Impact of\nLocal Lattice Disorder on Spin and Orbital Orders in\nCa2\u0000xSrxRuO 4, Phys. Rev. B 84, 064429 (2011).\n[67] G. Khaliullin, Excitonic Magnetism in Van Vleck-type d4\nMott Insulators , Phys. Rev. Lett. 111, 197201 (2013).\n[68] A. Akbari and G. Khaliullin, Magnetic Excitations in a\nSpin-Orbit-Coupled d4Mott Insulator on the Square Lat-\ntice, Phys. Rev. B 90, 035137 (2014).\n[69] C. N. Veenstra, Z.-H. Zhu, M. Raichle, B. M. Lud-\nbrook, A. Nicolaou, B. Slomski, G. Landolt, S. Kittaka,\nY. Maeno, J. H. Dil, I. S. El\fmov, M. W. Haverkort,\nand A. Damascelli, Spin-Orbital Entanglement and the\nBreakdown of Singlets and Triplets in Sr2RuO 4Revealed\nby Spin- and Angle-Resolved Photoemission Spectroscopy ,\nPhys. Rev. Lett. 112, 127002 (2014).\n[70] T. Tanaka and S. Ishihara, Dilution E\u000bects in Two-\nDimensional Quantum Orbital Systems , Phys. Rev. Lett.\n98, 256402 (2007).\n[71] T. F. Qi, O. B. Korneta, S. Parkin, L. E. De Long,\nP. Schlottmann, and G. Cao, Negative Volume Ther-\nmal Expansion via Orbital and Magnetic Orders in\nCa2Ru1\u0000xCrxO4(0< x < 0:13), Phys. Rev. Lett. 105,\n177203 (2010).\n[72] G. Cao, S. Chikara, X. N. Lin, E. Elhami, V. Durairaj,\nand P. Schlottmann, Itinerant Ferromagnetism to Insu-\nlating Antiferromagnetism: A Magnetic and Transport\nStudy of Single Crystal SrRu 1\u0000xMnxO3(0\u0014x<0:60),\nPhys. Rev. B 71, 035104 (2005).\n[73] M. A. Hossain, B. Bohnenbuck, Y. D. Chuang,\nM. W. Haverkort, I. S. El\fmov, A. Tanaka, A. G. Cruz\nGonzalez, Z. Hu, H.-J. Lin, C. T. Chen, R. Math-\nieu, Y. Tokura, Y. Yoshida, L. H. Tjeng, Z. Hus-\nsain, B. Keimer, G. A. Sawatzky, and A. Damascelli,\nMott versus Slater-type Metal-Insulator Iransition in Mn-\nSubstituted Sr3Ru2O7, Phys. Rev. B 86, 041102(R)\n(2012).\n[74] M. A. Hossain, Z. Hu, M. W. Haverkort, T. Bur-\nnus, C. F. Chang, S. Klein, J. D. Denlinger, H.-\nJ. Lin, C. T. Chen, R. Mathieu, Y. Kaneko, Y. Tokura,\nS. Satow, Y. Yoshida, H. Takagi, A. Tanaka, I. S. El\fmov,\nG. A. Sawatzky, L. H. Tjeng, and A. Damascelli, Crystal-\nField Level Inversion in Lightly Mn-Doped Sr3Ru2O7,Phys. Rev. Lett. 101, 016404 (2008).\n[75] D. Mesa, F. Ye, S. Chi, J. A. Fernandez-Baca, W. Tian,\nB. Hu, R. Jin, E. W. Plummer, and J. Zhang, Single-\nBilayer E-type Antiferromagnetism in Mn-Substituted\nSr3Ru2O7:Neutron Scattering Study , Phys. Rev. B 85,\n180410(R) (2012).\n[76] S. A. J. Kimber, J. A. Rodgers, H. Wu, C. A. Murray,\nD. N. Argyriou, A. N. Fitch, D. I. Khomskii, and J. P.\nAtt\feld, Metal-Insulator Transition and Orbital Order in\nPbRuO 3, Phys. Rev. Lett. 102, 046409 (2009).\n[77] J. F. Annett, G. Litak, B. L. Gy or\u000by, and\nK. I. Wysoki\u0013 nski, Spin-Orbit Coupling and Symmetry of\nthe Order Parameter in Strontium Ruthenate , Phys. Rev.\nB73, 134501 (2006).\n[78] A. B. Harris, T. Yildirim, A. Aharony, O. Entin-\nWohlman, and I. Ya. Korenblit, Unusual Symmetries in\nthe Kugel-Khomskii Hamiltonian , Phys. Rev. Lett. 91,\n087206 (2003).\n[79] M. Daghofer, K. Wohlfeld, A. M. Ole\u0013 s, E. Arrigoni, and\nP. Horsch, Absence of Hole Con\fnement in Transition-\nMetal Oxides with Orbital Degeneracy , Phys. Rev. Lett.\n100, 066403 (2008).\n[80] A. M. Ole\u0013 s, Antiferromagnetism and Correlation of Elec-\ntrons in Transition Metals , Phys. Rev. B 28, 327 (1983).\n[81] M. Cuoco, F. Forte, and C. Noce, Interplay of Coulomb\nInteractions and c-Axis Octahedra Distortions in Single-\nLayer Ruthenates , Phys. Rev. B 74, 195124 (2006).\n[82] P. Marra, K. Wohlfeld, and J. van den Brink, Unraveling\nOrbital Correlations with Magnetic Resonant Inelastic X-\nRay Scattering , Phys. Rev. Lett. 109, 117401 (2012).\n[83] M. A. Hossain, I. Zegkinoglou, Y.-D. Chuang, J. Geck,\nB. Bohnenbuck, A. G. Cruz Gonzalez, H.-H. Wu,\nC. Sch u\u0019ler-Langeheine, D. G. Hawthorn, J. D. Den-\nlinger, R. Mathieu, Y. Tokura, S. Satow, H. Takagi,\nY. Yoshida, Z. Hussain, B. Keimer, G. A. Sawatzky, and\nA. Damascelli, Electronic Superlattice Revealed by Reso-\nnant Scattering from Random Impurities in Sr3Ru2O7,\nSci. Rep. 3, 2299 (2013).\n[84] L. Limot, P. Mendels, G. Collin, C. Mondelli, B. Oulad-\ndiaf, H. Mutka, N. Blanchard, and M. Mekata, Suscep-\ntibility and Dilution E\u000bects of the kagom\u0013 e Bilayer Ge-\nometrically Frustrated Network: A Ga NMR study of\nSrCr 9pGa129pO19, Phys. Rev. B 65, 144447 (2002).\n[85] J. Bobro\u000b, N. La\rorencie, L. K. Alexander, A. V. Maha-\njan, B. Koteswararao, and P. Mendels, Impurity-Induced\nMagnetic Order in Low-Dimensional Spin-Gapped Mate-\nrials, Phys. Rev. Lett. 103, 047201 (2009).\n[86] A. Sen, K. Damle, and R. Moessner, Fractional Spin Tex-\ntures in the Frustrated Magnet SrCr 9pGa129pO19, Phys.\nRev. Lett. 106, 127203 (2011).\n[87] P. Bonf\u0012 a, P. Carretta, S. Sanna, G. Lamura, G. Prando,\nA. Martinelli, A. Palenzona, M. Tropeano, M. Putti, and\nR. De Renzi, Magnetic Properties of Spin-diluted Iron\nPnictides from \u0016SR and NMR in LaFe 1xRuxAsO, Phys.\nRev. B 85, 054518 (2012).\n[88] A. K. Paul, M. Reehuis, V. Ksenofontov, B. Yan,\nA. Hoser, D. M. T obbens, P. M. Abdala, P. Adler,\nM. Jansen, and C. Felser, Lattice Instability and Com-\npeting Spin Structures in the Double Perovskite Insulator\nSr2FeOsO 6, Phys. Rev. Lett. 111, 167205 (2013)."    },    {        "title": "2102.01400v2.Coupled_spin_orbital_fluctuations_in_a_three_orbital_model_for__4d__and__5d__oxides_with_electron_fillings__n_3_4_5_____Application_to___rm_NaOsO_3_____rm_Ca_2RuO_4___and___rm_Sr_2IrO_4_.pdf",        "content": "arXiv:2102.01400v2  [cond-mat.str-el]  20 Mar 2021Coupled spin-orbital fluctuations in a three orbital model f or4d\nand5doxides with electron fillings n= 3,4,5— Application to\nNaOsO 3, Ca2RuO4, and Sr 2IrO4\nShubhajyoti Mohapatra and Avinash Singh∗\nDepartment of Physics, Indian Institute of Technology, Kanpu r - 208016, India\n(Dated: March 23, 2021)\nA unified approach is presented for investigating coupled sp in-orbital fluctuations\nwithin a realistic three-orbital model for strongly spin-o rbit coupled systems with\nelectron fillings n= 3,4,5 in the t2gsector of dyz,dxz,dxyorbitals. A generalized\nfluctuation propagator is constructed which is consistent w ith the generalized self-\nconsistent Hartree-Fock approximation where all Coulomb i nteraction contributions\ninvolving orbital diagonal and off-diagonal spin and charge c ondensates are included.\nBesides the low-energy magnon, intermediate-energy orbit on and spin-orbiton, and\nhigh-energy spin-orbit exciton modes, the generalized spe ctral function also shows\nother high-energy excitations such as the Hund’s coupling i nduced gapped magnon\nmodes. We relate the characteristic features of the coupled spin-orbital excitations\nto the complex magnetic behavior resulting from the interpl ay between electronic\nbands, spin-orbit coupling, Coulomb interactions, and str uctural distortion effects,\nas realized in the compounds NaOsO 3, Ca2RuO4, and Sr 2IrO4.2\nI. INTRODUCTION\nThe 4dand 5dtransition metal (TM) oxides exhibit an unprecedented coupling bet ween\nspin, charge, orbital, and structural degrees of freedom. The c omplex interplay between\nthe different physical elements such as strong spin-orbit coupling ( SOC), Coulomb interac-\ntions, and structural distortions results in novel magnetic state s and unconventional collec-\ntive excitations.1–6In particular, the cubic structured NaOsO 3and perovskite structured\nCa2RuO4and Sr 2IrO4compounds, corresponding to dnelectronic configuration of the TM\nion with electron fillings n=3,4,5 in the t 2gsector, respectively, are at the emerging research\nfrontier as they provide versatile platform for the exploration of S OC-driven phenomena\ninvolving collective electronic and magnetic behavior including coupled s pin-orbital excita-\ntions.\nThedifferent physical elements giverisetoarichvarietyofnontrivia l microscopic features\nwhich contribute to the complex interplay. These include spin-orbita l-entangled states,\nbandnarrowing, spin-orbit gap, andexplicit spin-rotation-symmet ry breaking (dueto SOC),\nelectronic band narrowing due to reduced effective hopping (octah edral tilting and rotation),\ncrystal field induced tetragonal splitting (octahedral compress ion), orbital mixing (SOC\nand octahedral tilting, rotation) which self consistently generate s induced SOC terms and\norbitalmomentinteractionfromtheCoulombinteractionterms, sig nificantlyweakerelectron\ncorrelation term Ucompared to 3 dorbitals and therefore critical contribution of Hund’s\ncoupling to local magnetic moment. These microscopic features con tribute to the complex\ninterplay in different ways for electron fillings n=3,4,5, resulting in significantly different\nmacroscopic properties of the three compounds, which are briefly reviewed below along\nwith experimental observations about the collective and coupled sp in-orbital excitations as\nobtained from recent resonant inelastic X-ray scattering (RIXS) studies.\nThe nominally orbitally quenched d3compound NaOsO 3undergoes a metal-insulator\ntransition (MIT) ( TMI=TN= 410 K) that is closely related to the onset of long-range\nantiferromagnetic (AFM) order.7–10Various mechanisms, such as Slater-like, magnetic Lif-\nshitz transition, and AFM band insulator have been proposed to exp lain this unusual and\nintriguing nature of the MIT.8,11–14Interplay of electronic correlations, Hund’s coupling,\nand octahedral tilting and rotation induced band narrowing near th e Fermi level in this\nweakly correlated compound results in the weakly insulating state wit h G-type AFM or-3\nder, with magnetic anisotropy and large magnon gap resulting from in terplay of SOC, band\nstructure, and the tetragonal splitting.14,15The OsL3resonant edge RIXS measurements\nat room temperature show four inelastic peak features below 1.5 eV , which have been in-\nterpreted to correspond to the strongly gapped ( ∼58 meV) dispersive magnon excitations\nwith bandwidth ∼100 meV, excitations (centered at ∼1 eV) within the t2gmanifold, and\nexcitations from t2gtoegstates and ligand-to-metal charge transfer for the remaining tw o\nhigher-energypeaks.13,16,17Theintensity andpositionsofthethreehigh-energypeaksappear\nto be essentially temperature independent.\nThe nominally spin S=1d4compound Ca 2RuO4undergoes a MIT at TMI=357 K and\nmagnetic transition at TN=110 K ( ≪TMI) via a structural phase transition involving a\ncompressive tetragonal distortion, tilt, and rotation of the RuO 6octahedra.18–21The low-\ntemperature AFM insulating phase is thus characterized by highly dis torted octahedra\nwith nominally filled xyorbital and half-filled yz,xzorbitals.22–24This transition has also\nbeen identified in pressure,25–27chemical substitution,28–30strain,31and electrical current\nstudies,32,33and highlights the complex interplay between SOC, Coulomb interactio ns, and\nstructural distortions.\nInelastic neutron scattering (INS)34–36and Raman37studies on Ca 2RuO4have revealed\nunconventional low-energy ( ∼50 and 80 meV) excitations interpreted as gapped trans-\nverse magnon modes and possibly soft longitudinal (“Higgs-like”) or two-magnon excitation\nmodes. From both Ru L3-edge and oxygen K-edge RIXS studies, multiple nontrivial exci-\ntations within the t2gmanifold were observed recently below 1 eV.38–40Two low-energy ( ∼\n80 and 350 meV) and two high-energy ( ∼750 meV and 1 eV) excitations were identified\nwithin the limited energy resolution of RIXS. From the incident angle an d polarisation de-\npendence of the RIXS spectra, the orbital character of the 80 m eV peak was inferred to be\nmixture of xyandxz/yzstates, whereas the 0.4 eV peak was linked to unoccupied xz/yz\nstates. Guided by phenomenological spin models, the low-energy ex citations (consisting of\nmultiple branches) were interpreted as composite spin-orbital exc itations (also termed as\n“spin orbitons”).\nFinally, SOC induced novel Mott insulating state is realized in the d5compound\nSr2IrO4,41,42where band narrowing of the spin-orbital-entangled electronic sta tes near the\nFermi level plays a critical role in the insulating behavior. The AFM insu lating ground\nstate is characterized by the correlation induced insulating gap with in the nominally J=1/24\nbandsemerging fromtheKramersdoublet, which areseparatedfr omthebands ofthe J=3/2\nquartet by energy 3 λ/2, whereλis the SOC strength. The RIXS spectra show low-energy\ndispersive magnon excitations (up to 200 meV), further resolved in to two gapped magnon\nmodes with energy gaps ∼40 meV and 3 meV at the Γ point corresponding to out-of-\nplane and in-plane fluctuation modes, respectively.43–46Weak electron correlation effect and\nmixing between the J=1/2 and 3/2 sectors were identified as contributing significantly to\nthe strong zone-boundary magnon dispersion as measured in RIXS studies.47In addition,\nhigh-energy dispersive spin-orbit exciton modes have also been rev ealed in RIXS studies in\nthe energy range 0.4-0.8 eV.48This distinctive mode is also referred to as the spin-orbiton\nmode,49,50and has been attributed to the correlated motion of electron-hole pair excitations\nacross the renormalized spin-orbit gap between the J=1/2 and 3/2 bands.51\nMost of the theoretical studies involving magnetic anisotropy effec ts and excitations in\nabovesystems havemainlyfocusedonphenomenological spinmodels withdifferent exchange\ninteractions obtained as fitting parameters to the experimental s pectra. However, the in-\nterpretation of experimental data remains incomplete since the ch aracter of the effective\nspins, the microscopic origin of their interactions, and the microsco pic nature of the mag-\nnetic excitations are still debated.2–6Realistic information about the spin-orbital character\nof both low and high-energy collective excitations, as inferred from the study of coupled\nspin-orbital excitations, is clearly important since the spin and orbit al degrees of freedom\nare explicitly coupled, and both are controlled by the different physic al elements such as\nSOC, Coulomb interaction terms, tetragonal compression induced crystal-field splitting be-\ntweenxyandyz,xzorbitals, octahedral tilting and rotation induced orbital mixing hopp ing\nterms, and band physics.\nDue to the intimately intertwined roles of the different physical eleme nts, a unified ap-\nproach is therefore required for the realistic modeling of these sys tems in which all physical\nelements are treated on an equal footing. The generalized self-co nsistent approximation ap-\npliedrecentlytothe n= 4compoundCa 2RuO4providessuch aunifiedapproach.52Involving\ntheself-consistent determinationofmagneticorderwithinathree -orbitalinteractingelectron\nmodel including all orbital-diagonal and off-diagonal spin and charge condensates generated\nby the different Coulomb interaction terms, this approach explicitly in corporates the com-\nplex interplay and accounts for the observed behavior including the tetragonal distortion\ninduced magnetic reorientation transition, orbital moment interac tion induced orbital gap,5\nSOC and octahedral tilting induced easy-axis anisotropy, and Coulo mb interaction induced\nanisotropic SOC renormalization. Extension to the n= 5 compound Sr 2IrO4,53provides\nconfirmation of the Hund’s coupling induced easy-plane magnetic anis otropy, which is re-\nponsible for the ∼40 meV magnon gap measured for the out-of-plane fluctuation mod e.46\nTowards a generalized non-perturbative formalism unifying the mag netic order and\nanisotropy effects on one hand and collective excitations on the oth er, the natural exten-\nsion of the above generalized condensate approach is therefore t o consider the generalized\nfluctuation propagator in terms of the generalized spin ( ψ†\nµ[σα]ψν) and charge ( ψ†\nµ[1]ψν) op-\nerators in the pure spin-orbital basis of the t 2gorbitalsµ,ν=yz,xz,xy and spin components\nα=x,y,z. The generalized operators include the normal ( µ=ν) spin and charge opera-\ntors as well as the orbital off-diagonal ( µ/negationslash=ν) cases which are related to the generalized\nspin-orbit coupling terms ( LαSβ, whereα,β=x,y,z) and the orbital angular momentum\noperatorsLα. Constructing the generalized fluctuation propagator as above w ill ensure that\nthis scheme is fully consistent with the generalized self-consistent a pproach involving the\ngeneralized condensates.\nThe different components of the generalized fluctuation propagat or will therefore natu-\nrallyincludespin-orbitonsandorbitons, corresponding tothespin- orbital(LαSβ)andorbital\n(Lα) moment fluctuations, besides the normal spin andchargefluctua tions. The normal spin\nfluctuations will include in-phase and out-of phase fluctuations with respect to different or-\nbitals, the latter being strongly gapped due to Hund’s coupling. The s pin-orbitons will\ninclude the spin-orbit excitons measured in RIXS studies of Sr 2IrO4.\nThe structure of this paper is as below. The three-orbital model w ithin the t 2gsector\n(including SOC, hopping, Coulomb interaction, and structural disto rtion terms), and the\ngeneralized self-consistent formalism including orbital diagonal and off-diagonal condensates\nare reviewed in Sec. II and III. After introducing the generalized fl uctuation propagator in\nSec. IV, results of the calculated fluctuation spectral functions are presented for the cases\nn= 3,4,5 (corresponding to the three compounds NaOsO 3, Ca2RuO4, Sr2IrO4) in Sections\nV, VI, VII. Finally, conclusions are presented in Sec. VIII. The bas is-resolved contributions\nto the total spectral function showing the detailed spin-orbital c haracter of the collective\nexcitations are presented in the Appendix.6\nII. THREE ORBITAL MODEL WITH SOC AND COULOMB INTERACTIONS\nIn the three-orbital ( µ=yz,xz,xy ), two-spin ( σ=↑,↓) basis defined with respect to a\ncommon spin-orbital coordinate axes (Fig. 1), we consider the Ham iltonianH=Hband+\nHcf+Hint+HSOCwithin the t2gmanifold. For the band and crystal field terms together,\nwe consider:\nHband+cf=/summationdisplay\nkσsψ†\nkσs\n\nǫyz\nk′0 0\n0ǫxz\nk′0\n0 0ǫxy\nk′+ǫxy\nδss′+\nǫyz\nkǫyz|xz\nkǫyz|xy\nk\n−ǫyz|xz\nkǫxz\nkǫxz|xy\nk\n−ǫyz|xy\nk−ǫxz|xy\nkǫxy\nk\nδ¯ss′\nψkσs′\n(1)\nin the composite three-orbital, two-sublattice ( s,s′= A,B) basis. Here the energy offset\nǫxy(relative to the degenerate yz/xzorbitals) represents the tetragonal distortion induced\ncrystal field effect. The band dispersion terms in the two groups co rrespond to hopping\nterms connecting the same and opposite sublattice(s), and are giv en by:\nǫxy\nk=−2t1(coskx+cosky)\nǫxy\nk′=−4t2coskxcosky−2t3(cos2kx+cos2ky)\nǫyz\nk=−2t5coskx−2t4cosky\nǫxz\nk=−2t4coskx−2t5cosky\nǫyz|xz\nk=−2tm1(coskx+cosky)\nǫxz|xy\nk=−2tm2(2coskx+cosky)\nǫyz|xy\nk=−2tm3(coskx+2cosky). (2)\nHeret1,t2,t3are respectively the first, second, and third neighbor hopping ter ms for\nthexyorbital. For the yz(xz) orbital,t4andt5are the nearest-neighbor (NN) hopping\nterms iny(x) andx(y) directions, respectively, corresponding to πandδorbital overlaps.\nOctahedral rotation and tilting induced orbital mixings are represe nted by the NN hopping\ntermstm1(betweenyzandxz) andtm2,tm3(betweenxyandxz,yz). In then= 4 case\ncorresponding to the Ca 2RuO4compound, we have taken hopping parameter values: ( t1,t2,\nt3,t4,t5)=(−1.0,0.5,0,−1.0,0.2),orbital mixing hopping terms: tm1=0.2andtm2=tm3=0.15\n(≈0.2/√\n2), andǫxy=−0.8, all in units of the realistic hopping energy scale |t1|=150\nmeV.54–56The choice tm2=tm3corresponds to the octahedral tilting axis oriented along7\n(a)(b)\nFIG. 1: (a) The common spin-orbital coordinate axes ( x−y) along the Ru-O-Ru directions, shown\nalong with the crystal axes a,b. (b) Octahedral tilting about the crystal aaxis is resolved along\nthex,yaxes, resulting in orbital mixing hopping terms between the xyandyz,xzorbitals.\nthe±(−ˆx+ ˆy) direction, which is equivalent to the crystal ∓adirection (Fig. 1). The tm1\nandtm2,m3values taken above approximately correspond to octahedral rot ation and tilting\nangles of about 12◦(≈0.2 rad) as reported in experimental studies.26\nFor the on-site Coulomb interaction terms in the t2gbasis (µ,ν=yz,xz,xy ), we consider:\nHint=U/summationdisplay\ni,µniµ↑niµ↓+U′/summationdisplay\ni,µ<ν,σniµσniνσ+(U′−JH)/summationdisplay\ni,µ<ν,σniµσniνσ\n+JH/summationdisplay\ni,µ/negationslash=νa†\niµ↑a†\niν↓aiµ↓aiν↑+JP/summationdisplay\ni,µ/negationslash=νa†\niµ↑a†\niµ↓aiν↓aiν↑\n=U/summationdisplay\ni,µniµ↑niµ↓+U′′/summationdisplay\ni,µ<νniµniν−2JH/summationdisplay\ni,µ<νSiµ.Siν+JP/summationdisplay\ni,µ/negationslash=νa†\niµ↑a†\niµ↓aiν↓aiν↑(3)\nincluding the intra-orbital ( U) and inter-orbital ( U′) density interaction terms, the Hund’s\ncoupling term ( JH), and the pair hopping interaction term ( JP), withU′′≡U′−JH/2 =\nU−5JH/2 from the spherical symmetry condition U′=U−2JH. Herea†\niµσandaiµσare\nthe electron creation and annihilation operators for site i, orbitalµ, spinσ=↑,↓. The\ndensity operator niµσ=a†\niµσaiµσ, total density operator niµ=niµ↑+niµ↓=ψ†\niµψiµ, and spin\ndensity operator Siµ=ψ†\niµσψiµin terms of the electron field operator ψ†\niµ= (a†\niµ↑a†\niµ↓). All\ninteraction terms above are SU(2) invariant and thus possess spin rotation symmetry.\nFinally, for the bare spin-orbit coupling term (for site i), we consider the spin-space8\nrepresentation:\nHSOC(i) =−λL.S=−λ(LzSz+LxSx+LySy)\n=\n/parenleftig\nψ†\nyz↑ψ†\nyz↓/parenrightig/parenleftig\niσzλ/2/parenrightig\nψxz↑\nψxz↓\n+/parenleftig\nψ†\nxz↑ψ†\nxz↓/parenrightig/parenleftig\niσxλ/2/parenrightig\nψxy↑\nψxy↓\n\n+/parenleftig\nψ†\nxy↑ψ†\nxy↓/parenrightig/parenleftig\niσyλ/2/parenrightig\nψyz↑\nψyz↓\n\n+H.c. (4)\nwhich explicitly breaks SU(2) spin rotation symmetry and therefore generates anisotropic\nmagnetic interactions from its interplay with other Hamiltonian terms . Here we have used\nthe matrix representation:\nLz=\n0−i0\ni0 0\n0 0 0\n, Lx=\n0 0 0\n0 0−i\n0i0\n, Ly=\n0 0i\n0 0 0\n−i0 0\n, (5)\nfor the orbital angular momentum operators in the three-orbital (yz,xz,xy ) basis.\nAs the orbital “hopping” terms in Eq. (4) have the same form as spin -dependent hopping\ntermsiσ.t′\nij, carrying out the strong-coupling expansion57for the−λLzSzterm to second\norder inλyields the anisotropic diagonal (AD) intra-site interactions:\n[H(2)\neff](z)\nAD(i) =4(λ/2)2\nU/bracketleftbig\nSz\nyzSz\nxz−(Sx\nyzSx\nxz+Sy\nyzSy\nxz)/bracketrightbig\n(6)\nbetweenyz,xzmoments if these orbitals arenominally half-filled, as in thecase ofCa 2RuO4.\nThis term explicitly yields preferential x−yplane ordering (easy-plane anisotropy) for\nparallelyz,xzmoments, as enforced by the relatively stronger Hund’s coupling.\nSimilarly, from the strong coupling expansion for the other two SOC t erms, we obtain\nadditional anisotropic interaction terms which are shown below to yie ldC4symmetric easy-\naxis anisotropy within the easy plane. From the −λLxSxand−λLySyterms, we obtain:\n[H(2)\neff](x,y)\nAD(i) =4(λ/2)2\nU/bracketleftbig\nSx\nxzSx\nxy−(Sy\nxzSy\nxy+Sz\nxzSz\nxy)/bracketrightbig\n+4(λ/2)2\nU/bracketleftbig\nSy\nxySy\nyz−(Sx\nxySx\nyz+Sz\nxySz\nyz)/bracketrightbig\n(7)\nNeglecting the terms involving the Szcomponents which are suppressed by the easy-plane\nanisotropy discussed above, we obtain:\n[H(2)\neff](x,y)\nAD(i) =−4(λ/2)2\nU/bracketleftbig\nSx\nxy(Sx\nyz−Sx\nxz)+Sy\nxy(Sy\nxz−Sy\nyz)/bracketrightbig\n=−4(λ/2)2\nUfxyS2[sin2φsinφc] (8)9\nwhere the spin components are expressed as: Sx\nxy=fxyScosφ,Sx\nyz=Scos(φ−φc),Sx\nxz=\nScos(φ+φc) (and similarly for the ycomponents) in terms of the overall orientation angle φ\nof the magnetic order and the relative canting angle 2 φcbetween the yz,xzmoments. Here\nthe factorfxy<1 represents the reduced moment for the xyorbital.\nThe above expression shows the composite orientation and canting angle dependence of\nthe anisotropic interaction energy having the C4symmetry. Minimum energy is obtained\nat orientations φ=nπ/4 (wheren= 1,3,5,7) since the canting angle has the approximate\nfunctional form φc≈φmax\ncsin2φin terms of the orientation φ. Thus, while the easy-plane\nanisotropy involves only the yz,xzmoments, the xymoment plays a crucial role in the\neasy-axis anisotropy, which is directly relevant for NaOsO 3(xyorbital is also nominally\nhalf-filled), but also for Ca 2RuO4with the factor fxyas incorporated above.\nFor later reference, we note here that condensates of the orbit al off-diagonal (OOD) one-\nbody operators as in Eq. (4) directly yield physical quantities such a s orbital magnetic\nmoments and spin-orbital correlations:\n/angbracketleftLα/angbracketright=−i/bracketleftbig\n/angbracketleftψ†\nµψν/angbracketright−/angbracketleftψ†\nµψν/angbracketright∗/bracketrightbig\n= 2 Im/angbracketleftψ†\nµψν/angbracketright\n/angbracketleftLαSα/angbracketright=−i/bracketleftbig\n/angbracketleftψ†\nµσαψν/angbracketright−/angbracketleftψ†\nµσαψν/angbracketright∗/bracketrightbig\n/2 = Im/angbracketleftψ†\nµσαψν/angbracketright\nλint\nα= (U′′−JH/2)/angbracketleftLαSα/angbracketright= (U′′−JH/2)Im/angbracketleftψ†\nµσαψν/angbracketright (9)\nwhere the orbital pair ( µ,ν) corresponds to the component α=x,y,z, and the last equation\nyields the interaction induced SOC renormalization, as discussed in th e next section.\nIII. SELF-CONSISTENT DETERMINATION OF MAGNETIC ORDER\nWe consider the various contributions from the Coulomb interaction terms (Eq. 3) in the\nHF approximation, focussing first on terms with normal (orbital dia gonal) spin and charge\ncondensates. The resulting local spin and charge terms can be writ ten as:\n[HHF\nint]normal=/summationdisplay\niµψ†\niµ[−σ.∆iµ+Eiµ1]ψiµ (10)\nwhere the spin and charge fields are self-consistently determined f rom:\n2∆α\niµ=U/angbracketleftσα\niµ/angbracketright+JH/summationdisplay\nν<µ/angbracketleftσα\niν/angbracketright(α=x,y,z)\nEiµ=U/angbracketleftniµ/angbracketright\n2+U′′/summationdisplay\nν<µ/angbracketleftniν/angbracketright (11)10\nin terms of the local charge density /angbracketleftniµ/angbracketrightand the spin density components /angbracketleftσα\niµ/angbracketright.\nThere are additional contributions resulting from orbital off-diago nal (OOD) spin and\ncharge condensates which are finite due to orbital mixing induced by SOC and structural\ndistortions (octahedral tilting and rotation). The contributions c orresponding to different\nCoulomb interaction terms are summarized in Appendix A, and can be g rouped in analogy\nwith Eq. (10) as:\n[HHF\nint]OOD=/summationdisplay\ni,µ<νψ†\niµ[−σ.∆iµν+Eiµν1]ψiν+H.c. (12)\nwhere the orbital off-diagonal spin and charge fields are self-cons istently determined from:\n∆iµν=/parenleftbiggU′′\n2+JH\n4/parenrightbigg\n/angbracketleftσiνµ/angbracketright+/parenleftbiggJP\n2/parenrightbigg\n/angbracketleftσiµν/angbracketright\nEiµν=/parenleftbigg\n−U′′\n2+3JH\n4/parenrightbigg\n/angbracketleftniνµ/angbracketright+/parenleftbiggJP\n2/parenrightbigg\n/angbracketleftniµν/angbracketright (13)\nin terms of the corresponding condensates /angbracketleftσiµν/angbracketright ≡ /angbracketleftψ†\niµσψiν/angbracketrightand/angbracketleftniµν/angbracketright ≡ /angbracketleftψ†\niµ1ψiν/angbracketright.\nThe spin andcharge condensates inEqs. 11and 13 areevaluated us ing the eigenfunctions\n(φk) and eigenvalues ( Ek) of the full Hamiltonian in the given basis including the interaction\ncontributions [ HHF\nint] (Eqs. 10 and 12) using:\n/angbracketleftσα\niµν/angbracketright ≡ /angbracketleftψ†\niµσαψiν/angbracketright=Ek<EF/summationdisplay\nk(φ∗\nkµs↑φ∗\nkµs↓)[σα]\nφkνs↑\nφkνs↓\n (14)\nfor siteion thes=A/Bsublattice, and similarly for the charge condensates /angbracketleftniµν/angbracketright ≡\n/angbracketleftψ†\niµ1ψiν/angbracketrightwith the Pauli matrices [ σα] replaced by the unit matrix [ 1]. The normal spin and\ncharge condensates correspond to ν=µ. For each orbital pair ( µ,ν) = (yz,xz), (xz,xy),\n(xy,yz), there are three components ( α=x,y,z) for the spin condensates /angbracketleftψ†\nµσαψν/angbracketrightand\none charge condensate /angbracketleftψ†\nµ1ψν/angbracketright. This is analogous to the three-plus-one normal spin and\ncharge condensates for each of the three orbitals µ=yz,xz,xy .\nThe above additional terms involving orbital off-diagonal condensa tes contribute to or-\nbital physics. Thus, the charge terms lead to coupling of orbital an gular momentum opera-\ntors to weak orbital fields, the spin terms result in interaction-indu ced SOC renormalization\nas given in Eq. (9), and the self consistently determined renormalize d SOC values are\nobtained as:\nλα=λ+λint\nα (15)11\nforthethreecomponents α=x,y,z. Resultsoftheselfconsistentdeterminationofmagnetic\norder including all orbital diagonal and off-diagonal spin and charge condensates have been\npresented for the 4d4compound Ca 2RuO4recently,52illustrating the rich interplay between\ndifferent physical elements.\nIV. GENERALIZED FLUCTUATION PROPAGATOR\nSinceallgeneralizedspin /angbracketleftψ†\nµσψν/angbracketrightandcharge /angbracketleftψ†\nµψν/angbracketrightcondensateswereincludedintheself\nconsistent determination of magnetic order, the fluctuation prop agator must also be defined\nin terms of the generalized operators. We therefore consider the time-ordered generalized\nfluctuation propagator:\n[χ(q,ω)] =/integraldisplay\ndt/summationdisplay\nieiω(t−t′)e−iq.(ri−rj)×/angbracketleftΨ0|T[σα\nµν(i,t)σα′\nµ′ν′(j,t′)]|Ψ0/angbracketright (16)\nin the self-consistent AFM ground state |Ψ0/angbracketright, where the generalized spin-charge operators\nat lattice sites i,jare defined as σα\nµν=ψ†\nµσαψν, which include both the orbital diagonal\n(µ=ν) and off-diagonal ( µ/negationslash=ν) cases, as well as the spin ( α=x,y,z) and charge ( α=c)\noperators, with σαdefined as Pauli matrices for α=x,y,zand unit matrix for α=c.\nIn the random phase approximation (RPA), the generalized fluctua tion propagator is\nobtained as:\n[χ(q,ω)]RPA=2[χ0(q,ω)]\n1−[U][χ0(q,ω)](17)\nin terms of the bare particle-hole propagator [ χ0(q,ω)] which is evaluated by integrating out\nthe electronic degrees of freedom:\n[χ0(q,ω)]µ′ν′α′s′\nµναs=1\n2/summationdisplay\nk/bracketleftigg\n/angbracketleftk|σα\nµν|k−q/angbracketrights/angbracketleftk|σα′\nµ′ν′|k−q/angbracketright∗\ns′\nE⊕\nk−q−E⊖\nk+ω−iη+/angbracketleftk|σα\nµν|k−q/angbracketrights/angbracketleftk|σα′\nµ′ν′|k−q/angbracketright∗\ns′\nE⊕\nk−E⊖\nk−q−ω−iη/bracketrightigg\n(18)\nThe matrix elements in the above expression are evaluated using the eigenvectors of the HF\nHamiltonian in the self-consistent AFM state:\n/angbracketleftk|σα\nµν|k−q/angbracketrights= (φ∗\nkµ↑sφ∗\nkµ↓s)[σα]\nφk−qν↑s\nφk−qν↓s\n (19)12\nand the superscripts ⊕(⊖) refer to particle (hole) states above (below) the Fermi energy.\nThe subscripts s,s′indicate thetwo (A/B) sublattices. Inthe compositespin-charge- orbital-\nsublattice ( µναs) basis, the [ χ0(q,ω)] matrix is of order 72 ×72, and the form of the [ U]\nmatrix in the RPA expression (Eq. 17) is given in Appendix B.\nThe spectral function of the excitations will be determined from:\nAq(ω) =1\nπIm Tr[χ(q,ω)]RPA (20)\nusing the RPA expression for [ χ(q,ω)]. When the collective excitation energies lie within\nthe AFM band gap, it is convenient to consider the symmetric form of the denominator in\nthe RPA expression (Eq. 17):\n[U][χ0(q,ω)][U]−[U] (21)\nand in terms of the real eigenvalues λq(ω) of this Hermitian matrix, the magnon energies\nωqfor momentum qare determined by solving for the zeroes:\nλq(ω=ωq) = 0 (22)\ncorresponding to the poles in the propagator.\nResults of the calculated spectral function will be discussed in the s ubsequent sections\nfor different electron filling cases ( n= 3,4,5) with applications to corresponding 4 dand\n5dtransition metal compounds. Broadly, our investigation of the gen eralized fluctuation\npropagator will provide information about the dominantly spin, orbit al, and spin-orbital\nexcitations, as the generalized spin and charge operators ψ†\nµσαψνinclude spin ( µ=ν,\nα=x,y,z), orbital (µ/negationslash=ν,α=c), and spin-orbital ( µ/negationslash=ν,α=x,y,z) cases. Also\nincluded will be the high-energy spin-orbit exciton modes involving par ticle-hole excitations\nacross the renormalized spin-orbit gap between spin-orbital enta ngled states of different J\nsectors, as in the n= 5 case relevant for the Sr 2IrO4compound.\nV.n= 3— APPLICATION TO NaOsO 3\nThe strongly spin-orbit coupled orthorhomic structured 5 d3osmium compound NaOsO 3,\nwith nominally three electrons in the Os t2gsector, exhibits several novel electronic and\nmagnetic properties. These include a G-type antiferromagnetic (A FM) structure with spins13\noriented along the caxis, a significantly reduced magnetic moment ∼1µBas measured from\nneutron scattering, a continuous metal-insulator transition (MIT ) that coincides with the\nAFM transition ( TN=TMIT= 410 K) as seen in neutron and X-ray scattering, and a large\nmagnon gap of 58 meV as seen in resonant inelastic X-ray scattering (RIXS) measurements\nindicating strong magnetic anisotropy.8–10,16\nTwo different mechanisms contributing to SOC-induced easy-plane a nisotropy and large\nmagnon gap for out-of-plane fluctuation modes were identified for the weakly correlated\n5d3compound NaOsO 3in terms of a simplified picture involving only the normal spin and\ncharge densities.14,15Both essential ingredients — (i) small moment disparity between yz,xz\nandxyorbitals and (ii) spin-charge coupling effect in presence of tetragon al splitting — are\nintrinsically present in the considered three-orbital model on the s quare lattice. A realistic\nrepresentation of magnetic anisotropy in NaOsO 3is therefore provided by the considered\nmodel, while maintaining uniformity of lattice structure across the n= 3,4,5 cases consid-\nered in order to keep the focus on coupled spin-orbital fluctuation s.\nThe first mechanism involves the SOC-induced anisotropic interactio n terms as in Eq.\n(6) resulting from the three SOC terms −λLαSαforα=x,y,z. Due to the small moment\ndisparitymyz,xz>mxyresulting from the broader xyband, the interaction term in Eq. (6)\ndominates over the other two terms, leading to the easy-plane anis otropy for parallel yz,xz\nmoments enforced by the Hund’s coupling. With increasing U, this effect weakens as the\nmoments saturate myz,xz,xy≈1 in the large Ulimit. In the second mechanism, the SOC\ninduced decreasing xyorbital density nxywith spin rotation from zdirection to x−yplane\ncouples to the tetragonal distortion term, and for positive ǫxythe energy is minimized for\nspin orientation in the x−yplane.\nWe will consider the parameter set values U= 4,JH=U/5,U′′=U−5JH/2, bare\nSOC value λ=1.0, andǫxy= 0.5 unless otherwise indicated, with the hopping energy scale\n|t1|=300 meV. Thus, U= 1.2 eV,λ=0.3 eV,ǫxy= 0.15 eV, which are realistic values for the\nNaOsO 3compound. Initially, we will also set tm1,m2,m3= 0 for simplicity, and focus on the\neasy-plane anisotropy and large magnon gap for out-of-plane fluc tuations.\nSelf consistent determination of magnetic order using the generaliz ed approach discussed\nin Sec. III confirms the easy-plane anisotropy. Starting in nearly zdirection, the AFM order\ndirection self consistently approaches the x−yplane in a few hundred iterations. Initially,\nwe will discuss magnetic excitations in the self consistent state with A FM order along the14\n(π/2,π/2) (π,0) ( π,π) (0,0) 0 50 100 150 200 250 300ω (meV)\n 0.01 0.1 1 10 100\n(a)\n 0 50 100 150 200 250 300\n(π/2,π/2) (π,0) ( π,π) (π/2,π/2)(b)ωq (meV)\nFIG. 2: (a) Low energy part of the calculated spectral functi on from the generalized fluctuation\npropagator shows the magnon excitations in the self-consis tent state with planar AFM order, and\n(b) magnon dispersion showing the gapless and gapped modes c orresponding to in-plane and out-\nof-plane fluctuations.\nˆxor ˆydirections. Although these orientations correspond to metastab le states as discussed\nlater, they provide convenient test cases for explicitly confirming t he gapless in-plane and\ngapped out-of-plane magnon modes in the generalized fluctuation p ropagator calculation.\nThe low-energy part of the calculated spectral function using Eq. (20) is shown in the\nFig. 2(a) as an intensity plot for qalong symmetry directions of the Brillouin zone. The\ngapless and gapped modes corresponding to in-plane and out-of-p lane fluctuations reflect\nthe easy-plane magnetic anisotropy. The calculated gap energy 60 meV is close to the\nmeasured spin wave gap of 58 meV in NaOsO 3. Also shown for comparison in Fig. 2(b)\nis the magnon dispersion calculated from the poles of the RPA propag ator as described in\nSec. IV. Focussing on the magnon gap in Fig. 2(b), which provides a m easure of the SOC\ninduced easy-plane anisotropy, effects of various physical quant ities are shown in Fig. 3.\nThegaplessGoldstonemodecorrespondingtoin-planerotationofA FMorderingdirection\nin thex−yplane involves only small changes in spin densities /angbracketleftψ†\nµσαψµ/angbracketrightforα=x,yand\nµ=yz,xz, and also in generalized spin densities /angbracketleftψ†\nµσαψν/angbracketrightforα=x,yandµ=yz,xzwith\nfixedν=xy. For example, the magnetization values mx\nyz= 0.82 andmx\nxz= 0.84 change\ntomy\nyz= 0.84 andmy\nxz= 0.82 when the ordering is rotated from xtoydirection. Thus,\nthe Goldstone mode is nearly pure spin mode and the small orbital cha racter reflects the\neffectively suppressed spin-orbital entangement in the n= 3 AFM state. In contrast, the15\n 0 20 40 60 80 100\n 0 0.06 0.12 0.18 0.24 0.30 0.36(a)magnon gap (meV)\nSOC (eV) 0 20 40 60 80 100\n 0.9 1.2 1.5 1.8 2.1 2.4(b)magnon gap (meV)\nU (eV) 0 20 40 60 80 100\n-0.15-0.1-0.05 0 0.05 0.1 0.15 0.2 0.25(c)magnon gap (meV)\nεxy (eV)\nFIG. 3: Variation of the calculated magnon gap showing effects of (a) SOC, (b) Hubbard U, and\n(c) tetragonal distortion ǫxy, on the easy-plane magnetic anisotropy.\nn= 5 case corresponding to Sr 2IrO4shows strongly coupled spin-orbital character of the\nGoldstone mode (Appendix C) due to the extreme spin-orbital enta nglement.\nWe now consider the easy-axis anisotropy effects in our self consist ent determination of\nmagnetic order. With respect to the AFM order orientation (azimut hal angleφ) within the\neasy (x−y) plane, we find an easy-axis anisotropy along the diagonal orientat ionsφ=nπ/4\n(n= 1,3,5,7) even for no octahedral tilting. This anisotropy is due to the orien tation\nand canting angle dependent anisotropic interaction (Eq. 8) as disc ussed in Sec. II. The\nanisotropic interaction energy vanishes for φalong thex,yaxes (hence the gapless in-plane\nmode in Fig. 2), and is significant near the diagonal orientations, res ulting in easy-axis\nanisotropy and small relative canting between yz,xzmoments which is explicitly confirmed\nin our self-consistent calculation.\nThe resulting C4symmetry of the easy-axis ±(ˆx±ˆy) is reduced to C2symmetry ±(ˆx−ˆy)\nin the presence of octahedral tilting. The important anisotropy eff ects of the octahedral\ntilting induced inter-site DM interactions are discussed below. We find that the DM axis\nlies along the crystal baxis, leading to easy axis direction along the crystal aaxis. Both\nthese directions are interchanged in comparison to the Ca 2RuO4case, which follows from a\nsubtle difference in the present n= 3 case as explained below.\nFollowing the analysis carried out for the Ca 2RuO4compound,52within the usual strong-\ncoupling expansion in terms of the normal ( t) and spin-dependent ( t′\nx,t′\ny) hopping terms\ninduced by the combination of SOC and orbital mixing hopping terms tm2,m3due to octa-\nhedral tilting, the DM interaction terms generated in the effective s pin model are obtained16\nTABLE I: Self consistently determined magnetization and de nsity values for the three orbitals ( µ)\non the two sublattices ( s), showing easy-axis anisotropy along the crystal aaxis due to octahedral\ntilting induced DM interaction. Here tm2,m3= 0.15.\nµ(s)mx\nµmy\nµmz\nµnµ\nyz(A) 0.598 −0.557 0.006 1.012\nxz(A) 0.557 −0.598−0.006 1.012\nxy(A) 0.541 −0.541 0.0 0.977µ(s)mx\nµmy\nµmz\nµnµ\nyz(B)−0.598 0.557 0.006 1.012\nxz(B)−0.557 0.598 −0.006 1.012\nxy(B)−0.541 0.541 0.0 0.977\nTABLE II: Self consistently determined renormalized SOC va luesλα=λ+λint\nαand the orbital\nmagnetic moments /angbracketleftLα/angbracketrightforα=x,y,zon the two sublattices. Bare SOC value λ=1.0.\ns λ xλyλz/angbracketleftLx/angbracketright /angbracketleftLy/angbracketright /angbracketleftLz/angbracketright\nA 1.179 1.179 1.364 0.032 −0.032 0.0\nB 1.179 1.179 1.364 −0.032 0.032 0.0\nas:\n[H(2)\neff](x,y)\nDM=8tt′\nx\nU/summationdisplay\n/angbracketlefti,j/angbracketrightxˆx.(Si,xz×Sj,xz)+8tt′\ny\nU/summationdisplay\n/angbracketlefti,j/angbracketrightyˆy.(Si,yz×Sj,yz)\n≈8t|t′\nx|\nU/summationdisplay\n/angbracketlefti,j/angbracketright(ˆx+ ˆy).(Si,yz×Sj,yz) (23)\nwhere we have taken t′\nx=−t′\ny=−ive andSx\ni,xz=Sx\ni,yz(due to Hund’s coupling) as earlier,\nbut withSz\ni,xz=−Sz\ni,yzfor then= 3 case as obtained in our self consistent calculation which\nis discussed below. The effective DM axis (ˆ x+ ˆy) is thus along the crystal baxis (Fig. 1)\nfor theyzorbital, resulting in easy-axis anisotropy along the crystal adirection, as well as\nspin canting about the DM axis in the zdirection.\nResults for various physical quantities are shown in Tables I and II. Starting with initial\norientation along the ˆ xor ˆydirections, the AFM order direction self consistently approaches\nthe easy-axis direction in a few hundred iterations, explicitly exhibitin g the strong easy-axis\nanisotropywithintheeasy( x−y)planeduetotheoctahedraltiltinginducedDMinteraction,17\n-3-2-1 0 1 2\n(0,0) ( π,0) (π,π) (0,0) (0, π) (π,0)yz xz xy Ek - EF (eV)\n 0 50 100 150 200 250 300ωq (meV)\n(π/2,π/2) (π,0) ( π,π) (π/2,π/2)\nFIG. 4: (a) Calculated orbital resolved electronic band str ucture in the self-consistent state with\nAFMorderalongthecrystal aaxisduetooctahedraltiltinginducedDMinteraction. Here tm2,m3=\n0.15. Colors indicate dominant orbital weight: red ( yz), green ( xz), blue ( xy). (b) Magnon\ndispersion for the magnetic order as given in Table I, showin g that both in-plane and out-of-plane\nmodes are appreciably gapped due to the easy-axis and easy-p lane anisotropies.\nalong with small spin canting in the zdirection about the DM axis. The small moment\ndisparitymyz,xz>mxyand the negligible orbital moments can also be seen here explicitly.\nThe renormalized SOC strength λzis enhanced relative to the other two components, which\nfurther reduces the SOC induced frustration in this system with no minally one electron in\neach of the three orbitals.\nWithoctahedral tilting included, theorbital resolved electronic ban dstructure intheself-\nconsistent AFM state (Fig. 4(a)) shows the AFM band gap between valence and conduction\nbands, SOC induced orbital mixing and band splittings, the fine splittin g due to octahedral\ntilting, andtheasymmetricbandwidthfor xyorbitalbandscharacteristicofthe2ndneighbor\nhopping term t2which connects the same magnetic sublattice. The calculated magno n\ndispersion evaluated using Eq. 22 is shown in Fig. 4(b). As expected, both in-plane and\nout-of-plane magnon modes are gapped due to the easy-axis and e asy-plane anisotropies\ndiscussed above.\nThe high energy part of the spectral function is shown in the series of panels in Fig. 5\nfor different SOC strengths. The two groups of modes here corre spond to: (i) the Hund’s\ncoupling induced gapped magnon modes for out-of-phase spin fluct uations (the two disper-\nsive modes starting at energies 0.7 and 0.8 eV from the left edge in pan el (a)), and (ii)18\n(π/2,π/2) (π,0) ( π,π) (0,0) 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4ω (eV)\n 0.01 0.1 1 10 100\nλ = 0(a)\n(π/2,π/2) (π,0) ( π,π) (0,0) 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4ω (eV)\n 0.01 0.1 1 10 100\nλ = 0.5(b)\n(π/2,π/2) (π,0) ( π,π) (0,0) 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4ω (eV)\n 0.01 0.1 1 10 100\nλ = 1.0(c)\n(π/2,π/2) (π,0) ( π,π) (0,0) 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4ω (eV)\n 0.01 0.1 1 10 100\nλ = εxy = 0 (d)\nFIG. 5: Gapped magnon modes and dominantly magnetic exciton modes for the n= 3 case seen in\nthe high-energy part of the spectral function calculated in the self consistent AFM state including\noctahedral tilting, for different SOC ( λ) values shown in the panels.\nthe spin-orbiton modes (starting at energy below 0.6 eV) which are in ter-orbital magnetic\nexcitons corresponding to the lowest-energy particle-hole excita tions across the AFM band\ngap involving yz/xzorbitals (particle) and xyorbital (hole) states (Fig. 4(a)). Through\nthe usual resonant scattering mechanism, these modes are pulled down in energy below the\ncontinuum by the U′′interaction term, and form well defined propagating modes.\nThe spin-orbiton mode involving xyorbital shifts to higher energy when ǫxydecreases to\nzero (panel (d)) which lowers the dominantly xyvalence band (Fig. 4(a)) and thus increases\nthe particle-hole excitation energy. The splitting of the exciton mod es in panel (c) is due to\nthe SOC induced splitting of electronic bands as seen in Fig. 4(a), whic h is then reflected\nin the particle-hole excitation energies. The combination of orbitals f or these exciton modes\nindicates that LxandLycomponents of the orbital angular momentum are involved in these\ncoupled spin-orbital fluctuations. There is an additional spin-orbit on mode involving only\nyz,xzorbitals (and Lzcomponent) which is formed at higher energy near 0.8 eV (flat band\nnear the left edge iin panel (a)). With increasing SOC, the high-ener gy modes involving19\nyz,xzorbitals acquire significant spin-orbit exciton character.\nVI.n= 5— APPLICATION TO Sr2IrO4\nThe perovskite structured 5 d5compound Sr 2IrO4exhibits an AFM insulating state due\nto strong SOC induced splitting of the t2gstates, with four electrons in the nominally\nfilled and non-magnetic J=3/2 sector and one electron in the nominally half filled and\nmagnetically active J=1/2 sector. The SOC induced splitting of 3 λ/2 between states of the\ntwo total angular momentum sectors, strong spin-orbital entan glement, and band narrowing\nof states in the J=1/2 sector, all of these play a crucial role in the stabilization of the AFM\ninsulator state. Both low-energy magnon excitations and high-ene rgy spin-orbit excitons\nacrosstherenormalizedspin-orbit gaphave beenintensively studie dusing RIXSexperiments\nand variety of theoretical approaches.43,46,47,51,53\nIn this case, we have taken realistic parameter values U= 3,JH=U/7, bare SOC\nvalueλ= 1.35, andǫxy=−0.5 for simplicity, along with hopping terms: ( t1,t2,t3,t4,\nt5,tm1)=(-1.0, 0.5, 0.25, -1.0, 0.0, 0.2), all in units of the realistic hopping en ergy scale\n|t1|=290 meV. The self consistently determined results for various phy sical quantities are\ngiven in Table III for magnetic order in the xdirection. All ordering directions within the\nx−yplane are nearly equivalent. Besides the dominant Hund’s coupling indu ced easy-plane\nanisotropy,53there is an extremely weak easy-axis anisotropy which will be discuss ed at the\nend of this section. The octahedral rotation induces small in-plane canting of spins but the\ncanting axis is free to orient in any direction. The strong Coulomb inte raction induced SOC\nrenormalization by nearly 2/3 (Table IV) agrees with the pseudo-or bital based approach.51\nTABLE III: Self consistently determined magnetization and density values, showing small spin\ncanting about the zaxis due to octahedral rotation induced DM interaction. Her etm1= 0.2.\nµ(s)mx\nµmy\nµmz\nµnµ\nyz(A) 0.186 −0.052 0 1.653\nxz(A)−0.185 0.049 0 1.654\nxy(A)−0.172−0.047 0 1.693µ(s)mx\nµmy\nµmz\nµnµ\nyz(B)−0.186−0.052 0 1.653\nxz(B) 0.185 0.049 0 1.654\nxy(B) 0.172 −0.047 0 1.69320\nTABLE IV: Self consistently determined renormalized SOC va luesλα=λ+λint\nαand the orbital\nmagnetic moments /angbracketleftLα/angbracketrightforα=x,y,zon the two sublattices. Bare SOC value λ=1.35.\ns λ xλyλz/angbracketleftLx/angbracketright /angbracketleftLy/angbracketright /angbracketleftLz/angbracketright\nA 1.882 1.882 1.871 0.367 0.091 0\nB 1.882 1.882 1.871 −0.367 0.091 0\nThe strong orbital moments and their correlation with the magnetic order direction (Table\nIV) reflect the strong SOC induced spin-orbital entanglement.\nThe low energy part of the spectral function (Fig. 6(a)) clearly sh ows the gapless and\ngapped modes corresponding to in-plane and out-of-plane fluctua tions, consistent with the\neasy-plane anisotropy. The magnon gap ≈45 meV is close to the result obtained using the\npseudo-orbital based approach,53and in agreement with recent experiments.46It should be\nnoted that along with the full generalized spin sector, the orbital o ff-diagonal charge sector\n(ψ†\nµ1ψν)relatedtotheorbitalmomentoperators Lx,y,zwasincludedintheabovecalculations\nwhich allows for the accompanying transverse fluctuations of orbit al moments. Indeed, the\nexactly gapless Goldstone mode seen in Fig. 6(a) is obtained only if the ψ†\nµ1ψνsector is\nincluded, indicating the coupled spin-orbital nature of the Goldston e mode, as illustrated in\n(π/2,π/2) (π,0) ( π,π) (0,0) 0 50 100 150 200 250 300ω (meV)\n 0.01 0.1 1 10 100\n(a)\n(π/2,π/2) (π,0) ( π,π) (0,0) 300 400 500 600 700 800ω (meV)\n 0.01 0.1 1 10 100\n(b)\nFIG. 6: The spectral function in the self-consistent state f or then= 5 case with planar AFM\norder including octahedral rotation, showing the (a) gaple ss and gapped modes corresponding to\nin-plane and out-of-plane fluctuations and (b) the spin-orb it exciton modes near 500 meV and 300\nmeV in the high-energy part.21\n(π/2,π/2) (π,0) ( π,π) (0,0) 0 200 400 600 800 1000 1200ω (meV)\n 0.01 0.1 1 10 100\n(a)JH = 0\n(π/2,π/2) (π,0) ( π,π) (0,0) 0 0.4 0.8 1.2 1.6 2 2.4ω (eV)\n 0.01 0.1 1 10 100\n(b)\nλ = 5, JH = U/7\n(π/2,π/2) (π,0) ( π,π) (0,0) 1.9 2 2.1 2.2 2.3ω (eV)\n 0.01 0.1 1 10 100\n(c)\nλ = 5, JH = U/7\nFIG. 7: The spin-orbit exciton modes ( n= 5) for special cases showing (a) no splitting in the weak\nbranch (∼400 meV) for Hund’s coupling JH= 0, (b) disappearance of the weak branch for large\nSOC value λ= 5, and (c) expanded view of the multiple exciton modes ( ∼2 eV) in case (b). Here\noctahedral rotation is included in all three cases.\nAppendix C showing the detailed spin-orbital composition.\nFig. 6(b) shows the spin-orbit exciton modes ( ∼500 meV) involving particle-hole excita-\ntions between the J=1/2 and 3/2 sectors, which matches closely with results obtained u sing\nthe pseudo-orbital based approach.51As discussed in the previous ( n= 3) case, collective\nmodes arise from particle-hole excitations which are converted to w ell defined propagating\nmodes split off from the continuum by the Coulomb interaction induced resonant scattering\nmechanism. The significantly weaker modes ( ∼300 meV) just below the particle-hole con-\ntinuum for the nominally J= 1/2 sector are also spin-orbit exciton modes. The splitting\nseen beyond ( π,0) vanishes for JH= 0, as seen in Fig. 7(a). The weak intensity corresponds\nto the small J= 3/2 character (mainly mJ=±3/2) in the nominally J= 1/2 bands due\nto strong mixing between the two sectors induced by the band (hop ping) terms. For large\nSOC strength λ, the weak exciton modes disappear (Fig. 7(b)), confirming the abo ve pic-\nture. Thus, the (low) intensity of the weak exciton modes provides a direct measure of the\nmixing between the J= 1/2 and 3/2 sectors. The fine splitting of exciton bands in Fig.\n7(c) corresponds to four possible mJvalues (±3/2,±1/2) for the hole in the J= 3/2 sector\nand the exciton hopping terms connecting the two sublattices.\nWe now discuss the extremely weak easy-axis anisotropy which leads to preferred isospin\n(J= 1/2) orientation along the diagonal directions ±(ˆx±ˆy) within the easy plane. Fig.\n8(a) shows the small magnon gap ( ≈3 meV) for the in-plane magnon mode induced by the\nHund’s coupling JHdue to the extremely weak spin twisting as shown in Fig. 8(b) which\nresults in an easy-axis anisotropy with C4symmetry. Here the parameter set is same as22\n 0 2 4 6 8 10 12 14\n 0 0.02  0.04  0.06  0.08in-plane mode(a)ωq (meV)\nqx = qy\n(b)\nisospin\nFIG. 8: (a) Magnon energies for isospin order along ˆ x+ˆydirection showing the small magnon gap\n(≈3 meV) for the in-plane fluctuation mode. (b) The isospin and yz,xz,xy moment orientations\nfortheideal spin-orbital entangled state, whichis extrem ely weakly perturbedbyfinite JHresulting\nin slight twisting of the yz,xzmoments as indicated, leading to the easy-axis anisotropy w ithC4\nsymmetry. The isospin easy axes are along φ=nπ/4 wheren= 1,3,5,7.\nearlier including the octahedral rotation which only weakly enhances the magnon gap. The\nabove weak perturbative effect of JHonthestrongly spin-orbital entangled statecorresponds\nto the opposite end of the competition between SOC and JHas compared to the n= 3 case\ndiscussed in Sec. V.\nVII.n= 4— APPLICATION TO Ca2RuO4\nFor moderate tetragonal distortion ( ǫxy≈ −1), thexyorbital in the 4 d4compound\nCa2RuO4is nominally doubly occupied and magnetically inactive, while the nominally h alf-\nfilledandmagnetically active yz,xzorbitalsyieldaneffectively two-orbitalmagneticsystem.\nHund’s coupling between the two S= 1/2 spins results in low-lying (in-phase) and apprecia-\nbly gapped (out-of-phase) spin fluctuation modes. The in-phase m odes of the yz,xzorbital\nS= 1/2 spins correspond to an effective S= 1 spin system. However, the rich interplay be-\ntween SOC, Coulomb interaction, octahedral rotations, and tetr agonal distortion results in\ncomplex magnetic behaviour which crucially involves the xyorbital and is therefore beyond\nthe above simplistic picture.\nTreating all the different physical elements on the same footing with in the unified frame-\nwork of the generalized self-consistent approach explicitly shows t he variety of physical23\n(π/2,π/2) (π,0) ( π,π) (0,0) 0 20 40 60 80 100 120 140 160 180ω (meV)\n 0.01 0.1 1 10 100\n(a)\n(π/2,π/2) (π,0) ( π,π) (0,0) 250 300 350 400 450 500ω (meV)\n 0.01 0.1 1 10 100\n(b)\nFIG. 9: The generalized fluctuation spectral function for th en= 4 case, showing coupled spin-\norbital excitations including low-energy magnon modes (be low∼60 meV), intermediate-energy\norbiton (100 and 140 meV) and spin-orbiton (300 and 350 meV) m odes, and high-energy spin-\norbit exciton (425 meV) modes.\neffects arising from the rich interplay in Ca 2RuO4. These include: SOC induced easy-plane\nand easy-axis anisotropies similar to the n=3 case, octahedral tilting induced reduction of\neasy-axis anisotropyfrom C4toC2symmetry, spin-orbital coupling induced orbital magnetic\nmoments, Coulomb interaction induced strongly anisotropic SOC ren ormalization, decreas-\ning tetragonal distortion induced magnetic reorientation transitio n from planar AFM order\nto FM (z) order, and orbital moment interaction induced orbital gap.52Stable FM and\nAFM metallic states were also obtained near the magnetic phase boun dary separating the\ntwo magnetic orders. The self-consistent determination of magne tic order has also explicitly\nshown the coupled nature of spin and orbital fluctuations, as refle cted in the ferro and an-\ntiferro orbital fluctuations associated with in-phase and out-of- phase spin twisting modes,\nhighlighting the strong deviation from conventional Heisenberg beh aviour in effective spin\nmodels, as discussed recently to account for the magnetic excitat ion measurements in INS\nexperiments on Ca 2RuO4.35\nIn the following, we will take the same parameter set as considered in the self-consistent\nstudy,52along with U= 8 andJH=U/5 in the energy scale unit (150 meV), so that\nU= 1.2 eV,U′′=U/2 = 0.6 eV, andJH= 0.24 eV. These are comparable to reported\nvalues extracted from RIXS ( JH= 0.34 eV) and ARPES ( JH= 0.4 eV) studies.24,40The\nhopping parameter values considered are as given in Sec. II, and th e bare SOC value λ= 1.24\nFig. 9 shows the calculated generalized fluctuation spectral funct ion. Several well de-\nfined propagating modes are seen here including: (i) the low-energy (below∼60 meV)\ndominantly spin (magnon) excitations involving the magnetically active yz,xzorbitals and\ncorresponding to in-plane and out-of-plane fluctuations which are gapped due to the mag-\nnetic anisotropies, (ii) the intermediate-energy (100 and 140 meV) dominantly orbital exci-\ntations (orbitons) involving particle-hole excitations between xy(hole) andyz,xz(particle)\nstates, (iii) the intermediate-energy (300 and 350 meV) dominantly spin-orbital excitations\n(spin-orbitons) involving xy(hole) and yz,xz(particle) states, and (iv) the high-energy\n(425 meV) dominantly spin-orbital excitations (spin-orbit excitons ) involving particle-hole\nexcitations between yz,xz(hole) and yz,xz(particle) states of nominally different Jsec-\ntors. The SOC-induced spin-orbital entangled Jstates are strongly renormalized by the\ntetragonal splitting and the electronic correlation induced stagge red field.\nThe spin-orbital characterization of the various collective excitat ions mentioned above is\ninferred from the basis-resolved contributions to the total spec tral functions which explicitly\nshow the relative spin-orbital composition of the various excitation s (Appendix C). The\npresence of sharply defined collective excitations for the magnon, orbiton, and spin-orbiton\nmodes which are clearly separated from the particle-hole continuum highlights the rich spin-\norbital physics in the n= 4 case corresponding to the Ca 2RuO4compound. Many of our\ncalculated magnon spectra features such as the magnon gaps for in-plane and out-of-plane\nmodes, weak dispersive nature along the magnetic zone boundary, as well as the overall\nmagnon energy scale are in excellent agreement with the INS study.34,35The orbiton mode\nenergy scale is also qualitatively comparable to the composite excitat ion peaks obtained\naround 80 meV in Raman and RIXS studies.37–40The calculated spin-orbiton and spin-orbit\nexciton energies are also in agreement with the excitation peaks obt ained around 300-350\nmeV energy range and 400 meV in RIXS studies. We also obtained excit ations in the high-\nenergy range 750-800 meV and 900 meV (not shown), which are com parable to the peaks\nobtained around 750 meV and 1000 meV in RIXS studies.25\nVIII. CONCLUSIONS\nFollowing up on the generalized self-consistent approach including or bital off-diagonal\nspin and charge condensates, investigation of the generalized fluc tuation propagator reveals\nthecomposite spin-orbital character ofthe different types ofco llective excitations instrongly\nspin-orbit coupled systems. A realistic representation of magnetic anisotropy effects due to\nthe interplay of SOC, Coulomb interaction, and structural distort ion terms was included\nin the three-orbital model, while maintaining uniformity of lattice stru cture in order to\nfocus on the coupled spin-orbital excitations. Our unified investiga tion of the three electron\nfilling cases n= 3,4,5 corresponding to the three compounds NaOsO 3, Ca2RuO4, Sr2IrO4\nprovides deep insight into how the spin-orbital physics in the magnet ic ground state is\nreflected in the collective excitations. The calculated spectral fun ctions show well defined\npropagating modes corresponding to dominantly spin (magnon), or bital (orbiton), and spin-\norbital (spin-orbiton) excitations, along with the spin-orbit excito n modes involving spin-\norbital excitations between states of different Jsectors induced by the spin-orbit coupling.\nAppendix A: Orbital off-diagonal condensates in the HF approximation\nTheadditionalcontributionsintheHFapproximationarisingfromthe orbitaloff-diagonal\nspin and charge condensates are given below. For the density, Hun d’s coupling, and pair\nhopping interaction terms in Eq. 3, we obtain (for site i):\nU′′/summationdisplay\nµ<νnµnν→ −U′′\n2/summationdisplay\nµ<ν[nµν/angbracketleftnνµ/angbracketright+σµν./angbracketleftσνµ/angbracketright]+H.c.\n−2JH/summationdisplay\nµ<νSµ.Sν→JH\n4/summationdisplay\nµ<ν[3nµν/angbracketleftnνµ/angbracketright−σµν./angbracketleftσνµ/angbracketright]+H.c.\nJP/summationdisplay\nµ/negationslash=νa†\nµ↑a†\nµ↓aν↓aν↑→JP\n2/summationdisplay\nµ<ν[nµν/angbracketleftnµν/angbracketright−σµν./angbracketleftσµν/angbracketright]+H.c. (A1)\nin terms of the orbital off-diagonal spin ( σµν=ψ†\nµσψν) and charge ( nµν=ψ†\nµ1ψν) oper-\nators. The orbital off-diagonal condensates are finite due to the SOC-induced spin-orbital\ncorrelations. These additional terms in the HF theory explicitly pres erve the SU(2) spin\nrotation symmetry of the various Coulomb interaction terms.\nCollecting all the spin and charge terms together, we obtain the orb ital off-diagonal26\n(OOD) contributions of the Coulomb interaction terms:\n[HHF\nint]OOD=/summationdisplay\nµ<ν/bracketleftbigg/parenleftbigg\n−U′′\n2+3JH\n4/parenrightbigg\nnµν/angbracketleftnνµ/angbracketright+/parenleftbiggJP\n2/parenrightbigg\nnµν/angbracketleftnµν/angbracketright\n−/parenleftbiggU′′\n2+JH\n4/parenrightbigg\nσµν./angbracketleftσνµ/angbracketright−/parenleftbiggJP\n2/parenrightbigg\nσµν./angbracketleftσµν/angbracketright/bracketrightbigg\n+H.c. (A2)\nAppendix B: Coulomb interaction matrix elements in the orbital-pair basis\nCorrespondingtotheaboveHFcontributionsintheorbitaloff-diag onalsector, weexpress\nthe Coulomb interactions in terms of the generalized spin and charge operators (for site i):\n[Hint]OOD=/summationdisplay\nµ<ν/bracketleftbigg/parenleftbigg\n−U′′\n2+3JH\n4/parenrightbigg\nnµνn†\nµν−/parenleftbiggU′′\n2+JH\n4/parenrightbigg\nσµν.σ†\nµν/bracketrightbigg\n+/summationdisplay\nµ<ν/bracketleftbigg/parenleftbiggJP\n4/parenrightbigg\nnµνn†\nνµ−/parenleftbiggJP\n4/parenrightbigg\nσµν.σ†\nνµ+H.c./bracketrightbigg\n(B1)\nwheren†\nµν=nνµandσ†\nµν=σνµ. The above form shows that only the pair-hopping\ninteraction terms ( JP) are off-diagonal in the orbital-pair ( µν) basis. We will use the above\nCoulomb interaction terms in the orbital off-diagonal sector in the R PA series in order to\nensure consistency with the self-consistent determination of mag netic order including the\norbital off-diagonal condensates.\nThe Coulomb interaction terms in the orbital diagonal sector can be cast in a similar\nform:\n[Hint]OD=/summationdisplay\nµ/bracketleftbigg/parenleftbigg\n−U\n4/parenrightbigg\nσµ.σµ+/parenleftbiggU\n4/parenrightbigg\nnµnµ/bracketrightbigg\n+/summationdisplay\nµ<ν/bracketleftbigg/parenleftbigg\n−2JH\n4/parenrightbigg\nσµ.σν+U′′nµnν/bracketrightbigg\n(B2)\nwhich include the Hubbard, Hund’s coupling, and density interaction t erms.\nThe form of the [ U] matrix used in the RPA series Eq. (17) is now discussed below. In\nthe composite spin-charge-orbital-sublattice ( µναs) basis, the [ U] matrix is diagonal in spin,\ncharge, and sublattice sectors. There are two possible cases invo lving the orbital-pair ( µν)\nbasis. In the case µ=ν, the [U] matrices in the spin ( α=x,y,z) and charge ( α=c) sectors\nare obtained as:\n[U]µ′µ′α′=α\nµµα=x,y,z=\nU JHJH\nJHU JH\nJHJHU\n[U]µ′µ′α′=α\nµµα=c=\n−U−2U′′−2U′′\n−2U′′−U−2U′′\n−2U′′−2U′′−U\n(B3)27\nFIG. 10: The basis-resolved contributions to the total spec tral function for the low-energy magnon\n(left panel) and intermediate-energy orbiton (center and r ight panels) modes, showing dominantly\nspin (µ=ν,α=x,y,z) and orbital ( µ/negationslash=ν,α=c) character of the fluctuation modes, respectively.\ncorresponding to the interaction terms (Eq. B2) for the normal s pin and charge density\noperators. Similarly, for the six orbital-pair cases ( µ,ν) corresponding to µ/negationslash=ν, the [U]\nmatrix elements in the spin ( α=x,y,z) and charge ( α=c) sectors are obtained as:\n[U]µνα\nµνα=x,y,z=U′′+JH/2 [U]µνα\nµνα=c=U′′−3JH/2\n[U]νµα\nµνα=x,y,z=JP [U]νµα\nµνα=c=−JP (B4)\ncorresponding to the interaction terms (Eq. B1) involving the orbit al off-diagonal spin and\ncharge operators.\nAppendix C: Basis-resolved contributions to the total spectral function\nThe detailed spin-orbital character of the collective excitations ca n be identified from the\nbasis-resolved contributions to the total spectral functions. T his is illustrated here for the28\nFIG. 11: The basis-resolved contributions to the total spec tral function for the intermediate-\nenergy spin-orbiton (left and center panels) and high-ener gy spin-orbit exciton (right panel) modes,\nshowing dominantly spin-orbital character ( µ/negationslash=ν,α=x,y,z) involving xyandyz,xzorbitals (left\nand center panels) and yz,xzorbitals (right panel).\nexcitations shown in Fig. 9 for the n= 4 case corresponding to the Ca 2RuO4compound.\nFig. 10 shows dominantly spin excitations involving yz,xzorbitals for the magnon modes\n(below 60 meV) and dominantly orbital excitations involving xyandyz,xzorbitals for\nthe orbiton modes (100 and 140 meV). Similarly, Fig. 11 shows dominan tly spin-orbital\nexcitations involving xyandyz,xzorbitals for the spin-orbiton modes (300 and 350 meV),\nand dominantly spin-orbital excitations involving yz,xzorbitals for the spin-orbit exciton\nmodes (425 meV).29\nFIG. 12: The extreme spin-orbital-entanglement induced co rrespondence between (a) magnetic\nordering directions, (b) sign of magnetic moments for the th ree orbitals, and (c) orbital current\ninduced orbital moments for the three orbitals, for the n= 5 case corresponding to Sr 2IrO4.\n 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5\nx y z cω = 0 meV\n(a)π-1Im[χ(q,ω)]µναµναµν=yz yz\nxz xz\nxy xy\nyz xz\nxz xyxy yz\nxz yz\nxy xz\nyz xyq =(0,0)\n 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5\nx y z cω = 46 meV\n(b)π-1Im[χ(q,ω)]µναµναq =(0,0)\nFIG. 13: The basis-resolved contributions to the total spec tral function for the (a) gapless in-plane\nmagnon mode and (b) gapped out-of-plane magnon mode for the n= 5 case corresponding to\nSr2IrO4with extreme spin-orbital entanglement.\nSimilarly, for the n= 5 case corresponding to Sr 2IrO4, the detailed spin-orbital character\nof the Goldstone mode and gapped mode at q= (0,0) seen in Fig. 6 is shown in Fig.\n13, explicitly illustrating the effect of extreme spin-orbital entangle ment and the resulting\ncorrespondence (Fig. 12) between magnetic ordering directions, spin moments, and orbital\nmoments.30\n∗Electronic address: avinas@iitk.ac.in\n1W. Witczak-Krempa, G. Chen, Y. B. Kim, and L. Balents, Correlated Quantum Phenomena in\nthe Strong Spin-Orbit Regime , Annu. Rev. Condens. Matter Phys. 5, 57 (2014).\n2J. G. Rau, E. Kin-Ho Lee, and H.-Y. Kee, Spin-Orbit Physics Giving Rise to Novel Phases in\nCorrelated Systems: Iridates and Related Materials , Annu. Rev. Condens. Matter Phys. 7, 195\n(2016).\n3G. Cao and P. Schlottmann, The Challenge of Spin-Orbit-Tuned Ground States in Iridates: A\nKey Issues Review , Rep. Prog. Phys. 81, 042502 (2018).\n4G. Zhang and E. Pavarini, Spin-Orbit and Coulomb Effects in Single-Layered Ruthenates , Phys.\nStatus Solidi RRL 12, 1800211 (2018).\n5J. Bertinshaw, Y. K. Kim, G. Khaliullin, and B. J. Kim, Square Lattice Iridates , Annu. Rev.\nCondens. Matter Phys. 10, 315 (2019).\n6G. Zhang and E. Pavarini, Higgs Mode and Stability of xy-orbital Ordering in Ca2RuO4, Phys.\nRev. B101, 205128 (2020).\n7Y. G. Shi, Y. F. Guo, S. Yu, M. Arai, A. A. Belik, A. Sato, K. Yama ura, E. Takayama-\nMuromachi, H. F. Tian, H. X. Yang, J. Q. Li, T. Varga, J. F. Mitc hell, and S. Okamoto,\nContinuous Metal-Insulator Transition of the Antiferromagn etic Perovskite NaOsO 3, Phys.Rev.\nB80, 161104(R) (2009).\n8S. Calder, V. O. Garlea, D. F. McMorrow, M. D. Lumsden, M. B. St one, J. C. Lang, J.-W.\nKim, J. A. Schlueter, Y. G. Shi, K. Yamaura, Y. S. Sun, Y. Tsuji moto, and A. D. Christian-\nson,Magnetically Driven Metal-Insulator Transition in NaOsO 3, Phys. Rev. Lett. 108, 257209\n(2012).\n9Y. Du, X. Wan, L. Sheng, J. Dong, and S. Y. Savrasov, Electronic Structure and Magnetic\nProperties of NaOsO 3, Phys. Rev. B 85, 174424 (2012).\n10M.-C. Jung, Y.-J. Song, K.-W. Lee, and W. E. Pickett, Structural and Correlation effects in the\nItinerant Insulating Antiferromagnetic Perovskite NaOsO 3, Phys. Rev. B 87, 115119 (2013).\n11I. Lo Vecchio, A. Perucchi, P. Di Pietro, O. Limaj, U. Schade, Y. Sun, M. Arai, K. Yamaura,\nand S. Lupi, Infrared evidence of a Slater Metal-Insulator Transition in NaOsO 3, Sci. Rep. 3,\n2990 (2013).31\n12B. Kim, P. Liu, Z. Erg¨ onenc, A. Toschi, S. Khmelevskyi, and C . Franchini, Lifshitz Transition\nDriven by Spin Fluctuations and Spin-Orbit Renormalizatio n inNaOsO 3, Phys. Rev. B 94,\n241113(R) (2016).\n13J.G. Vale, S.Calder, C.Donnerer, D. Pincini, Y. G. Shi, Y. Ts ujimoto, K. Yamaura, M. Moretti\nSala, J. van den Brink, A. D. Christianson, and D. F. McMorrow ,Evolution of the Magnetic\nExcitations in NaOsO 3through its Metal-Insulator Transition , Phys. Rev. Lett. 120, 227203\n(2018).\n14S. Mohapatra, C. Bhandari, S. Satpathy, and A. Singh, Effect of Structural Distortion on the\nElectronic Band Structure of NaOsO 3studied within Density Functional Theory and a Three-\nOrbital Model , Phys. Rev. B 97, 155154 (2018).\n15A. Singh, S. Mohapatra, C. Bhandari, and S. Satpathy, Spin-Orbit Coupling Induced Magnetic\nAnisotropy and Large Spin Wave Gap in NaOsO 3, J. Phys. Commun. 2, 115016 (2018).\n16S. Calder, J. G. Vale, N. Bogdanov, C. Donnerer, D. Pincini, M . Moretti Sala, X. Liu, M. H.\nUpton, D. Casa, Y. G. Shi, Y. Tsujimoto, K. Yamaura, J. P. Hill , J. van den Brink, D. F.\nMcMorrow, and A. D. Christianson, Strongly Gapped Spin-Wave Excitation in the Insulating\nPhase of NaOsO 3, Phys. Rev. B 95, 020413(R) (2017).\n17J.G. Vale, S.Calder, C.Donnerer, D. Pincini, Y. G. Shi, Y. Ts ujimoto, K. Yamaura, M. Moretti\nSala, J. van den Brink, A. D. Christianson, and D. F. McMorrow ,Crossover from Itinerant to\nLocalized Magnetic Excitations Through the Metal-Insulator Transition in NaOsO 3, Phys. Rev.\nB97, 184429 (2018).\n18S. Nakatsuji, S.-i. Ikeda, and Y. Maeno, Ca 2RuO4: New Mott Insulators of Layered Ruthenate ,\nJ. Phys. Soc. Jpn 66(7), 1868 (1997).\n19M.Braden, G.Andr´ e, S.Nakatsuji, andY. Maeno, Crystal and Magnetic Structure of Ca2RuO4:\nMagnetoelastic Coupling and the Metal-Insulator Transition , Phys. Rev. B 58, 847 (1998).\n20C. S. Alexander, G. Cao, V. Dobrosavljevic, S. McCall, J. E. C row, E. Lochner, and R. P.\nGuertin, Destruction of the Mott Insulating Ground State of Ca2RuO4by a Structural Transi-\ntion, Phys. Rev. B 60, R8422 (1999).\n21O. Friedt, M. Braden, G. Andr´ e, P. Adelmann, S. Nakatsuji, a nd Y. Maeno, Structural and\nMagnetic Aspects of the Metal-Insulator Transition in Ca2−xSrxRuO4, Phys. Rev. B 63, 174432\n(2001).\n22E. Gorelov, M. Karolak, T. O. Wehling, F. Lechermann, A. I. Li chtenstein, and E. Pavarini,32\nNature of the Mott Transition in Ca2RuO4, Phys. Rev. Lett. 104, 226401 (2010).\n23G. Zhang and E. Pavarini, Mott transition, Spin-Orbit Effects, and Magnetism in Ca2RuO4,\nPhys. Rev. B 95, 075145 (2017).\n24D. Sutter, C. G. Fatuzzo, S. Moser, M. Kim, R. Fittipaldi, A. V ecchione, V. Granata, Y. Sassa,\nF. Cossalter, G. Gatti, M. Grioni, H. M. Rønnow, N. C. Plumb, C . E. Matt, M. Shi, M. Hoesch,\nT. K. Kim, T.-R. Chang, H.-T. Jeng, C. Jozwiak, A. Bostwick, E . Rotenberg, A. Georges, T.\nNeupert, and J. Chang, Hallmarks of Hund’s Coupling in the Mott Insulator Ca2RuO4, Nat.\nComm.8, 15176 (2017).\n25F. Nakamura, T. Goko, M. Ito, T. Fujita, S. Nakatsuji, H. Fuka zawa, Y. Maeno, P. Alireza, D.\nForsythe, and S. R. Julian, From Mott Insulator to Ferromagnetic Metal: A Pressure Study of\nCa2RuO4, Phys. Rev. B 65, 220402(R) (2002).\n26P. Steffens, O. Friedt, P. Alireza, W. G. Marshall, W. Schmidt, F. Nakamura, S. Nakatsuji, Y.\nMaeno, R. Lengsdorf, M. M. Abd-Elmeguid, and M. Braden, High-pressure Diffraction Studies\nonCa2RuO4, Phys. Rev. B 72, 094104 (2005).\n27H. Taniguchi, K. Nishimura, R. Ishikawa, S. Yonezawa, S. K. G oh, F. Nakamura, and Y.\nMaeno,Anisotropic Uniaxial Pressure Response of the Mott Insulato rCa2RuO4, Phys. Rev. B\n88, 205111 (2013).\n28S. Nakatsuji and Y. Maeno, Quasi-Two-Dimensional Mott Transition System Ca2−xSrxRuO4,\nPhys. Rev. Lett. 84, 2666 (2000).\n29Z. Fang and K. Terakura. Magnetic Phase Diagram of Ca2−xSrxRuO4Governed by Structural\nDistortions , Phys. Rev. B 64, 020509 (2001).\n30P. Steffens, O. Friedt, Y. Sidis, P. Link, J. Kulda, K. Schmalzl , S. Nakatsuji, and M. Braden,\nMagnetic Excitations in the Metallic Single-Layer Ruthena tesCa2−xSrxRuO4Studied by Inelas-\ntic Neutron Scattering , Phys. Rev. B 83, 054429 (2011).\n31C. Dietl, S. K. Sinha, G. Christiani, Y. Khaydukov, T. Keller , D. Putzky, S. Ibrahimkutty, P.\nWochner, G. Logvenov, P. A. van Aken, B. J. Kim, and B. Keimer, Tailoring the Electronic\nProperties of Ca2RuO4via Epitaxial Strain , Appl. Phys. Lett. 112, 031902 (2018).\n32F. Nakamura, M. Sakaki, Y. Yamanaka, S. Tamaru, T. Suzuki, an d Y. Maeno, Electric-Field-\nInduced Metal Maintained by Current of the Mott Insulator Ca2RuO4, Sci. Rep. 3, 2536 (2013).\n33R. Okazaki, Y. Nishina, Y. Yasui, F. Nakamura, T. Suzuki, and I. Terasaki, Current-Induced\nGap Suppression in the Mott Insulator Ca2RuO4, J. Phys. Soc. Jpn. 82, 103702 (2013).33\n34S. Kunkem¨ oller, D. Khomskii, P. Steffens, A. Piovano, A. A. Nu groho, and M. Braden, Highly\nAnisotropic Magnon Dispersion in Ca2RuO4: Evidence for Strong Spin-Orbit Coupling , Phys.\nRev. Lett. 115, 247201 (2015).\n35A. Jain, M. Krautloher, J. Porras, G. H. Ryu, D. P. Chen, D. L. A bernathy, J. T. Park, A.\nIvanov, J. Chaloupka, G. Khaliullin, B. Keimer, and B. J. Kim ,Higgs Mode and Its Decay in a\nTwo-Dimensional Antiferromagnet , Nat. Phys. 13, 633 (2017).\n36S. Kunkem¨ oller, E. Komleva, S. V. Streltsov, S. Hoffmann, D. I . Khomskii, P. Steffens, Y. Sidis,\nK. Schmalzl, and M. Braden, Magnon dispersion in Ca2Ru1−xTixO4: Impact of Spin-Orbit\nCoupling and Oxygen Moments , Phys. Rev. B 95, 214408 (2017).\n37S.-M. Souliou, J. Chaloupka, G. Khaliullin, G. Ryu, A. Jain, B. J. Kim, M. Le Tacon, and B.\nKeimer, Raman Scattering from Higgs Mode Oscillations in the Two-Dim ensional Antiferro-\nmagnetCa2RuO4, Phys. Rev. Lett. 119, 067201 (2017).\n38C. G. Fatuzzo, M. Dantz, S. Fatale, P. Olalde-Velasco, N. E. S haik, B. Dalla Piazza, S. Toth,\nJ. Pelliciari, R. Fittipaldi, A. Vecchione, N. Kikugawa, J. S. Brooks, H. M. Rønnow, M. Grioni,\nCh. R¨ uegg, T. Schmitt, and J. Chang, Spin-Orbit-Induced Orbital Excitations in Sr2RuO4and\nCa2RuO4: A resonant Inelastic X-Ray Scattering Study , Phys. Rev. B 91, 155104 (2015).\n39L. Das, F. Forte, R. Fittipaldi, C. G. Fatuzzo, V. Granata, O. Ivashko, M. Horio, F. Schindler,\nM. Dantz, Yi Tseng, D. E. McNally, H. M. Rønnow, W. Wan, N. B. Ch ristensen, J. Pelliciari, P.\nOlalde-Velasco, N. Kikugawa, T. Neupert, A. Vecchione, T. S chmitt, M. Cuoco, and J. Chang,\nSpin-Orbital Excitations in Ca2RuO4Revealed by Resonant Inelastic X-Ray Scattering , Phys.\nRev. X8, 011048 (2018).\n40H. Gretarsson, H. Suzuki, Hoon Kim, K. Ueda, M. Krautloher, B . J. Kim, H. Yava¸ s, G. Khali-\nullin, and B. Keimer, Observation of Spin-Orbit Excitations and Hund’s Multiple ts inCa2RuO4,\nPhys. Rev. B 100, 045123 (2019).\n41B. J. Kim, H. Jin, S. J. Moon, J.-Y. Kim, B.-G. Park, C. S. Leem, J. Yu, T. W. Noh, C.\nKim, S.-J. Oh, J.-H. Park, V. Durairaj, G. Cao and E. Rotenber g,NovelJeff= 1/2Mott State\nInduced by Relativistic Spin-Orbit Coupling in Sr2IrO4, Phys. Rev. Lett. 101, 076402 (2008).\n42B.J.Kim, H.Ohsumi,T.Komesu, S.Sakai, T.Morita, H. Takagi andT.Arima, Phase-Sensitive\nObservation of a Spin-Orbital Mott State in Sr2IrO4, Science 323, 1329–1332 (2009).\n43J. Kim, D. Casa, M. H. Upton, T. Gog, Y.-J. Kim, J. F. Mitchell, M. van Veenendaal, M.\nDaghofer, J.vandenBrink, G.Khaliullin, andB.J.Kim, Magnetic Excitation Spectra of Sr2IrO434\nProbed by Resonant Inelastic X-Ray Scattering: Establishin g Links to Cuprate Superconductors ,\nPhys. Rev. Lett. 108, 177003 (2012).\n44J-i Igarashi and T. Nagao, Interplay between Hund’s Coupling and Spin-Orbit Interaction on\nElementary Excitations in Sr2IrO4, J. Phys. Soc. Jpn. 83, 053709 (2014).\n45D. Pincini, J. G. Vale, C. Donnerer, A. de la Torre, E. C. Hunte r, R. Perry, M. Moretti Sala,\nF. Baumberger, and D. F. McMorrow, Anisotropic Exchange and Spin-Wave Damping in Pure\nand Electron-Doped Sr2IrO4, Phys. Rev. B 96, 075162 (2017).\n46J. Porras, J. Bertinshaw, H. Liu, G. Khaliullin, N. H. Sung, J .-W. Kim, S. Francoual, P.\nSteffens, G. Deng, M. Moretti Sala, A. Effimenko, A. Said, D. Casa , X. Huang, T. Gog, J.\nKim, B. Keimer, and B. J. Kim, Pseudospin-Lattice Coupling in the Spin-Orbit Mott Insulato r\nSr2IrO4, Phys. Rev. B 99, 085125 (2019).\n47S. Mohapatra, J. van den Brink, and A. Singh, Magnetic Excitations in a Three-Orbital Model\nfor the Strongly Spin-Orbit Coupled Iridates: Effect of Mixing Between the J= 1/2and3/2\nSectors, Phys. Rev. B 95, 094435 (2017).\n48J. Kim, M. Daghofer, A. H. Said, T. Gog, J. van den Brink, G. Kha liullin, and B. J. Kim,\nExcitonic Quasiparticles in a Spin-Orbit Mott Insulator , Nat. Commun. 5, 4453 (2014).\n49O. Krupin, G. L. Dakovski, B. J. Kim, J. Kim, S. Mishra, Yi-De C huang, C. R. Serrao, W.-S.\nLee, W. F. Schlotter, M. P. Minitti, D. Zhu, D. Fritz, M. Choll et, R. Ramesh, S. L. Molodtsov,\nand J. J. Turner, Ultrafast Dynamics of Localized Magnetic Moments in the Unc onventional\nMott Insulator Sr2IrO4, J. Phys.: Condens. Matter 28, 32LT01 (2016).\n50M. Souri, B. H. Kim, J. H. Gruenewald, J. G. Connell, J. Thomps on, J. Nichols, J. Terzic, B. I.\nMin, G. Cao, J. W. Brill, and A. Seo, Optical Signatures of Spin-Orbit Exciton in Bandwidth-\nControlled Sr2IrO4Epitaxial Films via High-Concentration Ca and Ba Doping , Phys. Rev. B\n95, 235125 (2017).\n51S. Mohapatra and A. Singh, Correlated Motion of Particle-Hole Excitations Across the R enor-\nmalized Spin-Orbit Gap in Sr2IrO4, J. Magn. Magn. Mater 512, 166997 (2020).\n52S. Mohapatra and A. Singh, Magnetic Reorientation Transition in a Three Orbital Model f or\nCa2RuO4— Interplay of Spin-Orbit Coupling, Tetragonal Distortion, a nd Coulomb Interac-\ntions, J. Phys.: Condens. Matter 32, 485805 (2020).\n53S. Mohapatra and A. Singh, Pseudo-Spin Rotation Symmetry Breaking by Coulomb Interacti on\nterms in Spin-Orbit Coupled Systems , J. Phys.: Condens. Matter 33, 065802 (2021).35\n54G. Khaliullin, Excitonic Magnetism in Van Vleck–type d4Mott Insulators , Phys. Rev. Lett.\n111, 197201 (2013).\n55A. Akbari and G. Khaliullin, Magnetic Excitations in a Spin-Orbit-Coupled d4Mott Insulator\non the Square Lattice , Phys. Rev. B 90, 035137 (2014).\n56T. Feldmaier, P. Strobel, M. Schmid, P. Hansmann, and M. Dagh ofer,Excitonic Magnetism\nat the Intersection of Spin-Orbit Coupling and Crystal-Field S plitting, Phys. Rev. Research 2,\n033201 (2020).\n57S.MohapatraandA.Singh, Spin Waves and Stability of Zigzag Order in the Hubbard Model with\nSpin-Dependent Hopping Terms: Application to the Honeycom b Lattice Compounds Na2IrO3\nandα-RuCl3, J. Magn. Magn. Mater 479, 229 (2019).(π/2,π/2) (π,0) ( π,π) (0,0) 600 650 700 750 800 850 900 950 1000ω (meV)\n 0.01 0.1 1 10 100\n(c)"    },    {        "title": "1205.6629v1.Conservation_law_in_noncommutative_geometry____Application_to_spin_orbit_coupled_systems.pdf",        "content": "arXiv:1205.6629v1  [math-ph]  30 May 2012Conservation law in noncommutative geometry\n– Application to spin-orbit coupled systems\nNaoyuki Sugimoto1and Naoto Nagaosa1,2\n1Cross-correlated Materials Research Group (CMRG) and Corr elated\nElectron Research Group (CERG), RIKEN, Saitama 351-0198, J apan\n2Department of Applied Physics University of Tokyo, Tokyo 11 3-8656, Japan\nThe quantization scheme by noncommutative geometry develo ped in string theory is applied to\nestablish the conservation law of twisted spin and spin curr ent densities in the spin-orbit coupled\nsystems. Starting from the pedagogical introduction to Hop f algebra and deformation quantization,\nthe detailed derivation of the conservation law is given.2\nCONTENTS\nI. Introduction 3\nII. Noether’s theorem in field theory 4\nA. Conventional formulation of Noether’s theorem 4\nB. Generalization of Noether’s theorem 7\nIII. Hopf algebra 8\n1. Algebra 8\n2. Coalgebra 10\n3. Dual-algebra and Hopf algebra 11\nIV. Deformation quantization 12\nA. Wigner representation 13\nB. Star product 14\n1. Cohomology equation 16\n2.L∞algebra 18\nC. Topological string theory 20\n1. Ghost fields and anti-fields 21\n2. Condition of gauge invariance of classical action 22\n3. Gauge invariance in path integral 23\nD. Equivalence between deformation quantization and topological s tring theory 25\n1. Path integral as L∞map 25\n2. Perturbation theory 26\nE. Diagram rules of deformation quantization 27\nF. Gauge invariant star product 29\nV. Twisted spin 31\nA. Derivation of a twisted spin in Wigner space 31\nB. Rashba-Dresselhaus model 33\nVI. Conclusions 35\nReferences 363\nI. INTRODUCTION\nElectrons are described by the Dirac equation where the U(1) Maxw ell electromagnetic field (emf) Aµis coupled\nto the charge current jµas described by the Lagrangian (in the natural unit where /planckover2pi1=c= 1;µ= 0,1,2,3) [1]\nL=¯ψ[iγµˆDµ−m]ψ. (1)\nwhereˆDµ=∂µ−ieAµis the covariant derivative, mis the electron mass. Note that the spin is encoded by 4\ncomponent nature of the spinors ψand¯ψ=ψ†γ0and the 4×4 gamma matrices γµ, but the charge and charge\ncurrent alone determine the electromagnetic properties of the ele ctrons, which are given by\njµ=−∂L\n∂Aµ=−e¯ψγµψ. (2)\nIn condensed matter physics, on the other hand, the low energy p henomena compared with the mass gap 2 mc2∼\n106eVare considered, and only the positive energy states described by t he two-component spinor are relevant. Then,\nthe relativistic spin-orbit interaction originates when the negative e nergy states (positron stats) are projected out\nto derive the effective Hamiltonian or Lagrangian. The projection to a subspace of the Hilbert space leads to the\nnontrivial geometrical structure which is often described by the g auge theory. This is also the case for the Dirac\nequation, and the resultant gauge field is SU(2) non-Abelian gauge fi eld corresponding to the Zeeman effect (time-\ncomponent) and the spin-orbit interaction (spatial components) as described below.\nThe effective Lagrangianforthe positiveenergystatescan be der ivedby the expansionwith respect to 1 /(mc2) [2–4]\nL= iψ†D0ψ+ψ†D2\n2mψ+1\n2mψ†/bracketleftbigg\neqσaA·Aa+q2\n4Aa·Aa/bracketrightbigg\nψ, (3)\nwhereψis now the two-component spinor and D0=∂0+ieA0+iqAa\n0σa\n2, andDi=∂i−ieAi−iqAa\niσa\n2(i= 1,2,3)\nare the gauge covariant derivatives with qbeing the quantity proportional to the Bohr magneton [2, 4]. Aµis the\nMaxwell emf, and the SU(2) gauge potential are defined as\nAa\n0=Ba\nAa\ni=ǫiaℓEℓ, (4)\nandσx,y,zrepresent the Pauli matrices. The SU(2) gauge field is coupled to th e 4-component spin current\nja\n0=ψσaψ,\nja\ni=1\n2mi[ψ†σaDiψ−Diψ†σaψ]. (5)\nNamely, the Zeeman coupling and the spin-orbit interaction can be re garded as the gauge coupling between the 4-spin\ncurrent and the SU(2) gauge potential. (The spin current is the te nsor quantity with one suffix for the direction\nof the spin polarization while the other for the direction of the flow.) N ote that the system has no SU(2) gauge\nsymmetry since the “vector potential” Aa\nµis given by the physical field strength BandE, i.e., the relation ∂µAa\nµ= 0\nautomatically holds. This fact is connected to the absence of the co nservation law for the spin density and spin\ncurrent density in the presence of the relativistic spin-orbit intera ction. In the spherically symmetric systems, the\ntotal angular momentum, i.e., the sum of the orbital and spin angular momenta, is conserved, but the rotational\nsymmetry is usually broken by the periodic or disorder potential A0in condensed matter systems. Therefore, it is\nusually assumed that the conservation law of spin is lost by the spin-o rbit interaction.\nHowever, it is noted that the spin and spin current densities are “co variantly” conserved as described by the\n“continuity equation” [2–4]\nD0Ja\n0+D·Ja= 0. (6)\nreplacing the usual derivative ∂µby the covariant derivative Dµ. This suggest that the conservation law holds in\nthe co-moving frame, but the crucial issue is how to translate this la w to the laboratory frame, which is the issue\naddressed in this paper. Note again that the SU(2) gauge symmetr y is absent in the present problem, and hence the\nLagrangian like tr( FµνFµν), which usually leads to the generalized Maxwell equation and also to t he conservation\nlaw of 4 spin current including both the matter field and gauge field [1], is missing. Instead, we will regard Aa\nµas the\nfrozen background gauge field, and focus on the quantum dynamic s of noninteracting electrons only.4\nIn this paper, we derive the hidden conservation law by defining the “ twisted” spin and spin current densities which\nsatisfy the continuity equation with the usual derivative ∂µ. The description is intended to be pedagogical and self-\ncontained. For this purpose, the theoretical techniques develop ed in high energy physics is useful. The essential idea\nis to take into account the effect of the background gauge field in te rms of the noncommutative geometry generalizing\nthe concept of “product”. This is achieved by extending the usual Lie algebra to Hopf algebra.\nUsually, a conservation law is derived from symmetry of an action, i.e., Noether’s theorem. The symmetry in the\nnoncommutative geometry is called as a “twisted” symmetry, and th is symmetry and the corresponding generalized\nNoether’s theorem have been studied in the high energy physics. Se iberg and Witten proposed that an equivalence of\na certain string theory and a certain field theory in noncommutative geometry [5]. Since then, the noncommutative\ngeometry have been attracted many researchers. On the other hand, it is known that the Poincaresymmetry is broken\nin a field theory on a noncommutative geometry. It is a serious proble m because the energy and momentum cannot\nbe defined. M. Chaichian, et al. proposed the twisted symmetry in the Minkowski spacetime, and a lleged that the\ntwisted Poincare symmetry is substituted for the Poincare symmet ry [6, 7]. Moreover, G. Amelino-Camelia, et al.\ndiscussed Noether’s theorem in the noncommutative geometry [8, 9 ].\nAs we will discuss in detail later, a certain type ofa noncommutative g eometry space is equal to a spin-orbit coupled\nsystem. Therefore, a global SU(2) gauge symmetry in the noncom mutative geometry space gives a Noether current\ncorresponding to the “twisted” spin and spin current in the spin-or bit coupling system. This enables us to derive the\ngeneralized Noether’s theorem for the twisted spin and spin curren t densities.\nNow some remarks about the application is in order. Spintronics is an e merging field of electronics where the role\nof charge and charge current are replaced by the spin and spin cur rent aiming at the low energy cost functions [10].\nThe relativistic spin-orbit interaction plays the key role there since it enables the manipulation of spins by the electric\nfield. However, this very spin-orbit interaction introduces the spin relaxation which destroys the spin information in\nsharp contrast to the case of charge where the information is pro tected by the conservation law. Therefore, it has\nbeen believed that the spintronics is possible in a short time-scale or t he small size devices. The discovery of the\nconservation law of twisted spin and spin current densities means th at the quantum information of spin is preserved\nby this hidden conservation law, and could be recovered. Actually, it has been recently predicted that the adiabatic\nchange in the spin-orbit interaction leads to the recovery of the sp in moment called spin-orbit echo [11]. Therefore,\nthe conservation law of the twisted spin and spin current densities is directly related to the applications in spintronics.\nThe plan ofthis paper follows (see Fig. 1). In section II, we reviewth e conventionalNoether’s theorem, and describe\nbriefly its generalization to motivate the use of Hopf algebra and def ormation quantization. In section III, the Hopf\nalgebra is introduced, and section IV gives the explanation of the de formation quantization with the star product.\nThe gauge interaction is compactly taken into account in the definitio n of the star product. These two sections are\nsort of short review for the self-containedness and do not conta in any original results except the derivation of the star\nproduct with gauge interaction. Section V is the main body of this pap er. By combining the Hopf algebra and the\ndeformation quantization, we present the derivation of the conse rved twisted spin and spin current densities. Section\nVI is a brief summary of the paper and contains the possible new direc tions for future studies. The readers familiar\nwith the noncommutative geometry and deformation quantization c an skip sections III, IV, and directly go to section\nV.\nII. NOETHER’S THEOREM IN FIELD THEORY\nIn this section, we discuss Noether’s theorem [12], and its generaliza tion as a motivation to introduce the Hopf\nalgebra and deformation quantization. In section IIA, we will recall Noether theorem, and rewrite it using the so-\ncalled “coproduct”, which is an element of the Hopf algebra. In sect ion IIB, we will sketch a derivation of generalized\nNoether theorem.\nA. Conventional formulation of Noether’s theorem\nWe start with the action Igiven by\nI=/integraldisplay\nΩdDimxL(x)\n=/integraldisplay\ndDimxhΩ(x)L(x), (7)5\nFIG. 1. Flows of derivation of generalized Noether’s theore m. Roman numerals and capital letters in boxes represent\nsection and subsection numbers, respectively. A generaliz ation of the Noether’s theorem is achieved through Hopf alge bra and\ndeformation quantization (section V). Hopf algebra appear to characterize feature of an infinitesimal transformed var iation\noperator (sections II and III). The SU(2) gauge structure is embedded in the star product (section IV).\nwhere Ω represents a range of the spacetime coordinate x(≡(x0,xi)≡(ct,x)) with a dimension Dim, i.e., (Dim −1)\nis the dimension of the space, Ldescribes a Lagrangian density, and\nhΩ(x) =/braceleftbigg\n1 for{x|x∈Ω}\n0 for{x|x /∈Ω}.; (8)\ncrepresents light speed. We introduce a field φrwith internal degree of freedom r, and infinitesimal transformations:\nxµ/ma√sto→(x′)µ:=xµ+δζxµ, (9)\nφr(x)/ma√sto→φ′\nr(x′) :=φr(x)+δζφr(x), (10)\nwhere we characterize the transformations by the subscript; sp ecifically,ζrepresents a general infinitesimal transfor-\nmation. Hereafter, we will employ Einstein summation convention, i.e., aµbµ≡aµbµ≡/summationtextDim−1\nµ=0ηµνaµbµwith vectors\naµandbµ(µ= 0,1,...,(Dim−1)), and the Minkowski metric: ηµν:= diag(−1,1,1,...,1/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nDim−1).\nWe define the variation operator of the action as follow:\nδζI:=/integraldisplay\nΩ′dDimx′L′(x′)−/integraldisplay\nΩdDimxL(x)\n=/integraldisplay\ndDimx′hΩ′(x′)L′(x′)−/integraldisplay\ndDimxhΩ(x)L(x), (11)\nwhere we characterize this variation by ζ, because this variation is derived from the infinitesimal transforma tions Eqs.\n(9) and (10). Since the integration variable x′can be replaced by x, Eq. (11) is\nδζI=/integraldisplay\ndDimxhΩ′(x)L′(x)−/integraldisplay\ndDimxhΩ(x)L(x)\n=/integraldisplay\ndDimx(hΩ′(x)−hΩ(x))L′(x)+/integraldisplay\ndDimxhΩ(x)[L′(x)−L(x)]\n=/integraldisplay\ndDimxhδΩ(x)L′(x)+/integraldisplay\ndDimxhΩ(x)[L′(x)−L(x)], (12)\nwhereδΩ := Ω′−Ω andhδΩ=−(∂µhΩ)δζxµ+O((δζx)2). Therefore, we obtain the following equation through partial\nintegration:\nδζI=/integraldisplay\ndDimxhΩ(x)/bracketleftbig\n∂µ(L(x)δζxµ)+δL\nζL(x)/bracketrightbig\n+O((δζx)2), (13)6\nwhere we have introduced the so-called Lie derivative:\nδL\nζφr(x) :=φ′\nr(x)−φr(x)\n=δζφr(x)−(∂µφr)δζxµ+O(δζx2), (14)\nand we replacedL′byLdue toL′δζxµ=Lδζxµ+O(δζx2).\nHereafter, we assume that the action is invariant under the infinite simal transformations Eqs. (9) and (10). In the\ncase where the Lagrangian density is a function of φrand∂µφr, i.e.,L(x) =L[φr(x),∂µφr(x)], the Lie derivative of\nthe Lagrangian is given by\nδL\nζL:=L′(x)−L(x)\n=L[φ′\nr(x),∂µφ′\nr(x)]−L[φr(x),∂µφr(x)]\n=/braceleftbigg∂L\n∂φrδL\nζφr+∂L\n∂(∂µφr)∂µδL\nζφr/bracerightbigg\n=/parenleftbigg∂L\n∂φr−∂µ∂L\n∂(∂µφr)/parenrightbigg\nδL\nζφr+∂µ/parenleftbigg∂L\n∂(∂µφr)δL\nζφr/parenrightbigg\n, (15)\nand the variation of the action is calculated by\nδζI=/integraldisplay\nΩdDimx/braceleftbigg/parenleftbigg∂L\n∂φr−∂µ∂L\n∂(∂µφr)/parenrightbigg\nδL\nζφr+∂µ/parenleftbigg\nLδζxµ+∂L\n∂(∂µφr)δL\nζφr/parenrightbigg/bracerightbigg\n. (16)\nIf we require that δζxandδL\nζφrvanish on the surface ∂Ω, we obtain the Euler-Lagrange equation. On the other\nhand, if we require that fields φrsatisfy the Euler-Lagrange equation, we obtain continuity equatio n∂µjµ= 0 with a\nNoether current\njµ:=/parenleftbigg\nLδζxµ+∂L\n∂(∂µφr)δL\nζφr/parenrightbigg\n. (17)\nHereafter let us discuss an infinitesimal global U(1) ×SU(2) gauge transformation and infinitesimal translation and\nrotation transformations, which are denoted by χin this paper. Variations in terms of χare defined by\nδχxµ:= Γµ\nνxν, (18)\nδχφr:= iϑµν(ξµν)r′\nrφr′ (19)\nwith an infinitesimal parameter ϑµν, and symmetry generators Γµ\nνand (ξµν)r′\nr.\n1. Forthe globalU(1) ×SU(2) gaugetransformation, Γµ\nν≡0,ϑµν≡ϑµδµν, and (ξµν)r′\nr≡δµν(ˆsµ)r′\nr(µ,ν= 0,1,2,3;\nr,r′= 1,2), where ˆs0:=/planckover2pi1/2, and ˆs1,2,3:=/planckover2pi1ˆσx,y,z/2 with the Planck constant h= 2π/planckover2pi1and Pauli matrices\nˆσx,y,z.\n2. For the translation, Γµ\nν≡εµδµ\nν,ϑµν≡εµδµνand (ξµν)r′\nr≡ˆpµδµνδr′\nrwith an infinitesimal parameter εµand the\nmomentum operator ˆ pµ=−i/planckover2pi1∂µ(µ,ν= 1,2,3;r,r′= 1,2).\n3. For the rotation, Γµ\nν≡ωµ\nν,ϑµν≡ωµν, and (ξµν)r′\nr≡δr′\nrxµˆpν, which corresponds to the angular momentum\ntensor (µ,ν= 1,2,3;r,r′= 1,2).\nFor these transformations, equation ∂µδχxµ= 0 is satisfied. This can be seen explicitly as follows. The variations\nof space coordinates of the global U(1) ×SU(2) and the translation transformations are given by δχxµ= 0 orδχxµ=\nconstant, respectively, and thus ∂µδχxµ= 0 is trivial. The variation of the rotation transformation is given by\nδχxµ=ωµ\nνxν, therefore∂µδχxµ=∂µωµ\nνxν=ωµ\nµ= 0.\nWe consider a variation of the Lagrangian density;\nδζL:=L′(x′)−L(x)\n=L′(x′)−L(x′)+L(x′)−L(x)\n=δL\nζL(x′)+δζxµ∂µL+O((δζx)2). (20)\nNote that Eq. (20) is correct for any infinitesimal transformation . Here we consider the global U(1) ×SU(2) gauge\ntransformation and/or the translation and rotation transforma tionsδχ. Because∂µδχxµ= 0, we obtain the following\nequation:\nδχL=δL\nχL(x)+∂µ(L(x)δχxµ)+O((δx)2). (21)7\nFrom Eqs.(13) and (21), one can see\nδ(ζ=χ)I=/integraldisplay\ndDimxδχL(x), (22)\nwhere (ζ=χ) denotes that the type of the variation in Eq. (13) is restricted to the global U(1)×SU(2) or Poincare\ntransformations. (For simplicity we omitted the subscript Ω in the int egral). Finally, for ζ=χ, the variation of the\naction is equal to the variation of Lagrangian. This fact will be used la ter in section V where the variation of the\nLagrangian density instead of the action will be considered.\nB. Generalization of Noether’s theorem\nNow, we wouldlike to introduce a Hopfalgebraforthe purposeofgen eralizingNoether’s theorem [8, 9, 13]. At first,\nwerewriteNoether’stheoreminsectionIIbyusingtheHopfalgebra ,andnext, weintroduceatwistedsymmetry [6,7].\nFor simplicity, we only consider the global U(1) ×SU(2) gauge symmetry and the Poincare symmetry. We assume tha t\nthe Lagrangian density is written as\nL(x) =ψ†(x)ˆL(x)ψ(x) (23)\nwith a field ψ:=/parenleftbigg\nψ1\nψ2/parenrightbigg\n, a Hermitian conjugate ψ†≡/parenleftbig\nψ1,ψ2/parenrightbig\n, and an single-particle Lagrangian density operator\nˆL, which is a 2×2 matrix; the overline represents the complex conjugate. The act ion can be rewritten as\nI=/integraldisplay\ndDimx1dDimx2ψ†(x2)δ(x2−x1)ˆL(x1)ψ(x1)\n= tr/integraldisplay\ndDimx1dDimx2δ(x2−x1)ˆL(x1)ψ(x1)ψ†(x2)\n= tr/integraldisplay\ndDimx1/braceleftbigg/integraldisplay\ndDimx2˜L(x1,x2)G(x2,x1)/bracerightbigg\n= tr/integraldisplay\ndDimx1/braceleftbigg\nlim\nx3→x1(˜L∗CG)(x1,x3)/bracerightbigg\n, (24)\nwhere “tr” represents the trace in the spin space, G(x1,x2) :=ψ(x1)ψ†(x2),˜L(x1,x2) :=δ(x1−x2)ˆL(x2), and∗C\nrepresents the convolution integral:\n(f∗Cg) :=/integraldisplay\ndDimx3f(x1,x3)g(x3,x2) (25)\nwith smooth two-variable functions fandg.\nThe variation operator δχof the action can be also rewritten as\nδχI= tr/integraldisplay\ndDimx1dDimx2˜L(x2,x1)/bracketleftbig\niϑξψ(x1)ψ†(x2)−ψ(x1)ψ†(x2)iϑξ/bracketrightbig\n= tr/integraldisplay\ndDimx1dDimx2/bracketleftig\n˜L(x2,x1)iϑξG(x1,x2)−iϑξ˜L(x2,x1)G(x1,x2)/bracketrightig\n(26)\nwithϑξ≡ϑµν(ξµν); in addition, we assumed that the single-particle Lagrangian densit y operator is invariant under\nthe infinitesimal transformation δχ.\nHere, we introduce Grassmann numbers θ1andθ2; an integral is defined by/integraltext\ndθi(θj) =δij. The variation of the8\nright-hand side of Eq.(26) can be rewritten as follow:\nδχI=−itr/integraldisplay\ndθ1dθ2dDimx1dDimx2/bracketleftig\nθ1˜L(x2,x1)ϑθ2ξG(x1,x2)+ϑθ2ξθ1˜L(x2,x1)G(x1,x2)/bracketrightig\n=−itr/integraldisplay\ndθ1dθ2dDimx1dDimx2µ◦(µ⊗id)\n◦/bracketleftig/parenleftig\nθ1˜L(x2,x1)⊗ϑθ2ξ+ϑθ2ξ⊗θ1˜L(x2,x1)/parenrightig\n⊗G(x1,x2)/bracketrightig\n=−itr/integraldisplay\ndθ1dθ2dDimx1dDimx2µ◦(µ⊗id)/bracketleftig\n△(ϑθ2ξ)◦/parenleftig\nθ1˜L(x2,x1)⊗G(x1,x2)/parenrightig/bracketrightig\n≡ˆTr/bracketleftig\n△(ϑθ2ξ)◦((θ1˜L)⊗G)/bracketrightig\n, (27)\nwhere⊗and◦represent a tensor product and a product of operators, respe ctively. The operator µdenotes the\ntransformation of the tensor product to the usual product µ:x⊗y/ma√sto→xy, and△represents a coproduct:\n△(ζ) :=ζ⊗id+id⊗ζ, (28)\nwhereζand id represent a certain operator and the identity map, respect ively. These operators constitutes the Hopf\nalgebraaswillbeexplainedinthenextsection. Moreover,wehavede finedˆTr :=−itr/integraltext\ndθ1dθ2dDimx1dDimx2µ◦(µ⊗id).\nWe emphasize here that the variation is written by the coproduct △, which is important to formulate the generalized\nNoether theorem in the presence of the gauge potential. The copr oduct determines an operation rule of a variation\noperator; for example, the coproduct (28) represents the Leib niz rule. A twisted symmetry transformation is given\nby deformation of the coproduct.\nWe now sketch the concept of the twisted symmetry in deformation quantization [6, 7]. First, we assume that\nthe variation of action δζI0is zero, i.e., ζrepresents the symmetry transformation of the system corres ponding to\nthe actionI0. Next, we consider the action IAwith external gauge fields A. Usually, external gauge fields breaks\nsymmetries of I0, i.e.,δζIA/ne}ationslash= 0. Here we introduce a map: F(0/mapsto→A):I0/ma√sto→IA, which will be defined in section\nIVF. The basic idea is to generalize the ”product” taking into accoun t the gauge interaction. Using this map, the\nvariation is rewritten as δζF(0/mapsto→A)I0/ne}ationslash= 0. On the other hand, when the twisted symmetry δt\nζ:=F(0/mapsto→A)δζF−1\n(0/mapsto→A)can\nbe defined, we obtain the following equation:\nδt\nχIA=F(0/mapsto→A)δχF−1\n(0/mapsto→A)IA\n=F(0/mapsto→A)δχF−1\n(0/mapsto→A)F(0/mapsto→A)I0\n=F(0/mapsto→A)δχI0\n= 0. (29)\nNamely,δt\nχcorresponds to a symmetry with external gauge fields. In the exp ression for the variation of action in\nterms of the Hopf algebra Eq.(27), we can replace ∆ by ∆tcorresponding to the change from δχtoδt\nχas shown in\nsection V. This is achieved by using the Hopf algebra and the deforma tion quantization, which will be explained in\nsections III and IV, respectively. Therefore, we can generalize t he Noether’s theorem and derive the conservation law\neven in the presence of the gauge field A.\nIII. HOPF ALGEBRA\nHere we introduce a Hopf algebra. First, we rewrite the algebra usin g tensor and linear maps. Secondly, a coalgebra\nis defined using diagrams corresponding to the algebra. Finally, we de fine a dual-algebra and Hopf algebra.\n1. Algebra\nWe define the algebra as a k-vector space Vhaving product µand unitε. Here,krepresents a field such as the\ncomplex number or real number. In this paper, we consider Vas the space of functions or operators. A space of\nlinear maps from a vector space V1to a vector space V2is written as Hom( V1,V2).\nA productµis a bilinear map: µ∈Hom(V/circlemultiplytextV,V), i.e.,\nµ:V/circlemultiplydisplay\nV→V,(x,y)/ma√sto→xy, (30)9\nand a unit is a linear map: ε∈Hom(k,V), i.e.,\nε:k→V, α/ma√sto→α·1 (31)\nwithx,y,xy∈Vandα∈k. Hereµandεsatisfies\nµ((x+y)⊗z) =µ(x⊗z)+µ(y⊗z), µ(x⊗(y+z)) =µ(x⊗y)+µ(x⊗z), (32)\nµ(αx⊗y) =αµ(x⊗y), µ(x⊗αy) =αµ(x⊗y), (33)\nε(α+β) =ε(α)+ε(β) (34)\nwithx,y,z∈Vandα,β∈k.\nThe product µhas the association property, which is written as µ◦(id⊗µ) =µ◦(µ⊗id). Because the left-hand\nside and the right-hand side of the previous equation give the followin g equations:\nµ◦(id⊗µ)(x⊗y⊗z) =µ(x⊗(yz)) =x(yz) (35)\nand\nµ◦(µ⊗id)(x⊗y⊗z) =µ((xy⊗z)) = (xy)z, (36)\nfor allx,y,z,xy,yz,xyz ∈V, thenµ◦(id⊗µ) =µ◦(µ⊗id) is equal to the association property x(yz) = (xy)z. This\nproperty is illustrated as the following diagram:\nV/circlemultiplytextV/circlemultiplytextV V/circlemultiplytextV\nV/circlemultiplytextV V❄✲\n✲❄µ⊗id\nid⊗µ µ\nµ/clockwise\nHere/clockwisedenotes that this graph is the commutative diagram.\nThe unitεsatisfies the following equation: µ◦(ε⊗id) =µ◦(id⊗ε). Since the left-hand side and the right-hand\nside of the previous equation give the following equations\nµ◦(ε⊗id)(α⊗x) =µ⊗(α1V⊗x) =α1Vx=αx (37)\nand\nµ◦(id⊗ε)(x⊗α) =µ◦(x⊗α1V) =αx1V=αx (38)\nfor allx∈Vandα∈k, and∃1V∈V, then the unit can be written as µ◦(id⊗ε) =µ◦(ε⊗id). Note that\nV∼k/circlemultiplytextV∼V/circlemultiplytextk, where∼represents the equivalence relation, i.e., a∼bdenotes that aandbare identified. This\nproperty is illustrated as:\nk/circlemultiplytextV V/circlemultiplytextk V/circlemultiplytextV\nV✲\n❄✛\n❍❍❍❍❍❍❍ ❥✟✟✟✟✟✟✟ ✙ε⊗id id ⊗ε\nµ\n∼ ∼\nAlgebra is defined as a set ( V,µ,ε).10\n2. Coalgebra\nA coalgebra is defined by reversing the direction of the arrows in the diagrams corresponding to the algebra. Thus,\nwe will define a coproduct △∈Hom(V,V/circlemultiplytextV) and counit η∈Hom(V,V) with ak-vector space V.\nA coproduct is a bilinear map from VtoV/circlemultiplytextV:\n△:V→V/circlemultiplydisplay\nV, (39)\nand satisfies co-association property:\nV/circlemultiplytextV/circlemultiplytextV V/circlemultiplytextV\nV/circlemultiplytextV V✻✛\n✛✻△⊗id\nid⊗△ △\n△/clockwise\nNamely,\n(id⊗△)◦△= (△⊗id)◦△ (40)\n(Compare the diagram corresponding to the association property and that corresponding to the co-association prop-\nerty).\nA counitηis a linear map from Vto fieldk:\nη:V→k, (41)\nand satisfies the following diagram:\nk/circlemultiplytextV V/circlemultiplytextk V/circlemultiplytextV\nV✛\n✻✲\n❍❍❍❍❍❍❍ ❨\n✟✟✟✟✟✟✟ ✯η⊗id id ⊗η\n△∼ ∼\nNamely,\n(η⊗id)◦△= (id⊗η)◦△, (42)\nwhereV∼k/circlemultiplytextV∼V/circlemultiplytextk.\nSince△andηare linear maps,△andηsatisfy\n△(x+y) =△(x)+△(y),△(αx) =α△(x), (43)\nη(x+y) =η(x)+η(y), η(αx) =αη(x) (44)\nwithx,y∈Vandα∈k. Note that V∼k/circlemultiplytextV∼V/circlemultiplytextkandV/circlemultiplytextV∼k/circlemultiplytextV/circlemultiplytextV∼V/circlemultiplytextk/circlemultiplytextV∼V/circlemultiplytextV/circlemultiplytextk.\nA coalgebra is defined as a set ( V,△,η). For example, in the vector space D≡k/circleplustextk∂:={a0+a1∂|a0,a1∈k},\nwe define a coproduct △D(∂) =∂⊗1 + 1⊗∂and△D(1) = 1⊗1, and a counit ηD(∂) = 0 and ηD(1) = 1.\nThe set (D,△D,ηD) is coalgebra, because this set satisfies the equations: ( △D⊗id)◦△D= (id⊗△D)◦△Dand\n(ηD⊗id)◦△D= (id⊗ηD)◦△D. Because the coproduct and counit are linear map, we only check th e above equations\nwith respect to x= 1 and∂.\nForx= 1,\n(△D⊗id)◦△D(1) =△D(1)⊗1 = 1⊗1⊗1, (45)11\nand\n(id⊗△D)◦△D(1) = 1⊗△D(1) = 1⊗1⊗1. (46)\nTherefore, (△D⊗id)◦△D(1) = (id⊗△D)◦△D(1). Moreover,\n(ηD⊗id)◦△D(1) =ηD(1)⊗1 = 1⊗1, (47)\nand\n(id⊗ηD)◦△D(1) = 1⊗ηD(1) = 1⊗1. (48)\nTherefore ( ηD⊗id)◦△D(1) = (id⊗ηD)◦△D(1).\nForx=∂,\n(△D⊗id)◦△D(∂) =△D(∂)⊗1+△D(1)⊗∂=∂⊗1⊗1+1⊗∂⊗1+1⊗1⊗∂, (49)\nand\n(id⊗△D)◦△D(∂) =∂⊗△D(1)+1⊗△D(∂) =∂⊗1⊗1+1⊗∂⊗1+1⊗1⊗∂. (50)\nTherefore, (id⊗△D)◦△D(∂) = (△D⊗id)◦△D(∂). Finally,\n(ηD⊗id)◦△D(∂) =ηD(∂)⊗1+ηD(1)⊗∂= 1⊗∂=∂, (51)\nand\n(id⊗ηD)◦△D(∂) =∂⊗ηD(1)+1⊗ηD(∂) =∂⊗1 =∂. (52)\nTherefore, ( ηD⊗id)◦△D(∂) = (id⊗ηD)◦△D(∂). Namely, the set ( D,△D,ηD) is the coalgebra. Note that △D(1)\ncorresponds to the product with a constant: a(fg) =a1(fg) =a(1f1g) =aµ◦△D(1)(f⊗g), where we have used the\ncoproduct△D(1) = 1⊗1atthefinalequalsign. Here f,andgaresmoothfunctions, 1isincludedinthefunctionspace,\nanda∈k.△D(∂) represents the Leibniz rule: ∂(fg) = (∂f)g+f∂(g) =µ◦(∂⊗1+1⊗∂)◦(f⊗g) =µ◦△D(∂)(f⊗g),\nwhere we have used the coproduct △D(∂) = 1⊗∂+∂⊗1 at the last equal sign. ηD(1) andηD(∂) represent the\nfiltering action to a constant function: 1 a=a=ηD(1)aand∂(a) = 0 =ηD(∂)a, respectively.\n3. Dual-algebra and Hopf algebra\nA dual-algebra is the set of an algebra and a coalgebra, i.e., the set of (V,µ,ε,△,η). On a dual-algebra, we define\na∗-product as\nf∗g=µ◦(f⊗g)◦△ (53)\nwithf,g∈Hom(V,V). We define an antipode S∈Hom(V,V) which satisfies the following equation:\nµ◦(id⊗S)◦△=µ◦(S⊗id)◦△=ε◦η, (54)\nwhereε◦ηcorresponds to the identity mapping, i.e., Sis an inverse of unit. For example, SDin the set\n(D,µD,εD,△D,ηD) is defined as SD(1) = 1 and SD(∂) =−∂.\nForx= 1,\nµD◦(id⊗SD)◦△D(1) =µD◦(1⊗1) = 1, (55)\nand\nµD◦(SD⊗id)◦△D(1) =µD◦(1⊗1) = 1. (56)\nTherefore, we obtain µD◦(id⊗SD)◦△D(1) =µD◦(SD⊗id)◦△D(1) =εD◦ηD. For∂,\nµD◦(id⊗SD)◦△D(∂) =µD◦(∂⊗1−1⊗∂) = 0, (57)12\nand\nµD◦(SD⊗id)◦△D(∂) =µD◦(−∂⊗1+1⊗∂) = 0. (58)\nTherefore, we obtain µD◦(id⊗SD)◦△D(∂) =µD◦(SD⊗id)◦△D(∂) =εD◦ηD(∂). Namely, ( D,µD,εD,△D,ηD)\nis the Hopf algebra.\nA dual-algebra with an antipode S, i.e., (V,µ,ε,△,η,S), is called a Hopf algebra.\nBy using the approach similar to a coproduct and counit, we can defin e a codifferential operator Q∈Hom(V,V)\nfrom a diagram of the differential ∂∈Hom(V,V). The differential ∂is the linear map:\n∂:V→V, (59)\nand satisfies Leibniz rule\n∂◦µ=µ◦(id⊗∂+∂⊗id), (60)\nwhich is illustrated as\nV V\nV/circlemultiplytextV V/circlemultiplytextV✻✛\n✛✻∂\nµ µ\n(id⊗∂+∂⊗id)/clockwise\nA codifferential operator Qis a linear map; Q:V→V, and satisfies the following diagrams:\nV V\nV/circlemultiplytextV V/circlemultiplytextV❄✲\n✲❄Q\n△ △\n(id⊗Q+Q⊗id)/clockwise\nNamely, a codifferential operator Qsatisfies△◦Q= (id⊗Q+Q⊗id)◦△. In section IVB2, the codifferential\noperator will be introduced.\nIV. DEFORMATION QUANTIZATION\nIn this section, we explain the deformation quantization using the no ncommutative product encoding the commu-\ntation relationships. At first, in section IVA, we introduce the so-c alled Wigner representation and Wigner space,\nand show that a product in the Wigner space is noncommutative. This product is called Moyal product and it\nguarantees the commutation relationship of the coordinate and ca nonical momentum. Next, we add spin functions\nand background gauge fields to the Wigner space, and rewrite the c oordinates of Wigner space as a set of spacetime\ncoordinates X, mechanical momenta p, and spins s:= (sx,sy,sz) (pincludes the background gauge fields). To gen-\neralize the Moyal product for the deformed Wigner space, which is a set of function defined on ( X,p,s), we explain\nthe general constructing method of the noncommutative produc t in section IVB; the noncommutative product is\nthe generalized Moyal product, which is called “star product”. This constructing method is given as a map from a\nPoisson bracket in the Wigner space to the noncommutative produc t (see section IVB), and we see the condition of\nthis deformation quantization map in section IVB. This map is describe d by the path integral of a two-dimensional\nfield theory, which is called the topological string theory. In section IVC, we explain this topological string theory,\nand in section IVD, we discuss the perturbative treatment of this t heory. In section IVE, we summarize the diagram\ntechnique. Finally, in section IVF, we construct the star product in (X,p,s) space. We note that the star product\nguarantees the background gauge structure.13\nA. Wigner representation\nWe start with the introduction of the Wigner representation. From Equation (24), a natural product is the\nconvolution integral:\n(f∗Cg)(x1,x2) :=/integraldisplay\ndDimx3f(x1,x3)g(x3,x2), (61)\nwheref,g∈Gwith a two spacetime arguments function space G. Here we introduce the center of mass coordinate X\nand the relative coordinate ξas follows:\nX≡(T,X) := ((t1+t2)/2,(x1+x2)/2), (62)\nξ≡(ξt,ξ) := (t1−t2,x1−x2). (63)\nMoreover we employ the following Fourier transformation:\nFT:f(x1,x2)/ma√sto→f(X,p) =/integraldisplay\ndDimξe−ipµξµ//planckover2pi1f(X+ξ/2,X−ξ/2). (64)\nNow we define the Wigner space: W:={FT[f]|f∈G}[14]. In this space, the convolution is transformed to the\nso-called Moyal product [15, 16]:\n(f ⋆Mg)(X,p) :=f(X,p)ei/planckover2pi1\n2/parenleftBig← −∂X− →∂pν−← −∂p− →∂X/parenrightBig\ng(X,p), (65)\nbecause\nF−1\nT[f ⋆Mg] =/integraldisplaydDimp\n(2π/planckover2pi1)Dimeipνξν//planckover2pi1/braceleftbigg\nf(X,p)ei/planckover2pi1\n2/parenleftBig← −∂Xν− →∂pν−← −∂pν− →∂Xν/parenrightBig\ng(X,p)/bracerightbigg\n=/integraldisplaydDimp\n(2π/planckover2pi1)DimdDimξ1dDimξ2eipνξν//planckover2pi1/braceleftig\ne−ipνξ1//planckover2pi1f(X+ξ1/2,X−ξ1/2)\n×ei/planckover2pi1\n2/parenleftBig← −∂Xν− →∂pν−← −∂pν− →∂Xν/parenrightBig\ne−ipνξν\n2//planckover2pi1g(X+ξ2/2,X−ξ2/2)/bracerightig\n=/integraldisplaydDimp\n(2π/planckover2pi1)DimdDimξ1dDimξ2eipν(ξν−ξν\n1−ξν\n2)//planckover2pi1\n×f(X+ξ1/2,X−ξ1/2)e1\n2/parenleftBig← −∂Xνξν\n2−ξν\n1− →∂Xν/parenrightBig\ng(X+ξ2/2,X−ξ2/2)\n=/integraldisplay\ndDimξ1dDimξ2δ(ξ−ξ1−ξ2)\n×f/parenleftbigg\nX+ξ1+ξ2\n2,X−ξ1−ξ2\n2/parenrightbigg\ng/parenleftbigg\nX−ξ1−ξ2\n2,X−ξ1+ξ2\n2/parenrightbigg\n=/integraldisplay\ndDimx+dDimx−δ(ξ−x+)f/parenleftig\nX+x+\n2,x−/parenrightig\ng/parenleftig\nx−,X−x+\n2/parenrightig\n=/integraldisplay\ndDimx−f(X+ξ/2,x−)g(x−,X−ξ/2)\n=f∗Cg (66)\nwithξ1+ξ2≡x+andξ1−ξ2≡2(X−x−).\nIn the Wigner space, the position operator ˆ xµ=xµand the momentum operator ˆ pµ=−i/planckover2pi1∂µbecomesXµ⋆Mand\npµ⋆Mbecause\nFT[ˆxµ\n1g(x1,x2)] =/integraldisplay\ndDimξe−i\n/planckover2pi1pνξν(Xµ+ξµ/2)g(X+ξ/2,X−ξ/2)=/parenleftbigg\nXµ+i/planckover2pi1\n2∂pµ/parenrightbigg\ng(X,p) =Xµ⋆Mg(X,p),(67)\nFT[ˆxµ\n2g(x1,x2)] =/integraldisplay\ndDimξe−i\n/planckover2pi1pνξνg(X+ξ/2,X−ξ/2)(Xµ−ξµ/2) =g(X,p)⋆MXµ, (68)\nFT[(ˆp1)µg(x1,x2)] =/integraldisplay\ndDimξe−i\n/planckover2pi1pνξν/planckover2pi1\ni∂xµ\n1g(x1,x2) =pµ⋆g(X,p), (69)14\nand\nFT[(ˆp2)µg(x1,x2)] =/integraldisplay\ndDimξe−i\n/planckover2pi1pνξν/planckover2pi1\ni∂xµ\n2g(x1,x2) =g(X,p)⋆pµ. (70)\nThe commutation relationship of operators is [ Xµ,pν]⋆M:=Xµ⋆Mpν−pν⋆MXµ= i/planckover2pi1δµ\nν, which corresponds to the\ncanonical commutation relationship of operators: [ˆ xµ,ˆpν] = i/planckover2pi1δµ\nν.\nTo add the spin arguments in W, we will employ the following bilinear map:\nFM/ma√sto→FA=0:=ei/planckover2pi1\n2(∂Xµ⊗∂pµ−∂pµ⊗∂Xµ)+i\n2ǫabcsa∂sb⊗∂sc(71)\nwith∂saf:=faandf≡f0+/summationtext\na=x,y,zsafa. Note that the spin operator ˆs:= (ˆsx,ˆsy,ˆsz) is characterized by the\ncommutation relation [ˆ sa,ˆsb] = iǫabcˆsc(a,b,c=x,y,z) with the Levi-Civita tensor ǫabc, and the star product (71)\nreproduces the relation, i.e., the operator ( sa⋆) satisfies [sa,sb]⋆= iǫabcsc.\nTo obtain the map F(0/mapsto→A):I0/ma√sto→IA, we introduce the variables transformation ( Xµ,pµ,s)/ma√sto→(Xµ,ˆpµ,s) where\nˆpµ=pµ−qAa\nµ(Xν)sa+eAµ (72)\nwithq=|e|/mc2, the electric charge −e=−|e|, a U(1) gauge field Aµ, and a SU(2) gauge field Aa\nµ. Their fields are\ntreated as real numbers, and the integral over pµcan be replaced by an integral over ˆ pµ. This transformation induces\nthe following transformations of differential operators:\n∂Xµ⊗∂pµ−∂pµ⊗∂Xµ/ma√sto→∂Xµ⊗∂ˆpµ−∂ˆpµ⊗∂Xµ+q/parenleftig\n∂XµˆAν−∂XνˆAµ/parenrightig\n∂ˆpµ⊗∂ˆpν, (73)\nǫabcsa∂sb⊗∂sc/ma√sto→ǫabcsa∂sb⊗∂sc−qǫabcAb\nµsa∂ˆpµ⊗∂sc−qǫabcAc\nµsa∂sb⊗∂ˆpµ\n+q2ǫabcsaAb\nµAc\nν∂ˆpµ⊗∂ˆpν, (74)\nwhereˆAµ:=Aa\nµsa−(e/q)Aµ.\nWe expandFA=0in terms of /planckover2pi1as\nFA=0=∞/summationdisplay\nn=0/parenleftbiggi/planckover2pi1\n2/parenrightbiggn\nFn\nA=0. (75)\nWe define the bilinear map FAcorresponding to the commutation relation in terms of the phase sp ace (Xµ,ˆpµ,s),\nand expand it in terms of /planckover2pi1as\nFA=∞/summationdisplay\nn=0/parenleftbiggi/planckover2pi1\n2/parenrightbiggn\nFn\nA. (76)\nFrom Eqs. (73) and (74), F1\nAis given as follows:\nF1\nA=∂Xµ⊗∂ˆpµ−∂ˆpµ⊗∂Xµ+qˆFµν∂ˆpµ⊗∂ˆpν+ǫabcsa∂sb⊗∂sc\n−qǫabcsaAb\nµ∂ˆpµ⊗∂sc+qǫabcsaAb\nµ∂sc⊗∂ˆpµ (77)\nwithˆFµν:=∂XµˆAν−∂XνˆAµ+(q//planckover2pi1)εabcsaˆAb\nµˆAb\nν. Note that µ◦F1\nAis the Poisson bracket.\nA constitution method of higher order terms Fn\nAwithn>1 is called a deformation quantization, which is given by\nKontsevich [17], as will be described in the next subsection.\nB. Star product\nIn this subsection, we explain the Kontsevich’s deformation quantiz ation method [17]. We define a star product as\nf ⋆g≡µ◦FA(f⊗g) =f·g+∞/summationdisplay\nn=1νnβn(f⊗g) (78)\nwithν= i/planckover2pi1/2 [18, 19]. Here βn∈Hom(Vf⊗Vf,Vf) is called the two-cochain ( Vfrepresents the function space). We\nrequire that the star product satisfies the association property (f ⋆g)⋆h=f ⋆(g⋆h), which limits forms of Fn≥1\nA15\nL∞L∞\nd.g.L.aαβαβ\n/g54\n/g54 /g37/g372/g5422/g372/g40 /g40(a) (b) (c)\nFIG. 2. Steps of the derivation of the deformation quantizat ion. (a): The image of the deformation quantization, which i s the\nmap from T2with the Jacobi identity to C2with the association property. (b): Enlargement of algebra s. The two vector space\nT2and two cochain space C2generalize to multi-vector space Tand cochain space C, respectively. These spaces are compiled\nin the d.g.L.a; finally, L∞algebra is introduced by using the d.g.L.a. (c): The deforma tion quantization is redefined as the\nmap on the L∞algebra.\nandβn≥1. We note that the association property is necessary for the exist ence of the inverse with respect to the star\nproduct. For example, the inverse of the Lagrangian is a Green fun ction, which always exists as ψψ†with a wave\nfunctionψ.\nNow, we define a p-cochain spaceCp:= Hom(V⊗p\nf,Vf) withV⊗p\nf≡Vf⊗Vf⊗···⊗Vf/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\np, whereVfrepresents a\nfunction space such as the Wigner space W; we define a multi-vector space Tk:= Γ(M,/logicalandtextkTM), whereMrepresents\na manifold such as a classical phase space (dimension d),TM:=/uniontext\np∈MTpMdenotes a tangent vector bundle with a\ntangent vector space TpM≡{/summationtextd\niai(x)∂xi}atp∈M(xis a coordinate at p;airepresents a certain coefficient),/logicalandtextk\ndenotes ak-th completely antisymmetric tensor product, (for example, ∂i∧∂j=1\n2!(∂i⊗∂j−∂j⊗∂i)∈/logicalandtext2TM), and\nΓ represents the section; for example, Γ( M,TM) is defined as a set of tangent vector at each position p∈M. The\nPoisson bracket{f,g}≡α(f⊗g) :=αij(x)(∂i∧∂j)(f⊗g) is element ofT1, whereαij=−αjiis called the Poisson\nstructure (i,j= 1,2,···,d).\nThe deformation quantization is the constitution method of higher o rder cochains βn≥2∈C2from the Poisson\nbracketα∈T2. In other words, the deformation quantization is the following map F:\nF:T2→C2\nα/ma√sto→β≡/summationdisplay\nn≥1νnβn, (79)\nwhereαsatisfies the Jacobi identity and βsatisfies the association property, as shown in Fig. 2(a).\nIn the following sections, we will generalize the two-cochain C2and the second order differential operator T2to the\nso-calledL∞algebra(the definition is given in section IVB2). In the section IVB1, we will introduce the two-cochain\nC2and second order differential operator T2, and thep-cochainCpandk-th order differential operator Tk. We will\nshow that these operators satisfy certain conditions, and CpandTkare embedded in a differential graded Lie algebra\n(d.g.L.a) (the definition is shown in section IVB1). Moreover, in sectio n IVB2, the d.g.L.a will be embedded in the\nL∞algebra (see Fig. 2(b)). In the L∞algebra, the Jacobi identity and the association property are com piled in the\nfollowing equation\nQ(eγ) = 0, (80)\nwhereγ=αorβ, andQis called the codifferential operator, which will be introduced in sectio n IVB2. Namely,\nin theL∞algebra, the deformation quantization is a map from α∈T2toβ∈C2holding the solution of Eq. (80)\n(Figure 2(c)). Such a map is uniquely determined in the L∞algebra.\nIn this paper, we will identify the tensor product ⊗with the direct product ×, i.e.,V1/circlemultiplytextV2∼V1×V2:f⊗g∼(f,g)\nwithf∈V1andg∈V2(a∼bdenotes that aandbare identified; ( f,g) represents the ordered pair, i.e., it is a set of\nfandg, and (a,b)/ne}ationslash= (b,a)).16\n1. Cohomology equation\nFrom Eq. (78), the association property is given by the following equ ation:\n/summationdisplay\ni+j=m\ni,j≥0βi(βj(f,g),h)) =/summationdisplay\ni+j=m\ni,j≥0βi(f,βj(g,h)) (81)\nwithβ0(f,g)≡f·g. (The symbol “·” represents the usual commutative product, and βj∈C2, j= 0,1,···.) Because\nβ1is the Poisson bracket, which is bi-linear differential operator, we de fineβj(∈C2, j= 2,3,···) as a differential\noperator on a manifold M; moreover, we also assume that p-cochains are differential operators and products of\nfunctions.\nHere,AandCk(A;A) represent a space of smooth functions on a manifold Mand a space of multilinear differential\nmaps fromA⊗ktoA, respectively. Degree of βk∈Ck(A;A) is defined by\ndeg(βk) :=kfork≥2. (82)\nNow, we introduce a coboundary operator ∂C:Ck(A;A)→Ck+1(A;A) [20, 21];\n(∂Cβk)(f0,···,fk/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nk+1) :=f0βk(f1,···,fk/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nk)+k/summationdisplay\nr=1(−1)rβk(f0,···,fr−1·fr,···fk/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nk)\n+(−1)k−1βk(f0,···,fk−1/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nk)fk (83)\nwithβk∈Ck(A;A); note that ∂2\nC= 0, and thus, ∂Cis the boundary operator. The Gerstenhaber bracket is defined\nas [,]C:Ck(A;A)⊗Ck′(A;A)→Ck+k′−1(A;A) [22]:\n[βk,βk′]C(f0,f1,···,fk+k′−2/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nk+k′−1)\n:=k−1/summationdisplay\nr=0(−1)r(k′−1)βk(f0,···,fr−1,βk′(fr,···,fr+k′−1),fr+k′,···,fk+k′−2/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright\nk)\n−k′−1/summationdisplay\nr=0(−1)(k−1)(r+k′−1)βk′(f0,···,fr−1,βk(fr,···,fr+k−1),fr+k,···,fk+k′−2/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright\nk′),\n(84)\nwhereβk∈Ck(A;A) andβk′∈Ck′(A;A). Note that ∂2\nC= 0, and thus, ∂Cis the boundary operator.\nBy using the coboundary operator and the Gerstenhaber bracke t, Eq. (81) is rewritten as\n∂Cβm=−1\n2/summationdisplay\ni+j=m\ni,j≥0[βi,βj]C (85)\nwithβj∈C2(A;A) (j= 1,2,···). For example, Eq. (81) for m= 0,1,2 is given as:\n(f·g)·h=f·(h·g) for m= 0,\n{f·g,h}+{f,g}·h={f,g·h}+f·{g,h} form= 1,\nβ2(f·g,h)+{{f,g},h}+β2(f,g)·h=β2(f,g·h)+{f,{g,h}}+f·β2(g,h) form= 2,(86)\nwhere we have used β0(f,g) :=f·gandβ1(f,g)≡{f,g}. The coboundary operator for β∈C2(A;A) is given by:\n(∂Cβ)(f,g,h) =f·β(g,h)−β(f·g,h)+β(f,g·h)−β(f,g)·h; (87)\nmoreover, the Gerstenhaber bracket in terms of βi,βj∈C2(A;A) is given by\n[βi,βj]C(f,g,h) =βi(βj(f,g),h)−βi(f,βj(g,h))+βj(βi(f,g),h)−βj(f,βi(g,h)). (88)\nUsing the above Eqs. (86-88), we can check the equivalence betwe en Eq. (81) and Eq. (85).17\nEquation (85) is called the cohomology equation, and the star produ ct is constructed by using solutions of the\ncohomology equation. If we add Eq. (85) with respect to m= 0,1,2,···, we obtain the following equation:\n∂Cβ+1\n2[β,β]C= 0 (89)\nwithβ≡/summationtext∞\nj=0βj;β,βj∈C2(A;A),j= 0,1,2,....\nHere, we identify the vector fields ∂i,∂j∈TMwith anti-commuting numbers ˜ ηi,˜ηj(˜ηi˜ηj=−˜ηj˜ηi),i,j= 1,2,...,d;\nthus the Poisson bracket αij(∂i∧∂j)/2 is rewritten by α=αij˜ηi˜ηj/2. Now, we define the Batalin-Vilkovisky (BV)\nbracket:\n[α1,α2]BV:=−d/summationdisplay\ni=1/parenleftigg\nα1← −∂\n∂xi− →∂α2\n∂˜ηi−α1← −∂\n∂˜ηi− →∂α2\n∂xi/parenrightigg\n(90)\nwithα1,α2∈T2. By using the BV bracket, the Jacobi identity is rewritten as\n∂BVα+1\n2[α,α]BV= 0, (91)\nwithα1,α2∈T2; forα=αij˜ηi˜ηj,α← −∂/∂xl=− →∂α/∂xl:= (∂xlαij)˜ηi˜ηjand− →∂α/∂˜ηl=−α← −∂/∂˜ηl:=αij(δil˜ηj−˜ηiδjl).\nBy using the BV bracket, the Jacobi identity is rewritten as\n∂BVα+1\n2[α,α]BV= 0, (92)\nwhere∂BV≡0, i.e.,∂2\nBV= 0.\nNow, we generalize the differential ∂BVand BV-bracket [ ,]BVforαk∈Tkandαk′∈Tk′as follows (Tk≡\nΓ(M,/logicalandtextkTM)):\n∂BV:Tk→Tk+1\n∂BV:= 0, (93)\n[,]BV:Tk/circlemultiplydisplay\nTk′→Tk+k′−1\n[αk,αk′]BV:=−d/summationdisplay\ni=1/parenleftigg\nαk← −∂\n∂˜ηi− →∂αk′\n∂xi−αk← −∂\n∂xi− →∂αk′\n∂˜ηi/parenrightigg\n(94)\nwithαk= (αk)i1,···,ik(x)ηi1∧···∧ηik∼(αk)i1,···,ik(x)˜ηi1···˜ηik, andαk′= (αk′)i0,···,ik′(x)ηi0∧···∧ηik′∼\n(αk′)i0,···,ik′(x)˜ηi0···˜ηik′; degree ofα∈Tkis defined by\ndeg(α) =k−1, α∈Tk. (95)\nThe cochain algebra is defined by the set of the differential operato r∂C, the Gerstenhaber bracket [ ,]Cand\nC:=/circleplustext∞\nk=2Ck, i.e., (∂C,[,]C,C); in addition, the multi-vector algebra is defined by the set of the diff erential operator\n∂BV:= 0, BV bracket [ ,]BVandT:=/circleplustext∞\nk=1Tk, i.e., (∂BV,[,]BV,T). The cochain algebra and the multi-vector\nalgebra satisfy the following common relations:\n∂2= 0, (96)\n∂[γ1,γ2] = [∂γ1,γ2]+(−1)deg(γ1)[γ1,∂γ2], (97)\n[γ1,γ2] =−(−1)deg(γ1)deg(γ2)[γ2,γ1] (98)\n[γ1,[γ2,γ3]]+(−1)deg(γ3)(deg(γ1)+deg(γ2))[γ3,[γ1,γ2]]+(−1)deg(γ1)(deg(γ2)+deg(γ3))[γ2,[γ3,γ1]] = 0 (99)\nwithγ1,γ2,γ3∈G≡(CorT),∂≡∂(CorBV), and [,]≡[,](CorBV). Therefore, the two algebra can be compiled\nin the so-called the differential graded Lie algebra (d.g.L.a) ( ∂,[,],G), whereG:=/circleplustext∞\nk=1Gkis a graded k-vector18\nspace withGkhas a degree deg( x)∈Z(x∈Gk;Zis the set of integers), and d.g.L.a. has the linear operator ∂and\nthe bi-linear operator [ ,]:\n∂:Gk→Gl, xk∈Gk, xl∈Gl,\ndeg(∂xk) = deg(xk)+1 = deg( xl), (100)\n[,] :Gk/circlemultiplydisplay\nGl→Gm, xk∈Gk, xl∈Gl, xm∈Gm,\ndeg([xk,xl]) = deg(xk)+deg(xl) = deg(xm), (101)\nwhere∂and [,] satisfy Eqs. (96), (97), (98) and (99). In d.g.L.a., Eqs. (89) and (92) are compiled in the so-called\nMaurer-Cartan equation [23]:\n∂γ+1\n2[γ,γ] = 0 (102)\nwithγ∈G. Therefore, the deformation quantization Fis a map:\nF:G→G, γ1/ma√sto→γ2,\n∂γi+1\n2[γi,γi] = 0, i= 1,2. (103)\nNamely, the deformation quantization is a map holding a solution of the Maurer-Cartan equation (102). In the\nsection IVB2, we will introduce a L∞algebra, and will redefine the deformation quantization; in the L∞algebra,\nthe Maurer-Cartan equation (102) is rewritten as Q(eγ) = 0 (Qandeγwill be defined in IVB2).\n2.L∞algebra\nNow we define a commutative graded coalgebra C(V).\nFirst, wedefineaset( V,△,τ,Q), whereV:=/circleplustext\nn=1,2,···V⊗nwithagraded k-vectorspace V⊗n(n= 1,2,...),△and\nQrepresent the coproduct and codifferential operator, respect ively; moreover, τdenotes cocommutation (definition\nis given later). The coproduct, cocommutation and codifferential o perator satisfy the following equations:\n(△⊗id)◦△= (id⊗△)◦△, (104)\nτ△=△, (105)\n△◦Q= (id⊗Q+Q⊗id)◦△, (106)\nτ(x⊗y) := (−1)degco(x)degco(y)y⊗x, (107)\nwith degco(x) := deg(x)−1, wherex∈V⊗deg(x)andy∈V⊗deg(y).Qrepresents a codifferential operator adding one\ndegree:Q∈Hom(V⊗m,V⊗(m+1)) with degco(Q(x)) = degco(x)+1 forx∈V⊗m,∃m∈Z+(the explicit form of Qis\ngiven later; Z+:={i|i>0, i∈Z}).\nBy usingτ, we define the commutative graded coalgebra C(V) from (V,△,τ,Q); the identify relation ∼is defined as\nx⊗y∼(−1)degco(x)degco(y)y⊗x, i.e.,x⊗yand (−1)degco(x)degco(y)y⊗xare identified. Now, we define the commutative\ngraded tensor algebra:\nC(V) :=V/∼ ≡{[x]|x∈V}, (108)\nwhere [x] ={y|y∈V, x∼y}, and degco(x1⊗x2⊗···⊗xn) = degco(x1) + degco(x2) +···+ degco(xn) with\nx1⊗x2⊗···⊗xn∈V⊗degco(x1)⊗V⊗degco(x2)⊗···⊗V⊗degco(xn); a product inC(V) is defined by xy:= [x⊗y].\nNamely, inC(V),\nx1x2···xixi+1···xn= (−1)degco(xi)degco(xi+1)x1x2···xi+1xi···xn (109)\nwithn≥2. (Let us recall that the derivation of the exterior algebra from t he tensor space;VandC(V) correspond\nto the tensor space and the exterior algebra, respectively.)\nMoreover, in the case that Q2= 0, the commutative graded coalgebra C(V) is called the L∞algebra. For the L∞\nalgebra, the coproduct and codifferential operator are uniquely d etermined by using multilinear operators:\nlk: (V⊗k∈C(V)))→V∈C(V) (110)\ndegco(lk(x1···xk)) = degco(x1)+···+degco(xk)+1 (111)19\nas follows:\n△(x1···xn) =/summationdisplay\nσn−1/summationdisplay\nk=1ε(σ)\nk!(n−k)!(xσ(1)···xσ(k))⊗(xσ(k+1)···xσ(n)), (112)\nQ=∞/summationdisplay\nk=1Qk, (113)\nQk(x1···xn) =/summationdisplay\nσε(σ)\nk!(n−k)!lk(xσ(1)···xσ(k))⊗xσ(k+1)⊗···⊗xσ(n),\n(114)\nwhereε(σ) represents a sign with a replacement σ:x1x2···xn/ma√sto→xσ(1)xσ(2)···xσ(n). From the condition Q2= 0,\nwe can identify ( l1, l2) with (∂,[,]) in d.g.L.a. If we put l3=l4=···= 0,Q(eα) = 0 forα∈Vis equal to the\nMaurer-Cartan equation Eq. (102) in d.g.L.a [17], where\neα≡1+α+1\n2!α⊗α+··· (115)\nwithα⊗n⊗1≡1⊗α⊗n≡α⊗nforn= 1,2,···. Therefore, the deformation quantization is a map:\nF:C(V)→C(V),\nγ1/ma√sto→γ2 (116)\nwith\nQ(eγi) = 0, i= 1,2. (117)\nTo constitute such a map F, we introduce the L∞mapF, which is defined as the following map holding degrees\nof coalgebra:\nF:C(V)→C(V), v1,v2∈C(V),\nv1/ma√sto→v2,\ndegco(v1) = degco(v2); (118)\nmoreover, the L∞map satisfies the following equations:\n△◦F= (F⊗F)◦△, (119)\nQ◦F=F◦Q. (120)\nA form of such a map is limited as [17]:\nF=F1+1\n2!F2+1\n3!F3+···, (121)\nFl:C(V)→V⊗l(⊂C(V))\nFl(x1···xn) =/summationdisplay\nσ/summationdisplay\nn1,···,nl≥1\nn1+···+nl=nε(σ)\nn1!···nl!\n·Fn1(xσ(1)···xσ(n1))⊗···⊗ Fnl(xσ(n−nl+1)···xσ(n)), (122)\nwhere Fnis a map fromC(V) toV(⊂C(V)) holding degrees;\nFn: V⊗n(⊂C(V))→V(⊂C(V))\nx1⊗···⊗xn/ma√sto→x′,\ndegco(x1)+···+degco(xn) = degco(x′). (123)\nHere we define β:=/summationtext∞\nn=11\nn!Fn(α···α), which satisfies F(eα) =eβ. The map Fholds solutions ofMaurer-Cartan\nequationsQ(eα) = 0 andQ/parenleftbig\neβ/parenrightbig\n= 0; from\nQ(eβ)≡Q◦F(eα) (124)20\nand the definition of the L∞map:Q◦F=F◦Q, we obtain the following equation:\nQ(eβ) =F◦Q(eα) = 0, (125)\nwhich means that the L∞map transfers a solution of the Maurer-Cartan equation from ano ther solution.\nNow, we return to the deformation quantization. The multi-vector spaceT, is embedded in C(V);C(V) =\n(T,△T,τ,QT), where△T(x1x2) :=x1∧x2forx1,x2∈T, (QT)1:=∂BV≡0, (QT)2:= [,]BV, and (QT)l:= 0 for\nl= 3,4,...;τreplaces the wedge product “ ∧” with the product “ ·”. For the cochain space C, it is also embedded in\nC(V);C(V) = (C,△C,τ,QC), where△C(x1x2) :=x1∧x2forx1,x2∈C, (QC)1:=∂C, (QC)2:= [,]C, and (QC)l:= 0\nforl= 3,4,....\nThe star product is given by f ⋆g=f·g+β(f⊗g), which is identified as the map F0+F1withF0:=µ◦.\nHere we summarize the main results of the succeeding sections witho ut explaining their derivations. The map\nF1is given by a path integral of a topological field theory having super fi elds:X:= (X1,...,XN); and scalar\nfields:ψ:= (ψ1,...,ψN),λ:= (λ1,...,λN), andγ:= (γ1,...,γN); and one-form fields: θ:= (θ1,θ2,...,θm),\nA:= (A1,...,A N),A+:= (A+1,...,A+N), andη:= (η1,...,ηN); and Grassmann fields ci:= (c1,...,cN); on a disk\nΣ ={z|z=u+iv, u,v∈R, v≥0}[17, 24, 25]. These fields are defined in section IVC. Using these field s, the\nmapFn:V⊗n→Vis given as follows:\nFn(α1,···,αn)(f1⊗···⊗fm)(x) =/integraldisplay\nei\n/planckover2pi1S0\nghi\n/planckover2pi1Sα1···i\n/planckover2pi1SαnOx(f1,...,f m) (126)\nfor any function f1,...,f m, which depend on x; in this paper, xrepresents the coordinate in the classical phase space.\nHereα1,α2,...,α n∈V, andmis defined by degco(αi)+2, which is common and independent of i(i= 1,2,...,n).\nThe operatorOxis defined as\nOx(f1,...,f m) :=/integraldisplay\n[f1(X(t1,θ1))···fm(X(tm,θm))]δx(ψ(∞)) (127)\n≡/integraldisplay\n1=t1>t2>···>tm=0f1(ψ(t1))m−1/productdisplay\nk=2∂ik/bracketleftbig\nf(ψ(tk))A+ik(tk)/bracketrightbig\nfm(ψ(0))δx(ψ(∞)) (128)\nform,δx(ψ(t)) :=/producttextd\ni=1δ(ψi−xi)γi(t), andt∈∂Σ, where\nS0\ngh:=/integraldisplay\nΣ/bracketleftbig\nAi∧dψi−∗Hdγi∧dci−λid∗HAi/bracketrightbig\n(129)\nwith a Hodge operator ∗H:∧k→∧2−k, (k= 0,1,2); we will introduce the explicit definition in section IVD2.\nMoreover, for αr:=αi1,···,inrr(X)∂i1∧···∧∂inr(nr>1 is an integer number; degco=nr−2),\nSαr:=/parenleftbigg/integraldisplay\nΣ/integraldisplay\nd2θ1\nnrαi1···inrr(X)ηi1···ηinr/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nΦ∗=∂ϕ, (130)\nwhere the subscript Φ∗=∂ϕmeans that the fields ( X,η,A+) go to (ψ,A,0). These results lead to the diagram\ntechnique in section IVE and the explicit expression of the star prod uct in section IVF.\nC. Topological string theory\nIn this section, we expound the fields: A,ψ,c,γ,λ,θ,η,A+, andX. The simplest topological string theory is\ndefined the following action:\nS0:=/integraldisplay\nΣd2σǫµνAµ,i∂νψi=−1\n2/integraldisplay\nΣd2σǫµνFµν,iψi(131)\nwith local coordinates σ= (σ1,σ2) on a disk Σ (we consider that the disk is the upper-half plain in the com plex\none, i.e., Σ :={z|z=u+iw;w≥0;u,w∈R}), whereAµ,i(σ) andψi(σ) are U(1) gauge fields and scalar fields,\nrespectively; Fµν,i(σ) is a gauge strength ( µ= 1,2 andi= 1,...,N). The other fields c,γ,λ,θ,η, andA+are\nintroduced in section IVC1; we discuss the gauge fixing method using the so-called BV-BRST formalism [26, 27]\n(where the BV refers to Batalin and Vilkovisky; BEST refers to Becc hi, Rouet, Stora and Tyutin). In section IVC2,\nwe discuss the gauge invariance of the path integral, and introduce the SD operator. In section IVC3, we see that\ncorrespondence of the deformation quantization and topological string theory.21\n1. Ghost fields and anti-fields\nHere, we quantize the action (131) using the path integral. Roughly speaking, the path integralis the Gaussintegral\naround a solution of an equation of motion. In many cases, a genera l actionShas no inverse. Therefore, we will add\nsome extra fields, and obtain the action Sghhaving inverse, which is called as the quantized action.\nNow, we discuss a general field theory. We assume that a general a ctionSis a function of fields φi, i.e.,S=S[φi];\neach fieldφiis labeled by a certain integer number, which is called as a ghost number gh(φi) (it is defined below).\nφi\nCdenotes that the fields fixed on the solution of the classical kinetic e quation:δS0/δφi= 0, and the subscript of\nthe fields represents a number of fields. Because the Gauss integr al is an inverse of a Hessian, a rank of the Hessian\nshould be equal to the number of the fields. Here a Hessian is defined by:\nK[φi,φj] :=− →δ\nδφiS← −δ\nδφj, (132)\nwhere− →δ\nδφiφj1φj2···φin:=δj1\niφj2···φj2+(−1)j1iφj1δj2\ni···φjn\ni+···+(−1)i(j1+j2+···+j(n−1))φj1φj2···φj(n−1)δjn\ni, and\nφi1φi2···φin← −δ\nδφj:=φi1φi2···φi(n−1)δin\nj+(−1)injφi1φi2···δi(n−1)\njφin+(−1)(i2+···+in)jδi1\njφi2···φin; for a boson φi,i\nis a ghost number gh( φi); for a fermion φi,iis gh(φi)+1.\nWe define the rank of the Hessian Kand the number of the fields φiby♯Kand♯φi, respectively. Generally\nspeaking,♯K <♯φi, because an action has some symmetries δRφi:=Ri\njφjwith nontrivial symmetry generators Ri\nj,\nwhere is satisfies the following equation:\nS← −δ\nδφiRi\nj= 0 (133)\nwithRi\nj|φk=φkc/ne}ationslash= 0. The nontrivial symmetry generator decrease the rank of Hes sian from the number of fields. To\ndefine the path integral, we should add ( ♯φi−♯K) virtual fields [26–28]. The additional fields are called as ghost fields\nΦα1and antifields Φ∗\nαl(l= 0,1), and these fields are labeled by ghost numbers. For Φα1, the ghost number is defined\nby gh(Φα1) := 1. The fields and ghost fields have antifields Φ∗\nαl. The antifields corresponding to φ≡Φα0and Φα1\nare described as Φ∗\nα0and Φ∗\nα1, respectively. The ghost number of Φ∗\nαlis defined by gh(Φ∗\nαl) =−l−1. Statistics of\nthe anti-fields is opposite of fields, i.e., if the fields are fermions(boso ns), the anti-fields are bosons(fermions). (Here\nwe only consider the so-called irreducible theory. For a general the ory, see references [26–28].)\nUsing these fields, we will transform the action S[Φα0]/ma√sto→Sgh[Ψ], where Ψ := (Φαl,Φ∗\nαl) withl= 0,1 and\nαl∈Z+:={i|i>0, i∈Z}(Zrepresents the set of integers), Φα0:=φiare fields, Φα1represents ghost fields, and\nΦ∗\nαldenotes anti-fields of the fields Φαl. Hereafter we write a function space created by the fields and ant i-fields as\nC(Ψ). It is known that Sghis given by\nSgh=S+Φ∗\nα0Rα0α1Φα1+O(Ψ3). (134)\nNote that the anti-fields will be fixed, and ♯Ψ =/summationtext\nl♯Φαl(see section IVC3).\nFor the topological string theory, the fields φαare U(1) gauge fields Ai,µand scalar fields ψiwithi= 1,...,Nand\nµ= 1,2; namely, Φα0≡φα:= (Ai,µ,ψi). Since♯Ai,µ= 2Nand♯ψi=N, the fields number ♯φαis 3N. The action\n(131) has the U(1) gauge invariance:\nδ0Aµ,i=∂µδj\niχj, (135)\nδ0ψi= 0, (136)\nδ0χi= 0 (137)\nwithχirepresentsa scalarfunction ( i= 1,...,N). Therefore, the topologicalstring theory has 2 Nlinear-independent\nnontrivial symmetry generators. Here we replace the scalar fields\nchiiwith ghost fields ci(BRST transformation). Moreover, we add antifields A∗\ni,µ; since the gauge transformation\ndoes not connect to ψand the other fields, we does not add ψ∗(the space of fields and ghost fields has 2 Nsymmetry\ngenerators, and the space of anti-fields and the anti-ghosts also have 2Nsymmetry generators corresponding to U(1)\ngauge symmetry; see Figure 3):\nR(µ,i)\nβ=∂µδi\nβ,(β= 1,2,...,N). (138)22\nFIG. 3. The Hessian matrix: K[Ψα(σ1),Ψβ(σ2)] :=δ\nδΨα(σ2)Sδ\nδΨβ(σ1)The first column and first raw represent the right-hand\nside and the left-hand side of variation functions, respect ively.ˆ∂σj:=δ(σ−σ1)∂σjδ(σ−σ2) represents a non-trivia Noether\ncurrent ( j= 1,2). The Hessian is block diagonal matrix; the ranks of the upp er left and the lower right parts are 2 = 3 −1.\nTherefore, the total rank of the Hessian is 2+2 = 4 ( iis fixed).\nIn this case, ♯K(φα,φβ) = 3N−2N; on the other hand, the action is a function of 3 Nfields (Aµ,i,ψi),Nghost fields\nciand 2Nanti-fieldsA∗\ni,µ. Therefore, a rank of the Hessian corresponding to ( S0)ghis calculated by\nrankK(Ψ,Ψ)|Ψc= rankK(φ,φ)|Ψc+♯ci+♯A∗\ni,µ\n=N+N+2N\n= 4N. (139)\nSince♯Φ = 4N(antifields will be fixed), the field number of the path-integral of ( S0)ghis equal to the rank of the\nHessian of ( S0)gh; hence, the path-integral of the action ( S0)ghbecome well-defined.\nFinally, the gauge invariance action is written by\n(S0)gh=S0+/integraldisplay\nΣ(Ai)+∧δ0Ai (140)\nwithAi:=Aµ,idσµandδ0Aµ,i=∂µδj\nicj, where we define Φ+\nαusing a Hodge operator ∗H: Φ+\nα≡ ∗HΦ∗\nα(the\ndefinition of the Hodge operator is depend on the geometry of the d isk Σ; we will introduce the explicit definition in\nsection IVD2), which is also called as the anti-field.\n2. Condition of gauge invariance of classical action\nIn this section, we will add interaction terms: Sgh:= (S0)gh+g(S1)gh+···, wheregrepresents an expansion\nparameter, and we will see that Sghis uniquely fixed except a certain two form αby a gauge invariance condition.\nNote thatαsatisfies the Jacobi’s identity. Therefore, we can identify αwith the Poisson bracket.\nFirst we discuss the gauge invariant condition. If we identify the field s and anti-fields with coordinates qand\ncanonical momentum p, i.e., (Φαi,Φ∗\nαi)↔(qαi,pαi), and we also identify the action Sand the Hamiltonian H:\nS↔H. In the analytical mechanics, δam:={H,}represents a transform along the surface H(q,p) = constant,\ni.e.,δamholds the Hamiltonian. Similarly, we can define a gauge transformation , which holds the action S, using the\nPoisson bracket in the two-dimension field theory. It is known as the Batalin-Vilkovisky (BV) bracket [26, 27]; the\ndefinitions of the bracket are\n{f,g}BV:=/summationdisplay\nαi\ni=0,1,.../parenleftbiggδf\nδΦαiδg\nδΦ∗αi−δf\nδΦ∗αiδg\nδΦαi/parenrightbigg\n(141)\nwithf, g∈C(Ψ).\nThe BV bracket has the ghost number 1, then a BV-BRST operator δBV:={S,}BVadds one ghost number. The\nBV bracket satisfies the following equations:\n{f,g}BV=−(−1)(gh(f)+1)(gh( g)+1){g,f}BV, (142)\n(−1)(gh(f)+1)(gh( h)+1){f,{g,h}BV}BV+cyclic = 0 , (143)\n{f,gh}BV={f,g}BVh+(−1)(gh(f)−1)gh(g)g{f,h}BV (144)23\nwithf, g, h∈C(Ψ).\nUsing the BV-BRST operator, the gauge invariance of action Sis written as δBVS= 0, i.e.,\n{S,S}BV= 0, (145)\nwhich is called the classical master equation. We use this equation and Eqs. (142) and (143); we obtain δ2\nBV= 0,\nwhich corresponds to the condition of the BRST operator: δ2\nBRST= 0 (δBRSTis the BRST operator). Therefore, the\nDV-BRST operator is the generalized BRST one.\nNext, we discuss generalization of the topological field theory. Let us write a generalized action Sghas\nSgh= (S0)gh+g(S1)gh+g2(S2)gh+···, (146)\nwheregis an expansion parameter. Using gauge invariance condition (145), (Sn)gh(n= 1,2,···) is given by a\nsolution of the following equation:\n∂n\n∂gn{Sgh,Sgh}BV/vextendsingle/vextendsingle/vextendsingle/vextendsingle\ng=0= 0. (147)\nThe general solution is given by [25]\n(S1)gh=/integraldisplay\nΣd2σ/bracketleftigg\n1\n2αij(AiAj−2ψ+\nicj)+∂αij\n∂ψk/parenleftbigg1\n2(c+)kcicj−(A+)kAicj/parenrightbigg\n+1\n4∂2αij\n∂ψk∂ψl(A+)k(A+)lcicj/bracketrightigg\n, (148)\n(Sn>1)gh= 0 (149)\nwith (A+)i≡ ∗HA∗\nµ,i=dσµεµνA∗\nν,i,ψ+\ni≡∗Hψ∗\ni=εµνdσµ∧dσνψ∗\niand (c+)i≡∗H(c∗)i=εµνdσµ∧dσν(c∗)i\n(εµν=−ενµ, ε12= 1), where αijis a function of ψ, and satisfies the following equation:\n∂αij\n∂ψmαmk+∂αjk\n∂ψmαmi+∂αkl\n∂ψmαmj= 0. (150)\nHere, if we identify ψiwithxi, this equation is the Jacobi identity of Poisson bracket. Therefor e, we can identify the\nPoisson bracket with the topological string theory.\n3. Gauge invariance in path integral\nNow we discuss the path integral of the topological string theory/integraltext\nDΦV(Ψ) withV(Ψ) =Oei\n/planckover2pi1S, and an observable\nquantity operator O. Note that this path-integral does not include integrals in terms of the anti-fields. Therefore,\nwe must fix the anti-fields; then, we consider that the anti-field Φ∗is a function of the field Φ, i.e., Φ∗= Ω(Φ) and\nΩ∈C(Φ = Ψ) Namely, the path integral is defined by\n/integraldisplay\nDΦV(Ψ)/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nΦ∗=Ω. (151)\nA choice of Ω(Φ) is corresponding to the gauge fixing in the gauge the ory. The path integral must be independent\nto the gauge choice (gauge invariance). To obtain a gauge invariant condition, we take the variation of the path\nintegral in terms of anti-fields, and obtain the following gauge invaria nt condition [29]:\n△SDV(Ψ) = 0, (152)\nwhere we have introduced the Schwinger-Dyson (SD) operator:\n△SD:=/summationdisplay\nαl(−1)αlδ\nδΦαlδ\nδΦ∗αl, (153)24\nwhere (−1)αlis defined as follows: if Φαlrepresents a boson, ( −1)αl= (−1)gh(Φαl); if Φαlrepresents a fermion,\n(−1)αl= (−1)(gh(Φαl)+1). Equation (152) is called the quantum master equation. It is known t hat the following two\nconditions are equivalence:\n△SDV(Ψ) = 0⇐⇒Ω =− →δϕ\nδΦa,∃ϕ, (154)\nwhereϕis called the gauge-fixing fermion (an example will be shown later).\nTo performthe path integral, we generalizethe classicalaction Sghtoa quantum action W=Sgh+i/planckover2pi1W1+(i/planckover2pi1)2W2+\n···. The correction terms Wn(n= 1,2,...) are calculated from the master equation:\n△SDei\n/planckover2pi1W= 0, (155)\nor\n{Sgh,Sgh}BV= 0, (156)\n{W1,Sgh}BV+i/planckover2pi1△SDSgh= 0, (157)\n{W2,Sgh}BV+i/planckover2pi1△SDW1+1\n2{W1,W1}BV= 0, (158)\n···\nIn the case where △SDSgh= 0, we can put W1=W2=···= 0. Fortunately, the topological string theory satisfies\n△SDSgh= 0. Therefore, we do not have to be concerned about the quantu m correction of the action.\nFinally, we consider the gauge fixing. Here we employ the Lorentz gau ge:\nd∗HAi= 0, (159)\nand we add the integral of the Lorentz gauge to Sgh. However, the path integral should hold gauge invariance, i.e., the\npath integral should be independent of gauge fixing term. Then, th e gauge fixing can be written gauge-fixed fermion:\nϕ:=/integraldisplay\nΣγi(d∗HAi) =−/integraldisplay\nΣdγi∗HAi, (160)\nwhere we introduced Nfieldsγi(i= 1,2,...,N), and anti-fields γ+\niare given by\nγ+\ni=− →∂ϕ\n∂γi=d∗HAi. (161)\nNow, we employ the Lagrange multiplier method, and introduce Nscalar fields λi. The gauge-fixed action is written\nby\nSgf=Sgh−/integraldisplay\nΣγid∗HAi (162)\n=Sgh−/integraldisplay\nΣλiγ+\ni. (163)\nThe other anti-fields are also fixed by this gauge-fixing fermion:\nψ+\ni=c+\ni=λ+\ni= 0, (164)\nA+\ni=∗Hdγi. (165)\nGauge fixed action Sgfis written by\nSgf=/integraldisplay\nΣ/bracketleftigg\nAi∧dψi+1\n2αijAi∧Aj−∗Hdγi∧/parenleftbigg\ndci+∂αkl\n∂ψiAkcl/parenrightbigg\n−1\n4∗Hdγi∧∗Hdγj∂2αkl\n∂ψi∂ψjckcl−λid∗HAi/bracketrightigg\n. (166)25\nHere we perform the following variable transformations:\nXi:=ψi+θµA∗\nµ−1\n2θµθνc+i\nµν, (167)\nηi:=ci+θµAi,µ+1\n2θµθνψ+\ni,µν, (168)\nwhereθµϑν=−θνθµ; gh(θµ) = 1. For any scalar field f(u) (u∈Σ),˜f(u,θ) :=f(u)+θµf(1)\nµ(u)+1\n2θµθνf(2)\nµνis called\nas the super field, where f(1)andf(2)represent a one-form field and a two-form field, respectively.\nBy using the super fields, the gauge fixed action Sgfcan be rewritten as\nSgf=/integraldisplay\nΣ/integraldisplay\nd2θ/bracketleftbigg\nηiDXi−λid∗HAi+1\n2αij(X)ηiηj/bracketrightbigg\n, (169)\nwhereD:=θµ∂\n∂uµ. This is the final result in this section. Hereafter, we write S0\ngf:=/integraltext\nΣ/bracketleftbig\nηiDXi−λid∗HAi/bracketrightbig\nand\nS1\ngf:=/integraltext\nΣαijηiηj/2.\nD. Equivalence between deformation quantization and topol ogical string theory\nWereturntothediscussionaboutthedeformationquantization. H ereweseethatthe equivalenceofthedeformation\nquantization and the topological string theory, and introduce the perturbation theory of the topological string theory,\nwhich is equal to Kontsevich’s deformation quantization [17].\n1. Path integral as L∞map\nHere we summarize correspondence between Path integral with L∞map.\nFirst we note that the map: α:=α(x)µνηµην/2/ma√sto→Sα:=S1\ngf=/integraltext\nΣα(x)µνηµην/2 is isomorphic, because\n{Sα1,Sα2}BV=S{α1,α2}BV.\nSD operator satisfies the conditions of codifferential operator QinL∞algebra, where the vector space and the\ndegree of the space correspond to C(Ψ) and the ghost number, respectively.\nThe path integral/integraltext\nei\n/planckover2pi1S0\ngfgives the deformation quantization F0+F1. The master equation\nQei\n/planckover2pi1Sα= 0 (170)\nwithQ=△SDis corresponding to the L∞map’s condition QF= 0\nForαr:=αi1,i2,···,imηi1ηi2···ηim/m! with a positive integer m,Fn:V⊗n\n1⊗→V2is given by\nFn(α1,...,α n)(f1⊗···⊗fm)(x) :=/integraldisplay\nei\n/planckover2pi1S0i\n/planckover2pi1Sα1···i\n/planckover2pi1SαnO(f1,...,f m), (171)\nwhereSαris the expansion of Sα, and is defined as\nSαr:=/parenleftbigg/integraldisplay\nΣ1\nm!αi1···im(X)ηi1···ηim/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nΦ∗=∂ϕ, (172)\nandOis chosen to satisfy\nQ◦F=F◦Q, (173)\nwhere F:=F0+F1+F2+···. We putOas follow:\nO(f1,...,f m) =/integraldisplay\nBm[X(t1,θ1))···fm(X(tm,θm))](m−2)δx(X(∞)), (174)\n≡/integraldisplay\n1=t1>t2>···>tm=0f1(ψ(1))m−1/productdisplay\nk=2∂ik/bracketleftbig\nf(ψ(tk))A+ik(tk)/bracketrightbig\nfm(ψ(0))δx(ψ(∞)) (175)26\nwhere the subscript ( m−2) denotes that ( m−2) forms are picked up from the products of super fields, and Bm\nrepresents the surface of the disk Σ, i.e., tis the parameter specifying the position on the boundary ∂Σ (1 =t1>\nt2>···>tm−1>tm= 0).\nTo be exact, the action and fields include gauge fixing terms, ghost fi elds and anti-field. Finally, the deformation\nquantization is given as follow:\n(f ⋆g)(x) =/integraldisplay\nDΦf(ψ(1))g(ψ(0))δ(xi−ψi(∞))ei\n/planckover2pi1Sgf. (176)\n2. Perturbation theory\nNowweseethattheperturbationtheoryofthetopologicalstring theory. First, wewritetheactionas Sgf=S0\ngf+S1\ngf.\nThe first term is defined as\nS0\ngf=/integraldisplay\nΣ/bracketleftbig\nAi∧(dξi+∗Hdλi)+cid∗Hdγi/bracketrightbig\n, (177)\nwhereξi≡ψi−xi, and we have expanded ψiaroundxi. The path integral of an observable quantity /an}bracketle{tO/an}bracketri}htis given by\n/integraldisplay\nei\n/planckover2pi1SgfO=∞/summationdisplay\nn=0in\n/planckover2pi1nn!/integraldisplay\nei\n/planckover2pi1S0\ngf(S1\ngf)nO, (178)\nwhere/integraltext\n:=/integraltext\nDξDADcDγDλ. This expansion corresponds to the summation of all diagrams by th e contractions of\nall pairs in terms of fields and ghost fields. From equation (177), pro pagators are inverses of\nd⊕∗Hd, d∗Hd. (179)\nHere we assume that the disk is the upper complex plane: Σ = {z|z=u+iv, u,v∈R, v≥0}withi2=−1, and\nthe boundary is ∂Σ ={z|z=u, u∈R}. (Rrepresents the real number space, and zdenotes a complex number.)\nThe Hodge operator ∗His defined by\n/braceleftbigg\n∗Hdu=dv\n∗Hdv=−du/ma√sto−→/braceleftbigg\n∗Hdz=−idz\n∗Hdz=idz, (180)\nwherezrepresents the complex conjugate of z. Moreover,\ndz=du∂\n∂u+dv∂\n∂v=dz∂\n∂z+dz∂\n∂z, (181)\nδz(w) :=δ(w−z)duw∧dvw,/integraldisplay\nδz(w) = 1, (182)\nwherew∈Cwith the complex number plane C, andw≡uw+ivw.\nNow, we calculate Green functions of d⊕∗Hdandd∗Hd, because the Green functions are inversesof these operators:\nDwG(z,w) = i/planckover2pi1δz(w), (183)\nwhereDw=dw⊕∗Hdwordw∗Hdw. The solution depends on the boundary condition. In the case that zandw\nsatisfy the Neumann boundary condition, a solution is a function of\nφh(z,w) :=1\n2ilog(z−w)(z−w)\n(z−w)(z−w). (184)\nOn the other hand, zandwsatisfy the Dirichlet boundary condition, a solution is a function of\nψh(z,w) := log/vextendsingle/vextendsingle/vextendsingle/vextendsinglez−w\nz−w/vextendsingle/vextendsingle/vextendsingle/vextendsingle. (185)\nTheNeumannboundaryconditionis0 = ∂u1G(z,w)|u2=0,andtheDirichletboundaryconditionis0 = ∂u2G(z,w)|u1=0.27\nThe propagators are given by\n/an}bracketle{tγk(w)cj(z)/an}bracketri}ht=i/planckover2pi1\n2πδk\njψh(z,w), (186)\n/an}bracketle{tξk(w)Aj(z)/an}bracketri}ht=i/planckover2pi1\n2πδk\njdzφh(z,w), (187)\n/an}bracketle{t(∗Hdγk)(w)cj(z)/an}bracketri}ht=i/planckover2pi1\n2πδk\njδwφh(z,w), (188)\nand so on. From these propagators, we can obtain diagram rules co rresponding to the deformation quantization. In\nsection IVE, we will introduce exact diagram rules.\nTo obtain the star product, we choice\nOx=f(X(1))g(X(0))δx(ψ(∞)) (189)\nE. Diagram rules of deformation quantization\nFrom the perturbation theory of the topological string theory, w e can obtain the following diagram rules of the star\nproduct, which is first given by Kontsevich [17, 30]:\n(f ⋆g)(x) =f(x)g(x)+∞/summationdisplay\nn=1/parenleftbiggi/planckover2pi1\n2/parenrightbiggn/summationdisplay\nΓ∈GnwΓBΓ,α(f,g). (190)\nwhere Γ,BΓ,α(f,g) andwΓare defined as follows:\nDefinition. 1 Gnis a set of the graphs Γwhich have n+ 2vertices and 2nedges. Vertices are labeled by symbols\n“1”, “2”,..., “n”, “L”, and “R”. Edges are labeled by symbol (k,v), wherek= 1,2,...,n,v= 1,2,...,n,L,R , and\nk/ne}ationslash=v.(k,v)represents the edge which starts at “ k” and ends at “ v”. There are two edges starting from each vertex\nwithk= 1,2,...,n;LandRare the exception, i.e., they act only as the end points of the edges. Hereafter, VΓand\nEΓrepresent the set of the vertices and the edges, respectivel y.\nDefinition. 2 BΓ,α(f,g)is the operator defined by:\nBΓ,α(f,g) :=/summationdisplay\nI:EΓ→{i1,i2,···,i2n}\nn/productdisplay\nk=1\n/productdisplay\ne∈EΓ,e=(k,∗)∂I(e)\nαI((k,v1\nk),(k,v2\nk))\n×\n\n\n/productdisplay\ne∈EΓ,e=(∗,L)∂I(e)\nf\n×\n\n/productdisplay\ne∈EΓ,e=(∗,R)∂I(e)\ng\n, (191)\nwhere,Iis a map from the list of edges ((k,v1,2\nk)),k= 1,2,...,nto integer numbers {i1,i2,···,i2n}. Here1≤in≤d;\ndrepresents a dimension of the manifold M.BΓ,α(f,g)corresponds to the graph Γin the following way: The vertices\n“1”, “2”,..., “n”, correspond to the Poisson structure αij.RandLcorrespond to the functions fandg, respectively.\nThe edgee= (k,v)represents the differential operator ∂(iorj)acting on the vertex v.\nThe simplest diagram for n= 1is shown in Fig. 4(a), which corresponds to the Poisson bracket: {f,g}=/summationtext\ni1,i2αi1i2(∂xi1f)(∂xi2g). The higher order terms are the generalizations of this Pois son bracket.\nFigure4(b)shows a graph Γex.2withn= 2corresponding to the list of edges\n((1,L),(1,R),(2,R),(2,3)); (192)\nin addition, the operator BΓex.2,αis given by\n(f,g)/ma√sto→/summationdisplay\ni1,···,i4(∂xi3αi1i2)αi3i4(∂xi1f)(∂xi2∂xi4g). (193)\nDefinition. 3 We put the coordinates for the vertices in the upper-half com plex planeH+:={z∈C|Im(z)>0}\n(Crepresents the complex plain; Im(z)denotes the imaginary part of z). Therefore, RandLare put at 0and1,\nrespectively. We associate a weight wΓwith each graph Γ∈Gnas\nwΓ:=1\nn!(2π)2n/integraldisplay\nHnn/logicalanddisplay\nk=1/parenleftig\ndφh\n(k,v1\nk)∧dφh\n(k,v2\nk)/parenrightig\n, (194)28\nFIG. 4. (a): The graph Γ ex.1∈G1corresponding to Poisson bracket. (b): A graph Γ ∈G2correspond to the list of edges:\n((1,L),(1,R),(2,R),(2,1))/mapsto→ {i1,i2,i3,i4}.\nwhereφis defined by\nφh\n(k,v):=1\n2iLog/parenleftbigg(q−p)(¯q−p)\n(q−¯p)(¯q−¯p)/parenrightbigg\n. (195)\npandqare the coordinates of the vertexes “ k” and “v”, respectively. ¯prepresents the complex conjugate of p∈C.\nHndenotes the space of configurations of nnumbered pair-wise distinct points on H+:\nHn:={(p1,···,pn)|pk∈H+, pk/ne}ationslash=plfork/ne}ationslash=l}. (196)\nHere we assume that H+has the metric:\nds2= (d(Re(p))2+d(Im(p))2)/(Im(p))2, (197)\nwithp∈H+;φh(p,q)is the angle which is defined by (p,q)and(∞,p), i.e.,φh(p,q) =∠pq∞with the metric (197).\nFor example, wΓex.1corresponding to Fig. 4(a)is calculated as:\nwΓex.1=2\n1!(2π)2/integraldisplay\nH1d1\n2iLog/parenleftbiggp2\np2/parenrightbigg\n∧d1\n2iLog/parenleftbigg(1−p)2\n(1−p)2/parenrightbigg\n= 1, (198)\nwhere we have included the factor “ 2” arising from the interchange between two edges in Γ.wΓcorresponding to the\nFig.4(b)is\nwΓ(b)=1\n2!(2π)4/integraldisplay\nH2d1\niLog/parenleftbiggp1\np1/parenrightbigg\n∧d1\niLog/parenleftbigg1−p1\n1−p1/parenrightbigg\n∧d1\niLog/parenleftbiggp2\np2/parenrightbigg\n∧d1\n2iLog/parenleftbigg(p1−p2)(p1−p2)\n(p1−p2)(p1−p2)/parenrightbigg\n=1\n2!(2π)4/integraldisplay\nH2d1\niLog/parenleftbiggp1\np1/parenrightbigg\n∧d1\niLog/parenleftbigg1−p1\n1−p1/parenrightbigg\n∧d(2arg(p2))∧d|p2|∂\n∂|p2|1\n2iLog/parenleftbigg(p1−p2)(p1−p2)\n(p1−p2)(p1−p2)/parenrightbigg\n=1\n2!(2π)4/integraldisplay\nH2d1\niLog/parenleftbiggp1\np1/parenrightbigg\n∧d1\niLog/parenleftbigg1−p1\n1−p1/parenrightbigg\n∧d1\niLog/parenleftbiggp2\np2/parenrightbigg\n∧d1\niLog/parenleftbigg1−p2\n1−p2/parenrightbigg\n=w2\n1\n2!\n=1\n2, (199)\nwherep1andp2are the coordinates of vertexes “ 1” and “2”, respectively. Here, we have used the following facts:\n/integraldisplay∞\n0d|p2|∂|p2|Log/parenleftbigg(p1−p2)(p1−p2)\n(p1−p2)(p1−p2)/parenrightbigg\n= lim\nΛ→∞Log/parenleftbigg(p1−Λeiarg(p2))(p1−Λeiarg(p2))\n(p1−Λe−iarg(p2))(p1−Λe−iarg(p2))/parenrightbigg\n= lim\nΛ→∞Log/parenleftbigg(1−Λeiarg(p2))(1−Λeiarg(p2))\n(1−Λe−iarg(p2))(1−Λe−iarg(p2))/parenrightbigg\n,\n/integraldisplay\n|p1|>ΛdLog/parenleftbiggp1\np1/parenrightbigg\n∧dLog/parenleftbigg1−p1\n1−p1/parenrightbigg\nΛ→∞−→/integraldisplay\n|p1|>ΛdLog/parenleftbiggp1\np1/parenrightbigg\n∧dLog/parenleftbiggp1\np1/parenrightbigg\n= 0. (200)\nGenerally speaking, the integrals are entangled for n≥3graphs, and the weight of these are not so easy to evaluate\nas Eq.(199).\nNote that the above diagram rules also define the twisted element as the following relation: ( f⋆g)≡µ◦F(f⊗g).29\nFIG. 5. A four vertexes graph, where the white circle and the w hite square represent αAandαF, respectively; the dotted\narrow, waved arrow, and real arrow represent ∂p,∂s, and∂X, respectively.\nF. Gauge invariant star product\nFrom Eq. (77), the Poisson structure corresponding to our mode l is\nαij=\n0ηµν0\n−ηµν−qˆFµν−qǫabcsaAb\nµ\n0qǫabcsaAb\nµǫabcsc\n, (201)\nwhere the symbols iandjrepresent indexes of the phase space ( TX,ωp,s). We separate the Poisson structure as\nfollows:\nα:=\n0ηµν0\n−ηµν0 0\n0 0 0\n+\n0 0 0\n0 0−qǫabcAb\nµsa\n0qǫabcsaAb\nµǫabcsa\n+\n0 0 0\n0qˆFµν0\n0 0 0\n\n≡α0+αA+αF. (202)\nHere, forf=f0+faσa,∂saf:=fa(a=x,y,z), wheref0,x,y,zare functions Xandp. Becauseα0is constant and\nαAandαFare functions of Xµands, and any function fis written as f=f0+/summationtext\na=x,y,zfasa(f0,aonly depends on\nXandp), then we obtain additional diagram rules:\nA1. Two edges starting from αFconnect with both vertices “ L” and “R”.\nA2. At least one edge from vertices α0orαFconnect with vertices “ L” or “R”.\nA3. A number of the edges entering αAis one or zero.\nWe also separate the graph Γ into Γ α0, ΓαAand Γ αF. Here, we define the numbers of vertices α0,αA, andαF\nasnα0,nαA, andnαF, respectively. Γ αFis the graph consisted by vertices corresponding to αF, and “L” and “R”,\nand edges starting from these vertices. We consider Γ αFas a cluster, and define Γ αAas the graph consisted by the\nvertices corresponding to αA, which acts on the cluster corresponding to Γ αF. Γα0is the rest of the graph Γ without\nΓαAand Γ αF. Here, we label vertexes Γ αF, ΓαAand Γ α0by “k= 1−nαF”, “k= (nαF+ 1)−(nαF+nαA)” and\n“k= (nαF+nαA+1)−(nαF+nαA+nα0)”, respectively. The edge starting from “ k” and ending to “ v1,2\nk” represents\n(k,v1,2\nk).\nNext, we calculate weight wnαFand the operator BΓαF,αFcorresponding to Γ αF, and later those for Γ αAor0.\nSeparation of graph Γ\nWe now sketch the proof of wΓBΓ,α=wnα0BΓα0,α0·wnαABΓαA,αA·wnαFBΓαF,αF, wherewnαa=wnαa\n1\nnαa!fora=\n0,A,F, andw1is given by Eq. (198).\nFrom the additional rule A1, each operator corresponding to vert exesαFand edges ( αF,LorR) acts onfandg\nindependently. Thus wαF∼wnαF\n1. Secondly we consider the graph which consists of four vertexes c orresponding to\nαA,αF, and “L”and“R” as shownin Fig. 5. We alsoassume that one edge ofthe vertex corr espondingto αAconnects\nwith a vertex corresponding to αF. In this case, from additional diagram rule A3, another edge of the vertex has to\nconnect with “ L” or “R”. Since we can exchange the role “ R” and “L” by the variable transformation p/ma√sto→1−p,\n(p∈H+), we assume that one edge of the vertex corresponding to αAconnect with “ L”. The weight wΓin this case is30\nFIG. 6. This figure shows the calculation method of the graph ( a), where the dotted arrow and real arrow represent the\nderivative with respect to pandX, respectively, and the white circle and the white triangle r epresent α0andαF, respectively.\nWe rewrite the graph ( a) as the graph ( c) which is given by the cluster represented by the big circle a nd the operators into it,\nwhere the big circle represents the graph ( b).\ngiven by Eq. (199), i.e., the integrals for the weight is given by replacin g coordinate of the vertex corresponding to αF\nwith coordinate of “ R” inH+. This result can be expanded to every graph though a graph include s the vertices α0.\nFor example, we illustrate the calculation of a six vertices graph, whic h only includes α0andαF, in Fig. 6. At first\nwe make the cluster having only vertices αF,fandg(fig. 6(b)), which is corresponding to the following operator:\nwnαF\nnαF!αi1i2\nFαi3i4\nF(∂pi1∂pi3f)(∂pi2∂pi4g). The edges from the vertices act on the cluster independently (fi g. 6(c)); we\nobtain the following operator:wnα0\nnα0!wnαF\nnαF!αj1j2\n0αj3j4\n0(∂Xj1αi1i2\nF)(∂Xj3αi3i4\nF)(∂Xj2∂pi1∂pi3f)(∂Xj4∂pi2∂pi4g).\nThe position of each vertex corresponding to αAandαFcan be move independently in integrals, and the entangled\nintegral does not appear. Therefore the weight wnαAof a graph Γ αA∼wnαA\n1only depends on the number of vertexes\ncorrespondingto αAandαF, andwΓ=wnαA·wnαFholds generally. From additional rule A2, we can similarly discuss\naboutagraphΓ α0, andobtain wnα0∼wnα0\n1. Finally,wecancountthecombinationof nα0,nαAandnαF, anditisgiven\nby(nα0+nαA+nαF)!\n(nαA+nαF)!nα0!·(nαA+nαF)!\nnαA!nαF!. Therefore we obtain the Eq: wΓBΓ,α=wnα0BΓα0,α0·wnαABΓαA,αA·wnαFBΓαF,αF.\nThe summation of each graph is easy, and we can derive the star pro duct:f ⋆g=µ◦FA(f⊗g), where twisted\nelementFAis written as follow:\nFA= exp/braceleftbiggi/planckover2pi1\n2/parenleftbig\n∂Xµ⊗∂pµ−∂pµ⊗∂Xµ/parenrightbig/bracerightbigg\n◦exp/braceleftbiggi/planckover2pi1\n2εijksk/parenleftbig\nAi\nµ∂pµ⊗∂sj+∂si⊗∂sj−∂sj⊗Ai\nµ⊗∂pµ/parenrightbig/bracerightbigg\n◦exp/braceleftbiggi/planckover2pi1\n2/parenleftbig\nFa\nµνsa+Fµν/parenrightbig\n∂pµ⊗∂pν/bracerightbigg\n. (203)\nBecause the action IAincluding a global U(1) ×SU(2) gauge field, the action IAis written as IA=FA◦F−1\n0I0,\nthus, the mapF0/mapsto→A:I0/ma√sto→IAis given byF0/mapsto→A=FA◦F−1\n0, i.e.,\nF(0/mapsto→A)= exp/braceleftbiggi/planckover2pi1\n2/parenleftbig\n∂Xµ⊗∂pµ−∂pµ⊗∂Xµ/parenrightbig/bracerightbigg\n◦exp/braceleftbiggi/planckover2pi1\n2εijksk/parenleftbig\nAi\nµ∂pµ⊗∂sj+∂si⊗∂sj−∂sj⊗Ai\nµ⊗∂pµ/parenrightbig/bracerightbigg\n◦exp/braceleftbiggi/planckover2pi1\n2/parenleftbig\nFa\nµνsa+Fµν/parenrightbig\n∂pµ⊗∂pν/bracerightbigg\n◦exp/braceleftbigg\n−i/planckover2pi1\n2εijksk∂si⊗∂sj/bracerightbigg\n◦exp/braceleftbigg\n−i/planckover2pi1\n2/parenleftbig\n∂Xµ⊗∂pµ−∂Xµ⊗∂pµ/parenrightbig/bracerightbigg\n. (204)\nThe inverse map is given by the replacement of i by −i in the map (204).31\nV. TWISTED SPIN\nIn this section, we will derive the twisted spin density, which corresp onds to the spin density in commutative\nspacetime without the background SU(2) gauge field.\nFirst, we will derive a general form of the twisted spin current in sec tion VA, which is written by using the twisted\nvariation operator. This operator is constituted of the coproduc t and twisted element; the coproduct reflects the\naction rule of the global SU(2) gauge symmetry generator, and th e twisted element represents gauge structure of the\nbackground gauge fields.\nIn section VB, at first, we will calculate the twisted spin density of th e so-called Rashba-Dresselhaus model in\nthe Wigner representation using the general form of the twisted s pin density, and next, we will find the twisted spin\noperator in real spacetime using correspondence between opera tors in commutative spacetime and noncommutative\nphasespace.\nA. Derivation of a twisted spin in Wigner space\nThe Lagrangian density in the Wigner space is given by\nL(X,p) =/parenleftbigg\np0−p2\n2m/parenrightbigg\n⋆/parenleftbig\nψψ†/parenrightbig\n, (205)\nwheremis the electric mass, f ⋆g:=µ◦FA(f⊗g) for any functions fandg.\nThe variation corresponding to the infinitesimal global SU(2) gauge transformation is defined as\nδsaψ= iϑsaψ, (206)\nδsaψ†=−iψ†sa, (207)\nδsaxµ= 0, (208)\nwhereθrepresents an infinitesimal parameter. Therefore, the variation of the Lagrangian density L(X,p) :=ˆL⋆ψψ†\nis given by\nδsa(L(X,p)) :=ˆL⋆iθsa⋆ψψ†−ˆL⋆ψψ†⋆iθsa(209)\nHere we introduce the Grassmann numbers λ1,2,3(λ3:=λ1λ2), the product µ, and the coproduct △η, where the\ncoproduct satisfies\n△η(f) :=f⊗η+η⊗f (210)\nfor any functions fand operator η. The equation (209) can be rewritten as\nδsa(L) =/integraldisplay\ndλ3µ◦FA/parenleftig\nˆL⊗/parenleftbig\niλ1θsa⋆λ2ψψ†+λ2ψψ†⋆iλ1θsa/parenrightbig/parenrightig\n=/integraldisplay\ndλ3µ◦FA/parenleftig\nˆL⊗µ◦FA△iλ1θsa/parenrightig\n=/integraldisplay\ndλ3µ◦FA(id⊗µ)◦(id⊗FA)◦(id⊗△iλ1θsa)◦(ˆL⊗λ2ψψ†)\n= i/integraldisplay\ndλ3µ◦(id⊗µ)◦(id⊗△)FA◦(id⊗FA)\n◦(id◦θ1/2⊗θ1/2)◦(id⊗△λ1sa)◦(ˆL⊗λ2ψψ†), (211)\nwhere we used/integraltextdλiλj=δij(i,j= 1,2,3) with the Kronecker delta δijHere we introduce the following symbols:\nˆµ:=µ◦(id⊗µ), (212)\nˆFA:= (id⊗△)FA◦(id⊗FA), (213)\nˆθ:=θ1/2⊗θ1/2, (214)\nˆ△λ1sa:= (id⊗△λ1sa), (215)32\nwhere the coproduct in the differential operator space is defined a s\n△(dn) :=/summationdisplay\ni+j=n\ni≥0, j≥0di⊗dj. (216)\nHere vectors{d0,d1,...}corresponding to following operators: d0:= id anddn:= (1/n!)(∂n/∂pn), ord0:= id and\ndn:= (1/n!)(∂n/∂xn) (n= 1,2,...), anddlare bases of a vector space B(k) :=/circleplustext∞\nl=0kdl(krepresents a scalar). The\ncoproduct△in the vector space B(k) satisfies the coassociation law: ( △⊗id)⊗△= (id⊗△)◦△because\n(△⊗id)◦△(dn) =/summationdisplay\ni+j=n\ni≥0, j≥0△(di)⊗dj\n=/summationdisplay\ni+j=n\ni≥0, j≥0\n/summationdisplay\nk+l=i\nk≥0, l≥0dk⊗dl⊗dj\n\n=/summationdisplay\nk+l+j=n\nk≥0, l≥0, j≥0dk⊗dl⊗dj (217)\nand\n(id⊗△)◦△(dn) =/summationdisplay\ni+j=n\ni≥0, j≥0di⊗△(dj)\n=/summationdisplay\ni+k+l=n\ni≥0, k≥0, l≥0di⊗dk⊗dl. (218)\nIt represents the Leibniz rule with respect to the differential oper ator∂µ. For instance, (id ⊗△)(∂µ⊗∂ν) :=\n∂µ⊗△(∂ν) =∂µ⊗∂ν⊗id+∂µ⊗id⊗∂ν; it corresponds to the following calculation:\n∂µf·∂ν(g·h) =∂µf·∂νg·h+∂µf·g·∂νh. (219)\nThe variation (211) is rewritten as\nδsaL(X,p) = i/integraldisplay\ndλ3ˆµ◦ˆFA◦ˆθ◦ˆ△λ1sa◦(ˆL⊗λ2ψψ†). (220)\nIf we replace ˆ△t\nλ1sa:=ˆF−1\nAˆ△λ1saF0withˆ△λ1sain equation (220), the integrals in terms of xandpof the\nright-hand side of Eq. (220) become zero because ˆFA◦ˆ△t\nλ1sadoes not include the SU(2) field, which breaks\nthe global SU(2) gauge symmetry, in the case that the parameter θis constant. Therefore, for the action S:=/integraltext\ndDimXdDimpL(X,p)/(2π/planckover2pi1)Dim,\nδt\nsaS:= tr/integraldisplay/integraldisplay/integraldisplay\ndλ3dDimXdDimp\n(2π/planckover2pi1)Dimˆµ◦ˆFA◦ˆθ◦ˆ△t\nλ1sa(ˆL⊗λ2ψψ†) (221)\nis the infinitesimal SU(2) gauge transformation with background SU (2) gauge fields.\nBecauseδt\nsaS= 0, we can write\nδt\nsaS=/integraldisplay\ndDimXθ/parenleftbig\n∂µjt\nµ/parenrightbig\n. (222)\nIn the case that the infinitesimal parameter depends on the space time coordinate, this equation can be written as\nδt\nsaS=/integraldisplay\ndDimXθ(X)/parenleftbig\n∂µjt\nµ/parenrightbig\n=−/integraldisplay\ndDimX/parenleftbigg∂θ(X′)\n∂X′µ/parenrightbigg\njt\nµ(X′). (223)33\nTherefore, we obtain the twisted Noether current\njt\nµ=−δt\nsaS\nδ(∂µθ(X)). (224)\nIn particular, the twisted spin\nSt\na=/integraldisplay\ndXjt\n0\n=/integraldisplay\ndXδt\nsaS\nδ(∂Tθ)(225)\nis conserved quantity. Here, we assumed that the SU(2) gauge is s tatic one. However, we do not use this condition in\nthe derivation of the twisted Noether charge and current density . Then, we can derive the virtual twisted spin with a\ntime-dependent SU(2) gauge: ˜St\na. In this case, we only use the time-dependent SU(2) gauge field str engthFa\nµν, which\nhas non-zero space-time components Fa\n0i(i= 1,2,...,Dim−1).\nHere, we discuss the adiabaticity of the twisted spin. In the case th at SU(2) gauge fields have time dependence, the\ntwisted spin is not conserved. Now, we assume that Aa\nµ=λ(t)Ca\nµ(a=x,y,z) with constant fields Ca\nµ= (0,Ca);λ(t)\nis an adequate slowly function dependent on time. Because ˜St\naincludesF−1\nµν∼1\n(1+(˙λ)2C·C)/parenleftbigg\n0 ˙λC\n−˙λCλ−2[Ci,Cj]−1/parenrightbigg\nwith˙λ≡dλ/dt, the difference between ˜St\naandSt\nacomes from only that between inverse of field strength: ∆ F−1∼\n1\n1+(˙λ)2λ−2[Ci,Cj]−1−λ−2[Ci,Cj]−1∼(˙λ/λ)2[Ci,Cj]−1. Therefore we obtain\ndSt\na\ndt=O(˙λ2). (226)\nThis means that St\nais the adiabatic invariance. Namely, for the infinitely slow change in λ(t) during the time period\nT(→∞),St\naremains constant while ∆ λ=λ(T)−λ(0) is finite. This fact is essential for the spin-orbit echo proposed\nin [11].\nB. Rashba-Dresselhaus model\nHere we apply the formalism developed so far to an explicit model, i.e., th e so-called Rashba-Dresselhaus model\ngiven by\nH=ˆp2\n2m+α(ˆpxˆσy−ˆpyˆσx)+β(ˆpxˆσx−ˆpyˆσy)+V(ˆx) (227)\nwith a potential V(ˆx), whereαandβare the Rashba and Dresselhaus parameters, respectively. Comp leting square\nin terms of ˆ p, we obtain Ax\nx=−2mβ/(/planckover2pi1q),Ay\nx=−2mα/(/planckover2pi1q),Ax\ny= 2mα/(/planckover2pi1q),Ay\ny= 2mβ/(/planckover2pi1q),A0=m(α2+β2)/e,\nandAz\nx,y=Ax,y\nz=Az\nz=Ax,y,z\n0=Ax,y,z= 0, where q=|e|/(mc2).\nTo calculate the twisted symmetry generator ˆ△t(λ2sa), we first consider the ˆF0. (id⊗△)F0is given by\n(id⊗△)F0= exp/braceleftbiggi/planckover2pi1\n2/parenleftig\n∂Xµ⊗1⊗∂pµ+∂Xµ⊗∂pµ⊗1−∂pµ⊗1⊗∂Xµ−∂pµ⊗∂Xµ⊗1/parenrightig\n+i\n2εijk/parenleftig\nsk∂si⊗1⊗∂sj+sk∂si⊗∂sj⊗1/parenrightig/bracerightbigg\n, (228)34\nandˆF0is given by\nˆF0= exp/braceleftbiggi/planckover2pi1\n2/parenleftig\n∂Xµ⊗1⊗∂pµ+∂Xµ⊗∂pµ⊗1−∂pµ⊗1⊗∂Xµ−∂pµ⊗∂Xµ⊗1/parenrightig\n+i\n2εijk/parenleftig\nsk∂si⊗1⊗∂sj+sk∂si⊗∂sj⊗1/parenrightig/bracerightbigg\n◦exp/braceleftbiggi/planckover2pi1\n2/parenleftig\n1⊗∂Xµ⊗∂pµ−1⊗∂pµ⊗∂Xµ/parenrightig\n+i\n2εijk/parenleftig\n1⊗sk∂si⊗∂sj/parenrightig/bracerightbigg\n= exp/braceleftbiggi/planckover2pi1\n2/parenleftig\n1⊗∂Xµ⊗∂pµ+∂Xµ⊗1⊗∂pµ+∂Xµ⊗∂pµ⊗1\n−1⊗∂pµ⊗∂Xµ−∂pµ⊗1⊗∂Xµ−∂pµ⊗∂Xµ⊗1/parenrightig\n+i\n2εijk/parenleftig\n1⊗sk∂si⊗∂sj+sk∂si⊗1⊗∂sj+sk∂si⊗∂sj⊗1/parenrightig/bracerightbigg\n≡G0. (229)\nSimilarly, ˆFAis given by\nˆFA= exp/braceleftbiggi/planckover2pi1\n2/parenleftig\n1⊗∂Xµ⊗∂pµ+∂Xµ⊗1⊗∂pµ+∂Xµ⊗∂pµ⊗1\n−1⊗∂pµ⊗∂Xµ−∂pµ⊗1⊗∂Xµ−∂pµ⊗∂Xµ⊗1/parenrightig/bracerightig\n◦exp/braceleftbiggi\n2εijk/parenleftig\n1⊗sk∂si⊗∂sj+sk∂si⊗1⊗∂sj+sk∂si⊗∂sj⊗1\n+1⊗skAi\nµ∂pµ⊗∂sj+skAi\nµ∂pµ⊗1⊗∂sj+skAi\nµ∂pµ⊗∂sj⊗1\n−1⊗∂sj⊗skAi\nµ∂pµ−∂sj⊗1⊗skAi\nµ∂pµ−∂sj⊗skAi\nµ∂pµ⊗1/parenrightig/bracerightig\n◦exp/braceleftbiggi/planckover2pi1\n2/parenleftig\n1⊗ˆFµν∂pµ⊗∂pν+ˆFµν∂pµ⊗1⊗∂pν+ˆFµν∂pµ⊗∂pν⊗1/parenrightig/bracerightbigg\n≡GA\nXp◦GA\nsp◦GA\npp. (230)\nWe note that the operators G0andGA\nXp,sp,pphave each inverse operator, which are denoted by G0andGA\nXp,sp,pp,\nrespectively. Here, the overline −represents the complex conjugate.\nBecause the twisted variation is ˆ µ◦ˆFA◦ˆθ◦ˆF−1\nA◦ˆ△λ1sa◦F0(L⊗λ2ψψ†), the infinitesimal parameter ˆθbecomes\nan operator ˆFA◦ˆθ◦ˆF−1\nA. It is calculated by using the operator formula\neBCe−B=∞/summationdisplay\nn=01\nn![B,[B,···[B,/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nnC]···]] (231)\nfor any operators BandC. In the calculation, one will use the following formula in midstream:\n/summationdisplay\nl=0Dl\n(l+1)!=/integraldisplay1\n0dλeλD, (232)\n/summationdisplay\nl=0Dl\n(l+2)!=/integraldisplay1\n0dλ/integraldisplayλ\n0dλ′eλ′D, (233)\nand so on, for any operator D.\nFrom these results of the calculations, we obtain the twisted spin as follow\nst\na=µ◦FA◦(sa⊗id)◦˜Υ◦(id⊗G)◦˜Υ†, (234)35\nwhere˜Υ is defined as\n˜Υ :=1\n2/bracketleftbigg\nei\n2(α+β)σ−/parenleftbig\n∂px+∂py/parenrightbig\n⊗ei\n2(α−β)σ+/parenleftbig\n∂px−∂py/parenrightbig/bracketrightbigg\n◦/bracketleftigg\ne−i\n2/parenleftbig\n∂X⊗∂Y+∂Y⊗∂X/parenrightig\n◦/parenleftbig\n1\n2m(α2−β2)⊗σz/parenrightbig\n+e−i\n2/parenleftbig\n∂X⊗∂Y+∂Y⊗∂X/parenrightig\n◦/parenleftbig\nσz⊗1\n2m(α2−β2)/parenrightbig/bracketrightigg\n+ei\n2(α+β)σ−/parenleftbig\n∂px+∂py/parenrightbig\n⊗sin2/parenleftigg\ni/planckover2pi1/parenleftbig\n∂X−∂Y/parenrightbig\n8√\n2m(α+β)/parenrightigg\n,(235)\nwhereσ±:=σx±σy.\nFinally, we will rewrite the twisted spin as an operator form in commuta tive spacetime. Roughly speaking, the\noperator in the commutative spacetime. and the one in the noncomm utative Wigner space have the following rela-\ntions (the left-hand side represents operators on the Wigner spa ce; the right-hand side represents operators on the\ncommutative spacetime):\nXµ⋆⇔ˆxµ, (236)\npµ⋆⇔ˆpµ, (237)\nsa⋆⇔ˆsa, (238)\ni/planckover2pi1∂pµ⋆⇔ˆxµ, (239)\n−i/planckover2pi1∂Xµ⋆⇔ˆpµ, (240)\nbecause [Xµ,pν]⋆:=Xµ⋆pν−pν⋆Xµ= i/planckover2pi1δµ\nνis equal to the commutation of the operator form: [ˆ xµ,ˆpν] = i/planckover2pi1δµ\nν. The\nequivalence of st\naand the twisted spin in the operator form on the commutative space time ˆst\nacan be confirmed using\nthe Wigner transformation in terms of ˆ st\na.\nThe operator form of the twisted spin is given by\nst\na=/planckover2pi1\n2ψ†Υ†σaΥψ, (241)\nwhere\nΥ = lim\nx′→x1\n2ei\n2m(α+β)σ−x+/bracketleftbigg\ne−i\n2m(α−β)σ+x′\n−e−i\n2σz\n2m2(α2−β2)/parenleftbig← −∂x′∂y+← −∂y′∂x/parenrightbig\n+e−i\n2σz\n2m2(α2−β2)/parenleftbig\n∂x′∂y+∂y′∂x/parenrightbig\ne−i\n2m(α−β)σ+x′\n−+2sin2/parenleftbiggi/planckover2pi1(∂x−∂y)\n8√\n2m(α+β)/parenrightbigg/bracketrightbigg\n(242)\nwithx±:=x±y.\nThis operator in Eq. (241), when integrated over X, is the conserved quantity for any potential configuration V(ˆx)\nas long asα,βare static and the electron-electron interaction is neglected.\nVI. CONCLUSIONS\nIn this paper, we have derived the conservation of the twisted spin and spin current densities. Also the adiabatic\ninvariant nature of the total twisted spin integrated over the spa ce is shown. Here we remark about the limit of\nvalidity of this conservation law. First, we neglected the dynamics of the electromagnetic field Aµwhich leads to\nthe electron-electron interaction. This leads to the inelastic electr on scattering, which is not included in the present\nanalysis, and most likely gives rise to the spin relaxation. This inelastic s cattering causes the energy relaxation and\nhence the memory of the spin will be totally lost after the inelastic lifet ime. This situation is analogous to the two\nrelaxation times T1andT2in spin echo in NMR and ESR. Namely, the phase relaxation time T2is usually much\nshorter than the energy relaxationtime T1, and the spin echo is possible for T <T 1. Similar story applies to spin-orbit\necho [11] where the recoveryof the spin moment is possible only within the inelastic lifetime of the spins. However, the\ngeneralization of the present study to the dynamical Aµis a difficult but important issue left for future investigations.\nAlso the effect of the higher order terms in 1 /(mc2) in the derivation of the effective Lagrangian from Dirac theory\nrequires to be scrutinized.\nAnother direction is to explore the twisted conserved quantities in t he non-equilibrium states. Under the static\nelectric field, the system is usually in the current flowing steady stat e. Usually this situation is described by the\nlinear response theory, but the far from equilibrium states can in pr inciple be described by the non-commutative\ngeometry [16, 30]. The nonperturbative effects in this non-equilibriu m states are the challenge for theories, and\ndeserve the further investigations.36\nThe author thanks Y.S. Wu, F.C. Zhang, K. Richter, V. Krueckl, J. N itta, and S. Onoda for useful discussions. This\nworkwassupportedbyPriorityAreaGrants, Grant-in-Aidsunder the Grantnumber21244054,StrategicInternational\nCooperative Program (Joint Research Type) from Japan Science a nd Technology Agency, and by Funding Program\nfor World-Leading Innovative R and D on Science and Technology (FI RST Program).\n[1] M. E. Peshkin and D. V. Schroeder. Introduction to Quantum Field Theory . Addison-Wesley, New York (1995).\n[2] J. Fr¨ ohlich and U. M. Studer. Gauge invariance and curre nt algebra in nonrelativistic many-body theory. Rev. Mod. P hys.,\n65, 733–802 (1993).\n[3] G. Volovik. The Universe in a Helium Droplet . Oxford University Press, Oxford, U.K. (2003).\n[4] B. Leurs, Z. Nazario, D. Santiago and J. Zaanen. Non-Abel ian hydrodynamics and the flow of spin in spinorbit coupled\nsubstances. Annals of Physics, 323(4), 907 – 945 (2008).\n[5] N. Seiberg and E. Witten. String theory and noncommutati ve geometry. Journal of High Energy Physics, 09, 032 (1999).\n[6] M. Chaichian, P. Kulish, K. Nishijima and A. Tureanu. On a Lorentz-invariant interpretation of noncommutative spac e-\ntime and its implications on noncommutative QFT. Physics Le tters B, 604(1-2), 98–102 (2004).\n[7] M. Chaichian, P. Preˇ snajder and A. Tureanu. New Concept of Relativistic Invariance in Noncommutative Space-Time:\nTwisted Poincar´ e Symmetry and Its Implications. Phys. Rev . Lett.,94, 151602 (2005).\n[8] G. Amelino-Camelia, G. Gubitosi, A. Marcian` o, P. Marti netti, F. Mercati, D. Pranzetti and R. A. Tacchi. First Resul ts\nof the Noether Theorem\nfor Hopf-Algebra Spacetime Symmetries. Progress of Theore tical Physics Supplement, 171, 65–78 (2007).\n[9] G. Amelino-Camelia, G. Gubitosi, A. Marcian` o, P. Marti netti and F. Mercati. A no-pure-boost uncertainty principl e from\nspacetime noncommutativity. Physics Letters B, 671(2), 298–302 (2009).\n[10] S. Maekawa. Concepts in Spin Electronics . OXFORD SCIENCE PUBLICATIONS (2006).\n[11] N. Sugimoto and N. Nagaosa. Spin-orbit echo. To apper in Science.\n[12] E. L. Hill. Hamilton’s Principle and the Conservation T heorems of Mathematical Physics. Rev. Mod. Phys., 23, 253–260\n(1951).\n[13] C. Gonera, P. Kosi´ nski, P. Ma´ slanka and S. Giller. Spa ce-time symmetry of noncommutative field theory. Physics Le tters\nB,622(1-2), 192–197 (2005).\n[14] E. Wigner. On the Quantum Correction For Thermodynamic Equilibrium. Phys. Rev., 40, 749–759 (1932).\n[15] J. E. Moyal. Mathematical Proceedings of the Cambridge Philosophical Society. Proc. Cambridge Phil. Soc., 45, 99–124\n(1949).\n[16] S. Onoda, N. Sugimoto and N. Nagaosa. Theory of Non-Equi libirum States\nDriven by Constant Electromagnetic Fields. Progress of The oretical Physics, 116(1), 61–86 (2006).\n[17] M. Kontsevich. Deformation Quantization of Poisson Ma nifolds. Letters in Mathematical Physics, 66, 157–216 (2003).\n[18] F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz and D. S ternheimer. Deformation theory and quantization. I. Defor ma-\ntions of symplectic structures. Annals of Physics, 111(1), 61 – 110 (1978).\n[19] F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz and D. S ternheimer. Deformation theory and quantization. II. Phys ical\napplications. Annals of Physics, 111(1), 111 – 151 (1978).\n[20] G. Hochschild. On the Cohomology Groups of an Associati ve Algebra. The Annals of Mathematics, 46(1), pp. 58–67\n(1945).\n[21] D. Happel. Hochschild cohomology of finitedimensional algebras. In M.-P. Malliavin, editor, S´ eminaire d’Alg` ebre Paul\nDubreil et Marie-Paul Malliavin , vol. 1404 of Lecture Notes in Mathematics , pages 108–126. Springer Berlin / Heidelberg.\nISBN 978-3-540-51812-9 (1989).\n[22] M. Gerstenhaber. The Cohomology Structure of an Associ ative Ring. The Annals of Mathematics, 78(2), 267–288 (1963).\n[23] E. Cartan. Sur la structure des groupes infinis de transf ormation. Annales Scientifiques de l’ ´Ecole Normale Sup´ erieure,\n22, 219–308 (0012-9593).\n[24] A. S. Cattaneo and G. Felder. A Path Integral Approach to the Kontsevich Quantization Formula. Communications in\nMathematical Physics, 212, 591–611 (2000).\n[25] N. Ikeda and I. Ken-iti. Dimensional reduction of nonli near gauge theories. Journal of High Energy Physics, 09, 030\n(2004).\n[26] I. A. Batalin and G. A. Vilkovisky. Gauge algebra and qua ntization. Physics Letters B, 102(1), 27–31 (1981).\n[27] I. A. Batalin and G. A. Vilkovisky. Quantization of gaug e theories with linearly dependent generators. Phys. Rev. D ,28,\n2567–2582 (1983).\n[28] J. Alfaro and P. Damgaard. Origin of antifields in the Bat alin-Vilkovisky lagrangian formalism. Nuclear Physics B, 404(3),\n751–793 (1993).\n[29] I. Batalin and G. Vilkovisky. Relativistic S-matrix of dynamical systems with boson and fermion constraints. Phys ics\nLetters B, 69(3), 309–312 (1977).\n[30] N. Sugimoto, S. Onoda and N. Nagaosa. Gauge Covariant Fo rmulation of the Wigner Representation through Deformatio n\nQuantization. Progress of Theoretical Physics, 117(3), 415–429 (2007)."    },    {        "title": "1304.2766v1.Spin_orbital_liquids_in_non_Kramers_magnet_on_Kagome_lattice.pdf",        "content": "Spin-orbital liquids in non-Kramers magnet on Kagome lattice\nRobert Schaffer1, Subhro Bhattacharjee1;2, and Yong Baek Kim1;3\n1Department of Physics and Center for Quantum Materials,\nUniversity of Toronto, Toronto, Ontario M5S 1A7, Canada.\n2Department of Physics and Astronomy, McMaster University, Hamilton, Ontario L8S 4M1, Canada.\n3School of Physics, Korea Institute for Advanced Study, Seoul 130-722, Korea.\n(Dated: June 21, 2018)\nLocalized magnetic moments with crystal-field doublet or pseudo-spin 1/2 may arise in correlated insulators\nwith even number of electrons and strong spin-orbit coupling. Such a non-Kramers pseudo-spin 1/2 is the\nconsequence of crystalline symmetries as opposed to the Kramers doublet arising from time-reversal invariance,\nand is necessarily a composite of spin and orbital degrees of freedom. We investigate possible spin-orbital\nliquids with fermionic spinons for such non-Kramers pseudo-spin 1/2 systems on the Kagome lattice. Using the\nprojective symmetry group analysis, we find tennew phases that are not allowed in the corresponding Kramers\nsystems. These new phases are allowed due to unusual action of the time reversal operation on non-Kramers\npseudo-spins. We compute the spin-spin dynamic structure factor that shows characteristic features of these\nnon-Kramers spin-orbital liquids arising from their unusual coupling to neutrons, which is therefore relevant for\nneutron scattering experiments. We also point out possible anomalous broadening of Raman scattering intensity\nthat may serve as a signature experimental feature for gapless non-Kramers spin-orbital liquids.\nPACS numbers:\nI. INTRODUCTION\nThe low energy magnetic degrees of freedom of a Mott\ninsulator, in the presence of strong spin-orbit coupling, are\ndescribed by states with entangled spin and orbital wave\nfunctions.1,2In certain crystalline materials, for ions with\neven numbers of electrons, a low energy spin-orbit entan-\ngled “pseudo-spin”-1/2 may emerge, which is not protected\nby time-reversal symmetry (Kramers degeneracy)12but rather\nby the crystal symmetries.3,4Various phases of such non-\nKramers pseudo-spin systems on geometrically frustrated lat-\ntices, particularly various quantum paramagnetic phases, are\nof much recent theoretical and experimental interest in the\ncontext of a number of rare earth materials including frus-\ntrated pyrochlores5–9and heavy fermion systems.10,11\nIn this paper, we explore novel spin-orbital liquids that may\nemerge in these systems due to the unusual transformation of\nthe non-Kramers pseudo-spins under the time reversal trans-\nformation. Contrary to Kramers spin-1/2, where the spins\ntransform as S!\u0000Sunder time reversal,12here only one\ncomponent of the pseudo-spin operators changes sign under\ntime reversal:f\u001b1;\u001b2;\u001b3g!f\u001b1;\u001b2;\u0000\u001b3g.3,4This is be-\ncause, due to the nature of the wave-function content, the\n\u001b3component of the pseudo-spin carries a dipolar magnetic\nmoment while the other two components carry quadrupolar\nmoments of the underlying electrons. Hence the time rever-\nsal operator for the non-Kramers pseudo-spins is given by\nT=\u001b1K(whereKis the complex conjugation operator),\nwhich allows for new spin-orbital liquid phases. Since the\nmagnetic degrees of freedom are composed out of wave func-\ntions with entangled spin and orbital components, we prefer\nto refer the above quantum paramagnetic states as spin-orbital\nliquids, rather than spin liquids.\nSince the degeneracy of the non-Kramers doublet is pro-\ntected by crystal symmetries, the transformation properties of\nthe pseudo-spin under various lattice symmetries intimatelydepend on the content of the wave-functions that make up\nthe doublet. To this end, we focus our attention on the ex-\nample of Praseodymium ions (Pr3+) in a localD3denviron-\nment, which is a well known non-Kramers ion that occurs in\na number of materials with interesting properties.5–7Such an\nenvironment typically occurs in Praseodymium pyrochlores\ngiven by the generic formulae Pr 2TM2O7, where TM(= Zr,\nSn, Hf, or Ir) is a transition metal. In these compounds, the\nPr3+ions host a pair of 4 felectrons which form a J= 4\nground state manifold with S= 1andL= 5, as expected due\nto Hund’s rules. In terms of this local environment we have\na nine fold degeneracy of the electronic states.4This degener-\nacy is broken by the crystalline electric field. The oxygen and\nTM ions form a D3dlocal symmetry environment around the\nPr3+ions, splitting the nine fold degeneracy. A standard anal-\nysis of the symmetries of this system (see appendix A) shows\nthat theJ= 4 manifold splits into three doublets and three\nsinglets ( \u0000j=4= 3Eg+ 2A1g+A2g) out of which one of the\ndoublets is found to have the lowest energy, usually well sepa-\nrated from the other crystal field states.4This doublet (details\nin Appendix A), formed out of a linear combination of the\nJz=\u00064withJz=\u00061andJz=\u00062states, is given by\nj\u0006i=\u000bjm=\u00064i\u0006\fjm=\u00061i\u0000\rjm=\u00072i:(1)\nThe non-Kramers nature of this doublet is evident from the\nnature of the “spin” raising and lowering operators within\nthe doublet manifold; the projection of the angular momen-\ntum raising and lowering operators to the space of doublets\nis zero (PJ\u0006Pj\u001bi= 0 wherePprojects into the doublet\nmanifold). However, the projection of the Jzoperator to this\nmanifold is non-zero, and describes the z component of the\npseudo-spin ( \u001b3). In addition, there is a non-trivial projection\nof the quadrupole operators fJ\u0006;Jzgin this manifold. These\nhave off-diagonal matrix elements, and are identified with the\npseudo-spin raising and lowering operators ( \u001b\u0006=\u001b1\u0006i\u001b2).\nIn a pyrochlore lattice the local D3daxes point to the centre\nof the tetrahedra.4On looking at the pyrochlore lattice alongarXiv:1304.2766v1  [cond-mat.str-el]  9 Apr 20132\nthe [111] direction, it is found to be made out of alternate\nlayers of Kagome and triangular lattices. For each Kagome\nlayer (shown in Fig. 1) the local D3daxes make an angle of\ncos\u00001(p\n2=3)with the plane of the Kagome layer. We imag-\nine replacing the Pr3+ions from the triangular lattice layer\nwith non-magnetic ions so as to obtain decoupled Kagome\nlayers with Pr3+ions on the sites. The resulting structure is\nobtained in the same spirit as the now well-known Kagome\ncompound Herbertsmithite was envisioned. As long as the lo-\ncal crystal field has D3dsymmetry, the doublet remains well\ndefined. A suitable candidate non-magnetic ion may be iso-\nvalent but non-magnetic La3+. Notice that the most extended\norbitals in both cases are the fifth shell orbitals and the crystal\nfield at each Pr3+site is mainly determined by the surrounding\noxygens and the transition metal element. Hence, we expect\nthat the splitting of the non-Kramers doublet due to the above\nsubstitution would be very small and the doublet will remain\nwell defined. In this work we shall consider such a Kagome\nlattice layer and analyze possible Z2spin-orbital liquids, with\ngapped or gapless fermionic spinons .\nThe rest of the paper is organized as follows. In Sec. II, we\nbegin with a discussion of the symmetries of the non-Kramers\nsystem on a Kagome lattice and write down the most general\npseudo-spin model with pseudo-spin exchange interactions up\nto second nearest neighbours. In Sec. III formulate the pro-\njective symmetry group (PSG) analysis for singlet and triplet\ndecouplings. Using this we demonstrate that the non-Kramers\ntransformation of our pseudo-spin degrees of freedom under\ntime reversal leads to a set of ten spin-orbital liquids which\ncannot be realized in the Kramers case. In Sec. IV we de-\nrive the dynamic spin-spin structure factor for a representa-\ntive spin liquid for the case of both Kramers and non-Kramers\ndoublets, demonstrating that experimentally measurable prop-\nerties of these two types of spin-orbital liquids differ qualita-\ntively. Finally, in Sec. V, we discuss our results, and propose\nan experimental test which can detect a non-Kramers spin-\norbital liquid. The details of various calculations are discussed\nin different appendices.\nII. SYMMETRIES AND THE PSEUDO-SPIN\nHAMILTONIAN\nSince the local D3daxes of the three sites in the Kagome\nunit cell differ from each other a general pseudo-spin Hamil-\ntonian is not symmetric under continuous global pseudo-spin\nrotations. However, it is symmetric under various symmetry\ntransformations of the Kagome lattice as well as time rever-\nsal symmetry. Such symmetry transformations play a major\nrole in the remainder of our analysis. We start by describing\nthe effect of various lattice symmetry transformations on the\nnon-Kramers doublet.\nWe consider the symmetry operations that generate the\nspace group of the above Kagome lattice. These are (as shown\nin Fig. 2(a))\n\u000fT1,T2: generate the two lattice translations.\n\u000f\u001b=C0\n2I: (not to be confused with the pseudo-spin\nFIG. 1: A Kagome layer, in the pyrochlore lattice environment. We\nconsider sites labelled z and z’ replaced by non-magnetic ions, de-\ncoupling the Kagome layers. The local axis at the u,v and w sites\npoint towards the center of the tetrahedron on which these lie.\nu\nvw\nFIG. 2: (color online) (a) The symmetries of the Kagome lattice.\nAlso shown are the labels for the sublattices and the orientation of the\nlocal z-axis. (b) Nearest and next nearest neighbour bonds. Colors\nrefer to the phases \u001er;r0and\u001e0\nr;r0, with these being 0 on blue bonds,\n1 on green bonds and 2 on red bonds.\noperators which come with a superscript) where Iis the\nthree dimensional inversion operator about a plaquette\ncenter andC0\n2refers to a two-fold rotation about a line\njoining two opposite sites on the plaquette.\n\u000fS6=C2\n3I: whereC3is the threefold rotation op-\nerator about the center of a hexagonal plaquette of the\nKagome lattice.\n\u000fT=\u001b1K: Time reversal.\nHere, we consider a three dimensional inversion operator\nsince the local D3daxes point out of the Kagome plane. The\nabove symmetries act non-trivially on the pseudo-spin degrees\nof freedom, as well as the lattice degrees of freedom. The ac-\ntion of the symmetry transformations on the pseudo-spin op-\nerators is given by,\nS6:f\u001b3;\u001b+;\u001b\u0000g!f\u001b3;\u0016!\u001b+;!\u001b\u0000g;\nT:f\u001b3;\u001b+;\u001b\u0000g!f\u0000\u001b3;\u001b\u0000;\u001b+g;\nC0\n2:f\u001b3;\u001b+;\u001b\u0000g!f\u0000\u001b3;\u001b\u0000;\u001b+g;\nT1:f\u001b3;\u001b+;\u001b\u0000g!f\u001b3;\u001b+;\u001b\u0000g;\nT2:f\u001b3;\u001b+;\u001b\u0000g!f\u001b3;\u001b+;\u001b\u0000g; (2)\n(!= \u0016!\u00001=ei2\u0019\n3). Operationally their action on the doublet\n(j+i j\u0000i )can be written in form of 2\u00022matrices. The\ntranslationsT1;T2act trivially on the pseudo-spin degrees of3\nfreedom, and the remaining operators act as\nT=\u001b1K; \u001b =\u001b1; S 6=\u0014\n\u0016!0\n0!\u0015\n; (3)\nwhereKrefers to complex conjugation. The above expres-\nsions can be derived by examining the effect of these operators\non the wave-function describing the doublet (Eq. 1).\nWe can now write down the most generic pseudo-spin\nHamiltonian allowed by the above lattice symmetries that is\nbilinear in pseudo-spin operators. The form of the time-\nreversal symmetry restricts our attention to those products\nwhich are formed by a pair of \u001b3operators or those which\nmix the pseudo-spin raising and lowering operators. Any\nterm which mixes \u001b3and\u001b\u0006changes sign under the sym-\nmetry, and can thus be excluded. Under the C3transforma-\ntion about a site, the terms C3:\u001b3\nr\u001b3\nr0!\u001b3\nC3(r)\u001b3\nC3(r0)and\nC3:\u001b+\nr\u001b\u0000\nr0!\u001b+\nC3(r)\u001b\u0000\nC3(r0). However, the term \u001b+\nr\u001b+\nr0(and\nits Hermitian conjugate) gain additional phase factors when\ntransformed; under the C3symmetry transformation, this term\nbecomesC3:\u001b+\nr\u001b+\nr0!\u0016!\u001b+\nC3(r)\u001b+\nC3(r0). In addition, un-\nder the\u001bsymmetry, this term transforms as \u001b:\u001b+\nr\u001b+\nr0!\n\u001b\u0000\n\u001b(r)\u001b\u0000\n\u001b(r0). Thus the Hamiltonian with spin-spin exchange\ninteractions up to next-nearest neighbour is given by\nHeff=JnnX\nhr;r0i[\u001b3\nr\u001b3\nr0+ 2(\u000e\u001b+\nr\u001b\u0000\nr0+h:c:)\n+2q(e2\u0019i\u001er;r 0\n3\u001b+\nr\u001b+\nr0+h:c:)]\n+JnnnX\nhhr;r0ii[\u001b3\nr\u001b3\nr0+ 2(\u000e0\u001b+\nr\u001b\u0000\nr0+h:c:)\n+2q0(e2\u0019i\u001e0\nr;r 0\n3\u001b+\nr\u001b+\nr0+h:c:)]; (4)\nwhere\u001eand\u001e0take values 0, 1 and 2 depending on the bonds\non which they are defined (Fig. 2(b)).\nIII. SPINON REPRESENTATION OF THE PSEUDO-SPINS\nAND PSG ANALYSIS\nHaving written down the pseudo-spin Hamiltonian, we now\ndiscuss the possible spin-orbital liquid phases. We do this in\ntwo stages in the following sub-sections.\nA. Slave fermion representation and spinon decoupling\nIn order to understand these phases, we will use the\nfermionic slave-particle decomposition of the pseudo-spin op-\nerators. At this point, we note that the pseudo-spins satisfy\nS= 1=2representations of a “SU(2)” algebra among their\ngenerators (not to be confused with the regular spin rotation\nsymmetry). We represent the pseudo-spin degrees of freedom\nin terms of a fermion bilinear. This is very similar to usual\nslave fermion construction for spin liquids13,14. We take\n\u001b\u0016\nj=1\n2fy\nj\u000b[\u001a\u0016]\u000b\ffj\f; (5)where\u000b;\f=\";#is defined along the local zaxis andfy(f)\nis anS= 1=2fermionic creation (annihilation) operator. Fol-\nlowing standard nomenclature, we refer to the f(fy)as the\nspinon annihilation (creation) operator, and note that these\nsatisfy standard fermionic anti-commutation relations. The\nabove spinon representation, along with the single occupancy\nconstraint\nfy\ni\"fi\"+fy\ni#fi#= 1; (6)\nform a faithful representation of the pseudo-spin-1/2 Hilbert\nspace. The above representation of the pseudo-spins, when\nused in Eq. 4, leads to a quartic spinon Hamiltonian. Fol-\nlowing standard procedure,13,14this is then decomposed us-\ning auxiliary fields into a quadratic spinon Hamiltonian (af-\nter writing down the corresponding Eucledian action). The\nmean field description of the phases is then characterized by\nthe possible saddle point values of the auxiliary fields. There\nare eight such auxiliary fields per bond, corresponding to\n\u001fij=hfy\ni\u000bfj\u000bi\u0003;\u0011ij=hfi\u000b\u0002\ni\u001c2\u0003\n\u000b\ffj\fi\u0003; (7a)\nEa\nij=hfy\ni\u000b[\u001ca]\u000b\ffj\fi\u0003;Da\nij=hfi\u000b\u0002\ni\u001c2\u001ca\u0003\n\u000b\ffi\fi\u0003;\n(7b)\nwhere\u001ca(a= 1;2;3) are the Pauli matrices. While Eq. 7a\nrepresents the usual singlet spinon hopping (particle-hole) and\npairing (particle-particle) channels, Eq. 7b represents the cor-\nresponding triplet decoupling channels. Since the Hamilto-\nnian (Eq. 4) does not have pseudo-spin rotation symmetry,\nboth the singlet and the triplet decouplings are necessary.16,17\nFrom this decoupling, we obtain a mean-field Hamiltonian\nwhich is quadratic in the spinon operators. We write this com-\npactly in the following form17(subject to the constraint Eq.6)\nH0=X\nijJij~fy\niUij~fj; (8)\n~fy\ni=h\nfy\ni\"fi#fy\ni#\u0000fi\"i\n; (9)\nUij=\u0018\u000b\f\nij\u0006\u000b\u0000\f; (10)\n\u0006\u000b=\u001a\u000b\nI; \u0000\f=I\n\u001c\f; (11)\nwhere\u001a\u000bare the Identity (for \u000b= 0) and Pauli matrices\n(\u000b= 1;2;3) acting on pseudo-spin degrees of freedom, and\n\u001c\u000brepresents the same in the gauge space. We immediately\nnote that\n\u0002\n\u0006\u000b;\u0000\f\u0003\n= 08\u000b;\f: (12)\nThe requirement that our H0be Hermitian restricts the coeffi-\ncients\u0018ijto satisfy\n\u001800\nij;\u0018ab\nij2=;\u0018a0\nij;\u00180b\nij2<: (13)\nfora;b2 f 1;2;3g. The relations between \u0018ijs and\nf\u001fij;\u0011ij;Eij;Dijgare given in Appendix C.17As a straight\nforward extension of the SU(2)gauge theory formulation for4\nspin liquids,13,18we find that H0is invariant under the gauge\ntransformation\n~fj!Wj~fj; (14)\nUij!WiUijWy\nj; (15)\nwhere theWimatrices are SU(2) matrices of the form Wi=\nei~\u0000\u0001~ ai(~\u0000\u0011(\u00001;\u00002;\u00003)). Noting that the physical pseudo-\nspin operators are given by\n~ \u001bi=1\n4~fy\ni~\u0006~fi; (16)\nEq. 12 shows that the spin operators, as expected, are gauge\ninvariant. It is useful to define the “ \u0006-components” of the Uij\nmatrices as follows:\nUij=V\u000b\nij\u0006\u000b; (17)\nwhere\nV\u000b\nij=\u0018\u000b\f\nij\u0000\f=\u0014J\u000b\nij0\n0J\u000b\nij\u0015\n; (18)\nand\nJ\u000b\nij=\u0014\u0018\u000b0\nij+\u0018\u000b3\nij\u0018\u000b1\nij\u0000i\u0018\u000b2\nij\n\u0018\u000b1\nij+i\u0018\u000b2\nij\u0018\u000b0\nij\u0000\u0018\u000b3\nij\u0015\n: (19)\nUnder global spin rotations the fermions transform as\n~fi!V~fi; (20)\nwhere V is an SU(2) matrix of the form V=ei~\u0006\u0001~b(~\u0006\u0011\nf\u00061;\u00062;\u00063g). So whileV0\nij(the singlet hopping and pairing)\nis invariant under spin rotation, fV1\nij;V2\nij;V3\nijgtransforms as\na vector as expected since they represent triplet hopping and\npairing amplitudes.\nB. PSG Classification\nWe now classify the non-Kramers spin-orbital liquids based\non projective representation similar to that of the conven-\ntional quantum spin liquids.13Each spin-orbital liquid ground\nstate of the quadratic Hamiltonian (Eq 11) is character-\nized by the mean field parameters (eight on each bond,\n\u001f;\u0011;E1;E2;E3;D2;D2;D3, or equivalently Uij). However,\ndue to the gauge redundancy of the spinon parametrization\n(as shown in Eq. 15), a general mean-field ansatz need not\nbe invariant under the symmetry transformations on their own\nbut may be transformed to a gauge equivalent form without\nbreaking the symmetry. Therefore, we must consider its trans-\nformation properties under a projective representation of the\nsymmetry group.13For this, we need to know the various pro-\njective representations of the lattice symmetries of the Hamil-\ntonian (Eq. 4) in order to classify different spin-orbital liquid\nstates.\nOperationally, we need to find different possible sets of\ngauge transformations fGGgwhich act in combination withthe symmetry transformations fSGgsuch that the mean-field\nansatzUijis invariant under such a combined transformation.\nIn the case of spin rotation invariant spin-liquids (where only\nthe singlet channels \u001fand\u0011are present), the above statement\nis equivalent to demanding the following invariance:\nUij= [GSS]Uij[GSS]y=GS(i)US(i)S(j)Gy\nS(j);(21)\nwhereS2SGis a symmetry transformation and GS2GG\nis the corresponding gauge transformation. The different pos-\nsiblefGSj8S2SGggive the possible algebraic PSGs that\ncan characterize the different spin-orbital liquid phases. To\nobtain the different PSGs, we start with various lattice sym-\nmetries of the Hamiltonian. The action of various lattice\ntransformations15is given by\nT1:(x;y;s )!(x+ 1;y;s);\nT2:(x;y;s )!(x;y+ 1;s);\n\u001b:(x;y;u )!(y;x;u );\n(x;y;v )!(y;x;w );\n(x;y;w )!(y;x;v );\nS6:(x;y;u )!(\u0000y\u00001;x+y+ 1;v);\n(x;y;v )!(\u0000y;x+y;w);\n(x;y;w )!(\u0000y\u00001;x+y;u); (22)\nwhere (x;y)denotes the lattice coordinates and s2fu;v;wg\ndenotes the sub-lattice index (see figure 2).\nIn terms of the symmetries of the Kagome lattice, these op-\nerators obey the following conditions\nT2=\u001b2= (S6)6=e;\ng\u00001T\u00001gT=e8g2SG;\nT\u00001\n2T\u00001\n1T2T1=e;\n\u001b\u00001T\u00001\n1\u001bT2=e;\n\u001b\u00001T\u00001\n2\u001bT1=e;\nS\u00001\n6T\u00001\n2S6T1=e;\nS\u00001\n6T\u00001\n2T1S6T2=e;\n\u001b\u00001S6\u001bS6=e: (23)\nIn addition, these commutation relations are valid in terms of\nthe operations on the pseudo-spin degrees of freedom, as can\nbe verified from Eq. 3.\nIn addition to the conditions in Eq. 23, the Hamiltonian is\ntrivially invariant under the identity transformation. The in-\nvariant gauge group (IGG) of an ansatz is defined as the set of\nall pure gauge transformations GIsuch thatGI:Uij!Uij.\nThe nature of such pure gauge transformations immediately\ndictates the nature of the low energy fluctuations about the\nmean field state. If these fluctuations do not destabilize the\nmean-field state, we get stable spin liquid phases whose low\nenergy properties are controlled by the IGG. Accordingly,\nspin liquids obtained within projective classification are pri-\nmarily labelled by their IGGs and we have Z2;U(1)and\nSU(2)spin liquids corresponding to IGGs of Z2;U(1)and5\nSU(2)respectively. In this work we concentrate on the set of\nZ2“spin liquids” (spin-orbital liquids with a Z2IGG).\nWe now focus on the PSG classification. As shown in Eq.\n2, in the present case, the pseudo-spins transform non-trivially\nunder different lattice symmetry transformations. Due to the\npresence of the triplet decoupling channels the non-Kramers\ndoublet transforms non-trivially under lattice symmetries (Eq.\n3). Thus, the invariance condition on the Uijs is not given by\nEq. 21, but by a more general condition\nUij= [GSS]Uij[GSS]y=GS(i)\u001eS\u0002\nUS(i)S(j)\u0003\nGy\nS(j):\n(24)\nHere\n\u001eS\u0002\nUS(i)S(j)\u0003\n=DSUS(i)S(j)Dy\nS; (25)\nandDSgenerates the pseudo-spin rotation associated with the\nsymmetry transformation ( S) on the doublet. The matrices\nDShave the form\nDS6=\u00001\n2\u00060\u0000ip\n3\n2\u00063; (26)\nD\u001b=DT=i\u0003\u00061;DT1=DT2= \u00060: (27)\nUnder these constraints, we must determine the relations\nbetween the gauge transformation matrices GS(i)for our set\nof ansatz. The additional spin transformation (Eq. 25) does\nnot affect the structure of the gauge transformations, as the\ngauge and spin portions of our ansatz are naturally separate\n(Eq. 12). In particular, we can choose to define our gauge\ntransformations such that\nGS:Uij=GS:\u0018\u000b\f\nij\u0006\u000b\u0000\f!\u0018\u000b\f\nij\u0006\u000bGy\nS(i)\u0000\fGS(j);\n(28)\nS:Uij=S:\u0018\u000b\f\nij\u0006\u000b\u0000\f!\u0018\u000b\f\nS(i)S(j)DS\u0006\u000bDy\nS\u0000\f;(29)\nwhere we have used the notation GS:Uij\u0011Gy\nS(i)UijGS(j)\nand so forth. As a result, we can build on the general con-\nstruction of Lu et al.15to derive the form of the gauge trans-\nformation matrices. The details are given in Appendix B.\nA major difference arises when examining the set of alge-\nbraic PSGs for Z2spin liquids found on the Kagome lattice\ndue to the difference between the structure of the time re-\nversal symmetry operation on the Kramers and non-Kramers\npseudo-spin- 1=2s. In the present case, we find there are 30\ninvariant PSGs leading to thirty possible spin-orbital liquids.\nThis is in contrast with the Kramers case analysed by Lu et\nal.,15where tenof the algebraic PSGs cannot be realized as\ninvariant PSGs, as all bonds in these ansatz are predicted to\nvanish identically due to the form of the time reversal oper-\nator, and hence there are only twenty possible spin liquids.\nHowever, with the inclusion of spin triplet terms and the non-\nKramers form of our time reversal operator, these ansatz are\nnow realizable as invariant PSGs as well. The time reversal\noperator, as defined in Appendix B, acts as\nT:\u0018\u000b\f\nij\u0006\u000b\u0000\f!~\u0018\u000b\f\nij\u0006\u000b\u0000\f; (30)where ~\u0018\u000b\f=\u0018\u000b\fif\u000b2f1;2gand~\u0018\u000b\f=\u0000\u0018\u000b\fif\u000b2f0;3g.\nThe projective implementation of the time-reversal symmetry\ncondition (Eq. 23) takes the form (see Appendix B)\n[GT(i)]2=\u0011TI8i; (31)\nwhereGT(i)is the gauge transformation associated with time\nreversal operation and \u0011T=\u00061for aZ2IGG.\nTherefore, the terms allowed by the time reversal symmetry\nto be non zero are, for \u0011T= 1,\n\u001810;\u001811;\u001812;\u001813;\u001820;\u001821;\u001822;\u001823; (32)\nand for\u0011T=\u00001, with the choice GT(i) =i\u00001(see appendix\nB),\n\u001802;\u001803;\u001810;\u001811;\u001820;\u001821;\u001832;\u001833: (33)\nThis contrasts with the case of Kramers doublets, in which no\nterms are allowed for \u0011T=\u00001, and for\u0011T=\u00001the allowed\nterms are\n\u001802;\u001803;\u001812;\u001813;\u001822;\u001823;\u001832;\u001833: (34)\nFurther restrictions on the allowed terms on each link arise\nfrom the form of the gauge transformations defined for the\nsymmetry transformations. All nearest neighbour bonds can\nthen be generated from Uijdefined on a single bond, by per-\nforming appropriate symmetry operations.\nUsing the methods outlined in earlier works (Ref. 13, 15)\nwe find the minimum set of parameters required to stabilize Z2\nspin-orbital liquids. We take into consideration up to second\nneighbour hopping and pairing amplitudes (both singlet and\ntriplet channels). The results are listed in Table I.\nThe spin-orbital liquids listed from 21\u000030are not allowed\nin the case of Kramers doublets and, as pointed out before,\ntheir existence is solely due to the unusual action of the time-\nreversal symmetry operator on the non-Kramers spins. Hence\nthese tenspin-orbital liquids are qualitatively new phases that\nmay appear in these systems. Of these ten phases, only two\n(labelled as 21and22in Table I) require next nearest neigh-\nbour amplitudes to obtain a Z2spin-orbital liquid. For the\nother eight , nearest neighbour amplitudes are already suffi-\ncient to stabilize a Z2spin-orbital liquid.\nIt is interesting to note (see below) that bond-pseudo-spin-\nnematic order (Eq. 35 and Eq. 36) can signal spontaneous\ntime-reversal symmetry breaking. Generally, since the triplet\ndecouplings are present, the bond nematic order parameter for\nthe pseudo-spins21,22\nQ\u000b\f\nij=h\u0010\nS\u000b\niS\f\nj+S\f\niS\u000b\nj\u0011\n=2\u0000\u000e\u000b\f(~Si\u0001~Sj)=3i;(35)\nas well as vector chirality order\n~Jij=h~Si\u0002~Sji; (36)\nare non zero. Since the underlying Hamiltonian Eq. 4) gener-\nally does not have pseudo-spin rotation symmetry, the above\nnon-zero expectation values do not spontaneously break any6\nTABLE I: Symmetry allowed terms: We list the terms allowed to be non-zero by symmetry, for the 30 PSGs determined by Yuan-Ming Lu et\nal15. The PSGs listed together are those with \u001112=\u00061and all other factors equal. Included are terms allowed on nearest and next-nearest\nneighbour bonds, as well as chemical potential terms \u0000which can be non zero on all sites for certain spin-orbital liquids. Also included is the\ndistance of bond up to which we must include in order to gap out the gauge fluctuations to Z2via the Anderson-Higgs mechanism13. Only\nPSGs 9 and 10 can not host Z2spin-orbital liquids with up to second nearest neighbour bonds.\nNo. \u0003s n.n. n.n.n. Z2\n1-2 \u00002;\u00003\u001810;\u001821;\u001802;\u001803;\u001832;\u001833\u001810;\u001821;\u001802;\u001803;\u001832;\u001833n.n.\n3-4 0\u001810;\u001821;\u001802;\u001803;\u001832;\u001833\u001810;\u001821n.n.\n5-6 \u00003\u001810;\u001821;\u001802;\u001803;\u001832;\u001833\u001810;\u001821;\u001803;\u001833n.n.\n7-8 0 \u001811;\u001820\u001811;\u001820;\u001802;\u001803;\u001832;\u001833n.n.n.\n9-10 0 \u001811;\u001820\u001811;\u001820-\n11-12 0 \u001811;\u001820\u001810;\u001811;\u001802;\u001832n.n.n.\n13-14 \u00003\u001810;\u001811;\u001803;\u001833\u001810;\u001821;\u001802;\u001803;\u001832;\u001833n.n.\n15-16 \u00003\u001810;\u001811;\u001803;\u001833\u001810;\u001821;\u001803;\u001833n.n.\n17-18 0\u001810;\u001811;\u001803;\u001833\u001810;\u001811;\u001802;\u001832n.n.\n19-20 0\u001810;\u001811;\u001803;\u001833\u001810;\u001821n.n.\n21-22 0\u001810;\u001821;\u001822;\u001823\u001810;\u001821;\u001822;\u001823n.n.n.\n23-24 0\u001810;\u001821;\u001822;\u001823\u001810;\u001811;\u001812;\u001823n.n.\n25-26 0\u001811;\u001812;\u001813;\u001820\u001813;\u001820;\u001821;\u001822n.n.\n27-28 0\u001811;\u001812;\u001813;\u001820\u001810;\u001811;\u001813;\u001822n.n.\n29-30 0\u001811;\u001812;\u001813;\u001820\u001811;\u001812;\u001813;\u001820n.n.\npseudo-spin rotation symmetry. However, because of the un-\nusual transformation property of the non-Kramers pseudo-\nspins under time reversal, the operators corresponding to\nQ13\nij;Q23\nij;J1\nij;J2\nijare odd under time reversal, a symmetry of\nthe pseudo-spin Hamiltonian. Hence if any of the above op-\nerators gain a non-zero expectation value in the ground state,\nthen the corresponding spin-orbital liquid breaks time rever-\nsal symmetry. While this can occur in principle, we check\nexplicitly (see Appendix C) that in all the spin-orbital liquids\ndiscussed above, the expectation values of these operators are\nidentically zero. This provides a non-trivial consistency check\non our PSG calculations.\nWe now briefly dicuss the effect of the fluctuations about\nthe mean-field states. In the absence of pairing channels (both\nsinglet and triplet) the gauge group is U(1). In this case, the\nfluctuations of the gauge field about the mean field (Eq. 15)\nare related to the scalar pseudo-spin chirality ~S1\u0001~S2\u0002~S3,\nwhere the three sites form a triangle.19Such fluctuations are\ngapless in a U(1)spin liquid. It is interesting to note that the\nscalar spin-chirality is odd under time-reversal symmetry and\nit has been proposed that such fluctuations can be detected\nin neutron scattering experiments in presence of spin rotation\nsymmetry breaking.20In the present case, however, due to the\npresence of spinon pairing, the gauge group is broken down\ntoZ2and the above gauge fluctuations are rendered gapped\nthrough Anderson-Higg’s mechanism.13\nIn addition to the above gauge fluctuations, because of thetriplet decouplings which break pseudo-spin rotational sym-\nmetry, there are bond quadrupolar fluctuations of the pseudo-\nspinsQ\u000b\f\nij(Eq. 35), as well as vector chirality fluctuations ~Jij\n(Eq. 36)21,22on the bonds. These nematic and vector chirality\nfluctuations are gapped because the underlying pseudo-spin\nHamiltonian (Eq. 4) breaks pseudo-spin-rotation symmetry.\nHowever, we note that because of the unusual transformation\nof the non-Kramers pseudo-spins under time reversal (only\nthez\u0000component of pseudo-spins being odd under time re-\nversal),Q13\nij;Q23\nij;J1\nijandJ2\nijare odd under time reversal.\nHence, while their mean field expectation values are zero (see\nabove), the fluctuations of these quantities can in principle lin-\nearly couple to the neutrons in addition to the z\u0000component\nof the pseudo-spins.\nHaving identified the possible Z2spin-orbital liquids, we\ncan now study typical dynamic structure factors for these\nspin-orbital liquids. In the next section we examine the typi-\ncal spinon band structure for different spin-orbital liquids ob-\ntained above and find their dynamic spin structure factor.7\nFIG. 3: The spin structure factor for an ansatz in spin liquid 17, with\nthe spin variables transforming as a Kramers doublet.\nFIG. 4: The spin structure factor for an ansatz in spin liquid 17, with\nthe spin variables transforming as a non-Kramers doublet.\nIV . DYNAMIC SPIN STRUCTURE FACTOR\nWe compute the dynamic spin structure factor\nS(q;!) =Zdt\n2\u0019ei!tX\nijeiq\u0001(ri\u0000rj)X\na=1;2;3h\u001ba\ni(t)\u001ba\nj(0)i;\n(37)\nfor an example ansatz of our spin liquid candidates, in order to\ndemonstrate the qualitative differences between the Kramers\nand non-Kramers spin-orbital liquids. In the above equation,\nthe pseudo-spin variables are defined in a global basis (with\nthe z-axis perpendicular to the Kagome plane). In computing\nthe structure factor for the non-Kramers example, we include\nonly the\u001b3components of the pseudo-spin operator in the lo-\ncal basis, since only the z-components carry magnetic dipole\nmoment (see discussion before). Hence, only this component\ncouples linearly to neutrons in a neutron scattering experi-\nment.\nEq. 37 fails to be periodic in the first Brillouin zone of\nthe Kagome lattice16, as the term ri\u0000rjin eq. 37 is a half-\ninteger multiple of the primitive lattice vectors when the sub-\nlattices of sites i and j are not equal. As such, we examine the\nstructure factor in the extended brillouin zone, which consists\nof those momenta of length up to double that of those in thefirst brillouin zone. We plot the structure factor along the cut\n\u0000!M0!K0!\u0000, whereM0=2MandK0=2K. We\nexamine the structure factors for two ansatz of spin liquid #\n17 which has both Kramers and non-Kramers analogues.\nAs expected, we find that the structure factor has greater\nweight in the case of a Kramers spin liquid. This is par-\ntially due to the fact that the moment of the scattering par-\nticle couples with all components of the spin, rather than sim-\nply thez-component. In addition, we note that the presence\nof terms allowed in the non-Kramers spin-orbital liquid in-\nduce the formation of a gap, which is absent for the Kramers\ncase with up to second nearest neighbour singlet and triplet\nterms in this particular spin-orbital liquid. Qualitative and\nquantitative differences such as these, which can be observed\nin these structure factors between Kramers and non-Kramers\nspin-orbital liquids, provides one possible distinguishing ex-\nperimental signature of these states. We shall not pursue this\nin detail in the present work.\nV . DISCUSSION AND POSSIBLE EXPERIMENTAL\nSIGNATURE OF NON-KRAMERS SPIN-ORBITAL LIQUIDS\nIn this work, we have outlined the possible Z2spin-orbital\nliquids, with gapped or gapless fermionic spinons, that can be\nobtained in a system of non-Kramers pseudo-spin-1/2s on a\nKagome lattice of Pr+3ions. We find a total of thirty , 10 more\nthan in the case of corresponding Kramers system, allowed\nwithin PSG analysis in presence of time reversal symmetry.\nThe larger number of spin-orbital liquids is a result of the dif-\nference in the action of the time-reversal operator, when real-\nized projectively. We note that the spin-spin dynamic struc-\nture factor can bear important signatures of a non-Kramers\nspin-orbital liquid when compared to their Kramers counter-\nparts. Our analysis of the number of invariant PSGs leading to\npossibly different spin-orbital liquids that may be realizable in\nother lattice geometries will form interesting future directions.\nWe now briefly discuss an experiment that can play an im-\nportant role in determining non-Kramers spin-orbital liquids.\nSince the non-Kramers doublets are protected by crystalline\nsymmetries, lattice strains can linearly couple to the pseudo-\nspins. As we discussed, the transverse ( xandy) components\nof the pseudo-spins f\u001b1;\u001b2gcarry quadrupolar moments and\nhence are even under the time reversal transformation. Fur-\nther, they transform under an Egirreducible representation\nof the local D3dcrystal field. Hence any lattice strain which\nhas this symmetry can linearly couple to the above two trans-\nverse components. It turns out that in the crystal type that\nwe are concerned, there is indeed such a mode related to the\ndistortion of the oxygen octahedra. Symmetry considerations\nshow that the linear coupling is of the form Eg1\u001b1+Eg2\u001b2\n(fEg1;Eg2gbeing the two components of the distortion in\nthe local basis). The above mode is Raman active. For\na spin-liquid, we expect that as the temperature is lowered,\nthe spinons become more prominent as deconfined quasipar-\nticles. So the Raman active phonon can efficiently decay into\nthe spinons due to the above coupling channel. If the spin\nliquid is gapless, then this will lead to anomalous broaden-8\ning of the above Raman mode as the temperature is lowered,\nwhich, if observed, can be an experimental signature of the\nnon-Kramers spin-orbital liquid. The above coupling is for-\nbidden in Kramers doublets by time-reversal symmetry and\nhence no such anomalous broadening is expected.\nAcknowledgments\nWe thank T. Dodds, SungBin Lee, A. Paramekanti and J.\nRau for insightful discussions. This research was supported\nby the NSERC, CIFAR, and Centre for Quantum Materials at\nthe University of Toronto.\nAppendix A: Crystal Field Effects\nIn this appendix, we explore the breaking of the J= 4spin\ndegeneracy by the crystalline electric field. The oxygen and\nTM ions form a D3dlocal symmetry environment around the\nPr3+ions, splitting the ground state degeneracy of the elec-\ntrons. This symmetry group contains 6 classes of elements: E,\n2C3,3C0\n2,i,2S6, and 3\u001bd, where theC3are rotations by 2\u0019=3\nabout the local z axis, the C0\n2are rotations by \u0019about axis per-\npendicular to the local z axis, iis inversion, S6is a rotation\nby4\u0019=3combined with inversion and \u001bdis a reflection about\nthe plane connecting one corner and the opposing plane, run-\nning through the Prmolecule about which this is measured\n(or, equivalently, a rotation about the x axis combined with\ninversion). For our J=4 manifold, these have characters given\nby\n\u001f(4)(E) = 2\u00034 + 1 = 9 = \u001f(4)(i) (A1)\n\u001f(4)(C3) =\u001f(4)(2\u0019\n3) =sin(3\u0019)\nsin(\u0019=3)= 0 =\u001f(4)(S6)(A2)\n\u001f(4)(\u001bd) =\u001f(4)(\u0019) =sin(9\u0019=2)\nsin(\u0019=2)= 1 =\u001f(4)(C0\n2)(A3)\nwhere the latter equalities are given by the fact that our J=4\nmanifold is inversion symmetric. Thus, decomposing this in\nterms ofD3dirreps, our l=4 manifold splits into a sum of\ndoublet and singlet manifolds as\n\u0000l=4= 3Eg+ 2A1g+A2g: (A4)\nTo examine this further, we need to consider the matrix ele-\nments of the crystal field potential between the states of differ-\nent angular momenta. We know that this potential must be in-\nvariant under all group operations of D3d, so we can examine\nthe transformation properties of individual matrix elements,\nhmjVjm0i. Under the C3operation, these states of fixed m\ntransform as\nC3jmi=e2\u0019im\n3jmi=!mjmi (!=e2\u0019i\n3) (A5)\nand thus the matrix elements transform as\nC3:hmjVjm0i!hmj(C3)\u00001VC3jm0i=!m0\u0000mhmjVjm0i:\n(A6)By requiring that this matrix be invariant under this transfor-\nmation, we can see that this potential only contains matrix\nelements for mixing of states which have the z-component of\nangular momentum which differ by 3. Thus, our eigenstates\nare mixtures of the jm= 4i,jm= 1i, andjm=\u00002istates,\nof thejm= 3i,jm= 0i, andjm=\u00003istates, and of the\njm=\u00004i,jm=\u00001i, andjm= 2istates.\nIn addition to this, we have the transformation properties\nTjmi= (\u00001)mj\u0000mi (A7)\nand\n\u001bjmi= (\u00001)mj\u0000mi (A8)\n(where the operators for time reversal and reflection are\nbolded for future clarity). Inversion acts trivially on these\nstates, as we have total angular momentum even. Thus our\ntime-reversal and lattice reflection (about one axis) symme-\ntries give us doublet states of eigenstates \u000bjm= 4i+\fjm=\n1i\u0000\rjm=\u00002iand\u000bjm=\u00004i\u0000\fjm=\u00001i\u0000\rjm= 2i\n(with\u000b,\f,\r2< in order to respect the time reversal symme-\ntry) for the three eigenstates of V in these sectors. The eigen-\nstates of thejm= 3i,jm= 0i, andjm=\u00003iportion of V\nmust therefore split into three singlet states, by our represen-\ntation theory argument A4. Due to the expected strong Ising\nterm in our potential, we expect the eigenstate with maximal J\nto be the ground state, meaning that to analyze the properties\nof this ground state we are interested in a single doublet state,\none with large \u000b(close to one). We will restrict ourselves\nto this manifold from this point forward, and define the two\nstates in this doublet as\nj+i=\u000bjm= 4i+\fjm= 1i\u0000\rjm=\u00002i (A9)\nj\u0000i=\u000bjm=\u00004i\u0000\fjm=\u00001i\u0000\rjm= 2i:(A10)\nWe shall also refer to states of angular momentum jm=ni\nasjnifor simplicity of notation.\nAppendix B: Gauge transformations\nWe begin by describing the action of time reversal on our\nansatz. The operation is antiunitary, and must be combined\nwith a spin transformation \u001b1in the case of non-Kramers dou-\nblets. As a result, the operation acts as T:\u0018\u000b\f\nij\u0006\u000b\u0000\f!\n\u0018\u000b\f\u0003\nij\u00061\u0006\u000b\u0003\u00061\u0000\f\u0003. However, we can simplify this consid-\nerably by performing a gauge transformation in addition to\nthe above transformation, which yields the same transforma-\ntion on any physical variables. The gauge transformation we\nperform isi\u00002, which changes the form of the time reversal\noperation to T:\u0018\u000b\f\nij\u0006\u000b\u0000\f!\u0018\u000b\f\u0003\nij\u00061\u0006\u000b\u0003\u00061\u00002\u0000\f\u0003\u00002=\n~\u0018\u000b\f\nij\u0006\u000b\u0000\f, where ~\u0018\u000b\f=\u0018\u000b\fif\u000b2f1;2gand~\u0018\u000b\f=\u0000\u0018\u000b\fif\n\u000b2f0;3g.\nOn the Kagome lattice, the allowed form of the gauge trans-\nformations has been determined by Yuan-Ming Lu et al.15For\ncompleteness, we will reproduce that calculation, valid also9\nfor our spin triplet ansatz, here. The relations between the\ngauge transformation matrices,\n[GT(i)]2=\u0011TI; (B1)\nG\u001b(\u001b(i))G\u001b(i) =\u0011\u001bI; (B2)\nGy\nT1(i)Gy\nT(i)GT1(i)GT(T\u00001\n1(i)) =\u0011T1TI; (B3)\nGy\nT2(i)Gy\nT(i)GT2(i)GT(T\u00001\n2(i)) =\u0011T2TI; (B4)\nGy\n\u001b(i)Gy\nT(i)G\u001b(i)GT(\u001b\u00001(i)) =\u0011\u001bTI; (B5)\nGy\nS6(i)Gy\nT(i)GS6(i)GT(S\u00001\n6(i)) =\u0011S6TI; (B6)\nGy\nT2(T\u00001\n1(i))Gy\nT1(i)GT2(i)GT1(T\u00001\n2(i)) =\u001112I; (B7)\nGS6(S\u00001\n6(i))GS6(S\u00002\n6(i))GS6(S3\n6(i))\n\u0002GS6(S2\n6(i))GS6(S6(i))GS6(i) =\u0011S6I; (B8)\nGy\n\u001b(T\u00001\n2(i))Gy\nT2(i)G\u001b(i)GT1(\u001b(i)) =\u0011\u001bT1I; (B9)\nGy\n\u001b(T\u00001\n1(i))Gy\nT1(i)G\u001b(i)GT2(\u001b(i)) =\u0011\u001bT2I; (B10)\nGy\n\u001b(S6(i))GS6(S6(i))G\u001b(i)GS6(\u001b(i)) =\u0011\u001bS6I; (B11)\nGy\nS6(T\u00001\n2(i))Gy\nT2(i)GS6(i)GT1(S\u00001\n6(i)) =\u0011S6T1I;(B12)\nGy\nS6(T\u00001\n2T1(i))Gy\nT2(T1(i))GT1(T1(i))\nGS6(i)GT2(S\u00001\n6(i)) =\u0011S6T2I; (B13)\nare valid for our case as well, due to the decoupling of spin\nand gauge portions of our ansatz. In the above, the relations\nare valid for all lattice sites i= (x;y;s ), I is the 4x4 identity\nmatrix, and the GSmatrices are gauge transformation matri-\nces generated by exponentiation of the \u0000matrices. The \u0011’s\nare\u00061, the choice of which characterize different spin liquid\nstates. In deriving this form of the commutation relations, we\nhave included a gauge transformation i\u00002in our definition of\nthe time reversal operator, as this simplifies the effect of the\noperator on the mean field ansatz.\nWe turn next to the calculation of the gauge transforma-\ntions. We look first at the gauge transformations associated\nwith the translations. We can perform a site dependent gauge\ntransformation W(i), under which the gauge transformations\nassociated with the translational symmetries transform as\nGT1(i)!W(i)GT1(i)Wy(i\u0000^x) (B14)\nGT2(i)!W(i)GT2(i)Wy(i\u0000^y): (B15)\nAs such, we can choose a gauge transformation W(i) to sim-\nplify the form of GT1andGT2. Using such a transformation,\nalong with condition B7, we can restrict the form of these\ngauge transformations to be\nGT1(i) =\u0011iy\n12I GT2(i) =I: (B16)\nTo preserve this choice, we can now only perform gauge\ntransformations which are equivalent on all lattice positions\n(W(x;y;s ) =W(s)) or transformations which change the\nshown matrices by an IGG transformation.\nNext, we look at adding the reflection symmetry \u001b. Given\nour formulae for GT1andGT2, along with the relations be-tween the gauge transformations, we have that\nGy\n\u001b(T\u00001\n2(i))G\u001b(i)\u0011x\n12=\u0011\u001bT1I (B17)\nGy\n\u001b(T\u00001\n1(i))G\u001b(i)\u0011y\n12=\u0011\u001bT2I: (B18)\nDefiningG\u001b(0,0,s) =g\u001b(s), we have, by repeated application\nof the above,\nG\u001b(0;y;s) =\u0011y\n\u001bT1g\u001b(s) (B19)\nG\u001b(x;y;s ) =\u0011y\n\u001bT1\u0011xy\n12\u0011x\n\u001bT2g\u001b(s): (B20)\nNext, using\nG\u001b(\u001b(i))G\u001b(i) =\u0011\u001bI (B21)\nwe find that\n\u0011\u001bI=G\u001b(y;x;\u001b (s))G\u001b(x;y;s ) (B22)\n= (\u0011\u001bT1\u0011\u001bT2)x+yg\u001b(\u001b(s))g\u001b(s): (B23)\nSince this is true for all x and y, \u0011\u001bT1\u0011\u001bT2= 1 and thus\n\u0011\u001bT1=\u0011\u001bT2andg\u001b(\u001b(s))g\u001b(s) =\u0011\u001bI(where\u001b(u) =\nu;\u001b(v) =wand\u001b(w) =v). Our final form for the gauge\ntransformation is\nG\u001b(x;y;s ) =\u0011x+y\n\u001bT1\u0011xy\n12g\u001b(s): (B24)\nNext we look at adding the S6symmetry to our calcula-\ntion. We can do an IGG transformation, taking GT1(T1(i))\nto\u0011S6T2GT1(T1(i)), with the net effect being that \u0011S6T2be-\ncomes one (previous calculations are unaffected). We now\nhave that\nGy\nS6(T\u00001\n2T1(i))GS6(i)\u0011y\n12=I (B25)\nGy\nS6(T\u00001\n2(i))GS6(i)\u0011\u0000x\u00001\n12 =\u0011S6T1I (s=u;v)(B26)\nGy\nS6(T\u00001\n2(i))GS6(i)\u0011\u0000x\n12=\u0011S6T1I (s=w):(B27)\nDefiningGS6(0;0;s) =gS6(s), we find that\nGS6(n;\u0000n;s) =\u0011n(n\u00001)=2\n12gS6(s) (B28)\nGS6(x;y;s ) =\u0011x(x\u00001)=2+y+xy\n12 \u0011x+y\nS6T1gS6(s) (s=u;v)\n(B29)\nGS6(x;y;s ) =\u0011x(x\u00001)=2+xy\n12 \u0011x+y\nS6T1gS6(s) (s=w):\n(B30)\nUsing the commutation relation between the \u001bandS6gauge\ntransformations, we find that\n\u0011\u001bS6I=\u0011y\n\u001bT1\u0011y\n12\u0011y\nS6T1gy\n\u001b(v)gS6(v)g\u001b(u)gS6(u) (B31)\n=\u0011y\n\u001bT1\u0011y\n12\u0011y\nS6T1\u0011\u001bg\u001b(w)gS6(v)g\u001b(u)gS6(u)(B32)\ngiving us that \u0011\u001bT1\u001112\u0011S6T1 = 1 and\ng\u001b(u)gS6(u)g\u001b(w)gS6(v) =\u0011\u001bS6\u0011\u001bI. A similar calcu-\nlation on a different sublattice gives us\n\u0011\u001bS6I=\u0011y\n\u001bT1\u0011y\n12\u0011y\nS6T1gy\n\u001b(w)gS6(w)g\u001b(v)gS6(w) (B33)\n=\u0011y\n\u001bT1\u0011y\n12\u0011y\nS6T1\u0011\u001bg\u001b(v)gS6(w)g\u001b(v)gS6(w)(B34)10\nTABLE II: We list the solutions of Eq. B43 - B54, along with a set of gauge transformations which realize these solutions.\nNo.\u0011T\u0011\u001bT\u0011S6T\u0011\u001b\u0011\u001bS6\u0011S6\u001112g\u001b(u)g\u001b(v)g\u001b(w)gS6(u)gS6(v)gS6(w)\n1,2 -1 1 1 1 1 \u00061\u00061 \u00000\u00000\u00000\u00000\u00000\u00000\n3,4 -1 1 1 1 -1 \u00071\u00061 \u00000\u00000\u00000\u00000-\u00000i\u00001\n5,6 -1 1 -1 1 -1 \u00071\u00061 \u00000\u00000\u00000i\u00003i\u00003i\u00003\n7,8 -1 1 1 -1 -1 \u00071\u00061i\u00001\u00000-\u00000\u00000i\u00001\u00000\n9,10 -1 1 1 -1 1 \u00061\u00061i\u00001\u00000-\u00000\u00000-i\u00001i\u00001\n11,12 -1 1 -1 -1 1 \u00071\u00061i\u00001\u00000-\u00000i\u00003-i\u00002i\u00003\n13,14 -1 -1 -1 -1 -1 \u00071\u00061i\u00003i\u00003i\u00003i\u00003i\u00003i\u00003\n15,16 -1 -1 1 -1 1 \u00061\u00061i\u00003i\u00003i\u00003\u00000\u00000\u00000\n17,18 -1 -1 1 -1 1 \u00071\u00061i\u00003i\u00003i\u00003\u00000\u00000i\u00001\n19,20 -1 -1 -1 -1 1 \u00071\u00061i\u00003i\u00003i\u00003i\u00003-i\u00003i\u00003\n21,22 1 1 1 1 1 \u00061\u00061 \u00000\u00000\u00000\u00000\u00000\u00000\n23,24 1 1 1 1 -1 \u00071\u00061 \u00000\u00000\u00000\u00000-\u00000i\u00003\n25,26 1 1 1 -1 -1 \u00071\u00061i\u00003\u00000-\u00000\u00000i\u00003\u00000\n27,28 1 1 1 -1 1 \u00071\u00061i\u00003\u00000-\u00000\u00000-i\u00003i\u00001\n29,30 1 1 1 -1 1 \u00061\u00061i\u00003\u00000-\u00000\u00000-i\u00003i\u00003\ngiving us (g\u001b(v)gS6(w))2=\u0011\u001bS6\u0011\u001bI. AZ2(IGG) gauge\ntransformation of the form W(x;y;s ) =\u0011y\n\u001bT1changes\u0011\u001bT1\nto 1. Using the cyclic relation of the gauge transformations\nrelated to the S6operators, we find\n\u0011S6I=\u001112(gS6(w)gS6(v)gS6(u))2(B35)\ngiving us that\n[gS6(w)gS6(v)gS6(u)]2=\u0011S6\u001112I: (B36)\nNext we turn to the time reversal symmetry. Similar meth-\nods to the above give us that\n[GT(i)]2=\u0011TI (B37)\nGy\nT(i)GT(i+ ^x) =\u0011T1TI (B38)\nGy\nT(i)GT(i+ ^y) =\u0011T2TI: (B39)\nThe first of these relations tells us that GT(i)is either the\nidentity (for \u0011T= 1) ori~ a\u0001~ \u0014(for\u0011T=\u00001, wherej~ aj= 1.\nDefiningGT(0;0;s) =gT(s),\nGT(x;y;s ) =\u0011x\nT1T\u0011y\nT2TgT(s) (B40)\nand further, using the commutation relations between the \u001b\nandTgauge transformations and the S6andTgauge trans-\nformations,\ngy\n\u001b(s)gy\nT(s)g\u001b(s)gT(\u001b(s))\u0011x+y\nT1T\u0011x+y\nT2T=\u0011\u001bTI(B41)\ngy\nS6(s)gy\nT(s)gS6(s)gT(S\u00001\n6(s))\u0011f1(i)\nT1T\u0011f2(i)\nT2T=\u0011S6TI:(B42)Because this is true for all x and y, and f1(i)is not equal to\nf2(i),\u0011T1T=\u0011T2T= 1. IfGT(i) =i~ a\u0001~ \u0014, we perform a\ngauge transformation W on GT(i)such thatWyGT(i)W=\ni\u00141(as this is the same on all sites, it does not affect our gauge\nfixing for the translation gauge transformations). Collecting\nthe necessary results for further use,\nGT1(x;y;s ) =\u0011y\n12I (B43)\nGT2(x;y;s ) =I (B44)\nG\u001b(x;y;s ) =\u0011xy\n12g\u001b(s) (B45)\nGS6(x;y;s ) =\u0011xy+(x+1)x=2\n12 gS6(s)s=u;v (B46)\nGS6(x;y;s ) =\u0011xy+x+y+(x+1)x=2\n12 gS6(s)s=w(B47)\nGT(s) =I=gT(s)\u0011T= 1 (B48)\nGT(s) =i\u00001=gT(s)\u0011T=\u00001 (B49)\ng\u001b(\u001b(s))g\u001b(s) =\u0011\u001bI (B50)\ng\u001b(u)gS6(u)g\u001b(w)gS6(v) = (g\u001b(v)gS6(w))2=\u0011\u001bS6\u0011\u001bI\n(B51)\n(gS6(w)gS6(v)gS6(u))2=\u0011S6\u001112I (B52)\ng\u001b(s)gT(\u001b(s)) =\u0011\u001bTgT(s)g\u001b(s) (B53)\ngS6(s)gT(S\u00001\n6(s)) =\u0011S6TgT(s)gS6(s): (B54)\nWe also have the gauge freedom left to perform a gauge ro-\ntation arbitrarily at all positions for \u0011T= 1 or an arbitrary\ngauge rotation about the x axis for \u0011T=\u00001.\nThe solution to the above equations is derived in detail by\nLuet al.15and as such we simply list the results in table II.\nThe basic method of obtaining these solutions is as follows:11\nfor each choice of Z2parameter set, we determine whether\nthere is a choice of gauge matrices fgSgwhich satisfy the\nequations B43 - B54. In order to do so, we determine the\nallowed forms of the gSmatrices from the equations, then use\nthe gauge freedom on each site to fix the form of these. Of\nparticular not is the fact that in the consistency equations for\nthegmatrices, the terms \u001112and\u0011S6only appear multiplied\ntogether, meaning that for any choice of the gauge matrices\ngSwe can choose \u001112=\u00061, which fixes the form of \u0011S6.\nAppendix C: Relation among the mean-field paramters\nThe relation among the different singlet and triplet param-\neters in terms of \u0018ijis given by\n\u001fij=\u001800\nij+\u001803\nij;\u0011ij=\u0000\u001801\nij+i\u001802\nij;\nE1\nij=\u001810\nij+\u001813\nij;E2\nij=\u001820\nij+\u001823\nij;E3\nij=\u001830\nij+\u001833\nij\nD1\nij=\u0000\u001811\nij+i\u001812\nij;D2\nij=\u0000\u001821\nij+i\u001822\nij;D3\nij=\u0000\u001831\nij+i\u001832\nij\n(C1)\nUsing these, we can derive the form of the bond nematic\norder parameter and vector chirality order parameters, which\nare given in terms of the mean field parameters21as\nQ\u0016;\u0017\nij=\u00001\n2\u0000\nE\u0016\nijE\u0003\u0017\nij\u00001\n3\u000e\u0016;\u0017j~Eijj2\u0001\n+h:c:\n\u00001\n2\u0000\nD\u0016\nijD\u0003\u0017\nij\u00001\n3\u000e\u0016;\u0017j~Dijj2\u0001\n+h:c:\nJ\u0015\nij=i\n2\u0000\n\u001fijE\u0003\u0015\nij\u0000\u001f\u0003\nijE\u0015\nij\u0001\n+i\n2\u0000\n\u0011ijD\u0003\u0015\nij\u0000\u0011\u0003\nijD\u0015\nij\u0001\n(C2)where our definition of \u0011ijdiffers by a factor of (-1) from that\nof the cited work. We rewrite this in terms of our variables,\nfinding\nQ\u0016\u0017\nij=\u0000\u0018\u00160\nij\u0018\u00170\nij+X\na\u0018\u0016a\nij\u0018\u0017a\nij\n+\u000e\u0016\u0017\n3X\nb\u0000\n(\u0018b0\nij)2\u0000X\na(\u0018ba\nij)2)\nJ\u0015\nij=i(\u001800\nij\u0018\u00150\nij\u0000X\na\u00180a\nij\u0018\u0015a\nij) (C3)\nIn particular, we find that J1,J2,Q13andQ23must be zero\nfor all non-Kramers spin liquids, as the terms allowed by sym-\nmetry in Eq. 32 and 33 do not allow non-zero values for these\norder parameters.\n1P. Fazekas, Lecture Notes on Electron Correlation and Magnetism\n(Series in Modern Condensed Matter Physics) , World Scientific\nPub Co Inc (1999).\n2K. Yosida, Theory of Magnetism , Springer (2001).\n3B. Bleaney and H. E. D. Scovil, Phil. Mag. 43, 999 (1952).\n4Shigeki Onoda and Yoichi Tanaka, Phys. Rev. B 83, 094411(R)\n(2011).\n5K. Matsuhira, Y . Hinatsu, K. Tenya1, H. Amitsuka and T. Sakak-\nibara, J. Phys. Soc. Jpn. 71, 1576 (2002).\n6K. Matsuhira, C. Sekine, C. Paulsen, M. Wakeshima, Y . Hinatsu,\nT. Kitazawa, Y . Kiuchi, Z. Hiroi, and S. Takagi, J. Phys. Conf. Ser.\n145, 012031 (2009).\n7S. Nakatsuji, Y . Machida, Y . Maeno, T. Tayama, T. Sakakibara, J.\nvan Duijn, L. Balicas, J. N. Millican, R. T. Macaluso, and J. Y .\nChan, Phys. Rev. Lett. 96, 087204 (2006).\n8SB. Lee, S. Onoda, and L. Balents, Phys. Rev. B 86, 104412\n(2012).\n9R. Flint, and T. Senthil, arXiv:1301.0815 (2013).\n10P. Chandra, P. Coleman, and R. Flint, Nature 493, 621 (2013).\n11O. Suzuki, H. S. Suzuki, H. Kitazawa, G. Kido, T. Ueno, T. Ya-maguchi, Y . Nemoto, T. Goto, J. Phys. Soc. Japan 75, 013704\n(2006).\n12H. A. Kramers, Proc. Amsterdam Acad. 33, 959 (1930).\n13X. G. Wen, Phys. Rev. B 65, 165113 (2002); X.-G. Wen, Quantum\nField Theory of Many-Body Systems. Oxford University Press,\nOxford (2004).\n14P. W. Anderson, G. Baskaran, Z. Zou, and T. Hsu, Phys. Rev. Lett.\n58, 2790 (1987).\n15Yuan-Ming Lu, Ying Ran, and Patrick A. Lee, Phys. Rev. B 83,\n224412 (2011).\n16T. Dodds, S. Bhattacharjee, and Y . B. Kim, arXiv:1303.1154\n17R. Schaffer, S. Bhattacharjee, and Y . B. Kim, Phys. Rev. B 86,\n224417 (2012).\n18I. Affleck, and J. B. Marston, Phys. Rev. B 37, 3774 (1988).\n19P. A. Lee, and N. Nagaosa, Phys. Rev. B 46, 5621 (1992).\n20P. A. Lee, and N. Nagaosa, Phys. Rev. B 87, 064423 (2013).\n21R. Shindou, and T. Momoi, Phys. Rev. B 80, 064410 (2009).\n22S. Bhattacharjee, Y . B. Kim, S.-S. Lee, and D.-H. Lee, Phys. Rev.\nB 85, 224428 (2012)."    },    {        "title": "1609.03141v1.Spin_Orbit_Coupling_Induced_Spin_Squeezing_in_Three_Component_Bose_Gases.pdf",        "content": "Spin-Orbit Coupling Induced Spin Squeezing in Three-Component Bose Gases\nX. Y. Huang,1F. X. Sun,1W. Zhang,2, 3,\u0003Q. Y. He,1, 4,yand C. P. Sun5,z\n1State Key Laboratory of Mesoscopic Physics, School of Physics, Peking University,\nCollaborative Innovation Center of Quantum Matter, Beijing 100871, China\n2Department of Physics, Renmin University of China, Beijing 100872, China\n3Beijing Key Laboratory of Opto-electronic Functional Materials and Micro-nano Devices,\nRenmin University of China, Beijing 100872, China\n4Collaborative Innovation Center of Extreme Optics,\nShanxi University, Taiyuan, Shanxi 030006, China\n5Beijing Computational Science Research Center, Beijing 100084, China\nWeobservespinsqueezinginthree-componentBosegaseswhereallthreehyperfinestatesarecoupled\nby synthetic spin-orbit coupling. This phenomenon is a direct consequence of spin-orbit coupling, as\ncan be seen clearly from an effective spin Hamiltonian. By solving this effective model analytically\nwith the aid of a Holstein-Primakoff transformation for spin-1 system in the low excitation limit,\nwe conclude that the spin-nematic squeezing, a novel category of spin squeezing existing exclusively\nin large spin systems, is enhanced with increasing spin-orbit intensity and effective Zeeman field,\nwhich correspond to Rabi frequency \nRand two-photon detuning \u000ewithin the Raman scheme\nfor synthetic spin-orbit coupling, respectively. These trends of dependence are in clear contrast\nto spin-orbit coupling induced spin squeezing in spin-1/2 systems. We also analyze the effects of\nharmonic trap and interaction with realistic experimental parameters numerically, and find that a\nstrong harmonic trap favors spin-nematic squeezing. We further show spin-nematic squeezing can be\ninterpreted as two-mode entanglement or two-spin squeezing at low excitation. Our findings can be\nobserved in87Rb gases with existing techniques of synthetic spin-orbit coupling and spin-selectively\nimaging.\nI. INTRODUCTION\nSpin squeezing is an important resource which has\nmany potential applications not only in quantum metrol-\nogy and atom interferometers [1–5], but also in many\naspects of quantum information due to its close rela-\ntion with quantum entanglement [6–10]. In conventional\nexperiments, squeezing is usually achieved via the non-\nlinearity induced by the inter-particle interaction [3–5].\nAs an example, spin squeezing has been obtained ex-\nperimentally in a Bose-Einstein condensate (BEC) of a\nthree-component Bose gas [11]. However, the intensity\nof spin squeezing in these experiments crucially depends\non the interaction between atoms. In cold atom exper-\niments, the background interaction is usually very weak\nsuch that the observation of squeezing is relatively hard.\nAlthough there are some techniques to enhance the in-\nteraction, e.g., by tuning the state-dependent microwave\npotentials [4], or through a magnetic Feshbach resonance\nin alkali atoms [5, 12], the side effects of decoherence,\nsevere atom loss and dynamical instability induced by\nstrong interaction still hinder the achievement of strong\nspin squeezing.\nThe experimental realization of synthetic spin-orbit\ncoupling (SOC) in ultracold atomic gases [13–15] has at-\ntracted much attention, partly due to its close relation\n\u0003wzhangl@ruc.edu.cn\nyqiongyihe@pku.edu.cn\nzcpsun@csrc.ac.cnto exotic many-particle states and novel excitations [16–\n18]. Recently, theoreticalstudieshaveproposedtorealize\nspin squeezing in two-component BEC by synthetic spin-\norbit coupling (SOC) [19, 20]. It has been shown that the\npresence of SOC will induce an effective spin-spin inter-\naction which can lead to spin squeezing. However, there\nare two disadvantages of these proposals. First, the syn-\nthetic SOC requires a Raman transition between two hy-\nperfine states. The Rabi frequency of this Raman transi-\ntion is detrimental to spin squeezing, i.e., a stronger SOC\nleads to a weaker squeezing. Besides, the two-photon de-\ntuning of this Raman transition is also unfavorable such\nthat best squeezing will be achieved when the detuning is\nzero. Nonetheless, in realistic experiments one would en-\ncounter severe heating effect when the detuning is tuned\non resonance.\nIn this paper, we study spin squeezing in a three-\ncomponent Bose gases where all three hyperfine states\nare coupled by spin-orbit coupling induced by Raman\ntransitions. As a result, this system has pseudo-spin-1,\nandthespinoperatorshereinmustbedescribedbySU(3)\nspin matrices, i.e., the Gell-Mann matrices. These Gell-\nMann matrices span an eight-dimensional spin hyper-\nspace, with three of them are usually refereed as spin\nvectors, and the other five as nematic tensors [21]. The\nsqueezed spin operators hence can be categorized into\nthree types, including the spin-spin squeezing, nematic-\nnematic squeezing, and spin-nematic squeezing. Here,\nwe focus on the spin-nematic squeezing, as it is a novel\ntypeofsqueezingwhichexistsexclusivelyinsystemswith\nlarge spins. We find that the presence of SOC can induce\nspin-nematic squeezing, which can be further enhancedarXiv:1609.03141v1  [quant-ph]  11 Sep 20162\nbyincreasingtheSOCintensityorreducingthequadratic\nZeeman splitting. These trends of dependence can be\nunderstood from an effective Hamiltonian, in which the\nRabi frequency and quadratic Zeeman splitting corre-\nspond to effective Zeeman fields in the spin and nematic\nsectors, respectively, hence causing opposite effects on\nvarious types of spin squeezing. More importantly, we\nfind that the squeezing is favored by two-photon detun-\ning of the Raman transition within a fairly large param-\neter regime, which is beneficial for experimental realiza-\ntions to avoid severe heating effect. When the system ex-\nhibits spin-nematic squeezing in the low excitation limit,\nwe also find two-mode entanglement [22] and two-spin\nsqueezing [23]inthesystem. Wefurtherstudytheeffects\nof an external trapping potential and inter-atomic inter-\naction which are present in realistic experimental situa-\ntionsbynumericallyanalysis, andconcludethatthespin-\nnematic squeezing is favored by stronger trapping poten-\ntials. Finally, we discuss possible detection scheme via\na spin-selective imaging technique and a radio-frequency\n(RF) rotation of the spin axes [11].\nThe remainder of this paper is organized as follows. In\nSec. II, we introduce the system under investigation and\ndiscuss the single-particle spectra. We then derive an ef-\nfectivespinHamiltonianfromwhichitcanbeseenclearly\nthat SOC induces an effective spin-spin interaction. We\nthen analyze the spin-nematic squeezing and its depen-\ndence of various factors in Sec. III. Finally, we discuss\npossible experimental detection scheme and summarize\nin Sec. IV.\nII. SINGLE-PARTICLE SPECTRA AND\nEFFECTIVE HAMILTONIAN\nSpin-orbit coupled three-component Bose gas can be\ngeneralized by counter-propagating Raman lasers along\n^xto couple the three hyperfine states with momentum\ntransfer of the Raman process 2kr. The non-interacting\nHamiltonian can be written in the matrix form as [24]\nH=0\nBBBBB@(kx+2kr)2\n2\u0000\u000e\nR=2 0\n\nR=2k2\nx\n2\u0000\u000f \nR=2\n0 \n R=2(kx\u00002kr)2\n2+\u000e1\nCCCCCA+k2\n?\n2;\n(1)\nwherek?=q\nk2y+k2zis the transverse momentum, \u000eis\nthe two-photon detuning from the Raman resonance, \u000fis\nthe quadratic Zeeman shift induced by the magnetic field\nalong ^y, and \nRrepresentstheRabifrequencyoftheRa-\nman transition. Notice that throughout the manuscript,\nwe use the natural units of }=m= 1, and define krand\nthe recoil energy Er=k2\nr=2as the units of momentum\nand energy, respectively.\nThe single-particle dispersion can be obtained by di-\nagonalizing th non-interacting Hamiltonian of Eq. (1).\n1minimum\n1minimum2minima\n2minima3minimaæcritical point\nÑdEr=1HaL\n0 1 2 3 4 5 6-6-4-20246\nÑWRErÑeEr\n1minimum1minimum\n1minimum2minima\n2minimaæ\næcritical point\ncritical point\nÑeEr=6HbL\n0 2 4 6 8-10-50510\nÑWRErÑdEr\nHcL\nÑWREr=76543210\n-4-2024-4-3-2-1012\nkxkrEEr\nÑeEr=6543210-1-2-3HdL\n-4-2024-6-4-202\nkxkrEErFigure 1. (Color online) (a-b) Single-particle phase diagrams\nof a three-component Bose gas with one-dimensional SOC in\nthe (a) \nR–\u000fplane with \u000e= 1and (b) \nR–\u000eplane with\n\u000f= 6. The lowest branch of the single-particle dispersion\nspectrum acquires either one, two, or three local minima in\ndifferent parameter regimes separated by solid lines. On the\ndashed lines within regions of multiple minima, two of the\nlocal minima are degenerate. Typical examples for the lowest\nbranch of dispersion curves by changing (c) Rabi frequency\n\nRwith\u000e= 1and\u000f= 0and (d) quadratic Zeeman energy \u000f\nwith\u000e= 1and\nR= 2.\nThe resulting spectra has three branches, among which\nthelowestonecanhavethreeminima,twominima,orone\nminimum depending on the combination of parameters.\nIn Figs. 1(a) and 1(b), we show the parameter regions ex-\nhibitingdifferentstructuresforthecaseof ~\u000e=Er= 1and\n~\u000f=Er= 6respectively. From Fig. 1(a), we can identify\nvarious regions where the lowest branch of single-particle\ndispersion acquires 1, 2, or 3 minima. Specifically, for\nthe case of a large positive quadratic Zeeman splitting\n\u000f, thej0istate is far detuned from the other two high-\nlying hyperfine states, so that the spectrum has only one\nminimum. On the other hand, if \u000fis large negative, the\nj0istate becomes the high-lying state and the system\nessentially turn into a spin-1/2 Bose gas where the two\nj\u00061ispin components are spin-orbit coupled via virtual\nprocesses involving the j0istate. As a result, the single-\nparticle dispersion can have either two or one minima,\ndepending on the SOC intensity \nRand two-photon de-\ntuning\u000e. For the case of intermediate j\u000fj, all three hyper-\nfine states are spin-orbit coupled and the shape of spec-\ntrum is sensitively dependent on all parameters. Typical\nexamples of dispersion spectra along the kxaxis showing\none minimum, two minima, and three minima, as well\nas the trends of evolution depending on \nRand\u000fare\nillustrated in Figs. 1(c) and 1(d), respectively.\nAs the spin operators in spin-1/2 systems all belong\nto the SU(2) group, those in spin-1 systems discussed3\nhere are elements in the SU(3) group. The SU(3) group\nis locally isomorphic to the O(8) group, which has eight\nlinearly independent observables as generators. These\ngeneratorscanbegroupedintotwotypes, includingthree\nspin vectors (or angular momentum operators) and five\nnematic tensors. The irreducible matrix representations\nof these observables are given by [21]\nJx=1p\n20\n@0 1 0\n1 0 1\n0 1 01\nA,Jy=ip\n20\n@0\u00001 0\n1 0\u00001\n0 1 01\nA,\nJz=0\n@1 0 0\n0 0 0\n0 0\u000011\nA,Qxy=i0\n@0 0\u00001\n0 0 0\n1 0 01\nA,\nQyz=ip\n20\n@0\u00001 0\n1 0 1\n0\u00001 01\nA,Qzx=1p\n20\n@0 1 0\n1 0\u00001\n0\u00001 01\nA,\nD=0\n@0 0 1\n0 0 0\n1 0 01\nA,Y=1p\n30\n@1 0 0\n0\u00002 0\n0 0 11\nA:\nThe commutators between these spin operators can then\nbe classified into three categories: [Jy;Jz] =iJxas\nspin-spin group, [Qxy;Qxz] =iJx,[Qyz;D] =iJx,\nand [Qyz;Y] =p\n3iJxas nematic-nematic group, and\n[Jx;Qyz] =i(p\n3Y+D),[Jy;Qzx] =i(\u0000p\n3Y+D)as\nspin-nematic group.\nTo study the effective spin-spin interaction induced by\nSOC,aswellastheinducedspinsqueezingeffect, nextwe\nderive an effective spin model. To facilitate the deriva-\ntion, we impose a weak harmonic trap V(x) =!2\nxx2=2 +\n!2\nyy2=2 +!2\nzz2=2. We will find that the resulting form\nof the effective model does not depend on the absolute\nvalue of trapping frequency, hence incorporate solely the\neffect of SOC. In the presence of such an auxiliary trap-\nping potential, we can quantize the motional degrees of\nfreedom along the trapping direction to a discrete energy\nspectrum. In particular, by introducing the bosonic op-\neratorsa\u0011p\n!x=2(x+ikx=!x),b\u0011p\n!y=2(y+iky=!y),\nc\u0011p\n!z=2(z+ikz=!z)and the collective spin operators\nFs=x;y;z =PN\ni=1Ji;s,FY=PN\ni=1Yi, the Hamiltonian of\nEq. (1) for N-particle can be rewritten as\n~H=!xNaya+N4k2\nr\u0000\u000f\n3+\nRp\n2Fx\n+ikrp\n2!x(ay\u0000a)Fz\u0000\u000eFz+2k2\nr+\u000fp\n3FY:(2)\nHere we ignore !yNbyb+!zNcycsince the boson modes\niny,zdirection do not interact with the ultracold atoms.\nEmploying the unitary transformation U= exp[iG(ay+a)Fz]withG=p\n2=!xkr=N, the Hamiltonian thus can\nbe transformed as\n~H0=!xNaya\u0000qF2\nz\u0000\u000eFz+2k2\nr+\u000fp\n3FY\n+\nRp\n2fFxcos[G(ay+a)]\u0000Fysin[G(ay+a)]g;(3)\nwhereq= 4k2\nr=N= 8Er=N. Notice that the term of\nN(4k2\nr\u0000\u000f)=3has been dropped out as the zero-point\nenergy.\nFor a BEC, the expectation value of hayaiis in the\norder ofNfor the ground state, and about unity for\nexcited states. Considering the prefactor of 1=Nin the\ndefinition of G, the leading order of the arguments in the\ncosine and sine functions in Eq. (3) are 1=p\nN, which\nis negligible for systems of large particle number. As a\nresult, we can approximate the cosine and sine functions\ntothezerothorder,andtheHamiltonianEq. (3)becomes\nseparable in spatial and spin degrees of freedom, leading\nto an effective spin Hamiltonian\nHe\u000b=\u0000qF2\nz+\nRp\n2Fx\u0000\u000eFz+4Er+\u000fp\n3FY:(4)\nOne can see clearly that an effective spin-spin interac-\ntion emerges as a result of SOC, and the Rabi frequency\n\nR, two-photon detuning \u000e, and the quadratic Zeeman\nsplitting\u000fact as effective Zeeman fields along different\ndirections in the eight-dimensional spin hyperspace.\nIII. SPIN-NEMATIC SQUEEZING\nWith the aid of the effective spin model of Eq. (4), we\ncan study the spin squeezing in the underlying system.\nAs the commutators relation between spin and nematic\noperators are not present in the spin-1/2 case, next we\nfocus on spin squeezing of this type. The method can be\nstraightforwardly applied to the spin-spin and nematic-\nnematic commutators, and the results are qualitatively\nconsistent with the findings for the spin-spin case in spin-\n1/2 system with SOC [19, 20].\nThe spin model of Eq. (4) can not be solved analyt-\nically due to the presence of nonlinear interaction. In\nthe low excitation limit, however, we can introduce the\nHolstein-Primakoff transformation for spin-1 systems\nFx\u00111p\n2\u0010\nby\n1N0\n0+N0\n0b\u00001+ h:c:\u0011\n;\nFy\u00111p\n2i\u0010\nby\n1N0\n0+N0\n0b\u00001\u0000h:c:\u0011\n; (5)\nwhereN0\n0\u0011q\nN\u0000by\n1b1\u0000by\n\u00001b\u00001, and the operators b1\nandb\u00001representing spin flipping processes between the\ninternal levelsj\u00061iandj0i, represented by the bosonic\nmodesa\u00061anda0. For the case that most of the par-\nticles remain in the mode a0, i.e.,hay\n0a0i 'Nand4\nhby\n\u00061b\u00061i\u001cN, the operators b\u00061=a\u00061ay\n0=p\nNare effec-\ntive bosonic modes satisfying the bosonic commutation\nrelationsh\nb\u000b;by\n\fi\n=\u000e\u000b\fwith\u000b;\f =\u00061. Within the\nassumption that the majority of the particles are resid-\ning in thej0istate, or equivalently the excitations to\nthej\u00061istates are rare, we can rewrite the bosonic\noperators as a mean-field value plus some fluctuations\nb\u00061=p\nN\f\u00061+\u000eb\u00061. The ground state energy can\nthe be obtained by minimizing the energy functional\nE(\f1;\f\u00001). As in this the low excitation limit, nearly\nall the spins are polarized in FYdirection, which means\f\f\n\u0006p\n3FY+FD\u000b\f\f\u00192N, the squeezing parameter is then\ngiven by [25]\n\u0018x\u0011min\u0000\n42Jn?\u0001\nJ=2\u001942Fx=N; (6)\nHere,Jis the expectation value of mean spin, Jn?is a\nspin component along the direction perpendicular to the\nmean spin direction. So in our case, it is clear that \u0018x\ncan be obtained by calculated the variance of Fx, one has\nspin squeezing in the spin-nematic channel as \u0018x<1.\nWe first discuss the case of zero two-photon detuning\n\u000e= 0, and show in Fig. 2 the spin-nematic squeezing pa-\nrameter as functions of Rabi frequency \nRand quadratic\nZeeman splitting \u000f. One can see clearly that the ground\nstateis aspinsqueezed stateundertheeffect ofSOC.Im-\nportantly, as shown in Fig. 2(a), spin-nematic squeezing\ncan be enhanced with increasing \nR. This behavior is in\nstark contrast to the case of spin-1/2 systems, where the\nspin-spin squeezing is favored by decreasing \nR[19, 20].\nWe then extend the discussion to the more general case\nof a nonzero two-photon detuning \u000e6= 0. This scenario is\nexperimentally relevant because a severe heating effect is\nusually present as the Raman transition is on-resonance.\nAs shown in Fig. 3(a), a finite \u000efavors spin-nematic\nsqueezingwithinafairlylargeregionof j~\u000e=Erj<5. This\nresult can be understood by analyzing the single-particle\nHamiltonian of Eq. 1, where \u000eand\u000fare energy offsets\nof the diagonal elements. As Raman transitions will be\nenhanced when difference states are near resonance, spin\nsqueezing will be favored when the absolute value of \u000e\nis close to\u000f. To further clarify this argument, we ana-\nlyze the atom populations of different ground states with\nchanging\u000e. As shown in Fig. 3(b), the presence of a fi-\nnite\u000ewill enhance the transition between the j0istate\nand one of thej\u00061istates, while the transition to the\notherj\u00061istate is reduced. Notice that this behavior\nis very different from the spin-1/2 case, where the two\nspin components are moved away from each other with\nincreasing\u000e, leading to an effectively weaker SOC.\nThe dependences of spin-nematic squeezing on the var-\nious parameters of \nR,\u000fand\u000ecan also be interpreted\nfrom the effective spin model of Eq. (4), within which the\nthree parameters correspond to effective Zeeman fields\nalong theFx,FY, andFydirections, respectively. Con-\nsidering that in the low excitation limit nearly all spins\nare polarized along the FYdirection, a stronger Zeeman\nHaL\n012340.70.80.91.0\nÑWRErxx\nHbL\n2468100.70.80.91.0\nÑeErxxFigure 2. (Color online) Spin-nematic squeezing parameter \u0018x\nas a function of (a) Rabi frequency \nRwith\u000e= 0and\u000f= 6\nand (b) quadratic Zeeman splitting \u000fwith\u000e= 0and\nR= 2.\nIn both figures, results obtained from the effective spin model\nEq.(4) are illustrated by blue solid lines, in comparison to the\nnumerical solutions of the GP equation for a pancake-shaped\ntrapwith!x=!y= 50Hz,!z= 1500Hz(blackdashed), and\nfor a cigar-shaped trap with !x=!y= 5000Hz,!z= 1500\nHz (red dotted). Here, we consider a gas of87Rb atoms in\ntheF= 1manifold with background interaction and total\nparticle number N= 105.\nfield along the same direction, i.e., a larger value of \u000f,\nwill further intensify the polarization so that the effec-\ntive spin-spin interaction becomes relatively weak, lead-\ning to a less spin-nematic squeezing effect. On the other\nhand, effective Zeeman fields along the perpendicular di-\nrections, either FxorFz, will tilt the spin polarization\nfrom theFYaxis slightly but the effect spin-spin interac-\ntion is enhanced obviously, resulting an increased squeez-\ning parameter as in Eq. (6).\nIn addition to the spin-nematic squeezing, we notice\nthat in the low excitation limit with the majority of par-\nticles residing in the j0istate, the two effective bosonic\nmodesb1andb\u00001can be entangled, which is referred\nas two-mode entanglement. A sufficient criterion for en-\ntanglement between the modes b1andb\u00001from the spin\nsqueezing parameters is then given by [22]\n\u0018\u0012\nDCZ= (\u0018\u0012\n++\u0018\u0012+\u0019=2\n\u0000 )=2<1; (7)\nwhere\u0018\u0012\n\u0006\u0019h\u00012F\u0012\n\u0006i=Nrepresentsthevarianceofquadra-\nture phase amplitudes which depends on the parameter5\nHaL\n-5.0-2.50.02.55.00.750.800.850.900.951.00\nÑdErxx\nHbL\n-10-505100.000.200.400.600.801.00\nÑdErr\nFigure 3. (Color online)(a) Variations of spin-nematic squeez-\ning parameter \u0018as a function of two-photon detuning \u000ewith\n\nR= 2and\u000f= 6. Analytic result obtained from the effective\nspin model Eq. (4) within low-density excitation approxima-\ntion (blue solid) is compared with numerical solutions of the\nGP equation for a pancake-shaped trap with !x=!y= 50\nHz,!z= 1500Hz (black dashed), and for a cigar-shaped trap\nwith!x=!y= 5000Hz,!z= 1500Hz (red dotted). (b)\nAtom number fractions of the j\u00001i(black dotted),j0i(blue\ndashed), andj+ 1i(red solid) states.\n\u0012, andwehaveusethedefinitions F\u0012\n+= cos\u0012Fx+sin\u0012Fyz\nandF\u0012\n\u0000= cos\u0012Fzx+ sin\u0012Fyin this system. Here, the\ncollective nematic operators are Fyz=PN\ni=1Qyzand\nFzx=PN\ni=1Qzx. Figure 4(a) shows that \u0018\u0012\nDCZreaches\nits minimum for \u0012=n\u0019withnan integer, and the en-\ntanglement is enhanced by Raman transition.\nAnother representation of two-mode entanglement is\ncalled two-spin squeezing, which is defined by dividing\nthe spin-1 space into three subspaces pseudospins (each\nof spin-1/2) U,VandTassociated the three relative\nnumber differences of particles N+1\u0000N0,N\u00001\u0000N0, and\nN+1\u0000N\u00001in three-component labeled by f+1;\u00001;0g\n[23]. Two-spin squeezing parameter is given to describe\nthe correlation between the spin subspace U(the spin\nflipping process between internal levels j+ 1iandj0i)\nandV(the spin flipping process between internal levels\nj\u00001iandj0i) [23]\n\u0018\u0012\nUV=\u00012F\u0012\n++ \u00012F\u0012+\u0019=2\n\u0000p\n3jhFYij<1; (8)\nIn Fig. 4(b), one can see clearly that the optimal correla-\n0p\n2p3p\n22p0.920.961.00\nqxDCZqHaL\n0 1 2 3 40.700.800.901.00\nÑWRErxDCZ0\n0p\n2p3p\n22p0.961.001.04\nqxUVqHbL\n0 1 2 3 40.850.900.951.00\nÑWRErxuv0Figure 4. (Color online) (a) Two-mode entanglement param-\neter\u00180\nDCZand (b) two-spin squeezing parameter \u00180\nUVver-\nsus Rabi frequency \nRwith other parameters being \u000e= 0\nand\u000f= 6. Analytic result obtained from the effective spin\nmodel Eq. (4) within low-density excitation approximation\n(blue solid) is compared with numerical solutions of the GP\nequation for a pancake-shaped trap with !x=!y= 50Hz,\n!z= 1500Hz (black dashed), and for a cigar-shaped trap\nwith!x=!y= 5000Hz,!z= 1500Hz (red dotted). The\ninsets show the squeezing parameters as functions of \u0012. No-\ntice that the optimal squeezing in both criteria are obtained\nwhen\u0012=n\u0019withnan integer.\ntion is obtained when \u0012=n\u0019withnan integer, and in-\ncreases with the the Raman transition. When comparing\nspin-nematic squeezing parameter (Fig. 2(a)) with these\ntwo criterions (Fig. 4), we find that the effect of squeez-\ning in spin-nematic channel is another representation of\nthe correlation between two spin subspaces and entangle-\nment between two effective modes in the low excitation\nlimit.\nFinally, we notice that in realistic experiments, one\nalso needs to take the effects of inter-atomic interaction\nand a global harmonic trap into consideration. Taking\n87Rb as a particular example, the interaction among the\nthree hyperfine states of the ground state manifold can\nbe categorized into two groups, depending on the total\nangularmomentumofthetwocollidingatoms. Theback-\nground scattering lengths are taken as as0= 101:8a0for\nF= 0, andas2= 100:4a0forF= 2, wherea0denotes the\nBohr radius [26]. For the effects of trapping potentials,\nwe consider two types of global harmonic traps includ-\ning a pancake-shaped quasi-two-dimensional trap with6\n!x=!y= 50Hz and!z= 1500Hz, and a cigar-shaped\nthree-dimensional trap with !x=!y= 5000 Hz and\n!z= 1500Hz.\nBynumericallysolvingtheGross-Pitaevski(GP)equa-\ntion for a total number of N= 105atoms, we obtain\nthe ground state of the system, and calculate the spin-\nnematic squeezing parameter \u0018x, the two-mode entangle-\nment parameter \u0018\u0012\nDCA, and the two-spin squeezing pa-\nrameter\u0018\u0012\nUV. The corresponding results are shown in\nFigs.2, 3and4. Bycomparingthenumericalresultswith\nthe outcome from the effective spin model, we conclude\nthat the effective model Eq. (4) is qualitatively valid in\nthelowexcitationlimit. Ontheotherhand, astronghar-\nmonic trap can cause sizable increment on spin-nematic\nsqueezing and two-mode entanglement. This observation\ncan be understood by noticing that in the presence of\na strong harmonic trap, the particles will be more con-\ndensed with a higher number density at the trap center.\nAs a result, the inter-particle interaction has stronger\neffect and causes better spin-nematic squeezing and two-\nmode entanglement.\nIV. EXPERIMENTAL DETECTION AND\nCONCLUSION\nWe have shown that an effective spin-spin interac-\ntion can be induced in spin-orbit coupled spin-1 BEC,\nwhichcanproduceaspecialkindofsqueezingcalledspin-\nnematicsqueezing. Thistypeofspinsqueezingcanbeen-\nhanced by increasing Raman transition intensity and de-creasing quadratic Zeeman splitting. More importantly,\nthe squeezing is favored by a finite two-photon detuning\nin a fairly large parameter regime, which could be bene-\nficial for experiments to reduce heating effect. These be-\nhaviors are in clear contrast to the spin squeezing within\nspin-orbit coupled spin-1/2 systems, where the trends\nof dependence on Raman transition intensity and two-\nphoton detuning are opposite. We also observe SOC\ninduced two-mode entanglement and two-spin squeezing\nin such a system, and investigate their dependence on\nRaman transition intensity. We further analyze the ef-\nfects of inter-particle interaction and external harmonic\ntrap by numerically solving the GP equation, and find\ngood agreement with approximate solutions of the effec-\ntive spin model.\nIn order to detect such an exotic type of spin squeezing\nin this system, one may need to rotate Jxinto the easily\nmeasuredJzdirection by applying a \u0019=2radio-frequency\n(RF) rotation about the Jyaxis. This operation can be\naccomplished with a two-turn coil on the experimental\ny-axis driven at the frequency splitting of the mFstates.\nThen, we can measure the variance of spin via a spin-\nselective imaging technique.\nACKNOWLEDGMENTS\nThis work is supported by NSFC (11274009, 11622428,\n11274025,11434011,11522436,61475006,and61675007),\nNKBRP (2013CB922000) and the Research Funds of\nRenmin University of China (10XNL016, 16XNLQ03).\n[1] D. J. Wineland, J. J. Bollinger, W. M. Itano, and D. J.\nHeinzen, Phys. Rev. A 50, 67 (1994).\n[2] M.KitagawaandM.Ueda, Phys.Rev.A 47, 5138(1993).\n[3] J. Estève, C. Gross, A. Weller, S. Giovanazzi, and M. K.\nOberthaler, Nature 455, 1216 (2008).\n[4] M. F. Riedel, P. Böhi, Y. Li, T. W. Hänsch, A. Sinatra,\nand P. Treutlein, Nature 464, 1170 (2010).\n[5] C. Gross, T. Zibold, E. Nicklas, J. Estève, and M. K.\nOberthaler, Nature 464, 1165 (2010).\n[6] H.F.HofmannandS.Takeuchi, Phys.Rev.A 68, 032103\n(2003).\n[7] G. Tóth, C. Knapp, O. Gühne, and H. J. Briegel, Phys.\nRev. A 79, 042334 (2009).\n[8] E. G. Cavalcanti, P. D. Drummond, H. A. Bachor, and\nM. D. Reid, Opt. Express 17, 18693 (2009).\n[9] M. D. Reid, P. D. Drummond, W. P. Bowen, E. G. Cav-\nalcanti, P. K. Lam, H. A. Bachor, U. L. Andersen, and\nG. Leuchs, Rev. Mod. Phys. 81, 1727 (2009).\n[10] E. G. Cavalcanti, S. J. Jones, H. M. Wiseman, and M.\nD. Reid, Phys. Rev. A 80, 032112 (2009).\n[11] C. D. Hamley, C. S. Gerving, T. M. Hoang, E. M. Book-\njans, and M. S. Chapman, Nat. Phys. 8, 305 (2012).\n[12] C. Chin, R. Grimm, P. Julienne, and E. Tiesinga, Rev.\nMod. Phys. 82, 1225 (2010).[13] Y. J. Lin, K. Jiménez-García, and I. B. Spielman, Nature\n471, 83 (2011).\n[14] P. Wang, Z. Q. Yu, Z. Fu, J. Miao, L. Huang, S. Chai, H.\nZhai, and J. Zhang, Phys. Rev. Lett. 109, 095301 (2012).\n[15] L. W. Cheuk, A. T. Sommer, Z. Hadzibabic, T. Yefsah,\nW. S. Bakr, and M. W. Zwierlein, Phys. Rev. Lett. 109,\n095302 (2012).\n[16] H. Zhai, Rep. Prog. Phys. 78, 026001 (2015).\n[17] W. Yi, W. Zhang, and X. Cui, Sci. China: Phys. Mech.\nAstron. 58, 014201 (2015).\n[18] J. Zhang, H. Hu, X. J. Liu, and H. Pu, Annu. Rev. Cold\nAt. Mol. 2, 81 (2014).\n[19] J. Lian, L. Yu, J.-Q. Liang, G. Chen, and S. Jia, Sci.\nRep.3, 3166 (2013).\n[20] Y. Huang and Z.-D. Hu, Sci. Rep. 5, 8006 (2015).\n[21] E. Yukawa, M. Ueda, and K. Nemoto, Phys. Rev. A 88,\n033629 (2013).\n[22] L. M. Duan, J. I. Cirac, and P. Zoller, Phys. Rev. A 65,\n033619 (2002).\n[23] Özgür E. Müstecaphoˇ glu, M. Zhang, and L. You, Phys.\nRev. A 66, 033611 (2002).\n[24] Z. Lan and P. Ohberg, Phys. Rev. A 89, 023630 (2014).\n[25] J. Ma, X. Wang, C. P. Sun, and F. Nori, Phys. Rep. 509,\n89 (2011).\n[26] W. Zhang, D. L. Zhou, M. S. Chang, M. S. Chapman,\nand L. You, Phys. Rev. A 72, 013602 (2005)."    },    {        "title": "2201.11823v2.Superfluid_transition_temperature_and_fluctuation_theory_of_spin_orbit_and_Rabi_coupled_fermions_with_tunable_interactions.pdf",        "content": "Super\ruid transition temperature and \ructuation theory of\nspin-orbit and Rabi coupled fermions with tunable interactions\nPhilip D. Powell,1, 2Gordon Baym,2and C. A. R. S\u0013 a de Melo3\n1Lawrence Livermore National Laboratory, 7000 East Avenue, Livermore, California 94550, USA\n2Department of Physics, University of Illinois at Urbana-Champaign,\n1110 W. Green Street, Urbana, Illinois 61801, USA\n3School of Physics, Georgia Institute of Technology, 837 State Street, Atlanta, Georgia 30332, USA\n(Dated: June 6, 2022)\nWe obtain the super\ruid transition temperature of equal Rashba-Dresselhaus spin-orbit- and\nRabi-coupled Fermi super\ruids, from the Bardeen-Cooper-Schrie\u000ber (BCS) to Bose-Einstein con-\ndensate (BEC) regimes in three dimensions for tunable s-wave interactions. In the presence of Rabi\ncoupling, we \fnd that spin-orbit coupling enhances (reduces) the critical temperature in the BEC\n(BCS) limit. For \fxed interactions, we show that spin-orbit coupling can convert a \frst-order (dis-\ncontinuous) phase transition into a second-order (continuous) phase transition, as a function of Rabi\ncoupling. We derive the Ginzburg-Landau free energy to sixth power in the super\ruid order pa-\nrameter to describe both continuous and discontinuous phase transitions as a function of spin-orbit\nand Rabi couplings. Lastly, we develop a time-dependent Ginzburg-Landau \ructuation theory for\nan arbitrary mixture of Rashba and Dresselhaus spin-orbit couplings at any interaction strength.\nI. INTRODUCTION\nThe ability to simulate magnetic and other external\n\felds [1{12] in cold atomic gases has created the oppor-\ntunity to explore a wide variety of new interactions and\ncomplex phase structures otherwise inaccessible in the\nlaboratory. Moreover, the capacity to generate these syn-\nthetic \felds in both bosonic and fermionic systems, and\nto continuously tune two-body interactions by means of a\nFeshbach resonance, has opened up a wonderland of tun-\nable systems, previously restricted to theorists' dreams.\nFor example, the possibility of simulating quantum chro-\nmodynamics (QCD) on an optical lattice [13{16] is a tan-\ntalizing prospect for researchers whose current theoreti-\ncal tools remain limited by QCD's non-perturbative char-\nacter and the restriction of lattice techniques to near-zero\nchemical potential.\nPrevious theoretical analyses of three-dimensional\nspin-orbit-coupled Fermi gases (e.g.,6Li,40K) have fo-\ncused mainly on the zero-temperature limit, in which sev-\neral exotic phases characterized by unconventional pair-\ning are expected to emerge [17{22]. However, the Ra-\nman laser platforms currently employed to produce syn-\nthetic spin-orbit \felds also induce heating that prevents\nthe realization of temperatures su\u000eciently low to observe\nthe super\ruid transition in either the weakly coupled\nBardeen-Cooper-Schrie\u000ber (BCS) or the strongly coupled\nBose-Einstein condensate (BEC) regimes [5, 7]. Thus,\nwhile two-body bound states (Feshbach molecules) have\nbeen observed in the BEC limit of40K [7, 23], the obser-\nvation of super\ruid states remain elusive. Future exper-\niments, however, may break this impasse by employing\na new platform currently under development|the radio-\nfrequency atom chip|which avoids heating of the atom\ncloud entirely [24]. While rf atom chips are somewhat\nmore restricted than the Raman scheme in the maximum\nobtainable spin-orbit coupling, its potential to reach su-per\ruid temperatures is leading to its adoption in the\nnext generation of experiments probing the topological\nsuper\ruid phases of spin-orbit-coupled fermions [25].\nOne class of systems of particular interest in the con-\ntext of quantum simulation is that of Rashba-Dresselhaus\nspin-orbit-coupled gases [17{22, 26, 27]. These systems\nare intriguing both because they re\rect physics studied\nextensively in the context of semiconductors [28, 29], and\nbecause they provide a platform for realizing tunable\nnon-Abelian \felds in the laboratory. Thus, while the\nholy grail of a full optical simulation of QCD remains\nyears in the future, there do exist notable analogies be-\ntween quark matter and cold atomic systems (e.g., non-\nAbelian \felds, evolution between strongly and weakly\ncoupled limits) within near-term experimental reach [30{\n33]. Investigations of spin-orbit-coupled ultracold gases\nhave also included optical lattices [34{43], thus enlarg-\ning the number of possible physical systems that can be\naccessible experimentally.\nTo date, most experimental realizations of these\nsystems have adopted equal Rashba-Dresselhaus cou-\nplings [4{6, 44], but systems exhibiting Rashba-only cou-\nplings have also been created [11, 45, 47]. Other ex-\nperiments have generated spin-orbit coupling dynami-\ncally [48] or even created three-dimensional spin-orbit\ncoupling [49]. Due to the versatility of Rashba-\nDresselhaus coupled systems, the ability to realize these\nsystems in the laboratory, and the myriad technical\nchallenges inherent in reaching arbitrarily low temper-\natures, it is increasingly important to provide a theoret-\nical framework for guiding and testing these simulators\nagainst experimental probes at realistic (nonzero) tem-\nperatures.\nThis problem bears a close relation to spin-orbit cou-\npling in solids, where the role of the Rabi frequency is\nplayed by an external Zeeman magnetic \feld. While a\nmean-\feld treatment describes well the evolution from\nthe BCS to the BEC regime at zero temperature [50, 51],arXiv:2201.11823v2  [cond-mat.quant-gas]  2 Jun 20222\nthis order of approximation fails to describe the correct\ncritical temperature of the system in the BEC regime\nbecause the physics of two-body bound states, i.e., Fes-\nhbach molecules, is not captured when the pairing order\nparameter goes to zero [52]. To remedy this problem, we\ninclude the e\u000bects of order-parameter \ructuations in the\nthermodynamic potential.\nIn this paper, we investigate the impact of a speci\fc\nclass of spin-orbit coupling, namely, an equal mixture\nof Rasha and Dresselhaus terms, on the super\ruid tran-\nsition temperature of a three-dimensional Rabi-coupled\nFermi gas, but also give general results for an arbitrary\nmixture of Rashba and Dresselhaus components. This\npaper is the longer version of our preliminary work [53].\nWe stress that the present results are applicable to both\nneutral cold atomic and charged condensed-matter sys-\ntems. We show that spin-orbit coupling, in the presence\nof a Rabi \feld (or Zeeman \feld, in solids), enhances the\ncritical temperature of the super\ruid in the BEC regime\nand converts a discontinuous \frst-order phase transition\ninto a continuous second-order transition, as a function of\nthe Rabi frequency for given two-body interactions. We\nanalyze the nature of the phase transition in terms of the\nGinzburg-Landau free energy, calculating it to the sixth\npower of the super\ruid order parameter, as required to\ndescribe both discontinuous transitions as a function of\nthe spin-orbit coupling, Rabi frequency, and two-body\ninteractions.\nThis paper is organized as follows. In Sec. II, we de-\nscribe the Hamiltonian and action for three-dimensional\nFermi gases in the presence of a general Rashba-\nDresselhaus spin-orbit coupling, Rabi \feld, and tunable\ns-wave interactions. We also obtain the inverse Green\noperator that is used in the calculation of the thermo-\ndynamic potential and Ginzburg-Landau theory of sub-\nsequent sections. In Sec. III, we analyze the thermo-\ndynamic potential across the entire BCS-to-BEC evolu-\ntion, including contributions from both the mean-\feld\nand Gaussian \ructuations, and obtain the order parame-\nter and number equations. In Sec. IV, we study the com-\nbined e\u000bects of Rabi \felds and spin-orbit coupling on the\nsuper\ruid critical temperature, constructing the \fnite-\ntemperature phase diagram versus Rabi \felds and scat-\ntering parameter. In Sec. V, we present the Ginzburg-\nLandau (GL) theory for the super\ruid order parameter\nand investigate further corrections to the critical tem-\nperature in the BEC limit by including interactions be-\ntween bosonic bound states. The GL action is obtained\nto sixth order in the order parameter to allow for the exis-\ntence of discontinuous (\frst-order) phase transitions. In\nSec. VI, we compare our work on the experimentally rel-\nevant equal Rashba-Dresselhaus spin-orbit coupling with\nearlier work that has considered di\u000berent forms of the-\noretically motivated spin-orbit couplings. In Sec. VII,\nwe conclude and look toward the future of experimental\nwork in this \feld.\nIn the interest of readability, we relegate a number of\ndetailed calculations to appendices. In Appendix A, wediscuss the Hamiltonian and e\u000bective Lagrangian for a\ngeneral Rashba-Dresselhaus spin-orbit coupling. In Ap-\npendix B, we analyze the saddle-point approximation for\ngeneral Rashba-Dresselhaus spin-orbit coupling. In Ap-\npendix C, we derive the modi\fed number equation, in-\ncluding the contribution arising from Gaussian \ructua-\ntions, which renormalizes the chemical potential obtained\nat the saddle-point level. In Appendix D, using a gen-\neral Rashba-Dresselhaus spin-orbit coupling, we obtain\nexpressions for the coe\u000ecients of the Ginzburg-Landau\ntheory up to sixth order in order parameter.\nII. HAMILTONIAN AND ACTION\nThroughout this paper, we adopt units in which ~=\nkB= 1. The Hamiltonian density of a three-dimensional\nFermi gas in the presence of Rashba-Dresselhaus spin-\norbit coupling and Rabi \feld is\nH(r) =Hk(r) +Hso(r) +HI(r)\u0000\u0016n(r): (1)\nThe \frst term in Eq. (1) is the kinetic energy,\nHk(r) =X\ns y\ns(r)^k2\n2m s(r); (2)\nwhere ^k=\u0000iris the momentum operator,  s(r) is the\nfermion \feld at position rwith (real or pseudo-) spin s\nand massm. The second term is the spin-orbit interac-\ntion,\nHso(r) =X\nss0 y\ns(r)h\nHso(^k)i\nss0 s0(r); (3)\nwith the spin-orbit coupling matrix in momentum ( k)\nspace being\nHso(^k) =\u0014\nm(^kx\u001bx+\u0011^ky\u001by)\u0000\nR\n2\u001bz; (4)\nwhere (\u001bx;\u001by;\u001bz) are the Pauli matrices in spin space, \u0014\nis the momentum transfer to the atoms in a two-photon\nRaman process [7] or on a radio frequency atom chip [24],\n\u0011is the anisotropy of the Rashba-Dresselhaus \feld, and\n\nRis the Rabi frequency. The third term is the two-body\ns-wave contact interaction,\nHI(r) =\u0000g y\n\"(r) y\n#(r) #(r) \"(r); (5)\nwhereg>0 corresponds to a constant attraction between\nopposite spins. Finally, \u0016is the chemical potential and\nn(r) =P\ns y\ns(r) s(r) is the local density. While the gen-\neral Rashba-Dresselhaus spin-orbit coupling is discussed\nin Appendix A, in what follows we focus on the more\nexperimentally relevant situation of equal Rashba and\nDresselhaus couplings ( \u0011= 0).\nStandard manipulations (see Appendix A) lead to the\nLagrangian density,\nL(r;\u001c) =1\n2\ty(r;\u001c)G\u00001(^k;\u001c)\t(r;\u001c) +1\ngj\u0001(r;\u001c)j2\n+K(^k)\u000e(r\u0000r0); (6)3\nwhere\u001c=itis the imaginary time, \t = (  \" # y\n\" y\n#)T\nis the Nambu spinor, K(^k) =^k2=2m\u0000\u0016is the kinetic\nenergy operator with respect to the chemical potential,\nand \u0001( r;\u001c) =\u0000gh #(r;\u001c) \"(r;\u001c)iis the pairing \feld\ndescribing the formation of pairs of two fermions with\nopposite spins. Note that \u0016includes the overall positive\nshift\u00142=2min the single-particle kinetic energies due to\nspin-orbit coupling. The inverse Green's operator ap-\npearing in Eq. (6) is\nG\u00001(^k;\u001c) =0\nBB@@\u001c\u0000K\"\u0000i\u0014^kx=m 0\u0000\u0001\ni\u0014^kx=m @\u001c\u0000K# \u0001 0\n0 \u0001\u0003@\u001c+K\"\u0000i\u0014^kx=m\n\u0000\u0001\u00030i\u0014^kx=m @\u001c+K#1\nCCA;\n(7)\nwhereK\";#=K(^k)\u0007\nR=2;are the kinetic energy terms\nshifted by the Rabi coupling.\nAs noted above, a mean-\feld treatment of this La-\ngrangian fails to correctly describe the super\ruid critical\ntemperature in the BEC regime. However, the inclu-\nsion of Gaussian \ructuations of \u0001 captures the e\u000bects of\ntwo-body bound states and leads to a physical super\ruid\ntransition temperature. It is to this task that we now\nturn.\nIII. THERMODYNAMIC POTENTIAL\nThe system's partition function may be expressed in\nterms of the functional integral,\nZ=Z\nD\u0001D\u0001\u0003D\tD\tye\u0000S; (8)\nwhere the Euclidean action is\nS=Z\f\n0d\u001cZ\nd3rL(r;\u001c); (9)\n\f= 1=Tis the inverse temperature, and the Lagrangian\ndensity is given by Eq. (6). Integrating over the fermion\n\felds yields the thermodynamic potential,\n\n =\u0000TlnZ= \n 0+ \nF; (10)\nwhere \n 0=\u0000TlnZ0=TS0is the mean-\feld (saddle-\npoint) contribution, for which \u0001( r;\u001c) = \u0001 0, and\n\nF=\u0000TlnZFis the contribution arising from order-\nparameter \ructuations. Detailed derivations of the ther-\nmodynamic potential for a general Rashba-Dresselhaus\nspin-orbit coupling, as well as the associated order pa-\nrameter and number equations, are given in Appen-\ndices B and C. The contributions to the thermody-\nnamic potential for the experimentally relevant situa-\ntion of equal Rashba-Dresselhaus spin-orbit coupling are\ndiscussed below in Sec. III A at the mean-\feld and in\nSec. III B at the Gaussian \ructuation level.A. Mean-Field Approximation\nThe mean-\feld, or saddle-point, term in the thermo-\ndynamic potential is\n\n0=Vj\u00010j2\ng\u0000T\n2X\nk;jlnh\n1 +e\u0000\fEj(k)i\n+X\nk\u0018k;(11)\nwhere\u0018k=\"k\u0000\u0016,\"k=k2=2m, and theEj(k), with\nj=f1;2;3;4g, are the eigenvalues of the momentum\nspace Nambu Hamiltonian matrix,\nH0(k) =@\u001c\u0000G\u00001(k;\u001c)j\u0001=\u0001 0; (12)\nwhere the operator @\u001c=I@\u001c, andIis the identity matrix.\nThe \frst set of eigenvalues,\nE1;2(k) =2\n4\u00102\nk\u00062s\nE2\n0;kh2\nk\u0000\u0012\u0014kx\nm\u00132\nj\u00010j23\n51=2\n(13)\ndescribe quasiparticle excitations, with the plus (+) as-\nsociated with E1and the minus ( \u0000) withE2. The\nsecond set of eigenvalues, E3;4(k) =\u0000E2;1(k);corre-\nsponds to quasiholes. Further, \u00102\nk=E2\n0;k+h2\nk, where\nE0;k=p\n\u00182\nk+j\u00010j2;andhk=p\n(\u0014kx=m)2+ \n2\nR=4 is\nthe magnitude of the combined spin-orbit and Rabi cou-\nplings.\nWe express the two-body interaction parameter gin\nterms of the renormalized s-wave scattering length as\nvia the relation [52]\n1\ng=\u0000m\n4\u0019as+1\nVX\nk1\n2\"k: (14)\nNote thatasis thes-wave scattering length in the absence\nof spin-orbit and Rabi \felds. It is, of course, possible to\nexpressg, and all subsequent relations, in terms of a scat-\ntering length which is renormalized by the presence of the\nspin-orbit and Rabi \felds [54, 55], but for both simplic-\nity and the sake of referring to the more experimentally\naccessible quantity, we do not do so here.\nThe order-parameter equation is obtained from the\nsaddle-point condition \u000e\n0=\u000e\u0001\u0003\n0jT;V;\u0016 = 0, leading to\nm\n4\u0019as=1\n2VX\nk\u00141\n\"k\u0000A+(k)\u0000\n2\nR\n4\u0018khkA\u0000(k)\u0015\n;(15)\nwhere we introduced the notation\nA\u0006(k) =1\u00002n1(k)\n2E1(k)\u00061\u00002n2(k)\n2E2(k); (16)\nwithnj(k) = 1=\u0002\ne\fEj(k)+ 1\u0003\nbeing the Fermi function.\nIn addition, the particle number at the saddle point N0=\n\u0000@\n0=@\u0016jT;V;is given by\nN0=X\nk\u001a\n1\u0000\u0018k\u0014\nA+(k) +(\u0014kx=m)2\n\u0018khkA\u0000(k)\u0015\u001b\n:(17)4\nThe mean-\feld temperature T0is determined by solv-\ning Eq. (15) for the given \u0016. The corresponding number\nof particles is given by Eq. (17). This mean-\feld treat-\nment leads to a transition temperature \u0018e1=kFas, where\nkFis the Fermi momentum. This result gives the correct\ntransition temperature on the BCS limit; however, it is\nunphysical on the BEC regime for kFas!0. In order to\n\fnd a physical result, we need to include order-parameter\n\ructuations, which we now do.\nB. Gaussian Fluctuations\nIn discussing Gaussian \ructuations, we concentrate on\nequal Rasha-Dresselhaus couplings, leaving details for\ngeneral Rashba-Dresselhaus coupling to Appendix C.\nTo obtain the correct super\ruid transition temperature\nin the BEC limit we must include the physics of two-body\nbound states near the transition, as described by the two-\nparticleT-matrix [56, 57]. Accounting for all two-particle\nchannels, the T-matrix calculation leads to a two-particle\nscattering amplitude \u0000, where\n\u0000\u00001(q;z) =m\n4\u0019as\u00001\n2VX\nk\u00141\n\"k+2X\ni;j=1\u000bijWij\u0015\n; (18)\nzis the complex frequency and\nWij=1\u0000ni(k)\u0000nj(k+q)\nz\u0000Ei(k)\u0000Ej(k+q): (19)\nAt the super\ruid phase boundary \u0001 0!0, the eigenval-\nues appearing in Eq. (19) reduce to E1;2(k) =jj\u0018kj\u0006hkj,\nbut it is straightforward to show that ignoring the abso-\nlute values does not result in any change in either the\nmean-\feld order parameter or number equation. Mean-\nwhile, the coe\u000ecients\n\u000b11=\u000b22=jukuk+q\u0000vkv\u0003\nk+qj2; (20)\n\u000b12=\u000b21=jukvk+q+uk+qvkj2; (21)\nare the coherence factors associated with the quasi-\nparticle amplitudes for \u0001 0= 0:\nuk=s\n1\n2\u0012\n1 +\nR\n2hk\u0013\n; v k=is\n1\n2\u0012\n1\u0000\nR\n2hk\u0013\n:(22)\nThe Gaussian \ructuation correction to the thermody-\nnamic potential is\n\nF=\u0000TX\nq;iqnln [\f\u0000(q;iqn)=V]: (23)\nover the entire BCS-to-BEC evolution. The \ructuation\ncontribution to the particle number is therefore NF=\n\u0000@\nF=@\u0016jT;V, where\nNF=X\nqZ1\n\u00001d!\n\u0019nB(!)\u0014@\u000e(q;!)\n@\u0016\u0000@\u000e(q;0)\n@\u0016\u0015\nT;V;\n(24)with the phase shift \u000e(q;!) de\fned via the relation\n\u0000(q;!\u0006i\u000f) =j\u0000(q;!)je\u0006i\u000e(q;!): (25)\nWhen two-body states are present, the \ructuation con-\ntribution can be written as NF=Nsc+Nb, where\nNsc=X\nqZ1\n!tp(q)d!\n\u0019nB(!)\u0014@\u000e(q;!)\n@\u0016\u0000@\u000e(q;0)\n@\u0016\u0015\nT;V\n(26)\nis the number of particles in scattering states, and !tp(q)\nis the two-particle continuum threshold corresponding to\nthe branch point of \u0000\u00001(q;z) [56, 58],\nNb= 2X\nqnB(Ebs(q)\u00002\u0016); (27)\nis the number of fermions in bound states, where\nnB(!) = 1=(e\f!\u00001) is the Bose distribution function,\nandEbs(q) is the energy of the bound states obtained\nfrom \u0000\u00001(q;z=E\u00002\u0016) = 0, corresponding to a pole\nin the scattering amplitude \u0000( q;z). In the limit of large\nand negative fermion chemical potential, the system be-\ncomes non-degenerate and \u0000\u00001(q;z) = 0 becomes the\nexact eigenvalue equation for the two-body bound state\nin the presence of spin-orbit and Rabi coupling [23]. The\ntotal number of fermions, as a function of \u0016, thus be-\ncomes\nN=N0+NF; (28)\nwhereN0is given in Eq. (17) and NFis the sum of Nsc\nandNb, as discussed above [52, 56].\nIV. CRITICAL TEMPERATURE\nWe calculate numerically the transition temperature\nTcbetween the normal and uniform super\ruid states, as\na function of the scattering parameter 1 =kFas, by simul-\ntaneously solving the order parameter and number equa-\ntions (15) and (28). The solutions correspond to the min-\nima of the free energy, F= \n +\u0016N. We do not discuss\nthe cases of Fulde-Ferrell [59] or Larkin-Ovchinnikov [60]\nnonuniform super\ruid phases since they only exist over a\nvery narrow region of the phase diagram deep in the BCS\nregime [59, 60], which is not experimentally accessible for\nultracold fermions.\nFigure 1, in which we scale temperatures by the Fermi\ntemperature TF=k2\nF=2m, shows the e\u000bects of spin-\norbit and Rabi couplings on Tc. The solid (black) line\nin Fig. 1(a) shows Tcversus 1=kFasfor zero Rabi cou-\npling (\nR= 0) and zero spin-orbit coupling \u0014. If\n\nR= 0, the spin-orbit coupling can be removed by a\nsimple gauge transformation, and thus plays no role. In\nthis situation, the pairing is purely s-wave. The dashed\n(blue) line shows Tcfor \nR6= 0, with vanishing equal\nRashba-Dresselhaus spin-orbit coupling. We see that for\n\fxed interaction strength, the pair-breaking e\u000bect of the5\nFIG. 1. (Color online) (a) The super\ruid transition tem-\nperatureTc=TF, whereTFis the Fermi temperature, vs the\nscattering parameter 1 =kFasfor equal Rashba-Dresselhaus\nspin-orbit coupling and two di\u000berent Rabi coupling strengths,\n\nR= 0 and\"F. For \n R= 0 [solid (black) curve], Tcis the\nsame as for zero spin-orbit coupling since the equal spin-orbit\n\feld can be gauged away. The dashed (blue) line shows Tc\nfor zero spin-orbit coupling, with \n R=\"F, while the dot-\nted (green) line shows Tcfor \n R=\"Fand ~\u0014=\u0014=kF= 0:5.\n(b)Tcis drawn at unitarity, 1 =kFas= 0, and in the inset\nat 1=kFas=\u00002:0, as a function of e\nR= \n R=\"F. The solid\n(red) curves represent ~ \u0014= 0 and the dotted (blue) curves\nrepresent ~\u0014= 0:5. Across the dotted (red) curves, the phase\ntransition is \frst order.\nRabi coupling suppresses super\ruidity, compared with\n\nR= 0; the Rabi \feld here plays the pair-breaking role\nof the Zeeman \feld in an superconductor.\nWith both spin-orbit and Rabi couplings present, the\ntwo-particle pairing is no longer purely singlet s-wave,\nbut obtains a triplet p-wave component; the admixture\nstabilizes the super\ruid phase, as shown by the dotted\n(green) line. The latter curve shows that in the BEC\nregime with large positive 1 =kFas, the super\ruid tran-\nsition temperature is enhanced by the presence of spin-\norbit and Rabi couplings, a consequence of the reduction\nFIG. 2. (Color online) Chemical potential at the super\ruid\ncritical temperature ( Tc) for ~\u0014=\u0014=kF= 0:5 and various Rabi\n\felds, e\nR= \nR=\"F.\nof the bosonic e\u000bective mass in the xdirection below 2 m.\nHowever, for su\u000eciently large \n R, the geometric mean\nbosonic mass MBincreases above 2 mandTcdecreases.\nThis renormalization of the mass of the bosons can be\ntraced back to a change in the energy dispersion of the\nfermions when both spin-orbit coupling and Rabi \felds\nare present.\nFigure 1(b) shows Tcversus \nRfor \fxed 1=kFas, both\nwith and without equal Rashba-Dresselhaus spin-orbit\ncoupling at \u0014= 0:5kF. When both \u0014andTare zero,\nsuper\ruidity is destroyed at a critical value of \n Rcorre-\nsponding to the Clogston limit [61]. At low temperature,\nthe phase transition to the normal state is \frst order\nbecause the Rabi coupling is su\u000eciently large to break\nsinglet Cooper pairs. However, at higher temperatures\nthe singlet s-wave super\ruid starts to become polarized\nby thermally excited quasiparticles that produce a para-\nmagnetic response. Thus, above the characteristic tem-\nperature indicated by the large (red) dots, the transition\nbecomes second order, as pointed out by Sarma [62]. The\nchange in the transition order occurs not only for \u0014= 0,\nbut also for nonzero values of \u0014both in the BCS regime\nand near unitarity, depending on the choice of parame-\nters, as illustrated in Fig. 1(b).\nThe critical temperature for \u00146= 0 vanishes only\nasymptotically in the limit of large \n R. We note that\nfor \nR=EFand\u0014= 0, the transition from the super-\n\ruid to the normal state is continuous at unitarity, but\nvery close to a discontinuous transition. In the range\n1:05.\nR=EF.1:10, numerical uncertainties as \u0014!0\nprevent us from predicting exactly whether the transition\nat unitarity is continuous or discontinuous.\nFigure 2 shows \u0016(Tc) for \fxed spin-orbit coupling and\nseveral Rabi couplings. The solid (black) curve, which\nrepresents the situation in which no Rabi \feld is present,\nis equivalent to the situation in which spin-orbit coupling\nis also absent, as noted in the discussion of Fig. 1. It is6\nFIG. 3. (Color online) Phase diagram of critical temperature\nTc=TFvs 1=kFasand \n R=\"Ffor equal Rashba-Dresselhaus\ncoupling\u0014=kF= 0:5. The \fnite-temperature uniform super-\n\ruid phases re\rect those at T= 0 shown in the background.\nThese phases are distinguished by the number of rings (line\nnodes) in the quasiparticle excitation spectrum [i.e., where\nE2(k) = 0] and type of gap: (1) direct gapped super\ruid with\nzero rings (magenta diamonds), (2) indirect gapped super\ruid\nwith zero rings (red circles), (3) gapless super\ruid with two\nrings (blue square), and (4) gapless one-ring super\ruid (green\nstars).\nevident that while the Rabi \feld reduces the chemical\npotential in the BCS limit, it also shifts the onset of the\nsystem's evolution to the BEC limit to larger inverse scat-\ntering lengths, and produces a non-monotonic behavior\nof\u0016(Tc) near unitarity.\nFigure 3 shows Tcfor equal Rashba-Dresselhaus cou-\npling\u0014= 0:5kF, as a function of Rabi \feld and\nscattering parameter. We also superpose the zero-\ntemperature phase diagram to illustrate the di\u000berent su-\nper\ruid ground states of this system. According to the\nzeros of the lowest quasiparticle energy E2(k), the uni-\nform super\ruid phases that emerge are [21] direct gapped\nwith zero rings (line nodes), indirectly gapped with zero\nrings, gapless with one ring, and gapless with two rings.\nFigure 4 shows the fractional number Nb=Nof bound\nfermions at Tcas a function of 1 =kFasfor two sets of\nexternal \felds. In the BCS (BEC) regime, the rela-\ntive contribution to Nis dominated by unbound (bound)\nfermions. The main e\u000bect of spin-orbit and Rabi \felds on\nNb=Nis to shift the location where the two-body bound\nstates emerge. For \fxed spin-orbit coupling (Rabi \feld)\nand increasing Rabi \feld (spin-orbit coupling), two-body\nbound states emerge at larger (smaller) scattering pa-\nrameters. These shifts are in agreement with the cal-\nculated shifts in binding energies of Feshbach molecules\nin the presence of equal Rashba-Dresselhaus spin-orbit\ncoupling and Rabi \felds [23].\nFIG. 4. (Color online) Fractional number Nb=Nof bound\nfermions as a function of the interaction parameter 1 =kFas,\nfor equal Rashba-Dresselhaus coupling \u0014=kF= 0:5 and Rabi\nfrequencies e\nR= \n R=\"F= 0 (black solid line) and e\nR=\n\nR=\"F= 2 (red dot-dashed line).\nV. GINZBURG-LANDAU THEORY\nTo further elucidate the e\u000bects of \ructuations on the\norder of the super\ruid transition, as well as to assess the\nimpact of spin-orbit and Rabi couplings near the crit-\nical temperature, we now derive the Ginzburg-Landau\ndescription of the free energy near the transition. In the\nlimit of small order parameter, the \ructuation action SF\ncan be expanded in powers of the order parameter \u0001( q)\nbeyond Gaussian order. The expansion of SFto quar-\ntic order is su\u000ecient to describe the continuous (second-\norder) transition in Tcversus 1=kFasin the absence of a\nRabi \feld [52]. However, to correctly describe the \frst-\norder transition [61, 62] at low temperature (Fig. 1), it is\nnecessary to expand the free energy to sixth order in \u0001.\nThe quadratic (Gaussian-order) term in the action is\nSG=\fVX\nqj\u0001qj2\n\u0000(q;z): (29)\nFor an order parameter varying slowly in space and time,\nwe may expand \u0000\u00001as\n\u0000\u00001(q;z) =a+X\n`c`q2\n`\n2m\u0000d0z+\u0001\u0001\u0001; (30)\nwith the sum over `=fx;y;zg. The full result, as a\nfunctional of \u0001( r;\u001c), has the form\nSF=Z\f\n0d\u001cZ\nd3r\u0010\nd0\u0001\u0003@\n@\u001c\u0001 +aj\u0001j2\n+X\n`c`jr`\u0001j2\n2m+b\n2j\u0001j4+f\n3j\u0001j6\u0011\n:(31)\nThe full time-dependent Ginzburg-Landau action de-\nscribes systems in and near equilibrium (e.g., with col-7\nlective modes). The imaginary part of d0measures the\nnon-conservation of j\u0001j2in time (i.e., the Cooper pair\nlifetime). Details of the derivation of SFare found in\nAppendix D.\nWe are interested in systems at thermodynamic equi-\nlibrium, where the order parameter is independent of\ntime, that is, \u0001( r;\u001c) = \u0001( r). In this situation, mini-\nmizing the free energy TSFwith respect to \u0001\u0003yields the\nGinzburg-Landau equation,\n \n\u0000X\n`c`r2\n`\n2m+bj\u0001(r)j2+fj\u0001(r)j4+a!\n\u0001(r) = 0:\n(32)\nForb > 0, the system undergoes a continuous phase\ntransition when achanges sign. However, when b <0,\nthe system is unstable in the absence of f. Forb <0\nanda > 0, a \frst-order phase transition occurs when\n3b2= 16af. Positivefstabilizes the system even when\nb<0.\nIn the BEC regime, where d0is purely real, we de\fne\nan e\u000bective bosonic wave function \t( r) =pd0\u0001(r) to re-\ncast Eq. (32) in the form of the Gross-Pitaevskii equation\nfor a dilute Bose gas,\n \n\u0000X\n`r2\n`\n2M`+U2j\t(r)j2+U3j\t(r)j4\u0000\u0016B!\n\t(r) = 0:\n(33)\nHere,\u0016B=\u0000a=d0is the bosonic chemical potential,\nM`=m(d0=c`) are the anisotropic bosonic masses, and\nU2=b=d2\n0andU3=f=d3\n0represent contact interac-\ntions of two and three bosons. In the BEC regime,\nthese terms are always positive, leading to a dilute gas\nof stable bosons. The boson chemical potential \u0016Bis\n\u00192\u0016+Eb<0, whereEb=\u0000Ebs(q=0) is the two-\nbody binding energy in the presence of spin-orbit cou-\npling and Rabi frequency, obtained from the condition\n\u0000\u00001(q;E\u00002\u0016) = 0, discussed earlier.\nThe anisotropy of the e\u000bective bosonic masses, Mx6=\nMy=Mz\u0011M?, stems from the anisotropy of the equal\nRashba-Dresselhaus spin-orbit coupling, which together\nwith the Rabi coupling modi\fes the dispersion of the\nconstituent fermions along the xdirection. In the limit\nkFas\u001c1, the many-body e\u000bective masses reduce to\nthose obtained by expanding the two-body binding en-\nergy,Ebs(q)\u0019\u0000Eb+P\n`q2\n`=2M`;and agree with known\nresults [23]. However, for 1 =kFas.2, many-body and\nthermal e\u000bects produce deviations from the two-body re-\nsult.\nIn the absence of two- and three-body boson-boson\ninteractions, U2andU3, we directly obtain an analytic\nexpression for Tcin the Bose limit from Eq. (27),\nTc=2\u0019\nMB\u0012nB\n\u0010(3=2)\u00132=3\n; (34)\nwithMB= (MxM2\n?)1=3, by noting that \u0016B= 0 or\nEbs(q=0)\u00002\u0016= 0, and using the condition that nB'n=2 [with corrections exponentially small in (1 =kFas)2],\nwherenBis the density of bosons. In the BEC regime,\nthe results shown in Fig. 1 include the e\u000bects of the mass\nanisotropy, but do not include the e\u000bects of boson-boson\ninteractions.\nTo account for boson-boson interactions, we adopt the\nHamiltonian of Eq. (33) with U26= 0, but with U3= 0,\nand apply the method developed in Ref. [63] to show that\nthese interactions further increase TBEC to\nTc(aB) = (1 +\r)TBEC; (35)\nwhere\r=\u0015n1=3\nBaB. Here,aBis thes-wave boson-\nboson scattering length, \u0015is a dimensionless constant\n\u00181, and we use the relation U2= 4\u0019aB=MB. Since\nnB=k3\nF=6\u00192and the boson-boson scattering length is\naB=U2MB=4\u0019, we have\r=~\u0015fMBeU2;wherefMB=\nMB=2m;eU2=U2k3\nF=\"F;and~\u0015=\u0015=4(6\u00195)1=3\u0019\u0015=50:\nFor \fxed 1=kFas,Tcis enhanced by the spin-orbit \feld,\na \nR-dependent decrease in the e\u000bective boson mass MB\n(\u001810-15%), as well as a stabilizing boson-boson repulsion\nU2(\u00182-3%), for the parameters used in Fig. 1.\nIn closing our discussion of the strongly bound BEC\nlimit, we note that in the absence of spin-orbit coupling,\na Gaussian-order calculation of the two-boson scattering\nlength yields the erroneous Born approximation result\naB= 2as. However, an analysis of the T-matrix beyond\nGaussian order, which includes the e\u000bects of two-body\nbound states, obtains the correct result aB= 0:6asat\nvery low densities [64] and agrees with four-body calcu-\nlations [65]. The same method can be used to estimate U2\noraBbeyond the Born approximation discussed above.\nNevertheless, while the precise quantitative relation be-\ntweenaBandasin the presence of spin-orbit coupling is\nyet unknown, the trend of increasing Tcdue to spin-orbit\ncoupling has been clearly shown.\nVI. COMPARISON TO EARLIER WORK\nIn this section, we brie\ry compare our results with ear-\nlier investigations of di\u000berent types of theoretically mo-\ntivated spin-orbit couplings, worked in di\u000berent dimen-\nsions, or at zero temperature. Our results focus mainly\non an analysis of the critical super\ruid temperature and\nthe e\u000bects thereon of order-parameter \ructuations for\na three-dimensional Fermi gas in the presence of equal\nRashba-Dresselhaus spin-orbit coupling and Rabi \felds.\nThe appendices consider the more general situation of\narbitrary Rashba and Dresselhaus components.\nSeveral works have analyzed the e\u000bects of spin-orbit-\ncoupled fermions in three dimensions at zero tempera-\nture [17{22, 66{69]. While some authors have described\nthe situation of Rashba-only couplings [17{19, 66], others\nhave assessed the case of equal Rashba and Dresselhaus\ncomponents [21, 22] or a general mixture of the two [20].\nIt has been demonstrated that in the absence of a Rabi\n\feld, the zero-temperature evolution from BCS to BEC8\nsuper\ruidity is a crossover for s-wave systems, not only\nfor Rashba-only couplings [17{20, 66], but also for ar-\nbitrary Rashba and Dresselhaus components [20]. This\nresult directly follows from the fact that the quasiparti-\ncle excitation spectrum remains fully gapped throughout\nthe evolution.\nIn contrast, the addition of a Rabi \feld gives rise\nto topological phase transitions for Rashba-only cou-\nplings [17] and equal Rashba and Dresselhaus compo-\nnents [21, 22], a situation which certainly persists for\ngeneral Rashba-Dresselhaus couplings. The simultane-\nous presence of a general Rashba-Dresselhaus spin-orbit\ncoupling and Rabi \felds leads to a qualitative change in\nthe quasiparticle excitation spectrum and to the emer-\ngence of topological super\ruid phases [17, 21, 22]. Two-\ndimensional systems have also been investigated at zero\ntemperature, where topological phase transitions have\nbeen identi\fed for Rashba-only [70] and equal Rashba-\nDresselhaus [71] couplings, in the presence of a Rabi \feld.\nWhile early papers in this \feld focused mainly\non the zero-temperature limit, progress toward \fnite-\ntemperature theories was made \frst in two dimen-\nsions [72, 73] and later in three dimensions [74{76].\nThe e\u000bects of a general Rashba-Dresselhaus spin-orbit\ncoupling and Rabi \feld on the Berezenskii-Kosterlitz-\nThouless transition were thoroughly investigated for\ntwo-dimensional Fermi gases at \fnite temperatures [72,\n73], including both Rashba-only and equal Rashba-\nDresselhaus spin-orbit couplings as examples.\nThe super\ruid critical temperature in three dimen-\nsions was investigated using a spherical (3D) spin-orbit\ncoupling\u0015k\u0001\u001bin the absence of a Rabi \feld [74, 75],\nand also for Rashba-only (2D) couplings in the pres-\nence of a Rabi \feld [76]. In a recent review article [77],\nthe critical temperature throughout the BCS-BEC evo-\nlution was discussed both in the absence [52] and pres-\nence [53] of Rashba-Dresselhaus spin-orbit coupling. In\nSecs. 5 and 6 of this review, the authors describe the\nsame method and expressions we obtained in our earlier\npreliminary work [53] for the analytical relations required\nto obtain the critical temperature at the Gaussian order;\nthey include, however, only the contribution of bound\nstates discussed earlier in the literature for Rashba-only\nspin-orbit coupling without Rabi \felds [18]. In contrast,\nhere we develop a complete Gaussian theory to compute\nthe super\ruid critical temperature of a three-dimensional\nFermi gas in the presence of both a general Rashba-\nDresselhaus (2D) spin-orbit coupling and Rabi \felds. We\nfocus our numerical calculations on the speci\fc situation\nof equal Rashba-Dresselhaus components, which is eas-\nier to achieve experimentally in the context of ultracold\natoms. Our key results, already announced in our earlier\nwork [53], include the contributions of bound and scat-\ntering states at the Gaussian level. As seen in Fig. 4\nof this present paper, there is a wide region of interac-\ntion parameters for which the contribution of scattering\nstates cannot be neglected. Furthermore, unlike previ-\nous work [74{77], we provide a comprehensive analysisof the Ginzburg-Landau \ructuation theory and include\nthe e\u000bects of boson-boson interactions on the super\ruid\ncritical temperature in the BEC regime.\nVII. CONCLUSION\nWe have evaluated the super\ruid critical tempera-\nture throughout the BCS-to-BEC evolution of three-\ndimensional Fermi gases in the presence of equal Rashba-\nDresselhaus spin-orbit couplings, Rabi \felds, and tun-\nables-wave interactions. Furthermore, we have devel-\noped the Ginzburg-Landau theory up to sixth power in\nthe order parameter to elucidate the origin of \frst-order\nphase transitions when the spin-orbit \feld is absent and\nthe Rabi \feld is su\u000eciently large. Lastly, in the appen-\ndices, we have presented the \fnite-temperature theory of\ns-wave interacting fermions in the presence of a generic\nRashba-Dresselhaus coupling and external Rabi \felds,\nas well as the corresponding time-dependent Ginzburd-\nLandau theory near the super\ruid critical temperature.\nACKNOWLEDGMENTS\nWe thank I. B. Spielman for discussions. The re-\nsearch of P.D.P. was supported in part by NSF Grant\nNo. PHY1305891 and that of G.B. by NSF Grants\nNo. PHY1305891 and No. PHY1714042. Both G.B.\nand C.A.R. SdM. thank the Aspen Center for Physics,\nsupported by NSF Grants No. PHY1066292 and No.\nPHY1607611, where part of this work was done. This\nwork was performed under the auspices of the U.S. De-\npartment of Energy by Lawrence Livermore National\nLaboratory under Contract No. DE-AC52-07NA27344.\nAppendix A: Hamiltonian and e\u000bective Lagrangian\nfor general Rashba-Dresselhaus spin-orbit coupling\nIn this appendix, we consider a larger class of spin-\ncoupled fermions in three dimensions with a general\nRashba-Dresselhaus (GRD) coupling. The Hamiltoninan\ndensity for equal Rashba-Dresselhaus (ERD) discussed\nin Sec. II is a particular case of the general Rashba-\nDresselhaus Hamiltonian density,\nH(r) =H0(r) +Hso(r) +HI(r): (A1)\nAdopting units in which ~=kB= 1, the independent-\nparticle Hamiltonian density without spin-orbit coupling\nis\nH0(r) =X\n\u000b\u0012jr \u000b(r)j2\n2m\u000b\u0000\u0016\u000b y\n\u000b(r) \u000b(r)\u0013\n;(A2)\nwhere \u000b,m\u000b, and\u0016\u000bare the fermion \feld operator,\nmass, and chemical potentials for internal state \u000b, re-\nspectively. The spin-orbit Hamiltonian can be written9\nas\nHso(r) =\u0000X\ni\u000b\f y\n\u000b(r)\u001bi;\u000b\fhi(r) \f(r); (A3)\nwhere the\u001biare the Pauli matrices in isospin (internal\nstate) space and h= (hx;hy;hz) includes both the spin-\norbit coupling and Zeeman \felds. Finally, we consider a\ntwo-bodys-wave contact interaction,\nHI(r) =\u0000g y\n\"(r) y\n#(r) #(r) \"(r); (A4)\nwhereg>0 corresponds to an attractive interaction.\nBy introducing the pairing \feld \u0001( r;\u001c) =\n\u0000gh #(r;\u001c) \"(r;\u001c)i;we remove the quartic inter-\naction and obtain the Lagrangian density,\nL(r;\u001c) =1\n2\ty(r;\u001c)G\u00001(^k;\u001c)\t(r;\u001c) +j\u0001(r;\u001c)j2\ng\n+eK+(^k)\u000e(r\u0000r0); (A5)\nwhere we introduced the momentum operator ^k=\u0000ir,\nthe Nambu spinor \t = (  \" # y\n\" y\n#)T, and de\fned\neK\u0006= (eK\"\u0006eK#)=2:Here,eK\"=K\"\u0000hz;andeK#=\nK#+hz;withK\u000b(^k) =^k2=(2m\u000b)\u0000\u0016\u000bbeing the kinetic\nenergy operator of internal state \u000bwith respect to its\nchemical potential. Lastly, the inverse Green's operator\nappearing in Eq. (A5) is\nG\u00001(^k;\u001c) =0\nBB@@\u001c\u0000eK\"h\u0003\n? 0\u0000\u0001\nh?@\u001c\u0000eK# \u0001 0\n0 \u0001\u0003@\u001c+eK\"\u0000h?\n\u0000\u0001\u00030\u0000h\u0003\n?@\u001c+eK#1\nCCA;\n(A6)\nwhereh?(^k) =hx(^k) +ihy(^k) plays the role of the spin-\norbit coupling, and hzis the Zeeman \feld along the z\ndirection.\nTo make progress, we expand the order parameter\nabout its saddle-point (mean-\feld) value \u0001 0by writ-\ning \u0001( r;\u001c) = \u0001 0+\u0011(r;\u001c):Next, we integrate over the\nfermionic \felds and use the decomposition G\u00001(^k;\u001c) =\nG\u00001\n0(^k;\u001c)+G\u00001\nF(^k;\u001c);where G\u00001\n0(^k;\u001c) is the mean-\feld\nGreen's operator, given by Eq. (A6) with \u0001( r;\u001c) = \u0001 0,\nandG\u00001\nF(^k;\u001c) is the contribution to the inverse Green's\noperator arising from \ructuations. These steps yield the\nsaddle-point Lagrangian density,\nL0(r;\u001c) =\u0000T\n2VTr ln(\fG\u00001\n0) +j\u00010j2\ng+eK+(^k)\u000e(r\u0000r0);\n(A7)\nand the \ructuation contribution,\nLF(r;\u001c) =\u0000T\n2VTr ln( I+G0G\u00001\nF) + \u0003( r;\u001c) +j\u0011(r;\u001c)j2\ng;\n(A8)\nresulting in the e\u000bective Lagrangian density Le\u000b(r;\u001c) =\nL0(r;\u001c) +LF(r;\u001c):In the expressions above, we workin a volume Vand take traces over both discrete and\ncontinuous indices. Notice that the term \u0003( r;\u001c) =\n[\u00010\u0011\u0003(r;\u001c) + \u0001\u0003\n0\u0011(r;\u001c)]=gin the \ructuation Lagrangian\ncancels out the linear terms in \u0011and\u0011\u0003when the loga-\nrithm is expanded, due to the saddle point condition\n\u000eS0\n\u000e\u0001\u0003\n0= 0; (A9)\nwhereS0=R\f\n0d\u001cd3rL0(r;\u001c) is the saddle-point action.\nAppendix B: Saddle Point Approximation for\ngeneral Rashba-Dresselhaus spin-orbit coupling\nWe \frst analyze the saddle-point contribution. The\nsaddle-point thermodynamic potential \n 0=\u0000TlnZ0\ncan be obtained for the saddle-point partition function\nZ=e\u0000S0as \n 0=TS0. Transforming the saddle-point\nLagrangianL0from Eq. (A7) into momentum space and\nintegrating over spatial coordinates and imaginary time\nleads to the saddle-point thermodynamic potential,\n\n0=Vj\u00010j2\ng\u0000T\n2X\nk;jln(1+e\u0000\fEk;j)+X\nkeK+(k);(B1)\nwhereK\u000b(k) =k2=2m\u000b\u0000\u0016\u000band the eigenvalues Ek;j\nare the poles of G0(k;z), withj=f1;2;3;4g.\nNext, we restrict our analysis to mass balanced sys-\ntems (m\"=m#) in di\u000busive equilibrium ( \u0016\"=\u0016#).\nWe also consider the general Rashba-Dresselhaus (GRD)\nspin-orbit \feld h?(k) =\u0014(kx+i\u0011ky)=m;where\u0014and\u0011\nare the magnitude and anisotropy of the spin-orbit cou-\npling, respectively. Note that this form is equivalent to\nanother common form of the Rashba-Dresselhaus cou-\npling found in the literature [21, 22]: hso=hR+hD\nwhere hR=vR(kx^y\u0000ky^x) and hD=vD(kx^y+ky^x).\nThe two forms are related via a momentum-space ro-\ntation and the correspondences \u0014=m(vR+vD) and\n\u0011= (vR\u0000vD)=(vR+vD). The equal Rashba-Dresselhaus\nlimit (ERD) corresponds to vR=vD=v, leading to\n\u0011= 0 and\u0014= 2mv. The speci\fc case of equal Rashba-\nDresselhaus spin-orbit coupling discussed in the main\npart of the paper corresponds to the case where \u0011= 0,\nthat is,h?(k) =\u0014kx=m:\nFor the general Rashba-Dresselhaus case, the four\neigenvalues are\nE1;2(k) =h\n\u00102\nk\u00062q\nE2\n0;kh2\nk\u0000j\u00010j2jh?(k)j2i1=2\n;(B2)\nE3;4(k) =\u0000E2;1(k); (B3)\nwhere the + (\u0000) sign within the outermost square root\ncorresponds to E1(E2), and the functions inside the\nsquare roots are \u00102\nk=E2\n0;k+h2\nk, with contributions\nE0;k=q\n\u00182\nk+j\u00010j2; (B4)\nhk=p\njh?(k)j2+h2z; (B5)10\nwhere\u0018k=\"k\u0000\u0016;and\"k=k2=2m:The order-\nparameter equation is found from the saddle point con-\ndition\u000e\n0=\u000e\u0001\u0003\n0jT;V;\u0016 = 0. At the phase boundary be-\ntween the super\ruid and normal phases, \u0001 0!0, and\nthe order-parameter equation becomes\nm\n4\u0019as=1\n2VX\nk\u00141\n\"k\u0000tanh(\fE1=2)\n2E1\u0000tanh(\fE2=2)\n2E2\n\u0000h2\nz\n\u0018khk\u0012tanh(\fE1=2)\n2E1\u0000tanh(\fE2=2)\n2E2\u0013\u0015\n;\n(B6)\nafter expressing the interaction parameter gin terms of\nthes-wave scattering length via the relation\n1\ng=\u0000m\n4\u0019as+1\nVX\nk1\n2\"k: (B7)\nWe note that asis thes-wave scattering length in the\nabsence of spin-orbit and Zeeman \felds. It is, of course,\npossible to express all relations obtained in terms of a\nscattering length which is renormalized by the presence\nof the spin-orbit and Rabi \felds [54, 55]. However, in ad-\ndition to complicating our already cumbersome expres-\nsions, it would make reference to a quantity that is more\ndi\u000ecult to measure experimentally and that would hide\nthe explicit dependence of the properties that we analyze\nin terms of the spin-orbit and Rabi \felds, so we do not\nconsider such complications here. Note that since \u0001 0= 0\nat the phase boundary, the eigenvalues in Eq. (B2) re-\nduce toE1(k) =\f\f\fj\u0018kj+hk\f\f\f,E2(k) =\f\f\fj\u0018kj\u0000hk\f\f\f, which is\nthe absolute value of the normal-state energy dispersions.\nHowever, it is straightforward to show that ignoring the\nabsolute values does not result in any change in either the\nmean-\feld order parameter given by Eq. (B6) or number\nequation shown in Eq. (B8), when \u0001 0!0.\nThe saddle-point critical temperature T0is determined\nby solving Eq. (B6) subject to the thermodynamic con-\nstraintN0=\u0000@\n0=@\u0016jT;V;which yields\nN0=X\nk\u001a\n1\u0000\u0018k\u00141\n\"k+tanh(\fE1=2)\n2E1+tanh(\fE2=2)\n2E2\n+jh?(k)j2\n\u0018khk\u0012tanh(\fE1=2)\n2E1\u0000tanh(\fE2=2)\n2E2\u0013\u0015\u001b\n:\n(B8)\nA mean-\feld description of the system, which involves\na simultaneous solution of Eqs. (B6) and (B8), yields\nthe asymptotically correct description of the system\nin the BCS limit; however, such a description fails\nmiserably in the BEC regime where it does not ac-\ncount for the formation of two-body bound states.\nThe general Rashba-Dresselhaus spin-orbit saddle-point\nequations (B6) and (B8) reduce to the equal Rashba-\nDresselhaus equations (15) and (17) of the main part\nof the paper with the explicit use of hz= \nR=2 and\nh?(k) =\u0014kx=m, where \n Ris the Rabi coupling.Appendix C: Derivation of the modi\fed number\nequation with Gaussian \ructuations\nWe begin by deriving the modi\fed number equation\narising from Gaussian \ructuations of the order parame-\nter near the super\ruid phase boundary. The \ructuation\nthermodynamic potential \n Fresults from the Gaussian\nintegration of the \felds \u0011(r;\u001c) and\u0011\u0003(r;\u001c) in the \ructua-\ntion partition function ZF=R\nd\u0011\u0003d\u0011e\u0000SF, where the ac-\ntionSF=R\nd\u001c\f\n0R\nd3rLF(r;\u001c) is calculated to quadratic\norder. The contribution to the thermodynamic potential\ndue to Gaussian \ructuations is\n\nF=\u0000TX\niqn;qln [\f\u0000(q;iqn)=V] (C1)\nwhereqn= 2\u0019nT are the bosonic Matsubara frequencies\nand \u0000( q;iqn) is directly related to the pair \ructuation\npropagator \u001fpair(q;iqn) =V\u0000\u00001(q;iqn):\nThe Matsubara sum can be evaluated via contour in-\ntegration,\n\nF=\u0000TX\nqI\nCdz\n2\u0019inB(z) ln [\f\u0000(q;z)=V]; (C2)\nwherenB(z) = 1=(ez\u00001) is the Bose function and the\ncountourCencloses all of the Matsubara poles of the\nBose function. Next, we deform the contour around the\nMatsubara frequencies towards in\fnity, taking into ac-\ncount the branch cut and the possibility of poles coming\nfrom the logarithmic term inside the countour integral.\nWe take the branch cut to be along the real axis, then\nadd and subtract the pole at iqn= 0 to obtain\n\nF=\u0000TX\nqZ1\n\u00001d!\n\u0019nB(!) [\u000e(q;!)\u0000\u000e(q;0)];(C3)\nwhere the phase shift \u000e(q;!) is de\fned via \u0000( q;!\u0006i\u000f) =\nj\u0000(q;!)je\u0006i\u000e(q;!);and arises from the contour segments\nabove and below the real axis.\nThe thermodynamic identity N=\u0000@\n=@\u0016jT;Vthen\nyields to the \ructuation correction,\nNF=TX\nqZ1\n\u00001d!\n\u0019nB(!)\u0014@\u000e(q;!)\n@\u0016\u0000@\u000e(q;0)\n@\u0016\u0015\n;\n(C4)\nto the the saddle-point number equation, and has a sim-\nilar analytical structure as in the case without spin-orbit\nand Zeeman \felds [52, 56]. Thus, we can write the \f-\nnal number equation at the critical temperature Tcas\nN=N0+NF. Since the phase shift \u000e(q;z) vanishes ev-\nerywhere that \u0000( q;z) is analytic, the only contributions\nto Eq. (C4) arise from a possible isolated pole at !p(q)\nand a branch cut extending from the two-particle contin-\nuum threshold !tp(q) = minfi;j;kg[Ei(k) +Ej(k+q)] to\nz!1 along the positive real axis. The explicit form of\n\u0000(q;z) can be extracted from Eq. (D15) of Appendix D.\nWhen there is a pole corresponding to the emergence of\na two-body bound state, we can explicitly write \u0000( q;z)\u001811\nR(q)=(z\u0000!p(q));from which we obtain @\u000e(q;!)=@\u0016=\n2\u000e(z\u0000!p(q));leading to the bound state density\nNb= 2X\nqnB(!p(q)); (C5)\nwhere the energy !p(q) must lie below the two-particle\ncontinuum threshold !tp(q). The factor of 2, which arises\nnaturally, is due to the two fermions comprising a bosonic\nmolecule. Naturally, the presence of this term in the\n\ructuation-modi\fed number equation is dependent upon\nthe existence of such a pole, that is, a molecular bound\nstate. These bound states correspond to the Feshbach\nmolecules in the presence of spin-orbit coupling and Zee-\nman \felds [7, 23].\nHaving extracted the pole contribution to Eq. (C4),\nwhen it exists, the remaining integral over the branch\ncut corresponds to scattering state fermions,\nNsc=TX\nqZ1\n!tp(q)d!\n\u0019nB(!)\u0014@\u000e(q;!)\n@\u0016\u0000@\u000e(q;0)\n@\u0016\u0015\n;\n(C6)\nwhose energy is larger than the minimum energy !tp(q)\nof two free fermions. Thus, when bound states are\npresent, we arrive at the modi\fed number equation,\nN=N0+Nsc+Nb (C7)\nwhereN0is the number of free fermions obtained from\nthe saddle-point analysis in Eq. (B8), and NbandNsc\nare the bound state and scattering contributions given in\nEqs. (C5) and (C6), respectively. These general results\nare particularized to the equal Rashba-Dresselhaus case\nin Sec. III B of this paper.\nThe number of unbound states Nuis then easily seen\nto beNu=N0+Nsc, that is, the sum of the free-fermion\n(N0) and scattering ( Nsc) contributions. Naturally, the\nnumber of unbound states is also equal to the total num-\nber of states, N, minus the number of bound states, Nb,\nthat is,Nu=N\u0000Nb.\nAppendix D: Derivation of Ginzburg-Landau\ncoe\u000ecients for general Rashba-Dresselhaus\nspin-orbit coupling\nNext, we derive explicit expressions for the coe\u000e-\ncients of the time-dependent Ginzburg-Landau theory\nvalid near the critical temperature of the super\ruid. We\nstart from the \ructuation Lagrangian,\nLF(r;\u001c) =\u0000T\n2VTr ln( I+G0G\u00001\nF) + \u0003( r;\u001c) +j\u0011(r;\u001c)j2\ng;\n(D1)\nin a volume V, and take the traces over both discrete\nand continuous indices. Notice that the term \u0003( r;\u001c) =\n[\u00010\u0011\u0003(r;\u001c) + \u0001\u0003\n0\u0011(r;\u001c)]=gin the \ructuation Lagrangian\ncancels out the linear terms in \u0011and\u0011\u0003when the log-\narithm is expanded, due to the saddle-point condition.Since the expansion is performed near Tc, we take the\nsaddle-point order parameter \u0001 0!0 and rede\fne the\n\ructuation \feld as \u0011(r;\u001c) = \u0001( r;\u001c) to obtain\nLF(r;\u001c) =j\u0001j2\ng\u0000T\n2VTr ln( I+G0[0]G\u00001\nF[\u0001]):(D2)\nNotice that the arguments in G0[0] and G\u00001\nF[\u0001] represent\nthe values of \u0001 0= 0 and\u0011= \u0001, respectively.\nWe expand the logarithm to sixth order in \u0001 to obtain\nLF(r;\u001c) =j\u0001j2\ng+T\n2VTr\u00141\n2(G0G\u00001\nF)2+1\n4(G0G\u00001\nF)4\n+1\n6(G0G\u00001\nF)6+:::\u0015\n; (D3)\nwhere the higher-order odd (cubic and quintic) terms in\nthe order-parameter amplitudes expansion can be shown\nto vanish due to conservation laws and energy or momen-\ntum considerations.\nThe traces can be evaluated explicitly by using the\nmomentum-space inverse single-particle Green's function\nG\u00001\n0(k;k0) =\u0012A\u00001(k) 0\n0\u0000\u0002\nA\u00001(\u0000k)\u0003T\u0013\n\u000ekk0;(D4)\nderived from Eq. (A6). Here, we use the shorthand no-\ntationk\u0011(i!;k), where!n= 2\u0019nT are bosonic Mat-\nsubara frequencies and de\fne the 2 \u00022 matrix,\nA\u00001(k) =\u0012\ni!n\u0000eK\"(k)h\u0003\n?(k)\nh?(k)i!n\u0000eK#(k)\u0013\n; (D5)\nwhereeK\"=\u0018k\u0000hz,eK#=\u0018k+hz, with\u0018k=k2=2m\u0000\u0016\nthe kinetic energy relative to the chemical potential, hz\nthe external Zeeman \feld, and h?(k) =hx(k) +ihy(k)\nthe spin-orbit \feld. We also de\fne the \ructuation con-\ntribution to the inverse Green's function,\nG\u00001\nF(k;k0) =\u00120\u0000i\u001by\u0001k\u0000k0\ni\u001by\u0001y\nk0\u0000k0\u0013\n; (D6)\nwhere\u001byis the second Pauli matrix in isospin (internal\nstate) space and\n\u0001k=\f\nVZ\f\n0d\u001cZ\nd3rei(k\u0001r\u0000!\u001c)\u0001(r) (D7)\nis the Fourier transform of \u0001( r), withr\u0011(r;\u001c), and also\nhas dimensions of energy. Recall that we set ~=kB= 1,\nsuch that energy, frequency and temperature have the\nsame units.\nInversion of Eq. (D4) yields\nG0(k;k0) =\u0012A(k) 0\n0\u0000[A(\u0000k)]T\u0013\n\u000ekk0; (D8)\nwhere the matrix A(k) is\nA(k) =1\ndet[A\u00001(k)]\u0012\ni!n\u0000eK#(k)\u0000h\u0003\n?(k)\n\u0000h?(k)i!n\u0000eK\"k)\u0013\n:\n(D9)12\nwith det[ A\u00001(k)] =Q2\nj=1[i!n\u0000Ej(k)] and where the\nindependent-particle eigenvalues Ej(k) are two of the\npoles of G0(k;k). These poles are exactly the gen-\neral eigenvalues described in Eqs. (B2) in the limit of\n\u00010!0. Note that setting \u0001 0= 0 in the general eigen-\nvalue expressions yields E1;2(k) =jj\u0018kj\u0006hkj. The other\nset of poles of G0(k;k) corresponds to the eigenvalues\nE3;4(k) =\u0000E2;1(k) found from det\u0002\nA\u00001(\u0000k)\u0003T= 0.\nUsing Eq. (D3) to write the \ructuation action as SF=R\f\n0d\u001cR\nd3rLF(r;\u001c);results in\nSF=\fVX\nqj\u0001qj2\n\u0000(q)+\fV\n2X\nq1;q2;q3b1;2;3\u00011\u0001\u0003\n2\u00013\u0001\u0003\n1\u00002+3\n+\fV\n3X\nq1\u0001\u0001\u0001q5f1\u0001\u0001\u00015\u00011\u0001\u0003\n2\u00013\u0001\u0003\n4\u00015\u0001\u0003\n1\u00002+3\u00004+5;(D10)\nwhere summation over q\u0011(iqn;q) indicates sums over\nboth the bosonic Matsubara frequencies qn= 2\u0019nT and\nmomentum q. Here, we used the shorthand notation\nj\u0011qjto represent the labels of \u0001 qjor \u0001\u0003\nqj.\nThe quadratic order appearing in Eq. (D10) arises\nfrom the termsj\u0001(r;\u001c)j2=gand (T=2V)Tr(G0G\u00001\nF)2=2\nin Eq. (D3), and is directly related to the pair propaga-\ntor\u001fpair(q) =V\u0000\u00001(q), with\n\u0000\u00001(q) =1\ng\u0000T\n2VX\nkTr\u0002\nA(k)A\u00001(q\u0000k)\u0003\ndet[A\u00001(q\u0000k)];(D11)\nwhere we use the identity \u001byA\u001by= det( A)(AT)\u00001:\nThe fourth-order contribution arises from1\n4(G0G\u00001\nF)4and leads to\nb(q1;q2;q3) =T\n2VX\nkTr\u0002\nA(k)A\u00001(q1\u0000k)A(k\u0000q1+q2)A\u00001(q1\u0000q2+q3\u0000k)\u0003\ndet [A\u00001(q1\u0000k)] det [ A\u00001(q1\u0000q2+q3\u0000k)]; (D12)\nwhile the sixth order contribution emergences from1\n6(G0G\u00001\nF)6, giving\nf(q1;\u0001\u0001\u0001;q5) =T\n2VX\nkdet [A(q1\u0000k)] det [ A(q1\u0000q2+q3\u0000k)] det [ A(q1\u0000q2+q3\u0000q4+q5\u0000k)]\n\u0002Tr\u0014\nA(k)A\u00001(q1\u0000k)A(k\u0000q1+q2)A\u00001(q1\u0000q2+q3\u0000k)\n\u0002A(k\u0000q1+q2\u0000q3+q4)A\u00001(q1\u0000q2+q3\u0000q4+q5\u0000k)\u0015\n: (D13)\nEvaluating the expressions given in Eqs. (D11)\nthrough (D13) requires us to perform summations over\nMatsubara frequencies of the type\nTX\ni!n1\ni!n\u0006E(k)=(\nn(k) if \\+\"\n1\u0000n(k) if \\\u0000\";(D14)\nwheren(k) = 1=\u0002\ne\fE(k)+ 1\u0003\nis the Fermi function. For\nthe quadratic term, we obtain the result\n\u0000\u00001(q;iqn) =\u0000m\n4\u0019as+1\n2VX\nk\u00141\n\"k\n+2X\ni;j=1\u000bij(k;q)Wij(k;q;iqn)\u0015\n;(D15)\nwhere the functions in the last term are\nWij(k;q;iqn) =1\u0000ni(k)\u0000nj(k+q)\niqn\u0000Ei(k)\u0000Ej(k+q); (D16)corresponding to the contribution of bubble diagrams to\nthe pair susceptibility. The coherence factors are\n\u000b11(k;q) =jukuk+q\u0000vkv\u0003\nk+qj2; (D17)\n\u000b12(k;q) =jukvk+q+uk+qvkj2; (D18)\nwith\u000b11(k;q) =\u000b22(k;q) and\u000b12(k;q) =\u000b21(k;q);\nwhere the quasiparticle amplitudes are\nuk=s\n1\n2\u0012\n1 +hz\nhk\u0013\n; (D19)\nvk=ei\u0012ks\n1\n2\u0012\n1\u0000hz\nhk\u0013\n: (D20)\nThe angle\u0012kis the phase associated with the spin-orbit\n\feldh?(k) =jh?(k)jei\u0012k;and we replaced the interac-\ntion parameter gby thes-wave scattering length asvia13\nEq. (B7), recalling that \"k=k2=2m. The phase and\nmodulus of h?(k) are\n\u0012k= arctan\u0012\u0011ky\nkx\u0013\n; (D21)\njh?(k)j=j\u0014j\nmq\nk2x+\u0011k2y; (D22)\nand the total e\u000bective \feld is\nhk=p\nh2z+jh?(k)j2: (D23)Since we are interested only in the long-wavelength\nand low-frequency regime, we perform an analytic con-\ntinuation to real frequencies iqn=!+i\u000eafter calculat-\ning the Matsubara sums for all coe\u000ecients appearing in\nEq. (D10) and perform a small momentum qand low-\nfrequency!expansion resulting in the Ginzburg-Landau\naction,\nSF=SGL=\fVX\nq \na+X\n`c`q2\n`\n2m\u0000d0!!\nj\u0001qj2+\fV\n2X\nq1;q2;q3b(q1;q2;q3)\u0001q1\u0001\u0003\nq2\u0001q3\u0001\u0003\nq1\u0000q2+q3\n+\fV\n3X\nq1\u0001\u0001\u0001q5f(q1;q2;q3;q4;q5)\u0001q1\u0001\u0003\nq2\u0001q3\u0001\u0003\nq4\u0001q5\u0001\u0003\nq1\u0000q2+q3\u0000q4+q5: (D24)\nHere, the label `appearing explicitly in the termP\n`c`q2\n`=(2m) represents the spatial directions fx;y;zg,\nwhile theqj's in the sums correspond to ( qj;!j) and\nthe summationsP\nqjrepresent integrals \fVR\nd!jR\nd3qj,\nwherejlabels a fermion pair and can take values in the\nsetf1;2;3;4;5g. In the expression above, we used the\nresult\n\u0000\u00001(q;!) =a+X\n`c`q2\n`\n2m\u0000d0!+\u0001\u0001\u0001 (D25)\nfor the analytically continued expression of \u0000\u00001(q;iqn)\nappearing in Eq. (D15). To write the coe\u000ecients above\nin a more compact notation, we de\fne\nXi=Xi(k) = tanh [\fEi(k)=2]; (D26)\nYi=Yi(k) = sech2[\fEi(k)=2]: (D27)\nThe frequency- and momentum-independent coe\u000ecient\nis\na=\u0000m\n4\u0019as+1\nVX\nk\u00141\n2\"k\u0000\u0012X1\n4E1+X2\n4E2\u0013\n\u0000h2\nz\n\u0018khk\u0012X1\n4E1\u0000X2\n4E2\u0013\u0015\n;(D28)whereE1=E1(k) andE2=E2(k). The coe\u000ecient\nd0=dR+idImultiplying the linear term in frequency\nhas a real component given by\ndR=1\n2VPX\nk2X\ni;j=1\u000bij(k;0)1\u0000ni(k)\u0000nj(k)\n[Ei(k) +Ej(k)]2:(D29)\nUsing the explicit forms of the coherence factors ukand\nvkthat de\fne \u000bij(k;q=0), the above expression can be\nrewritten as\ndR=1\n2VPX\nk\u0014\u0012\n1 +h2\nz\n\u00182\nk\u0013\u0012X1\n4E2\n1+X2\n4E2\n2\u0013\n+2h2\nz\n\u0018khk\u0012X1\n4E2\n1\u0000X2\n4E2\n2\u0013\u0015\n;(D30)\nwhich de\fnes the time scale for temporal oscillations of\nthe order parameter. Here, the symbol Pdenotes the\nprincipal value, and the coe\u000ecient dRis obtained from\nRe\u0002\n\u0000\u00001(q=0;!+i\u000e)\u0003\n=\u0000m\n4\u0019as+1\n2VX\nk2\n41\n\"k+P2X\ni;j=1\u000bij(k;q=0)1\u0000ni(k)\u0000nj(k)\n!\u0000Ei(k)\u0000Ej(k)3\n5: (D31)\nThe imaginary component of the coe\u000ecient dhas the form\ndI=\u0019\n2VX\nk2X\ni;j=1\u000bij(k;0) [1\u0000ni(k)\u0000nj(k)]\u000e0(Ei(k) +Ej(k)); (D32)14\nwhere the derivative of the delta function is \u000e0(\u0015) =@\u000e(x+\u0015)=@xjx=0:Using again the expressions of the coherence\nfactorsukandvkleads to\ndI=\u0019\n2VX\nk\u001a\n(X1+X2)\u000e0(2\u0018k) +jh?j2\nh2\nk\u0014\nX1\u000e0(2E1) +X2\u000e0(2E2)\u0000(X1+X2)\u000e0(2\u0018k)\u0015\u001b\n; (D33)\nwhich determines the lifetime of fermion pairs. This result originates from\nIm\u0002\n\u0000\u00001(q=0;!+i\u000e)\u0003\n=\u0000\u0019\n2VX\nk2X\ni;j=1\u000bij(k;q=0) [1\u0000ni(k)\u0000nj(k)]\u000e(!\u0000Ei(k)\u0000Ej(k)); (D34)\nwhich immediately reveals that below the two-particle\nthreshold!tp(q=0) = minfi;j;kg[Ei(k) +Ej(k)] at\ncenter-of-mass momentum q=0, the lifetime of the pairs\nis in\fnitely long due to the emergence of stable two-body\nbound states. Note that collisions between bound states\nare not yet included.\nThe expressions for the c`coe\u000ecients appearing in\nEq. (D25) are quite long and complex. Since these coef-\n\fcients are responsible for the mass renormalization and\nanisotropy within the Ginzburg-Landau theory, we out-\nline below their derivation in detail. These coe\u000ecients\ncan be obtained from the last term in Eq. (D15), which\nwe de\fne as\nF(q) =1\n2VX\nk2X\ni;j=1\u000bij(k;q)Wij(k;q;iqn= 0):(D35)\nThe relation between c`and the function F(q) de\fned\nabove is\nc`=m\u0014@2F(q)\n@q2\n`\u0015\nq=0: (D36)\nA more explicit form of c`is obtained by analyzing the\nsymmetry properties of F(q) under inversion and re\rec-\ntion symmetries. To make these properties clear, we\nrewrite the summand in Eq. (D35) by making use of the\ntransformation k!k\u0000q=2. This procedure leads to the\nsymmetric form,\nF(q) =1\n2VX\nk2X\ni;j=1e\u000bij(k\u0000;k+)fWij[Ei(k\u0000);Ej(k+)]:\n(D37)\nHere, k+=k+q=2 and k\u0000=k\u0000q=2 are new momentum\nlabels, and\ne\u000b11(k\u0000;k+) =juk\u0000uk+\u0000vk\u0000v\u0003\nk+j2; (D38)\ne\u000b12(k\u0000;k+) =juk\u0000vk+\u0000vk\u0000uk+j2(D39)\nare coherence factors, with e\u000b11(k\u0000;k+) =e\u000b22(k\u0000;k+)\nande\u000b12(k\u0000;k+) =e\u000b21(k\u0000;k+) The functions uk\u0006and\nvk\u0006are de\fned in Eqs. (D19) and (D20). It is now very\neasy to show that e\u000bij(k\u0000;k+) =e\u000bij(k+;k\u0000), that is,\ne\u000bij(k\u0000;k+) is an even function of q, since taking q!\u0000q\nleads to k\u0000!k+andk+!k\u0000leavinge\u000bijinvariant.It is also clear, from its de\fnition, that e\u000bijis symmetric\nin the band indices fi;jg. Furthermore, the function\nfWij[Ei(k\u0000);Ej(k+)] =Nij\nDij; (D40)\nde\fned above, is the ratio between the numerator,\nNij= tanh [\fEi(k\u0000)=2] + tanh [\fEj(k+)=2];(D41)\nrepresenting the Fermi occupations and the denominator,\nDij= 2 [Ei(k\u0000) +Ej(k+)]; (D42)\nrepresenting the sum of the quasi-particle excitation\nenergies. To elliminate the Fermi distributions ni(k)\nin the numerator, we used the relation 1 \u00002ni(k) =\ntanh [\fEi(k\u0000)=2]. Notice that fWij[Ei(k\u0000);Ej(k+)] is\nnot generally symmetric under inversion q! \u0000 q,\nthat is, under the transformation k\u0000!k+and\nk+!k\u0000. This means that fWij[Ei(k\u0000);Ej(k+)]6=\nfWij[Ei(k+);Ej(k\u0000)], unless when i=j, where\nit is trivially an even function of q. However,\nfWij[Ei(k\u0000);Ej(k+)] is always symmetric under simul-\ntaneous momentum inversion ( q!\u0000q) and band index\nexchange, that is,\nfWij[Ei(k\u0000);Ej(k+)] =fWji[Ej(k+);Ei(k\u0000)] (D43)\nfor anyfi;jg. This property will be used later to write a\n\fnal expression for c`. Next, we write\n\u0014@2F(q)\n@q2\n`\u0015\nq=0=1\n2VX\nk2X\ni;j=1Fij; (D44)\nwhere the function inside the summation is\nFij=\"\n@2e\u000bij\n@q2\n`fWij+\u000bij@2fWij\n@q2\n`#\nq=0: (D45)\nNotice the absence of terms containing the product of\nthe \frst-order derivatives of e\u000bijandfWij. These terms\nvanish due to parity since e\u000bijis an even function of q,\nleading to [@e\u000bij=@q`]q=0= 0. The last expression can be15\nfurther developed upon summation over the band indices,\nleading to\n\u0014@2F(q)\n@q2\n`\u0015\nq=0=A+B: (D46)\nThe \frst contribution is given by\nA=1\n2VX\nk\u0014@2e\u000b11\n@q2\n`fWdi+@2e\u000b12\n@q2\n`fWod\u0015\nq=0;(D47)\nand contains the second derivatives of e\u000bijand the sym-\nmetric terms\nfWdi=\u0010\nfW11+fW22\u0011\n; (D48)\nfWod=\u0010\nfW12+fW21\u0011\n; (D49)The second contribution is given by\nB=1\n2VX\nk\"\ne\u000b11@2fWdi\n@q2\n`+e\u000b12@2fWod\n@q2\n`#\nq=0:(D50)\nNext, we explicitly write e\u000bij,fWijand their second\nderivatives with respect to q`atq=0. We start with\nh\nfWiji\nq=0=Xi+Xj\n2 [Ei+Ej](D51)\nand for the second derivative, we write\n\"\n@2fWij\n@q2\n`#\nq=0=\u00141\nDij@2Nij\n@q2\n`\u0015\nq=0\u0000\"\n2\nD2\nij@Dij\n@q`@Nij\n@q`#\nq=0+\"\n2Nij\nD3\nij\u0012@Dij\n@q`\u00132#\nq=0\u0000\"\nNij\nD2\nij@2Dij\n@q2\n`#\nq=0:\nEach one of the four terms in the above expression is\nevaluated at q=0and can be written in terms of speci\fc\nexpressions that are given below. The numerator is\n[Nij]q=0=Xi+Xj; (D52)\nthe \frst derivative of Nijis\n\u0014@Nij\n@q`\u0015\nq=0=Y2\nj\n4T@Ej\n@k`\u0000Y2\ni\n4T@Ei\n@k`; (D53)\nand the second derivative of Nijis\n\u0014@2Nij\n@q2\n`\u0015\nq=0=\u0000XjY2\nj\n8T2\u0012@Ej\n@k`\u00132\n+Yi\n8T@2Ei\n@k2\n`\n\u0000XiY2\ni\n8T2\u0012@Ei\n@k`\u00132\n+Y2\ni\n8T@2Ei\n@k2\n`:(D54)\nThe denominator Dijand its \frst derivative are\n[Dij]q=0= 2(Ei+Ej); (D55)\n\u0014@Dij\n@q`\u0015\nq=0=@Ej\n@k`\u0000@Ei\n@k`; (D56)\nwhile the second derivative of Dijis\n\u0014@2Dij\n@q2\n`\u0015\nq=0=1\n2\u0014@2Ei\n@k2\n`+@2Ej\n@k2\n`\u0015\n: (D57)\nWhen the order parameter is zero, that is, j\u00010j= 0, the\nenergiesE1(k) andE2(k) become\nE1(k) =\f\f\fj\u0018kj+hk\f\f\f (D58)\nE2(k) =\f\f\fj\u0018kj\u0000hk\f\f\f: (D59)The \frst derivatives of these energies are\n@E1(k)\n@k`=S1(k)k`\nm+@hk\n@k`; (D60)\n@E2(k)\n@k`=S2(k)k`\nm\u0000@hk\n@k`; (D61)\nwith the functions S1(k) = sgn [j\u0018kj+hk] sgn [\u0018k] and\nS2(k) = sgn [j\u0018kj\u0000hk] sgn [\u0018k]:The derivative of the ef-\nfective Zeeman \feld is\n@hk\n@k`=1\nhk\u00142\nm2(kx\u000e`x+\u0011ky\u000e`y): (D62)\nThe second derivatives of the energies are\n@2E1(k)\n@k2\n`=S1(k)\nm+@2hk\n@k2\n`(D63)\n@2E2(k)\n@k2\n`=S2(k)\nm\u0000@2hk\n@k2\n`; (D64)\nwhere the second derivative of the e\u000bective \feld is\n@2hk\n@k2\n`=1\nhk\u00142\nm2\u0014\n(\u000e`x+\u0011\u000e`y)\u00001\nh2\nk\u00142\nm2\u0000\nk2\nx\u000e`x+\u00112k2\ny\u000e`y\u0001\u0015\n:\n(D65)\nSince the diagonal elements fWiiare even functions of\nqand so areNiiandDii, their expressions are simpler\nthan in the general case discussed above, because the \frst\norder derivatives of NiiandDiivanish. The surviving\nterms involve only the second derivatives of NiiandDii\nleading to the expression\n\"\n@2fWii\n@q2\n`#\nq=0=\u00141\nDii@2Nii\n@q2\n`\u0015\nq=0\u0000\u0014Nii\nD2\nii@2Dii\n@q2\n`\u0015\nq=0:\n(D66)16\nHere, the numerator and denominator functions are\n[Nii]q=0= 2Xiand [Dii]q=0= 4Ei; (D67)\nwhile their second derivatives are\n\u0014@2Nii\n@q2\n`\u0015\nq=0=\u0000XiY2\ni\n4T2\u0012@Ei\n@k`\u00132\n+Y2\ni\n4T@2Ei\n@k2\n`;(D68)\n\u0014@2Dii\n@q2\n`\u0015\nq=0=@2Ei\n@k2\n`: (D69)\nThe next step in obtaining the c`coe\u000ecients is to an-\nalyze the functions e\u000bijand their second derivatives. We\nbegin by writing e\u000b11atq=0:\n[e\u000b11]q=0=\f\fu2\nk\u0000jvkj2\f\f2=h2\nz\nh2\nk: (D70)\nTo investigate the second derivative of e\u000b11, we write\ne\u000b11=\r11\r\u0003\n11; (D71)\nwhere the complex function is given by\n\r11=uk\u0000uk+\u0000vk\u0000vk+: (D72)\nIn this case, we write the \frst derivative of e\u000b11as\n@e\u000b11\n@q`=@\r11\n@q`\r\u0003\n11+\r11@\r\u0003\n11\n@q`(D73)\nand the second derivative as\n@2e\u000b11\n@q2\n`=@2\r11\n@q2\n`\r\u0003\n11+ 2@\r11\n@q`@\r\u0003\n11\n@q`+\r11@2\r\u0003\n11\n@q2\n`:(D74)\nTo explore the symmetry with respect to q, we express\n\r11in terms of its odd and even components via the re-\nlation\r11=\r11;e+\r11;o, where the even component\n\r11;e= [\r11(q) +\r11(\u0000q)]=2 is\n\r11;e=uk\u0000uk+\u0000jvk\u0000jjvk+jcos\u0000\n\u0012k\u0000\u0000\u0012k+\u0001\n(D75)\nand the odd component \r11;o= [\r11(q)\u0000\r11(\u0000q)]=2 is\n\r11;o=ijvk+jjvk\u0000jsin\u0000\n\u0012k+\u0000\u0012k\u0000\u0001\n: (D76)\nExpressed via the even \r11;eand odd\r11;ocomponents,\nthe second derivative in Eq. (D74) is\n@2e\u000b11\n@q2\n`=@2\r11;e\n@q2\n`\r\u0003\n11:e+ 2@\r11;o\n@q`@\r\u0003\n11;o\n@q`+\r11;e@2\r\u0003\n11;e\n@q2\n`:\n(D77)\nNotice that the even component is purely real, that is,\n\r\u0003\n11;e=\r11;e, and that the odd component is purely imag-\ninary,\r\u0003\n11;o=\u0000\r11;o. Use of this property leads to\n@2e\u000b11\n@q2\n`= 2\r11;e@2\r11;e\n@q2\n`\u00002\u0012@\r11;o\n@q`\u00132\n: (D78)The contribution from the even term \r11;eis\n[\r11;e]q=0=u2\nk\u0000jvkj2=hz\nhk; (D79)\nand from its second derivative is\n\u0014@2\r11;e\n@q2\n`\u0015\nq=0=1\n2\u0012@jvkj\n@k`\u00132\n\u00001\n2jvkj@2jvkj\n@k2\n`+jvkj2\u0012@\u0012k\n@k`\u00132\n;\n(D80)\nwhile the contribution from the odd term \r11;ois\n\u0014@\r11;o\n@q`\u0015\nq=0=ijvkj2@\u0012k\n@k`: (D81)\nNow, we turn our attention to e\u000b12and its second\nderivative. From Eq. (D39), we notice that \r12is ex-\nplicitly odd in qbecause\r12(q) =\u0000\r12(\u0000q), since the\noperation q!\u0000qtakes k\u0000!k+and vice versa, leading\nto\n[e\u000b12]q=0= 0: (D82)\nTo calculate the second derivative of e\u000b12, we write\ne\u000b12=\r12\r\u0003\n12; (D83)\nwhere the complex function\n\r12=uk\u0000vk+\u0000vk\u0000uk+: (D84)\nWe relate @2e\u000b12=@q2\n`to\r12and its \frst and second\nderivatives via\n@2e\u000b12\n@q2\n`=@2\r12\n@q2\n`\r\u0003\n12+ 2@\r12\n@q`@\r\u0003\n12\n@q`+\r12@2\r\u0003\n12\n@q2\n`:(D85)\nGiven that [ \r12]q=0= 0 and [\r\u0003\n12]q=0= 0, the expression\nabove simpli\fes to\n\u0014@2e\u000b12\n@q2\n`\u0015\nq=0= 2\u0014@\r12\n@q`@\r\u0003\n12\n@q`\u0015\nq=0= [\u0003`(q)]2;(D86)\nwhere we used the expressions\n\u0014@\r12\n@q`\u0015\nq=0=ei\u0012k\u0003`(k) (D87)\nfor the derivatives of \r12atq=0with the function\n\u0003`(k) =uk@jvkj\n@k`\u0000jvkj@uk\n@k`+ukjvkj@\u0012k\n@k`: (D88)\nThe last information needed is the derivatives of uk,\njvkj, and\u0012k, which are given by\n@uk\n@k`=\u00001\n2hz\nh3\nk\u00142\nm2(kx\u000e`x+\u0011ky\u000e`y)\n(1 +hz=hk)1=2; (D89)\n@jvkj\n@k`=1\n2hz\nh3\nk\u00142\nm2(kx\u000e`x+\u0011ky\u000e`y)\n(1\u0000hz=hk)1=2; (D90)\n@\u0012k\n@k`=\u0011(kx\u000e`y\u0000ky\u000e`x)\nk2x+\u00112k2y: (D91)17\nThe long steps discussed above complete the derivation of\nall the functions needed to compute the c`coe\u000ecients for\nan arbitrary spin-orbit coupling, expressed as a general\nlinear combination of Rashba and Dresselhaus terms.\nAs announced earlier, the calculation of c`, de\fned in\nEq. (D36), is indeed very long and requires the use of\nall the expressions given from Eq. (D37) to Eq. (D91).\nDespite this complexity, that are a few important com-\nments about the symmetries of the c`coe\u000ecients that\nare worth mentioning. Given that c`determines the\nmass anisotropies in the Ginzburg-Landau (GL) theory,\nwe discuss next the anisotropies of c`as a function of\nthe spin-orbit coupling parameters \u0014and\u0011. First, in\nthe limit of zero spin-orbit coupling, where \u0014and\u0011are\nequal to zero, all the c`coe\u000ecients are identical re\rect-\ning the isotropy of the system, that is, cx=cy=cz\nand reduce to previously known results [52]. In this\ncase, the GL e\u000bective masses m`=mdR=c`are isotropic:\nmx=my=mz. Second, in the limit of \u00146= 0 and\u0011=\u00061, the spin-orbit coupling has the same strength\nalong thexandydirections, and thus for the Rashba\n(\u0011= 1) or Dresselhaus ( \u0011=\u00001) cases, the coe\u000ecients\nobey the relation cx=cy6=cz. This leads to e\u000bective\nmassesmx=my6=mz. Third, in the limit \u00146= 0, but\n\u0011= 0, corresponding to the ERD case, the coe\u000ecients\nhave the symmetry cx6=cy=cz. Now the e\u000bective\nmasses obey the relation mx6=my=mz. Finally, in\nthe case where \u00146= 0, and 06=j\u0011j<1, all thec`coe\u000e-\ncients are di\u000berent, that is, cx6=cy6=cz. Therefore, the\ne\u000bective masses are also di\u000berent in all three directions:\nmx6=my6=mz.\nFollowing an analogous procedure, we analyze the co-\ne\u000ecientsb(q1;q2;q3), ande(q1;q2;q3;q4;q5) with allqi=\n(0;0), and de\fne\nZij=Xi+\fEiYj=2: (D92)\nUsing the notation b(0;0;0) =b(0), we obtain\nb(0) =1\n8VX\nk\u0014\u0012\n1 +h4\nz\n\u00182\nkh2\nk\u0013\u0012Z11\nE3\n1+Z22\nE3\n2\u0013\n+2h2\nz\n\u0018khk\u0012Z11\nE3\n1\u0000Z22\nE3\n2\u0013\n+h4\nz\n\u00183\nkh3\nk\u0012X1\nE1\u0000X2\nE2\u0013\u0015\n; (D93)\nwhich is a measure of the local interaction between two pairing \felds. Using the notation f(0;0;0;0;0) =f(0), we\nobtain\nf(0) =3\n32VX\nk\u0014\n\u0000\u0012\n1 +3h4\nz\n\u00182\nkh2\nk\u0013\u0012Z11\nE5\n1+Z22\nE5\n2\u0013\n\u0000h2\nz\n\u0018khk\u0012\n3 +h4\nz\n\u00182\nkh2\nk\u0013\u0012Z11\nE5\n1\u0000Z22\nE5\n2\u0013\n\u0000h6\nz\n\u00184\nkh4\nk\u0012Z11\nE3\n1+Z22\nE3\n2\u0013\n\u0000h4\nz\n\u00183\nkh3\nk\u0012Z11\nE3\n1\u0000Z22\nE3\n2\u0013\n+\f2\n6\u0012X1Y1\nE3\n1+X2Y2\nE3\n2\u0013\n+\f2h2\nz\n6\u0018khk\u0012X1Y1\nE3\n1\u0000X2Y2\nE3\n2\u0013\n\u0000h6\nz\n\u00185\nkh5\nk\u0012X1\nE1\u0000X2\nE2\u0013\u0015\n; (D94)\nwhich is a measure of the local interaction between three pairing \felds. It is important to mention that in the absence\nof spin-orbit and Zeeman \felds, the Ginzburg-Landau coe\u000ecients obtained above reduce to those reported in the\nliterature [52].\nAs we proceed to explicitly write the Ginzburg-Landau\naction and Lagrangian density, we emphasize that in con-\ntrast to the standard crossover that one observes in the\nabsence of an external Zeeman \feld [52], for \fxed hz6= 0\nit is possible for the system to undergo a \frst-order phase\ntransition with increasing 1 =kFas. The same applies for\n\fxed 1=kFaswith increasing hz. Thus, while an expan-\nsion ofSFto quartic order is su\u000ecient when no Zeeman\n\felds are present, when Zeeman \felds are turned on, the\nfourth-order coe\u000ecient b(0) =bmay become negative.\nSuch a situation requires the analysis of the sixth-order\ncoe\u000ecientf(0) =fto describe this \frst-order transition\ncorrectly and to stabilize the theory since f >0.\nThe Ginzburg-Landau action in Euclidean space can\nbe written asSGL=R\ndtR\nd3rLGL(r);wherer\u0011(r;t).Here, the Lagrangian density is\nLGL(r)=aj\u0001(r)j2+b\n2j\u0001(r)j4+f\n3j\u0001(r)j6\n+X\n`c`jr`\u0001(r)j2\n2m\u0000id0\u0001\u0003(r)@\u0001(r)\n@t;(D95)\nwhere`=fx;y;zg,b=b(0) andf=f(0). A variation\nofSGLwith respect to \u0001\u0003(r) via\u000eSGL=\u000e\u0001\u0003(r) = 0 yields\nthe time-dependent Ginzburg-Landau (TDGL) equation,\n \n\u0000id0@\n@t\u0000X\n`c`r2\n`\n2m+bj\u0001j2+fj\u0001j4+a!\n\u0001(r) = 0\n(D96)\nwith cubic and quintic terms, where \u0001 = \u0001( r) are de-\npendent on space and time. This equation describes the\nspatio-temporal behavior of the order parameter \u0001( r;t)\nin the long-wavelength and long-time regime.18\nIn the static homogeneous case with b>0, Eq. (D96)\nreduces to either the trivial (normal-state) solution \u0001 =\n0 whena > 0 or to the nontrivial (super\ruid state)\nj\u0001j=p\njaj=b, whena < 0. The coe\u000ecient dprovides\nthe timescale of the TDGL equation, and thereby deter-\nmines the lifetime associated with the pairing \feld \u0001( r).\nThis can be seen directly by again considering the ho-\nmogeneous case to linear order in \u0001( r), in which case\nthe TDGL equation has the solution \u0001( t)\u0019\u0001(0)eiat=d 0:\nThis last expression can be rewritten more explicitly\nas \u0001(t)\u0019\u0001(0)e\u0000i!0te\u0000t=\u001c0;where!0=jajdR=jd0j2is\nthe oscillation frequency of the pairing \feld, and \u001c0=\njd0j2=(jajdI) is the lifetime of the pairs, where both dR\nanddIare positive de\fnite, that is, dR>0 anddI>0.\nIn the BEC regime, where stable two-body bound\nstates exist, the imaginary part of d0vanishes (dI= 0),\nand the lifetime time of the pairs is in\fnitely long. In\nthis case,d0=dRand we can de\fne the e\u000bective bosonic\nwave function \t =pdR\u0001 to recast Eq. (D96) in the form\nof the Gross-Pitaevskii equation,\n \n\u0000i@\n@t\u0000X\n`r2\n`\n2M`+U2j\tj2+U3j\tj4\u0000\u0016B!\n\t(r) = 0;\n(D97)with cubic and quintic nonlinearities, where \t = \t( r),\nto describe a dilute Bose gas. Here, \u0016B=\u0000a=dRis\nthe bosonic chemical potential, M`=m(dR=c`) are the\nanisotropic masses of the bosons, and U2=b=d2\nRand\nU3=f=d3\nRrepresent contact interactions of two and\nthree bosons, respectively. In the Bose regime, the life-\ntime\u001cof the composite boson is \u001c/1=dI!1 and the\ninteractions U2andU3are always repulsive, thus leading\nto a system consisting of a dilute gas of stable bosons.\nIn this regime, the chemical potential of the bosons is\n\u0016B\u00192\u0016+Eb<0, whereEbis the two-body bound state\nenergy in the presence of spin-orbit coupling and Zeeman\n\felds obtained from the condition \u0000\u00001(q;E\u00002\u0016) = 0\ndiscussed in the main text. Notice that when \u0016B!0\u0000,\nin the absence of boson-boson interactions, the bosons\ncondense.\n[1] Y-J. Lin, R. L. Compton, K. Jimin\u0013 ez-Garc\u0013 \u0010a, J. V. Porto,\nand I. B. Spielman, Synthetic magnetic \felds for ultra-\ncold neutral atoms, Nature (London) 462, 628 (2009).\n[2] Y-J. Lin, R. L. Compton, A. R. Perry, W. D. Phillips, J.\nV. Porto, and I. B. Spielman, Bose-Einstein condensate\nin a uniform light-induced vector potential, Phys. Rev.\nLett. 102, 130401 (2009).\n[3] C. J. Kennedy, W. C. Burton, W. C. Chung and W.\nKetterle, Observation of Bose{Einstein condensation in\na strong synthetic magnetic \feld, Nature Physics, 11,\n859 (2015).\n[4] Y.-J. Lin, K. Jim\u0013 enez-Garc\u0013 \u0010a and I. B. Spielman,\nSpin{orbit-coupled Bose{Einstein condensates, Nature\n471, 83 (2011).\n[5] L. W. Cheuk, A. T. Sommer, Z. Hadzibabic, T. Yefsah,\nW. S. Bakr, and M. W. Zwierlein, Spin-injection spec-\ntroscopy of a spin-orbit coupled Fermi gas, Phys. Rev.\nLett. 109, 095302 (2012).\n[6] P. Wang, Z.-Q. Yu, Z, Fu, J, Miao, L, Huang, S. Chai, H.\nZhai, and J. Zhang, Spin-orbit coupled degenerate Fermi\ngases, Phys. Rev. Lett. 109, 095301 (2012).\n[7] R. A. Williams, M. C. Beeler, L. J. LeBlanc, K. Jim\u0013 enez-\nGarc\u0013 \u0010a and I. B. Spielman, Raman-Induced Interactions\nin a Single-Component Fermi Gas Near an s-Wave Fesh-\nbach Resonance, Phys. Rev. Lett. 111, 095301 (2013).\n[8] V. Galitski and I. B. Spielman, Spin{orbit coupling in\nquantum gases, Nature 494, 49 (2013).\n[9] Z. Fu, L. Huang, Z. Meng, P. Wang, L. Zhang, S. Zhang,\nH. Zhai, P. Zhang, and J. Zhang, Production of Feshbach\nmolecules induced by spin-orbit coupling in Fermi gases,\nNat. Phys. 10, 110 (2014).\n[10] M. Mancini, G. Pagano, G. Cappellini, L. Livi, M. Rider,J. Catani, C. Sias, P. Zoller, M. Inguscio, M. Dalmonte,\nand L. Fallani, Observation of chiral edge states with\nneutral fermions in synthetic Hall ribbons, Science 349,\n1510 (2015).\n[11] L. Huang, Z. Meng, P. Wang, P. Peng, S.-L. Zhang, L.\nChen, D. Li, Q. Zhou, and J. Zhang, Experimental real-\nization of two-dimensional synthetic spin{orbit coupling\nin ultracold Fermi gases, Nature Physics 12, 540 (2016).\n[12] Z. Wu, L.Zhang, W. Sun, X.-T. Xu, B.-Z. Wang, S.-C. Ji,\nY. Deng, S. Chen, X.-J. Liu, and J.-W. Pan, Realization\nof two-dimensional spin-orbit coupling for Bose-Einstein\ncondensates Science 354, 83 (2016).\n[13] J. I. Cirac and P. Zoller, Goals and opportunities in quan-\ntum simulation, Nature Phys. 8, 264 (2012).\n[14] U.-J. Wiese, Ultracold quantum gases and lattice sys-\ntems: Quantum simulation of lattice gauge theories, An-\nnalen der Physik 525, 777 (2013).\n[15] E. Zohar, J. I. Cirac, and B. Reznik, Quantum simula-\ntions of lattice gauge theories using ultracold atoms in\noptical lattices, Rep. Prog. Phys. 79, 1 (2015).\n[16] M. Dalmonte and S. Montangero, Lattice gauge theory\nsimulations in the quantum information era, Contemp.\nPhys. 57, 388 (2016).\n[17] M. Gong, S. Tewari, and C. Zhang, BCS-BEC crossover\nand topological phase transition in 3D spin-orbit cou-\npled degenerate Fermi gases Phys. Rev. Lett. 107, 195303\n(2011).\n[18] Z.-Q. Yu and H. Zhai, Spin-orbit coupled Fermi gases\nacross a Feshbach resonance, Phys. Rev. Lett. 107,\n195305 (2011).\n[19] H. Hu, L. Jiang, X.-J. Liu, and H. Pu, Probing\nanisotropic super\ruidity in atomic Fermi gases with19\nRashba spin-orbit coupling, Phys. Rev. Lett. 107, 195304\n(2011).\n[20] L. Han and C. A. R. S\u0013 a de Melo, Evolution from BCS to\nBEC super\ruidity in the presence of spin-orbit coupling,\nPhys. Rev. A 85, 011606(R) (2012).\n[21] K. Seo, L. Han, and C. A. R. S\u0013 a de Melo, Emergence\nof Majorana and Dirac particles in ultracold fermions\nvia tunable interactions, spin-orbit e\u000bects, and Zeeman\n\felds, Phys. Rev. Lett. 109, 105303 (2012).\n[22] K. Seo, L. Han, and C. A. R. S\u0013 a de Melo, Topological\nphase transitions in ultracold Fermi super\ruids: The evo-\nlution from Bardeen-Cooper-Schrie\u000ber to Bose-Einstein-\ncondensate super\ruids under arti\fcial spin-orbit \felds,\nPhys. Rev. A 85, 033601 (2012).\n[23] D. M. Kurkcuoglu and C. A. R. S\u0013 a de Melo, Formation of\nFeshbach molecules in the presence of arti\fcial spin-orbit\ncoupling and Zeeman \felds, Phys. Rev. A 93, 023611\n(2016).\n[24] N. Goldman, I. Satija, P. Nikolic, A. Bermudez, M. A.\nMartin-Delgado, M. Lewenstein, and I. B. Spielman, Re-\nalistic time-reversal invariant topological Insulators with\nneutral atoms, Phys. Rev. Lett. 105, 255302 (2010).\n[25] I. B. Spielman, private communication.\n[26] G. Juzeli unas, J. Ruseckas, and J. Dalibard, Generalized\nRashba-Dresselhaus spin-orbit coupling for cold atoms,\nPhys. Rev. A 81, 053403 (2010).\n[27] D. L. Campbell, G. Juzeli unas, and I. Spielman, Realistic\nRashba and Dresselhaus spin-orbit coupling for neutral\natoms Phys. Rev. A 84, 025602 (2011).\n[28] G. Dresselhaus, Spin-orbit coupling e\u000bects in zinc blende\nstructures Phys. Rev. 100, 580 (1955).\n[29] E. I. Rashba, Properties of semiconductors with an ex-\ntremum loop: I. Cyclotron and combinational resonance\nin a magnetic \feld perpendicular to the plane of the loop,\nSov. Phys. Solid State 2, 1224 (1960)\n[30] C. A. R. S\u0013 a de Melo, When fermions become bosons:\nPairing in ultracold gases, Physics Today vol. 61issue\n10, page 45 (2008).\n[31] T. Ozawa and G. Baym, Population imbalance and pair-\ning in the BCS-BEC crossover of three-component ultra-\ncold fermions, Phys. Rev. A 82, 063615 (2010).\n[32] P. D. Powell, From Quarks to Cold Atoms: The Phases of\nStrongly-Interacting Systems, Ph.D. Thesis, University\nof Illinois at Urbana-Champaign (2013).\n[33] D. M. Kurkcuoglu and C. A. R. S\u0013 a de Melo, Color super-\n\ruidity of neutral ultracold fermions in the presence of\ncolor-\rip and color-orbit \felds, Phys. Rev. A 97, 023632\n(2018)\n[34] Daisuke Yamamoto, I. B. Spielman, and C. A. R. S\u0013 a de\nMelo, Quantum phases of two-component bosons with\nspin-orbit coupling in optical lattices, Phys. Rev. A 96,\n061603(R) (2017).\n[35] S. L. Bromley, S. Kolkowitz, T. Bothwell, D. Kedar, A.\nSafavi-Naini, M. L. Wall, C. Salomon, A. M. Rey, and\nJ. Ye, Dynamics of interacting fermions under spin{orbit\ncoupling in an optical lattice clock, Nature Physics 14,\n399 (2018).\n[36] Yu Yi-Xiang, Fadi Sun, and Jinwu Ye, Class of topo-\nlogical phase transitions of Rashba spin-orbit coupled\nfermions on a square lattice, Phys. Rev. B 98, 174506\n(2018).\n[37] M. Mamaev, R. Blatt, J Ye, and A. M. Rey, Cluster\nstate generation with spin-orbit coupled fermionic atoms\nin optical lattices, Phys. Rev. Lett. 122, 160402 (2019).[38] Baihua Gong, Shuai Li, Xin-Hui Zhang, Bo Liu, and Wei\nYi, Bloch bound state of spin-orbit-coupled fermions in\nan optical lattice, Phys. Rev. A 99, 012703 (2019).\n[39] Man Hon Yau and C. A. R. S\u0013 a de Melo Chern-number\nspectrum of ultracold fermions in optical lattices tuned\nindependently via arti\fcial magnetic, Zeeman, and spin-\norbit \felds, Phys. Rev. A 99, 043625 (2019)\n[40] Wei Jia, Zhi-Hao Huang, Xian Wei, Qing Zhao, and\nXiong-Jun Liu, Topological super\ruids for spin-orbit cou-\npled ultracold Fermi gases, Phys. Rev. B 99, 094520\n(2019).\n[41] Majid Kheirkhah, Zhongbo Yan, Yuki Nagai, and Frank\nMarsiglio, First- and second-order topological supercon-\nductivity and temperature-driven topological phase tran-\nsitions in the extended hubbard model with spin-orbit\ncoupling, Phys. Rev. Lett. 125, 017001 (2020).\n[42] Urs Gebert, Bernhard Irsigler, and Walter Hofstetter,\nLocal Chern marker of smoothly con\fned Hofstadter\nfermions, Phys. Rev. A 101, 063606 (2020).\n[43] Irakli Titvinidze, Julian Legendre, Maarten Grothus,\nBernhard Irsigler, Karyn Le Hur, and Walter Hofstet-\nter, Spin-orbit coupling in the kagome lattice with \rux\nand time-reversal symmetry, Phys. Rev. B 103, 195105\n(2021).\n[44] R. A. Williams, L. J. LeBlanc, K. Jim\u0013 enez-Garc\u0013 \u0010a, M. C.\nBeeler, A. R. Perry, W. D. Phillips, and I. B. Spielman,\nSynthetic partial waves in ultracold atomic collisions, Sci-\nence335, 314 (2012).\n[45] Z. Meng, L. Huang, P. Peng, D. Li, L. Chen, Y. Xu,\nC. Zhang, P. Wang, and J. Zhang Experimental obser-\nvation of a topological band gap opening in ultracold\nFermi gases with two-dimensional spin-orbit coupling,\nPhys. Rev. Lett. 117, 235304 (2016).\n[46] L. Huang, Z. Meng, P. Wang, P. Peng, S.-L. Zhang, L.\nChen, D. Li, Q. Zhou, and J. Zhang, Experimental real-\nization of two-dimensional synthetic spin{orbit coupling\nin ultracold Fermi gases, Nature Physics 12, 540 (2016).\n[47] A. Vald\u0013 es-Curiel, D. Trypogeorgos, Q.-Y. Liang, R. P.\nAnderson, and I. B. Spielman, Topological features with-\nout a lattice in Rashba spin-orbit coupled atoms, Nature\nCommunications 12, 593 (2021).\n[48] Ronen M. Kroeze, Yudan Guo, and Benjamin L. Lev,\nDynamical spin-orbit coupling of a quantum gas Phys.\nRev. Lett. 123, 160404 (2019).\n[49] Zong-Yao Wang, Xiang-Can Cheng, Bao-Zong Wang,\nJin-Yi Zhang, Yue-Hui Lu, Chang-Rui Yi, Sen Niu, You-\njin Deng, Xiong-Jun Liu, Shuai Chen, and Jian-Wei Pan,\nRealization of an ideal Weyl semimetal band in a quan-\ntum gas with 3D spin-orbit coupling, Science 372, 271\n(2021).\n[50] A. J. Leggett in Modern Trends in the Theory of Con-\ndensed Matter , pp. 13{27, edited by A. Pekalski and R.\nPrzystawa, Springer-Verlag, Berlin (1980).\n[51] J. R. Engelbrecht, M. Randeria, and C. A. R. S\u0013 a de Melo,\nBCS to Bose crossover: Broken-symmetry state, Phys.\nRev. B 55, 15153 (1997).\n[52] C. A. R. S\u0013 a de Melo, M. Randeria, and J. Engelbrecht,\nCrossover from BCS to Bose superconductivity: Transi-\ntion temperature and time-dependent Ginzburg-Landau\ntheory, Phys. Rev. Lett. 71, 3202 (1993).\n[53] P. D. Powell, G. Baym, C. A. R. S\u0013 a de Melo, Super\ruid\ntransition temperature of spin-orbit and Rabi coupled\nfermions with tunable interactions, arXiv:1709.07042v1\n(2017).20\n[54] S. Gopalakrishnan, A. Lamacraft, and P. M. Gold-\nbart, Universal phase structure of dilute Bose gases with\nRashba spin-orbit coupling, Phys. Rev. A 84, 061604\n(2011).\n[55] T. Ozawa, Topics in Multi-component Ultracold Gases\nand Gauge Fields, Ph.D. Thesis, University of Illinois at\nUrbana-Champaign (2012).\n[56] P. Nozi\u0012 eres and S. Schmitt-Rink, Bose condensation in\nan attractive fermion gas: From weak to strong coupling\nsuperconductivity, J. Low Temp. Phys. 59, 195 (1985).\n[57] Z. Yu and G. Baym, Spin correlation functions in ul-\ntracold paired atomic-fermion systems: Sum rules, self-\nconsistent approximations, and mean \felds, Phys. Rev.\nA73, 063601 (2006).\n[58] Z. Yu, G. Baym, and C. J. Pethick, Calculating energy\nshifts in terms of phase shifts, J. Phys. B: At. Mol. Opt.\nPhys. 44, 195207 (2011).\n[59] P. Fulde and R. A. Ferrell, Superconductivity in a strong\nspin-exchange \feld, Phys. Rev. 135, A550 (1964).\n[60] A. I. Larkin and Y. N. Ovchinnikov, Nonuniform state of\nsuperconductors, Sov. Phys. JETP 20, 762 (1965).\n[61] A. M. Clogston, Upper limit for the critical \feld in hard\nsuperconductors, Phys. Rev. Lett. 9, 266 (1962).\n[62] G. Sarma, On the in\ruence of a uniform exchange \feld\nacting on the spins of the conduction electrons in a su-\nperconductor, J. Phys. Chem. Sol. 24, 1029 (1963).\n[63] G. Baym, J.-P. Blaizot, M. Holzmann, F. Lalo e and D.\nVautherin, The transition temperature of the dilute in-\nteracting Bose gas, Phys. Rev. Lett. 83, 1703 (1999).\n[64] M. Iskin and C. A. R. S\u0013 a de Melo, Fermi-Fermi mixtures\nin the strong-attraction limit, Phys. Rev. A 77, 013625\n(2008).\n[65] D. S. Petrov, C. Salomon, and G. V. Shlyapnikov, Di-\natomic molecules in ultracold Fermi gases: Novel com-\nposite bosons, J. Phys. B 38, S645 (2005).\n[66] J. P. Vyasanakere, S. Zhang, and V. B. Shenoy, BCS-\nBEC crossover induced by a synthetic non-Abelian gauge\n\feld, Phys, Rev. B 84, 014512 (2011)\n[67] X.-J. Feng and L. Yin, Phase diagram of a spin-orbit\ncoupled dipolar Fermi gas at T= 0 K, Chin. Phys. Lett.\n37, 020301 (2020).[68] L. Dell'Anna, G. Mazzarella, and L. Salasnich, Conden-\nsate fraction of a resonant Fermi gas with spin-orbit cou-\npling in three and two dimensions, Physical Review A\n84, 033633 (2011).\n[69] L. Dell'Anna, G. Mazzarella, and L. Salasnich, Tuning\nRashba and Dresselhaus spin-orbit couplings: E\u000bects on\nsinglet and triplet condensation with Fermi atoms, Phys-\nical Review A 86, 053632 (2012).\n[70] S. Tewari, T. D, Stanescu, J. D. Sau and S. Das\nSarma, Topologically non-trivial superconductivity in\nspin{orbit-coupled systems: Bulk phases and quantum\nphase transitions, New Journal of Physics, 13, 065004\n(2011).\n[71] Li Han and C. A. R. S\u0013 a de Melo, Ultra-cold fermions\nin the \ratland: Evolution from BCS to Bose super\ruid-\nity in two-dimensions with spin-orbit and Zeeman \felds,\narXiv:1206.4984v1 (2012)\n[72] J. P. A. Devreese, J. Tempere, and C. A. R. S\u0013 a de\nMelo E\u000bects of spin-orbit coupling on the Berezinskii-\nKosterlitz-Thouless transition and the vortex-antivortex\nstructure in two-dimensional Fermi gases, Phys. Rev.\nLett. 113, 165304 (2014).\n[73] J. P. A. Devreese, J. Tempere, and C. A. R. S\u0013 a de Melo\nQuantum phase transitions and Berezinskii-Kosterlitz-\nThouless temperature in a two-dimensional spin-orbit-\ncoupled Fermi gas, Phys. Rev. A 92, 043618 (2015)\n[74] J. P. Vyasanakere and V. B. Shenoy, Fluctuation theory\nof Rashba Fermi gases: Gaussian and beyond, Phys. Rev.\nB.92, 121111(R) (2015).\n[75] T. Yamaguchi, D. Inotani, and Y. Ohashi, Rashbon\nbound states associated with a spherical spin{orbit cou-\npling in an ultracold Fermi gas with an s-wave interac-\ntion, J. Low Temp. Phys. 183, 161 (2016).\n[76] B. M. Anderson, C.-T. Wu, R. Boyack, and K. Levin\nTopological e\u000bects on transition temperatures and re-\nsponse functions in three-dimensional Fermi super\ruids,\nPhys. Rev. B 92, 134523 (2015).\n[77] L. Dell'Anna, and S. Grava, Critical temperature in the\nBCS-BEC crossover with spin-orbit coupling, Condens.\nMatter 6, 16 (2021)."    },    {        "title": "1110.6364v1.Artificial_spin_orbit_coupling_in_ultra_cold_Fermi_superfluids.pdf",        "content": "arXiv:1110.6364v1  [cond-mat.quant-gas]  28 Oct 2011Artificial spin-orbit coupling in ultra-cold Fermi superflu ids\nKangjun Seo, Li Han and C. A. R. S´ a de Melo\nSchool of Physics, Georgia Institute of Technology, Atlant a, Georgia 30332, USA\n(Dated: November 23, 2018)\nThe control and understanding of interactions in many parti cle systems has been a major chal-\nlenge in contemporary science, from atomic to condensed mat ter and astrophysics. One of the\nmost intriguing types of interactions is the so-called spin -orbit coupling - the coupling between the\nspin (rotation) of a particle and its momentum (orbital moti on), which is omnipresent both in the\nmacroscopic and microscopic world. In astrophysics, the sp in-orbit coupling is responsible for the\nsynchronization of the rotation (spinning) of the Moon and i ts orbit around Earth, such that we\ncan only see one face of our natural satellite. In atomic phys ics, the spin-orbit coupling of electrons\norbiting around the nucleus gives rise to the atom’s fine stru cture (small shifts in its energy levels).\nIn condensed matter physics, spin-orbit effects are respons ible for exotic electronic phenomena in\nsemiconductors (topological insulators) and in supercond uctors without inversion symmetry. Al-\nthough spin-orbit coupling is ubiquitous in nature, it was n ot possible to control it in any area of\nphysics, until it was demonstrated in a breakthrough experi ment [1] that the spin of an atom could\nbe coupled to its center-of-mass motion by dressing twoatom ic spin states with apair of laser beams.\nThis unprecedented engineered spin-orbit coupling was pro duced in ultra-cold bosonic atoms, but\ncan also be created for ultra-cold fermionic atoms [1–3]. In anticipation of experiments, we develop\na theory for interacting fermions in the presence of spin-or bit coupling and Zeeman fields, and show\nthat many new superfluids phases, which are topological in na ture, emerge. Depending on values\nof spin-orbit coupling, Zeeman fields, and interactions, in itially gapped s-wave superfluids acquire\np-wave,d-wave,f-wave and higher angular momentum components, which produc e zeros in the\nexcitation spectrum, rendering the superfluid gapless. Sev eral multi-critical points, which separate\ntopological superfluid phases from normal or non-uniform, a re accessible depending on spin-orbit\ncoupling, Zeeman fields or interactions, setting the stage f or the study of tunable topological super-\nfluids.\nPACS numbers: 03.75.Ss, 67.85.Lm, 67.85.-d\nThe effects of spin-orbit coupling in few body systems\nlike the Earth-Moon complex in astrophysics or the elec-\ntron spin and its orbital motion around the nucleus in\nisolated atoms of atomic physics are reasonably well un-\nderstood due to the simplificity of these systems. How-\never, in the setting of many identical particles, spin-orbit\neffects have revealed quite interesting surprises recently\nrunning from topological insulators in semiconductors [4]\nto exotic superconductivity [5] and non-equillibrium ef-\nfects [6] depending on the precise form of the spin-orbit\ncoupling. In atomic physics the coupling arises from the\ninteraction of the magnetic moment of the electron and\na magnetic field, present in the frame of electron, due\nto the electric field of the nucleus. Similarly in con-\ndensed matter physics, the coupling arises from the mag-\nnetic moment mof electrons, which move in the back-\nground of ions. In the electron’s reference frame, these\nions are responsible for a magnetic field B, which de-\npends on the electron’s momentum kand couple to elec-\ntron’s spin. The resulting spin-orbit coupling has the\nformHSO=−m·B=−/summationtext\njhj(k)σj,whereσjrep-\nresents the Pauli matrices and hj(k) describes the j-th\ncomponent ( j=x,y,z) of the effective magnetic field\nvectorh. For some materials hcan take the Dressel-\nhaus [7] form hD(k) =vD(kyˆx+kxˆy),the Rashba [8]\nformhR(k) =vR(−kyˆx+kxˆy),or more generally a lin-\near combination of the two h⊥(k) =hD(k)+hR(k).In\nall these situations the type of spin-orbit coupling cannot be changed arbitrarily and the magnitude can not\nbe tuned from weak to strong, making the experimental\ncontrol of spin-orbit effects very difficult.\nRecently, however, it has been demonstrated experi-\nmentally that spin-orbit coupling can be engineered in\na ultra-cold gas of bosonic atoms in their Bose-Einstein\ncondensatephase[1], whenapairofRamanlaserscreates\na coupling between two internal spin states of the atoms\nand its center-of-mass motion (momentum). Thus far,\nthe type of spin-orbit field that has been created in the\nlaboratory [1] has the equal-Rashba-Dresselhaus (ERD)\nformh⊥(k) =hERD(k) =vkxˆy, wherevR=vD=v/2.\nOther forms of spin-orbit fields require additional lasers\nand create further experimental difficulties [9]. In ultra-\ncoldbosonsthemomentum-dependent ERDcouplinghas\nbeen created in conjunction with uniform Zeeman terms,\nwhich are independent of momentum, along the z axis\n(controlled by the Raman coupling Ω R), and along the\ny-axis (controled by the detuning δ). The simultaneous\npresence of hz,hyandhERD(k) leads to the Zeeman-\nspin-orbit (ZSO) Hamiltonian\nHZSO(k) =−hzσz−hyσy−hERD(k)σy\nfor an atom with center-of-mass momentum kand spin\nbasis| ↑/an}b∇acket∇i}ht,| ↓/an}b∇acket∇i}ht. The fields hz=−ΩR/2,hy=−δ/2 and\nhERD=vkxˆycan be controlled independently, and thus\ncan be used as tunable parameters to explore the avail-\nable phase space and to investigate phase transitions, as2\nachieved in the experiment involving a bosonic isotope of\nRubidium (87Rb). Although current experiments have\nfocused on Bose atoms, there is no fundamental reason\nthat impeeds the realization of a similar set up for Fermi\natoms [1–3] designed to study fermionic superfluidity [3].\nConsidering possible experiments with fermionic atoms\nsuch as6Li,40K, we discuss in this letter phase diagrams,\ntopological phase transitions, spectroscopic and thermo-\ndynamic properties at zero and finite temperatures dur-\ning the evolution from BCS to BEC superfluidity in the\npresence of controllable Zeeman and spin-orbit fields in\nthree dimensions.\nTo investigate artificial spin-orbit and Zeeman fields in\nultra-cold Fermi superfluids, we start from the Hamilto-\nnian density\nH(r) =H0(r)+HI(r), (1)\nwhere the single-particle term is simply\nH0(r) =/summationdisplay\nαβψ†\nα(r)\nˆKαδαβ−/summationdisplay\njˆhj(r)σj,αβ\nψβ(r).\n(2)\nHere,ˆKα=−∇2/(2m)−µαis the kinetic energyin refer-\nence to the chemical potential µα,ˆhj(r) is the combined\neffective field including Zeeman and spin-orbit compo-\nnents along the j-direction ( j=x,y,z), andψ†\nα(r) are\ncreation operators for fermions with spin αat position\nr. Notice that we allow the chemical potential µ↑to be\ndifferent from µ↓, such that the number of fermions N↑\nwith spin ↑may be different from the number of fermions\nwith spin ↓. The interaction term is\nHI(r) =−gψ†\n↑(r)ψ†\n↓(r)ψ↓(r)ψ↑(r), (3)\nwheregrepresents a contact interaction that can be\nexpressed in terms of the scattering length via the\nLippman-Schwinger relation V/g=−Vm/(4πas) +/summationtext\nk1/(2ǫk).The introduction of the averagepairing field\n∆(r)≡g/an}b∇acketle{tψ↓(r)ψ↑(r)/an}b∇acket∇i}ht ≈∆0and its spatio-temporal fluc-\ntuationη(r,τ) produce a complete theory for superfluid-\nity in this system.\nFrom now on, we focus on the experimental case where\na) the Raman detuning is zero ( δ= 0) indicating that\nthere is no component of the Zeeman field along the ydi-\nrection; b) the Raman coupling Ω Ris non-zero meaning\nthat a Zeeman component along the zdirection exists,\nthat is,hz=−ΩR/2; and c) the spin-orbit field has com-\nponentshy(k) andhx(k) alongtheyandxdirections. To\nstart our discussion, we neglect fluctuations, and trans-\nformH0(r) into momentum space as H0(k). Using the\nbasisψ†\n↑(k)|0/an}b∇acket∇i}ht ≡ |k↑/an}b∇acket∇i}ht, ψ†\n↓(k)|0/an}b∇acket∇i}ht ≡ |k↓/an}b∇acket∇i}ht,where|0/an}b∇acket∇i}htis\nthe vacuum state, the Fourier-transformed Hamiltonian\nH0(k) becomes the matrix\nH0(k) =K+(k)1+K−σz−hzσz−hy(k)σy−hx(k)σx,\nSuch matrix can be diagonalized in the helicity basis\nΦ†\n⇑(k)|0/an}b∇acket∇i}ht ≡ |k⇑/an}b∇acket∇i}ht,Φ†\n⇓(k)|0/an}b∇acket∇i}ht ≡ |k⇓/an}b∇acket∇i}ht,where the spins⇑and⇓are aligned or antialigned with respect to the\neffective magnetic field heff(k) =h/bardbl(k) +h⊥(k).Here,\nK+(k) = (K↑+K↓)/2 =ǫk−µ+,is a measure of the\naverage kinetic energy ǫk=k2/2min relation to the\naverage chemical potential µ+= (µ↑+µ↓)/2.While\nh⊥(k) =hx(k)ˆx+hy(k)ˆyis the spin-orbit field and\nh/bardbl(k) = (hz−K−)ˆzis the effective Zeeman field, with\nK−= (K↑−K↓)/2 =−µ−whereµ−= (µ↑−µ↓)/2 is\nthe internal Zeeman field due to initial population im-\nbalance, and hzis the external Zeeman field. When\nthere is no population imbalance the internal Zeeman\nfield isµ−= 0, and we have only hz. In general,\nthe eigenvalues of the Hamiltonian matrix H0(k) are\nξ⇑(k) =K+(k)−|heff(k)|andξ⇓(k) =K+(k)+|heff(k)|,\nwhere|heff(k)|=/radicalbig\n(µ−+hz)2+|h⊥(k)|2is the magni-\ntude of the effective magnetic field, with the transverse\ncomponent being expressedin termsof the complex func-\ntionh⊥(k) =hx(k) +ihy(k).In the limit where the in-\nternalµ−and external hzZeeman fields vanish and the\nspin-orbit field is null ( h⊥= 0), the energies of the helic-\nity bands are identical ξ⇑(k) =ξ⇓(k) producing no effect\nin the original energy dispersions [10].\nWhen interactions are added to the problem, pair-\ning can occur within the same helicity band (intra-\nhelicity pairing) or between two different helicity bands\n(inter-helicity pairing). This leads to a tensor order\nparameter for superfluidity that has four components\n∆⇑⇑(k) =−∆T(k)e−iϕ,corresponding to the helicity\nprojectionλ= +1; ∆ ⇑⇓(k) =−∆S(k),and ∆ ⇓⇑(k) =\n∆S(k),corresponding to helicity projection λ= 0; and\n∆⇓⇓(k) =−∆T(k)eiϕ,corresponding to helicity pro-\njectionλ=−1. The phase ϕ(k) is defined from\nthe amplitude-phase representation of the complex spin-\norbit fieldh⊥(k) =|h⊥(k)|eiϕ(k),while the amplitude\n∆T(k) = ∆0|h⊥(k)|/|heff(k)|for helicities λ=±1aredi-\nrectly proportional to the scalar order parameter ∆ 0and\nto the relative magnitude of the spin-orbit field |h⊥(k)|\nwith respect to the magnitude of the effective magnetic\nfield|heff(k)|. Additionally, ∆ Thas the simple physi-\ncal interpretation of being the triplet component of the\norder parameter in the helicity basis, which is induced\nby the presence of non-zero spin-orbit field h⊥, but van-\nishes when h⊥= 0. Analogously the amplitude ∆ S(k) =\n∆0h/bardbl(k)/|heff(k)|for helicity λ= 0 are directly propor-\ntional to the scalar order parameter ∆ 0and to the rela-\ntive magnitude of the total Zeeman field h/bardbl(k) =µ−+hz\nwith respect to the magnitude of the effective magnetic\nfield|heff(k)|. Additionally, ∆ Shas the simple physical\ninterpretationofbeingthesingletcomponentofthe order\nparameter in the helicity basis. It is interesting to note\nthe relation |∆T(k)|2+|∆S(k)|2=|∆0|2,which, for fixed\n|∆0|, shows that as |∆S(k)|increases, |∆T(k)|decreases\nand vice-versa. Such relation indicates that the singlet\nand triplet channels are not separable in the presence of\nspin-orbitcoupling. Furthermore, the orderparameterin\nthe triplet sector ∆ ⇑⇑(k) and ∆ ⇓⇓(k) contains not only\np-wave,but also f-waveand evenhigherodd angularmo-\nmentum contributions, as long as the total Zeeman field3\nµ−+hzis non-zero. Similarly, the orderparameterin the\nsinglet sector ∆ ⇑⇓(k) and ∆ ⇓⇑(k) contains not only only\ns-wave, but also d-wave and even higher even angular\nmomentum contributions, as long as the total Zeeman\nfieldµ−+hzis non-zero. Higher angular momentum\npairing in the helicity basis, occurs because the original\nlocal (zero-ranged) interaction in the original ( ↑,↓) spin\nbasis is transformed into a finite-ranged interaction in\nthe helicity basis ( ⇑,⇓). In the limiting case of zero total\nZeemanfield µ−+hz= 0, the singletcomponentvanishes\n(∆S(k) = 0), while the triplet component becomes inde-\npendent of momentum (∆ T(k) = ∆ 0), leading to order\nparameter ∆ ⇑⇑(k) =−h∗\n⊥(k), and ∆ ⇓⇓(k) =−h⊥(k)\nwhich contains only p-wave contributions [11], since the\ncomponents of h⊥(k) depend linearly on momentum k.\nThe eigenvalues Ej(k) of the Hamiltonian including\nthe order parameter contribution emerge from the diago-\nnalization ofa 4 ×4 matrix (see supplementary material).\nThe two eigenvalues for quasiparticles are\nE1(k) =/radicalbigg/parenleftig\nξh−−/radicalig\nξ2\nh++|∆S(k)|2/parenrightig2\n+|∆T(k)|2,(4)\ncorresponding to the highest-energy quasiparticle band,\nand\nE2(k) =/radicalbigg/parenleftig\nξh−+/radicalig\nξ2\nh++|∆S(k)|2/parenrightig2\n+|∆T(k)|2,(5)\ncorresponding to the lowest-energy quasiparticle band,\nwhile the eigenvalues for quasiholes are E3(k) =−E2(k)\nfor highest-energy quasihole band and E4(k) =−E1(k)\nfor the lowest-energy quasihole band. The energy ξh−=\n[ξ⇑(k)−ξ⇓(k)]/2 is momentum-dependent, corresponds\nto the average energy difference between the helicity\nbands and can be written as ξh−=−|heff(k)|,while\nthe energy ξh+= [ξ⇑(k)+ξ⇓(k)]/2 is also momentum\ndependent, corresponds to the averaged energy sum of\nthe helicity bands and can be written as ξh+=K+(k) =\nǫk−µ+.\nThere are a few important points to notice about the\nexcitation spectrum of this system. First, notice that\nE1(k)> E2(k)≥0. Second, that the eigenergies are\nsymmetric about zero, such that we can regard quasi-\nholes (negative energy solutions) as anti-quasiparticles.\nThird, that only E2(k) can have zeros (nodal regions)\ncorrespondingto the locus in momentum space satisfying\nthe following conditions: a) ξh−=−/radicalig\nξ2\nh++|∆S(k)|2,\nwhich corresponds physically to the equality between the\neffective magnetic field energy |heff(k)|and the excita-\ntion energy for the singlet component/radicalig\nξ2\nh++|∆S(k)|2;\nand b)|∆T(k)|= 0,corresponding to zeros of the triplet\ncomponent of the order parameter in momentum space.\nSinceE2(k)< E1(k), and only E2(k) can have ze-\nros, the low energy physics is dominated by this ein-\ngenvalue. In the case of equal Rashba-Dresselhaus\n(ERD) where h⊥(k) =v|kx|, zeros ofE2(k) can oc-\ncur whenkx= 0, leading to the following cases: (a)two possible lines (rings) of nodes at ( k2\ny+k2\nz)/(2m) =\nµ++/radicalbig\n(µ−+hz)2−|∆0|2for the outer ring, and ( k2\ny+\nk2\nz)/(2m) =µ+−/radicalbig\n(µ−+hz)2−|∆0|2forthe innerring,\nwhen (µ−+hz)2− |∆0|2>0; (b) doubly-degenerate\nline of nodes ( k2\ny+k2\nz)/(2m) =µ+forµ+>0, doubly-\ndegenerate point nodes for µ+= 0, or no-line of nodes\nforµ+<0, when (µ−+hz)2− |∆0|2= 0; (c) no\nline of nodes when ( µ−+hz)2− |∆0|2<0. In ad-\ndition, case (a) can be refined into cases (a2), (a1)\nand (a0). In case (a2), two rings indeed exist pro-\nvided that µ+>/radicalbig\n(µ−+hz)2−|∆0|2. However, the\ninner ring disappears when µ+=/radicalbig\n(µ−+hz)2−|∆0|2.\nIn case (a1), there is only one ring when |µ+|</radicalbig\n(µ−+hz)2−|∆0|2,In case (a0), the outer ring dis-\nappears at µ+=−/radicalbig\n(µ−+hz)2−|∆0|2, and forµ+<\n−/radicalbig\n(µ−+hz)2−|∆0|2no rings exist.\nWe choose our momentum, energy and velocity scales\nthrough the Fermi momentum kF+defined from the to-\ntal density of fermions n+=n↑+n↓=k3\nF+/(3π2).This\nchoice leads to the Fermi energy ǫF+=k2\nF+/2mand to\nthe Fermi velocity vF+=kF+/m, as energy and veloc-\nity scales respectively. In Fig. 1, we show the phase\ndiagram of Zeeman field hz/ǫF+versus chemical poten-\ntialµ+/ǫF+describing possible superfluid phases accord-\ning to their quasiparticle excitation spectrum. We la-\nbel the uniform superfluid phases with zero, one or two\nrings of nodes as US-0, US-1, and US-2, respectively.\nNon-uniform (NU) phases also emerge in regions where\nuniform phases are thermodynamically unstable. The\nUS-2/US-1 phase boundary is determined by the condi-\ntionµ+=/radicalbig\n(µ−+hz)2−|∆0|2, when|µ−+hz|>|∆0|;\nthe US-0/US-2 boundary is determined by the Clogston-\nlike condition |(µ−+hz)|=|∆0|whenµ+>0, where\nthe gapped US-0 phase disappears leading to the gap-\nless US-2 phase; and the US-0/US-1 phase boundary\nis determined by µ+=−/radicalbig\n(µ−+hz)2−|∆0|2, when\n|µ−+hz|>|∆0|. Furthermore, with the US-0 bound-\naries, a crossover line between an indirectly gapped and\na directly gapped US-0 phase occurs at µ+= 0. Lastly,\nsome important multi-critical points arise at the inter-\nsections of phase boundaries. First the point µ+= 0 and\n|(µ−+hz)|=|∆0|corresponds to a tri-critical point for\nphasesUS-0, US-1, and US-2. Second, the point |∆0|= 0\nandµ+=|(µ−+hz)|corresponds to a tri-critical point\nfor phases N, US-1 and US-2. In the limit where both\nµ−andhzvanish no phase transitions take place and the\nproblem is reduced to a crossover [12–14].\nIn the US-1 and US-2 phases near the zeros of E2(k),\nquasiparticles have linear dispersion and behave as Dirac\nfermions. Such change in nodal structures is associated\nwith bulk topological phase transitions of the Lifshitz\nclassasnotedfor p-wave[15]and d-wave[16,17]superflu-\nids. Such Lifshitz topological phase transitions are possi-\nble here because the spin-orbit coupling field induces the\ntriplet component of the order parameter ∆ T(k). The\nloss of nodal regions correspond to annihilation of Dirac4\na\nN\nNUS-1 \nUS-1 NU \nNU hz / !F+ \n\"+ / !F+ -2 -1 0 10123\n-3 -2 -1 b\n-2 -1 0 10123\n-3 -2 -1 N\nNUS-1 \nUS-1 Indirect\nUS-0 Direct\nUS-0 \n\"+ / !F+ Indirect\nUS-0 NU \nNU hz / !F+ \nDirect\nUS-0 US-2 \nUS-2 \nFIG. 1: Phase diagram of Zeeman field hz/ǫF+versus chem-\nical potential µ+/ǫF+for a)v/vF+= 0 and b) v/vF+= 0.28\nidentifying uniform superfluid phases US-0 (gapped), US-1\n(gapless with one ring of nodes), and US-2 (gapless with\ntwo-rings of nodes). The NU region corresponds to unsta-\nble uniform superfluids which may include phase separation\nand/or a modulated superfluid (supersolid). Solid lines rep -\nresent phase boundaries, while the dashed line represents t he\ncrossover from the direct-gap to the indirect-gap US-0 phas e.\nquasiparticles with opposite momenta, which lead to the\ndisappearance of rings. The transition from phase US-\n2 to indirect gapped US-0 occurs through the merger of\nthe two-rings at the phase boundary followed by the im-\nmediate opening of the indirect gap at finite momentum.\nHowever, the transition from phase US-2 to US-1 corre-\nsponds to the disappearance of the inner ring through\nthe origin of momenta, similarly the transition from US-\n1 to the directly gappped US-0 corresponds to the dis-\nappearance of the last ring also through the origin of\nmomenta. In the case of Rashba-only coupling rings\nof nodes are absent and it is possible to have at most\nnodal points [18, 19]. The last two phase transitions\nare special because the zero-momentum quasiparticles at\nthesephaseboundariescorrespondtotrueMajoranazero\nenergy modes if the phase ϕ(k) of the spin-orbit field\nh⊥(k) =|h⊥(k)|eiϕ(k)and the phase θ(k) of the order\nparameter ∆ 0=|∆0|eiθ(k)have opposite phases at zero\nmomentum: ϕ(0) =−θ(0) [mod(2π)]. This can be seen\nfrom an analysis of the quasiparticle eigenfunction\nΦ2(k) =u1(k)ψk↑+u2(k)ψk↓+u3(k)ψ†\n−k↑+u4(k)ψ†\n−k↓\ncorresponding to the eigenvalue E2(k). The emergence\nofzero-energyMajoranafermionsrequiresthequasiparti-\ncle to be its own anti-quasiparticle: Φ†\n2(k) = Φ2(k). This\ncanonlyhappenatzeromomentum k=0,wheretheam-\nplitudesu1(0) =u∗\n3(0) andu2(0) =u∗\n4(0). Such require-\nment leads to the conditions µ2\n+= (µ−+hz)2+|∆0|2,\nandϕ(0) =−θ(0) [mod(2π)], showing that Majorana\nfermions can exist only at the US-0/US-1 and US-2/US-\n1 phase boundaries. It is important to emphasize that\nthe Majorana fermions found here exist in the bulk, and\nthus their emergence or disappeareance affect bulk ther-modynamic properties, unlike Majorana fermions found\nat the edge (surfaces) of topological insulators and some\ntopological superfluids. The common ground between\nbulk and surface Majorana fermions is that both exist at\nboundaries: the bulk Majorana zero-energy modes may\nexist at the phase boundaries between two topologically\ndistinct superfluid phases, while surface Majorana zero-\nenergy modes may exist at the spatial boundaries of a\ntopologically non-trivial superfluid.\nIt is evident that the transition between different su-\nperfluid phases occurs without a change in symmetry in\nthe orderparameter∆ 0, and thus violatesthe symmetry-\nbased Landau classification of phase transitions. In the\npresentcase, the simultaneousexistenceofspin-orbitand\nZeeman fields (internal or external) couple the singlet\n∆S(k) and triplet ∆ T(k) channels and all the super-\nfluid phases US-0, US-1 and US-2 just have different\nweights from each order parameter component. How-\never a finer classification based on topological charges\ncan be made via the construction of topological invari-\nants. Since the superfluid phases US-0, US-1, US-2 are\ncharacterized by different excitation spectra correspond-\ning to the eigenvalues of the Hamiltonian matrix includ-\ning interactions H(k), we can use the resolvent matrix\nR(ω,k) = [−ω1+H(k)]−1and the methods of algebraic\ntopology [20] to construct the topological invariant\nℓ=/integraldisplay\nDdSγ\n24π2ǫµνλγTr/bracketleftbig\nΛkµΛkνΛkλ/bracketrightbig\n,\nwhereΛkµ=R∂kµR−1.The topological invariant is\nℓ= 0 in the gapped US-0 phase, is ℓ= 1 in the gap-\nless US-1 phase and ℓ= 2 in the gapless US-2 phase,\nshowing that, for ERD spin-orbit coupling, ℓcounts the\nnumber of rings of zero-energy excitations in each super-\nfluid phase. The integral above has a hyper-surface mea-\nsuredSγandadomain Dthat enclosesthe regionofzeros\nofω=Ej(k) = 0. Here µ,ν,λ,γ run from 0 to 3, and kµ\nhas components k0=ω,k1=kx,k2=ky, andk3=kz.\nThe topological invariant measures the flux of the four-\ndimensional vector Fγ=ǫµνλγTr/bracketleftbig\nΛkµΛkνΛkλ/bracketrightbig\n/24π2,\nthrough a hypercube including the singular region of the\nresolvent matrix R(ω,k), much in the same way that the\nflux of the electric field Ein Gauss’ law of classical elec-\ntromagnetism measures the electric charge qenclosed by\na Gaussian surface:/contintegraltext\ndS·(ǫ0E) =q. Thus, the topologi-\ncal invariant defined above defines the topological charge\nof fermionic excitations, in the same sense as Gauss’ law\nfor the electric flux defines the electric charge.\nA full phase diagram can be constructed only upon\nverification of thermodynamic stability of all the pro-\nposed phases. For this purpose it becomes imper-\native to investigate the maximum entropy condition\n(see supplementary material). Independent of any\nmicroscopic approximations, the necessary and suffi-\ncient conditions for thermodynamic stability of a given\nphase are: positive isovolumetric heat capacity CV=\nT(∂S/∂T)V,{Nα}≥0; positive chemical susceptibility\nmatrixξαβ= (∂µα/∂Nβ)T,V,i.e, eigenvalues of the5\nmatrix [ξ] are both positive; and positive bulk mod-\nulusB= 1/κTor isothermal compressibility κT=\n−V−1(∂V/∂P)T,{Nα}.Using these conditions, we con-\nstruct the full phase diagramdescribed in Fig. 1 for equal\nRasha-Dresselhaus (ERD) spin-orbit coupling. The re-\ngions, where the uniform superfluid phases are unsta-\nble are labeled by the abbreviation NU to indicate that\nnon-uniform phases such as phase separation or modu-\nlated superfluid (supersolid) may emerge. In Fig. 2, we\nshowthephasediagramofZeemanfield hz/ǫF+versusin-\nteraction parameter 1 /(kF+as), for population balanced\nfermions, where the number of spin-up fermions N↑is\nequal to the number of spin-down fermions N↓.\nv / v_F = 0.141 \nb\nN\n0 1 2 3 4-1 -2 2468\n0hz / !F+ \nUS-2 Indirect US-0 Direct \nUS-0 US-1 \n1/k F+ asNU \nv / v_F = 0.283 \nc\nN\nUS-1 Direct \nUS-0 \nUS-2 Indirect US-0 \n0 1 2 3 4-1 -2 2468\n0hz / !F+ \n1/k F+ asNU v / v_F = 0.424 \n0 1 2 3 4-1 -2 2468\n0d\nNDirect \nUS-0 \nUS-2 Indirect US-0 hz / !F+ \n1/k F+ asUS-1 \nNU v / v_F = 0 \na\nN\n0 1 2 3 4-1 -2 2468\n0hz / !F+ \nIndirect US-0 Direct \nUS-0 \n1/k F+ asNU US-1 \nFIG.2: Phasediagram ofZeemanfield hz/ǫF+versusinterac-\ntion 1/(kF+as) showing uniform superfluid phases US-0, US-\n1, and US-2, and non-uniform (NU) region for a) v/vF+= 0;\nb)v/vF+= 0.14; c)v/vF+= 0.28; d)v/vF+= 0.56.\nSolid lines are phase boundaries, the dashed line indicates\na crossover from the indirect- to direct-gapped US-0.\nSince these superfluid phases exhibit major changes in\nmomentum-frequency space as evidenced by their single\nparticle excitation spectrum, it is important to explore\nadditional spectroscopic quantitities to characterize fur-\nther the nature of these phases and the phase transitions\nbetween them. An important quantity is the 4 ×4 resol-\nvent matrix\nR(iω,k) =/parenleftbiggG(iω,k)F(iω,k)\nF†(iω,k)G(iω,k)/parenrightbigg\n,(6)\nfrom where the spectral density Aα(ω,k) =\n−(1/π)ImGαα(iω=ω+iδ,k) for spin α=↑,↓can\nbe extracted. The spectral function Aα(ω,k) in the\nplane of momenta ky-kzwithkx= 0 and frequency\nω= 0 reveals the existence of rings of zero-energy\nexcitations in the US-1 and US-2 phases. The density of\nstatesDα(ω) =/summationtext\nkAα(ω,k) for spinαas a function of\nfrequencyωis also an important spectroscopic quantitywhich is shown in Fig. 3 along with excitation spectra\nEj(k) for phases US-1 and US-2 at fixed ERD spin-orbit\ncouplingv/vF+= 0.28. The parameters used for phase\nUS-1 arehz/ǫF+= 0.5 and 1/(kF+as) =−0.4, while for\nphase US-2 they are hz/ǫF+= 2.0 and 1/(kF+as) = 1.0.\nNotice that, even though the excitation spectrum Ej(k)\nis symmetric, the coherence factors appearing in the\nmatrixGare not, such that the density of states Dα(ω)\nis not an even function of ω, and thus it is not particle-\nhole symmetric. The main feature of Dα(ω) at low\nfrequencies is the linear behavior due to the existence\nof Dirac quasiparticles and quasiholes in the US-1 and\nUS-2 phases, which are absent in the direct-gap and the\nindirect-gap US-0 phases. The peaks and structures in\nDα(ω) mostly emerge due to the maxima and minima of\nEj(k). Notice that for finite Zeeman field hz, the density\nof states D↑(ω)/ne}ationslash=D↓(ω) because the induced population\nimbalanceP= (N↑−N↓)/(N↑+N↓) is non-zero. For\nthe US-2 case shown in Fig. 3b, the induced population\nimbalanceP≪1 sincehz/ǫF+is small, while for the\nUS-1 case shown in Fig. 3e, P≈1 as the spins are\nalmost fully polarized since hz/ǫF+is large.\n!!\"# $$\"# %!!\"& !!\"' !!\"' !\"& \n!!\"# $$\"# %!!\"& !!\"' !!\"' !\"& \n!\"#$%!!&' !!&% !!&% !&' \n!!\"# $$\"# %!!\"& !!\"' !!\"' !\"& \n!!\"# $$\"# %!!\"& !!\"' !!\"' !\"& \n!!\"# $$\"# %!!\"& !!\"' !!\"' !\"& |k x| / k F+ Ej / !F+ \n|k y| / k F+ Ej / !F+ \n|k x| / k F+ Ej / !F+ \n|k y| / k F+ Ej / !F+ D( \") !F+ \" / !F+ \nD( \") !F+ \" / !F+ a b c\nf e d\nFIG. 3: Energy spectrum and density of states in phase US-2\nareshown ina), b), c)for hz/ǫF+= 0.5and1/(kF+as) =−0.4\nand in phase US-1 are shown in d), e), f) for hz/ǫF+= 2.0\nand 1/(kF+as) = 1.0. Energies Ej(kx,0,0) versus |kx|in a)\nand d); frequency ωversus density of states D↑(ω) (dashed),\nD↓(ω) (dot-dashed), and their sum D(ω) (solid) in b) and e);\nenergies Ej(0,ky,0) versus |ky|in c) and f).\nIn summary, we have discussed the effects of spin-orbit\nand Zeeman fields in ultra-cold Fermi superfluids, ob-\ntained the phase diagrams of Zeeman field versus inter-\naction parameter or versus chemical potential, and iden-\ntified several bulk topological phase transitions between\ngapped and gapless superfluids as well as a variety of\nmulti-critical points. We haveshown that the presenceof\nsimultaneousZeeman and spin-orbitfields induces higher6\nangular momentum pairing, as manifested in the emer-\ngence of momentum dependence ofthe singlet and triplet\ncomponents of order parameter expressed in the helicity\nbasis. Finally, we have characterized topological phases\nand phase transitions between them through their exci-\ntation spectra (existence of Dirac quasiparticles or Majo-\nrana zero-energy modes), topological charges, and spec-\ntroscopic and thermodynamic properties, such as densityof states and isothermal compressibility.\nAcknowledgments\nWe thank ARO (W911NF-09-1-0220) for support.\n[1] Y. J. Lin, K. Jimenez-Garcia, and I. B. Spielman, Nature\n471, 83-86 (2011).\n[2] X. J. Liu, M. F. Borunda, X. Liu, and J. Sinova, Phys.\nRev. Lett. 102, 046402 (2009).\n[3] M. Chapman and C. S´ a de Melo, Nature 471, 41-42\n(2011).\n[4] C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 146802\n(2005).\n[5] L. P. Gor’kov and E. I. Rashba, Phys. Rev. Lett. 87,\n037004 (2001).\n[6] T. D. Stanescu, C. Zhang, and V. Galitski, Phys. Rev.\nLett.99110403 (2007).\n[7] G. Dresselhaus, Phys. Rev. 100, 580 (1955).\n[8] Y. A. Bychkov and E. I. Rashba, J. Phys. C 17, 6039\n(1984).\n[9] J. Dalibard, F. Gerbier, G. Juzeliunas, and P. Ohberg,\narXiv:1008.5378 (2010).\n[10] J. P. Vyasanakere, S. Zhang, and V. B. Shenoy,\narXiv:1104.5633 (2011).\n[11] C. Zhang, S. Tewari, R. M. Lutchyn, and S. Das Sarma,\nPhys. Rev. Lett. 101160401 (2008).\n[12] Z.-Q. Yu and H. Zhai, arXiv:1105.2250 (2011).\n[13] H. Hu, L. Jiang, X. Jiu, and H. Pu, arXiv:1105.2488\n(2011).\n[14] L. Han and C. A. R. S´ a de Melo, arXiv:1106.3613 (2011).\n[15] G. E. Volovik, World Scientific, Singapore (1992).\n[16] R. D. Duncan and C. A. R. S´ a de Melo, Phys. Rev. B\n62, 9675 (2000).\n[17] S. S. Botelho and C. A. R. S´ a de Melo, Phys. Rev. B 71,\n134507 (2005).\n[18] M. Gong, S. Tewari, and C. Zhang, arXiv:1105.1796\n(2011).\n[19] M. Iskin and A. L. Subasi, arXiv:1106.0473 (2011).\n[20] M. Nakahara, Adam Hilger, Bristol (1990).\nI. ARTIFICIAL SPIN-ORBIT COUPLING IN\nULTRA-COLD FERMI SUPERFLUIDS:\n(SUPPLEMENTARY MATERIAL)\nThe method used to study the spin-orbit and Zeeman\neffects in ultra-cold Fermi superfluids is the functional\nintegral method and its saddle-point approximation in\nconjunction with fluctuation effects. To describe the\nthermodynamic phases and the corresponding phase dia-\ngram in terms of the interactions, Zeeman and spin-orbit\nfields, we calculate partition function at temperature TZ=/integraltext\nD[ψ,ψ†]exp/parenleftbig\n−S[ψ,ψ†]/parenrightbig\nwith action\nS[ψ,ψ†] =/integraldisplay\ndτdr/bracketleftigg/summationdisplay\nαψ†\nα(r,τ)∂\n∂τψα(r,τ)+H(r,τ)/bracketrightigg\n,\nwhere the Hamiltonian density is given in Eq. (1).\nUsing the standard Hubbard-Stratanovich transfor-\nmation that introduces the pairing field ∆( r,τ) =\ng/an}b∇acketle{tψ↓(r,τ)ψ↑(r,τ)/an}b∇acket∇i}htand integrating over the fermion vari-\nables lead to the effective action\nSeff=/integraldisplay\ndτdr/bracketleftbigg|∆(r,τ)|2\ng−T\n2VlndetM\nT+/tildewideK+δ(r−r′)/bracketrightbigg\n,\nwhere/tildewideK+= (/tildewideK↑+/tildewideK↓)/2.The matrix Mis\nM=\n∂τ+/tildewideK↑−h⊥0−∆\n−h∗\n⊥∂τ+/tildewideK↓∆ 0\n0 ∆†∂τ−/tildewideK↑h∗\n⊥\n−∆†0h⊥∂τ−/tildewideK↓\n,(7)\nwhereh⊥=hx−ihycorresponds to the transverse com-\nponent of the spin-orbit field, hzto the parallel com-\nponent with respect to the quantization axis z,/tildewideK↑=\nˆK↑−hz, and/tildewideK↓=ˆK↓+hz.\nTo make progress, we use the saddle point approxi-\nmation ∆( r,τ) = ∆ 0+η(r,τ),and write M=Msp+\nMf. The matrix Mspis obtained via the saddle point\n∆(r,τ)→∆0which takes M→Msp, and the fluctua-\ntion matrix Mf=M−Mspdepends only on η(r,τ) and\nits Hermitian conjugate. Thus, we write the effective ac-\ntion asSeff=Ssp+Sf. The first term is\nSsp=V\nT|∆0|2\ng−1\n2/summationdisplay\nk,iωn,jln/bracketleftbigg−iωn+Ej(k)\nT/bracketrightbigg\n+/summationdisplay\nk/tildewideK+\nT,\nin momentum-frequency coordinates ( k,iωn), where\nωn= (2n+1)πT. Here,Ej(k) are the eigenvalues of\nHsp=\n/tildewideK↑(k)−h⊥(k) 0 −∆0\n−h∗\n⊥(k)/tildewideK↓(k) ∆ 0 0\n0 ∆†\n0−/tildewideK↑(−k)h∗\n⊥(−k)\n−∆†\n00h⊥(−k)−/tildewideK↓(−k)\n,\n(8)\nwhich describes the Hamiltonian of elemen-\ntary excitations in the four-dimensional basis7\nΨ†=/braceleftig\nψ†\n↑(k),ψ†\n↓(k),ψ↑(−k),ψ↓(−k)/bracerightig\n.The fluctu-\nation action is\nSf=/integraldisplay\ndτdr/bracketleftbigg|η(r,τ)|2\ng−T\n2Vlndet/parenleftbig\n1+M−1\nspMf/parenrightbig/bracketrightbigg\n.\nThe spin-orbit field is h⊥(k) =hR(k) +hD(k),\nwherehR(k) =vR(−kyˆx+kxˆy) is of Rashba-type and\nhD(k) =vD(kyˆx+kxˆy) is of Dresselhaus-type, has\nmagnitude |h⊥(k)|=/radicalig\n(vD−vR)2k2y+(vD+vR)2k2x.\nFor Rashba-only (RO) ( vD= 0) and for equal Rashba-\nDresselhaus(ERD) couplings( vR=vD=v/2), the mag-\nnitude of the transverse fields are |h⊥(k)|=vR/radicalig\nk2x+k2y\n(vR>0) andh⊥(k) =v|kx|(v>0), respectively.\nThe Hamiltonian in the helicity basis Φ = UΨ, where\nUis the unitary matrix that diagonalizes the Hamilto-\nnian in the normal state, is\n/tildewideHsp(k) =\nξ⇑(k) 0 ∆ ⇑⇑(k) ∆⇑⇓(k)\n0ξ⇓(k) ∆⇓⇑(k) ∆⇓⇓(k)\n∆∗\n⇑⇑(k) ∆∗\n⇑⇓(k)−ξ⇑(k) 0\n∆∗\n⇑⇓(k) ∆∗\n⇓⇓(k) 0 −ξ⇓(k)\n.\nThe components of the order parameter in the helic-\nity basis are given by ∆ ⇑⇑(k) = ∆ T(k)e−iϕk,and\n∆⇓⇓(k) =−∆T(k)eiϕkfor the triplet channel and by\n∆⇑⇓(k) =−∆S(k) and ∆ ⇓⇑(k) = ∆ S(k) for the sin-\nglet channel. The eigenvalues of Hsp(k) for quasiparti-\nclesE1(k),E2(k) are listed in Eqs. (4) and (5), while\nthe eigenvalues for quasiholes are E3(k) =−E2(k), and\nE4(k) =−E1(k).\nThe thermodynamic potential is Ω = Ω sp+Ωf, where\nΩsp=V|∆0|2\ng−T\n2/summationdisplay\nk,jln{1+exp[−Ej(k)/T]}+/summationdisplay\nk¯K+,\nwith¯K+=/bracketleftig\n/tildewideK↑(−k)+/tildewideK↓(−k)/bracketrightig\n/2 is the saddle\npoint contribution and Ω f=−TlnZf, withZf=/integraltext\nD[¯η,η]exp[−Sf(¯η,η)] is the fluctuation contribution.\nThe order parameter is determined via the minimization\nof Ωspwith respect to |∆0|2, leading to\nV\ng=−1\n2/summationdisplay\nk,jnF[Ej(k)]∂Ej(k)\n∂|∆0|2, (9)\nwherenF[Ej(k)] = 1/(exp[Ej(k)/T] + 1) is the Fermi\nfunction for energy Ej(k). The contact interaction gis\nexpressed in terms of the scattering parameter asvia the\nLippman-Schwinger relation discussed in the main text.\nThe total number of particles N+=N↑+N↓is defined\nfromthethermodynamicrelation N+=−(∂Ω/∂µ+)T,V,\nand can be written as\nN+=Nsp+Nf. (10)\nThe saddle point contribution is\nNsp=−/parenleftbigg∂Ωsp\n∂µ+/parenrightbigg\nT,V=1\n2/summationdisplay\nk\n1−/summationdisplay\njnF[Ej(k)]∂Ej(k)\n∂µ+\n,and the fluctuation contribution is Nf=\n−(∂Ωf/∂µ+,)T,Vleading to\nNf=T\nZf/integraldisplay\nD[¯η,η]exp[−Sf(¯η,η)]/parenleftbigg\n−∂Sf(¯η,η)\n∂µ+/parenrightbigg\n,\nwith the partial derivative being\n∂SF(¯η,η)\n∂µ+=−T\n2VTr/bracketleftbigg/parenleftbig\n1+M−1\nspMf/parenrightbig−1∂\n∂µ+/parenleftbig\nM−1\nspMf/parenrightbig/bracketrightbigg\n.\nKnowledge of the thermodynamic potential Ω, of the\norder parameter Eq. (9) and number Eq. (10) provides\na complete theory for spectroscopic and thermodynamic\nproperties of attractive ultra-cold fermions in the pres-\nence of Zeeman and spin-orbit fields. Representative\nSaddle point solutions for chemical potential µ+and or-\nder parameter amplitude |∆0|as a function of 1 /(kF+as)\nin the equal Rashba-Dresselhaus (ERD) case ( v/vF+=\n0.28) are shown in Fig. 4 for hz/ǫF+= 0,0.5,1.0,2.0.\nThese parameters are used to obtain the phase diagrams\ndescribed in Figs. 1 and 2 in combination with an anal-\nysis of the excitation spectrum Ej(k) given in Eqs. (4)\nand (5) and the thermodynamic stability conditions for\nall the uniform superfluid phases: directly or indirectly\ngapped superfluid with zero nodal rings (US-0); gapless\nsuperfluid with one ring of nodes (US-1); and gapless\nsuperfluid with two rings of nodes (US-2).\n!!!\"#\"!$%#\"!$\n!!!\"#\"!$%!%!!#!\n1/k F+ as 1/k F+ as!+ / \"F+ \n|#0| / \"F+ b a\nFIG. 4: a) Chemical potential µ+/ǫF+and b) order pa-\nrameter amplitude |∆0|/ǫF+versus interaction parameter\n1/(kF+as) for spin-orbit parameter v/vF+= 0.28 and val-\nues of the Zeeman field hz/ǫF+= 0 (solid); hz/ǫF+= 0.5\n(dashed); hz/ǫF+= 1.0 (dotted); and hz/ǫF+= 2.0 (dot-\ndashed).\nA thermodynamic stability analysis of all proposed\nphases can be performed by investigating the maximum\nentropy condition. The total change in entropy due to\nthermodynamic fluctuations, irrespective to any approx-\nimations imposed on the microscopic Hamiltonian, can\nbe written as\n∆Stot=−1\n2T(∆T∆S−∆P∆V+∆µα∆Nα),\nwhere the repeated αindex indicates summation, and\nthe condition ∆ Stot≤0 guarantees that the entropy is\nmaximum. Considering the entropy Sto be a function\nof temperature T, number of particles Nαand volume8\nV, we can elliminate the fluctuations ∆ S, ∆P, and ∆µα\nin favor of fluctuations ∆ T, ∆Vand ∆Nα, and show\nthat the fluctuations ∆ Tare statistically independent of\n∆Nαand ∆V, while fluctuations ∆ Nαand ∆Vare not.\nThe first condition for thermodynamic stability leads\nto the requirement that the isovolumetric heat capacity\nCV=T(∂S/∂T)V,{Nα}≥0.Additional conditions are\ndirectly related to number ∆ Nαand volume ∆ Vfluctu-\nations. They require the chemical susceptibility matrix\nξαβ= (∂µα/∂Nβ)T,Vto be positive definite, i.e, that\nits eigenvalues are both positive. This is guaranteed by\ndet[ξ] =ξ↑↑ξ↓↓−ξ↑↓ξ↓↑>0 andξ↑↑>0. The last con-\ndition for thermodynamic stability is that the bulk mod-\nulusB= 1/κTor the isothermal compressibility κT=\n−V−1(∂V/∂P)T,{Nα},are positive. Since the number\n∆Nαand volume ∆ Vfluctuations are not statistically\nindependent, the bulk modulus is related to the matrix\n[ξ] viaV/κT=N2\n↑ξ↑↑+N↑N↓ξ↑↓+N↓N↑ξ↓↑+N2\n↓ξ↓↓.\nThe positivity of the volumetric specific heat CV, chemi-\ncal susceptibility matrix [ ξ] and bulk modulus B= 1/κT\nare the necessary and sufficient conditions for thermody-\nnamic stability, which must be satisfied irrespective of\napproximations used at the microscopic level.\n!!!\"#\"!$%#%&\"! \"' \n!!!\"#\"!$%#%&\"! \"' !!!\"#\"!$%#%&\"! \"' \n!!!\"#\"!$%#%&\"! \"' !T \"F+ \n!T \"F+ !T \"F+ \n!T \"F+ 1/k F+ as 1/k F+ as\n1/k F+ as 1/k F+ asd cb a\n0.5 11.5 025 50 \n!!\"# $!#! !#! !!\"# !!\"$# !\"% !#! !#! \nFIG. 5: Isothermal compressibility ¯ κT= (N2\n+)V−1κT=\n(∂N+/∂µ+)T,Vin units of 3 N+/(4ǫF+) versus interaction\n1/(kF+as) at spin-orbit coupling v/vF+= 0.28 for the val-\nues of the Zeeman field a) hz/ǫF+= 0; b) hz/ǫF+= 0.5;\nc)hz/ǫF+= 1.0; and d) hz/ǫF+= 2.0. Insets show regions\nwhere the compressibility is large.\nFurther characterization of phases US-0, US-1 and US-2 is made via thermodynamic prop-\nerties such as the isothermal compressibility\nκT= (V/N2\n+)(∂N+/∂µ+)T,V,which is shown in\nFig. 5 versus 1 /(kF+as) for the values of the Zee-\nman field hz/ǫF+= 0,0.5,1.0,2.0 and spin-orbit\ncouplingv/vF+= 0.28. Notice the negative re-\ngions ofκTindicating that the uniform superfluid\nphases are unstable, and its discontinuities at phase\nboundaries. The normal state compressibility κTor\n¯κT= (N2\n+)V−1κT= (∂N+/∂µ+)T,Vcan be obtained\nanalytically for arbitrary Zeeman hzand spin-orbit\nparametervin the BCS limit where 1 /kF+as→ −∞as\n¯κT=3N+\n4ǫF+/summationdisplay\nj=±/bracketleftbigg\nAj+/bracketleftbigg\n˜µ+−A2\nj+/radicalig\n˜h2z+2˜vA2\nj/bracketrightbigg∂Aj\n∂˜µ+/bracketrightbigg\n,\n(11)\nwhere the auxiliary function Ajis\nA±=/radicaligg\n(˜µ++ ˜v)±/radicalbigg\n(˜µ++ ˜v)2−/parenleftig\n˜µ2\n+−˜h2z/parenrightig\nand its derivative is\n∂A±\n∂˜µ+=/bracketleftbigg\n1±˜v//radicalig\n(˜µ++ ˜v)2−(˜µ2\n+−˜h2z)/bracketrightbigg\n/(2A±)\nwith ˜µ+=µ+/ǫF+,˜hz=hz/ǫF+, and ˜v=v/(2ǫF+).\nNotice that, as hz→0 and ˜v→0,A±→√˜µ+and\n¯κT→(3N+)/(2ǫF+) is reduced to the standard result,\nsince ˜µ+→1. In addition, κTor ¯κTcan be obtained\nanalytically in the BEC limit where 1 /kF+as→+∞.\nWhenhzandvare zero, then\n¯κT=3N+\n2ǫF+π\nkF+as(12)\ncan also be written in terms of bosonic properties\n1\nV/parenleftbigg∂N+\n∂µ+/parenrightbigg\nT,V=1\nπ/parenleftbiggmB\naB/parenrightbigg\n, (13)\nwheremB= 2mis the boson mass and aB= 2asin\nthe boson-boson interaction. In the case where hz/ne}ationslash= 0\nandv/ne}ationslash= 0, a similar expression can be derived for\nV−1(∂N+/∂µ+)T,Vbut the effective boson mass mB=\n2mf(hz,v), and the effective boson-boson interaction\naB= 2asg(hz,v) are now functions of hzandv. Notice\nthat the ratio mB/aBin the BEC limit can be directly\nextracted from the behavior of ¯ κTfor large 1/(kF+as)."    },    {        "title": "2201.06265v2.Spin_orbit_coupled_superconductivity_with_spin_singlet_non_unitary_pairing.pdf",        "content": "Spin-orbit-coupled superconductivity with spin-singlet non-unitary pairing\nMeng Zeng,1Dong-Hui Xu,2, 3Zi-Ming Wang,2, 3and Lun-Hui Hu4, 5,\u0003\n1Department of Physics, University of California, San Diego, California 92093, USA\n2Department of Physics and Chongqing Key Laboratory for Strongly Coupled Physics,\nChongqing University, Chongqing 400044, China\n3Center of Quantum Materials and Devices, Chongqing University, Chongqing 400044, China\n4Department of Physics, the Pennsylvania State University, University Park, PA, 16802, USA\n5Department of Physics and Astronomy, University of Tennessee, Knoxville, Tennessee 37996, USA\nThe gap functions for a single-band model for unconventional superconductivity are distinguished\nby their unitary or non-unitary forms. Here we generalize this classi\fcation to a two-band super-\nconductor with two nearly degenerate orbitals. We focus on spin-singlet pairings and investigate the\ne\u000bects of the atomic spin-orbit coupling (SOC) on superconductivity which is a driving force behind\nthe discovery of a new spin-orbit-coupled non-unitary superconductor. Multi-orbital e\u000bects like or-\nbital hybridization and strain induced anisotropy will also be considered. The spin-orbit-coupled\nnon-unitary superconductor has three main features. First, the atomic SOC locks the electron\nspins to be out-of-plane, leading to a new Type II Ising superconductor with a large in-plane upper\ncritical \feld beyond the conventional Pauli limit. Second, it provides a promising platform to re-\nalize the topological chiral or helical Majorana edge state even without external magnetic \felds or\nZeeman \felds. More surprisingly, a spin-polarized superconducting state could be generated by spin-\nsinglet non-unitary pairings when time-reversal symmetry is spontaneously broken, which serves as\na smoking gun to detect this exotic state by measuring the spin-resolved density of states. Our\nwork indicates the essential roles of orbital-triplet pairings in both unconventional and topological\nsuperconductivity.\nI. INTRODUCTION\nIn condensed matter physics, research on unconven-\ntional superconductivity [1, 2] remains a crucial topic\nand continues to uncover new questions and challenges\nin both theory and experiment, since the discovery of the\nheavy-fermion superconductors (SCs) [3] and the d-wave\npairing states in high-temperature cuprate SCs [4{7]. In\naddition to the anisotropic gap functions (e.g., p;d;f;g -\nwave...), the sublattice or orbital-dependent pairings [8{\n10] are shown to be an alternative avenue to real-\nize unconventional SCs. They might be realized in\nmulti-orbital correlated electronic systems, whose candi-\ndate materials include iron-based SCs [11{22], Cu-doped\nBi2Se3[23, 24], half-Heusler compounds [25{34], and pos-\nsibly Sr 2RuO 4[35{40] etc. In particular, considering\nthe atomic orbital degrees of freedom, the classi\fcation\nof unconventional pairing states could be signi\fcantly\nenriched. Among them, SCs with spontaneous time-\nreversal symmetry (TRS) breaking is of special interest,\nin which two mutually exclusive quantum phenomena,\nspin magnetism, and superconductivity may coexist with\neach other peacefully[41{46].\nOn the other hand, the orbital multiplicity could also\ngive rise to non-unitary pairings, which again include\nboth time-reversal breaking (TRB) and time-reversal\ninvariant (TRI) pairings. Very recently, prior studies\nhave demonstrated the existence of spin-singlet non-\nunitary pairing states that break the inversion symme-\ntry in Dirac materials [9]. One aim of this work is the\n\u0003hu.lunhui.zju@gmail.comgeneralization of unitary and non-unitary gap functions\nin a two-band SC while preserving inversion symme-\ntry, which is possible exactly due to the multi-orbital\ndegrees of freedom [47]. We focus on a system with\ntwo nearly degenerate orbitals and \fnd that the non-\nunitary pairing state is generally a mixed superconduct-\ning state with both orbital-independent pairings and\norbital-dependent pairings. Recently, the interplay be-\ntween orbital-independent pairings and spin-orbit cou-\npling (SOC) has been shown to demonstrate the intrigu-\ning phenomenon of a large in-plane upper critical \feld\ncompared with the Pauli paramagnetic \feld for a two-\ndimensional SC. For example, the Type I Ising super-\nconductivity in monolayer MoS 2[48, 49] and NbSe 2[50]\nand the Type II Ising superconductivity in monolayer\nstanene [51]. Therefore, the interplay of atomic SOC\nand the multi-orbital pairing could potentially give rise\nto exciting physics. However, to the best of our knowl-\nedge, the in\ruence of the atomic SOC on the orbital-\ndependent pairings remains unsolved. Furthermore, the\nmulti-orbital nature also gives rise to possible orbital hy-\nbridization e\u000bects and provides an experimentally con-\ntrollable handle using lattice strains, both of which could\nlead to orbital anisotropy and could potentially change\nthe pairing symmetry. In particular, lattice strain has\nbeen a useful experimental tool to study unconventional\nsuperconductors [52{54] and has even been proposed to\ninduce the elusive charge-4e phase [55]. We will be do-\ning an extensive investigation on all the aforementioned\nmulti-orbital e\u000bects.\nAnother topic of this work is concerned with the co-\nexistence of TRB pairings and spin magnetism even in\na spin-singlet SC. It is well-known that spin-polarizationarXiv:2201.06265v2  [cond-mat.supr-con]  1 Mar 20232\n(SP) can be generated by nonunitary spin-triplet super-\nconductivity, which is believed to be the case for LaNiC 2\n[56] and LaNiGa 2[57, 58]. More recently, the coexis-\ntence of magnetism and spin-singlet superconductivity\nis experimentally suggested in multi-orbital SCs, such\nas iron-based superconductors [59, 60] and LaPt 3P [61].\nTherefore, in addition to the spin-triplet theory, it will be\ninteresting to examine how SP develops in multi-orbital\nspin-singlet SCs as spontaneous TRS breaking in the ab-\nsence of external magnetic \felds or Zeeman \felds.\nIn this work, we address the above two major issues by\nstudying a two-band SC with two atomic orbitals (e.g.,\ndxzanddyz). We start with the construction of a k\u0001p\nmodel Hamiltonian on a square lattice with applied lat-\ntice strain. The breaking of C4vdown toC2point group\ngenerally leads to the degeneracy lifting of dxzanddyz.\nBased on this model, we study the stability of supercon-\nductivity and the realization of 2D topological supercon-\nductors in both class D and DIII. First and foremost, the\nin\ruence of atomic SOC is studied, which gives birth to\na new spin-orbit-coupled SC. This exotic state shows the\nfollowing features: \frstly, a large Pauli-limit violation\nis found for the orbital-independent pairing part, which\nbelongs to the Type II Ising superconductivity. Further-\nmore, the orbital-dependent pairing part also shows a\nweak Pauli-limit violation even though it does not belong\nto the family of Ising SCs. Secondly, topological super-\nconductivity can be realized with a physical set of pa-\nrameters even in the absence of external magnetic \felds\nor Zeeman \felds. In addition, a spin-polarized super-\nconducting state could be energetically favored with the\nspontaneous breaking of time-reversal symmetry. Our\nwork implies a new mechanism for the establishment of\nspin magnetism in the spin-singlet SC. In the end, we also\ndiscuss how to detect this e\u000bect by spin-resolved scanning\ntunneling microscopy measurements.\nThe paper is organized as follows: in section II, we\ndiscuss a two-orbital normal-state Hamiltonian on a 2D\nsquare lattice and also its variants caused by applied in-\nplane strain e\u000bects, then we show the spin-singlet unitary\nor non-unitary pairing states with or without TRS. The\nstrain e\u000bect on pairing symmetries is also studied based\non a weak-coupling theory. In section III, the e\u000bects of\natomic SOC on such pairing states are extensively stud-\nied, as well as the in-plane paramagnetic depairing e\u000bect.\nBesides, the topological superconductivity is studied in\nsection IV even in the absence of external magnetic \felds\nor Zeeman \felds, after which we consider the spontaneous\nTRB e\u000bects in section V and show that spin-singlet SC-\ninduced spin magnetism could emerge in the presence of\norbital SOC. In the end, a brief discussion and conclu-\nsion are given in section VI. We will also brie\ry comment\non a very recent experiment [62], demonstrating that a\nfully gapped superconductor becomes a nodal phase by\nsubstituting S into single-layer FeSe/SrTiO 3.II. MODEL HAMILTONIAN\nIn this section, we \frst discuss the normal-state Hamil-\ntonian that will be used throughout this work for an\nelectronic system consisting of both spin and two locally\ndegenerate atomic orbitals (e.g., dxzanddyz) on a 2D\nsquare lattice. We assume each unit cell contains only one\natom, so there is no sublattice degree of freedom. The\norbital degeneracy can be reduced by applying the in-\nplane lattice strain because the original C4vpoint group\nis reduced down to its subgroup C2vfor strain\u001b10;\u001b01or\n\u001b11(A more generic strain would reduce the symmetry\ndirectly to C2). Here\u001bn1n2represents the strain tensor\nwhose form will be given later. We will apply the sym-\nmetry analysis to construct the strained Hamiltonian in\nthe spirit of k\u0001ptheory. Then, we discuss the pairing\nHamiltonian and the corresponding classi\fcation of spin-\nsinglet pairing symmetries including non-unitary pairing\nstates. The strain e\u000bect is also investigated on the super-\nconducting pairing symmetries based on a weak-coupling\nscheme [10].\nA. Normal-state Hamiltonian\nIn this subsection, we construct the two-orbital\nnormal-state Hamiltonian H0(k) with lattice strain-\ninduced symmetry-breaking terms. Before that, We \frst\nshowH0(k) in the absence of external lattice strains.\nFor a square lattice as illustrated in Fig. 1 (a), it owns\ntheC4vpoint group that is generated by two symme-\ntry operators: a fourfold rotation symmetry around the\n^z-axisC4z: (x;y)!(y;\u0000x) and a mirror re\rection\nabout the ^y\u0000^zplaneMx: (x;y)!(\u0000x;y). Other\nsymmetries can be generated by multiplications, such\nas the mirror re\rection about the (^ x+ ^y)\u0000^zplane\nMx+y: (x;y)!(y;x) is given by C4z\u0002Mx. In the\nabsence of Rashba spin-orbit coupling (SOC), the sys-\ntem also harbors inversion symmetry I, enlarging the\nsymmetry group to D4h=C4v\nfE;Ig. In the spirit of\nk\u0001pexpansion around the \u0000 point or the Mpoint, we\nconsider a two-orbital system described by the inversion-\nsymmetric Hamiltonian in two dimensions (2D),\nH0(k) =\u000f(k)\u001c0\u001b0+\u0015soc\u001c2\u001b3+\u0015o[go(k)\u0001\u001c]\u001b0;(1)\nwhere the basis is made of fdxz;dyzg-orbitals y\nk=\n(cy\ndxz;\"(k);cy\ndxz;#(k);cy\ndyz;\"(k);cy\ndyz;#(k)). Herecyis the\ncreation operator of electrons, \u001cand\u001bare Pauli ma-\ntrices acting on the orbital and spin subspace, respec-\ntively, and \u001c0,\u001b0are 2-by-2 identity matrices. Besides,\n\u000f(k) =\u0000(k2\nx+k2\ny)=2m\u0000\u0016is the band energy measured\nrelative to the chemical potential \u0016,mis the e\u000bective\nmass,\u0015socis the atomic SOC [63{65] and \u0015ocharacter-\nizes the strength of orbital hybridization. This model\ncould describe the two hole pockets of iron-based super-\nconductors [66, 67]. Moreover, the \frst two components\nofgo(k) are for the inter-orbital hopping term, while the3\n𝑀𝑥\n (a)\n𝐶4𝑧 𝑥𝑦\n𝑑𝑥𝑧orbital\n++−\n−𝑥𝑧𝑑𝑦𝑧orbital\n++−\n−𝑦𝑧𝑪𝟒𝒗𝑪𝟐𝒗\n𝑪𝟐𝒗\n𝑪𝟐𝑪𝟐𝒗𝑴𝒙𝑴𝒚 Basis Orbitals\n𝐴1++𝑥2,𝑦2\n𝐴2−−𝑥𝑦\n𝐵1−+𝑥𝑧𝑑𝑥𝑧\n𝐵2+−𝑦𝑧𝑑𝑦𝑧Strain𝜎10(b)\n𝑀𝑥𝑀𝑦\n(c)\nStrain𝜎11\n(d)𝑪𝟐𝒗𝑴𝒙′𝑴𝒚′ Basis Orbitals\n𝐴1++𝑥′2,𝑦′2\n𝐴2−−𝑥′𝑦′\n𝐵1−+𝑥′𝑧𝑑𝑥′𝑧\n𝐵2+−𝑦′𝑧𝑑𝑦′𝑧\nStrain𝜎1𝛿𝐶2𝑧\n𝐶2𝑧\n𝐶2𝑧𝑪𝟐𝑪𝟐𝒛 Basis Orbitals\n𝐴1+𝑥2,𝑦2,𝑥𝑦\n𝐵1−𝑦𝑧,𝑥𝑧𝑑𝑥𝑧,𝑑𝑦𝑧Strain𝑥𝑦\n𝑥𝑦\nFIG. 1. The strain e\u000bect on a two-dimensional square lattice. In the absence of lattice strain, (a) shows the square lattice\nowing theC4vpoint group that is generated by C4zandMx. We consider the normal-state Hamiltonian with dxz;dyz-orbitals.\nInversion symmetry ( I) is broken by growing crystal samples on an insulating substrate. The in-plane strain e\u000bects on the\nsquare lattice are illustrated in (b-d) for applied strain along di\u000berent directions. (b) shows that the ^ xor ^y-axis strain breaks\nthe square lattice into the rectangular lattice with two independent mirror re\rection symmetries MxandMy, obeying the\nsubgroupC2vofC4v. TheC2vpoint group contains four one-dimensional irreducible representations (irrep.) A1;A2;B1;B2.\n(c) shows that the strain along the ^ x+ ^y-direction also reduces the C4vdown toC2v. (d) represents a general case, where the\nsubgroupC2is preserved that only has A1andB1irreps.\nthird term is for the anisotropic e\u000bective mass, explained\nbelow in detail.\nTheC4v(orD4h) point group restricts go(k) =\n(aokxky;0;k2\nx\u0000k2\ny), whereao= 2 is a symmetric case\nthat increases the C4zto a continues rotational sym-\nmetry about the ^ z-axis. To be precise, the g1-term,\n2\u0015okxky\u001c1\u001b0, is attributed to the inter-orbital hopping\nintegral along the \u0006^x\u0006^ydirections,\n\u0015o\n2(cy\ndxz;\u001b(ix;iy)cdyz;\u001b(ix+ 1;iy+ 1)\n+cy\ndxz;\u001b(ix;iy)cdyz;\u001b(ix\u00001;iy\u00001)\n\u0000cy\ndxz;\u001b(ix;iy)cdyz;\u001b(ix+ 1;iy\u00001)\n\u0000cy\ndxz;\u001b(ix;iy)cdyz;\u001b(ix\u00001;iy+ 1) + h.c.) ;(2)\nwhere (ix;iy) represents the lattice site. In addition, the\ng3-term,\u0015o(k2\nx\u0000k2\ny)\u001c3\u001b0, causes the anisotropic e\u000bective\nmasses. For example, the e\u000bective mass of the dxzorbital\nis1\n1=m\u00002\u0015oalong the ^x-axis while that is1\n1=m+2\u0015oalong\nthe ^y-axis. This means that the hopping integrals aredi\u000berent along ^ xand ^ydirections,\n(1\n2m\u0000\u0015o)cy\ndxz;\u001b(ix;iy)cdxz;\u001b(ix+ 1;iy)\n+(1\n2m+\u0015o)cy\ndxz;\u001b(ix;iy)cdxz;\u001b(ix;iy+ 1)\n+(1\n2m+\u0015o)cy\ndyz;\u001b(ix;iy)cdyz;\u001b(ix+ 1;iy)\n+(1\n2m\u0000\u0015o)cy\ndyz;\u001b(ix;iy)cdyz;\u001b(ix;iy+ 1) + h.c.:(3)\nIn this work, we focus on a negative e\u000bective mass case\nby choosing 1 =m\u00062\u0015o>0. However, using a posi-\ntive e\u000bective mass does not change our main conclusion.\nMoreover, our results can be generally applied to other\nsystems with two orbitals px;py, once it satis\fes the C4v\npoint group.\nThe time-reversal symmetry operator is presented as\nT=i\u001c0\u001b2KwithKbeing the complex conjugate. And\nthe inversion symmetry is presented as I=\u001c0\u001b0. It is\neasy to show Eq. (1) is invariant under both TandI.\nHowever, inversion can be broken by growing the sample\non insulating substrates, the asymmetric Rashba SOC is\ndescribed by\nHR(k) =\u0015R\u001c0[gR(k)\u0001\u001b]; (4)4\nwhere\u0015Ris the strength of the Rashba SOC with\ngR(k) = (\u0000ky;kx;0) as required by the C4vpoint group.\nNext, we consider the lattice strain e\u000bect on the two-\ndimensional crystal with a square lattice, as summarized\nin Fig. 1 (b-d). The in-plane strain e\u000bect is characterized\nby the 2-by-2 strain tensor \u001bwhose elements are de\fned\nas\u001bij=1\n2\u0000\n@xiuj+@xjui\u0001\n, whereuiis the displacement\natralong the ^eidirection. Even though it is an abuse of\nnotation, it should be self-evident that the \u001bhere does\nnot represent the Pauli matrices. The strain tensor \u001bcan\nbe parametrized as the following\n\u001b\u001e=\u0012cos2\u001ecos\u001esin\u001e\ncos\u001esin\u001e sin2\u001e\u0013\n; (5)\nwhere\u001eis the polar angle with respect to the ^ x-axis. For\nthe\u001e= 0 (\u0019=2) case, the compressive or tensile strain\napplied along the ^ x-axis (^y-axis) makes the square lattice\nas a rectangular lattice, as illustrated in Fig. 1 (b). And\nthe\u001e=\u0019=4 case is for the shear strain along the (^ x+ ^y)-\ndirection in Fig. 1 (c). All the above cases reduce the\nC4vpoint group into its subgroup C2vthat is generated\nby two independent mirror re\rections. Otherwise, it is\ngenerally reduced to C2. The irreducible representations\nforC2vandC2are shown in Fig. 1 (b-d). Based on the\nstandard symmetry analysis, to the leading order, the\nstrained Hamiltonian is given by\nHstr=tstr[sin(2\u001e)\u001c1+ cos(2\u001e)\u001c3]\u001b0; (6)\nwhere both tstrand\u001ecan be controlled in experi-\nments [68]. AndHstrcan be absorbed into the go-vector\nin Eq. (1), renormalizing the orbital hybridization as ex-\npected. Furthermore, one can check that Hstrpreserves\nbothTandI, but explicitly breaks the C4z=i\u001c2ei\u0019\n4\u001b3\nbecause of [Hstr;C4z]6= 0. Interestingly, the orbital tex-\nture on the Fermi surface can be engineered by strain,\nand its e\u000bect on superconducting pairing symmetries is\nbrie\ry discussed in the Appendix C.\nTherefore, a strained normal-state Hamiltonian is\nHN(k) =H0(k) +HR(k) +Hstr; (7)\nwhich will be used throughout this work. The Rashba\nSOC induced spin-splitting bands are considered only\nwhen we discuss the topological superconducting phases\nin section IV and V, even though the normal-state Hamil-\ntonianHN(k) is topologically trivial. For the supercon-\nducting states, we focus on the inversion symmetric pair-\nings (i.e., spin-singlet s-wave pairing) and their response\nto applied strains or in-plane magnetic \felds.\nIn the absence of Rashba SOC, the band structures of\nHN(k) in Eq. (7) are given by\nE\u0006(k) =\u00001\n2m(k2\nx+k2\ny)\u0006q\n\u00152soc+ ~g2\n1+ ~g2\n3;(8)\nwhere we de\fne the strained orbital hybridization ~gvec-\ntor with ~g1=ao\u0015okxky+tstrsin(2\u001e) and ~g3=\u0015o(k2\nx\u0000\nk2\ny)+tstrcos(2\u001e). Each band has two-fold degeneracy, en-\nforced by the presence of both TandI. At the \u0000 point\n𝑘𝑥𝑘𝑦\n𝑘𝑥𝑘𝑦(a) (b)FIG. 2. The lattice strain e\u000bect on the Fermi surfaces of the\nnormal-state Hamiltonian without Rashba SOC. (a) shows\nthe two Fermi surfaces without lattice strain (i.e., tstr= 0),\nthusC4z-symmetric energy contours are formed. (b) shows\nthe breaking of C4zby lattice strain with tstr= 0:4 and\n\u001e= 0, onlyC2z-symmetric energy contours appear. Other\nparameters used here are m= 0:5;a0= 1;\u0015o= 0:4,\u0015R= 0\nand\u0016=\u00000:5.\n(kx=ky= 0),E\u0000\n\u0006=\u0006p\n\u00152soc+t2\nstr. The two Fermi sur-\nfaces with and without strain are numerically calculated\nand shown in Fig. 2, where we choose \u0016<\u0000p\n\u00152soc+t2\nstr.\nThese are two hole pockets because of the negative e\u000bec-\ntive mass of both orbitals. The Fermi surfaces in Fig. 2\n(a) areC4-symmetric ( tstr= 0), while those in Fig. 2\n(b) are only C2-symmetric due to the symmetry break-\ning of lattice strains. Please note that there is only one\nFermi surface when j\u0016j<p\n\u00152soc+t2\nstr, which is a neces-\nsary condition to realize topological superconductors as\nwe will discuss in Sec. IV.\nB. Review of singlet-triplet mixed pairings\nBefore discussing the possible unconventional pairing\nsymmetry forHNin Eq. (7), we brie\ry review both uni-\ntary and non-unitary gap functions for a single-band SC\nin the absence of inversion symmetry. In this case, a\nsinglet-triplet mixed pairing potential is given by\n\u0001(k) = [\u0001s s(k)\u001b0+ \u0001t(ds(k)\u0001\u001b)] (i\u001b2);(9)\nwhere \u001bare Pauli matrices acting in the spin subspace.\nHere s(k) represents even-parity spin-singlet pairings\nand the odd-parity ds(k) is for the spin-triplet ds-vector.\nPhysically, the unitary SC has only one superconducting\ngap like in the conventional BCS theory, while a two-\ngap feature comes into being by the non-unitary pairing\npotential. More explicitly, the unitary or non-unitary is\nde\fned by whether the following is proportional to the\nidentity matrix \u001b0:\n\u0001(k)\u0001y(k) =j\u0001sj2 2\ns+j\u0001tj2jdsj2\n+ 2Re[\u0001s\u0001\u0003\nt sd\u0003\ns]\u0001\u001b+ij\u0001tj2(ds\u0002d\u0003\ns)\u0001\u001b;(10)\nTherefore, Eq. (10) gives rise to a possible classi\fcation\nby assuming a non-vanishing \u0001 s2Rand a proper choice5\nPairings TRS Unitary Δ𝑠 Δ𝑜 𝒅𝑜\nTRI unitary Yes YesReal\nZeroZero\nRealReal\nTRI non -unitary Yes No Real Real Real\nTRB unitary No Yes RealPurely \nimaginaryReal\nTRB non -unitary No No Real Real Complex\nTABLE I. The four pairing states classi\fed by time-reversal\nsymmetry and unitary for a spin-singlet superconductor with\nboth orbital-independent pairing \u0001 sand orbital-dependent\npairing \u0001 oanddo.\nof a global phase. In principle, there are four possible\nphases, including the TRI non-unitary SCs (\u0001 t2R;ds2\nR), the TRB unitary SCs (\u0001 t\u0018i;ds2R), and the TRB\nnon-unitary SCs (\u0001 t2R;ds2C). On the other hand,\nthe TRI unitary SCs are achieved only with \u0001 s= 0 or\n\u0001t= 0 and real ds, meaning a purely spin-singlet SC\nor a purely spin-triplet SC. These states might be distin-\nguished in experiments, for example, the TRB unitary\npairing state might induce a spontaneous magnetization\nwith the help of Rashba spin-orbit coupling [69], which\ncan be detected by \u0016SR [70].\nAs we know, the spin-singlet pairings do not coexist\nwith the spin-triplet pairings in the presence of inversion\nsymmetry (e.g. centrosymmetric SCs). Roughly speak-\ning, it seems out of the question to realize non-unitary\npairing states in purely spin-singlet SCs. However, this\nis a challenge but not an impossibility for an SC with\nmulti-orbitals, which is one of the aims of this work. In\nthe following, we will discuss how to generalize the classi-\n\fcation of TRI or TRB and unitary or non-unitary pair-\ning states to a spin-singlet SC with two atomic orbitals\nin the presence of inversion symmetry. The four cases\nare summarized in Table I.\nC. Pairing Hamiltonian of a two-orbital model\nWe now consider the pairing Hamiltonian for HNin\nEq. (7). By ignoring the \ructuations, the mean-\feld pair-\ning Hamiltonian is generally given by,\nH\u0001=X\nk;s1a;s2b\u0001a;b\ns1;s2(k)cy\nas1(k)cy\nbs2(\u0000k) + h.c.;(11)\nwheres1;s2are index for spin and a;bare for orbitals. As\nstudied in Ref. [10], the orbital-triplet pairing is robust\neven against orbital hybridization and electron-electron\ninteractions. Thus, we consider both orbital-independent\nand orbital-dependent pairings for generality. In analogy\nto spin-triplet SCs, we use an orbital do(k)-vector for\nthe spin-singlet orbital-dependent pairing potential [12],\nwhich takes the generic form\n\u0001tot(k) = [\u0001s\ts(k)\u001c0+ \u0001o(do(k)\u0001\u001c)] (i\u001b2);(12)where \u0001 sand \u0001oare pairing strengths in orbital-\nindependent and orbital-dependent channels, respec-\ntively. The Fermi statistics requires that \t s(k) =\n\ts(\u0000k), while the three components of dosatisfy\nd1;3\no(k) =d1;3\no(\u0000k) andd2\no(k) =\u0000d2\no(\u0000k). Namely,\nd2\no(k) represents odd-parity spin-singlet orbital-singlet\npairings and the others are for even-parity spin-singlet\norbital-triplet pairings. The pairing potential presented\nin this form is quite convenient, similar to the spin-triplet\ncase [10, 12, 14, 71]. The bene\fts of this form in Eq. (12)\nwill be shown when we discuss the mixture of orbital-\nindependent and orbital-dependent pairings. Combining\nEq. (12) with Eq. (1), the Bogoliubov-de-Gennes (BdG)\nHamiltonian is\nHBdG(k) =\u0012HN(k) \u0001tot(k)\n\u0001y\ntot(k)\u0000H\u0003\nN(\u0000k)\u0013\n; (13)\nwhich is based on the Nambu basis (  y\nk; T\n\u0000k). Same\nwith the spin case in Eq. (10), the non-unitarity of a spin-\nsinglet pairing potential de\fned in Eq. (12) is determined\nby whether \u0001 tot(k)\u0001y\ntot(k) is proportional to an identity\nmatrix. More explicitly we have\n\u0001tot(k)\u0001y\ntot(k) =j\u0001sj2 2\ns\u001c0\u001b0+j\u0001oj2jdoj2\u001c0\u001b0\n+ 2Rej\u0001s\u0001\u0003\no sd\u0003\noj\u0001\u001c\u001b0+ij\u0001oj2(do\u0002d\u0003\no)\u0001\u001c\u001b0;(14)\nwhich could also exhibit four general possibilities: time-\nreversal-invariant (TRI) or time-reversal-breaking (TRB)\nand unitary or non-unitary SCs, with a simple replace-\nmentf\u0001t;dsg!f \u0001o;dog. In the absence of band split-\ntings, i.e.,\u0015soc=\u0015o=\u0015R=tstr= 0 as an illustration,\nthe superconducting excitation gaps on the Fermi sur-\nfaces of a TRI unitary SC are\nE\u0014;\u0017(k) =\u0014p\n\u000f2(k) + (\u0001s s(k) +\u0017\u0001ojdoj)2;(15)\nwith\u0014;\u0017=\u0006. It is similar to the superconducting gaps\nfor non-unitary spin-triplet SCs [1]. Moreover, the two-\ngap feature indicates the non-unitarity of the supercon-\nducting states, which implies the possibility of a nodal\nSC as long as \u0001 s s(k)\u0006\u0001ojdoj= 0 is satis\fed on the\nFermi surfaces. And, the nodal quasi-particle states can\nbe experimentally detected by measuring speci\fc heat,\nLondon penetration depths, \u0016SR, NMR, etc. As a re-\nsult, this provides possible evidence to get a sight of TRI\nnon-unitary phases in real materials (e.g. centrosymmet-\nric SCs). Furthermore, the above conclusion is still valid\nwhen we turn on \u0015soc,\u0015o, andtstr.\nIII. THE PAULI LIMIT VIOLATION: A LARGE\nIN-PLANE UPPER CRITICAL FIELD\nIn this section, we study the Pauli limit violation for\nthe spin-singlet TRI non-unitary SC against an in-plane\nmagnetic \feld (e.g. Hc2;k> HP). For a 2D crystalline\nSC or a thin \flm SC, the realization of superconducting\nstates that are resilient to a strong external magnetic \feld6\nhas remained a signi\fcant pursuit, namely, the pairing\nmechanism can remarkably enlarge the in-plane upper\ncritical \feld. Along this crucial research direction, one\nrecent breakthrough has been the identi\fcation of \\Ising\npairing\" formed with the help of Ising-type spin-orbit\ncoupling (SOC), which breaks the SU(2) spin rotation\nand pins the electron spins to the out-of-plane direction.\nDepending on whether the inversion symmetry is broken\nor not by the Ising-type SOC, the Ising pairing is clas-\nsi\fed as Type I (broken) and Type II (preserved) Ising\nsuperconductivity, where the breaking of Cooper pairs is\ndi\u000ecult under an in-plane magnetic \feld.\nTo demonstrate the underlying physics, in the fol-\nlowing, we consider the interplay between atomic SOC\n\u0015soc6= 0 and spin-singlet TRI non-unitary pairing state.\nThus, we consider the pairing potential\n\u0001tot=\u0002\n\u0001s\u001c0+ \u0001o(d1\no\u001c1+d3\no\u001c3)\u0003\n(i\u001b2); (16)\nwhere \u0001s, \u0001o,d1\no, andd3\noare all real constant. This can\nbe realized once we have on-site attractive interactions in\nboth orbital channels. Another reason for studying the\natomic SOC is that it is not negligible in many real ma-\nterials. It is interesting to note that the strength of SOC\ncan be tuned in experiments, for example, by substituting\nS into single-layer FeSe/SrTiO 3[62] or growing a super-\nconductor/topological insulator heterostructure [72].\nWithout loss of generality, the direction of the mag-\nnetic \feld can be taken to be the x-direction, i.e., H=\n(Hx;0;0) withHx\u00150. Therefore, the normal Hamilto-\nnian becomes\nHN(k) +h\u001c0\u001b1; (17)\nwhere the \frst part is given by Eq. (7) and h=1\n2g\u0016BHx\nis the Zeeman energy with g= 2 the electron's g-factor.\nTo explicitly investigate the violation of the Pauli limit\nfor the spin-orbit coupled SCs, we calculate the in-plane\nupper critical magnetic \feld normalized to the Pauli-\nlimit paramagnetic \feld Hc2;k=HPas a function of the\nnormalized temperature Tc=T0, by solving the linearized\ngap equation. Here HP= 1:86T0represents the Pauli\nlimit withT0the critical temperature in the absence of\nan external magnetic \feld.\nFollowing the standard BCS decoupling scheme [10],\nwe \frst solve Tcfor the orbital-independent pairing chan-\nnel by solving the linearized gap equation, v0\u001fs(T)\u00001 =\n0, wherev0is e\u000bective attractive interaction and the su-\nperconductivity susceptibility \u001fs(T) is de\fned by\n\u001fs(T) =\u00001\n\fX\nk;!nTrh\nGe(k;i!n)Gh(\u0000k;i!n)i\n;(18)\nwhere the conventional s-wave pairing with  s(k) = 1\nis considered for Eq. (16). Here Ge(k;i!n) = [i!n\u0000\nH0(k)]\u00001is the Matsubara Green's function for elec-\ntrons and that for holes is de\fned as Gh(k;i!n) =\n\u0000\u001b2G\u0003\ne(k;i!n)\u001b2. Here\f= 1=kBTand!n= (2n+\n1)\u0019=\fwithninteger. Likewise, for the orbital-dependentpairing channels, the superconductivity susceptibility\n\u001fo(T) is de\fned as\n\u001fo(T) =\u00001\n\fX\nk;!nTrh\n(do(k)\u0001\u001c)yGe(k;i!n)\n\u0002(do(k)\u0001\u001c)Gh(\u0000k;i!n)i\n;(19)\nwhere the orbital-dependent pairing ( Agrepresentation)\nwith the vector-form as do= (d1\no;0;d3\no) for Eq. (16) is\nused for the Tccalculations. However, the momentum-\ndependent do-vector does not a\u000bect the formalism and\nmain results, as we will discuss in the appendix C.\nThe coupling between orbital-independent and orbital-\ndependent channels leads to a high-order correction ( \u0018\n\u00152k2\nF=\u00162, with\u0015being the coupling strength of the e\u000bec-\ntive~gin the Hamiltonian representing orbital hybridiza-\ntion and strain ), which can be ignored once \u0015\u001c\u0016=kF.\nA. Type II Ising superconductivity\nIn this subsection, we \frst consider the orbital-\nindependent pairing state (i.e., \u0001 s6= 0 and \u0001 o= 0) and\nshow it is a Type II Ising SC protected from the out-\nof-plane spin polarization by the atomic SOC \u0015soc\u001c2\u001b3.\nTo demonstrate that, one generally needs to investigate\nthe e\u000bects of atomic SOC on the SC Tcas a function of\nthe in-plane magnetic \feld hbased on Eq. (18), in the\npresence of both orbital hybridization \u0015oand straintstr.\nAs de\fned in Eq. (8), the e\u000bects of orbital hybridiza-\ntion and lattice strain on the system can be captured by\nan e\u000bective ~g\u0011(ao\u0015okxky+tstrsin(2\u001e);\u0015o(k2\nx\u0000k2\ny) +\ntstrcos(2\u001e)). The case with tstr= 0 has been studied in\nRef. [73], however, the strain e\u000bect on the type II Ising\nSC has not been explored yet. To reveal the pure role of\nlattice strains, we consider kFto be close to the \u0000-point\nso that the k-dependent hybridization part is dominated\nby the strain part for generic \u001e. Therefore, we focus on\n~g=tstr(sin 2\u001e;cos 2\u001e) in the following discussions.\nAfter a straightforward calculation (see details in Ap-\npendix B), the superconductivity susceptibility \u001fs(T) in\nEq. (18) is calculated as\n\u001fs(T) =\u001f0(T) +N0fs(T;\u0015soc;tstr;h); (20)\nwithN0is the DOS near the Fermi surface and the pair-\nbreaking term is given by\nfs(T;\u0015soc;tstr;h) =1\n2[C0(T;\u001a\u0000) +C0(T;\u001a+)]\n+ [C0(T;\u001a\u0000)\u0000C0(T;\u001a+)]\u0012\u00152\nsoc+t2\nstr\u0000h2\n2E+E\u0000\u0013\n;(21)\nwhereE\u0006\u0011p\n\u00152soc+ (tstr\u0006h)2,\u001a\u0006=1\n2(E+\u0006E\u0000), and\n\u001f0(T) =N0ln\u0010\n2e\r!D\n\u0019kBT\u0011\nis the superconducting suscep-\ntibility when \u0015soc;tstr;h= 0. Here \r= 0:57721\u0001\u0001\u0001is7\nλsoc=0,tstr=0λsoc=1.5,tstr=20.00.20.40.60.81.00.00.20.40.60.81.0λsoc=0,tstr=0λsoc=1.5,tstr=0λsoc=1.5,tstr=10.00.20.40.60.81.00.00.20.40.60.81.0\nTc/T0Tc/T0Tc/T0(a)(b)(c)λsoc/tstrhhtstr=0.10tstr=0.12tstr=0.160.00.51.01.52.02.50.750.800.850.900.951.00\nFIG. 3. The pair-breaking e\u000bects. (a) A signi\fcant Pauli limit violation is due to the atomic SOC for the orbital-independent\npairing with \u0001 s= 1. However, the lattice strain might slightly suppress the Hc2by comparing the blue and red curves. (b) A\nweak Pauli limit violation due to the atomic SOC for the orbital-dependent pairing with \u0001 o= 1. (c) The suppression of Tcby\natomic SOC for orbital-dependent pairing at zero external magnetic \felds with \u0001 o= 1. For the three \fgures here, we have set\nthe strain parameter \u001e=\u0019\n8, i.e. ~g= (p\n2\n2;0;p\n2\n2).\nthe Euler-Mascheroni constant. Furthermore, the kernel\nfunction of the pair-breaking term fsis given by\nC0(T;E) = Re\u0014\n (0)(1\n2)\u0000 (0)(1\n2+iE\n2\u0019kBT)\u0015\n;(22)\nwith (0)(z) being the digamma function. Note that\nC0(T;E)\u00140 and it monotonically decreases as Ein-\ncreases, indicating the reduction of Tc. Namely,C0(T;E)\ngets smaller for a larger E.\nWe \frst discuss the simplest case with \u0015soc=tstr= 0,\nwhere the pair-breaking function becomes fs(T;0;0;h) =\nC0(T;h), which just leads to the Pauli limit Hc2;k\u0019\nHP= 1:86Tc, as shown in Fig. 3(a). Furthermore, we\nturn on\u0015socwhile take the tstr!0 limit, the pair-\nbreaking term in Eq. (21) is reduced to\nfs(T;\u0015soc;0;h) =C0\u0010\nT;p\n\u00152soc+h2\u0011h2\n\u00152soc+h2;(23)\nwhich reproduces the same results of Type II Ising super-\nconductors in Ref. [73]. Under a relatively weak magnetic\n\feld (h\u001c\u0015soc), the factor h2=(\u00152\nsoc+h2)\u001c1 leads to\nfs(T;\u0015soc;0;h)!0, which in turn induces a large in-\nplaneHc2;k=HP.\nNext, we investigate the e\u000bect of lattice strain tstron\nthe in-plane upper critical \feld Hc2;k. Interestingly, tstr\nwould generally instead reduce Hc2;k. To see it explicitly,\nwe expand the pair-breaking function fsin Eq. (21) up\nto the leading order of t2\nstr,\nfs(T;\u0015soc;tstr;h)\u0019fs(T;\u0015soc;0;h)\n+F(T;\u0015soc;h)t2\nstr+O(t4\nstr);(24)\nwhereF(T;\u0015soc;h) is given in Appendix B and we \fnd\nit is always negative (i.e., F(T;\u0015soc;h)<0). In addition\nto the \frst term fs(T;\u0015soc;0;h) discussed in Eq. (23),\nthe second term F(T;\u0015soc;h)t2\nstralso serves as a pair-\nbreaking e\u000bect on Tcat non-zero \feld. Therefore, the\nsecond\u0015o-term further reduces Tc, leading to the reduc-\ntion of the in-plane upper critical \feld.We then numerically con\frm the above discussions.\nWe solve the linearized gap equation v0\u001fs(T)\u00001 = 0 and\narrive at log( Tc=T0) =fs(Tc;\u0015soc;tstr;h), from which\nTc=T0is numerically calculated in Fig. 3 (a). Here T0is\nthe critical temperature at zero external magnetic \felds.\nThe Pauli limit corresponds to T0(\u0015soc= 0;tstr= 0;h=\n0). The non-monotonic behavior of the curves at small\nTc=T0(.0:5, i.e. dashed line) from solving the linearized\ngap equation calls for a comment. In the small temper-\nature range, the transition by tuning the \feld strength\nbecomes the \frst order supercooling transition [74]. Here\nwe mainly focus on the solid line part, which is second\norder and gives the critical \feld Hc2. We see that in\ngeneral there is a Pauli limit violation for non-zero \u0015soc\nandtstr. Furthermore, by comparing the two cases with\n\u0015soc= 1:5;tstr= 0 and\u0015soc= 1:5;tstr= 1, we con\frm\nthe above approximated analysis. We believe the strain\ne\u000bect on the type II Ising SC will be tested in experiments\nsoon.\nB.Hc2;kfor orbital-dependent pairings\nIn this subsection, we further study the in\ruence of\nthe atomic SOC \u0015socon the paramagnetic pair-breaking\ne\u000bect for orbital-dependent pairings (i.e., \u0001 s= 0 and\n\u0001o6= 0). We \fnd a weak enhancement of the in-plane\nupper critical \feld Hc2;kcompared with the Pauli limit.\nFollowing the criteria of the orbital do-vector in Ref. [10]\n(also discussed in the Appendix C), we take doto be\nparallel to the vector ~gby assuming \u0015soc\u001ctstr, which\nleads to the maximal condensation energy. This would be\njusti\fed in the next subsection. After a straightforward\ncalculation (see details in Appendix B), the superconduc-\ntivity susceptibility \u001fo(T) in Eq. (19) is calculated as,\n\u001fo(T) =\u001f0(T) +N0fo(T;\u0015soc;tstr;h); (25)8\nwhere the pair-breaking term is given by\nfo(T;\u0015soc;tstr;h) =1\n2[C0(T;\u001a\u0000) +C0(T;\u001a+)]\n+ [C0(T;\u001a\u0000)\u0000C0(T;\u001a+)]\u0012t2\nstr\u0000\u00152\nsoc\u0000h2\n2E+E\u0000\u0013\n;(26)\nwhich di\u000bers from fs(T;\u0015soc;tstr;h) for orbital-\nindependent pairings in Eq. (21). The only di\u000berence\nbetween them lies in the factor ( t2\nstr\u0000\u00152\nsoc\u0000h2)=2E+E\u0000,\ncompared with that of fs(T;\u0015soc;tstr;h) (i.e.\n(t2\nstr+\u00152\nsoc\u0000h2)=2E+E\u0000), which leads to a com-\npletely distinct superconducting state, demonstrated as\nfollows.\nTo understand Eq. (26), we \frst discuss the simplest\ncase with\u0015soc=tstr= 0, where the pair-breaking func-\ntion becomes f(T;0;0;h) =C0(T;h), which just leads\nto the Pauli limit Hc2;k\u0019HP= 1:86Tc, as shown in\nFig. 3(b). Likewise, when \u0015soc= 0 andtstr6= 0, the pair-\nbreaking function again simpli\fes to C0(T;h). Therefore,\nthe Pauli limit of the in-plane upper critical \feld is not\na\u000bected by tstritself. Physically, this is because spin\nand orbital degrees of freedom are completely decoupled\nin this case, and it has also been shown that a similar or-\nbital e\u000bect does not suppress Tcwhen dok~g[10], which\nis what we assumed here.\nOn the other hand, if we turn on merely the atomic\nSOC\u0015soc6= 0 while keeping tstr= 0, the pair-breaking\nfunction is given by\nfo(T;\u0015soc;0;h) =C0(T;p\n\u00152soc+h2); (27)\nwhich leads to the reduction of the upper critical\n\feld, i.e., Hc2;k< HP, because of f(T;\u0015soc;0;h)<\nf(T;0;0;h)<0. Remarkably, we \fnd that the atomic\nSOC also plays a similar role of \\magnetic \feld\" to sup-\npress the orbital-dependent pairing, as discussed in the\nnext subsection. Thus, it does not belong to the family\nof Ising SCs, which makes the orbital-dependent pair-\ning signi\fcantly di\u000berent from the orbital-independent\npairings. Moreover, their di\u000berent dependence on the in-\nplane magnetic \feld might also be tested in experiments,\nwhich is beyond this work and left for future work. This\nalso indicates the di\u000berence between orbital-triplet SC\nand spin-triplet SC in responses to Zeeman \felds.\nHowever, it is surprising to notice that there is a weak\nenhancement of the in-plane upper critical \feld Hc2;kfor\nthe case with both tstr6= 0 and\u0015soc6= 0. Solving the\ngap equation v0\u001fo(T)\u00001 = 0, we obtain\nln\u0012Tc\nT0\u0013\n=fo(T;\u0015soc;tstr;h): (28)\nFig. 3 (b) shows how Tc=T0changes with the applied\nin-plane magnetic \feld, where the Pauli limit curve cor-\nresponds to \u0015soc;tstr= 0. When both the atomic SOC\nand strain are included, the critical \feld Hc2exceeds the\nPauli limit by a small margin. Therefore, a spin-orbit-\ncoupled SC with spin-singlet non-unitary pairing sym-\nmetries does not belong to the reported family of Ising\nsuperconductivity.C. Atomic SOC induced zero-\feld Pauli limit\nAs mentioned above, the atomic SOC breaks the spin\ndegeneracy, which generally suppresses the even parity\norbital-dependent pairings, in the case with \u0001 s= 0 and\n\u0001o6= 0. Thus, the robustness of such pairings in the\npresence of atomic SOC is the preliminary issue that\nwe need to address. And we \fnd that the spin-singlet\norbital-dependent pairing is also prevalent in solid-state\nsystems when the energy scale of atomic SOC is smaller\nthan that of the orbital hybridization or external strain.\nIn this case, we focus on the zero magnetic \feld limit.\nUsing the general results from the calculations in the pre-\nvious section, we have\nln\u0012Tc\nT0\u0013\n=fo(T;\u0015soc;tstr;h= 0)\n=C0\u0012\nT;q\nt2\nstr+\u00152soc\u0013\u00152\nsoc\nt2\nstr+\u00152soc;(29)\nwhereC0(T;E) is de\fned in Eq. (22). In the case of\n\u0015soc= 0, it can be been that Tc(tstr) =T0(tstr= 0),\ni.e. the superconducting Tcis not suppressed by stain\nor the orbital hybridization when the orbital do-vector is\nparallel to ~g[10]. However, in the presence of non-zero\natomic SOC \u0015soc, theTcwill be suppressed even when\ndok~gis satis\fed. Fig. 3 (c) shows the behavior of Tcas\na function of the \u0015soc=tstrfor two di\u000berent values of tstr.\nWe see the suppression of Tcas long as\u0015soc6= 0, and the\nsuppression is more prominent when tstris larger.\nTo understand the suppression of orbital-dependent\npairings by the atomic SOC, we take the tstr= 0 limit.\nEq. (29) leads to\nln\u0012Tc\nT0\u0013\n=C0(T;\u0015soc); (30)\nwhich implies that \u0015socplays the same role of \\magnetic\n\feld\" that suppresses the Tcof the orbital-dependent\npairing states. And \u0015soc\u0018HProughly measures the\nzero-\feld \\Pauli-limit\" of the orbital-dependent pairing\nstates. We dub this new e\u000bect as zero-\feld Pauli limit for\norbital-dependent pairings induced by the atomic SOC,\nwhich can serve as the preliminary analysis of whether\norbital-dependent pairings exist or not in real materials\nby simply calculating \u0015soc=Tc.\nMotivated by this observation, we notice that the nor-\nmal Hamiltonian given in Eq. (7) satis\fes [ HN(k);\u001c2] = 0\nwith both\u0015o!0 andtstr!0. It stands for the U(1)\nrotation in the orbital subspace. As a result, we can\nproject the normal Hamiltonian HN(k) in Eq. (7) into\nblock-diagonal form corresponding to the \u00061 eigenvalues\nof\u001c2by using the basis transformation\nU=\u001b0\n1p\n2\u0014\n1\u0000i\n1i\u0015\n: (31)\nThe new basis is given by\n~\ty(k) = (cy\n+;\";cy\n+;#;cy\n\u0000;#;cy\n\u0000;\"); (32)9\nwherecy\n\u0006;s\u00111p\n2(cy\ndxz;s\u0007icy\ndyz;s). On this basis, the\nnormal Hamiltonian is given by\nH0=H+\n0\bH\u0000\n0; (33)\nwhereH\u0006\n0are given by\nH\u0006\n0=\u000f(k)\u0007\u0015soc\u001b3: (34)\nNote that the time-reversal transforms H\u0006\n0(k) to\nH\u0007\n0(\u0000k). Explicitly, the atomic SOC is indeed a \\mag-\nnetic \feld\" in each subspace, while it switches signs in\nthe two subspaces to conserve TRS.\nNext, we project the pairing Hamiltonian to the new\nbasis, and we \fnd that it also decouples as\nH\u0001=H+\n\u0001\bH\u0000\n\u0001; (35)\nwhereH\u0006\n\u0001are given by\nH\u0006\n\u0001= 2\u0001\u0006h\ncy\n\u0006;\"(k)cy\n\u0006;#(\u0000k)\u0000(\"$# )i\n+ h.c.;(36)\nwhere \u0001\u0006\u0011\u0001o(\u0007id1\no+d3\no) are the gap strengths in each\nsubspace. In each subspace, it resembles an s-wave su-\nperconductor under an e\u000bective \\magnetic \feld\" of the\natomic SOC along the out-of-plane direction. It natu-\nrally explains the \\zero-\feld Pauli-limit\" pair-breaking\ne\u000bect of atomic SOC on the orbital-dependent pairings\nwith thetstr!0 limit. As a brief conclusion, our results\ndemonstrate that the spin-singlet orbital-dependent pair-\nings occur only in weak atomic SOC electronic systems.\nIV. 2D HELICAL SUPERCONDUCTIVITY\nIn the above sections, the spin-orbit-coupled SCs\nconcerning inversion symmetry have been comprehen-\nsively studied. In addition to that, it will be natural\nto ask if there exist more interesting superconducting\nstates (e.g. topological phases) by including an inversion-\nsymmetry breaking to the normal Hamiltonian in Eq. (7),\nnamely,\u0015R6= 0. For this purpose, in this section, we\nfocus on the Rashba SOC and explore its e\u000bect on the\nspin-orbit-coupled SCs, especially the orbital-dependent\npairings. Even though the 2D bulk SC or thin \flm\nSC preserves the inversion symmetry, a Rashba SOC\nappears near an interface between the superconducting\nlayer and the insulating substrate. Remarkably, we \fnd\na TRI topological SC (helical TSC) phase generated by\nthe interplay between the two types of SOC (atomic\nand Rashba) and spin-singlet orbital-dependent pairings.\nSince TRS is preserved, it belongs to Class DIII accord-\ning to the ten-fold classi\fcation. On the boundary of\nthe interface, there exists a pair of helical Majorana edge\nstates [75{84].\nTo explore the topological phases, we consider the\nnormal-state Hamiltonian in Eq. (7), and the TRI spin-\nsinglet non-unitary pairing symmetry in Eq. (16) for theBdG Hamiltonian (13), namely, a real orbital do-vector\nis assumed for the orbital-dependent pairings.\nIn thetstr!0 and \u0001s!0 limit, the bulk band gap\ncloses only at the \u0000-point for \u0016\u0006\nc=\u0006p\n\u00152soc\u00004j\u0001oj2\nwhile no gap-closing happens at other TRI momenta,\nleading to a topological phase transition. Thus, we con-\nclude that the topological conditions are \u0016\u0000\nc< \u0016 < \u0016+\nc\nand an arbitrary orbital do-vector. In Appendix D, we\nshow theZ2topological invariant can be analytically\nmapped to a BdG-version spin Chern number, similar\nto the spin Chern number in the 2D topological insula-\ntors. As mentioned in Sec. III (c), the conservation of \u001c2,\nthe U(1) symmetry in the orbital subspace, leads to the\ndecomposition of the BdG Hamiltonian into two blocks\nfor di\u000berent eigenvalues of \u001c2. In each subspace, we can\nde\fne the BdG Chern number as\nC\u0006=1\n2\u0019X\n\flled bandsZ\nBZdk\u0001h\u001e\u0006\nn(k)jirkj\u001e\u0006\nn(k)i;(37)\nwithj\u001e\u0006\nnibeing the energy eigenstate of H\u0006\nBdG(see the\ndetails in Appendix D). Then the Z2topological invari-\nant, in this case, is then explicitly given by,\n\u0017\u0011C+\u0000C\u0000\n2; (38)\nwhereC\u0006are the Chern numbers of the \u0006channels.\n\u0017= 1 corresponds to the TSC phase, shown in Fig. 4\n(a). Based on the analysis for the topological condition,\nwe learn that \u0001 oshould be smaller than \u0015soc. How-\never, as shown in Sec. III, the atomic SOC actually will\nreduce the Tcof orbital-dependent pairings, which set\na guideline to a physically realizable set of parameters,\nT0\u001d\u0015soc\u001d\u0001o, beyond the BCS theory (\u0001 o\u00181:76T0).\nFor example, the monolayer FeSe superconductor \flms\non di\u000berent substrates achieve a very high critical tem-\nperatureT0\u001870 K [85].\nAs for a more general case with non-zero \u0015o,tstrand\n\u0001s, the BdG Hamiltonian can no longer be decomposed\ninto two decoupled blocks, hence the Chern number ap-\nproach fails to characterize the Z2invariant. However,\nthe more general Wilson-loop approach still works (see\ndetails in Appendix E). In general, the Z2-type topolog-\nical invariant of helical superconductivity could be char-\nacterized by the Wilson loop spectrum [86, 87], shown in\nFig. 4 (b), which demonstrates the non-trivial Z2index.\nTo verify the helical topological nature, we calculate the\nedge spectrum in a semi-in\fnite geometry with kybe-\ning a good quantum number. Fig. 4 (c) con\frms clearly\nthat there is a pair of 1D helical Majorana edge modes\n(MEMs) propagating on the boundary of the 2D system.\nV. TRB NON-UNITARY SUPERCONDUCTOR\nSo far, the TRI non-unitary pairing states are investi-\ngated, which exhibit the Pauli-limit violation for in-plane\nupper critical \feld and topological phases. Furthermore,10\n0.40.20-0.2-0.40.3-0.30(c)Eky/π-1-0.500.51-0.500.511.5-1-0.50110.5-0.51.500.5(a)μChern numberky−ππ0−π0πθ(b)\n-0kx-0.500.5vys=0 ;  0=0 ;\nFIG. 4. Topological helical superconductivity for spin-singlet orbital-dependent pairing in the presence of Rashba SOC. (a)\nTheZ2index is calculated by decoupling the BdG Hamiltonian into two chiral blocks when \u0001 s= 0 andtstr= 0. The other\nparameters used: m= 0:5,\u0016= 0:2,\u0015soc= 0:4,\u0015R= 1, \u0001o= 0:1,do= (1;0;1). (b) The Wilson loop calculation of the Z2\ninvariant for \u0001 s= 0:05,tstr= 0:1,\u001e=\u0019\n8andgo= (1;0;1). The other parameters remain the same as those in (a). The\nspectrum of edge states in (c) shows two counter-propagating Majorana edge states of the helical TSC.\nin this section, we study the TRB non-unitary pairing\nstates characterized by a complex do-vector when both\n\u0001sand \u0001oare real. As it is well known, the experiments\nby zero-\feld muon-spin relaxation ( \u0016SR) and the polar\nKerr e\u000bect (PKE) can provide strong evidence for the\nobservation of spontaneous magnetization or spin polar-\nization in the superconducting states, which indicates a\nTRB superconducting pairing symmetry. On the theory\nside, the non-unitary spin-triplet pairing potentials are\nalways adopted to explain the experiments. However, for\na spin-singlet SC, a theory with TRB pairing-induced\nspin-magnetization is in great demand. Addressing this\ncrucial issue is one of the aims of this work, and we \fnd\nthat a spin-singlet TRB non-unitary SCs supports a TRB\natomic orbital polarization, which in turn would give rise\nto spin polarization in the presence of atomic SOC.\nA. 2D chiral TSC\nWe \frst explore the possible 2D chiral topological\nphases by considering the simplest case with \u0015o=\ntstr= \u0001s= 0 to demonstrate the essential physics.\nFor the TRB non-unitary pairing, a complex orbital\ndo-vector can be generally parameterized as do=\n(cos\u0012;0;ei\u001esin\u0012). And the relative phase \u001e=\u0006\u0019=2 is\nenergetically favored by minimizing the free energy.\nAt the \u0000 point, the bulk gap closes at \u0016\u0006\nc;i=\n\u0006p\n\u00152soc\u00004j\u0001ij2, wherei= 1;2 and \u0001 1;2=i\u0001o(sin\u0012\u0006\ncos\u0012). Due to TRB, \u0016\u0006\nc;16=\u0016\u0006\nc;2. Accordingly, we semi-\nqualitatively map out the phase diagram in Fig. 5 by tun-\ning\u0012and\u0016, and label the di\u000berent phase regions by the\nnumber of Majorana edge modes (MEMs), denoted as Q.\nWhenj\u0016j>maxfj\u0016c;1j;j\u0016c;2jg, the topologically trivial\nphase is achieved with Q= 0. As for minfj\u0016c;1j;j\u0016c;2jg<\nj\u0016j<maxfj\u0016c;1j;j\u0016c;2jg, there is only one MEM on the\nboundary, corresponding to the Q= 1 regions [88, 89].\nWhenj\u0016j<minfj\u0016c;1j;j\u0016c;2jg, there areQ= 2 MEMs.\n0.00.51.01.52.02.53.0-0.55-0.35-0.150.050.250.45\nθμπ02π0-0.60.60.3-0.3μ/u1D4AC=2/u1D4AC=2/u1D4AC=2/u1D4AC=1/u1D4AC=1/u1D4AC=1/u1D4AC=1Trivial SC\nTrivial SCθFIG. 5. Topological chiral superconductivity. We plot the\nphase diagram in terms of the number of MEMs ( Q) of the\nTSC. Parameters used: \u0001 o= 0:14,\u0015soc= 0:4,\u001e=\u0006\u0019=2,\n\u0015o= 0,tstr= 0 and \u0001 s= 0.\nThe chiral TSC might be detected by anomalous thermal\nHall conductivity Kxy=Q\n2\u0019T\n6[90].\nB. Atomic orbital polarization and spin\npolarization\nNext, we show how spin-singlet TRB non-unitary pair-\ning can induce spin polarization, and discuss how to\nidentify such pairings by using spin-polarized scanning\ntunneling microscopy measurements. We assume a TRB\ncomplex orbital do-vector and \fnd that it can generate\nthe orbital orderings as\nMo=\u0000i\r1=\u000bM(do\u0002d\u0003\no); (39)\nof which the y-component breaks TRS shown in\nFig. 6 (a). More precisely, we \fnd that My\no/11\n(a)𝑴𝑜∝𝑖𝒅𝑜×𝒅𝑜∗𝒅𝑜𝒅𝑜∗\n(a)TRS-breaking OP(c)(b)𝐸𝑘𝑥𝑘𝑥𝜔𝐷−𝐷+\nTRS-breaking OP(b)Two orbitals{dxz,dyz}{dxy,dyz}{dxz,dxy}SP direction-axisz-axisy-axisxλsocL⋅σ\n-0.5 0 0.50123456(c)Orbital DOS (a.u.)κ=+1κ=−1E-0.5 0 0.50123456Spin DOS (a.u.)σ=↓σ=↑(d)\nE\nFIG. 6. (a) Schematic diagram showing the TRB orbital po-\nlarization (OP) induced by complex do-vector. (b) Spin could\nbe polarized in di\u000berent directions based on the two active\norbitals involved in the pairing. (c) Orbital DOS projected\ninto the chiral \u0014=\u00061 basis, showing a two-gap feature due\nto TRB. (d) The corresponding spin DOS, shifted relative to\nthe fermi level due to the non-zero e\u000bective Zeeman \feld from\nthe OP. Parameters used: m= 0:5;\u0016=\u00002;\u0015R= 0;\u0015soc=\n0:2;\u0001o= 0:4;\u0015o= 0;tstr= 0;do= (1;0;ei\u0019=10).\nP\nk;\u001bh^ndxz+idyz;\u001b(k)\u0000^ndxz\u0000idyz;\u001b(k)i6= 0 indicates the\natomic orbital-polarization (OP) (see Appendix F for de-\ntails). Here, ^ nis the density operator of electrons. Once\nMy\nodevelops a \fnite value, it leads to orbital-polarized\nDOS and two distinct superconducting gaps of the quasi-\nparticle spectrum (Fig. 6 (c), more details below). There-\nfore, the orbital degree of freedom in spin-singlet SCs\nplays a similar role as the spin degree of freedom of spin-\ntriplet SCs.\nOnce the atomic SOC is present, spin-polarization\n(SP) could be induced indirectly. A possible Ginzburg-\nLandau term could be\n\u0001F=\u000bsjMsj2+\rsocMz\nsMy\no; (40)\nwith\u000bs>0 and\rsoc6= 0. Here, Mz\ns/P\nk;\u001ch^n\u001c;\"\u0000\n^n\u001c;#i. Therefore, the complex orbital do-vector can be\nidenti\fed by the spin-resolved density of states (DOS) for\nspin-singlet superconductors.Minimizing Eq. (40)directly\nleads toMz\ns=\rsocMy\no=Mz\ns, which indicates the OP-\ninduced spin magnetism. In addition, the direction of\nSP can be also aligned to xoryaxes, discussed later.\nTo verify the above analysis, we numerically solve the\nBdG Hamiltonian (13), HBdGjEn(k)i=En(k)jEn(k)i,\nwhere the n-th eigenstate is given by jEn(k)i=\n(un\ndxz;\";un\ndxz;#;vn\ndxz;\";vn\ndxz;#;un\ndyz;\";un\ndyz;#;vn\ndyz;\";vn\ndyz;#)T.\nThus, the atomic-orbital and spin-resolved DOS can becalculated as the following,\nD\u0014\norbit(E) =X\n\u001b;n;kjun\n\u0014;\u001bj2\u000e(E\u0000En(k));\nD\u001b\nspin(E) =X\n\u001c;n;kjun\n\u001c;\u001bj2\u000e(E\u0000En(k));(41)\nwhereun\n\u0014;\u001b=1p\n2(un\ndxz;\u001b\u0000i\u0014un\ndyz;\u001b) and\u0014=\u00061 for\ndxz\u0006idyzorbitals. In Fig. 6 (c), D+1\norbit6=D\u00001\norbitin-\ndicates that the DOS is orbital-polarized. Remarkably,\nwe also have D\"\nspin6=D#\nspindue to coupling between elec-\ntron spin and atomic orbitals, shown in Fig. 6 (d). The\ndi\u000berence in orbital DOS acts as an e\u000bective Zeeman \feld\nfor the electron spins, hence shifting the spin DOS rela-\ntive to the fermi level in opposite directions for up spin\nand down spin. This interesting phenomenon is quite\ndi\u000berent from spin-triplet SCs. In TRB spin-triplet SCs,\nThe spin-up channel and spin-down channel will form\ndi\u000berent symmetric gaps in spin DOS, similar to the two\norbital channels in Fig. 6 (c) for our case. Therefore,\nthe spin DOS pro\fles are distinct in the two cases. As\na result, the spin-resolved DOS, which can be probed by\nspin-resolved STM [91] and muon-spin relaxation [92, 93],\ncan serve as a smoking gun evidence to identify TRB due\nto complex orbital do-vector in multi-orbital SCs.\nVI. DISCUSSIONS AND CONCLUSIONS\nIn the end, we brie\ry discuss the direction of spin-\npolarization induced by atomic orbital-polarization, sum-\nmarized in Fig. 6 (b). We consider the three-dimensional\nsubspace of t2gorbitals spanned by fdyz;dxz;dxyg, where\nthe matrix form of the angular momentum operators L\nreads [63],\nLx=0\n@0 0 0\n0 0i\n0\u0000i01\nA; Ly=0\n@0 0\u0000i\n0 0 0\ni0 01\nA; Lz=0\n@0i0\n\u0000i0 0\n0 0 01\nA;\n(42)\nwhich satisfy the commutation relation [ Lm;Ln] =\n\u0000i\u000fmnlLl. Therefore, the spin-orbit coupling for a system\nwith thet2gorbitals is given by,\nHsoc=\u0015socL\u0001\u001b: (43)\nThen, let us consider a two-orbital system, the above\nSOC Hamiltonian will be reduced to,\n8\n><\n>:Forfdyz;dxzg:Hsoc=\u0000\u0015soc\u001c2\u001b3;\nForfdyz;dxyg:Hsoc=\u0015soc\u001c2\u001b2;\nForfdxz;dxyg:Hsoc=\u0000\u0015soc\u001c2\u001b1:(44)\nTherefore, in the above three cases, the spin-polarization\nis pointed to z;y;x -axis, respectively. Because the atomic12\norbital polarization is induced by the complex orbital do-\nvector as (0 ;My\no;0)/id\u0003\no\u0002do.\nTo summarize, we establish a phenomenological the-\nory for spin-singlet two-band SCs and discuss the distinct\nfeatures of both TRI non-unitary pairings and TRB non-\nunitary pairings by studying the e\u000bects of atomic spin-\norbit coupling (SOC), lattice strain e\u000bect, and Rashba\nSOC. Practically, we demonstrate that the stability of\norbital-dependent pairing states could give birth to the\nnon-unitary pairing states in a purely spin-singlet SC.\nRemarkably, the interplay between atomic SOC and\norbital-dependent pairings is also investigated and we\n\fnd a new spin-orbit-coupled SC with spin-singlet non-\nunitary pairing. For this exotic state, there are mainly\nthree features. Firstly, the atomic SOC could enlarge\nthe in-plane upper critical \feld compared to the Pauli\nlimit. A new e\u000bect dubbed as \\zero-\feld Pauli limit\"\nfor orbital-dependent pairings is discovered. Secondly,\ntopological chiral or helical superconductivity could berealized even in the absence of external magnetic \felds\nor Zeeman \felds. Furthermore, a spontaneous TRB\nSC could even generate a spin-polarized superconducting\nstate that can be detected by measuring the spin-resolved\ndensity of states. We hope our theory leads to a deeper\nunderstanding of spin-singlet non-unitary SCs.\nOur theory might have potential applications to the\nintriguing Sr 2SuO 4[94, 95], LaNiGa 2[58], iron-based SCs\n[59, 60] and ultra-cold atomic systems with large spin\nalkali and alkaline-earth fermions [96{100].\nVII. ACKNOWLEDGMENTS\nWe thank J.-L. Lado, R.-X. Zhang and C.-X. Liu\nfor helpful discussions. We especially acknowledge J.-\nL. Lado's careful reading of the manuscript. D.-H.X. was\nsupported by the NSFC (under Grant Nos. 12074108 and\n11704106).\n[1] M. Sigrist and K. Ueda, Rev. Mod. Phys. 63, 239 (1991).\n[2] V. P. Mineev, K. Samokhin, and L. Landau, Introduc-\ntion to unconventional superconductivity (CRC Press,\n1999).\n[3] F. Steglich, J. Aarts, C. D. Bredl, W. Lieke,\nD. Meschede, W. Franz, and H. Sch afer, Phys. Rev.\nLett. 43, 1892 (1979).\n[4] J. G. Bednorz and K. A. M \u0013'uller, Z. Physik B - Con-\ndensed Matter 64, 189 (1986).\n[5] F. C. Zhang and T. M. Rice, Phys. Rev. B 37, 3759\n(1988).\n[6] P. W. Anderson, P. Lee, M. Randeria, T. Rice,\nN. Trivedi, and F. Zhang, Journal of Physics: Con-\ndensed Matter 16, R755 (2004).\n[7] D. J. Scalapino, Rev. Mod. Phys. 84, 1383 (2012).\n[8] P. M. R. Brydon, D. S. L. Abergel, D. F. Agterberg,\nand V. M. Yakovenko, Phys. Rev. X 9, 031025 (2019).\n[9] T. M. R. Wolf, M. F. Holst, M. Sigrist, and J. L. Lado,\nPhys. Rev. Res. 4, L012036 (2022).\n[10] M. Zeng, D.-H. Xu, Z.-M. Wang, L.-H. Hu, and F.-\nC. Zhang, arXiv e-prints , arXiv:2109.06039 (2021),\narXiv:2109.06039 [cond-mat.supr-con].\n[11] X. Dai, Z. Fang, Y. Zhou, and F.-C. Zhang, Phys. Rev.\nLett. 101, 057008 (2008).\n[12] T. T. Ong and P. Coleman, Phys. Rev. Lett. 111,\n217003 (2013).\n[13] Z. P. Yin, K. Haule, and G. Kotliar, Nature Physics 10,\n845 (2014).\n[14] T. Ong, P. Coleman, and J. Schmalian, Proceedings of\nthe National Academy of Sciences 113, 5486 (2016).\n[15] R. Nourafkan, G. Kotliar, and A.-M. S. Tremblay, Phys.\nRev. Lett. 117, 137001 (2016).\n[16] A. V. Chubukov, O. Vafek, and R. M. Fernandes, Phys.\nRev. B 94, 174518 (2016).\n[17] M. Yi, Y. Zhang, Z.-X. Shen, and D. Lu, npj Quantum\nMaterials 2, 1 (2017).\n[18] E. M. Nica, R. Yu, and Q. Si, npj Quantum Materials\n2, 1 (2017).[19] P. O. Sprau, A. Kostin, A. Kreisel, A. E. B ohmer,\nV. Taufour, P. C. Can\feld, S. Mukherjee, P. J.\nHirschfeld, B. M. Andersen, and J. S. Davis, Science\n357, 75 (2017).\n[20] H. Hu, R. Yu, E. M. Nica, J.-X. Zhu, and Q. Si, Phys.\nRev. B 98, 220503 (2018).\n[21] W. Chen and W. Huang, Phys. Rev. Research 3,\nL042018 (2021).\n[22] E. M. Nica and Q. Si, npj Quantum Materials 6, 1\n(2021).\n[23] L. A. Wray, S.-Y. Xu, Y. Xia, Y. San Hor, D. Qian,\nA. V. Fedorov, H. Lin, A. Bansil, R. J. Cava, and M. Z.\nHasan, Nature Physics 6, 855 (2010).\n[24] L. Fu and E. Berg, Phys. Rev. Lett. 105, 097001 (2010).\n[25] P. M. R. Brydon, L. Wang, M. Weinert, and D. F. Agter-\nberg, Phys. Rev. Lett. 116, 177001 (2016).\n[26] W. Yang, Y. Li, and C. Wu, Phys. Rev. Lett. 117,\n075301 (2016).\n[27] D. F. Agterberg, P. M. R. Brydon, and C. Timm, Phys.\nRev. Lett. 118, 127001 (2017).\n[28] L. Savary, J. Ruhman, J. W. F. Venderbos, L. Fu, and\nP. A. Lee, Phys. Rev. B 96, 214514 (2017).\n[29] W. Yang, T. Xiang, and C. Wu, Phys. Rev. B 96, 144514\n(2017).\n[30] C. Timm, A. P. Schnyder, D. F. Agterberg, and P. M. R.\nBrydon, Phys. Rev. B 96, 094526 (2017).\n[31] J. Yu and C.-X. Liu, Phys. Rev. B 98, 104514 (2018).\n[32] I. Boettcher and I. F. Herbut, Phys. Rev. Lett. 120,\n057002 (2018).\n[33] B. Roy, S. A. A. Ghorashi, M. S. Foster, and A. H.\nNevidomskyy, Phys. Rev. B 99, 054505 (2019).\n[34] H. Kim, K. Wang, Y. Nakajima, R. Hu, S. Ziemak,\nP. Syers, L. Wang, H. Hodovanets, J. D. Denlinger,\nP. M. Brydon, et al. , Science advances 4, eaao4513\n(2018).\n[35] D. F. Agterberg, T. M. Rice, and M. Sigrist, Phys. Rev.\nLett. 78, 3374 (1997).\n[36] T. Takimoto, Phys. Rev. B 62, R14641 (2000).13\n[37] W. Huang, Y. Zhou, and H. Yao, Phys. Rev. B 100,\n134506 (2019).\n[38] Z.-M. Wang, M. Zeng, C. Lu, L.-H. Hu, and D.-H. Xu,\nIn preparation (2022).\n[39] J. Clepkens, A. W. Lindquist, X. Liu, and H.-Y. Kee,\nPhys. Rev. B 104, 104512 (2021).\n[40] A. Ramires and M. Sigrist, Phys. Rev. B 100, 104501\n(2019).\n[41] W. Yang, C. Xu, and C. Wu, Phys. Rev. Research 2,\n042047 (2020).\n[42] N. F. Q. Yuan, W.-Y. He, and K. T. Law, Phys. Rev.\nB95, 201109 (2017).\n[43] L. Chirolli, F. de Juan, and F. Guinea, Phys. Rev. B\n95, 201110 (2017).\n[44] A. Robins and P. Brydon, Journal of Physics: Con-\ndensed Matter 30, 405602 (2018).\n[45] J. L. Lado and M. Sigrist, Phys. Rev. Research 1, 033107\n(2019).\n[46] L.-H. Hu, P. D. Johnson, and C. Wu, Phys. Rev. Re-\nsearch 2, 022021 (2020).\n[47] A. Ramires, Journal of Physics: Condensed Matter 34,\n304001 (2022).\n[48] Y. Saito, Y. Nakamura, M. S. Bahramy, Y. Kohama,\nJ. Ye, Y. Kasahara, Y. Nakagawa, M. Onga, M. Toku-\nnaga, T. Nojima, et al. , Nature Physics 12, 144 (2016).\n[49] J. Lu, O. Zheliuk, I. Leermakers, N. F. Yuan, U. Zeitler,\nK. T. Law, and J. Ye, Science 350, 1353 (2015).\n[50] X. Xi, Z. Wang, W. Zhao, J.-H. Park, K. T. Law,\nH. Berger, L. Forr\u0013 o, J. Shan, and K. F. Mak, Nature\nPhysics 12, 139 (2016).\n[51] J. Falson, Y. Xu, M. Liao, Y. Zang, K. Zhu, C. Wang,\nZ. Zhang, H. Liu, W. Duan, K. He, et al. , Science 367,\n1454 (2020).\n[52] J. P. Ruf, H. Paik, N. J. Schreiber, H. P. Nair, L. Miao,\nJ. K. Kawasaki, J. N. Nelson, B. D. Faeth, Y. Lee, B. H.\nGoodge, et al. , Nature Communications 12, 59 (2021).\n[53] S. Beck, A. Hampel, M. Zingl, C. Timm, and\nA. Ramires, Phys. Rev. Res. 4, 023060 (2022).\n[54] K. Ahadi, L. Galletti, Y. Li, S. Salmani-Rezaie, W. Wu,\nand S. Stemmer, Science advances 5, eaaw0120 (2019).\n[55] R. M. Fernandes and L. Fu, Phys. Rev. Lett. 127,\n047001 (2021).\n[56] A. D. Hillier, J. Quintanilla, and R. Cywinski, Phys.\nRev. Lett. 102, 117007 (2009).\n[57] A. D. Hillier, J. Quintanilla, B. Mazidian, J. F. Annett,\nand R. Cywinski, Phys. Rev. Lett. 109, 097001 (2012).\n[58] Z. F. Weng, J. L. Zhang, M. Smidman, T. Shang,\nJ. Quintanilla, J. F. Annett, M. Nicklas, G. M. Pang,\nL. Jiao, W. B. Jiang, Y. Chen, F. Steglich, and H. Q.\nYuan, Phys. Rev. Lett. 117, 027001 (2016).\n[59] V. Grinenko, R. Sarkar, K. Kihou, C. Lee, I. Morozov,\nS. Aswartham, B. B uchner, P. Chekhonin, W. Skrotzki,\nK. Nenkov, et al. , Nature Physics , 1 (2020).\n[60] N. Zaki, G. Gu, A. Tsvelik, C. Wu, and P. D. Johnson,\nProceedings of the National Academy of Sciences 118\n(2021).\n[61] P. K. Biswas, S. K. Ghosh, J. Z. Zhao, D. A. Mayoh,\nN. D. Zhigadlo, X. Xu, C. Baines, A. D. Hillier, G. Bal-\nakrishnan, and M. R. Lees, Nature communications 12,\n2504 (2021).\n[62] Q. Zou, B. D. Oli, H. Zhang, T. Shishidou,\nD. Agterberg, M. Weinert, and L. Li, arXiv preprint\narXiv:2212.13603 (2022).[63] W.-C. Lee, D. P. Arovas, and C. Wu, Phys. Rev. B 81,\n184403 (2010).\n[64] J. Clepkens, A. W. Lindquist, and H.-Y. Kee, Phys.\nRev. Research 3, 013001 (2021).\n[65] J. B oker, P. A. Volkov, P. J. Hirschfeld, and I. Eremin,\nNew Journal of Physics 21, 083021 (2019).\n[66] S. Raghu, X.-L. Qi, C.-X. Liu, D. J. Scalapino, and S.-C.\nZhang, Phys. Rev. B 77, 220503 (2008).\n[67] M. Yi, Y. Zhang, Z.-X. Shen, and D. Lu, npj Quantum\nMaterials 2, 57 (2017).\n[68] C. Guo, L. Hu, C. Putzke, J. Diaz, X. Huang, K. Manna,\nF.-R. Fan, C. Shekhar, Y. Sun, C. Felser, et al. , Nature\nPhysics , 1 (2022).\n[69] L.-H. Hu, X. Wang, and T. Shang, Phys. Rev. B 104,\n054520 (2021).\n[70] T. Shang, J. Zhao, L.-H. Hu, J. Ma, D. J. Gawryluk,\nX. Zhu, H. Zhang, Z. Zhen, B. Yu, Y. Xu, et al. , Science\nadvances 8, eabq6589 (2022).\n[71] P. A. Frigeri, D. F. Agterberg, A. Koga, and M. Sigrist,\nPhys. Rev. Lett. 92, 097001 (2004).\n[72] H. Yi, L.-H. Hu, Y. Wang, R. Xiao, J. Cai, D. R. Hickey,\nC. Dong, Y.-F. Zhao, L.-J. Zhou, R. Zhang, et al. , Na-\nture Materials , 1 (2022).\n[73] C. Wang, B. Lian, X. Guo, J. Mao, Z. Zhang, D. Zhang,\nB.-L. Gu, Y. Xu, and W. Duan, Phys. Rev. Lett. 123,\n126402 (2019).\n[74] K. Maki and T. Tsuneto, Progress of Theoretical\nPhysics 31, 945 (1964).\n[75] X.-L. Qi, T. L. Hughes, S. Raghu, and S.-C. Zhang,\nPhys. Rev. Lett. 102, 187001 (2009).\n[76] C.-X. Liu and B. Trauzettel, Phys. Rev. B 83, 220510\n(2011).\n[77] S. Nakosai, Y. Tanaka, and N. Nagaosa, Phys. Rev.\nLett. 108, 147003 (2012).\n[78] S. Deng, L. Viola, and G. Ortiz, Phys. Rev. Lett. 108,\n036803 (2012).\n[79] F. Zhang, C. L. Kane, and E. J. Mele, Phys. Rev. Lett.\n111, 056402 (2013).\n[80] J. Wang, Y. Xu, and S.-C. Zhang, Phys. Rev. B 90,\n054503 (2014).\n[81] A. Haim and Y. Oreg, Physics Reports 825, 1 (2019).\n[82] O. E. Casas, L. Arrachea, W. J. Herrera, and A. L.\nYeyati, Phys. Rev. B 99, 161301 (2019).\n[83] Y. Volpez, D. Loss, and J. Klinovaja, Phys. Rev. Re-\nsearch 2, 023415 (2020).\n[84] R.-X. Zhang and S. Das Sarma, Phys. Rev. Lett. 126,\n137001 (2021).\n[85] J.-F. Ge, Z.-L. Liu, C. Liu, C.-L. Gao, D. Qian, Q.-K.\nXue, Y. Liu, and J.-F. Jia, Nature Materials 14, 285\n(2014).\n[86] R. Yu, X. L. Qi, A. Bernevig, Z. Fang, and X. Dai, Phys.\nRev. B 84, 075119 (2011).\n[87] W. A. Benalcazar, B. A. Bernevig, and T. L. Hughes,\nPhys. Rev. B 96, 245115 (2017).\n[88] X.-L. Qi, T. L. Hughes, and S.-C. Zhang, Phys. Rev. B\n82, 184516 (2010).\n[89] J. D. Sau, R. M. Lutchyn, S. Tewari, and S. Das Sarma,\nPhys. Rev. Lett. 104, 040502 (2010).\n[90] T. Meng and L. Balents, Phys. Rev. B 86, 054504\n(2012).\n[91] R. Wiesendanger, Reviews of Modern Physics 81, 1495\n(2009).14\n[92] G. Csire, J. Annett, J. Quintanilla, and B. \u0013Ujfalussy,\nMagnetically-textured superconductivity in elemental\nrhenium (2021), arXiv:2005.05702 [cond-mat.supr-con].\n[93] T. Shang, M. Smidman, S. K. Ghosh, C. Baines, L. J.\nChang, D. J. Gawryluk, J. A. T. Barker, R. P. Singh,\nD. M. Paul, G. Balakrishnan, E. Pomjakushina, M. Shi,\nM. Medarde, A. D. Hillier, H. Q. Yuan, J. Quintanilla,\nJ. Mesot, and T. Shiroka, Phys. Rev. Lett. 121, 257002\n(2018).\n[94] G. M. Luke, Y. Fudamoto, K. Kojima, M. Larkin,\nJ. Merrin, B. Nachumi, Y. Uemura, Y. Maeno, Z. Mao,\nY. Mori, et al. , Nature 394, 558 (1998).\n[95] J. Xia, Y. Maeno, P. T. Beyersdorf, M. M. Fejer, and\nA. Kapitulnik, Phys. Rev. Lett. 97, 167002 (2006).\n[96] T.-L. Ho and S. Yip, Phys. Rev. Lett. 82, 247 (1999).\n[97] C. Wu, J.-p. Hu, and S.-c. Zhang, Phys. Rev. Lett. 91,\n186402 (2003).\n[98] B. J. DeSalvo, M. Yan, P. G. Mickelson, Y. N. Mar-\ntinez de Escobar, and T. C. Killian, Phys. Rev. Lett.\n105, 030402 (2010).\n[99] S. Taie, Y. Takasu, S. Sugawa, R. Yamazaki, T. Tsuji-\nmoto, R. Murakami, and Y. Takahashi, Phys. Rev. Lett.\n105, 190401 (2010).\n[100] A. V. Gorshkov, M. Hermele, V. Gurarie, C. Xu, P. S.\nJulienne, J. Ye, P. Zoller, E. Demler, M. D. Lukin, and\nA. Rey, Nature physics 6, 289 (2010).\n[101] Y. Wang and L. Fu, Phys. Rev. Lett. 119, 187003\n(2017).\n[102] C.-X. Liu, Phys. Rev. Lett. 118, 087001 (2017).\n[103] L.-H. Hu, C.-X. Liu, and F.-C. Zhang, Communications\nPhysics 2, 1 (2019).\n[104] S. Ryu, A. P. Schnyder, A. Furusaki, and A. W. Ludwig,\nNew Journal of Physics 12, 065010 (2010).\n[105] C.-K. Chiu, J. C. Y. Teo, A. P. Schnyder, and S. Ryu,\nRev. Mod. Phys. 88, 035005 (2016).15\nAppendix A: Toy model for two-band\nsuperconducting phase diagrams\nIn this part of the appendix, we explore a possible su-\nperconducting phase diagram including the non-unitary\npairing states in the GL framework. Here we assume a\ntwo-band SC with\n\u0001tot=\u0002\n\u0001s\u001c0+ \u0001o(d1\no\u001c1+d3\no\u001c3)\u0003\n(i\u001b2): (A1)\nIn terms of the superconducting order parameters\nf\u0001s;\u0001o;do= (d1\no;0;d3\no)gand the order parameter for\nthe orbital orderings Mo/P\nk;\u001bhcy\na\u001b(k)\u001cabcb\u001b(k)i, the\ntotal GL free energy can be constructed to address the\nhomogeneous superconducting phase without external\nmagnetic \felds,\nF[\u0001s;\u0001o;do;Mo] =F0+Fb+Fo; (A2)\nwhere\nF0=1\n2\u000b(T)j\u0001oj2+1\n2\u000b0(T)j\u0001sj2+1\n2\u000bMjMoj2\n+1\n4\fj\u0001oj4+1\n4\f0j\u0001sj4+\f00j\u0001sj2j\u0001oj2\n+\fojd1\noj4+\f0\nojd3\noj4;(A3)\nwherejdoj= 1 is adopted, \u000b(T) =\u000b0(T=Tc1\u00001),\n\u000b0(T) =\u000b0\n0(T=Tc2\u00001) and the coe\u000ecients \u000b0,\u000b0\n0,\u000bM,\f,\n\f0,\f00,\fo,\f0\noare all positive. Tc1;Tc2are critical tem-\nperatures in orbital-dependent and orbital-independent\nchannels respectively, which are in general di\u000berent from\neach other. And \u000bM>0 means that there is no spon-\ntaneous atomic orbital polarization. In the supercon-\nducting state with both non-zero \u0001 sand \u0001odeveloped\nalready, additionally, there are two possible ways to pur-\nsue the spontaneous TRB, denoted as FbandFo. Firstly,\nwe consider theFbterm\nFb=b1\u0001\u0003\ns\u0001o+b2(\u0001\u0003\ns\u0001o)2+ h.c.; (A4)\nwhere the sign of b2determines the breaking of TRS. Here\nwe focus on the generic case where \u0001 sand \u0001obelong to\ndi\u000berent symmetry representations so that there is no\nlinear order coupling between them, i.e. b1= 0. Given\nb1= 0 andb2>0, we have a \u0012o=\u0006\u0019=2 relative phase\ndi\u000berence between \u0001 sand \u0001oei\u0012o[101], which gives to\nthe achievement of the TRB unitary pairing state (\u0001 s2\nR;\u0001o\u0018i;do2R).\nMore generally, a TRB non-unitary SC arises from\nthe non-zero bilinear b1-term, which is symmetry-allowedonly when \u0001 sand \u0001obelong to the same symmetry rep-\nresentation of the crystalline symmetry group. Namely,\nthe case with b16= 0 andb2>0 can pin the phase\ndi\u000berence\u0012oto an arbitrary nonzero value, i.e., \u0012o2\n(0;\u0019). Then, this case can also give rise to TRB non-\nunitary pairing with (\u0001 s2R;\u0001t2C;do2R) or\n(\u0001s2R;\u0001t2C;do2C). On the other hand, the\nb2<0 situation makes TRI non-unitary pairing states\n(\u0001s2R;\u0001o2R;do2R).\nHowever, even in the case with b2<0, we still have an\nalternative approach to reach TRB pairing states, driven\nby theFoterm\nFo=\r0jdo\u0002d\u0003\noj2+i\r1Mo\u0001(do\u0002d\u0003\no) + h.c.;(A5)\nwhere the sign of \r0identi\fes the TRB due to a complex\ndo. In particular \r0<0 results in a TRB non-unitary\nstate (\u0001s2R;\u0001o2R;do2C).\nWe summarize many of the possible interesting super-\nconducting phases in Fig. 7, which schematically shows a\nsuperconducting phase diagram as a function of b2and\r0\nby settingb1= 0, i.e. the generic case where \u0001 o;\u0001sbe-\nlong to di\u000berent representations. Notice that this phase\ndiagram characterized by b2and\r0does not contain the\nTRI unitary pairing phase.\nb2γ0(TRB + NU)(TRB + NU)(TRB + U)(TRI + NU)Δo∝i,do∈ℂΔo∈ℝ,do∈ℂΔo∝i,do∈ℝΔo∈ℝ,do∈ℝ0\nFIG. 7. Schematic superconducting phase diagrams on the\nb2-\r0plane when b1= 0 and \u0001 sis real and non-zero. Here,\nTRB and TRI are short for TR-breaking and TR-invariant,\nrespectively; U and NU represent unitary and non-unitary,\nrespectively.\nAppendix B: Derivation of Tcfrom linearized gap\nequation\nStarting from the generic Hamiltonian, containing atomic SOC, generic ~g= (~g1;0;~g2) withj~gj= 1 and in-plane\nmagnetic \feld,\nH0(k) =\u000f(k) +\u0015soc\u001b3\u001c2+\u0015(~g1\u001c1+ ~g3\u001c3) +h\u001b1: (B1)16\nThe Matsubara Green's function for electrons is\nGe(k;i!n) = [i!n\u0000H 0(k)]\u00001\n=P\u0000\u0000\u0000\ni!n\u0000\u000fk+E\u0000+P+\u0000+\ni!n\u0000\u000fk+E++P\u0000+\u0000\ni!n\u0000\u000fk\u0000E\u0000+P+++\ni!n\u0000\u000fk\u0000E+;(B2)\nwhere the projection operator\nP\u000b\f\r=1\n4[1 +\u000b(~g1\u001b1\u001c1+ ~g3\u001b1\u001c3)]\u0001[1 +\f\nE\r(\u0015soc\u001b3\u001c2+\u0015o(~g1\u001c1+ ~g3\u001c3) +h\u001b1)]; (B3)\nwith\u000b;\f;\r2f+;\u0000gandE\r=p\n\u00152soc+ (\u0015+\rh)2. The Green's function for hole is Gh(k;i!n) =\u0000G\u0003\ne(k;i!n). Here\n!n= (2n+ 1)\u0019kBT.\nThe linearized gap equation is given by\n\u0001a;b\ns1;s2(k) =\u00001\n\fX\n!nX\ns0\n1a0;s0\n2b0Vs1a;s2b\ns0\n1a0;s0\n2b0(k;k0)\u0002[Ge(k0;i!n)\u0001(k0)Gh(\u0000k0;i!n)]s0\n1a0;s0\n2b0; (B4)\nwhere the generic attractive interaction can be expanded as\nVs1a;s2b\ns0\n1a0;s0\n2b0(k;k0) =\u0000v0X\n\u0000;m[d\u0000;m\no(k)\u0001\u001ci\u001b2]s1a;s2b[d\u0000;m\no(k0)\u0001\u001ci\u001b2]s0\n1a0;s0\n2b0; (B5)\nwherev0>0 and \u0000 labels the irreducible representation with m-dimension of crystalline groups. The linearized gap\nequation is reduced to v0\u001f(T)\u00001 = 0 where \u001f(T) is the superconductivity susceptibility. We have\n•For orbital-independent pairing:\n\u001f(T)s=\u00001\n\fX\nk;!nTr\u0002\n( s(k)i\u001b2)yGe(k;i!n)( s(k)i\u001b2)Gh(\u0000k;i!n)\u0003\n: (B6)\n•For orbital-dependent pairing:\n\u001f(T)o=\u00001\n\fX\nk;!nTr\u0002\n(do(k)\u0001\u001ci\u001b2)yGe(k;i!n)(do(k)\u0001\u001ci\u001b2)Gh(\u0000k;i!n)\u0003\n: (B7)\nThen we take the standard replacement,\nX\nk;!n!N0\n4Z+!D\n\u0000!Dd\u000fZZ\nSd\nX\n!n; (B8)\nwhereN0is the density of states at Fermi surface, \n is the solid angle of kon Fermi surfaces and !Dthe Deybe\nfrequency. We will also be making use of,\n\u0000N0\n\fZ+!D\n\u0000!DX\n!nd\u000fG+\ne(k;i!n)G+\nh(k;i!n) =\u0000N0\n\fZ+!D\n\u0000!DX\n!nG\u0000\ne(k;i!n)G\u0000\nh(k;i!n) =\u001f0(T); (B9)\n\u0000N0\n\fZ+!D\n\u0000!DX\n!nd\u000fG\u0000\ne(k;i!n)G+\nh(k;i!n) =\u0000N0\n\fZ+!D\n\u0000!DX\n!nG+\ne(k;i!n)G\u0000\nh(k;i!n) =\u001f0(T) +N0C0(T);(B10)\nwhere\u001f0(T) =N0ln\u0010\n2e\r!D\n\u0019kBT\u0011\n,\r= 0:57721\u0001\u0001\u0001the Euler-Mascheroni constant and C0(T) = Re[ (0)(1\n2)\u0000 (0)(1\n2+\niE(k)\n2\u0019kBT)] with (0)(z) being the digamma function.\nFor orbital-independent pairing considered in the main text \u0001 s\u001c0i\u001b2, we have\n\u001fs(T) =\u001f0(T) +N0\n2\u0014\nC0\u0012\nT;E+\u0000E\u0000\n2\u0013\n+C0\u0012\nT;E++E\u0000\n2\u0013\u0015\n+N0\n2\u0014\nC0\u0012\nT;E+\u0000E\u0000\n2\u0013\n\u0000C0\u0012\nT;E++E\u0000\n2\u0013\u0015\n\u0002\u00152+\u00152\nsoc\u0000h2\nE+E\u0000\n\u0011\u001f0(T) +N0fs(T;\u0015soc;\u0015;h):(B11)17\nIn order to look at the e\u000bect of \u0015on the Pauli limit, we could Taylor expand fs(T;\u0015soc;\u0015;h) for small\u0015:\nfs(T;\u0015soc;\u0015;h) =fs(T;\u0015soc;0;h) +F(T;\u0015soc;h)\u00152+O(\u00154); (B12)\nwith\nF(T;\u0015soc;h) = (2)(1\n2)\u00152\nsoch2\n4\u0019k2\nBT2(\u00152soc+h2)2\n\u0000Ref (0)(1\n2)\u0000 (0)(1\n2+ip\n\u00152soc+h2\n2\u0019kBT)g4\u00152\nsoch2\n(\u00152soc+h2)3\n+ Imf (1)(1\n2+ip\n\u00152soc+h2\n2\u0019kBT)g\u00152\nsoch2\n2\u0019kBT(\u00152soc+h2)5=2:(B13)\nThis is used in the main text.\nFor orbital-dependent pairing \u0001 o(d1\u001c1+d3\u001c3)i\u001b2withdo=~g, we have\n\u001fo(T) =\u001f0(T) +N0\n2\u0014\nC0\u0012\nT;E+\u0000E\u0000\n2\u0013\n+C0\u0012\nT;E++E\u0000\n2\u0013\u0015\n+N0\n2\u0014\nC0\u0012\nT;E+\u0000E\u0000\n2\u0013\n\u0000C0\u0012\nT;E++E\u0000\n2\u0013\u0015\n\u0002\u00152\u0000\u00152\nsoc\u0000h2\nE+E\u0000\n\u0011\u001f0(T) +N0fo(T;\u0015soc;\u0015;h):(B14)\nAppendix C: Strain e\u000bect on Tcand pairing\nsymmetry\nThe strain e\u000bect characterized by Eq. (6) in the\nmain text can be absorbed into the orbital hybridiza-\ntion vector goand gives rise to an e\u000bective ~g\u0011go+\ntstr=\u0015o(sin 2\u001e;0;cos 2\u001e). Then in the absence of SOC\nterms, the corrected critical temperature Tcdue to the\nstrain and hybridization e\u000bects is perturbatively given by\nln\u0012Tc\nT0\u0013\n=Z Z\nSd\nC0(T0)\u0010\njdoj2\u0000jdo\u0001^~gj2\u0011\n;(C1)\nwhereT0is the critical temperature without strain or hy-\nbridization and the integration is over the solid angle of\nkover the Fermi surface. Similar to previous discussions,\nthe strain generally suppresses the critical temperature\nwhen ~gis not exactly parallel to do, as shown in Fig. 8\n(a). For non-zero strain, the Tcis not suppressed when\ndojj~g. Fig. 8 (b) shows the symmetry breaking pattern of\nthejdoj, which is proportional to the SC gap (the propor-\ntionality constant has been normalized to 1 in the \fgure),\naround the Fermi surface. The strain would reduce the\nsymmetry from C4toC2, as expected.\nAppendix D: TSC with \u0001s= 0;\u0015o= 0\nTo demonstrate the topology, we also show a simple\ncase with \u0001 s= 0 and\u0015o= 0, where the Z2can be\ncharacterized analytically.\nIn this section, we focus on the simpli\fed case without\norbital independent pairing or orbital hybridization. InFig. 4 (c), we calculate the edge spectrum with kxbe-\ning a good quantum number in a semi-in\fnite geometry,\nand it shows the corresponding bulk band structure to-\ngether with two counter-propagating MEMs. The bulk\ntopology of the 2D helical TSC phase is characterized by\ntheZ2topological invariant \u0017, which can be extracted by\ncalculating the Wilson-loop spectrum. And, \u0017= 1 mod\n2 characterizes the helical TSC. In Fig. 4 (b), we plot\nthe evolution of \u0012as a function of ky, and the winding\npattern indicates the topological Z2invariant\u0017= 1.\nOn the other hand, with \u0001 s= 0, which is the case\nif we only consider on-site attractive interactions be-\ntween electrons [102, 103], the BdG Hamiltonian (13)\ncan be decomposed into two orbital subspaces that are\nrelated through time-reversal transformation. Each of\nthese blocks has a well-de\fned Chern number because\neach block alone breaks TRS. The two Chern numbers\ncan then be used to de\fne the Z2invariant of the whole\nBdG system. The detailed procedures are the following.\nFor the normal Hamiltonian given in Eq. (1), we have\n[H0;\u001c2] = 0. As a result, we can project the normal\nHamiltonianH0in Eq. (1) into block-diagonal form cor-\nresponding to the \u00061 eigenvalues of \u001c2by using the basis\ntransformationU=\u001b0\n1p\n2\u0014\n1\u0000i\n1i\u0015\n. The new basis is\ngiven by\n~\ty(k) = (cy\n+;\";cy\n+;#;cy\n\u0000;#;cy\n\u0000;\"); (D1)\nwherecy\n\u0006;s\u00111p\n2(cy\ndxz;s\u0007icy\ndyz;s). On this basis, the\nnormal Hamiltonian is given by\nH0=H+\n0\bH\u0000\n0; (D2)18\nTc/T0t/tstr(a)tstr/λokF2=0tstr/λokF2=1tstr/λokF2=20.00.51.01.52.00.50.60.70.80.91.0\n|do(θ)|θ(b)tstr/λokF2=0tstr/λokF2=0.30π2π0.00.51.01.52.0\nFIG. 8. (a) shows the suppression of Tcfor di\u000berent strain strengths. Here do=go+t\n\u0015ogstrwhereas ~g=go+tstr\n\u0015ogstr.\n(b) shows the symmetry breaking of the SC gap from C4toC2due to the existence of the external strain. We have chosen\ngo= (3kxky;0;k2\nx\u0000k2\ny) and the strain parameter \u001e= 0 in gstr.\nwhereH\u0006\n0are given by\nH\u0006\n0=\u000f(k) +\u0015R(kx\u001b2\u0000ky\u001b1)\u0007\u0015soc\u001b3: (D3)\nNote that the time-reversal transforms H\u0006\n0(k) to\nH\u0007\n0(\u0000k). In the new basis the pairing Hamiltonian also\ndecouples asH\u0001=H+\n\u0001\bH\u0000\n\u0001withH\u0006\n\u0001given by\nH\u0006\n\u0001= 2\u0001\u0006h\ncy\n\u0006;\"(k)cy\n\u0006;#(\u0000k)\u0000(\"$# )i\n+ h.c.;(D4)\nwhere \u0001\u0006\u0011\u0001o(\u0007id1\no+d3\no) are the gap strengths in each\nsubspace. Therefore, the Bogoliubov de-Gennes (BDG)\nHamiltonian takes the following block-diagonal form,\nHBdG=H+\nBdG\bH\u0000\nBdG; (D5)\nwhere\nH\u0006\nBdG(k) = (\u000f(k)\u0007\u0015soc\u001b3)\r3+\u0015R(kx\u001b2\r3\u0000ky\u001b1\r0)\n\u00062d1\u001b2\r1\u00002d3\u001b2\r2; (D6)\nwith\r\u0016being the Pauli matrices in the particle-\nhole space. The Nambu basis is \ty\n\u0006(k) =\n(cy\n\u0006;\"(k);cy\n\u0006;#(k);c\u0006;\"(\u0000k);c\u0006;#(\u0000k)). Each subspace\nhas its own particle-hole symmetry.\nBy symmetry, the 2D BdG Hamiltonian in Eq. (D5)\nbelongs to Class DIII of the A-Z classi\fcation[104, 105]\nfor topological insulators and superconductors because\nboth TRS and particle-hole symmetry are preserved.\nHowever, it is not the case for our model. The BdG\nHamiltonian here could exhibit topological states with Z2\ntype topological invariant, which can be de\fned as the\nfollowing. In each subspace, we de\fne the BdG Chern\nnumber as\nC\u0006=1\n2\u0019X\n\flled bandsZ\nBZdk\u0001h\u001e\u0006\nn(k)jirkj\u001e\u0006\nn(k)i;(D7)\nwithj\u001e\u0006\nnibeing the energy eigenstate of H\u0006\nBdG. Then\ntheZ2invariant, in this case, is then explicitly given by,\n\u0017\u0011C+\u0000C\u0000\n2; (D8)\nwhereC\u0006are the Chern numbers of the \u0006channels. This\nhas been discussed in the main text.Appendix E: Wilson loop calculation for Z2TSC\nIn the thermodynamics limit, the Wilson loop operator\nalong a closed path pis expressed as\nWp=Pexp\u0014\niI\npA(k)dk\u0015\n; (E1)\nwherePmeans path ordering and A(k) is the non-\nAbelian Berry connection\nAnm(k) =ih\u001en(k)jrkj\u001em(k)i; (E2)\nwithj\u001em;n(k)ithe occupied eigenstates. The Wilson line\nelement is de\fned as\nGnm(k) =h\u001en(k+ \u0001k)j\u001em(k)i; (E3)\nwhere the k= (kx;ky), and \u0001 k= (0;2\u0019=Ny) is the\nsteps. In the discrete case, the Wilson loop operator\non a path along kyfrom the initial point kto the \fnal\npoint k+ (0;2\u0019) can be written as Wy;k=G(k+ (Ny\u0000\n1)\u0001k)G(k+(Ny\u00002)\u0001k):::G(k+\u0001k)G(k), which satis\fes\nthe eigenvalue equation\nWy;kj\u0017j\ny;ki=ei2\u0019\u0017j\ny(kx)j\u0017j\ny;ki (E4)\nThe phase of eigenvalue \u0012= 2\u0019\u0017j\ny(kx) is the Wannier\nfunction center.\nAppendix F: Spin and orbital magnetizations: M s\nand Mo\nIn this section, we show the de\fnition of spin and or-\nbital magnetization at the mean-\feld level. The spin\nmagnetization in orbital-inactive systems takes the form\nMs/X\nk;s1;s2hcy\ns1(k)\u001bs1s2cs2(k)i; (F1)19\nwhich tells us the magnetic moments generated by spin\npolarization. Similarly, the orbital magnetization in\norbital-active system is given by\nMo/X\nk;s;a;bhcy\ns;a(k)\u001cabcs;b(k)i: (F2)\nThe di\u000berent components of the orbital magnetization\nvector represent di\u000berent orders in the SC ground state.More speci\fcally, we have\nMx\no=X\nk;shcy\ns;dxzcs;dyz+cy\ns;dyzcs;dxzi; (F3)\nMy\no=\u0000iX\nk;shcy\ns;dxzcs;dyz\u0000cy\ns;dyzcs;dxzi (F4)\n=1\n2X\nk;sh^ns;dxz+idyz\u0000^ns;dxz\u0000idyzi; (F5)\nMz\no=X\nk;shcy\ns;dxzcs;dxz\u0000cy\ns;dyzcs;dyzi: (F6)\nWe see that Mx;z\nobreaks theC4rotation symmetry and\nMy\nobreaks TRS. In our work, we only consider the pos-\nsibility of spontaneous TRS breaking, thus the Mx;z\no\nwill not couple to the superconducting order parameters,\nwhich are required to be invariant under Cn. BecauseMy\no\nbreaks TRS so that it could be coupled to the supercon-\nducting order parameters, which spontaneously breaks\nTRS. This is one of the main results of our work,\n(0;My\no;0)/id\u0003\no\u0002do; (F7)\nwhere the complex orbital do-vector breaks TRS."    },    {        "title": "1212.0420v5.Normal_state_properties_of_spin_orbit_coupled_Fermi_gases_in_the_upper_branch_of_energy_spectrum.pdf",        "content": "arXiv:1212.0420v5  [cond-mat.quant-gas]  2 May 2013Normal state properties of spin-orbit coupled Fermi gases i n the upper branch of\nenergy spectrum\nXiao-Lu Yu, Shang-Shun Zhang, and Wu-Ming Liu\nBeijing National Laboratory for Condensed Matter Physics,\nInstitute of Physics, Chinese Academy of Sciences, Beijing 100190, China\n(Dated: June 17, 2021)\nWe investigate normal state properties of spin-orbit coupl ed Fermi gases with repulsive s-wave\ninteraction, in the absence of molecule formation, i.e., in the so-called “upper branch”. Within\nthe framework of random phase approximation, we derive anal ytical expressions for the quasi-\nparticle lifetime τs, the effective mass m∗\ns, and the Green’s function renormalization factor Zsin the\npresence of Rashba spin-orbit coupling. In contrast to spin -orbit coupled electron gas with Coulomb\ninteraction, we show that the modifications are dependent on the Rashba band index s, and occur\nin the first order of the spin-orbit coupling strength. We als o calculate experimental observable such\nas spectral weight, density of state and specific heat, which exhibit significant differences from their\ncounterparts without spin-orbit coupling. We expect our mi croscopic calculations of these Fermi\nliquid parameters would have the immediate applications to the spin-orbit coupled Fermi gases in\nthe upper branch of the energy spectrum.\nPACS numbers: 03.75.Ss, 05.30.Fk, 67.85.Lm\nI. INTRODUCTION\nMotivated by the recent success on the evidence of\nStoner ferromagnetism for the repulsive Fermi gas in the\nupper branch of the energy spectrum [1], there has been\nincreasing interest in the nature of uncondensed Fermi\ngas (free of molecules) within the repulsively interacting\nregime [2], which naturedly becomes the well-controlled\nplatform for simulating Landau Fermi liquid. Much of\nthe interest in ultracold atomic gases comes from their\namazing tunability. A wide range of atomic physics and\nquantum optics technology provides unprecedented ma-\nnipulationofavarietyofintriguingquantumphenomena.\nBased on the Berry phase effect [3] and its non-Abelian\ngeneralization [4], Spielman’s group in NIST has suc-\ncessfully generated a synthetic external Abelian or non-\nAbelian gauge potential coupled to neutral atoms. Re-\ncent experiments have realized the atomic40K [5] or6Li\n[6] gases with spin-orbit coupling (SOC). These achieve-\nments will open a whole new avenue in cold atom physics\n[7–10].\nThe effect of SOC in fermionic systems has conse-\nquently become an important issue in recent years, and\nattracts a great deal of attentions in ultracold Fermi\ngases. Most of existing works are devoted to the effect of\nSOC on the superfluid state with negative s-wave scat-\ntering length [11–13] and the Bardeen-Cooper-Schrieffer\n(BCS) to Bose-Einstein condensation (BEC) crossover\n[14–17]. Furthermore, the SOC give rise to a variety of\ntopological phases, such as the quantum spin Hall state\nand the topological superfluid [18–21]. In addition to the\nstudy of SOC effect on these symmetry-breakingor topo-\nlogical phases, the normal state contains various poten-\ntial instabilities and deserves attention. The considera-\ntion of SOC systems in the framework of Fermi liquid\ntheory is therefore desirable. As is well known, Lan-\ndau’s Fermi-liquid theory provides a phenomenologicalapproach to describe the properties of strongly interact-\ning fermions. Fermi liquid parameters characterizing the\nrenormalizedmany-bodyeffective interactionshaveto be\ndetermined through experimental results.\nThe purpose of this paper is to study key normal-state\npropertiesoftwodimensional(2D)FermigaseswithSOC\nin the repulsive regime—their quasi-particle lifetime, ef-\nfective massand Green’s function renormalizationfactor.\nFollowing previous studies of Landau’s Fermi-liquid the-\nory including SOC [22, 23], we are attempted to build\na microscopic foundation of phenomenological parame-\nters. Therefore within the framework of random phase\napproximation (RPA), we derive analytical expressions\nfor the quasi-particle lifetime τs, the effective mass m∗\ns,\nand the Green’s function renormalization factor Zsfor\na 2D Fermi gases with repulsive s-wave interaction in\nthe presence of Rashba SOC. To make contact with cur-\nrent experiments directly, we also calculate experimental\nobservable such as spectral weight, density of state and\nspecific heat, and discuss their corresponding experimen-\ntal signatures. We shall show that the modifications are\ndependent on the Rashba band index sdenoting the two\ndirectionsofthe eigenspinorsofthe RashbaHamiltonian.\nThe paper is organizedasfollows. The model Hamilto-\nnian and the renormalizations due to s-wave interaction\nin the presence of SOC is discussed in Sec. II. In Sec.\nIII, Starting from the RPA self-energy of the SOC Fermi\nliquid, we derived all the analytical formula of the Fermi\nliquid parameters in presence of SOC. The experimen-\ntal observable quantities such as the spectral function,\ndensity of states and specific heat are calculated. Sec.\nIV is devoted to discuss the experimental measurements\nof these fundamental parameters and their correspond-\ning experimental signatures. The comparisons with the\nordinary Fermi liquid are also presented.2\nII. PERTUBATIVE THEORY OF 2D FERMI\nGASES WITH s-WAVE INTERACTION IN THE\nPRESENCE OF RASHBA SOC\nA. Model Hamiltonian\nWe consider a 2D spin-1/2 fermionic system with\nRashba-type SOC and s-wave interaction, described by\nthe model Hamiltonian\nH=H0+HI. (1)\nThe non-interacting part H0reads as,\nH0=/summationdisplay\npc†\np[p2\n2m+α(ˆ z×p)·σ−µ]cp,(2)\nwherecp= (cp,↑,cp,↓)T,µ=k2\nF/2mis the chemical po-\ntential,αrepresents SOC strength and kFis the Fermi\nmomentum in the absence of Rashba-type SOC. The re-\nduced Plank constant /planckover2pi1is taken as 1 in this paper. The\nnon-interacting Hamiltonian H0can be diagonalized in\nthe helicity bases\n|k,s/angb∇acket∇ight=1√\n2/parenleftbigg1\niseiφ(k)/parenrightbigg\n,s=±1, (3)\nwhereφ(k) = arctan( ky/kx) andsis the helicity of\nFermi surfaces, which represents that the in-plane spin\nis right-handed or left-handed to the momentum. The\ndispersion relations for two helical branches are ξk,s=\n(k2+2skR|k|−k2\nF)/2m, wherekR=mαcorresponds to\nthe recoil momentum in experiments [5, 6]. The Fermi\nsurfaces are given by ξk,s= 0, which yields two Fermi\nmomenta ks=κkF−skRwithκ=/radicalbig\n1+γ2. We have\ndefined the dimensionless SOC strength γ=kR/kF. Re-\ncently, the experimental realization of the SOC degen-\nerated Fermi gases have been reported [5–7]. By apply-\ning a pair of laser beams to the ultracold40K or6Li\natoms trapped in a anisotropic harmony trap, the equal\nweight combination of the Rashba-type and Dresselhaus-\ntype SOC is generated. Their elegant experiments are\nperformed in the weakly repulsive regime, which could\nprovide the possibilities to study the SOC degenerated\nFermi gases in the normal state.\nThe interacting part reads as\nHI= 2g/integraldisplayd2kd2pd2q\n(2π)6c†\nk+q,↑c†\np−q,↓cp,↓ck,↑,(4)\nwhereg= 2πNas/3√\n2πmζz, which is controlled by the\ns-wave scattering length as. HereNis the total atom\nnumber, ζz=/radicalbig\n1/mωzis the confinement scale of the\natomic cloud in the ˆ zdirection perpendicular to the 2D\nplane, and mis the mass for the ultracold atoms. Notice\nthat the universal properties of the low-energy interac-\ntion among ultra-cold atoms depend only on the scat-\ntering length as[24–32]. We focus on the normal stateregime of Fermi atomic gas in this work assuming posi-\ntive scattering length, which could be reached in the up-\nper branch of Feshbach resonance. We note that the gas\nof dimers and the repulsive gas of atoms represent two\ndifferent branches of the many-body system, both corre-\nsponding to positive values of the scattering length [33].\nThe atomic repulsive gas configuration has been experi-\nmentally achieved by ramping up adiabatically the value\nof the scattering length, starting from the value a= 0\n[34].\nB. Renormalizations due to the s-wave interaction.\nLet’s consider the problem in the helicity bases |k,s/angb∇acket∇ight.\nThe non-interacting Green’s function is\nG0\ns(k,ω) =1\nω−ξk,s+isgn(ω)0+. (5)\nThe Dyson’s equation expresses the relation between\nthe non-interacting and interacting Green’s functions in\nterms of the self-energy Σ sas\nGs(k,ω) =1\nω−ξk,s−Σs(k,ω). (6)\nAll the many-body physics is contained in the self-energy\nΣs. The poles of interacting Green’s function Gs(k,ω)\ngive the quasi-particle excitations, of which lifetime τs=\n1/Γscan be obtained from the imaginary part of self-\nenergy as\nΓs(k) =−2ImΣs(k,ξk,s). (7)\nThe real part of the self-energy gives a modification of\ndispersion relations. At low temperature, the properties\nof the low energy excitation in the vicinity of the Fermi\nsurface is essential. Thus we can expand the real part of\nself-energy to the first order of ωand|k|−ksas\nReΣs(k,ω) = ReΣ s(ks,0)+ω∂ωReΣs(ks,ω)|0\n+(k−ks)·∇kReΣs(k,0)|ks.(8)\nThe interacting Green’s function now becomes\nGs(k,ω) =Zs\nω−ξ∗\nk,s+i(1/2)Γs(k),(9)\nwhereξ∗\nk,sisthemodified energydispersion. TheGreen’s\nfunction acquires a renormalized factor\nZs=1\n1−As, (10)\nwhereAs=∂ωReΣs(ks,ω)|0. The Fermi velocity vks=\n∂ξ∗\nk,s/∂k|kscan be calculated as follows\n∂ξ∗\nk,s\n∂k||k|=ks=Zs(κkF/m+∂kReΣs(k,0)|ks).(11)3\nk\rsq,q 0\nk-q \rrk\rs\np+q,p +q 0 0 \np,p 0=\n+ + + ...q,q 0,k -q 0 0\nk-q \rr ,k -q 0 0\nFIG. 1. Feynman diagrams for the self-energy of the SOC\nFermi liquid in the presence of s-wave interaction. The Feyn-\nman rules are defined under the helicity bases. The label s\nandrdenote the helicity index. The self-energy is calculated\nwithin the framework of RPA [35].\nFornon-interactingFermigas, the Fermivelocityis v0\nks=\nκkF/m. Therefore we introduce the effective mass via\n∂ξ∗\nk,s/∂k||k|=ks=κkF/m∗\ns. The effective mass in terms\nof self-energy is\nm∗\ns\nm=1\nZs/parenleftbigg\n1+m\nκkF∂kReΣR\ns(ks,0)/parenrightbigg−1\n.(12)\nThe Eqs. (7), (10) and (12) are our starting points\nof microscopic calculations of normal states properties,\nwhich embody the main properties of a quasi-particle in\nthe Landau theory of Fermi liquids.\nIII. SOC FERMI LIQUID PARAMETERS WITH\nREPULSIVE s-WAVE INTERACTION\nA. RPA Self-energy\nTo investigate the renormalization effects in two\nRashba energy bands separately, it is convenient to work\nin the helicity bases. The interacting part of the Hamil-\ntonian in the helicity bases is rewritten as\nHI=/summationdisplay\nk,p,qVss′;rr′(k,p,q)ϕ†\nk+q,s′ϕ†\np−q,r′ϕp,rϕk,s,(13)\nwhere the interaction vertex Vss′;rr′(k,p,q) =\ngfss′(θk,θk+q)frr′(θp,θp−q). Due to the presence\nof SOC, the spin is locked to momentum. The interac-\ntion vertex acquires an overlap factor fss′(θk,θk+q) and\nfrr′(θp,θp−q), which is defined by\nfss′(θk,θp) =1\n2(1+ss′ei[θk−θp]),(14)\nwhereθkandθpare the azimythal angles of kandpre-\nspectively. WithintheframeworkofRPA,theinteractionPath C1Path CRe[ω] i Im[ ω] \nRe[ω] i Im[ ω] \nPath C2\nFIG. 2. (Color online). (a): The contours of integration in\nthe complex ωplane for the self-energy given byEq. (19). For\nintermediate states with the energy 0 < ξk−q,r< ω, the pole\nof the propagator falls in the first quadrant. (b): Schematic of\nthe deformation of the contour into the imaginary axis. The\nself-energy given by Eq. (19) is equal to the integration alo ng\nthe path C1 with an additional contribution of the residue as\nshown by the integral path C2.\nvertex is modified as (see Fig. 1)\nVRPA\nss′,rr′(k,p,q,ω) =g\nǫ(q,ω)fss′(θk,θk+q)frr′(θp,θp−q),\n(15)\nwhere the dielectric function ǫ(q,ω) = 1+gχ(q,ω) and\nχ(q,ω) is the bare density-density susceptibility of non-\ninteracting SOC Fermi gas. In the long wavelength and\nlow frequency limit, the susceptibility can be carried out\nas\nReχ(y) =m\nπ/bracketleftBigg\n1−|y|/radicalbig\ny2−1Θ(|y|−1)/bracketrightBigg\n,\nImχ(y) =m\nπy/radicalbig\n1−y2Θ(1−|y|), (16)\nwherey=mω/κk F|q|. It can be seen from Eq. (16) that\nthe susceptibility satisfies the following relations\nχ∗(q,ω) =χ(−q,−ω) =χ(q,−ω).(17)\nIt’simportanttonoticethat thebaresusceptibilityisreal\nfor|ω|> vF/radicalbig\n1+γ2|q|. For|ω|< vF/radicalbig\n1+γ2|q|, the\nbaresusceptibilityinEq. (16)containsanimaginarypart\nwhich represents the absorptive behavior of the medium.\nThisimaginarypartisresponsibleforthefinitelifetimeof\nthe quasi-particle in the medium. Along the imaginary\naxis, the analytical formula of χ(q,ω) is much simpler\n(see Fig. 2) as\nχ(q,iω) =m\nπ[1−|y|/radicalbig\ny2+1]. (18)\nThe RPA self-energy (see Fig. 1) is\nΣs(k,ω) =i/integraldisplay\nCd2qdq0\n(2π)3/summationdisplay\nrgFsr\nǫ(q,q0)G0\nr(k−q,ω−q0),\n(19)4\nwhere the Fsris the overlap factor\nFsr=1+srcos(θk−θk−q)\n2. (20)\nThe integral path Cis shown in Fig. 2 (a). After defor-\nmation of contour path in Fig. 2 (b), the pole gives rise\nto a residue contribution\nΣpole\ns(k,ω) =−/summationdisplay\nr/integraldisplay\nDrd2q\n(2π)2g\nǫFsr,(21)\nwhere the region of integration Driskr<|k−q|<\n|k|−(r−s)kRandǫ= 1+gχ0(q,ω−ξk−q,r). The line\nintegral along the imaginary axis is\nΣline\ns(k,ω) =−/integraldisplay∞\n−∞d2qdq0\n(2π)3/summationdisplay\nrgFsr\nǫ(q,iq0)1\nω−iq0−ξk−qr.\n(22)\nThe total self-energy is given by\nΣs= Σpole\ns+Σline\ns. (23)\nThe advantage of this decomposition is that the imagi-\nnary part is given by the contribution of residue, and the\nreal part is mainly determined by the line integral.\nB. Quasi-particle lifetime\nThe quasi-particle lifetime can be calculated by the\nimaginary part of self-energy, which comes from the con-\ntribution of residue in Eq. (21). At zero temperature\nand for quasi-particle ξk,s>0, the imaginary part of the\nself-energy reads\nΓs(k) =2/summationdisplay\nq,rΘ(ξk−q,r)Θ(ξk,s−ξk−q,r)\n×ImVRPA\nsr;sr(k,k−q,q;ξk,s−ξk−q,r).(24)\nThe imaginary part of the RPA vertex in the helicity\nbases is\nImVRPA\nsr;rs(k,k−q,q,k;w) =gFsrIm1\nǫ(y),(25)\nwherew=ξk,s−ξk−q,randy=w/vF|q|. Sincethe main\ncontribution of integral comes from the forward scatter-\ning, i.e., y≪1, the density-density susceptibility can be\nexpanded about y= 0. One finds that the susceptibility\nin this neighborhood is\nχ(y) =m\nπ(1+iy)+O(y2). (26)\nThe imaginary part of RPA vertex goes to\nImVRPA\nsr;rs(k,k−q,q;w)≃ −m2g2\n(mg+π)2π\nmyFsr.(27)(k-ks)/k Fτs-1 (units of εF)\nγ=0.5γ=0 (ordinary Fermi liquid )\n0.00 0.02 0.04 0.06 0.08 0.100.000.050.100.150.20\nFIG. 3. (Color online). The inverse of the lifetime τsfor\n40K ultracold atoms as a function of the momentum kin the\nvicinity of the Fermi surface. The lifetime of quasi-partic le is\nenhanced due to the presence of SOC. The parameters taken\nhere are: the number of atoms is about 104,kR=h/λwith\nλ= 773nm, γ= 0.5, trap frequency ωz= 2π×400Hz, and\nas= 32a0, where a0is the Bohr Radius. The unit ǫF=\n/planckover2pi1×0.21MHz.\nSubstituting Eq. (27) into Eq. (24), the inverse lifetime\nΓs(k) can be evaluated as\nΓs(k) =−m2g2ǫF\nπ(mg+π)2δ2/braceleftbigg\nlnδ\n8−1\n2−γ2lnγ\n4/bracerightbigg\n,(28)\nwhereδ= (k−ks)/kF. The result for ordinary Fermi\nliquid in the presence of s-wave repulsive interaction can\nbe obtained by taking the limit γ→0 in Eq. (28). The\neffect of SOC is shown in Fig. 3 via the comparison with\nthe ordinary Fermi liquid with the same strength of s-\nwave repulsive interaction, where we can conclude that\nthe quasi-particleis much stablerin the presence ofSOC.\nC. Green’s function renormalization factor\nOur starting point is the real part of the self-energy\nwhich contains two parts: one is the residue contribution\ngiven by Eq. (21), the other is the integral along the\nimaginary axis given by Eq. (22). Thus we want to\nevaluate\nAs=Apole\ns+Aline\ns\n=∂ξΣpole\ns(ks,ξ)|ξ=0+∂ξΣline\ns(ks,ξ)|ξ=0.(29)\nGiven the rotation symmetry, we only need to consider\nks=ksex. The first term in Eq. (29) is\nApole\ns=/summationdisplay\nr/integraldisplayd2q\n(2π)2δ(ξks−q,r)g\nǫFsr.(30)5\n50 100 150 200Analytical \nNumerical γ=0 \nFIG. 4. (Color online). Renormalization factor Zsas func-\ntions of scattering length aswith the same parameters in Fig.\n3. The black thick line represents the renormalization fact or\nof ordinary Fermi gases ( γ= 0). The red (dashed) and blue\n(dotted) lines represent the analytical results for the SOC\nFermi gas given by Eq. (34). The discrete points are evalu-\nated numerically.\nThe second term in Eq. (29) can be integrated by parts,\nwhich gives two parts: the first one reads as\nAline\ns=−/summationdisplay\nr/integraldisplayd2q\n(2π)2δ(ξks−q,r)g\nǫFsr,(31)\nwhich cancels the residue contribution given by Eq. (30).\nThus the final result of Eq. (29) can be expressed as\nAs=−mgk2\nF\n(2π)3k2s/integraldisplay2π\n0dφ/integraldisplay∞\n0d¯y/integraldisplay∞\n0dx/summationdisplay\nrImfsr(x,¯y,φ),\n(32)\nwhere we have defined ¯ y=mw/qk s,x=q/2ksand\nfsr(x,¯y,φ) =Fsr\ni¯y−µ(x,φ)1\nǫ2∂ǫ\n∂¯y. (33)\nHereµs,r(x,φ) =mξk−q,r/|q|ks, and the overlap fac-\ntorFsr(x,φ) = 1/2 +sr(1−2xcosφ)/2l, withl=\n|k−q|/ks=/radicalbig\n1−4xcosφ+4x2.\nFromEqs. (10), (32)and(33), wecanobtaintherenor-\nmalization factor straightforwardly\nZ−1\ns= 1+m2g2\n8π(mg+π)1\n(κ−sγ)2\n= 1+m2g2\n8π(mg+π)(1+sγ+O(γ2)).(34)\nThe renormalization factor turns out to be dependent on\nthe helicity s, and the leading correction due to SOC is\nO(γ), which is different from the results of two dimen-\nsional electron gas (2DEG) with SOC in semiconductors\n[36]. We show the analytical results along with the nu-\nmerical calculation for Zsas a function of the s-wave\nscattering length in Fig. 4 and the strength of SOC in\nFig. 5. The Green’s function renormalization factor for\nordinaryFermiliquid isalsoshownforcomparisonin Fig.γAnalytical\nNumerical\nFIG. 5. (Color online). Renormalization factor Zsas func-\ntions of dimensionless SOC strength γwith the same param-\neters in Fig. 3. The red (dashed) and blue (dotted) lines\nrepresent the analytical results for the SOC Fermi gas given\nby Eq. (34). The discrete points are evaluated numerically.\n4. We can see from Fig. 5 that the renormalization fac-\ntor is reduced for the s= +1 branch while enhanced for\nthes=−1 branch with increasing strength of SOC.\nD. Effective mass\nThe effective mass can be evaluated by the real part\nof the static self-energy in the vicinity of the Fermi sur-\nface from Eq.(12). In contrast to the calculation of the\nrenormalization factor Zs, the contribution of residue is\nirrelevant now. The correction of the effective mass is\nisotropic due to the rotation symmetry. Without loss of\ngeneralities, we assume ks=ksexin the following. We\nbegin with\n∂kReΣs(k,0)||k|=ks= Re/integraldisplay∞\n−∞d2qdw\n(2π)3\n×∂k/summationdisplay\nr1\niw+ξk−q,rVRPA\nsr;sr(k,q,iw).(35)\nThe interaction vertex is dependent on the external mo-\nmentum kbecause of the overlap of the helical eigen-\nstates. It is instructive to consider some special cases.\nFor weak SOC ( γ≪1), the integration in Eq. (35) can\nbe expanded to\n∂kReΣs(k,0)||k|=ks=kFg\n4πsγ+O(γ2).(36)\nIn contrast to the SOC Fermi liquid with Coulomb inter-\naction [36], our result is band dependent and has a first\norder correction γ. The effective mass reads as\nm∗\ns\nm=Z−1\ns(1+smg\nmg+πγ\n4κ)−1. (37)\nFor strong SOC ( γ∼1), we show the numerical results\nalong with the analytical results of Eq. (37) in Fig. 6.6\nAnalytical\nNumerical\n10 20 30 40           0γ=0\nFIG. 6. (Color online). Effective mass as functions of s-wave\nscattering length aswith the same parameters as in Fig. 3.\nThe red (dashed) and blue (dotted) line denote the case for\ns= +1 and s=−1 respectively. The solid triangle and circle\npoints are the corresponding numerical results. The black\nline represents the analytical results for ordinary Fermi l iquid\n(γ= 0). The many-body modifications of the effective mass\nare dependent on the helical bands.\nThe effect of SOC is shown in Fig. 7, from which we\ncan see that the effective mass for the s= +1 branch\nis enhanced while the s=−1 branch is reduced with\nincreasing strength of SOC.\nE. Spectral function, density of state and specific\nheat\nA close related quantity is the spectral function\nA(k,ω), which is the imaginary part of the single par-\nticle Green’s function [39, 40]\nA(k,ω) =−1\nπImGret(k,ω), (38)\nwhereGret(k,ω) is the retarded Green’s function. It\ncanbe straightforwardlyevaluatedfromthe time-ordered\nGreen’s function G(k,ω) as\nImGret(k,ω) = ImG(k,ω)sign(ω),\nReGret(k,ω) = ReG(k,ω). (39)\nWith the great improvements in the spectroscopic tech-\nnique, it has been possible to directly measure the\nlow-energy spectral weight function of a 2D system in\nthe momentum-resolvedradiofrequency(rf) experiments\n[42–44]. In Fig. 8, we show the spectral functions of the\ntwo Rashba bands separately. The spectral functions at\ntheFermisurfacesasshowninFig. 8(a)and(b)(vertical\narrow) have the form\nA(ks,ω) =Zsδ(ω). (40)\nFig. 8 (c) and (d) are density plots of the spectral func-\ntions, which could be compared with the results of the\nmomentum-resolved rf spectroscopy in current experi-γ0.25 0.5 0.75 1.0 0Analytical\nNumerical\nFIG. 7. (Color online). Effective mass as functions of the\nstrength of SOC γwith the same parameters as in Fig. 3.\nThe red (dashed) and blue (dotted) line denote the case for\ns= +1 and s=−1 respectively. The solid triangle and circle\npoints are the corresponding numerical results.\nments.\nGiven the spectralfunctions, we canobtain the density\nof states (DOS) of the fermionic system with SOC via\n[39, 40]\nρ(ω) =/summationdisplay\nsρs(ω) =/summationdisplay\ns/integraldisplayd2k\n(2π)2As(k,ω).(41)\nFor the non-interacting case, the DOS now becomes\nρ0(ω) =\n\n0, ω < −κ2k2\nF\n2m,\nm\nπγ√\n2mω/k2\nF+κ2,−κ2k2\nF\n2m< ω <−k2\nF\n2m,\nm\nπ, ω > −k2\nF\n2m.(42)\nThe DOS is modified in the presence of s-wave in-\nteraction. On the Fermi surfaces, for non-interacting\nSOC Fermi gas, the DOS in units of m/πis given by:\nρ0\n+1(EF) = 0.28,ρ0\n−1(EF) = 0.72, and ρ0(EF) = 1\nforγ= 0.5. For the interacting SOC Fermi gas, the\nDOS is evaluated numerically as follow: ρ+1(EF) = 1.37,\nρ−1(EF) = 0.83 andρ(EF) = 2.20, where all the param-\neters are the same with Fig. 3.\nThe quasi-particles from two helical bands both con-\ntribute to the specific heat. At low temperature, the\nspecific heat is proportional to the DOS on the Fermi\nsurfaces and the temperature T. Thus the ratio between\nthe specific heats at low temperature is\ncv\nc0v=ρ(EF)\nρ0(EF)= 2.20, (43)\nwherec0\nvis the specific heat of SOC Fermi gases without\ninteractions.7\n(a)                                                       (b)\n(c)                                                       (d)ω/ εFs=+1\n0.2 0.050.5\u0011\u000f\u0013\u0014 δ(ω) \u0011\u000f\u0017\u0018 δ(ω) \ns=+1 s=-1\n\u0011\u0013\u0011 1.01.52.02.5\n0.01.0\n0.02.03.04.05.06.0\ns=-1ω/εF ω/ εF\nk/kF\u0012\u0011 3\nA(k, ω)/ εF \u0012\u0011 3\n0.0 0.05 0.1 0.0 0.1 0.2 \n\u0011\u0013\u0011 A(k, ω)/ εF ω/εF \nk/kF\nFIG. 8. (Color online). Zero temperature spectral func-\ntion at different values of ( k−ks)/kFare shown at (a) and\n(b). All the parameters are the same with Fig. 3. (a)\nis fors= +1 and (b) is for s=−1 respectively. The\neight peaks, from left to right, correspond to ( k−ks)/kF=\n−0.01,−0.075,−0.05,−0.025,0.025,0.05,0.075,0.01. The\nvertical arrows in (a) and (b) denote δfunctions at ω= 0\nwith weights 0 .23 and 0 .67 respectively. (c) and (d) are den-\nsity plots of the spectral function for the same parameters a s\nabove, which correspond to s= +1 and s=−1 respectively.\nThe white dashed line denotes the modified single particle\ndispersion.\nIV. DISCUSSIONS AND SUMMARIES\nWehaveobtainedvariousnormalstatequantitiesof2D\nSOC Fermi gases in the presence of repulsive s-wave in-\nteraction. Ultimately, to makecontact with experiments,\ntwo practical considerations warrant mention. (i): The\nrepulsive s-wave interaction can be achieved on the up-\nper branchof a Feshbach resonance. One problem should\nbe considered is that the upper branch of a Feshbach\nresonance is an excited branch, and will decay to the\nBEC molecule state due to inelastic three-body collisions\n[37]. However, with small scattering length, the decay\nrate is well suppressed [1, 38] and the system may be\nmetastable for observation. (ii): Recently, the SOC de-\ngenerate Fermi gases have been realized in the ultra-cold\natom systems[5, 6]. The effective SOC generatedin their\nexperimental schemes is an equal weight combination of\nRashba-type and Dresselhaus-type SOC. In this paper,\nwe have investigated the case of Rashba-type SOC. The\nDresselhaus-type SOC is presented in Appendix, which\nis demonstrated to give the same normal state properties\nas the Rashba case.\nIn current experiments, we consider the following typ-\nical experimental parameters for quasi-2D systems. One\ncan trap about 104 40K atoms within a pancake-shaped\nharmonic potential with the trap frequency chosen as\n2π×(10,10,400)Hz along the (ˆ x,ˆy,ˆz) direction. The\nsystem size can be estimated as (37 .8,37.8,5.98)µm.The other related parameters are taken as: as= 32a0,\nγ= 0.5,kR= 2π/λ= 8.128×106m. The Fermi liquid\nparameters are listed in Table. I under this typical ex-\nperimental setup. For comparison, we give the results of\nordinary Fermi liquid and the 2DEG in semiconductors\ntogether. We should notice that the single particle spec-\ntral function captures the valuable information for low\nenergy excitations of the Fermi liquid and can be mea-\nsured by means of mometnum-resolved rf experiments\n[41–44]. The rf spectroscopy is a technique used to probe\natomic correlation by exciting atoms from occupied hy-\nperfine states to another (usually empty) reference hy-\nperfine state. The single-particle spectral function is ob-\ntained in experiment through the mometnum-resolvedrf-\ntransfer strength. As a result, the inferences drawn from\nthe spectral function in the vicinity of the Fermi surfaces\ncan be used to determine the Fermi liquid parametersde-\nscribingthe lowenergybehaviorsofthenormalstate. Up\nto now, the rf experiment is the most promising method\nin ultracold atomic gases to measure these Fermi liquid\nparameters.\nTABLE I. Normal state properties for SOC Fermi liquid ( γ=\n0.5), ordinary Fermi liquid ( γ= 0), and 2DEG ( γ= 0.051) in\nsemiconductor. All other parameters used here are the same\nwith Fig. 3.\nγ= 0.5 γ= 0 2DEG( γ= 0.051)†\n1/τ††\ns0.73kHz 0 .67kHz 55 .36GHz\nZsZ+1= 0.23,\nZ−1= 0.67.0.96 0 .97\nm∗\ns/mm∗\n+1/m= 4.88,\nm∗\n−1/m= 1.16.1.04 0 .98\n†The results for 2DEG in InGaAs are taken from Ref.\n[36]. Compared with the SOC Fermi liquid in the pres-\nence ofs-wave interaction, the results for the 2DEG\nwith Coulomb interaction are independent on the en-\nergy band, and the leading order correction relative to\nthe SOC strength is γ2.\n††The values of the inverse lifetime are evaluated at ( k−\nk±1)/kF= 0.01. Compared with 2DEG in the typical\nsemiconductor, the quasi-particle in ultracold atomic\ngases is much stabler.\nIn summary, we studied the normal state properties\nof the SOC Fermi gas with repulsive s-wave interaction.\nThe quasi-particle lifetime τs, the renormalization factor\nZs, the effective mass m∗\ns/mare calculated, which em-\nbody the main properties of a quasi-particle. To make\ncontactwithexperimentsdirectly,wecalculatedthespec-\ntral function A(k,ω), density of state ρ(ω), and the spe-\ncificheat cvatlowtemperature. Thesequantitiesprovide\na good description of the low energy physics with SOC\nands-wave interaction, which are measurable in current\nexperiments. The analytical and numerical results show8\nthat the normal state properties are distinct for the two\nenergy bands, and the leading correction relative to the\nordinary Fermi liquid with s-wave interaction is on the\norder of γ, which are strikingly different from the SOC\nFermi liquid with Coulomb interaction [36]. We expect\nour microscopic calculations of the Fermi liquid param-\neters and related quantities would have the immediate\napplicability to the SOC Fermi gasesin the upper branch\nof the energy spectrum.\nACKNOWLEDGMENTS\nWe acknowledge helpful discussions with Jinwu\nYe, Han Pu, Congjun Wu, and Hui Hu. This\nwork was supported by the NKBRSFC under Grants\nNo. 2009CB930701, No. 2010CB922904, No.\n2011CB921502, and No. 2012CB821300, NSFC under\nGrantsNo. 10934010,and NSFC-RGC under Grants No.\n11061160490 and No. 1386-N-HKU748/10.\nAppendix A: Relationships between the Rashba\nSOC and Dresselhaus SOC\nIn this paper, the normal state properties of the Fermi\nliquid with Rashba-type SOC is studied. Now, we would\nlike to demonstrate that another type of SOC, namely\nthe Dresselhaus-type, gives exactly the same results for\nthe normal state properties considered here.\nWe start with the single particle Hamiltonian with\nDresselhaus SOC [7]\nHD=p2\n2m+α(−pyσx−pxσy)−µ.(A1)\nThe helicity bases with Dresselhaus SOC are\n|p,s/angb∇acket∇ightD=−1√\n2/parenleftBigg\n1\nise−iφ(p)/parenrightBigg\n,s=±1,(A2)\nwhere the subscript Drepresents the Dresselhaus-type\nSOC. The starting point of the microscopic calculation\nis the self-energy Σ sfor each band. To illustrate the\nrelationships of the two types of SOC, it is essential to\nderive the relationships of the Feynman rules between\nthe two cases. For the Dresselhaus-type SOC, the non-\ninteracting Green’s function and the interaction vertex\nin the helicity bases reads\nGD\ns(k,ω) =1\nω−ξk,s+isgn(ω)0+,(A3)\nVD\nss′;rr′(k,p,q) =gfD\nss′(θk,θk+q)fD\nrr′(θp,θp−q).(A4)\nThe energy spectrum of the Dresselhaus-type SOC is the\nsame with the Rashba-type. Thus the single particle re-\ntarded Green’s function within the Dresselhaus represen-\ntation is the same with the result for Rashba-type SOCgiven in Eq. (5). The overlap factor fD\nss′(θk,θk+q) for\nthe Dresselhaus-type SOC is given by\nfD\nss′(θk,θk+q) =1\n2(1+ss′e−i[φ(k)−φ(k+q)]).(A5)\nCompared with Eq. (14), we find the overlap factors are\nconjugated for the two cases\nfD\nss′(θk,θk+q) =fR\nss′(θk,θk+q)∗.(A6)\nThe bare susceptibility in Matsubara formalism is\ngiven by\nχD(q,iωn)=kBT/summationdisplay\nk,iωms,rGD\ns(k,iωm)GD\nr(k−q,iωm−iωn)FD\nsr,\n(A7)\nwhere the factor\nFD\nsr=1+srcosθ\n2(A8)\nis the same with the Rashba case in Eq. (20). So the\nbare susceptibility χDfor the Dresselhaus type SOC is\nequal to the case for the Rashba type SOC. The self-\nenergy is given by Eq. (21) and (22). The integrand\nfunction includes the following factors: the single parti-\ncle’s Green’s function, the bare susceptibility χD(q,iωn)\nand the overlapfactor FD\nsr, which areall demonstratedto\nbe the same for the two types of SOC. Therefore we con-\nclude that the normal state properties calculated here,\nsuch as the quasi-particle lifetime τs(k), the renormal-\nization factor Zsand the effective mass m∗\ns/mare all the\nsame exactly for the two types of SOC.\nAppendix B: spectral function in the spin\nrepresentation\nIn this work, we studied the microscopic parameters\nsuch asτk,s,Zsandm∗\ns/mand associated experimental\nobservable such as As(k,ω),ρ(EF) andcvin the helicity\nrepresentation. Theoreticallyspeaking, the helicity bases\nprovides a clear representation for the description of the\nmicroscopicpicture ofthe singleparticle excitation ofthe\nrepulsiveSOC Fermi gas. In the normalstate regime, the\nquasi-particle in the weakly repulsive SOC Fermi gas can\nbe adiabatically connected with the particle with helicity\nin the non-interacting theory. In this Appendix, we will\nalso present the main results in the spin representation\nin our manuscript, since experimentalists find it handier\nto work with spin representation. In the following, we\nwill transform the results to the spin representation.\nIn the spin representation, the Green’s functions of the\nSOC Fermi gas has the of 2 ×2 matrix form. The non-\ninteracting form is given by\nG0(k,ω)α,β=/summationdisplay\ns1\nω−ξk,s+isgn(ω)0+Ps(k),(B1)9\nwherePs(k) = [1+s(ˆz׈k)·σ]/2 is the projection opera-\ntor to the helicity bases. Considering the s-wave interac-\ntion, there is a modification of the quasi-particle disper-\nsionξ∗\nk,s, and a finite lifetime of the quasi-particle τk,s\ncorresponding to the imaginary part of the self-energy.\nIn the weakly repulsive regime, the polarization of the\nquasi-particle is hold because of the stability from its\nnon-trivial topology structure, such that we obtain the\nmany-body Green’s function as\nGα,β(k,ω) =/summationdisplay\ns1\nω−ξ∗\nk,s−isgn(ω)Γk,s\n2Ps(k).(B2)\nBased on this form of Green’s function modified with s-\nwave interaction, we could obtain the spectral function\nin the spin representation,\nAα,β(k,ω) =/summationdisplay\nsPs(k)α,βAs(k,ω),(B3)whereAsis the spectral function in the helicity bases\nrepresentation. We could find that\nA↑,↑(k,ω) =A↓,↓(k,ω) =1\n2(A+(k,ω)+A−(k,ω)),(B4)\nA↑,↓(k,ω) =A↓,↑(k,ω)∗\n=−ie−iθk\n2(A+(k,ω)−A−(k,ω)).(B5)\nThe elements A↑,↑,A↓,↓are real and equal to each other,\nwhile the off-diagonal elements A↑,↓,A↑,↓have an angle\ndependence e−iθk,eiθk, which is the very effect of spin-\norbit coupling.\nThe other qualities such as the density of states at the\nFermi energy, and the low temperature specific heat are\nboth independent to the representation we use. There-\nfore they could be compared with experiment straight-\nforwardly.\n[1] G. B. Jo, Y. R. Lee, J. H. Choi, C. A. Christensen, T. H.\nKim, J. H. Thywissen, D. E. Pritchard, and W. Ketterle,\nScience325, 1521 (2009).\n[2] V. B. Shenoy and T.-L. Ho, Phys. Rev. Lett. 107, 210401\n(2011).\n[3] F. Wilczek and A. Zee, Proc. R. Soc. Lond. A 392, 45\n(1984).\n[4] F. Wilczek and A. Zee, Phys. Rev. Lett. 52, 2111 (1984).\n[5] P. Wang, Z. Q. Yu, Z. Fu, J. Miao, L. Huang, S. Chai, H.\nZhai, and J. Zhang, Phys. Rev. Lett. 109, 095301 (2012).\n[6] L. W. Cheuk, A. T. Sommer, Z. Hadzibabic, T. Yefsah,\nW. S. Bakr, and M. W. Zwierlein, Phys. Rev. Lett. 109,\n095302 (2012).\n[7] Y.-J. Lin, K. Jimenez-Garcia, and I. B. Spielman, Nature\n(London) 471, 83 (2011).\n[8] Y.-J. Lin, R. L. Compton, A. R. Perry, W. D. Phillips,\nJ. V. Porto, and I. B. Spielman, Phys. Rev. Lett. 102,\n130401 (2009).\n[9] Y. J. Lin, R. L. Compton, K. Jimnez-Garca, J. V. Porto,\nand I. B. Spielman, Nature 462, 628 (2009).\n[10] Y.-J. Lin, R. L. Compton, K. Jimenez-Garcia, W. D.\nPhillips, J. V. Porto, and I. B. Spielman, Nat. Phys. 7,\n531 (2011).\n[11] Z. Q. Yu and H. Zhai, Phys. Rev. Lett. 107, 195305\n(2011).\n[12] H. Hu, L. Jiang, X. J. Liu, and H. Pu, Phys. Rev. Lett.\n107, 195304 (2011).\n[13] R. Liao, Y. Yi-Xiang, and W. M. Liu, Phys. Rev. Lett.\n108, 080406 (2012).\n[14] J. P. Vyasanakere, S. Zhang, and V. B. Shenoy, Phys.\nRev. B84, 014512 (2011).\n[15] L. He and X. G. Huang, Phys. Rev. Lett. 108, 145302\n(2012).\n[16] L. Han and C. A. R. S´ ade Melo, Phys. Rev. A 85,\n011606(R) (2012).\n[17] G. Chen, M. Gong, and C. Zhang, Phys. Rev. A 85,\n013601 (2012).\n[18] A. Kubasiak, P. Massignan, and M. Lewenstein, Euro-\nphys. Lett. 92, 46004 (2010).\n[19] A. Bermudez, N. Goldman, A. Kubasiak, M. Lewenstein,and M. A. Martin-Delgado, New J. Phys. 12, 033041\n(2010).\n[20] G. Liu, S. L. Zhu, S. Jiang, F. Sun, and W. M. Liu, Phys.\nRev. A82, 053605 (2010).\n[21] J. D. Sau, R. Sensarma, S. Powell, I. B. Spielman, and\nS. Das Sarma, Phys. Rev. B 83, 140510 (2011).\n[22] T. Fujita and K. F. Quader , Phys. Rev. B 36, 5152\n(1987).\n[23] Y. Li and C. Wu, Phys. Rev. B 85, 205126 (2012).\n[24] N. F. Mott and H. S. W. Massey, The Theory of Atomic\nCollisions , 3rd ed. (Oxford University Press, London,\n1965).\n[25] A. Messiah, Quantum Mechanics (Wiley, New York,\n1962).\n[26] E. Braaten and H. W. Hammer, Phys. Rep. 428, 259\n(2006).\n[27] T. K¨ ohler and K. G´ oral, Rev. Mod. Phys. 78, 1311\n(2006).\n[28] C. Chin, R. Grimm, P. Julienne, and E. Tiesinga, Rev.\nMod. Phys. 82, 1225 (2010).\n[29] S. Inouye, M. R. Andrews, J. Stenger, H.-J. Miesner, D.\nM. Stamper-Kurn, and W. Ketterle, Nature (London)\n392, 151 (1998).\n[30] P. Courteille, R. S. Freeland, and D. J. Heinzen, F. A.\nvan Abeelen, and B. J. Verhaar, Phys. Rev. Lett. 81, 69\n(1998).\n[31] J. L. Roberts, N. R. Claussen, J. P. Burke, C. H. Greene,\nE. A. Cornell, and C. E. Wieman, Phys. Rev. Lett. 81,\n5109 (1998).\n[32] V. Vuleti´ c, A. J. Kerman, C. Chin, and S. Chu, Phys.\nRev. Lett. 82, 1406 (1999).\n[33] L. Pricoupenko and Y. Castin, Phys. Rev. A 69,\n051601(R) (2004).\n[34] T. Bourdel, J. Cubizolles, L. Khaykovich, K. M. F.\nMagalh˜ aes, S. J. J. M. F. Kokkelmans, G. V. Shlyap-\nnikov, and C. Salomon, Phys. Rev. Lett. 91, 020402\n(2003).\n[35] J. W. Negele and H. Orland, Quantum Many-Particle\nSystems (Addison-Wesley, New York, 1988).\n[36] D. S. Saraga and Daniel Loss, Phys. Rev. B 72, 19531910\n(2005).\n[37] D. S. Petrov, Phys. Rev. A 67, 010703 (2003).\n[38] J. P. DIncao and B. D. Esry, Phys. Rev. Lett. 94, 213201\n(2005).\n[39] G. D. Mahan, Many-Particle Physics (Plenum, New\nYork, 1990).\n[40] A.L.FetterandJ. D.Walecka, Quantum Theory of Many\nParticle Systems (McGraw Hill, New York, 1971).\n[41] H. Hu, H. Pu, J. Zhang, S. G. Peng, and X. J. Liu,\narXiv:1208.5841.[42] C. Chin, M. Bartenstein, A. Altmeyer, S. Riedl, S.\nJochim, J. H. Denschlag and R. Grimm, Science 305,\n1128 (2004).\n[43] C. H. Schunck, Y. Shin, A. Schirotzek, M. W. Zwierlein,\nand W. Ketterle, Science 316, 867 (2007).\n[44] M. Veillette, E. G. Moon, A. Lamacraft, L. Radzihovsky,\nS. Sachdev, and D. E. Sheehy, Phys. Rev. A 78, 033614\n(2008)."    },    {        "title": "0905.1389v1.Spin_state_transition_and_phase_separation_in_multi_orbital_Hubbard_model.pdf",        "content": "arXiv:0905.1389v1  [cond-mat.str-el]  9 May 2009Spin-state transitionand phaseseparationinmulti-orbit al Hubbard model\nRyo Suzuki1†, Tsutomu Watanabe2∗, and Sumio Ishihara1\n1Department of Physics, Tohoku University, Sendai 980-8578 , Japan\n2Institute of Multidisciplinary Research for Advanced Mate rials, Tohoku University, Sendai 980-8577, Japan\n(Dated: December 4, 2018)\nWestudyspin-statetransitionandphaseseparationinvolv ingthistransitionbasedonthemilti-orbitalHubbard\nmodel. Multiple spinstatesare realizedbychanging theene rgy separationbetweenthe twoorbitalsandthe on-\nsiteHundcoupling. ByutilizingthevariationalMonte-Car losimulation,weanalyzetheelectronicandmagnetic\nstructures in hole doped and undoped states. Electronic pha se separation occurs between the low-spin band\ninsulating state and the high-spin ferromagnetic metallic one. Difference of the band widths inthe two orbitals\nisof prime importance for the spin-state transitionandthe phase separation.\nPACS numbers: 75.25.+z, 71.70.-d,71.30.+h\nNovel electric and magnetic phenomena observed in cor-\nrelated electron systems are responsible for competition a nd\ncooperationbetweenmulti-electronicphaseswithdelicat een-\nergybalance. These areowingto the internaldegreesof free -\ndom of electrons, i.e. spin, charge and orbital, under stron g\nelectron correlation,and their couplingwith crystal latt ice.1,2\nInsome transition-metalions, thereis anadditionaldegre eof\nfreedom, termed the spin-state degree of freedom, i.e. mul-\ntiple spin states due to the different electron configuratio ns\nin a single ion. One prototypical example is the perovskite\ncobaltites R1−xAxCoO3(R: rare earth ion, A: alkaline earth\nion) where transitions between the multiple spin states occ ur\nby changing carrier concentration, temperature and so on. I n\nCo3+with thed6configuration, there are three possible spin\nstates, the high-spin (HS) state (e2\ngt4\n2g)with an amplitude of\nS=2, the intermediate-spin (IS) one (e1\ngt5\n2g)withS=1, and\nthelow-spin(LS)one (t6\n2g)withS=0.\nSeveralmagnetic,electricandtransportmeasurementshav e\nbeen carried out in the insulating and metallic cobaltites. It\nis known that LaCoO 3is a non-magnetic LS band-insulator\n(BI) at low temperatures, although there is still controver sy\nin the spin-state transition and the IS state at finite temper -\nature.3,4,5,6,7In high hole doping region of x>0.3−0.4 in\nLa1−xSrxCoO3,theferromagnetic(FM)metallicstatewasex-\nperimentally confirmed. In the lightly hole doped region be-\ntween the two, a number of inhomogeneousfeatures in mag-\nnetic, electric and lattice structures have been reported e x-\nperimentally. Spatial segregation of hole-rich FM regions\nand hole-poor insulating ones have been suggested by the\nneutron diffraction, the electron microscopy, NMR and so\non.8,9,10,11Magnetic/non-magnetic clusters have been found\nby the small-angle and inelastic neutron scattering experi -\nments.12,13,14It is widely believed that the observed giant\nmagneto-resistance effect in the lightly doped region resu lts\nfromtheelectronicandmagneticinhomogeneity.12\nElectronic phase separation (PS) phenomena in transition-\nmetal compounds have been studied extensively and inten-\nsively, in particular, in the high Tc superconducting cupra tes\nand the colossal magnetoresistive manganites.15,16,17,18In\nthese materials, the long-rangespin/orbitalordersin the Mott\ninsulating phases and their melting by carrier doping are of\nessence in the electronic PS. The exchangeenergyfor the lo-calizedspins/orbitalsandthekineticonefortheitineran telec-\ntrons are gained in spatially separate regions. On the other\nhand, in the present case, the non-magnetic band insulator i s\nrealizedintheinsulatingphase,andthespin-statetransi tionis\nbrought about by carrier doping. Thus, the present phenom-\nena belong to a new class of the electronic PS in correlated\nsystem, although only a little theoretical studies have bee n\ndone until now. In this paper, we address the issues of the\nspin-state transitionand the PS associated with thistrans ition\nby analyzing the multi-orbital Hubbard model. We examine\nthe electronic structures in hole doped and undoped systems\nby utilizing the variational Monte-Carlo (VMC) method. We\nfindthat,betweenthenon-magneticBIandtheHSFMmetal,\ntheelectronicPS isrealized. We claimthat the differentba nd\nwidthsplayanessentialrolein thepresentelectronicPS.\nWe set up a minimal model, the two-orbital Hubbard\nmodel,19,20,21,22where the spin-state degrees of freedom and\na transition between them are able to be examined. In each\nsite in a crystal lattice, we introduce two orbitals, termed A\nand B, which represent one of the egandt2gorbitals, respec-\ntively. Anisotropic shape of the orbital wave function is no t\nconcerned. An energy difference between the two orbitals is\ndenotedby Δ≡εA−εB>0 whereεA(εB) is the level energy\nforA(B).Whentheelectronnumberpersiteistwo,thelowest\ntwo electronic states in a single site are |B2/angbracketrightand|A1B1/angbracketrightwith\ntripletspinstate whicharetermedtheLSandHSstatesinthe\npresent model, respectively. The explicit form of the model\nHamiltonianisgivenby\nH=Δ∑\niσc†\niAσciAσ−∑\n/angbracketleftij/angbracketrightγσtγ/parenleftBig\nc†\niγσcjγσ+H.c./parenrightBig\n+U∑\niγniγ↑niγ↓+U′∑\niσσ′niAσniBσ′\n−J∑\niσσ′c†\niAσciBσc†\niBσ′ciAσ′−J′∑\niγc†\niγ↑ci¯γ↑c†\niγ↓ci¯γ↓,(1)\nwhereciγσis the annihilation operator of an electron at site i\nwith orbital γ(=A,B)andspinσ(=↑,↓), andniγσ≡c†\niγσciγσ\nis the number operator. A subscript ¯γtakesA(B), whenγis\nB(A). Weassumethatthetransferintegralisdiagonalwithre-\nspectto theorbitalsand |tA|>|tB|, bothofwhichare justified\ninperovskitecobaltites. Inmostofthenumericalcalculat ions,\na relation tB/tA=1/4is chosen. As the intra-site electronin-2\nteractions, we introduce the intra- and inter-orbital Coul omb\ninteractions, UandU′, respectively,the Hundcoupling Jand\nthepair-hopping J′. Therelations U=U′+2JandJ=J′sat-\nisfiedinanisolatedionareassumed. Inaddition,weintrodu ce\ntherelation U=4Jinthenumericalcalculation.\nWe adopt the VMC method where the electron correlation\nis treated in an unbiased manner and simulations in a large\nclustersizearepossible. Forsimplicityandalimitationi nthe\ncomputerresource, we introducetwo-dimensionalsquare la t-\ntices with a system size of N≡L2(L≤6) and the periodic\nand anti-periodic boundary conditions. The number of elec-\ntron isNe, and the hole concentrationper site measured from\nNe=2Nisdenotedas x≡(2N−Ne)/N. Thevariationalwave\nfunction is given as a product form of Ψ=G|Φ/angbracketrightwhereGis\nthe correlation factor and |Φ/angbracketrightis the one-bodywave function.\nThe two types of the wave function are considered in |Φ/angbracketright:\ntheSlaterdeterminantobtainedbythesecondterminEq.(1) ,\nand that for the HS antiferromagnetic (AFM) order given by\napplying the Hartree-Fock approximationto the third term i n\nEq. (1). In the latter, the AFM order parameteris treated as a\nvariationalparameter. WeassumetheGutzwiller-typecorr ela-\ntion factor Πil(1−ξlPil)wherelindicatesthe local electron\nconfigurations, Pilis the projection operator at site ifor the\nconfiguration l, andξlis the variational parameter. Here we\nintroducethe10variationalparametersforthe10inequiva lent\nelectron configurations in a single site.23The fixed-sampling\nmethod is used to optimize the variational parameters.24In\naddition to the standard VMC method, we improve the vari-\national wave function by estimating analytically the weigh ts\nfor the configurations which are sampled by the MC simu-\nlations. This method is valid for the LS state and reduces the\nCPUtimebymorethanoneorder. Inmostofthecalculations,\n104−105MCsamplesareadoptedformeasurements.\nWe start from the case at x=0 where the average\nelectron number per site is two. The electronic states\nobtained by the simulation are monitored by the total\nspin amplitude defined by S2= (1/N)∑i/angbracketleftS2\ni/angbracketrightwhereSi=\n∑γSiγ= (1/2)∑ss′γc†\niγsσss′ciγs′is the spin operator with the\nPauli matrices σ, the spin correlation function Sγ(q) =\n(4/N)∑ijeiq·(ri−rj)/angbracketleftSz\niγSz\njγ/angbracketright, and the momentum-distribution\nfunction nγ(k) = (1/2)∑σ/angbracketleftc†\nkγσckγσ/angbracketrightwhereckγσis the\nFourier transform of ciγσ. Size dependences of S2andSγ(q)\ninL=4−8 are within a few percent. We obtain the three\nphases, the HS Mott insulator (MI), the LS BI and the metal-\nlic (ML) phase. In the HS-MI phase, S2is about 1.6 being\nabout 80% of the maximumvalue for S=1. A sharp peak in\nSγ(q)atq= (π,π)and no discontinuity in nγ(k)imply that\nthis is the AFM MI. In the LS-BI phase, nA(k) [nB(k)]is al-\nmostzero(one)inallmomenta,and S2≃0. IntheMLphase,\ndiscontinuous jumps are observed in both nA(k)andnB(k).\nThe electron(hole)fermi surface is located around k=(0,0)\n[(π,π)]in the A (B) band;this is a semi metal. A valueof S2\nisabout0.3,andnoremarkablestructureis seenin Sγ(q).\nThephasediagramat x=0ispresentedinFig.1. Theerror\nbars imply the upper and lower bounds of the phase bound-\nary, and symbols are plotted at the middle of the bars. In the\nregion of large Δ(J), the LS-BI (HS-MI) phase is realized,\n\u0001 \u0002 \u0003 \u0000 \u0004 \u0005\n\u0006\u0007\b\t\n\u000b \f \r \u000e \u000f \u0010\n\u0011\u0012\u0013\n\u0014\u0015 \u0016 \u0017\u0018\u0019 \u001a \u001b\u001c\n\u001d \u001e\u001f  ! \" # $ %\n& ' (\n)*\n+ ,-\n. / 0 12 34 56 7 8 9 :\n; < =>?\n@ AB\nC D\nFIG.1: (coloronline)Phasediagramsat x=0. Aratiooftheelectron\ntransfers is taken to be tB/tA=1/4 in (a) and tB/tA=1 in (b). In\n(b),filledsquares andopen circles are forthe results obtai ned bythe\nVMC method and the previous DMFT one in Ref. 19, respectively .\nBroken curves are guides for eyes. Stars represent the param eters\nwhere the carrier dopings are examined.\nand between the two with small ΔandJ, the ML phase ap-\npears. To compare the present results with the previous ones\ncalculatedbythedynamical-meanfieldtheory(DMFT),19we\npresent,inFig.1(b),thephasediagramwherethetwotransf er\nintegrals are chosen to be equal, i.e. tB/tA=1. Although the\nglobal features in the phase diagrams are the same with each\nother,theHS-MIphaseobtainedbytheVMCmethodappears\nin a broader parameter region than that in DMFT, in partic-\nular, near the boundary of the HS-MI and ML phases. This\nis because the AFM long-range order in the HS-MI phase is\ntreatedproperlyintheVMCmethod. We haveconfirmedthat\nthephaseboundariesobtainedbytheVMCmethodwherethe\nAFM orderisnotconsideredalmostreproducetheDMFT re-\nsults.\nNow we show the results at finite x. Holes are introduced\ninto the LS-BI phase near the phase boundary with the pa-\nrametervaluesof (Δ/tA,J/tA)=(12.2,4)and(8.25,2.5)[see\nFig. 1]. By changingthe initial conditionsin the VMC simu-\nlation,weobtainthefollowingfourstates: i)theLS-MLsta te\nwherenA(k)is almost zero in all k, and the fermi surface is\nlocated in the B band around k= (π,π), ii) the FM HS-ML\nstate where nB(k)is about 1 /2 in allk, the fermi surface is\nin the A band, and Sγ(q)has a sharp peak at q= (0,0), iii)\ntheAFMHS-MLstatewherethefermisurfaceexistsintheA\nband around k= (π,0), andSγ(q)has a peak at q= (π,π),\nand iv) the mixed state where the wave function is a linear-\ncombinationoftheLS-MLandFM HS-MLstates.\nInFig.2(a),theenergyexpectationvalues E≡/angbracketleftH/angbracketrightforthe\nseveral states in (Δ/tA,J/tA) = (12.2,4)are plotted as func-\ntionsofx. Thetransferintegralsarechosentobe tB/tA=1/4.\nToshowthenumericaldataclearly,weplot E′=(E/tA)+Cx\nwith a numerical constant C, instead of E. This transforma-\ntiondoesnotaffecttheMaxwell’sconstructionintroduced be-\nlow. The results in the AFM HS-ML are not plotted, because\nof their higher energy values than others. We also present,\nin Fig. 3, a ratio of the LS sites to the LS and HS ones in3E F GH\nI J K LMN OPQ\nR ST U VW\nX Y Z [ \\] ^_`\na b c d\ne f gh i j\nk l m no p q rs t u v\nw x y z { | } ~          \n    \n\n\n      \n¡\n¢FIG.2: (coloronline) Holeconcentrationdependences ofth eenergy\nexpectations for several states at (Δ/tA,J/tA)=(12.2,4)in (a), and\nthose at(Δ/tA,J/tA)=(8.25,2.5)in (b). Broken lines are given by\nthe Maxwell’s construction. Aratioof the electrontransfe rs is taken\nto betB/tA=1/4. A constant parameter Cin the definition of E′is\ntaken tobe 8.2in(a)and 5.25in(b).\nthe mixedstates defined by RLS=nLS/(nLS+nHS). HerenLS\n(nHS) is a number of the sites where the LS (HS) state is re-\nalized. As shown in Fig. 2(a), the LS state, where holes are\ndopedinto the B band,is destabilizedmonotonicallywith in -\ncreasingx. On the other side, in a region of x>0.5, the\nFM HS-ML state is realized. In between the two regions,\nthe mixed state is the lowest energy state. The mixed state is\nsmoothlyconnectedtotheLSandHSonesinthelowandhigh\nxregions, respectively. As shown in Fig. 3, a discontinuous\njumpinthemixedstate isseenaround x=0.25;thesystemis\nchangedfromthe LS dominantmixedstate into the HS dom-\ninant one with x. It is noticeable that the E′versusxcurve in\nthemixedstateisconvexintheregionof0 <x<0.33. Thatis,\nbyfollowingtheMaxwell’sconstruction,thePSoftheLS-BI\nandtheFMHS dominantmixedstatesismorestabilizedthan\nthe homogeneous phase in this region of x. In the Fig. 2(b),\nwe show the results in (Δ/tA,J/tA) = (8.25,2.5)where the\nsystem at x=0 is close to the ML phase [see Fig. 1(a)]. The\nPS appears,butits regionisshrunken.\nThe magnetization per site in the lowest energy state de-\nfined by M(x) = (1/N)/angbracketleft∑iSz\ni/angbracketrightis plotted in Fig. 3. A zero\nmagnetization at x=0 reflects the LS-BI ground state. In a\nhighdopedregionof x>0.33,the magnetizationdata almost\nfollow a relation M(x)≃(1+x)/2: the system is expected\nto consist of the N/2 HS sites, the (1/2−x)NLS ones, and\n£ ¤ ¥ ¦ §\n¨ © ª«\n¬  ® ¯ °± ² ³´\nµ ¶· ¸ ¹ º» ¼ ½ ¾¿ À Á ÂÃ Ä Å Æ\nÇ È É Ê Ë Ì Í Î Ï Ð Ñ Ò Ó Ô Õ Ö × Ø\nÙ Ú ÛÜ Ý Þß à áâ ã äå æ çè é ê\nëì\ní\nîï\nFIG. 3: (color online) A ratio of the LS sites to the LS and HS\nones in the mixed state, RLS, and magnetization M(x)as func-\ntions of the hole concentration x. A broken line connecting data at\nM(x=0)andM(x=0.33)is drawn by the Maxwell’s rule. For\ncomparison, we plot a M(x) =x/2 curve which is expected from\nthe hole doping in the LS-BI phase. Parameters are chosen to b e\n(Δ/tA,J/tA)=(12.2,4)andtB/tA=1/4.ð ñ ò ó ô õ ö ÷ ø ù ú û ü ý þ ÿ \u0001 \u0000\n\u0003 \u0002\u0004 \u0005\n\u0006\u0007\b\t\n\n\u000b\n\f \r \u000e \u000f\u0010 \u0011 \u0012 \u0013 \u0014\u0015 \u0016\n\u0017\u0018\n\u0019 \u001a\u001b\n\u001c \u001d\nFIG.4: (coloronline)Hole concentrationdependences ofth eenergy\nexpectations for several states where the electron transfe r integrals\narechosentobeequalas tB/tA=1. Otherparametersaretakentobe\n(Δ/tA,J/tA)=(12.2,4),andaconstantparameter Cinthedefinition\nofE′istaken tobe 8.\nthexNsingly electron occupiedones. In this scheme, we ob-\ntainRLS=(1−2x)/(2−2x)which is consistent with the nu-\nmerical data of RLSinx>0.33. Between x=0 and 0.33,\nwherethe PS is realized, M(x=0)andM(x=0.33)are con-\nnectedbyastraightlineaccordingtothevolume-fractionr ule\nin the Maxwell’s construction. The slope of M(x)is about\nthree times higher than M(x) =x/2 which is expected in the\nholedopingintotheLS-BIphase. Thisisqualitativelycons is-\ntent with the experimental observations in the magnetizati on\nwheredopedholesinducehighspin value.3,25\nWe now address an origin of the electronic PS where the\nspin-state degree of freedom is concerned. In Fig. 4, we\npresent the hole concentration dependence of the energy ex-\npectationswherethebandwidthsaresettobeequalwitheach\nother,tB/tA=1. AswellasthecalculationinFig.2(a),theen-\nergyparametersaretakentobe (Δ/tA,J/tA)=(12.2,4)which4\u001e\u001f\n !\n\"#\n$ % &' ( )*+ , - . / 0 1 2 3 4 5 6\n789\n:;\n<=\n> ? @A B CDE F G H I J K L M N O P\nQ\nFIG.5: (color online) Schematic density-of-states inthe L S-BIstate\natx=0and that inthe HS-ML one ina highhole doped region.\nisclosetotheLS-HSphaseboundaryat x=0[see Fig.1(b)].\nThemixedstateisnotobtainedinthesimulation. Inallregi on\nofxuptox=0.45,theLSstateisthelowestgroundstate,and\nneither the spin-state transition nor the PS occur. The diff er-\nenceofthebandwidthsinthetwoorbitalsisofessenceinthe\nelectronicPS phenomena.\nToclarifythemechanismofPSfurthermore,schematicpic-\nturesof the density of states (DOS) in the LS-BI at x=0 and\nthe FM HS-ML in a high hole doped region are presented in\nFig. 5. For simplicity, detailed shapes of DOS are not taken\ninto account. In LS-BI state at x=0, the fermi level is lo-\ncatedinside ofthe bandgapbetweenthe A andB bands. The\nbandwidthintheAbandislargerthanthatinB.Ontheother\nhand, in the FM HS-ML state which is realized in x/greaterorsimilar0.5 in\nFig. 2(a), the system is a doped MI with ferromagnetic spin\npolarization. The fermi level is located in the A band. Be-\ncause of the large band width in the A band, there is a large\nkineticenergygainincomparisonwiththedopedLS-BIstate\nwherethefermilevelislocatedintheBbandintherigidband\nscheme. Thiskineticenergygainistheoriginofthespinsta te\ntransitionbydoping. ItisshowninFig.4that,whentheequa l\nband widths are assumed, the E′v.s.xcurvesfor the LS-ML\nand FM HS-ML states are almost parallel and do not crosswith each other. This data implies that there is no differenc e\nin the kinetic energy gains for the two states, when the band\nwidths are assumed to be equal. The present PS phenomena\narealso attributedto thisbandwidthdifferenceasfollows . In\nthe rigid-band sense, by doping of holes in the LS-BI state,\nthefermilevelfallsintothetopoftheBbandfromthemiddle\nof the gap in Fig. 5(a). If we suppose that this state is real-\nized in a low xregion and is transferred into the FM HS-ML\nstate shown in Fig. 5(b) with increasing x, the fermi level is\nincreased with increasing hole concentration because of th e\ndifferentbandwidths. Thisis nothingbutthe negativechar ge\ncompressibility κ=(∂µ/∂x)<0withthechemicalpotential\nµ,i.e. appearanceoftheelectronicPS.\nFinally, we discuss implications of the perovskite\ncobaltites. The obtained PS between the insulating nonmag-\nnetic state and the hole-rich FM one is qualitatively consis -\ntent with the inhomogeneity suggested by a number of ex-\nperiments. The PS and the spin-state transition are attribu ted\nto the band-width difference of the two bands correspond-\ning to the egandt2gbands in the perovskite cobaltites. This\nelectronic PS is robust by changingthe model parameter val-\nues, except for tB/tA, when the non-doped system is located\nnear the phase boundarybetween the LS-BI and HS-MI. The\npresent phenomena are different from the previous PS’s dis-\ncussed in the high-Tc cuprates and the manganiteswhere the\nlong-range spin/orbital orders are realized in the MI’s; th e\nspatialsegregationsoccurbetweenthelong-rangeordered MI\nand the ML states where the superexchange interaction en-\nergyand the kinetic one of dopedholes are separately gained\nin the different spatial regions. Our scenario of the PS base d\non the band-widthdifferencemay be checkedexperimentally\nby adjusting the tolerance factor, i.e. the Co-O-Co bond an-\ngle; the smaller tolerance factor implies the smaller (larg er)\nbandwidthin the eg(t2g)orbitals,and suppressionof the PS.\nDetailed values of xwhere the PS is realized, and a typical\nsize of the clusters remain as questions. Several factors no t\nconsidered here, the intermediate-spin state, the long-ra nge\nCoulombinteraction,thelatticevolumedependingonthesp in\nstates, andso on,are requiredtoanswerthese questions.\nAuthors would like to thank H. Yokoyama and\nH. Takashima for their valuable discussions. This work\nwas supported by JSPS KAKENHI, TOKUTEI from\nMEXT, and Grand challenges in next-generation integrated\nnanoscience.\n1S. Maekawa, et al. Physics of TransitionMetal Oxides , (Springer\nVerlag, Berlin,2004), andreferences therein.\n2M. Imada, et al.Rev. Mod. Phys. 70, 1039 (1998).\n3S.Yamaguchi et al.Phys.Rev. B 53, R2926 (1996).\n4M. A. Korotin, etal.Phys.Rev. B 54, 5309 (1996).\n5M. W.Haverkort, et al.Phys. Rev. Lett. 97, 176405 (2006).\n6S.Noguchi, et al.Phys. Rev. B 66, 094404 (2002).\n7Y. Kobayashi, et al.Phys.Rev. B 72, 174405 (2005).\n8M. Itoh,et al.J. Phys.Soc.Jpn. 63, 1486 (1994).\n9R. Caciuffo, etal.Phys.Rev. B 59, 1068 (1999).\n10P.L.Kuhns, et al.Phys.Rev. Lett. 91, 127202 (2003).11A. Ghoshray, etal.Phys.Rev. B 69, 064424 (2004).\n12J. Wu,et al.Phys. Rev. Lett. 94, 037201 (2005).\n13D. Phelan, et al.Phys. Rev. Lett. 96, 027201 (2006).\n14D. Phelan, et al.Phys. Rev. Lett. 97, 235501 (2006).\n15E.L.Nagaev, Phys.State.Sol.(b) 186, 9(1994).\n16E. Dagotto, The Physics of Manganites and RelatedCompounds ,\n(Springer-Verlag,Berlin2003).\n17S.Okamoto, etal.Phys.Rev. B 61, 451(2000).\n18K. I.Kugel, etal.Phys.Rev. Lett.95, 267210 (2005)\n19P.Werner,and A.J. Millis,Phys.Rev. Lett. 99, 126405 (2007).\n20K. SanoandY. Ono, J.Phys. Soc.Jpn. 72, 1847 (2003).5\n21K.Kobayashi,andH.Yokoyama,PhysicaC 445-448,162(2006).\n22K. Kubo, Phys.Rev. B 79, 020407 (2009).\n23The 10 inequivalent configurations considered here are |0/angbracketright,|Aσ/angbracketright,\n|Bσ/angbracketright,|AσB¯σ/angbracketright,|AσBσ/angbracketright,|B↑B↓/angbracketright,|A↑A↓/angbracketright,|AσB↑B↓/angbracketright,|A↑A↓Bσ/angbracketright,\n|A↑A↓B↑B↓/angbracketrightwithσ=↑and↓, whereAσ(Bσ)impies that the A\n(B) orbital isoccupied by the up(down) spinelectron.\n24C. J.Umrigar, et al.Phys. Rev. Lett 60, 1719 (1988).25J. Okamoto, et al.Phys. Rev. B 62, 4455 (2000).\n†Presentaddress: ThebankofTokyo-MitsubishiUFJ,Tokyo,\nJapan.\n∗Presentaddress: ChibaInstituteofTechnology,Tsudanuma ,\nChiba275-0016,Japan."    },    {        "title": "1807.04579v1.Spin_Phonon_coupling_parameters_from_maximally_localized_Wannier_functions_and_first_principles_electronic_structure__the_case_of_durene_single_crystal.pdf",        "content": "Spin-phonon coupling parameters from maximally localized Wannier functions and\n\frst principles electronic structure: the case of durene single crystal\nSubhayan Roychoudhury and Stefano Sanvito\nSchool of Physics and CRANN Institute, Trinity College, Dublin 2, Ireland\nSpin-orbit interaction is an important vehicle for spin relaxation. At \fnite temperature lattice\nvibrations modulate the spin-orbit interaction and thus generate a mechanism for spin-phonon\ncoupling, which needs to be incorporated in any quantitative analysis of spin transport. Starting\nfrom a density functional theory ab initio electronic structure, we calculate spin-phonon matrix\nelements over the basis of maximally localized Wannier functions. Such coupling terms form an\ne\u000bective Hamiltonian to be used to extract thermodynamic quantities, within a multiscale approach\nparticularly suitable for organic crystals. The symmetry of the various matrix elements are analyzed\nby using the \u0000-point phonon modes of a one-dimensional chain of Pb atoms. Then the method\nis employed to extract the spin-phonon coupling of solid durene, a high-mobility crystal organic\nsemiconducting. Owing to the small masses of carbon and hydrogen spin-orbit is weak in durene\nand so is the spin-phonon coupling. Most importantly we demonstrate that the largest contribution\nto the spin-phonon interaction originates from Holstein-like phonons, namely from internal molecular\nvibrations.\nI. INTRODUCTION\nIn a non-magnetic material the electrical resistance ex-\nperienced by a charge carrier is independent of its spin.\nIn contrast, when the material is magnetic the resistance\ntypically depends on the relative orientation of the car-\nrier spin and the local magnetization1. This observa-\ntion inspired the advent of the \feld of spin-electronics\nor spintronics2, which concerns the injection, manipu-\nlation and detection of spins in a solid-state environ-\nment. A prototype spintronics device, the spin-valve3,\nconsists of two ferromagnetic layers sandwiching a non-\nmagnetic spacer4, which can display a metallic5,6, insu-\nlating7or semiconducting8,9electronic structure. The\ncarriers, which are spin-polarized by one ferromagnet,\ntravel through the spacer to the other ferromagnet. If the\nspin direction is maintained during such transfer, then\nthe total resistance of the device will depend on the mu-\ntual orientation of the magnetization vectors of the two\nferromagnets. It is then crucial to understand how the\nspin direction evolves during the motion of the carriers\nthrough the spacer, and in particular to understand how\nthis is preserved.\nThere are multiple possible sources of spin relaxation\nin a material, such as the presence of impurities, hyper-\n\fne interaction and spin-orbit (SO) coupling. A theoret-\nical description of all such phenomena is needed for an\naccurate evaluation of the quantities related to spin re-\nlaxation. The relative dominance of one interaction over\nthe others is typically highly dependent on the speci\fc\nmaterial. In this work, we shall focus on SO interac-\ntion, more speci\fcally on the modulation of such inter-\naction due to lattice vibrations. The spin of an electron\ninteracts with the magnetic \feld generated by the rel-\native motion of the nucleus about the electron, giving\nrise to SO interaction. At \fnite temperature the atoms\nof a solid vibrate with respect to their equilibrium posi-\ntions with the amplitudes of such vibrations increasingwith temperature. Such vibrations, the phonons, change\nthe potential felt by the electrons, including the compo-\nnent due to SO coupling10. This e\u000bectively generates a\nmechanism for spin-phonon coupling11, which is key for\nthe calculation of quantities related to spin-relaxation in\nmany systems. It must be noted that in current liter-\nature the term `spin-phonon' coupling has been used to\ndenote di\u000berent e\u000bects. For instance in the study of mul-\ntiferroic compounds `spin-phonon coupling' indicates the\nmodulation of the phonon frequencies due to changes in\nthe magnetic ordering.12{16Here we are interested in the\nopposite, namely in the change of electronic structure\nbrought by the vibrations, in particular for the case of\norganic crystals.\nRecent years have witnessed a growing interest in ex-\nploring the possibility of using organic crystal semicon-\nductors for electronic and spintronic applications17{21.\nThis stems from the high degree of mechanical \rexibility,\nthe light weight and the ease of synthesis and patterning\nthat characterize organic compounds. In these systems\ncovalently bonded organic molecules are held together\nby weak van der Waals interactions. Due to the weak\nbonds between the individual molecules, vibrational mo-\ntions are prominent in organic crystals and the coupling\nof the vibrations to the charge carriers plays a crucial\nrole22in the transport properties of such materials.\nThe presence of experimental evidence in support of\ndi\u000berent transport regimes23{27has generated a signif-\nicant debate on whether the transport in organic crys-\ntals is dominated by delocalized band-like transport, as\nin covalently bonded inorganic semiconductors, by local-\nized hopping, or by a combination of both. This can\nvery well depend on the speci\fc crystal and the exper-\nimental conditions, such as the temperature. Typically,\nin organic crystals the vibrational degrees of freedom\nare thought to introduce signi\fcant dynamical disorder28\nand thereby have paramount in\ruence on the transport\nproperties. Since the typical energies associated to lat-\ntice vibrations in organics are of the same order of magni-arXiv:1807.04579v1  [cond-mat.mtrl-sci]  12 Jul 20182\ntude of the electronic bandwidth, the coupling between\ncarriers and phonons can not be treated by perturba-\ntion theory. Thus, in general, formulating a complete\ntheoretical framework for the description of transport in\norganic crystals is more challenging than that for cova-\nlently bonded inorganic semiconductors22,29. Even more\ncomplex is the situation concerning spin transport, for\nwhich the theoretical description often relies on param-\neters extracted from experiments30, or on approximate\nspin Hamiltonians31\nOne viable option towards a complete ab initio de-\nscription of spin transport consists in constructing a mul-\ntiscale approach, where information about the electronic\nand vibrational properties calculated with \frst-principles\ntechniques are mapped onto an e\u000bective Hamiltonian re-\ntaining only the relevant degrees of freedom. For instance\nthis is the strategy for constructing e\u000bective giant-spin\nHamiltonians with spin-phonon coupling for the study of\nspin relaxation in molecular magnets32,33. The approach\npresented here instead consists in projecting the elec-\ntronic structure over appropriately chosen maximally lo-\ncalized Wannier functions (MLWFs)34, which e\u000bectively\nde\fne a tight-binding (TB) Hamiltonian. In a previous\nwork35we have described a computationally convenient\nscheme for extracting the SO coupling matrix elements\nfor MLWFs. Here we extend the method to the com-\nputation of the spin-phonon matrix elements. Our de-\nrived Hamiltonian can be readily used to compute spin-\ntransport quantities, such as the spin relaxation length.\nThe paper is organized as follows. In the next section\nwe introduce our computational approach and describe\nthe speci\fc implementation used. Then we present our\nresults. We analyze \frst the symmetry of the various\nmatrix elements by considering the simple case of a lin-\near atomic chain of Pb atoms. Then we move to the\nmost complex case of the durene crystal, a popular high-\nmobility organic semiconductor. Finally we conclude.\nII. METHOD\nWannier functions, which form the basis functions\nof the proposed TB Hamiltonian, are essentially the\nweighted Fourier transforms of the Bloch states of a crys-\ntal. From a set of N0isolated Bloch states, fj mkig,\nwhich for instance can be the Kohn-Sham (KS) eigen-\nstates of a DFT calculation, one can obtain N0Wannier\nfunctions. The n-th Wannier ket centred at the lattice\nsiteR,jwnRi, is found from the prescription,\njwnRi=V\n(2\u0019)3Z\nBZ\"NX\nm=1Uk\nmnj mki#\ne\u0000ik:Rdk;(1)\nwhereVis the volume of the primitive cell, j mkiis\nthem-th bloch vector, and the integration is performed\nover the \frst Brillouin zone (BZ). Here Ukis a unitary\noperator that mixes the Bloch states. In order to \fx the\ngauge choice brought by Ukone minimizes the spread ofa Wannier function, which is de\fned as\n\n =X\nn\u0002\nhwn0jr2jwn0i\u0000jhwn0jrjwn0ij2\u0003\n:(2)\nSuch choice de\fnes the so-called MLWFs36. We use the\ncodewannier9037for construction of such MLWFs.\nSince the TB Hamiltonian operator, ^H, depends on the\nionic positions, the ionic motions give rise to changes in\n^H. In addition, since the MLWFs are constructed from\nthe Bloch states, which themselves depend on the ionic\ncoordinates, lattice vibrations result in a change of the\nMLWFs as well. Therefore, the change in the Hamilto-\nnian matrix elements due to the ionic motion, namely the\nonsite energies and hopping integrals, originates from the\ncombined action of 1) the change in ^Hand 2) the change\nin the MLWFs basis. Hence, in the MLWFs TB picture\nthe variation of the matrix element, \"nm, due to an ionic\ndisplacement, is given by\n\u0001\"nm=hwf\nnj^Hfjwf\nmi\u0000hwi\nnj^Hijwi\nmi; (3)\nwherewi\nm(wf\nm) and ^Hi(^Hf) are the initial (\fnal)\nMLWF and the Hamiltonian operator, respectively.\nEq. (3) describes the variation of an onsite energy or\na hopping integral depending on whether jwmiandjwni\nare located on the same site or at di\u000berent sites. Since\nany general lattice vibration can be expanded as a linear\ncombination of normal modes, one is typically interested\nin calculating \u0001 \"nmdue to vibrations along the normal\nmode coordinates. In order to quantify the rate of such\nchange, we de\fne the electron-phonon coupling parame-\nter,g\u0015\nmn, for the\u0015-th phonon mode as the rate of change,\n\u0001\"mn, of\"mnwith respect to a displacement \u0001 Q\u0015along\nsuch normal mode, namely\ng\u0015\nmn=@\"mn\n@Q\f\f\f\f\nQ!Q+\u0001Q\u0015: (4)\nHereQdescribes the system's geometry, so that Q!\nQ+ \u0001Q\u0015indicates that the partial derivative is to be\ntaken with respect to the atomic displacement along the\nphonon eigenvector corresponding to the mode \u0015.\nThis coupling constant is fundamentally di\u000berent from\nthat de\fned in a conventional TB formulation. In that\ncase the electron-phonon coupling is simply de\fned as\n\u000b\u0015\nnm=@\u0010\nh\u001ei\nnj^Hf\u0000^Hij\u001ei\nmi\u0011\n@Q\f\f\f\f\nQ!Q+\u0001Q\u0015;(5)\nwherej\u001ei\nniis then-th basis function before the motion.\nNote that, at variance with Eq. (3), which takes into ac-\ncount both the changes in the operator and the basis set,\nin Eq. (5) only the Hamiltonian operator is modi\fed and\nthe matrix element is evaluated with respect to the basis\nset corresponding to the equilibrium structure. For the\nremaining of this paper, unless stated otherwise, electron-\nphonon coupling will always denote the \frst description,\ni.e. theg\u0015\nmns of Eq. (4). The e\u000bect of such coupling on3\ncharge transport has been the subject of many previous\ninvestigations.38{41\nAs all matrix elements, also those associated to the\nSO coupling depend on the ionic coordinates. In a pre-\nvious paper35we have described a method to calculate\nthe SO matrix elements associated to the MLWFs ba-\nsis,hws1\nmRj^VSOjws2\nnR0i, from those computed over the spin-\npolarized Bloch states, h s1\nm;kj^VSOj s2\nn;k0i(the superscript\ndenotes the magnetic spin quantum number). Note that\nhere the MLWFs computed in absence of SO coupling\nare used as basis functions, since they span the entire\nrelevant Hilbert space. The term h s1\nm;kj^VSOj s2\nn;k0ican\nbe, in principle, calculated from any DFT implementa-\ntion that incorporates SO coupling. Our choice is the\nsiesta code42, which uses an on-site approximation43for\nthe SO coupling and gives the SO elements in terms of\na set of localized atomic orbitals fj\u001es\n\u0016;Rlig44. Hence, the\nbasic \rowchart for such calculation follows the general\nprescription\nh\u001es1\n\u0016;Rjj^VSOj\u001es2\n\u0017;Rli!h s1\nm;kj^VSOj s2\nn;k0i!hws1\nmRj^VSOjws2\nnR0i;\n(6)\nnamely from the SO matrix elements calculated for the\nsiesta local orbitals one computes those over the Bloch\nfunctions and then the ones over the MLWFs.\nOnce the matrix elements hws1\nmRj^VSOjws2\nnR0iare known,\nit is possible to determine the spin-phonon coupling by\nfollowing a prescription similar to that used for comput-\ning the electron-phonon coupling in Eq. (4),\ngs1s2(\u0015)\nm;n =@\"s1s2\n(SO)mn\n@Q\f\f\f\f\nQ!Q+\u0001Q\u0015; (7)\nwhere\"s1s2\n(SO)mnis the SO matrix element between the ML-\nWFsjws1miandjws2ni,Qdenotes the atomic positions\nand \u0001Qrefers to an in\fnitesimal displacement of the\ncoordinates along the \u0015-th phonon mode. As noted ear-\nlier, a change in atomic coordinates results in a change\nin the MLWFs and such change must be taken into ac-\ncount when calculating the di\u000berence in the SO elements\n\u0001\"s1s2\n(SO)mn. We use the same symbol gto denote both\nthe electron-phonon and the spin-phonon coupling, since\nthey can be distinguished by the presence or absence of\nthe spin indices.\nIn practice, when calculating both the electron-phonon\nand the spin-phonon coupling each atom iin the unit cell\nis in\fnitesimally displaced by \u0001 Q\u0015ei\n\u0015along the direc-\ntion of the corresponding phonon eigenvector, ei\n\u0015. Then\nthe electron-phonon (spin-phonon) coupling is calculated\nas \u0001\"mn=\u0001Q\u0015(\u0001\"s1s2\n(SO)mn=\u0001Q\u0015), i.e. from \fnite di\u000ber-\nences. If \u0001Q\u0015is too large, then the harmonic approxima-\ntion, which is the basis of this approach, breaks down. In\ncontrast, if \u0001 Q\u0015is too small, then the quantity will have\na signi\fcant numerical error. Hence, for any system stud-\nied, one must evaluate the coupling term for a range of\n\u0001Q\u0015and, from a plot of coupling terms vs \u0001 Q\u0015, choose\nΓ π /a-30-20-100E-EF (eV)\nOmitted bands\nIncluded bandsFIG. 1. Band structure of a diatomic Pb chain calculated\nwith a minimal basis set in siesta . The black and the red\nlines correspond to bands omitted from and included in the\nconstruction of the MLWFs, respectively.\nthe most suitable value of \u0001 Q\u0015. It is important to note\nthat the coupling terms so de\fned have the dimension of\nenergy/length. This is consistent with the semiclassical\nTB Hamiltonian used, for example in Ref.45, for treat-\ning transport in organic crystals with signi\fcant dynamic\ndisorder. However, various other de\fnitions and dimen-\nsions for the electron-phonon coupling can be found in\nliterature.41,46{48\nIII. RESULTS AND DISCUSSION\nA. One Dimensional Pb Chain\nA linear chain of Pb atoms with a diatomic unit cell has\n6 phonon modes for each wave-vector, q. For simplicity\nwe restrict our calculations to the \u0000-point, q=0, so that\nequivalent atoms in all unit cells have the same displace-\nments with respect to their equilibrium positions. Since\nfor the acoustic modes there is no relative displacement\nbetween the atoms of a unit cell, we are left with three\noptical modes of vibration as shown in the bottom panel\nof Fig. 2. The electronic band structure of a diatomic\nPb chain calculated with a single-zeta basis functions is\nshown in Fig. 1. Note that two of the bands marked in\nred are composed mostly of porbitals\u0019-bonding and are\ndoubly degenerate. Thus, as expected, the band struc-\nture contains 8 bands in total. The MLWFs are con-\nstructed by omitting the lowest two bands (mostly made\nofs-orbitals) and retaining the remaining 6 bands. This\ngives us six MLWFs per unit cell, three centred on each\natom. For each of the three modes, we evaluate the cou-4\nFIG. 2. The unit cell of the Pb chain containing two atoms.\nThe \fgures in the top panel show isovalue plots of the three\nMLWFs (from left to right: jw1;0i,jw2;0iandjw3;0i) cen-\ntred on the \frst atom. The bottom panels indicate the direc-\ntions of the atomic motion corresponding to the three phonon\nmodes (mode 1, mode 2 and mode 3, from left to right).\npling matrix elements between the MLWFs of the same\nunit cell for a range of \u0001 Q\u0015. By analysing these results\nwe \fnd that \u0001 Q\u0015= 0:03 is an acceptable value for such\nfractional displacement.\nThe top panel of Fig. 2 shows the MLWFs correspond-\ning to the \frst atom of the unit cell at the equilibrium\ngeometry. From this \fgure one can see that jw1;0i,jw2;0i\nandjw3;0iclosely resemble the porbitals of the \frst atom,\nwhich we can denote arbitrarily (the de\fnition of the axes\nis arbitary) as pz,pxandpy, respectively. By symmetry,\njw4;0i,jw5;0i,jw6;0ican be associated with the pz,px\nandpyorbitals located on the second atom. However, it\nis important to note that such similarity between the ML-\nWFs and the orbital angular momentum eigenstates does\nnot mean that they are equivalent . In order to appreciate\nthis point, note that\n\u000fhwi;0jwj;0i= 0;8i6=jbut this is not necessarily\ntrue forhpm;1jpn;2i, wherejpm;1iandjpn;2iare or-\nbital angular momentum eigenkets centred on the\n\frst and the second atom, respectively.\n\u000fWhen an atom is displaced from its equilibrium\nposition, the porbitals (e.g. the basis orbitals of\nsiesta ) experience a rigid shift only, but do not\nchange in shape. In contrast, the MLWFs change\nin shape along with being displaced.\n\u000fMost importantly, in the on-site SO approximation\nused in siesta , the hopping term for SO coupling,\ni.e. the SO matrix element between two orbitals lo-\ncated on two di\u000berent atoms, is always zero. As for\nthe on-site term, the SO matrix element between\ntwo orbitals of the same atom is independent of the\nposition of the other atom. Thus, the spin-phonon\nmatrix elements are always zero, when calculatedMode Element value(meV/ \u0017A)\nMode 1 [ w3jw4] -0.85\nMode 2 [ w1jw4] 4.03\n[w2jw5] -1.51\n[w3jw6] -1.51\nMode 3 [ w2jw4] -0.85\nTABLE I. The non-vanishing electron-phonon coupling ma-\ntrix elements for the \u0000-point phonon modes of the Pb chain\nwith a diatomic unit cell. [ w\u0016jw\u0017] denotes the electron-\nphonon coupling matrix element between the MLWFs jw\u0016i\nandjw\u0017i. One must keep in mind that the matrix elements\nare real and the remaining non-vanishing ones not reported in\nthe table can be found from the relation [ w\u0016jw\u0017] = [w\u0017jw\u0016].\nSee Fig. 2 for a diagram of the modes and the MLWFs.\nMode Element Value(meV/ \u0017A)\nMode 1 [ w\"\n1jw\"\n5] (0.0,-0.07)\n[w\"\n2jw\"\n4] (0.0,0.07)\n[w\"\n2jw#\n6] (-0.19,0.0)\n[w\"\n3jw#\n5] (0.19,0.0)\nMode 2 [ w\"\n1jw#\n5] (0.05,0.0)\n[w\"\n2jw#\n4] (-0.05,0.0)\n[w\"\n1jw#\n6] (0.0,-0.05)\n[w\"\n3jw#\n4] (0.0,0.05)\nMode 3 [ w\"\n1jw\"\n6] (0.0,0.07)\n[w\"\n3jw\"\n4] (0.0,-0.07)\n[w\"\n2jw#\n6] (0.0,-0.19)\n[w\"\n3jw#\n5] (0.00,0.19)\nTABLE II. Spin-phonon coupling matrix elements for the \u0000-\npoint phonon modes of the Pb chain with diatomic unit cell.\n[ws1\u0016jws2\u0017] denotes the complex spin-phonon coupling matrix\nelement between the MLWFs jws1\u0016iandjws2\u0017i. The remain-\ning non-vanishing matrix elements can be found from the re-\nlations in Eq. (8). The phonon modes and the MLWFs are\nshown in Fig. 2\n.\nwith the on-site SO approximation over the siesta\nbasis set. This is not the case for the MLWFs. Even\nwhen used in conjunction with an on-site SO ap-\nproximation, the spin-phonon coupling is typically\nnon-zero for a MLWF basis owing to the change in\nthe basis functions upon ionic displacement.\nBefore calculating the spin-phonon coupling, let us\ntake a brief look at the electron-phonon coupling ma-\ntrix elements for the three phonon modes. The non-zero\nmatrix elements are presented in Tab. I for each of the\nnormal modes. It is interesting to note that the change\nin overlap between the associated ` p' orbitals due to the\natomic displacements corresponding to the normal modes\ncan be intuitively expected to have the same trend as\nthe electron-phonon coupling matrix elements calculated\nwith respect to the MLWFs (since the MLWFs closely\nresembleporbitals). For example, for an atomic motion\nalong mode 3 (see Fig. 2), hpy;1jpz;2imust be zero, since\njpz;2ihas always equal overlap with the positive and neg-5\native lobe ofjpy;1i. Keeping in mind that modes 1, 2 and\n3 correspond, respectively, to a motion in the y,zandx\ndirection, one can easily show that\n\u000f\u0001hpz;1jpz;2imode:2>\u0001hpz;1jpx;2imode:3,\n\u000f\u0001hpz;1jpx;2imode:3\n= \u0001hpx;1jpz;2imode:3\n= \u0001hpy;1jpz;2imode:1,\n\u000f\u0001hpx;1jpy;2imode:1\n= \u0001hpx;1jpz;2imode:2\n= \u0001hpy;1jpz;2imode:3\n= 0\nwhere \u0001 denotes a change in the overlap of the orbitals\ndue to their corresponding atomic motion.\nNow we proceed to present our results for the spin-\nphonon coupling. At variance with the electron-phonon\ncoupling matrix elements, the spin-phonon ones are not\nnecessarily real valued. For each mode of the three\nmodes, the inequivalent non-zero spin-phonon coupling\nmatrix elements are tabulated in Tab. II. We denote the\nspin-phonon matrix element between jws1\u0016iandjws2\u0017ias\n[ws1\u0016jws2\u0017]. All other (equivalent) non-zero spin-phonon\nmatrix elements can be found from those presented in\nTab. II by using the following relations\n[w\"\n\u0016jw#\n\u0017] =\u0000[w#\n\u0016jw\"\n\u0017]\u0003;\n[w\"\n\u0016jw#\n\u0017] = [w#\n\u0017jw\"\n\u0016]\u0003;\n=[w\"\n\u0016jw\"\n\u0017] =\u0000=[w#\n\u0016jw#\n\u0017]: (8)\nAlso, from the symmetry of the MLWFs, it is easy to\nshow that\n[w\"\n1jw\"\n5]Mode1 =\u0000[w\"\n2jw\"\n4]Mode1; (9)\n[w\"\n1jw\"\n6]Mode3 =\u0000[w\"\n3jw\"\n4]Mode3: (10)\nWe have noted that in the on-site approximation, the\nspin-phonon coupling (according to our de\fnition) of the\nPb chain should be zero, when calculated over the siesta\nbasis set. However, if such on-site approximation is re-\nlaxed, one will be able to determine a number of ana-\nlytical expressions for these coupling elements in terms\nof the change in orbital overlaps. It is interesting to\nnote that the analytical expressions calculated in this\nway share many qualitative similarities with those pre-\nsented in Tab. II. We summarize the \fndings of this sec-\ntion by noting that the spin-phonon couplings matrix ele-\nments corresponding to the two equivalent normal modes\nshow the expected symmetry. We have also seen that the\nnon-zero spin-phonon coupling matrix elements for mode\n2 are, in general smaller than those for the symmetry-\nequivalent modes 1 and 3.\n-15-10-505\nΓZ Z'Y ΓB B' Y A'Energy (eV)\n-7.5-7.0-6.5-6.0\nΓZ Z'Y ΓB B' Y A'Energy (eV)\nkPlot from SIESTA\nPlot from MLWFsFIG. 3. Band structure of the durene crystal. Panel (a) shows\nall the occupied and many unoccupied bands. MLWFs are\nconstructed from the 4 highest occupied bands, which are\nplotted in black. Panel (b) shows the magni\fed structure of\nthese 4 bands plotted with siesta (green line) and obtained\nfrom the MLWFs computed with wannier90 (red circle).\nB. Durene Crystal\nFinally we are in the position to discuss the spin-\nphonon coupling in a real organic crystal, namely in\ndurene. In an electron-phonon or spin-phonon coupling\ncalculation, one needs to make sure that the construc-\ntion of the MLWFs converges to a global minimum, oth-\nerwise the various displaced geometries may correspond\nto di\u000berent local minima resulting in the description of\na di\u000berent energy landscape. Typically, a MLWF calcu-\nlation with dense k-mesh is likely to converge to a local\nminimum, while a calculation with coarse k-mesh has a\nhigher probability of giving the global minimum (\u0000-point\ncalculation always converges to the global minimum).\nHowever, a coarse k-mesh translates in a small period\nfor the Born-Von-Karman boundary conditions, i.e. a\npoorer description of the crystal. In our calculation, we6\n(b)(c)(d)(a)\nFIG. 4. Isovalue plots for MLWFs of the four topmost va-\nlence bands of a durene crystal. Panels (a), (b), (c) and (d)\ncorrespond to jw1;0i,jw2;0i,jw3;0iandjw4;0irespectively.\nuse a 4\u00024\u00024k-grid and construct the MLWFs from\nthe top four valence bands. This enables the calculation\nto converge to a global minimum, identi\fed by vanish-\ning or negligible imaginary elements in the Hamiltonian\nmatrix. In Fig. 3(a) we show a plot of the durene band-\nstructure (within a large energy window) and in Fig. 3(b),\nthe bandstructure corresponding to the four bands used\ntoconstruct MLWFs. These are plotted from the DFT\nsiesta eigenvalues and by diagonalizing the tight-binding\nHamiltonian constructed over the MLWFs.\nSince the unit cell of durene contains two molecules,\nthe four valence bands give us four MLWFs per unit cell,\nso that each molecule has associated two MLWFs. In\nFig. 4 we show an isovalue plot of the 4 MLWFs cor-\nresponding to R=0. We see that unlike jw3;0iand\njw4;0i, which are situated on the same molecule, jw1;0i\nandjw2;0iare on di\u000berent but equivalent molecules dis-\nplaced by a primitive lattice vector a2. Thus,jw1;0iand\njw2;R0iare on the same molecule for R0=\u0000a2, where\nfa1;a2;a3gis the set of primitive vectors. This means\nthat for our tight-binding picture hw1;0j^Hjw2;0icorre-\nsponds to a non-local (hopping) matrix element, whereas\nhw1;0j^Hjw2;R0iis a local (on-site) energy term. In the fol-\nlowing, we shall calculate the electron-phonon and spin-\nphonon coupling corresponding to various modes of the\ndurene crystal and compare: 1) the relative contribution\nof the di\u000berent modes, 2) for each mode, the relative con-\ntribution of the local and non-local terms.\nSince the unit cell contains two molecules, each with 24\natoms (48 atoms in the unit cell), a \u0000-point phonon calcu-\nlation will give us 144 modes, with 141 being non-trivial.\nAmong these, 12 will be predominantly intermolecular\nmodes (3 translational and 9 rotational modes, where the\nmolecules move rigidly with respect to each other) and\nthe remaining ones will be of predominantly intramolecu-\nlar nature. Here we shall consider only the phonon modes\nwith an energy less than 75 meV, as the modes with\nhigher energy are accessible only at high temperature41.\nFIG. 5. Histogram of the e\u000bective electron-phonon coupling\nas a function of the phonons energy. The local and the non-\nlocal contributions are denoted by green and red bars, respec-\ntively.\nThus, we take into account 25 modes, of which the \frst\n12 are intermolecular (these are lower in energy) and the\nrest are symmetry inequivalent intramolecular ones49.\nIn order to compare the contributions of the di\u000ber-\nent phonon modes and of the local (Holstein-type) and\nnon-local (Peierls-type) contributions, we calculate the\nfollowing e\u000bective electron-phonon coupling parameters\nGL\n\u0015=X\nm;njg\u0015\nmnj2; (11)\nwheremandnare functions centred on same molecule,\nand\nGN\n\u0015=X\nm6=njg\u0015\nmnj2; (12)\nwheremandnare on di\u000berent molecules.\nHere the superscripts L and N stand for Local and\nNon-local, respectively. A crucial point to be noted for\ntreating bulk crystals is that in wannier90 , the direct\nlattice points, where the MLWFs are calculated, are the\nlattice points of the Wigner-Seitz cell about the cell ori-\ngin,R=0. Typically, one should expect the number\nof such lattice points to be the same as the number of\nk-points in reciprocal space. However, in a 3-D crystal it\nis possible to have lattice points, which are equidistant\nfrom the R=0cell and (say) nnumber of other cells.\nThis means that such lattice point is shared by Wigner-\nSeitz cells of n+ 1 cells. In this case, this degenerate\nlattice point is taken into consideration by wannier90 ,\nbut a degeneracy weight of 1 =(n+ 1) is associated with\nit. Consequently in further calculations (such as the band\nstructure interpolation), its contribution carries a factor\nof 1=(n+ 1). Keeping this in mind, we multiply the con-\ntributions from the MLWFs of degenerate direct lattice7\npoints by their corresponding weighting factors. Fig. 5\nshows a histogram of the G\u0015terms as function of the\nphonons energy. It must be kept in mind that the cou-\npling matrix elements are strongly dependent on the ML-\nWFs. Therefore, constructing Wannier functions from a\ndi\u000berent set of Bloch states can in principle result in dif-\nferent values of G\u0015. We see that in our case, most of\nthe modes with high G\u0015(=GL\n\u0015+GN\n\u0015) are located at high\nphonon energies. Also, the electron-phonon couplings for\nmodes with lower G\u0015are dominated by the non-local con-\ntributions, while those with higher G\u0015are dominated by\nlocal contributions.\nConcerning the spin-phonon coupling, we can de\fne\nspin-dependent G\u0015terms, namely the e\u000bective spin-\nphonon couplings,\nGL(s1s2)\n\u0015=X\nm;njgs1s2(\u0015)\nmnj2; (13)\nwheremandnare on same molecule and\nGN(s1s2)\n\u0015=X\nm6=njgs1s2(\u0015)\nmnj2; (14)\nwheremandnare on di\u000berent molecules.\nIn Fig. 6, we plot these e\u000bective spin-phonon coupling\nterms, and we break down the local and non-local con-\ntributions. The top and the bottom panels correspond\nto the (s1=\";s2=\") and (s1=\";s2=#) case, respec-\ntively. As expected, the spin-phonon coupling terms are\nextremely small (about four orders of magnitude smaller\nthan those of the Pb chain), owing to the small atomic\nmasses in the crystal (the SOC is small). As in the\ncase of the electron-phonon interaction, the e\u000bective spin-\nphonon coupling terms are dominated by non-local con-\ntributions for low G(s1s2)\n\u0015=GL(s1s2)\n\u0015+GN(s1s2)\n\u0015and by\nlocal contributions for high G\u0015. We also see that the\nspin-phonon coupling (for same spin, as well as for dif-\nferent spins) is very small for the \frst few modes, which\nrepresent intermolecular motions. This is fully consis-\ntent with the short-ranged nature of SO coupling. An\nimportant message emerging from these results is that\nphonon modes having high e\u000bective electron-phonon cou-\npling do not necessarily have high e\u000bective spin-phonon\ncoupling, and vice-versa. This means that the knowledge\nof the phonon spectrum says little a priori about the\nspin-phonon coupling, so that any quantitative theory of\nspin relaxation cannot proceed unless a detail analysis\nalong the lines outlined here is performed.\nIn conclusion, we have discovered that both the\nelectron-phonon and the spin-phonon coupling constants\nare, in general, dominated by the local modes, as ex-\npected by the short-range nature of the SOC. However,\nmodes with very small e\u000bective coupling tend to have a\nlarger relative contribution arising from non-local modes.\nNo apparent correlation can be found between the ef-\nfective coupling constants pertaining to various phonon\nmodes for the electron-phonon coupling and those for the\nspin-phonon coupling.\nFIG. 6. Histogram plot of the e\u000bective spin-phonon coupling\nparameters, G(s1s2)\n\u0015 , as a function of the phonons energy. The\ntop panel corresponds to the s 1= s2case (same spins), while\nthe bottom one corresponds to s 16= s2(di\u000berent spins). The\nlocal and the non-local contributions are denoted by green\nand red bars respectively.\nIV. CONCLUSION\nBased on our previous work concerning the calculation\nof the SO matrix elements with respect to MLWFs basis\nsets, we have presented calculations of the spin-phonon\ncoupling matrix elements of periodic systems. We note\nthat, in order to be useful in a multiscale approach based\non an e\u000bective Hamiltonian, the electron-phonon and the\nspin-phonon coupling are not to be calculated in terms\nof a \fxed set of MLWFs. Instead, one must take into\naccount the change in the MLWFs as a result of the\nionic motions. The coupling matrix elements for a given\nphonon mode are calculated by displacing atoms from\nthe ground state geometry along that phonon eigenvec-\ntor and by taking \fnite di\u000berences. For phonon modes at\nthe \u0000-point, we have calculated the electron-phonon and8\nspin-phonon coupling elements of a 1D chain of Pb atoms\nwith two atoms per unit cell and of a bulk durene crys-\ntal. This latter is a widely-studied and well-known or-\nganic semiconductor. The spin-phonon coupling matrix\nelements of the Pb chain obey the expected symmetry\nrelations. For durene we have observed that, in general,\nthe spin-phonon coupling is dominated by local contribu-\ntions (Holstein-modes), although, for phonon modes with\na small net e\u000bective coupling, the non-local part seems\nto dominate. Our calculations of spin-phonon coupling\nmatrix elements are expected to be valuable in the con-\nstruction of a e\u000bective Hamiltonians to be used for com-\nputing transport-related quantities. This is particularlywelcome in the case of organic crystals, where ab initio\ncomputation of transport properties is a challenging task.\nV. ACKNOWLEDGEMENTS\nThis work is supported by the European Research\nCouncil, Quest project. Computational resources have\nbeen provided by the supercomputer facilities at the Trin-\nity Center for High Performance Computing (TCHPC)\nand at the Irish Center for High End Computing\n(ICHEC). We thank Carlo Motta for providing the \u0000-\npoint phonon eigenvalues and eigenvectors of durene\ncrystal used in Ref.41.\n1N. F. Mott, Proc. Royal Soc. A 153, 699 (1936).\n2S. A. Wolf, D. D. Awschalom, R. A. Buhrman, J. M.\nDaughton, S. von Moln\u0013 ar, M. L. Roukes, A. Y. Chtchelka-\nnova, and D. M. Treger, Science 294, 1488 (2001).\n3B. Dieny, V. S. Speriosu, S. S. P. Parkin, B. A. Gurney,\nD. R. Wilhoit, and D. Mauri, Phys. Rev. B 43, 1297\n(1991).\n4S. Sanvito, Nature Mater. 10, 484 (2011).\n5M. N. Baibich, J. M. Broto, A. Fert, F. N. Van Dau,\nF. Petro\u000b, P. Etienne, G. Creuzet, A. Friederich, and\nJ. Chazelas, Phys. Rev. Lett. 61, 2472 (1988).\n6G. Binasch, P. Gr unberg, F. Saurenbach, and W. Zinn,\nPhys. Rev. B 39, 4828 (1989).\n7J. S. Moodera, L. R. Kinder, T. M. Wong, and R. Meser-\nvey, Phys. Rev. Lett. 74, 3273 (1995).\n8J. Fabian, A. Matos-Abiague, C. Ertler, P. Stano, and\nI. Zutic, Acta Phys. Slovaca 57, 565 (2007).\n9D. D. Awschalom and M. E. Flatt\u0013 e, Nature Phys. 3, 153\n(2007).\n10F. P. Marn and H. Suhl, Phys. Rev. Lett. 63, 442 (1989).\n11H. Ochoa, A. H. Castro Neto, V. I. Fal'ko, and F. Guinea,\nPhys. Rev. B 86, 245411 (2012).\n12T. Birol and C. J. Fennie, Phys. Rev. B 88, 094103 (2013).\n13J. H. Lee and K. M. Rabe, Phys. Rev. B 84, 104440 (2011).\n14J. Hong, A. Stroppa, J. \u0013I~ niguez, S. Picozzi, and D. Van-\nderbilt, Phys. Rev. B 85, 054417 (2012).\n15T. Birol, N. A. Benedek, H. Das, A. L. Wysocki, A. T.\nMulder, B. M. Abbett, E. H. Smith, S. Ghosh, and C. J.\nFennie, Curr. Opin. Solid State Mater. Sci. 16, 227 (2012).\n16M. Mochizuki, N. Furukawa, and N. Nagaosa, Phys. Rev.\nB84, 144409 (2011).\n17G. Malliaras and R. Friend, Phys. Today 58, 53 (2005).\n18Nature Mater. 12, 591 (2013), editorial.\n19S. R. Forrest, Nature 428, 911 (2004).\n20S. Sanvito, Chem. Soc. Rev. 40, 3336 (2011).\n21S. Wang, A. Wittmann, K. Kang, S. Schott, G. Schwe-\nicher, R. D. Pietro, J. Wunderlich, D. Venkateshvaran,\nM. Cubukcu, and H. Sirringhaus, in 2017 IEEE Inter-\nnational Magnetics Conference (INTERMAG) (2017) pp.\n1{1.\n22A. Troisi, Chem. Soc. Rev. 40, 2347 (2011).\n23C. Dimitrakopoulos and P. Malenfant, Adv. Mater. 14, 99\n(2002).24O. D. Jurchescu, J. Baas, and T. T. M. Palstra, Appl.\nPhys. Lett. 84, 3061 (2004).\n25O. Ostroverkhova, D. G. Cooke, F. A. Hegmann, J. E.\nAnthony, V. Podzorov, M. E. Gershenson, O. D. Jurchescu,\nand T. T. M. Palstra, Appl. Phys. Lett. 88, 162101 (2006).\n26K. Marumoto, S.-i. Kuroda, T. Takenobu, and Y. Iwasa,\nPhys. Rev. Lett. 97, 256603 (2006).\n27T. Sakanoue and H. Sirringhaus, Nature Mater. 9, 736\n(2010).\n28A. Troisi and G. Orlandi, J. Phys. Chem. A 110, 4065\n(2006).\n29K. Hannewald and P. A. Bobbert, Phys. Rev. B 69, 075212\n(2004).\n30P. A. Bobbert, W. Wagemans, F. W. A. van Oost,\nB. Koopmans, and M. Wohlgenannt, Phys. Rev. Lett. 102,\n156604 (2009).\n31S. Bhattacharya, A. Akande, and S. Sanvito, Chem. Com-\nmun. 50, 6626 (2014).\n32A. Lunghi, F. Totti, R. Sessoli, and S. Sanvito, Nature\nCommun. 8, 14620 (2017).\n33A. Lunghi, F. Totti, S. Sanvito, and R. Sessoli, Chem. Sci.\n8, 6051 (2017).\n34N. Marzari, A. A. Mosto\f, J. R. Yates, I. Souza, and\nD. Vanderbilt, Rev. Mod. Phys. 84, 1419 (2012).\n35S. Roychoudhury and S. Sanvito, Phys. Rev. B 95, 085126\n(2017).\n36N. Marzari and D. Vanderbilt, Phys. Rev. B 56, 12847\n(1997).\n37A. A. Mosto\f, J. R. Yates, Y.-S. Lee, I. Souza, D. Vander-\nbilt, and N. Marzari, Comput. Phys. Commun. 178, 685\n(2008).\n38F. Ortmann, K. Hannewald, and F. Bechstedt, phys. sta-\ntus solidi B 245, 825 (2008).\n39F. Ortmann, K. Hannewald, and F. Bechstedt, Appl.\nPhys. Lett. 93, 222105 (2008).\n40H. Ishii, N. Kobayashi, and K. Hirose, Phys. Rev. B 95,\n035433 (2017).\n41C. Motta and S. Sanvito, J. Chem. Theo. Comp. 10, 4624\n(2014).\n42J. M. Soler, E. Artacho, J. D. Gale, A. Garc\u0012 \u0010a, J. Junquera,\nP. Ordejon, and D. Sanchez-Portal, J. Phys. Condens.\nMatter 14, 2745 (2002).\n43In the on-site approximation a SO coupling matrix element\nis neglected unless both orbitals and the SO pseudopoten-9\ntial are on the same atom.\n44L. Fern\u0013 andez-Seivane, M. A. Oliveira, S. Sanvito, and\nJ. Ferrer, J. Phys. Condens. Matter 18, 7999 (2006).\n45A. Troisi and G. Orlandi, Phys. Rev. Lett. 96, 086601\n(2006).\n46K. Kaasbjerg, K. S. Thygesen, and K. W. Jacobsen, Phys.\nRev. B 85, 115317 (2012).\n47A. Girlando, L. Grisanti, M. Masino, I. Bilotti, A. Bril-\nlante, R. G. Della Valle, and E. Venuti, Phys. Rev. B 82,035208 (2010).\n48V. Coropceanu, J. Cornil, D. A. da Silva Filho, Y. Olivier,\nR. Silbey, and J.-L. Br\u0013 edas, Chem. Rev. 107, 926 (2007).\n49The phonon spectrum is calculated with FHI-AIMS. Cal-\nculation courtesy: Dr. Carlo Motta."    },    {        "title": "1012.4757v1.Spin_dephasing_and_pumping_in_graphene_due_to_random_spin_orbit_interaction.pdf",        "content": "arXiv:1012.4757v1  [cond-mat.mes-hall]  21 Dec 2010Spin dephasing and pumping in graphene due to random spin-or bit interaction\nV. K. Dugaev1,2, E. Ya. Sherman3,4, and J. Barna´ s5∗\n1Department of Physics, Rzesz´ ow University of Technology,\nal. Powsta´ nc´ ow Warszawy 6, 35-959 Rzesz´ ow, Poland\n2Department of Physics and CFIF, Instituto Superior T´ ecnic o,\nTU Lisbon, Av. Rovisco Pais, 1049-001 Lisbon, Portugal\n3Department of Physical Chemistry, Universidad del Pa´ ıs Va sco UPV-EHU, 48080 Bilbao, Spain\n4IKERBASQUE, Basque Foundation for Science, 48011, Bilbao, Spain\n5Institute of Molecular Physics, Polish Academy of Sciences ,\nul. Smoluchowskiego 17, 60-179 Pozna´ n, Poland\n(Dated: April 25, 2022)\nWe consider spin effects related to the random spin-orbit int eraction in graphene. Such a random\ninteraction can result from the presence of ripples and/or o ther inhomogeneities at the graphene\nsurface. We show that the random spin-orbit interaction gen erally reduces the spin dephasing\n(relaxation) time, even if the interaction vanishes on aver age. Moreover, the random spin-orbit\ncoupling also allows for spin manipulation with an external electric field. Due to the spin-flip\ninterband as well as intraband optical transitions, the spi n density can be effectively generated by\nperiodic electric field in a relatively broad range of freque ncies.\nPACS numbers: 72.25.Hg,72.25.Rb,81.05.ue,85.75.-d\nI. INTRODUCTION\nGraphene is currently attracting much attention as a\nnew excellent material for modern electronics1–3. The\nnatural two-dimensionality of graphene matches per-\nfectly to the dominating planar technology of other semi-\nconducting materials, and correspondingly gives the way\nto creatingnewhybrid systems. However,the moststrik-\ning properties of graphene are not directly related to its\ntwo-dimensionality. Due to the bandstructure effects,\nelectrons in pure graphene can be described by the rela-\ntivistic Dirac Hamiltonian, leading to the linear electron\nenergy spectrum near the Dirac points. As a result, the\nelectronic and transport properties of graphene are sig-\nnificantly different from those of any other metallic or\nsemiconducting material,3–6except (to some extent) its\nparent material – the clean graphite.7\nIt has been also suggested that graphene may have\ngood perspectives as a new material for applications\nin spintronics.8–11The intrinsic spin-orbit interaction\nin graphene is usually very small, and therefore one\ncan expect extremely long spin dephasing (relaxation)\ntime.12–16Thus, spin injected to graphene, for instance\nfrom ferromagnetic contacts, can maintain its coherence\nfor a relatively long time. Experiments demonstrate\nspin relaxation times for various graphene-based systems\nspanned over several orders of magnitude17–21with some\nof them being much shorter than expected.17,18The rea-\nson of this contradiction is not quite clear, and several\ndifferent explanations of these observations have been\nalready put forward.22–24In this paper we present an-\n∗Also at Faculty of Physics, Adam Mickiewicz University, ul.\nUmultowska 85, 61-614 Pozna´ n, Polandother model based on the random Rashba spin-orbit\ninteraction.25,26Physical origin of such random spin-\norbit interaction can be related to the ripples existing\nat the surface of graphene27–29and/or to some impuri-\nties adsorbedat the surface, which randomlyenhance the\nmagnitude of spin-orbit coupling as compared to that in\nthe clean graphene.23\nOne of the key issues in spintronics (including\ngraphene-based spintronics) is the possibility of spin ma-\nnipulation with an external electric/optical field. This\nincludes spin generation, spin rotation, spin switching,\netc. Here we consider the possibility of spin pumping in\ngrapheneusingtheideaofcombinedresonanceinsystems\nwith Rashba spin-orbit interaction.30,31The possibility\nof spin manipulation using optical excitation32is based\non various mechanisms of spin-orbit interaction in semi-\nconductor systems. In particular, the spin polarization\nappears in systems with regular spin-orbit coupling, sub-\nject to periodic electric field.33–35It has been shown re-\ncently, that the randomspin-orbit interaction also can be\napplied to generate spin polarization in symmetric semi-\nconductor quantum wells.36In this paper we show that\nsimilar method can be used to generate spin polarization\nin graphene with random Rashba spin-orbit interaction.\nTo do this we analyze the intensity of optically-induced\nspin-flip transitions assuming two-dimensional massless\nDirac model of electron energy spectrum in graphene,\nand calculate the magnitude of spin-polarization induced\nby the optical pumping.\nThe paper is structured as follows. In section 2 we\ndescribe the model Hamiltonian assumed for graphene.\nSpindephasingduetotherandomspin-orbitRashbacou-\npling is calculated in section 3. In turn, spin pumping by\nan external electric field is considered in section 4. Final\nconclusions are presented in section 5.2\nII. MODEL\nTo describe electrons and holes in the vicinity of Dirac\npoints we use the model Hamiltonian H0which is suffi-\ncient when considering the effects related to low-energy\nelectron and hole excitations. We also include the spin-\ndependent perturbation in the form of a spatially fluc-\ntuating Rashba spin-orbit interaction, Hso. Thus, the\nsystem Hamiltonian can be written as (we use system of\nunits with ¯h≡1)\nH=H0+Hso, (1)\nH0=vτ·k, (2)\nHso=λ(r)\nv/parenleftbigg∂H0\n∂kxσy−∂H0\n∂kyσx/parenrightbigg\n=λ(r) (τxσy−τyσx), (3)\nwherevis the electron velocity, λ(r) is the random spin-\norbit parameter, r= (x,y) is the two-dimensional coor-\ndinate, and τandσare the Pauli matrices acting in the\nsublattice and spin spaces, respectively.37Equations (2)\nand (3) show that spin-orbit coupling can be described\nby a conventional Rashba Hamiltonian, proportional to\nvxσy−vyσx, where the velocity components vx,vyare, in\ngeneral, obtained with the unperturbed Hamiltonian H0.\nIt is well known that there is an intrinsic (internal) spin-\norbit coupling in graphene, which is related to relativistic\ncorrections to the crystal field of the corresponding lat-\ntice. In addition, a spatially uniform Rashba field can\nbe induced by the substrate on which the graphene sheet\nis located. The reported results suggest that these inter-\nactions, which can be considered as independent sources\nof spin relaxation, are either very weak15,38or do not\ncontribute to the dephasing rate by symmetry reasons.24\nTherefore, we will neglect them in our considerationsand\nwill briefly discuss their role in the following.\nThe Schr¨ odinger equation, ( H0−ε)ψk= 0, for the\npseudospinor components of the wavefunction ψkis\n/parenleftbigg\n−ε vk−\nvk+−ε/parenrightbigg/parenleftbigg\nϕk\nχk/parenrightbigg\n= 0, (4)\nwherek±=kx±iky. The normalized solutions corre-\nsponding to the eigenstates εk=±vkof Hamiltonian H0\ncan be written in the form\nψkσ±(r) =eik·r\n√\n2/parenleftbigg\n|1σ/angbracketright±k+\nk|2σ/angbracketright/parenrightbigg\n,(5)\nwhere the ±signs correspond to the states in upper and\nlower branches, respectively.\nWe assume that the average value of spin-orbit inter-\naction vanishes, while the spatial fluctuation of λ(r) can\nbe described by the correlation function F(r−r′) of a\ncertain form,\n/angbracketleftλ(r)/angbracketright= 0, (6)\nC(r−r′)≡ /angbracketleftλ(r)λ(r′)/angbracketright=/angbracketleftbig\nλ2/angbracketrightbig\nF(r−r′).(7)When calculating spin dephasing, one can consider\nonly the electron states corresponding to the upper\nbranch (conduction band) of the energy spectrum, εk=\nvk. The spin-flip scattering from the random potential\ndetermines the spin relaxation in this particular band,\nwhile from symmetry of the system follows that spin de-\nphasing in the lower (valence) band is the same. The\nintraband matrix elements of the random spin-orbit in-\nteraction (3) in the basis of wavefunctions (5) for the\nconduction band form the following matrix in the spin\nsubspace\nVkk′=λkk′/parenleftbigg\n0−ik−/k\nik′\n+/k′0/parenrightbigg\n, (8)\nwhereλkk′is the Fourier component of the random spin-\norbit coupling. Since scattering from the random spin-\norbit potential is elastic, only the intraband transitions\ncontribute to the spin relaxation.\nIII. SPIN DEPHASING\nTo demonstrate how the random spin-orbit coupling\nworks in graphene and how its effects can be observed\nin experiment, as well as to compare graphene and con-\nventional two-dimensional semiconductor structures, we\ncalculate in this Section the corresponding spin dephas-\ning time. For this purpose we use the kinetic equation for\nthe density matrix (Wigner distribution function),36,40\n∂ρk\n∂t= Stρk. (9)\nThe collision integral St ρkon the right-hand side of this\nequation is due to the spin-flip scattering from the ran-\ndom spin-orbit interaction,\nStρk=π/summationdisplay\nk′(2Vkk′ρk′Vk′k−Vkk′Vk′kρk−ρkVkk′Vk′k)\n×δ(εk−εk′). (10)\nWe assume the following form of the density matrix:\nρk=ρ0k+skσz, (11)\nwhere the first term ρ0kcorresponds to the spin-\nunpolarized equilibrium state. On substituting (8) and\n(11) into Eq. (10) we find\nStρk=−2πσz/summationdisplay\nk′C(q)(sk+sk′)δ(εk−εk′),(12)\nwhereC(q)≡/angbracketleftbig\nλ2\nkk′/angbracketrightbig\nandq=k′−k. Assuming that\nskdoes not depend on the point at the Fermi surface we\nobtain\nStρk=−4kσzsk\nπv/integraldisplay2k\n0C(q)/radicalbig\n4k2−q2dq. (13)3\n20 30 40 50 60\nR (nm)0.30.40.50.60.70.80.91\nk = 106 /cmτs /τs0\nFIG. 1: Spindephasingtimeas afunctionofthecharacterist ic\nrangeRof the random spin-orbit fluctuations.\nFordefiniteness, we assumethat the characteristicspa-\ntial range of the random spin-orbit fluctuations is R, and\ntakeC(q) in the following form:\nC(q) = 2π/angbracketleftλ2/angbracketrightR2e−qR, (14)\nsatisfying the normalization condition\n/integraldisplay\nC(q)d2q\n(2π)2=/angbracketleftλ2/angbracketright. (15)\nThe resulting spin relaxation rate is not strongly sensi-\ntive to the shape of the correlator. However, the ap-\nplicability of the approach based on Eq.(12) depends on\nthe ratio of electron mean free path ℓtoR, being valid\nonly ifℓ/R≫1. In such a case, typically realized in\ngraphene, the electron spin experiences indeed random,\nweakly correlated in time fluctuations of the spin-orbit\ncoupling. In addition, we can safely neglect the effect of\nthe random spin-orbit coupling on the momentum relax-\nation rate. Finally, for the spin dephasing time we obtain\nthe following expression:\n1\nτs\nk=8k\nv/angbracketleftbig\nλ2/angbracketrightbig\nR2/integraldisplay2kR\n0e−xdx√\n4k2R2−x2\n=4πk\nv/angbracketleftbig\nλ2/angbracketrightbig\nR2[I0(2kR)−L0(2kR)],(16)\nwhereI0(x)andL0(x)arethemodifiedBesselandStruve\nfunctions of zeroth order, respectively.\nIn the limiting semiclassical ( kR≫1) and quantum\n(kR≪1) cases we find\n1\nτs\nk≃4R\nv/angbracketleftbig\nλ2/angbracketrightbig\n\n1, kR≫1,\nπkR, kR ≪1.(17)\nForkR≫1, the result in Eq.(17) can be interpreted as\nthe special realization of the Dyakonov-Perel’ spin relax-\nation mechanism. To see this, we note that the electron\nspin rotates at the rate Ω ∼/angbracketleftbig\nλ2/angbracketrightbig1/2, with the preces-\nsion direction changing randomly at the timescale of the\norder of time that electron needs to pass through one do-\nmain of the size R, i.e.,τR∼R/v.The resulting spinrelaxation rate 1 /τs\nkis of the order of Ω2τR.It is worth\nmentioning that at given spatial and energy scale of the\nfluctuating spin-orbit field, the decrease in the electron\nfree path and in the momentum relaxation time leads to\nthe decrease in the spin dephasing rate: if ℓ≪R, elec-\ntron spin interacts with the local rather than with the\nrapidly changing random field, and spin relaxation rate\nbecomes of the order of/angbracketleftbig\nλ2/angbracketrightbig\nτ, whereτis the momen-\ntum relaxation time. This agrees qualitatively with the\nobservations of Ref.[20], however, a quantitative compar-\nison needs a more detailed analysis.\nTaking examples with typical values v= 108cm/s,\nR∼50 nm, and /angbracketleftλ2/angbracketright ∼500µeV2, similar to what can\nbe expected from Ref.[12], we obtain τs\nkless than or of\nthe order of 10 ns. As one can see from Eq.(17), the\nspin relaxation for small kR≪1 is suppressed, as can\nbe understood in terms of the averaging of the random\nfield over a large 1 /k2≫R2area. The full k-dependence\nin Eq.(17) implies that the spin dephasing rate is pro-\nportional to n1/2at low carrier concentrations nand\nis independent of nat higher ones. Therefore, at the\ncharge neutrality point, where n= 0, the spin relaxation\nvanishes, in agreement with the observations of Ref.[21].\nMoreover, our approach qualitatively agrees with the in-\ncrease in the spin relaxation time in the bilayer graphene\ncomparedtothesinglelayerone21: thetransverserigidity\nof the bilayer can be larger, thus suppressing formation\nof the long-range ripples, and, as a result, the random\nspin-orbit coupling.\nEquation (17) shows that, as far as the spin dephasing\nis considered, the only difference between graphene and\nconventional semiconductors25,26,36is related to the fact\nthat the electron velocity is constant for the former case\nand is proportional to the momentum in the latter one.\nAswewill seein the nextSection, this difference becomes\ncrucial for the spin pumping processes.\nThe dependence of spin relaxation time, calculated\nfrom Eq. (16) as a function of the characteristic domain\nsizeRof the random spin-orbit interaction is presented\nin Fig. 1, were τs0is defined as τ−1\ns0≡8/angbracketleftbig\nλ2/angbracketrightbig\n/vk. The\ncurves corresponding to different values of k>106cm−1\nwould be practically indistinguishable in this figure.\nHere several comments on the numerical values of spin\nrelaxation are in order. The values observed in experi-\nments on spin injection from ferromagnetic contacts17,18\nare of the order of 10−10s, two orders of magnitude less\nthan our estimate which does not take into account ex-\nplicitly the role of the Si-based substrates. The effect of\nthe SiO 2substrate, including the contributions from im-\npurities and electron-phonon coupling, was thoroughly\nanalyzed in Ref.[23]. However, the obtained dephas-\ning rates were well below the experimental values and\nalso below the estimate obtained here, leading the au-\nthors of Ref.[23] to the suggestion of an important role\nof heavy adatoms in the spin-orbit coupling.39On the\nother hand, it was shown that the spin dephasing rate\ncanbestronglyinfluencedatrelativelyhightemperatures\nby the electron-electron collisions.24However, including4\nthese collisions does not bring theoretical values closer to\nthe experimental ones.\nThe discrepancies between theory and experiment and\nbetween experimental data obtained on different systems\ncall for a more detailed analysis of the experimental sit-\nuation, including the dependence of spin relaxation time\non the device functional properties and the experimental\ntechniques applied.\nIV. SPIN PUMPING\nLet us consider now spin pumping by an external elec-\ntromagnetic periodic field corresponding to the vector\npotential A(t). We assume that the system described by\nEqs. (1)-(3) is additionally in a constant magnetic field\nB. For simplicity, we consider the Voigt geometry with\nthe field in the graphene plane, so that the effects of Lan-\ndau quantization are absent. Thus, the Hamiltonian H0\nincludes now the constant field and can be written as\nH0=vτ·k+∆σx, (18)\nwhere 2∆ = gµBBis the spin splitting and the magnetic\nfield is orientedalongthe x−axis (the electronLand´ efac-\ntor for graphene is g= 2). The induced spin polarization\nis opposite to the direction of the magnetic field.\nSince we are interested in real transitions in which the\nenergy is conserved, we consider interaction with a single\ncomponent of the periodic electromagnetic field, A(t) =\nAe−iωt,which enters in the gauge-invariant form\nHA=−e\nc∂H\n∂k·A(t) =−ev\ncτ·A(t),(19)\nand in the following we treat the term HAas a small\nperturbation.\nThe absorption in a periodic field (probability of field-\ninduced transitions in unit time) can be written as\nI(ω) = ReTr/summationdisplay\nk/integraldisplaydε\n2πHAGk(ε+ω)HAGk(ε),(20)\nwhereGk(ε) is the Green function. In the absence of\nspin-orbit interaction, the absorption (20) does not in-\nclude any spin-flip transitions. We can account for the\nspin-orbit interaction (3) in the second order perturba-\ntion theory, including the corresponding matrix elements\nas shown in Fig. 2. This means that we do not con-\nsiderperturbationtermsasthe self-energywithin asingle\nGreen function assuming that they are already included\nin the electron relaxation rate.\nThus, in the second order perturbation theory with re-\nspect to the random spin-orbit interaction Hsowe obtain\nI(ω) = ReTr/summationdisplay\nkk′/integraldisplaydε\n2πHAG0\nk(ε+ω)Hkk′\nsoG0\nk′(ε+ω)\n×HAG0\nk′(ε)Hk′k\nsoG0\nk(ε), (21)FIG. 2: (Color online.) Feynman diagrams for the light ab-\nsorption in the second order perturbation theory with respe ct\nto the random spin-orbit interaction. The coupling and elec -\ntromagnetic vertices are shown as white and filled circles, r e-\nspectively.\nFIG. 3: (Color online.) The energy spectrum and indirect\nspin-flip transitions in graphene in a uniform magnetic field\nB. Long and short arrows correspond to the interband and\nintraband transitions, respectively.\nwhere Green’s function G0\nk(ε) = diag/braceleftBig\nG0\nk↑(ε), G0\nk↓(ε)/bracerightBig\ncorresponds to the Hamiltonian H0of Eq. (18),\nG0\nkσ(ε) =ε+vτ·k+∆σ+µ\n(ε−ε1kσ+µ+iδsgnε)(ε−ε2kσ+µ+iδsgnε),\n(22)\nwithσ=±1 corresponding to the spins oriented along\nand opposite to the x−axis, respectively, ε(1,2)kσ=\n±vk+∆σ,δbeing the half of the momentum relaxation\nrate,δ= 1/2τ, andµdenoting the chemical potential.\nDiagrams in Fig. 2 show the qualitative difference\nbetween the graphene and semiconductor quantum well\nwith respect to the effects of random spin-orbit coupling.\nIn semiconductors, the external electric field is explicitly\ncoupled to the anomalous spin-dependent term in the\nelectron velocity, which is random, and therefore the di-\nagrams describing the corresponding transitions include\nonly two Green functions. In graphene, due to the ab-\nsence of randomness-originated term in the Hamiltonian\nHA,four Green functions are required to take into ac-\ncountthe randomcontributionofthe spin-orbitcoupling.\nThis situation, in some sense, is more close to what is ob-\nserved in the conventionalkinetic theory of normal metal5\nconductivity, where the coupling to the external field\ndoes not depend on the randomness explicitly, and the\nadditional disorder effects appear due to the self-energy\nand/or due to the vertex corrections, as in Fig. 2.\nUpon calculating contributions from the diagrams of\nFig. 2, one finds the total rate of spin-flip and spin-\nconserving transitions due to the random spin-orbit cou-\npling in the form\nIrso(ω) =e2A2\nc2ReTr/summationdisplay\nσσ′/summationdisplay\nkk′v2C(q)/integraldisplaydε\n2π(τ·nA)\n×G0\nkσ(ε+ω)τ−G0\nk′σ′(ε+ω)(τ·nA)\n×G0\nk′σ′(ε)τ+G0\nkσ(ε), (23)\nwhereτ±=τx±iτy,andnAdescribes the direction of\nA. In the following we assume linear polarization of\nlight,A= (A,0). After calculating the trace and in-\ntegrating over energy εin Eq. (23) one finds a rather\ncumbersome expression (see Appendix) consisting of sev-\neral terms, each of them corresponding to transitions be-\ntween certain branches of the spectrum (Fig. 3). For\ndefiniteness, we locate the chemical potential in the va-\nlence band. Correspondingly, only the transitions from\nthe bands (2 ↑) and (2 ↓) to the unoccupied states in\nthe bands (1 ↑), (1↓), (2↑) and (2 ↓) are possible. We\nwill concentrate on the optically induced spin-flip transi-\ntions contributing to the optically-generated spin pump-\ning. Hence, wedonotconsiderspin-conservedtransitions\ncontributing to the usual Drude conductivity.41–43\nA. Interband spin-flip transitions\nLet us consider first the spin-flip transitions from the\nvalence to conduction bands, such as k2→k′\n1.For con-\nvenience we introduce the parameter kj= (k,σx)j,de-\nscribing the momentum and spin projection for an elec-\ntron in the subband j.The corresponding expression for\nthe transition rate can be obtained from the equations\npresented in the Appendix, and considerably simplified\nby taking into account that: (i) in the nonsingular terms\nεk1−εk2can be substituted by ω,(ii) in the semiclassical\nlimitkR≫1 the energy change due to the change in the\nmomentumissmallcomparedto1 /τ, and(iii)thephoton\nenergy is much larger than the characteristic low-energy\nscale parameters, i.e., ω≫1/τ,andω≫∆. We mention\nthat the linear in ∆ terms have to be kept despite ∆ ≪ω\nsince the resulting spin pumping rate, determined by the\ncontributions of both initial spin states, is linear in ∆.\nThe expressions for the parameters K2σ(ω), which deter-\nmine the spin-flip rate as introduced in the Appendix,\nEq.(A2), can be then simplified, and as a result one ob-\ntains the following formula for the spin-flip rate in the\nrelevant frequency domain:\nI2σ→1σ′(ω) = 4πσe2A2\nc2v4\nω/summationdisplay\nkk′C(q)/bracketleftBig\nf(εk2)−f(εk′\n1)/bracketrightBig0.020.040.060.080.11010K    [s-3eV-2]\n5 10 15 20\nphoton energy [meV]00.020.040.061010 K    [s-3eV-2]\nR=20 nmR=30 nm\nR=20 nm\nR=10 nmR=30 nmR=10 nm(a)\n(b)2σ 2σ\nFIG. 4: (Color online.) The parameter K2↓(ω) for the rate\nof the spin-flip interband transitions in the high-frequenc y\ndomain (a) and K2↑(b) for different values of the range pa-\nrameter Rdescribing characteristic size of the fluctuations in\nspin-orbit interaction and /angbracketleftλ2/angbracketright= 100µeV2.\n×k2−kqcosϕ\nεk2+ω−εk1δ/parenleftBig\nεk2+ω−εk′\n1/parenrightBig\n/parenleftBig\nεk2+ω−εk′\n2/parenrightBig/parenleftBig\nεk2−εk′\n2/parenrightBig.(24)\nThe transitions are constrained due to the δ-function\nin Eq. (24) corresponding to the energy conservation\nwith the change in the momentum q(here we use ϕ=\ncos−1(k,q))\nδ/parenleftBig\nεk2+ω−εk′\n1/parenrightBig\n=|ω−vk−2σ∆|δ(ϕ−ϕ0)\nv2kq|sinϕ0|,(25)\nwhereϕ0is a solution of the equation\nvk+2σ∆−ω+v/radicalbig\nk2−2kqcosϕ+q2= 0,(26)\nwhich gives us the condition for a minimum value of mo-\nmentum in Eq. (26), vkmin=ω−2σ∆. The energy con-\nservation determines the angle ϕ0between the vectors k\nandqas\ncosϕ0=q\n2k+ω−2σ∆\nvq/parenleftbigg\n1−ω−2σ∆\n2vk/parenrightbigg\n.(27)\nThe usual condition of |cosϕ0|<1 leads to the following\nrestrictions in the integration over qin Eq. (24):\na) ifω/2−σ∆<vk<ω −2σ∆ thenvk−|vk+2σ∆−\nω|<vq<vk +|vk+2σ∆−ω|,\nb) ifω/2−σ∆>vkthen−vk+|vk+2σ∆−ω|<vq<\nvk+|vk+2σ∆−ω|.\nAccounting for all these conditions, the integral over\nkandqin Eq. (24) can be calculated numerically. We\nusedthe followingparameters: µ=−3meV, correspond-\ning to the Fermi momentum kF≈5×104cm−1and6\n5 10 15 20\nphoton energy [meV]00.010.020.030.041010  K2 [s-3eV-2] \nR = 10 nm R = 30 nm \nR = 20 nm \nFIG. 5: (Color online.) The parameter K2(ω)≡K2↓(ω)−\nK2↑(ω) for the spin injection rate for different R. Other pa-\nrameters are the same as Fig.4.\nhole concentration ≈4×108cm−2, ∆ = 1 meV, and\n1/τ= 1 meV. These parameters correspond to a rather\nclean graphene and strong Zeeman splitting of the bands\n(magnetic field of 17 T). For numerical accuracy, the cal-\nculations were performed with exact Eqs.(A4) and (A6).\nThe results of numerical calculations for the interband\nspin-flip transitions are presented in Fig. 4. Figure 5\npresents the quantity describing the total spin injection\nrate for the interband transitions in this frequency do-\nmain,K2(ω)≡K2↓(ω)−K2↑(ω). The positive sign of\nK2(ω) corresponds to the fact that the absolute value of\nspin density decreases due to the pumping. The peaks\ncorrespondtotheabsorptionedgeofdirectopticaltransi-\ntions, where the transition probability rapidly increases,\nand the increase at large frequencies is due to the linear\nenergy dependence of the density of states.\nB. Intraband transitions in the hole subband\nUsing the general formulas (A4) and (A6) we can\nwrite down the expression for the spin-flip intraband\ntransitions within the valence band in a low-frequency\nregionω≪ |µ|. Such transitions are associated with\na relatively large change of the momentum in the ab-\nsorption process, and one can expect that they give a\nsmaller transition rate. Upon taking into account that\nεk1−εk2=−2µ≫ω,and/vextendsingle/vextendsingle/vextendsingleεk2−εk′\n2/vextendsingle/vextendsingle/vextendsingle≪ |µ|, we find\nI2σ→2σ′(ω) =−4πσe2A2\nc2v4τ2\nµ2(1+ω2τ2)\n×/summationdisplay\nkk′C(q)/bracketleftBig\nf(εk2)−f(εk′\n2)/bracketrightBig\n×εk2εk′\n2(k2−kqcosϕ)δ/parenleftBig\nεk2+ω−εk′\n2/parenrightBig\n(εk2+ω−εk′\n1)(εk2−εk′\n1).(28)\nTheδ-function can be presented in the form of Eq.(25),\nwhereϕ0is a solution of the equation\nvk+2σ∆−ω−v/radicalbig\nk2−2kqcosϕ+q2= 0.(29)00.0050.010.0151010  K    [s-3eV-2]\n0 2 4 68\nphoton energy [meV]00.0040.0081010 K    [s-3eV-2] \nR=40 nm\nR=20 nmR=30 nmR=40 nm(a)\n(b)30 nm\n20 nm2σ 2σ\nFIG. 6: (Color online.) The parameter K2↓(ω) for the rate\nof the spin-flip intraband transitions in the low-frequency do-\nmain (a) and K2↑(ω) (b) for different values of the range pa-\nrameter R. Other parameters are the same as Fig.4.\n0 2 4 68\nphoton energy [meV]00.0040.0081010  K2 [s-3eV-2]\nR=20 nm R=40 nm \nFIG. 7: (Color online.) The parameter K2(ω)≡K2↓(ω)−\nK2↑(ω) for the spin injection rate for different Rin the low-\nfrequency domain. Other parameters are the same as Fig.4.\nUnmarked line corresponds to R= 30 nm.\nEquations (29) and (26) are similar, with\nthe important difference in the sign in front of/radicalbig\nk2−2kqcosϕ+q2.In the case of interband tran-\nsitions this corresponds to the transitions occurring\natω≈2vk,while the intraband transitions occur at\nlower frequencies determined by the possible momentum\ntransfer due to the randomness of the spin-orbit cou-\npling. The solution exists only for vk > ω−2σ∆ and\ngives Eq.(27). However, the condition |cosϕ0|<1 leads\nhere to a different restriction. Since only the condition\nω−2σ∆<2vkis consistent with vk > ω−2σ∆, the\nmomentum qshould be then in the single range of\nvk− |vk+ 2σ∆−ω|< vq < vk +|vk+ 2σ∆−ω|, in\ncontrast to the case of interband transitions.\nTheresultsofcalculationsfortheintrabandtransitions\nare presented in Fig. 6. The intensity of such processes\nis relatively small compared to the interband transitions,7\nand they can be seen only at low photon energies. Fig-\nure 7 corresponds to spin injection rate by the intraband\ntransitions, K2(ω).\nV. CONCLUSIONS\nWe haveconsidered certainspin effects associatedwith\nrandomspin-orbitinteractioningraphene. First,wehave\ncalculated the corresponding spin relaxation time, and\nbelieve that this mechanism can be dominating when the\namplitude offluctuationsin spin-orbitinteractionis large\nenough. This may happen in the presence of surface rip-\nples with short wavelengths. The other possibility can be\nrelated to the absorbed impurities at both surfaces of a\nfree-standinggraphene. One can expect especially strong\nrandom spin-orbit coupling for heavy impurity atoms.\nThe second effect concerns the possibility of spin\npumping by an external electromagnetic field. The re-\nsults of our calculations show that graphene can be used\nas a material, in which the electron spin density can be\ngenerated by the optical pumping. The mechanism of\npumping here is related to the spin-flip transitions asso-\nciated with the random Rashba spin-orbit interaction.\nAcknowledgements\nThis work is partly supported by the FCT Grant\nPTDC/FIS/70843/2006 in Portugal and by the PolishMinistry of Science and Higher Education as a research\nproject in years 2007 – 2010. This work of EYS was sup-\nported by the University of Basque Country UPV/EHU\ngrant GIU07/40, MCI of Spain grant FIS2009-12773-\nC02-01, and ”Grupos Consolidados UPV/EHU del Gob-\nierno Vasco” grant IT-472-10.\nAppendix A: Formula for the absorption rate\nUsing (22) and (23), after calculating the trace and\nintegrating over energy ε, we find out that spin-flip\nprocesses can be characterized by the initial state as\nI1↓(ω),I2↓(ω),I1↑(ω),andI2↑(ω) with the corresponding\ntransition rate Ijσ(ω) defined as\nIjσ(ω)≡16e2A2\nc2Kjσ(ω), (A1)\nKjσ(ω)≡ −Im/integraldisplay\nv4C(q)Jjσ(q,ω)\nω+iδd2kd2k′\n(2π)4,(A2)\ndescribing transitions from the band and spin states cor-\nresponding to the subscript jσ. HereJjσ(q,ω) have the\nform:\nJ1↓(q,ω) = [f(ε1k↓)−f(ε1k↓+ω)](ε2\n1k↓−∆2+ε1k↓ω)(k2−kqcosϕ)\n(ε1k↓−ε1k′↑)(ε1k↓−ε2k↓)(ε1k↓−ε2k′↑)\n×1\n(ε1k↓+ω−ε1k′↑+iδ)(ε1k↓+ω−ε2k↓+iδ)(ε1k↓+ω−ε2k′↑+iδ), (A3)\nJ2↓(q,ω) = [f(ε2k↓)−f(ε2k↓+ω)](ε2\n2k↓−∆2+ε2k↓ω)(k2−kqcosϕ)\n(ε2k↓−ε2k′↑)(ε2k↓−ε1k↓)(ε2k↓−ε1k′↑)\n×1\n(ε2k↓+ω−ε2k′↑+iδ)(ε2k↓+ω−ε1k↓+iδ)(ε2k↓+ω−ε1k′↑+iδ), (A4)\nJ1↑(q,ω) = [f(ε1k↑)−f(ε1k↑+ω)](ε2\n1k↑−∆2+ε1k↑ω)(k2−kqcosϕ)\n(ε1k↑−ε1k′↓)(ε1k↑−ε2k↑)(ε1k↑−ε2k′↓)\n×1\n(ε1k↑+ω−ε1k′↓+iδ)(ε1k↑+ω−ε2k↑+iδ)(ε1k↑+ω−ε2k′↓+iδ), (A5)\nJ2↑(q,ω) = [f(ε2k↑)−f(ε2k↑+ω)](ε2\n2k↑−∆2+ε2k↑ω)(k2−kqcosϕ)\n(ε2k↑−ε2k′↓)(ε2k↑−ε1k↑)(ε2k↑−ε1k′↓)\n×1\n(ε2k↑+ω−ε2k′↓+iδ)(ε2k↑+ω−ε1k↑+iδ)(ε2k↑+ω−ε1k′↓+iδ). (A6)\nAlthough the expressions seem to be long, all Jjσ(q,ω)\nterms have the same simple structure. They con-tain energy-differencedenominatorscorrespondingto the8\ntransitions from initial jσstates to all allowed final\nstates, and the corresponding resonant terms. Here the\nsingle allowed spin-conserving transition arising due to\ntheHAterm in Eq.(19) is the momentum-conserving as\nwell, while the two transitions caused by random spin-orbit term Hk′k\nsoare off-diagonal both in the spin and\nmomentum subspaces, as illustrated in Fig.2. Taking the\nimaginary part in each of these terms we get δ−functions\ncorresponding to the energy conservation for the transi-\ntions to different bands.\n1K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y.\nZhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov,\nScience306, 666 (2004).\n2K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M.\nI. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A.\nFirsov, Nature (London) 438, 197 (2005).\n3A. K. Geim, K. S. Novoselov, Nature Mater. 6, 183 (2007).\n4A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S.\nNovoselov, and A. K. Geim, Rev. Mod. Phys. 81, 109\n(2009).\n5N. M. R. Peres, J. Phys. Condens. Matter 21, 323201\n(2009).\n6Analysis of the bandstructure in the presence of spin-orbit\ncoupling can be found in P. Rakyta, A. Korm´ anyos, and\nJ. Cserti, Phys. Rev. B 82, 113405 (2010).\n7The Dirac-like electron spectrum in single carbon layers\nwas established by P. R. Wallace Phys. Rev. 71, 622\n(1947) and observed experimentally in graphite (I. A.\nLuk’yanchuk and Y. Kopelevich, Phys. Rev. Lett. 93,\n166402 (2004)). Also, the quantum Hall effects in graphene\nand clean graphite are similar: I. A. Luk’yanchuk and Y.\nKopelevich, Phys. Rev. Lett. 97, 256801 (2006).\n8M. Nishioka and A. M. Goldman, Appl. Phys. Lett. 90,\n252505 (2007).\n9S. Cho, Y. F. Chen, and M. S. Fuhrer, Appl. Phys. Lett.\n91, 123105 (2007).\n10F. S. M. Guimar˜ aes, A. T. Costa, R. B. Muniz, and M. S.\nFerreira, Phys. Rev. B 81, 233402 (2010).\n11O. V. Yazyev, Rep. Prog. Phys. 73, 056501 (2010).\n12D. Huertas-Hernando, F. Guinea, and A. Brataas, Phys.\nRev. B74, 155426 (2006).\n13H. Min, J. E. Hill, N. A.Sinitsyn, B. R.Sahu, L. Kleinman,\nand A. H. MacDonald, Phys. Rev. B 74, 165310 (2006).\n14Y. Yao, F. Ye, X. L. Qi, S. C. Zhang, and Z. Fang, Phys.\nRev. B75, 041401(R) (2007).\n15M. Gmitra, S. Konschuh, C. Ertler, C. Ambrosch-Draxl,\nand J. Fabian, Phys. Rev. B 80, 235431 (2009).\n16Spin relaxation times in carbon nanotubes, which can be\nconsidered as the rolled graphene sheets are indeed long\nenough to allow information transfer (L. E. Hueso, J. M.\nPruneda, V. Ferrari, G. Burnell, J. P. Vald´ es-Herrera, B.\nD. Simons, P. B. Littlewood, E. Artacho, A. Fert, and N.\nD. Mathur, Nature 445, 410 (2007)) due to the very weak\nspin-orbit coupling. However, theoretical analysis shows\nthat the spin resonance caused by external electric field\ncan be observable in these systems: A. De Martino, R. Eg-\nger, K. Hallberg, and C. A. Balseiro, Phys. Rev. Lett. 88,\n206402 (2002).\n17N. Tombros, C. Jozsa, M. Popinciuc, H. T. Jonkman, and\nB. J. van Wees, Nature (London) 448, 571 (2007).\n18N. Tombros, S. Tanabe, A. Veligura, C. Jozsa, M. Popin-\nciuc, H. T. Jonkman, and B. J. van Wees, Phys. Rev. Lett.\n101, 046601 (2008).\n19T. Maassen, F. K. Dejene, M. H. D. Guimar˜ aes, C. J´ ozsa,and B. J. van Wees, preprint arXiv:1012.0526.\n20T.-Y. Yang, J. Balakrishnan, F. Volmer, A. Avsar, M.\nJaiswal, J. Samm, S.R. Ali, A.Pachoud, M. Zeng, M.\nPopinciuc, G. G¨ untherodt, B. Beschoten, B. ¨Ozyilmaz,\npreprint arXiv:1012.1156.\n21Wei Han and Roland K. Kawakami, preprint\narXiv:1012.3435.\n22D. Huertas-Hernando, F. Guinea, and A. Brataas, Eur.\nPhys. J. Special Topics 148, 177 (2007); D. Huertas-\nHernando, F. Guinea, and A. Brataas, Phys. Rev. Lett.\n103, 146801 (2009).\n23C. Ertler, S. Konschuh, M. Gmitra, and J. Fabian, Phys.\nRev. B80, 041405(R) (2009).\n24Y. Zhou and M. W. Wu, Phys. Rev. B 82, 085304 (2010).\n25E. Ya. Sherman, Phys. Rev. B 67, 161303(R) (2003).\n26M. M. Glazov and E. Ya. Sherman, Phys. Rev. B 71,\n241312(R) (2005).\n27J. C. Meyer, A. K. Geim, M. I. Katsnelson, K. S.\nNovoselov, T. J. Booth, and S. Roth, Nature (London)\n446, 60 (2007).\n28M. Ishigami, J. H. Chen, W. G. Cullen, M. S. Fuhrer, and\nE. D. Williams, Nano Lett. 7, 6 (2007).\n29T. O. Wehling, A. V. Balatsky, A. M. Tsvelik, M. I. Kat-\nsnelson, and A. I. Lichtenstein, EPL 84, 17003 (2008).\n30E. I. Rashba and V. I. Sheka, Fizika Tverdogo Tela 3, 2369\n(1961 [Sov. Phys. Solid State 3, 1718 (1962)].\n31E. I. Rashba and Al. L. Efros, Phys. Rev. Lett. 91, 126405\n(2003); Appl. Phys. Lett. 83, 5295 (2003).\n32Optical orientation , ed. by F. Mayer and B. Zakharchenya\n(North-Holland, Amsterdam, 1984).\n33S. A. Tarasenko, Phys. Rev. B 73, 115317 (2006).\n34O. E. Raichev, Phys. Rev. B 75, 205340 (2007).\n35A. A. Perov, L. V. Solnyshkova, and D. V. Khomitsky,\nPhys. Rev. B 82, 165328 (2010); D. V. Khomitsky, Phys.\nRev. B79, 205401 (2009).\n36V.K.Dugaev, E.Ya.Sherman, V.I.Ivanov,andJ. Barna´ s,\nPhys. Rev. B 80, 081301(R) (2009).\n37C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 226801\n(2005).\n38The influence of the substrate on the spin-orbit coupling\nis weak for graphene on the top of SiO 2and SiC. An\nexception is given by graphene on the Ni/Au substrate,\nwhere, duetothelarge atomic numberofAu, inducedspin-\norbit coupling is very strong: A. Varykhalov, J. Sanchez-\nBarriga, A. M. Shikin, C. Biswas, E. Vescovo, A. Rybkin,\nD.Marchenko, andO.Rader, Phys.Rev.Lett. 101, 157601\n(2008) However, this extreme case is not of our interest\nhere.\n39The detailed analysis done by A. H. Castro Neto and F.\nGuinea, Phys. Rev. Lett. 103, 026804 (2009) shows that\ntaking intoaccount the bond hybridization by the adatoms\ncan lead to the spin relaxation rate of the order of the\nobserved experimentally.\n40S. A. Tarasenko, JETP Letters 84, (2006).9\n41A. B. Kuzmenko, I. Crassee, D. van der Marel, P. Blake,\nand K. S. Novoselov, Phys. Rev. B 80, 165406 (2009).\n42Effects of spin-orbit coupling for the optical properties in\ndisorder-free graphene were considered by P. Ingenhoven,\nJ. Z. Bern´ ad, U. Z¨ ulicke, and R. Egger, Phys. Rev. B 81,035421 (2010)\n43Magnetic-field inducedelectron spin resonance in graphene\nat frequency 2∆ was analyzed in B. D´ ora, F. Mur´ anyi and\nF. Simon, EPL 9217002 (2010)."    },    {        "title": "1110.6661v2.Quasiparticle_velocities_in_2D_electron_hole_liquids_with_spin_orbit_coupling.pdf",        "content": "arXiv:1110.6661v2  [cond-mat.mes-hall]  2 Mar 2012Quasiparticle velocities in 2D electron/hole liquids with spin-orbit coupling\nD. Aasen, Stefano Chesi, and W. A. Coish\nDepartment of Physics, McGill University, Montr´ eal, Qu´ e bec H3A 2T8, Canada\n(Dated: October 30, 2018)\nWe study the influence of spin-orbit interactions on quasipa rticle dispersions in two-dimensional\nelectron and heavy-hole liquids in III-V semiconductors. T o obtain closed-form analytical results,\nwe restrict ourselves to spin-orbit interactions with isot ropic spectrum and work within the screened\nHartree-Fock approximation, valid in the high-density lim it. For electrons having a linear-in-\nmomentum Rashba (or, equivalently, Dresselhaus) spin-orb it interaction, we show that the screened\nHartree-Fock approximation recovers known results based o n the random-phase approximation and\nwe extend those results to higher order in the spin-orbit cou pling. While the well-studied case of\nelectrons leads only to a weak modification of quasiparticle properties in the presence of the linear-\nin-momentum spin-orbit interaction, we find two important d istinctions for hole systems (with a\nleading nonlinear-in-momentum spin-orbit interaction). First, the group velocities associated with\nthe two hole-spin branches acquire a significant difference i n the presence of spin-orbit interactions,\nallowing for the creation of spin-polarized wavepackets in zero magnetic field. Second, we find that\nthe interplay of Coulomb and spin-orbit interactions is sig nificantly more important for holes than\nfor electrons and can be probed through the quasiparticle gr oup velocities. These effects should\nbe directly observable in magnetotransport, Raman scatter ing, and femtosecond-resolved Faraday\nrotation measurements. Our results are in agreement with a g eneral argument on the velocities,\nwhich we formulate for an arbitrary choice of the spin-orbit coupling.\nPACS numbers: 71.10.-w, 71.70.Ej, 71.45.Gm, 73.61.Ey\nI. INTRODUCTION\nSemiconductor heterostructures offer the possibility of\nforming two-dimensional liquids with tunable sheet den-\nsityns. In an idealized model where the carriers have\nparabolic dispersion with band mass mand interact with\nCoulomb forces,1the only relevant quantity is the di-\nmensionless Wigner-Seitz radius rs= 1//radicalbig\nπa2\nBns, where\naB=/planckover2pi12ǫr/me2is the effective Bohr radius ( ǫris the di-\nelectric constant). Since rsserves as the interaction pa-\nrameter, changing nsallows for a systematic study of the\neffects of the Coulomb interaction. In particular, proper-\nties of quasiparticle excitations such as their dispersion\nand lifetime are significantly modified due to electron-\nelectron interactions.1\nGreat attention has been paid in recent years to band-\nstructure effects involving the spin degree of freedom.2\nThe strength and form of spin-orbit interaction (SOI)\ncan be controlled in two-dimensional liquids through the\nchoice of materials, the type of carriers (electrons/holes),\nand details of the confinement potential. For example, it\nis possible to change the coupling constant with external\ngates.3–5In addition to detailed studies of single-particle\nproperties, the problem of understanding the effects of\nSOI in the presence of Coulomb interactions is a topic of\nongoing investigations.\nQuasiparticle properties in the presence of SOI6–10\nhavebeenexaminedprimarilyaccountingforRashba11,12\nand/orDresselhaus SOI,13,14which are dominant in elec-\ntronic systems. The SOI results in two distinct spin sub-\nbands, with two associated Fermi surfaces. The effects\non quasiparticles are usually very small; at each of the\ntwo Fermi surfaces the quasiparticle dispersion6,7,10and\nlifetime7–9are almost unaffected by SOI, except in thecase of very large SOI coupling.9,15In fact, it was found\nwith Rashba SOI that the corrections to these quanti-\nties linear in the SOI coupling are absent.7Although ex-\nplicit calculations are performed within perturbative ap-\nproximation schemes, notably the random-phase approx-\nimation (RPA),7–9the SOI leading-order cancellation is\nvalid non-perturbatively (to all orders in rs).16Similar\narguments hold for other physical quantities.16,17For ex-\nample, values of the ground-state energy obtained with\nMonte Carlo simulations18for up to rs= 20 could be\nreproduced with excellent accuracy by simply neglecting\nSOIcorrectionstotheexchange-correlationenergy.19No-\nticeable exceptions exhibiting larger SOI effects are spin-\ntextured broken symmetry phases,20,21non-analytic cor-\nrections to the spin susceptibility,22,23and the plasmon\ndispersion.10All these examples involve the presence of\nspin polarization (either directly20–22or indirectly10), in\nwhich case the arguments of Ref. 16 do not apply.\nAnother interesting situation occurs when the SOI\nhas a nonlinear dependence on momentum, thus cannot\nbe written as a spin-dependent gauge potential.17,24,25\nThen, the approximate cancellations mentioned above\nare not expected. Winkler has shown that the domi-\nnant SOI induced by heterostructure asymmetry is cubic\nin momentum for heavy holes in III-V semiconductors.26\nThis theoretical analysis was later shown to be in good\nagreement with magnetoresistance experiments.27Re-\ncently, the relevance of this cubic-in-momentum model\nwas supported by the anomalous sign and magnetic-\nfield dependence of spin polarization in quantum point\ncontacts.28,29SOIquadraticinmomentumcanalsobein-\nduced for heavy holes by an in-plane magnetic field.30,31\nThat these band-structure effects can substantially\nmodify standard many-body results is confirmed by re-2\ncent Shubnikov-de Haas oscillation measurements in low-\ndensity hole systems.33,34For example, a surprisingly\nsmall Coulomb enhancement of the g-factor has been\nreported33and puzzling results have also been obtained\nfor the effective masses m±of the two hole-spin sub-\nbands (σ=±).34A mechanism for the small g-factor\nenhancement was suggested in Ref. 31: if SOI strongly\ndistorts the groundstate spin structure, the exchangeen-\nergy becomes ineffective in promoting full polarization of\nthe hole system.\nIn this paper, we focus on the effective masses m±.\nWe note that m±are directly related to the quasiparti-\ncle group velocities v±at the Fermi surfaces. With this\nin mind, we find it more transparentto discussthe effects\nof SOI and electron-electron interactions on m±in terms\nofwavepacketmotion, asillustratedin Fig.1. Ifanunpo-\nlarized wavepacket is injected at the Fermi energy (with\naverage momentum along a given direction), the sub-\nsequent motion is very different depending on whether\nthe SOI is linear ( n= 1) or non-linear ( n= 2,3). In\nthe former case ( n= 1), corresponding to electrons, we\nhavev+≃v−and the motion is essentially equivalent to\nthe case without SOI. In contrast, for holes with strong\nSOI,v+∝negationslash=v−and the two spin components become spa-\ntially separated. The separation between the two spin\ncomponents can become quite sizable, and it should be\npossible to observe such an effect with, e.g., Faraday-\nrotation imaging techniques.35,36Electron-electron inter-\nactions, in addition to modifying the average velocity\nv= (v++v−)/2 (an effect which is well-known without\nSOI1,37,38), are also reflected on the velocity difference\n(v+−v−) between the two spin branches.\nWith these motivations in mind, we pursue a study of\nthe quasiparticle group velocity in the presence of linear\nRashbaand non-linearSOI.An accurateanalyticaltreat-\nment of the electron-electron interactions can be carried\nout at high density, and we restrict ourselves to this in-\nteresting limit. The regime of strong electron-electron\ninteractions is much more difficult to treat (see Ref. 38\nfor a Monte Carlo study without SOI) but it represents\na relevant topic for future investigations ( rs∼6−12 in\nRefs. 33 and 34).\nThis paper is organized as follows: In Sec. II we in-\ntroduce a model Hamiltonian including a generalized\nSOI31and we briefly review its non-interacting proper-\nties, demonstrating that injected holes will separate into\nspin-polarized wavepackets. The Coulomb interaction is\ntreated in Sec. III by extending the classic treatment\nof Ref. 39. We describe several results of this screened\nHartree-Fock approach in detail, focusing on the quasi-\nparticle group velocities and the interplay of spin-orbit\nandCoulombinteractioneffects. Adiscussionofthesere-\nsults is givenin Sec. IV. In Appendix A, wegivea general\nargument showing that corrections to the velocity from\nany linear-in-momentum SOI can always be neglected to\nlowest order. Finally, a number of technical details are\nprovided in Appendices B and C./LParen1a/RParen1electrons/LParen1n/Equal1/RParen1\nvΤ\nvFΤ\n/CapDelΤa∆\nt/Equal0 t/EqualΤ/Minus/Plus/LParen1b/RParen1holes/LParen1n/Equal2,3/RParen1\nFIG. 1. (Color online) Motion of a wavepacket for a fixed\ntimeτ. In (a) the SOI is not seen, because the difference\nin the velocities v±of the two spin components is too small.\nPanel (a) applies without SOI or to electrons with Rashba\nSOI (n= 1). In (b) we show the effect of the non-linear\nSOI present in hole systems ( n= 2,3). Since the two spin\nbranches have significantly different velocities v±, there is an\nappreciable separation ∆ = |v+−v−|τ. We also illustrate\nthe propagation of non-interacting wavepackets (dashed) f or\nthe same time τ. The effect of interactions (at high density)\nis to enhance both the average velocity ( v > v F) and the\nseparation of spin-components: ∆ > δ=|v0\n+−v0\n−|τ.\nII. NONINTERACTING PROBLEM\nIn the high-density limit, rsis small and the system is\nwell-describedbyanon-interactingsingle-particleHamil-\ntonian. We consider here the following model,31includ-\ning a generalized SOI with a linear-, quadratic-, and\ncubic-in-momentum dependence for n= 1,2,3:\nH0=p2\n2m+iγpn\n−σ+−pn\n+σ−\n2, (1)\nwherepis the momentum operator, mthe band mass,\np±=px±ipy,σ±=σx±iσy, andγthe generalized\nspin-orbit coupling, with σthe vector of Pauli matrices.\nThe physical justification of this model has been given in\nRef. 31: the n= 1 Hamiltonian contains the Rashba SOI\npresent in electronic systems,11,12whilen= 3 includes\nthe analogous term generated by an asymmetric confine-\nment potential for holes.26,27,29Finally, the n= 2 case is\nalsorelevantforholes, inthepresenceofanin-planemag-\nnetic field.30,31While SOI terms of different form gener-\nally coexist (see Appendix A), we assume here that one\nnvalue is dominant. This greatly simplifies the problem\nby preserving the isotropy of the electron liquid in the x-\ny plane and is a good approximation for several relevant\nsituations. For example, it was found for holes26,27,29\nthat the n= 3 term can be much larger than corrections\nto the SOI due to bulk inversion asymmetry.2,32Diago-3\nnalizingH0yields the energy spectrum\nE0\nσ(k) =/planckover2pi12k2\n2m+σγ/planckover2pi1nkn, (2)\nwithσ=±labeling the two chiral spin branches. The\ncorresponding eigenfunctions are31\nϕkσ(r) =eik·r\n√\n2L2/parenleftbigg1\niσeinθk/parenrightbigg\n, (3)\nwherekis a wavevector in the x-y plane, θkis the angle\nkmakes with the x-axis, and Lis the linear size of the\nsystem.\nIt is useful at this point to introduce a dimensionless\nquantity gcharacterizing the strength of the spin-orbit\ncoupling:31\ng=γ/planckover2pi1nkn\nF\nEF, (4)\nwhereEF=/planckover2pi12k2\nF/2mis the Fermi energy without SOI,\nwritten in terms of the Fermi wavevector kF=√2πns=√\n2/(aBrs). While γhas different physical dimensions\nfor each value, n= 1,2,3, the dimensionless coupling\ngalways gives the ratio of the spin-orbit energy to the\nkineticenergy. Thecoupling gthusplaysaroleanalogous\nto that of rsfor the Coulomb interaction. Taking γto be\nindependent of the density (this is not always the case2),\nthen from Eq. (4) we have g∝kn−2\nF∝r2−n\ns(since\nEF∝k2\nF). This suggests that in the high-density limit\n(rs→0), the effects of SOI are suppressed for electrons\n(n= 1), but remain constant ( n= 2) or are enhanced\n(n= 3) for holes. This simple estimate already indicates\na qualitative difference for holes relative to electrons. We\nwill see that this difference is indeed significant in the\nfollowing sections.\nInthepresenceofSOIthetwospinbands( σ=±)have\ndifferent densities n±, giving the total 2D sheet density\nns=n++n−. Keeping ns(thuskF) fixed gives a con-\nstraintonthe Fermiwavevectors k±fortheσ=±bands,\nk2\n++k2\n−= 2k2\nF. We can then characterize the solution\nto this equation with a single parameter χ:\nk±=kF/radicalbig\n1∓χ. (5)\nThe parameter χ= (n−−n+)/nsgives the chirality\nand is determined by both the SOI and electron-electron\ninteraction.40We will assume for definiteness that γ≥0\nsuch that χ≥0 andk+≤k−. For a fixed generalized\nspin-orbit coupling γ, the non-interacting value of χcan\nbe determined by setting the Fermi energies of the two\nbands equal, i.e., E0\n+(k+) =E0\n−(k−). This equation im-\nmediately gives a relationship between gandχ,\ng=2χ\n(1+χ)n/2+(1−χ)n/2. (6)\nWe denote the solution of Eq. (6) by χ0(g), which gives\nthe non-interacting Fermi wavevectors k0\n±=kF√1∓χ0.Explicit expressions for χ0(g) are given in Ref. 31. We\nonly cite here the small- gbehavior, which is easily found\nfrom Eq. (6):\nχ0(g)≃g. (7)\nWe are mainly interested here in the properties of the\nquasiparticles and, in particular, their group velocities\nv±. The fact that the dispersion relation (2) is a function\nonly of the magnitude kis a consequence of the model\nbeing isotropic with respect to k, which allows us to dis-\ncuss the magnitude of the group velocity on the Fermi\nsurfaces,\nv0\n±=1\n/planckover2pi1∂E0\n±(k)\n∂k/vextendsingle/vextendsingle/vextendsingle\nk=k0\n±=/planckover2pi1k0\n±\nm±nγ(/planckover2pi1k0\n±)n−1.(8)\nTheaboveexpressioncanbeevaluatedexplicitlyinterms\nof the non-interacting chirality χ0(g):\nv0\n±\nvF=/radicalbig\n1∓χ0(g)±n\n2g[1∓χ0(g)](n−1)/2\n≃1±g\n2(n−1), (9)\nwherevF=/planckover2pi1kF/mistheFermivelocityintheabsenceof\nSOIandinthesecondlinewehaveexpandedtheresultto\nlowest order in g, by making use of Eq. (7). Equation (9)\nshows that there is no relative difference in group veloc-\nity for the σ=±bands when n= 1. In contrast, when\nn= 2,3 a sizable correction linear in gis present. This\ndifference reflects itself on the evolution of an initially\nunpolarized wavepacket injected at the Fermi surface,\nschematically illustrated in Fig. 1. While for electrons\nthe wavepacket remains unpolarized, for holes the two\nspin components spatially separate with time. Electron-\nelectron interactions modify the non-interacting veloci-\ntiesv0\n±, but the qualitative difference introduced by SOI\nbetween electrons ( n= 1) and holes ( n= 2,3) remains\nessentially unchanged. A similar behavior holds for other\nproperties of the electron liquid as well:16vanishing cor-\nrections to lowest order in gwere found for the quasi-\nparticle lifetime,7,9the occupation,31and the exchange-\ncorrelation energy,19if only Rashba SOI ( n= 1) is in-\ncluded.\nFinally, anotherrelevantquasiparticleobservableisthe\neffective mass. This has recently been measured in hole\nsystems through experiments on quantum oscillations.34\nThis physical quantity is simply given by m±=/planckover2pi1k±/v±\nand is thus essentially equivalent to v±.\nIII. SCREENED HARTREE-FOCK\nAPPROXIMATION\nRealistically, charged particles interact through the\nCoulomb potential so it is interesting to understand how\nthe presence of SOI modifies the behavior of the quasi-\nparticles. The fully interacting Hamiltonian is given by:\nH=/summationdisplay\niH(i)\n0+1\n2/summationdisplay\ni/negationslash=je2\nǫr|ri−rj|,(10)4\nwhereH(i)\n0(for electron i) is as in Eq. (1) and the pres-\nence in (10) of a uniform neutralizing background is un-\nderstood. Although many sophisticated techniques exist\nto approach this problem,1,38the simplest approxima-\ntion to the quasiparticle self-energy is obtained by only\nincluding the exchange contribution:\nEσ(k) =E0\nσ(k)+Σx\nσ(k), (11)\nwhere\nΣx\nσ(k) =−/summationdisplay\nk′σ′(1+σσ′cosnθ′)\n2L2nk′σ′V(|k−k′|).(12)\nHere,nk′±= Θ(k±−k′) is the occupation at T= 0\nfor theσ=±band, respectively, with Θ( x) the Heavi-\nside step function. The first factor in the summation of\nEq. (12), involving the angle θ′between k′andk, arises\nfrom the scalar product of the non-interacting spinors\n[Eq. (3)], and takes into account the specific nature of\nthe spin-orbit interaction ( n= 1,2,3).\nTo lowest order, V(q) is the Fourier transform of the\nbare Coulomb potential, 2 πe2/(ǫrq). As is well known,1\nthis form of the Coulomb interaction leads to an unphys-\nical divergence in the quasiparticle velocity. By consider-\ning an infinite resummation in perturbation theory, the\nscreening of the Coulomb interaction removes the diver-\ngence, e.g., in the RPA approximation. Finally, by ap-\nproximating the dielectric function in the effective inter-\naction by its zero-frequency long-wavelength limit, the\nRPA self-energy gives Eq. (12) with\nV(q) =2πe2/ǫr\nq+√\n2rskF. (13)\nThis screened Hartree-Fock approximation with SOI,6,10\nnotwithstanding its simplicity, becomes accurate in the\nhigh-density limit.\nA. Renormalized occupation\nBy including the SOI, we can verify that Eq. (12) gives\nthe correct high-density behavior for the Fermi wavevec-\ntorsk±. These are modified by electron-electron inter-\nactions from their non-interacting values k0\n±.31,41From\nEq. (12), k±can be obtained by equating the chemical\npotentials in the two spin branches:\nE+(k+) =E−(k−). (14)\nAfter taking the continuum limit, this equation is rewrit-\nten in dimensionless form as follows:\n(yn\n++yn\n−)g=2χ+rs√\n2/summationdisplay\nσσ′/integraldisplay2π\n0dθ\n2π/integraldisplayyσ′\n0dy\n×y(σ+σ′cosnθ)/radicalbig\ny2+y2σ−2yyσcosθ+√\n2rs,(15)where we have rescaled the wavevectors k=kFyand\ndefinedy±=√1∓χ. The integral on the right-hand\nside is the correction from the exchange term and we\nhave verified that Eq. (15) gives Eq. (6) for rs= 0. We\nnote that, for a given value of the SOI, χ(rs,g) enters in\na rather complicated way in Eq. (15), being involved in\nthe integration limits of the exchange term as well as the\nintegrand. In practice, instead of solving Eq. (15) for χ,\nit is convenient to evaluate gfor a given value of χand\nnumerically invert the function g(rs,χ).\nThe values of k±=kFy±were obtained in Ref. 31\nthrough a different procedure, i.e., by minimizing the to-\ntal energy (including the exchange contribution) of non-\ninteracting states.42Although both methods are unreli-\nable atrs>1, they both become accurate in the high-\ndensity limit, rs<1. In fact, the only difference in the\ntwo approaches is due to the presence of the Thomas-\nFermiscreeningwavevectorinEq.(15)and, byneglecting√\n2rsin the denominator of the second line, the equation\nfrom the variational treatment is recovered. In particu-\nlar, expanding Eq. (15) at small rsandggives the same\nresult found in Ref. 31:\nχ(rs,g)≃g\n1−√\n2rs\nπn/summationdisplay\nj=01\n2j−1\n.(16)\nDetails of the derivation of Eq. (16) are given in Ap-\npendix B. Two salient features of Eq. (16) are:31(i) For\nn= 1 there is no correction to the noninteracting result\nχ(rs,g)≃g. On the other hand, the linear dependence\nongisactually modified byelectron-electroninteractions\natn= 2,3. (ii) The effect of the electron-electron inter-\nactions is a reduction of χ(rs,g) from the non-interacting\nvalue. This result could be rather surprising, having in\nmind the well-known enhancement of spin polarization\ncaused by the exchange energy1(when the spin-splitting\nis generated by a magnetic field). However, χdoes not\ncorrespondhere to a real spin-polarization, which is zero.\nInstead, χis simply related to the population difference\nof the two chiral spin subbands.\nB. Quasiparticle velocity\nInthescreenedHartree-Fockapproximation,thegroup\nvelocities at the Fermi surfaces are given by:\nv±=1\n/planckover2pi1∂Ek±\n∂k/vextendsingle/vextendsingle/vextendsingle\nk=k±\n=/planckover2pi1k±\nm±nγ(/planckover2pi1k±)n−1\n−/summationdisplay\nk′σ′(1±σ′cosnθ′)\n2L2nk′σ′/bracketleftbigg∂\n∂kV(|k−k′|)/bracketrightbigg\nk=k±.(17)\nIn general, we can discuss all corrections to v±by intro-\nducing the following notation:\nv±\nvF= 1+δv(rs)+δv0\n±(g)+δv±(rs,g),(18)5\nwhereδv(rs) is the (spin-independent) correction due to\nelectron-electron interactions at g= 0, which has been\nthe subject of many theoretical and experimental studies\n(see, e.g., Ref. 1, 34, 37–39, and references therein). In\nthe approximation (17), it is given by39\nδv(rs) =−√\n2rs\nπ+r2\ns\n2+rs(1−r2\ns)√\n2πcosh−1(√\n2/rs)/radicalbig\n1−r2s/2.(19)\nAs is known,1this approximation gives the correct lead-\ning behavior at small rs:δv≃ −(rslnrs)/(√\n2π). The\nsecond nontrivial term in Eq. (18) is the non-interacting\ncorrection purely due to SOI\nδv0\n±(g) =v0\n±(g)\nvF−1, (20)\nwhich only depends on g[see Eq. (9)]. Finally, δv±(rs,g)\ncollects all remaining corrections.\nA pictorial representation of the physical meaning of\nthe three terms is shown in Fig. 1 for an unpolarized\nwavepacket injected at the Fermi energy. At high den-\nsity, as seen in Eq. (19) and illustrated in Fig. 1(a), the\ngroup velocity is larger than without electron-electron\ninteractions. In Fig. 1(b) we depict the generic situ-\nation with SOI. In the presence of SOI, the two spin\nbranches have different group velocities. An initially un-\npolarized wavepacket then splits into its two spin com-\nponents. Both the SOI and electron-electron interactions\ninfluence the relative velocity ( v+−v−). The separation\nafter a time τisδ=|v0\n+−v0\n−|τfor the non-interacting\ncase and is modified by δv±with electron-electron inter-\nactions: ∆ = |v+−v−|τ[see Fig. 1(b)].\nThe presence of ‘interference’ terms, δv±(rs,g), be-\ncomes clear from Eq. (17). These terms are due to the\ninterplay of many-body interactions with SOI. A first\ncontribution to δv±(rs,g), which we refer to as the ‘self-\nenergy contribution’, comes directly from the exchange\nintegral [third line of Eq. (17)]: due to the presence of\ntwo distinct Fermi wavevectors k±, the result obviously\ncontainscorrectionstoEq.(19)whichdependon g(inad-\ndition to rs). A second contribution to δv±(rs,g) comes\nindirectly from the non-interacting part and we refer to\nit as the ‘repopulation contribution’. Since the Fermi\nwavevectors k±are modified from the non-interacting\nvaluesk0\n±, the second line of Eq. (17) gives a result\ndistinct from v0\n±. The sign of this repopulation con-\ntribution is easily found by noting that, as discussed in\nSec.IIIA, theexchangeenergyreducesthe valueof χ(for\nn= 2,3). This corresponds to an increase (decrease) of\nk+(k−), i.e., a positive (negative) correction to v+(v−).\nThe effect would be to enhance the difference in veloc-\nity between the two branches, as illustrated in Fig. 1(b)\n(∆> δ). However, to establish the ultimate form and\nsign ofδv±(rs,g) requires a detailed calculation of both\nself-energy and repopulation contributions, which is pre-\nsented below for some interesting cases. The total spin-\ndependent part of the velocity, δv0\n±+δv±, can then be\ncompared to the simple non-interacting effect, δv0\n±.C.n= 1: higher-order corrections in g\nWe begin the analysis of Eq. (17) by rewriting it in a\nmore explicit way. To this end, we use\n∂\n∂kV(|k−k′|) =−ˆk·∂\n∂k′V(|k−k′|),(21)\nwhereˆk=k/k. This allows us to integrate Eq. (17) by\nparts, which leads to:\nv±\nvF=y±±g\n2nyn−1\n±\n+rs/summationdisplay\nσ/integraldisplay2π\n0dθ′\n8π√\n2cosθ′(1±σcosnθ′)yσ/radicalBig\ny2\n±+y2σ−2y±yσcosθ′+rs√\n2\n±rs/integraldisplay2π\n0dθ′\n8π/integraldisplayy−\ny+√\n2nsinnθ′sinθ′dy′\n/radicalBig\ny2\n±+y′2−2y′y±cosθ′+rs√\n2\n=L1+L2+L3. (22)\nNotice that, in the integration by parts of Eq. (17) in the\ncontinuum limit, two types of terms enter: those corre-\nsponding to the second line of Eq. (22) ( L2) involve the\nderivative of nk′±. This results in a delta function in the\ndk′integral, which can then be easily evaluated. Thus,\nonly the integral in dθ′is left. The second type of term\ninvolves the derivative of cos nθ′:\nˆk·∂\n∂k′cosnθ′=n\nk′sinnθ′sinθ′, (23)\nand indeed this angular factor appears in the third line\nof Eq. (22) ( L3).\nEq. (22) can alwaysbe evaluated numerically, after ob-\ntainingthevaluesof χ(thusy±=√1∓χ)fromEq.(15).\nBy specializing to the small g,rslimit for n= 1 SOI, it is\nknown that the linear-in- gcorrection to v±vanishes.7,16\nThus, an expansion to second order in ghas to be per-\nformed, which has been done in Ref. 7 in the context of\nthe RPA treatment of the quasiparticle properties. To\nverify the validity of the simpler screened Hartree-Fock\nprocedure,itisinterestingtoperformthesameexpansion\nfor our Eq. (22).\nIn fact, the two calculations bear some similarities\nsinceL2is the same as the boundary term B(u→0+)\nboundaryof\nthe RPA treatment, see Eq. (69) of Ref. 7. Thus, we can\nborrow the expansion in small g,rs:\nL2≃ −rs√\n2π/parenleftbigg\nlnrs\n2√\n2+2±2\n3g−g2\n8lng/parenrightbigg\n; (n= 1),\n(24)\nwhere terms of order O(r2\ns,rsg2) have been omitted. To\nthe same order of approximation, we have for L1:\nL1≃1−g2\n8; (n= 1), (25)\nwhich can also be obtained by expanding Eq. (9) (with\nn= 1). Since χonly receives O(rsg3lng) corrections6\nfromelectron-electroninteractions,19itissufficienttouse\nthe non-interacting value, χ0(g), to this order of approx-\nimation. Thus, the repopulation induced by electron-\nelectron interactions has a negligible effect on v±in this\ncase. The situation will be different for n= 2,3. Ex-\npanding L3for small rs,gyields\nL3≃ ±√\n2rs\n3πg; (n= 1), (26)\nwhichcancelsthelinear-in- gtermofEq.(24),asexpected\nforn= 1.\nWe note that in the screened Hartree-Fock approxi-\nmation we are able to obtain an analytic result for the\nleadingterm of L3, and somedetails ofthe derivationcan\nbe found in Appendix C. In contrast, in the RPA treat-\nment of Ref. 7 it was not possible to expand the more\ninvolved corresponding term, Bint, in a fully analytic\nfashion. The cancellation of the linear-in- gcontribution\nwas indicated by a general argument7,16and confirmed\nthrough numerical study. Additionally, the absence of\nhigher-order terms which modify Eq. (24) was inferred\nnumerically in Ref. 7. Although the final results of both\napproaches (RPA and screened Hartree-Fock) agree, the\nsituation is clearly more satisfactory within the frame-\nwork of Eq. (22), since cancellation of the linear term in\nEq. (24) can be checked exactly and the expansion can\nbe carried out to higher order systematically. The final\nresult, computed to higher order in g, reads (for n= 1):\nv±\nvF≃1+δv(rs)+δv0\n±(g)\n+√\n2rs\n16π/bracketleftbigg\ng2/parenleftbigg\nlng\n8+3\n2/parenrightbigg\n±g3\n6/parenleftbigg\nlng7\n85+319\n20/parenrightbigg/bracketrightbigg\n.(27)\nThe second line of Eq. (27) represents the expansion of\nδv±(rs,g) including all terms up to O(rsg3). Forn= 1,\nthenon-interactingresultgivesthesamevelocityforboth\nspin branches [ δv0\n±(g) is actually independent of ±for\nthis particular model]. The O(rsg3) term in Eq. (27) is\ntherefore quite interesting. It shows that a small differ-\nenceinvelocityexists. Thisisagenuineeffectofelectron-\nelectron interactions.\nThe accuracy of Eq. (27) can be seen in Fig. 2: it\nbecomes accurate at very small values of rs(as shown\nin the inset) while at larger more realistic values of rs\nit still gives the correct magnitude of the effect. The\nnumerical example of Fig. 2 also shows that δv±(rs,g)\nis very small. In fact, it is generally much smaller than\nthe non-interacting correction: since δv0\n±(g)≃ −g2/8\n[see Eq. (25)] the rsg2lngterm becomes larger only if\nrslng≫1. This condition is only satisfied at extremely\nsmall values of g(ifrsis small as well) at which SOI\neffectsarehardlyofanyrelevance. Thus, Eq.(27)implies\nthat the effects of SOI and electron-electron interactions\nare essentially decoupled for n= 1. This picture changes\nsubstantially for hole systems ( n= 2,3), as we show in\nthe next section./Minus\n/Plus /Plus0/Minus5 10/Minus5\n0.00 0.02 0.04 0.06 0.08 0.10/Minus0.006/Minus0.005/Minus0.004/Minus0.003/Minus0.002/Minus0.0010.000\ng102∆v/PlusMinus\n0 0.05 0.1\nFIG. 2. (Color online) Thick solid lines: the corrections\nδv±(rs,g) evaluated by numerical integration of Eq. (22).\nDashed lines: approximation to δv±(rs,g), given by the sec-\nond line of of Eq. (27). In the main plot, we take rs= 0.1, for\nwhich Eq. (27) is not very accurate. Very good agreement is\nobtained at small values of rs(inset, with rs= 0.001). The\nthinner solid line in the main plot is the total correction du e\nto spin-orbit coupling: δv0\n±(g) +δv±(rs,g). It is dominated\nby the non-interacting effect and the spin-splitting is not v is-\nible. The value of δv0\n±(g) atg= 0.1, outside the plot range,\nis∼1% (all corrections are in units of vF).\nD. Corrections to the velocity for n= 2,3\nWe now apply the discussion of the previous section to\nthe SOI more appropriate for holes and point out some\nimportant differences. Expansions for small rsandgare\ngiven in Appendix C. From Eqs. (C16) and (C17) and\nusingχ=g+O(rsg,g3) [Eq. (16)], we find the following\nexpressions for the self-energy contribution. For n= 2:\nL2+L3≃δv(rs)+√\n2rs\n4π/bracketleftBig\n±g\n3+g2/parenleftBig\nlng\n8+2/parenrightBig/bracketrightBig\n,(28)\nand forn= 3:\nL2+L3≃δv(rs)+√\n2rs\n4π/bracketleftbigg\n±8\n15g+g2\n4/parenleftbigg\n9lng\n8+613\n30/parenrightbigg/bracketrightbigg\n.\n(29)\nAt variance with the case of n= 1, the linear-in- gterm\ndoes not vanish here. Thus, we find an appreciable cor-\nrection to the velocity. This correction has opposite sign\nin the two branches and is positive for the + (higher-\nenergy) branch.\nA second contribution to the g-linear correction comes\nfromL1. Expanding L1in terms of χand using Eq. (16)\ngives, for n= 2,\nL1≃1+δv0\n±(g)+√\n2rs\n4π/parenleftbigg\n±2\n3g+g2/parenrightbigg\n; (n= 2),(30)\nand forn= 3:\nL1≃1+δv0\n±(g)+√\n2rs\n4π/parenleftbigg\n±16\n15g+32\n15g2/parenrightbigg\n; (n= 3).\n(31)7\n/Plus\n/Minus/Minus\n/Plus\n0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35/Minus0.20.00.20.40.60.81.0\ng102∆v/PlusMinus\nFIG. 3. (Color online) Plot of δv±(rs,g) forn= 3 (solid\ncurves) and n= 1 (dashed curves) as a function of the SOI\nstrength g. We have taken rs= 0.3 in both cases. ±indicate\nthe spin branch of each curve.\nThus, the repopulation contribution is present in this\ncase and has the sign discussed at the end of Sec. IIIB\n(it is positive for the + branch).\nAs it turns out, the self-energy, repopulation, and non-\ninteracting contributions to the velocity have the same\nsign. The three contributions therefore have a cooper-\native effect in enhancing the difference in velocity be-\ntween the two spin branches. Of course, based on the\nhigh-density theory presented here, we cannot tell if this\nconclusion holds at all densities. We also note that the g-\nlinear term of the self-energy correction [Eq. (28) or (29)]\nis always half of the corresponding repopulation correc-\ntion [Eq. (30) or (31)]. Again, we have not investigated\nif this curious relation only occurs within this approxi-\nmation scheme or if it is more general.\nFinally, we give the complete result for n= 2\nv±\nvF≃1+δv(rs)+δv0\n±(g)\n+√\n2rs\n4π/bracketleftBig\n±g+g2/parenleftBig\nlng\n8+3/parenrightBig/bracketrightBig\n; (n= 2),(32)\nand forn= 3\nv±\nvF≃1+δv(rs)+δv0\n±(g)\n+√\n2rs\n4π/bracketleftbigg\n±8\n5g+g2/parenleftbigg\n9lng\n8+869\n120/parenrightbigg/bracketrightbigg\n; (n= 3),(33)\nand show two numerical examples in Figs. 3 and 4. Fig.\n3 is a comparison of δv±(rs,g) forn= 1 and n= 3: it\nis clear that the dependence on gis very weak for the\nelectron case ( n= 1) and the magnitude of δv±(rs,g)\nis much larger for holes ( n= 3) with SOI of comparable\nstrength. Fig. 4 shows the separation ∆ between the two\nspin components of an initially unpolarized wave packet\nfor a fixed travel distance of 1 µm:\n∆ = 2|v+−v−|\nv++v−×(1µm). (34)0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.350.00.20.40.60.8\ng/CapDelΤa/LParen1Μm/RParen1/CapDelΤa\n/CapDelΤa0\n1Μm\nFIG. 4. (Color online) Thick solid line: separation ∆ be-\ntween the two spin components of a hole wavepacket ( n= 3)\nat a fixed drift distance of 1 µm, see Eq. (34). The dashed\nline is the non-interacting value ∆ 0. The thin solid line is\nn= 1, indistinguishable from ∆ = 0. We have assumed here\nrs= 0.3. The inset schematically illustrates the definitions of\n∆ and ∆ 0.\nIn addition to including both n= 1 and n= 3, in\nFig. 4 we plot the non-interacting value ∆ 0, obtained\nby substituting v±→v0\n±in Eq. (34). For n= 1 the\nnon-interacting velocities v0\n±are the same (∆ 0= 0).\nThe effect of electron-electron interactions is not visi-\nble. Thus, the wavepacket remains essentially unsplit\nand unpolarized (∆ ≃0) and the only significant influ-\nence is on the averagevelocity( v++v−)/2, from Eq. (19)\n[see Fig. 1(a)]. In contrast, a large splitting is found\nforn= 3 where ∆ ,∆0can reach a large fraction of the\ntraveling distance. Since spin-polarized ballistic trans-\nport is observed in two-dimensional hole systems on µm-\nscales (e.g., in spin-focusing experiments28,29) and the\ntypical wavepacket traveling time is 1 −50 ps (depend-\ning on the density), Fig. 4 suggests that a direct op-\ntical imaging of the wavepacket separation should be\nwithin reach of femtosecond-resolved Faraday rotation\nmeasurements.35,36\nAs for electron-electron interaction effects, we see in\nFig. 4 that a visible difference between ∆ and ∆ 0exists.\nThe difference is quite small, due to the fact that we are\nconsidering here the weak-coupling limit, and all the in-\nteraction corrections are proportional to rs<1. ∆0is\nmodified here by ∼2−3% and we collect some represen-\ntative values in Table I. It can be seen in Table I that,\nwhile ∆ 0does not change with rs, the interaction correc-\ntion ∆−∆0grows at lower densities (see Table I). Since\nexperiments on hole systems can reach values as large as\nrs= 6−12,33,34it is reasonable to expect significant ef-\nfects from δv±in this low-density regime. As a reference,\nin electron systems, δvchanges from ∼5% to−30% for\nrs∼1 to 6.37,38\nIt might be surprising to see ∆ <∆0in Fig. 4. This is\ndue to the interplay of two competing interaction effects.\nIt is true that, as discussed already, the difference be-\ntweenv±is enhanced by δv±. The two spin components8\nSOIgrs∆0(nm)∆−∆0(nm)\nn=10.050.50 0.001\nn=30.10.1203 -4.7\nn=30.10.3203 -6.2\nTABLE I. Separation of wavepackets after 1 µm drift dis-\ntance for electrons ( n= 1) and holes ( n= 3). The non-\ninteracting value and the electron-electron interaction c orrec-\ntion are listed, see Eq. (34) and Fig. 4. Typical values for\ng(∼χ) are/lessorsimilar0.05 for electrons2,4and/lessorsimilar0.2 for holes.2,27\nWe have assumed in this table that typical values of gare\nindependent of rsand have used high-density values rs<1.\ntherefore split faster in the interacting case. However,\nthe situation shown in Fig. 4 is distinct from that shown\nin Fig. 1 since a constant traveling distance (and not\ntimeτ) is assumed. At high density the mean velocity\n(v++v−)/2 is greater for the interacting gas ( δv >0),\nwhich allows the wavepackets less time to separate. As\nit turns out, the latter effect is dominant in Fig. 4.\nInterestingly, the sign of δvchanges at low density.1\nThis would imply a cooperative effect of δv±andδvon\n∆ ifδv±does not change sign. Unfortunately, to infer\nthe behavior of δv±at low density requires a much more\nsophisticated approach.\nIV. CONCLUSION\nWe have presented a theory of the quasiparticle group\nvelocity at high density, in the presence of SOI of dif-\nferent types. Contrasting the behavior of electron and\nhole systems, we find several intriguing differences. We\nhave shown explicitly that the lowest-order cancellations\nof SOI effects occur only for the electronic case, when\nthe SOI is approximately linear in momentum (e.g., a\nstrong Rashba or Dresslhaus SOI is present). On the\nother hand, SOI terms non-linear in pare often domi-\nnant in hole systems.26,27,29Thus, larger effects of the\nSOI and a non-trivial interplay with electron-electron in-\nteractions are expected for holes on general grounds.\nAsanimportantmotivationforfuturetheoreticalstud-\nies, hole liquids can be realized in the laboratory with\nstrong SOI and large values of rs. For example, the spin-\nsubband population difference at zero field is of order\n15−20% in Ref. 27, with ns≃2−4×1010cm−2. With\na hole effective mass m≃0.2m0in GaAs these densities\ncorrespond to rs≃9−12. For electrons, materials with\nstrong SOI typically have small effective masses, which\nresults in much lower values of rs. A diluted electron liq-\nuid with ns≃2×1010cm−2givesrs≃1.2, using InAs\nparameters ( m= 0.023m0).\nDiscussing the large- rsregime of holes would require\nextending many-body perturbation theory7–9or Monte\nCarlo18approaches, so far only applied to linear SOI.\nThe high-density regime studied here would represent a\nwell-controlled limit of these theories for the quasiparti-cle dispersion. In addition to being relevant for trans-\nport measurements of the effective mass, the significant\ndifference in group velocity at the Fermi surface of the\ntwo spin branches could also be addressed by Raman\nscattering experiments, demonstrated for electron sys-\ntems in Ref. 43, or via time-resolved Faraday-rotation\ndetection of the spin-polarization.35,36A similar discus-\nsion should hold with n= 2,3 for other physical observ-\nables and many-body effects. For example, studying the\ncompressibility1,44in the presence of SOI and extend-\ning then= 1 discussion of the quasiparticle lifetime7–9\nwould also be topics of interest. Finally, the problem of\nincluding in our framework a more general form of SOI\nthan Eq. (1) is clearly of practical relevance. However, as\ndiscussed in Appendix A, we expect that our qualitative\npicture on the different role of linear and non-linear SOI\nremains valid also in this more general situation.\nACKNOWLEDGMENTS\nWe thank J. P. Eisenstein and D. L. Maslov for use-\nful discussions and acknowledge financial support from\nCIFAR, NSERC, and FQRNT.\nAppendix A: General spin-orbit coupling\nA general SOI contains terms with linear and non-\nlinear dependence on momentum, and does not neces-\nsarily have the isotropic form assumed in Eq. (1). For\ndefiniteness, we suppose that there is no magnetic field,\nso that quadratic terms are not present:\nH=/summationdisplay\nip2\ni\n2m+Hso\n1+Hso\n3+Hel−el,(A1)\nwhereHso\n1,Hso\n3, andHel−elgive, respectively, the linear-\nin-momentum spin-orbit, the cubic-in-momentum spin-\norbit, and electron-electron interactions. To show that\nHso\n1always has a small effect, we consider the uni-\ntary transformation H′=e−SHeSwithSdefined by\n[S,/summationtext\nip2\ni/2m] =Hso\n1, giving\nH′≃p2\n2m+Hso\n3+Hel−el−[S,Hso\n1+Hso\n3].(A2)\nIn the same transformed frame, the velocity operator of\nelectron iis given by v′\ni=e−S(∂H/∂pi)eS, which yields\nv′\ni≃pi\nm+∂Hso\n3\n∂pi−[S,∂Hso\n1\n∂pi+∂Hso\n3\n∂pi].(A3)\nIn deriving Eq. (A2) we have used the fact that\n[S,Hel−el] = 0. For example, for an isotropic SOI as\nin Eq. (1),\nS=imγ\n/planckover2pi1/summationdisplay\ni(xiσy,i−yiσx,i). (A4)9\nThe property [ S,Hel−el] = 0 is valid for a general\nlinear-in-momentum SOI, including a combination of\nRashba and Dresselhaus SOI.24However, the same iden-\ntity [S,Hel−el] = 0 does not hold for a transformation\nwith [S,/summationtext\nip2\ni/2m] =Hso\n3, i.e., a transformation that\nis designed to remove the non-linear component from\nthe non-interacting Hamiltonian. In writing Eq. (A3),\nwe have used [ S,pi/m] =∂Hso\n1/∂pi, which implies a\ncancellation of the Hso\n1contribution to v′\nito lowest or-\nder. Again, this cancellation is only valid for linear-in-\nmomentum SOI.\nBy introducing dimensionless couplings g1,3associated\nwithHso\n1,3, in direct analogy with Eq. (4), we see that the\ncommutators in Eqs. (A2) and (A3) are of quadratic or\nbilinear order in the couplings ( ∼g2\n1and∼g1g3). This\nindicates on general grounds that Hso\n3has the largest ef-\nfect on the quasiparticle velocity if g1/lessorsimilarg3≪1. In\nthis case, if we are interested in lowest-order effects,\nwe can neglect both anticommutators in Eqs. (A2) and\n(A3), which is equivalent to neglecting Hso\n1in the origi-\nnal Hamiltonian (A1). Thus, to leading order all results\nwe report for the quasiparticle velocities due to a pure\ncubic-in-momentum spin-orbit interaction also apply in\nthe case of a mixed linear-plus-cubic spin-orbit interac-\ntion (with the caveat that we consider only the isotropic\nform of cubic SOI).\nAppendix B: Derivation of Eq. (16)\nAs discussed in the text, at high density we can neglect\nthe√\n2rsin the integrand of Eq. (15) (second line). For\nsmallg, thevalueof χisalsosmallandwecanperforman\nexpansion of the exchange integral. First notice that the\nconstanttermat χ= 0ismissing,becausetheintegration\nlimits simply become y±= 1 and the integrand vanishes\nupon the summation on σ,σ′. Therefore we only need to\ncompute the linear term in χ:\n/bracketleftBigg\n∂\n∂χ/summationdisplay\nσσ′/integraldisplay2π\n0dθ\n2π/integraldisplayyσ′\n0(σ+σ′cosnθ)ydy/radicalbig\ny2+y2σ−2yyσcosθ/bracketrightBigg\nχ=0\n=/integraldisplay2π\n0dθ\n2π/bracketleftBigg/integraldisplay1\n02(1−ycosθ)ydy\n(1+y2−2ycosθ)3\n2−cosnθ\nsinθ\n2/bracketrightBigg\n,(B1)\nand after evaluating the dyintegral in the square paren-\nthesis, Eq. (B1) gives\n/integraldisplay2π\n0dθ\n2π/bracketleftBigg\n2+2cosθ−cosnθ−1\nsinθ\n2\n−2ln/parenleftBigg\n1+1\nsinθ\n2/parenrightBigg\ncosθ/bracketrightBigg\n=4\nπn/summationdisplay\nj=01\n2j−1.(B2)\nEquation (16) is then easily obtained from Eq. (15) by\nneglecting all the cubic terms in the small parameters\ng,χ,rs[e.g., the left side of Eq. (15) is ( y2\n++y2\n−)g=\n2g+O(gχ2)].Appendix C: Small rs,gexpansions\nWe give in this appendix some details on the expan-\nsions of Eq. (22):\nvF±\nvF=y±±g\n2nyn−1\n±+√\n2rs\n16π(I1+I2+I3),(C1)\nwhere we have split L2in its two contributions ( I1,2refer\ntoσ=±) andI3corresponds to L3:\nI1=/integraldisplay2π\n0dθ√\n2y±cosθ(1+cosnθ)√\n2y±sinθ/2+rs, (C2)\nI2=/integraldisplay2π\n0dθ√\n2y∓cosθ(1−cosnθ)/radicalbig\n1−y+y−cosθ+rs, (C3)\nI3=±/integraldisplay2π\n0dθ/integraldisplayy−\ny+dy2nsinnθsinθ/radicalBig\ny2\n±+y2−2yy±cosθ+rs√\n2,(C4)\nNotice that these integrals have an explicit dependence\nonrsandy±=√1∓χ. So, it is easier to perform first\nthe expansion in the two small parameters rs,χ. The\nfinal results in the main text are given in terms of the\nphysical couplings of the hamiltonian: rsandg. Those\nfinal expression are easily obtained by substituting the\nvalue of χin terms of rsandg(χ≃gin first approxi-\nmation).\nThe first integral, Eq. (C2), can be evaluated exactly.\nIn particular for n= 1 we obtain\nI1=−40\n3+8πδ(1−δ2)+16δ2\n+8(1−3δ2+2δ4)tanh−1√\n1−δ2\n√\n1−δ2(C5)\nwhereδ=rs/√\n2y±. This expression can then be easily\nexpanded in rs,χand an analogousprocedure is followed\nforn= 2,3. To lowest-order in rs, we can set rs= 0 in\nI2andI3. Similarly to the I1angular integral above,\nthedθintegrals of I2andI3atrs= 0 can be computed\nanalytically for n= 1,2,3. ForI2this yields directly the\ndesired function of χ. ForI3we still need to perform a\nlast integration in dy. Since the integration region is of\nsize∼χaroundy= 1, we can expand the integrand in\nthesmallparameter( y−1)andperformtheintegrationin\ndyorder-by-order, which allows us to extract the leading\nterms of the expansion in χ. Forn= 1 all this gives\nδI1≃ ∓4χ−2χ2∓4\n3χ3, (C6)\nδI2≃ ∓4\n3χ+χ2/parenleftbigg\nlnχ\n8+13\n6/parenrightbigg\n±χ3\n2/parenleftbigg\nlnχ\n8+3\n2/parenrightbigg\n,(C7)\nδI3≃ ±16\n3χ+4\n3χ2±χ3/parenleftbigg2\n3lnχ\n16+389\n120/parenrightbigg\n,(C8)\nwhere only the corrections δIα=Iα(χ)−Iα(χ= 0) are\nlisted, sincetermsindependenton χsimplygivethesmall\nrsexpansion of the well known Eq. (19). Here, terms of10\norderO(rsχ,χ4) are omitted, while it is interesting to\nkeep the O(χ3) terms, since they give the leading spin\nsplitting. Indeed, itiseasilycheckedthatthelinearterms\ncancel\n3/summationdisplay\nα=1δIα≃χ2/parenleftbigg3\n2+lnχ\n8/parenrightbigg\n±χ3\n6/parenleftbigg\nlnχ7\n85+319\n20/parenrightbigg\n,(C9)\nwhich immediately leads to Eq. (27).\nForn= 2,3 we can proceed in a similar way. The spin\nsplitting appears now already to linear order in χ. By\nkeeping the first subleading correction in χwe have for\nn= 2:\nδI1≃ ∓4χ−2χ2, (C10)\nδI2≃ ±16\n15χ+χ2/parenleftbigg\n4lnχ\n8+134\n15/parenrightbigg\n,(C11)\nδI3≃ ±64\n15χ+16\n15χ2, (C12)\nand forn= 3:\nδI1≃ ∓4χ−2χ2, (C13)\nδI2≃ ±212\n105χ+χ2/parenleftbigg\n9lnχ\n8+899\n42/parenrightbigg\n,(C14)\nδI3≃ ±144\n35χ+36\n35χ2. (C15)The final results are for n= 2:\n3/summationdisplay\nα=1δIα≃ ±4\n3χ+4χ2/parenleftBig\nlnχ\n8+2/parenrightBig\n,(C16)\nand forn= 3:\n3/summationdisplay\nα=1δIα≃ ±32\n15χ+χ2/parenleftbigg\n9lnχ\n8+613\n30/parenrightbigg\n.(C17)\nFrom Eqs. (C16) and (C17) we immediately obtain\nEqs. (28) and (29).\n1F. Giuliani and G. Vignale, Quantum Theory of the\nElectron Liquid (Cambridge University Press, Cambridge,\n2005).\n2R. Winkler, Spin-Orbit Coupling Effects in Two-\nDimensional Electron and Hole Systems (Springer, Berlin,\n2003).\n3S. Datta and B. Das, Appl. Phys. Lett. 56, 665 (1989).\n4J. Nitta, T. Akazaki, H. Takayanagi, and T. Enoki, Phys.\nRev. Lett. 78, 1335 (1997).\n5H. C. Koo, J. H. Kwon, J. Eom, J. Chang, S. H. Han, and\nM. Johnson, Science 325, 1515 (2009).\n6G.-H. Chen and M. E. Raikh, Phys. Rev. B 60, 4826\n(1999).\n7D. S. Saraga and D. Loss,, Phys. Rev.B 72, 195319 (2005).\n8I. A. Nechaev, M. F. Jensen, E. D. L. Rienks, V. M. Silkin,\nP. M. Echenique, E. V. Chulkov, and P. Hofmann, Phys.\nRev. B80, 113402 (2009).\n9I. A. Nechaev, P. M. Echenique, and E. V. Chulkov, Phys.\nRev. B81, 195112 (2010).\n10A. Agarwal, S. Chesi, T. Jungwirth, J. Sinova, G. Vignale,\nand M. Polini, Phys. Rev. B 83, 115135 (2011).\n11Y. A. Bychkov and E. Rashba, JETP Lett. 39, 78 (1984).\n12Y. A. Bychkov and E. Rashba, J. Phys. C 17, 6039 (1984).\n13G. Dresselhaus, Phys. Rev. 100, 580 (1955).\n14M. I. D’yakonov and V.Yu. Kachorovskii, Sov. Phys. Semi-\ncond.20, 110 (1986).\n15D. H. Berman and M. E. Flatt´ e, Phys. Rev. Lett. 105,\n157202 (2010).\n16S. Chesi and G. F. Giuliani, Phys. Rev. B 83, 235308\n(2011).17S. H. Abedinpour, G. Vignale, and I. V. Tokatly, Phys.\nRev. B81, 125123 (2010).\n18A. Ambrosetti, F. Pederiva, E. Lipparini, and S. Gandolfi,\nPhys. Rev. B 80, 125306 (2009).\n19S. Chesi and G. F. Giuliani, Phys. Rev. B 83, 235309\n(2011).\n20G. F. Giuliani and S. Chesi, Proceedings of Highlights in\nthe Quantum Theory of Condensed Matter (Edizioni della\nNormale, Pisa, 2005); S. Chesi, G. Simion, and G. F.\nGiuliani, e-print arXiv:cond-mat/0702060; S. Chesi, Ph.D .\nthesis, Purdue University, 2007.\n21L. O. Juri and P. I. Tamborenea, Phys. Rev. B 77, 233310\n(2008).\n22R. A.˙Zak, D. L. Maslov, and D. Loss, Phys. Rev. B 82,\n115415 (2010).\n23R. A.˙Zak, D. L. Maslov, and D. Loss, arXiv:1112.4786.\n24I. L. Aleiner and V. I. Falko, Phys. Rev. Lett. 87, 256801\n(2001).\n25W. A. Coish, V. N. Golovach, J. C. Egues, and D. Loss,\nphys. stat. sol (b) 243, 3658 (2006).\n26R. Winkler, Phys. Rev. B 62, 4245 (2000).\n27R. Winkler, H. Noh, E. Tutuc, and M. Shayegan, Phys.\nRev. B65, 155303 (2002).\n28L. P. Rokhinson, V. Larkina, Y. B. Lyanda-Geller, L. N.\nPfeiffer, and K. W. West, Phys. Rev. Lett. 93, 146601\n(2004).\n29S. Chesi, G. F. Giuliani, L. P. Rokhinson, L. N. Pfeiffer,\nand K. W. West, Phys. Rev. Lett. 106, 236601 (2011).\n30D. V. Bulaev and D. Loss, Phys. Rev. Lett. 98, 097202\n(2007).11\n31S. Chesi and G. F. Giuliani, Phys. Rev. B 75, 155305\n(2007).\n32E. I. Rashba and E. Ya. Sherman, Phys. Lett. A 129, 175\n(1988).\n33R. Winkler, E. Tutuc, S. J. Papadakis, S. Melinte, M.\nShayegan, D. Wasserman, and S. A. Lyon, Phys. Rev. B\n72, 195321 (2005).\n34Y. T. Chiu, M. Padmanabhan, T. Gokmen, J. Sha-\nbani, E. Tutuc, M. Shayegan, and R. Winkler, e-print\narXiv:1106.4608.\n35S. A. Crooker, J. J. Baumberg, F. Flack, N. Samarth, and\nD. D. Awschalom, Phys. Rev. Lett. 77, 2814 (1996).\n36J. M. Kikkawa and D. D. Awschalom, Phys. Rev. Lett. 80,\n4313 (1998).\n37Y.-W. Tan, J. Zhu, H. L. Stormer, L. N. Pfeiffer, K. W.\nBaldwin, and K. W. West, Phys. Rev. Lett. 94, 016405(2005).\n38N. D. Drummond and R. J. Needs, Phys. Rev. B 80,\n245104 (2009).\n39J. F. Janak, Phys. Rev. 178, 1416 (1969).\n40Todeal with extremelylow densities atwhich onlyone spin\nbandisoccupiedandtheoccupationisaringinmomentum\nspace, it is also useful to generalize χto values larger than\none.19However, we do not consider this low-density regime\nin this paper.\n41J. Schliemann, Phys. Rev. B 74, 045214 (2006).\n42The method of Ref. 31 is slightly preferable, in that it leads\nto variational wavefunctions with a lower total energy.\n43B. Jusserand, D. Richards, H. Peric, and B. Etienne, Phys.\nRev. Lett. 69, 848 (1992).\n44J. P. Eisenstein, L. N. Pfeiffer, and K. W. West, Phys. Rev.\nLett.68, 674 (1992)."    },    {        "title": "2108.06202v2.Coupling_the_Higgs_mode_and_ferromagnetic_resonance_in_spin_split_superconductors_with_Rashba_spin_orbit_coupling.pdf",        "content": "Coupling the Higgs mode and ferromagnetic resonance in spin-split superconductors\nwith Rashba spin-orbit coupling\nYao Lu,1Risto Ojaj arvi,1P. Virtanen,1M.A. Silaev,1, 2, 3and Tero T. Heikkil a1\n1Department of Physics and Nanoscience Center, University of Jyv askyl a,\nP.O. Box 35 (YFL), FI-40014 University of Jyv askyl a, Finland\n2Moscow Institute of Physics and Technology, Dolgoprudny, 141700 Russia\n3Institute for Physics of Microstructures, Russian Academy of Sciences, 603950 Nizhny Novgorod, GSP-105, Russia\n(Dated: February 22, 2022)\nWe show that the Higgs mode of superconductors can couple with spin dynamics in the presence\nof a static spin-splitting \feld and Rashba spin-orbit coupling. The Higgs-spin coupling dramatically\nmodi\fes the spin susceptibility near the superconducting critical temperature and consequently\nenhances the spin pumping e\u000bect in a ferromagnetic insulator/superconductor bilayer system. We\nshow that this e\u000bect can be detected by measuring the magnon transmission rate and the magnon-\ninduced voltage generated by the inverse spin Hall e\u000bect.\nSuperconductors (SC) with broken U(1) symmetry\nhost two kinds of collective modes associated with the\norder parameter \ructuations: the phase mode and the\namplitude mode. Coupled to a dynamical gauge \feld,\nthe phase mode is lifted up to the plasma frequency [1]\ndue to the Anderson{Higgs mechanism [2, 3]. The other\ncollective mode in SC is the amplitude mode [4, 5] with\nan energy gap of 2\u0001, called the Higgs mode by anal-\nogy with the Higgs boson [3] in particle physics. It was\ncommonly believed that unlike the phase mode the Higgs\nmode usually does not couple linearly to any experimen-\ntal probe. That is why in earlier experiments, the Higgs\nmode was only observed in charge-density-wave (CDW)\ncoexisting systems [6{11]. With the advance of terahertz\nspectroscopy technique [12] it became possible to inves-\ntigate the Higgs mode through the nonlinear light{Higgs\ncoupling [13{17]. In these experiments, the perturba-\ntion of the order parameter is proportional to the square\nof the external electromagnetic \feld \u000e\u0001/E2, so very\nstrong laser pulses are required.\nRecently, it has been shown that in the presence of a\nsupercurrent the Higgs resonance can actually contribute\nto the total admittance Y\ndue to the linear coupling of\nthe Higgs mode and the external electromagnetic \feld\n[18{22]. This can be understood from a symmetry ar-\ngument. Suppose the external electric \feld is linearly\npolarized in the xdirectionE= ^xExei\nt. The linear\ncoupling of the Higgs mode and the external \feld is rep-\nresented by the susceptibility \u001f\u0001E=\u0000@2S\n@\u0001@Eobtained\nfrom the action Sdescribing the electron system con-\ntaining the pair potential \feld \u0001. Without a supercur-\nrent, the system preserves the inversion symmetry ( ^I)\nand the mirror symmetry in the xdirection ( ^Mx). On\nthe other hand \u001f\u0001Eis odd under both these operations\nbecauseEchanges sign under ^Iand ^Mxwhereas \u0001 re-\nmains the same. Therefore \u001f\u0001Ehas to vanish. In the\npresence of a supercurrent, the inversion symmetry and\nthe mirror symmetry are both broken and there is no re-\nstriction for \u001f\u0001Exfrom these symmetries, so \u001f\u0001Ecan be\nFIG. 1. System under consideration. A superconductor thin\n\flm is placed on the top of a FI with in-plane magnetization.\nThe SC and FI are coupled via spin exchange interaction. The\nmagnon in FI can be injected into SC in a process known as\nthe spin pumping e\u000bect. For magnon frequency \n = 2\u0001 0the\nSC Higgs mode greatly increases the spin pumping.\nnonzero. This symmetry argument also explains why the\nHiggs mode does not couple with an external \feld in the\ndirection perpendicular to the supercurrent.\nNow a natural question arises: without a supercurrent\ndoes the Higgs mode couple linearly with other exter-\nnal probes, such as spin exchange \felds? As we show in\nthis Letter it does. The above discussion indicates that\nthe decoupling of the Higgs mode is protected by cer-\ntain symmetries. In order to couple the Higgs mode to\nan external \feld one needs to break these symmetries.\nHere we show how it happens in a ferromagnetic insu-\nlator (FI)/superconductor (SC) bilayer system (Fig. 1).\nMagnons with momentum qand frequency \n in the FI\ncan be injected into the SC in a process known as spin\npumping [23{29]. We predict that the Higgs mode in\nthe SC couples linearly with the magnon mode in the\nFI in the presence of Rashba spin-orbit coupling and the\nmagnetic proximity e\u000bect into the SC. In this system\nthe symmetries protecting Higgs-spin decoupling are bro-\nken: in particular, the (spin) rotation symmetry and thearXiv:2108.06202v2  [cond-mat.supr-con]  21 Feb 20222\ntime-reversal symmetry. Near the critical temperature,\nsuperconductivity is suppressed and \u0001 0becomes compa-\nrable with the magnon frequency \n. When the magnon\nfrequency matches the Higgs frequency \n M= 2\u0001 0, the\nHiggs mode is activated and the magnon absorption is\nhugely enhanced which can be detected through the in-\nverse spin Hall e\u000bect (iSHE) [30{32]. This e\u000bect can pos-\nsibly explain the voltage peak observed in the experiment\n[33].\nWe consider a SC/FI bilayer in which the FI and the\nSC are coupled via the exchange interaction as shown in\nFig. 1. For simplicity, we assume that the thickness d\nof the SC \flm is much smaller than the spin relaxation\nlength and the coherence length so that we consider it as\na 2D system. The magnetization of the FI can be written\nasm=m0+m\n, wherem0is the static manetization\npolarized in the zdirection and m\nis the dynamical\ncomponent perpendicular to m0. When magnons (spin\nwaves) are excited in the FI, they can be injected into\nthe SC in a process known as the spin pumping e\u000bect.\nThe DC interface spin current \rowing from the FI into\nthe SC is polarized in the zdirection and given by [34]\nIz=X\n\n;q\u00002JsdIm[~\u001fss(\n;q)]m2\n\n;q; (1)\nwhereJsdis the exchange coupling strength and\nm\nis the Fourier amplitude of m\n. ~\u001fss(\n;q) is\nthe total dynamical spin susceptibility ~ \u001fss(\n;q) =\nS+(\n;q)=h+(\n;q), whereSis the dynamical spin of\nthe SC,his the proximity induced exchange \feld h=\nJsdm=d[35] and for a vector A= (Ax;Ay;Az) the\u0006\ncomponent is de\fned as A\u0006=Ax\u0006iAy. One can see\nthat for a \fxed Jsd, the e\u000eciency of the magnon injection\nis soley determined by ~ \u001fss(\n;q). The spin susceptibility\nof superconductors has been extensively studied [29, 36].\nHowever the previous theories, based on the static mean-\n\feld description, failed to explain the peak of the iSHE\nsignal observed in the spin Seebeck experiment [33]. In\nthis work, we start with the general partition function of\nthe SC,Z=R\nD[\u0016\t;\t;\u0016\u0001;\u0001]e\u0000Sobtained by performing\nthe Hubbard-Stratonovich transformation. The action S\nis given by\nS=\fX\nK;Q\u0016\tK(\u0000i!+\u000fk\u0000h\u0001\u001b) \tK+ \u0001Q\tK+Q\t\u0000K\n+\u0016\u0001\u0000Q\u0016\tK\u0016\t\u0000K\u0000Q+\u0016\u0001\u0000Q\u0001Q\nU;(2)\nHereK= (!;k) andQ= (\n;q) are the four-momenta\nof the electrons and magnons, respectively. != (2n+\n1)\u0019Tand \n = 2n\u0019T are the Matsubara frequencies with\nn2Zand\f= 1=T.\u000fkis the energy dispersion of the\nelectron in the normal state, his the proximity induced\nexchange \feld, and Uis the BCS interaction. In the\nmean-\feld theory, one can ignore the path integral over\u0001 and replace it by its saddle point value \u0001 0which is\ndetermined by the minimization of the action@S\n@\u0001j\u0001=\u0001 0=\n0 after integrating out the fermion \felds.\nTo include the Higgs mode, we go beyond the mean-\n\feld theory and write the order parameter as \u0001 = \u0001 0+\u0011,\nwhere\u0011is the deviation of \u0001 from its saddle point value\n\u00010. Here we only consider the amplitude \ructuation of\n\u0001, so\u0011is real. Expanding the action to the second order\nin\u0011and the strength of the external Zeeman \feld h\u0006\ngivesS=S0\u0000S2with [37]\nS2=\fX\nQ\u0002\u0011(\u0000Q)h\u0000(\u0000Q)\u0003\u0014\n\u0000\u001f\u00001\n\u0001\u0001\u001f\u0001s\n\u001fs\u0001\u001fss\u0015\u0014\u0011(Q)\nh+(Q)\u0015\n:\n(3)\nHere, all the susceptibilities are functions of Q.S0is the\nmean-\feld action without the external \feld. In usual su-\nperconductors the o\u000b-diagonal susceptibilities \u001f\u0001sand\n\u001fs\u0001vanish as required by the time-reversal symmetry\nand the (spin) rotation symmetry because these oper-\nations change the sign of h+but have no e\u000bect on \u0011\n[38, 39]. In the system under consideration, the proxim-\nity induced static exchange \feld breaks the time-reversal\nsymmetry and RSOC breaks the (spin) rotation symme-\ntry. Thus the pair-spin susceptibility does not have to\nvanish, allowing for a nonzero Higgs{spin coupling.\nThen it is straightforward to calculate the total spin\nsusceptibility ~ \u001fssby integrating out the \u0011\feld\n~\u001fss=\u001fss\u0000\u001fs\u0001\u001f\u0001\u0001\u001f\u0001s: (4)\nThe imaginary part of \u001f\u0001\u0001is sharply peak at the Higgs\nfrequency \n = 2\u0001 dramatically modifying the total spin\nsusceptibility.\nPhenomenological theory . Before we go to the detailed\ncalculations, we use a simple phenomenological theory\nto illustrate the e\u000bect of RSOC. It has been shown that\nRSOC can induce a Dzyaloshinskii-Moriya (DM) interac-\ntion in superconductors described by the DM free energy\n[40]\nFDM=X\niZ\ndrj\u0001j2d\u000b;i\u0001(h\u0002rih); (5)\nwhere both \u0001 = \u0001( r) andh=h(r) are position depen-\ndent.d\u000b;iis the DM vector proportional to the strength\nof spin-orbit coupling \u000b. For RSOC d\u000b/\u000b[\u001bx;\u0000\u001bz],\nwhere\u000bis the spin-orbit coupling strength and \u001bis\nthe Pauli matrix acting on the spin space. To \fnd\nthe pair spin susceptibility we write \u0001 = \u0001 0+\u0011(t),\nh=h0^z+h+(t)(^x+i^y), where ^nis the unit vector\nin thendirection with n=x;y;z , and generalize the\nDM free energy to the time dependent DM action. Here\nwe consider the case where the spin wave is propagating\nin thezdirectionh+(t;r) =P\n\n;qzh+(^x+i^y)ei(\nt\u0000qzz).\nFocusing on the \frst order terms in \u0011(t) andh+(t) and3\nFourier transforming them to momentum and frequency\nspace, the DM action can be written as\nSDM1=\fX\n\n;qziqz\u00010h0h+(\n;qz)\u0011(\n;qz)~d\u000b;z(\n;qz)\n\u0001(i^x\u0000^y);(6)\nwhere ~d\u000b;iis the dynamical DM vector, which has the\nsame \fniteness and spin structure as d\u000b;ifrom symmetry\nanalysis. From the above expression, one can see that\nthe Higgs mode couples linearly with the spin degree of\nfreedom in the presence of RSOC.\nSpin susceptibility . We adopt the quasiclassical ap-\nproximation to systematically evaluate the susceptibili-\nties. In the di\u000busive limit, this system can be described\nby the Usadel equation [18, 36, 41{45]\nFIG. 2. Imaginary part of the pair susceptibility. This can\nbe interpreted as the spectral weight of the Higgs mode. A\nsigni\fcant peak emerges when the driving frequency matches\nthe Higgs frequency \n = 2\u0001 0. The inset shows the height\nof the Higgs peak PHas a function of the inverse of the mo-\nmentum q. Parameters: \u0001 0= 0:8\u0001T0,h0= 0:5\u0001T0with\n\u0001T0\u0011\u00010(T= 0).\n\u0000if\u001c3@t;^gg=D~r\u0010\n^g~r^g\u0011\n\u0000i[H0;^g] +h\nXei(\nt\u0000qzz);^gi\n:\n(7)\nHere ^gis the quasiclassical Green function, D=vF\u001c2=3\nis the di\u000busion constant and \u001cis the disorder scat-\ntering time. H0=\u0000ih0\u001b3+ \u0001 0\u001c1, whereh0is the\nproximity induced e\u000bective static exchange \feld and \u001ci\nis the Pauli matrix acting on the particle-hole space.\n~r= (~rz;~rx) is the covariant derivative de\fned by\n~rz\u0001=rz+i\u000b[\u001bx;\u0001],~rx\u0001=rx\u0000i\u000b[\u001bz;\u0001]. The Usadel\nequation is supplemented by the normalization condition\n^g2= 1. In the quasiclassical approximation the approxi-\nmate PH symmetry of the full Hamiltonian becomes ex-act. In the linear response theory, the external oscillat-\ning \feldXis small and can be treated as a perturbation.\nThus we can write the quasiclassical Green function as\n^g= ^g0ei!(t1\u0000t2)+ ^gXei(!+\n)t1\u0000i!t2\u0000iqzz, where ^g0is the\nstatic Green function and ^ gXis the perturbation of the\nGreen function describing the response to the external\n\feld. Solving the Usadel equation we obtain the quasi-\nclassical Green function, the anomalous Green function\nF=NeTr [\u001c1^g]=4iand the\u001b+component of spin in the\nSChsi=NeTr [\u001b\u0000\u001c3^g]=4i, whereNeis the electron den-\nsity of states at the Fermi surface and Tr is the trace.\nThe susceptibilities can be evaluated as\n^\u001f=\u0014\n\u001f\u00001\n\u0001\u0001\u001f\u0001s\n\u001fs\u0001\u001fss\u0015\n=\"@F\n@\u0011+1\nU@F\n@h+\n@hsi\n@\u0011@hsi\n@h+#\n: (8)\nLet us \frst set X= \u00010\u001c1and consider the pair suscep-\ntibility. We assume the RSOC is weak and treat \u000bas a\nperturbation. At q= 0 and 0th order in \u000b, we have\n\u001f\u0001\u0001(i\n) =\"\nNeT\n2X\n!;\u001b4\u00012+ \n2\ns\u001b(!)(4!2\u0000\n2)#\u00001\n;(9)\nwheres\u001b(!) =p\n(!+i\u001bh)2+ \u00012, with\u001b=\u00061. To get\nthe pair susceptibility as a function of real frequency, we\nneed to perform an analytical continuation [38]. Thus\ni\n is replaced by \n + i0+. One can see that the \u001f\u0001\u0001is\npeaked at the Higgs frequency \n = 2\u0001.\nWe numerically calculate \u001f\u0001\u0001with \fnite momentum\nand show the results in Fig. 2 [38, 46]. One can see that\nthe imaginary part of the inverse of the pair suscepti-\nbility exhibits a sharp peak when the driving frequency\nequals 2\u0001 0. With a \fnite momentum, the Higgs mode is\ndamped in the sense that the peak in the Higgs spectrum\nhas a \fnite height and width.\nFIG. 3. Real part (a) and imaginary part (b) of pair-spin\nsusceptibility. The solid line is the approximate result calcu-\nlated from Eq. (12) and the circles show the numerical solu-\ntion from Eq. (7). Parameters used here are: \n = 0 :8\u0001T0\nfor the blue lines, \n = \u0001 T0for the red lines, h0= 0:5\u0001T0,\nDq2\nz=D\u000b2= 0:01\u0001 T0.\nTo study the response of this system to the external\nexchange \feld we set X=h+\u001b+\u001c3. Again we treat \u000bas\na perturbation and write the Green function as\n^g= ^g0ei!(t1\u0000t2)+ (^gh0+ ^gh\u000b)ei(!+\n)t1\u0000i!t2\u0000iqzz;(10)4\nwhere ^gh0is 0th order in \u000band ^gh\u000bis \frst order in \u000b.\nThe 0th order solution in \u000bis given by [38]\n^gh0= ^gh00\n\u001b+=i[\u001c3\u0000^g\"(1)\u001c3^g#(2)]h\n\u001b+\ns\"(1) +s#(2);(11)\nwhere ^g\"=# =(!\u0006ih0)\u001c3+\u0001\u001c1\ns\"=#ands\"=# =p\n(!\u0006ih0)2+ \u00012. ^gh00is a 2\u00022 matrix in the\nparticle-hole space. Without doing detailed calculations,\none can immediately see that \u001f\u0001shas to vanish without\nRSOC because ^ ghhas no\u001b0component. In this case\nthe external exchange \feld cannot activate the Higgs\nmode. To get a \fnite pair-spin susceptibility we need to\nconsider the \frst order terms in \u000bwhich break the spin\nrotation symmetry. The \frst order solution in \u000byields\n^gh\u000b= diag(^gh\u000b\";^gh\u000b#) with\n^gh\u000b\"=#= 2iD\u000b^g0\"=#\u0002\n^gh00;^g0\"=#\u0003\ns\"=#(!1) +s\"=#(!2): (12)\nFIG. 4. (a) Total spin susceptibility as a function of tem-\nperature with a \fxed frequency. (b) Total spin susceptibility\nas a function of frequency with a \fxed temperature. The\ntwo temperatures have been chosen so that \u0001( T1) = 0:2\u0001T0\nand \u0001(T2) = 0:1\u0001T0. The Higgs peak thus shows up when\n\n = 2\u0001(T). The parameters used here are: h0= 0:5\u0001T0,\nDq2\nz=D\u000b2= 0:01\u0001 T0.\nSince the 0th order term does not contribute to the\npair-spin susceptibility, we have \u001f\u0001s= Tr[\u001c1^gh\u000b]=4ih+.\nWe compare this analytical result with the non-\nperturbative numerical solution of the Usadel equation\nin Fig. 3. It shows that the perturbative approach is ac-\ncurate at high temperatures when D\u000b2\u001c\u00010;T, and\ncaptures the qualitative behavior of \u001f\u0001salso at the low\ntemperatures. Another feature of this pair spin suscepti-\nbility is that at a lower frequency (\n = 0 :8\u0001T0),\u001f\u0001sissuppressed at low temperatures because the spin excita-\ntion is frozen by the pair gap at low temperatures. On\nthe other hand, at higher frequency (\n = \u0001 T0),\u001f\u0001sis\nslightly enhanced at low temperatures.\nWe can also get the bare spin susceptibility from ^ gh0,\n\u001fss= Tr[\u001b\u0000\u001c3^gh0]=4ih+. Then it is straightforward\nto calculate the total spin susceptibility according to\nEq. (4). The results are shown in Fig. 4. The total\nspin susceptibility exhibits a signi\fcant peak near criti-\ncal temperature. This is a signature of the Higgs mode\nwith the frequency \n = 2\u0001 0. The dependence of the to-\ntal susceptibility on the strength of RSOC is studied in\nthe supplementary information [38]. The details depend\nsensitively on the amount of disorder, as in the disordered\ncase increasing RSOC leads to a stronger spin relaxation.\nWe note that even though the pair-spin susceptibility is\nlinear in momentum qz, the magnon momentum need\nnot be large for the detection of the Higgs mode. This is\nbecause the spectral weight of the Higgs mode is propor-\ntional to 1=q2\nzat the Higgs frequency, so that the height\nof the peak in the total spin susceptibility is independent\nof the magnon momentum.\nExperimental detection . We propose that the Higgs\nmode in Rashba superconductors can be detected in the\nspin pumping experiment as shown in Fig. 1. Magnons\nin the FI with momentum qand frequency \n are injected\nfrom one side of FI and propagate in the zdirection to-\nwards the other end. Due to the spin pumping e\u000bect,\npart of the magnons can be absorbed by the SC on top\nof it and converted to quasiparticles. This spin injection\ncauses a spin current Is\rowing in the out-of-plane di-\nrection. In the presence of RSOC, Isis converted into a\ncharge current Ievia the iSHE Ie=\u0012z\nxzIs, where\u0012is the\nspin Hall angle [47]. When the width of the SC is smaller\nthan the charge imbalance length the non-equilibrium\ncharge accumulation cannot be totally relaxed resulting\ninto a \fnite resistance \u001aof the SC. Therefore a voltage\ncan be measured across the SC, given by\nV=\u0012z\nxz\u001aX\n\n;q\u00002JsdIm[~\u001fss(\n;q)]m2\n\n: (13)\nThus by tuning the temperature or the frequency of\nmagnon, one can observe a peak in the voltage [33].\nMeanwhile we can also obtain the magnon absorption\nrate de\fned as the energy of the absorbed magnons di-\nvided by time\nW= 2\nX\n\n;q\u00002JsdIm[~\u001fss(\n;q)]m2\n\n: (14)\nThis magnon absorption rate results in a dip in the\nmagnon transmission rate which is experimentally mea-\nsurable.\nConclusion . In this Letter, we consider a FI/SC bi-\nlayer with RSOC in the bulk of the SC. Using symme-5\ntry arguments and microscopic theory, we show that the\nHiggs mode in the SC couples linearly with an exter-\nnal exchange \feld. This Higgs{spin coupling hugely en-\nhances the total spin susceptibility near a critical phase\ntransition point, which can be detected using iSHE or\nvia strong frequency dependent changes in the magnon\ntransmission. Note that in this work, we consider the dif-\nfusive limit where the disorder strength is stronger than\nthe RSOC and exchange \feld. However, our conclusion\non Higgs{spin coupling should still be valid in the case\nof strong RSOC. In fact, we expect that the coupling is\nmuch stronger with strong SOC in the clean limit. In\nthe di\u000busive limit, the RSOC together with disorder ef-\nfectively generate spin relaxation which reduces the prox-\nimity induced exchange \feld suppressing the Higgs{spin\ncoupling. On the other hand, in the clean case without\ndisorder this e\u000bect is absent and hence the Higgs{spin\ncoupling can be stronger. We also compare the Higgs\nmode in the di\u000busive limit and ballistic limit in the sup-\nplemental material [38].\nThis work was supported by Jenny and Antti Wi-\nhuri Foundation, and the Academy of Finland Project\n317118 and the European Union's Horizon 2020 Research\nand Innovation Framework Programme under Grant No.\n800923 (SUPERTED).\n[1] N. Bittner, D. Einzel, L. Klam, and D. Manske, Phys.\nRev. Lett. 115, 227002 (2015).\n[2] P. W. Anderson, Phys. Rev. 130, 439 (1963).\n[3] P. W. Higgs, Phys. Rev. Lett. 13, 508 (1964).\n[4] I. Kulik, O. Entin-Wohlman, and R. Orbach, J. Low\nTemp. Phys. 43, 591 (1981).\n[5] A. Pashkin and A. Leitenstorfer, Science 345, 1121\n(2014).\n[6] P. B. Littlewood and C. M. Varma, Phys. Rev. B 26,\n4883 (1982).\n[7] P. B. Littlewood and C. M. Varma, Phys. Rev. Lett. 47,\n811 (1981).\n[8] M.-A. M\u0013 easson, Y. Gallais, M. Cazayous, B. Clair,\nP. Rodiere, L. Cario, and A. Sacuto, Phys. Rev. B 89,\n060503(R) (2014).\n[9] R. Grasset, Y. Gallais, A. Sacuto, M. Cazayous,\nS. Ma~ nas-Valero, E. Coronado, and M.-A. M\u0013 easson,\nPhys. Rev. Lett. 122, 127001 (2019).\n[10] R. Grasset, T. Cea, Y. Gallais, M. Cazayous, A. Sacuto,\nL. Cario, L. Benfatto, and M.-A. M\u0013 easson, Phys. Rev. B\n97, 094502 (2018).\n[11] T. Cea and L. Benfatto, Phys. Rev. B 90, 224515 (2014).\n[12] D. N. Basov, R. D. Averitt, D. vanderMarel, M. Dressel,\nand K. Haule, Rev. Mod. Phys. 83, 471 (2011).\n[13] R. Matsunaga, Y. I. Hamada, K. Makise, Y. Uzawa,\nH. Terai, Z. Wang, and R. Shimano, Phys. Rev. Lett.\n111, 057002 (2013).\n[14] R. Matsunaga, N. Tsuji, H. Fujita, A. Sugioka,\nK. Makise, Y. Uzawa, H. Terai, Z. Wang, H. Aoki, and\nR. Shimano, Science 345, 1145 (2014).\n[15] M. Beck, I. Rousseau, M. Klammer, P. Leiderer, M. Mit-tendor\u000b, S. Winnerl, M. Helm, G. N. Gol'tsman, and\nJ. Demsar, Phys. Rev. Lett. 110, 267003 (2013).\n[16] M. Silaev, Phys. Rev. B 99, 224511 (2019).\n[17] M. A. Silaev, R. Ojaj arvi, and T. T. Heikkil a, Phys. Rev.\nResearch 2, 033416 (2020).\n[18] A. Moor, A. F. Volkov, and K. B. Efetov, Phys. Rev.\nLett. 118, 047001 (2017).\n[19] S. Nakamura, Y. Iida, Y. Murotani, R. Matsunaga,\nH. Terai, and R. Shimano, Phys. Rev. Lett. 122, 257001\n(2019).\n[20] S. Nakamura, K. Katsumi, H. Terai, and R. Shimano,\nPhys. Rev. Lett. 125, 097004 (2020).\n[21] M. Puviani, L. Schwarz, X.-X. Zhang, S. Kaiser, and\nD. Manske, Phys. Rev. B 101, 220507(R) (2020).\n[22] Z. M. Raines, A. A. Allocca, M. Hafezi, and V. M. Gal-\nitski, Phys. Rev. Research 2, 013143 (2020).\n[23] I. \u0014Zuti\u0013 c, J. Fabian, and S. Das Sarma, Rev. Mod. Phys.\n76, 323 (2004).\n[24] J. Linder and J. W. Robinson, Nat. Phys. 11, 307 (2015).\n[25] Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys.\nRev. Lett. 88, 117601 (2002).\n[26] E. Saitoh, M. Ueda, H. Miyajima, and G. Tatara, Appl.\nPhys. Lett. 88, 182509 (2006).\n[27] K. Ando, S. Takahashi, J. Ieda, Y. Kajiwara,\nH. Nakayama, T. Yoshino, K. Harii, Y. Fujikawa,\nM. Matsuo, S. Maekawa, et al. , J. Appl. Phys. 109,\n103913 (2011).\n[28] Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys.\nRev. B 66, 224403 (2002).\n[29] T. Kato, Y. Ohnuma, M. Matsuo, J. Rech, T. Jonck-\nheere, and T. Martin, Phys. Rev. B 99, 144411 (2019).\n[30] S. Takahashi and S. Maekawa, Phys. Rev. Lett. 88,\n116601 (2002).\n[31] S. Takahashi and S. Maekawa, Sci. Technol. Adv. Mater.\n9, 014105 (2008).\n[32] T. Wakamura, H. Akaike, Y. Omori, Y. Niimi, S. Taka-\nhashi, A. Fujimaki, S. Maekawa, and Y. Otani, Nat.\nMater. 14, 675 (2015).\n[33] K.-R. Jeon, J.-C. Jeon, X. Zhou, A. Migliorini, J. Yoon,\nand S. S. Parkin, ACS Nano 14, 15874 (2020).\n[34] R. Ojaj arvi, T. T. Heikkil a, P. Virtanen, and M. A.\nSilaev, Phys. Rev. B 103, 224524 (2021).\n[35] T. T. Heikkil a, M. A. Silaev, P. Virtanen, and F. S. Berg-\neret, Progress in Surface Science 94, 100540 (2019).\n[36] M. A. Silaev, Phys. Rev. B 102, 144521 (2020).\n[37] T. Cea, C. Castellani, G. Seibold, and L. Benfatto, Phys-\nical review letters 115, 157002 (2015).\n[38] Supplementary material includes details of the symmetry\noperators and derivation of the susceptibilities.\n[39] In general h+is inhomogeneous and the gradient of h+\nbreaks the mirror symmetry locally. Thus we do not need\nto consider the mirror symmetry of the SC.\n[40] M. A. Silaev, D. Rabinovich, and I. Bobkova, arXiv\npreprint arXiv:2108.08862 (2021).\n[41] K. D. Usadel, Phys. Rev. Lett. 25, 507 (1970).\n[42] W. Belzig, F. K. Wilhelm, C. Bruder, G. Sch on, and\nA. D. Zaikin, Superlatt. and Microstruc. 25, 1251 (1999).\n[43] F. S. Bergeret, A. F. Volkov, and K. B. Efetov, Rev. Mod.\nPhys. 77, 1321 (2005).\n[44] F. S. Bergeret and I. V. Tokatly, Phys. Rev. B 89, 134517\n(2014).\n[45] F. S. Bergeret and I. V. Tokatly, Phys. Rev. Lett. 110,\n117003 (2013).\n[46] The code used to obtain the results in this manuscript6\ncan be found at https://gitlab.jyu.\f/jyucmt/sssc-higgs- rashba .\n[47] I. V. Tokatly, Phys. Rev. B 96, 060502(R) (2017)."    },    {        "title": "0806.0420v1.Spin_Orbit_Coupling_in_an_f_electron_Tight_Binding_Model.pdf",        "content": "arXiv:0806.0420v1  [cond-mat.mtrl-sci]  3 Jun 2008Spin-Orbit Coupling in an f-electron Tight-Binding Model\nM. D. Jones\nDepartment of Physics and Center for Computational Researc h,\nUniversity at Buffalo, The State University of New York, Buffalo , NY 14260∗\nR. C. Albers\nTheoretical Division, Los Alamos National Laboratory,\nLos Alamos, NM 87501†\n(Dated: October 31, 2018)\nWe extend a tight-binding method to include the effects of spi n-orbit coupling, and apply it to\nthe study of the electronic properties of the actinide eleme nts Th, U, and Pu. These tight-binding\nparameters are determined for the fcc crystal structure usi ng the equivalent equilibrium volumes.\nIn terms of the single particle energies and the electronic d ensity of states, the overall quality of\nthe tight-binding representation is excellent and of the sa me quality as without spin-orbit coupling.\nThe values of the optimized tight-binding spin-orbit coupl ing parameters are comparable to those\ndetermined from purely atomic calculations.\nPACS numbers: 71.15.Ap, 71.15.Nc, 71.15.Rf, 71.20.Gj,71. 70.Ej\nI. INTRODUCTION\nThe accurate determination of inter-atomic forces is\ncrucial for almost all aspects of modeling the fundamen-\ntal behavior of materials. Whether one is interested in\nstatic equilibrium properties using Monte Carlo meth-\nods, or time dependent phenomena using molecular dy-\nnamics, the essential feature remains the origin, appli-\ncability, and transferability of the forces acting on the\nfundamental unit being modeled (atoms or molecules in\nmost cases). First principles methods based on density\nfunctional theory have gained wide acceptance for their\nease of use, relatively accurate determination of funda-\nmental properties, and high transferability. These tech-\nniques, however, are limited in their application by cur-\nrent computing technology to systems of a few hundred\natoms or less (most commonly a few dozen atoms). Po-\ntentials that are classically derived (i.e., pair potentials)\nlack directional bonding (or at best add some bond angle\ninformation) and other quantum mechanical effects but\nare computationally far more tractable for larger simu-\nlations. Recent advances in tight-binding (TB) theory,\nwhich include directional bonding, but treat only the\nmost important valence electrons shells, therefore show a\ngreat deal of promise.\nTB models have become a useful method for the\ncomputational modeling of materials properties thanks\nto their ability to incorporate quantum mechanics\nin a greatly simplified theoretical treatment, making\nlarge accurate simulations possible on modern digital\ncomputers1,2. Another advantage of these TB models\nis their ability to treat a general class of problems that\ninclude directional bonding between valence electrons, of\nparticularimportance fortransition metal and f-electron\nmaterials. Finally, TB models are widely used in many-\nbodyformalismsfortheone-electronpartofthe Hamilto-\nnian. It is therefore a useful representation of the band-\nstructure for a more sophisticated treatment ofelectroniccorrelation, and has so been used3, for example, in dy-\nnamical mean-field theory applications for Pu.\nIn this report we present recent developments towards\na transferable tight-binding total energy technique appli-\ncable to heavy metals. With the addition of spin-orbit\ncoupling effects for angular momentum up to (and in-\ncluding)f-character, we demonstrate the applicability of\nthis technique for the element Pu, of particular interest\nfor its position near the half-filling point of the 5 fsub-\nshell in the actinide sequence and the boundary between\nlocalized and delocalized f-electrons4.\nII. TB METHOD\nThe TB model used in this report is similar to that\nused in the handbook by Papaconstantopoulos5. We\nhave extended the calculations to include f-electrons6\nand spin-orbit coupling7. As such, in this report we will\nelaborate only on those aspects of the technique that\nare unique to this work. A very brief recapitulation of\nthe underlying TB method and its approximations is in-\ncluded to create the proper context for the addition of\nf-electrons and spin-orbit coupling.\nThe Slater-Koster method8consists of solving the sec-\nular equation,\nHψi,v=ǫi,vSψi,v, (1)\nfor the single-particle eigenvalues and orbitals, under the\nfollowing restrictions: terms involving more than two\ncenters are ignored, terms where the orbitals are on the\nsame atomic site are taken as constants, and the result-\ning reduced set of matrix elements are treated as variable\nparameters. The Hamiltonian, H, includes the labels for\norbitals having generic quantum numbers α,βlocalized\non atomsi,j, where the effective potential is assumed\nto be spherical, and can be represented as a sum over2\natomic centers,\nHαi,βj=/angbracketleftigg\nα,i/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle−∇2+/summationdisplay\nkVeff\nk/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleβ,j/angbracketrightigg\n,(2)\nwhich we further decompose into “on-site” and “inter-site” terms,\nHαi,βj=eαδαβδij+Eαi,βj/negationslash=i, (3)\nwhere the on-siteterms, eα, representterms in which two\norbitals share the same atomic site, and\nEαi,βj/negationslash=i=/summationdisplay\nneik·(Rn+bj−bi)/integraldisplay\ndrψα(r−Rn−bi)Hψβ(r−bj), (4)\nare the remaining energy integrals involving orbitals lo-\ncated on different atomic sites, and we have used transla-\ntional invariance to reduce the number of sums over bra-\nvais lattice points {Rn}, and the bidenote atomic basis\nvectors within the repeated lattice cells. Note that terms\nwhich have both orbitals located on the same site, but\nthe effective potential ( Veff) on other sites have been ig-\nnored. These contributions are typically taken to be “en-\nvironmental”correctionstotheon-siteterms, andarenot\naccounted for in the usual Slater-Koster formalism. For\nthe inter-site terms, the two center approximation also\nconsists of ignoring these additional terms in which the\neffective potential, Veff, does not lie on one of the atomic\nsites. Once this approximation has been made, the inter-\natomic (i/ne}ationslash=j) matrix elements reduce to a simple sum\nover angular functions, Gll′m(Ωi,j), and functions which\ndepend only upon the magnitude of the distances be-\ntween atoms,\nHαi,βj=/summationdisplay\nhll′m(rij)Gll′m(Ωi,j), (5)\nwherewe havenowadoptedthe usualconventionofusing\nthe familiar l,mangular momentum quantum numbers,\nand the axis connecting the atoms is the quantization\naxis. Anequivalentexpressionfor sll′mtermsexistswhen\nnon-orthogonal orbitals are used. The basis set used for\ntheαandβquantum states are the cubic harmonics9\nwhose functional forms are given in Table I (with ap-\npropriate normalization factors) where |±/an}b∇acket∇i}htdenotes the\nspin-state, which we will need for spin-orbit coupling.\nThe Slater-Kostertables for the sp3d5matrix elements\ncan be found in standard references10, and we have used\nthe tabulated results of Takegahara et al.11for the addi-\ntional matrix elements involving f-electrons. Typical TB\napplications are then reduced to using TB as an interpo-\nlation scheme; the matrix elements ( hll′m,sll′mandeα)\nare determined by fitting to ab-initio calculated quanti-\nties such as the total energy and band energies.\nIn this study we restrict ourselves to the determina-\ntion of optimal TB parameters at the neighbor distances\nin the face-centered cubic crystal structure (often used\nas a surrogate for the more complex ground state crystal\nstructure of the actinides) near the equilibrium volume.Suchtabulationshavebeenextensivelyused5inthestudy\nof materials with lower atomic number. To the best of\nour knowledge this is the first time that such parame-\nters have been presented for light actinide elements that\ninclude the f-electron orbitals (although similar param-\neters have been determined for the elements Ac and Th\nin ansp3d5basis5). The TB parameter values so derived\nare available (on request) from the authors.\nA. Spin-orbit coupling\nThe primary impact of spin-orbit coupling is to non-\ntrivially couple electrons of different spin states, thus\ndoubling the size of the TB Hamiltonian. The spin-orbit\ncontribution to the Hamiltonian is given by\nHso=ξ(r)L·S, (6)\nwhereξ(r) = (α2/(2r))(∂V/∂r),Vis the total (crystal)\npotential. We neglect contributions from more than one\ncenter. A new Hamiltonian matrix can then be defined\nin terms of the spinless one,\nH=H+Hso=/parenleftbigg\nH+1\n2ξLz1\n2ξL−\n1\n2ξL+H−1\n2ξLz/parenrightbigg\n(7)\nwhere\nξnl=/planckover2pi1/integraldisplay∞\n0ξ(r)/bracketleftbig\nR0\nnl(r)/bracketrightbig2r2dr, (8)\nis the spin-orbit coupling parameter between orbitals of\norbital angular momentum land primary quantum num-\nbernlocated on the same atom, L±are the usual raising\nand lowering operators, and Lzthe azimuthal angular\nmomentum operator,\nL±Ylm(θ,φ) =/planckover2pi1/radicalbig\nl(l+1)−m(m±1)Ylm±1\nLzYlm(θ,φ) =/planckover2pi1mYlm.\nThe functions R0\nnl(r) are the non-relativistic radial wave\nfunctions. The spin-orbit contributions to the Hamilto-\nnian matrix can then be expressed in term of the TB3\nTABLE I: TB basis functions used for an sp3d5f7calculation. Note that fl(r) = 1/rl.\nl=0 l=1 l=2 l=3\n|s±/angbracketright=p\n1/4π|±/angbracketright | p1±/angbracketright=p\n3/4πf1(r)x|±/angbracketright | d1±/angbracketright=p\n5/16πf2(r)xy|±/angbracketright | f1±/angbracketright= 2p\n105/16πf3(r)xyz|±/angbracketright\n|p2±/angbracketright=p\n3/4πf1(r)y|±/angbracketright | d2±/angbracketright= 2p\n15/16πf2(r)yz|±/angbracketright | f2±/angbracketright=p\n7/16πf3(r)x(5x2−3r2)|±/angbracketright\n|p3±/angbracketright=p\n3/4πf1(r)z|±/angbracketright | d3±/angbracketright= 2p\n15/16πf2(r)zx|±/angbracketright | f3±/angbracketright=p\n7/16πf3(r)y(5y2−3r2)|±/angbracketright\n|d4±/angbracketright=p\n15/16πf2(r)(x2−y2)|±/angbracketright | f4±/angbracketright=p\n7/16πf3(r)z(5z2−3r2)|±/angbracketright\n|d5±/angbracketright=p\n5/16πf2(r)(3z2−r2)|±/angbracketright | f5±/angbracketright=p\n105/16πf3(r)x(y2−z2)|±/angbracketright\n|f6±/angbracketright=p\n105/16πf3(r)y(z2−x2)|±/angbracketright\n|f7±/angbracketright=p\n105/16πf3(r)z(x2−y2)|±/angbracketright\nbasis functions listed in Table I. Rather than list contri-\nbutions for the 32x32 matrix, here we list the matrices in\nthe sub-blocks corresponding to each orbital angular mo-\nmentum. The panddcontributionshavebeen previously\ndiscussed in relation to the tight-binding formalism12,13;to the best of our knowledge no fcontribution has yet\nappeared in the literature. For completeness we detail\nthe spin-orbit contribution for all values of the angular\nmomentum up to l= 3.\nHso\np=ξnp\n2\n0−i0 0 0 1\ni0 0 0 0 −i\n0 0 0 −1i0\n0 0−1 0i0\n0 0−i−i0 0\n1i0 0 0 0\n, (9)\nHso\nd=ξnd\n2\n0 0 0 2 i0 0 1 −i0 0\n0 0i0 0 −1 0 0 −i−i√\n3\n0−i0 0 0 i0 0 −1√\n3\n−2i0 0 0 0 0 i1 0 0\n0 0 0 0 0 0 i√\n3−√\n3 0 0\n0−1−i0 0 0 0 0 −2i0\n1 0 0 −i−i√\n3 0 0 −i0 0\ni0 0 1 −√\n3 0i0 0 0\n0i−1 0 0 2 i0 0 0 0\n0i√\n3√\n3 0 0 0 0 0 0 0\n, (10)\nHso\nf=ξnf\n4\n0 0 0 0 0 0 2 i0 0 0 0 2 i2 0\n0 03i\n20 0it0 0 0 0 −3\n20 0t\n0−3i\n20 0it0 0 0 0 03i\n20 0it\n0 0 0 0 0 0 0 03\n2−3i\n20t it 0\n0 0−it0 0−i\n20−2i0 0−t0 01\n2\n0−it0 0i\n20 0−2 0 0 −it0 0−i\n2\n−2i0 0 0 0 0 0 0 −t−it0−1\n2i\n20\n0 0 0 0 2 i−2 0 0 0 0 0 0 0 −2i\n0 0 03\n20 0−t0 0−3i\n20 0−it0\n0 0 03i\n20 0it03i\n20 0−it0 0\n0−3\n2−3i\n20−t it 0 0 0 0 0 0 0 0\n−2i0 0t0 0−1\n20 0it0 0i\n20\n2 0 0 −it0 0−i\n20it0 0−i\n20 0\n0t−it01\n2i\n20 2i0 0 0 0 0 0\n, (11)\nwheret=√\n15/2. B. Fitting the Parameters\nThe values of the TB parameters were determined us-\ning standard non-linear least squares optimization rou-4\ntinesbymatchingenergybandvaluesderivedfromhighly\naccurate first principles density functional theory (DFT)\ncalculations14. The technique is described in detail in\na previous work6, where the DFT calculations in this\ncase used a generalized gradient approximation DFT\nfunctional15, and the improved tetrahedron scheme16for\nBrillouin zone integrations. In this study we use as a\nstarting point high quality fits to the scalar-relativistic\nenergy bands and approximate atomic values of the spin-\norbit parameters. The first step is to then use this fit for\nfitting the relativistic energy bands including spin-orbit\ncoupling. Successive optimization steps then relax only\nthe spin-orbit coupling paramaters (step 1), the remain-\ning on-site parameters (step 2), and finally the inter-site\nterms (step 3). The fit quality through these steps is\nshown in Figure 1. Note that the quality of the final fit\nis comparable to the original fit quality (open symbols\nat step 3) when only scalar-relativistic effects were taken\ninto account.\n1 2 3\nOptimization Step00.1Average rms Fitting Errors [eV]Th\nU\nPu\nFIG. 1: TB fit quality in terms of the cumulative root mean\nsquare (rms) errors at various steps of the optimization pro ce-\ndure. Step 1 relaxes the spin-orbit parametes ( ξnl), 2 relaxes\nthe remaining on-site parameters, and 3 is a full relaxation of\nall parameters. Open symbols at Step 3 indicate the original\nscalar-relativistic fit quality. Note that the cumulative r ms\nerror is over all of the fitted bands (20 bands for Th, U, and\nPu). Although the spin-orbit coupling is an atomic quantity ,\nthe improvement of our results in step 3 (which relaxes inter -\nsite parameters) indicates some environmental effects shou ld\nalso be taken into account.\nIII. APPLICATION TO THE LIGHT\nACTINIDES, TH, U, AND PU\nA. Energy bands including spin-orbit coupling\nThe first comparison between the TB fit and FLAPW\ncalculations are the energy bands shown in Figure 2.\nNote the excellent agreement between the two sets ofcalculations (the cumulative root mean square error in\nthe TB fits to the first 20 energy bands in the irreducible\nBrillouinzoneis0.013,0.013,and0.072Ry, respectively).\nAlso note that we have included the “semi-core” 6 p\n(a)Th\n(b)U\n(c)Pu\nFIG. 2: TB energybands for Th ( a= 9.61), U(a= 8.22), and\nPu (a= 8.14), shown in comparison with FLAPW valence\nenergy bands (dotted lines). Note the excellent agreement.\nThe abscissa for each calculation has been shifted such that\nthe Fermi energy is at zero. Higher valence states (above the\nfirst 20) are not fit, hence the poorer fit quality well above\nthe Fermi level.\nstates in the fit to better fix the available pstates in the\nTB basis. To expand the energy scale comparing the va-\nlence bands, the fit quality for the semi-core 6p states is5\nFIG. 3: TB energy bands (dashed lines) for Pu semi-core 6p\nstates, compared with FLAPW values (solid lines).\nshown separately in Figure 3 for Pu (all three elements\nhave similarexcellent fit quality for the more localized 6p\nstates). Note that higher energy bands (well above the\nFermi level) are not fit, hence the larger discrepancies for\nthose levels.\nB. Density of states including spin-orbit coupling\nWe also compare the total density of states (DOS) be-\ntween TB and FLAPW methods in Figure 4.\nThe TB method shown in the figure used a simple\nFermi-Dirac temperature smearing method (with kBT=\n500) for integrating over the irreducible wedge of the\nBrillouin zone, while the FLAPW calculations used the\nimproved tetrahedron16method with Gaussian smear-\ning. From the comparison between the TB and FLAPW\nmethods shown in the above figure, we note that the\nagreementisexcellent,withallmajorfeaturesintheDOS\nreproduced by the TB calculations. There is a slight re-\nduction in the height of some of the larger peaks in the\nDOS for the TB technique, most likely due to the inabil-\nity of the temperature smearing technique to represent\nthe finer grained features as well as the improved tetra-\nhedron method does.\nC. Spin-orbit coupling terms\nIt is interesting to compare the spin-orbit coupling pa-\nrameters,ξnl, predicted by TB theory for the various\nvalence shells relative to the values predicted by accu-\nrate Hartree-Fock-Slatercalculations ofisolated atoms19.\nThis comparison is shown in Table II.\nNote the overall agreement between the TB fitted pa-\nrameters and the atomic values. The overall shift of a\nfew tenths of an eV for the TB values is interesting, and\nthis trend could be representative of crystal field effects\n(this speculation could be checked by performing equiv-\nalent fits at different densities). Equivalently, one can-24 -20 -16 -12 -8 -4 0 4 8\nε−EF [eV]0481216Total DOS [states/eV]FLAPW\nTB\n(a)Th\n-28 -24 -20 -16 -12 -8 -4 0 4 8\nε−EF [eV]04812Total DOS [states/eV]FLAPW\nTB\n(b)U\n-28 -24 -20 -16 -12 -8 -4 0 4 8\nε−EF [eV]04812Total DOS [states/eV]FLAPW\nTB\n(c)Pu\nFIG. 4: TB (dotted lines) and FLAPW (solid lines) total\nDOS, including spin-orbit coupling. Note that the TB calcu-\nlation is in quite good agreement with the FLAPW results,\ndespite using a different BZ integration method.The absciss a\nfor each calculation has been shifted such that the Fermi en-\nergy is at zero.\ncompare the spin-orbit splitting of the electronic energy\nlevels with the purely atomic case. This comparison is\nalso shown in Table II.6\nTABLE II: Values of spin-orbit coupling strength, ξnl, and spin-orbit splittings, ∆ nl= (2l+ 1)ξnl/2, for the various valence\nelectron shells predicted by the TB fit compared with purely a tomic values using relativistic density functional theory (DFT)17,\na Dirac-Slater atomic code (DIRAC)18, and relativistic Hartree-Fock-Slater (HFS)19atomic calculations. Dashed entries are\nused for orbitals not populated in the atomic calculations. Values are in eV.\nMethod ξ6p ∆6p ξ5d ∆5d ξ5f ∆5f\nTh\nDIRAC 5.29 7.94 0.20 0.51 0.19 0.66\nDFT 5.24 7.86 0.21 0.52 – –\nHFS 4.09 6.14 0.30 0.75 – –\nTB 4.19 6.29 0.20 0.51 0.18 0.62\nU\nDIRAC 5.96 8.94 0.19 0.47 0.24 0.83\nDFT 5.90 8.85 0.20 0.50 0.24 0.84\nHFS 4.38 6.57 0.30 0.75 0.35 1.24\nTB 4.64 6.96 0.23 0.58 0.42 1.48\nPu\nDIRAC 6.92 10.38 0.20 0.51 0.31 1.10\nDFT – – – – – –\nHFS 4.60 6.90 – – 0.41 1.43\nTB 5.23 7.84 0.59 1.46 0.54 1.90\nIV. CONCLUSIONS\nWe have included f-electron and spin-orbit effects in\na standard tight-binding method for solids in order to\nadvance simpler simulation methods that are capable of\nthe accuracy of more expensive, full-potential density-\nfunctional techniques. We have applied this TB tech-\nnique to elemental fcc Th, U, and Pu, and have achieved\nexcellent agreement with the electronic properties pre-\ndicted using a highly accurate FLAPW method. The\nfitted spin-orbit coupling parameters match very well\nthe values independently predicted by atomic electronic\nstructure calculations. This methodology bodes well for\nfurther TB investigations, especially for the study of de-\nfects, phonons, and dynamical properties. In future work\nwe intend to develop a more transferable model based ona TB total energy formalism6, which should allow the\nstraightforward calculation of detailed materials proper-\nties.\nAcknowledgments\nThis work was carried out under the auspices of the\nNationalNuclearSecurityAdministrationofthe U.S.De-\npartment of Energy at Los Alamos National Laboratory\nunder Contract No. DE-AC52-06NA25396. Calculations\nwere performed at the Los Alamos National Laboratory\nand the Center for Computational Research at SUNY–\nBuffalo. FLAPW calculations were performed using the\nWien2k package14. We thank Jian-Xin Zhu for providing\nhelpful remarks.\n∗Electronic address: jonesm@ccr.buffalo.edu\n†Electronic address: rca@lanl.gov\n1C. M. Goringe, D. R. Bowler, and E. Hernandez, Rep.\nProg. Phys. 60, 1447 (1997).\n2D. A. Papaconstantopoulos and M. J. Mehl, J. Phys.: Con-\ndens. Matter 15, R413 (2003).\n3J.-X. Zhu, A. K. McMahan, M. D. Jones, T. Durakiewicz,\nJ. J. Joyce, J. M. Wills, and R. C. Albers, Phys. Rev. B\n76, 245118 (2007).\n4R. C. Albers, Nature 410, 759 (2001).\n5D. A. Papaconstantopoulos, Handbook of the Band Struc-\nture of Elemental Solids (Plenum Press, New York, 1986).\n6M. D. Jones and R. C. Albers, Phys. Rev. B 66, 134105\n(2002).\n7M. Lach-hab, M. J. Mehl, and D. A. Papaconstantopoulos,\nJ. Phys. Chem. Solids 63, 833 (2002).\n8J. C. Slater and G. F. Koster, Phys. Rev. 94, 1498 (1954).\n9F. von Der Lage and H. A. Bethe, Phys. Rev. 71, 612\n(1947).\n10W. A. Harrison, Electronic Structure and the Properties of\nSolids(Freeman, San Francisco, CA, USA, 1980).\n11K. Takegahara, Y. Aoki, and A. Yanase, J. Phys. C 13,583 (1980).\n12J. Friedel, P. Lenglart, and G. Leman, J. Phys. Chem.\nSolids25, 781 (1964).\n13D. J. Chadi, Phys. Rev. B 16, 790 (1977).\n14P. Blaha, K. Schwarz, G. K. H. Madsen, D. Kvasnicka, and\nJ. Luitz, WIEN2K, An Augmented Plane Wave + Local\nOrbitals Program for Calculating Crystal Properties (Karl-\nheinz Schwartz, Techn. Universitt Wien, Austria, 2001.\nISBN 3-9501031-1-2).\n15J. P. Perdew, S. Burke, and M. Ernzerhof, Phys. Rev. Lett.\n77, 3865 (1996).\n16P. E. Bl¨ ochl, O. Jepsen, and O. K. Andersen, Phys. Rev.\nB49, 16223 (1994).\n17S. Kotochigova, Z. H. Levine, E. L. Shirley, M. D. Stiles,\nand C. W. Clark, http://math.nist.gov/DFTdata (1996).\n18ADF2004.01, SCM, Theoretical Chemistry, Vrije\nUniversiteit, Amsterdam, The Netherlands,\nhttp://www.scm.com .\n19F. Herman and S. Skillman, Atomic Structure Calculations\n(Prentice-Hall, Englewood Cliffs, NJ, USA, 1963)."    },    {        "title": "1811.09088v1.Enhanced_Rashba_spin_orbit_coupling_in_core_shell_nanowires_by_the_interfacial_effect.pdf",        "content": "Enhanced Rashba spin-orbit coupling in core-shell nanowires by the\ninterfacial e\u000bect\nPawe l W\u0013 ojcik,1,a)Andrea Bertoni,2,b)and Guido Goldoni3, 2,c)\n1)AGH University of Science and Technology, Faculty of Physics and Applied Computer Science, 30-059 Krakow,\nAl. Mickiewicza 30, Poland\n2)S3, Istituto Nanoscienze-CNR, Via Campi 213/a, 41125 Modena, Italy\n3)Department of Physics, Informatics and Mathematics, University od Modena and Reggio Emilia,\nItaly\n(Dated: November 26, 2018)\nWe report on ~k\u0001~ pcalculations of Rashba spin-orbit coupling controlled by external gates in InAs/InAsP\ncore-shell nanowires. We show that charge spilling in the barrier material allows for a stronger symmetry\nbreaking than in homoegenous nano-materials, inducing a speci\fc interface-related contribution to spin-orbit\ncoupling. Our results qualitatively agree with recent experiments [S. Futhemeier et al. , Nat. Commun. 7,\n12413 (2016)] and suggest additional wavefunction engineering strategies to enhance and control spin-orbit\ncoupling.\nUnderstanding and controlling spin-orbit coupling\n(SOC) is critical in semiconductor physics. In particular,\nin semiconductor nanowires (NWs)1{7SOC is essential\nfor to the development of a suitable hardware for topo-\nlogical quantum computation8,9, with qubits encoded in\nzero-mode Majorana states which are supported by hy-\nbrid semiconductor-superconductor NWs10{15. Among\nother parameters, qubit protection at su\u000eciently high\ntemperatures relies on a large SOC which determines the\ntopological gap. Additionally, electrical control of SOC\nis necessary in the realization of spintronic devices16{26.\nSOC arises from the absence of inversion symmetry\nof the electrostatic potential. In semiconductor NWs,\ntypically having a prismatic shape, \fnite SOC may be\ninduced by distorting the quantum con\fnement (Rashba\nSOC27) by means of external gates, with the advantage\nof electrical control. A lattice contribution (Dresselhaus\nSOC28) is typically small and may vanish in speci\fc crys-\ntallographic directions - for zincblende NWs, the Dres-\nselhaus term vanishes along [111] due to the inversion\nsymmetry.\nThe Rashba SOC constant \u000bRhas been investigated\nexperimentally in homogeneous NWs based on the strong\nSOC materials InSb29,30and InAs31{35. Recently5, we\nreported on a ~k\u0001~ papproach applied to homogeneous\nNWs which predicts \u000bRfrom compositional and struc-\ntural parameters only. Our calculations performed for\nInSb NWs5and InAs NWs36generally con\frm recent\nexperiments in homogeneous NWs29,31, exposing values\nof\u000bRexceeding by one order of magnitude those re-\nported for 2D analogous planar systems37{39. Moreover,\n\u000bRproved to be strongly tunable with external gates in\nsamples and con\fgurations which can be routinely real-\nized with current technology.\na)Electronic mail: pawel.wojcik@\fs.agh.edu.pl\nb)Electronic mail: andrea.bertoni@nano.cnr.it\nc)Electronic mail: guido.goldoni@unimore.itFor a quantitative prediction of SOC, it is necessary to\ntake into account valence-to-conduction band coupling,\nthe explicit geometry and crystal structure of the NW,\nand the electron gas distribution which, in turn, must\nbe self-consistently determined by quantum con\fnement\ne\u000bects, interaction with dopants and electron-electron in-\nteraction. Indeed, in NWs the electron gas localization,\nand ensuing SOC, is a non-trivial result of competing en-\nergy contributions. As a function of doping concentration\nand ensuing free charge density, the electron gas evolves\nfrom a broad cylindrical distribution in the NW core (low\ndensity regime) to coupled quasi-1D and quasi-2D chan-\nnels at the NW edges (large density regime)40{43. Until\npolygonal symmetry holds, \u000bR= 0 regardless. However,\nexternal gates easily remove the symmetry; again, how\n\u000bRmoves from zero under the in\ruence of the external\ngates strongly depends on the charge density regime5.\nIn this Letter we extend and apply the ~k\u0001~ papproach to\ncore-shell NWs (CSNWs) and expose a novel mechanism\nthrough which SOC can be further tailored, and possibly\nenhanced. Epitaxially overgrown shells are often used\nin NW technology, either as a passivating layer improv-\ning optical performance44, or as a technique to engineer\nradial heterostructures45. Here we show that CSNWs\nFigure 1. Schematics of a InAs/InAsP CSNW grown along\n[111] with a bottom gate.arXiv:1811.09088v1  [cond-mat.mes-hall]  22 Nov 20182\nallow for an increased \rexibility in distorting the elec-\ntron gas of the NWs, giving rise to a speci\fc, interfacial\nSOC contribution46,47which substantially increases the\ntotal SOC. We make the case for InAs/InAsP CSNWs,\na systems of speci\fc interest in photonics48and electri-\ncal engineering49. Our results qualitatively agree with\nthe recent experiments by Furthmeier et al. in Ref. 50,\nwhere the enhancement of SO coupling was measured in\nGaAs/AlGaAs CSNW, and establish a strategy to in-\ncrease the SOC in Majorana InAs NWs.\nWe consider CSNWs with hexagonal cross-section51\ngrown along [111] (see Fig. 1), assuming in-wire trans-\nlational invariance along z. The used ~k\u0001~ papproach is\ndescribed in full in Ref. 5; here we focus on generaliza-\ntions required to account for the contribution of the in-\nternal heterointerface. The 8 \u00028 Kane Hamiltonian is\ngiven by16\nH8\u00028=\u0012HcHcv\nHy\ncvHv\u0013\n; (1)\nwhereHcandHvare the diagonal matrices correspond-\ning to the conduction (\u0000 6c) and valence (\u0000 8v, \u00007v) bands\nwhose expressions are given in Ref. 5. Using the pertur-\nbative transformation H(E) =Hc+Hcv(Hv\u0000E)\u00001Hy\ncv,\nthe Hamiltonian (1) reduces to a 2 \u00022 e\u000bective Hamil-\ntonian for the conduction band electrons. Emphasizing\nthe dependence of material parameters on the position,\n~ \u001a= (x;y),\nH=\u0014\n\u0000~2\n2r2D1\nm\u0003(~ \u001a)r2D+~2k2\nz\n2m\u0003(~ \u001a)+Ec(~ \u001a) +V(~ \u001a)\u0015\n\u000212\u00022+ [^\u000bx(~ \u001a)\u001bx+ ^\u000by(~ \u001a)\u001by]kz;(2)\nwhere\u001bx(y)are the Pauli matrices and m\u0003is the e\u000bective\nmass given by\n1\nm\u0003(~ \u001a)=1\nm0+2P2\n3~2\u00122\nE0(~ \u001a)+1\nE0(~ \u001a) + \u0001 0(~ \u001a)\u0013\n;(3)\nwherePis the conduction-to-valance band coupling pa-\nrameter.\nIn Eq. (2), ^ \u000bx, ^\u000byare the SOC operators\n^\u000bx=i\n3P2^ky\f(~ \u001a)\u0000i\n3P2\f(~ \u001a)^ky; (4)\n^\u000by=i\n3P2^kx\f(~ \u001a)\u0000i\n3P2\f(~ \u001a)^kx; (5)\nand\f(~ \u001a) is a material-dependent coe\u000ecient obtained as\nfollows. In the i-th layer\n\fi(~ \u001a) =1\nEc;i+V(~ \u001a)\u0000E0;i\u0000E(6)\n\u00001\nEc;i+V(~ \u001a)\u0000E0;i\u0000\u00010;i\u0000E;\nwhereEc,E0and \u0001 0are the conduction band edge, the\nenergy gap and the split-o\u000b band gap, respectively. As-\nsuming that the above parameters change as a step-likefunction at the interfaces\n\f(~ \u001a) =X\ni[\fi(~ \u001a)\u0000\fi+1(~ \u001a)]\ni(~ \u001a); (7)\nwhere the sum is carried out over all the layers, and \n i(~ \u001a)\nis the shape function, which for the hexagonal section is\ngiven by\n\ni(~ \u001a) = [\u0012(x+xi)\u0000\u0012(x\u0000xi)][\u0012(y+yi)\u0000\u0012(y\u0000yi)]\n\u0002[\u0012(x\u0000y+xi)\u0000\u0012(x\u0000y\u0000xi)]; (8)\nwhere\u0012is the Heaviside'a function and ( xi;yi) denotes\nthe position of the ( i)-th interface. Further Taylor ex-\npansion gives\n\fi(~ \u001a)\u0019\u00121\nE0;i+ \u0001 0i\u00001\nE0;i\u0013\n(9)\n+ \n1\nE2\n0;i\u00001\n(E0;i+ \u0001 0;i)2!\n(Ec;i+V(x;y)\u0000E):\nSubstituting (9) into Eqs. (4) and (5), the Rashba cou-\npling constants can be written as\n\u000bx(y)(~ \u001a) =\u000bV\nx(y)(~ \u001a) +\u000bint\nx(y)(~ \u001a); (10)\ni.e., the sum of the SOC induced by the electrostatic\npotential asymmetry,\n\u000bV\nx(y)(~ \u001a)\u0019X\ni1\n3P2 \n1\nE2\n0;i\u00001\n(E0;i+ \u0001 0;i)2!\n@V(~ \u001a)\n@y(x);\n(11)\nand the interface SOC, related to the electric \feld at the\ninterfaces between shells,\n\u000bint\nx(y)(~ \u001a)\u0019X\ni1\n3P2\u0010\n~\fi\u0000~\fi+1\u0011@\ni(~ \u001a)\n@y(x); (12)\nwith ~\fi=1\nE0;i+\u00010;i\u00001\nE0;i:\nProjecting the 3D Hamiltonian (2) on the basis of in-\nwire states n(~ \u001a) exp(ikzz), where the envelope functions\n n(~ \u001a) are determined by the strong con\fnement in the\nlateral direction, leads to SOC matrix elements\n\u000b\r;nm\nx(y)=Z Z\n n(~ \u001a)\u000b\r\nx(y)(~ \u001a) m(~ \u001a)d~ \u001a; (13)\nwhere\ridenti\fes the electrostatic ( \r=V) or the inter-\nfacial (\r=int) contribution.\nFor the NW in Fig. 1 with a single bottom gate,\n\u000b\r;nn\ny= 0 due to inversion symmetry about y. Moreover,\nhere we focus on the lowest intra-subband coe\u000ecient,\nn= 1. Below we discuss the SOC constant \u000bR=\u000b11\nx\nand corresponding interfacial and electrostatic compo-\nnents,\u000b\r\nR=\u000bV;11\nxand\u000b\r\nR=\u000bint;11\nx, respectively.\nThe electronic states in the CSNW section,  n(~ \u001a),\nare calculated by a mean-\feld self-consistent Sch odinger-\nPoisson approach40. We neglect the exchange-correlation\npotential which is substantially smaller than the Hartree3\nInAs InAs 0:9P0:1\nm\u0003[m0]0.0265 0.0308\nEc[eV]0.252 0.3\nE0[eV]0.42 0.5\n\u00010[eV]0.38 0.35\nTable I. Bulk parameters used in calculations56.\npotential40,52,53. The gradient of the self-consistent po-\ntentialV(~ \u001a) and the corresponding envelope functions\n n(~ \u001a) are \fnally used to determine \u000bRfrom Eq. (13).\nMaterial parameters mismatch at the interfaces is\ntaken into account solving the eigenproblem H n=E n\nwith boundary conditions46,54\n (i)\nn(~ \u001ak) = (j)\nn(~ \u001ak) (14)\n~2\n2m\u0003(i)r2D (i)\nn(~ \u001ak)\u0000~2\n2m\u0003(j)r2D (j)\nn(~ \u001ak) (15)\n+[\f(j)(~ \u001ak)\u0000\f(i)(~ \u001ak)](\u001bx+\u001by)kz (i)\nn(~ \u001ak) = 0;\nwhere~ \u001akis the position of the interface between i-th and\nj-th shells. Equations (14), (15), depend on both the po-\ntentialV(~ \u001a) at the interface and the energy E. We elim-\ninate this dependence neglecting the term proportional\nto (Ec;i+V(~ \u001a)\u0000E) in the Taylor series, Eq. (9). Then,\nthe interface contributions (12) are determined fully by\nmaterial parameters. This assumption, justi\fed when\n\f(j)\u0000\f(i)is small, neglects the SOC related to the motion\nof electron in the ~ \u001aplane, which in general contributes\nto the SOC by the boundary conditions.\nBelow we investigate a InAs 50 nm-wide core (mea-\nsured facet-to-facet) surrounded by a 30 nm InAs 1\u0000xPx\nshell, withx= 0:1 which allows to neglect strain-induced\nSOC.55Furthermore, as shown below, interfacial SOC\nis enhanced by the easy penetration of envelope func-\ntions in low band o\u000bset barriers, here only 48 meV\nhigh. Simulations have been carried out for a temper-\natureT= 4:2 K, in the constant electron concentration\nregime. The parameters adopted are given in Tab. I. P\nis assumed to be constant throughout the materials and\nEP(InAs) = 2m0P2=~2= 21:5 eV.\nThe calculated SOC coe\u000ecients for the InAs/InAsP\nCSNW of Fig. 1 as a function of the back gate voltage is\nreported in Fig. 2(a). The SOC constant is trivially zero\nifVg= 0, due to the overall inversion symmetry. At any\n\fnite voltage the inversion symmetry is removed, hence\n\u000bR6= 0. As shown in Fig. 2(a), the total \u000bRensues from\ntwo di\u000berent contributons, namely interfacial and elec-\ntrostatic, whose magnitude is of the same order. It is\nthus crucial to include both of them in the assessment of\nSOC in CSNWs. Note that the electrostatic component\nalmost coincides with the value for an InAs NW with\nthe same geometry, but no overgrown shell57. However,\nfor this speci\fc nanostructure, the largest part of \u000bRis\ndue to the interfacial contribution, which is \u001950% larger\nthan the electrostatic one. While the ratio between the\ntwo contributions is nearly independent of Vg[see the in-\nFigure 2. (a) Lines: total, electrostatic and interfacial SOC\nconstants vsgate voltage Vg, according to labels. Dots: to-\ntal SOC constant for an equivalent homegeneous InAs NW.\nInset: ratio between interfacial and electrostatic components,\n\u000bint\nR=\u000bV\nR. Results for ne= 107cm\u00001. See text for structure\nand material parameters. (b) Electrostatic and interfacial\nSOC constants vsVgaroundVg= 0 showing shooting up\nof SO couplings for higher electron density.\nset in Fig. 2(a)], they are both strongly anisotropic with\nrespect to the \feld direction. This is due to the di\u000berent\ne\u000bects on the charge density, as discussed in Ref. 5. This\ne\u000bect can also be grasped from the probability distribu-\ntion reported in Fig. 3 (top and bottom rows). Indeed,\nthe positive Vgpushes electron states towards the in-\nterface opposite to the gate, where the gradient of the\nself-consistent \feld is low. On the other hand, at Vg<0\nelectrons are pulled to the region of the nearest interface\nwith the stronger electric \feld, additionally strengthened\nby the electron-electron interaction5.\nFigure 3. Top row: Square of the ground state envelope func-\ntionj 1(x;y)j2. Middle row: linear density of the interfacial\nSOC constants at interfaces. Bottom row: self-consistent po-\ntential pro\fle (black line) and j 1(x;y)j2(red line) along the\nfacet-to-facet dashed line marked in the top-middle panel. Re-\nsults at selected gate voltages Vg=\u00000:1;0;0:1 V for the same\nstructure as in Fig. 2.\nThe value of \u000bint\nRdepends on the penetration of the4\nwave function into the interfaces. As shown in Fig. 3\n(middle row) the linear density of interfacial SOC at the\ninterfaces \u0016\u000bint\nR= 1(~ \u001a)\u000bint\nR(~ \u001a) 1(~ \u001a) is \fnite almost ev-\nerywhere, but it has opposite sign at opposite facets.58\nFor a centro-symmetric system ( Vg= 0) the overall value\nis zero, since opposite contributions cancel out exactly.\nWe stress a remarkable di\u000berence between CSNWs and\nanalogous planar structures. In a planar asymmetric\nquantum well, for example, \u000bR6= 0. In a CSNW with an\nembedded quantum well, however, the overall symmetry\nis recovered even if each facet of the quantum well is indi-\nvidually asymmetric. Therefore, opposite segments have\nopposite Rashba contributions and compensate. How-\never, any asymmetric gate potential unbalances opposite\ncontributions, the total e\u000bect being related to the amount\nof envelope function at the interface.\nFigure 4. Electrostatic \u000bV\nRand interfacial \u000bint\nRcontributions\nof SOC constant vselectron density ne. Results for Vg=\n\u00000:1 V. Inset: ratio \u000bint\nR=\u000bV\nRvsne.\nNote the almost linear increase of \u000bRwithVg. This\nbehaviour is observed in a relatively small charge density\nregime: the average Coulomb energy is small, most of\nthe charge is located in the core, and it is relatively rigid\nto an applied transverse electric \feld. At larger densi-\nties, however, charge moves at the interfaces to minimize\nCoulomb interaction40, with negligible tunneling energy\nbetween opposite facets. In this regime, the symmetric\ncharge density distribution is unstable and it is easily dis-\ntorted by an electric \feld5. Accordingly, SOC constant\nshoots around Vg= 0 as soon as the gate is switched on\n- see Fig. 2(b).\nAs we show in Fig. 4, both SOC components substan-\ntially increase in intensity with charge density, while their\nratio is weakly a\u000bected, it being rather a property of the\nnano-material (band parameters and band o\u000bset). This\nis explicitly shown in Fig. 5, where the two contributions\nare plot vsthe stechiometric fraction x. At lowx, pene-\ntration is very large, and the interfacial e\u000bect is dominant\n(of course for x= 0 the heterostructure is an homoge-\nneous NW with a larger diameter), while as x= 0:15 the\ntwo contributions are comparable, as also shown in theinset.\nFigure 5. The interfacial \u000bint\nR(blue circles) and electrostatic\n\u000bV\nR(red circles) SOC constants vsInAs 1\u0000xPxalloy composi-\ntion,x. Inset:\u000bint\nR=\u000bV\nRvsx. Results for Vg=\u00000:1 V and\nne= 107cm\u00001.\nTo summarize, we have shown that Rashba SOC in\nCSNWs is increased by the e\u000bect of the radial heteroint-\nerface, and its control via external metallic gates may be\nhighly improved by this interfacial e\u000bect. Although we\ndid not attempt to optimize \u000bRin the many parameter\nspace allowed by CSNWs, our results suggest that a gen-\neral strategy to enhance SOC in CSNWs relies on a mod-\ni\fcation of the compositional structure exploiting asym-\nmetric penetration of the wave function into the shell\nlayer.\nThis work was partially supported by the AGH UST\nstatutory tasks No. 11.11.220.01/2 within subsidy of the\nMinistry of Science and Higher Education and in part\nby PL-Grid Infrastructure. P.W. was supported by Na-\ntional Science Centre, Poland (NCN) according to deci-\nsion 2017/26/D/ST3/00109.\nREFERENCES\n1I. A. Kokurin, Physica E 74, 264 (2015).\n2I. A. Kokurin, Solid State. Commun. 195, 49 (2014).\n3M. Kammermeier, P. Wenk, J. Schliemann, S. Heedt, and\nT. Sch apers, Phys. Rev. B 93, 205306 (2016).\n4G. W. Winkler, D. Varjas, R. Skolasinski, A. A. Soluyanov,\nM. Troyer, and M. Wimmer, Phys. Rev. Lett. 119, 037701\n(2017).\n5P. W\u0013 ojcik, A. Bertoni, and G. Goldoni, Phys. Rev. B 97, 165401\n(2018).\n6T. Campos, P. E. Faria Junior, M. Gmitra, G. M. Sipahi, and\nJ. Fabian, Phys. Rev. B 97, 245402 (2018).\n7J.-W. Luo, S.-S. Li, and A. Zunger, Phys. Rev. Lett. 119, 126401\n(2017).\n8J. Alicea, Phys. Rev. B 81, 125318 (2010).\n9J. Sau, Physics 10, 68 (2017).\n10A. Kitaev, Ann. Phys. 303, 2 (2003).\n11J. D. Sau, S. Tewari, and S. Das Sarma, Phys. Rev. B 85, 1\n(2012).5\n12V. Mourik, K. Zuo, S. M. Frolov, S. R. Plissard, E. P. A. M.\nBakkers, and L. P. Kouwenhoven, Science 336, 1003 (2012).\n13J. Mastom aki, S. Roddaro, M. Rocci, V. Zannier, D. Ercolani,\nL. Sorba, I. J. Maasilta, N. Ligato, A. Fornieri, E. Strambini,\nand F. Giazotto, Nano Research 10, 3468 (2017).\n14S. M. Albrecht, A. P. Higginbotham, M. Madsen, F. Kuemmeth,\nT. S. Jespersen, J. Nyg\u0017 aard, P. Krogstrup, and C. M. Marcus,\nNature 531, 206 (2016).\n15A. Manolescu, A. Sitek, J. Osca, L. m. c. Serra, V. Gudmundsson,\nand T. D. Stanescu, Phys. Rev. B 96, 125435 (2017).\n16J. Fabian, A. Matos-Abiague, C. Ertler, P. Stano, and I. \u0014Zuti\u0013 c,\nActa Physica Slovaca 57, 565 (2007).\n17S. Datta and B. Das, Appl. Phys. Lett. 56, 665 (1990).\n18J. Schliemann, J. C. Egues, and D. Loss, Phys. Rev. Lett. 90,\n146801 (2003).\n19P. W\u0013 ojcik and J. Adamowski, Semicond. Sci. Technol. 31, 115012\n(2016).\n20P. W\u0013 ojcik, J. Adamowski, B. J. Spisak, and M. Wo loszyn, J.\nAppl. Phys. 115, 104310 (2014).\n21P. W\u0013 ojcik and J. Adamowski, Sci Rep 7, 45346 (2017).\n22P. P. Das, J. Bhandari, N abd Wan, M. Cahay, K. B. Chetry,\nR. S. Newrock, and S. T. Herbert, Nanotechnology 23, 215201\n(2012).\n23P. Debray, S. Rahman, and M. Muhammad, Nature Nano Tech-\nnol.4, 759 (2009).\n24M. Kohda, S. Nakamura, Y. Nishihara, K. Kobayashi, T. Ono,\nJ. O. Ohe, Y. Tokura, T. Mineno, and J. Nitta, Nature Com-\nmunications 3, 1082 (2012).\n25F. Rossella, A. Bertoni, D. Ercolani, M. Rontani, L. Sorba,\nF. Beltram, and S. Roddaro, Nat. Nanotechnol. 9, 997 (2014).\n26A. Iorio, M. Rocci, L. Bours, M. Carrega, V. Zannier, L. Sorba,\nS. Roddaro, F. Giazotto, and E. Strambini, ArXiv e-prints\n(2018), arXiv:1807.04344 [cond-mat.mes-hall].\n27E. I. Rashba, Sov. Phys. Solid State 2, 1109 (1960).\n28G. Dresselhaus, Phys. Rev. 100, 580 (1955).\n29I. van Weperen, B. Tarasinski, D. Eeltink, V. S. Pribiag, S. R.\nPlissard, E. P. A. M. Bakkers, L. P. Kouwenhoven, and M. Wim-\nmer, Phys. Rev. B 91, 201413(R) (2015).\n30J. Kammhuber, M. C. Cassidy, F. Pei, M. P. Nowak, A. Vuik,\nD. Car, S. R. Plissard, E. P. A. M. Bakkers, M. Wimmer, and\nL. P. Kouwenhoven, Nat Commun. 8, 478 (2017).\n31Z. Scher ubl, G. m. H. F ul op, M. H. Madsen, J. Nyg\u0017 ard, and\nS. Csonka, Phys. Rev. B 94, 035444 (2016).\n32A. E. Hansen, M. T. Bj ork, C. Fasth, C. Thelander, and\nL. Samuelson, Phys. Rev. B 71, 205328 (2005).\n33S. Dhara, H. S. Solanki, V. Singh, A. Narayanan, P. Chaudhari,\nM. Gokhale, A. Bhattacharya, and M. M. Deshmukh, Phys. Rev.\nB79, 121311 (2009).\n34P. Roulleau, T. Choi, S. Riedi, T. Heinzel, I. Shorubalko, T. Ihn,\nand K. Ensslin, Phys. Rev. B 81, 155449 (2010).\n35S. Est\u0013 evez Hern\u0013 andez, M. Akabori, K. Sladek, C. Volk, S. Alagha,\nH. Hardtdegen, M. G. Pala, N. Demarina, D. Gr utzmacher, and\nT. Sch apers, Phys. Rev. B 82, 235303 (2010).\n36P. W\u0013 ojcik, A. Bertoni, and G. Goldoni, (2018), unpublished.\n37R. L. Kallaher, J. J. Heremans, N. Goel, S. J. Chung, and M. B.\nSantos, Phys. Rev. B 81, 0035335 (2010).\n38R. L. Kallaher, J. J. Heremans, N. Goel, S. J. Chung, and M. B.\nSantos, Phys. Rev. B 81, 075303 (2010).\n39F. Herling, C. Morrison, C. S. Knox, S. Zhang, O. Newell, M. My-\nronov, E. H. Lin\feld, and C. H. Marrows, Phys. Rev. B 95,\n155307 (2017).40A. Bertoni, M. Royo, F. Mahawish, and G. Goldoni, Phys. Rev.\nB84, 205323 (2011).\n41S. Funk, M. Royo, I. Zardo, D. Rudolph, S. Morktter, B. Mayer,\nJ. Becker, A. Bechtold, S. Matich, M. Dblinger, M. Bichler,\nG. Koblmller, J. J. Finley, A. Bertoni, G. Goldoni, and G. Ab-\nstreiter, Nano Letters 13, 6189 (2013).\n42J. Jadczak, P. Plochocka, A. Mitioglu, I. Breslavetz, M. Royo,\nA. Bertoni, G. Goldoni, T. Smolenski, P. Kossacki, A. Kretinin,\nH. Shtrikman, and D. K. Maude, Nano Letters 14, 2807 (2014).\n43M. Royo, A. Bertoni, and G. Goldoni, Phys. Rev. B 89, 155416\n(2014).\n44F. Jabeen, S. Rubini, V. Grillo, L. Felisari, and F. Martelli,\nAppl. Phys. Lett. 93, 083117 (2008).\n45D. Spirkoska, J. Arbiol, A. Gustafsson, S. Conesa-Boj, F. Glas,\nI. Zardo, M. Heigoldt, M. H. Gass, A. L. Bleloch, S. Estrade,\nM. Kaniber, J. Rossler, F. Peiro, J. R. Morante, G. Abstreiter,\nL. Samuelson, and A. Fontcuberta i Morral, Phys. Rev. B 80,\n245325 (2009).\n46Z. A. Devizorova and V. A. Volkov, JETP Lett. 98, 101 (2013).\n47E. L. Ivchenko, A. Y. Kaminski, and U. R ossler, Phys. Rev. B\n54, 5852 (1996).\n48J. Treu, M. Bormann, H. Schmeiduch, M. Dblinger, S. Morktter,\nS. Matich, P. Wiecha, K. Saller, B. Mayer, M. Bichler, M.-C.\nAmann, J. J. Finley, G. Abstreiter, and G. Koblmller, Nano\nLetters 13, 6070 (2013).\n49X. Liu, P. Liu, H. Huang, C. Chen, T. Jin, Y. Zhang, X. Huang,\nZ. Jin, X. Li, and Z. Tang, Nanotechnology 24, 245306 (2013).\n50S. Furthmeier, F. Dirnberger, M. Gmitra, A. Bayer, M. Forsch,\nJ. Hubmann, C. Sch uller, E. Reiger, J. Fabian, T. Korn, and\nD. Bougeard, Nat Commun. 7, 12413 (2016).\n51A. Sitek, M. Urbaneja Torres, K. Torfason, V. Gudmundsson,\nA. Bertoni, and A. Manolescu, Nano Letters 18, 2581 (2018).\n52B. M. Wong, F. Lonard, Q. Li, and G. T. Wang, Nano Letters\n11, 3074 (2011).\n53M. Royo, C. Segarra, A. Bertoni, G. Goldoni, and J. Planelles,\nPhys. Rev. B 91, 115440 (2015).\n54E. A. de Andrada e Silva, G. C. La Rocca, and F. Bassani, Phys.\nRev. B 55, 16293 (1997).\n55Lattice mismatch and strain \feld are additional sources of SOC,\nwhich are neglected for the present lattice matched heterostruc-\ntures. We also neglect a SOC contribution resulting from the\ninterface inversion asymmetry related to di\u000berent atomic termi-\nnation at opposite interfaces59,60which is small. Moreover, ad-\nditional calculations with slight smearing of the potential at the\ninterface instead of the step like function did not lead to signi\f-\ncantly di\u000berent results.\n56I. Vurgaftman, J. R. Meyer, and L. R. Ram-Mohan, J. Appl.\nPhys. 89, 5815 (2001).\n57For the model CSNW under consideration, the gate is attached\ndirectly to the NW, so that \u000bRis an upper bound value, while\nany dielectric interlayer would decrease \u000bR. For the NW with no\nshell, the gate has been placed at the same distance from the\ncore as in the case with the shell, in other to compare the two\ncalculations.\n58The sign of the SOC linear density on each facet is decided by the\nchoice of the reference frame. Since ^ \u000bxappears combined with\nthe spin matrices in the Hamiltonian (2), the overall sign of the\nSOC term is una\u000bected by a change in the reference frame.\n59E. L. Ivchenko, A. Y. Kaminski, and U. R ossler, Phys. Rev. B\n54, 5852 (1996).\n60U. R ossler and J. Kainz, Solid State Commun. 121, 313 (2002)."    },    {        "title": "0902.3244v1.Impurity_induced_spin_orbit_coupling_in_graphene.pdf",        "content": "arXiv:0902.3244v1  [cond-mat.mtrl-sci]  19 Feb 2009Impurity induced spin-orbit coupling in graphene\nA. H. Castro Neto1and F. Guinea2\n1Department of Physics, Boston University, 590 Commonwealt h Ave., Boston MA 02215, USA\n2Instituto de Ciencia de Materiales de Madrid, CSIC, Cantobl anco E28049 Madrid, Spain\nWe study the effect of impurities in inducing spin-orbit coup ling in graphene. We show that the\nsp3distortion induced by an impurity can lead to a large increas e in the spin-orbit coupling with a\nvalue comparable to the one found in diamond and other zinc-b lende semiconductors. The spin-flip\nscattering produced by the impurity leads to spin scatterin g lengths of the order found in recent\nexperiments. Our results indicate that the spin-orbit coup ling can be controlled via the impurity\ncoverage.\nPACS numbers: 81.05.Uw,71.70.Ej,71.55Ak,72.10.Fk\nSince the discovery of graphene in 2004 [1] much has\nbeen written about its extraordinary charge transport\nproperties [2, 3], such as sub-micron electron mean-free\npaths, that derive from the specificity of the carbon σ-\nbonds against atomic substitution by extrinsic atoms.\nHowever,beinganopensurface,itisrelativelyeasytohy-\nbridizethegraphene’s pzorbitalswith impuritieswithdi-\nrect consequences in its transport properties [4, 5]. This\ncapability for hybridization with external atoms, such as\nhydrogen (the so-called graphane), has been shown to be\ncontrollable and reversible [6] leading to new doors to\ncontrol graphene’s properties.\nMuch less has been said about the spin-related trans-\nport properties such as spin relaxation, although recent\nexperiments show that the spin diffusion length scales\n[7, 8] are much shorter than what one would expect from\nstandard spin-orbit (SO) scattering mechanisms in a sp2\nbonded system [9]. In fact, atomic SO coupling in flat\ngraphene is a very weak second order process since it af-\nfects the πorbitals only through virtual transitions into\nthe deep σbands [10]. Nevertheless, it would be very in-\nteresting if one could enhance SO interactions because of\nthe prediction of the quantum spin Hall effect in the hon-\neycomb lattice [11] and its relation to the field of topo-\nlogical insulators [12].\nIn this paper we argue that impurities (adatoms), such\nas hydrogen, can lead to a strong enhancement of the\nSO coupling due to the lattice distortions that they in-\nduce. In fact, it is well known that atoms that hybridize\ndirectly with a carbon atom induce a distortion of the\ngraphene lattice from sp2to sp3[13]. By doing that, the\nelectronicenergyis loweredand the path wayto chemical\nreactionis enhanced. Nevertheless, it hasbeen knownfor\nquite sometime [14] that in diamond, a purely sp3carbon\nbonded system, spin orbit coupling plays an important\nrole in the band structure since it is a first order effect, of\nthe order of the atomic SO interaction, ∆at\nso≈10 meV,\nin carbon [15]. Here we show that the impurity induced\nsp3distortion of the flat graphene lattice lead to a signif-\nicant enhancement of the SO coupling, explaining recent\nexperiments [7, 8] in terms of the Elliot-Yafet mechanism\nforspin relaxation[16, 17] due to presenceofunavoidableenvironmental impurities in the experiment. Moreover,\nour predictions can be checked in a controllable way in\ngraphane [6] by the control of the hydrogen coverage.\nWe assume that the carbon atom attached to an impu-\nrity is raised above the plane defined by its three carbon\nneighbors (see Fig. 1). The local orbital basis at the po-\nsition of the impurity (which is assumed to be located at\nthe origin, Ri=0= 0) can be written as:\n|πi=0/angbracketright=A|s/angbracketright+/radicalbig\n1−A2|pz/angbracketright,\n|σ1,i=0/angbracketright=/radicalbigg\n1−A2\n3|s/angbracketright−A√\n3|pz/angbracketright+/radicalbigg\n2\n3|px/angbracketright,\n|σ2,i=0/angbracketright=/radicalbigg\n1−A2\n3|s/angbracketright−A√\n3|pz/angbracketright−1√\n6|px/angbracketright+1√\n2|py/angbracketright,\n|σ3,i=0/angbracketright=/radicalbigg\n1−A2\n3|s/angbracketright−A√\n3|pz/angbracketright−1√\n6|px/angbracketright−1√\n2|py/angbracketright,\n(1)\nwhere|s/angbracketright, and|px,y,z/angbracketright, are the local atomic orbitals.\nNotice that this choice of orbitals interpolates between\nthe sp2configuration, A= 0, to the sp3configuration,\nA= 1/2. The angle θbetween the new σorbitalsand the\ndirection normal to the plane is cos( θ) =−A/√\nA2+2.\nThe energy of the state |πi/angbracketright,ǫπ, and the energy of the\nthree degenerate states |σa,i/angbracketright,ǫσ(a= 1,2,3), are given\nby (see Fig. 2):\nǫπ(A) =A2ǫs+(1−A2)ǫp, (2)\nǫσ(A) = (1−A2)ǫs/3+(2+ A2)ǫp/3,(3)\nwhereǫs≈ −19.38 eV (ǫp≈ −11.07 eV) is the energy\nof thes(p) orbital [18]. At the impurity site one has\nA≈1/2 while away from the impurity A= 0.\nThe Hamiltonian of the problem can be written as,\nH=Hπ+Hσ+δH, whereHπ(Hσ) describes the π-\nband (σ-band) of flat graphene, and δHdescribes the\nlocalchange in the hopping energies due to the presence2\n(b)(a)\n(a)\n (b)\nFIG. 1: (Color online). Top: Top view of the graphene lattice\nwith its orbitals. The orbitals associated with the impurit y\nand lattice distortion are shown in solid black. (a) sp3or-\nbital at impurity position; (b) sp2orbital of the flat graphene\nlattice.\nof the impurity and sp3distortion:\nδH=/summationdisplay\nα=↑,↓/braceleftBig\nǫIc†\nIαcIα+tC−Ic†\nIαcπα0\n+δǫπc†\nπα0cπα0+δǫσ/summationdisplay\na=1,2,3c†\nσaα0cσiα0\n+Vπσc†\nπα0(cσ1α0+cσ2α0+cσ3α0)+h.c./bracerightBig\n(4)\nwhere\nVπσ(A) =A/radicalbigg\n1−A2\n3(ǫs−ǫp), (5)\ncI,α(c†\nI,α) annihilates (creates) an electron at the impu-\nrity, and cπαi(cσaαi) annihilates an electron at a carbon\nsite in an orbital π(σa) at position Riwith spin α,ǫIis\nthe electron energy in the impurity, and tC−Ithe tunnel-\ning energy between the carbon and impurity, δǫπ(A) =\nǫπ(A)−ǫπ(A= 0), and δǫσ(A) =ǫσ(A)−ǫσ(A= 0).\nIn (4) we have not included the change in the hopping\nbetween σa,0orbitals(thechangeinenergyduetothedis-\ntortion is −A2(ǫs−ǫp)/3) and the inter-atomic hopping\nterms. In this way, we have simplified the calculations\nand the interpretation of the results. The inclusion of\nthe other terms do not modify our conclusions.\nThe atomic spin orbit coupling, Hat\nso= ∆at\nsoL·S, in-\nduces transitions between porbitals of different spin pro-\njection [10]. In flat graphene ( A= 0), it leads to tran-\nsitions between the πandσbands. The change in theground state energy in this case is rather small and given\nby: (∆at\nso)2/(ǫπ(A= 0)−ǫσ(A= 0))≈10−2meV [10].\nHowever, the perturbation described by (4) leads to a di-\nrect local hybridization Vπσbetween the πandσbands\nthat modifies the effective SO coupling acting on the π\nelectrons. The propagator of πelectrons from position\nRiwith spin αtoRjwith spin βcan be written as:\n/angbracketleftπi,α|(ǫ−H)−1|πj,β/angbracketright ≈ /angbracketleftπi,α|(ǫ−Hπ)−1|π0,α/angbracketright\n×/angbracketleftπ0,α|δH|¯σ0,α/angbracketright×/angbracketleft¯σ0,α|(ǫ−Hσ)−1|¯¯σk,α/angbracketright\n×/angbracketleft¯¯σk,α|Hat\nso|πk,β/angbracketright/angbracketleftπk,β|(ǫ−Hπ)−1|πj,β/angbracketright(6)\nwhere|¯σ0,α/angbracketright= [|σ10,α/angbracketright+|σ20,α/angbracketright+|σ30,α/angbracketright]/√\n3and¯¯σj,α/angbracketright=\n[|σ1j,α/angbracketright+eiφ|σ2j,α/angbracketright+e2iφ|σ3j,α/angbracketright]/√\n3 where φ= 2π/3.\nThe propagator in (6) can be understood as arising from\nan effective non-local SO coupling within the πband\nwhich goes as:\n∆I\nso(0,i)≈Vπσ/angbracketleft¯σ0,α|(ǫ−Hσ)−1|¯¯σi,α/angbracketright∆at\nso,(7)\nwhich allows us to estimate the local value of the SO\ncoupling as:\n∆I\nso(A)\n∆atso≈A/radicalbig\n3(1−A2). (8)\nAs shown in Fig. 2 the value of the SO coupling depends\non the angle (i.e., the value of A) associated with the\ndistortion of the carbon atom away from the graphene\nplane. Notice that for the sp2case (A= 0) this term\nvanishes indicating that SO only contributes in second\norder in ∆at\nso, while for the sp3case (A= 1/2), the SO\ncoupling is approximately 75% of the atomic value ( ≈7\nmeV). Also observe that the dependence on the distance\nfrom the location of the hydrogen atom is determined by\nthe Green’s function Gσ(0,Rj)) =/angbracketleft¯σ0|(ǫ− Hσ)−1|¯¯σj/angbracketright.\nThisfunction, calculatedforthesimplified modelofthe σ\nbands discussed in ref. [10], showsa significant dispersion\nin Fourier space, ranging from a maximum at the Γ point\nto zero at the KandK′points. Hence, the range of\nGσ(0,R)shouldbe ofthe orderofafew latticeconstants.\nBased on the previous results we can now calculate\nthe effect of the impurity induced SO coupling in the\ntransport properties. Firstly, we linearize the πband\naround the K and K’ points in the Brillouin zone and\nfind the 2D Dirac spectrum [3]: ǫ±,k=±vFkwherevF\n(≈106m/s) is the Fermi-Dirac velocity. In this long\nwavelength limit the impurity potential induced by (7)\nhascylindricalsymmetryandwecanuseadecomposition\nof the wavefunction in terms of radial harmonics [19, 20,\n21, 22, 23]. A similar analysis, for a system with SO\ninteraction in the bulk has been studied in ref. [9]. We\ndescribe the potential scattering by boundary conditions\nsuch as one of the components of the spinor vanishes at\na distance r=R1(of the order of the Bohr radius) of\nthe impurity [24]. A Rashba-like SO interaction exists in3\nthe region R1≤r≤R2(region I), and there is neither\npotential nor spin orbit interaction for r > R2, region II\n(R2if of the order of the carbon-carbon distance).\nThe wavefunctions in region I can be written as a su-\nperposition of angular harmonics:\nΨn(r,θ)≡A+/bracketleftbigg/parenleftbiggc+Jn(k+r)einθ\nic−Jn+1(k+r)ei(n+1)θ/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/arrowtp/angbracketrightBig\n+\n+/parenleftbigg\nic−Jn+1(k+r)ei(n+1)θ\n−c+Jn+2(k+r)ei(n+2)θ/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/arrowbt/angbracketrightBig/bracketrightbigg\n+\n+B+/bracketleftbigg/parenleftbiggc+Yn(k+r)einθ\nic−Yn+1(k+r)ei(n+1)θ/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/arrowtp/angbracketrightBig\n+\n+/parenleftbiggic−Yn+1(k+r)ei(n+1)θ\n−c+Yn+2(k+r)ei(n+2)θ/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/arrowbt/angbracketrightBig/bracketrightbigg\n+\n+A−/bracketleftbigg/parenleftbiggc′\n−Jn(k−r)eiθ\nic′\n+Jn+1(k−r)ei(n+1)θ/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/arrowtp/angbracketrightBig\n−\n−/parenleftbiggic′\n+Jn+1(k−r)ei(n+1)θ\n−c′\n−Jn+2(k−r)ei(n+2)θ/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/arrowbt/angbracketrightBig/bracketrightbigg\n+\n+B−/bracketleftbigg/parenleftbiggc′\n−Yn(k−r)einθ\nic′\n+Yn+1(k−r)ei(n+1)θ/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/arrowtp/angbracketrightBig\n−\n−/parenleftbigg\nic′\n+Yn+1(k−r)ei(n+1)θ\n−c′\n−Yn+2(k−r)ei(n+2)θ/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/arrowbt/angbracketrightBig/bracketrightbigg\n(9)\nwhere| ↑/angbracketrightand| ↓/angbracketrightare the spin states. The functionsJn(x),Yn(x) are Bessel functions of order n, and:\nǫ=±∆I\nso/2+/radicalBig\nv2\nFk2\n±+(∆Iso/2)2 (10)\nc±=/radicalbigg\n1/2±∆Iso/(4/radicalBig\nv2\nFk2\n++(∆Iso/2)2) (11)\nc′\n±=/radicalbigg\n1/2±∆Iso/(4/radicalBig\nv2\nFk2\n−+(∆Iso/2)2) (12)\nǫis the energy of the scattered electron ( k±is defined\nthrough (10)).\nThe wavefunctions outside the region affected by the\nimpurity, r > R2, can be written as:\nΨn(r,θ)≡/parenleftbiggJn(kr)einθ\niJn+1(kr)ei(n+1)θ/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/arrowtp/angbracketrightBig\n+\n+C↑/parenleftbiggYn(kr)einθ\niYn+1(kr)ei(n+1)θ/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/arrowtp/angbracketrightBig\n+\n+C↓/parenleftbiggYn+1(kr)ei(n+1)θ\niYn+2(kr)ei(n+2)θ/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/arrowbt/angbracketrightBig\n(13)\nand:ǫ=vFk. The boundary conditions at r=R1and\nr=R2lead to the equations:\nc+A+Jn(k+R1)+c+B+Yn(k+R1)+c′\n−A−Jn(k−R1)+c′\n−B−Yn(k−R1) = 0\nc−A+Jn+1(k+R1)+c−B+Yn+1(k+R1)+c′\n+A−Jn+1(k−R1)+c′\n+B−Yn+1(k−R1) = 0\nc+A+Jn(k+R2)+c+B+Yn(k+R2)+c′\n−A−Jn(k−R2)+c′\n−B−Yn(k−R2) =Jn(kR2)+C↑Yn(kR2)\nc−A+Jn+1(k+R2)+c−B+Yn+1(k+R2)+c′\n+A−Jn+1(k−R2)+c′\n+B−Yn+1(k−R2) =Jn+1(kR2)+C↑Yn+1(kR2)\nc−A+Jn+1(k+R2)+c−B+Yn+1(k+R2)−c′\n+A−Jn+1(k−R2)−c′\n+B−Yn+1(k−R2) =C↓Yn+1(kR2)\nc+A+Jn+2(k+R2)+c+B+Yn+2(k+R2)−c′\n−A−Jn+2(k−R2)−c′\n−B−Yn+2(k−R2) =C↓Yn+2(kR2) (14)\nThese six equations allow us to obtain the coefficients\nA±,B±,C↑andC↓. In the absence of the SO inter-\naction, we have A+=A−,B+=B−,C↓= 0 and\nC↑=−Jn(kR1)/Yn(kR1).\nWe show in Fig. 3 the results for the cross section for\nspin flip processes, determined by |C↓|2/kF. The main\ncontribution arises from the n= 0 channel. For compari-\nson, the elastic cross section, calculated in the same way,\nisσel≈k−1\nF. This is about three of magnitude larger\nthan the spin-flip cross section due to the spin orbit cou-\npling. Hence, the spin relaxation length is 103times the\nelastic mean free path [9]. We obtain a mean free path\nof about 1 µm, in reasonable agreement with the exper-\nimental results in ref. [7]. This value depends quadrat-\nically on ∆I\nso(A). For a finite, but small, concentrationof impurities, our results scale with the impurity concen-\ntration and hence the spin flip processes should increase\nroughly linearly with impurity coverage in transport ex-\nperiments in systems like graphane [6].\nInsummary, wehaveshownthat the impurityinduced,\nlattice driven, SO coupling in graphene can be of the\norder of the atomic spin orbit coupling and compara-\nble to what is found in diamond and zinc-blend semi-\nconductors. The value of the SO coupling depends on\nhow much the carbon atom which is hybridized with\nthe impurity displaces from the plane inducing a sp3hy-\nbridization. Wehavecalculatedthespin-flipcrosssection\ndue to SO coupling for the impurity and shown that it\nagrees with recent experiments. This results indicates\nthat there are substantial amounts of hybridized impu-4\n0.0 0.1 0.2 0.3 0.4 0.5/MinuΣ13.5/MinuΣ13.0/MinuΣ12.5/MinuΣ12.0/MinuΣ11.5/MinuΣ11.0\nAΕΠ,ΕΣ/LParen1eV/RParen1\n0.0 0.1 0.2 0.3 0.4 0.50.00.10.20.30.40.50.60.7\nA/CaΠDΕltaSOI/Slash1/CaΠDΕltaSOat\nFIG. 2: (Color online).Top: Energy (in eV) of the π(blue)\nandσ(red)bandsas afunctionof Aaccordingto(3); Bottom:\nRelative value of the SO coupling at the impurity site relati ve\nto the atomic value in carbon as a function of Aaccording to\n(8).\n4/MultiΠly10128/MultiΠly1012Ρ/LParen1cm/MinuΣ2/RParen12/MultiΠly10/MinuΣ34/MultiΠly10/MinuΣ3Σso/LParen1nm/RParen1\nFIG. 3: (Color online). Cross section for a spin flip process\nfor a defect as described in the text. The parameters used are\nR1= 1˚AR2= 2˚A and ∆I\nso= 1meV (blue) and ∆I\nso= 2meV\n(red) .\nrities in graphene, even under ultra-clean high vacuumconditions. Experiments where the impurity coverage is\nwell controlled can provide a “smoking-gun” test of our\npredictions.\nWe thank illuminating discussions with D. Huertas-\nHernando and A. Brataas. AHCN acknowledges the par-\ntial support of the U.S. Department of Energy under\ngrant DE-FG02-08ER46512. FG acknowledges support\nfrom MEC (Spain) through grant FIS2005-05478-C02-\n01 and CONSOLIDER CSD2007-00010, by the Comu-\nnidad de Madrid, through CITECNOMIK, CM2006-S-\n0505-ESP-0337.\n[1] K. S. Novoselov et al., Science 306, 666 (2004).\n[2] A. K. Geim and K. S. Novoselov, Nature Materials 6, 183\n(2007).\n[3] A. H. Castro Neto et al., Rev. Mod. Phys. 81, 109 (2009).\n[4] J. H. Chen et al., Nat. Phys. 4, 377 (2008).\n[5] P. Blake et al.(2008), arXiv:0810.4706.\n[6] D. C. Elias et al., Science 323, 610 (2009).\n[7] N. Tombros et al., Nature 448, 571 (2007).\n[8] N. Tombros et al., Phys. Rev. Lett. 101, 046601 (2008).\n[9] D. Huertas-Hernando, F. Guinea, and A. Brataas (2008),\narXiv:0812.1921.\n[10] D. Huertas-Hernando, F. Guinea, and A. Brataas, Phys.\nRev. B74, 155426 (2006).\n[11] C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 226801\n(2005).\n[12] C. Kane and E. Mele, Science 314, 1692 (2006).\n[13] E. J. Duplock, M. Scheffler, and P. J. Lindan, Phys. Rev.\nLett.92, 225502 (2004).\n[14] P. Y. Yu and M. Cardona, Fundamentals of Semiconduc-\ntors: Physics and Materials Properties (Springer, New\nYork, 2005).\n[15] J. Serrano, M. Cardona, and T. Ruf, Solid St. Commun.\n113, 411 (2000).\n[16] P. G. Elliot, Phys. Rev. 96, 266 (1954).\n[17] Y. Yafet, in Solid State Physics, vol 13 , edited by ed. by\nF. Seitz and D. Turnbull (Academic, New York, 1963).\n[18] W. A. Harrison, Solid State Theory (Dover, New York,\n1980).\n[19] P. M. Ostrovsky, I. V. Gornyi, and A. D. Mirlin, Phys.\nRev. B74, 235443 (2006).\n[20] M. Hentschel and F. Guinea, Phys. Rev. B 76, 115407\n(2007).\n[21] D. S. Novikov, Phys. Rev. B 76, 245435 (2007).\n[22] M. I. Katsnelson and K. S. Novoselov, Solid State Com-\nmun.143, 3 (2007).\n[23] F. Guinea, Journ. Low Temp. Phys. 153, 359 (2008).\n[24] V. M. Pereira et al., Phys. Rev. Lett. 96, 036801 (2006)."    },    {        "title": "2303.11687v1.Intrinsic_Magnon_Orbital_Hall_Effect_in_Honeycomb_Antiferromagnets.pdf",        "content": "Intrinsic Magnon Orbital Hall Effect in Honeycomb Antiferromagnets\nGyungchoon Go,1Daehyeon An,1Hyun-Woo Lee,2and Se Kwon Kim1\n1Department of Physics, Korea Advanced Institute of Science and Technology, Daejeon 34141, Korea\n2Department of Physics, Pohang University of Science and Technology, Pohang 37673, Korea\nWetheoreticallyinvestigatethetransportofmagnonorbitalsinahoneycombantiferromagnet. Wefindthatthe\nmagnon orbital Berry curvature is finite even without spin-orbit coupling and thus the resultant magnon orbital\nHall effect is an intrinsic property of the honeycomb antiferromagnet rooted only in the exchange interaction\nand the lattice structure. Due to the intrinsic nature of the magnon orbital Hall effect, the magnon orbital\nNernst conductivity is estimated to be orders of magnitude larger than the predicted values of the magnon spin\nNernst conductivity that requires finite spin-orbit coupling. For the experimental detection of the predicted\nmagnonorbitalHalleffect,weinvokethemagnetoelectriceffectthatcouplesthemagnonorbitalandtheelectric\npolarization,whichallowsustodetectthemagnonorbitalaccumulationthroughthelocalvoltagemeasurement.\nOur results pave a way for a deeper understanding of the topological transport of the magnon orbitals and also\nits utilization for low-power magnon-based orbitronics, namely magnon orbitronics.\nIntroduction. —Thecollectivelow-energyexcitationsofthe\nordered materials are of great interest in condensed matter\nphysics. One of the representative examples is a quantum of\nspin waves, called a magnon which is a charge-neutral bo-\nson in magnetic materials. Magnons have been intensively\nstudied for technological applications since they can real-\nize Joule-heating-free information transport and processing\nas well as wave-based computing [1]. In addition, for fun-\ndamental interest, various topological properties of magnon\nbands have been investigated in the context of the magnon\nHall effect [2–7] and the spin Nernst effect [8, 9]. According\nto the existing theories, the finite Hall response of magnons\ncanoccurasmanifiestationsofspin-orbitcouplingthroughthe\nDzyaloshinskii-Moriyainteraction(DMI)[5–9]orthroughthe\nmagnon-phononcoupling[10–13]. TheHallresponsecanalso\noccurinspintexturesystems[2,3]withthescalarspinchiral-\nity, which acts as an effective spin-orbit coupling.\nIn electronics systems, on the other hand, there have been\nstudies showing that electrons can exhibit a Hall effect with-\noutspin-orbit coupling, owing to their orbital degree of free-\ndom [14–17]. This discovery evoked a surge of interest in\nelectron-orbital transport phenomena such as the orbital Hall\neffect[18–22]andtheorbitaltorque[23–25]. Moreover,there\nhave been theoretical suggestions that orbital-dependent elec-\ntron transport critically affects electron spin dynamics when\nspin-orbit coupling is present. For instance, it has been sug-\ngested [16–18] that the orbital Hall effect may play a crucial\nrole in the spin Hall effect.\nMotivated by the aforementioned advancement of our un-\nderstandingofelectronorbitals,theorbitalmotionofmagnons\nhasstartedgatheringattentionrecentlyinmagnetismandspin-\ntronics. For example, the circulating magnonic modes have\nbeen investigated in confined geometries such as whisper-\ning gallery mode cavities [26–28], magnetic nanocylinders\nand nanotubes [29–32]. Also, the orbital magnetization of\nmagnonshasbeenpointedoutastheoriginofweakferromag-\nnetisminanoncollinearkagomeantiferromagnet(AFM)with\nthe DMI [33]. Recently, the orbital-angular-momentum tex-\nturesofthemagnonbandshavebeenrealizedincollinearmag-\nnetswithnontrivialnetworksofexchangeinteraction[34,35].Furthermore, inspired by achievements in topological meta-\nmaterials [36–38], topological magnonic modes carrying the\nmagnon current circulation have been demonstrated in hon-\neycomb magnets with exchange-interaction modulation [39].\nDespitethestronginterestinmagnonorbitals,studiesontheir\ntransportpropertiesareverylimited[33]. Inparticular,itisan\nopen question whether the magnon orbital degree of freedom\ncan induce a Hall phenomenon withoutspin-orbit coupling,\njust as its electron counterpart can.\nIn this Letter, we answer this question by investigating the\ntransport of magnon orbitals. For the model system of 2D\nhoneycomb AFMs, we demonstrate that the magnon orbitals\ncan exhibit a Hall effect, namely the magnon orbital Hall ef-\nfect,withoutspin-orbitcoupling. Themagnonorbitalmoment\nrepresentsamagnoncurrentcirculationasshowninFig.1(a).\nWe find that application of a longitudinal temperature gradi-\nent drives thermal magnons to opposite transverse directions\ndepending on their orbital characters [see Fig. 1(b)], giving\nrisetoamagnonHallphenomenon,themagnonorbitalNernst\neffect. TheestimatedmagnitudeofthemagnonorbitalNernst\nconductivityisorders-of-magnitudelargerthantheknownval-\nues of the magnon spin Nernst conductivity induced by the\nDzyaloshinskii-Moriya interaction [8], revealing the hitherto\nunrecognizedroleofthemagnonorbitalinmagnontransport.\nToproposeanexperimentalmethodfordetectingtheaccumu-\nlation of the magnon orbital at the sample edges induced by\nthemagnonorbitalHalleffect,weinvokethemagnetoelectric\neffect by which a magnonic spin current induces an electric\ndipole moment. Since the magnon orbital moment can be\nregarded as a magnonic spin-current circulation, the afore-\nmentioned magnetoelectric effect dictates that the magnon\norbital moment should engender a polarization charge in a\ntwo-dimensional space [4, 40, 41] [see Fig. 1(c)] and thus\nthe magnon orbital accumulation should be accompanied by\nthe accumulation of the polarization charge. We estimate the\nelectricvoltageprofileinducedbythemagnonorbitalmoment\naccumulation,whichisshowntobewithincurrentexperimen-\ntal reach. Owing to strenuous efforts to realize magnetism in\nvarious 2D magnetic crystals, our proposal can be tested in\na number of transition metal compounds that are known toarXiv:2303.11687v1  [cond-mat.mes-hall]  21 Mar 20232\n𝐿𝐿<0 𝐿𝐿>0\nright -circular left-circular−𝛻𝛻𝑇𝑇right -circular\nmagnon\nleft-circular\nmagnon𝐿𝐿>0\n𝑆𝑆=+ℏelectric \npolarization (P )(a) (b) (c)\nmagnon+\n+\n+\n++\n+−\n−\n−−−\n−\n𝑞𝑞pol=−∇�𝐏𝐏\npolarization \ncharge (𝑞𝑞pol)\n𝑆𝑆=−ℏ\n𝑆𝑆𝐴𝐴 𝑆𝑆𝐵𝐵\nright -handed (𝛽𝛽)\n𝑆𝑆=+ℏ\nleft-handed (𝛼𝛼 )𝑆𝑆𝐴𝐴 𝑆𝑆𝐵𝐵\nFIG. 1. (a) Schematics of the magnon spin ( S) and magnon orbital ( L) in the honeycomb AFM. The magnon spin and the magnon orbital are\ndeterminedby,respectively,theprecessionsofconstituentspinswithinthesitesandtheintersitehopping. (b)Schematicsofthemagnonorbital\nNernsteffect,whereatemperaturegradient rTinducesanetmagnon-orbitalcurrentinatransversedirectionconsistingofoppositely-moving\nright-circularmagnonsandleft-circularmagnons. (c)Schematicsofthepolarization Pinducedbythecirculatingmagnonicspincurrentinthe\ncase of orbital L> 0and spinS= +~(left) and the corresponding polarization charge qpol=\u0000r\u0001P(right). The sign of the polarization\ncharge is determined by the product of the sign of the magnon spin Sand the direction of the spin current circulation, i.e., the sign of L.\nhost honeycomb AFMs [ e:g:;MPX3(M= Fe, Ni, Mn; X=S,\nSe)] [42–45].\nModel construction. —Here we consider a 2D AFM on a\nhoneycomb lattice\nH=JX\nhi;jiSi\u0001Sj\u0000KX\ni(Si;z)2+g\u0016BBX\niSi;z;(1)\nwhereJ(>0)istheantiferromagneticexchangecouplingand\nK(>0)is the easy-axis anisotropy, gis the g-factor, \u0016Bis\ntheBohrmagneton,and Bistheappliedmagneticfield. Note\nthat our model does not include the DMI which comes from\nspin-orbitcouplingandthusdoesnotexhibitthemagnonspin\nNernsteffect[6–9]. InthisLetter,wefocusonthecasewhere\na ground state is the collinear Néel state along the z-axis.\nPerforming the Holstein-Primakoff transformation and taking\nthe Fourier transformation, we have\nH=1\n2X\nk y\nkHk k;  k= (ak;bk;ay\n\u0000k;by\n\u0000k)T;(2)\nwith the momentum-space Hamiltonian\nHk=JS0\nBB@3 +\u0014+ 0 0 fk\n0 3 +\u0014\u0000f\u0003\nk 0\n0fk 3 +\u0014+ 0\nf\u0003\nk 0 0 3 + \u0014\u00001\nCCA;(3)\nwhere\u0014\u0006= (2K\u0006g\u0016BB)=Jandfk=P\njeik\u0001ajwith\na1=a\n2(p\n3;\u00001),a2=a(0;1), and a3=\u0000a\n2(p\n3;1).\nThe magnon excitations can be described by the general-\nized Bogoliubov-de Gennes equation in the particle-hole\nspace representation [11, 46]. In this representation, the\npseudo-energy-eigenvalue satisfies \u0016\u000fn;k= (\u001b3\u000fk)nn, where\n\u001b3= diag(1;1;\u00001;\u00001)is the Pauli matrix acting on the\nparticle-hole space and \u000f(k)are the magnon bands given by\n\u000f\u000b=\f\nk=\u000f0\nk\u0006g\u0016BBS; (4)where\u000f0\nk=JSp\n(3 +\u0014)2\u0000jfkj2and\u0014= 2K=J(see Sup-\nplemental Material for calculation details). Here, the indices\n\u000band\fstand for two magnonic bands with opposite spin an-\ngularmomenta[seeFig1(a)]. Thetopologicalpropertyofthe\nmagnonic statejniis characterized by the Berry curvature:\n\nn(k) =@An\ny(k)\n@kx\u0000@An\nx(k)\n@ky; (5)\nwhere An(k) =hnj\u001b3i@kjni=hnj\u001b3jniis the Berry connec-\ntion. Figures 2(a-d) show the magnon band structures \u000fn\nkand\nthe corresponding Berry curvatures \nn(k)withn=\u000b;\f.\nIn the honeycomb AFM, the broken inversion symmetry al-\nlows a non-zero Berry curvature without the DMI. Also, be-\ncauseofthecombinedsymmetryofthetime-reversal( T)and\na180\u000espin rotation around the x-axis (Cx) of the Hamil-\ntonian, the energy spectra are even in momentum space\n(\u000fk=\u000f\u0000k), whereas the magnon Berry curvatures are odd\n[\nn(k) =\u0000\nn(\u0000k)][8]. Therefore, the momentum-space\nintegration of the magnon Berry curvature weighed by the\nBose-Einsteindistributioniszeroforeachband,indicatingthe\nabsence of the Hall transport of magnons and their spins.\nMagnon orbital Hall effect. —Although the momentum-\nspaceintegrationofthemagnonBerrycurvaturesvanishes,the\nnonvanishingand k-oddstructureoftheBerrycurvatureopens\nup a possibility for topological transport of certain quantities.\nIf there is a momentum-dependent quantity whose profile is\nalso odd in k, its Hall effect can be present. We show below\nthat this is indeed the case for the magnon orbital moment\nsince it holds the same symmetry property in the momentum\nspace as the Berry curvature [34], i.e., k-odd in the presence\nof theTCxsymmetry with broken inversion symmetry.\nFrom the orbital moment operator ^L=1\n4(r\u0002v\u0000v\u0002r),\nwereadthematrixelementofthemagnonorbitalmoment[21,3\n0 𝜋𝜋 −𝜋𝜋−𝜋𝜋0𝜋𝜋\n0 𝜋𝜋 −𝜋𝜋−𝜋𝜋0𝜋𝜋\n−0.10.1\nΩ𝛼𝛼(𝐤𝐤) Ω𝛽𝛽(𝐤𝐤)(c) (d)\n0 𝜋𝜋 −𝜋𝜋−𝜋𝜋0𝜋𝜋\n0 𝜋𝜋 −𝜋𝜋−𝜋𝜋0𝜋𝜋0\n−0.03 Ω𝛼𝛼𝐿𝐿(𝐤𝐤) Ω𝛽𝛽𝐿𝐿(𝐤𝐤)(g) (h)\n𝐸𝐸(meV)\n012\n0 𝜋𝜋 −𝜋𝜋−𝜋𝜋0𝜋𝜋\n𝜖𝜖𝐤𝐤𝛽𝛽\n0 𝜋𝜋 −𝜋𝜋−𝜋𝜋0𝜋𝜋\n𝜖𝜖𝐤𝐤𝛼𝛼\n0 𝜋𝜋 −𝜋𝜋−𝜋𝜋0𝜋𝜋\n0 𝜋𝜋 −𝜋𝜋−𝜋𝜋0𝜋𝜋\n−0.40.4𝐿𝐿(ℏ/𝑚𝑚eff)\n𝐿𝐿𝛼𝛼(𝐤𝐤) 𝐿𝐿𝛽𝛽(𝐤𝐤)(a) (b)\n(e) (f)\nFIG.2. (a,b)Magnonbandstructures[Eq.(4)],(c,d)theBerrycurvatures[Eq.(5)],(e,f)theequilibriumorbitalmomentstructures[Eq.(6)],\nand (g, h) the orbital Berry curvatures [Eq. (7)] of two magnonic states denoted by \u000band\f. For material parameters, we take J= 1:54meV\nandKS= 0:0086meV, andg\u0016BB=J = 0:25. For (e) and (f), me\u000bis the magnon effective mass at the Dirac points.\n22]\nhnj^Ljmi=hnj\u0012r\u0002v\u0000v\u0002r\n4\u0013\njmi\n=1\n2~Imh@knj\u0002H kj@kmi\n\u00001\n4~(\u0016\u000fn;k+ \u0016\u000fm;k)Imh@knj\u001b3j@kmi:(6)\nBy taking the diagonal element of (6), one recovers the well-\nknown formula of the intrinsic orbital moment [47–49]. The\nmagnonorbitalmomentprofiles Ln(k) =hnj^Lzjniofthetwo\nmagnonsareshowninFig.2(e)and(f)[50]. Inagreementwith\nthe momentum-space texture of the magnon orbital angular\nmomentum in Ref. [34, 35], the evaluated magnon orbital\nmoment has the C3rotation symmetry with k-odd structure.\nBecauseLn\nz(k)=\u0000Ln\nz(\u0000k)and\u000fk=\u000f\u0000k, the total magnon\norbital moment is zero in equilibrium. However, our system\ncanexhibittheintrinsicmagnonorbitalHalltransportbecause\nbothLn\nz(k)and\nn\nz(k)are odd in k, and thus their product\nLn\nz(k)\nn\nz(k)isk-even. Analogous to the generalized Berry\ncurvature[46,51],wewritetheorbitalBerrycurvaturewhich\ncharacterizes the magnon orbital Hall transport as follows:\n\nL\nn(k) =X\nm6=n(\u001b3)mm(\u001b3)nn2~2Im\u0002\nhnjjL\nz;yjmihmjvxjni\u0003\n(\u0016\u000fn;k\u0000\u0016\u000fm;k)2;\n(7)\nwherejL\nz;y= (vy\u001b3^Lz+^Lz\u001b3vy)=4is the magnon orbital\ncurrent operator, and vi=1\n~@Hk\n@kiis the velocity operator.\nNote the summation is performed in the particle-hole space.\nTheprofilesoftheorbitalBerrycurvaturesofthetwomagnon\nmodesareshowninFig.2(g)and(h). Asexpected,theprofiles\nareevenin kandthustheirmomentum-spaceintegrationsare\nfinite,indicatingtheexistenceoftheHalleffectofthemagnonorbitals. We emphasize that our model has no spin-orbit cou-\npling term such as the DMI. Also, the orbital Berry curvature\nremainsfinitewhenboth BandKapproachzeroaslongasthe\nantiferromagnetic ground state is maintained. Therefore, the\nmagnon orbital Berry curvature and the resultant Hall effect\nareintrinsicpropertiesofthehoneycombAFMthatoriginated\nsolely from the exchange interaction and the lattice geometry.\nThe magnon orbital Berry curvature leads to the transverse\nmagnonorbitalcurrentinresponsetoanexternalperturbation,\nnamely the magnon orbital Hall effect. The linear response\nequation of the transverse magnon orbital current driven by\na temperature gradient is given by (JL\nz)y=\u0000\u000bL\nz@xT[46],\nwhere\u000bL\nz=\u000bL\nz;\u000b+\u000bL\nz;\fis the magnon orbital Nernst con-\nductivitywith \u000bL\nz;n=2kB\n~VP\nkc1(\u001an)\nL\nn(k),wherekBisthe\nBoltzmann constant, c1(\u001a) = (1 +\u001a)ln(1 +\u001a)\u0000\u001aln\u001a, and\n\u001an= (e\u000fn=kBT\u00001)\u00001is the Bose-Einstein distribution. In\nFig. 3(a), we show the orbital Nernst conductivity for differ-\nent temperatures by using the material parameters of MnPS 3:\nJ= 1:54meV andKS= 0:0086meV [52]. To compare\nthe orbital Nernst conductivity with the spin Nernst conduc-\ntivity,wetaketheconstanteffectivemassapproximationatthe\nDirac points where the Berry curvatures are maximized. The\nmagnon orbital Nernst conductivity is estimated to be about\n10\u00002kB. This value is 103times larger than the estimated\nvalues of the magnon spin Nernst conductivity of honeycomb\nAFMsinthepresenceoftheDMI[8]. Thisisourmainresult:\nThe honeycomb AFM exhibits the magnon orbital Hall ef-\nfect without spin-orbit coupling and, therefore, its magnitude\nis orders-of-magnitude larger than the magnon spin Nernst\nconductivity that requires spin-orbit coupling. Note that the\norbital Nernst conductivity is almost independent of the mag-\nnetic field in Fig. 3(a), because the magnon eigenstates are\nunaffected by the magnetic field.\nMagnon-orbital-induced polarization. —The magnon or-4\nbital Hall effect induces the accumulation of the magnon or-\nbital at the edges of a system. To propose an experimental\nmethod to detect the magnon orbital accumulation, here we\ndevelop a phenomenological model for the transverse voltage\nprofileinducedbythelongitudinaltemperaturegradientviathe\nmagnonorbitalNernsteffect. Weemphasizethatthefollowing\ntheoryisqualitativeinnatureandthusintendedtoprovidethe\norder-of-magnitude estimation, not quantitative predictions.\nTobeginwith, letusreviewtherelation betweenthespincur-\nrentandthepolarization. Thespincurrentfromanoncollinear\nspin configuration is known to induce an electric polarization\nin magnetic materials by the combined action of the atomic\nspin-orbit interaction and the orbital hybridization [40, 41]:\nP=\u0000ea\nESOe12\u0002Is; (8)\nwhereeis the magnitude of the electron charge, ais the dis-\ntance between the two sites, e12is the unit vector connecting\ntwo sites, Is=J(S1\u0002S2)is the spin current from site 1\nto site 2, and the energy scale ESOis inversely proportional\nto the spin-orbit coupling strength. In the ground state of a\ncollinearmagnet,thereisnospincurrent (Is= 0)andthusno\nelectric polarization ( P= 0). However, the magnon consists\nof spatially-varying noncollinear deviations from the ground\nstate. Therefore,amagnoncurrentinacollinearmagnetgives\nrise to a finite spin current Is[53, 54]. For the magnonic\nspin current, we can invoke Eq. (8) to compute the induced\npolarization since the characteristic time scale of the magnon\nisgenerallymuchlongerthanthatoftheelectronhoppingpro-\ncess. The spin current carried by a single magnon is given by\nIs=\u0000S(v=a)^z, which leads to the electric polarization\nP=eS\nESO(v\u0002^z); (9)\nwhereS=\u0006~is the magnon spin and v=ve12is the\nmagnon velocity. By considering the typical energy scale of\nESO, it has been predicted in Refs. [40, 41, 55] that a mea-\nsurable electric polarization can be induced by a magnonic\nspincurrentinmagneticmaterials[56]. Thismagnetoelectric\neffect allows us to relate the magnon orbital motion (i.e., the\ncirculatingmagnonicspincurrent)andthepolarizationcharge\ndensity. For the qualitative understanding, we schematically\ndepicttheelectricpolarizationproducedbythemagnonicspin-\ncurrentcirculationaroundahexagoninahoneycomblatticein\nFig. 1(c). The magnonic spin-current circulation induces the\nelectricpolarizationpointingoutwardorinward(andtherefore\nthe positive or negative polarization charge density), depend-\ning on the product of the magnon-spin sign and the magnon-\norbital-moment sign.\nNowletusconsiderthesituationwherethenonequilibrium\naccumulationofthemagnonorbitalmomentisgeneratedatthe\nedgesofthesamplebyatemperaturegradientviathemagnon\norbital Nernst effect. In the absence of an external field along\nthez-direction, there would be a finite accumulation of the\nmagnon orbital moment at the edges, but there would be no\n0 0.1 0.20123\n(a) (b)\n0 0.1 0.20123\n10 K30 K50 K70 K\n10 K30 K50 K70 KFIG. 3. (a) Magnetic-field dependence of the magnon orbital Nernst\nconductivity \u000bL\nzand(b)thespin-polarizedcomponentofthemagnon\norbital Nernst conductivity \u000bL\nz;s=\u000bL\nz;\u000b\u0000\u000bL\nz;\f. See the main text\nfor the details.\ninduced electric polarization, for spin-up magnons and spin-\ndown magnons are equally populated and thus their contribu-\ntions to the electric polarization cancel each other. However,\nwhen we apply an external magnetic field, spin-up magnons\nandspin-downmagnonsarepopulatedunequallyandthusthe\nnetspindensityofmagnonsbecomesfinite. Consequently,the\nmagnonorbitalaccumulationandalsothemagnonorbitalHall\ncurrentarespin-polarizedinthepresenceoftheexternalfield.\nInparticular,thespin-polarizedcomponentofthemagnonor-\nbital Nernst conductivity \u000bL\nz;s(=\u000bL\nz;\u000b\u0000\u000bL\nz;\f) is zero when\nthe magnetic field is zero and becomes finite as the magnetic\nfield is applied as shown in Fig. 3(b).\nInstead of the magnon orbital accumulation, what is di-\nrectly related to the observable electric polarization is the\nspin-polarized magnon orbital accumulation \u001aL\ns=\u001aL\n\u000b\u0000\u001aL\n\f.\nTo estimate the spin-polarized magnon orbital accumulation\nthatisinducedbythemagnonorbitalNernsteffect,weusethe\nphenomenological drift-diffusion formalism by following the\npreviousstudiesonelectronorbitaltransport[20,57,58]. For\nparameters, we use g\u0016BB=J = 0:25,ESO= 3eV,T= 70\nK and@xT= 1K/\u0016m with the constant magnon relaxation\ntime\u001c= 30ns and the magnon diffusion length \u0015= 20\nnm. The considered system size is 1 \u0016m\u00021\u0016m. Figure 4(a)\nshows the resultant spin-polarized magnon orbital accumula-\ntion along the y-direction. We also numerically compute the\nelectric voltage profile induced by the spin-polarized magnon\norbitalaccumulationbasedonasimplifiedmodelfortheinho-\nmogeneous magnon orbital accumulation (see Supplemental\nMaterialfordetailedcalculation),whichisshowninFig.4(b).\nNote that the estimated electric voltage at the edges is about\n0:1\u0016Vwhichiswithinthecurrentexperimentalcapacity. The\naccumulation of the magnon orbital moment gives rise to the\nelectricvoltageprofileviathemagnetoelectriceffect,bywhich\nwe can probe the proposed magnon orbital Nernst effect.\nDiscussion. —In this Letter, we have investigated the trans-\nport of magnon orbital moments in a honeycomb AFM. The\nk-oddstructuresofboththeBerrycurvatureandthemagnon-\norbital texture lead to the k-even structure of the magnon\norbitalBerrycurvature,whichgivesrisetothemagnonorbital\nNernst effect after momentum-space integration. We empha-\nsize that the magnon orbital Berry curvature does not require\nspin-orbit coupling and thus is an intrinsic property of the5\n(a) (b)\n0 0.05 0.10.9 0.95 1−0.10.1\n0V (𝜇𝜇V)\ny(𝜇𝜇m)0 0.05 0.1\ny(𝜇𝜇m)0.9 0.95 1𝜌𝜌𝑠𝑠𝐿𝐿(10−4ℏ/nm2)\n-6-3036\nFIG.4. (a)Theprofileofthespin-polarizednonequilibriummagnon\norbital accumulation \u001aL\nsin theydirection and (b) the corresponding\nvoltageVinduced by the nonequilibrium polarization. The consid-\neredsystemsizeis1 \u0016m\u00021\u0016mandatemperaturegradientisapplied\nin thexdirection. See the main text for the details.\nhoneycomb AFM originating solely from the exchange inter-\nactionandthelatticegeometry. Asaresult,themagnonorbital\nNernst effect is predicted to be orders-of-magnitudes stronger\nthan the magnon spin Nernst effect that requires spin-orbit\ncoupling. AlthoughherewefocusonthemagnonorbitalHall\neffectinthehoneycombAFM,wenotethatthemagnonorbital\nHall effect is generally expected to be present in systems with\nbroken inversion symmetry such as honeycomb and Kagome\nferromagnets with DMI. We also remark that, although we\nhave considered a temperature gradient as a means to drive a\nmagnontransport,onecanalsouseelectronicmeanstopump\nmagnons by using, e.g., the spin Hall effects [59, 60].\nFor an experimental scheme, we have shown that the\nmagnonorbitalaccumulationcanbedetectedthroughtheelec-\ntric voltage profile by invoking the magnetoelectric effect. In\nparticular, our theory for the magnon orbital Hall effect pre-\ndicts that, upon the application of a longitudinal temperature\ngradient, the electric voltage profile is developed in the trans-\nverse direction. This phenomenon has the same symmetry as\nthe Nernst effect in metallic layers. A remarkable feature of\nour results is that the electric voltage is not induced by the\nconduction electrons but by the circulating spin current asso-\nciatedwiththeorbitalmotionofmagnons. Weherenotethat,\nin addition to the electric polarization, there are several other\ndegrees of freedom that are expected to couple with magnon\norbital motions such as photons, phonons, and spin angular\nmomenta as mentioned in Ref. [34], which may lead us to\na new detection scheme for magnon orbitals. For the future\noutlook,consideringaplethoraofmagnon-spin-relatedeffects\nstudied in magnonics and spintronics, it is expected that there\naremanymagnonorbital-relatedeffectsthatawaitthediscov-\neryinvolving,e.g.,theinterplayofthemagnonorbitalandthe\nelectron orbital.\nWe thank Kyung-Jin Lee and Giovanni Vignale for use-\nful discussions. G.G. acknowledges support by the National\nResearch Foundation of Korea (NRF-2022R1C1C2006578).\nH.W.L was supported by Samsung Science and Technol-\nogy Foundation (SSTF-BA-1501-51). S.K.K. was supported\nby Samsung Science and Technology Foundation (SSTF-\nBA2202-04), Brain Pool Plus Program through the National\nResearch Foundation of Korea funded by the Ministry of Sci-\nence and ICT (NRF-2020H1D3A2A03099291), and NationalResearch Foundation of Korea funded by the Korea Govern-\nment via the SRC Center for Quantum Coherence in Con-\ndensed Matter (NRF-RS-2023-00207732).\n[1] A.V.Chumak,V.I.Vasyuchka,A.A.Serga,andB.Hillebrands,\nMagnon spintronics, Nat. Phys. 11, 453 (2015).\n[2] H. Katsura, N. Nagaosa, and P. A. Lee, Theory of the Thermal\nHall Effect in Quantum Magnets, Phys. Rev. Lett. 104, 066403\n(2010).\n[3] J. H. Han and H. Lee, Spin Chirality and Hall-Like Transport\nPhenomena of Spin Excitations, J. Phys. Soc. Jpn. 86, 011007\n(2017).\n[4] R. Matsumoto and S. Murakami, Theoretical Prediction of a\nRotatingMagnonWavePacketinFerromagnets,Phys.Rev.Lett.\n106, 197202 (2011).\n[5] Y. Onose, T. Ideue, H. Katsura, Y. Shiomi, N. Nagaosa, and\nY.Tokura,ObservationoftheMagnonHallEffect,Science 329,\n297 (2010).\n[6] S.K.Kim,H.Ochoa,R.Zarzuela,andY.Tserkovnyak,Realiza-\ntionoftheHaldane-Kane-MeleModelinaSystemofLocalized\nSpins, Phys. Rev. Lett. 117, 227201 (2016).\n[7] S. A. Owerre, Topological honeycomb magnon Hall effect: A\ncalculation of thermal Hall conductivity of magnetic spin exci-\ntations, J. Appl. Phys. 120, 043903 (2016).\n[8] R. Cheng, S. Okamoto, and D. Xiao, Spin Nernst Effect of\nMagnons in Collinear Antiferromagnets, Phys. Rev. Lett. 117,\n217202 (2016).\n[9] V. A. Zyuzin and A. A. Kovalev, Magnon Spin Nernst Effect in\nAntiferromagnets, Phys. Rev. Lett. 117, 217203 (2016).\n[10] R. Takahashi and N. Nagaosa, Berry Curvature in Magnon-\nPhonon Hybrid Systems, Phys. Rev. Lett. 117, 217205 (2016).\n[11] X. Zhang, Y. Zhang, S. Okamoto, and D. Xiao, Thermal Hall\nEffectInducedbyMagnon-PhononInteractions,Phys.Rev.Lett.\n123, 167202 (2019).\n[12] G. Go, S. K. Kim, and K.-J. Lee, Topological Magnon-Phonon\nHybrid Excitations in Two-Dimensional Ferromagnets with\nTunable Chern Numbers, Phys. Rev. Lett. 123, 237207 (2019).\n[13] S. Park, N. Nagaosa, and B.-J. Yang, Thermal Hall Effect, Spin\nNernstEffect,andSpinDensityInducedbyaThermalGradient\nin Collinear Ferrimagnets from Magnon–Phonon Interaction,\nNano Lett. 20, 2741 (2020).\n[14] T. Tanaka, H. Kontani, M. Naito, T. Naito, D. S. Hirashima,\nK. Yamada, and J. Inoue, Intrinsic spin Hall effect and orbital\nHall effect in 4dand5dtransition metals, Phys. Rev. B 77,\n165117 (2008).\n[15] H. Kontani, T. Tanaka, D. S. Hirashima, K. Yamada, and J. In-\noue, Giant Intrinsic Spin and Orbital Hall Effects in Sr2MO4\n(M= Ru, Rh, Mo), Phys. Rev. Lett. 100, 096601 (2008).\n[16] H. Kontani, T. Tanaka, D. S. Hirashima, K. Yamada, and J. In-\noue, Giant Orbital Hall Effect in Transition Metals: Origin of\nLarge Spin and Anomalous Hall Effects, Phys. Rev. Lett. 102,\n016601 (2009).\n[17] T.TanakaandH.Kontani,IntrinsicspinandorbitalHalleffects\nin heavy-fermion systems, Phys. Rev. B 81, 224401 (2010).\n[18] D.Go,D.Jo,C.Kim,andH.-W.Lee,IntrinsicSpinandOrbital\nHallEffectsfromOrbitalTexture,Phys.Rev.Lett. 121,086602\n(2018).\n[19] D. Jo, D. Go, and H.-W. Lee, Gigantic intrinsic orbital Hall\neffects in weakly spin-orbit coupled metals, Phys. Rev. B 98,\n214405 (2018).6\n[20] G.SalaandP.Gambardella,GiantorbitalHalleffectandorbital-\nto-spin conversion in 3d,5d, and 4fmetallic heterostructures,\nPhys. Rev. Res. 4, 033037 (2022).\n[21] S.BhowalandG.Vignale,OrbitalHalleffectasanalternativeto\nvalleyHalleffectingappedgraphene,Phys.Rev.B 103,195309\n(2021).\n[22] A.Pezo,D.GarcíaOvalle,andA.Manchon,OrbitalHalleffect\nincrystals: Interatomicversusintra-atomiccontributions,Phys.\nRev. B106, 104414 (2022).\n[23] D. Go and H.-W. Lee, Orbital torque: Torque generation by\norbital current injection, Phys. Rev. Res. 2, 013177 (2020).\n[24] D. Lee, D. Go, H.-J. Park, W. Jeong, H.-W. Ko, D. Yun, D. Jo,\nS. Lee, G. Go, J. H. Oh, K.-J. Kim, B.-G. Park, B.-C. Min,\nH. C. Koo, H.-W. Lee, O. Lee, and K.-J. Lee, Orbital torque in\nmagnetic bilayers, Nat. Commun. 12, 6710 (2021).\n[25] H. Hayashi, D. Jo, D. Go, Y. Mokrousov, H.-W. Lee, and\nK. Ando, Observation of long-range orbital transport and gi-\nant orbital torque, arXiv:2202.13896.\n[26] J.A.Haigh,A.Nunnenkamp,A.J.Ramsay,andA.J.Ferguson,\nTriple-Resonant Brillouin Light Scattering in Magneto-Optical\nCavities, Phys. Rev. Lett. 117, 133602 (2016).\n[27] S. Sharma, Y. M. Blanter, and G. E. W. Bauer, Light scattering\nby magnons in whispering gallery mode cavities, Phys. Rev. B\n96, 094412 (2017).\n[28] A. Osada, A. Gloppe, Y. Nakamura, and K. Usami, Orbital\nangular momentum conservation in Brillouin light scattering\nwithinaferromagneticsphere,NewJ.Phys. 20,103018(2018).\n[29] Y. Jiang, H. Y. Yuan, Z.-X. Li, Z. Wang, H. W. Zhang, Y. Cao,\nandP.Yan,TwistedMagnonasaMagneticTweezer,Phys.Rev.\nLett.124, 217204 (2020).\n[30] A.González,P.Landeros,andÁ.S.Núñez,Spinwavespectrum\nofmagneticnanotubes,J.Magn.Magn.Mater. 322,530 (2010).\n[31] S. Lee and S. K. Kim, Orbital angular momentum and current-\ninduced motion of a topologically textured domain wall in a\nferromagnetic nanotube, Phys. Rev. B 104, L140401 (2021).\n[32] S. Lee and S. K. Kim, Generation of Magnon Orbital Angular\nMomentum by a Skyrmion-Textured Domain Wall in a Ferro-\nmagnetic Nanotube, Front. Phys. 10, 858614 (2022).\n[33] R.R.Neumann,A.Mook,J.Henk,andI.Mertig,OrbitalMag-\nneticMomentofMagnons,Phys.Rev.Lett. 125,117209(2020).\n[34] R. S. Fishman, J. S. Gardner, and S. Okamoto, Orbital Angular\nMomentum of Magnons in Collinear Magnets, Phys. Rev. Lett.\n129, 167202 (2022).\n[35] R.S.Fishman,L.Lindsay,andS.Okamoto,Exactresultsforthe\norbital angular momentum of magnons on honeycomb lattices,\nJ. Phys. Condens. Matter 35, 015801 (2022).\n[36] L.-H.WuandX.Hu,SchemeforAchievingaTopologicalPho-\ntonicCrystalbyUsingDielectricMaterial,Phys.Rev.Lett. 114,\n223901 (2015).\n[37] C. He, X. Ni, H. Ge, X.-C. Sun, Y.-B. Chen, M.-H. Lu, X.-P.\nLiu, and Y.-F. Chen, Acoustic topological insulator and robust\none-way sound transport, Nat. Phys. 12, 1124 (2016).\n[38] Y. Li, Y. Sun, W. Zhu, Z. Guo, J. Jiang, T. Kariyado, H. Chen,\nand X. Hu, Topological LC-circuits based on microstrips and\nobservationofelectromagneticmodeswithorbitalangularmo-\nmentum, Nat. Commun. 9, 4598 (2018).\n[39] H.Huang,T.Kariyado,andX.Hu,Topologicalmagnonmodes\nonhoneycomblatticewithcouplingtextures,Sci.Rep. 12,6257\n(2022).\n[40] H. Katsura, N. Nagaosa, and A. V. Balatsky, Spin Current and\nMagnetoelectric Effect in Noncollinear Magnets, Phys. Rev.\nLett.95, 057205 (2005).\n[41] T. Liu and G. Vignale, Electric Control of Spin Currents and\nSpin-Wave Logic, Phys. Rev. Lett. 106, 247203 (2011).[42] X.Wang,K.Du,Y.Y.F.Liu,P.Hu,J.Zhang,Q.Zhang,M.H.S.\nOwen, X. Lu, C. K. Gan, P. Sengupta, C. Kloc, and Q. Xiong,\nRaman spectroscopy of atomically thin two-dimensional mag-\nnetic iron phosphorus trisulfide (FePS3) crystals, 2d Mater. 3,\n031009 (2016).\n[43] K. Kim, S. Y. Lim, J. Kim, J.-U. Lee, S. Lee, P. Kim, K. Park,\nS. Son, C.-H. Park, J.-G. Park, and H. Cheong, Antiferromag-\nnetic ordering in van der Waals 2D magnetic material MnPS3\nprobed by Raman spectroscopy, 2d Mater. 6, 041001 (2019).\n[44] G. Le Flem, R. Brec, G. Ouvard, A. Louisy, and P. Segransan,\nMagneticinteractionsinthelayercompoundsMPX3(M=Mn,\nFe, Ni; X = S, Se), J. Phys. Chem. Solids 43, 455 (1982).\n[45] X. Jiang, Q. Liu, J. Xing, N. Liu, Y. Guo, Z. Liu, and J. Zhao,\nRecent progress on 2D magnets: Fundamental mechanism,\nstructural design and modification, Appl. Phys. Rev. 8, 031305\n(2021).\n[46] B. Li, S. Sandhoefner, and A. A. Kovalev, Intrinsic spin Nernst\neffectofmagnonsinanoncollinearantiferromagnet,Phys.Rev.\nRes.2, 013079 (2020).\n[47] D. Xiao, J. Shi, and Q. Niu, Berry phase correction to electron\ndensity of states in solids, Phys. Rev. Lett. 95, 137204 (2005).\n[48] T.Thonhauser,D.Ceresoli,D.Vanderbilt,andR.Resta,Orbital\nmagnetizationinperiodicinsulators,Phys.Rev.Lett. 95,137205\n(2005).\n[49] J. Shi, G. Vignale, D. Xiao, and Q. Niu, Quantum theory of or-\nbitalmagnetizationanditsgeneralizationtointeractingsystems,\nPhys. Rev. Lett. 99, 197202 (2007).\n[50] Notethattheorbitalmoment \u0018r\u0002vdiffersfromtheorbitalan-\ngularmomentum \u0018r\u0002pbydimensionofmassintheelectronic\ncase [21].\n[51] S.ParkandB.-J.Yang,PhononAngularMomentumHallEffect,\nNano Lett. 20, 7694 (2020).\n[52] A. R. Wildes, B. Roessli, B. Lebech, and K. W. Godfrey, Spin\nwavesandthecriticalbehaviourofthemagnetizationin,J.Phys.\nCondens. Matter 10, 6417 (1998).\n[53] F. Schütz, M. Kollar, and P. Kopietz, Persistent Spin Currents\nin Mesoscopic Heisenberg Rings, Phys. Rev. Lett. 91, 017205\n(2003).\n[54] F. Schütz, M. Kollar, and P. Kopietz, Persistent spin currents\ninmesoscopichaldane-gapspinrings,Phys.Rev.B 69,035313\n(2004).\n[55] J. Xia, X. Zhang, X. Liu, Y. Zhou, and M. Ezawa, Univer-\nsal quantum computation based on nanoscale skyrmion helic-\nity qubits in frustrated magnets, Phys. Rev. Lett. 130, 106701\n(2023).\n[56] The electric polarization can be also caused by the Lorentz\ntransformation of the magnetic dipole from a moving frame\nto the laboratory P=v\u0002\u0016\nc2=g\n2e~\nmec2(v\u0002^z), where vand\n\u0016(=g\u0016Bz)are the velocity and the magnetic moment of the\nmagnetic dipole, respectively, and meis the electron mass [53,\n61, 62]. However, this effect is extremely small because the\ndenominator mec2is order of MeV.\n[57] Y.-G. Choi, D. Jo, K.-H. Ko, D. Go, K.-H. Kim, H. G. Park,\nC.Kim,B.-C.Min,G.-M.Choi,andH.-W.Lee,Observationof\nthe orbital Hall effect in a light metal Ti, arXiv:2109.14847.\n[58] S. Han, H.-W. Lee, and K.-W. Kim, Orbital Dynamics in Cen-\ntrosymmetric Systems, Phys. Rev. Lett. 128, 176601 (2022).\n[59] L. J. Cornelissen, J. Liu, R. A. Duine, J. B. Youssef, and B. J.\nvanWees,Long-distancetransportofmagnonspininformation\ninamagneticinsulatoratroomtemperature,Nat.Phys. 11,1022\n(2015).\n[60] S.T.B.Goennenwein,R.Schlitz,M.Pernpeintner,K.Ganzhorn,\nM. Althammer, R. Gross, and H. Huebl, Non-local magnetore-7\nsistanceinYIG/Ptnanostructures,Appl.Phys.Lett. 107,172405\n(2015).\n[61] J.E.Hirsch,OverlookedcontributiontotheHalleffectinferro-\nmagnetic metals, Phys. Rev. B 60, 14787 (1999).[62] F. Meier and D. Loss, Magnetization Transport and Quantized\nSpin Conductance, Phys. Rev. Lett. 90, 167204 (2003)."    },    {        "title": "1411.3950v1.Spin_orbit_coupling_and_chaotic_rotation_for_eccentric_coorbital_bodies.pdf",        "content": "Complex Planetary Systems\nProceedings IAU Symposium No. 310, 2014\nZ. Knezevic & A. Lema^ \u0010trec\r2014 International Astronomical Union\nDOI: 00.0000/X000000000000000X\nSpin-orbit coupling and chaotic rotation for\neccentric coorbital bodies\nAdrien Leleu1, Philippe Robutel1and Alexandre C.M. Correia2;1\n1IMCCE, Observatoire de Paris, CNRS, UPMC Univ. Paris 06, Univ. Lille 1,\n77Av.Denfert-Rochereau, 75014 Paris, France\nemail: aleleu@imcce.fr, robutel@imcce.fr\n2Departamento de F\u0013 \u0010sica, I3N, Universidade de Aveiro, Campus de Santiago, 3810-193 Aveiro,\nPortugal\nemail: correia@ua.pt\nAbstract. The presence of a co-orbital companion induces the splitting of the well known\nKeplerian spin-orbit resonances. It leads to chaotic rotation when those resonances overlap.\nKeywords. Coorbitals, Rotation, Resonance, Spin-orbit resonance.\n1. Introduction and Notations\nGiven an asymmetric body on a circular orbit, denoting \u0012its rotation angle in the\nplane with respect to the inertial frame, the only possible spin-orbit resonance is the\nsynchronous one _\u0012=n,nbeing the mean motion of the orbit. On an Keplerian eccentric\norbit, Wisdom et al. (1984) showed that there is a whole family of spin-orbit eccen-\ntric resonances, the main ones being _\u0012=pn=2 wherepis an integer. In 2013, Correia\nand Robutel showed that in the circular case, the presence of a coorbital companion in-\nduced a splitting of the synchronous resonance, forming a family of co-orbital spin-orbit\nresonances of the form _\u0012=n\u0006k\u0017=2,\u0017being the libration frequency in the coorbital res-\nonance. Inside this resonance, the di\u000berence of the mean anomaly of the two coorbitals,\ndenoted by \u0010, librates around a value close to \u0006\u0019=3 (around the L4 or L5 Lagrangian\nequilibrium - tadpole con\fguration), around \u0019(encompassing L3, L4 and L5 - horseshoe\ncon\fguration) or 0 (quasi-satellite) con\fguration. We generalize the results of Correia\nand Robutel (2013) from the case of circular co-orbital orbits to eccentric ones.\n2. Rotation\nThe rotation angle \u0012satis\fes the di\u000berential equation:\n\u0012+\u001b2\n2\u0010a\nr\u00113\nsin 2(\u0012\u0000f) = 0;with\u001b=nr\n3(B\u0000A)\nC; (2.1)\nwhereA<B <C are the internal momenta of the body, ( r;f) the polar coordinates\nof the center of the studied body and aits instantaneous semi-major axis.\nLet us consider that the orbit is quasi-periodic. As a consequence, the elliptic elements\nof the body can be expended in Fourier series whose frequencies are the fundamental\nfrequencies of the planetary system. In other words the time-dependent quantity\u0000a\nr\u00013ei2f\nthat appears in equation (2.1) reads:\n\u0010a\nr\u00113\nei2f=X\nj>0\u001aje(i\u0011jt+\u001ej): (2.2)\n1arXiv:1411.3950v1  [astro-ph.EP]  14 Nov 20142 Adrien Leleu, Philippe Robutel & Alexandre C.M. Correia\nWhere\u0011jare linear combinations with integer coe\u000ecients of the fundamental frequen-\ncies of the orbital motion (here nand\u0017) and\u001ejtheir phases. Thus (2.1) becomes:\n\u0012=\u0000\u001b2\n2X\nj>0\u001ajsin (2\u0012+\u0011jt+\u001ej): (2.3)\nFor a Keplerian circular orbit, the only spin orbit resonance possible is the synchronous\none, since\u001a0= 1,\u00110= 2n, and\u001aj=\u0011j= 0 forj >0. In the general Keplerian case we\nhave the spin-orbit eccentric resonances, \u0011j=pnand the\u001ajare the Hansen coe\u000ecients\nX\u00003;2\np(e) (see Wisdom et al. ). For the circular coorbital case, Correia and Robutel (2013)\nshowed that a whole family results from the splitting of the synchronous resonance of\nthe form\u0011j= 2n\u0006k\u0017. For small amplitudes of libration around L4 or L5 (tadpole), the\nwidth of the resonant island decreases as kincreases.\nIn the eccentric coorbital case, each eccentric spin-orbit resonance of the Keplerian\ncase splits in resonant multiplets which are centred in _\u0012=pn=2\u0006k\u0017=2. For relatively\nlow amplitude of libration of \u0010, the width of the resonant island decreases as kincreases,\nsee Figure 1 (left). But for higher amplitude, especially for horseshoe orbit, the main\nresonant island may not be located at k= 0. In Figure 1 (right), the main islands are\nlocated at _\u0012= 3n=2\u00065\u0017=2 and _\u0012= 3n=2\u00066\u0017=2. These islands overlap, giving rise to\nchaotic motion for the spin, while the island located at _\u0012= 3n=2 is much thinner.\nFigure 1. Poincar\u0013 e surface of section in the plane ( \u0012\u0000t3n\n2;_\u0012=n) near the 3 =2 spin-orbit eccen-\ntric resonance. (left): \u0010max\u0000\u0010min= 35\u000e- tadpole con\fguration. (right): \u0010max\u0000\u0010min= 336\u000e\nhorseshoe con\fguration.\n3. Conclusion\nThe coorbital spin-orbit resonances populate the phase space between the eccentric\nresonances. Generalised chaotic rotation can be achieved when harmonics of co-orbital\nspin-orbit resonances overlap each other, which is a di\u000berent mechanism than the one\ndescribed by Wisdom et al. (1984), where the eccentricity harmonics overlap.\nReferences\nCorreia, A.C.M. C.M., & Robutel, P. 2013, Spin-orbit coupling and chaotic rotation for coorbital\nbodies in quasi-circular orbits AJ, 779, 20\nWisdom, J., Peale, S. J. & Mignard, F. 1984, The chaotic rotation of Hyperion, Icarus , 58, 137"    },    {        "title": "1505.07667v1.Intrinsic_spin_Hall_effect_in_systems_with_striped_spin_orbit_coupling.pdf",        "content": "Intrinsic spin Hall e\u000bect in systems with striped spin-orbit coupling\nG otz Seibold,1Sergio Caprara,2Marco Grilli,2and Roberto Raimondi3\n1Institut f ur Physik, BTU Cottbus-Senftenberg, PBox 101344, 03013 Cottbus, Germany\n2Dipartimento di Fisica - Universit\u0012 a di Roma Sapienza, piazzale Aldo Moro 5, I-00185 Roma, Italy\n3Dipartimento di Matematica e Fisica, Universit\u0012 a Roma Tre, Via della Vasca Navale 84, 00146 Rome, Italy\nThe Rashba spin-orbit coupling arising from structure inversion asymmetry couples spin and\nmomentum degrees of freedom providing a suitable (and very intensively investigated) environment\nfor spintronic e\u000bects and devices. Here we show that in the presence of strong disorder, non-\nhomogeneity in the spin-orbit coupling gives rise to a \fnite spin Hall conductivity in contrast with\nthe corresponding case of a homogeneous linear spin-orbit coupling. In particular, we examine the\ninhomogeneity arising from a striped structure for a two-dimensional electron gas, a\u000becting both\ndensity and Rashba spin-orbit coupling. We suggest that this situation can be realized at oxide\ninterfaces with periodic top gating.\nPACS numbers: 72.25.-b, 75.76.+j, 72.25.Rb, 72.15.Gd\nThe spin Hall e\u000bect (SHE) [1] is the generation of a\ntransverse spin current by an applied electric \feld with\nthe current spin polarization being perpendicular to both\nthe \feld and the current \row. Since the SHE allows the\ncontrol of the spin degrees of freedom even without exter-\nnal magnetic \felds (see e.g. Refs. [2, 3]), it has become\na central topic in present spintronics research. [4, 5] The\nmicroscopic origin of the SHE lies in the spin-orbit cou-\npling (SOC), which in solid-state systems may be due to\nthe potential of the ionic cores of the host lattice, the po-\ntential of the impurities and the con\fnement potential of\nthe device structure. In a two-dimensional electron gas\n(2DEG), Bychkov and Rashba [6] have proposed that the\nlack of inversion symmetry along the direction perpen-\ndicular to the gas plane leads to a momentum-dependent\nspin splitting usually described by the so-called Rashba\nHamiltonian\nH=p2\n2m+\u000b\u001c\u0002z\u0001p; (1)\nwhere pis the momentum operator for the motion along\nthe plane hosting the 2DEG, say the xy plane, zis a\nunit vector perpendicular to it, \u001c= (\u001cx;\u001cy;\u001cz) is the\nvector of the Pauli matrices, and \u000bis a coupling con-\nstant whose strength depends on both the SOC of the\nmaterial and the \feld responsible for the parity break-\ning. The Hamiltonian (1), which has been extensively\nused in the study of the 2DEG in semiconducting sys-\ntems, has been recently applied also to interface states\nbetween di\u000berent metals [7] and between two insulating\noxides [9{12]. In the latter systems, higher mobilities,\ncarrier concentration and SOC strengths have led to the\nexpectation of observing stronger SOC-induced e\u000bects.\nThe Hamiltonian (1) is deceptively simple, as one realizes\nwhen considering transport phenomena. In particular,\nthe intrinsic universal SHE proposed in Ref. [13] turned\nout to be a non stationary e\u000bect, while under station-\nary conditions cancellations occur, leading to a vanish-\ning spin Hall conductivity (SHC) \u001bsH, i.e., the coe\u000ecientrelating the z-spin current in the ydirection to the ap-\nplied electric \feld, Jz\ny=\u001bsHEx. Here, we show that\nthis is only true for a spatially homogeneous \u000b, while\nconsidering a space-dependent \u000b(x;y) opens the way to\na substantial SHE under stationary conditions, even in\nthe presence of strong disorder. We shall \frst discuss\nfrom a general perspective how this comes about, and\nshall afterwards demonstrate numerically the e\u000bect in the\npresence of a spatially modulated SOC as it could be re-\nalized in the 2DEG at the interface of a LaAlO 3/SrTiO 3\n(LAO/STO) heterostructure, schematically depicted in\nFig. 1. It is experimentally established that the Rashba\nSOC increases when the electron density in the 2DEG\nof these heterostructures is increased [8{10, 12]. Since\nthe local electric \feld determining \u000bis tightly related to\nthe electron density [14], one can naturally infer that the\nstructure in Fig. 1 produces a modulation of the Rashba\nSOC.\ndw\nJz\ny\n2DEG\nSTOLAO\nEx\nFIG. 1. Schematic view of a possible device in which the\nSHE is enforced in the 2DEG at the interface of a LAO/STO\nheterostructure. The yellow stripes represent top-gating elec-\ntrodes of width wand interspacing d.\n| General arguments | The interplay of the intrinsic\nRashba SOC with the scattering from impurities makes\nthe dynamics of charge and spin degrees of freedom in-\ntrinsically coupled in a subtle way. This is especially evi-arXiv:1505.07667v1  [cond-mat.mes-hall]  28 May 20152\ndent in the vanishing of the SHC with homogeneity in the\nSOC. Notice that the spin current is a tensor quantity de-\npending on the \row direction (lower index) and spin po-\nlarization axis (upper index) and hence the spin current\nand the electric \feld are related by a tensor of third rank\n\u001ba\nij. For \fxed polarization a=z, Onsager's relations\nrequire the antisymmetry property \u001bz\nyx=\u0000\u001bz\nxy=\u001bsH.\nThe vanishing of the SHC manifests via an exact\ncompensation of the contribution originally proposed by\nSinova et al. [13] by a further contribution, which arises\nby the coupling between the spin current and the spin\npolarization, which is induced in the plane perpendic-\nularly to the applied electric \feld. This latter e\u000bect\nwas almost simultaneously proposed by Edelstein [15]\nand by Aronov and Lyanda-Geller [16]. It consists of a\nnon-equilibrium spin polarization due to the electric \feld\nSy\n0=\u0000e\u000bN 0\u001cExfor a \feldExalong thexaxis. Heree\nis the unit charge ( \u0000efor electrons), N0=m=2\u0019~2is the\ndensity of states per spin of the 2DEG described by the\n\frst term of Eq. (1) and \u001cis the elastic relaxation time\nintroduced by impurity scattering. In such a 2DEG the\nstandard Drude formula can be written via the Einstein\nrelation as \u001b= 2e2N0D, with the di\u000busion coe\u000ecient\nD=v2\nF\u001c=2,vFbeing the Fermi velocity related to the\nFermi energy EF=mv2\nF=2. To understand the origin of\nthe compensation mentioned above, it is useful to start\nfrom a property of the Hamiltonian (1) \frst pointed out\nby Dimitrova [17], which relates the time derivative of\ntheSyspin polarization to the spin current\n@tSy=\u00002m\u000b~\u00001Jz\ny: (2)\nNotice that such a relation is not changed by disorder\nscattering as long as the latter is spin independent. Dis-\norder is necessary to guarantee a stationary state which\nimplies the left-hand side of Eq. (2) to be zero. Obvi-\nously, the corresponding vanishing of the right-hand side\nentails a vanishing SHC when \u000bis a constant. The spe-\nci\fc way in which the vanishing of the SHC occurs in\na disordered 2DEG via the so-called vertex corrections\n[18{21] can be heuristically understood by describing the\ncoupling between spin current and spin polarization as a\ngeneralized di\u000busion in spin space. By dimensional ar-\nguments spin current and spin density must be related\nby the factor LSO=\u001cDP, whereLSO=~=(2m\u000b) is the\nspin-orbit precession length originating from the di\u000ber-\nence of the Fermi momenta of the two branches of the\nspectrum of Eq. (1), while \u001cDPis the Dyakonov-Perel\nspin relaxation time due to the interplay of SOC and dis-\norder scattering. In a disordered 2DEG, \u001cDPis related\ntoLSOby the di\u000busion coe\u000ecient L2\nSO=D\u001cDP, so that\none obtains for the spin current\nJz\ny= 2m\u000b~\u00001DSy+\u001bsH\n0Ex: (3)\nIn the above equation, which can be rigorously derived\n[22, 23], the quantity \u001bsH\n0is the intrinsic contribution ofRef. [13] in the di\u000busive regime \u000bpF\u001c=~\u001c1, whereas\nthe term proportional to Sycorresponds to the vertex-\ncorrection contribution mentioned above. Given the ex-\npression for \u001bsH\n0= (e=8\u0019)(2\u001c=\u001cDP) derived in Ref. 13 it\nis now apparent that, if we replace Sywith the Edelstein\nresult, the spin current in Eq. (3) vanishes, consistently\nwith the stationarity requirement derived from Eq. (2).\nThe key observation is that this compensation does not\nnecessarily occur for an inhomogeneous SOC, where it is\nno longer possible to express the time derivative of the\nspin polarization in terms of the spin current. In such a\nsituation a spin current becomes possible under station-\nary conditions.\nIn order to illustrate the physical mechanism by which\nan inhomogeneous Rashba SOC leads to a \fnite SHE it\nis useful to consider a single-interface problem, which is\ndescribed by Eq. (1) with the replacement \u000b!\u000b(x) =\n\u0012(x)\u000b++\u0012(\u0000x)\u000b\u0000(with\u000b+>\u000b\u0000). Clearly as x!\u00061 ,\none recovers the uniform case with complete cancellation\nof the spin current for the Rashba model with couplings\n\u000b\u0006. On both sides of the interface, the y-spin polariza-\ntion obeys a di\u000busion-like equation with L\u0006, the corre-\nsponding spin-orbit lengths in the two regions. One can\nthen seek a solution of the form\nSy(x) =\u0012(x)\u0010\nS0;++\u000es+e\u0000x=L+\u0011\n+\u0012(\u0000x)\u0010\nS0;\u0000+\u000es\u0000ex=L+\u0011\n;\nwhereS0;\u0006are the asymptotic values of the y-spin po-\nlarization at\u00061. The constants \u000es\u0006must be deter-\nmined by imposing the appropriate boundary conditions\natx= 0. As a result, the spin current is exponentially\nlocalized near the interface, where the spin polarization\nSy(x) must interpolate between the two asymptotic val-\nues and there is no longer complete cancellation between\nthe two terms of Eq. (3).\nOne can imagine to generalize this analysis to a series\nof interfaces apt to describe a periodic modulation of the\nSOC. Expectedly, if the spin-orbit length is larger than\nthe distance between two successive interfaces, the spin\nHall current should become practically uniform.\n| The model and its numerical solution | The previ-\nous arguments within the di\u000busive limit are now substan-\ntiated by numerical results for a microscopic 2D lattice\nmodel (size Nx\u0002Ny) with inhomogeneous Rashba SOC\ndescribed by the Hamiltonian\nH=X\nij\u001btijcy\ni\u001bcj\u001b+X\ni\u001b(Vi\u0000\u0016)cy\ni\u001bci\u001b+HRSO;(4)\nwherecy\ni\u001b(ci\u001b) creates (annihilates) an electron with spin\nprojection\u001bon the site identi\fed by the lattice vector Ri,\nthe \frst term describes the kinetic energy of electrons on\na square lattice (lattice constant a, only nearest-neighbor\nhopping:tij\u0011\u0000tforjRi\u0000Rjj=a) and in the second3\ntermViis a local disorder potential with a \rat distribu-\ntion\u0000V0\u0014Vi\u0014V0, and\u0016is the chemical potential. The\nlast term is the lattice Rashba SOC,\nHRSO=\u0000iX\ni\u001b\u001b0\u000bi;i+xh\ncy\ni\u001b\u001cy\n\u001b\u001b0ci+x;\u001b0\u0000cy\ni+x;\u001b\u001cy\n\u001b0\u001bci;\u001bi\n+iX\ni\u001b\u001b0\u000bi;i+yh\ncy\ni\u001b\u001cx\n\u001b\u001b0ci+y;\u001b0\u0000cy\ni+y;\u001b\u001cx\n\u001b0\u001bci;\u001bi\nand the coupling constants \u000bi;i+x=y>0 are now de\fned\non the bonds. Note that for constant \u000b\u0011\u000bi;i+x=yEq.\n(4) takes the usual form [2nd term in Eq. (1)] in momen-\ntum space [24].\nOne can show [30] that a \\continuity equation\" for the\nlocal spin density Sy\ni, (a dot stands for time derivative)\n_Sy\ni+ [div Jy]i+\u000bi;i+yJz\ni;i+y+\u000bi\u0000y;iJz\ni\u0000y;i= 0;(5)\nholds. For a homogeneous Rashba SOC, where\n[divJy]i= 0, Eq. (5) corresponds to Eq. (2) and im-\nplies that the total z-spin current has to vanish under\nstationary conditions. On the contrary, when \u000bvaries in\nspace, a cancellation occurs [30] between div Jyand the\nlast two terms of Eq. (5), so that the stationarity condi-\ntion _S= 0 can be ful\flled without the vanishing of Jz.\nThis is also clear for a system with periodic boundary\nconditions, where the total divergence of any current has\nto vanish, i.e.,\n\u0000X\ni_Sy\ni=X\ni\b\n\u000bi;i+yJz\ni;i+y+\u000bi\u0000y;iJz\ni\u0000y;i\t\n:(6)\nClearly, the left-hand side can vanish without implying\nJz= 0, because, if \u000bis inhomogeneous, Eq. (6) can\nbe ful\flled with alternating signs of Jzin regions with\ndi\u000berent\u000b[see the top panel of Fig. 4 in Ref. 30].\nWe exemplify the situation for an inhomogeneous\nRashba SOC which varies along the xdirection form-\ning a superlattice with d= 20aandw= 10a[see Fig. 1].\nIn the regions of width d,\u000b=a0is smaller than in the\nregions of width w, where\u000b=a1>a 0. The inhomogene-\nity in\u000bi;i+x;yleads to a concomitant charge modulation\nwhich is shown in Fig. 2(a). We have diagonalized the\nHamiltonian (4) and calculated the SHC from the Kubo\nformula (see, e.g., Ref. 13) \u001bsH=P\nij\u001bsH\nijwith\n\u001bsH\nij\u00112\nNX\nEn<EF\nEm>EFImhnjjz\ni;i+yjmihmjjch\nj;j+xjni\n(En\u0000Em)2+\u00112:(7)\nHere we have taken the limit of zero temperature and\n\u0011!0 is a small regularization term which can be inter-\npreted as an inverse electric-\feld turn-on time [27] . Note\nthat for the striped system one has already spin currents\nJz;0\ni;i+y\rowing in the ground state [25] (see, e.g., Fig. 3 in\nRef. 30) whereas the electrically induced spin current is\nJz;ind\ni;i+y=P\nj\u001bsH\nijEx. Thus Eq. (6) can be split into a\n-5 -4 -3 -2 -1 0\nchem. pot.00.511.522.5σsH, γ  [1/8π]10 20 30x00.0020.0040.006\nn(x)\n10 20 3000.20.40.60.81α(x)d w\nσsH=γ (hom.)σsH (stripe)\nγ (stripe)FIG. 2. Main panel: Spin Hall coe\u000ecient \u001bsHand \\station-\narity\" parameter \r[both in units of 1 =(8\u0019)] as a function of\n\u0016for a homogeneous system with \u000b= 0:5t(black solid line)\nand a striped system ( \u001bsH: red solid and \r: blue dashed) with\nmodulated Rashba SOC as shown in the inset. Here the (red)\ndashed line displays the variation of \u000b(x)\u0011\u000bi;i+x=\u000bi;i+y\nalong thexdirection for stripes along the ydirection and\nwidthw= 10aseparated by a distance d= 20a. The Rashba\nSOC on the stripes is \u000b(x)\u0011\u000b1= 0:8twhile between the\nstripes\u000b(x)\u0011\u000b0= 0:2t. The black solid line (dots) in the in-\nset reports the charge pro\fle at chemical potential \u0016=\u00004:3t.\nSystem size: 3060 \u00023060 sites.\n0 0.02 0.04 0.06n00.51σsH, γ  [1/8π]α0=0.2, α1=0.8 \nα0=0.3, α1=0.7\nα0=0.4, α0=0.6\n0 0.02 0.04 0.06 0.08 0.1n00.20.40.60.8σsH, γ   [1/8π]∆µ=0\n∆µ=0.5a) b)\nFIG. 3. a): \u001bsH(thick lines) and \r(thin lines) as a function\nof densityn(average number of electrons per lattice site) for\na striped system and parameters are indicated in the panel.\nb):\u001bsH(thick lines) and \r(thin lines) as a function of density\nbut now with an additional modulation of the local chemical\npotential\u0016loc\nixwhich is set to \u00000:5ton the stripes ( a0= 0:2t,\na1= 0:8t).\nground-state contribution (for which of course _Sy\ni= 0)\nand a linear-response part. For the latter we de\fne the\nquantity\r= 2P\nij\u000bi;i+y\u001bsH\nijwhich therefore also de-\nscribes the linear response ofP\ni_Sy\ni=\u0000\rExto the ap-\nplied electric \feld and which in the following will be used\nto quantify the \\stationarity\" of the solution. In fact,\nFig. 2 demonstrates that for a constant \u000b= 0:5t(black\nsolid line)\u001bsHcoincides with \rand therefore the \fnite\n\u001bsHis a non-stationary result. On the other hand, the\nsame panel also reports the results for the case a0= 0:2t\nanda1= 0:8t. In this case one can see that for a non-\nnegligible range of chemical potential near the bottom of4\nthe band a substantial \u001bsH(red solid curve) is present\nwhile\r= 0 (blue dashed curve), marking the occurrence\nof a SHE in stationary conditions. This indicates the rel-\nevant role of those states that are still extended along the\nydirection, while they are nearly localized inside the po-\ntential wells arising from the modulation of \u000b. As shown\nin Fig. 3(a), this situation occurs for increasingly large\ndensity ranges by increasing the inhomogeneity of \u000b. We\nhave also checked that the stationarity is not only global\nbut that in the low density regime _Si\u00190 is ful\flled at\neach lattice site.\nWe also investigated the e\u000bect of an inhomogeneous\nchemical potential as it is induced by the striped gating\nof Fig. 1. In particular, we shift the chemical potential\ndownwards by \u0001 \u0016= 0:5ton the sites below the gate (the\nregions of width wwith\u000b=a1= 0:8t. From Fig. 3(b)\none sees that, although \u001bsHis reduced, it still remains\nsubstantial and the density range with a stationary SHE\nis even extended (black dashed curves). On the contrary\none can see [30] that, in the absence of an inhomogeneous\nRashba SOC, a simple charge modulation does not pro-\nduce any stationary SHE.\n-4.3 -4.2 -4.1 -4 -3.9 -3.8\nchemical potential00.51σsH [1/8π]V0/t=0\nV0/t=0.5\nV0/t=0.2\nV0/t=1.0\n-4.2 -4 -3.8 -3.600.05\nγ [1/8π]-4.2 -4 -3.8 -3.600.51\nσsH[1/8π]η/t=0\nη/t=0.02a)b)\nc)V0/t=0.5\nFIG. 4. Main panel (a): Spin Hall conductivity as a function\nof chemical potential \u0016and various values of disorder. Panel\n(b):\u001bsHas a function of \u0016including error bars for V0=t= 0:5\nand two values of \u0011=t= 0 (circles) and \u0011=t= 0:02 (triangles).\nPanel (c) reports the behavior of the \\stationarity\" parameter\n\r. Results in panels (a,c) are obtained for \u0011= 0.\nFinally, we address the quite important issue of the\nrobustness of SHE in the presence of disorder. Previ-\nous analyses [26, 27] showed that the SHC for a linear\nRashba SOC is rapidly destroyed by disorder. This is\neasily understood because in the homogeneous Rashba\nSOC the SHE is a non-stationary e\u000bect which cannot\nsurvive the relaxation e\u000bect of disorder scattering. Here,\ninstead, when \r= 0 the SHE is present in a stationary\nstate and disorder is much less e\u000bective in spoiling it.\nIn the presence of a random potential of \fnite width V0\nthe calculations can only be carried out on smaller lat-\ntices (40\u000240 sites) where we consider stripes of widthw= 4aand distance d= 4a(see Fig. 1). We follow the\nprocedure described in Ref. 26 and diagonalize the Hamil-\ntonian for di\u000berent disorder con\fgurations and di\u000berent\ntwisted boundary conditions. For each concentration we\nconsider 250 random boundary phases and 50 disorder\ncon\fgurations.\nAs a striking result (main panel of Fig. 4) we \fnd\nthat the average SHC at low densities ( \u0016 <\u00004t) is not\na\u000bected by disorder and only gets suppressed when the\nchemical potential is within the range of band states ex-\ntended both along xandydirections. In contrast, and as\nmentioned above, \u001bsHvanishes for a homogeneous, lin-\near Rashba SOC [26, 27] in the presence of disorder and\nfor\u0011!0.\nIt has been pointed out in Ref. [27] that the evalua-\ntion of\u001bsHon \fnite lattices and taking the limit \u0011!0\nis complicated by strong \ructuations. These strong vari-\nances in the SHC are exempli\fed in panel (b) of Fig. 4\nforV0=t= 0:5 where we also show the corresponding re-\nsult for\u0011=t= 0:02. The SHC in the low density regime\nis not dependent on the small \u0011value but one observes a\nlarge reduction in the variance which becomes of the or-\nder of the symbol size. We therefore can safely conclude\nthat our \fnite size results support a \fnite SHC at low\ndensities even for strongly disordered systems.\nNaturally, the system gets more stationary with disor-\nder, as it is shown in panel (c) of Fig. 4, where a small\nresidual value of \rfor the clean striped system (black\ncurve) is suppressed for all \u0016's. Again, for \u0016<\u00004tthis\ndoes not imply the vanishing of \u001bsH, as would be the\ncase for homogeneous Rashba SOC.\n| Discussion and conclusions | The above analy-\nsis clearly demonstrates that a system with modulated\nRashba SOC can sustain a \fnite SHE in stationary con-\nditions. This occurs for a limited density range, when the\nchemical potential falls in a region where the states are\nstrongly a\u000bected by the modulated \u000b(x) and are almost\nlocalized in the bottom of the modulating potential (in\nthe direction of the modulation; the states are extended\nin the perpendicular direction). Therefore the response\nof the charge current Jch\nxto the electric \feld along the\nmodulation direction is strongly suppressed which can\nlead to large spin Hall angles eJz\ny=Jch\nxfor the striped\nsystem. It is important to note that this e\u000bect is due to\nthe modulation of the Rashba SOC and cannot arise in\na \\conventional\" charge density wave. In fact, for con-\nstant\u000band independent of the electronic structure Eq.\n(6) predicts the vanishing of the SHE under stationary\nconditions.\nThe implementation of this analysis in a real system is\nfor sure a challenging task for several reasons. First of all\nthe top gating structuring has to be sharp enough to pro-\nduce a su\u000eciently sharp spatial modulation of the 2DEG\nbelow the LAO layer (which is at least 20 nm thick): if\nthe modulation of the SOC is not sharp enough on the\nLSOscale, the 2DEG would feel a nearly uniform \u000band5\nthe SHE is expected to vanish. One should also con-\nsider that our analysis is based on a simple one-band\nmodel, while the SOC in the 2DEG in the LAO/STO\ninvolves several bands [14, 31, 32]. Of course, the basic\nideas of this work could be tested and hopefully imple-\nmented in other, perhaps simpler, systems involving het-\nerostructures of semiconductors with modulated Rashba\nSOC which have been already discussed in the literature\nin di\u000berent contexts [33{35].\nWe acknowledge interesting discussions with N.\nBergeal, V. Brosco, and J. Lesueur. G.S. acknowl-\nedges support from the Deutsche Forschungsgemein-\nschaft. M.G. and S.C. acknowledge \fnancial support\nfrom Sapienza University of Rome, project Awards No.\nC26H13KZS9.\nAPPENDIX\nRashba systems in the di\u000busive limit\nTo gain insight on the numerical results discussed in\nthe main text, we discuss here a continuum Rashba model\nin the di\u000busive limit. The corresponding di\u000busion equa-\ntions can be derived from a microscopic model by using\ne.g. the Keldysh technique. Such a derivation is, for in-\nstance given in Ref. 36. For the following discussion,\nhowever, one does not need to know such a microscopic\nderivation in detail. One main advantage of the di\u000busion\nequation description is that it contains all the important\naspects of the Rashba model and allows an almost an-\nalytic treatment, which helps in elucidating the physics.\nWe \frst provide the di\u000busion equation description for the\nuniform case. This is very standard and a recent discus-\nsion can be found in Ref. 23. Then we describe the\nsingle-interface problem as the simplest realization of a\nnon-uniform Rashba system with two regions with dif-\nferent Rashba SOC. A subsequent subsection reports the\ntwo-interface problem.\nThe uniform case\nIn the presence of the Rashba spin-orbit coupling, the\nspin polarization along the y direction obeys the follow-\ning equation in the di\u000busive regime (see Ref. 36 for a\nderivation)\n\u0000@tsy+D@2\nxsy=1\n\u001cDP(sy\u0000s0); (8)\nwhereD=v2\nF\u001c=2 is the standard di\u000busion coe\u000ecient.\n\u001c\u00001\nDP= (2m\u000b)2Dis the inverse Dyakonov-Perel spin re-laxation time. s0=\u0000eN0\u000b\u001cExis the non-equilibrium\nspin polarization induced by the electric \feld Exapplied\nalong the x direction. Such a non-equilibrium polariza-\ntion is sometimes called the Edelstein e\u000bect or the spin-\ngalvanic e\u000bect. Here N0=m=2\u0019~2is the 2D density\nof states and, in the following, we take units such that\n~= 1. We consider only the dependence on the x di-\nrection in the di\u000busion equation to make contact with\nthe numerical calculation. In stationary and uniform cir-\ncumstances, we must have sy=s0. This leads to the\nvanishing of the spin current Jz\ny, as it is well known in\nthe Rashba model (see for instance Ref. 23). To see this,\nconsider that the spin current is given by two terms: a\n\"drift-like\" Hall term and a \"di\u000busion-like\" one. The\ndrift term corresponds to the calculation of Ref. 13, i.e.\nit is just the Drude formula for the spin Hall conductiv-\nity. As for the ordinary Hall conductivity, it is non-zero\nand \fnite even in the absence of disorder. The di\u000busion\nterm is usually expected to vanish in uniform situations.\nHowever as derived in Ref. 22 and used in Ref. 23, the\nRashba interaction can be described in terms of a SU(2)\nvector potential, which then introduces covariant deriva-\ntives. The latter are de\fned by\nh\n~risia\n=risa\u0000\u000fabcAb\nisc(9)\nwhereAa\niis the SU(2) vector potential. In the Rashba\ncase,Ax\ny=\u0000Ay\nx= 2m\u000b. The key observation is that the\ndi\u000busion term related to the covariant derivative of the\nspin density is present even in the uniform case. Speci\f-\ncally the spin Hall current we are interested in reads\nJz\ny=\u001bsH\n0Ex+D\nLSOsy; (10)\nwhereLSO= (2m\u000b)\u00001is the spin-orbit length. Notice\nthe relation L2\nSO=D\u001cs, which amounts to say that the\nDyakonov-Perel relaxation time is the time over which\nelectrons di\u000buse over a spin-orbit length. \u001bsH\n0corre-\nsponds to the expression given in Ref. 13, i.e. the Drude\nformula evaluated without vertex corrections, of the spin\nHall conductivity\n\u001bsH\n0=e\n8\u00192\u001c\n\u001cDP: (11)\nIn the di\u000busive regime \u000bpF\u001c\u001c1, one has \u001cDP\u001d\u001c,\nwhereas in the ballistic one \u000bpF\u001c\u001d1\u001cDP\u0018\u001c. Notice\nthat, by using Eq.(11) and the expression for s0, the spin\ncurrent (10) can also be written as\nJz\ny=1\n2m\u000b1\n\u001cs(sy\u0000s0): (12)\nBy identifying the y-polarized spin current \rowing along\nthe x direction\nJy\nx=\u0000D@xsy(13)6\nthe di\u000busion equation becomes\n@tsy+@xJy\nx=\u00002m\u000bJz\ny=\u00001\nLSOJz\ny (14)\nwhich is the continuity-like equation for syshowing how\nthe torque term associated to syis expressed in terms of\nJz\ny.\nSingle-interface problem\nIn this Section we consider a static time-independent\nsituation. The idea is to analyze a single-interface prob-\nlem in order to understand the supercell numerical calcu-\nlation. We then assume the following expression for the\nRashba coe\u000ecient\n\u000b(x) =\u0012(x)\u000b++\u0012(\u0000x)\u000b\u0000: (15)\nClearly at\u00061, one recovers the uniform case with com-\nplete cancellation of the spin current for the Rashba\nmodel with couplings \u000b\u0006. The strategy is to solve the\ndi\u000busion equation in the two regions and connect them\nvia the appropriate boundary conditions at x= 0. We\nseek a solution of the form\nsy(x) =\u0012(x)\u0010\ns0;++\u000es+e\u0000x=L+\u0011\n+\u0012(\u0000x)\u0010\ns0;\u0000+\u000es\u0000ex=L+\u0011\n: (16)\nIn the above s0;\u0006are the asymptotic values of the y-\nspin polarization at \u00061. AlsoL\u0006are the corresponding\nspin-orbit lengths in the two regions. The z-polarized\nspin current in the two regions reads\nJz\ny;\u0006(x) =\u001bsH\n0;\u0006Ex+D\u0006\nL\u0006\u0010\ns0;\u0006+\u000es\u0006e\u0007x=L\u0006\u0011\n=D\u0006\nL\u0006\u000es\u0006e\u0007x=L\u0006; (17)\nwhere in the last step we used the fact that the constant\nterms cancel in each region. D\u0006may di\u000ber in the two\nregion because via the Fermi velocity they depend on\nthe electron density. Eq.(17) shows that in the interface\nregion there can be a spin current di\u000berent form zero. To\nevaluate it we need to know the two values \u000es\u0006. To this\nend we use the continuity of the spin density syand of\nthe spin current Jy\nxat the interface. Continuity of the\nspin density gives\n\u000es\u0000\u0000\u000es+= \u0001s0; (18)\nwhere \u0001s0=s0;+\u0000s0;\u0000=\u0000eN0\u001c(\u000b+\u0000\u000b\u0000)Ex. Conti-\nnuity of the spin current instead gives\nD\u0000\nL\u0000\u000es\u0000+D+\nL+\u000es+= 0: (19)After solving the system for \u000es\u0000and\u000es+,we get,\n\u000es\u0000= \u0001s0D\u0000\nL\u00001\nD+L\u00001\n++D\u0000L\u00001\n\u0000(20)\n\u000es+=\u0000\u0001s0D+\nL+1\nD+L\u00001\n++D\u0000L\u00001\n\u0000: (21)\nFrom this it is clear that the spin Hall current averaged\nover the spin-orbit length\n1\nL\u0000Z0\n\u00001dx Jz\ny;\u0000+1\nL+Z1\n0dx Jz\ny;+= 0; (22)\nwhich is the analog of Eq.(14). Furthermore the left-hand\nside correspond to the quantity \rin the main text. On\nthe other hand we have that the total z-polarized spin\ncurrent is di\u000berent from zero\nZ0\n\u00001dx Jz\ny;\u0000+Z1\n0dx Jz\ny;+= (23)\n\u00002eN0D+D\u0000m(\u000b+\u0000\u000b\u0000)2\u001c\nD+L\u00001\n++D\u0000L\u00001\n\u0000Ex\u0011\u001bsHLeffEx;\nwhereLeffis an e\u000bective length determined in terms of\nL\u0006andD\u0006. The spin Hall current is localized at the\ninterface within a distance of order L\u0006.\nThe above calculation of a single interface suggests\nthat the calculation for the periodic modulation de-\nscribed in the main text can be analyzed in terms of a\nseries of interfaces. The spin current \rows in the interface\nregions. However, by making the interface separation of\nthe order of the spin relaxation length, one may have a\nspin current \fnite everywhere.\nTwo-interface problem\nHere we consider the problem with two interfaces with\nthe model given by\n\u000b(x) =\u000b+\u0012(l\u0000jxj) +\u000b\u0000\u0012(jxj\u0000l): (24)\nThe solution for the spin density sy(x) is of the form\nsy(x) =\u0012(x\u0000l)h\ns0;\u0000+\u000esRe\u0000(x\u0000l)=L\u0000i\n+\u0012(\u0000x\u0000l)h\ns0;\u0000+\u000esLe(x+l)=L\u0000i\n+\u0012(l\u0000jxj)\u0014\ns0;+\u0000\u0001s0cosh(x=L +)\ncosh(l=L +)\n+\u000esLsinh((l\u0000x)=L+)\nsinh(2l=L +)+\u000esRsinh((l+x)=L+)\nsinh(2l=L +)\u0015\nwhere the continuity of syhas already been implemented.\nThe two constants \u000esRand\u000esLmust be determined by\nimposing the conservation of the longitudinal y-polarized\nspin current Jy\nxas in the single-interface problem.7\nAfter going through steps as for the single-interface we\nget\nZ1\n\u00001dx Jz\ny(x) = (25)\n\u00004eN0D+D\u0000m(\u000b+\u0000\u000b\u0000)2\u001ctanh(l=L +)\nD+L\u00001\n+tanh(l=L +) +D\u0000L\u00001\n\u0000Ex:\nThe above equation has a simple interpretation in the\nlimitl!1 , when the two interfaces are far apart. In\nthis limit the two interfaces are independent. The total\nspin current is the sum of the spin currents \rowing at\nthe two interfaces. Eq.(25) reduces indeed to twice the\ncontribution (23) of a single interface. On the other hand\nwhenl\u0018L+the two interfaces interact and the spin Hall\ncurrent in non zero everywhere.\nContinuity equations for the Rashba lattice model\nWe provide here a discussion to clarify the role of an in-\nhomogeneous Rashba SOC to allow for a SHE in station-\nary conditions. The electron spin Sobeys the equation\nof motion\ndS\ndt=\u0000i[S;H] (26)\nand for the following it is convenient to separate the com-\nmutator into a term related to the divergence of spin\ncurrents and a 'rest' G\n[S;H] =\u0000idivJ+iG: (27)\nAs a result Eq. (26) can be interpreted in terms of a\ncontinuity equation\nG\u000b=divJ\u000b+dS\u000b\ndt(28)\nwhere Gacts as 'source' term which in general is \fnite\ndue to the non-conservation of spin.\nFrom evaluation of the commutators one \fnds the re-\nlation\nGy\ni=\u0000\u000bi;i+yJz\ni;i+y\u0000\u000bi\u0000y;iJz\ni\u0000y;i:; (29)\ni.e. a torque for the y-component of the spin is associated\nwith a z-spin current when the Rashba SOC \u000b6= 0. Upon\ncombining Eq. (29) with the y-component of Eq. (28)\none \fnds\ndSy\ni\ndt+ [divJy]i+\u000bi;i+yJz\ni;i+y+\u000bi\u0000y;iJz\ni\u0000y;i= 0 (30)\nand in particular for a system with periodic boundaries\nX\ni\u001adSy\ni\ndt+\u000bi;i+yJz\ni;i+y+\u000bi\u0000y;iJz\ni\u0000y;i\u001b\n= 0 (31)since the total divergence of any current has to vanish.\nFor a homogeneous SOC coupling Eq. (31) implies that\nthe total z-spin current has to vanish under stationary\nconditions.\nWe exemplify the situation for a inhomogeneous\nRashba SOC which varies along the x-direction as\n\u000bi;i+x=1\n2\u0014\na0+a1+ (a0\u0000a1) sgn(sin2\u0019ix\n2L)\u0015\n(32)\n\u000bi;i+y=\u000bi;i+x; (33)\ni.e. one has stripes of width Lwith Rashba SOC a0alter-\nnating with L-wide stripes having coupling a1(cf. Fig.\n5). The inhomogeneity in \u000bi;i+xleads to a concomitant\ncharge modulation which is shown in Fig. 6 for the case\nL= 4. Clearly the charge accumulates in regions with\na0a0a0\na1a1a1\n2 L\nFIG. 5. Stripe-like modulation of the Rashba SOC along the\nx-direction. Stripes of width Land coupling a0alternate with\nstripes of the same width and coupling a1as indicated by thin\nand thick bonds, respectively.\nlarge Rashba SOC leading to a CDW pro\fle. We can\ntherefore view this model as an 'e\u000bective' model for a\ndensity dependent Rashba SOC.\n0 5 10 15 20 25\nsite ix00.040.080.120.160.2charge density\n0 5 10 15 20 2500.20.40.60.81\nRashba coupling [t]\nFIG. 6. Modulation of the coupling constant \u000bi;i+x(red) and\ncharge density along the x-direction for the L= 4 stripe-like\nRashba SOC. Particle concentration: n= 0:07.\nFig. 7 shows the currents \rowing in the (stationary)\nground state. Thus from Eq. (28) the torques Gy\niare\ncompletely determined by the divergence of the Jyspin\ncurrents which are shown by squares in the top panel\nof Fig. 7. As a consequence of Eq. (29) a large z-spin8\ncurrent is \rowing on sites where also the y-torque is large\nin contrast to a homogeneous system where Jz= 0. In\nthe ground state the total y-torqueP\niGy\nivanishes so\nthat from Eq. (29) one obtains\n0 =X\ni\u0002\n\u000bi;i+yJz\ni;i+y+\u000bi\u0000y;iJz\ni\u0000y;i\u0003\n: (34)\n 0 5 10 15 20 25\n 0  5  10  15  20  25\n 0 5 10 15 20 25\n 0  5  10  15  20  25\nFIG. 7. Spin currents (arrows) and torques (squares) in the\nground state for a system with alternating L= 4 stripes with\nRashba couplings a0= 0:2 anda1= 0:8, respectively. Top\npanel: y-components; Bottom panel: z-components. Particle\nconcentration: n= 0:07.\nThe total number of a0;1-stripes for a Nx\u0002Nylattice\nisnstr=Nx=(2L). Denote with Jz\n0;1the total z-spin\ncurrent \rowing along the bonds of the a0;1-stripes. Then\nwe can rewrite Eq. (34) as\n0 =a0Jz\n0+a1Jz\n1\u0000!Jz\n1=\u0000a0\na1Jz\n0 (35)\nand the total z-spin current of the system is thus givenby\nJz\ntot=nstr(Jz\n0+Jz\n1) =nstrJz\n0\u0012\n1\u0000a0\na1\u0013\n: (36)\nNow we can draw the following conclusions: The mod-\nulated Rashba SOC causes local torques Gy\niwhich are\nrelated to local \rows of z-spin currents. Provided that\nthe total y-torque vanishes the system thus exhibits a net\n\row ofJz\ntotfora06=a1in the ground state.\nThe same analysis can now be applied in the presence\nof an electric \feld Ex=\u0000@tAxwhich couples to the sys-\ntem via the charge current, i.e. H0=\u0000eP\nijch\ni;i+xAx(i;t)\nandjch\ni;i+x. Obviously Eq. (30) holds also in the result-\ning non-equilibrium situation which we evaluate in linear\nresponse. Each operator ^Oiin Eq. (30) reacts to the\nelectric \feld according to ^Oi(!) =ie\u0003ij(!)Ej(!)=!and\nthe correlation function is given by\n\u0003ij(!) =\u0000i\nNZ1\n\u00001dt\u0002(t\u0000t0)ei!(t\u0000t0)hh\n^Oi(t);jch\nj;j+x(t0)i\ni:\n0 5 10 15 20 25\nsite ix-5e-0505e-050.0001Jz(ix) / E  ; dSy(ix) / dt / E\n0 5 10 15 20 250.20.40.60.8\nRashba coupling [t]\n0 5 10 15 20 25\nsite ix-0.0002-0.000100.00010.0002Jy(ix) / E  ; div Jy / E; γ(ix) / EJy\ndiv Jy\nγ\n0 5 10 15 20 250.20.40.60.8\nRashba coupling [t]\nFIG. 8. Top panel: Modulation of the coupling constant\n\u000bi;i+x(red) and induced Jz-current (black, circles) for a cut\nalong the x-direction and L= 4 stripe-like SOC. The blue\nline (squares) shows the temporal change of the y-spin com-\nponentdSy(ix)=dt. Bottom panel: Induced Jyspin current\n(black, open circles) and corresponding divergence (black,\nfull circles). The blue line (squares) shows the response of\n\rix= 2\u000bi;i+yJz\ni;i+yalong the same cut. Particle concentra-\ntion:n= 0:07.\nThe top panel of Fig. 8 shows the induced Jzspin\ncurrent together with the induced temporal change of9\ndSy(ix)=dtalong a cut in x-direction of a L= 4 striped\nsystem. The dominant contribution to Jzcomes from the\nboundary regions between small and large \u000bstripes which\ngives rise to a \fnite spin Hall conductivity. Moreover, in\ncontrast to the homogeneous case, where the induced Jz\ncurrent and dSy(ix)=dtare equal\n@Sy\n@t=\u00002m\u000b\n~Jz\ny: (37)\n(although with opposite sign), we now observe a much\nmore stationary behavior. The reason can be deduced\nfrom the bottom panel which reports the x-dependence\nof the induced Jyspin current along with its diver-\ngence, i.e. [ divJy]i\u0011Jy\nix;ix +1(ix)\u0000Jy\nix\u00001;ix. It turns\nout that the spatial behavior of the contribution \rix=\n2\u000bi;i+yJz\ni;i+y(ix) is similar to [ divJy]ibut opposite in sign\nwhich from Eq. (30) is responsible for the small value of\ndSy(ix)=dt.\nFrom the above considerations we see that in a ho-\nmogeneous Rashba system a \fnite spin Hall conductivity\nnecessarily implies a non-stationary situation with the lo-\ncal accumulation of Syspin density. On the other hand\nan inhomogeneous Rashba coupling partially shifts this\ntime dependence to a \fnite divergence of the Jyspin\ncurrents which is stationary due to non-conservation of\nspin.\n[1] M. I. Dyakonov and V. I. Perel, Phys. Lett. A 35, 459\n(1971).\n[2] G. Vignale, J. Supercond. Nov. Magn. 23, 3 (2010).\n[3] T. Jungwirth, J. Wunderlich, and K. Olejn\u0013 \u0010k, Nat. Ma-\nterials 11, 382 (2012).\n[4] I. \u0014Zuti\u0013 c, J. Fabian, and S. Das Sarma, Rev. Mod. Phys.\n76, 323 (2004).\n[5] S. Maekawa, H. Adachi, K. Uchida, J. Ieda, and E.\nSaitoh, J. Phys. Soc. Japan 82, 102002 (2013).\n[6] Yu. A. Bychkov and E. I. Rashba, J Phys C: Solid State\nPhys. 17, 6039 (1984).\n[7] J. C. Rojas S\u0013 anchez, L. Vila, G. Desfonds, S. Gambarelli,\nJ. P. Attan\u0013 e, J. M. De Teresa, C. Mag\u0013 en, and A. Fert,\nNat. Commun. 4, 2944 (2013).\n[8] J. Nitta, T. Akazaki, and H. Takayanagi, Phys. Rev. Lett.\n78, 1335 (1997).\n[9] A. D. Caviglia, M. Gabay, S. Gariglio, N. Reyren, C.\nCancellieri, and J.-M. Triscone, Phys. Rev. Lett. 104,\n126803 (2010).\n[10] A. Fete, S. Gariglio, A. D. Caviglia, J.-M. Triscone, and\nM. Gabay, Phys. Rev. B 86, 201105 (2012).[11] Lorien H. Hayden, R. Raimondi, M. E. Flatt, and G.\nVignale Phys. Rev. B 88, 075405 (2013).\n[12] S. Hurand, A. Jouan, C. Feuillet-Palma, G. Singh, J. Bis-\ncaras, E. Lesne, N. Reyen, Thales, C. Ulysse, X. Lafosse,\nM. Pannetier-Lecoeur, S. Caprara, M. Grilli, J. Lesueur,\nN. Bergeal, arXiv:1503.00967 .\n[13] Jairo Sinova, Dimitrie Culcer, Q. Niu, N. A. Sinitsyn, T.\nJungwirth, and A. H. MacDonald Phys. Rev. Lett. 92,\n126603 (2004).\n[14] S. Caprara, F. Peronaci, and M. Grilli, Phys. Rev. Lett.\n109, 196401 (2012); D. Bucheli, M. Grilli, F. Peronaci,\nG. Seibold, and S. Caprara, Phys. Rev. B 89, 195448\n(2014).\n[15] V. M. Edelstein, Solid State Communications 73, 233\n(1990).\n[16] A. G. Aronov and Yu. B. Lyanda-Geller, JETP Lett. 50,\n431 (1989).\n[17] Ol'ga V. Dimitrova, Phys. Rev. B 71, 245327 (2005).\n[18] J. I. Inoue, G. E. W. Bauer, and L. W. Molenkamp, Phys.\nRev. B 70, 041303(R) (2004).\n[19] E. G. Mishchenko, A. V. Shytov, and B. I. Halperin,\nPhys. Rev. Lett. 93, 226602 (2004).\n[20] R. Raimondi and P. Schwab, Phys. Rev. B 71, 033311\n(2005).\n[21] A. Khaetskii, Phys. Rev. Lett. 96, 056602 (2006).\n[22] C. Gorini, P. Schwab, R. Raimondi and A.L. Shelankov,\nPhys. Rev. B 82, 195316 (2010).\n[23] R. Raimondi, P. Schwab, C. Gorini, G. Vignale, Ann.\nPhys. (Berlin) 524, 153 (2012).\n[24] G. Seibold, D. Bucheli, S. Caprara, and M. Grilli, Euro-\nphys. Lett. 109, 17006 (2015).\n[25] E. B. Sonin, Phys. Rev. B 76, 033306 (2007).\n[26] D. N. Sheng, L. Sheng, Z. Y. Weng, and F. D. M. Hal-\ndane, Phys. Rev. B 72, 153307 (2005).\n[27] K. Nomura, Jairo Sinova, N. A. Sinitsyn, and A. H. Mac-\nDonald, Phys. Rev. B 72, 165316 (2005).\n[28] A. Agarwal, Stefano Chesi, T. Jungwirth, Jairo Sinova,\nG. Vignale, and Marco Polini, Phys. Rev. B 83, 115135\n(2011).\n[29] K. C. Hass, B. Velicky, and H. Ehrenreich, Phys. Rev. B\n29, 3697 (1984).\n[30] See Supplemental Material at http://link.aps.org/ sup-\nplemental/..... for a detailed description of the single and\ndouble interface case in the di\u000busive limit, for the deriva-\ntion of a continuity equation on a lattice in the pres-\nence of inhomogeneous Rashba SOC. Numerical results\nfor a spin and spin-current distribution on a lattice with\nstriped Rashba SOC are also reported.\n[31] G. Khalsa, B. Lee, and A. H. MacDonald, Phys. Rev. B\n88, 041302(R) (2013).\n[32] Z. Zhong, A. T\u0013 oth, and K. Held, Phys. Rev. B 87,\n161102(R) (2013).\n[33] X. F. Wang, Phys. Rev. B 69, 035302 (2004).\n[34] G. I. Japaridze, Henrik Johannesson, and Alvaro Ferraz,\nPhys. Rev. B 80, 041308(R) (2009).\n[35] Mariana Malard, Inna Grusha, G. I. Japaridze, and Hen-\nrik Johannesson, Phys. Rev. B 84, 075466 (2011).\n[36] R. Raimondi et al. Phys. Rev. B 74, 035340 (2006)."    },    {        "title": "1206.1062v1.Von_Neumann_Entropy_Spectra_and_Entangled_Excitations_in_Spin_Orbital_Models.pdf",        "content": "arXiv:1206.1062v1  [cond-mat.str-el]  4 Jun 2012VonNeumann Entropy Spectra and Entangled Excitations inSp in-Orbital Models\nWen-Long You,1,2Andrzej M. Ole´ s,1,3and Peter Horsch1\n1Max-Planck-Institut f¨ ur Festk¨ orperforschung, Heisenb ergstrasse 1, D-70569 Stuttgart, Germany\n2School of Physical Science and Technology, Soochow Univers ity, Suzhou, Jiangsu 215006, People’s Republic of China\n3Marian Smoluchowski Institute of Physics, Jagellonian Uni versity, Reymonta 4, PL-30059 Krak´ ow, Poland\n(Dated: October 21, 2018)\nWe consider the low-energy excitations of one-dimensional spin-orbital models which consist of spinwaves,\norbital waves, and joint spin-orbital excitations. Among t he latter we identify strongly entangled spin-orbital\nbound states which appear as peaks in the von Neumann entropy (vNE) spectral function introduced in this\nwork. Thestrongentanglement ofboundstatesismanifested byauniversallogarithmicscalingofthevNEwith\nsystemsize,whilethevNEofotherspin-orbitalexcitation ssaturates. Wesuggestthatspin-orbitalentanglement\ncan be experimentally explored by the measurement of the dyn amical spin-orbital correlations using resonant\ninelasticx-ray scattering, where strongspin-orbit coupl ing associatedwiththe core hole plays a role.\nPACS numbers: 75.10.Jm,03.65.Ud, 03.67.Mn,75.25.Dk\nIntroduction.— The spin-orbital interplay is one of the im-\nportant topics in the theory of strongly correlated electro ns\n[1]. In many cases, the intertwined spin-orbital interacti on is\ndecoupledby mean-field approximation,and the spin and or-\nbital dynamicsare independentfromeach other. Thusa spin-\nonly Heisenberg model can be derived by averaging over the\norbitalstate,whichsuccessfullyexplainsmagnetismando pti-\ncalexcitationsinsomematerials,forinstanceinLaMnO 3[2].\nBut in others,especiallyin t2gsystems[3], the orbitaldegen-\neracy plays an indispensable role in understanding the low-\nenergy properties in the Mott insulators of transition meta l\noxides (TMOs), such as LaTiO 3[4], LaVO 3and YVO 3[5],\nand also in recently discussed RbO 2[6]. The well known\ncasesarealsostrongspin-orbitcouplingwhichleadstoloc ally\nentangled states [7], and entanglement on the superexchang e\nbondsin K 3Cu2F7[8]. For such models, the mean-field-type\napproximation and the decoupling of composite spin-orbita l\ncorrelations fail and generate uncontrolled errors, even w hen\nthe orbitals are polarized [9]. The strong spin-orbital fluc tu-\nations on the exchange bondswill induce the violation of the\nGoodenough-Kanamori rules [10]. Furthermore, the flavors\nmayformexoticcompositespin-orbitalexcitations.\nModel and system.— A paradigmatic model derived for a\nTMO in Mott-insulating limit is the one-dimensional (1D)\nspin-orbitalHamiltonian,whichreads\nH=−J/summationdisplay\ni/parenleftBig\n/vectorSi·/vectorSi+1+x/parenrightBig/parenleftBig\n/vectorTi·/vectorTi+1+y/parenrightBig\n,(1)\nwhere/vectorSiand/vectorTiare spin-1/2 and pseudospin-1/2 operators\nrepresenting the spin and orbital degrees of freedom locate d\nat sitei, respectively,and we set below J= 1. It is proposed\nthat ultracold fermions in zig-zag optical lattices can rep ro-\nduce an effectivespin-orbitalmodel[11]. For general {x,y},\nthe model (1) has an SU(2) ⊗SU(2) symmetry. An additional\nZ2bisymmetry occurs by interchanging spin and orbital op-\nerators when x=y. In the case of x=y=1\n4, Hamiltonian\n(1)reducestoaSU(4)symmetricmodel,whichisexactlysol-\nuble by the Bethe ansatz [12, 13]. There are three Goldstone\nmodes corresponding to separate spin and orbital excitatio n,aswellascompositespin-orbitalexcitationsincaseof J <0,\nin contrast to a quadratic dependence of the energy upon the\nmomentum in the long-wave limit for J >0. The spectra\nof elementary excitations are commonly not analytically so l-\nuble away from the SU(4) point. We will, however, show\nthat the low-energy excitations can be analytically obtain ed\nin some specific phases in the case when J >0, and this\noffers a platform to study the spin-orbital entanglement. I n\nthis Letter, we go beyond the ideas developed for spin sys-\ntems[14]. We demonstratethatspin-orbitalentanglemente n-\ntropy clearly distinguishes weakly correlated spin-orbit al ex-\ncitations from bound states and resonances by its magnitude\nanddistinct scaling behavior. We proposehow to connectthe\nentanglemententropywith experimentallyobservablequan ti-\ntiesofrecentlydevelopedspectroscopies.\nvonNeumannentropy.— Currently,conceptsfromquantum\ninformation theory are being studied with the aim to explore\nmany-body theory from another perspective and vice versa.\nA particularly fruitful direction is using quantum entangl e-\nment to shed light on exotic quantum phases [15, 16]. En-\ntanglement entropy even distinguishes phases in the absenc e\nof conventional order parameters [17]. In general a many-\nbody quantum system is subdivided into AandBparts and\ntheentanglemententropyisthevonNeumannentropy(vNE),\nSvN=−Tr{ρAlog2ρA},whereρA=TrB{ρ}isthereduced\ndensity matrix of the subspace Aandρis the full density\nmatrix. The vNE is bounded, SvN≤log2dimρA, and easy\nto calculate. Experimental determination appears harder, yet\nthereareproposalsinvolvingtransportmeasurementsinqu an-\ntumpointcontacts[18].\nInterestinglythevNEscalesproportionallytotheboundar y\nofthesubregionobtainedbythespatialpartitioning[19]. The\ndependenceoftheboundaryorarealawcanbetracedbackto\nstudyofblackholephysics[20]andwasextensivelyexploit ed\nfor 1D spin chains [21]. If the block Ais of length lin a sys-\ntemoflength Lwithperiodicboundarycondition,thevNEof\ngappedgroundstates is boundedas Sl=O(1), while a loga-\nrithmic scaling Sl=clog2l+O(1) (L≫l≫1)has been\nprovento be universal propertyof the gaplessphases in crit i-2\nFIG.1: (color online). Spin-orbital entanglement SvNinthe ground-\nstateofthespin-orbitalmodel(1)asafunctionof xandyandsystem\nsizeL= 8. The(red)dashedlinesmarkthecriticallinesdetermined\nby the fidelity susceptibility (see text). The two-site confi gurations\nin phases I-IV are shown on the left. The two orbitals per site are\ndegenerate (their splittingis onlyfor clarityof presenta tion).\ncal systems by the underlyingconformalfield theory [22]. A\nviolationofthe arealaw is expectedforthe low-lyingexcit ed\nstates of critical chains [23]. To date, measurements of the\nvNE for subdivisionof degrees of freedomother than in spa-\ntialsegmentationhavenotbeenfullyexplored. Inacomposi te\nsystem containing spin and orbital operators, the decompos i-\ntionofdifferentflavorsretainsthereal-spacesymmetries .\nPhase diagram.— A quantum phase transition (QPT) is\nidentified as a point of nonanalyticity of the groundstate an d\nassociated expectationvaluesin the thermodynamiclimit. To\nshedlightonthephaseboundaries,wefirst considertwosite s\n[24],H12=−1\n4(/vectorS2\n12−/vectorS2\n1−/vectorS2\n2+2x)(T2\n12−/vectorT2\n1−/vectorT2\n2+2y),\nwhere/vectorS12=/vectorS1+/vectorS2and/vectorT12=/vectorT1+/vectorT2. A pair of spins\n(orbitals)canformeither a singlet with S12= 0(T12= 0)or\na triplet with S12= 1(T12= 1), and various combinations\nof quantumnumberscorrespondto differentphases shown in\nFig. 1. In phase I, the state with S12= 1 =T12has the\nlowest energy, and thus the energy per bond is eI\nB≥exy=\n−(x+1/4)(y+1/4). For a largersystem with Lbonds, we\nhaveEI\n0(H)≥Lexy. On the other hand, takinga ferro-ferro\nstate|0∝an}bracketri}htas a variational state, EI\n0(H)≤Lexy. Therefore,\ntheenergyofphaseIisexactly EI\n0(H) =Lexyandtheferro-\nferrostate isthecorrespondinggroundstate.\nWithout prior knowledge of order parameter, variouschar-\nacterizations from the perspective of quantum information\ntheory can be used to identify phase boundaries. One often\nusedtoolisthevNE[25]. Tracingorbitaldegreesoffreedom ,\nwe obtained the spin-orbital vNE SvNfor the ground state of\nL= 8chain in the Hilbert subspace of Sz=Tz= 0[25].\nHowever, here we find that the vNE of the ground state does\nnotdistinguishphaseIfromphaseIIorIV—allthreephases\nhavingSvN= 0(seeFig. 1). Thereforeweusethequantumfi-\ndelitytoquantifythephasediagram[26]. Thefidelitydefine d/s48 /s49 /s50 /s51/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48\n/s48/s46/s50/s53 /s48/s46/s53/s48 /s48/s46/s55/s53 /s49/s46/s48/s48/s48/s49/s50/s51/s52/s32/s32/s40/s81 /s41\n/s81/s40/s97/s41\n/s32/s32\n/s120/s32/s81/s61/s48/s46/s50/s32\n/s32/s81/s61/s48/s46/s53 /s32\n/s32/s81/s61/s48/s46/s56/s32/s40/s98/s41\nFIG.2: (color online). (a) Energyspectra of 40-site spin-o rbital sys-\ntem atx=y= 1/4. Dashed lines in the spin-orbital continuum\ndenote the spin, orbital and OBS excitation, all degenerate ; the (red)\nsolid line below corresponds to the BS. (b) The decay rate Γof the\nOBSfor different momenta Qwithy=xatL→ ∞.\nasfollows, F(λ,δλ) =|∝an}bracketle{tΨ0(λ)|Ψ0(λ+δλ)∝an}bracketri}ht|, istakenalong\na certain path {x(λ),y(λ)}and reveals all phase boundaries.\nThe fidelity susceptibility, χF≡ −(2lnF)/(δλ)2|δλ→0, ex-\nhibits a peak at the critical point, and can be treated as a ver -\nsatile order parameter in distinguishing ground states [27 ]. It\nsignals the phase boundaries shown in Fig. 1. Remarkably,\nthe phase diagram found from the fidelity susceptibility for\nlargersystemsisthe sameastheonefor L= 2.\nExcitations.— In phase I of Fig. 1, with boundaries given\nby:x+y=1\n2,x=−1\n4andy=−1\n4, the spins and orbitals\nare fully polarized,and the ferro-ferrogroundstate |0∝an}bracketri}htis dis-\nentangled, i.e., can be factorized into spin and orbital sec tor.\nIt is now interesting to ask whether: (i) the vanishing spin-\norbitalentanglementinthegroundstateimpliesasuppress ion\nof joint spin-orbital quantum fluctuations, and (ii) collec tive\nspin-orbital excitations can form. Using equation of motio n\nmethod one finds spin (magnon) excitations with dispersion\nωs(q) = (1\n4+y)(1−cosq),andorbital(orbiton)excitations,\nωt(q) = (1\n4+x)(1−cosq)[28]. Thestabilityoftheorbitons\n(magnons) implies that x >−1\n4(y >−1\n4), and determines\nthe QPT between phasesI and II (IV), respectively,while the\nspin-orbitalcouplingonlyrenormalizesthespectra.\nFor our purpose, it is straightforward to consider the prop-\nagation of a pair of magnon and orbiton along the ferro-ferro\nchain, by simultaneously exciting a single spin and a single\norbital. The translation symmetry imposes that total momen -\ntumQ= 2mπ/L(m= 0,···,L−1)is conserved during\nscattering. The scattering of magnon and orbiton with initi al\n(final)momenta {Q\n2−q,Q\n2+q}({Q\n2−q′,Q\n2+q′})andtotal\nmomentum Qisrepresentedbythe Green’sfunction[29],\nG(Q,ω) =1\nL/summationdisplay\nq,q′/angbracketleftBig/angbracketleftBig\nS+\nQ\n2−q′T+\nQ\n2+q′|S−\nQ\n2−qT−\nQ\n2+q/angbracketrightBig/angbracketrightBig\n,(2)\nfor a combined spin ( S−\nQ\n2−q) and orbital ( T−\nQ\n2+q) excitation.3\n/s48 /s49 /s50 /s51/s81/s61/s48/s46/s48\n/s81/s61/s48/s46/s50\n/s81/s61/s48/s46/s52\n/s81/s61/s48/s46/s54\n/s81/s61/s48/s46/s56\n/s48/s50/s52\n/s83\n/s118/s78\nFIG. 3: (color online). The vNE distribution of 400-site spi n-orbital\nsysteminsubspace PST= 1forx=y= 1/2anddifferentmomenta\nQ. Isolated vertical lines indicate the BS, with dispersion g iven by\nthe (red) dashed line. The OBSinthe center of spectra is damp ed.\nThe spin-orbital continuum is given by Ω(Q,q) =ωs(Q\n2−\nq) +ωt(Q\n2+q). In the noninteracting case, the Green’s\nfunction exhibits square-root singularities at the edges o f the\ncontinuum [30]. Due to residual, attractive interactions s pin-\norbital bound states (BSs) are shifted outside the continuu m\n[24,31,32], see Fig. 2(a). Thecollectivemodeis determine d\nby1 +1\n2π/integraltextπ\n−πdq(cosQ\n2−cosq)2/[ω−Ω(Q,q)] = 0. The\nanalytic solution of this equation is tedious but straightf or-\nward. The collective BS with dispersion ωBS(Q)is well sep-\narated from the spinon-orbitoncontinuum[Fig. 2(a)] at lar ge\nQ. In the long-wave limit the BS energy coincides with the\nArovas-Auerbach line [33], i.e., the boundary of the contin -\nuum,yetthe BS remainsundampedfor x+y >1\n2.\nInaddition,acollectivemodeofspin-orbitalresonances,\n|Ψ(Q)∝an}bracketri}ht=1√\nL/summationdisplay\nm,lal(Q)eiQmS−\nmT−\nm+l|0∝an}bracketri}ht,(3)\noccurs inside the continuum. Here 0≤l≤L−1de-\nnotes the distance between spin and orbital flips. Remark-\nably, the spin and orbital flips are glued together at the same\nsite with al(Q) =δl,0at the SU(4) point [28]. This cou-\npled on-site BS (OBS) is a coherent superposition of local\nmodes, all of them with equal weight. It has dispersion\nωOBS(Q) =x+y−1\n2cosQ, which is degenerate with both\nωs(Q)andωt(Q)atx=y=1\n4, see Fig. 2(a). This is remi-\nniscentofthedegeneracyofthethreeGoldstonemodesat the\nSU(4) point for J=−1[12, 13]. Moving away from the\nSU(4) point,the OBS decaysdueto residualinteractionsint o\nmagnon-orbitonpairs, and the mean separation ξof spin and\norbital excitations increases, i.e., al(Q)∼exp(−l/ξ), lead-\ning in the thermodynamic limit to a finite linewidth defined\nbyΓ =ImG−1(Q,ω)[34]. Thedecayrateofthespin-orbital\nOBS increases with growing x >1\n4and also for decreasing\nmomenta Q,asseeninFig. 2(b).\nEntropy spectral function.— To investigate the degree of\nentanglementofexcitedstates,weintroducethevNEspectr al/s55 /s56 /s57 /s49/s48/s53/s54/s55/s56/s57/s49/s48\n/s48/s46/s48/s48/s49/s48 /s48/s46/s48/s48/s49/s53 /s48/s46/s48/s48/s50/s48/s51/s46/s54/s52/s46/s48/s52/s46/s52/s40/s97/s41\n/s32/s32/s83\n/s118/s78\n/s108/s111/s103\n/s50/s76/s32/s120/s61/s48/s46/s50/s53/s32/s121/s61/s48/s46/s50/s53/s32\n/s32/s120/s61/s48/s46/s53/s48/s32/s121/s61/s48/s46/s51/s48/s32\n/s32/s120/s61/s48/s46/s53/s48/s32/s121/s61/s48/s46/s53/s48/s32/s81/s61/s48/s46/s56\n/s81/s61/s48/s46/s54/s81/s61/s48/s46/s50\n/s32/s32/s83\n/s118/s78\n/s49/s47/s76/s40/s98/s41\n/s52 /s54 /s56/s52/s54/s56/s83\n/s118/s78\n/s108/s111/s103\n/s50/s76\nFIG.4: (color online). (a) Scalingbehavior of entanglemen t entropy\nSvNof the spin-orbital BSs for Q= 0.8π. Lines represent loga-\nrithmicfits SvN= log2L+c0,withc0=−0.659,−1.059,−1.251,\nrespectively. (b)Thescalingbehaviorofentanglementent ropyofthe\nOBS forx=y= 1/2. Lines are fitted by SvN=c1/L+c0, with\nc0(c1) = 3.69 (380.5), 3.37 (138.4) and 3.31 (47.6) for Q= 0.8π,\n0.6πand0.2π. The inset shows the logarithmic behavior of SvNfor\nthe OBSwith Q= 0.8πandx=y= 1/4.\nfunctionin theLehmannrepresentation,\nSvN(Q,ω) =−/summationdisplay\nnTr{ρ(µ)\nslog2ρ(µ)\ns}δ{ω−ωn(Q)},(4)\nwhere(µ) = (Q,ωn)denote momentum and excitation en-\nergy, and ρ(µ)\ns=Tro|Ψn(Q)∝an}bracketri}ht∝an}bracketle{tΨn(Q)|is obtained by tracing\nthe orbital degreesof freedom. Let us first consider the sym-\nmetric case, i.e., x=y. The Hilbert space can be divided\ninto two subspaces characterized by the parity PSTof the in-\nterchangeof S↔T, which is odd or even. Translation sym-\nmetry allows us to express the reduceddensity matrix ρsin a\nblock-diagonalform,whereeachblockcorrespondstoanirr e-\nduciblerepresentationlabeledbytotalmomentum Qandpar-\nity of exchangesymmetry PST. The vNE can be obtained by\ndiagonalizing separately these blocks. In particular, the non-\ndegenerate eigenstates with odd parity can be explicitly ca st\nin the form1√\n2(S−\nQ/2−qT−\nQ/2+q−S−\nQ/2+qT−\nQ/2−q)|0∝an}bracketri}ht. Con-\nsequently, the singlet-like pair results in SvN= 1. For other\nspin-orbital eigenstates with PST= 1,SL≥1, except the\npure spin and orbital waves. Interestingly, we find that the\nparity is still conserved in subspace Q= 0forx∝ne}ationslash=y. The\nstrongly entangled spin-orbital BSs are reflected by peaks i n\nthe von Neumann spectra SvN(ω), shown in Fig. 3. As mo-\nmentumQdecreases, the OBS-peak in the center of spectra\ngetsbroader,implyingashorterlifetime.\nInspection of vNE spectra shows that the entanglement\nreaches a local maximum at the BSs. Finite size scaling\nof vNE of spin-orbital BSs reveals the asymptotic logarith-\nmic scaling SvN= log2L+c0shown in Fig. 4(a). The\nsame logarithmic scaling is found for the OBS at the SU(4)\npointx=y=1\n4, as seen in the inset of Fig. 4(b). How-\never, far away from the SU(4) point the scaling is entirely4\n/s48 /s49 /s50/s48/s49/s48/s50/s48\n/s49 /s50 /s51 /s49 /s50 /s51/s48/s51/s54/s32/s32/s65\n/s48/s40/s81/s44 /s41/s32\n/s40/s99/s41 /s40/s98/s41 /s40/s97/s41/s32/s65\n/s49/s43/s40/s81/s44 /s41/s32\nFIG.5: (coloronline). Thespectralfunctionofthe on-site excitation\nA0(Q,ω)for: (a)x=y= 1/4, (b)x=y= 1/2; (c) the nearest-\nneighbor A1+(Q,ω)forx=y= 1/2. The momenta range from\nπ/10(bottom) to 9π/10(top); the peak broadening is η= 0.01.\nDashed(red)anddotted(green)linescorrespondtotheBSan dOBS,\nwhile graydash-dot lines indicate the boundaries of the con tinuum.\ndifferent and the entropy of the OBS scales as a power law,\nSvN=c1/L+c0,asseeninFig. 4(b). Thischangeofscaling\nfromlogarithmictopowerlawin 1/Liscontrolledbythecor-\nrelation length ξmeasuring the average distance of spin and\norbital excitations in the OBS wave function (3). From Eq.\n(3) andal(Q)∼exp(−l/ξ)we obtain,\nSvN≃log2{L/(1+ξ)}, (5)\nwhich yields log2Latx=y= 1/4whereξ= 0. Asξ\nincreases the correction to the vNE is ∝ −log2(1 +ξ). Far\nawayfromtheSU(4)point,theOBSisdampedand ξbecomes\nextensive, i.e., ξ/L≈˜c0−˜c1/L, and the vNE approachesa\nfinite value with a correction ∝1/Las shown in Fig. 4(b).\nThisclosecorrespondenceofthevNEofboundstatesandthe\ncorrelation length ξsuggests to use the dynamic spin-orbital\ncorrelation function as a probe of spin-orbital entangleme nt\nandasaqualitativemeasureofthe vNEspectra.\nSpectralfunctions.— Returningto TMOs, onerealizes that\njoint spin-orbital excitations are not created in the ferro -ferro\nground state in photoemission spectroscopy because of spin -\nconservation. Onthecontrary,resonantinelasticx-raysc atter-\ning(RIXS)[35–39]isinprincipleabletomeasurethespectr al\nfunctionofthe coupledspin-orbitalexcitationsat distan cel,\nAl(Q,ω) =1\nπlim\nη→0Im/angbracketleftbigg\n0/vextendsingle/vextendsingle/vextendsingle/vextendsingleΓ(l)†\nQ1\nω+E0−H−iηΓ(l)\nQ/vextendsingle/vextendsingle/vextendsingle/vextendsingle0/angbracketrightbigg\n.\n(6)\nHereΓ(0)\nQ=1√\nL/summationtext\njeiQjS−\njT−\njis the local excitation op-\nerator for an on-site spin-orbital excitation. We use as wel l\nΓ(1±)\nQ=1√\n2L/summationtext\njeiQj(S−\nj+1±S−\nj−1)T−\njfor the nearest-\nneighborexcitation. IntheRIXSprocessanelectronwithsp in\nup is excited by the incoming x-rays from a deep-lying core\nlevel into the valence shell. For the time of its existence th ecore hole generates a Coulomb potential and a strong spin-\norbit coupling that allows for the non-conservation of spin .\nNext the hole is filled by an electron from the occupied va-\nlence band under the emission of an x-ray. This RIXS pro-\ncess creates a joint spin-valence excitation with momentum\nQin−Qoutand energy ωin−ωout, which can unveil the spec-\ntralfunctionofthespin-orbitalexcitation.\nThe on-site spectral function A0(Q,ω)shown in Figs.\n5(a,b)highlightstheOBS.AttheSU(4)point[Fig. 5(a)]ita p-\npearsasa δ-function, A0(Q,ω) =δ{ω−ωOBS(Q)},whereas\nin Fig. 5(b) the OBS is damped and its intensity decreases\nstrongly with Q. In the latter figure the BS at the low en-\nergysideofthecontinuumappearsasweakadditionalfeatur e,\nwhile it is absent in (a), i.e., at the SU(4) point. The neares t\nneighborspectralfunction A1+(Q,ω)inFig. 5(c)showsboth\nthe spin-orbital continuum and the BS outside of the contin-\nuum. Notably, comparing with the vNE spectral function in\nFig. 3, we find the same characteristic energies and similar\nintensity features as in the RIXS spectra. The spectral func -\ntion provides information of various correlations, which a re\ningredientsto derivethereduceddensitymatrices[40].\nSummary.— In this Letter, we study a spin-orbital system\nand extend the analysis of entanglement to excited states by\nintroducing the vNE spectral function. Our study demon-\nstrates that even in cases where the ground state of a spin-\norbitalchainisfullydisentangled,e.g.,intheferro-fer rostate,\n(i) the spin-orbital excitations are in general entangled, (ii)\nmaximal spin-orbital entanglement occurs for BSs which ap-\npearassharppeaksinthevNEspectra,and(iii)thevNEofun-\ndamped BSs exhibits a logarithmic dependence on the chain\nlengthL. We propose to study the dynamic spin-orbital cor-\nrelation functionas a qualitativemeasure of the vNE spectr a,\nandsuggestto usehereRIXS asa promisingtechnique.\nW-L.Y. acknowledges support by the National Natural\nScience Foundation of China (NSFC) under Grant No.\n11004144. A.M.O. acknowledges support by the Polish Na-\ntionalScienceCenter(NCN)underProjectNo. N202069639.\n[1] Y. Tokura andN. Nagaosa, Science 288, 462 (2000).\n[2] L.F. Feiner and A.M. Ole´ s, Phys. Rev. B 59, 3295 (1999);\nN.N. Kovaleva, A.M. Ole´ s, A.M. Balbashov, A. Maljuk, D.N.\nArgyriou, G. Khaliullin, and B. Keimer, ibid.81, 235130\n(2010).\n[3] SergeyKrivenko, Phys.Rev. B 85, 064406 (2012).\n[4] G. Khaliullin and S. Maekawa, Phys. Rev. Lett. 85, 3950\n(2000).\n[5] PeterHorsch,AndrzejM.Ole´ s,LouisFelixFeiner,andG iniyat\nKhaliullin, Phys.Rev. Lett. 100, 167205 (2008).\n[6] K. Wohlfeld, M. Daghofer, and A.M. Ole´ s, Europhys. Lett .96,\n27001 (2011).\n[7] G. Jackeli and G. Khaliullin, Phys. Rev. Lett. 102, 017205\n(2009).\n[8] W.Brzezicki and A.M.Ole´ s, Phys.Rev. B 83, 214408 (2011).\n[9] K. Wohlfeld, M. Daghofer, S. Nishimoto, G. Khaliullin, a nd\nJ. van denBrink, Phys.Rev. Lett. 107, 147201 (2011).5\n[10] AndrzejM.Ole´ s,PeterHorsch,LouisFelixFeiner,and Giniyat\nKhaliullin,Phys.Rev. Lett. 96, 147205 (2006).\n[11] G. Sun, G. Jackeli, L. Santos, and T. Vekua, arXiv: 1112. 5082\n(unpublished).\n[12] You-Quan Li, Michael Ma, Da-Ning Shi, and Fu-Chun Zhang ,\nPhys.Rev. B 60, 12781 (1999).\n[13] B.Sutherland, Phys.Rev. B 12, 3795 (1975).\n[14] Vincenzo Alba, Maurizio Fagotti, and Pasquale Calabre se,\nJ.Stat.Mech. 10, P10020 (2009).\n[15] Hui Li and F.D.M. Haldane, Phys. Rev. Lett. 101, 010504\n(2008).\n[16] Hong Yao and Xiao-Liang Qi, Phys. Rev. Lett. 105, 080501\n(2010).\n[17] A.Kitaevand J.Preskill,Phys.Rev. Lett. 96, 110404 (2006).\n[18] I.KlichandL.Levitov, Phys. Rev. Lett. 102, 100502 (2009).\n[19] J.Eisert,M. CramerandM.B.Plenio, Rev.Mod. Phys. 82, 277\n(2010).\n[20] Luca Bombelli, Rabinder K. Koul, Joohan Lee, and Rafael D.\nSorkin, Phys. Rev. D 34, 373 (1986); Mark Srednicki, Phys.\nRev. Lett. 71, 666 (1993); S.W. Hawking, J. Maldacena, A.\nStrominger,J. HighEnergyPhys. 05, 001 (2001).\n[21] LuigiAmico,RosarioFazio,Andreas Osterloh,andVlat koVe-\ndral,Rev. Mod. Phys. 80, 517 (2008).\n[22] Christoph Holzhey, Finn Larsen, and Frank Wilczek, Nuc l.\nPhys.B424, 443 (1994).\n[23] Llu´ ısMasanes, Phys.Rev. A 80, 052104 (2009).\n[24] J. van den Brink, W. Stekelenburg, D.I. Khomskii, G.A.\nSawatzky, andK. I.Kugel, Phys.Rev. B 58, 10276 (1998).\n[25] Yan Chen, Z.D. Wang, Y.Q. Li, and F.C. Zhang, Phys. Rev. B\n75, 195113 (2007).[26] Shi-JianGu, Int.J. Mod. Phys.B 24, 4371 (2010).\n[27] Wen-LongYou,Ying-WaiLi,andShi-JianGu,Phys.Rev.E 76,\n022101 (2007).\n[28] Alexander Herzog, Peter Horsch, Andrzej M. Ole´ s, and J esko\nSirker, Phys.Rev. B 83, 245130 (2011).\n[29] Michael Wortis,Phys.Rev. 132, 85(1963).\n[30] T. Schneider, Phys.Rev. B 24, 5327 (1981).\n[31] JanBała,AndrzejM.Ole´ s,andGeorgeA.Sawatzky,Phys .Rev.\nB63, 134410 (2001).\n[32] S. Cojocaru and A. Ceulemans, Phys. Rev. B 67, 224413\n(2003).\n[33] D.P.Arovas and A.Auerbach, Phys.Rev. B 52, 10114 (1995).\n[34] Dysonequationreads G−1(Q,ω) =ω−ωOBS(Q)−Σ(Q,ω),\nwiththetotalself-energy Σ(Q,ω).TherealpartRe G−1(Q,ω)\nidentifies the effective energy pole, and the imaginary part of\nself-energy isthe inverse of lifetime τ,i.e.,Γ =ImΣ(Q,ω).\n[35] S.Ishihara andS.Maekawa, Phys.Rev. B 62, 2338 (2000).\n[36] M.W. Haverkort, Phys.Rev. Lett. 105, 167404 (2010).\n[37] L.J.P.Ament,M.vanVeenendaal,T.P.Devereaux, J.P.H ill,and\nJ. van denBrink, Rev. Mod. Phys. 83, 705(2011).\n[38] Filomena Forte, Luuk J.P. Ament, and Jeroen van den Brin k,\nPhys. Rev. Lett. 101, 106406 (2008).\n[39] J. Schlappa, K. Wohlfeld, K.J. Zhou, M. Mourigal, M.W.\nHaverkort, V. N. Strocov, L. Hozoi, C. Monney, S. Nishi-\nmoto, S. Singh, A. Revcolevschi, J.-S. Caux, L. Patthey, H.M .\nRønnow, J. van den Brink, and T. Schmitt, Nature 485, 82\n(2012).\n[40] I. Peschel andV. Eisler,J.Phys.A 42, 504003 (2009)."    },    {        "title": "2402.01080v1.Photonic_Spin_Orbit_Coupling_Induced_by_Deep_Subwavelength_Structured_Light.pdf",        "content": "Photonic Spin-Orbit Coupling Induced by Deep-Subwavelength Structured Light\nXin Zhang1,2, Guohua Liu1,2, Yanwen Hu1,2,3∗, Haolin Lin1,2, Zepei Zeng1,2,\nXiliang Zhang1,2, Zhen Li1,2,3, Zhenqiang Chen1,2,3, and Shenhe Fu1,2,3∗\n1Department of Optoelectronic Engineering, Jinan University, Guangzhou 510632, China\n2Guangdong Provincial Key Laboratory of Optical Fiber Sensing and Communications, Guangzhou 510632, China\n3Guangdong Provincial Engineering Research Center of Crystal and Laser Technology, Guangzhou 510632, China\nWe demonstrate both theoretically and experimentally beam-dependent photonic spin-orbit coupling\nin a two-wave mixing process described by an equivalent of the Pauli equation in quantum mechan-\nics. The considered structured light in the system is comprising a superposition of two orthogonal\nspin-orbit-coupled states defined as spin up and spin down equivalents. The spin-orbit coupling is\nmanifested by prominent pseudo spin precession as well as spin-transport-induced orbital angular\nmomentum generation in a photonic crystal film of wavelength thickness. The coupling effect is sig-\nnificantly enhanced by using a deep-subwavelength carrier envelope, different from previous studies\nwhich depend on materials. The beam-dependent coupling effect can find intriguing applications;\nfor instance, it is used in precisely measuring variation of light with spatial resolution up to 15 nm.\nI. INTRODUCTION\nSpin-orbit coupling (SOC), which refers to interaction of\na quantum particle’s spin with its momentum, is a fun-\ndamentally important concept. It has been extensively\ninvestigated in condensed matter physics [1, 2], atomic\nand molecular physics [3, 4] and contributes to exciting\nphenomena such as the spin Hall effect [5] and topolog-\nical insulators [6, 7]. Analogous photonic SOC is also\ndemonstrated in a variety of settings [8]. The photonic\nSOC refers to an interaction between the momentum of\nlight, which also includes spin angular momentum and\norbital angular momentum (SAM and OAM). Whereas\nthe SAM is associated with photon circular polarization\n[9], the OAM is relevant to a helical wavefront of light\ncharacterized by a topological number ℓ[10]. The pho-\ntonic SOC is crucial for the optical Hall effects [11–14],\nspin-to-orbital angular momentum conversions [15, 16],\nspin-orbit photonic devices [17–19], etc.\nThe SOC can be engineered in appropriately designed\nmaterials. For examples, engineering a tensional strain in\ngraphene shifts the electronic dispersions and induces a\ncontrollable vector potential for the electronic SOC [20–\n24]. Analogous strategy can be applied to engineer the\nphotonic SOC, by using strained evanescently coupled\nwaveguide arrays [25, 26]. Other approaches for manip-\nulating the photonic SOCs are demonstrated by appro-\npriately designing microcavities [27–32], metamaterials\n[33–36], photonic crystals [37–39], twisted optical fibers\n[40, 41], dual-core waveguides [42, 43], etc. The resultant\nSOCs are material-dependent, determined by geometric\nconfigurations of the materials which are often difficult\nto be tuned once fixed by designs. As a consequence,\na tunable photonic SOC process remains elusive. Re-\ncently, several engineered photonic SOC schemes have\nbeen reported, by either embedding a strained honey-\n∗Electronic address: huyanwen@jnu.edu.cn;fushenhe@jnu.edu.cncomb metasurface inside a cavity waveguide [44] or us-\ning an optical cavity filled with controllable liquid crys-\ntals [45]. However, the resultant photonic SOCs remain\nmaterial-dependent.\nIn this work, we report theoretically and experimen-\ntally a new mechanism for engineering the photonic\nSOC. We demonstrate this by exploiting analogy be-\ntween quantum description of a spin-1/2 system and a\nspin-orbit Hamiltonian derived for structured light in a\nphotonic crystal. The obtained Hamiltonian is closely\nrelevant to structured light, which means that the SOC\ncan be engineered by controlling carrier envelope rather\nthan the structures of materials. Strong SOC is achieved\nby using deep-subwavelength structured light, as man-\nifested by clear pseudo spin precessions. Although the\nstructured light has been extensively investigated in re-\ncent year [46–51], the dependence of the photonic SOC\non its spatial structure remains unnoticed.\nII. THEORETICAL MODEL\nWe consider a two-wave mixing process involving two\ninteracting photonic states. The SOC takes place in a\ncrystal, represented by its principal refractive index: nx,\nny, and nz. With an approximation of the slowly varying\nenvelope along optical axis z, a coupled-wave equation for\nthe process is given by [52]\n2iβx∂Ex\n∂z+n2\nx\nn2z∂2Ex\n∂x2+∂2Ex\n∂y2=γy∂2Ey\n∂y∂xexp (+ i∆β·z)\n2iβy∂Ey\n∂z+n2\ny\nn2z∂2Ey\n∂y2+∂2Ey\n∂x2=γx∂2Ex\n∂x∂yexp (−i∆β·z)(1)\nwhere Ex,yare linearly polarized fields, and βx,y=k0nx,y\ndenote their propagation constants. k0= 2π/λis free-\nspace wavenumber with λbeing the wavelength. ∆ β=\nβy−βxis a phase mismatch. We define γx,y= 1−n2\nx,y/n2\nz\nas coupling parameters, related to crystal’s polarity. The\nderivatives ∇2\nxyand∇2\nyxin Eq. (1) stem from the non-\nzero term ∇ ·E̸= 0, featuring origin of the SOC [41].arXiv:2402.01080v1  [physics.optics]  2 Feb 20242\nFIG. 1: (a) Geometrical representation of spin precession in the presence of SOC. ˆRandˆLdefine the spin-up and spin-down\nequivalents in the B2direction, respectively; whereas Φ +and Φ −denotes two spin eigenstates in the direction of synthetic\nfieldB. Spin precession is initiated by a mixing spin Φ = 1 /√\n2(Φ ++iΦ−) located at z0. (b) Bloch-sphere representation of\nspin-1/2 system, in the presence of external field B. Φ↑and Φ ↓are spin up and spin down in the zdirection, while Φ1/2\n+and\nΦ1/2\n−denotes eigenstates of the system, corresponding to direction of B. (c) Polarization states mapped on a longitude line of\nthe first-order ( ℓ= 1) sphere in (a). (d) Corresponding spin vectors to (c). (e) SOC strength as a function of beam width r0.\n(f)-(i) Theoretical results for the spin vectors under actions of LG beam with different widths.\nTo address the rapid oscillation terms exp( ±i∆β·z),\nwe transform the wave equation to a rotating form, by\ndefining\nEx=˜Axexp(+ i∆β·z/2)\nEy=˜Ayexp(−i∆β·z/2)(2)\nrespectively. Thus a Hamiltonian of the system is written\nas\nH=1\n2¯β\u0014−∇2\n⊥+ ¯γ∇2\nxx,0\n0,−∇2\n⊥+ ¯γ∇2\nyy\u0015\n+\u0014∆β/2,¯γ∇2\nyx/(2¯β)\n¯γ∇2\nxy/(2¯β),−∆β/2\u0015\n(3)\nwhere ∇2\n⊥=∇2\nxx+∇2\nyydenotes the Laplace operator.\nWe have assumed shallow crystal birefringence, namely\n¯β≈(βx+βy)/2 and ¯ γ≈(γx+γy)/2. The second term in\nEq. (3), which includes the derivative operators, couples\nthe two polarization components. It means that the SOC\nis related to spatial structure of light.\nWe study the beam-dependent SOC in a synthetic two-\nlevel spin-orbit system. We define right and left circularly\npolarized vortex states as spin up and spin down equiv-\nalents in the zdirection. They are written as [53–55]:\nˆR= exp(+ iℓϕ)(ˆx−iˆy)/√\n2\nˆL= exp( −iℓϕ)(ˆx+iˆy)/√\n2(4)\nrespectively, where ˆ xand ˆyare unit vectors and ϕ=\narctan( y/x). Since the pseudo spins are defined in thecircular basis, the Hamiltonian is modified by a trans-\nformation from the cartesian coordinate to the circular\nbasis, yielding\nH′=¯γ−2\n4¯β\u0014\n∇2\n⊥,0\n0,∇2\n⊥\u0015\n+\u00140,∆β/2−i¯γ∇2\nyx/(2¯β)\n∆β/2 +i¯γ∇2\nyx/(2¯β),0\u0015\n(5)\nGiven an overall field ˜A=˜A(x, y, z )(ΦRˆR+ΦLˆL), where\nΦRand Φ Lare weights on ˆRand ˆL, respectively, we\nreduce Eq. (1) to the Schr¨ odinger-like (Pauli) form\ni∂Φ(z)\n∂z=\u00121\n2MP2\n⊥˜A−1\n2σ·B\u0013\nΦ(z) (6)\nwhere Φ = (Φ R,ΦL)T,P2\n⊥= [−∇2\n⊥,0; 0,−∇2\n⊥], and\nM= 2¯β˜A/(2−¯γ). Here σis the Pauli matrix vec-\ntor. The SOC is described by a term −σ·B, where\nB1=−¯γ∇2\nxy˜A/(¯β˜A),B2= 0, and B3=−∆β. It is\nanalogous to a coupling form which describes interaction\nbetween a particle’s spin and its angular momentum in\na moving frame [1, 2]. More details refer to Appendix A.\nSince B2is zero, the vector Blies on the purely trans-\nverse B1B3plane. The SOC Hamiltonian admits eigen-\nstates that point to the vector Band comprise an equal\nsuperposition of ˆRandˆL, written as\nΦ+= 1/√\n2h\nˆR+ exp( iφ)ˆLi\nΦ−= 1/√\n2h\nˆR−exp(iφ)ˆLi (7)3\nFigure 1(a) visualize the eigenstates and the pure states\n(ˆRandˆL) in the Poincar´ e sphere, showing close analo-\ngies to Bloch-sphere representation of the spin-1/2 sys-\ntem [56, 57] [Fig. 1(b)]. The spin states exhibit cylin-\ndrically symmetric polarization distributions. As illus-\ntration, Fig. 1(c) displays typical polarizations of states\nmapped onto a longitude line in the first-order ( ℓ= 1)\nsphere; while Fig. 1(d) depicts their corresponding spin\nvectors, represented by an angle arccos( S2), where S2is\nvalue of polarization ellipticity. Since the state exhibits\nidentical polarization ellipticity in the transverse plane,\nthe resultant spin vectors are homogeneous.\nThe SOC term shows a dynamical effect, caused by\nthe propagation-variant envelope ˜A. This shows sharp\ncontrast to conventional ones which are usually being in-\ndependent terms. However, if optical diffraction is ne-\nglected, the dynamical behavior disappears and the SOC\nstrongly relies on the envelope. In this scenario, a rel-\nevant beam parameter becomes an important degree of\nfreedom for engineering the SOC. This is demonstrated\nin Fig. 1(e), showing close relationship between SOC\nstrength and beam width in the phase-matching condi-\ntion (∆ β=0). Here the Laguerre-Gaussian (LG) envelope\nis considered as:\n˜A(r) =r\nr0exp(−r2\nr2\n0) (8)\nwhere r= (x2+y2)1/2, and r0features the beam\nwidth. At the deep-subwavelenth region ( r0< λ/ 2),\nthe SOC strength is rapidly increasing with a slight\ndecrease of r0. It becomes relatively negligible when\nr0> λ. This relation suggests that shrinking light\nto deep-subwavelength scale significantly enhances the\nSOC. Although the derivations are based on the slowly\nvarying envelope approximation, the model can be ap-\nplied to deep-subwavelength regime at the early stage of\nspin evolution.\nTo demonstrate the deep-subwavelength-induced SOC,\nwe set the coupling length to be only one cycle ( z=λ),\nsuch that a moderate SOC cannot cause obvious spin\ntransport phenomenon. On the other hand, the SOC\nstrength can be maintained during beam propagation,\ndue to the short coupling length. This results in an adi-\nabatic spin evolution, represented as a spin precession\naround B, i.e.,\ndS\ndz=B×S (9)\nwhere S= (S1, S2, S3) is the state vector defined as\nSh= Φ†σhΦ (h= 1,2,3). The spin vector is therefore\ndescribed by S2. We initiate the spin precession from\na mixing state: Φ = 1 /√\n2[Φ++iΦ−]. Figure 1(f)-1(i)\ndisplay theoretical distributions of the spin vectors for\ndifferent beam parameters. Evidently, for r0=0.05 µm,\nthe spin rotates to an angle about -78o; By comparison,\nincreasing the parameter to r0= 0.13µm causes less\nsignificant spin precession, manifested by a spin rotation\nFIG. 2: (a) Experimental setup. BS: beam splitter; Q: q-\nplate; FL: flat lens; M: mirror; OB: objective lens; TL: tube\nlens; QWP: quarter wave plate; P: polarizer; CCD: charge\ncoupled device. The laser is operating at wavelength of\nλ= 632 .8 nm. The insert in (a) shows that an equatorial\nmixing spin with equal weight on Φ Rand Φ Lis adiabatically\nconverted to a pure spin down in the presence of the SOC. (b)\nLayout of the 60-nm-thick flat lens with NA=0.87. (c) Inten-\nsity distribution of the LG beam at the focal plane ( zf) of the\nflat lens. (d) Plane-wave interference and (e) y-polarization\ncomponent of beam at zf, indicating a generation of the ex-\npected spin state Φ = 1 /√\n2(Φ ++iΦ−). The scale bar in (c-e)\nis 250 nm. In color bar, L: low; H: high.\nangle about -12o. This indicates that the SOC strongly\ndepends on the carrier envelope. Figure 1(h, i) show that\na moderate SOC induced by the relatively larger envelope\ncannot cause spin precession.\nIII. EXPERIMENTAL RESULTS AND\nDISCUSSION\nExperiments are carried out to confirm the predictions.\nA crucial ingredient is to generate the required spin-orbit\nstate at the deep-subwavelength scale. This is challeng-\ning since the incident state cannot maintain its prop-\nerty after tightly focused by the high-numerical-aperture\n(NA) objective lens [58–60]. To overcome this problem,\nwe fabricate a topology-preserving high-NA flat lens (the\nthickness is 60 nm) according to a technique reported in\n[61]. The flat lens [the layout is shown in Fig. 2(b)] has\na NA up to 0.87 and a focal length of zf= 8µm. A sys-\ntem comprising an objective lens (150 ×, NA=0.9) and a\ntube lens is utilized to characterize the flat lens, see Fig.\n2(a). Figure 2(c) presents recorded intensity distribution\nof light at the focal plane. The focused LG beam exhibits\na parameter of r0≃0.32µm. The recorded regular inter-\nference [(Fig. 2(d)] and y-polarization component [(Fig.4\nFIG. 3: Experimental observation of spin rotation induced by\nthe deep-subwavelength LG beam ( r0= 0.32µm), as mani-\nfested by spin angular momentum conversion (flipping) from\nright-handed one (b) to the left-handed one (a). In compar-\nison, a larger LG beam parameter r0=2.2 µm is considered,\nresulting in balanced left-handed (c) and right-handed (d)\ncomponents. (e, f) The measured spin vectors before and af-\nter the crystal film, for (e) r0= 0.32µm, and (f) r0= 2.2\nµm. In the color bar (d), L: low; H: high.\n2(e)] suggest that the expected initial spin is generated.\nTheoretical derivation about topology-preserving prop-\nerty of the flat lens (Appendix B) further confirms the\ngeneration.\nAn experimental setup is built for measuring the spin\nprocession. A linearly polarized He-Ne laser ( λ= 632 .8\nnm) is divided by a beam splitter. A q-plate with a\ncharge of q= 1/2 is applied to transform the beam into\nexpected spin state carrier by the LG envelope. The pu-\nrity of the spin state from the q-plate is measured as\n95.2% (Appendix C). The LG beam is focused into deep-\nsubwavelength region by the flat lens. A c-cut lithium\nniobate crystal film (¯ γ=−0.08) with a thickness about\none wavelength is placed at the focal plane. The emerg-\ning beam, in the presence of the SOC, is expected to\naccumulate a non-trivial spin phenomenon [see the in-\nsert in Fig. 2(a)].\nFigure 3 presents measurements confirming the spin\nprecession. Since the spin is relevant to the circular po-\nlarization, we measure the right- (spin ↑) and left-handed\n(spin ↓) circular polarization components. These are\nachieved by rotating a quarter wave plate to an angle\nof−π/4 and + π/4 with respect to xaxis, respectively,\nwhile inserting a linear polarizer in front of the camera.\nFigure 3(a) and 3(b) depicts intensity distributions of\nΦLand Φ R, respectively. The measured Φ Lcomponent\nis stronger than the Φ Rone, indicating a spin precession\ntoward south pole of the sphere. Figure 3(e) shows the\nmeasured spin rotation by an angle of -5.2o, compared\nto the initial one [62]. This approximately matches to\nthe simulated result. However, for a larger parameter\n(r0= 2.2µm), the Φ Land Φ Rcomponents are approxi-\nmately identical [Fig. 3(c, d)], meaning that the induced\nSOC is insufficient to flip the spin [Fig. 3(f)]. Slight dif-\nference between the experiment and theory can be mainly\nattributed to the imperfect LG envelope that is closely\nrelevant to the derivative operator ∇2\nxy[62].\nFIG. 4: Observation of the orbital-angular-momentum state\ninduced by spin precession. (a, c) The experimentally mea-\nsured plane-wave interference patterns, for two different LG\nbeam parameters: (a) r0= 0.32µm, and (c) r0= 2.2µm.\n(b, d) The simulated [based on Eq. (1)] interference patterns\ncorresponding to the measurements in (a, c). Experimental\nconditions are kept the same as those in Fig. 3.\nWe observe non-trivial spin-precession phenomenon,\nmanifested by a generation of the photonic OAM. Ini-\ntially, both the SAM and OAM of state at the equator\nare zero. Under the action of the SOC, its intrinsic OAM\nand SAM are separated simultaneously. This non-trivial\nphenomenon is observed in Fig. 4(a), showing a clear\ndislocation in the plane-wave interference fringes for the\ndeep-subwavelength LG beam. This is a manifestation of\nwavefront helicity with a topological charge being ℓ= 1.\nThe spin precession accompanied by the OAM genera-\ntion confirms the phenomenon of spin-orbit separation.\nThis effect becomes negligible for larger envelope, since\nthe spin remains at its original position, as indicated by\nthe regular interference fringes [Fig. 4(c)]. Theoretical\nresults correspondingly shown in Fig. 4(b) and 4(d) are\nin accordance with the measurements.\nWe observe more prominent spin precession by con-\nsidering the Bessel structured light with deeper subwave-\nlength feature size. The carrier envelope is replaced by\n˜A(r) =Jℓ(r/r0) (10)\nwhere Jℓdenotes the Bessel function of order ℓ. In\npractice, we should properly truncate the ideal Bessel\nbeam by using a Gaussian factor. The resultant Bessel-\nGaussian (BG) profile exhibits nondiffracting property\nFIG. 5: Observation of the spin rotation by using the deep-\nsubwavelength BG beam ( r0= 0.12µm). (a, b) Experimen-\ntally measured intensity distributions of the left- and right-\nhanded circular polarizations. (c, d) Plane-wave interference\npatterns obtained both in experiment (c) and in simulation\n(d). (e, f) The measured output spin rotation in comparison\nwith the initial one: (e) experiment; (f) simulation.5\nFIG. 6: (a) The simulated [based on Eq. (6)] beam-dependent\nspin oscillatory modes. (b)(c) The simulated [based on Eq.\n(1)] phase distributions of the output light states from the\nbarium metaborate crystal film (¯ γ=−0.16), for (b) r0= 95\nnm, and (c) r0= 110 nm.\nover a certain distance. We generate this BG beam us-\ning a metasurface whose geometry exhibits cylindrical\nsymmetry. The highly localized BG beam is a result\nof in-phase interference of many high-spatial-frequency\nwaves [51]. We demonstrate result for a beam param-\neter of r0= 0.12µm, while maintaining other param-\neters unchanged. Similarly, an initial balance between\nthe left and right-handed components is broken by the\nSOC [Fig. 5(a, b)]. The output spin rotates to a larger\nangle of -17.1o, nearly in accordance with the theoreti-\ncal calculation [Fig. 5(f)]. The measured and simulated\ninterference patterns verify the spin-precession-induced\nOAM generation, see Fig. 5(c) and 5(d), respectively.\nFinally, we propose to using the beam-dependent SOC\nin precision measurement of slight variation of structured\nlight, with measurement accuracy up to 15 nm. This\nnanometric resolution is usually impossible to be reached\nby current optical detectors. It requires to realize rapid\noscillation between the spin up and spin down. Specifi-\ncally, we exploit the deep-subwavelength BG beam as car-\nrier envelope of the spin. In this scenario, the Pauli equa-\ntion [Eq. (6)] emulates a SOC process for the spin oscil-\nlation. Figure 6(a) depicts the SOC-supported spin har-\nmonic oscillations along with the coupling distance, for\ndifferent cases of beam widths. Obviously, the spin oscil-\nlation is very sensitive to the change of spatial structure\nof light, giving rising to ultrasensitive beam-dependent\noscillatory modes. As a result, a slight change of the\nbeam width leads to significant spin flipping. This al-\nlows to detect the spatial variation of light as small as\n15 nm. To verify the result, we present simulated out-\ncomes [see Fig. 6(b) and 6(c)], clearly showing opposite\nhelical wavefronts of the output states (corresponding to\nthe spin down and spin up), for r0=95 nm and r0=110\nnm. Note that one can further increase the measuring\nsensitivity by properly reducing the beam width.\nIV. CONCLUSION\nIn summary, we have demonstrated both theoretically\nand experimentally novel SOC phenomena, caused by\nthe deep-subwavelength spin-orbit structured light. Thisbeam-dependent SOC contrasts to those being material-\ndependent [44, 45]. The reported SOC is closely rele-\nvant to the spatial gradient of light field, hence it can be\nsignificantly enhanced by using the deep-subwavelength\ncarrier envelopes. We have qualitatively characterized\nthis effect, by measuring the spin precessions under dif-\nferent beam parameters. Particularly, based on the\ndeep-subwavelength Bessel beam, a significant spin ro-\ntation about -17.1o, accompanied by OAM generation,\nwas achieved within a coupling length of only one wave-\nlength. The influence of the phase mismatch on the\nbeam-dependent SOC was also discussed, see Appendix\nD. These fundamental SOC phenomena may find inter-\nesting applications in different areas [63–66]. As an ex-\nample, we have proposed to use such a strong SOC effect\nin the precise measurement of slight spatial change of\nlight with nanometric resolution.\nV. ACKNOWLEDGMENTS\nWe thank Boris Malomed from Tel Aviv University\nfor kind discussions about the SOC. This work was sup-\nported by the National Natural Science Fundation of\nChina (62175091, 12304358), and the Guangzhou science\nand technology project (202201020061).\nVI. APPENDIX A: ANALOGY OF SPIN-ORBIT\nCOUPLING IN SPIN-1/2 SYSTEM AND\nSYNTHETIC TWO-LEVEL SYSTEM\nThe spin-1/2 dynamics in the external vector field B\ncan be described by a Hamiltonian term H1/2=σ∗B,\nwhere σis the Pauli matrix vector. In a normalized form,\nit can be expressed as\nH1/2=1\n2\u0014\ncosθ sinθ·exp (−iφ)\nsinθ·exp (iφ) −cosθ\u0015\n(11)\nwhere θandφare two angles that define a normalized\n(unit) sphere. The vector Bthen possesses around the\nsphere, with direction determined by θandφ. This\nHamiltonian H1/2admits two spin eigenstates that point\nalong to B, written as\nΦ1/2\n+= cos\u0012θ\n2\u0013\nΦ↑+ exp( iφ) sin\u0012θ\n2\u0013\nΦ↓\nΦ1/2\n−= sin\u0012θ\n2\u0013\nΦ↑−exp(iφ) cos\u0012θ\n2\u0013\nΦ↓(12)\nwhere Φ ↑= [1 0]Tand Φ ↓= [0 −1]Tare spin-up and\nspin-down states defined in the zdirection. Figure 1(b)\ngeometrically depicts this picture onto a Bloch sphere.\nAll possible spins of the system can now be mapped onto\nthe sphere, with the spin up Φ ↑and spin down Φ ↓located\nat the north and south poles of the sphere, respectively.\nIn the presence of the external field B, the initial spin6\nTABLE I: Analogies between the presented synthetic spin-1/2 system in the higher-order optical regime and the spin-1/2\nsystem in the quantum mechanics. The direct analogies between these two different settings enable us to emulate intriguing\nspin transport phenomena in the presence of spin-orbit coupling.\nPhysical parameters Spin-1/2 system Synthetic spin-1/2 system\nSpins Φ ↑and Φ ↓ˆRandˆL\nEigenstates Φ1/2\n+and Φ1/2\n− Φ+and Φ −\nField vector B(real) B=(−¯γ/(¯β˜A)∇2\nxy˜A,0,−∆β)\nSpin-orbit coupling term H1/2=σ·B H SOC=σ·B\nSpace/time coordinates ( x, y, t ) ( x, y, z )\nMass m M = 2¯β˜A/(2−¯γ)\nMomentum operator P2\n⊥= [−∇2\n⊥,0; 0,−∇2\n⊥] P2\n⊥= [−∇2\n⊥,0; 0,−∇2\n⊥]\nprecesses around the vector B, giving rise to many in-\ntriguing spin transport phenomena such as the geometric\nphase.\nIn our case, we study spin-orbit coupling of structured\nlight in a photonic crystal. The structured light in the\nsystem is comprising a superposition of two orthogonal\nspin-orbit states with non-trivial topological structures.\nThey can be written as ˆR= exp ( ilϕ)(ˆx−iˆy)/√\n2 and\nˆL= exp ( −ilϕ)(ˆx+iˆy)/√\n2, respectively. These topolog-\nical states define the spin up and spin down equivalents\nalong the zaxis, respectively, but they are not eigenstates\nof the analogous spin-orbit Hamiltonian Hsoc=−σ∗B.\nIn the circular basis, a similar Hamiltonian matrix can\nbe written as\nHsoc=\u0014\nB2 B3−iB1\nB3+iB1−B2\u0015\n(13)\nIn our case, since B2is zero (see the main text), the\neffective vector Bobtained here lies on the purely trans-\nverse B1B3plane, as shown in Fig. 1(a). As a result,\nthe pseudospin eigenstates of Hsocthat point along this\ntransverse vector Bcomprise an equal superposition of\nˆRandˆL, express as\nΦ+= cos\u0010π\n4\u0011\nˆR+ exp( iφ) sin\u0010π\n4\u0011\nˆL\nΦ−= sin\u0010π\n4\u0011\nˆR−exp(iφ) cos\u0010π\n4\u0011\nˆL(14)\nWe can now interpret these eigenstates as a mixing of ˆR\nandˆL. Poincar´ e-sphere representation allows us to visu-\nalize these spin eigenstates as well as the pure states ˆR\nandˆL. Clearly, this is analogous to the Bloch-sphere rep-\nresentation for the spin-1/2 system. The spin-orbit cou-\npling makes this state evolves along the Poincar´ e sphere,\nwhich can be described by the synthetic Pauli equation,\ni∂Φ\n∂z=\u00121\n2MP2\n⊥˜A−1\n2σ·B\u0013\nΦ (15)\nTable I summaries analogous formulas between these two\nsystems.VII. APPENDIX B: THEORETICAL\nDERIVATION FOR THE\nTOPOLOGY-PRESERVING FLAT LENS\nIn this section, we theoretically prove that the flat lens\nused in the experiment does not change the spin-orbit\nproperty of the LG beam after tightly focusing. The flat\nlens is designed by an amplitude-only hologram gener-\nated from an interference between an angular cosine wave\nand a spherical wave (see ref. [61] in the text). When the\nLG beam ˜A(x, y) carrying a general spin state Φ passes\nthrough the flat lens, it is modulated in binary. As a re-\nsult, the light field behind the flat lens can be expressed\nas\nE(x, y, z = 0) = ˜A(x, y)∗t(x, y)\u0002\nΦx(ϕ)ˆx+Φy(ϕ)ˆy\u0003\n(16)\nwhere t(x, y) denotes transmission function of the flat\nlens and ϕ= arctan ( y/x). Within this initial condition,\nwe solve the diffractive problem according to the vectorial\nHelmholtz wave equation. The diffractive field at the\nfocal plane of the flat lens can be written as\nE(x, y, z f) =k\ni2πzfZZ\nE(x′, y′, z= 0)\nexp\u001aik\n2zf[(x−x′)2+ (y−y′)2]\u001b\ndx′dy′\n(17)\nNote that owing to the cylindrical symmetry of the flat\nlens (see the layout in the text, Fig. 2(b)), the transmis-\nsion function can be also given in a cylindrical form of\nt(r), where r= (x2+y2)1/2. In this case, the complex\namplitude of the initial field is separable in the polar co-\nordinates ( r, ϕ). We therefore rewrite the solution in the\ncylindrical coordinate system and deal with the integrals.\nWe finally obtain the analytical solution for the vectorial\nlight field at the focal plane, given by\nE(x, y, z f) =f(r)\u0002\nΦx(ϕ)ˆx+ Φy(ϕ)ˆy\u0003\n(18)7\nFIG. 7: Modal decomposition results. (a) An experimental\nsetup used to measure the purity of the first-order LG beam\nemerging from the q-plate. The LG beam is decomposed into\nLG basis modes. The linearly polarized He-Ne laser operating\nat the wavelength of 632.8 nm is considered. QWP, quarter\nwave plate; QP, q-plate with a topological number of q=\n1/2; SLM, spatial light modulator; BS, beam splitter; CCD,\ncharge-coupled device. (b) The modal decomposition results\nat the basis of LG modes with topological charge ranging from\nl= -5 to 5.\nwhere\nf(r) =−k\nzfZ∞\n0˜A(r′)t(r′)r′J1\u0012krr′\nzf\u0013\nexp\u0014ik\n2zf(r2+r′2)\u0015\ndr′(19)\nandJ1indicates the first-order Bessel function. It is ev-\nident that the diffractive field at the focal plane shares a\nsimilar analytic form to the initial one, except for that the\nenvelope becomes a z-dependent function. It indicates\nthat the flat lens can completely retain the initial spin\nstate when it is focused into the input end of the crystal.\nThe topology-preserving flat lens enables us to detect the\npseudo spin precession caused by the deep-subwavelength\nstructured light, which cannot be achieved by using the\nconventional high NA objective lens.\nVIII. APPENDIX C: PURITY MEASUREMENT\nOF THE FIRST-ORDER LG BEAM FROM THE\nQ-PLATE\nwe perform additional experiment to show that the\ngenerated first-order LG beam from the q-plate is of high\npurity, which is sufficiently enough to detect the photonic\nspin-orbit coupling effect. We utilize a modal decomposi-\ntion method [67, 68] to measure the purities of the output\nLG mode from the q-plate with a topological charge of\nq= 1/2, see an experimental setup in Fig. 7(a). Two\nquarter wave plates (QWPs) are used to select a proper\npolarization of the generated first-order ( l=1) LG beam\nthat matches to the spatial light modulator (SLM). A\ngroup of pure LG modes generated from digital holo-\ngrams by using the SLM are considered to decomposed\nthe LG beam. Fig. 7(b) shows the decomposing result\ndepicted in a histogram. It is seen that the measured pu-\nFIG. 8: Controllable spin-orbit coupling by engineering the\nphase mismatch in a c-cut electro-optic lithium niobate crys-\ntal. (a) Experimental scheme for observing the electrically en-\ngineered spin-orbit coupling. (b-f) Experimentally measured\nphotonic spin states at different applied voltages: (b) U=60\nV, (c) U=80 V, (d) U=100 V, (e) U=120 V, and (f) U=140\nV. (g) The measured topological charge as a function of ap-\nplied voltage. Sim: simulation; Expt: experiment. In this\nexperiment, the coupling length of the crystal is set to z=30\nmm.\nrity of the first-order LG beam from the q-plate is 95.2%.\nIX. APPENDIX D: ENGINEERING PHOTONIC\nSPIN-ORBIT COUPLING BY TUNING THE\nPHASE MISMATCH\nIn addition to the beam-dependent photonic spin-\norbit coupling which we have shown in the main text,\nwe perform additional experiments confirming that the\nspin-orbit coupling can be also controlled by engineering\nthe phase mismatch. To this end, we consider electri-\ncally tuning the phase mismatch in a c-cut electro-optic\nlithium niobate (LN) crystal, whose optical axis is in ac-\ncordance with propagation direction of the beam [see Fig.\n8(a)]. In the presence of transverse modulation, the phase\nmismatch can be written as\n∆β=−k0n3\noγ22U/d (20)\nwhere k0= 2π/λdenotes wavenumber in vacuum with\nλbeing wavelength, nois the ambient refractive index8\nof the crystal, Uis the applied voltage, dis the thick-\nness, and γ22= 6.8 pm /V is an electro-optic coefficient\nof the crystal. In this case, the external knob Uis uti-\nlized to finely tune the phase mismatch and the resulting\nspin-orbit coupling. We use the same experimental setup\nand obtain a voltage-dependent transition between differ-\nent spin states in the phase mismatching regime. Panels(b-f) in Fig. 8 show controllable spin states of light by\nvarying the applied voltage. Moreover, we perform de-\ntailed experiments to measure the topological charge of\nthe output light state as a function of voltage, see Fig.\n8(g). These results suggest another important degree of\nfreedom for engineering the spin-orbit coupling.\n[1] R. Winkler, “Spin-Orbit Coupling Effects in Two-\nDimensional Electron and Hole Systems,” Springer\nTracts Mod. Phys. 191, Springer, New York (2003).\n[2] M. Aidelsburger, S. Nascimbene, and N. Goldman, “Ar-\ntifical gauge fields in materials and engineered systems,”\nC. R. Phys. 19, 394 (2018).\n[3] Y. -J. Lin, K. Jim´ enez-Garc ´ia, and I. B. Spielman, “Spin-\norbit-coupled Bose-Einstein condensates,” Nature 471,\n83 (2011).\n[4] V. Galitski, and I. B. Spielman, “Spin-orbit coupling in\nquantum gases,” Nature 494, 49 (2013).\n[5] Y. K. Kato, R. C. Myers, A. C. Gossard, and D. D.\nAwschalom, “Observation of the spin Hall effect in semi-\nconductors,” Science 306, 1910 (2004).\n[6] M. Z. Hasan, and C. L. Kane, “Colloquium: Topological\ninsulators,” Rev. Mod. Phys. 82, 3045 (2010).\n[7] M. Hafezi, E. A. Demler, M. D. Lukin, and J. M. Taylor,\n“Robust optical delay lines with topologial protection,”\nNat. Phys. 7, 907 (2011).\n[8] K. Y. Bliokh, F. J. Rodr ´iguez-Fortu˜ no, F. Nori, and A. V.\nZayats, “Spin-orbit interactions of light,” Nat. Photonics\n9, 796 (2015).\n[9] A. T. O’Neil, I. MacVicar, L. Allen, and M. J. Pad-\ngett, “Intrinsic and Extrinsic Nature of the Orbital An-\ngular Momentum of a Light Beam,” Phys. Rev. Lett. 88,\n053601 (2002).\n[10] L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J.\nP. Woerdman, “Orbital angular momentum of light and\nthe transformation of Laguerre-Gaussian laser modes,”\nPhys. Rev. A 45, 8185 (1992).\n[11] M. Onoda, S. Murakami, and N. Nagaosa, “Hall Effect\nof Light,” Phys. Rev. Lett. 93, 083901 (2004).\n[12] A. Kavokin, G. Malpuech, and M. Glazov, “Optical Spin\nHall Effect,” Phys. Rev. Lett. 95, 136601 (2005).\n[13] O. Hosten, and P. Kwiat, “Observation of the Spin Hall\nEffect of Light via Weak Measurements,” Science 319,\n787 (2008).\n[14] S. Fu, C. Guo, G. Liu, Y. Li, H. Yin, Z. Li, and Z. Chen,\n“Spin-Orbit Optical Hall Effect,” Phys. Rev. Lett. 123,\n243904 (2019).\n[15] L. Marrucci, C. Manzo, and D. Paparo, “Optical Spin-\nto-Orbital Angular Momentum Conversion in Inhomoge-\nneous Anisotropic Media,” Phys. Rev. Lett. 96, 163905\n(2006).\n[16] Y. Zhao, J. S. Edgar, G. D. M. Jeffries, D. McGloin,\nand D. T. Chiu, “Spin-to-Orbital Angular Momentum\nConversion in a Strongly Focused Optical Beam,” Phys.\nRev. Lett. 99, 073901 (2007).\n[17] Y. Gorodetski, A. Niv, V. Kleiner, and E. Hasman,\n“Observation of the Spin-Based Plasmonic Effect in\nNanoscale Structures,” Phys. Rev. Lett. 101, 043903\n(2008).[18] N. Shitrit, I. Yulevich, E. Maguid, D. Ozeri, D. Veksler,\nV. Kleiner, and E. Hasman, “Spin-Optical Metamaterial\nRoute to Spin-Controlled Photonics,” Science 340, 724\n(2013).\n[19] L. Fang, H. Wang, Y. Liang, H. Cao, and J. Wang, “Spin-\nOrbit Mapping of Light,” Phys. Rev. Lett. 127, 233901\n(2021).\n[20] N. Levy, S. A. Burke, K. L. Meaker, M. Panlasigul, A.\nZettl, F. Guinea, A. H. C. Neto, and M. F. Crommie,\n“Strain-Induced Pseudo-Magnetic Feidls Greater Than\n300 Tesla in Graphene Nanobubbles,” Science 329, 544\n(2010).\n[21] T. Low, and F. Guinea, “Strain-Induced Pseudomagnetic\nField for Novel Graphene Electronics,” Nano Lett. 10,\n3551 (2010).\n[22] F. de Juan, A. Cortijo, M. A. H. Vozmediano, and A.\nCano, “Aharonov-Bohm interferences from local defor-\nmations in graphene,” Nat. Phys. 7, 810 (2011).\n[23] F. Guinea, M. I. Katsnelson, and A. K. Geim, “Energy\ngaps and a zero-field quantum Hall effect in graphene by\nstrain engineering,” Nat. Phys. 6, 30 (2010).\n[24] K. I. Bolotin, F. Ghahari, M. D. Shulman, H. L. Stormer,\nand P. Kim, “Observation of the fractional quantum Hall\neffect in graphene,” Nature 462, 196 (2009).\n[25] M. C. Rechtsman,J. M.Zeuner, A. T¨ unnermann ,S. Nolte,\nM. Segev, and A. Szameit, “Strain-induced pseudomag-\nnetic field and photonic Landau levels in dielectric struc-\ntures,” Nat. Photonics 7, 153 (2013).\n[26] Y. Lumer, M. A. Bandres, M. Heinrich, L. J. Maczewsky,\nH. Herzig-Sheinfux, A. Szameit, and M. Segev, “Light\nguiding by artificial gauge fields,” Nat. Photonics 13, 339\n(2019).\n[27] R. O. Umucal1lar, and I. Carusotto, “Artificial gauge\nfield for photons in coupled cavity arrays,” Phys. Rev. A\n84, 043804 (2011).\n[28] S. Longhi, “Effective magnetic field for photons in waveg-\nuide and coupled resonator lattices,” Opt. Lett. 38, 3570\n(2013).\n[29] K. Fang, Z. Yu, and S. Fan, “Realizing effective mag-\nnetic field for photons by controlling the phase of dy-\nnamic modulation,” Nat. Photonics 6, 782 (2012).\n[30] K. Fang, Z. Yu, and S. Fan, “Photonic Aharonov-Bohm\nEffect Based on Dynamic Modulation,” Phys. Rev. Lett.\n108, 153901 (2012).\n[31] K. Fang, and S. Fan, “Controlling the Flow of Light\nUsing the Inhomogeneous Effective Gauge Field that\nEmerges from Dynamic Modulation,” Phys. Rev. Lett.\n111, 203901 (2013).\n[32] M. Kr´ ol, K. Rechci´ nska, H. Sigurdsson, P. Oliwa,\nR. Mazur, P. Morawiak, W. Piecek, P. Kula, P. G.\nLagoudakis, M. Matuszewski, W. Bardyszewski, B.\nPietka, and J. Szczytko, “Realizing Optical Persistent9\nSpin Helix and Stern-Gerlach Defection in an Anisotropic\nLiquid Crystal Microcavity,” Phys. Rev. Lett. 127,\n190401 (2021).\n[33] F. Liu, and J. Li, “Gauge Field Optics with Anisotropic\nMedia,” Phys. Rev. Lett. 114, 103902 (2015).\n[34] V. G. Sala, D. D. Solnyshkov, I. Carusotto, T. Jacqmin,\nA. Lemaitre, H. Tercas, A. Nalitov, M. Abbarchi, E. Ga-\nlopin, I. Sagnes, J. Bloch, G. Malpuech, and A. Amo,\n“Spin-Orbit Coupling for Photons and Polaritons in Mi-\ncrostructures,” Phys. Rev. X 5, 011034 (2015).\n[35] S. Ma, H. Jia, Y. Bi, S. Ning, F. Guan, H. Liu, C. Wang,\nand S. Zhang, “Gauge Field Induced Chiral Zero Mode in\nFive-Dimensional Yang Monopole Metamaterials,” Phys.\nRev. Lett. 130, 243801 (2023).\n[36] Q. Yan, Z. Wang, D. Wang, R. Ma, C. Lu, G. Ma, X.\nHu, and Q. Gong, “Non-Abelian gauge field in optics,”\nAdv. Opt. Photonics 15, 907 (2023).\n[37] O. Yesharim, A. Karnieli, S. Jackel, G. Di Domenico, S.\nTrajtenberg-Mills, and A. Arie, “Observation of the all-\noptical Stern–Gerlach effect in nonlinear optics,” Nat.\nPhotonics 16, 582 (2022).\n[38] A. Karnieli, and A. Arie, “All-optical Stern-Gerlach ef-\nfect,” Phys. Rev. Lett. 120, 053901 (2018).\n[39] G. Liu, S. Fu, S. Zhu, H. Yin, Z. Li, and Z. Chen,\n“Higher-order optical rabi oscillations,” Fundamental\nResearch 3, 898 (2023).\n[40] A. Tomita, and R. Y. Chiao, “Observation of Berry’s\ntopological phase by use of an optical fiber,” Phys. Rev.\nLett. 57, 937 (1986).\n[41] V. S. Liberman, and B. Y. Zel’dovich, “Spin-orbit inter-\naction of a photon in an inhomogeneous medium,” Phys.\nRev. A 46, 5199 (1992).\n[42] Y. V. Kartashov, B. A. Malomed, V. V. Konotop, V. E.\nLobanov, and L. Torner, “Stabilization of spatiotemporal\nsolitons in Kerr media by dispersive coupling,” Opt. Lett.\n40, 1045 (2015).\n[43] H. Sakaguchi, and B. A. Malomed, “One- and two-\ndimensional solitons in PT-symmetric systems emulating\nspin-orbit coupling,” New. J. Phys. 18, 105005 (2016).\n[44] C. Mann, S. A. R. Horsley, and E. Mariani, “Tunable\npseudo-magnetic fields for polaritons in strained meta-\nsurfaces,” Nat. Photonics 14, 669 (2020).\n[45] K. Rechci´ nska, M. Kr´ ol, R. Mazur, P. Morawiak, R.\nMirek, K. Lempicka, W. Bardyszewski, M. Matuszewski,\nP. Kula, W. Piecek, P. G. Lagoudakis, B. Pietka, and\nJ. Szczytko, “Engineering spin-orbit synthetic Hamilto-\nnians in liquid-crystal optical cavities,” Science 366, 727\n(2019).\n[46] A. Forbes, M. de Oliveira, and M. R. Dennis, “Structured\nlight,” Nat. Photonics 15, 253 (2021).\n[47] Y. Shen, X. Wang, Z. Xie, C. Min, X. Fu, Q. Liu, M.\nGong, and X. Yuan, “Optical vortices 30 years on: OAM\nmanipulation from topological charge to multiple singu-\nlarities,” Light: Sci. Appl. 8, 90 (2019).\n[48] J. Jia, H. Lin, Y. Liao, Z. Li, Z. Chen, and S. Fu,\n“Pendulum-type light beams,” Optica 10, 90 (2023).\n[49] A. F. Koenderink, Andrea Al` u, and A. Polman,\n“Nanophotonics: Shrinking light-based technology,” Sci-\nence348, 516 (2015).\n[50] Y. Hu, S. Wang, J. Jia, S. Fu, H. Yin, Z. Li, and Z. Chen,\n“Optical superoscillatory waves without side lobes along\na symmetric cut,” Adv. Photonics 3, 045002 (2021).\n[51] Y. Hu, S. Fu, H. Yin, Z. Li, Z. Li, and Z. Chen, “Sub-\nwavelength generation of nondiffracting structured lightbeams,” Optica 7, 1261 (2020).\n[52] C. Guo, S. Fu, H. Lin, Z. Li, H. Yin, and Z. Chen, “Dy-\nnamic control of cylindrical vector beams via anisotropy,”\nOpt. Express 26, 18721 (2018).\n[53] Y. Hu, G. Mo, Z. Ma, S. Fu, S. Zhu, H. Yin, Z. Li,\nand Z. Chen, “Vector vortex state preservation in Fresnel\ncylindrical diffraction,” Opt. Lett. 46, 1313 (2021).\n[54] D. Naidoo, F. S. Roux, A. Dudley, I. Litvin, B. Piccirillo,\nL. Marrucci, and A. Forbes, “Controlled generation of\nhigher-order Poincar´ e sphere beams from a laser,” Nat.\nPhotonics 10, 327 (2016).\n[55] G. Milione, H. I. Sztul, D. A. Nolan, and R. R. Alfano,\n“Higher-Order Poincar´ e Sphere, Stokes Parameters, and\nthe Angular Momentum of Light,” Phys. Rev. Lett. 107,\n053601 (2011).\n[56] M. O. Scully, W. E. Lamb, and A. Barut, “Theory of the\nStern-Gerlach apparatus,” Found. Phys. 17, 575 (1987).\n[57] M. V. Berry, “The adiabatic phase and Pancharatnam’s\nphase for polarized light,” J. Mod. Opt. 34, 1401 (1987).\n[58] R. Dorn, S. Quabis, and G. Leuchs, “Sharper Focus for\na Radially Polarized Light Beam,” Phys. Rev. Lett. 91,\n233901 (2003).\n[59] X. Xie, Y. Chen, K. Yang, and J. Zhou, “Harnessing the\nPoint-Spread Function for High-Resolution Far-Field Op-\ntical Microscopy,” Phys. Rev. Lett. 113, 263901 (2014).\n[60] H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T.\nChong, “Creation of a needle of longitudinally polarized\nlight in vacuum using binary optics,” Nat. Photonics 2,\n501 (2008).\n[61] X. Zhang, Y. Hu, X. Zhang, S. Zhu, H. Yin, Z. Li, Z.\nChen, and S. Fu, “A Global Phase-Modulation Mech-\nanism for Flat-Lens Design,” Adv. Opt. Mater. 10,\n2202270 (2022).\n[62] In experiment, the spin rotation was obtained by mea-\nsuring S2=|ΦR|2− |ΦL|2. Ideally, the spatial distri-\nbution of S2is homogeneous; whereas in experiment it\nexhibits fluctuations due to the imperfect beam enve-\nlope. We therefore average it by using an expression:\n¯S2=PNx\nnPNy\nmS2,n,m/(NxNy).\n[63] M. Mirhosseini, M. Malik, Z. Shi, and R. W. Boyd, “Effi-\ncient separation of the orbital angular momentum eigen-\nstates of light,” Nat. Commun. 4, 2781 (2013).\n[64] J. Chen, C. Wan, and Q. Zhan, “Engineering photonic\nangular momentum with structured light: a reivew,”\nAdv. Photonics 3, 064001 (2021).\n[65] R. Vijay, C. Macklin, D. H. Slichter, S. J. Weber, K. W.\nMurch, R. Naik, A. N. Korotkov, and I. Siddiqi, “Stabi-\nlizing Rabi oscillations in a superconducting qubit using\nquantum feedback,” Nature 490, 77 (2012).\n[66] G. Cronenberg, P. Brax, H. Filter, P. Geltenbort, T.\nJenke, G. Pignol, M. Pitschmann, M. Thalhammer, and\nH. Abele, “Acoustic Rabi oscillations between gravita-\ntional quantum states and impact on symmetron dark\nenergy,” Nat. Phys. 14, 1022 (2018).\n[67] C. Schulze, A. Dudley, D. Flamm, M. Duparr´ e, and A.\nForbes, Measurement of the orbital angular momentum\ndensity of light by modal decomposition, New J. Phys.\n15, 073025 (2013).\n[68] D. Wei, Y. Cheng, R. Ni, Y. Zhang, X. Hu, S. Zhu,\nand M. Xiao, Generating controllable Laguerre-Gaussian\nlaser modes through intracavity spin-orbital angular mo-\nmentum conversion of light, Phys. Rev. Appl. 11, 014038\n(2019)."    },    {        "title": "1408.3753v1.Two_Dimensional_TaSe2_Metallic_Crystals__Spin_Orbit_Scattering_Length_and_Breakdown_Current_Density.pdf",        "content": "1 \n  \n \n \n \n \nTwo-Dimensional TaSe 2 Metallic Crystals:  \nSpin-Orbit S cattering Length and Breakdown Current D ensity  \n \nAdam T. Neal, Yuchen Du , Han Liu, Peide D. Ye*  \n \nSchool of Electrical and Computer Engineering and Birck Nanotechnology Center,  \nPurdue University, West Lafayette, IN  47907, USA  \n \n*correspondence  to:  yep@purdue.edu   2 \n Abstract  \nWe have determined the spin -orbit scattering length of two-dimensional layered \n2H-TaSe 2 metallic crystals by detailed characterization of the weak an ti-localization phenomena \nin this strong spin -orbit interaction  material.  By fitting the observed magneto -conductivity , the \nspin-orbit scattering length for 2H -TaSe 2 is determined to be 17 nm  in the few -layer films .  This \nsmall spin -orbit scattering lengt h is comparable to that of Pt, which is widely used to study the \nspin Hall effect, and indicates the potential of TaSe 2 for use in spin Hall effect devices .  In \naddition to strong spin -orbit coupling, a material must also support large charge currents to \nachieve spin -transfer -torque via the spin Hall effect.  Therefore,  we have characterized the room \ntemperature breakdown current density of TaSe 2 in air, where the best breakdown current density \nreaches 3.7×107 A/cm2.  This large breakdown current further indicates the potential of TaSe 2 for \nuse in spin -torque devices  and two -dimensional  device  interconnect applications.  \n \nKeywords: tantalum diselenide, transition metal dichalcogenide , spin -orbit scattering, weak \nanti-localization, breakdown current   3 \n Although studied for some time ,1,2 the transition metal dichalcogenide (TMD) family of \nmaterials has attracted increased attention  in the nanoelectronics community  due to their \ntwo-dimensional layered structure , following the prolific research into graphene .3-6  With \ngraphene’s zero bandgap limit ation for transistor technology, much  of the nanoelectronics \ncommunity’s interest in TMDs has bee n focused on the semiconductors , particularly MoS 2, with \nthe demonstration of single -layer and few-layer field-effect transistors .7-9  Metallic TMDs , on the \nother hand, have received much less attention in the nanoelectronics community thus far , but \nrecent works on exfoliated TaSe 2 indicate that interest is on the rise .10-12  Notably, single -layer \nTaSe 2 has been recently characterized  by Raman  spectroscopy .13  Historically, m etallic TMDs \nhave been  intensely  studied by material physicists  and condensed matter physicists  due to their \nsuperconducting1,2 and charge density wave14 properties , which remain acti ve areas of research .   \n Obviously , metallic TMDs are not suitable for field-effect  transistor channel material s, \nsimilar to graphene .  One possible  nanoelectronics  application, previously proposed for \ngraphene ,15 is the use of metallic TMDs as device interconnects  for an all two-dimensional ( 2D) \nmaterial logic technology.  Another application of metallic TMDs , particularly of 2H-TaSe 2 on \nwhich we will focus in this work, is  in spintronics devices.  Angle -resolved photoemission \nspectroscopy (ARPES ) measurements of 2H -TaSe 2 reveal a “d og-bone” like structure of the \nFermi surface  in the “normal” (not charge density wave) state,16,17 and this structure is attributed \nto strong spin -orbit coupling in TaSe 2.18  This strong spin -orbit coupling may make TaSe 2 an \nideal 2D material for generation of spin currents via the spin Hall effect.  Motivated by  these \npotential  applications of 2H -TaSe 2, we determine , for the first time,  the spin -orbit scattering \nlength of TaSe 2 by characterizing the weak anti -localizat ion phenomena in the material.   In \naddition to strong spin -orbit coupling, a material must also support large charge currents to 4 \n achieve spin -transfer -torque via the spin Hall effect.  The ability to conduct large charge currents \nis also important for the previously mentioned two -dimensio nal interconnect application.  \nTherefore , we have also characterized the breakdown current density of 2D TaSe 2 crystals for the \nfirst time.  \nResults and Discussion  \n First of all, it is important to establish the polytype of the TaSe 2 samples used in this \nwork.  The 1T polytype, with octahedral coordination, and the 2H polytype, with trigonal \nprismatic coordination, are the two most studied in the literature.  For the 1T polytype of TaSe 2, \nthe material is in the incommensurate charge d ensity wave state below 600 K and in the \ncommensurate charge density wave state below 473 K .14,19,20 The transition at 473K is \naccompanied by a stark discontinuity in the resistivity  as a function of temperature.  The 2H \npolytype does not  transition into the incommensurate charge density wave state until ~120K, and \nthe commensurate charge density wave set s in below 90K.  In contrast to the 1T polytype, there \nare no discontinuities in the resistivity as a function of temper ature, but there is a characteristic \nchange in the slope of the resistivity  versus  temperature  curve at onset of the incommensurate \ncharge density wave state at ~120 K .14,19,20  These properties of the resistivity versus  temperature \nallow one to distinguish between the two polytypes of TaSe 2 electrically.   Figure 1(d) shows the \nresistivity of the TaSe 2 used in this work as a function of temperature from 4 K to 300 K.  The \ncharacteristic change  in the slope of the resistivity versus  temperature curve, indicated by the \narrow in the figure, confirms that the TaSe 2 used in this work is of the 2H polytype .  The crystal \nstructure of 2H -TaSe 2 is shown in Figure 1 (a) -(c) with layered 2D structures as expected .  5 \n With the polytype of the TaSe 2 established, we determine the electrically active  thickness \nof the flake used to study spin -orbit coupling of TaSe 2.  Figure 2 (a) shows atomic force \nmicroscopy ( AFM ) image of the TaSe 2 device  with the AFM height measurement of the flake \noverlaid.  The 2D crystal has a physical thickness of ~12 nm. A calculation of the Hall coefficient  \nfrom Hall effect measurements  using the thickness  measured  by AFM  yields Hall coefficients \nwhich are much too large compared to those  published in the  literature .21-23  Because of this \ndiscrepancy, we conclude that the physical thickness of the flake as measure d by AFM is not \nelectrically active, perhaps due to oxidation of the top  and bottom  layers of the TaSe 2 flake  while \nexposed to air  for long periods  of time . We also note  that the devices with much thinner flakes \ncannot be m easured reliably. Considering these  observation s, we can estimate  the electrically \nactive thickness of our TaSe 2 flake as the Hall coefficients from the literature divided by the Hall \nslope  measured for our TaSe 2 device ,            \n  ⁄ .  The Hall slope     \n   is the slope of the \nHall resistance ,        ⁄, as a function of magnetic field .  The measured Hall slopes for this \ndevice  as a function of temperature  are shown in Figure 2(b).   We perform this thickness \nestimation at two  temperatures, T~5K and T~ 120K, where the Hall slope for our flake at 120K \nwas estimated by linear extrapolation using the measured Hall slopes in Figure 2(b).  The \nelectrically active thickness,     , is determined to be 0.81 nm and 0.88 nm for 5K and 120K , \nrespectively.  Therefore, from this estimation  via the Hall effect measurement , we conclude that \nonly one or two atomic layers of the TaSe 2 flake are electrically active.  The dependence of the \nmagneto -conductivity  on the angle of the magnetic field, shown in Figure 2(c), also confirms this \nclaim that the system studied is a two -dimensional  electron system . The resistivity and Hall \ncoefficient plotted in Figure 1 (d) and Figure 2(b)  were calculated using th e electric ally active \nthickness,      , that we have determined.  Note that the change sign change of the Hall coefficient 6 \n is expected and is related to the reconstruction of the Fermi surface as the charge density wave \nstate develops with decreasing temperature.21   \n We now study the spin-orbit  coupling  strength  of 2H-TaSe 2.  Indeed, the strong spin-orbit  \ncoupling indicated by the “dog’s bone” Fermi surface shape16,17 is confirmed by magneto -\ntransport measurements.  Figure 3(a) shows the  differential  sheet conductance of TaSe 2 as a \nfunction of magnetic field for various temperatures .  TaSe 2 exhibits a negative magneto -\nconductivity , characteristic of weak anti -localization, which indicates the strong spin-orbit  \ncoupling of TaSe 2.  A classical background has been subtracted from the data, determined by \nfitting the data at T = 8 K and B more th an one Tesla where the  localization phenomena is \nsuppressed.   To quantitatively determine the spin-orbit  scattering length, we must first  determine \nthe dimensionality of the weak anti -localization phenomena in the system.  Figure 2(c) shows the \ndifferential magneto -conductivity  for different angles between the magnetic field and the sample.  \nIn this case no classical background is subtracted.   The angular dependence shows that the weak \nanti-localization phenomenon behaves two dimensionally.   The differential magneto -\nconductivity ,   , can be descri bed for 2D weak localization by the following equation:24-26 \n          \n    ( ( \n      )   \n  ( ( \n  )  ( \n       )))  (1) \n  ( )  ( \n   \n )   ( )             \n               \nwhere    is the valley degeneracy,      for the spin degeneracy,   the charge of an electron,   \nis Plan ck’s constant divided by   ,    the phase coherence length, and     the spin-orbit  \nscattering length.     and     are the free parameters which  allow fitting of the data.  The \nnumber of valleys,   , can be determined from ARPES performed on TaSe 2 in its commensurate 7 \n charge den sity wave state.17,21,27  In the commensurate charge density wave state, the TaSe 2 \nlattice is deformed, effectively increasing the period of the material system in real space.  This \nleads to a smaller Brillouin  zone in k -space compared to the undeformed material and also \ncauses reconstruction of the Fermi surface.  ARPES indicate s that there are three independent \nvalley’s in the charge density wave Brillouin  zone, therefore we choose      when fitting  the \nweak anti -localization data.   The solid orange line in Figure 3(a)  shows an example fit of the \nweak -anti-localization peak using Equation 1.  \n We can now determine the spin-orbit  scattering length of TaSe 2 by fitting the weak anti -\nlocalization data in Figure 3(a) , along with others not shown.  Figure 3(b) shows the phase \ncoherence length    and the spin-orbit  scattering length     determined from the fittings as a \nfunction of temperature.  We find that     is independent of temperature, while    decreases as \n   , which is consistent with dephasing due to electron -electron scattering  without too much \ndisorder .28  Because      in Figure 3(b) is independent of temperature, we take their average  and \ndetermine that the spin-orbit  scattering length       17 nm for 2H -TaSe 2. This length  is \ncomparable to the spin-orbit  scattering length of Pt (      12 nm)29, widely used to study the \nspin Hall effect, and indicates the potential of TaSe 2 for use in  2D spintronics devices . \nThe weak anti -localization is als o be suppressed by increasing bias  current as shown in \nFigure 4(a) .  The higher current bias adds energy to the sample, increasing the electron \ntemperature above that of the helium bath, leading to suppression of the weak anti -localization.  \nFigure 4(b)  shows the phase coherence length and spin-orbit  scattering length determined from \nthe data in Figure 4(a).  The phase coherence length decreases as       , which indicates that the \nelectron temperature increases as       considering        as previous ly determined .  The 8 \n same relationship between bias current and electron temperature has also been observed for a \ntwo-dimensional electron gas in the quantum Hall regiem.30 This further confirms that the \nstudied TaSe 2 device is , electrically , an atomic ally-thin material system.   \n Finally, we evaluate the breakdown current density of 2H-TaSe 2 in order to determine its \npotential to achieve spin Hall effect based spin -transfer -torque  by DC characterization  of the \ntotal 18 fabricated devices.  The average room temperature resistivity of the TaSe 2 flakes used \nfor the breakdown current measurements was 1.9×10-4 Ω·cm , determined using four terminal \nmeasurements of 8 of the devices.  The contact resistance of th e Ni/Au contact to the TaSe 2 \nflakes was also estimated by subtracting the four terminal resistance from the two terminal \nresistance and dividing by two.  The average contact resistance determined from the 8 four  \nterminal devices was 0.74 Ω·mm, which is one order of magnitude small er compared to \nmetal/MoS 2 contacts.31 This is because TaSe 2 is metallic and forms Ohmic contacts with metals \nwhile MoS 2 is a semiconductor and forms Schottky contacts with metals.  To ach ieve this low \ncontact resistance, the Ni/Au contact was deposited as soon as possible after  TaSe 2 exfoliation  to \nminimize surface oxidation and avoid oxide barriers between the Ni/Au contact and the TaSe 2.  \nFlakes were also stored in a nitrogen box before metal contact deposition to  help minimize the \nsurface oxidation.  Measurement of t he breakdown current density is  performed by continuously \nincreasing the bias voltage across the device until a decrease in current more  than one order of \nmagnitude is  observed.  Figure 5(a) shows the current density  versus  bias voltage data for the \ndevice showing the highest  breakdown current density observed among the 18 devices.  The \nbreakdown current de nsity i s taken as the current density immediately before the sharp decrease \nin current was observed.  Figure 5(b) shows the histogram of the breakdown currents determined \nfrom the 18 devices.  The maximum breakdown current density observed is 3.7×107 A/cm2, the 9 \n average 1.9×107 A/cm2, the minimum 0.5×107 A/cm2 and the standard deviation 0.8×107 A/cm2.  \nThe flake thickness as characterized by AFM was used when computing the breakdown current \ndensities , so these reported current densities could be slightly underestimated  if we consider the \nsurface oxidation .  These breakdown currents are comparable to the  charge currents used to \ninduce spin -transfer -torque via the spin Hall effect in Tantalum thin films,32 indicating th e \npossibility of TaSe 2 based 2D spin-torque devices.   The breakdown currents are about one order \nof magnitude less than those of graphene ;15,33 however , they are comparable to those of MoS 2.34 \nThese large breakdown currents  also indicate the potential of TaSe 2 as a 2D interconnect \nmaterial, particularly if used in conjunction semiconducting TMDs  where the similar crystal \nstructure may provide some  integration  advantage s.35 \nConclusions  \nIn conclusi on, we have determined the spin -orbit scattering length of 2H-TaSe 2 by \ndetailed characterization of the weak anti -localization phenomena in the material.  By fitting the \nobserved magneto -conductivity , the spin orbit scattering length for 2H -TaSe 2 is determined to be \n17 nm .  This small spin orbit scattering length is compa rable to that of Pt, which is widely used \nto study the spin Hall effect, and indicates the potential of 2D TaSe 2 for use in spin Hall effect \ndevices.  Additionally , we have characterized the room temperature breakdown current density \nof TaSe 2 in air, where  the best breakdown current density observed is 3.7×107 A/cm2.  This large \nbreakdown current  density  further  indicates the potential of TaSe 2 for use in 2D spin-torque \ndevices  and 2D device  interconnect applicat ions. \nMethods  10 \n  The 2D TaSe 2 devices were fabricated as follows.  A bulk Nanosurf TaSe 2 sample was \npurchased from nanoScience Instruments and confirmed by Raman characterization shown in \nFigure 1(e) . TaSe 2 flakes were prepared by the method of mechanical exfoliation using adhesive \ntape, depositing TaSe 2 flakes on an insulating substrate.  For the weak anti -localization \nmeasurements, the flakes were deposited on insulating SrTiO 3 substrate, while, for the \nbreakdown current measurements, flakes were deposited on 90 nm SiO 2 on Si substrate.  \nElectrical contacts were defined using an electron beam lithography and liftoff process.  The \nmetal contacts were deposited by electron beam evaporation.  For the weak anti -localization \nmeasurements, a 30nm/50nm Ni 0.8Fe0.2/Cu contact was used, and for the breakdown current \nmeasurements, a 30nm/50nm Ni/Au contact was used.  The weak anti -localization measurements \nwere carried out in a 3He cryostat with a superconducting magne t using a Stanford Research 830 \nlock-in amplifier.  Breakdown currents were measured at room temperature in air using a \nKeithley 4200 semiconductor characterization system.  \nThe authors declare no competing financial interest.  \nAcknowledgements  \n This materia l is based upon work partly supported by NSF under Grant CMMI -1120577 \nand SRC under Task 2396. A portion of the low temperature measurements was performed at the \nNational High Magnetic Field Laboratory, which is supported by National Science Foundation \nCooperative Agreement No. DMR -1157490, the State of Florida, and the U.S. Department of \nEnergy.  The authors thank Z. Luo, X. Xu, E. Palm, T. Murphy, J. -H. Park, and G. Jones for \nexperimental assistance.  \n  11 \n  \nFigure 1: (a) 3D view of the 2H -TaSe 2 crystal structure. (b) Side view of the TaSe 2 crystal \ncleaved at the (   ̅ ) face.  (c) Top view of the TaSe 2 crystal with lattice vectors shown. Purple  \nballs represent Ta atoms, while yellow balls represent Se atoms.   (d) Resistivity ρxx and sheet \nresistance  Rsheet as a function of temperate for the TaSe 2 flake used for weak anti -localization \nmeasurements.  The change in slope at ~120K indicates that the TaSe 2 is of the 2H polytype . (e) \nRaman characterization of the bulk TaSe 2 from which the flakes were exfol iated.  \n  \n100 150 200 250\n  Intensity [A.U.] Raman Shift [cm-1]E2gA1g\n0 100 200 300024xx [10-5.cm]\nTemperature [K]CDW Transition \nT~120K\n100300500\nRsheet [/]\na1 a2(a) (b)\n(c)\na3\n   ̅ face(d)\n(e)12 \n  \nFigure 2: (a) AFM image of the TaSe 2 device used to study the spin -orbit scattering via weak \nanti-localization measurements.  The AFM height measurement along the white line is overlaid . \n(b) Hall coefficient and Hall slope of TaSe 2 as a function of temperature.  The sign change \nresults from the reconstruction of the Fermi surface as the charge density wave state develops \nwith decreasing temperature. (c) Differential magneto -conductivity of TaSe 2 at T = 0.4 K with \nmagnetic field appl ied at different angles relative to the sample  surface.  Zero degrees indicate  \nthat the magnetic field is parallel to the TaSe 2 planes, while 90 degrees indicate that the magnetic \nfield is perpendicular to the TaSe 2 planes.   \n  \n0204060-10-505Rh [10-11 m3/C]\nTemperature [K]-10-505\nHall Slope [10-2 m2/C]\n-4-2024-10-8-6-4-202\n   90 30\n xx(B) - xx(0) [10-5 S]\nMagnetic Field [T]0(b) (c)\n(a)\n12 nm13 \n  \nFigure 3: (a) Differential  magneto -conductivity of 2H -TaSe 2 at temperatures from 1K to 8K.  \nThe negative magneto -conductivity shown in the figure is the characteristic of weak anti -\nlocalization and indicates the strong spin -orbit coupling of TaSe 2. (b) Phase coherence length    \n(black squares) and spin -orbit scattering length     (red circles) extracted from the weak anti -\nlocalization data from Figure 3(a).  The  black solid  line indicates the     power law decrease of \n  . \n  \n1 1010-810-7\n  L, Lso [meters]\nTemperature [K] L\n Lso\n-2 -1 0 1 2-6-4-202 1 K   2 K   3 K\n 5 K   8 K   1 K fitxx(B) - xx(0) [10-5 S]\nMagnetic Field [T](a) (b)14 \n  \nFigure 4: (a) Differential magneto -conductivity of TaSe 2 for different RMS bias currents as \nindicated in the figure.  The helium bath temperature was 0.4K for these measurements. (b) \nPhase coherence length    (black squares) and spin -orbit scattering length     (red circles) \nextracted from the weak anti -localization data from Figure 4(a).   The black solid  line indicates \nthe       power law decrease of    with increasing bias current.  \n  \n-2 -1 0 1 2-6-4-202 10µA     20 µA  50 µA\n 75 µA   100 µAxx(B) - xx(0) [10-5 S]\nMagnetic Field [T]\n1E-6 1E-5 1E-410-810-7\n  \n L\n LsoL, Lso [meters]\nCurrent [A](a) (b)15 \n  \nFigure 5: (a) DC current density versus  voltage characteristic of the TaSe 2 device which show s \nthe highest breakdown current. (b) Histogram of breakdown current densities for 18 TaSe 2 \ndevices measured in air at room temperature.  \n  \n0.00 0.75 1.50 2.25 3.00 3.7502468\n Count\nJbreakdown (107 A/cm2)\n0 1 2 3 4 501234\n Current Density (107 A/cm2)\nVoltage (V)(a) (b)16 \n References  \n \n(1) Hulliger,  F. Crystal Chemistry of the Chalcogenides and Pnictides  of the Transition \nElements. Structure and Bonding  1968 , 4, 83–229.  \n(2) Wilson,  J.; Yoffe,  A. The Transition Metal Dichalcogenides Discussion and \nInterpretation of the Observed Optical, Electrical and Structural Properties. Adv. Phys.  1969 , 18, \n193–335.  \n(3) Novoselov,  K. S.; Geim,  A. K.; Morozov,  S. V.; Jiang,  D.; Zhang,  Y.; Dubonos,  S. V.; \nGrigorieva,  I. V.; Firsov,  A. A. Electric Field Effect in Atomically Thin Carbon Films. Science  \n2004 , 306, 666 –669.  \n(4) Novoselov,  K. S.; Jiang,  D.; Schedin , F.; Booth,  T. J.; Khotkevich,  V. V.; \nMorozov,  S. V.; Geim,  A. K. Two -Dimensional Atomic Crystals. Proc. Natl. Acad. Sci. U.S.A.  \n2005 , 102, 10451 –10453.  \n(5) Novoselov,  K.; Geim,  A. K.; Morozov,  S.; Jiang,  D.; Grigorieva,  M. K.  I.; Dubonos,  S.; \nFirsov,  A. Two-Dimensional Gas of Massless Dirac Fermions in Graphene. Nature  2005 , 438, \n197–200.  \n(6) Zhang,  Y.; Tan,  Y.-W.; Stormer,  H. L.; Kim,  P. Experimental Observation of the \nQuantum Hall Effect and Berry’s Phase in Graphene. Nature  2005 , 438, 201 –204.  \n(7) Radisavljevic,  B.; Radenovic,  A.; Brivio,  J.; Giacometti,  V.; Kis,  A. Single -Layer MoS 2 \nTransistors. Nat. Nanotechnol.  2011 , 6, 147 –150.  \n(8) Ayari,  A.; Cobas,  E.; Ogundadegbe,  O.; Fuhrer,  M. S. Realization and Electrical \nCharacterization of Ultrathin Cryst als of Layered Transition -Metal Dichalcogenides. J. Appl. \nPhys.  2007 , 101, 014507.  \n(9) Liu, H.; Ye,  P. D. MoS 2 Dual -Gate MOSFET with Atomic -Layer -Deposited Al 2O3 as \nTop-Gate Dielectric. IEEE Electron Device Lett.  2012 , 33, 546 –548.  \n(10) Castellanos -Gomez,  A.; Navarro -Moratalla,  E.; Mokry,  G.; Quereda,  J.; Pinilla -\nCienfuegos,  E.; Agraït,  N.; Zant,  H.; Coronado,  E.; Steele,  G.; Rubio -Bollinger,  G. Fast and \nReliable Identification of Atomically Thin Layers of TaSe 2 Crystals. Nano Research  2013 , 6, \n191–199.  \n(11) Yan,  Z.; Jiang,  C.; Pope,  T. R.; Tsang,  C. F.; Stickney,  J. L.; Goli,  P.; Renteria,  J.; \nSalguero,  T. T.; Balandin,  A. A. Phonon and Ther mal Properties of Exfoliated TaS e2 Thin Films. \nJ. Appl.  Phys . 2013 , 114, 204301.  \n(12) Renteria,  J.; Samnakay , R.; Jiang,  C.; Pope,  T. R.; Goli,  P.; Yan,  Z.; Wickramaratne,  D.; \nSalguero,  T. T.; Khitun,  A. G.; Lake,  R. K.; Balandin,  A. A. All -metallic electrically gated 2H -\nTaSe 2 thin-film switches and logic circuits. J. Appl.  Phys . 2014 , 115, 034305.  \n(13) Hajiyev , P.; Cong,  C.; Qiu,  C.; Yu,  T. Contrast and Raman Spectroscopy Study of Single -\nAnd Few -Layered Charge Density Wave Material: 2H -TaSe 2. Sci. Rep.  2013 , 3, 2593.  \n(14) Wilson,  J.; Di  Salvo,  F.; Mahajan,  S. Charge -Density Waves and Superlattices in the \nMetal lic Layered Transition Metal Dichalcogenides. Adv. Phys.  1975 , 24, 117 –201.  \n(15) Murali,  R.; Yang,  Y.; Brenner,  K.; Beck,  T.; Meindl,  J. D. Breakdown Current Density of \nGraphene Nanoribbons. Appl. Phys. Lett.  2009 , 94, 243114 –243114.  \n(16) Rossnagel , K.; Rotenberg,  E.; Koh,  H.; Smith,  N. V.; Kipp,  L. Fermi Surface, Charge -\nDensity -Wave Gap, and Kinks in 2H -TaS 2. Phys. Rev. B  2005 , 72, 121103.  \n(17) Borisenko,  S. V.; Kordyuk,  A. A.; Yaresko,  A. N.; Zabolotnyy,  V. B.; Inosov,  D. S.; \nSchuster,  R.; Büchne r, B.; Weber,  R.; Follath,  R.; Patthey,  L.; Berger,  H. Pseudogap and Charge \nDensity Waves in Two Dimensions. Phys. Rev. Lett.  2008 , 100, 196402.  17 \n (18) Rossnagel,  K.; Smith,  N. V. Spin -Orbit Splitting, Fermi Surface Topology, and Charge -\nDensity -Wave Gapping  in 2H -TaSe 2. Phys. Rev. B  2007 , 76, 073102.  \n(19) Wilson,  J. A.; Di  Salvo,  F. J.; Mahajan,  S. Charge -Density Waves in Metallic, Layered, \nTransition -Metal Dichalcogenides. Phys. Rev. Lett.  1974 , 32, 882 –885.  \n(20) Craven,  R. A.; Meyer,  S. F. Specific Heat and Resistivity Near The Charge -Density -\nWave Phase Transitions in 2H -TaSe 2 And 2H -TaS 2. Phys. Rev. B  1977 , 16, 4583 –4593.  \n(21) Evtushinsky,  D. V.; Kordyuk,  A. A.; Zabolotnyy,  V. B.; Inosov,  D. S.; Büchner,  B.; \nBerger,  H.; Patthey,  L.; Follath,  R.; Borisen ko, S. V. Pseudogap -Driven Sign Reversal of the \nHall Effect. Phys. Rev. Lett.  2008 , 100, 236402.  \n(22) Lee, H.; Garcia,  M.; McKinzie,  H.; Wold,  A. The Low -Temperature Electrical And \nMagnetic Properties Of TaSe 2 And NbSe 2. J. Solid State Chem . 1970 , 1, 190 –194.  \n(23) Naito,  M.; Tanaka,  S. Electrical Transport Properties in 2H -NbS 2, -NbSe 2, -TaS 2 and -\nTaSe 2. J. Phys. Soc. Jpn.  1982 , 51, 219 –227.  \n(24) Fukuyama,  H. Non -Metallic Behaviors of Two -Dimensional Metals and Effect of \nIntervalley Impurity Scattering. Suppl. Prog. Theor. Phys.  1980 , 69, 220 –231.  \n(25) Fukuyama,  H. Effects of Intervalley Impurity Scattering on the Non -Metallic Behavior in \nTwo-Dimensional Systems. J. Phys. Soc. Jpn.  1980 , 49, 649 –651.  \n(26) Knap,  W.; Skierbiszewski,  C.; Zduniak,  A.; Litwi n-Staszewska,  E.; Bertho,  D.; \nKobbi,  F.; Robert,  J. L.; Pikus,  G. E.; Pikus,  F. G.; Iordanskii,  S. V.; Mosser,  V.; Zekentes,  K.; \nLyanda -Geller,  Y. B. Weak Antilocalization and Spin Precession in Quantum Wells. Phys. Rev. \nB 1996 , 53, 3912 –3924.  \n(27) Kuchin skii, E.; Nekrasov,  I.; Sadovskii,  M. Electronic Structure of Two -Dimensional \nHexagonal Diselenides: Charge Density Waves and Pseudogap Behavior. J. Exp. Theor. Phys.  \n2012 , 114, 671 –680.  \n(28) Fukuyama,  H.; Abrahams,  E. Inelastic Scattering Time in Two -Dimensional Disordered \nMetals. Phys. Rev. B  1983 , 27, 5976 –5980.  \n(29) Niimi,  Y.; Wei,  D.; Idzuchi,  H.; Wakamura,  T.; Kato,  T.; Otani,  Y. Experimental \nVerification of Comparability between Spin -Orbit and Spin -Diffusion Lengths. Phys. Rev. Lett.  \n2013 , 110, 016 805.  \n(30) Wei,  H. P.; Engel,  L. W.; Tsui,  D. C. Current Scaling in the Integer Quantum Hall Effect. \nPhys. Rev. B  1994 , 50, 14609 –14612.  \n(31) Liu, H.; Neal,  A. T.; Ye,  P. D. Channel Length Scaling of MoS 2 MOSFETs. ACS Nano  \n2012 , 6, 8563 –8569.  \n(32) Liu, L.; Pai,  C.-F.; Li,  Y.; Tseng,  H. W.; Ralph,  D. C.; Buhrman,  R. A. Spin -Torque \nSwitching with the Giant Spin Hall Effect of Tantalum. Science  2012 , 336, 555 –558.  \n(33) Liao,  A. D.; Wu,  J. Z.; Wang,  X.; Tahy,  K.; Jena,  D.; Dai,  H.; Pop,  E. Thermally Li mited \nCurrent Carrying Ability of Graphene Nanoribbons. Phys. Rev. Lett.  2011 , 106, 256801.  \n(34) Lembke,  D.; Kis,  A. Breakdown of High -Performance Monolayer MoS 2 Transistors. ACS \nNano  2012 , 6, 10070 –10075.  \n(35) Chhowalla,  M.; Shin,  H. S.; Eda,  G.; Li,  L.-J.; Loh,  K. P.; Zhang,  H. The Chemistry of \nTwo-Dimensional Layered Transition Metal Dichalcogenide Nanosheets. Nature C hemistry  \n2013 , 5, 263 –275.  \n \n  18 \n TOC  Graphics  \n \n-2 -1 0 1 2-6-4-202 1 K   2 K   3 K\n 5 K   8 K   1 K fitxx(B) - xx(0) [10-5 S]\nMagnetic Field [T]\n2H-TaSe2 "    },    {        "title": "1307.2363v1.Multi_Orbital_Superconductivity_in_SrTiO3__LaAlO3_Interface_and_SrTiO3_Surface.pdf",        "content": "arXiv:1307.2363v1  [cond-mat.supr-con]  9 Jul 2013Typeset withjpsj3.cls <ver.1.0> FullPaper\nMulti-OrbitalSuperconductivity inSrTiO 3/LaAlO 3Interface andSrTiO 3Surface\nYasuharuNAKAMURA1andYouichiYANASE1,2∗\n1Graduate School ofScience and Technology, NiigataUnivers ity, Niigata950-2181, Japan\n2Department of Physics, NiigataUniversity, Niigata950-21 81, Japan\n(Received April23, 2013; accepted May 28, 2013)\nWeinvestigate thesuperconductivity intwo-dimensional e lectronsystems formedinSrTiO 3nanostruc-\ntures. Our theoretical analysis is based on the three-orbit al model, which takes into account t2gorbitals of\nTi ions. Because of the interfacial breaking of mirror symme try, a Rashba-type antisymmetric spin-orbit\ncoupling arises from the cooperation of intersite and inter orbital hybridyzation and atomic LS coupling.\nThis model shows a characteristic spin texture and carrier d ensity dependence of Rashba spin-orbit cou-\npling through the orbital degree of freedom. Superconducti vity is mainly caused by heavy quasiparticles\nconsisting of d yzand dzxorbitals at high carrier densities. We find that the Rashba sp in-orbit coupling\nstabilizes a quasi-one-dimensional superconducting phas e caused by one of the d yzor dzxorbitals at high\nmagnetic fields along interfaces. This quasi-one-dimensio nal superconducting phase is protected against\nparamagnetic depairing e ffects by the Rashba spin-orbit coupling and realizes a large u pper critical field\nHc2beyond the Pauli-Clogston-Chandrasekhar limit. This findi ng is consistent with an extraordinarily\nlarge upper critical field observed in SrTiO 3/LaAlO 3interfaces and its carrier density dependence. The\npossible coexistence of superconductivity and ferromagne tism in SrTiO 3/LaAlO 3interfaces may also be\nattributedtothis quasi-one-dimensional superconductin g phase.\nKEYWORDS: Non-centrosymmetric superconductivity, two-d imensional electron gas, multi-orbital model\nTwo-dimensional conducting electron systems formed on\nSrTiO3heterostructures are attracting much attention. For\ninstance, electron gases with a high carrier density on the\norder of 1013cm−2have been realized in SrTiO 3/LaAlO 3\n(STO/LAO) interfaces,1)SrTiO3/LaTiO 3interfaces,2)SrTiO3\n(STO)surfaces,3)andδ-dopedSTO.4)Thediscoveryofsuper-\nconductivity,5)ferromagnetism,6–10)andtheircoexistence7–10)\nshed light on innovating phenomena in these systems. These\nquantum condensed phases are controlled by a gate volt-\nage through the change of carrier density.3,11–14)One of the\nkey issues is the role of Rashba-type antisymmetric spin-\norbit coupling15)arising from the interfacial breakingof mir-\nror symmetry, which may realize an exotic quantum con-\ndensed phase, such as non-centrosymmetric superconductiv -\nity,16)chiral magnetism,17)and their coexistent phase. In this\nresearch,we theoreticallystudy the non-centrosymmetric su-\nperconductivityrealized in STO nanostructuresfrom the mi -\ncroscopicpointofview.\nIt has been shown that a two-dimensional electron gas is\nconfined in a few TiO 2layers of the STO/LAO interface and\nSTO surface in the high-carrier-density region.3,18–23)The\nconduction bands mainly consist of three t 2gorbitals of Ti\nions.18–23)Although the degeneracy of t 2gorbitals signifi-\ncantly affects the band structure of two-dimensional electron\ngases,atheoryofsuperconductivitybasedonthemulti-orb ital\nmodel has not been conducted. Multiband models have been\nstudied,24,25)but the symmetry of t 2gorbitals is taken into\naccount in this study for the first time. We show that the\nsynergy of broken inversion symmetry and orbital degener-\nacystabilizesanintriguingsuperconductingphaseinthet wo-\ndimensionalelectrongases.\nOur study is based on a two-dimensional tight-binding\nmodel that reproduces the electronic structure of the\nSTO/LAO interface indicated by first principles band struc-\n∗E-mail address: yanase@phys.sc.niigata-u.ac.jpturecalculations20–23,26–28)andexperiments.18,19)Weherefo-\ncus on the STO/LAO interface, which has been intensively\ninvestigated,butourmainresultsarealsovalidforotherS TO\nheterostructures.Themodelisdescribedas\nH=H0+HI+HZ, (1)\nwherethesingle-particleHamiltonian H0is\nH0=Hkin+Hhyb+HCEF+Hodd+HLS, (2)\nHkin=/summationdisplay\nk/summationdisplay\nm=1,2,3/summationdisplay\ns=↑,↓(εm(k)−µ)c†\nk,msck,ms, (3)\nHhyb=/summationdisplay\nk/summationdisplay\ns=↑,↓[V(k)c†\nk,1sck,2s+h.c.], (4)\nHCEF=∆/summationdisplay\nin3i, (5)\nHodd=/summationdisplay\nk/summationdisplay\ns=↑,↓[Vx(k)c†\nk,1sck,3s+Vy(k)c†\nk,2sck,3s+h.c.],(6)\nHLS=λ/summationdisplay\niLi·Si. (7)\nWedenote(d yz,dzx,dxy)orbitalsusingtheindex m=(1,2,3),\nrespectively. The first term Hkindescribes the kinetic en-\nergy of each orbital and includes the chemical potential µ.\nHhybis the intersite hybridization term of d yzand dzxor-\nbitals.HCEFrepresents the crystal electric field of tetrago-\nnal systems. Because the mirror symmetry is broken near\nthe interface/surface, hybridization is allowed between d xy\nand dyz/dzxorbitals, and is represented by the “odd par-\nity hybridization term” Hodd. The atomic spin-orbit coupling\nterm (LS coupling term) of Ti ions is taken into account\ninHLS. We here adopt the tight-binding model reproduc-\ning first principles band structure calculations for STO het -\nerostructures,26–28)ε1(k)=−2t3coskx−2t2cosky,ε2(k)=\n−2t2coskx−2t3cosky,ε3(k)=−2t1(coskx+cosky)−\n12 J.Phys.Soc. Jpn. FullPaper Author Name\n4t4coskxcosky,V(k)=4t5sinkxsinky,Vx(k)=2itoddsinkx,\nandVy(k)=2itoddsinky. The same tight-binding model has\nbeen adopted for the study of surface spin-triplet supercon -\nductivity in Sr 2RuO4.29)Recent studies have examined the\nRashba-type antisymmetric spin-orbit coupling27,28,30)and\nmagnetotransport30)in STO/LAO interfaces on the basis of\nthismodel.\nInthispaper,wefocusontheroleofRashba-typeantisym-\nmetric spin-orbit coupling in the interface superconducti vity.\nIntheabovemodel,theRashbaspin-orbitcouplingisinduce d\nbythecombinationoftheoddparityhybridizationterm, Hodd,\nand the LS coupling term, HLS. The former arises from the\nparity mixing of local orbitals, which is a general source\nof antisymmetric spin-orbit coupling.31,32)For instance, the\nVx(k) (Vy(k)) term describes the mixing of d yz(dzx) and d xy\norbitals of Ti ions, which mainly occurs through the parity\nmixingwiththe p yorbital(p xorbital)onoxygenions.\nWe consider the s-wave superconductivity as expected in\nthe bulk STO.33)Unconventional pairing due to the electron\ncorrelationhasbeenstudied,34)however,wedonottouchthis\npossibility. Our reasonable assumption has been justified b y\ntherecentexperimentonsuperfluiddensity.35)Forsimplicity,\nwetakeintoaccounttheintraorbitalattractiveinteracti onU<\n0andtheinterorbitalattractiveinteraction U′<0inthespin-\nsingletchannel;\nHI=U/summationdisplay\ni/summationdisplay\nmni,m↑ni,m↓+U′/summationdisplay\ni/summationdisplay\nm/nequalm′ni,m↑ni,m′↓.(8)\nFor the discussion of the superconducting state in the mag-\nneticfield,we considertheZeemancouplingterm\nHZ=−/summationdisplay\nk/summationdisplay\nm/summationdisplay\ns,s′µBH·σss′c†\nk,msck,ms′,(9)\nin whichσis the Pauli matrix and µBis the Bohr magne-\nton. The orbital depairing e ffect arising from the coupling of\nelectron motion and vector potential is suppressed by the ge -\nometry when we consider the magnetic field parallel to the\ntwo-dimensional conducting plane, H/bardblˆx. The orbital po-\nlarization due to the magnetic field is also ignored since the\norbital moment along the plane vanishes for the degenerate\ndyz/dzxorbitals.\nNow, we formulate the linearized gap equation, by which\nwe determine the instability to the superconducting phase.\nFirst, we diagonalize the noninteracting Hamiltonian ( H0+\nHZ) using the unitary matrix ˆU(k)=/parenleftBig\nums,j(k)/parenrightBig\n. Thereby,\nthe basis changes as C†\nk= Γ†\nkU†(k), where C†\nk=\n(c†\nk,1↑,c†\nk,2↑,···,c†\nk,3↓) andΓ†\nk=(γ†\nk,1,γ†\nk,2,···,γ†\nk,6). With\nthe use of the operators of quasiparticles, γ†\nk,jandγk,j, the\nnoninteractingHamiltonianisdescribedas,\nH0+Hz=/summationdisplay\nk6/summationdisplay\nj=1Ej(k)γ†\nk,jγk,j, (10)\nwhereEj(k)is aquasiparticle’senergyand Ei(k)≥Ej(k)for\ni>j.\nNext, we introduce Matsubara Green functions in the or-\nbitalbasis,\nGm′s′,ms(k,iωl)=/integraldisplayβ\n0dτeiωlτ/angbracketleftck,m′s′(τ)c†\nk,ms(0)/angbracketright,(11)=6/summationdisplay\nj=11\niωl−Ej(k)um′s′,j(k)u∗\nms,j(k),(12)\nwhereωlis the Matsubara frequency. The linearized gap\nequation is obtained by looking at the divergence of the T-\nmatrix,ˆT(q),whichisgivenby\nˆT(q)=ˆT0(q)−ˆT(q)ˆHIˆT0(q). (13)\nThe wave vector qrepresents the total momentumof Cooper\npairs. In our model, the matrix element of the irreducible T-\nmatrixˆT0(q)isobtainedas\nT(mn,m′n′)\n0(q)\n=T/summationdisplay\nωl/summationdisplay\nk[Gm↑,m′↑(q/2+k,iωl)Gn↓,n′↓(q/2−k,−iωl)\n−Gm↑,n′↓(q/2+k,iωl)Gn↓,m′↑(q/2−k,−iωl)],(14)\nwhereTis the temperature. When we represent the T-matrix\nusing the basis ( mn)=(11,12,13,21,22,23,31,32,33), the\ninteraction term is represented by the 9 ×9 diagonal matrix,\nˆHI=(Umδmn)withUm=Uform=1,5,9 andUm=U′\nfor others. The superconducting transition occurs when the\nmaximum eigenvalue of the matrix, −ˆHIˆT0, is unity. Then,\nan element of the eigenvector( ψmn) is proportionalto the or-\nder parameter∆mn=−g/summationtext\nk/angbracketleftck,m↑c−k,n↓/angbracketright, whereg=Ufor\nm=nandg=U′form/nequaln. In what follows, we assume\na zero total momentum of Cooper pairs, namely, q=0. Al-\nthoughahelicalsuperconductingstatewith q/nequal0isstabilized\nin non-centrosymmetricsuperconductorsunder the magneti c\nfield,16,36)a finite momentum qdoes not play any important\nroleinthefollowingresults.Thisisbecausetheparamagne tic\ndepairing effect is suppressed by the orbital degree of free-\ndom,aswe showbelow.\nWe choosetheparameters\n(t1,t2,t3,t4,t5,∆)=(1.0,1.0,0.05,0.4,0.1,2.45),(15)\nsoastoreproducetheelectronicstructureoftwo-dimensio nal\nelectron gases.18–23,26–28)We choose the unit of energy as\nt1=1. Band structure calculations resulted in t1=300\nmeV,26)giving rise to an anisotropic Fermi velocity, vF=\n7×104−4×105m/s, forn=0.15. For the parameters in\neq. (15), the d xyorbital has a lower energy than the d yz/dzx\norbitals, as expected in STO heterostructures;18–23,26–28)the\nlevel splitting at the Γpoint is−2t2−2t3+4t1+4t4−∆=\n1.05∼300meV. The chemical potential µis determined so\nthat the mobile carrier density per Ti ion is n. Although an\nenormouscarrierdensityof3 .5×1014cm−2correspondingto\nn=0.5 at the STO/LAO interface was predicted by the “po-\nlarcatastrophe”mechanism,1)recentexperimentshaveshown\na rather low density of mobile carriers.11–14,18,19)One of our\npurposes is to clarify the carrier density dependence of the\nsuperconductingstate. ThesourcesofRashba spin-orbitco u-\npling are assumed to be todd=0.25 andλ=0.2 unless men-\ntioned otherwise explicitly. We here assume rather large va l-\nues oftoddandλso that the amplitude of Rashba spin-orbit\ncouplingα∼toddλ/∆is larger than the transition tempera-\nture of superconductivity. We assume attractive interacti ons\nU=U′so that the transition temperature at zero magnetic\nfield isTc=0.005=17 K. A large transition temperature\ncompared with the experimental Tc=0.3 K is assumed for\nthe accuracy of numerical calculation. Since we discuss theJ.Phys.Soc.Jpn. FullPaper Author Name 3\n-1.5-1-0.5 0 0.5 1 1.5 2 2.5 3\n-1.5  0 1.5Ej(k)\nkx\n 0 0.5 1 1.5 2 0 0.5 1 1.5 2 ky\n kx 0 0.5 1 1.5 2 0 0.5 1 1.5 2 ky\n kx(a)(b)\n(c) (d)\nn=0.05 n=0.2dxydyz /dzxdyz\n 0 0.05 0.1 0.15 0.2\n 0 0.2 0.4 0.6 0.8 1 α1, α2\nn α1\n α2\nFig. 1. (Color online) (a) Band structure of our model. We sho w the dis-\npersion relation Ej(k) fork=(kx,0). The origin is the chemical po-\ntentialµfor a carrier density n=0.15. (b) Carrier density dependence\nof spin-orbit coupling on the Fermi surface. We show α1=E2(kF2)−\nE1(kF2) (solid line) and α2=E4(kF4)−E3(kF4) (dashed line) with kFj\nbeing the Fermi wave number of the j-th band along the [100] axis.\n(c) and (d) show Fermi surfaces for n=0.05 and for n=0.2, re-\nspectively. Other parameters are assumed as ( t1,t2,t3,t4,t5,∆,todd,λ)=\n(1.0,1.0,0.05,0.4,0.1,2.45,0.25,0.2).\nnormalized µBHc2/Tc,thefollowingresultsarehardlyaltered\nbythemagnitudeof Tc.Asweshowelsewhere,thesupercon-\nductingphaseisalmostindependentofthe ratio U′/U.\nFigure 1(a) shows the band structure of our model. We\nsee the spin splitting caused by the Rashba spin-orbit cou-\npling. Because the Rashba spin-orbit coupling is enhanced\naround the band crossing points,31)the magnitude of spin\nsplitting shows a nonmonotonic carrier density dependence .\nFigure 1(b) shows the spin splitting in the lowest pair of\nbands [α1=E2(kF2)−E1(kF2)] and that in the second low-\nest pair of bands [ α2=E4(kF4)−E3(kF4)] as a function of\ncarrier density, where kFjis the Fermi wave number of the\nj-th band along the [100] axis. The nonmonotonic behavior\nof a spin splitting, α1, is consistent with experimental obser-\nvationsforSTO/LAOinterfaces.Theseeminglycontradictory\ncarrierdensitydependence13,14)ofRashbaspin-orbitcoupling\nisprobablycausedbythepeakof α1,aspointedoutbyZhong\net al.27)In our model, the Fermi level crosses the bottom of\nthe second lowest pair of bands [ E4(0)=E3(0)=0] at ap-\nproximately n=0.16. The Fermi surfaces for n=0.05 and\nn=0.2 are shown in Figs. 1(c) and 1(d), respectively. The\nisotropic Fermi surfaces mainly consist of the d xyorbital for\na low carrier density, n=0.05,while large anisotropicFermi\nsurfaces mainly consist of the d yz/dzxorbitals for a large car-\nrierdensity, n=0.2.\nFirst, we discuss the superconducting state at zero mag-\nnetic field. While the superconductivity is mainly caused by\nthe dxyorbital at low carrier densities, n<0.078,the intraor-\nbitalCooperpairingofd yzanddzxorbitalsisthemainsource\nof superconductivity at high carrier densities, n>0.078.\nThiscrossoverofthesuperconductingstatecoincideswith thechange of quasiparticles on the Fermi surfaces discussed fo r\nFigs. 1(c) and 1(d). When we assume the attractive interac-\ntionsU=U′independent of carrier density, the transition\ntemperature monotonically increases with increasing carr ier\ndensity. The nonmonotonic carrier density dependence ob-\nservedinexperiments11)isreproducedbyassumingadecreas-\ning function of U=U′against carrier density. In this study,\nWe avoid such a phenomenological assumption and discuss\nthe normalized values such as µBHc2/Tc. Note that the odd-\nparity hybridization toddand LS coupling λhardly affect the\nsuperconductingstate atzeromagneticfield.\n 0 2 4 6 8 10\n 0 0.2 0.4 0.6 0.8 1µBHc2/TC(a)\nn=0.06\nT/TCn=0.1\nn=0.12\nn=0.15\nn=0.2\n 0 2 4 6 8 10 12 14\n 0.05 0.1 0.15 0.2 0.25µBHc2/Tc\nn(todd,λ)=(0.25,0.2)(b)\n(todd,λ)=(0.0,0.2)\n(todd,λ)=(0.25,0.0)\n(todd,λ)=(0.0,0.0)\nFig. 2. (Coloronline)(a)Normalizeduppercriticalfield, µBHc2/Tc,forthe\nfield parallel to the [100] axis. Solid, dashed, and dash-dot ted lines show\nthe results for high carrier densities, n=0.12, 0.15, and 0.2, respectively.\nThe dotted line is obtained in the crossover region, n=0.1, while dash-\ntwo-dotted line assumes a low carrier density, n=0.06. Fermi surfaces\nmainly consist of the d xyorbital (d yz/dzxorbitals) in the low (high) carrier\ndensity region. The other parameters are the same as those in Fig. 1. (b)\nCarrierdensity dependence of µBHc2/Tcatthelowesttemperature T/Tc=\n0.05 [circles]. We also show the results for ( todd,λ)=(0.25,0) [squares],\n(0,0.2)[diamonds], and (0 ,0) [pluses] for comparison.\nOn the other hand, the Rashba spin-orbit coupling aris-\ning from the combination of toddandλleads to an intrigu-\ning superconducting phase in the magnetic field. Figure 2(a)\nshows the phase diagram against temperatures and magnetic\nfields for various carrier densities. We see an extraordinar ily\nlarge normalized upper critical field, µBHc2/Tc>9, beyond\nthe Pauli-Clogston-Chandrasekar limit, µBHc2/Tc=1.25,37)\naroundn=0.12−0.15.Ithasbeenshownthattheuppercriti-\ncalfieldisenhancedbytheRashbaspin-orbitcoupling,38)but\nthat the enhancement is minor in the canonical Rashba-type\nnon-centrosymmetric superconductors as µBHc2/Tc≈2.39)\nWe here find that the rather large enhancement of the upper4 J.Phys.Soc. Jpn. FullPaper Author Name\ncriticalfieldiscausedbythesynergyoftheRashbaspin-orb it\ncouplingand the orbitaldegreeof freedom.Indeed,whenwe\ndecrease the carrier density to n<0.08,the orbital degree of\nfreedom is quenched and the upper critical field is suddenly\ndecreased.\nAs shown in Fig. 2(b), the normalized upper critical field\nµBHc2/Tcshowsa broadpeak at approximately n=0.12and\ndecreases with increasing carrier density for n>0.12 ex-\ncept for a sharp enhancement at around n=0.16. The de-\ncrease inµBHc2/Tcis attributed to the decrease in Rashba\nspin-orbit coupling [see Fig. 1(b)]. A sharp peak at around\nn=0.16 is induced by the appearance of small Fermi sur-\nfaces around theΓpoint, that is, the Lifshitz transition. Be-\ncausetheg-factorofthisbandvanishesat k=(0,0)(Γpoint)\ninthepresenceofatomicLScoupling λ,Cooperpairinginthe\nsmall Fermi surfaces is not disturbed by the magnetic field.\nThus, a sharp enhancement of the normalized upper critical\nfield,µBHc2/Tc, is a signature of the Lifshitz transition. It\nwillbeinterestingtolookforthisLifshitztransitionsin cethe\nClass D topological superconducting phase is realized near\nthe Lifshitz transition by applying a magnetic field.40)Since\nthe renormalization of the g-factor is not due to the broken\ninversion symmetry, a sharp peak of µBHc2/Tcalso appears\nfor (todd,λ)=(0,0.2) [diamonds in Fig. 2(b)], for which the\nLifshitz transition occurs at approximately n=0.1. Aside\nfrom this peak, a small upper critical field below the Pauli-\nClogston-Chandrasekarlimitisobtainedwheneither toddorλ\nis zero, because the Rashba spin-orbit coupling vanishes. A s\nexpected, the normalized upper critical field increases as w e\nincreasetoddorλ. For instance, we obtain µBHc2/Tc∼4.9\nfor (todd,λ)=(0.1,0.2) andn=0.12, in agreement with the\nexperimentalresultofSTO /LAOinterfaces.13)\n 0 0.2 0.4 0.6 0.8 1\n 0 0.4 0.8 1.2\nµΒH/TC(1,1)\n(2,2)(a)|ψmn|\n 0 0.2 0.4 0.6 0.8 1\n 0 0.4 0.8 1.2 1.6\nµΒH/TC(b)|ψmn|\n 0 0.2 0.4 0.6 0.8 1\n 0 0.4 0.8 1.2\nµΒH/TC(c)|ψmn|\n 0 0.2 0.4 0.6 0.8 1\n 0 1 2 3 4 5 6 7 8 9\nµΒH/TC(d)|ψmn|(1,2)\n(1,3)\n(2,3)\n(3,3)\nFig. 3. (Color online) Magnetic field dependence of order par ameters for\nn=0.15.Weshowtheamplitude |ψmn|atthetransition temperature, which\nis proportional to the order parameters |∆mn|belowTc. The main compo-\nnents are|ψ11|(thick solid line) and |ψ22|(thick dashed line). The other\nsmallcomponentsareshownbythethinlines,asdescribedin Fig.3(d).We\nassume (todd,λ)=(a) (0,0), (b) (0,0.2), (c) (0.25,0), and (d) (0 .25,0.2).\nTheother parameters are the sameas those in Fig. 2.\nInordertoclarifytherolesoftheorbitaldegreeoffreedom ,\nwe show the magnetic field dependence of order parametersfor a high carrier density, n=0.15. When both odd parity\nhybridization, todd, and LS coupling, λ, are finite [Fig. 3(d)],\nthe magneticfield alongthe x-axissubstantiallyenhancesthe\nCooperpairsofthed yzorbitalrepresentedby |ψ11|whilethose\nof the d zxorbital (|ψ22|) are suppressed. This means that a\nquasi-one-dimensional superconducting state dominated b y\nthe dyzorbital is stabilized in the magnetic field. Since this\nhigh-fieldsuperconductingphaseisrobustagainstthepara m-\nagneticdepairinge ffect,alargeuppercriticalfieldisobtained,\nasshowninFig.2.ItshouldbestressedthattheRashbaspin-\norbit couplingplays an essential role in stabilizing the qu asi-\none-dimensionalsuperconductingphase. Indeed,we obtain a\nnearlyisotropictwo-dimensionalsuperconductingphasew ith\n|ψ11|∼|ψ22|when either the odd parity hybridization toddor\ntheLS coupling λiszero[Figs.3(a)-3(c)].\n 0 0.2 0.4 0.6 0.8 1 1.2 1.4\n 0 0.2 0.4 0.6 0.8 1 1.2 1.4ky\nkx\nFig. 4. (Color online) g-vector of the lowest band ( l=1),g1(k), which is\ndefined in eq. (18). Arrows show the direction of the g-vector ; the length\nofarrowsisproportional to theamplitude oftheg-vector. S olid lines show\ntheFermisurfacesfor n=0.15.Theotherparametersarethesameasthose\nin Fig. 1.\nWe here illustrate why the quasi-one-dimensional super-\nconducting phase is protected against the paramagnetic de-\npairing effect. For this purpose, we derive the Rashba spin-\norbit coupling in the band basis as we have performed in\nref. 29. We reduce the single-particle Hamiltonian H0to the\nthree-bandmodelas H0=Hband+HASOC, where\nHband=3/summationdisplay\nl=1/summationdisplay\nksξl(k)a†\nk,lsak,ls, (16)\nHASOC=3/summationdisplay\nl=1/summationdisplay\nkgl(k)·Sl(k), (17)\nandξl(k)=(E2l(k)+E2l−1(k))/2. The Rashba spin-orbit\ncouplingofthe l-thbandisrepresentedbytheg-vector\ngl(k)=(E2l(k)−E2l−1(k))Sav\n2l(k)/|Sav\n2l(k)|,(18)\nwhose direction is obtained by calculating the average\nSav\nj(k)=/angbracketleft/summationtext\nm/summationtext\nss′σss′c†\nk,msck,ms′/angbracketrightjfor thej-th eigenstate.\nFigure 4 shows the g-vector g1(k) and the Fermi surfaces\nforn=0.15. It is shown that the momentum dependence of\nthe g-vector is quite di fferent from an often-assumed form,\ng(k)=α/parenleftBig\nsinky,−sinkx,0/parenrightBig\n. This is the characteristic prop-\nerty of orbitally degenerate non-centrosymmetricsystems .29)J.Phys.Soc.Jpn. FullPaper Author Name 5\nIn the case of STO heterostructures, the g-vector is nearly\nparallel to the y-axis for kx>ky, while it is almost along\nthex-axis for kx<ky. The quasiparticles mainly consist of\nthe dyzorbital (d zxorbital) for the former (latter). Since the\nCooperpairingisdisturbedbytheparamagneticdepairinge f-\nfect when the g-vector is parallel to the magnetic field, the\nfield along the x-axis suppresses the Cooper pairs of the d zx\norbital.Ontheotherhand,theCooperpairsformedbythed yz\norbital are protected by the g-vector nearly perpendicular to\nthemagneticfield.Inthisway,thequasi-one-dimensionals u-\nperconductingphaseisstabilizedbytheorbitaldegreeoff ree-\ndom so as to avoid the paramagneticdepairinge ffect. This is\nanintuitiveexplanationforthelargeuppercriticalfields hown\ninFig. 2.\nFinally, we discuss experimental results of the supercon-\nducting phase in STO /LAO interfaces. The superconducting\ntransitiontemperatureshowsanon-monotoniccarrierdens ity\ndependence11)and its peak at around n=2×1013cm−2co-\nincides with the crossover from d xy-orbital-dominated Fermi\nsurfaces to d yz/dzx-orbital-dominated Fermi surfaces.19)The\nRashba spin-orbit coupling seems to have the maximum am-\nplitudeinthecrossoverregion,13,14)consistentwiththethree-\norbital tight-binding model adopted in this study. Interes t-\ningly, a large upper critical field, µBHc2/Tc≈4.2, beyond\nthePauli-Clogston-Chandrasekarlimithasbeenreportedf ora\nhighcarrierdensity n=3×1013cm−2closetothecrossover.13)\nThe decrease in the normalized upper critical field µBHc2/Tc\nwith increasing carrier density was also observed for n>\n3×1013cm−2.13)Thesebehaviorsareconsistentwithourfind-\ning in Fig. 2, although the signature of Lifshitz transition\nhas not been found. This agreement with experimental re-\nsults indicates that the quasi-one-dimensional supercond uct-\ning phase is realized in the STO /LAO interfaces with high\ncarrier densities. In contrast to the theoretical proposal for\na helical superconducting phase with a finite total momen-\ntumofCooperpairs,15)alargeuppercriticalfieldisattributed\nto the entanglement of orbitals and spins in our three-orbit al\nmodel. Indeed, we confirmed that the finite total momentum\nof Cooper pairs, namely, the finite qin the T-matrix, hardly\nchangesourresults.Thecoexistenceofsuperconductivity and\nferromagnetism7–10)may also be attributed to the quasi-one-\ndimensionalsuperconductingphaseprotectedagainstspin po-\nlarization. We would like to stress that such a spin-polariz ed\nsuperconducting state is hardly stabilized in the multiban d\nmodels,41,42)which phenomenologically assume the Rashba\nspin-orbitcouplingandneglecttheorbitaldegreeoffreed om.\nOur proposal for the quasi-one-dimensionalsuperconducti ng\nphase can be verified by experiments using a tilted magnetic\nfield. For instance, a vortex lattice structure elongated al ong\nthe[010]axiswillbeobservedinthefieldslightlytiltedfr om\nthe [100] axis to the [001] axis. As for a quantitative discus -\nsion, the crossover between low and high carrier density re-\ngions occurs in our model at around n=0.08, which cor-\nresponds to a carrier density of n=5×1013cm−2. This is\nin reasonable agreement with experimental carrier density of\nn=2×1013cm−2,19)anda discrepancyprobablyarisesfrom\nour inexact choice of tight-binding parameters. Note that a\nlarge upper critical field has been observed in δ-doped STO\nthin films.43)Although the global inversion symmetry is not\nbroken in this system, surface Rashba spin-orbit couplings\nplay a similar role to the spin-orbit coupling in this study, asdemonstrated for locally non-centrosymmetric supercondu c-\ntors.44)\nIn summary, we studied the superconductivity in the two-\ndimensional electron systems formed at the STO /LAO inter-\nface and STO surface. We analyzed the three-orbital model\ntaking into account t2gorbitals of Ti ions, and found that\nan unconventional structure of Rashba spin-orbit coupling\narisesfromtheorbitaldegeneracyandprotectsthequasi-o ne-\ndimensionalsuperconductingphaseagainstthe paramagnet ic\ndepairing effect. The orbital degree of freedom plays an es-\nsential role in the response to the magnetic field and leads to\nalargeuppercriticalfield.Thepeakoftheuppercriticalfie ld\nas a function of carrier density coincides with the crossove r\nfrom d xy-orbital-dominatedFermi surfaces to d yz/dzx-orbital-\ndominatedFermi surfaces. These observationsprovidea sys -\ntematic understanding of superconducting properties at th e\nSTO/LAOinterface.\nThe authors are grateful to S. Fujimoto and T. Shishido\nfor fruitful discussions. This work was supported by KAK-\nENHI(GrantNos.25103711,24740230,and23102709),and\nby a Grant for the Promotion of Niigata University Research\nProjects.\n1) A.Ohtomo and H.Y. Hwang: Nature 427(2004) 423.\n2) J. Biscaras, N. Bergeal, A. Kushwaha, T. Wolf, A. Rastogi, R.C. Bud-\nhani, and J.Lesueur: Nat. Commun. 1(2010) 89.\n3) K. Ueno, S. Nakamura, H. Shimotani, A. Ohtomo, N. Kimura, T . No-\njima, H.Aoki, Y.Iwasa, and M. Kawasaki: Nat. Mater. 7(2008) 855.\n4) Y. Kozuka, M. Kim, C. Bell, B. G. Kim, Y. Hikita, and H. Y. Hwa ng:\nNature462(2009) 487.\n5) N. Reyren, S. Thiel, A. D. Caviglia, L. Fitting Kourkoutis , G. Ham-\nmerl, C.Richter, C.W.Schneider, T.Kopp,A.-S.R¨ uetschi, D.Jaccard,\nM. Gabay, D.A. Muller, J.-M.Triscone, and J. Mannhart: Scie nce317\n(2007) 1196.\n6) A. Brinkman, M. Huijben, M. van Zalk, J. Huijben, U. Zeitle r, J.\nC. Maan, W. G. van der Wiel, G. Rijnders, D. H. A. Blank, and H.\nHilgenkamp: Nat. Mater. 6(2007) 493.\n7) D.A.Dikin, M.Mehta, C.W.Bark,C.M.Folkman, C.B.Eom,an d V.\nChandrasekhar: Phys.Rev. Lett. 107(2011) 056802.\n8) L. Li, C. Richter, J. Mannhart, and R. C. Ashoori: Nat. Phys .7(2011)\n762.\n9) J.A.Bert, B.Kalisky, C.Bell, M.Kim,Y.Hikita, H.Y.Hwan g, and K.\nA.Moler: Nat. Phys. 7(2011) 767.\n10) Ariando, X. Wang, G. Baskaran, Z. Q. Liu, J. Huijben, J. B. Yi, A.\nAnnadi, A. Roy Barman, A. Rusydi, S. Dhar, Y. P. Feng, J. Ding, H.\nHilgenkamp, and T.Venkatesan: Nat. Commun. 2(2011) 188.\n11) A. D. Caviglia, S. Gariglio, N. Reyren, D. Jaccard, T. Sch neider, M.\nGabay, S. Thiel, G.Hammerl, J.Mannhart, and J.-M.Triscone : Nature\n456(2008) 624.\n12) C. Bell, S. Harashima, Y. Kozuka, M. Kim, B. G. Kim, Y. Hiki ta, and\nH.Y. Hwang: Phys.Rev. Lett. 103(2009) 226802.\n13) M. Ben Shalom, M. Sachs, D. Rakhmilevitch, A. Palevski, a nd Y. Da-\ngan: Phys.Rev. Lett. 104(2010) 126802.\n14) A. D. Caviglia, M. Gabay, S. Gariglio, N. Reyren, C. Cance llieri, and\nJ.-M.Triscone: Phys.Rev. Lett. 104(2010) 126803.\n15) K. Michaeli, A. C. Potter, and P. A. Lee: Phys. Rev. Lett. 108(2012)\n117003.\n16)Non-Centrosymmetric Superconductors: Introduction and O verview,\ned. by E.Bauer and M.Sigrist (Springer-Verlag, Berlin, 201 2).\n17) S.Banerjee, O.Erten, and M. Randeria: arXiv:1303.3275 .\n18) A. F. Santander-Syro, O. Copie, T. Kondo, F. Fortuna, S. P ailh` es, R.\nWeht,X.G.Qiu,F.Bertran,A.Nicolaou,A.Taleb-Ibrahimi, P.LeF` evre,\nG.Herranz,M.Bibes,N.Reyren,Y.Apertet,P.Lecoeur,A.Ba rth´ el´ emy,\nand M.J.Rozenberg Nature 469(2011) 189.\n19) A.Joshua,S.Pecker,J.Ruhman,E.Altman,andS.Ilani:N at.Commun.\n3(2012) 1129.\n20) Z.S.Popovi´ c,S.Satpathy,andR.M.Martin:Phys.Rev.L ett.101(2008)6 J.Phys.Soc. Jpn. FullPaper Author Name\n256801.\n21) R. Pentcheva and W.Pickett: Phys.Rev. B 78(2008) 205106.\n22) P. Delugas, A. Filippetti, V. Fiorentini, D. I. Bilc, D. F ontaine, and P.\nGhosez: Phys.Rev. Lett. 106(2011) 166807.\n23) G.Khalsa and A.H.MacDonald: Phys.Rev. B 86(2012) 125121.\n24) Y. Mizohata, M. Ichioka, and K. Machida: Phys. Rev. B 87(2013)\n014505.\n25) R. M. Fernandes, J. T. Haraldsen, P. W¨ olfle, and A. V. Bala tsky: Phys.\nRev. B87(2013) 014510.\n26) M. Hirayama, T. Miyake, and M. Imada: J. Phys. Soc. Jpn. 81(2012)\n084708.\n27) Z.Zhong,A. T´ oth, and K.Held: Phys.Rev. B 87(2013) 161102.\n28) G.Khalsa, B.Lee, and A.H.MacDonald: arXiv:1301.2784.\n29) Y. Yanase: J. Phys. Soc. Jpn. 82(2013) 044711; Y. Yanase and H.\nHarima: Kotai-Butsuri 47(2012) No.3,1 [in Japanese].\n30) Y. Kim,R.M. Lutchyn, and C. Nayak: arXiv:1304.0464.\n31) Y. Yanase and M.Sigrist: J.Phys. Soc.Jpn. 77(2008) 124711.\n32) M.Nagano,A.Kodama,T.Shishidou,andT.Oguchi:J.Phys :Condens.\nMatter21(2009) 064239.\n33) G.Binnig, A.Barato ff, H.E.Hoenig,J.G.Bednorz: Phys.Rev.Lett. 45\n(1980) 1352.34) K. Yada, S. Onari, Y. Tanaka, and J. Inoue: Phys. Rev. B 80(2009)\n140509.\n35) J.A.Bert, K.C.Nowack, B.Kalisky,H.Noad,J.R.Kirtley , C.Bell, H.\nK. Sato, M. Hosoda, Y. Hikita1, H. Y. Hwang, and K. A. Moler: Ph ys.\nRev. B86(2012) 060503(R).\n36) K.Aoyama and M.Sigrist: Phys.Rev. Lett. 109(2012) 237007.\n37) B.S.Chandrasekhar:Appl.Phys.Lett. 1(1962)7;A.M.Clogston:Phys.\nRev. Lett. 9(1962) 266.\n38) P.A.Frigeri, D.F.Agterberg, A.Koga,and M.Sigrist: Ph ys.Rev.Lett.\n92(2004) 097001.\n39) Y. Yanase and M.Sigrist: J.Phys.Soc. Jpn. 76(2007) 124709.\n40) M. Sato, Y. Takahashi, and S. Fujimoto: Phys. Rev. Lett. 103(2009)\n020401.\n41) M.H.Fischer,S.Raghu,andE.-A.Kim:NewJ.Phys. 15(2013)023022.\n42) S. Caprara, F. Peronaci, and M. Grilli: Phys. Rev. Lett. 109(2012)\n196401.\n43) M. Kim,Y. Kozuka, C. Bell, Y. Hikita, and H.Y. Hwang: Phys . Rev. B\n86(2012) 085121.\n44) D. Maruyama, M. Sigrist, and Y. Yanase: J. Phys. Soc. Jpn. 81(2012)\n034702."    },    {        "title": "2001.08794v1.Bright_solitons_in_a_spin_tensor_momentum_coupled_Bose_Einstein_condensate.pdf",        "content": "arXiv:2001.08794v1  [cond-mat.quant-gas]  23 Jan 2020Bright solitons in a spin-tensor-momentum-coupled Bose-E instein condensate\nJie Sun,1Yuanyuan Chen,1,∗Xi Chen,1,2,†and Yongping Zhang1,‡\n1International Center of Quantum Artificial Intelligence fo r Science and Technology\n(QuArtist) and Department of Physics, Shanghai University , Shanghai 200444, China\n2Department of Physical Chemistry, University of the Basque Country UPV/EHU, Apartado 644, 48080 Bilbao, Spain\nSynthetic spin-tensor-momentum coupling has recently bee n proposed to realize in atomic Bose-\nEinstein condensates. Here we study bright solitons in Bose -Einstein condensates with spin-\ntensor-momentum coupling and spin-orbit coupling. The pro perties and dynamics of spin-tensor-\nmomentum-coupled and spin-orbit-coupled bright solitons are identified to be different. We con-\ntribute the difference to the different symmetries.\nI. INTRODUCTION\nIn ultracold neutral atoms, hyperfine spin states, cou-\npling to linear momentum [ 1–7] or orbital angular mo-\nmentum [ 8,9], are interesting and significant not only\nin fundamental phenomena of ultracold atoms and con-\ndensed matter physics, but also in the applications in\nquantum information processing, atom metrology and\natomitronics, with the current experimental progress.\nParticularly, the spin-orbit coupling (SOC) provides the\nunique dispersion relationship, exhibiting particular fea-\ntures without analogues in the case of without the SOC.\nThecompetitionbetweenatomicmany-bodyinteractions\nand the dispersion relation generates many fundamental\nground state phases [ 10–17] and exotic collective excita-\ntions [18,19] in spin-orbit-coupled Bose-Einsteinconden-\nsates (BECs).\nThe interplay between the nonlinearity stemming from\natomic interactions and dispersions also gives rise to the\nexistence of bright solitons which are spatially localized\nstates. The interested spin-orbit-coupled dispersions in-\nevitably change the existence and properties of bright\nsolitons [ 20,21]. In general, solitons follow the symme-\ntries of spin-orbit-coupled Hamiltonian, which provides\na deep insight into the searching of solitons. Moreover,\nthe dynamics of solitons is always accompanied by rich\nspin dynamics [ 22,23]. The lack of Galilean invariancein\nspin-orbit-coupled systems [ 24] makes that it is nontriv-\nial to find movable solitons, one can not directly obtain a\nmovable soliton from its stationary correspondence. Dif-\nferent aspects of bright solitons with the SOC have been\ninvestigated a lot [ 25–30], ranging from with long-ranged\ndipole interactions [ 31–33] to in optical lattices [ 34–40].\nVery recently, the generation of artificial spin-tensor-\nmomentum coupling (STMC) into an atomic BEC has\nbeen proposed [ 41]. Different from the usual spin-orbit\ncoupling where linear momentum is coupled with spin\nvectors, STMC is the interaction between linear momen-\ntum and spin tensors. Such emergent interaction can\n∗cyyuan@staff.shu.edu.cn\n†xchen@shu.edu.cn\n‡yongping11@t.shu.edu.cnbe applicable to the discovery of exotic topological mat-\nters [42,43].\nIn this paper, we investigate bright solitons in STMC\nBECs in which the three components of the ground hy-\npefine states of87Rb are utilized for experimental im-\nplementation. We first apply imaginary-time evolution\nmethod to study the stationary properties of STMC soli-\nton, and further explore the dynamics by using varia-\ntional method. By comparing with SOC bright soliton\nin Refs. [ 44–46], we conclude that the difference between\nSTMC and SOC bright solitons originates from the dif-\nferent symmetry relevant to spin rotation.\nThe paper will be organized as follows. In Sec. IIthe\nsystems and Hamiltonian are introduced for SOC and\nSTMC. Here we present both for completeness and fur-\nther comparison. Later, the bright solitons are discussed\nfor both STMC and SOC BECs in Sec. III, to clarify the\ndifference in the spin rotation and symmetry. Finally,\nconclusion are made in Sec. IV.\nII. MODEL AND HAMILTONIAN\nWe first consider the experiment of synthetic SOC in\nthree-component BECs [ 47,48], where the three hyper-\nfine states of87Rb atoms are utilized, with the energy\nsplitting by a bias magnetic field, as shown in Fig. 1(a,b).\nTo realize SOC, the atoms are dressed by two counter-\npropagating Raman laser beams, and the polarizations\nof lasers are arranged so that two-photon optical transi-\ntions can be induced, see Fig. 1(b). The transitions in\nthe basis of ( | ↑∝angbracketright=|1,−1∝angbracketright,|0∝angbracketright=|1,0∝angbracketright,| ↓∝angbracketright=|1,1∝angbracketright) are\nengineered as,\nHSOC\nRam= Ω\n0e−i2kRx0\nei2kRx0e−i2kRx\n0ei2kRx0\n,\nwhere Ω is the strength of two-photon Rabi coupling [ 49]\nandkRis the wavenumber of the Raman beams. Dur-\ning the transitions, there is a momentum exchange be-\ntween the atoms and lasers. Including kinetic energy, the\nHamiltonian becomes, HSOC=p2\nx/2m+HSOC\nRam,, with\nmbeing atomic mass and pxbeing momentum along\nthe direction of Raman lasers. To explicitly show the\nexistence of SOC, a unitary transformation is needed,2\nFIG. 1. Experimental schemes to realize the spin-orbit cou-\npling (a,b) and spin-tensor-momentum coupling (c,d). Thre e\nhyperfine states ( | ↑/angbracketright,|0/angbracketright,| ↓/angbracketright) are split by a bias magnetic\nfieldB0. In (a,b) two laser beams propagate oppositely to\ncouple|px−2/planckover2pi1kR,↑/angbracketright,|px,0/angbracketright,|px+2/planckover2pi1kR,↓/angbracketrightwithpxbeing mo-\nmentum along laser direction and quasimomentum 2 /planckover2pi1kRrel-\nevant to the wavenumber of lasers, the quasimoentum differ-\nence between hyperfine states constitutes the spin-orbit co u-\npling. In (c,d) two beams whoes polarizations are parallel t o\nthe bias magnetic field propagate along same direction and\nthe third beams in the opposite direction. They can couple\n|px−2/planckover2pi1kR,↑/angbracketright,|px,0/angbracketright,|px−2/planckover2pi1kR,↓/angbracketright.\nUSOC=ei2kRxFz, such that the Hamiltonian ˜HSOC=\nUSOCHSOCU−1\nSOCbecomes\n˜HSOC=p2\nx\n2m−4/planckover2pi1kRpxFz\n2m+4(/planckover2pi1kR)2F2\nz\n2m+√\n2ΩFx.(1)\nHere (Fx,Fy,Fz) are spin-1 Pauli matrices, and the SOC\n2/planckover2pi1kRpxFz/mis involved. Physically, the SOC means\nthatthereisaquasimomentumdifference −2/planckover2pi1kRbetween\nstates| ↑∝angbracketrightand|0∝angbracketright, and between |0∝angbracketrightand| ↓∝angbracketright.\nNext, the STMC can be introduced artificially by\ndressing the atoms with three Raman beams [ 41], see\nFig.1(c,d). Two of them with same linear polarization\npropagate along same direction, and the other propa-\ngates oppositely. The two-photon transitions accompa-\nnying momentum transfers become,\nHSTMC\nRam= Ω\n0e−i2kRx0\nei2kRx0ei2kRx\n0e−i2kRx0\n.\nNote that the difference between HSOC\nRamandHSTMC\nRam\nis very slight. To eliminate the spatial dependence in\nHSTMC\nRam, a unitary transformation USTMC=ei2kRxF2\nzis\nperformed, and the new total Hamiltonian ˜HSTMC=\nUSTMCHSTMCU−1\nSTMC, withHSTMC=p2\nx/2m+HSTMC\nRam\nis expressed as,\n˜HSTMC=p2\nx\n2m−4/planckover2pi1kRpxF2\nz\n2m+4(/planckover2pi1kR)2F2\nz\n2m+√\n2ΩFx.(2)The STMC takes a specific form as 2 /planckover2pi1kRpxF2\nz/m. From\nthe above equation, it is clear that such specific STMC is\njust a rearrangement of quasimomentum difference com-\nparing with the case of the SOC. The quasimomentum\ndifference between | ↑∝angbracketrightand|0∝angbracketrightis−2/planckover2pi1kR, while it is 2 /planckover2pi1kR\nbetween |0∝angbracketrightand| ↓∝angbracketright.\nIII. BRIGHT SOLITONS WITH STMC AND\nSOC\nNow, we arereadytostudy brightsolitonsin the BECs\nwith both the STMC and SOC whose experimental re-\nalizations are analyzed in the previous section II. We\nstart from the standard Gross-Pitaevskii (GP) equations\nand take into consideration the spin-tensor-momentum-\ncoupled and spin-orbit-coupled Hamiltonian in Eq. ( 1)\nand Eq. ( 2). The dimensionless GP equations for spin-\ntensor-momentum-coupled BEC are,\ni∂Ψ\n∂t= [−∂2\nx+(4i∂x+4+∆)F2\nz+√\n2ΩFx+Hint]Ψ,(3)\nwhile, the spin-orbit-coupled GP equations are\ni∂Ψ\n∂t= [−∂2\nx+4i∂xFz+(4+∆) F2\nz+√\n2ΩFx+Hint]Ψ.(4)\nIn the both equations, the units of energy, position co-\nordinate and time that we adopt are /planckover2pi12k2\nR/2m,1/kRand\n2m//planckover2pi1k2\nRrespectively. The additional term ∆ F2\nzorigi-\nnates from quadratic Zeeman effect. Three-component\nwave functions are Ψ = (Ψ ↑,Ψ0,Ψ↓)T, for convenience,\nin the following, we relabel the wave functions as Ψ =\n(Ψ1,Ψ2,Ψ3)T. In above equations, Hint=g0(|Ψ1|2+\n|Ψ2|2+|Ψ3|2), forsimplicity, weonlyconsiderinteractions\nhaving SU(3) symmetry. Since our aim is to investigate\nthe bright solitons, we focus on attractive interactions of\ng0<0.\nThe difference between the spin-tensor-momentum-\ncoupled and spin-orbit-coupled GP equations is the ap-\npearance of 4 i∂xF2\nzand 4i∂xFz. Such difference leads\nto different symmetries of GP equations, which affects\nthe properties of bright solitons. We find stationary\nbright solitons by the numerical calculation of GP equa-\ntions using the imaginary-time evolution method, be-\ncause of which, the soliton solutions belong to ground\nstates. Typical soliton profiles are demonstrated in\nFig.2. The upper panel is the profiles of spin-tensor-\nmomentum-coupled solitons, and the lower panel is that\nof spin-orbit-coupled solitons. For further comparison,\nwe adopt same parameters for the GP equations with\nSTMC and SOC. Our general observation is that the\nimaginarypartsof soliton wavefunctions for both STMC\nand SOC do not vanish. In contrast, the ground states\nof ordinary BECs (without STMC or SOC) are real-\nvalued with no node in wave functions [ 12]. This is\nthe unique feature of spin-orbit-coupled [ 21] and spin-\ntensor-momentum-coupled BECs. At first sight, the3\n/s45/s48/s46/s52/s45/s48/s46/s50/s48/s46/s48/s48/s46/s50\n/s45/s48/s46/s48/s53/s48/s46/s48/s48/s48/s46/s48/s53/s48/s46/s49/s48\n/s45/s48/s46/s52/s45/s48/s46/s50/s48/s46/s48/s48/s46/s50\n/s45/s49/s48 /s48 /s49/s48/s45/s48/s46/s52/s45/s48/s46/s50/s48/s46/s48/s48/s46/s50/s48/s46/s52\n/s45/s49/s48 /s48 /s49/s48/s45/s48/s46/s49/s48/s45/s48/s46/s48/s53/s48/s46/s48/s48/s48/s46/s48/s53/s48/s46/s49/s48\n/s45/s49/s48 /s48 /s49/s48/s45/s48/s46/s52/s45/s48/s46/s50/s48/s46/s48/s48/s46/s50/s48/s46/s52/s97/s49 /s97/s50 /s97/s51\n/s120/s98/s49\n/s120/s82/s101/s97/s108/s73/s109/s97/s103/s98/s50\n/s120/s89/s49/s89/s50/s89/s51\n/s98/s51\n/s45/s48/s46/s52/s45/s48/s46/s50/s48/s46/s48/s48/s46/s50\n/s45/s48/s46/s48/s53/s48/s46/s48/s48/s48/s46/s48/s53/s48/s46/s49/s48\n/s45/s48/s46/s52/s45/s48/s46/s50/s48/s46/s48/s48/s46/s50\n/s45/s49/s48 /s48 /s49/s48/s45/s48/s46/s52/s45/s48/s46/s50/s48/s46/s48/s48/s46/s50/s48/s46/s52\n/s45/s49/s48 /s48 /s49/s48/s45/s48/s46/s49/s48/s45/s48/s46/s48/s53/s48/s46/s48/s48/s48/s46/s48/s53/s48/s46/s49/s48\n/s45/s49/s48 /s48 /s49/s48/s45/s48/s46/s52/s45/s48/s46/s50/s48/s46/s48/s48/s46/s50/s48/s46/s52/s99/s49 /s99/s50 /s99/s51\n/s120/s100/s49\n/s120/s100/s50/s51\n/s120/s82/s101/s97/s108 /s73/s109/s97/s103/s89/s49/s89/s50/s89/s51\n/s100/s51\nFIG. 2. Profiles of the spin-tensor-momentum-coupled (uppe r panel) and spin-orbit-coupled (lower panel) bright solit ons. In\neach panel, the first (second) row is the real (imaginary) par ts of soliton wave functions Ψ = (Ψ 1,Ψ2,Ψ3)T. Solid-lines are\nsolutions from the imaginary-time evolution method and dot -lines are analytical solutions from the variational metho d. The\ndimensionless parameters are ∆ = −1,Ω = 0.5 andg0=−2.\nspin-tensor-momentum-coupled solitons share same pro-\nfiles with spin-orbit-coupled solitons, especially, the real\nparts of soliton wave functions are almost same. How-\never, there exists an apparent difference in the imaginary\nparts.\nOur solitons as ground states follow symmetries of the\nsystems. The stationary spin-tensor-momentum-coupled\nGP equations in Eq. ( 3) have a spin rotating symmetry,\nRSTMC=eiπFx=\n0 0−1\n0−1 0\n−1 0 0\n,(5)\nwhich rotates spins along the Fxaxis by the angle of π,\nand a joint parity symmetry,\nOSTMC=PK, (6)\nwithPandKbeing the parity and complex conjugate\noperators. The symmetry RSTMCis relevant to the spin\ntensorF2\nx, sinceF2\nx=1\n2(I−RSTMC). The eigen-equation\nisRSTMCΨ =±Ψ. For the +1 eigenstate, Ψ 2(x) = 0,\nwhich leads to ∝angbracketleftFx∝angbracketright= 0. Whereas, to minimize energy ofRabicouplingterm√\n2ΩFx, itispreferablethat ∝angbracketleftFx∝angbracketright<0.\nTherefore, bright solitons select the eigenstate with −1\neigenvalue, RSTMCΨ =−Ψ, the consequence of which is\nΨ1(x) = Ψ3(x). Fig.2demonstratesΨ 1(x) = Ψ3(x)from\nthe real and imaginary parts. The symmetry OSTMC\ndetermines that the parity of real parts of soliton wave\nfunctions Ψ 1,Ψ2and Ψ 3should be opposite to that of\nimaginary parts. The real parts are even and imaginary\nparts are odd, see Fig. 2.\nThe symmetry of the stationaryspin-orbit-coupledGP\nequations in Eq. ( 4) is slightly different from the case of\nthe STMC. The spin-orbit-coupled equations possess a\nparticular spin rotating symmetry,\nRSOC=PeiπFx=P\n0 0−1\n0−1 0\n−1 0 0\n,(7)\nwhich must be the joint of spin rotation and parity. The\nequations also have the symmetry PKwhich is same as\nthe spin-tensor-momentum-coupled case, so the parity of\nreal and imaginary parts of spin-orbit-coupled solitons\nare even and odd respectively, which can be confirmed4\n/s45/s52 /s48 /s52/s48/s46/s48/s49/s46/s48/s50/s46/s48\n/s45/s52 /s48 /s52/s48/s46/s48/s49/s46/s48/s50/s46/s48\n/s45/s52 /s48 /s52/s48/s46/s53/s49/s46/s48/s49/s46/s53\n/s45/s52 /s48 /s52/s45/s52/s46/s48/s45/s50/s46/s48/s48/s46/s48\n/s48 /s49 /s50/s49/s46/s56/s50/s46/s48\n/s48 /s49 /s50/s49/s46/s54/s49/s46/s56/s50/s46/s48\n/s48 /s49 /s50/s48/s46/s53/s48/s46/s54\n/s48 /s49 /s50/s45/s52/s46/s48/s45/s51/s46/s48/s68/s97/s49\n/s68/s107 /s49\n/s107 /s49/s107 /s50\n/s107 /s50/s115\n/s115/s69\n/s69/s97/s50\n/s68/s97/s51\n/s68/s97/s52\n/s87/s98/s49\n/s87/s98/s50\n/s87/s98/s51\n/s87/s98/s52\n/s45/s52 /s48 /s52/s48/s46/s48/s49/s46/s48/s50/s46/s48\n/s45/s52 /s48 /s52/s48/s46/s48/s49/s46/s48/s50/s46/s48\n/s45/s52 /s48 /s52/s48/s46/s53/s49/s46/s48/s49/s46/s53\n/s45/s52 /s48 /s52/s45/s52/s46/s48/s45/s50/s46/s48/s48/s46/s48\n/s48 /s49 /s50/s49/s46/s56/s50/s46/s48\n/s48 /s49 /s50/s49/s46/s54/s49/s46/s56/s50/s46/s48\n/s48 /s49 /s50/s48/s46/s53/s48/s46/s54/s48/s46/s55\n/s48 /s49 /s50/s45/s52/s46/s48/s45/s51/s46/s53/s45/s51/s46/s48/s68/s99/s49\n/s68/s107 /s49\n/s107 /s49/s107 /s50\n/s107 /s50/s115\n/s115/s69\n/s69/s99/s50\n/s68/s99/s51\n/s68/s99/s52\n/s87/s100/s49\n/s87/s100/s50\n/s87/s100/s51\n/s87/s100/s52\nFIG. 3. Features of the spin-tensor-momentum-coupled (upp er panel) and spin-orbit-coupled (lower panel) bright soli tons\ncharacterized from variational wave functions. The variat ional parameters k1,k2,σand total energy ESTMC,ESOCare a\nfunction of ∆ and Ω. Solid-lines are from the variational met hod and dots are from the imaginary-time evolution method. I n\nthe first (second) row of each panel, Ω = 1 (∆ = −3). The nonlinear coefficient g0=−2.\nfrom Fig. 2. The eigen-equation of RSOCisRSOCΨ(x) =\n±Ψ(x), taking into account the parity of real and imag-\ninary parts of wave functions, solitons choose the eigen\nstate with −1 eigenvalue, if they choose the state with\n+1 eigenvalue, then ∝angbracketleftFx∝angbracketright= 0, the Rabi coupling en-\nergy can not be minimized. With −1 eigenvalue, the\nsymmetry RSOCrequires that Ψ 1(x) = Ψ 3(−x) and\nΨ2(x) = Ψ 2(−x). Finally, because of the parity from\nPK, the real parts of Ψ 1(x) and Ψ 3(x) become equal and\nthe imaginary parts of Ψ 1(x) and Ψ 3(x) have opposite\nsigns, while the imaginary part of Ψ 2(x) must disappear.\nThe above symmetry analysis provides a deep insight\ninto the understanding of solitons. So, we are motivated\nto apply a variational function to stimulate correspond-\ning solitons as follows. For the spin-tensor-momentum-coupled soliton, the variational wave function is,\nΨSTMC=\nA[cos(k1x)+iρ0sin(k1x)]\nB[cos(k2x)+iρ1sin(k2x)]\nA[cos(k1x)+iρ0sin(k1x)]\nsech(σx).(8)\nThis trial wave function completely satisfies the sym-\nmetries of RSTMCandOSTMC. Variational parameters\nA,B,k 1,k2,ρ0,ρ1andσwould be determined by the\nminimization of the total energy ESTMC=/integraltext\ndx(E0+\n¯ESTMC), with the energy density,\nE0=|∂xΨ1|2+|∂xΨ2|2+|∂xΨ3|2+(∆+4)( |Ψ1|2\n+|Ψ3|2)+Ω(Ψ 1Ψ∗\n2+Ψ∗\n1Ψ2+Ψ2Ψ∗\n3+Ψ∗\n2Ψ3)\n+g0\n2/parenleftbig\n|Ψ1|2+|Ψ2|2+|Ψ3|2/parenrightbig2, (9)\nand\n¯ESTMC= 4i(Ψ∗\n1∂xΨ1+Ψ∗\n3∂xΨ3).(10)5\n/s48/s53/s48/s49/s48/s48/s48/s50/s48/s52/s48/s54/s48\n/s48/s46/s48/s48/s46/s50/s48/s46/s52\n/s116 /s124/s89 /s124/s50\n/s120/s97\n/s45/s50/s48/s48 /s45/s49/s48/s48 /s48 /s49/s48/s48 /s50/s48/s48/s48/s50/s48/s52/s48/s54/s48\n/s48/s46/s48/s48/s46/s50/s48/s46/s52\n/s116 /s124/s89 /s124/s50\n/s120/s98\nFIG. 4. The time evolution of an initial spin-tensor-moment um-coupled (a) and spin-orbit-coupled (b) solitons after s witching\noff the spin-tensor-momentum coupling and spin-orbit coupl ing respectively, |Ψ|2=|Ψ1|2+|Ψ2|2+|Ψ3|2. The parameters are\n∆ =−1,Ω = 0.5 andg0=−2.\nConsidering the symmetries of RSOCandPK, the vari-\national wave function for a spin-orbit-coupled soliton\nmight be,\nΨSOC=\nA[cos(k1x)+iρ0sin(k1x)]\nBcos(k2x)\nA[cos(k1x)−iρ0sin(k1x)]\nsech(σx).(11)\nAll unknown quantities appearing in above function\nshould be determined by the minimization of the energy\nESOC=/integraltext\ndx(E0+¯ESOC), here the spin-orbit-coupled\nenergy density is,\n¯ESOC= 4i(Ψ∗\n1∂xΨ1−Ψ∗\n3∂xΨ3).(12)\nThe results from variational approximation approch\nfor both spin-tensor-momentum-coupled and spin-orbit-\ncoupled solitons are shown by dot-lines in Fig. 2. Obvi-\nously, the variational wave functions are consistent with\nthe results from the imaginary-timeevolution method, as\ndiscussed before.\nWe characterizethe propertiesof brightsolitons by the\nvariational wavefunctions. The features are identified by\nthe dependence of k1,k2,σand the total energy ESTMC\nandESOCon the variables of ∆ and Ω. The results are\ndescribedin Fig. 3. The magnitudesof k1andk2arerele-\nvanttothenumberofnodesinsolitonprofiles. Thelarger\nk1andk2induce more oscillations in real and imaginary\nparts of soliton wave functions (see Fig. 2). This type\nof oscillation is the exotic properties of STMC (4 i∂xF2\nz)\nand SOC (4 i∂xFz). Because of the competition between\n4i∂xF2\nz(4i∂xFz) and (∆+4) F2\nzor√\n2ΩFx, large ∆ and\nΩ suppress the effect of the STMC and SOC, thus reduc-\ning the oscillation nodes. As a result, k1ork2decreases\nwith increasing ∆ or Ω. This somehow explains the ten-\ndency of lines in Fig. 3(a1-d2). Besides, the modification\nofk1andk2, the Rabi coupling√\n2ΩFxalso makes soli-ton wave packets more spatially localized to reduce os-\ncillations. Finally, as shown in Fig. 3(b3,d3), σincreases\nwhen increasing Ω. However, the dependence of σon ∆\nis not monotonous at all (see Fig. 3(a3,c3)), resulting in\ntwo obvious slopes in the total energy as a function of\n∆ in Fig. 3(a4,c4). Fig. 3(b4,d4) demonstrates that Ω\nalways reduces the total energy, due to the fact that the\nRabi coupling energy is proportional to ∝angbracketleftFx∝angbracketright, satisfying\n∝angbracketleftFx∝angbracketright<0.\nNext, we turn to the dynamics of spin-tensor-\nmomentum-coupled and spin-orbit-coupled solitons.\nTwo different kinds of dynamics are presented as follows.\nFirstofall,thequenchdynamicsisshowninFig. 4, where\nthe initial soliton states are evolved after switching off\nthe STMC or SOC, by solving the real time evolution\nof Eqs. ( 3) or Eq. ( 4) but without the STMC (4 i∂xF2\nz\nterm) or the SOC (4 i∂xFzterm). Fig. 4(a) and (b) cor-\nrespond to the spin-tensor-momentum-coupled and the\nspin-orbit-coupled solitons, respectively. After switch-\ning off the STMC or SOC, solitons are not stationary.\nThis provides clear evidence that the solitons are intrin-\nsically supported by the STMC or SOC. Interestingly,\nthe time evolution of the spin-tensor-momentum-coupled\nand spin-orbit-coupled solitons are much different. The\nspin-tensor-momentum-coupled soliton moves along one\ndirection, while the spin-orbit-coupled soliton splits into\ntwo parts with opposite velocities. This is because that\nthe initial soliton satisfies k1=k2,ρ0=ρ1= 1. There-\nfore, the spin-tensor-momentum-coupled soliton is the\nspatial confinement of a plane wave, after tuning off\nthe STMC, it moves in the direction of the plane wave.\nWhile, the spin-orbit-coupled solion includes two plane-\nwave modes due to the component Ψ 2∝cos(k2x) =\n(eik2x+e−ik2x)/2. The Rabi coupling transfers these\ntwo plane-wave modes into other components, leading to\nthe splitting of two branches during the evolution.6\n/s48 /s49/s48/s48 /s51/s48/s48 /s51/s53/s48 /s55/s53/s48 /s49/s51/s53/s48 /s50/s49/s53/s48 /s51/s49/s50/s48/s48/s53/s48/s48/s49/s48/s48/s48/s49/s53/s48/s48/s50/s48/s48/s48\n/s120/s116/s97\n/s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48 /s53/s48/s48 /s55/s48/s48 /s49/s49/s48/s48 /s49/s50/s48/s48/s48/s53/s48/s48/s49/s48/s48/s48/s49/s53/s48/s48/s50/s48/s48/s48\n/s116\n/s120/s98\nFIG. 5. The time evolution of an initial spin-tensor-moment um-coupled (a) and spin-orbit-coupled (b) initiated by a co nstant\nweak acceleration force which is implemented by adding a lin ear potential −0.001xinto GP equations in Eq. ( 3) and Eq. ( 4).The\nparameters are ∆ = −1,Ω = 0.5 andg0=−2.\nSecondly, we shall explore the acceleration of bright\nsolitons. We add a constant weak force to accelerate\nthe initially prepared soliton. The slow adiabatic ac-\nceleration connects moving bright solitons to station-\nary bright solitons [ 21]. Due to the lack of Galilean\ninvariance in spin-tensor-momentum-coupled and spin-\norbit-coupled systems, the profiles of moving solitons be-\ncomes different from these of stationary solitons, there-\nfore, they are changed during the acceleration, as illus-\ntrated in Fig. 5. The change of the spin-orbit-coupled\nsoliton is more pronounced than that of the spin-tensor-\nmomentum-coupled soliton (see Fig. 5). We provide a\nsimple insight into the understanding of such difference.\nThe moving bright soliton solutions should be\nΨ(x,t) = Φv(x−2vt,t)eivx−iv2t,(13)\nwithvbeing moving velocity. Substituting this ansatz\ninto GP equations in Eq. ( 3) and Eq. ( 4), the resulted\nequations for Φ v(x−2vt,t) are different from the orig-\ninal ones by additional appearing of −4vF2\nzand−4vFz\nrespectively for the spin-tensor-momentum-coupled and\nspin-orbit-coupled equations. The additional −4vF2\nz\ndoes not have an effect on the symmetry RSTMC, so\nthe moving spin-tensor-momentum-coupled bright soli-\ntons still possess RSTMC. In contrast, −4vFzfor the\nspin-orbit-coupled solitons breaks the symmetry RSOC.\nThe constant acceleration force linearly increases the ve-\nlocities of solitons. However, the symmetry RSTMCman-\nages to protect the profiles of bright soliton, by avoiding\nto dramatic change. The initial stationary spin-orbit-coupled bright soliton changes distinctly during the ac-\nceleration, since the symmetry of the stationary one is so\ndifferent from that of the moving one.\nIV. CONCLUSION\nWe systematically study bright solitons in three-\ncomponent BECs with the spin-tensor-momentum cou-\npling and spin-orbit coupling, motivated by the rapid\ndevelopment of the research field of spin-orbit-coupled\nultracoldatomic gasesand by the recent proposalto real-\nize the spin-tensor-momentum-coupled BEC. The slight\ndifference between the STMC and SOC leads to various\nsymmetries,whichgivesrisetodifferentprofilesofsoliton\nwave functions. Moreover, the dynamics of spin-tensor-\nmomentum-coupled and spin-orbit-coupled solitons are\ndifferent during the time-evolution, when they are ini-\ntiated by switching off the couplings or by a constant\nweak acceleration force. We conclude that all different\nproperties comes from different symmetries.\nACKNOWLEDGEMENT\nWe sincerely acknowlege Yong Xu, Biao Wu and Ray-\nKuang Lee for helpful discussions. The work supported\nby the NSF of China (Grant Nos. 11974235, 1174219\nand 11474193), the Thousand Young Talents Program\nof China, SMSTC (18010500400 and 18ZR1415500),7\nand the Eastern Scholar and Shuguang (Program No.\n17SG39) Program. XC also thanks the support fromRam´ on y Cajal program of the Spanish MINECO (RYC-\n2017-22482).\n[1]Y. -J. Lin, K. Jim´ enez-Garc´ ıa, and I. B. Spielman, Spin-\norbit-coupled Bose-Einstein condensates, Nature, 471,\n83 (2011).\n[2]J.-Y. Zhang, S.-C. Ji, Z. Chen, L. Zhang, Z.-D. Du,\nB. Yan, G.-S. Pan, B. Zhao, Y.-J. Deng, H. Zhai,\nS. Chen, and J.-W. Pan, Collective dipole oscilla-\ntions of a spin-orbit coupled Bose-Einstein condensate,\nPhys. Rev. Lett. 109, 115301 (2012).\n[3]P. Wang, Z.-Q. Yu, Z. Fu, J. Miao, L. Huang, S. Chai,\nH. Zhai, and J. Zhang, Spin-orbit coupled degenerate\nFermi gases, Phys. Rev. Lett. 109, 095301 (2012).\n[4]L. W. Cheuk, A. T. Sommer, Z. Hadzibabic, T. Yef-\nsah, W. S. Bakr, and M. W. Zwierlein, Spin-\ninjection spectroscopy of a spin-orbit coupled Fermi gas,\nPhys. Rev. Lett. 109, 095302 (2012).\n[5]C. Qu, C. Hamner, M. Gong, C. Zhang, and P. Engels,\nObservation of Zitterbewegung in a spin-orbit-coupled\nBose-Einstein condensate, Phys. Rev. A 88, 021604(R)\n(2013).\n[6]A. J. Olson, S.-J. Wang, R. J. Niffenegger, C.-H. Li,\nC. H. Greene, and Y. P. Chen, Tunable Landau-Zener\ntransitions in a spin-orbit-coupled Bose-Einstein conden -\nsate, Phys. Rev. A 90, 013616 (2014).\n[7]C. Hamner, C. Qu, Y. Zhang, J. J. Chang, M. Gong,\nC. Zhang, and P. Engels, Dicke-typephase transition in a\nspin-orbit-coupled Bose-Einstein condensate, Nat. Com-\nmun.5, 4023 (2014).\n[8]H.-R. Chen, K.-Y. Lin, P.-K. Chen, N.-C. Chiu,\nJ.-B. Wang, C.-A. Chen, P.-P. Huang, S.-K. Yip,\nY. Kawaguchi, and Y.-J. Lin, Phys. Rev. Lett. 121,\n113204 (2018). Spin-orbital-angular-momentum coupled\nBose-Einstein condensates.\n[9]D. Zhang, T. Gao, P. Zou, L. Kong, R. Li, X. Shen,\nX.-L. Chen, S.-G. Peng, M. Zhan, H. Pu, and K. Jiang,\nPhys. Rev. Lett. 122, 110402 (2019). Ground-state phase\ndiagram of a spin-orbital-angular-momentum coupled\nBose-Einstein condensate.\n[10]C. Wang, C. Gao, C.-M. Jian, and H. Zhai,\nSpin-orbit coupled spinor Bose-Einstein condensates,\nPhys. Rev. Lett. 105, 160403 (2010).\n[11]T.-L. Ho and S. Zhang, Bose-Einstein condensates with\nspin-orbit interaction, Phys. Rev. Lett. 107, 150403\n(2011).\n[12]C.-J. Wu, I. Mondragon-Shem, and X.-F. Zhou, Uncon-\nventional Bose-Einstein condensations from spin-orbit\ncoupling, Chin. Phys. Lett. 28, 097102 (2011).\n[13]S. Sinha, R. Nath, and L. Santos, Trapped two-\ndimensional condensates with synthetic spin-orbit cou-\npling, Phys. Rev. Lett. 107, 270401 (2011).\n[14]Y. Zhang, L. Mao, and C. Zhang, Mean-field dy-\nnamics of spin-orbit coupled Bose-Einstein condensates,\nPhys. Rev. Lett. 108, 035302 (2012).\n[15]H. Hu, B. Ramachandhran, H. Pu, and X.-J. Liu, Spin-\norbit coupled weakly interacting Bose-Einstein conden-\nsates in harmonic traps, Phys. Rev. Lett. 108, 010402\n(2012).[16]Y. Li, L. P. Pitaevskii, and S. Stringari, Quantum\ntricriticality and phase transitions in spin-orbit cou-\npled Bose-Einstein condensates, Phys. Rev. Lett. 108,\n225301(2012).\n[17]J. -R. Li, J. Lee, W. Huang, S. Burchesky, B. Shteynas,\nF. C ¸. Top, A. O. Jamison, and W. Ketterle, A stripe\nphase with supersolid properties in spin-orbit-coupled\nBose-Einstein condensates, Nature 543, 91 (2017).\n[18]M. A. Khamehchi, Y. Zhang, C. Hamner, Th. Busch,\nand P. Engels, Measurement of collective excitations in a\nspin-orbit-coupled Bose-Einstein condensate, Phys. Rev.\nA90, 063624 (2014).\n[19]S. -C. Ji, L. Zhang, X. -T. Xu, Z. Wu, Y. Deng, S. Chen,\nand J. -W. Pan, Softening of roton and phonon modes\nin a Bose-Einstein condensate with spin-orbit coupling,\nPhys. Rev. Lett. 114, 105301 (2015).\n[20]V. Achilleos, D. J. Frantzeskakis, P. G. Kevrekidis,\nand D. E. Pelinovsky, Matter-wave bright soli-\ntons in spin-orbit coupled Bose-Einstein condensates,\nPhys. Rev. Lett. 110, 264101 (2013).\n[21]Y. Xu, Y. Zhang, and B. Wu, Bright solitons\nin spin-orbit-coupled Bose-Einstein condensates,\nPhys. Rev. A 87, 013614 (2013).\n[22]O. Fialko, J. Brand, and U. Z¨ ulicke, Soliton magnetiza-\ntion dynamics in spin-orbit-coupled Bose-Einstein con-\ndensates, Phys. Rev. A 85, 051605(R) (2012).\n[23]L. Wen, Q. Sun, Y. Chen, D.-S. Wang, J. Hu, H. Chen,\nW.-M. Liu, G. Juzeli¯ unas, B. A. Malomed, and A.-C. Ji,\nMotion of solitons in one-dimensional spin-orbit-coupled\nBose-Einstein condensates, Phys. Rev. A 94, 061602(R)\n(2016).\n[24]Q. Zhu, C. Zhang, and B. Wu, Exotic superfluidity\nin spin-orbit coupled Bose-Einstein condensates, Euro-\nphys. Lett. 100, 50003 (2012).\n[25]L. Salasnich, W. B. Cardoso, and B. A. Mal-\nomed, Localized modes in quasi-two-dimensional Bose-\nEinstein condensates with spin-orbit and Rabi couplings,\nPhys. Rev. A 90, 033629 (2014).\n[26]Y.-C. Zhang, Z.-W. Zhou, B. A. Malomed, and H. Pu,\nStable solitons in three dimensional free space without\nthe ground state: self-trapped Bose-Einstein condensates\nwith spin-orbit coupling, Phys. Rev. Lett. 115, 253902\n(2015).\n[27]S. Gautam and S. K. Adhikari, Vector solitons in\na spin-orbit-coupled spin-2 Bose-Einstein condensate,\nPhys. Rev. A 91, 063617 (2015).\n[28]H. Sakaguchi, E. Ya. Sherman, and B. A. Malomed, Vor-\ntex solitons in two-dimensional spin-orbit-coupled Bose-\nEinstein condensates: Effects of the Rashba-Dresselhaus\ncoupling and Zeeman splitting, Phys. Rev. E 94, 032202\n(2016). .\n[29]H. Sakaguchi and B. A. Malomed, Flipping-shuttle os-\ncillations of bright one- and two-dimensional solitons in\nspin-orbit-coupled Bose-Einstein condensates with Rabi\nmixing, Phys. Rev. A 96, 043620 (2017).\n[30]Y. V. Kartashov and V. V. Konotop, Solitons in Bose-\nEinstein condensates with helicoidal spin-orbit coupling ,8\nPhys. Rev. Lett. 118, 190401 (2017).\n[31]Y. Xu, Y. Zhang, and C. Zhang, Bright solitons\nin a two-dimensional spin-orbit-coupled dipolar Bose-\nEinstein condensate, Phys. Rev. A 92, 013633 (2015).\n[32]X.Jiang, Z.Fan, Z.Chen, W.Pang, Y.Li, andB.A.Mal-\nomed, Two-dimensional solitons in dipolar Bose-Einstein\ncondensates with spin-orbit coupling, Phys. Rev. A 93,\n023633 (2016); Y. Li, Y. Liu, Z. Fan, W. Pang, S. Fu, and\nB. A. Malomed, Two-dimensional dipolar gap solitons in\nfree space with spin-orbit coupling, Phys. Rev. A 95,\n063613 (2017).\n[33]E. Chiquillo, Quasi-one-dimensional spin-orbit- and\nRabi-coupled bright dipolar Bose-Einstein-condensate\nsolitons, Phys. Rev. A 97013614 (2018).\n[34]Y. V. Kartashov, V. V. Konotop, and F. Kh. Abdullaev,\nGap solitons in a spin-orbit-coupled Bose-Einstein con-\ndensate, Phys. Rev. Lett. 111, 060402 (2013).\n[35]V. E. Lobanov, Y. V. Kartashov, and V. V. Kono-\ntop, Fundamental, multipole, and half-vortex gap soli-\ntons in spin-orbit coupled Bose-Einstein condensates,\nPhys. Rev. Lett. 112, 180403 (2014).\n[36]Y. Zhang, Y. Xu, and Th. Busch, Gap solitons in spin-\norbit-coupled Bose-Einstein condensates in optical lat-\ntices, Phys. Rev. A 91, 043629 (2015).\n[37]H. Sakaguchi and B. A. Malomed, One- and two-\ndimensional gap solitons in spin-orbit-coupled systems\nwith Zeeman splitting, Phys. Rev. A 97013607 (2018).\n[38]H. Li, S.-L. Xu, M. R. Beli´ c, and J.-X. Cheng,\nThree-dimensional solitons in Bose-Einstein condensates\nwith spin-orbit coupling and Bessel optical lattices,\nPhys. Rev. A 98033827 (2018).\n[39]Sh. Mardonov, V. V. Konotop, B. A. Malomed, M. Mod-\nugno, and E. Ya. Sherman, Spin-orbit-coupled soliton in\na random potential, Phys. Rev. A 98, 023604 (2018).\n[40]Y.V.KartashovandD.A.Zezyulin, Stablemultiringand\nrotating solitons in two-dimensional spin-orbit-coupled\nBose-Einstein condensates with a radially periodic po-\ntential, Phys. Rev. Lett. 122, 123201 (2019).\n[41]X.-W. Luo, K. Sun, and C. Zhang, Spin-tensor-\nmomentum-coupled Bose-Einstein condensates,\nPhys. Rev. Lett. 119, 193001 (2017).\n[42]H. Hu, J. Hou, F. Zhang, and C. Zhang, Topolog-\nical triply degenerate points induced by spin-tensor-\nmomentum couplings, Phys. Rev. Lett. 120, 240401\n(2018).\n[43]X. Zhou, X.-W. Luo, G. Chen, S. Jia, C. Zhang, Quan-\ntum spiral spin-tensor magnetism, arXiv: 1901.00575.\n[44]Y.-K. Liu and S.-J. Yang, Exact solitons and manifold\nmixing dynamics in the spin-orbit-coupled spinor con-\ndensates, Europhys. Lett. 108, 30004 (2014).\n[45]S. Gautam and S. K. Adhikari, Vortex-bright solitons in\na spin-orbit-coupled spin-1 condensate, Phys. Rev. A 95,\n013608 (2017); S. Gautam and S. K. Adhikari, Three-\ndimensional vortex-brightsolitons inaspin-orbit-coupl ed\nspin-1 condensate , Phys. Rev. A 97, 013629 (2018).\n[46]Y.-E. Li and J.-K. Xue, Stationary and moving solitons\nin spin-orbit-coupled spin-1 Bose-Einstein condensates,\nFront. Phys. 13, 130307 (2018).\n[47]Y.-J. Lin, R. L. Compton, A. R. Perry, W. D. Phillips,\nJ. V. Porto, and I. B. Spielman, Bose-Einstein Con-\ndensate in a Uniform Light-Induced Vector Potential,\nPhys. Rev. Lett. 102, 130401 (2009).\n[48]D. Campbell, R. Price, A. Putra, A. Vald´ es-Curiel,\nD. Trypogeorgos, and I. B. Spielman, Magnetic phases ofspin-1 spin-orbit-coupled Bose gases, Nat. Commun. 7,\n10897 (2016).\n[49]Y. Zhang, M. E. Mossman, Th. Busch, P. Engels, and\nC. Zhang, Properties of spin-orbit-coupled Bose-Einstein\ncondensates, Front. Phys. 11, 118103, (2016)."    },    {        "title": "1211.0867v2.Tailoring_spin_orbit_torque_in_diluted_magnetic_semiconductors.pdf",        "content": "arXiv:1211.0867v2  [cond-mat.mes-hall]  3 May 2013Tailoring spin-orbit torque in diluted magnetic semicondu ctors\nHang Li, Xuhui Wang, Fatih Doˇ gan, and Aurelien Manchon∗\nKing Abdullah University of Science and Technology (KAUST) ,\nPhysical Science and Engineering Division, Thuwal 23955-6 900, Saudi Arabia\n(Dated: July 20, 2018)\nWe study the spin orbit torque arising from an intrinsic linearDresselhaus spin-orbit coupling in\na single layer III-V diluted magnetic semiconductor. We inv estigate the transport properties and\nspin torque using the linear response theory and we report he re : (1) a strong correlation exists\nbetween the angular dependence of the torque and the anisotr opy of the Fermi surface; (2) the spin\norbit torque depends nonlinearly on the exchange coupling. Our findings suggest the possibility to\ntailor the spin orbit torque magnitude and angular dependen ce by structural design.\nPACS numbers: 72.25.Dc,72.20.My,75.50.Pp\nThe electrical manipulation of magnetization is cen-\ntral to spintronic devices such as high density mag-\nnetic random access memory,1for which the spin trans-\nfer torque provides an efficient magnetization switch-\ning mechanism.2,3Beside the conventional spin-transfer\ntorque, the concept of spin-orbit torque in both metal-\nlic systems and diluted magnetic semiconductors (DMS)\nhas been studied theoretically and experimentally.4–9In\nthe presence of a charge current, the spin-orbit coupling\nproduces an effective magnetic field which generates a\nnon-equilibrium spin density that in turn exerts a torque\non the magnetization.4–6Several experiments on magne-\ntization switching in strained (Ga,Mn)As have provided\nstrong indications that such a torque can be induced by\na Dresselhaus-type spin-orbit coupling, achieving criti-\ncal switching currents as low as 106A/cm2.7–9However,\nup to date very few efforts are devoted to the nature of\nthe spin-orbit torque in such a complex system and its\nmagnitude and angular dependence remain unaddressed.\nIn this Letter, we study the spin-orbit torque in a di-\nlutedmagneticsemiconductorsubmittedtoalinearDres-\nselhaus spin-orbitcoupling. We highlight two effects that\nhave not been discussed before. First, a strong correla-\ntion exists between the angular dependence of the torque\nand the anisotropyof the Fermi surface. Second, the spin\ntorquedepends nonlinearly onthe exchangecoupling. To\nillustrate the flexibility offered by DMS in tailoring the\nspin-orbit torque, we compare the torques obtained in\ntwo stereotypical materials, (Ga,Mn)As and (In,Mn)As.\nThe system under investigation is a uniformly mag-\nnetized single domain DMS film made of, for example,\n(Ga,Mn)As or (In,Mn)As. We assume the system is well\nbelow its critical temperature. An electric field is applied\nalong the ˆxdirection. It is worth pointing out that we\nconsiderhere a large-enoughsystem to allowus disregard\nany effects arising due to boundaries and confinement.\nWe use the six-band Kohn-Luttinger Hamiltonian todescribe the band structure of the DMS,9\nHKL=/planckover2pi12\n2m/bracketleftbigg\n(γ1+5\n2γ2)k2−2γ3(k·ˆJ)2\n+2(γ3−γ2)/summationdisplay\nik2\niˆJ2\ni/bracketrightBigg\n. (1)\nwhere the phenomenological Luttinger parameters γ1,2,3\ndetermine the band structure and the effective mass of\nvalence-band holes. γ3is the anisotropy parameter, ˆJis\nthe total angular momentum and kis the wave vector.\nThe bulk inversion asymmetry allows us to augment the\nKohn-Luttinger Hamiltonian by a strain-induced spin-\norbit coupling of the Dresselhaus type.5,7We assume the\ngrowth direction of (Ga,Mn)As is directed along the z-\naxis, two easy axes are pointed at xandy, respectively.10\nIn this case, the components of the strain tensor ǫxxand\nǫyyare identical. Consequently, we may have a linear\nDresselhaus spin-orbit coupling7\nHDSOC=β(ˆσxkx−ˆσyky), (2)\ngivenβthe coupling constant that is a function of the\naxial strain.7,11ˆσx(y)is the 6×6 spin matrix of holes and\nkx(y)is the wave vector.\nIn the DMS systems discussed here, we incorporate\na mean-field like exchange coupling to enable the spin\nangular momentum transfer between the hole spin ( ˆs=\n/planckover2pi1ˆσ/2) and the localized ( d-electron) magnetic moment\nˆΩof ionized Mn2+acceptors,12,13\nHex= 2JpdNMnSaˆΩ·ˆs//planckover2pi1 (3)\nwhereJpdis the antiferromagnetic coupling\nconstant.13,14HereSa= 5/2 is the spin of the ac-\nceptors. The hole spin operator, in the present six-band\nmodel, is a 6 ×6 matrix.13The concentration of the\nordered local Mn2+momentsNMn= 4x/a3is given as\na function of xthat defines the doping concentration of\nMn ion.ais the lattice constant. Therefore, the entire\nsystem is described by the total Hamiltonian\nHsys=HKL+Hex+HDSOC. (4)2\nIn order to calculate the spin torque, we determine the\nnonequilibrium spin densities S(of holes) as a linearre-\nsponse to an external electric field,5\nS=eEx1\nV/summationdisplay\nn,k1\n/planckover2pi1Γn,k∝angbracketleftˆv∝angbracketright∝angbracketleftˆs∝angbracketrightδ(En,k−EF).(5)\nwhereˆvis thevelocityoperator. InEq.(5), thescattering\nrate of hole carriers by Mn ions is obtained by Fermi’s\ngolden rule,12\nΓMn2+\nn,k=2π\n/planckover2pi1NMn/summationdisplay\nn′/integraldisplaydk′\n(2π)3/vextendsingle/vextendsingle/vextendsingleMk,k′\nn,n′/vextendsingle/vextendsingle/vextendsingle2\n×δ(En,k−En′,k′)(1−cosφk,k′),(6)\nwhereφk,k′is the angle between two wave vectors kand\nk′. The matrix element Mk,k′\nn,n′between two eigenstates\n(k,n) and (k′,n′) is\nMkk′\nn,n′=JpdSa∝angbracketleftψnk|ˆΩ·ˆs|ψn′k′∝angbracketright\n−e2\nǫ(|k−k′|2+p2)∝angbracketleftψnk|ψn′k′∝angbracketright.(7)\nHereǫis the dielectric constant of the host semiconduc-\ntorsandp=/radicalbig\ne2g/ǫistheThomas-Fermiscreeningwave\nvector, where gis the density of states at Fermi level. Fi-\nnally, we calculate the field like spin-orbit torque using4\nT=JexS׈Ω, (8)\nwhereJex≡JpdNMnSa. Throughout this Letter, the\nresults are given in terms of the torque efficiency T/eE.\nThe interband transitions, arising from distortions in the\ndistributionfunctioninducedbytheappliedelectricfield,\nare neglected in our calculation. This implies that the\ntorque extracted from the present model is expected to\naccommodate only a field-like component. The above\nprotocols based on linear response formalism allow us to\ninvestigate the spin-orbit torque for a wide range of DMS\nmaterial parameters.\nWe plot in Fig.1(a) the spin torque as a function of\nthe magnetization angle for different values of the band\nstructure anisotropy parameter γ3. The topology of the\nFermi surface can be modified by a linear combination of\nγ2andγ3: ifγ2=γ3∝negationslash= 0, the Fermi surface around the\nΓ point is spherical, as shown in Fig.1(c). In this spe-\ncial case, the angular dependence of the torque is simply\nproportional to cos θ[red curve in Fig.1(a)], as expected\nfrom the symmetry of the k-linear Dresselhaus Hamilto-\nnian, Eq. (2)4. Whenγ3∝negationslash=γ2, the Fermi surfacedeviates\nfromasphere[Fig.1(b)and(d)]and,correspondingly,the\nangular dependence of the torque deviates from a simple\ncosθfunction [i.e., curves corresponding to γ3= 1.0 and\nγ3= 2.93 in Fig.1(a)]. In a comparison to the spherical\ncase, the maximal value of the torque at θ= 0 is lower\nforγ3∝negationslash=γ2. As Eq.(5) indicates, in the linear response\ntreatment formulated here, the magnitude of the spin\ntorque is determined by the transport scattering timeand the expectation values of spin and velocity opera-\ntors of holes. Qualitatively, as the Fermi surface deviates\nfrom a sphere, the expectation value ∝angbracketleftˆsx∝angbracketrightof the heavy\nhole band, contributing the most to the spin torque, is\nlowered atθ= 0.\nMore specifically, as the Fermi surface warps, the an-\ngular dependence of the spin torque develops, in addition\nto the cosθenvelop function, an oscillation with a period\nthat is shorter than π. The period of these additional\noscillations increases as the Fermi surface becomes more\nanisotropic in k-space, see Fig. 1(b) and (d). To fur-\nther reveal the effect of band warping on spin torque,\nwe plotTy/cosθas a function of the magnetization an-\ngle in inset of Fig.1(a). When γ3= 2.0 (spherical Fermi\nsphere),Ty/cosθis a constant, for T∝cosθ. When\nγ3= 2.93 or 1.0, the transport scattering time of the\nhole carriers starts to develop an oscillating behavior in\nθ,15which eventually contributes to additional angular\ndependencies in the spin torque. The angular dependen-\ncies in spin-orbittorque shall be detectable by techniques\nsuch as spin-FMR9.\n/s48 /s51/s48 /s54/s48 /s57/s48 /s49/s50/s48 /s49/s53/s48 /s49/s56/s48/s45/s48/s46/s54/s45/s48/s46/s51/s48/s46/s48/s48/s46/s51/s48/s46/s54\n/s45/s50 /s45/s49 /s48 /s49 /s50/s45/s50/s45/s49/s48/s49/s50\n/s45/s50 /s45/s49 /s48 /s49 /s50/s45/s50/s45/s49/s48/s49/s50\n/s45/s50 /s45/s49 /s48 /s49 /s50/s45/s50/s45/s49/s48/s49/s50\n/s32/s32\n/s32\n/s100/s101/s103/s32/s84\n/s121/s40/s49/s48/s52\n/s47 /s109/s50\n/s41\n/s88/s90\n/s77\n/s121\n/s32/s32/s32\n/s32\n/s107\n/s120/s40/s110/s109/s45/s49\n/s41\n/s107\n/s120/s40/s110/s109/s45/s49\n/s41/s107\n/s121/s40/s110/s109/s45/s49\n/s41\n/s107\n/s120/s40/s110/s109/s45/s49\n/s41/s40/s98/s41/s40/s97/s41\n/s32/s32\n/s32/s40/s100/s41/s107\n/s121/s40/s110/s109/s45/s49\n/s41/s32 /s32/s32\n/s32\n/s32/s32 /s32/s32/s40/s99/s41/s107\n/s121/s40/s110/s109/s45/s49\n/s41\nFIG. 1. (Color online) (a)The y-component of the spin torque\nas a function of magnetization direction. Fermi surface int er-\nsection in the kz= 0 plane for (b) γ3= 1.0, (c)γ3= 2.0\nand (c)γ3= 2.93. The red, black, orange and blue contours\nstands for majority heavy hole, minority heavy hole, major-\nity light hole and minority light hole band, respectively. I n-\nset (a) depicts Ty/cosθas a function of magnetization di-\nrection. The others parameters are ( γ1,γ2) = (6.98,2.0),\nJpd= 55 meV nm3andp= 0.2 nm−3.\nIn Fig.2, we compare the angular dependence of\nspin torque ( Ty) for both (Ga,Mn)As and (In,Mn)As\nwhich are popular materials in experiments and device\nfabrication.16–18Although (In,Mn)As is, in terms of ex-\nchange coupling and general magnetic properties, rather\nsimilar to (Ga,Mn)As, the difference in band structures,\nlattice constants, and Fermi energies between these two\nmaterials gives rise to different density of states, strains,\nand transport scattering rates. For both materials, the3\nspin torque decrease monotonically as the angle θin-\ncreasesfrom 0 to π/2. Throughoutthe entire angle range\n[0,π], the amplitude of the torque in (In,Mn)As is twice\nlarger than that in (Ga,Mn)As. We mainly attribute this\nto two effects. First of all, the spin-orbit coupling con-\nstantβin (In,Mn)As is about twice as larger than that\nin (Ga,Mn)As. Second, for the same hole concentration,\nthe Fermi energy of (In,Mn)As is higher than that of\n(Ga,Mn)As.\n/s48 /s51/s48 /s54/s48 /s57/s48 /s49/s50/s48 /s49/s53/s48 /s49/s56/s48/s45/s48/s46/s56/s45/s48/s46/s52/s48/s46/s48/s48/s46/s52/s48/s46/s56\n/s32/s32/s32\n/s32/s71/s97/s77/s110/s65/s115\n/s32/s73/s110/s77/s110/s65/s115/s84\n/s121/s40/s49/s48/s52\n/s47 /s109/s50\n/s41\n/s100/s101/s103\nFIG. 2. (Color online) Torque Tyas afunctionof themagneti-\nzation direction for (Ga,Mn)As (black square) and (In,Mn)A s\n(red dots). For (Ga,Mn)As, ( γ1,γ2,γ3) = (6.98,2.0,2.93);\nfor (In,Mn)As, ( γ1,γ2,γ3) = (20.0,8.5,9.2). The strength\nof the spin-orbit coupling constant is: for (Ga,Mn)As, β=\n1.6meVnm; for (In,Mn)As, β= 3.3meVnm.19Theexchange\ncoupling constant Jpd= 55 meV nm3for (Ga,Mn)As20and\n39 meV nm3for (In,Mn)As.21\nIn the following, we further demonstrate a counter-\nintuitive feature that, in the DMS system considered in\nthis Letter, the spin orbit torque depends nonlinearly on\nthe exchange splitting. In Fig. 3(a), Tycomponent of\nthe spin torque is plotted as a function of the exchange\ncouplingJpd, for different values of β. In the weak ex-\nchange coupling regime, the electric generation of non\nequilibrium spin density dominates, then the leading role\nof exchange coupling is defined by its contribution to the\ntransport scattering rate. We provide a simple qualita-\ntive explanation on such a peculiar Jpddependence. Us-\ning a Born approximation, the scattering rate due to the\np−dinteraction is proportional to 1 /τJ=bJ2\npd, where\nparameter bisJpd- independent. When the nonmag-\nnetic scattering rate 1 /τ0is taken into account, i.e., the\nCoulomb interaction part in Eq.(7), the total scattering\ntime in Eq.(5) can be estimated as\n1\n/planckover2pi1Γ∝1\nbJ2\npd+1\nτ0, (9)\nwhich contributes to the torque by T∝Jpd/(/planckover2pi1Γ). This\nexplains the transition behavior, i.e., increases linearly\nthen decreases, in the moderate Jpdregime in Fig.3. As\nthe exchange coupling further increases, Eq.(9) is dom-\ninated by the spin-dependent scattering, therefore thescattering time 1 //planckover2pi1Γ∝1/J2\npd. Meanwhile, the energy\nsplitting due to the exchange coupling becomes signifi-\ncant, thus ∝angbracketleftˆs∝angbracketright ∝Jpd. In total, the spin torque is insensi-\ntivetoJpd, explainingtheflatcurveinthelargeexchange\ncoupling regime. In Fig. 3(b), we plot the influence of\nthe exchange coupling on the spin torque for two materi-\nals. In (In,Mn)As, mainly due to a largerFermi energyin\na comparison to (Ga,Mn)As, the peak of the spin torque\nshiftstowardsalarger Jpd. Thedependence ofthetorque\nas a function of the exchange in (In,Mn)As is more pro-\nnounced than in (Ga,Mn)As, due to a stronger spin-orbit\ncoupling.\n/s48 /s50/s48 /s52/s48 /s54/s48 /s56/s48 /s49/s48/s48 /s49/s50/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52\n/s48 /s50/s48 /s52/s48 /s54/s48 /s56/s48 /s49/s48/s48 /s49/s50/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56\n/s32/s32\n/s32/s74\n/s112/s100/s40/s109/s101/s86/s32/s110/s109/s51\n/s41/s32 /s32/s61/s32/s48/s46/s56/s32/s109/s101/s86/s32/s110/s109\n/s32 /s32/s61/s32/s49/s46/s54/s32/s109/s101/s86/s32/s110/s109\n/s32 /s32/s61/s32/s50/s46/s52/s32/s109/s101/s86/s32/s110/s109/s84\n/s121/s40/s49/s48/s52\n/s109/s50\n/s41\n/s40/s97/s41\n/s32/s32\n/s32/s40/s98/s41\n/s74\n/s112/s100/s40/s109/s101/s86/s32/s110/s109/s51\n/s41/s32/s71/s97/s77/s110/s65/s115\n/s32/s73/s110/s77/s110/s65/s115/s84\n/s121/s40/s49/s48/s52\n/s109/s50\n/s41\nFIG. 3. (Color online) The Tycomponent of the spin torque\nas a function of exchange coupling Jpd. (a)TyversusJpd\nat various values of β, for (Ga,Mn)As. (b) TyversusJpd,\nfor both (Ga,Mn)As and (In,Mn)As. The magnetization is\ndirected along the z-axis (θ= 0). The other parameters are\nthe same as those in Fig.2.\nThe possibility to engineer electronic properties by\ndoping is one of the defining features that make DMS\npromising for applications. Here, we focus on the doping\neffect which allows the spin torque to vary as a function\nof hole carrier concentration. In Fig. 4(a), the torque is\nplotted as a function of the hole concentration for differ-\nentβparameters. With the increase of the hole concen-\ntration, the torque increases due to an enhanced Fermi\nenergy. In the weak spin-orbit coupling regime (small β),\nthe torque as a function of the hole concentration ( p) fol-\nlows roughly the p1/3curve as shown in the inset in Fig.\n4(a). The spherical Fermi sphere approximation and a\nsimple parabolic dispersion relation allow for an analyti-\ncal expression ofthe spin torque, i.e., in the leading order4\ninβandJex,\nT=m∗\n/planckover2pi1βJex\nEFσD (10)\nwherem∗is the effective mass. The Fermi energy EF\nand the Drude conductivity are given by\nEF=/planckover2pi12\n2m∗(3π2p)2/3, σD=e2τ\nm∗p, (11)\nwhereτis the transport time. The last two relations im-\nmediately give rise to T∝p1/3. In the six-band model,\ntheFermisurfacedeviatesfromasphereand, asthevalue\nofβincreases, the spin-orbit coupling starts to modify\nthe density of states. Both effects render the torque-\nversus-hole concentration curve away from the p1/3de-\npendence. This effect is illustrated in Fig. 4(b). The\nformer (strong spin-orbit coupling) clearly deviates from\np1/3, whereas the latter (weak spin-orbit coupling) fol-\nlows the expected p1/3trend.\n/s48/s46/s48 /s48/s46/s52 /s48/s46/s56 /s49/s46/s50 /s49/s46/s54 /s50/s46/s48/s48/s46/s48/s48/s46/s52/s48/s46/s56/s49/s46/s50\n/s48/s46/s52 /s48/s46/s56 /s49/s46/s50 /s49/s46/s54 /s50/s46/s48/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48\n/s48/s46/s48 /s48/s46/s52 /s48/s46/s56 /s49/s46/s50 /s49/s46/s54 /s50/s46/s48/s48/s46/s48/s48/s46/s52/s48/s46/s56/s49/s46/s50/s49/s46/s54\n/s32/s32\n/s32/s40/s97/s41/s84\n/s121/s40/s49/s48/s52\n/s47 /s109/s50\n/s41/s84\n/s121/s47/s112/s49/s47/s51\n/s32/s32 /s32/s61/s32/s48/s46/s56/s109/s101/s86/s32/s110/s109\n/s32/s32 /s32/s61/s32/s49/s46/s54/s109/s101/s86/s32/s110/s109\n/s32/s32 /s32/s61/s32/s50/s46/s52/s109/s101/s86/s32/s110/s109\n/s80/s40/s110/s109/s45/s51\n/s41/s84\n/s121/s40/s49/s48/s52\n/s47 /s109/s50\n/s41/s32\n/s32\n/s32/s32\n/s32/s32\n/s32/s40/s98/s41\n/s80/s40/s110/s109/s45/s51\n/s41/s32/s71/s97/s77/s110/s65/s115\n/s32/s73/s110/s77/s110/s65/s115FIG. 4. (Color online) The y-component of the spin torque\nas a function of hole concentration. (a) The y-component\nof the spin torque versus hole concentration at different\nβ. (b) spin torque versus hole concentration in (Ga,Mn)As\nand (In,Mn)As. For (Ga,Mn)As, Jpd= 55 meV nm3; for\n(In,Mn)As, Jpd= 39 meV nm3. The other parameters are\nthe same as in Fig.3.\nIn conclusion, in a DMS system subscribing to a lin-\near Dresselhaus spin-orbit coupling, we have found that\nthe angular dependence of the spin-orbit torque has a\nstrong yet intriguing correlation with the anisotropy of\nthe Fermi surface. Our study also reveals a nonlinear\ndependence of the spin torque on the exchange coupling.\nFrom the perspective of material selection, for an equiv-\nalent set of parameters, the critical switching current\nneeded in (In,Mn)As is expected to be lower than that\nin (Ga,Mn)As. The results reported here shed light on\nthe design and applications of spintronic devices based\non DMS.\nWhereasthematerialsstudiedinthisworkhaveaZinc-\nBlende structure, DMS adopting a wurtzite structure,\nsuch as (Ga,Mn)N, might also be interesting candidates\nfor spin-orbittorque observationdue to their sizable bulk\nRashba spin-orbit coupling. However, these materials\nusuallypresent asignificantJahn-Tellerdistortionthat is\nlarge enough to suppress the spin-orbit coupling.22Fur-\nthermore, the formalism developed here applies to sys-\ntems possessing delocalized holes and long range Mn-Mn\ninteractionsand isnot adaptedto the localizedholescon-\ntrolling the magnetism in (Ga,Mn)N.\nWe are indebted to K. V´ yborn´ y and T. Jungwirth\nfor numerous stimulating discussions. F.D. acknowl-\nedges support from KAUST Academic Excellence Al-\nliance Grant N012509-00.\n∗aurelien.manchon@kaust.edu.sa\n1J. A. Katine, F. J. Albert, R. A. Buhrman, E. B. Myers,\nand D. C. Ralph, Phys. Rev. Lett. 84, 3149 (2000).\n2J. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996).\n3L. Berger, Phys. Rev. B 54, 9353 (1996).\n4A. Manchon and S. Zhang, Phys. Rev. B 78, 212405\n(2008); Phys. Rev. B 79, 212405 (2009).\n5I. Garate and A. H. MacDonald, Phys. Rev. B 80, 134403\n(2009).\n6K. M. D. Hals, A. Brataas and Y. Tserkovnyak Eur. Phys.\nLett.90, 47002 (2010).7A. Chernyshov, M. Overby, X. Liu, J. K. Furdyna, Y.\nLyanda-Geller, and L. P. Rokhinson, Nature Phys. 5, 656\n(2009).\n8M. Endo, F. Matsukura, and H. Ohno, Appl. Phys. Lett.\n97, 222501 (2010).\n9D. Fang, H. Kurebayashi, J. Wunderlich, K. V´ yborn´ y, L.\nP. Zˆ arbo, R. P. Campion, A. Casiraghi, B. L. Gallagher, T.\nJungwirth, and A. J. Ferguson, Nature Nanotech. 6, 413\n(2011).\n10U. Welp, V. K. Vlasko-Vlasov, X. Liu, J. K. Furdyna, and\nT. Wojtowicz, Phys. Rev. Lett. 90, 167206 (2003).5\n11B. A. Bernevig and S.-C. Zhang, Phys. Rev. B 72, 115204\n(2005).\n12T. Jungwirth, M. Abolfath, J. Sinova, J. Kuˇ cera, and A.\nH. MacDonald, Appl. Phys. Lett. 81, 4029 (2002).\n13M. Abolfath, T. Jungwirth, J. Brum, and A. H. MacDon-\nald, Phys. Rev. B 63, 054418 (2001).\n14J. van Bree, P. M. Koenraad, and J. Fern´ andez-Rossier,\nPhys. Rev. B 78, 165414 (2008).\n15A. W. Rushforth, K. V´ yborn´ y, C. S. King, K. W. Ed-\nmonds, R. P. Campion, C. T. Foxon, J. Wunderlich, A.\nC. Irvine, P. Vaˇ sek, V. Nov´ ak, K. Olejn´ ık, Jairo Sinova,\nT. Jungwirth, and B. L. Gallagher, Phys. Rev. Lett. 99,\n147207 (2007).16H. Ohno, H. Munekata, T. Penney, S. von Moln´ ar, and L.\nL. Chang, Phys. Rev. Lett. 68, 2664 (1992).\n17S. Koshihara, A. Oiwa, M. Hirasawa, S. Katsumoto, Y.\nIye, C. Urano, H. Takagi, and H. Munekata, Phys. Rev.\nLett.78, 4617 (1997).\n18T. Jungwirth, Qian Niu and A. H. MacDonald, Phys. Rev.\nLett.88, 207208 (2002).\n19J. Fabian, A. Matos-Abiague, C. Ertler, P. Stano, and I.\nZutic, Acta Phys. Slov. 57, 565 (2007).\n20H. Ohno, J. Magn. Magn. Mater. 200, 110 (1999).\n21J. Wang, Master’s thesis (Rice University, Houston, Texas,\n2002).\n22A. StroppaandG. Kresse, Phys. Rev.B 79, 201201 (2009)."    },    {        "title": "0708.2595v2.Manipulating_the_spin_texture_in_spin_orbit_superlattice_by_terahertz_radiation.pdf",        "content": "arXiv:0708.2595v2  [cond-mat.mes-hall]  2 Apr 2008Manipulating the spin texture in spin-orbit superlattice b y terahertz radiation\nD.V. Khomitsky∗\nDepartment of Physics, University of Nizhny Novgorod,\n23 Gagarin Avenue, 603950 Nizhny Novgorod, Russian Federat ion\n(Dated: October 23, 2018)\nThe spin texture in a gate-controlled one-dimensional supe rlattice with Rashba spin-orbit cou-\npling is studied in the presence of external terahertz radia tion causing the superlattice miniband\ntransitions. It is shown that the local distribution of the e xcited spin density can be modified by\nvarying the Fermi level of the electron gas and by changing th e radiation intensity and polarization,\nallowing the controlled coupling of spins and photons.\nPACS numbers: 72.25.Fe, 73.21.Cd, 78.67.Pt\nI. INTRODUCTION\nThe control of the spin degrees of freedom is one of the\nprimary goals of rapidly developing field of condensed\nmatter physics known as spintronics.1,2In addition to\nthe methods involving the magnetic field which effec-\ntively governs the spins, the application of non-magnetic\nspin systems can be considered by taking into account\nthe spin-orbit (SO) interaction. In most widely con-\nsidered two-dimensional semiconductor heterostructures\nthe SO interaction is usually dominated by the Rashba\ncoupling3coming from the structure inversion asymme-\ntry of confining potential and effective mass difference.\nThe value of Rashba coupling strength can be tuned by\nthe external gate voltage4and it reaches the value of\n2·10−11eVm in InAs-based structures.5One of the im-\nportant issues of spintronics is the interaction between\nspins and photons which is promising for further ap-\nplications in novel electronic and optical devices. The\nstudies of optical properties of SO semiconductor struc-\ntures have formed a fast growing field of studies during\nthe last decade. Some of the research topics included\nthe photogalvanic,6,7,9,10,11,12spin-galvanic8,9and spin-\nphotovoltaic13effects as well as optical spin orientation14\nand pure spin current generation.15,16The effects of ter-\nahertz radiation onto spin-split states in semiconduc-\ntors were also the subject of investigation.17,18Another\nimportant property of non-uniform spin distributions\nsuch as spin coherence standing waves was discussed by\nPershin19who found an increase of the spin relaxation\ntime in such structures, making them promising for spin-\ntronics applications. One of possible ways to create a\nnon-uniform spin distribution in a heterostructure is to\napply a metal-gated superlattice with tunable amplitude\nof electric potential to the two-dimensional electron gas\n(2DEG) with spin-orbit coupling. The quantum states\nand spin polarization in this system with standing spin\nwaveswerestudied previously,20andthe problemofscat-\ntering on such structure has been considered.21\nIn the present Brief Report we study the problem of\nthe excited spin density creation in a one-dimensional\nInAs-based superlattice with Rashba spin-orbit coupling\nby applying an external terahertz radiation. The direc-\ntion of photon propagation is chosen perpendicular to2DEG plane where both linear and circular polarizations\nare considered. It is shown that the excited spin texture\nis sensitive to the position of the Fermi level of the 2DEG\nand to the radiation intensity. The former can be tuned\nby the gate voltage, thus providing a new possible way\nto couple local excited distribution of spins and photons\nin 2DEG with SO interaction. The obtained results have\na qualitative character and are not restricted to one spe-\ncific type of semiconductor heterostructure, superlattice\nperiod, amplitude of periodic potential, Fermi level posi-\ntion, etc. Another important issue is the spin relaxation\nwhich tends to transform the excited spin distribution\nback to the equilibrium. The terahertz scale of excita-\ntionfrequenciesisatleastofanorderofmagnitudelarger\nthan the spin relaxationratesin InAs semiconductorhet-\nerostructures which can be estimated as 1 /τswhere the\nspin lifetime τsreaches there the values from 60 ps22to\n600 ps23and the spin relaxation time is further increased\nin non-homogeneous spin textures.19Thus, one can ex-\npect that the effects discussed in the currentBriefReport\ncan be experimentally observable.\nThis Brief Report is organized as follows. In Sec.II we\nbriefly describe quantum states in SO superlattice and\nin Sec.III we calculate the spatial distribution of the ex-\ncited spin density in a superlattice cell under terahertz\nradiationwith differentpositionsofFermilevel, radiation\nintensity and polarization. The conclusions are given in\nSec.IV.\nII. QUANTUM STATES IN SO SUPERLATTICE\nWe start with the brief description of the quantum\nstates of 2DEG with Rashba SO coupling and one-\ndimensional (1D) periodic superlattice potential in the\nabsence of external electromagnetic field.20The Hamil-\ntonian is the sum of the 2DEG kinetic energy operator\nin a single size quantization band with effective mass m,\nthe Rashba SO term with strength αand the periodic\nelectrostatic potential of the 1D superlattice:\nˆH=ˆp2\n2m+α(ˆσxˆpy−ˆσyˆpx)+V(x),(1)2\nwhere ¯h= 1 and the periodic potential is chosen in\nthe simplest form V(x) =V0cos2πx/awhereais the\nsuperlattice period and the amplitude V0is controlled by\nthe gate voltage. The eigenstates of Hamiltonian (1) are\ntwo-componentBlochspinorswith eigenvalueslabeledby\nthe quasimomentum kxin a one-dimensional Brillouin\nzone−π/a≤kx≤π/a, the momentum component ky,\nand the miniband index m:\nψmk=/summationdisplay\nλnam\nλn(k)eiknr\n√\n2/parenleftbigg1\nλeiθn/parenrightbigg\n, λ=±1.(2)\nHerekn=k+nb=/parenleftbig\nkx+2π\nan, ky/parenrightbig\nandθn=\narg[ky−iknx]. The energy spectrum of Hamiltonian (1)\nconsists of pairs of spin-split minibands determined by\nthe SO strength αseparated by the gaps of the order\nofV0. An example of the energy spectrum is shown in\nFig.1 for the four lowest minibands in the InAs 1D su-\nperlattice with Rashba constant α= 2·10−11eVm, the\nelectron effective mass m= 0.036m0, the superlattice\nperioda= 60 nm and the amplitude of the periodic po-\ntentialV0= 10 meV. It should be mentioned that the\nspectrum in Fig.1 is limited to the first Brillouin zone\nof the superlattice in the kxdirection while the cutoff in\nthekydirection is shown only to keep the limits along\nkxandkycomparable. The Fermi level position EF(Vg)\ncan be varied by tuning the gate voltage which controls\nthe concentration of the 2D electrons. This feature is\nshown in Fig.1 schematically by the arrows near EF(Vg)\nas well as the photon energy ¯ hω= 10 meV corresponding\ntoω/2π= 2.43 THz.\nIII. SPIN TEXTURE MANIPULATION\nThe Rashba SO coupling as well as the periodic su-\nperlattice potential cannot produce the net polarization\nof 2DEG. Moreover, the two-component eigenvectors of\nthe Rashba Hamiltonian describe the homogeneous lo-\ncal spin density σi=ψ†ˆσiψ, wherei=x,y,z. In the\npresence of an additional superlattice potential, how-\never, the local spin density for a given state ( kx,ky) can\nbe inhomogeneous,20as in the spin coherence standing\nwave,19which gives an idea to obtain a non-uniform spin\ndensitydistributionunderanexternalradiationwhichin-\nvolvesintransitionsthestateswithdifferent( kx,ky)with\na varying impact depending on the matrix elements. In\nthis Section we subject the 2DEG with Hamiltonian (1)\nto the electromagneticradiationpropagatingalong zaxis\nperpendicular to the 2DEG plane with the electric field\nof the radiation E(t) =eEωexp−iωt+ c.c.with ampli-\ntudeEω, frequency ωpolarization e= (ex,ey). When\nthe electromagnetic radiation is applied, the excited spin\ndensity rate Siat a given point in a real space can be\nfound in the following way:15\nFIG. 1: (color online) Energy spectrum of four lowest mini-\nbands in the InAs 1D superlattice with Rashba constant\nα= 2·10−11eVm, the electron effective mass m= 0.036\nm0, the superlattice period and amplitude a= 60 nm and\nV0= 10 meV. The Fermi level position EF(Vg) and the pho-\nton energy ¯ hω= 10 meV corresponding to ω/2π= 2.43 THz\nare shown schematically.\nSi=πe2E2\nω\nω2/integraldisplay\nd2k/summationdisplay\njlξjl\ni(k)¯ejel (3)\nwhere\nξjl\ni(k) =/summationdisplay\nc,m,m′/parenleftBig\nψ†\nm′ˆσiψm/parenrightBig\n¯vj\nm′cvl\nmc (4)\n×[δ(ωmc(k)−ω)+δ(ωm′c(k)−ω)]. (5)\nSince the structure is completely homogeneous in the\nydirection, the Sycomponent vanishes. The other com-\nponentsSxandSzof the excited non-equilibrium spin\ndensity can be nonzero at a given point in a superlattice\nevenforthe linear x-polarizedradiation. The coexistence\nof the axial vector components ( Sx,Sz) in the left side of\nEq.(3) together with the polar vector component exin\nthe right side is in agreement with the principles of mag-\nnetic crystal class analysis.24There is a mirror plane of\nreflectionσyin our system which changes yto−yand\nthus changes the sign of the magnetic moment, leaving\ntheexcomponent of the polarization unchanged. The\nelement of the magnetic crystal class, however, is applied\nonly as a combination σyRwhereRis the time rever-\nsal operator24which again changes the direction of the3\nmagnetic moment but does not change the polarization\ncomponent ex. As a result, the combination σyRleaves\nboth spin projections Sx,zand the polarization compo-\nnentexinvariant. Another restriction is the absence of\ntotal magnetic moment of the sample which means that\nbothSxandSzmust satisfy to the requirement of zero\nnet polarization:\n/integraldisplaya\n0Sx(x)dx=/integraldisplaya\n0Sz(x)dx= 0. (6)\nThe spin density distribution in the superlattice cell\nis found for three different polarizations: linear along x\naxis, linear along yaxis, and circular in the ( xy) plane\nof 2DEG. The Fermi level in Fig.1 varies from 4 to 20\nmeV counted from the bottom of the topmost partially\nfilled electron size quantization band. According to the\nspectrum in Fig.1, this variation fills gradually all four\nminibands shown there. The excitation energy is chosen\nto be ¯hω= 10 meV which corresponds to ω/2π= 2.43\nTHz and provides effective transitions between the occu-\npied and vacant minibands, as it can be seen in Fig.1.\nFirst, let us considerthe casesofthe excitationlinearly\npolarized along xaxis and the σ+- circular polarized one.\nThe excited non-equilibrium spin density (3) is shown\nin Fig.2 as a 2D vector field ( Sx(x),Sz(x)). Its magni-\ntude is proportional to the radiation intensity which is\n0.3W/cm2for the present case and the texture shape\nis varied with respect to the Fermi energy. The mag-\nnitude of the arrow length in Fig.2, i.e. the maximum\nexcited spin density can be obtained from the value of\nthe excited charge density nex. Taking the data from the\nexperiments with optical excitation17where the volume\nconcentration of the excited carriers reached 1016cm−3\none can estimate the excited surface concentration nexto\nbe oftheorderof1010...1011cm−2. Onecanseein Fig.2\nthat the spin textures aresimilar for x- andσ+radiation,\nas well as for σ−(not shown). The explanation is that all\nthese polarizationscontain the xcomponent oftransition\nmatrix elements which causes the most effective transi-\ntions in the x-oriented superlattice. The transformation\nof spin density distribution with the Fermi level position\nin Fig.2 is produced by gradual filling of the minibands.\nThe small amplitudes of spin density (at EF= 4 meV in\nFig.2) correspond to the small filling factor (see Fig.1).\nBy increasing the Fermi energy the complete filling of\ntwo lowest subbands (at EF= 8 meV and EF= 12 meV\nin Fig.2) is reached and, finally, the excited spin den-\nsity magnitude decreases again with complete filling of\nall four nearest miniband in Fig.1 (at EF= 16 meV and\nEF= 20 meV in Fig.2). It can be seen in Fig.2 that\nthe excited spin density texture have the primary spatial\nwavelength being close to the superlattice period a. This\nfeature can be explained by the structure of the matrix\nelement of the transitions which reaches maximum am-\nplitude atkx=±π/a, leading to the effective creation\nof spin texture (3) with the specific primary wavelength\nequal toa. Since the Fermi level position can be varied\nFIG. 2: Excited spin density distribution ( Sx(x),Sz(x)) along\nthe 1D superlattice elementary cell created under x- andσ+-\npolarized terahertz excitation with thefrequency ω/2π= 2.43\nTHz and with the intensity I= 0.3W/cm2.\nin experiments by tuning the gate voltage which controls\nthe concentration of 2DEG, our model predicts a possi-\nbleandrealisticmechanismforopticalcreationofvarious\nspin textures.\nAnother type of the considered polarization is the ra-\ndiation polarized along yaxis. The structure is homoge-\nneous in this direction and the only reason for the transi-\ntionprobabilitiestobenonzeroisthe non-parabolicchar-\nacter of energy spectrum as a function of kydue to the\ninterplay between SO and superlattice potential. How-\never, thisinterplayis reducedwith increasing kyandthus\none can expect the probabilities to be smaller for the y-\npolarized radiation than for the x-polarized one, making\nthe creation of the excited spin texture comparable to\nthe one in Fig.2 possible at higher intensities. This sug-\ngestion is confirmed by the spin textures in Fig.3 where4\nFIG. 3: Excited spin density distribution ( Sx(x),Sz(x)) along\nthe 1D superlattice elementary cell created under y-polarized\nterahertz excitation with the same parameters as in Fig.2 bu t\nat higher intensity I= 0.9W/cm2.\nthe similar spin textures as in Fig.2 are created at the\nintensity 0.9W/cm2which is three times greater than\nfor thex- andσ- polarized light. Nevertheless, all of the\nintensities considered in the paper are within the range\nof 0.5−1W/cm2which is accessible in modern experi-\nmental setups.9,10,12,16,17\nFurther investigations of gated 2DEG with SO inter-action would require the studies of the spin current con-\nventionally described by the operator ˆjs\nij= ¯h{ˆvi,σj}/4\nwhere ˆvi=∂ˆH/∂ki. The excited spin current distribu-\ntion can be analyzed under the same approach as the\nspin density, and the spin separation distance can be\ncalculated.15The investigations of spin current in our\nsystem deserves a separate detailed discussion and will\nbe performed in the forthcoming paper.\nIV. CONCLUSIONS\nWe have studied the excited spin texture distribution\nin 2DEG with Rashba spin-orbit interaction subject to\n1D tunable superlattice potential and illuminated by the\nterahertz radiation with different polarizations and in-\ntensities. It was found that in the absence of the net\npolarization the local excited spin texture can be effec-\ntively manipulated by varying the Fermi level position in\n2DEG as well as the intensity and polarization of the ra-\ndiation at fixed terahertz frequency. The effect of excited\nspin texture creation discussed in the paper has a quali-\ntative character and should be observable in a wide class\nof two-dimensional heterostructures where the spin-orbit\ncoupling energy is more pronounced than the tempera-\nture broadening or the smearing caused by edges, defects\nor impurities.\nAcknowledgments\nThe author thanks V.Ya. Demikhovskii and A.A.\nPerov for many helpful discussions. The work was sup-\nported by the RNP Program of the Ministry of Educa-\ntion and Science RF, by the RFBR, CRDF, and by the\nFoundation ”Dynasty” - ICFPM.\n∗Electronic address: khomitsky@phys.unn.ru\n1Semiconductor Spintronics and Quantum Computation ,\nedited by D.D. Awschalom, D. Loss, and N. Samarth,\nNanoscience and Technology (Springer, Berlin, 2002)\n2I. Zˇ uti´ c, J. Fabian, and S. Das Sarma, Rev. Mod. Phys.\n76, 323 (2004).\n3E.I. Rashba, Fiz. Tverd. Tela (Leningrad) 2, 1224 (1960)\n[Sov. Phys. Solid State 2, 1109 (1960)]; Y.A. Bychkov and\nE.I. Rashba, J. Phys. C 17, 6039 (1984).\n4J.B. Miller, D.M. Zumb¨ uhl, C.M. Marcus, Y.B. Lyanda-\nGeller, D. Goldhaber-Gordon, K. Campman, and A.C.\nGossard, Phys. Rev. Lett. 90, 076807 (2003).\n5D. Grundler, Phys. Rev. Lett. 84, 6074 (2000).\n6L.E. Golub, Phys. Rev. B 67, 235320 (2003).\n7S.D. Ganichev, V.V. Bel’kov, Petra Schneider, E.L.\nIvchenko, S.A. Tarasenko, W. Wegscheider, D. Weiss, D.\nSchuh, E.V. Beregulin, and W. Prettl, Phys. Rev. B 68,\n035319 (2003).8S.D. Ganichev, Petra Schneider, V.V. Bel’kov, E.L.\nIvchenko, S.A. Tarasenko, W. Wegscheider, D. Weiss, D.\nSchuh, B.N. Murdin, P.J. Phillips, C.R. Pidgeon, D.G.\nClarke, M. Merrick, P. Murzyn, E.V. Beregulin, and W.\nPrettl, Phys. Rev. B 68, 081302(R) (2003).\n9S. Giglberger, L.E. Golub, V.V. Bel’kov, S.N. Danilov, D.\nSchuh, C. Gerl, F. Rohlfing, J. Stahl, W. Wegscheider, D.\nWeiss, W. Prettl, and S.D. Ganichev, Phys. Rev. B 75,\n035327 (2007).\n10C.L. Yang, H.T. He, Lu Ding, L.J. Cui, Y.P. Zeng, J.N.\nWang, and W.K. Ge, Phys. Rev. Lett. 96, 186605 (2006).\n11Bin Zhou and Shun-Quing Shen, Phys. Rev. B 75, 045339\n(2007).\n12K.S. Cho, C.-T. Liang, Y.F. Chen, Y.Q. Tang, and B.\nShen, Phys. Rev. B 75, 085327 (2007).\n13Arkady Fedorov, Yuriy V. Pershin, and Carlo Piermaroc-\nchi, Phys. Rev. B 72, 245327 (2005).\n14S.A. Tarasenko, Phys. Rev. B 73, 115317 (2006).5\n15R.D.R. Bhat, F. Nastos, Ali Najmaie, and J.E. Sipe, Phys.\nRev. Lett. 94, 096603 (2005).\n16Hui Zhao, Xinyu Pan, Arthur L. Smirl, R.D.R. Bhat, Ali\nNajmaie, J.E. Sipe, and H.M. van Driel, Phys. Rev. B 72,\n201302(R) (2005).\n17J.T. Olesberg, Wayne H. Lau, Michael E. Flatt´ e, C. Yu,\nE. Altunkaya, E.M. Shaw, T.C. Hasenberg, and Thomas\nF. Boggess, Phys. Rev. B 64, 201301(R) (2001).\n18Jacob B. Khurgin, Phys. Rev. B 73, 033317 (2006).\n19Yuriy V. Pershin, Phys. Rev. B 71, 155317 (2005).\n20V.Ya. Demikhovskii and D.V. Khomitsky, JETP Letters\n83, iss.8, p.340 (2006) [Pis’ma v ZhETF 83, iss.8, p.399(2006)].\n21D.V. Khomitsky, Phys. Rev. B 76, 033404 (2007).\n22B.N. Murdin, K. Litvinenko, J. Allam, C.R. Pidgeon, M.\nBird, K. Morrison, T. Zhang, S.K. Clowes, W.R. Branford,\nJ. Harris, andL.F. Cohen, Phys.Rev.B 72, 085346 (2005).\n23K.C.Hall, K.G¨ undoˇ gdu, E.Altunkaya,W.H.Lau, Michael\nE. Flatt´ e, Thomas F. Boggess, J.J. Zinck, W.B. Barvosa-\nCarter, and S.L. Skeith, Phys. Rev. B 68, 115311 (2003).\n24L.D. Landau and E.M. Lifshitz, Electrodynamics of Con-\ntinuous Media , Pergamon, New York, 1984."    },    {        "title": "1012.5572v1.The_multi_state_CASPT2_spin_orbit_method.pdf",        "content": "arXiv:1012.5572v1  [physics.chem-ph]  27 Dec 2010The multi-state CASPT2 spin-orbit method\nZoila Barandiar´ an1,2and Luis Seijo1,2\n1Departamento de Qu´ ımica, Universidad Aut´ onoma de Madrid , 28049 Madrid, Spain\n2Instituto Universitario de Ciencia de Materiales Nicol´ as Cabrera,\nUniversidad Aut´ onoma de Madrid, 28049 Madrid, Spain\n(Dated: June 2, 2021)\nAbstract\nWe propose the multi-state complete-active-space second- order perturbation theory spin-orbit\nmethod (MS-CASPT2-SO) for electronic structurecalculati ons. It is a two-step spin-orbit coupling\nmethod that does not make use of energy shifts and that intrin sically guarantees the correct\ncharacters of the small space wave functions that are used to calculate the spin-orbit couplings, in\ncontrast with previous two-step methods.\nPACS numbers: 31.15.A-, 31.15.aj, 31.15.am\n1I. INTRODUCTION\nIn electronic structure two-step spin-orbit coupling methods, dy namic correlation is han-\ndled in the first step, using the spin-free part of the Hamiltonian and a large configurational\nspace in variational or perturbational schemes. Then, spin-orbit coupling is handled in the\nsecond step, using an effective Hamiltonian and a small configuration al space in spin-orbit\nconfiguration interaction (CI) calculations. In these methods, th e effective Hamiltonian con-\ntains explicit energy shifts, which are a mean to transfer dynamic co rrelation effects from\nthe first step to the second step in a simple and effective manner.1–3\nIt has been found that the energy shifts of the spin-orbit free lev els, which are driven\nby their energy order within each irreducible representation, can le ad to anomalous results\nwhen avoided crossings exist with significant change of character o f the wave functions at\neach side, which take place at different nuclear positions in the large a nd in the small\nelectronic configurational spaces. In these cases, the shifts mu st be assigned according to\nthe characters of the wave functions.4This usually implies analyses of wave functions in\nboth configurational spaces.\nThe ultimate reason behind these problems, which are present in the available two-step\nmethods,1–3is the different nature of the wave functions of the spin-free stat es in the large\nand in the small configurational spaces, so that, even when the av oided crossings do not\nexist or when they take place at the same nuclear positions in the larg e and in the small\nspaces, such a different nature makes the spin-orbit couplings calc ulated in the small space\nnot as accurate (meaning as close to the spin-orbit couplings calcula ted in the large space)\nas desired.\nHere, we propose the multi-state complete-active-space second -order perturbation theory\nspin-orbit method (MS-CASPT2-SO). It is a two-step method that does not make use of\nenergy shifts and that guarantees by construction the correct characters of the small space\nwave functions that are used to calculate the spin-orbit couplings.\n2II. THE MS-CASPT2-SO METHOD\nLet us assume we have a many electron system with a Hamiltonian ˆHwhich is made of\nthe addition of a spin-free contribution, ˆHSF, and a spin-orbit coupling contribution, ˆHSO:\nˆH=ˆHSF+ˆHSO. (1)\nIn the spin-orbit free MS-CASPT2-SO method, the procedure is init ially the same as\nthe MS-CASPT2 procedure:5,6Several state-average complete-active-space self-consisten t-\nfield SA-CASSCF (or CASCI) states are calculated, which define a re ference con-\nfigurational space called the Pspace. Let us collect them in the row vector\nΨCAS= (|ΨCAS\n1/angbracketright,|ΨCAS\n2/angbracketright,...,|ΨCAS\np/angbracketright), wherepis the total number of SA-CASSCF states.\nThese wave functions can be classified according to their values of s pin quantum numbers\nand symmetry group irreducible representations and subspecies, SMSΓγ, but we will omit\nthese labels here for simplicity.\nIn spin-orbit free MS-CASPT2 calculations, the SA-CASSCF wave fu nctions are used as\na basis to calculate the matrix of a spin-free second order effective Hamiltonian, ˆHSF,eff\n2nd,\nwhichisdefinedinEq.30ofRef.5anddependsonlyonthespin-freep artoftheHamiltonian,\nˆHSF. This matrix, which is HSF,eff,CAS\n2nd = ΨCAS†ˆHSF,eff\n2ndΨCAS, is diagonalized in order to\ncompute the MS-CASPT2 energies, as the eigenvalues, and the mod ified SA-CASSCF (or\nCASCI) states, as the eigenfunctions:\nHSF,eff,CAS\n2nd U=UEMS2, (2)\nwhereEMS2is a diagonal matrix with the MS-CASPT2 energies EMS2\n1,EMS2\n2,...,EMS2\np, as\nthe diagonal elements and Uis a unitary transformation of the original SA-CASSCF wave\nfunctions that preserves the SMSΓγvalues,\nΨCAS′= ΨCASU. (3)\nObviously, the modified SA-CASSCF wave functions ΨCAS′also span the Pspace. What is\nimportant isthat they aretheappropriatezeroth-order basis fo r asecond-order perturbation\ntheory treatment of the dynamic correlation that leads to the MS- CASPT2 energies6and\nthey have the appropriate characters in correspondence with th ese energies.\nIn the MS-CASPT2 spin-orbit calculations proposed here, we can fo llow two alternative\nprocedures that lead to the same result. Both of them are based o n the use of the spin-\n3dependent effective Hamiltonian that results from the addition of th e spin-orbit coupling\noperator to the spin-free effective Hamiltonian of the MS-CASPT2 m ethod,\nˆHeff=ˆHSF,eff\n2nd+ˆHSO. (4)\nIn the first procedure, which is a formal two-step procedure, th e regular spin-orbit free\nMS-CASPT2 calculation is completed and the modified SA-CASSCF wave functions ΨCAS′\nare used as a basis for the matrix representation of ˆHeff. The resulting matrix,\nHeff,CAS′= ΨCAS′†ˆHeffΨCAS′=EMS2+HSO,CAS′, (5)\nwithHSO,CAS′= ΨCAS′†ˆHSOΨCAS′, couples the modified SA-CASSCF states via spin-orbit\ncoupling. (It couples different SMSΓγblocks and it can be factorized according to double\ngroup irreducible representations.) Its diagonalization gives the fin al energies and wave\nfunctions:\nHeff,CAS′USO′=USO′EMS2−SO, (6)\nwhereEMS2−SOis a diagonal matrix with the MS-CASPT2-SO target energies\nEMS2−SO\n1,EMS2−SO\n2,...,EMS2−SO\npas the diagonal elements and USO′is a unitary trans-\nformation of the modified SA-CASSCF wave functions that couples t heSMSΓγvalues and\ngives the target spin-orbit wave functions,\nΨMS2−SO= ΨCAS′USO′. (7)\nAlternatively, in the second procedure, which is a formal one-step procedure, the original\nSA-CASSCF wave functions ΨCASare used as the basis for the matrix representation of\nˆHeff. In order to do this, the regular spin-orbit free MS-CASPT2 calcula tion does not need\nto be completed, but only the computation of the HSF,eff,CAS\n2nd matrix used in Eq. 2, plus\nthe addition of the matrix of ˆHSOin this basis ( HSO,CAS= ΨCAS†ˆHSOΨCAS):\nHeff,CAS= ΨCAS†ˆHeffΨCAS=HSF,eff,CAS\n2nd +HSO,CAS. (8)\nIts diagonalization gives the same target energies and wave functio ns as the first procedure,\nHeff,CASUSO=USOEMS2−SO, (9)\nwith\nΨMS2−SO= ΨCASUSO(10)\n4andUSO=UUSO′.\nThe present method is closely related with the restricted active spa ce state interaction\napproach with spin-orbit coupling of Ref. 3, SO-RASSI. The latter, when it is used in a\nSA-CASSCF/MS-CASPT2 context, corresponds to diagonalizing th e mixed effective Hamil-\ntonian matrix EMS2+HSO,CAS, with the spin-orbit free part of Eq. 5 and the spin-orbit\ncoupling part of Eq. 8. The results of both approaches are expect ed to be similar when the\nCAS and the CAS′wave functions (Eq. 3) are also similar, which is a common case. Differ-\nences should show up when the two sets of wave functions are not s o similar in a one-to-one\nbasis, for instance when the dynamic correlation switches the orde r of states. This is the\nbasic advantage of the present method. However, we must note t hat the SO-RASSI method\ncan be used together with general single-state andstate-avera ge RASSCF and CASSCF plus\nRASPT2 and single-state and multi-state CASPT2 spin-orbit free fr ameworks, whereas the\npresent one can only be used in the spin-orbit free framework of SA -CASSCF plus MS-\nCASPT2 calculations.\nLet us now justify why ˆHeff(Eq. 4) is the effective Hamiltonian of choice in the MS-\nCASPT2-SO method. For this purpose, we recall that the basic idea of two-step methods is\nto use a spin-orbit effective Hamiltonian made of a spin-free effective Hamiltonian (usually\nˆHSF+ˆHshift)plusthespin-orbitcoupling operator ˆHSO, andtochoosethespin-freeeffective\nHamiltonianbyimposing therequirement that, when used inthesmall s pacePofthesecond\nstep, it has the same eigenvalues that ˆHSFhas in the large space Gof the first step.1In\nthe particular case in which the first step is a MS-CASPT2 calculation a nd the small space\nof the second step is defined by the SA-CASSCF wave functions, ˆHSF,eff\n2ndis a spin-free\neffective Hamiltonian that fulfills such a condition. In consequence, ˆHeff(Eq. 4) is the\nproper spin-orbit effective Hamiltonian.\nIII. CONCLUSION\nAtwo-step spin-orbit coupling methodformulti-state complete-ac tive-space second-order\nperturbationtheorycalculationsMS-CASPT2isproposedwhichdoe snotmakeuseofenergy\nshifts. It intrinsically guarantees the correct characters of the small space wave functions\nused to calculate the spin-orbit couplings, in contrast with previous two-step spin-orbit\ncoupling methods, where it has to be checked externally.\n5Acknowledgments\nThisworkwas partlysupported byagrantfromMinisterio deCiencia e Innovaci´ on, Spain\n(Direcci´ on General de Programas y Transferencia de Conocimient o MAT2008-05379/MAT).\n1R. Llusar, M. Casarrubios, Z. Barandiar´ an, and L. Seijo, J. Chem. Phys. 105, 5321 (1996).\n2V. Vallet, L. Maron, C. Teichteil, and J.-P. Flament , J. Chem . Phys.113, 1391 (2000).\n3P. A. Malmqvist, B. O. Roos, and B. Schimmelpfennig, Chem. Ph ys. Lett. 357, 230 (2002).\n4G. S´ anchez-Sanz, Z. Barandiar´ an, and L. Seijo, Chem. Phys . Lett.498, 226 (2010).\n5J. Finley, P.- ˚A. Malmqvist, B. O. Roos and L. Serrano-Andr´ es, Chem. Phys. Lett.288, 299\n(1998).\n6A. Zaitsevskii and J. P. Malrieu, Chem. Phys. Lett. 233, 597 (1995).\n6"    },    {        "title": "1411.2990v1.Striped_Ferronematic_ground_states_in_a_spin_orbit_coupled__S_1__Bose_gas.pdf",        "content": "arXiv:1411.2990v1  [cond-mat.quant-gas]  11 Nov 2014Striped Ferronematic ground states in a spin-orbit coupled S= 1Bose gas\nStefan S. Natu,1,∗Xiaopeng Li,1and William S. Cole1\n1Condensed Matter Theory Center and Joint Quantum Institute , Department of Physics,\nUniversity of Maryland, College Park, Maryland 20742-4111 USA\nWe theoretically establish the mean-field phase diagram of a homogeneous spin-1, spin-orbit\ncoupled Bose gas as a function of the spin-dependent interac tion parameter, the Raman coupling\nstrength and the quadratic Zeeman shift. We findthat the inte rplay between spin-orbit coupling and\nspin-dependent interactions leads to the occurrence of fer romagnetic or ferronematic phases which\nalso break translational symmetry. For weak Raman coupling , increasing attractive spin-dependent\ninteractions (as in87Rb or7Li) induces a transition from a uniform to a stripe XY ferroma gnet\n(with no nematic order). For repulsive spin-dependent inte ractions however (as in23Na), we find a\ntransition from an XYspin spiral phase ( /an}bracketle{tSz/an}bracketri}ht= 0 and uniform total density) with uniaxial nematic\norder, to a biaxial ferronematic, where the total density, s pin vector and nematic director oscillate in\nreal space. We investigate the stability of these phases aga inst the quadratic Zeeman effect, which\ngenerally tends to favor uniform phases with either ferroma gnetic or nematic order but not both.\nWe discuss the relevance of our results to ongoing experimen ts on spin-orbit coupled, spinor Bose\ngases.\nINTRODUCTION\nThe interplay between competing orders such as su-\nperfluidity/superconductivity, magnetism, liquid crys-\ntallinity and density wave order is fundamental to the\nrich phenomenology of strongly correlated systems. A\ncandidate system for exploring this physics is a spin-\norbit coupled Bose condensate [1–4], where the coupling\nbetween spin and motional degrees of freedom can lead\nto a spin textured ground state which breaks rotational\nsymmetry in spin space [5–7], as well as translational\nsymmetry in real space [8, 9]. Indeed, for a pseudospin-\n1/2Bosesystem, varyingthespin-orbitcouplingstrength\ndrivesatransitionfromanunmagnetizedphasewithden-\nsitywaveordertoauniformmagnetizedphase, whichhas\nbeen studied both theoreticallyand experimentally[4, 9].\nMore recently, attention has turned towards exploring\nthe physics of large spin systems, which have no ana-\nlog in condensed matter, such as highly magnetic atoms\nlike Dysprosium, Erbium, Chromium [10–12], and alka-\nline earthatomswith SU(N) symmetry(See Ref. [13] and\nreferenes therein). The large spin nature of these atoms\nproduces a rich phase diagram with novel topological de-\nfects, where uniaxial and biaxial nematic and more ex-\notic platonic solid orders compete and complement con-\nventional magnetically ordered phases [14–16]. In the\npresence of spin-orbit coupling, the possibility of transla-\ntional symmetry breaking can lead to textured ground\nstates phases with intertwining magnetic and nematic\norder. Here we study the simplest, experimentally re-\nalizable system where such physics is manifest: a spin-1,\nspin-orbit coupled Bose gas [5, 17], finding a rich phase\ndiagram.\nOur main result is summarized in Fig. 1, which shows\nthe schematic, zero-temperaturephase diagramofa spin-\norbit coupled spin-1 Bose gas as a function of the spin-\ndependent interaction and the quadratic Zeeman energy,at fixed Raman coupling and spin-independent interac-\ntion strength. A new feature of the spin-orbit coupled\ngas is the appearance of translational symmetry break-\ning phases with simultaneous spin and nematic order,\nwhich are generically competing orders in this system\n[15, 16]. Weak repulsive spin-dependent interactions fa-\nvorauniaxialnematicferromagnet(ferronematic), which\nsupportsaspiralspintextureinthe x−yplane(UN+XY\nspiral). Large attractive spin-dependent interactions fa-\nvor a ferromagnetic stripe ( FMstripe), whereas large\nrepulsive spin-dependent interactions favor a biaxialfer-\nronematic stripe phase ( BNstripe) where the total den-\nsity, spin vector and nematic director oscillates in real\nspace.\nSPIN-1PHENOMENOLOGY\nIn the absence of spin-orbit coupling, the phase dia-\ngram of a spin-1 Bose gas has been well established the-\noretically [18–21] and experimentally [22–27]. Assum-\ning short-range (s-wave) interactions, spin rotation in-\nvariance and bosonic statistics forces two-body collisions\nto occur in the total spin-0 or spin-2 channels, producing\nthe interaction Hamiltonian [18, 19]:\nHint=1\n2/integraldisplay\nd3rψ†\nαψ†\nβψγψδ(c0δαδδβγ+c2Sαδ·Sβγ),(1)\nwhere the greek indices denote the hyperfine spin projec-\ntion, andψσ(r) is the the boson field operator. Unlike\nthepseudospin-1 /2case, thisHamiltonianhasSU(2)spin\nrotation invariance.\nThe two coupling constants, c0andc2represent\nspin-independent and spin-dependent interactions re-\nspectively, and Sis the vector {Sx,Sy,Sz}, whereSi\nare 3×3, spin-1 matrices. The interactions are expressed\nin terms of the microscopicscatteringlengths in the spin-\n0 (a0) and spin-2 ( a2) channels and atomic mass mas:2\nFM/UpTeeFM/VertBar1/VertBar1UN/UpTeeUN/UpTee UN/VertBar1/VertBar1/PlusXY spiralBN stripe\nFM/VertBar1/VertBar1stripeUN/UpTeeUN/VertBar1/VertBar1\n00\nq/Slash1c0n0c2/Slash1c0/Minus2/Minus1012\nk/Slash1k0/Minus2/Minus1012Ek/Slash1ER\n/Minus2/Minus1012\nFIG. 1: (Color Online) Schematic phase diagram for a\nspin-orbit coupled spin-1 Bose gas, as a function of spin-\ndependent interaction strength and quadratic Zeeman shift ,\nshowing translation symmetry breaking phases. The underly -\ning single-particle dispersion is shown above. For sufficien tly\nlarge attractive spin-dependent interactions, an XY spin d en-\nsity wave phase occurs, with oscillations in the total densi ty\n(FMstripe). For repulsive spin-dependent interactions, an\nXY spiral phase occurs, which simultaneously has uniaxial\nnematic order (UN), but the total density remains uniform.\nFor sufficiently large spin-dependent interactions, a biaxi al\nnematic phase ( BN) is present where the total density, spin\nvector and nematic director oscillate in real space. Suffi-\nciently large positive or negative quadratic Zeeman effect f a-\nvors homogeneous phases with either uniaxial nematic ( UN⊥\n(ψ={0,1,0}) orUN/bardbl(ψ=1√\n2{1,0,1})) or ferromagnetic\norder (FM⊥(ψ={1,0,0}) orFM/bardbl(ψ=1\n2{1,√\n2,1})), but\nnot both.\nc0= 4π(a0+ 2a2)/3mandc2= 4π(a2−a0)/3m[25].\nFor87Rb,c2/c0=−0.005, for23Na,c2/c0∼0.05 and\nfor7Li,c2/c0∼ −0.5 [25]. These interactions (and their\nsign) can however be tuned using optical Feshbach reso-\nnances [28].\nThe wave-function of a spin-1 Bose condensate is rep-\nresented as a spinor ψ=eiθ{ψ1,ψ0,ψ−1}, whereθrepre-\nsents the broken global U(1) gauge symmetry ofthe Bose\ncondensate, and {1,0,−1}label the three spin states.\nOwing to the structure of the spin-1 Pauli matrices, this\nsystem can exhibit both magnetic order, given by the\nvector order parameter /an}bracketle{tS/an}bracketri}ht=/an}bracketle{tψ|S|ψ/an}bracketri}ht/n, wherenis the\ndensity, or nematic order, described by the tensor Nµν=\nδµν−1\n2n/an}bracketle{tψ|(SµSν+SνSµ)|ψ/an}bracketri}ht, where{µ,ν} ∈ {x,y,z},\nandδµνdenotesthe identity matrix. However,aspointed\nout by Mueller [16], these orders are not independent of\noneanother, butrathercompeting. Diagonalizingthene-\nmatic tensor yields three distinct eigenvalues λ1,λ2,λ3,\nconstrained by λ1+λ2+λ3= 1. Auniaxial nematic has\nonenon-zeroeigenvalue,whilea biaxialnematichasthree\ndistinct eigenvalues. Attractive spin-dependent interac-\ntions (c2<0) favor a maximally ferromagnetic phase(/an}bracketle{tψ|S|ψ/an}bracketri}ht=ˆz), with no nematic order, which can be\nrepresented by unitary rotations of ψ=eiθU{1,0,0}T,\nwhereas repulsive spin-dependent interactions ( c2>0)\nfavor a uniaxial nematic with no spin order, represented\nby unitary rotations of ψ=eiθU{0,1,0}T[18].\nIn the presence of spin-orbit coupling, spin-rotation\nsymmetry is broken and the three spin states are no\nlonger degenerate at the single-particle level. We choose\nthespin-orbitcouplingtobeofequalRashba-Dresselhaus\ntype, whichwasrecentlyrealizedinexperiments[2–4,29–\n31], but generalized to the spin-1 case:\nHsoc=/planckover2pi12(kx−k0Sz)2\n2m+/planckover2pi12k2\n⊥\n2m+Ω\n2Sx+δ\n2Sz+q\n2S2\nz(2)\nwherek0is the wave-vector of the Raman beams, Ω is\nthe strength of the Raman coupling, δandqare the\nlinear and quadratic Zeeman effects respectively. It is\nconvenient to normalize energy by the recoil energy of\nthe Raman lasers ER=/planckover2pi12k2\n0\n2m. For simplicity, we neglect\nthe linear Zeeman effect term ( δ= 0) in this work, but\ngenerallyassumethat q/ne}ationslash= 0, andcantakeonpositiveand\nnegative values, which can be achieved using microwaves\n[32, 33].\nA detailed description of the single-particle physics of\na spin-1 spin-orbit coupled gas was recently given by Lan\nand¨Ohberg [17], and is not repeated here. For weak q\nand Ω, the low energy spectrum has three local minima\natk= 0,±k1, where 0 ≤k1/k0≤1 is determined by\ndiagonalizing Eq. 2, at fixed Ω and q. Increasing positive\nqresults in a single minimum at k= 0, whereas negative\nqproduces a two minimum structure, with the minima\nat±k1.\nThedispersionofthelowestbranch(toquadraticorder\ninkx) is obtained as\nǫ(kx) =/planckover2pi12k2\nx\nm/bracketleftbigg\n1/2−/planckover2pi12k2\n0\nm(˜q2+4Ω2)−1/21−z\n1+z/bracketrightbigg\n+O(k4\nx),\n(3)\nwith ˜q=q+/planckover2pi12k2\n0\nmandz=˜q√\n˜q2+4Ω2. The analytic form\nof the transition line from three to two minima easily\nfollows as/planckover2pi12k2\n0\nm(˜q2+4Ω2)−1/21−z\n1+z= 1/2.\nFrom the triple minimum structure of single-particle\ndispersion, we make a reasonable variational ansatz for\nthe condensate wave-function:\nψ=/radicalbigg\nN\nV(χ+eik1xφ++χ0φ0+χ−e−ik1xφ−),(4)\nwhereχ0,χ±are complex numbers, which are deter-\nmined variationallybelow, Nis the particle number, Vis\nthe volume, and φ±,φ0are the normalized single-particle\nspinor eigenstates at the minima ±k1,0 respectively. We\nfix the gaugechoice bychoosingthe eigenvectors( φ±,φ0)\nto be real, where the respective spin components obey\nφ±1\n±=φ∓1\n∓, andφ0\n+=φ0\n−. Particle number conservation3\nTABLE I: Orders in spin-orbit coupled spin-1 gas.\nOrder Symbol Order Parameter\nferromagnetic FM/bardbl/⊥ /an}bracketle{tSi/an}bracketri}ht /ne}ationslash= 0\nUniaxial nematic UN/bardbl/⊥λ1/ne}ationslash= 0,λ2=λ3= 0\nBiaxial nematic BN λ 1<λ2<λ3\nTranslation stripe, XYspiral /an}bracketle{tSi(r)/an}bracketri}ht ∼cos(k1r)\nn(r)∼cos(k2r)\nN=/integraltext\nd3rn(r) =/summationtext\nσ∈{−1,0,1}/integraltext\nd3r|ψσ(r)|2imposes the\nconstraint |χ+|2+|χ0|2+|χ−|2= 1.\nThe variational interaction energy takes a suggestive\nform:\nE=r(|χ+|2+|χ−|2)+gµν|χµ|2|χν|2+g3(χ∗\n+χ∗\n−χ0χ0+c.c.),\n(5)\nwhereris the kinetic term and gµνandg3are related\nto the original interaction parameters multiplied by form\nfactors proportional to the single-particle wave-functions\natthethreeminima. Theenergyisinvariantundertrans-\nformations UC(1) :χµ→eiθχµ,UA(1) :χµ→eiµθχµ\nandZ2:χµ→χ−µ. Heretheaxial UA(1)originatesfrom\ntranslational symmetry, and the Z2symmetry is related\nto reflection where spin transforms as a pseudovector.\nIt is important to emphasize that the Josephson term,\nproportional to g3is zero throughout the phase diagram\nof the spin-1 Bose gas without spin-orbit coupling. How-\never, asweshowhere, it playsacrucialroleinthe physics\nof the spin-orbit coupled gas.\nA new feature of the spin-orbit coupled Bose gas is\nthe possibility oftranslationalsymmetrybreakingphases\n[5, 8], which arise because bosons in different spin states\ncondense into finite momentum states. For the spin-1 /2\ncase, where bosons condense at two minima, the total\ndensity develops stripes at a wave-vector 2 k1[5, 8, 9],\nbut the spin density /an}bracketle{tS/an}bracketri}htremains uniform throughout the\nphase diagram [9, 34]. In the spin-1 case however, in\naddition to stripe structure in the density [5, 17], the\nsystem can display oscillations in the spin and nematic\norder parameters. This leads to a rich phase diagram,\nreproduced in Fig. 1, which we now discuss in detail. In\nTable I, we characterize the condensed phases we find, in\nterms of their order parameters.\nWe minimize the totalenergyperparticle(Eq. 5), with\nrespect to the complex variational parameters χ0,χ±.\nThe spin and nematic order parameters are then com-\nputed using the resulting mean-field ground state wave-\nfunction. Normalizing the energy to the laser recoil\nenergyER, and setting δ= 0, yields four dimension-\nless parameters Ω /ER,q/ER,c0n0/ER,c2n0/ER, where\nn0=N/Vis the total density. Throughout, we fix the\nRaman coupling Ω /ERsuch that, in the absence of a\nquadratic Zeeman effect, the low energy, single-particle\ndispersion has three local minima. The two minima atk=±k1are always degenerate in the absence of δ. For\nq>qc1>0 the three minimum structure disappears and\nonly a single minimum at k= 0 is present, whereas for\nq < qc2<0, the system only has two minima at finite\nk. We fix Ω but vary qto access both these regimes in\nparameter space [17].\nATTRACTIVE SPIN-DEPENDENT\nINTERACTIONS c2<0\nWe first consider the regime of attractive spin-\ndependent interactions, which corresponds to87Rb (as\nin the NIST experiments [2, 29]) and7Li. Absent spin-\norbitcoupling, thegroundstateisauniformferromagnet,\nwhich is of Ising type ( /an}bracketle{tS/an}bracketri}ht=ˆz) forq <0 and XY type\n(/an}bracketle{tS/an}bracketri}ht=ˆx) forq >0 (spin rotation symmetry is restored\natq= 0). For sufficiently large q >0, there is a sec-\nond order quantum phase transition to a polar ( UN⊥)\nphase, which has been investigated in detail recently (see\nRef. [25] and References therein).\nIn the presence of spin-orbit coupling, at q= 0, spin\nsymmetry is explicitly broken by the Rabi coupling (Ω)\nterm, whichpreferstoalignthetotalmagnetizationalong\nx. Absent interactions, the single particle wave function\nis centered around k= 0, and has a small but finite value\nof/an}bracketle{tSx/an}bracketri}ht, correspondingtothe explicitly brokensymmetry.\nUpon turning on c2, the minimum energy state is a ferro-\nmagnet with /an}bracketle{tS/an}bracketri}ht=ˆx. The wave-function for such a state\nrequires all three minima to be occupied, but the relative\nphaseθ++θ−−2θ0≈0, whereχ±0=|χ±,0|eiθ±,0. This\nstate is degenerate with the plane wave Ising ferromag-\nnet formed by occupying a single minimum at k=±k1\nwith respect to the spin-dependent interaction term, but\nhas stripes in the total density, and is thus penalized by\nthe repulsive density-density interactions (proportional\ntoc0). For weak |c2|therefore, an Ising plane wave phase\noccurs, which has no stripes in the total density. Upon\nincreasing |c2|however, the total energy can be lowered\nby aligning the spin along the xdirection, satisfying the\nRabi coupling term, at the expense of producing den-\nsity (and spin) stripes. This interaction driven transition\nfrom a uniform Ising ferromagnet to striped XY ferro-\nmagnet is a new feature of the spin-1, spin-orbit coupled\ngas [17].\nIn Fig. 2 we plot the critical value of |c2|/c0for the\nferromagnetic stripe phase as a function of the Rabi\nstrength Ω/ER. For our parameters, when Ω /ER>1,\nthe system enters the single minimum regime, and the\nstripe phase is destroyed. Although the stripe phase can\noccur for arbitrarily small spin dependent interactions, it\nshould be emphasized that the amplitude of the stripes\ndecreases with decreasing |c2|/c0, and may be hard to\nresolve experimentally, particularly for87Rb. The hori-\nzontal dashed line shows the spin-dependent interaction\nfor7Li [25]; all values of Rabi coupling below the line4\n00.20.40.60.8100.10.20.30.40.5\n/CapOmega/Slash1ER/VertBar1c2/VertBar1/Slash1c0\n/Minus10/Minus505100.00.51.01.5\nx/LParen11/Slash1k1/RParen1n/LParen1x/RParen1/Slash1n0\nFIG. 2: Critical attractive spin-dependent interaction |c2|/c0\nfor onset of stripe ferromagnetic phase as a function of Rabi\ncoupling Ω/ERatq= 0. We set c0n0/ER= 0.4. Dashed\nline shows |c2|/c0for7Li; all values of Ω /lessorsimilar0.9ERsupport the\nferromagnetic phase with stripes in the total density. Inse t\nshows the density in real space for Ω /ER= 0.8 at|c2|/c0\ncorresponding to7Li.\nshould exhibit a stripe phase. The inset shows the pro-\nnounced amplitude of the stripes for the7Li interaction\nparameters at Ω /ER= 0.8, which strongly supports the\nexperimental observability of this phase.\nAs shown in Fig. 1, the striped ferromagnetic phase\nis destroyed for positive and negative values of the\nquadratic Zeeman effect. For q >0, we find a transition\nfrom the stripe ferromagnetic phase to a polar conden-\nsate, which occurs roughly where the single-particle dis-\npersion enters the single minimum regime ( qc1), largely\nindependent of c2. Forq <0, the single-particle disper-\nsion has two minima, and a uniform Ising ferromagnetic\nphaseoccurswhereonlyoneofthese twodegeneratemin-\nima are occupied. The transition from the stripe ferro-\nmagnet to the uniform Ising ferromagnet where trans-\nlation symmetry is restored, depends on the interaction\nstrength and the magnitude of qas shown in Fig. 1.\nPOLAR REGIME: c2>0\nWe now turn our attention to the case of repulsive\nspin-dependent interactions. Absent spin-orbit coupling,\nrepulsivespin-dependentinteractionsyieldapolarphase,\nwhere/an}bracketle{tS/an}bracketri}ht= 0, but the system has uniaxial nematic or-\nder. In the presence of the Rabi term Ω, residual ferro-\nmagneticorderispresentevenfor c2>0, andgenerically,\nthe ground state is ferronematic.\nFor weak spin-dependent interactions and q >0, the\nsystem condenses at k= 0, and a uniform ferronematic\nphase is found. For q <0 however, the system con-\ndenses atk=±k1, and the phases of the condensate\nat these two points are such that the total density re-\nmains uniform, but the transverse spin density sponta-\nneously breaks translational symmetry and develops XY\nspin density wave order. Similar spiral phases have been\npredicted to occur in the spin-1 /2 system in the presence\nof Rashba spin-orbit coupling [5, 7].\nThis origin of the spin density wave can also be un-xz\n/Minus10/Minus50510/Minus0.6/Minus0.4/Minus0.20.00.2\nx/LParen11/Slash1k0/RParen1S/Slash1n0\nFIG. 3: Top: Spatial spin texture in the polar regime of a\nspin-orbit coupled spin-1 Bose gas for strong spin-depende nt\ninteractions c2/c0= 1.1, and Ω/ER= 0.8, which corresponds\nto the three minimum regime. We set q= 0 here, although\nthis phase is stable for moderate values of q(see Fig. 1). The\narrows indicate the projection of the spin in the x−zplane\nin spin space at each point in real space. Bottom: x(green\ndotted),y(red dashed) and z(black solid) components of\nspin in real space. Oscillations in the zdirection (at wave-\nvectork=k1) andx−ydirection (at two wave-vectors k=\nk1,2k1) are different with one another. The total density (not\nshown) also shows anharmonic oscillations at two dominant\nwave-vectors k=k1andk= 2k1. Note that due to the\nRaman coupling term, time reversal is explicitly broken, an d\nthe spatially averaged spin along the xdirection is finite.\nderstood as follows: For weak Raman coupling, the con-\ndensates at k=±k1,0 are closely related to the original\n±1,0 spin states. Thus by applying a gauge transforma-\ntion, whereby ψ±1→ψ±1e∓ik1x,ψ0→ψ0, we obtain a\nnon spin-orbit coupled Hamiltonian with a spatially os-\ncillating magnetic field along xof the form Ω Sxcos(k1x),\nleaving all other terms in the Hamiltonian unchanged.\nThis produces an XY spin density wave texture in real\nspace. Asqbecomes more and more negative, the ampli-\ntude of oscillation of the spiral goes to zero as Ω /|q|. The\nnematic tensor in this phase has only one non-zero eigen-\nvalue, which corresponds to a uniaxial nematic. Thus\nspin-orbitcouplingnaturallyleadstoferronematicphases\nwhich break translation symmetry.\nFor even larger c2, the situation becomes more exotic,\nand the system condenses into all three minima, even at\nq= 0. The relative phase of the condensates at the three\nminimaθ++θ−−2θ0≈π. This state has stripes in the\ntotal density [17], and concurrently, magnetic order in all\nx,y,zdirections, as shown in Fig. 3. Interestingly how-\never, the wavelengths of the oscillations in zand those\nin thex−ydirections are generally different from one\nanother.\nFurthermore, diagonalizing the nematic tensor for this\nsituationyieldsthreedistincteigenvalues,whichisa biax-\nialnematicphase. Owingtotheconstraint λ1+λ2+λ3=\n1, the degree of biaxiality can be found by taking the\ndifference between the largest two eigenvalues. As the5\ndensity, spin and nematic order parameters are not inde-\npendent of one another, all these orders simultaneously\noscillate in real space. In Fig. 4, we plot the spatial oscil-\nlations in the biaxiality and the total spin, the minimum\nin the biaxiality coincides with the minimum in the total\nspin, where a maximally uniaxial nematic phase occurs.\nEXPERIMENTAL RELEVANCE\nAs we show in Fig. 2, observing the ferromagnetic\nstripe phase for attractive spin-dependent interactions\nmay be challenging for87Rb, but feasible in7Li. Al-\nthough the contrast in the oscillations in the spin den-\nsity is large, experiments may not have enough spatial\nresolution to observe the individual oscillations, which\nwill further be smeared out by spatial averaging effects\nduring expansion and imaging. Martone et al.[35] have\nrecently proposed using Bragg spectroscopy to increase\nthe wavelength of the stripes, making them visible.\nThe XY spiral nematic state occurs even for weak, re-\npulsive spin-dependent interactions (as in23Na spinor\ncondensates) by simply tuning the quadratic Zeeman\nshift to take on negative values, which produces two\nglobal minima in the single-particle dispersion. The\ntransverse components of spin can then be probed in\nsituby applying spin echo radio-frequency pulses be-\ntween individual snapshots [36] to reveal the spin den-\nsity texture. Nematicity can be probed experimentally\nby directly measuring spin fluctuations /an}bracketle{tSµSν/an}bracketri}ht, averaged\novermanyshots[37]. Alternatively,opticalbirefringence,\nwhereby, the coupling between the nematic order param-\neter and the polarization of a probe beam leads to a ro-\ntation in the polarization of the light field can be used to\nmeasure the local nematic/biaxial order parameter [38].\nObserving the biaxial spin-density wave phase requires\nstrong spin-dependent interactions, which could be in-\nduced using optical or magnetic Feshbach resonances.\nThis however breaks the SU(2) spin rotation invariance\nwhich underpins Eq. 1, and may lead to a qualitatively\ndifferent phase diagram than that discussed here.\nCONCLUSIONS AND FUTURE DIRECTIONS\nIn conclusion, we find that the spin-1 spin-orbit cou-\npled Bose gas possesses a rich phase diagram with phases\nwhich break translational symmetry, spin symmetry and\npossess liquid crystalline order. Generally, we find that\nspin-orbit coupling intertwines magnetic and nematic or-\nder, giving rise to ferronematic phases that break trans-\nlationalsymmetry. In additionto the usualhomogeneous\npolarand ferromagneticphasesin the non spin-orbitcou-\npled spin-1 gas, we find three new phases: a ferromag-\nnetic stripe phase for attractive spin-dependent interac-\ntions with stripes in the total density and spin, an XY/Minus10/Minus505100.00.10.20.30.40.50.60.7\nx/LParen11/Slash1k0/RParen1Order\nFIG. 4: Total spin (red dashed) and biaxial order parameter\n(black, solid) shown as a function of space in the biaxial, sp in-\ndensity wave phase of a polar spin-orbit coupled gas. The\nminimum in the biaxiality corresponds to a minimum in the\ntotal spin, and here a nearly uniaxial nematic appears, whic h\nillustrates the competing nature of thenematic and spin ord er\nparameters.\nspiral ferronematic phase with uniform total density for\nweak repulsive spin-dependent interactions and negative\nquadratic Zeeman shift, and a biaxial ferronematic stripe\nphase, with spatial oscillations in the total density, spin\nvector and nematic director.\nA key difference between the spin-1 case from the\npseudo-spin 1 /2 counterpart is the appearance of spin\ndensity wavephases, even when the total density remains\nuniform for repulsive spin-dependent interactions. We\nalsoemphasize that although ferronematicphasesarebe-\nlieved to occur in dipolar fermions [11, 39] and high spin\nsystems such as spin-3 Chromium atoms [15], a crucial\ndifference here is that the ferronematic ground states we\nfind also break translationalsymmetry, due to the under-\nlying spin-orbit coupling. High spin spin-orbit coupled\nsystems thus offer unique insight into the interplay be-\ntweencompeting orders, which areubiquitous in strongly\ncorrelatedsystems. It will be extremelyinteresting to ex-\nplore generalizations of this work to even larger spin sys-\ntems, such asspin-orbitcoupled Dysprosiumand Erbium\natoms [11, 12]. Other exciting directions for future work\nare to explore the nature of the Goldstone modes asso-\nciated with broken translation symmetry [40], the topo-\nlogical defects that occur in these striped ferronematic\nphases [16, 41] and the restoration of different symme-\ntries at finite temperature [42–44].\nAcknowledgements.— We aregrateful to the JQI-NSF-\nPFC,AFOSR-MURI,andARO-MURI(Atomtronics)for\nsupport.\n∗Electronic address: snatu@umd.edu\n[1] V. Galitski and I. B. Spielman, Nature 49549 (2013).\n[2] Y. K Lin, K. Jimenez-Garcia and I. B. Spielman, Nature,\n47183 (2011).\n[3] J.-Y. Zhang et al., Phys. Rev. Lett. 109115301 (2012).\n[4] S.-C. Ji, et al.Nature Physics 10314 (2014).\n[5] C. Wang, C. Gao, C.-M. Jian and H. Zhai, Phys. Rev.6\nLett.105160403 (2010).\n[6] T. D. Stanescu, B. Anderson and V. Galitski, Phys. Rev.\nA78023616 (2008).\n[7] C. Wu, I. Mondragon-Shem and X.-F. Zhou, Chin. Phys.\nLett.28097102 (2011).\n[8] T-L. Ho, S. Zhang, Phys. Rev. Lett. 107150403 (2011).\n[9] Y. Li, L. P. Pitaevskii and S. Stringari, Phys. Rev. Lett.\n108225301 (2012).\n[10] A. Griesmaier, J. Werner, S. Hensler, J. Stuhler and T.\nPfau, Phys. Rev. Lett. 94160401 (2005).\n[11] K. Aikawa, S. Baier, A. Frisch, M. Mark, C. Ravensber-\ngen and F. Ferlaino, Science 3451484 (2014).\n[12] X. Cui, B. Lian, T.-L. Ho, B. L. Lev and H. Zhai, Phys.\nRev. A88011601 (R) (2013).\n[13] M. A. Cazalilla and A. M. Rey, eprint.arXiv: 1403.2792.\n[14] T.-L. Ho and B. Huang, eprint.arXiv:1401.4513.\n[15] R. B. Diener and T.-L. Ho, Phys. Rev. Lett. 96190405\n(2006).\n[16] E. J. Mueller, Phys. Rev. A 69033606 (2004).\n[17] Z. Lan and P. Ohberg, Phys. Rev. A 89023630 (2014).\n[18] T.-L. Ho, Phys. Rev. Lett. 81742 (1998).\n[19] T. Ohmi and K. Machida, J. Phys. Soc. Jpn. 671822\n(1998).\n[20] C. K. Law, H. Pu and N. P. Bigelow, Phys. Rev. Lett.\n815257 (1998).\n[21] S. Mukerjee, C. Xu and J.E. Moore, Phys. Rev. Lett. 97,\n120406 (2006).\n[22] J. Stenger et al.Nature396345-348 (1998).\n[23] D. M. Stamper-Kurn et al.Phys. Rev. Lett. 802027\n(1998).\n[24] L. E. Sadler et al.Nature443312-315 (2006).\n[25] D. M. Stamper-Kurn and M. Ueda, Rev. Mod. Phys. 85\n1191 (2013).\n[26] D. Jacob et al., Phys. Rev. A 86061601 (R) (2012).\n[27] M.-S. Chang et al.Phys. Rev. Lett. 92140403 (2004).\n[28] F. K. Fatemi, K. M. Jones and P. D. Lett, Phys. Rev.\nLett.854462 (2000).[29] Y.-J. Lin, R. L. Compton, A. R. Perry, W. D. Phillips,\nJ. V. Porto, and I. B. Spielman, Phys. Rev. Lett. 102\n130401 (2009).\n[30] P. Wang, Z-Q. Yu, Z. Fu, J. Miao, L. Huang, S. Chai, H.\nZhai and J. Zhang Phys. Rev. Lett. 109095301 (2012).\n[31] L. W. Cheuk, A. T. Sommer, Z. Hadzibabic, T. Yefsah,\nW. S. Bakr and M. W. Zwierlein Phys. Rev. Lett. 109\n095302 (2012).\n[32] F. Gerbier, A. Widera, S. F¨ olling, O. Mandel and I.\nBloch, Phys. Rev. A 73041602 (2006).\n[33] E. M.Boojkans, A.VinitandC.Raman, Phys.Rev.Lett.\n107195306 (2011).\n[34] In the presence of more exotic Rashba type spin-orbit\ncoupling, the spin density can display spiral order,\nwhichspontaneouslybreakstime-reversalsymmetry.(See\nRefs. [5, 7]) Here time-reveral symmetry is explicitly bro-\nken by the Raman coupling term (see Eq. 2)\n[35] G. I. Martone, Y. Li and S. Stringari, Phys. Rev. A 90\n041604 (R) (2014).\n[36] J. Guzman, G.-B. Jo, A. N. Wenz, K. W. Murch, C.\nK. Thomas and D. M. Stamper-Kurn, Phys. Rev. A 84\n063625 (2011).\n[37] C. D. Hamley, C. S. Gerving, T. M. Hoang, E. M. Book-\njans and M. S. Chapman, Nature Physics 8305 (2012).\n[38] I.CarusottoandE.J.Mueller, J.Phys.B 37S115(2004).\n[39] B. M. Fregoso and E. Fradkin, Phys. Rev. Lett. 103\n205301 (2009).\n[40] Y. Li, G. I. Martone, L. P. Pitaevskii and S. Stringari,\nPhys. Rev. Lett. 110235302 (2013).\n[41] C.-M Jian and H. Zhai, Phys. Rev. B 84060508 (R)\n(2011).\n[42] J. Radi´ c, S. S. Natu and V. Galitski, Phys. Rev. Lett.\n113185302 (2014).\n[43] S. S. Natu and E. J. Mueller, Phys. Rev. A 84053625\n(2011).\n[44] C. Hickey and A. Paramekanti, eprint arXiv:1409.1216."    },    {        "title": "1611.08424v2.Spin_orbit_Hamiltonian_for_organic_crystals_from_first_principles_electronic_structure_and_Wannier_functions.pdf",        "content": "Spin-orbit Hamiltonian for organic crystals from \frst principles electronic structure\nand Wannier functions\nSubhayan Roychoudhury and Stefano Sanvito\nSchool of Physics, AMBER and CRANN Institute, Trinity College Dublin, Dublin 2, Ireland\nSpin-orbit coupling in organic crystals is responsible for many spin-relaxation phenomena, going\nfrom spin di\u000busion to intersystem crossing. With the goal of constructing e\u000bective spin-orbit Hamil-\ntonians to be used in multiscale approaches to the thermodynamical properties of organic crystals,\nwe present a method that combines density functional theory with the construction of Wannier func-\ntions. In particular we show that the spin-orbit Hamiltonian constructed over maximally localised\nWannier functions can be computed by direct evaluation of the spin-orbit matrix elements over\nthe Wannier functions constructed in absence of spin-orbit interaction. This eliminates the prob-\nlem of computing the Wannier functions for almost degenerate bands, a problem always present\nwith the spin-orbit-split bands of organic crystals. Examples of the method are presented for iso-\nlated organic molecules, for mono-dimensional chains of Pb and C atoms and for triarylamine-based\none-dimansional single crystals.\nI. INTRODUCTION\nSpintronics devices operate by detecting the spin of\na carrier in the same way as a regular electronic device\nmeasures its electrical charge1. These devices are already\nthe state of the art in the design of magnetic sensors such\nas the magnetic read-head of hard-disk drives2, but also\nhave excellent prospect as logic gate elements3{6. Logic\ncircuits using the spin degree of freedom may o\u000ber low\nenergy consumption and high speed owing to the fact\nthat the dynamics of spins takes place at a much smaller\nenergy scale than that of the charge1,3.\nRecent years have also witnessed a marked increase\nin interest into investigations of organic molecules and\nmolecular crystals as materials platform, initially for elec-\ntronics7,8and lately also for spintronics9{11. The main\nreason behind such interest is that organic crystals, com-\ning in a wide chemical variety, are typically much more\n\rexible than their inorganic counterparts and they can\nexhibit an ample range of electronic properties, which are\nhighly tuneable in practice. For example, it is possible to\nchange the conductivity of organic polymers over \ffteen\norders of magnitude12. In addition to such extreme spec-\ntrum of physical/chemical properties organic materials\nare usually processed at low temperature. This is an ad-\nvantage over inorganic compounds, which translates into\na drastic reduction of the typical manufacturing and in-\nfrastructure costs13. Finally, speci\fc to spintronics is the\nfact that both the spin-orbit (SO) and hyper\fne interac-\ntion are very weak14in organic compounds, resulting in a\nweak spin scattering during the electron transport15{17.\nRegardless of the type of media used, either organic\nor inorganic, spintronics always concerns phenomena re-\nlated to the injection, manipulation and detection of\nspins into a solid state environment11. In the prototyp-\nical spintronic device, the spin-valve18, a non-magnetic\nspacer is sandwiched between two ferromagnents. Spins,\nwhich are initially aligned along the magnetization vector\nof the \frst ferromagnet, travel to the other ferromagnent\nthrough the spacer, and the resistance of the entire devicedepends on the relative orientation of the magnetization\nvectors of the two magnets. However, if the spin direc-\ntion is lost across the spacer, the resistance will become\nindependent of the magnetic con\fguration of the device.\nAs such, in order to measure any spin-dependent e\u000bect\none has to ensure that the charge carriers maintain their\nspin direction through the spacer. Notably, this require-\nment is not only demanded by spin-valves, but also by\nany devices based on spins. There are several mechanism\nfor spin-relaxation in the solid state19.\nIn an organic semiconductor (OSC) the unwanted spin-\nrelaxation can be caused by the presence of paramagnetic\nimpurities, by SO coupling and by hyper\fne interaction.\nIn general paramagnetic impurities can be controlled to\na very high degree of precision and they can be almost\ncompletely eliminated from an OSC during the chemi-\ncal synthesis20. The hyper\fne interaction instead can be\nusually considered small. This is because there are only a\nfew elements typically present in organic molecules with\nabundant isotopes baring nuclear spins. The most obvi-\nous exception is hydrogen. However, most of the OSC\ncrystals are \u0019-conjugated and the \u0019-states, responsible\nfor the extremal energy levels, and hence for the elec-\ntron transport, are usually delocalized. This means that\nthe overlap of the wave function over the H nuclei has\nto be considered small. Finally, also the SO coupling is\nweak owing to the fact that most of the atoms composing\norganic compounds are light.\nAs such, since all the non-spin-conserving interactions\nare weak in OSCs, it is not surprising that there is con-\ntradictory evidence concerning the interaction mostly re-\nsponsible for spin-di\u000busion in organic crystals. Con\rict-\ning experimental evidence exists supporting either the SO\ncoupling21,22or the hyper\fne interaction23,24, indicating\nthat the dominant mechanism may depend on the speci\fc\nmaterial under investigation. For this reason it is impor-\ntant to develop methods for determining the strength of\nboth the SO and the hyper\fne coupling in real materi-\nals. These can eventually be the basis for constructing\ne\u000bective Hamiltonians to be used for the evaluation ofarXiv:1611.08424v2  [cond-mat.mtrl-sci]  9 Feb 20172\nthe relevant thermodynamics quantities (e.g. the spin\ndi\u000busion length). Here we present one of such methods\nfor the case of the SO interaction.\nThe SO interaction is a relativistic e\u000bect arising from\nthe electron motion in the nuclear potential. In the elec-\ntron reference frame the nucleus moves and creates a\nmagnetic \feld, which in turn interacts with the electron\nspin. This is the spin-orbit coupling25. Since the SO\ninteraction allows the spin of an electron to change di-\nrection during the electron motion, it is an interaction re-\nsponsible for spin relaxation. In fact, there exist several\nSO-based microscopic theories of spin relaxation in solid\nstate systems19. In the case of inorganic semiconductors\nthese usually require knowledge of the band-structure of\nthe material, some information about its mobility and\nan estimate of the spin-orbit strength. In the case of\nOSCs the situation, however, is more complex, mostly\nbecause the transport mechanism is more di\u000ecult to de-\nscribe. Firstly, the band picture holds true only for a\nfew cases, while for many others one has to consider\nthe material as an ensemble of weakly coupled molecules\nwith a broad distribution of hopping integrals26. Sec-\nondly, the typical phonon energies are of the same order\nof magnitude of the electronic band width, indicating\nthat electron-phonon scattering cannot be treated as a\nperturbation of the band structure. For all these rea-\nsons the description of the thermodynamical properties\nof OSCs requires the construction of a multi-scale the-\nory, where the elementary electronic structure is mapped\nonto an e\u000bective Hamiltonian retaining only a handful of\nthe original degrees of freedom27. A rigorous and now\nstandard method for constructing such e\u000bective Hamil-\ntonian consists in calculating the band structure over a\nset of Wannier functions28,29. These can be constructed\nin a very general way as the Fourier transform of a linear\ncombination of Bloch states, where the linear combina-\ntion is taken so to minimize the spatial extension of the\nWannier functions. These are the so-called maximally\nlocalized Wannier fuctions (MLWFs)30,31.\nThe MLWF method performs best for well-isolated\nbands. This is indeed the case of OSCs, where often\nthe valence and conduction bands originate respectively\nfrom the highest occupied molecular orbital (HOMO)\nand the lowest unoccupied molecular orbital (LUMO) of\nthe gas-phase molecule. In fact, when the MLWF proce-\ndure is applied to such band structure one obtains Wan-\nnier orbitals almost identical to the molecule HOMO and\nLUMO27. Spin-orbit interaction, however, splits such\nwell-de\fned bands, and in OSCs the split is typically a\nfew tenths of \u0016eV. Thus, in this case, one has to apply the\nMLWF procedure to bands, which are indistinguishable\nat an energy scale larger then a few \u0016eV. In such con-\nditions the minimization becomes almost impossible to\nconverge, the MLWFs cannot be calculated for SO-split\nbands and an alternative scheme must be implemented.\nHere we describe a method for obtaining the SO ma-\ntrix elements with respect to the Wannier functions cal-\nculated in the absence of the SO interaction. Since theSO coupling in OSCs is weak, such spin-independent\nWannier functions represent a close approximation of\nthose that one could, at least in principle, obtain in the\npresence of the SO interaction. Furthermore, when the\nMLWF basis spans the same Hilbert space de\fned by all\nthe atomic orbitals relevant for describing a given bands\nmanifold, our method provides an accurate description\nof the system even in the case of heavy elements, i.e. for\nstrong spin-orbit interaction. In particular we implement\nour scheme together with the atomic-orbital, pseudopo-\ntential, density functional theory (DFT) code Siesta32.\nSiesta is used to generate the band structure in absence\nof the spin-orbit interaction and for calculating the SO\npotential, while the MLWF procedure is performed with\ntheWannier90 code33.\nThe paper is organized as follows. In the next section\nwe describe our method in detail, by starting from the\ngeneral idea and then going into the speci\fc numerical\nimplementation. A how-to work\row will also be pre-\nsented. Next we discuss results obtained for rather di-\nverse physical systems. Firstly, we evaluate the SO-split\nenergy eigenvalues of a plumbane molecule and show how\naccurately these match those obtained directly from DFT\nincluding SO interaction. Then, we apply our procedure\nto the calculation of the band structure of a chain of Pb\natoms, before moving to materials composed of light ele-\nments with low SO coupling. Here we will show that our\nmethod performs well for chains made of carbon atoms\nand of methane molecules. Finally we obtain the SO ma-\ntrix elements for the Wannier functions derived from the\nHOMO band of a triarylamine-based nanowire, a rela-\ntively well-known semiconducting material with potential\napplications in photo-voltaic34and spintronics.\nII. METHOD\nA. General idea\nHere we describe the idea behind our method, which is\ngeneral and does not depend on the speci\fc implementa-\ntion used for calculating the band structure. Consider a\nset ofN0isolated Bloch states, j mki, describing an in-\n\fnite lattice. These for instance can be the DFT Kohn-\nSham eigenstates of a crystal. One can then obtain the\nassociatedN0Wannier functions from the de\fnition,\njwnRi=V\n(2\u0019)3Z\nBZ2\n4N0X\nm=1Uk\nmnj mki3\n5e\u0000ik\u0001Rdk;(1)\nwherejwnRiis then-th Wannier vector centred at the\nlattice site R,Vis the volume of the primitive cell and\nthe integration is performed over the \frst Brillouin zone\n(BZ). In Eq. (1) Ukis a unitary operator that mixes the\nBloch states and hence de\fnes the speci\fc set of Wan-\nnier functions. A particularly convenient gauge choice for\nUkconsists in minimizing the Wannier functions spread,3\nwhich writes\n\n =X\nn\u0002\nhwn0jr2jwn0i\u0000jhwn0jrjwn0ij2\u0003\n:(2)\nSuch choice de\fnes the so-called maximally localized\nWannier functions (MLWFs).\nIn the absence of SO coupling a Wannier function of\nspins1is composed exclusively of Bloch states with the\nsame spin, s1. By moving from a continuos to a dis-\ncretek-point representation the spin-polarized version of\nEq. (1) becomes31\njws1\nnRi=1\nNX\nkX\nmUs1\nmn(k)j s1\nmkie\u0000ik\u0001R: (3)\nNote that this represents either a \fnite periodic lat-\ntice comprising Nunit cells or a sampling of Nuni-\nformly distributed k-points in the Brillouin zone of an\nin\fnite lattice. Here the Bloch states, which are nor-\nmalized within each unit cell according to the relation\nh s1\nmkj s2\nnk0i=N\u000em;n\u000ek;k0\u000es1;s2, obey to the condition\n pk(r1) = pk(rN+1), where pk(rm) denotes the Bloch\nfunction for the p-th band at the wavevector kand posi-\ntionrm.\nThe projection of a generic Bloch state onto a MLWF\nin the absence of SO coupling can be written as\nh s1\nqk0jws2\nnR2i=\n=1\nNX\nkX\nmUs2\nmn(k)h s1\nqk0j s2\nmkie\u0000ik\u0001R=\n=1\nNX\nkX\nmUs2\nmn(k)e\u0000ik:RN\u000eq;m\u000ek;k0\u000es1;s2=\n=Us2\nqn(k0)e\u0000ik0\u0001R\u000es1;s2:(4)\nHence a generic SO matrix element can be expanded over\nthe MLWF basis set as\nhws1\nmR1jVSOjws2\nnR2i=\n=1\nN2X\np;qX\nk1;k2hws1\nmR1j s1\npk1i(VSO)s1;s2\npk1;qk2h s2\nqk2jws2\nnR2i=\n=1\nN2X\np;qX\nk1;k2U\u0003(s1)\npm(k1)eik1\u0001R1(VSO)s1;s2\npk1;qk2\u0001\n\u0001Us2\nqn(k2)e\u0000ik2\u0001R2;\n(5)\nwhere\n(VSO)s1;s2\npk1;qk2=h s1\npk1jVSOj s2\nqk2i: (6)\nIt must be noted that in the absence of SO coupling,\nthe Bloch states are spin-degenerate, i.e. there are two\nstates corresponding to each spatial wave-function, one\nwith spin up,j \"(r)i=j (r)i\nj\"i , and one with spin\ndown,j #(r)i=j (r)i\nj#i . The same is true for the\nWannier functions, i.e. one has always the pair jw\"(r)i=jw(r)i\nj\"i ,jw#(r)i=jw(r)i\nj#i . In the presence of SO\ncoupling, spin mixing occurs and each Bloch and Wannier\nstate is, in general, a linear combination of both spin\nvectors. Since the Bloch states (or the Wannier ones)\nobtained in the absence of SO coupling form a complete\nbasis set in the Hilbert space, the SO coupling operator\ncan be written over such basis provided that one takes\nboth spins into account. Therefore we use such spin-\ndegenerate states as our basis for all calculations.\nB. Numerical Implementation\nThe derivation leading to Eq. (5) is general and the\n\fnal result is simply a matrix transformation of the SO\noperator from the basis of the Bloch states to that of\nWannier ones. Note that both basis sets are those calcu-\nlated in the absence of SO coupling, i.e. we have assumed\nthat the spatial part of the basis function is not modi\fed\nby the introduction of the SO interaction. For practical\npurposes we now we wish to re-write Eq. (5) in terms\nof a localized atomic-orbital basis set, i.e. we wish to\nmake our method applicable to \frst-principles DFT cal-\nculations implemented over local orbitals. In particular\nall the calculations that will follow use the Siesta pack-\nage, which expands the wave-function and all the opera-\ntors over a numerical atomic-orbital basis sets, fj\u001es\n\u0016;Rjig,\nwherej\u001es\n\u0016;Rjidenotes the \u0016-th atomic orbital ( \u0016is a\ncollective label for the principal and angular momentum\nquantum numbers) with spin sbelonging to the cell at the\nposition Rj.Siesta uses relativistic pseudopotentials to\ngenerate the spin-orbit matrix elements with respect to\nthe basis vectors and truncates the range of the SO inter-\naction to the on-site terms35. For a \fnite periodic lattice\ncomprising Nunit cells, a Bloch state is written with\nrespect to atomic orbitals as\nj pki=NX\nj=1eik\u0001Rj X\n\u0016C\u0016p(k)j\u001e\u0016;Rji!\n; (7)\nwhere the coe\u000ecients C\u0016p(k) are in general C-numbers.\nThis state is normalized over unit cell with the allowed k-\nvalues beingm\nNK, where Kis the reciprocal lattice vector\nandmis an integer.\nHence the SO matrix elements written with respect to\nthe spin-degenerate Bloch states calculated in absence of\nSO interaction are\nh s1\npk1jVSOj s2\nqk2i=X\nj;lei(k2\u0001Rl\u0000k1\u0001Rj)\u0001\n\u0001X\n\u0016;\u0017C\u0003s1\n\u0016p(k1)Cs2\n\u0017q(k2)h\u001es1\n\u0016;RjjVSOj\u001es2\n\u0017;Rli:(8)\nAs mentioned above Siesta neglects all the SO ma-\ntrix elements between atomic orbitals located at di\u000berent\natoms. This leads to the approximation\nh\u001es1\n\u0016;RjjVSOj\u001es2\n\u0017;Rli=h\u001es1\n\u0016jVSOj\u001es2\n\u0017i\u000eRj;Rl;(9)4\nso that Eq. (8) becomes\nh s1\npk1jVSOj s2\nqk2i=X\njei(k2\u0000k1)\u0001Rj\u0001\n\u0001X\n\u0016;\u0017C\u0003(s1)\n\u0016p(k1)C(s2)\n\u0017q(k2)h\u001es1\n\u0016jVSOj\u001es2\n\u0017i:(10)\nThis can be further simpli\fed by taking into account the\nrelation\nNX\nj=1ei(k1\u0000k2)\u0001Rj=N\u000ek1;k2; (11)\nwhich leads to the \fnal expression for the SO matrix\nelements\nh s1\npk1jVSOj s2\nqk2i=\n=NX\n\u0016;\u0017C\u0003(s1)\n\u0016p(k1)C(s2)\n\u0017q(k1)h\u001es1\n\u0016jVSOj\u001es2\n\u0017i\u000ek1;k2:\n(12)\nWith the result of Eq. (12) at hand we can now come\nback to the expression for the SO matrix elements written\nover the MLWFs computed in absence of spin-orbit [see\nEq. (5)]. In the case of the Siesta basis set this now\nreads\nhws1\nmR1jVSOjws2\nnR2i=\n=1\nNX\np;q;\u0016;\u0017X\nkC\u0003s1\n\u0016p(k)Cs2\n\u0017q(k)U\u0003(s1)\npm(k)Us2\nqn(k)\u0001\n\u0001eik\u0001(R1\u0000R2)h\u001es1\n\u0016jVSOj\u001es2\n\u0017i:(13)\nFinally, we go back to the continuous representation\n(N!1 ), where the sum over kis replaced by an inte-\ngral over the \frst Brillouin zone\nhws1\nmR1jVSOjws2\nnR2i=\n=V\n(2\u0019)3X\np;q;\u0016;\u0017Z\nBZC\u0003s1\n\u0016p(k)Cs2\n\u0017q(k)U\u0003s1\npm(k)Us2\nqn(k)\u0001\n\u0001eik\u0001(R1\u0000R2)h\u001es1\n\u0016jVSOj\u001es2\n\u0017idk:(14)\nTo summarize, our strategy consists in simply evaluat-\ning the SO matrix elements over the basis set of the ML-\nWFs constructed in the absence of SO interaction. These\nare by de\fnition spin-degenerate and they are in gen-\neral easy to compute since associated to well-separated\nbands. Our procedure thus avoids to run the minimiza-\ntion algorithm necessary to \fx the Wannier's gauge over\nthe SO-split bands, which in the case of OSCs have tiny\nsplits. Our method is exact in the case the MLWFs form\na complete set describing a particular bands manifold. In\nother circumstances they constitute a good approxima-\ntion, as long as the SO interaction is weak, namely when\nit does not change signi\fcantly the spatial shape of the\nWannier functions. However, for a material with strongSO coupling (eg. Pb), if the MLWFs under considera-\ntion do not span the entire Bloch states manifold, then\nthe SO-split eigenvalues calculated with our method will\nnot match those obtained directly with the \frst principles\ncalculation.\nC. Work\row\nThe following procedure is adopted when calculating\nthe SO-split band structures from the MLWFs Hamilto-\nnian. The results are then compared to the band struc-\nture obtained directly from Siesta including SO interac-\ntion.\n1. We \frst run a self-consistent non-collinear spin-\nDFTSiesta calculation and obtain the band struc-\nture.\n2. From the density matrix obtained at step (1), we\nrun a non self-consistent single-step Siesta calcu-\nlation including SO coupling. This gives us the ma-\ntrix elementsh\u001es1\u0016jVSOj\u001es2\u0017i. The band structure\nobtained in this calculation (from now on this is\ncalled the SO-DFT band structure) will be then\ncompared with that obtained over the MLWFs.\nNote that we do not perform the Siesta DFT cal-\nculation including spin-orbit interaction in a self-\nconsistent way. This is because the SO interaction\nchanges little the density matrix so that such cal-\nculation is often not necessary. Furthermore, as\nwe cannot run the MLWF calculation in a self-\nconsistent way over the SO interaction, consider-\ning non-self-consistent SO band structure at the\nSiesta level allows us to compare electronic struc-\ntures arising from identical charge densities.\n3. Since the current version of Wannier90 imple-\nmented for Siesta works only with collinear spins,\nwe run a regular self-consistent spin-polarized\nSiesta calculation. This gives us the coe\u000e-\ncientsCs\n\u0016n(k), which are spin-degenerate for a non-\nmagnetic material, C\"\n\u0016n(k) =C#\n\u0016n(k).\n4. We run a Wannier90 calculation to construct the\nMLWFs associated to the band structure com-\nputed at point (3). This returns us the uni-\ntary matrix, Us\npm(k), the Hamiltonian matrix el-\nementshws1\nmR1jH0jws2\nnR2i(H0is the Kohn-Sham\nHamiltonian in absence of SO interaction) and the\nphase factors36eik\u0001R. For a non-magnetic mate-\nrial the matrix elements of H0satisfy the relation\nhws1\nmR1jH0jws2\nnR2i=hwmR1jH0jwnR2i\u000es1;s2.\n5. Fromh\u001es1\u0016jVSOj\u001es2\u0017iand theCs\n\u0016n(k)'s we calcu-\nlate the matrix elements h s1\npkjVSOj s2\nqkiby using\nEq. (12).\n6. Next we transform the SO matrix ele-\nments constructed over the Bloch functions,5\nFIG. 1. (Color on line) Atomic structure of (a) a plumbane\nmolecule, (b) a chain of lead atoms and (c) a chain of methane\nmolecules. We have also calculated the electronic structure\nof a chain of C atoms, which is essentially identical to that\npresented in (b). Color code: Pb = grey, H = light blue, C\n= yellow.\nh s1\npkjVSOj s2\nqki, into their Wannier counterparts,\nhws1\nmR1jVSOjws2\nnR2i, by using Eq. (14).\n7. The \fnal complete Wannier Hamiltonian now reads\nhws1\nmR1jHjws2\nnR2i=hws1\nmR1jH0+VSOjws2\nnR2i; (15)\nand the associated band structure can be directly\ncompared with that computed at point (2) directly\nfromSiesta .\nIII. RESULTS AND DISCUSSION\nWe now present our results, which are discussed in the\nlight of the theory just described.\nA. Plumbane Molecule\nWe start our analysis by calculating the SO matrix\nelements and then the energy eigenvalues of a plumbane,\nPbH 4, molecule [see \fgure 1(a)]. Due to the presence of\nlead, the molecular eigenstates change signi\fcantly when\nthe SO interaction is switched on. For this non-periodic\nsystem the key relations in Eq. (12) and Eq. (5) reduce\nto\nh s1\npjVSOj s2\nqi=X\n\u0016;\u0017C\u0003s1\n\u0016pCs2\n\u0017qh\u001es1\n\u0016jVSOj\u001es2\n\u0017i(16)\nand\nhws1\nmjVSOjws2\nni=X\np;qU\u0003s1\npmUs2\nqnh s1\npjVSOj s2\nqi;\n(17)\nrespectively, where now the vectors  s\nnare simply the\neigenvectors with quantum number nand spins.\nIn Table I we report the \frst 10 energy eigenvalues of\nplumbane, calculated either with or without SO coupling.NonSO SO\nSiesta MLWF Siesta MLWF\n-33.93534 -33.93521 -33.93532 -33.93521\n-33.93530 -33.93521 -33.93528 -33.93521\n-13.02511 -13.02507 -14.69573 -14.69568\n-13.02511 -13.02507 -14.69573 -14.69568\n-13.02510 -13.02506 -12.64301 -12.64298\n-13.02509 -13.02506 -12.64301 -12.64298\n-13.02320 -13.02315 -12.64166 -12.64162\n-13.02318 -13.02315 -12.64165 -12.64162\n-5.75256 -5.75251 -5.75255 -5.75251\n-5.75245 -5.75251 -5.75245 -5.75251\nMRAD =4:320\u000210\u00006MRAD =3:998\u000210\u00006\nTABLE I. The 10 lowest energy eigenvalues of a plumbane\nmolecule calculated with (SO) and without (NonSO) spin-\norbit interaction. The \frst and third columns correspond to\nthe SO-DFT Siesta calculation, while the second and the\nfourth to the MLWFs one. The MRAD for both cases is\nreported in the last row.\nThese have been computed within the LDA (local den-\nsity approximation) and a double-zeta polarized basis set.\nThe table compares results obtained with our MLWFs\nprocedure to those computed with SO-DFT by Siesta .\nClearly in this case of a heavy ion the SO coupling\nchanges the eigenvalues appreciably, in particular in the\nspectral region around -13 eV. Such change is well cap-\ntured by our Wannier calculation, which returns energy\nlevels in close proximity to those computed with SO-DFT\nbySiesta . In order to estimate the error introduced by\nour method, we calculate the Mean Relative Absolute Dif-\nference (MRAD) , which we de\fne as1\nNPj\u000fs\ni\u0000\u000fw\nij\nj\u000fs\nijfor a\nset ofNeigenvalues ( i= 1;:::;N ), where\u000fs\niand\u000fw\niare\nthei-th eigenvalues calculated from Siesta and the ML-\nWFs, respectively. Notably the MRAD is rather small\nboth in the SO-free case and when the SO interaction\nis included. Most importantly, we can report that our\nprocedure to evaluate the SO matrix elements over the\nMLWFs basis clearly does not introduce any additional\nerror.\nBefore discussing some of the properties of the SO ma-\ntrix elements associated to this particular case of a \fnite\nmolecule, we wish to make a quick remark on the Wan-\nnier procedure adopted here. The eigenvalues reported\nin Table I are the ten with the lowest energies. However,\nin order to construct the MLWFs we have considered all\nthe states of the calculated Kohn-Sham spectrum. This\nmeans that, if our Siesta basis set describes PbH 4with\nNdistinct atomic orbitals, then the MLWFs constructed\nare 2N(the factor 2 accounts for the spin degeneracy).\nIn this case the original local orbital basis set and the\nconstructed MLWFs span the same Hilbert space and\nthe mapping is exact, whether or not the SO interaction\nis considered.\nIn most cases, however, one wants to construct the\nMLWFs by using only a subset of the spectrum, for in-\nstance the \frst N0eigenstates. Since in general the SO6\ninteraction mixes all states, there will be SO matrix ele-\nments between the selected N0states and the remaining\nN\u0000N0. This means that a MLWF basis constructed\nonly from the \frst N0eigenstates will not be able to pro-\nvide an accurate description of the SO-split spectrum.\nImportantly, one in general may expect that the SO in-\nteraction matrix elements between di\u000berent Kohn-Sham\norbitals,h s1pjVSOj s2qi, are smaller than those calcu-\nlated at the same orbital, h s1njVSOj s2ni. This is be-\ncause of the short-range of the SO interaction and the\nfact that the Kohn-Sham eigenstates are orthonormal. In\nthe case of light elements, i.e. for a weak SO potential,\none may completely neglect the o\u000b-diagonal SO matrix\nelements. This means that the SO spectrum constructed\nwith the MLWFs associated to the \frst N0eigenstates\nwill be approximately equal to the \frst N0eigenvalues\nof the MLWFs Hamiltonian constructed over the entire\nN-dimensional spectrum. Such property is particularly\nrelevant for OSCs, for which the SO interaction is weak.\nWe now move to discuss a general property of\nthe MLWF SO matrix elements, namely the relations\nhws\nmjVSOjws\nmi= 0 and<[hws\nmjVSOjws\nni] = 0. This\nmeans that the SO matrix elements for the same spin\nand the same Wannier function vanish, while those for\nthe same spin and di\u000berent Wannier functions are purely\nimaginary. This property can be understood from the\nfollowing argument. The SO coupling operator is VSO=P\nRjVRjLRj\u0001S, whereVRjis a scalar potential indepen-\ndent of spin, and LRjis the angular momentum operator\ncorresponding to the central potential of the atom at po-\nsition Rj. Here Sis the spin operator and the sum runs\nover all the atoms. By now expanding Sin terms of\nthe Pauli spin matrices one can see that for any vector\nj\rs\nii=j\rii\njsi, which can be written as a tensor prod-\nuct of a spin-independent part, j\rii, and a spinorjsi, the\nfollowing equality holds\nh\rs1\nmjL\u0001Sj\rs2\nni=1\n2h\nh\rmj^Lzj\rni\u000es1\"\u000es2\"+\n+h\rmj^L\u0000j\rni\u000es1\"\u000es2#+h\rmj^L+j\rni\u000es1#\u000es2\"+\n+h\rmj\u0000^Lzj\rni\u000es1#\u000es2#i\n:(18)\nEq. (18) can then be applied to both the Kohn-Sham\neigenstates and the MLWFs, since they are both written\nasj\rs\nii=j\rii\njsi.\nNow, the atomic orbitals used by Siesta have the fol-\nlowing form\nj\u001eii=jRni;lii\njli;Mii;(19)\nwherejRn;liis a radial numerical function, while the an-\ngular dependence is described by the real spherial har-\nmonic,37jl;Mi. It can be proved that the real spherical\nharmonics follow the relation\nhl;Mij^Lzjl;Mji=\u0000iMi\u000eMi;Mj:(20)\nSince any Kohn-Sham eigenstate, j s1pi, can be written\nasj\u001eii\njs1i, Eq. (18) implies that only the terms in\nΓπ/ a-25-20-15-10-505E-EF (eV)NonSO\nSO\nΓ\nπ/ a(a)(b) σ\nσ∗πFIG. 2. (Color on line) Bandstructure of a 1D Pb chain cal-\nculated (a) with Siesta and (b) by diagonalizing the Hamil-\ntonian matrix constructed over the MLWFs. Black and red\nlines are for the bands obtained without and with SO cou-\npling, respectively. The \u001b,\u001b\u0003and\u0019bands are identi\fed in\nthe picture.\n^Lz(or\u0000^Lz) contribute to the matrix element between\nsame spins,h s1pjL\u0001Sj s1pi. Eq. (20) together with the\nfact that the Kohn-Sham eigenstates are real for a \fnite\nmolecule further establishes that <[h pj^Lzj qi] = 0. As\na consequenceh mj^Lzj mi= 0. Finally, by keeping\nin mind that the unitary matrix elements transforming\nthe Kohn-Sham eigenstates into MLWFs are real for a\nmolecule, we have also\nhws1\nmjL\u0001Sjws1\nni=\u0006hwmj^Lzjwni=\n=X\np6=qUpmUqnh pj^Lzj qi;(21)\nwhich has to be imaginary. Thus we have\n<hws1mjVSOjws1ni= 0 andhws1mjVSOjws1mi= 0 since\nVSOmust have real expectation values.\nB. Lead Chain\nNext we move to calculating the SO matrix elements\nfor a periodic structure. In particular we look at a 1D\nchain of Pb atoms with a unit cell length of 2.55 \u0017A, which\nis the DFT equilibrium lattice constant obtained with\nthe LDA. Note that free-standing mono-dimensional Pb\nchains have been never reported in literature, although7\nthere are studies of low-dimensional Pb structures encap-\nsulated into zeolites38. Here, however, we do not seek at\ndescribing a real compound, but we rather take the 1D\nPb mono-atomic chain as a test-bench structure to apply\nour method to a periodic structure with a large SO cou-\npling. Also in this case we have constructed the MLWFs\nby taking the entire bands manifold and not a subset\nof it. For the DFT calculations we have considered a\nsimplesandpsingle-zeta basis set, which, in absence of\nSO interaction yields three bands with one of them being\ndoubly degenerate [see Fig. 2(a)]. The doubly-degenerate\nrelatively-\rat band just cuts across the Fermi energy, EF,\nand it is composed of the pyandpzorbitals orthogonal\nto the chain axis ( \u0019band). The other two bands are sp\nhybrid (\u001bbands). The lowest one at about 25 eV be-\nlowEFhas mainly scharacter (\u001bband), while the other\nmainlypx(\u001b\u0003band).\nSpin-orbit coupling lifts the degeneracy of the p-type\nband manifold, which is now composed of three distinct\nbands. In particular the degeneracy is lifted only in the\n\u0019band at the edge of the 1D Brillouin zone, while it\nalso involves the \u001bone close to the \u0000 point (after the\nband crossing). When the same band structure is calcu-\nlated from the MLWFs we obtain the plot of Fig. 2(b).\nThis is almost identical to that calculated with SO-DFT\ndemonstrating the accuracy of our method also for peri-\nodic system.\nIt must be noted that for a periodic structure the\nBloch state expansion coe\u000ecients, C\u0016p(k), and the el-\nements of the unitary matrix Uare complex and con-\nsequently the diagonal elements of VSOwith respect to\nWannier functions are not zero in general. However, as\nexpectedhws1\nmRjVSOjws2\nnR0itends to vanish as the sepa-\nrationjR\u0000R0jincreases. Furthermore, it is clear from\nEq. (18) that the SO matrix elements for Wannier func-\ntions should obey the spin-box anti-hermitian relation\nhws1\nmRjVSOjws2\nnR0i=\u0000hws2\nmRjVSOjws1\nnR0i\u0003:(22)\nThese two properties can be appreciated in Fig. 3, where\nwe plot the real [panel (a)] and imaginary [panel (b)]\npart ofhws1\nm0jVSOjws2\nnRifor some representative band\ncombinations, mandn, as a function of R.\nC. Carbon Chain\nNext we look at the case of a 1D mono-atomic car-\nbon chain with a LDA-relaxed interatomic distance of\ns1:3\u0017A. This has the same structure and electron count\nof the Pb chain, and the only di\u000berence concerns the fact\nthat the SO coupling in C is much smaller then that in\nPb. In this situation we expect that an accurate SO-split\nband structure can be obtained even when the MLWFs\nare constructed only for a limited number of bands and\nnot for the entire band manifold as in the case of Pb.\nThis time the DFT band structure is calculated at the\nLDA level over a double-zeta polarized (DZP) Siesta\nFIG. 3. (Color on line) The SO matrix elements of a chain\nof lead atoms calculated with respect to some representative\nWannier functions and plotted as a function of the site index,\ni.e. of the distance between the Wannier function. Panels (a)\nand (b) show the real and imaginary components respectively.\nbasis set, comprising 13 atomic orbitals per unit cell. In\ncontrast, the MLWFs are constructed only from the \frst\nfour bands, which are well isolated in energy from the\nrest and again describe the spbands with \u001band\u0019sym-\nmetry. Since the SO interaction in carbon is small (the\nband split is of the order of a few meV) it is impossible\nto visualize the e\u000bects of the SO interaction in a stan-\ndard band plot as that in Fig. 2. Hence, in Fig. 4 we plot\nthe di\u000berence between the band structure calculated in\nthe presence and in the absence of SO coupling. In par-\nticular we compare the bands calculated with SO-DFT\nbySiesta (left-hand side panels in Fig. 4), with those\nobtained with the MLWFs scheme described here (right-\nhand side panels in Fig. 4). In the \fgure we have labelled\nthe bands in order of increasing energy and neglecting the\nspin degeneracy. Thus, for instance, the  1and 2bands\ncorrespond to the two lowest \u001bspin sub-bands (note that\nthe band structure of the linear carbon chain is qualita-\ntively identical to that of the Pb one and we can use\nFig. 2 to identify the various bands).\nWe note that the lowest \u001bbands, de\fned as  1and 2,\ndo not split at all due to the SO interaction, exactly as\nin the case of Pb. This contrasts the behaviour of both\nthe\u0019( 3through 6) and\u001b\u0003( 7and 8) bands, which\ninstead are modi\fed by the SO interaction. Notably the\nchanges in energy of the eigenvalues is never larger then8\n0246ESO-ENSO(meV)ψ1\nψ2\nψ3\nψ4\n7.5\n-5-2.502.55ESO-ENSO(meV)ψ5\nψ6\nΓπ /a-7.5-5-2.50ESO-ENSO(meV) ψ7ψ8\nΓπ /a\nFIG. 4. (Color on line) Di\u000berence, ESO\u0000ENSO, between the\nband structure of chain of carbon atoms calculated with, ESO,\nand without, ESO, considering SO interaction. The bands\nare labelled in increasing energy order without taking into\naccount spin degeneracy. For instance the bands  1and 2\nare the two spin sub-bands corresponding to the \u001bband (see\nFig. 2 for notation). The left-hand side panels show results\nfor the SO-DFT calculations performed with Siesta , while\nthe right-hand side one, those obtained from the MLWFs.\n8 meV and it is perfectly reproduced by our MLWFs\nrepresentation. This demonstrates that truncating the\nbands selected for constructing the MLWFs is a possi-\nble procedure for materials where SO coupling is weak.\nHowever, we should note that the truncation still needs\nto be carefully chosen. Here for instance we have con-\nsidered all the 2 sand 2pbands and neglected those with\neither higher principal quantum number (e.g. 3 sand 3p)\nor higher angular momentum (e.g. bands with dsymme-\ntry originating from the p-polarized Siesta basis), which\nappear at much higher energies. Truncations, where one\nconsiders only a particular orbital of a given shell (say\nthepzorbital in an npshell), need to be carefully as-\nsessed since it is unlikely that a clear energy separation\nbetween the bands takes place.\nD. Methane Chain\nAs a \frst basic prototype of 1D organic molecular\ncrystal we perform calculations for a periodic chain of\nmethane molecules. We use a double-zeta polarized ba-\nsis set and a LDA-relaxed unit cell length of 3.45 \u0017A (the\ncell contains only one molecule). Similarly to the previ-\nous case, the MLWFs are constructed over only the low-\nest 4 bands (8 when considering the spin degeneracy).\nWhen compared to the bands of the carbon chain, those\nof methane are much narrower. This is expected, since\nthe bonding between the di\u000berent molecules is small. In\nFig. 5 we plot the di\u000berence between the eigenvalues\n(1D band structure) calculated with, ESO, and without,\nENSO, including SO interaction.\n00.51ESO-ENSO(meV)ψ1ψ2ψ3ψ4\nΓπ /a-101ESO-ENSO(meV)ψ5ψ6ψ7ψ8\nΓπ /a\nFIG. 5. (Color on line) Di\u000berence, ESO\u0000ENSO, between\nthe band structure of chain of methane molecules calculated\nwith,ESO, and without, ESO, considering SO interaction.\nThe bands are labelled in increasing energy order without\ntaking into account spin degeneracy. The left-hand side pan-\nels show results for the SO-DFT calculations performed with\nSiesta , while the right-hand side one, those obtained from the\nMLWFs. The inset shows an isovalue plot of one of the four\nMLWFs with the red and blue surfaces denoting positive and\nnegative isovalues, respectively. All the MLWFs have simi-\nlar structure and they resemble those of the isolated methane\nmolecule because of the small intermolecular chemical bond-\ning owing to the large separation.\nWhen SO interaction is included the spin-degeneracy is\nbroken and one has now eight bands. These are labeled\nas min Fig. 5 in increasing energy order. Again we\n\fnd no SO split for the lowermost band and then a split,\nwhich is signi\fcantly smaller than that found in the case\nof the C chain. This is likely to originate from the crystal\n\feld of the C atoms in CH 4, which is di\u000berent from that\nin the C chain (the C-C distance is di\u000berent and there are\nadditional C-H bonds). Again, as in the previous case, we\n\fnd that our MLWFs procedure perfectly reproduces the\nSO-DFT band structure, indicating that in this case of\nweak SO interaction band truncation does not introduce\nany signi\fcant error.\nE. Triarylamine Chain\nFinally we perform calculations for a real system,\nnamely for triarylamine-based molecular nanowires.\nThese can be experimentally grown through a photo-\nself-assembly process from the liquid phase39, and have\nbeen subject of numerous experimental and theoretical\nstudies34,40. In general, triarylamines can be used as\nmaterials for organic light emitting diodes, while their\nnanowire form appears to possess good transport and\nspin properties, making it a good platform for organic\nspintronics41. Triarylamine-based molecular nanowires\nself-assemble only when particular radicals are attached9\nFIG. 6. (Colour on line) Structure of the triarylamine\nmolecule (upper picture) and of the triarylamine-based\nnanowire investigated here. The radicals associated to the\ntriarylamine derivative are C 8H17, H and Cl, respectively.\nColour code: C=yellow, H=light blue, O=red, N=grey,\nCl=green.\nto the main triarylamine backbone and here we consider\nthe case of C 8H17, H and Cl radicals, corresponding to\nthe precursor 1of Ref. [39] (see upper panel in Fig. 6).\nThe nanowire then arranges in such a way to have the\ncentral N atoms aligned along the wire axis (see Fig. 6).\nIn general self-assembled triarylamine-based molecular\nnanowires appear slightly p-doped so that charge trans-\nport takes place in the HOMO-derived band. This is well\nisolated from the rest of the valence manifold and has a\nbandwidth of about 100 meV (see \fgure Fig. 7 for the\nband structure). Such band is almost entirely localized\non thepzorbital of the central N atoms ( pzis along the\nwire axis), a feature that has allowed us to construct a\npz-sp2model with the spin-orbit strength extracted from\nthat of an equivalent mono-atomic N chain. The model\nwas then used to calculate the temperature-dependent\nspin-di\u000busion length of such nanowires42. Here we wish\nto use our MLWFs method to extract the SO matrix ele-\nments of triarylamine-based molecular nanowires in their\nown chemical environment, i.e. without approximating\nthe backbone with a N atomic chain.\nFor this system we use a 1D lattice with LDA-\noptimized lattice spacing of 4.8 \u0017A and run the DFT cal-\nculations with double-zeta polarized basis and the LDA\nfunctional. The MLWFs are constructed by using only\nthe HOMO-derived valence band, i.e. we have a single\nspin-degenerate Wannier orbital. We can then drop the\nFIG. 7. (Color on line) Band structure of the 1D triarylamine-\nbased nanowire constructed with the precursor 1of Ref. [39].\nThis is plotted over the 1D Brillouin zone (Z= \u0019=awithathe\nlattice parameter). The Fermi level is marked with a dashed\nblack line and it is placed just above the HOMO-derived va-\nlence band (in red). The lower panel is a magni\fcation of\nthe valence band. Note the bandwidth of about 100 meV\nand the fact that the band has a cosine shape, \fngerprint of\na single-orbital nearest-neighbour tight-binding-like interac-\ntion. Only the HOMO band is considered when constructing\nthe MLWFs.\nband index and write the SO matrix elements as\nhws1\n0jVSOjws2\nRi\n=V\n(2\u0019)3Z\ndkU\u0003(k)U(k)e\u0000ik:Rh s1\nkjVSOj s2\nki\n=V\n(2\u0019)3Z\ndke\u0000ik:Rh s1\nkjVSOj s2\nki;(23)\nor in a discrete representation of the reciprocal space\nhws1\n0jVSOjws2\nRi=1\nNX\nke\u0000ik:Rh s1\nkjVSOj s2\nki(24)\nwhere the second equality comes from the unitarity of\nthe gauge transformation, U(k).\nIn Fig. 8 we plot the di\u000berence between the band struc-\nture computed by including SO interaction and those cal-\nculated without. Notably our MLWFs band structure is\nalmost identical to that computed directly with SO-DFT,\nagain demonstrating both the accuracy of our method\nand the appropriateness of the drastic band truncation\nused here. In this particular case the SO band split is\nmaximized half-way between the \u0000 point and the edge of\nthe 1D Brillouin zone, where it takes a value of approxi-\nmately 80\u0016eV. Clearly such split is orders of magnitude\nsmaller than the value that one can possibly calculate\nby a direct construction of the MLWFs from the SO-\nsplitted band structure. Note also that the SO split of10\nΓπ /a-0.04-0.0200.020.04ESO -ENonSO   (meV)\nDFT\nWANNIER\nFIG. 8. (Color on line) Plot of (E SO\u0000ENSO) as a function of k\nin arbitrary unit over a brillouin zone for the highest occupied\nband of a 1-d chain of triarylamine derivatives. The blue and\nthe red points correspond to calculations with Siesta and\nWannier90 respectively.\nthe valence band is calculated here approximately a fac-\ntor ten smaller than that estimated previously for a N\natomic chain42, indicating the importance of the details\nof the chemical environment in these calculations.\nFinally we take a closer look at the calculated SO ma-\ntrix elements. As mentioned earlier, in the Siesta on-\nsite approximation35only the matrix elements calculated\nover orbitals centred on the same atom do not vanish.\nAs a consequence the components hws1\nRjVSOjws2\nR0idrop\nto zero asjR\u0000R0jgets large. This can be clearly ap-\npreciated in Fig. 9(a) and Fig. 9(b), where we plot the\nSO matrix elements for same and di\u000berent spins, respec-\ntively.\nFrom Fig. 9(a) we can observe that <hws1\n0jVSOjws1\nRi\nvanishes for all R. This can be understood in the\nfollowing way. In general any expectation value of\nVSO,h s\nkjVSOj s\nki, has to be real. This is in\nfact anti-symmetric with respect to k, i.e we have\nh s\n0+kjVSOj s\n0+ki=\u0000h s\n0\u0000kjVSOj s\n0\u0000ki, where k=0\ndenotes the \u0000 point of the Brillouin zone. Addition-\nally,eik\u0001Rsatis\fes the relation ei(0+k)\u0001R=\u0002\nei(0\u0000k)\u0001R\u0003\u0003.\nHence, by performing the k-sum over \frst Brillouin zone\nwe can write\n<hws1\n0jVSOjws1\nRi=<X\nke\u0000ik:Rh s1\nkjVSOj s1\nki= 0;\n(25)\nwherehws1\n0jVSOjws1\n0iis the expectation value of VSO\nand must be real. This implies\nhws1\n0jVSOjws1\n0i= 0:(26)\nWe can also see from Fig. 9(b) that for triarylamine\nthe matrix elements hws1\nRjVSOjws2\nRiare almost zero for\ns16=s2. This follows directly from Eq. (18). In fact in\nthe particular case of triarylamine nanowires the Wannier\nFIG. 9. (Color on line) SO matrix elements of a triarylamine-\nbased nanowire calculated with respect to the Wannier func-\ntions obtained from the HOMO band. Panels (a) and (b)\ncorrespond to matrix elements calculated between for same\nand di\u000berent spins, respectively.\nfunctions are constructed from one band only. As such, in\norder to have a non-zero matrix element, hws1\nRjVSOjws2\nRi,\nwe must have non-zero values for hwRj^L\u0006jwRi. There-\nfore, the band under consideration must contain an\nappreciable mix of components of both the jl;piand\njl;p+ 1icomplex spherical harmonics for some landp.\nAs mentioned earlier, the triarylamine HOMO band is\ncomposed mostly of pzN orbitals. Hence, it has to be\nexpected that the hws1\nRjVSOjws2\nRimatrix elements are\nsmall.\nIV. CONCLUSION\nWe have presented an accurate method for obtaining\nthe SO matrix elements between the MLWFs constructed\nin absence of SO coupling. Our procedure, implemented\nwithin the atomic-orbital-based DFT code Siesta , allows\none to avoid the construction of the Wannier functions\nover the SO-split band structure. In some cases, in par-\nticular for organic crystals, such splits are tiny and a di-\nrect construction is numerically impossible. The method\nis then put to the test for a number of materials systems,\ngoing from isolated molecules, to atomic nanowires, to11\n1D molecular crystals. When the entire band manifold is\nused for constructing the MLWFs the mapping between\nBloch and Wannier orbitals is exact and the method can\nbe used for both light and heavy elements. In contrast\nfor weak spin-orbit interaction one can construct the ML-\nWFs on a subset of the states in the band structures\nwithout any loss of accuracy. As such our scheme ap-\npears as an important tool for constructing e\u000bective spin\nHamiltonians for organic materials to be used as input\nin a multiscale approach to the their thermodynamical\nproperties.ACKOWLEDGEMENT\nThis work is supported by the European Research\nCouncil, Quest project. Computational resources have\nbeen provided by the supercomputer facilities at the Trin-\nity Center for High Performance Computing (TCHPC)\nand at the Irish Center for High End Computing\n(ICHEC). Additionally, the authors would like to thank\nIvan Rungger and Carlo Motta for helpful discussions\nand Akinlolu Akande for providing the structure of the\ntriarylamine-based nanowire.\n1S. A. Wolf, D. D. Awschalom, R. A. Buhrman, J. M.\nDaughton, S. von Moln\u0013 ar, M. L. Roukes, A. Y. Chtchelka-\nnova, and D. M. Treger, Science 294, 1488 (2001).\n2S. Ornes, Proc. Natl. Acad. Sci. USA 110, 3710 (2013).\n3B. Behin-Aein, D. Datta, S. Salahuddin, and S. Datta,\nNature Nanotech. 5, 266 (2010).\n4G. Prinz and K. Hathaway, Phys Today 48, 24 (1995).\n5D. D. Awschalom and M. E. Flatt\u0013 e, Nature Physics 3, 153\n(2007).\n6I.\u0014Zuti\u0013 c and M. Fuhrer, Nature Physics 1, 85 (2005).\n7G. Horowitz, Organic Transistors , edited by H. Klauk\n(Wiley-VCH Verlag GmbH & Co. KGaA, 2006).\n8C. Joachim, J. K. Gimzewski, and A. Aviram, Nature\n408, 541 (2000).\n9V. Dediu, M. Murgia, F. Matacotta, C. Taliani, and\nS. Barbanera, Solid State Commun. 122, 181 (2002).\n10Z. H. Xiong, D. Wu, Z. Valy Vardeny, and J. Shi, Nature\n427, 821 (2004).\n11S. Sanvito, Chem. Soc. Rev. 40, 3336 (2011).\n12C. K. Chiang, C. R. Fincher, Y. W. Park, A. J. Heeger,\nH. Shirakawa, E. J. Louis, S. C. Gau, and A. G. MacDi-\narmid, Phys. Rev. Lett. 39, 1098 (1977).\n13S. R. Forrest, Nature 428, 911 (2004).\n14S. Sanvito and A. R. Rocha, J. Comput. Theor. Nanosci.\n3, 624 (2006).\n15S. Pramanik, C. G. Stefanita, S. Patibandla, S. Bandy-\nopadhyay, K. Garre, H. N., and M. Cahay, Nature Nan-\notech. 2, 216 (2007).\n16K. Tsukagoshi, B. W. Alphenaar, and H. Ago, Nature\n401, 572 (1999).\n17G. Szulczewski, S. Sanvito, and J. M. D. Coey, Nature\nMaterials 8, 693 (2009).\n18B. Dieny, V. Speriosu, B. Gurney, S. Parkin, D. Wilhoit,\nK. Roche, S. Metin, D. Peterson, and S. Nadimi, Journal\nof Magnetism and Magnetic Materials 93, 101 (1991).\n19I.\u0014Zuti\u0013 c, J. Fabian, and S. Das Sarma, Rev. Mod. Phys.\n76, 323 (2004).\n20V. I. Krinichnyi, Synth. Met. 108, 173 (2000).\n21S. Bandyopadhyay, Phys. Rev. B 81, 153202 (2010).\n22A. J. Drew et al. , Nature Materials 8, 109 (2009).\n23F. J. Wang, C. G. Yang, Z. V. Vardeny, and X. G. Li,\nPhys. Rev. B 75, 245324 (2007).\n24V. Dediu, L. E. Hueso, I. Bergenti, A. Riminucci, F. Bor-\ngatti, P. Graziosi, C. Newby, F. Casoli, M. P. De Jong,\nC. Taliani, and Y. Zhan, Phys. Rev. B 78, 115203 (2008).\n25C. Cohen-Tannoudji, B. Diu, and F. Lalo_ e, Quantum me-\nchanics. 2 , Textbook physics (John Wiley & Sons, 1977).26A. Troisi, J. Chem. Phys. 134, 034702 (2011).\n27C. Motta and S. Sanvito, J. Chem. Theo. Comp. 10, 4624\n(2014).\n28G. H. Wannier, Phys. Rev. 52, 191 (1937).\n29G. H. Wannier, Rev. Mod. Phys. 34, 645 (1962).\n30N. Marzari and D. Vanderbilt, Phys. Rev. B 56, 12847\n(1997).\n31N. Marzari, A. A. Mosto\f, J. R. Yates, I. Souza, and\nD. Vanderbilt, Rev. Mod. Phys. 84, 1419 (2012).\n32J. M. Soler, E. Artacho, J. D. Gale, A. Garc\u0013 \u0010a, J. Junquera,\nP. Ordej\u0013 on, and D. S\u0013 anchez-Portal, J. Phys. Condens.\nMatter 14, 2745 (2002).\n33A. A. Mosto\f, J. R. Yates, Y.-S. Lee, I. Souza, D. Vander-\nbilt, and N. Marzari, Comput. Phys. Commun. 178, 685\n(2008).\n34Z. Ning and H. Tian, Chem. Commun. 37, 5483 (2009).\n35L. Fern\u0013 andez-Seivane, M. A. Oliveira, S. Sanvito, and\nJ. Ferrer, J. Phys. Condens. Matter 18, 7999 (2006).\n36The correctness of the elements Us\npm(k) andeik\u0001Ris easily\nveri\fed by ensuring that the following relation is satis\fed\nhwmR1jwnR2i=1\nNX\npZ\nFBZdkhwmR1j pkih pkjwnR2i=\n=1\nNX\npZ\nFBZdkU\u0003\npm(k)Upn(k)eik\u0001(R1\u0000R2)=\n=\u000em;n\u000eR1;R2:\n(27)\n.\n37The real spherical harmonics are constructed from the\ncomplex ones,jl;mi, asjl;Mi=1p\n2[jl;mi+(\u00001)mjl;\u0000mi]\nandjl;\u0000Mi=1\nip\n2[jl;mi\u0000(\u00001)mjl;\u0000mi]. ForM= 0 the\nreal and complex spherical harmonics coincide.\n38S. Romanov, Journal of Physics: Condensed Matter 5,\n1081 (1993).\n39E. Moulin, F. Niess, M. Maaloum, E. Buhler, I. Nyrkova,\nand N. Giuseppone, Angew. Chem. Int. Ed. 49, 6974\n(2010).\n40V. Faramarzi, F. Niess, E. Moulin, M. Maaloum, J.-F.\nDayen, J.-B. Beaufrand, S. Zanettini, B. Doudin, and\nN. Giuseppone, Nature Chemistry 4, 485 (2012).\n41A. Akande, S. Bhattacharya, T. Cathcart, and S. Sanvito,\nJ. Chem. Phys 140, 074301 (2014).\n42S. Bhattacharya, A. Akande, and S. Sanvito, Chem. Com-\nmun. 50, 6626 (2014)."    },    {        "title": "1502.05683v1.Enhanced_photogalvanic_effect_in_graphene_due_to_Rashba_spin_orbit_coupling.pdf",        "content": "arXiv:1502.05683v1  [cond-mat.mes-hall]  19 Feb 2015Enhanced photogalvanic effect in graphene due to Rashba spin -orbit coupling\nM. Inglot,1V. K. Dugaev,1,2E. Ya. Sherman,3,4and J. Barna´ s5,6\n1Department of Physics, Rzesz´ ow University of Technology,\nal. Powsta´ nc´ ow Warszawy 6, 35-959 Rzesz´ ow, Poland\n2Departamento de F´ ısica and CeFEMA, Instituto Superior T´ e cnico,\nUniversidade de Lisboa, av. Rovisco Pais, 1049-001 Lisbon, Portugal\n3Department of Physical Chemistry, Universidad del Pa´ ıs Va sco UPV-EHU, 48080 Bilbao, Spain\n4IKERBASQUE Basque Foundation for Science, Bilbao, Spain\n5Faculty of Physics, Adam Mickiewicz University, ul. Umulto wska 85, 61-614 Pozna´ n, Poland\n6The Nano-Bio-Medical Centre, Umultowska 85, 61-614 Pozna´ n, Poland\n(Dated: August 12, 2021)\nWe analyze theoretically optical generation of a spin-pola rized charge current (photogalvanic\neffect)andspinpolarization ingraphene withRashbaspin-o rbitcoupling. Anexternalmagnetic field\nis applied in the graphene plane, which plays a crucial role i n the mechanism of current generation.\nWe predict a highly efficient resonant-like photogalvanic eff ect in a narrow frequency range which is\ndetermined by the magnetic field. A relatively less efficient p hotogalvanic effect appears in a broader\nfrequency range, determined by the electron concentration and spin-orbit coupling strength.\nPACS numbers: 72.25.Fe, 78.67.Wj, 81.05.ue\nIntroduction: Two-dimensional electron systems with\nspin-orbit (SO) coupling are currently of a broad interest\ndue to the coupled charge and spin dynamics, as revealed\nin a variety of spin related transport phenomena [1–4].\nOwing to the SO coupling, the spin dynamics canbe gen-\nerated, among others, by a low-frequency electric field [5]\naswell as opticallyby interband electronictransitions[6–\n8]. Moreover,an externalstatic magneticfield canenable\nthe current generation by light absorption, leading to a\nphotogalvanic effect [9].\nMany of the spin related phenomena, including also\nthe ones mentioned above, can be observed in two di-\nmensional graphene monolayers and other graphene-like\nmaterials like silicene for instance. The huge inter-\nest in graphene is related mainly to its natural two-\ndimensionality, very unusual electronic structure, and\nhigh electron mobility which ensures its excellent trans-\nport properties [10, 11]. Even though the intrinsic SO\ninteraction in free-standing graphene is negligibly small,\nthe Rashba spin-orbit coupling can be rather strong\nfor graphene deposited on certain heavy-element sub-\nstrates [12–14]. Since the electronic band structure of\ngraphene is significantly different from that of a sim-\nple two-dimensional electron gas (2DEG), and the spin-\norbit coupling creates a gap in the electronic spectrum,\ngraphene can reveal qualitatively new effects which can\nnot be observed in 2DEG in conventional semiconductor\nheterostructures. Additionally, the SO-related phenom-\nenaingraphenearealsoimportantfromthe pointofview\nof potential applications in all-graphene based spintron-\nics devices [15–17].\nIn this letter we predict an enhanced photogalvanic ef-\nfect in graphene. To do this we consider the charge and\nspin currents generated in graphene by optical pump-\ning in the infrared photon energy region, and show thatthe optical pumping can be used to generate in graphene\nnot only the spin density [18], but also a spin-polarized\nnet current. An external magnetic field applied in the\ngraphene plane plays an important role in the mecha-\nnism of current generation. We show that the efficiency\nof current generation per absorbed photon can be very\nhigh at certain conditions. Apart from this, we also show\nthat one can create spin density without creating electric\ncurrent, but not vice versa .\nModel:We consider low-energyelectronic spectrum of\ngraphene in the vicinity of the Dirac points [11]. Addi-\ntionally, we include the Rashba SO coupling [19] and the\nZeemanenergyin aweakexternal in-planemagnetic field\nB. The corresponding Hamiltonian can be then written\nin the form\nˆH=/planckover2pi1v0(±τxkx+τyky)+λ(±τxσy−τyσx)+∆\n2(b·σ),(1)\nwherev0≃108cm/s is the electron velocity in graphene,\nτxandτyare the Pauli matrices defined in the sublattice\nspace, ∆ ≡gBis the maximum Zeeman splitting, b≡\nB/B,while the + and −signs refer to the KandK′\nDirac points, respectively. Furthermore, g=gLµBwith\ngL= 2 being the Land´ e factor, and λ=α/2 withα\nstanding for the Rashba coupling constant [20].\nThe electronicspectrum correspondingto Hamiltonian\n(1) consists of four energy bands in each valley, Enk,\nwheren= 1 ton= 4 is the band index. The correspond-\ningspectrumfor B= 0isshowninFig.1a, andisgivenby\nthe formula En,k(B= 0) =∓λ±/radicalbig\nλ2+/planckover2pi12v2\n0k2. In turn,\nthe expectation value of the spin in the absence of mag-\nnetic field is oriented perpendicularly to the wavevector\nk[20], similarly as in a semiconductor-based 2DEG with\nRashba SO coupling. Contrary to the 2DEG, there are,\nhowever, no eigenstates of Hamiltonian (1) with a defi-\nnite eigenvalue of any spin component. This appears due2\nFigure 1: (Color online). (a) Schematic picture of the band\nstructure of graphene with Rashba spin-orbit interaction w ith\nall possible band transitions for a chosen chemical potenti al,\nindicated by the vertical arrows. Circles with crosses and\ndots inside correspond to the opposite spin orientations in\nthe subbands. (b) Dispersion of the low energy states for\nindicated wavevector orientations. The band index is also\nmarked at the plots. The assumed magnetic field B/bardblxis\nequal to 5 T.\nto the specific form of the SO coupling in graphene. The\ncorresponding spin components in the absence of mag-\nnetic field and for the wavevector k≡k(cosθ,sinθ) are\n/angbracketleftσx(k)/angbracketrightB=0=∓vk\nv0sinθ, (2a)\n/angbracketleftσy(k)/angbracketrightB=0=±vk\nv0cosθ, (2b)\nwherevk=v0×/planckover2pi1v0k//radicalbig\nλ2+/planckover2pi12v2\n0k2istheabsolutevalue\nof the electron velocity in the absence of magnetic field.\nHere the upper and lower signs correspond to the bands\n(1,4) and (2,3), respectively. Note, both spin compo-\nnents given by Eqs. (2a) and (2b) vanish in the limit of\nk= 0 [20] corresponding to the mixed rather than pure\ncharacter of the band states in the spin subspace.\nThe electronic spectrum presented in Fig.1a is signif-\nicantly modified by an external in-plane magnetic field.\nAssume this field is oriented along the axis x. The ex-\nact electronic spectrum can be then obtained by directdiagonalization of the Hamiltonian (1), and is shown in\nFig.1b for wavevectors along the axis xandy. Only the\nstates corresponding to the bands labeled in Fig.1a with\nthe index 2 and 3 are shown there. As one can note,\nfor one propagation orientation the electron bands are\nshifted vertically, i.e. to higher (lower) energy, while for\nthe second propagation orientation the bands are shifted\nhorizontally, i.e. to left (right) from the point k= 0.\nThe latter separation in the k-space of the bands 2 and\n3 is crucial for the enhanced photogalvanic effect. For a\nweak Zeeman energy, ∆ ≪α, and for electron momenta\nof our interest, k≫√\nα∆//planckover2pi1v0, the field-dependent cor-\nrection to the electron energy, calculatedby perturbation\ntheory, has the form\nEn,k(B)−En,k(B= 0) =∓∆\n2vk\nv0sinθ,(3)\nwhere again the upper (lower) sign corresponds to the\nbands (1,4) and (2,3), respectively.\nAssume now that the system is subject to electromag-\nnetic irradiation. Hamiltonian describing interaction of\nelectrons in graphene with the external periodic electro-\nmagnetic field, A(t) =A0e−iωt, takes the form\nˆHA=∓e\ncv0(τ·A). (4)\nAs in Eq. (1), different signs correspondhere to electrons\nwithin the KandK′valleys.\nThe injection rate of a quantity O, related to the inter-\nsubband optical transitions, can be calculated by using\nthe Fermi’s golden rule,\nO(ω) =/summationdisplay\nn,n′On→n′(ω),\nOn→n′(ω) =2π\n/planckover2pi1/integraldisplayd2k\n(2π)2/vextendsingle/vextendsingle/vextendsingle/angbracketleftΨnk|ˆHA|Ψn′k/angbracketright/vextendsingle/vextendsingle/vextendsingle2/hatwideOn→n′\n×δ(Enk+/planckover2pi1ω−En′k)f(Enk)[1−f(En′k)],(5)\nwheref(Enk) is the Fermi-Dirac distribution function.\nSince there are two valleys, KandK′, one needs to cal-\nculate contributions to O(ω) from both of them. The\nquantities of our interest here are:\n/hatwideOn→n′≡/hatwide11, (6)\nforthelightabsorption(where /hatwide11istheidentityoperator),\n/hatwideOn→n′≡ /angbracketleftΨn′k|σν|Ψn′k/angbracketright−/angbracketleftΨnk|σν|Ψnk/angbracketright,(7)\nfor the corresponding spin component injection, and\n/hatwideOn→n′≡ /angbracketleftΨn′k|ˆIi|Ψn′k/angbracketright−/angbracketleftΨnk|ˆIi|Ψnk/angbracketright(8)\nfor the current injection. Here, ˆIi≡eˆvi, whereeis the\nelectron charge, while ˆ vx≡ ±v0τxand ˆvy≡v0τy. The\ninjected spin current, in turn, can be calculated as [21]\n/hatwideOn→n′≡ /angbracketleftΨn′k|ˆJν\ni|Ψn′k/angbracketright−/angbracketleftΨnk|ˆJν\ni|Ψnk/angbracketright,(9)3\nFigure 2: (Color online) (a) Normalized charge current ˜Iy, in\nthe case of low temperature, T= 1 K (T= 10 K in the in-\nset). Rashba spin-orbit coupling strength is α= 4 meV (solid\nred line) and α= 13 meV (dashed blue line). The chemical\npotential is µ= 5 meV, Bis in the plane of graphene and\nalong the x-axis, while A||B. (b) Contributions of indicated\nintersubband transitions to the total current presented in (a)\nforα= 4 meV.\nwhereˆJν\ni= [σν,ˆvi]+/2. Below we concentrate on the\nresults for injection of charge current and spin density.\nResults: Using equations for the injection rate one\ncan calculate the charge current and spin polarization\ninduced by the optical pumping. Let us begin with the\nphotogalvanic effect, i.e. charge current generation. Re-\nsultsfortwodifferentpolarizationsoftheelectromagnetic\nfieldA(t) are presented in Figs. 2 and 3. Here the in-\njection efficiency ˜Iiis defined as ˜Ii≡Ii/ev0I0, where\nI0=πe2Q//planckover2pi1candQis the incident photon flux [22–24].\nThe transitions start at /planckover2pi1ω≈2µ−αifµ≥αand at\n/planckover2pi1ω≈2µotherwise. In both cases a strong narrow peak\nin the injection efficiency appears at a resonant energy\n/planckover2pi1ω≈2µ. This peak is remarkably higher for the electro-\nmagneticfieldpolarizedalongthestaticmagneticfield B,\ncompareFigs. 2 and 3. To understand originof this peak\nlat us consider transitions between the subbands marked\nwithn= 2 and n= 3. First, we determine the shape\nof the isoenergetical line corresponding to a given Fermi\nenergy,µ≫∆. For the band corresponding to n= 3,\none obtains from Eq. (3) the first-order correction to theFigure 3: (Color online) Normalized charge current ˜Iy, for a\nlow temperature, T= 1 K ( T= 10 K in the inset). Solid\nred line is for α= 4 meV and dashed blue line is for α= 13\nmeV. Chemical potential µ= 5 meV, Bis along the x-axis\nandA⊥B.\nFermi wavevector,\n/planckover2pi1kF=/radicalbig\nµ2+2λµ\nv0∓∆\n2v0sinθ, (10)\nwhichsets the followingboundariesforthe Fermi surface:\n−/radicalbig\nµ2+2λµ\nv0−∆\n2v0</planckover2pi1kF,y</radicalbig\nµ2+2λµ\nv0−∆\n2v0\n(11a)\n−/radicalbig\nµ2+2λµ\nv0</planckover2pi1kF,x</radicalbig\nµ2+2λµ\nv0. (11b)\nDue to the k-dependent Zeeman term, the Fermi sur-\nface becomes considerably deformed and anisotropic, as\nshown in the left panel of Fig.4. The maximum deforma-\ntion is independent of the chemical potential and spin-\norbit coupling. In turn, the resonance line determined by\nE3,k−E2,k=/planckover2pi1ωis still a circle given by the condition\n/planckover2pi1kω=/radicalbig\n/planckover2pi12ω2/4+λ/planckover2pi1ω\nv0. (12)\nA part of the resonance line is inside the occupied region.\nTherefore, we have an interval of the photon energies,\n(/planckover2pi1ω1,/planckover2pi1ω2), as shown in the right panel of Fig.4, where\nthe transitions occur for positive values of ky, while the\ntransitions with negative ky(which would compensate\npartly current) are forbidden. As a result, a very efficient\ncurrentinjection occurs in this photon energy window, as\nvisible in Fig. 2. In the considered regime of µ≫∆, this\nphoton energy interval is determined by the conditions\n/planckover2pi1ω1= 2µ−∆vkF\nv0, (13a)\n/planckover2pi1ω2= 2µ+∆vkF\nv0, (13b)4\nFigure 4: (Color online). Left: Schematic pictureof the Fer mi\nline (solid) and resonance line (dashed) for the chosen sub-\nbands. Optical transitions are possible only at the part of t he\ndashed line outside the filled area. Each transition generat es\na current of the order of 2 evkF, making the generation highly\nefficient. Right: Side view on the intersubband transitions,\nwhich can occur in the frequency interval ω1≤ω≤ω2.\nwhich results in the peak width given by the formula,\n/planckover2pi1(ω2−ω1) = 2/radicalbig\nµ2+2λµ\nµ+λ∆. (14)\nWith the increase in temperature to T >∆, this effect\nbecomessmearedoutbythermalbroadeningoftheFermi\ndistribution, and the injection rate decreases as shown in\nthe insets to Figs. 2 and 3.\nSimilar arguments can be also applied to the transi-\ntions between n= 1 and n= 4 subbands. As a result,\none gets a relatively small negative peak in the injection\nrate at/planckover2pi1ω≈2µ, see Fig. 2 (b). The weakness of this in-\njection channel is due to a relatively small Fermi velocity\nin the subband 4 at µ−α≪α, while its reversed sign is\ndue to the opposite spin orientation in these subbands,\nwhich results (similar to Eq. (11a) and Fig. 4) in a dif-\nferent shape of the Fermi surface, where the transitions\nbegin to occur at ky<0. In the limit α≪µ, the positive\nand negative contributions compensate each other, and\nthe current injection efficiency tends to zero, as expected\nin the absence of spin-orbit coupling.\nHaving discussed the strong peaks in the current injec-\ntion rate, let us consider now briefly the broad structure.\nIt is formed by momentum dependence of the matrix el-\nements and velocity, and has the efficiency of the order\nof ∆/α. The current injection stops at /planckover2pi1ω≈2µ+α,\nwherethecontributionsduetotransitionsbetween differ-\nent bands compensate each other. We also mention that\nforB/bardblx, the charge current has only the y-component\nfor both polarizations of the incident light.\nNow let us address briefly the problem of spin and spin\ncurrent injection. For both incident light polarizations\noneobtainsanetspinpolarizationalongthe xandyaxes.\nThe numerical results are presented in Fig. 5 (a) for the\ntotal spin polarization Sx, while Fig. 5(b) shows injected\nspin polarization associated with specific optical transi-\ntions. Physical mechanism of the optically injected spin\npolarization is rather clear, since the spin-flip transitionsFigure 5: (Color online). (a) Total injected normalized spi n\npolarization ˜Sx=/summationtext\nn,n′˜Sn→n′\nx. Here the Rashba SO cou-\nplingα= 4 meV (solid red line) and α= 13 meV (dashed\nred line), µ= 5 meV and B= 5 T. The orientations of A\nandBare parallel to the x−axis. (b) Transition-related spin\ninjection ˜Sn→n′\nx.\nare related to the above-mentioned fact that the eigen-\nstates of Hamiltonian (1) are not the spin eigenstates,\nand the broken in magnetic field time-reversal symme-\ntry allows one to inject spin density. Since the charge\ncurrent is along the y-axis, we obtain effectively a spin-\npolarized current transferring in-plane spin components\nin they−direction. As concerns the spin current defined\nin Eq. (9), it is symmetric with respect to the time re-\nversal and, therefore, magnetic field produces there only\nchanges proportional to B2.\nSummary: We have calculated optical injection of\ncharge current in graphene as the photogalvanic effect\ndue to spin-orbit coupling [25]. The current is injected\nonly in a finite range of infrared light frequencies, de-\ntermined by the chemical potential µand the spin-orbit\ncoupling strength. The striking feature of the injection\nis a narrow peak at the resonant frequency /planckover2pi1ω≈2µ,\nwhere the current injection can be very efficient. Com-\nparing the ω-dependence of the current and spin injec-\ntion, we conclude that, depending on the light frequency,\none caninject either spin-polarizednet electric currentor\nnet spin polarization without the current injection. This\nresult can be applied to a controllable current generation\nin spin-orbit coupled graphene.5\nAcknowledgements. This work is supported by the Na-\ntional Science Center in Poland under Grant No. DEC-\n2012/06/M/ST3/00042. The work of MI is supported\nby the project No. POIG.01.04.00-18-101/12. The\nwork of EYS was supported by the University of Basque\nCountry UPV/EHU under program UFI 11/55, Spanish\nMEC (FIS2012-36673-C03-01), and ”Grupos Consolida-\ndos UPV/EHU del Gobierno Vasco” (IT-472-10).\n[1]Spin Physics in Semiconductors , (M. I. Dyakonov, Ed.)\nSpringer Series in Solid-State Sciences (Springer, Berlin ,\n2008).\n[2] J. Sinova, S. O. Valenzuela, J. Wunderlich, C. H. Back,\nand T. Jungwirth, arXiv:1411.3249.\n[3] A. G. Aronov and Yu. B. Lyanda-Geller, JETP Lett. 50,\n431 (1989).\n[4] V. M. Edelstein, Solid State Communications 73233\n(1990).\n[5] E. I. Rashba and Al. L. Efros, Phys. Rev. Lett. 91,\n126405 (2003).\n[6] M. J. Stevens, A. L. Smirl, R. D. R. Bhat, A. Najmaie,\nJ. E. Sipe, and H. M. van Driel, Phys. Rev. Lett. 90,\n136603 (2003).\n[7] Y. K. Kato, R. C. Myers, A. C. Gossard and D. D.\nAwschalom, Phys. Rev. Lett. 93, 176601 (2004).\n[8] A. Y. Silov, P. A. Blajnov, J. H. Wolter, R. Hey, K. H.\nPloog, and N. S. Averkiev, Appl. Phys. Lett. 85, 5929\n(2004).\n[9] S. D. Ganichev, and W. Prettl, J. Phys. Cond. Matter.\n15, R935 (2003).[10] A. K. Geim and K. S. Novoselov, Nature Mater. 6, 183\n(2007).\n[11] M. I. Katsnelson, Graphene: Carbon in Two Dimensions\n(Cambridge Univ. Press, 2012).\n[12] Y. S. Dedkov, M. Fonin, U. R¨ udiger, and C. Laubschat,\nPhys. Rev. Lett. 100, 107602 (2008).\n[13] A. Varykhalov, J. S´ anchez-Barriga, A. M. Shikin, C.\nBiswas, E. Vescovo, A. Rybkin, D. Marchenko, and O.\nRader, Phys. Rev. Lett. 101, 157601 (2008).\n[14] M. Zarea andN. SandlerPhys. Rev.B 79, 165442 (2009).\n[15] B. Trauzettel, D. V. Bulaev, D. Loss, and G. Burkard,\nNature Phys. 3, 192 (2007).\n[16] W. Han, R. K. Kawakami, M. Gmitra, and J. Fabian,\nNature Nanotechnology 9794 (2014).\n[17] P. Seneor, B. Dlubak, M.-B. Martin, A. Anane, H. Jaf-\nfres, and A. Fert, MRS Bulletin 37, 1245 (2012).\n[18] M. Inglot, V. K. Dugaev, E. Y. Sherman, and J. Barna´ s,\nPhys. Rev. B 89, 155411 (2014).\n[19] C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 226801\n(2005).\n[20] E. I. Rashba, Phys. Rev. B 79, 161409 (2009).\n[21] J. RiouxandG. Burkard, Phys.Rev.B 90, 035210 (2014)\n[22] V. P. Gusynin and S. G. Sharapov, Phys. Rev. B 73,\n245411 (2006).\n[23] A. B. Kuzmenko, E. van Heumen, F. Carbone, and D.\nvan der Marel, Phys. Rev. Lett. 100, 117401 (2008).\n[24] P. Ingenhoven, J. Z. Bern´ ad, U. Z¨ ulicke, and R. Egger,\nPhys. Rev. B 81, 035421 (2010).\n[25] The proposed mechanism is qualitatively different from\nthe coherent control approach of D. Sun, C. Divin, J.\nRioux, J. E. Sipe, C. Berger, W. A. de Heer, P. N. First,\nand T. B. Norris, Nano Lett. 101293 (2010)."    },    {        "title": "1108.1806v3.Spin_Orbital_Locking__Emergent_Pseudo_Spin__and_Magnetic_order_in_Honeycomb_Lattice_Iridates.pdf",        "content": "arXiv:1108.1806v3  [cond-mat.str-el]  10 May 2012Spin-Orbital Locking, Emergent Pseudo-Spin, and Magnetic order in Honeycomb\nLattice Iridates.\nSubhro Bhattacharjee1,2, Sung-Sik Lee2,3, and Yong Baek Kim1,4\n1Department of Physics, University of Toronto, Toronto, Ont ario, Canada M5S 1A7.\n2Department of Physics & Astronomy, McMaster University, Ha milton, Ontario, Canada L8S 4M1.\n3Perimeter Institute for Theoretical Physics, Waterloo, On tario, Canada N2L 2Y5.\n4School of Physics, Korea Institute for Advanced Study, Seou l 130-722, Korea.\n(Dated: October 25, 2018)\nThe nature of the effective spin Hamiltonian and magnetic ord er in the honeycomb iridates is\nexplored by considering a trigonal crystal field effect and sp in-orbit coupling. Starting from a\nHubbard model, an effective spin Hamiltonian is derived in te rms of an emergent pseudo-spin-1/2\nmoment in the limit of large trigonal distortions and spin-o rbit coupling. The present pseudo-spins\narise from a spin-orbital locking and are different from the jeff= 1/2 moments that are obtained\nwhen the spin-orbit coupling dominates and trigonal distor tions are neglected. The resulting spin\nHamiltonian is anisotropic and frustrated by further neigh bour interactions. Mean field theory\nsuggests a ground state with 4-sublattice zig-zagmagnetic order in a parameter regime that can\nbe relevant to the honeycomb iridate compound Na 2IrO3, where similar magnetic ground state has\nrecently been observed. Various properties of the phase, th e spin-wave spectrum and experimental\nconsequences are discussed. The present approach contrast s with the recent proposals to understand\niridate compounds starting from the strong spin-orbit coup ling limit and neglecting non-cubiclattice\ndistortions.\nI. INTRODUCTION\nInterplay between strong spin-orbit (SO) coupling and electron-e lectron interaction in correlated electron systems\nhas been a recent subject of intensive study1–24. In particular, 5 dtransition metal (e.g. Iridium (Ir) or Osmium (Os))\noxides are regarded as ideal playgrounds for observing such coop erative effects1–24. Compared to 3 dtransition metal\noxides, the repulsive Coulomb energy scale in these systems is reduc ed by the much larger extent of 5 dorbitals, while\nthe SO coupling is enhanced due to high atomic number ( Z= 77 for Ir and z= 76 for Os). Moreover, owing to the\nextended 5 dorbitals, these systems are very sensitive to the crystal fields. A s a result, the energy scales mentioned\nabove often become comparable to each other, leading to a variety of competing phases. Precisely for this reason,\none expects to see newer emergent quantum phases in such syste ms. Indeed, there have been several theoretical\nproposals, in context of concrete experimental examples, for sp in liquids1,11,12,14,16,18,19, topological insulators10,19,20,\nWeyl semimetals21,22, novel magnetically ordered Mott insulators2–6,23,24and other related phases8,9,23in Iridium and\nOsmium oxides.\nA typical situation in the iridatesconsist of Ir+4atoms sitting in the octahedralcrystal field of a chalcogen, typica lly\noxygen or sulphur1–4,7. This octahedral crystal field splits the five 5 dorbitals of Ir into the doubly degenerate eg\norbitals and the triply degenerate t2gorbitals(each orbital has a further two-fold spin degeneracy). T heegorbitalsare\nhigher in energy with the energy difference being approximately 3 eV. There are 5 electrons in the outermost 5 dshell\nof Ir+4which occupy the low lying t2gorbitals and the low energy physics is effectively described by projec ting out\nthe empty egorbitals25. A characteristic feature of most of the approaches used to und erstand these compounds is to\ntreat the SO coupling as the strongest interactionat the atomic lev el;i.e., by consideringthe effect ofextremely strong\nSO coupling for electrons occupying the t2gorbitals. This decides the nature of the participating atomic orbitals in\nthe low energy effective theory. In this limit, the orbital angular mom entum, projected to the t2gmanifold, carries an\neffective orbital angular momentum leff= 1 with a negative SO coupling constant2,3,15. The projected SO coupling\nsplits the t2gmanifold into the lower jeff= 3/2 quadruplet and the upper jeff= 1/2 doublet. Out of the five valence\nelectrons, four fill up the quadruplet sector leaving the doublet se ctor half filled. Thus, in the limit of very strong SO\ncoupling, the half filled doublet sector emerge as the correct low ene rgy degrees of freedom. Considering the effect\nof coulomb repulsion within a Hubbard model description and perform ing strong coupling expansion, various spin\nHamiltonians for jeff= 1/2 are then derived within a strong-coupling perturbation theory11,20,23.\nIn this paper, we, however, consider a different limit where the oxyg en octahedra surrounding the Ir+4ions are\nhighly distorted. While the above scenario of half filled jeff= 1/2 orbitals is applicable to undistorted case, as we\nshall see, it breaks down in presence of strong distortions of the o ctahedra. In particular, we consider the effect of\ntrigonal distortions, which may be relevant for some of the iridate s ystems including the much debated honeycomb\nlattice iridate, Na 2IrO3. We show that, in this limit, a different “doublet” of orbitals emerge as the low energy degree\nof freedom. This doublet forms a pseudo-spin-1 /2that results from a kind of (physical)spin-orbital locking, so that the\nspin and orbital fluctuations are not separable (as discussed below ). We emphasize that this pseudo-spin is different2\nb\na\nFIG. 1: The zig-zagmagnetic structure as found in Ref.5. The magnetic unit cell has 4 sites.\nfrom the jeff= 1/2 doublet discussed above. The spin Hamiltonian for these pseudo-s pins (Eq. 6), on a honeycomb\nlattice, admits a 4-sub-lattice zig-zag(fig.I) pattern in a relevant parameter regime. Such magnetic order has been\nrecently observed in the experiments5,28on Na 2IrO3and hence our theory may be applicable to this material.\nThe distortion of the octahedron surrounding the Ir+4generates a new energy scale associated with the change in\nthe crystal field, which, as we shall see, competes with the SO coup ling. Several kinds of distortion may occur, of\nwhich we consider the trigonal distortions of the octahedron wher e it is stretched/compressedalong the body diagonal\nof the enclosing cube25. In the absence SO coupling, such trigonal crystal field splits the t2gmanifold into e′\ng(with\ntwo degenerate orbitals e′\n1gande′\n2g) and non-degenerate a1g(again there is an added two-fold spin degeneracy for\neach of these orbitals). The e′\nganda1glevels are respectively occupied by three and two electrons in Ir4+. For large\ntrigonal distortions, the splitting between them is big and the a1gorbitals can be projected out. Now, if one adds\nSO coupling, the low energy degrees of freedom is described by a sub space of the e′\ngorbitals which form an emergent\npseudo-spin-1/2 doublet out of |e′\n1g,↓/angbracketrightand|e′\n2g↑/angbracketrightstates, where ↑and↓represent the physical spin sz= 1/2,−1/2\n(the spins are quantized along the axis of trigonal distortion). The se pseudo-spin-1 /2 is different from the jeff= 1/2\nandjeff= 3/2 multiplets in the strong SO coupling limit as discussed above. Notice th e (physical)spin-orbital locking\nfor the pseudo-spins, as alluded above.\nThe two approaches, described in the last two paragraphs, of arr iving at the low energy manifold are mutually\nincompatible. This can be seen as follows: In presence of sizeable trig onal distortions the jeff= 1/2 andjeff= 3/2\nmultiplets mix with each other and can no longer serve as good low ener gy atomic orbitals. This dichotomy becomes\nquite evident in the recent studies of Na 2IrO3, where, the Ir+4form a honeycomb lattice. Taking into account the\nstrong SO coupling of Ir4+in Na2IrO3, proposed are a model for a topological insulator in the weak intera ction limit10\nand a Heisenberg-Kitaev (HK) model for a possible spin liquid phase in t he strong coupling limit11. These proposals\nprompted several experimental4–6and theoretical efforts12,26to understand the nature of the ground state in this\nmaterial. Subsequently, it was found that Na 2IrO3orders magnetically at low temperatures4. However, the magnetic\nmoments form a “zig-zag” pattern (fig. I) which is not consistent with the ones that would be obtained by addin g a\nweak interaction in a topological insulator (canted antiferromagne t)10or from a nearest neighbour HK model (spin\nliquid or the so-called stripe antiferromagnet)11. While recent studies27show that a ‘zig-zag’ order may be stabilized\nwithin the HK model by including substantial second and third neighbo ur antiferromagnetic interactions, it is hard\nto justify such large further neighbour exchanges without lattice distortions. (An alternate explanation that we do\nnot pursue here is significant charge fluctuations which would mean t hat the compound is close to metal-insulator\ntransition. Theresistivitydataseemstosupportthefactthat th iscompoundisagoodinsulator.4) Iflatticedistortions\nare responsible for the significant further neighbour exchanges, then, there would be sizeable distortion of the oxygen\noctahedra, which in turn may invalidate the above jeff= 1/2 picture and thus the basic paradigm of the HK model,\nby mixing the jeff=1/2and jeff= 3/2 subspaces. Recent finite temperature numerical calculations on the HK model26\nalso suggest possible inconsistencies with experiments on Na 2IrO3. This necessitates the need for a different starting\npoint, to explain the magnetic properties of Na 2IrO3.\nThe rest of this paper is organized as follows. In Section II, we derive the effective spin Hamiltonian in limit\nof the large trigonal distortion and large SO coupling. This is done by t aking the energy scale associated with\ntrigonal distortion to infinity first, followed by that of the SO energ y scale. This order of taking the limit gives a\nspin Hamiltonian in terms of emergent pseudo-spin −1/2, which is different from the HK model. This Hamiltonian3\nis both anisotropic and frustrated. It also has further neighbour interactions, the effect of which are enhanced due\nto anisotropy that makes some of the nearest neighbour bonds we aker. The origin of this anisotropy is trigonal\ndistortion. We argue that this limit may be more applicable for the comp ound Na 2IrO3. Having derived the spin\nHamiltonian, we calculate the phase diagram and the spin wave spectr um within mean field theory in Sec. III. We see\nthat the ‘zig-zag’ phase occurs in a relevant parameter regime. We also point out the experimental implications of our\ncalculations in context of Na 2IrO3. Finally we summarize the results in Sec. IV. The details of various calculations\nare given in various appendices.\nII. THE EFFECTIVE HAMILTONIAN\nIn the cubic environment the t2gorbitals are degenerate when there is no SO coupling. Trigonal disto rtion due to\ncompression or expansion along one of the four C3axes of IrO 6octahedra lifts this degeneracy. Although it is possible\nthat the axes of trigonal distortions are different in different octa hedra20, we find that uniform distortions are more\nconsistent with the experiments (see below) on Na 2IrO3. Hence we consider uniform trigonal distortion.\nA. The Trigonal Hamiltonian\nLet us denote the axis of this uniform trigonal distortion of the oct ahedron by the unit vector ˆ n=1√\n3[n1,n2,n3],\nwherenα=±1. Since there are 2 directions to each of the 4 trigonal axes we may choose a “gauge” to specify ˆ n. This\nis done by taking n1n2n3= +1. The Hamiltonian for trigonal distortion, when projected in the t2gsector, gives20(in\nour chosen gauge)\nHt2g\ntri=−/summationdisplay\ni∆tri\n3Ψ†\ni\n0n3n2\nn30n1\nn2n10\nΨi, (1)\nwhere Ψ†\ni= [d†\nyz,d†\nzx,d†\nxy] and ∆ triis the energy scale for trigonal distortion. The eigenstates are ( ω=eı2π/3)\n|a1g/angbracketright=1√\n3[n1|dyz/angbracketright+n2|dzx/angbracketright+n3|dxy/angbracketright],\n|e′\n1g/angbracketright=1√\n3/bracketleftbig\nωn1|dyz/angbracketright+ω2n2|dzx/angbracketright+n3|dxy/angbracketright/bracketrightbig\n,\n|e′\n2g/angbracketright=1√\n3/bracketleftbig\nω2n1|dyz/angbracketright+ωn2|dzx/angbracketright+n3|dxy/angbracketright/bracketrightbig\n. (2)\nThe trigonal distortion splits the t2gsector into the doubly degenerate e′\ngand the non-degenerate a1gwith energies\n∆tri/3 and−2∆tri/3 respectively.\nA description based on Hubbard model for the e′\ngorbitals may be systematically derived starting from the t2g\norbitals. This is done in A. This has the following general form\nH′=He′\ng\nSO−/summationdisplay\nij/summationdisplay\nM,M′/summationdisplay\nσ˜tiM;jM′e†\niMσeiM′σ+U\n2/summationdisplay\ni/summationdisplay\nM,M′/summationdisplay\nσσ′e†\niMσe†\niM′σ′eiM′σ′eiMσ,\n(3)\nwheree†\niMσis the electron creation operator in the e′\ngorbital (M= 1,2) with spin σ(=↑,↓);˜tiM;jM′are the effective\nhopping amplitudes within the subspace and Uis the effective onsite coulomb’s repulsion. We note that the Hund’s\ncoupling (which arises from the orbital dependence of the Coulomb r epulsion) for the t2gorbitals only renormalizes\nUin this restricted subspace. (See Afor details).\nB. The Projected SO coupling\nThe SO coupling, when projected in the e′\ngsubspace, yields a block diagonal form (see Bfor details):\nHe′\ng\nSO=−λˆn·/vector siτz\ni, (4)4\nFIG. 2: The e′\ngstates split by the SO coupling.\nwhere/vector siis the spin operator at the site i,λ≈500meVis the SO coupling parameter and τz= +1(−1) refers to the\ne′\n1g(e′\n2g) orbital.\nThus the projected SO interaction acts as a “Zeeman coupling” whe re the direction of the “magnetic field” is along\nthe trigonal axis or opposite to it13. Thus it is natural to choose the direction of spin quantization along the axis\nof trigonal distortion. This then gives the active atomic orbitals aft er incorporating the SO coupling. These active\norbitals are the Krammer’s doublet |e′\n1g,↓/angbracketrightand|e′\n2g,↑/angbracketrightas shown in Fig. 2.\nC. The Spin Hamiltonian\nHence the low energy physics may be described by considering only th e above atomic orbitals. The starting point\nfor the calculations is projection of the Hubbard model (Eq. 3) in the space spanned by the Krammers’s doublet\n|e′\n1g,↓/angbracketrightand|e′\n2g,↑/angbracketright. The bandwidth of this projected model is narrow and the effect of the Hubbard repulsion is\nimportant. Indeed it can easily render the system insulating. To cap ture the magnetic order in this Mott insulator,\nwe do a strong-coupling expansion in ˜t/Uto get an effective “pseudo-spin” model in terms of the pseudo-sp in-1/2\noperators,\nSα=1\n2e†\naρα\nabeb, (5)\nwhere,ρα(α=x,y,z) are the Pauli matrices and a,b= (e′\ng1;↓),(e′\ng2;↑). The “pseudo-spin” Hamiltonian has the\nfollowing form up to the quadratic order:\nH=/summationtext\n/angbracketleftij/angbracketrightJ(1)\nij/vectorSi·/vectorSj+/summationtext\n/angbracketleft/angbracketleftij/angbracketright/angbracketrightJ(2)\nij/vectorSi·/vectorSj+/summationtext\n/angbracketleft/angbracketleft/angbracketleftij/angbracketright/angbracketright/angbracketrightJ(3)\nij/vectorSi·/vectorSj\n+/summationtext\n/angbracketleftij/angbracketrightJ(z1)\nijSz\niSz\nj+/summationtext\n/angbracketleft/angbracketleftij/angbracketright/angbracketrightJ(z2)\nijSz\niSz\nj+/summationtext\n/angbracketleft/angbracketleft/angbracketleftij/angbracketright/angbracketright/angbracketrightJ(z3)\nijSz\niSz\nj. (6)\nHere/angbracketleftij/angbracketright,/angbracketleft/angbracketleftij/angbracketright/angbracketrightand/angbracketleft/angbracketleft/angbracketleftij/angbracketright/angbracketright/angbracketrightrefer to summation over first, second and third nearest neighbou rs (NNs) respectively.\nThe different exchange couplings are given in terms of the underlying parameters of the Hubbard model as\nJ(zα)\nij=8\nU(T(zα)\nij)2,\nJ(α)\nij=4\nU/bracketleftBig\n(T(0α)\nij)2−(T(zα)\nij)2/bracketrightBig\n=J(0α)\nij−1\n2J(zα)\nij (7)\nwhereα= 1,2,3 denotes that ijare first, second or third NNs, respectively and the last expressio n defines J(0α)\nij.\nT(0α)\nijandT(zα)\nijare given in terms of the hopping amplitudes (e.g. txy;yz\nijfrom the overlap of xyandyzorbitals) of\nthet2gorbitals as (details are given in C).\nT(0α)\nij=1\n3/bracketleftbig/parenleftbig\ntyz;yz\nij+txz;xz\nij+txy;xy\nij/parenrightbig/bracketrightbig\n−1\n6/bracketleftbig/parenleftbig\nn1/parenleftbig\ntxz;xy\nij+txy;xz\nij/parenrightbig\n+n2/parenleftbig\ntxy;yz\nij+tyz;xy\nij/parenrightbig\n+n3/parenleftbig\ntyz;xz\nij+txz;yz\nij/parenrightbig/parenrightbig/bracketrightbig\n(8)\nT(zα)\nij=1\n2√\n3/bracketleftbig\nn1/parenleftbig\ntxz;xy\nij−txy;xz\nij/parenrightbig\n+n2/parenleftbig\ntxy;yz\nij−tyz;xy\nij/parenrightbig\n+n3/parenleftbig\ntyz;xz\nij−txz;yz\nij/parenrightbig/bracketrightbig\n(9)5\nFIG. 3: Section of a honeycomb lattice (shaded in yellow). Ir sites (black) are connected by bonds (orange). The green arr ow\nis the the [ −1,−1,1] direction of trigonal distortion that makes an angle of ab out 19◦with the plane of the lattice pointing\ninside the plane.\nBefore moving on to the details of the spin Hamiltonian, we note that, on projecting to the subspace of |e′\n1g,↓/angbracketrightand\n|e′\n2g,↑/angbracketright, the spin and orbitals are no longer independent. Instead at every site there is a pseudo-spin-1 /2 degree of\nfreedom where the spin is lockedto the orbital wave function. This, we refer to as spin-orbital locking .\nIII. APPLICATION TO Na 2IrO3\nWe now apply the above results to the case of Na 2IrO3. The early X-Ray diffraction experiments4suggested a a\nmonoclinic C2/Cstructure for the compound and distorted IrO 6octahedra. However, more recent experiments see a\nbetter match for the X-Ray diffraction data with the space group C2/m.28,30They also unambiguously confirm the\npresence of uniform trigonal distortion of the IrO 6octahedra. However, the magnitude of such distortion is not clear\nat present. Further, experimental measurements suggest: (1 ) The magnetic transition occurs at TN= 15Kwhile the\nCurie-Weiss temperature is about Θ CW≈ −116K. This indicates presence of frustration. (2) The high temperatur e\nmagnetic susceptibility is anisotropic; the in-plane and out-of-plane susceptibilities are different. This may be due to\na trigonal distortion of the IrO 6octahedra4. (3) The magnetic specific heat is suppressed at low temperatures .4(4)\nRecent resonant X-ray scattering experiment5suggests that the magnetic order is collinear and have a 4-site unit c ell.\n(5) The magnetic moments have a large projection on the a-axis of the monoclinic crystal5. (6) A combination of\nthese experimental findings and density functional theory (DFT) calculations strongly suggest that a ‘zig-zag’ pattern\nfor the magnetic moments, as shown in Fig. Iin the ground state5, which has since been verified independently by\ntwo groups using Neutron scattering28,30.\nTaking these phenomenological suggestions, we try to apply the ab ove calculations to the case of Na 2IrO3. At\nthe outset, we must note that, in the above derivation of the spin H amiltonian we have assumed that the trigonal\ndistortion to be the largest energy scale followed by the SO coupling. While this extreme limit of projecting out\nthea1gorbitals most likely is not true for Na 2IrO3. However, we expect the real ground state to be adiabatically\nconnected to this limit. With this in mind, we now consider the case of Na 2IrO3.\nClearly, the exchanges (Eq. 7) depend both on the direction of the bond and the direction of the t rigonal distortion.\nSo it is important to ask about the direction of the latter. Comparing the crystallographic axes of Na 2IrO3, we find\nthat the direction [1 ,1,1] is perpendicular to the honeycomb plane while the other three dire ctions make an acute\nangle to it. In the monoclinic C2/mstructure, uniform trigonal distortion in these four directions ma y not cost the\nsame energy. In experiments5, the moments are seen to point along the a-axis of the monoclinic crystal which is\nparallel to the honeycomb plane. This, along with the fact that the m agnetic moment in our model is in the direction\nof ˆn(explained below) seems to suggest that ˆ n=1√\n3[−1,−1,1] is chosen in the compound (see Fig. 3). In the absence\nof a better theoretical understanding of the direction of the trig onal distortion, we take this as an input from the\nexperiments.\nTo identity different hopping paths (both direct and indirect), we co nsider various overlaps (see C) and find, while\nJ(3z)= 0,J(1z)/negationslash=J(2z)/negationslash= 0 are approximately (spatially-)isotropic and antiferromagnetic. For the exchanges of the\nHeisenberg terms, both J(2)andJ(3)are antiferromagnetic and isotropic (both of them result from indir ect hopping6\nFIG. 4: Mean field phase diagram for Eq. 6. The two axes are: x0=˜J(1)\nJ(1);y0=J(2)\nJ(1), where J(1)(˜J(1)) are related to the\nstrong(weak) NN exchange and J(2)is the 2ndand 3rdneighbour exchange (see Eq. 7). We take the Ising anisotropy to be 5%\nofJ(1). Note that, due to Ising anisotropies, one has zig-zag order aty0= 0.\nmediated by the Na s-orbitals and are expected to be comparable). For the NN Heisenberg exchanges, the couplings\nare antiferromagnetic, but, much more spatially anisotropic. We fin d that for the chosen direction of the trigonal\ndistortion, the coupling along one of the NN exchanges ( J(1)) (vizb1in Fig. 3) is different from the other two\nneighbours ( ˜J(1))(b2andb3in Fig.3).\nA. Mean-Field Theory and Magnetic Order\nWe now consider the mean field phase diagram for the above anisotro pic spin Hamiltonian. For J(1)being the\nlargest energy scale, the classical ground state for the model ca n be calculated within mean-field theory as a function\nofx0=˜J(1)/J(1)andy0=J(2)/J(1)(we have taken J(2)=J(3)). A representativemean-field phase diagram is shown\nin Fig.4. It shows a region of the parameter-space where the zig-zag ord er is stabilized29. The effect of the Ising\nanisotropies J(1z)andJ(2z)is to pin the magnetic orderingalongthe z-directionofthe pseudo-spinquantizationwhich\nis also the direction of the trigonal distortion ˆ n. They also gap out any Goldstone mode that arises from the orderin g\nof the pseudo-spins. The latter results in the exponential suppre ssion of the specific heat at low temperatures. The\nother competing phase with a collinear order is the regular two-subla ttice Neel phase.\nThe nature of the ground states may be understood from the follo wing arguments. In the presence of the ˆ nin\n[−1,−1,1] direction, the NN exchange coupling becomes anisotropic. When it is strong in one direction ( J(1)) and\nweak in two other directions ( ˜J(1)), for the bonds where the NN coupling becomes weak, the effects o f the small\nsecond and third neighbour interactions become significant. Since t he latter interactions are antiferromagnetic, they\nprefer anti-parallel alignment of the spins. As there are more seco nd and third neighbours, their cumulative effect can\nbe much stronger. This naturally leads to the zig-zag state. The NN antiferromagnetic interactions on the weaker\nbonds compete with the antiferromagnetic second and third neighb our interactions and frustrates the magnet. This\nsuppresses the magnetic ordering temperature far below the Cur ie-Weiss temperature.\nB. The spectrum for Spin-orbital waves\nThe low energy excitations about this magnetically ordered zig-zag s tate are gapped spin-orbital waves. Signatures\nof such excitations may be seen in future resonant X-Ray scatter ing experiments. It is important to note that this\n“pseudo-spin” waves actually contain both orbital and the spin com ponents due to the spin-orbital locking.\nWe calculate the dispersion of such spin-orbital waves to quadratic order using the well-known Holstein-Primakoff\nmethods. Thedetailsarediscussedin D. Arepresentativespin wavespectrum in thezig-zagphaseisshown inFig.5(a)\nand5(b). The spectrum is gapped and the bottom of the spin-wave dispersio n has some characteristic momentum\ndependence.\nC. Experimental Implications\nApart from the already discussed exponential suppression of low t emperature magnetic specific heat, the above\ncalculation predicts an interesting feature in the magnetic suscept ibility. The relation between the magnetic moment7\n(a)\n(b)\nFIG. 5: The “pseudo-spin” wave spectrum (contours of both th e bands are shown in (a) and a section is shown in (b)). The\nvalues used for the parameters are same as that used for the ca lculation of the mean field phase diagram (Fig. 4). We note\nthat, as expected, the spectrum is gapped.\nand the pseudo-spins is\n/vectorMi=−4µBˆnSz\ni, (10)\nwhereµBis the Bohr magneton. This follows from the twin facts that, in e′\ngsubspace, the angular momentum\ntransverse to ˆ nis quenched and the spins are locked to the orbitals with the axis of qu antization being ˆ nin our\npseudo-spin sector (see E). Thus, the magnetization is sensitive to the z-component of the pseudo-spin (the direction\nof which is shown in Fig. I). Indeed the magnetization has the largest projection along the a-axis of the monoclinic\ncrystal. This was seen in experiments5and was the motivation for choosing the [ −1,−1,1] direction for the trigonal\ndistortion. Along two other axes [ −1,1,−1]and [1,−1,−1], a large component of in-plane magnetization exists, but in\ndifferent directions. Finally the direction [1 ,1,1] is perpendicular to the honeycomb plane and leads to magnetizatio n\nin the same [1,1,1] direction. While this does not appear to be the case f or Na2IrO3, this may be more relevant for the\nless-distorted compound Li 2IrO3(see below). Eq. 10suggests that the magnetic susceptibility is highly anisotropic\nand depends on the cosine of the angle between the direction of mag netic field and ˆ n. Indeed signatures of such\nanisotropy have been already seen in experiments4. We emphasize that within this picture, the in-plane susceptibility\nalso varies with the direction of the magnetic field. So the ratio χ⊥/χ/bardblcan be lesser or greater than 1. The current\nexperiments4does nottell the in-planedirectionofmagneticfield andhence wecan notcommentonthe ratiopresently.\nHowever, the above picture is strictly based on atomic orbitals. One generally expects that there is also hybridization\nof the Ir d-orbitals with the oxygen p-orbitals. Such hybridization will contribute to a non-zero isotropic component\nto the susceptibility24. Also, as remarked earlier, in the actual compound, the SO coupling scaling may not be very\nsmall compared to the trigonal distortion limit scale. Additional pert urbation coming from the mixing with the a1g\norbitals will also contribute to decrease the anisotropy of the susc eptibility.8\nIV. SUMMARY AND CONCLUSION\nIn this paper, we have studied the effect of trigonal distortion and SO coupling and applied it to the case of the\nhoneycomb lattice compound Na 2IrO3. We find that, in the limit of large trigonal distortion and SO coupling, a\npseudo-spin-1 /2 degree of freedom emerges. Low energy Hamiltonian, in terms of t his pseudo-spin gives a ‘zig-zag’\nmagnetic order as seen in the recent experiments on Na 2IrO3. We have also calculated the low energy spin-wave\nspectrum and elucidated various properties of the compound that has been observed in experiments. The pseudo-spin\ncouples the physical spin and the orbitals in a non-trivial manner, sig natures of this may be seen in future inelastic\nX-ray resonance experiments probing the low energy excitations.\nWhile very recent experiments28,30clearly indicate presence of trigonal distortions, their magnitude is yet not\nconfirmed. On the other hand, the only available numerical estimate of the energy scale for trigonal distortion comes\nfrom the DFT calculations by Jin et al.13(based on C2/Cstructure). It suggests ∆ tri≈600meV. While, it is not\nclear if such a large value is in confomity with the experiments, at pres ent, the detection of trigonal distortion in\nexperiments is highly encouraging from the perspective of the pres ent calculations.\nIn these lights of the above calculations, it is tempting to predict the case of Li 2IrO3where recent experiments\nsuggest a more isotropic honeycomb lattice6,31. A possibility is that sizeable trigonal distortion is also present in\nLi2IrO3(so that the above discussion holds), but, the axis is perpendicular to the plane. What may be the fallouts in\nsuch a case ? Our present analysis would then suggest that the ant iferromagnetic exchanges are isotropic and equally\nstrong for the three NNs. This would develop 2-sublattice Neel ord er in the pseudo-spins with the magnetic moments\nbeing perpendicular to the plane. Also the further neighbour excha nges are rather weak (compared to Na 2IrO3) and\nhence frustration is quite small. Indeed recent experiments see or dering very close to the Curie-Weiss temperature,\nthe later being calculated from the high temperature magnetic susc eptibility data6,31. However, present experiments\ndo not rule out the possibility of small or no trigonal distortions in Li 2IrO3, in which case the limit of HK model11,26\nmay be appropriate.\nAcknowledgments\nWe acknowledge useful discussion with H. Gretarsson, R. Comin, S. Furukawa, H. Jin, C. H. Kim, Y.-J. Kim, W.\nWitczak-Krempa, H. Takagi. YBK thanks the Aspen Center for Phy sics, where parts of the research were done. This\nwork was supported by the NSERC, Canadian Institute for Advanc ed Research, and CanadaResearch Chair program.\nAppendix A: The microscopic model for Na 2IrO3\nThe generic Hubbard model (for the t2gorbitals) including the trigonal distortions, Hund’s coupling and the S O\ncoupling is\nH=−λ/summationtext\ni/vectorli·/vector si+Ht2g\ntri+/summationtext\nij/summationtext\nmm′/summationtext\nσσ′/parenleftBig\ntm;m′\nijd†\nimσdjm′σ′/parenrightBig\n+1\n2/summationtext\ni/summationtext\nmm′/summationtext\nσσ′Umm′d†\nimσd†\nim′σ′dim′σ′dimσ. (A1)\nHerem,m′=yz,xz,xy andσ=↑,↓andHt2g\ntriis given by Eq. 1. We note that the hopping is diagonal in spin space\nand in the cubic harmonic basis all hopping are real. Also, the hopping c ontain both the direct and indirect (through\nOxygen and Sodium) paths. We have taken Hund’s coupling into accou nt through Umm′, though this is expected to\nbe small in 5 dtransition metals. To a very good approximation the form of Umm′is given by\nUmm′≡\nU0U0−JHU0−JH\nU0−JHU0U0−JH\nU0−JHU0−JHU0\n, (A2)\nwhere the basis is given, as before, by Ψ†\ni= [d†\nyz,d†\nzx,d†\nxy].U0andJHare the intra orbital Coulomb repulsion and\nHund’s coupling term respectively.\nThe transformation between the operators in the trigonal basis, Φ†=/bracketleftBig\na†\n1g,e′†\n1g,e′†\n2g/bracketrightBig\n, andt2gbasis, Ψ†=/bracketleftbig\nd†\nyz,d†\nzx,d†\nxy/bracketrightbig\n, is given by Ψ m=Tm,MΦM. The transformation matrix is given by\nTm,M=1√\n3\nn1n1ω n1ω2\nn2n2ω2n2ω\nn3n3n3\n. (A3)9\nThe transformations for the hopping amplitudes and repulsion term are then given by\n˜tiM;jM′ =/summationtext\nm,m′T∗\nm,Mtm;m′\nijTm′,M′;\n˜UM1M2=/summationtext\nm,m′Umm′/parenleftbig\nT∗\nmM1TmM1/parenrightbig/parenleftbig\nT∗\nm′M2Tm′M2/parenrightbig\n. (A4)\nNotice that there are contributions to tm;m′\nijfrom both direct and indirect exchanges for the first, second and third\nneighbours, as confirmed from the DFT calculations by H. Jin et al.13. These show that there are contributions from\nboth direct and indirect hoppings for the first, second and third ne arest neighbours. Projecting them into the e′\ng\norbitals we get the effective hopping amplitudes which are then used in Eq.3. As for the Coulomb repulsion term,\nwe find that it has the following form\n˜UM1M2=U\n1 1 1\n1 1 1\n1 1 1\n, (A5)\nwhereU=U0−2JH/3. This form is then used in Eq. 3. The reason for this special form of ˜UM1M2lies in the fact\nthat the e′\ngorbitals have equal weight of the three t2gorbitals (see the wave functions in Eq. 2).\nAppendix B: Projection of Spin-Orbit coupling to the e′\ngsubspace\nThe SO coupling, when projected to the t2gorbitals give\nHt2g\nSO=−λ/vectorl·/vector s, (B1)\nwhere/vectorlis al= 1 angular momentum operator. We can re-write the t2gcubic harmonics in terms of the spherical\nharmonics of the effective l= 1 angular momentum operator. These are given by:\n|dyz/angbracketright=1√\n2[|1,−1/angbracketright−|1,+1/angbracketright];\n|dzx/angbracketright=ı√\n2[|1,−1/angbracketright+|1,−1/angbracketright];\n|dxy/angbracketright =|1,0/angbracketright (B2)\nThe projector for the e′\ngspace is: Pe′\ng=|e′\n1/angbracketright/angbracketlefte′\n1|+|e′\n2/angbracketright/angbracketlefte′\n2|. It turns out that /vectorl·/vector sis block diagonal in this subspace.\nHence\n/vectorl·/vector s=|e′\n1/angbracketright/angbracketlefte′\n1|/vectorl·/vector s|e′\n1/angbracketright/angbracketlefte′\n1|+|e′\n2/angbracketright/angbracketlefte′\n2|/vectorl·/vector s|e′\n2/angbracketright/angbracketlefte′\n2| (B3)\nMaking the “gauge” choice we get\n/angbracketlefte′\n1|/vectorl·/vector s|e′\n1/angbracketright= ˆn·/vector s;/angbracketlefte′\n2|/vectorl·/vector s|e′\n2/angbracketright=−ˆn·/vector s (B4)\nAppendix C: The hopping parameters\n1. Nearest neighbours\nThe nearest neighbours are shown in Fig. 6(a). There are two different processes contributing to the hopping.: 1 )\nthe direct hopping between the Ir atoms and 2) the indirect hopping between the Ir atoms mediated by the oxygen\natoms. In presence of the trigonal distortion which has a compone nt along the honeycomb plane (like in this case\n[−1,−1,1]) the magnitudes of the different hopping parameters are differen t in different directions (for both direct\nand indirect hopping). The results are shown in Table I.\nWe shall make an approximation here. We shall leave out the direction al dependence of the magnitudes on the\ndirection. The argument is that the essential directional depende nce due to the trigonal distortion has been taken\ncare of by the parameter ∆ 1. When the DFT13results are used to find the tight-binding parameters29, it is found\nthat (they use ∆ 1= 0) (here tdd1andtdd2are direct hopping and t0is the indirect hopping respectively.) tdd1=\n−0.5eV;tdd2= 0.15eV;t0= 0.25eV.10\n(a)\n (b)\n (c)\nFIG. 6: The 3 nearest neighbours (a), six 2ndnearest neighbours (b) and three 3rdnearest neighbours (c) of the central site.\nThe nomenclature has been used to label the hoppings.\n(a)NN:tam;b1m′\nm′\\mdxy dyz dzx\ndxytdd1(b1) - -\ndyz- tdd2(b1) −tdd2(b1)+t0(b1)+∆ 1(b1)\ndzx-−tdd2(b1)+t0(b1)−∆1(b1) tdd2(b1)\n(b)NN:tam;b2m′\nm′\\mdxy dyz dzx\ndxy tdd2(b2) -−tdd2(b2)+t0(b2)+∆ 1(b2)\ndyz - tdd1(b2) -\ndzx−tdd2(b2)+t0(b2)−∆1(b2)- tdd2(b2)\n(c)NN:tam;b3m′\nm′\\mdxy dyz dzx\ndxy tdd2(b3) −tdd2(b3)+t0(b3)+∆ 1(b3)-\ndyz−tdd2(b3)+t0(b3)−∆1(b3) tdd2(b3) -\ndzx - - tdd1(b3)\nTABLE I: The hopping paths (both direct and indirect) in the t2gbasis.\nPerforming the transformation to the e′\ngbasis, we have\nT(01)\nab1=1\n3[tdd1+2tdd2+(tdd2−t0)n3],\nT(02)\nab2=1\n3[tdd1+2tdd2+(tdd2−t0)n1],\nT(03)\nab3=1\n3[tdd1+2tdd2+(tdd2−t0)n2]. (C1)\nand\nT(z1)\nab1=−∆1√\n3n3\nT(z1)\nab2=−∆1√\n3n1\nT(z1)\nab3=−∆1√\n3n2 (C2)\nHence,\nJ(0)\nab1=4\n3U/bracketleftBig\n1\n3[tdd1+2tdd2+(tdd2−t0)n3]2−(∆1)2/bracketrightBig\n,\nJ(0)\nab2=4\n3U/bracketleftBig\n1\n3[tdd1+2tdd2+(tdd2−t0)n1]2−(∆1)2/bracketrightBig\n,\nJ(0)\nab3=4\n3U/bracketleftBig\n1\n3[tdd1+2tdd2+(tdd2−t0)n2]2−(∆1)2/bracketrightBig\n. (C3)11\nJ(1z)\nab1=8(∆1)2\n3U;\nJ(1z)\nab2=8(∆1)2\n3U;\nJ(1z)\nab1=8(∆1)2\n3U; (C4)\nwhere we have taken the direction of the trigonal distortion is take n to be uniform.\n2. Second nearest neighbour\nThese are shown in Fig. 6(b). These indirect hoppings are mediated by the Na atoms. In general, in presence of the\ntrigonal distortion in the [ −1,−1,1] direction, the magnitude of the hopping amplitudes are also direct ion dependent.\nHowever, since the magnitudes themselves are expected to be sma ll we shall neglect such directional dependence in\nthe magnitudes. The result is summarized in Table II.\n(a)NNN: tam;a1m′/tam;a4m′\nm′\\mdxydyzdzx\ndxy-t2+∆2-\ndyzt2−∆2--\ndzx- --(b)NNN: tam;a2m′/tam;a5m′\nm′\\mdxydyzdzx\ndxy--t2+∆2\ndyz---\ndzxt2−∆2--(c)NNN: tam;a3m′/tam;a6m′\nm′\\mdxydyzdzx\ndxy-- -\ndyz--t2+∆2\ndzx-t2−∆2-\nTABLE II: Hopping paths for the second nearest neighbours\nSo for the e′\ngbasis, we have\nT(02)\na,a1=T(02)\na,a4=−t2\n3n2, T(z2)\na,a1=T(z2)\na,a4=−∆2√\n3n2\nT(02)\na,a2=T(02)\na,a5=−t2\n3n1, T(z2)\na,a2=T(z2)\na,a5=−∆2√\n3n1\nT(02)\na,a3=T(02)\na,a6=−t2\n3n3, T(z2)\na,a3=T(z2)\na,a6=−∆2√\n3n3 (C5)\nFor example, tight binding fit of the DFT data uses only t2and finds t2≈ −0.075eV13,29. Therefore we have:\nJ(2)\na,aα=4\n3U/bracketleftbigg(t2)2\n3−(∆2)2/bracketrightbigg\n, J(2z)\na,aα=8(∆2)2\n3U. (C6)\n3. Third nearest neighbour\nThe third nearest neighbours are listed in Fig. 6(c). The hopping to the third nearest neighbour is mediated by the\nNa atoms. Again we shall neglect the directional dependence and ta ke these to be in the magnitudes of the hoping\namplitudes. The result is summarized in table III.\n(a)NNNN: tam;b′\n1m′\nm′\\mdxydyzdzx\ndxyt3(b′\n1)--\ndyz---\ndzx---(b)NNNN: tam;b′\n2m′\nm′\\mdxydyzdzx\ndxy---\ndyz-t3(b′\n2)-\ndzx---(c)NNNN: tam;b′\n3m′\nm′\\mdxydyzdzx\ndxy---\ndyz---\ndzx--t3(b′\n3)\nTABLE III: The hoppings for the third nearest neighbours\nTight-binding fit to the DFT results13,29indeed show that this hopping energy scale is of the order of\nt3(b′\nα) =tn≈ −0.075eV (C7)\nTherefore we have:\nT(03)\nab′α=tn\n3, T(z3)\nab′\n1= 0; (C8)12\nor,\nJ(3)\nij=4[tn]2\n9U;J(3z)\nij= 0; (C9)\nAppendix D: Spin Wave Spectrum\nTo calculate the spin wave spectrum for the zig-zag state we use th e usual Holstein-Primakoff method suited to\ncollinear ordering which may alternate in direction. More precisely we in troduce:\nSz=S−a†a;S+=√\n2Sa;S−=√\n2Sa†(D1)\nfor one direction and\nSz=−S+a†a;S+=√\n2Sa†;S−=√\n2Sa (D2)\nfor the other direction. Since there are4 sites per unit cell (refer Fig. 1(a) ofthe main text) the quadraticHamiltonian\nis a 8×8 matrix given by:\nHQ=Hcl+Hsp, (D3)\nwhereHclis the classical part dealt in the previous section. The spin wave Hamilt onian has the following form\nHsp=S\n2/summationdisplay\nkΨ†\nkHkΨk+Hs (D4)\nHere Ψ†\nk=/bracketleftBig\na†\nk,1,a†\nk,2,a†\nk,3,a†\nk,4,a−k,1,a−k,2,a−k,3,a−k,4/bracketrightBig\n(the subscript 1 ,2,3,4 refers to the four sites in the unit\ncell as shown in Fig. 1(a) of the main text) and\nHs=−S\n2[(1−2x+5y)−(2δ2−δ1)]NCell; (D5)\nHk =/bracketleftBigg\nAkBk\nB†\nkAk/bracketrightBigg\n(D6)\nwhereNcellis the number of unit cells and\nAk=\nχk0 0ηk\n0χkφk0\n0φ∗\nkχk0\nη∗\nk0 0χk\n;Bk=\n0ξkρk0\nξ∗\nk0 0ρk\nρ∗\nk0 0ξk\n0ρ∗\nkξk0\n(D7)\nwhere,\nχk= (2δ2−δ1)+(1−2x+5y+ycoskx); (D8)\nηk =xeıky/parenleftbig\n1+eıkx/parenrightbig\n; (D9)\nφk =x/parenleftbig\n1+eıkx/parenrightbig\n; (D10)\nξk = (1+2ycoskx+ye−ıky); (D11)\nρk =y(1+eıkx)(1+e−ıky). (D12)\nNow following usual methods we diagonalize\n/bracketleftBigg\nAkBk\n−B†\nk−Ak/bracketrightBigg\n(D13)\nto get the spin wave spectrum as plotted in Fig. 3(a) and 3(b) of the main text.13\nAppendix E: Projection of Zeeman term in the t2gand/braceleftbig\n|e′\n1g↓/a\\gbracketright,|e′\n2g↑/a\\gbracketright/bracerightbig\nsubspaces.\nThe Zeeman coupling term, when projected to the t2gspace, gives\nHt2g\nZ=µB/parenleftBig\n−/vectorl+2/vector s/parenrightBig\n·/vectorB. (E1)\nThus the magnetization after projection is given by:\n/vectorMt2g=µB/parenleftBig\n−/vectorl+2/vector s/parenrightBig\n(E2)\nThis when projected to the subspace |e′\n1,↑/angbracketrightand|e′\n2,↓/angbracketrightgives (using the Block diagonal property of the orbital angular\nmomentum as above):\n˜HZ= 4µBSzˆn·/vectorB (E3)\nwhere/vectorS(note that this is in upper case compared to the physical spin writte n lower case) is the emergent pseudo-\nspin-1/2 per site. This is the emergent degree of freedom at low ene rgies. Clearly, the magnetic moment is then given\nby Eq. 7 of the main text.\n1Y. Okamoto, M. Nohara, H. Aruga-Katori, and H. Takagi, Phys. Rev. Lett. 99, 137207 (2007).\n2B. J. Kim, et al., Phys. Rev. Lett. 101, 076402 (2008).\n3B. J. Kim, H. Ohsumi, T. Komesu, S. Sakai, T. Morita, H. Takagi , and T. Arima, Science 323, 1329 (2009).\n4Y. Singh and P. Gegenwart, Phys. Rev. B 82, 064412 (2010).\n5X. Liu, T. Berlijn, W.-G. Yin, W. Ku, A. Tsvelik, Y.-J. Kim, H. Gretarsson, Y. Singh, P. Gegenwart, and J. P. Hill, Phys.\nRev. B 83, 220403(R) (2011).\n6Y. Singh, S. Manni, J. Reuther, T. Berlijn, R. Thomale, W. Ku, S. Trebst and P. Gegenwart, Phys. Rev. Lett. 108, 127203\n(2012).\n7S. Nakatsuji Y. Machida, Y. Maeno, T. Tayama, T. Sakakibara, J. van Duijn, L. Balicas, J. N. Millican, R. T. Macaluso,\nand Julia Y. Chan, Phys. Rev. Lett. 96, 087204 (2006).\n8S. Zhao, J. M. Mackie, D. E. MacLaughlin, O. O. Bernal, J. J. Is hikawa, Y. Ohta, and S. Nakatsuji, Phys. Rev. B 83,\n180402(R) (2011).\n9F. F. Tafti, J. J. Ishikawa, A. McCollam, S. Nakatsuji, and S. R. Julian, Phys. Rev. B 85, 205104 (2012).\n10A. Shitade et al., Phys. Rev. Lett. 102, 256403 (2009).\n11J. Chaloupka, G. Jackeli, and G. Khaliullin, Phys. Rev. Lett . 105, 027204 (2010); G. Jackeli and G. Khaliullin, Phys. Rev .\nLett. 102, 017205 (2009).\n12H.-C. Jiang, Z.-C. Gu, X.-L. Qi, and S. Trebst, Phys. Rev. B 83 , 245104 (2011).\n13H. Jin, H. Kim, H. Jeong, C. H. Kim, and J. Yu, arXiv:0907.0743 (unpublished) (2009); C. H. Kim, H. S. Kim, H. Jeong,\nH. Jin, and J. Yu, Phys. Rev. Lett. 108, 106401 (2012).\n14M. J. Lawler, A. Paramekanti, Y. B. Kim, and L. Balents, Phys. Rev. Lett. 101, 197202 (2008).\n15G. Chen and L. Balents, Phys. Rev. B 78, 094403 (2008).\n16M. R. Norman and T. Micklitz, Phys. Rev. Lett. 102, 067204 (20 09).\n17D. Podolsky, A. Paramekanti, Y. B. Kim, T. Senthil, Phys. Rev . Lett. 102, 186401 (2009).\n18D. Podolsky, Y. B. Kim, Phys. Rev. B 83, 054401 (2011).\n19D. Pesin and L. Balents, Nat Phys 6, 376 (2010).\n20B.-J. Yang and Y. B. Kim, Phys. Rev. B 82, 085111 (2010).\n21X. Wan, A. Turner, A. Vishwanath, S. Y. Savrasov, Phys. Rev. B 83, 205101 (2011); X. Wan, A. Vishwanath, S. Y. Savrasov,\nPhys. Rev. Lett. 108, 146601 (2012).\n22W. Witczak-Krempa, and Y. B. Kim, Phys. Rev. B 85, 045124 (201 2).\n23F. Wang, and T. Senthil, Phys. Rev. Lett. 106, 136402 (2011).\n24G. Chen, R. Pereira, and L. Balents, Phys. Rev. B 82, 174440 (2 010).\n25P. Fazekas, Lecture notes on electron correlation and magnetism . (Singapore, World Scientific, 1999).\n26J. Reuther, R. Thomale, S. Trebst, Phys. Rev. B 84, 100406(R) (2011).\n27I Kimchi, Y.-Z. You, Phys. Rev. B 84, 180407(R) (2011).\n28S. K. Choi, R. Coldea, A. N. Kolmogorov, T. Lancaster, I. I. Ma zin, S. J. Blundell, P. G. Radaelli, Yogesh Singh, P.\nGegenwart, K. R. Choi, S.-W. Cheong, P. J. Baker, C. Stock, an d J. Taylor, Phys. Rev. Lett. 108, 127204 (2012).\n29For example, the parameters calculated from DFT13fall within this regime (Private Communication with H. Jin a nd C. H.\nKim).\n30Feng Ye, S. Chi, H. Cao, B. C. Chakoumakos, J. A. F-Baca, R. Cus telcean, T. Qi, O. B. Korneta, G. Cao, arXiv:1202.3995\n(unpublished) (2012).\n31H. Takagi (Private Communication)."    },    {        "title": "1702.03199v1.Orbital_and_spin_order_in_spin_orbit_coupled__d_1__and__d_2__double_perovskites.pdf",        "content": "Orbital and spin order in spin-orbit coupled d1and d2double perovskites\nChristopher Svoboda,1Mohit Randeria,1and Nandini Trivedi1\n1Department of Physics, The Ohio State University, Columbus, Ohio 43210, USA\n(Dated: February 13, 2017)\nWe consider strongly spin-orbit coupled double perovskites A 2BB'O 6with B' magnetic ions in\neitherd1ord2electronic con\fguration and non-magnetic B ions. We provide insights into several\nexperimental puzzles, such as the predominance of ferromagnetism in d1versus antiferromagnetism\nind2systems, the appearance of negative Curie-Weiss temperatures for ferromagnetic materials and\nthe size of e\u000bective magnetic moments. We develop and solve a microscopic model with both spin and\norbital degrees of freedom within the Mott insulating regime at \fnite temperature using mean \feld\ntheory. The interplay between anisotropic orbital degrees of freedom and spin-orbit coupling results\nin complex ground states in both d1andd2systems. We show that the ordering of orbital degrees of\nfreedom in d1systems results in coplanar canted ferromagnetic and 4-sublattice antiferromagnetic\nstructures. In d2systems we \fnd additional colinear antiferromagnetic and ferromagnetic phases\nnot appearing in d1systems. At \fnite temperatures, we \fnd that orbital ordering driven by both\nsuperexchange and Coulomb interactions may occur at much higher temperatures compared to\nmagnetic order and leads to distinct deviations from Curie-Weiss law.\nI. INTRODUCTION\nStrong SOC in correlated materials has provided a\nplatform for quantum spin liquids, Weyl semimetals, and\nan ongoing search for high Tcsuperconductivity in the\niridates.1,2Among the strongly spin-orbit coupled mate-\nrials include 4 dand 5ddouble perovskites A 2BB'O 6with\nelectron counts d1-d5on the magnetic B' ion. Here we re-\nstrict our discussion to Bsite ions that are non-magnetic.\nDue to large distances between the magnetic ions, these\nmaterials are often Mott insulators and present a promis-\ning class of materials to explore the interplay of strong\ncorrelations and spin-orbit coupling. Additionally, the\nmagnetic sites form an FCC lattice leading to frustrated\nmagnetism.\nEach electron count carries a di\u000berent total angular\nmomentum quantum number providing a new platform\nfor studying novel magnetism. The half \flled t2gshells\nofd3ions result in an e\u000bective spin-3/2 model which\nare nominally expected to be described as a classically\nfrustrated spin systems.3,4In the opposite limit, d5sys-\ntems withj= 1=2 are both intrinsically more quantum\nand are protected from local distortions by time rever-\nsal symmetry. These systems may o\u000ber a route to real-\nizing Kitaev physics and more generally spin liquids in\nthree dimensions.5,6Thed4case is especially unique since\nspin-orbit coupling dictates that local moments should\nbe absent and magnetism is forbidden. However several\ntheoretical7{10and experimental11{13studies have exam-\nined the possibility of inducing local moments through\nsuperexchange interactions.\nThed1andd2electron counts stand out in that they\ncombine aspects of the former three electron counts and\nwill be the focus of this paper. First, they possess local\nangular momenta large enough to support quadrupolar\norder. Second, they possess unquenched orbital degrees\nof freedom that result in highly anisotropic interactions\nbetween magnetic ions.14,15Both of these aspects will al-\nlow for the orbital degrees of freedom to play a signi\fcantrole in determining the spin, orbital, and spin-orbital or-\ndering.\nWhile both electron counts have similar potential, ex-\nperimental observations of magnetic properties of the d1\ndouble perovskites have drawn signi\fcant interest. The\n4d1compound Ba 2YMoO 6shows no long range magnetic\norder down to 2K despite having a large Curie-Weiss\ntemperature \u0012=\u0000160Kand retaining cubic symmetry\nwhich leads to the conclusion that the ground state con-\nsists of valence bonds16{20. Among the 5 d1compounds\nare ferromagnetic Ba 2NaOsO 621{24, Ba 2MgReO 625,26,\nand Ba 2ZnReO 626which is unusual since ferromagnetism\nin Mott insulators is uncommon. There are two addi-\ntional twists to the story: \frst, negative Curie-Weiss\ntemperatures have been observed in these ferromag-\nnets, and, second, Ba 2LiOsO 6is antiferromagnetic de-\nspite sharing the same cubic structure as Ba 2NaOsO 6.21\nThed2compounds o\u000ber a similar platform to search\nfor unusual magnetism, however experimental stud-\nies seem to suggest that antiferromagnetic interactions\nare more prevalent in d2systems. Phase transitions\nto antiferromagnetic order are reported in Ca 3OsO 627,\nBa2CaOsO 628, and Sr 2MgOsO 629,30while glass-like tran-\nsitions are reported in Ba 2YReO 631, Ca 2MgOsO 629,\nand Sr 2YReO 632. There are also several alleged sin-\nglet ground states: La 2LiReO 631, SrLaMReO 633, and\nSr2InReO 632.\nMany theoretical investigations have been undertaken\nto understand the magnetism in both d1andd2double\nperovskites. In the limit of large spin-orbit coupling, the\nspinS= 1=2 and orbital Le\u000b=\u00001 angular momenta\nadd to a total angular momenta of j= 3=2. Within\nthejj-coupling scheme, magnetic moments are identi-\ncally zero due to cancellation of the spin and orbital\nmoments, M= 2S\u0000L= 0. On the other hand, d2\nsystems allow for a nonzero moment of M=p\n6\n2\u0016BJ\nfor totalJ= 2 within the LS-coupling scheme. How-\never both systems are experimentally observed to be\nmagnetic. Density functional theory studies have re-arXiv:1702.03199v1  [cond-mat.str-el]  10 Feb 20172\ncently revealed the importance of oxygen hybridization\nin suppressing the orbital moment so that a large non-\nzero moment results.34,35Other density functional theory\nstudies have pointed out that spin-orbit coupling and\nhybridized orbitals play a major role in opening a gap\nwithin DFT+U.36{38\nModel approaches have shed some light on the nature\nof the magnetically ordered states by using spin-orbital\nHamiltonians39, projecting spin-orbital Hamiltonians to\nthe lowest energy total angular momentum multiplet40,41\nor lowest energy doublet42, and other approaches43. In\nboth electron counts, Chen et. al.40,41\fnd canted ferro-\nmagnetism accompanied by quadrupolar order occupies\na majority of parameter space. Additionally they \fnd a\nnovel non-colinear antiferromagnetic phase in d2, but not\nd1, which was recently found in d1as the most energet-\nically favorable antiferromagnetic state.39Proposals for\nboth valence bond ground states39,40and quantum spin\nliquids40,44also exist.\nYet many puzzles remain unsolved. Despite predic-\ntions for canted ferromagnetic phases40,41in bothd1and\nd2, many ferromagnetic d1systems exist but few d2fer-\nromagnets exist. Furthermore, the physical origin of neg-\native Curie-Weiss temperatures in these ferromagnets is\nstill not understood, and there are multiple studies try-\ning to reproduce the magnitudes of the e\u000bective Curie\nmoments experimentally measured.\nHere we study magnetic models for the d1andd2cubic\ndouble perovskites with strong spin-orbit coupling with\nboth spin and orbital degrees of freedom at \fnite temper-\nature. While we are focusing on applications to ordering\nin 5dcubic double perovskites, our results may also ap-\nply to 4dcompounds and non-cubic double perovskites\nas well. Despite the greater complexity than the J= 3=2\nandJ= 2 multipolar descriptions, the spin-orbital pic-\nture actually leads to an intuitive and qualitative under-\nstanding of several aspects of the phenomenology in these\ndouble perovskites. In our study of magnetically ordered\nphases, we arrive at several conclusion which we now list.\nFirst, our results emphasize the importance of the\norbital degrees of freedom and anisotropic interactions\nthat accompany them. In particular, we show that the\nanisotropic interactions result in orbital order that sta-\nbilizes exotic magnetic order. The orbital (quadrupolar)\nordering temperature scale is set both by superexchange\ninteractions and by inter-site Coulomb repulsion, and, in\nseveral cases, the orbital ordering temperature can be\nmuch larger than the magnetic ordering temperature.\nSecond, although we start with the same electronic\nmodel for both d1andd2systems, the energetics of the\nground states strongly depend on the electron count.\nThis is re\rected in how the spin and orbital degrees of\nfreedom order and provides a qualitative understanding\nfor why ferromagnetism has been repeatedly observed in\nd1systems while antiferromagnetic interactions remain\nprevalent in d2systems.\nThird, the onset of orbital order causes changes in mag-\nnetic susceptibility resulting in non-Curie-Weiss behav-ior. Our model gives the appearance of a negative Curie-\nWeiss temperature for the ferromagnetic phase while still\nretaining a properly diverging susceptibility at the ferro-\nmagnetic transition.\nFourth, if orbital order occurs, hybridization with oxy-\ngen alone does not reproduce the experimentally deter-\nmined values of the magnetic moments in d1systems.\nCorrections are necessary which may arise from dynami-\ncal Jahn-Teller e\u000bects34and more generally with mixing\nof thej= 3=2 andj= 1=2 states as we propose. Charge\ntransfer from oxygen might also be considered for sys-\ntems where the Curie moment has measured be in excess\nof 1\u0016B.45\nLastly, we outline where our calculations stand with\nrespect to other work. First, our zero temperature phase\ndiagram for d1contains a 4-sublattice antiferromagnetic\nphase and a canted ferromagnetic phase which share un-\nderlying orbital ordering patterns. Our \fndings are com-\npatible with those of Romh\u0013 anyi et. al.39, and we further\nprovide a clear interpretation of why these orbital order-\ning patterns occur, how they dictate the magnetic order-\ning, and then extend our calculations to \fnite tempera-\nture. Like Chen et. al., we \fnd that orbital ordering can\noccur at temperatures much higher than the magnetic\nordering temperature, however, it leads to a di\u000berent in-\nterpretation of the negative Curie-Weiss temperature in\nd1ferromagnets. Furthermore, our spin-orbital approach\nincludes mixing between the j= 3=2 andj= 1=2 states\ninduced by orbital order and intermediate spin-orbit cou-\npling energy scales. Second, our zero temperature phase\ndiagram for d2di\u000bers remarkably from that of Chen et.\nal.41which we discuss in detail in later sections. How-\never, the most signi\fcant di\u000berence is in the energetics\nof antiferromagnetism versus ferromagnetism which gives\na qualitative explanation for the broadly observed dif-\nferences in ordering between 5 d1and 5d2compounds.\nFinally, we do not consider valence bond or spin liquid\nphases in this work although both may be applicable to\nd1andd2systems.\nExperimentally, many of our \fndings can be tested us-\ning multiple probes. At the orbital ordering temperature,\nthere will be second order phase transition with a signa-\nture in heat capacity as well as changes in the magnetic\nsusceptibility which are relevant for both powder sam-\nples and single crystals. NMR/NQR has recently found\nevidence of time-reversal invariant order above the mag-\nnetic ordering temperature in Ba 2NaOsO 6.24Resonant\nX-ray scattering may also provide crucial insights into\nthis hidden order as it is sensitive to orbital occupancy.\nWe show that time-reversal invariant orbital order oc-\ncurs in both ferromagnetic and antiferromagnetic phases\nwe \fnd, and we suggest that experimental probes which\nare sensitive to such order should also be pointed at the\nantiferromagnetic compounds as well.3\nII.d1DOUBLE PEROVSKITES\nHere we develop a spin-orbital model for the d1dou-\nble perovskites with magnetic B' ions with spin-orbit\ncoupling featuring both spin-orbital superexchange and\ninter-site Coulomb repulsion between B' ions. We then\nsolve the model within mean \feld theory at both zero\ntemperature and \fnite temperature. At zero tempera-\nture, we \fnd phases with orbital order and show how this\nordering restricts the magnetic order. At \fnite temper-\nature, we examine how orbital order modi\fes magnetic\nsusceptibility and the Curie-Weiss parameters.\nA. Model\nIn the presence of cubic symmetry, the magnetic B0\nions form an FCC lattice and contain one electron in the\noutermostdshell. The \fve degenerate levels are split by\nthe octahedral crystal \feld into the higher energy egor-\nbitals and lower t2gorbitals so that the t2gshell contains\none electron. The electronic structure for the t2gorbitals\nmay be approximated by a nearest neighbor tight-binding\nmodel where only one of the three orbitals interacts along\neach direction.\nHTB=\u0000tX\n\u000bX\nhiji2\u000bcy\ni;\u000bcj;\u000b+ h:c: (1)\nHere the sum over \u000bis over allyz,zx, andxyplanes in\nthe FCC lattice. For B' sites in plane \u000b, the\u000borbital on\nsiteioverlaps with the \u000borbital on site j. Each\u000borbital\nhas four neighboring \u000borbitals in its plane plane giving\na total of twelve relevant B' neighbors per B' site. In\naddition to the tight-binding term, the unquenched t2g\norbital angular momentum L= 1 results in a spin-orbit\ncoupling on each B0ion2HSO=\u0000\u0015P\niLi\u0001Si. Here the\norbitalL= 1 and spin S= 1=2 operators both satisfy\nthe usual commutation relations for angular momentum\n(ie.L\u0002L=i~L).\nThe on-site multi-orbital Coulomb interaction is given\nbyHU=P\niH(i)\nU\nH(i)\nU=U\u00003JH\n2Ni(Ni\u00001) +JH\u00005\n2Ni\u00002S2\ni\u00001\n2L2\ni\u0001\n(2)\nwhereUis the Coulomb repulsion and JHis Hund's\ncoupling.46Being in the Mott limit, we calculate the\ne\u000bective spin-orbital superexchange Hamiltonian within\nsecond order perturbation theory. The superexchange\nHamiltonian is given by the following\nHSE=\u0000JSE\n4X\n\u000bX\nhiji2\u000b\b\nr1(3\n4+Si\u0001Sj)(n\u000b\ni\u0000n\u000b\nj)2\n+(1\n4\u0000Si\u0001Sj)\u0002\nr2(n\u000b\ni+n\u000b\nj)2+4\n3(r3\u0000r2)n\u000b\nin\u000b\nj\u0003\t(3)\nwhereJSE= 4t2=Uis the superexchange strength and\nr1= (1\u00003\u0011)\u00001,r2= (1\u0000\u0011)\u00001, andr3= (1+2\u0011)\u00001with\n\u0011=JH=U.15Here thet2gorbital electron occupationnumbers are written as n\u000b\ni=cy\ni;\u000bci;\u000bThe top line of\nequation (3) contributes a ferromagnetic spin interaction\nwhich requires that one of the two orbitals along a bond\nis occupied while the other is unoccupied. The bottom\nline of equation (3) contributes an antiferromagnetic spin\ninteraction which is maximized when both orbitals along\na bond are occupied. The strength of Hund's coupling,\nJH=U, determines the strength of the two interactions\nrelative to each other.\nDue to the large spatial extent of 5 dorbitals from\nstrong oxygen hybridization, we include a term account-\ning for the Coulomb repulsion between orbitals on di\u000ber-\nent sites.40Let (\u000b;\f;\r ) be a cyclic permutation of the\nt2gorbitals (yz;zx;xy ). The repulsion is described by\nthe following:\nHV=VX\n\u000bX\nhiji2\u000bh\n9\n4n\u000b\nin\u000b\nj\u00004\n3(n\f\ni\u0000n\r\ni)(n\f\nj\u0000n\r\nj)i\n(4)\nWhile the coe\u000ecients in (4) are only quantitatively cor-\nrect in the limit of quadrupolar interactions, they qualita-\ntively capture the correct repulsion. For example, within\nthexyplane, a pair of xyorbitals repel each other more\nthan anxyandyzorbital.\nThe total e\u000bective magnetic interaction then reads\nH=HSO+HSE+HV. Of the three parameters, spin-\norbit coupling has the largest energy scale \u0015\u00190:4 eV\nfor the 5doxides while superexchange and and intersite\nCoulomb repulsion are taken to have energy scales on the\norder of tens of meV. For 4 doxides, the spin-orbit energy\nscale is reduced to 0 :1\u00000:2 eV so that mixing between\nthej= 3=2 andj= 1=2 states is likely to occur. While\nour spin-orbit superexchange interaction is calculated in\ntheLS-coupling scheme, recent evidence suggests that\nthe true picture for the 5 doxides lies between the LS\nandjjlimits.47\nWe decouple HSEandHVinto all possible on-site mean\n\felds, i.e. Sin\u000b\niSjn\u000b\nj!Sin\u000b\nihSjn\u000b\nji+hSin\u000b\niiSjn\u000b\nj\u0000\nhSin\u000b\niihSjn\u000b\nji. Since the FCC lattice is not bipartite, we\ndecouple into four inequivalent sites shown in Fig. 1(a)\nwhere each set of four inequivalent neighbors forms a\ntetrahedron. Since the mean \felds need not factor into\nthe product of spins and orbitals, hSin\u000b\nii 6=hSiihn\u000b\nii,\nthere are a total of 15 mean \felds per site comprised of\nthree spin operators, three orbital operators, and prod-\nucts of the spin and orbital operators. Applying the con-\nstraint that one electron resides on each site, there are 11\nindependent mean \felds per site giving a total of 44 mean\n\felds in the tetrahedron. We then numerically solve for\nthe lowest energy solutions of the mean \feld equations.\nB. Zero Temperature Mean Field Theory\nIn the limit where spin-orbit coupling \u0015is the domi-\nnant energy scale, the magnetically ordered phases can\nbe characterized by an arrangement of ordered J= 3=2\nmultipoles.40However, when JSEand\u0015are comparable,4\n0.00 0.05 0.10 0.15 0.200.00.20.40.60.8\nkBT/λnyz\nnzxnxy\nMTo Tc\n0.0 0.1 0.2 0.3 0.4 0.50.000.050.100.150.20\nμeff(μB)kBTo/λ\n←0.02=V/λ←0.040.06→AFM\n4-sublatticeCanted\nFM\nV/λ=0\n0.01\n0.02\n0.03\n0.0 0.2 0.4 0.6 0.80.100.120.140.160.180.20\nJSE/λJH/U\nLSLS(a) (b)\n(c)\n(d) (e)\nFIG. 1. (a) FCC lattice decoupled into four inequivalent sites shown by four di\u000berent colors. (b) The orbital ordering\npattern driven by both JSEandVconstrains the direction of orbital angular momentum. Deviations from the j= 3=2 limit\nproduce a net magnetic moment Mas the spin and orbital components separate. (c) The zero temperature phase diagram\nshows phases where the moments in each plane of the page (eg. plane containing yellow and black sites) are collinear and the\nmoments between planes are at approximately 90 degrees due to the orbital ordering pattern. Increasing inter-orbital repulsion\nVbetween sites reduces minimum strength of Hund's coupling required to induce FM. (d) Mean \feld values for the bottom\nsites (black, yellow) are shown as a function of temperature. The nyzorbital (red) has the largest occupancy followed by the\nxyorbital (blue). (e) With JSE= 0, we calculate the orbital ordering temperature Toand e\u000bective Curie moment enhancement\n\u0016e\u000bfor di\u000berent values of V.\na multipolar description within the j= 3=2 states breaks\ndown and consequently both spin and orbital parts must\nbe considered independently. Furthermore, the orbital\ncontributions come in the forms of both orbital occu-\npancyn\u000band orbital angular momentum L. Sincen\u000b,\nL, andSare coupled, there is competition between order\nparameters which results in non-trivial ordering.\nThe zero temperature phase diagram is shown in\nFig. 1(c) as a function of the strength of Hund's cou-\npling\u0011=JH=Uand superexchange JSE=\u0015. Large val-\nues of\u0011support a canted ferromagnetic (FM) structure\nwhile smaller values support an antiferromagnetic (AFM)\nstructure. The spin-1/2 and orbital-1 angular momenta\norder parameters hSiandhLiare shown for each of the\nfour inequivalent sites from Fig. 1(a). In both phases, one\nof the three directions has no ordered angular momenta,\ne.g.hLzi=hSzi= 0, so that both magnetic structures\nare co-planar. Both phases feature some separation of\nthe ordered spin and orbital moments which increases as\na function of JSE=\u0015. To understand why these magnetic\nstructures emerge, we examine the orbital occupancy or-\nder parameters, n\u000b, separately from the magnetic order\nparameters. In both the FM and AFM phases, there is an\norbital ordering pattern pictured in Fig. 1(b). The two\nsites in the lower plane of Fig. 1(b) have the yzorbital(red) with the highest electron occupancy while the xy\norbital (blue) receives the second highest and the zxor-\nbital receives the lowest (green, not pictured). The two\nsites in the upper plane have identical ordering except\nthe roles of the yzandzxorbitals are reversed. Qualita-\ntively this orbital ordering pattern is favored by both the\nHVandHSEterms which pushes electrons onto orbitals\nthat have small overlaps. This allows the electron on a\ngreen orbital to hop onto an unoccupied green orbital in\nthe plane directly above or below (and similarly for red\norbitals). Since these mechanisms work to suppress the\noverlap of half \flled orbitals, ferromagnetic interactions\nmay become energetically favorable. A derivation of the\nmean \feld solution for HVis provided in Appendix A.\nOnce orbital order sets in, the allowed magnetic phases\nare restricted by the direction of orbital angular momen-\ntum. Full orbital polarization is time-reversal invariant\nand would not allow orbital magnetic order. However\nFig. 1(d) shows that each site has at least two orbitals\nwith non-negligible occupancy which allows for the devel-\nopment of an orbital moment. Thus the direction of the\norbital moment is determined by the direction common\nto the two planes of occupied orbitals with the overall\nsign of the direction (e.g. + xor\u0000x) left undetermined.\nFigure 1(c) shows that the orbital angular momenta be-5\ntween planes are close to 90 degrees apart for both FM\nand AFM phases. As spin and orbital angular momen-\ntum are coupled together, the spin moments will select\nwhich direction the orbital moments choose (i.e. + xor\n\u0000x). The decision to enter an FM or AFM state is then\ndetermined by the spin interactions characterized both\nby the strength of \u0011=JH=Uand the magnitude of the\norbital order parameter. If \u0011is large, then ferromag-\nnetic spin interactions follow and result in both the spin\nand orbital degrees of freedom aligning within each xy\nplane producing a net canted FM structure. If \u0011is small,\nthen antiferromagnetic spin interactions follow which re-\nsult in the 4-sublattice AFM structure. We note that\nthe Goodenough-Kanamori-Anderson rules48{50are not\nenough to determine whether FM or AFM is favored,\nand the interplay between spin-orbit coupling and the\nanisotropic orbital degrees of freedom play a crucial role\nin tipping the energy balance one way or the other.\nThere are two additional factors that determine if the\nFM or AFM state is selected. The dominant e\u000bect is\nthe degree of orbital polarization. When the strength\nof inter-orbital repulsion Vis increased, the tendency\nfor orbitals to order becomes stronger. This disfavors\nthe overlap of half \flled orbitals causing AFM superex-\nchange, and hence promotes FM superexchange. Figure\n1(c) shows a dramatic shift toward FM when a small V\ninteraction is included. The second e\u000bect comes from the\nseparation of spin and orbital degrees of freedom. When\nJSEbecomes comparable to \u0015, the spin moments can\npartially break away from the orbital moments tending\nmore toward a regular spin FM state instead of a canted\nspin FM state. Since a spin AFM state does not bene-\n\ft from this separation to the same extent, FM becomes\nincreasingly energetically favorable.\nDimer phases have been proposed39,40and o\u000ber a way\nto explain the absence of magnetic order in d1materials.\nHowever when \u0015=JSEis large, these dimer phases only\noccur at very small values of \u0011=JH=U.39Furthermore,\norbital repulsion Vacts to further suppress dimerization.\nSince our focus is on the magnetically ordered phases of\nthese double perovskites, we will not pursue these possi-\nbilities in this work.\nC. Finite Temperature Mean Field Theory\nWe now examine the model at \fnite temperature. Fig-\nure 1(d) shows a characteristic order parameter vs tem-\nperature curve. At high temperatures all order parame-\nters are trivial and each orbital occupancy takes a value\nofnyz=nzx=nxy= 1=3. As temperature is low-\nered, the \frst transition is to a time reversal invariant\norbitally ordered state [see Fig. 1(b)] at temperature To\nwhose scale is set both by VandJSE. AtTo, the entropy\nreleased is from orbital degeneracy, even when V= 0.\nBelow the second transition at a temperature Tcwhose\nenergy scale is set only by JSE, time reversal symmetry\nis broken on each site with the development of magneticorder, and the remaining entropy is released.\nThe fundamental question arises of how large the ex-\nchange interaction JSEand repulsion Vare in materi-\nals systems. Fits to experimental susceptibility21show\nBa2LiOsO 6and Ba 2NaOsO 6have relatively small Curie-\nWeiss temperatures of \u0012=\u000040:5 K and\u0012=\u000032:5 K\nrespectively indicating that JSEin cubic 5d1double per-\novskites is weak. However integrated heat capacity22of\nBa2NaOsO 6shows an entropy release just short of Rln 2\natTcconsistent with the splitting of a local Kramer's\ndoublet with no further anomalies in heat capacity up to\n300 K. This suggests To\u001dTcso thatVis the most rele-\nvant parameter for determining the properties well above\nTc.\nSince the onset of orbital order necessarily alters the\nangular momenta available to order and respond to an\napplied magnetic \feld, we calculate how the e\u000bective\nCurie-Weiss constant depends on orbital ordering. Using\nJSE= 0, we calculate the temperature dependent suscep-\ntibility within mean \feld theory as a function of tempera-\nture for di\u000berent values of V=\u0015. For each value of V=\u0015we\ncalculate both the orbital ordering temperature Toand\nthe e\u000bective Curie moment \u0016e\u000b=g\u0016Bp\nJ(J+ 1) from\na \ft to low temperature inverse susceptibility. Fig. 1(e)\ngives numerical results from our mean \feld theory that\nshows a linear relationship between Toand\u0016e\u000b. In the\nabsence of orbital order, the projection of the magneti-\nzation operator to the J= 3=2 space is identically zero.\nHowever once orbital order sets in, the j= 1=2 com-\nponents of the wavefunction get mixed with the j= 3=2\ncomponents. The matrix elements that connect these two\nJspaces then acquire expectation values and allow the\ne\u000bective Curie moment to become non-zero. An approx-\nimate derivation of this relation is provided in Appendix\nA.\nIn addition to the perturbative separation of LandS\ndue to mixing of the Jmultiplets, hybridization with oxy-\ngen has been shown to greatly reduce the orbital contri-\nbution to the moment.34,35Here the magnetization oper-\nator assumes the form M= 2S\u0000\rLwhere\r= 0:536 and\nresults in an e\u000bective Curie moment of 0 :60\u0016Bcompared\nto an experimental value of 0 :67\u0016B.21However the onset\nof quadrupolar order within the j= 3=2 states results in\na reduction of the nominal 0 :60\u0016Bvalue. In general, the\nprojection of a linear combination of the nyz,nzx, and\nnxyoperators to the j= 3=2 states is (up to a constant\nshift) a linear combination of the operators J2\nx\u0000J2\nyand\nJ2\nz. By projecting to the lowest energy doublet induced\nby these operators, we may calculate the gfactors for this\npseudo-spin 1/2 space. While the gfactors are di\u000berent\nin the three cubic directions due to the anisotropic na-\nture of quadrupolar order, the sum of the squares is a con-\nstant, and the powder average is g2=1\n3(g2\nx+g2\ny+g2\nz) = 3.\nThen splitting of the j= 3=2 states reduces the Curie\nmoment by a factor of ( gp\n3=4)=(p\n15=4) =p\n3=5 which\nmakes the calculated moment 0 :47\u0016B. We \fnd that mix-\ning between the j= 3=2 andj= 1=2 states brings the\ncalculated moment closer to experimental values.6\nwithout\nhybridizationwith\nhybridization\n0 0.04 0.08 0.12012345\nkBT/λχ\nTo0 0.04 0.08 0.120246810\nkBT/λχ\nTo\n0 0.04 0.08 0.1200.20.40.60.81\nkBT/λχ-1\nTo0 0.04 0.08 0.120.0.10.20.30.40.5\nkBT/λχ-1\nTo\nFIG. 2. Typical susceptibility, \u001f=1\n3(\u001fxx+\u001fyy+\u001fzz), and\ninverse susceptibility are plotted against temperature. The\nsusceptibility curves are shown both without the correction\ndue to hybridization, \r= 1, and with the correction, \r=\n0:536. We have chosen JSE= 0 and left V\fnite to illustrate\nthe consequence of orbital order on the susceptibility. By\nchoosingJSE= 0, we show that although Tc= 0 whileTo 6= 0,\nthe \ftted Curie-Weiss temperature appears to be negative.\nNote that a single Curie-Weiss \ft cannot span the entire range\nbelowTo.\nThere are more consequences of orbital ordering that\nare particularly important for the magnetic susceptibil-\nity of this spin-orbital system. The orbital order re-\nduces symmetry of the system and causes susceptibility\nto become anisotropic. Since the orbital ordering pat-\ntern tends to push angular momentum into the order-\ning planes, susceptibility is enhanced in these two di-\nrections while reduced in the third direction. Although\nanisotropic susceptibility is expected once cubic symme-\ntry is broken, it is an easy test to determine at what tem-\nperature orbital order occurs. However this is yet a more\nimportant e\u000bect. When orbital order sets in at To, the ef-\nfective moment changes as the orbital degrees of freedom\ntend toward a (partially) quenched state which results\nin an e\u000bective moment which changes with temperature.\nThe non-Curie-Weiss behavior will be critical when inter-\npreting the observed negative Curie-Weiss temperatures\nin 5d1ferromagnetic compounds.\nWithin our mean \feld theory, we now calculate the\nsusceptibility without the hybridization correction \rand\nwith the hybridization correction to show this e\u000bect. Forclarity, we set JSE= 0 to isolate the contributions from\norbital order from those of magnetic interactions. Fig. 2\nshows that below the orbital ordering temperature, the\nsusceptibility deviates from the Curie-Weiss law. How-\never the data below Tocan be \ft over a large range to\ngive a negative Curie-Weiss intercept despite the absence\nof magnetic interactions. Although the region where the\n\ft works the best is just below Towhere the orbital oc-\ncupation is rapidly changing, there is a quantitative ex-\nplanation for this.\nWe consider the case without hybridization where the\ne\u000bective moment for the j= 3=2 states is identically zero.\nWhen orbital order occurs, there is mixing between the\nj= 3=2 andj= 1=2 states proportional to Vh\u000eni=\u0015.\nThen below To, the e\u000bective magnetization operator for\nthe lowest energy Kramer's doublet increases in a way\nproportional toh\u000enidue to the matrix elements between\nj= 3=2 andj= 1=2. The e\u000bective Curie moment goes as\nthe square of magnetization and thus the enhancement\nis of orderh\u000eni2. Since orbital order below Toscales as\nh\u000eni/jTo\u0000Tj1=2within mean \feld theory, the e\u000bective\nCurie moment gains a contribution scaling as jTo\u0000Tjjust\nbelowTo. At temperatures far away from Tc, the leading\ncorrection to susceptibility and and inverse susceptibil-\nity is linear leading to the appearance of a Curie-Weiss\nlaw. We note, however, that this is arti\fcial and is not\nindicative of the physical magnetic interactions.\nDespite using mean \feld critical exponents, qualita-\ntively we have understood how deviations from the Curie-\nWeiss law occur from changing orbital occupancy. Be-\ncause we have used a simple model consisting of only \u0015\nandVwith a-priori knowledge of the ideal Curie-Weiss\ntemperature of zero, we have been able to clearly in-\nterpret the non-Curie-Weiss susceptibility. However the\n\ftting procedure must be performed with some caution\nsince both the \ft region and the chosen value of \u001f0(tem-\nperature independent term) determine the reported \u0012CW\nand\u0016e\u000b. In fact, experimental behavior may deviate\neven more strongly due to the quantitative details of how\norbital occupancies change with temperature. In partic-\nular, coupling between orbitals and phonons may be a\ncrucial aspect here.34\nReference 40 claimed negative Curie-Weiss tempera-\ntures were achievable in their model for ferromagnetic\nground states, although this crucial result was not ex-\nplicitly shown. Reference 26 has reproduced that model\nunder the circumstances necessary to generate ferromag-\nnets with negative Curie Weiss temperatures, and they\n\fnd jump discontinuities (\fnite-to-in\fnite) in the mag-\nnetic susceptibility at Tc. Such jump discontinuities are\nnot seen in Ba 2NaOsO 6, Ba 2MgReO 6, or Ba 2ZnReO 6.\nWe note that our mechanism for shifting the Curie-Weiss\ntemperature is free from these discontinuities and fea-\ntures a properly diverging susceptibility at Tcfor the fer-\nromagnetic phase, thereby providing a more accurate and\nnatural description of the transition.7\nIII.d2DOUBLE PEROVSKITES\nHere we will modify the d1spin-orbital model to ac-\ncommodate two electrons. Again, we then solve the\nmodel within mean \feld theory at both zero tempera-\nture and \fnite temperature. At zero temperature, we\n\fnd new orbital phases not found in our d1phase di-\nagram. For completeness, we show susceptibilities and\norbital occupancies at \fnite temperature.\nA. Model\nOur model for d2is constructed from the same consid-\nerations used in d1only changing the electron count. The\ntight-binding model HTB, the inter-site orbital repulsion\nHV, and the on-site Coulomb interaction HUare valid for\nthed2model without modi\fcation. However spin-orbit\ncoupling and superexchange will change since the total\nspin and orbital angular momentum on each site are now\ncomposed of two electrons. In the Mott limit, Hund's\nrules are enforced by HUresulting in a total spin S= 1\nand total orbital angular momentum L= 1 on each lat-\ntice site. Within this space, the spin-orbit interaction\ntakes the form H0\nSO=\u0000\u0015\n2P\niLi\u0001Si. The superexchange\nHamiltonian is given by the following\nH0\nSE=\u0000JSE\n12X\n\u000bX\nhiji2\u000b\b\nr1(2 +Si\u0001Sj)(n\u000b\ni\u0000n\u000b\nj)2\n(1\u0000Si\u0001Sj)\u0002\n(n\u000b\ni+n\u000b\nj)2+ (3\n2r3\u00005\n2)n\u000b\nin\u000b\nj\u0003\t(5)\nwhere the de\fnitions of JSE,r1, andr3correspond to\nthose used previously. As before, the top line in (5) gives\na ferromagnetic spin interaction when only one of the two\ninteracting orbitals is occupied while the second line gives\nan antiferromagnetic spin interaction which is maximized\nwhen two half \flled orbitals overlap. The total e\u000bective\nmagnetic interaction then reads H0=H0\nSO+H0\nSE+HV.\nWe decouple H0\nSEandHVinto all possible on-site mean\n\felds using four inequivalent sites as before and then\nsolve the mean \feld equations numerically.\nB. Zero Temperature Mean Field Theory\nThe zero temperature phase diagram is shown in\nFig. 3(b) as a function of the strength of Hund's cou-\npling\u0011=JH=Uand superexchange JSE=\u0015. In the limit\nof large spin-orbit coupling and the absence of inter-site\norbital repulsion, the ground state is predominantly AFM\nwith the moment aligning parallel to the [110] direction\nwithin a plane and antiparallel to the [110] direction in\nthe next plane. To see why this phase occupies such a\nlarge region of phase space, we analyze the orbital struc-\nture that accompanies it, as shown in Fig. 3(a). On each\nsite, one electron moves onto the yzorbital and the other\nonto thezxorbital. In this con\fguration both occupiedorbitals overlap with occupied orbitals on neighboring\nsites and unoccupied orbitals overlap with other unoccu-\npied orbitals so that AFM superexchange is maximized.\nThese orbitally controlled AFM interactions then take\nplace between planes and not within planes resulting in\nAFM between planes while FM interactions prevail in\neach plane. Since this this orbital pattern is compatible\nwith tetragonal distortion, as observed in Sr 2MgOsO 630,\nwe expect nominally cubic crystal structures to distort.\nThe next phase we \fnd is the AFM 4-sublattice copla-\nnar structure previously found in the d1phase diagram.\nAs before, the orbital degrees of freedom are closely\naligned with the directions perpendicular to the occupied\norbitals, and the spin and orbital moments perturbatively\nseparate from each other with increasing superexchange.\nIt is worth noting that in this region of the phase dia-\ngram, the next lowest energy phase is AFM [100] that\ncan become a competitive ground state upon inclusion of\nanisotropy.\nFor large superexchange and Hund's coupling, we \fnd\na ferromagnetic phase with ordering along the [100] di-\nrection that is best characterized as a \\3-up, 1-down\"\ncollinear structure where three of the four moments order\nparallel to each other along the chosen direction and the\nfourth moment orders anti-parallel to the other three. It\nis worth noting that the second most energetically favor-\nable phase in this region of the phase diagram is another\n\\3-up, 1-down\" structure where each moment is either\napproximately parallel or antiparallel to the [110] direc-\ntion. The energy di\u000berence between the FM [100] and\nFM [110] phases is negligible and either phase is a suit-\nable ground state. In addition to these two FM phases,\nwe \fnd a canted FM solution to the mean \feld equa-\ntions with the same orbital ordering pattern as the d1\ncanted FM phase. However it is higher enough in energy\nto rule out as a viable ground state and consequently is\nnot shown in the phase diagram.\nUnlike the AFM [110] and AFM 4-sublattice struc-\ntures, the FM/AFM [100] structures features an approx-\nimately higher degree of degeneracy due to the orbital\ndegrees of freedom. Like the AFM 4-sublattice orbital\nstructure, the FM/AFM [100] orbital structure tends to\nminimize repulsion between orbitals. Of the four tetra-\nhedral sites, three of them are able to minimize the re-\npulsion and allow occupied orbitals to hop to unoccupied\norbitals. While the repulsion is minimized between those\nthree sites, this forces occupied orbitals on each site to\npoint toward the fourth site. Figure 3(a) shows that this\nfourth site in the FM/AFM [100] orbital pattern chooses\none of the orbitals to have a majority occupancy (solid\ncolor) and the other two orbitals to have minority occu-\npancies (semi-transparent colors). In the FM [110] phase,\na similar situation occurs with the main di\u000berence being\nthat now two orbitals have majority occupancy and one\norbital has minority occupancy. Before magnetic order\nsets in, the degeneracy is approximately extensive as the\nfourth site on every tetrahedron in the lattice has local\norbital frustration.8\nAFM 110AFM 4-sublattice\n(AFM 100)FM 100\n(FM 110)\nV/λ=0\n0.01\n0.02\n0.03SL\n0.0 0.2 0.4 0.6 0.80.000.050.100.150.20\nJSE/λJH/U(a) (b)\nFIG. 3. (a) Orbital ordering patterns are shown for each type of magnetic order. Orbitals shown in solid colors represent\nthe most occupied orbitals while orbitals not shown or shown transparently have lower occupancy. (b) The zero temperature\nphase diagram shows three ground state phases: AFM with moments (anti)parallel to [110], AFM 4-sublattice structure, and\nFM with moments parallel to [100]. Phases shown in parenthesis (AFM [100], FM [110]) show the next lowest energy phase in\neach region.\nWhen inter-site orbital repulsion HVis included, the\nphase boundaries shift. The most dramatic e\u000bect is the\nrecession of the boundary between AFM [110] and the\nAFM 4-sublattice structure. This becomes apparent by\ncomparing the orbital con\fgurations of the two phases as\nthe AFM [110] structure maximizes the number of AFM\nsinglets which are penalized by the orbital repulsion. Un-\nlike in the d1situation, we \fnd that the inclusion of V\ndoes not enhance FM. While the FM/AFM [100] and FM\n[110] orbital structures are much more compatible with\nHVthan the AFM [110] structure, the AFM 4-sublattice\nstructure still dominates. We note that unlike the d1\ncase, canted FM is not favorable here due to the electron\ncount. The d1case relies on pushing the large majority\nof the electron weight onto one orbital while retaining\na smaller occupancy on a second orbital to generate an\norbital moment. However in d2, this second orbital must\nalso be occupied which consequently induces AFM inter-\nactions within each horizontal plane.\nAlthough we have focused on spin-orbital magnetic or-\nder, it is necessary to remark that exotic singlet ground\nstates are also possible. The Kramer's theorem guaran-\ntees that trivial ionic singlets will not occur in d1systems,\nand therefore the experimental observation of singlet be-\nhavior is an indication of a non-trivial ground state. Such\nconsiderations do not apply to d2, and experimental ob-\nservations of singlet behavior may arise from trivial local\nmagnetic singlets. Consequently this local non-magnetic\nsinglet possibility must \frst be ruled out when searchingfor exotic singlet behavior.\nC. Finite Temperature Mean Field Theory\nHere we consider the model at \fnite temperature. Fig-\nure 4 shows orbital occupations and inverse magnetic\nsusceptibility as a function of temperature for the three\nground state phases from the previous section. At high\ntemperature, the orbitals have a uniform occupancy of\nnyz=nzx=nxy= 2=3. There is a temperature To\nwhere time-reversal invariant order sets in through the\norbitals and second temperature Tcwhere magnetic or-\nder sets in. In the case of the AFM [110] phase, Fig. 4(a)\nshows the two ordering temperatures coincide and that\nthe electrons are pushed onto the nyzandnzxorbitals to\nmaximize antiferromagnetic superexchange. This is dif-\nferent from the orbital ordering previously reported be-\ncause this ordering maximizes orbital repulsion instead of\nminimizing it, so orbital order itself is not favorable and\nis entirely driven by antiferromagnetic superexchange. In\nthis situation, the Curie-Weiss law with a negative Curie-\nWeiss temperature occurs as expected.\nThe transition to an AFM 4-sublattice structure is\nshown in Fig. 4(b). Above Tosusceptibility follows the\nCurie-Weiss law with a negative Curie-Weiss constant.\nBelowTothe orbital occupancies change along with the\ninverse susceptibility to deviate from the high tempera-\nture behavior. Just below To, susceptibility may be \ft9\n(a)AFM 110\n(b)AFM 4-sublattice\n(c)FM 100\n0.00 0.02 0.04 0.06 0.08 0.100.000.050.100.150.200.250.30\nkBT/λ(χ-χ 0)-1(a.u.)0.00.20.40.60.81.0\nnyz,nzx,nxynyz,nzx\nnxy\nTc0.00 0.02 0.04 0.06 0.08 0.100.000.050.100.15\nkBT/λ(χ-χ 0)-1(a.u.)0.00.20.40.60.81.0\nnyz,nzx,nxynyz\nnxy\nnzx\nTc To0.00 0.05 0.10 0.15 0.200.000.050.100.150.20\nkBT/λ(χ-χ 0)-1(a.u.)0.00.20.40.60.81.0\nnyz,nzx,nxynyznxy\nnzx\nTc To\nFIG. 4. Characteristic inverse susceptibility (blue) and orbital occupation (purple) curves are plotted against temperature for\nthe three phases in Fig. 3: (a) AFM [110], (b) AFM 4-sublattice, and (c) FM [100]. Susceptibility is averaged over all three\ndirections, \u001f\u00001= 3(\u001fxx+\u001fyy+\u001fzz)\u00001, and all sites in the tetrahedra. Orbital occupancies are shown for the site pictured\nabove each plot.\nto another Curie-Weiss law with another negative Curie-\nWeiss constant. Similarly to the d1case, there is still\ndeviation from the Curie-Weiss law in this regime, how-\never, the deviations are smaller and so is the enhance-\nment of the e\u000bective magnetic moment due to mixing of\ntheJ= 2 states with higher energy multiplets. But we\nnote that when JSE= 0, we still \fnd the appearance of\na negative Curie-Weiss constant due to non-Curie-Weiss\nsusceptibility as we did in the d1model.\nFinally, the transition to an FM [100] structure is\nshown in Fig. 4(c). Deviations from the Curie-Weiss\nlaw are seen below To, and the sign of the Curie-Weiss\nconstant can switch from negative to positive depending\nwhich region \ftted. Unlike the other phases, magnetic or-\nder appears at Tcwith a \frst-order transition marked by\nthe jumps in orbital occupancy and susceptibility. This\narises from competition between having the most ener-\ngetically favorable orbital structure at high temperature\nand the most energetically favorable magnetic structure\nat low temperature.\nAs in the d1case, we compare values of the the-\noretical moments to those from experiment. Oxygen\nhybridization will result in a Curie moment of \u0016e\u000b=p\n6(1\u0000\r=2)\u0016B. Assuming almost half of the moment\nresides on oxygen, the calculated moment is then \u0016e\u000b\u0019\n1:8\u0016B. This is close to the experimentally observed mo-\nments in Sr 2MgOsO 6and Ca 2MgOsO 6(both 1:87\u0016B)29\nbut further o\u000b from those of Ba 2YReO 6(1:93\u0016B)31and\nLa2LiReO 6(1:97\u0016B)31.\nIV. CONCLUSIONS\nWe have studied spin-orbital models for both d1and\nd2double perovskites where the B' ions are magnetic and\nhave strong spin-orbit coupling. We found several non-\ntrivial magnetically ordered phases characterized both byordering of the spin/orbital angular momentum and or-\ndering of the orbitals. This orbital ordering shows why\nferromagnetism is energetically favorable in these sys-\ntems when electron count is d1but not when it is d2,\nparticularly at large spin-orbit coupling. Additionally,\nordering of the orbital degrees of freedom can produce\nnon-Curie-Weiss behavior which can lead to the appear-\nance of a negative Curie-Weiss in the canted ferromag-\nnetic phase. We emphasize that examination of the spin\nand orbital degrees of freedom separately gives an en-\nhanced qualitative understanding of the magnetism for\nthis class of spin-orbit coupled double perovskites.\nV. ACKNOWLEDGEMENTS\nWe thank Patrick Woodward and Jie Xiong for their\nuseful discussions. We acknowledge the support of the\nCenter for Emergent Materials, an NSF MRSEC, under\nAward Number DMR-1420451.\nAppendix A: \u0016e\u000benhancement and Toford1model\nTo obtain the orbital ordering temperature Toand the\ne\u000bective moment \u0016e\u000bas a function of V=\u0015, we will solve\nthe mean \feld equations for HV+HSOanalytically. The\nrelevant mean \feld parameters for the four sites from\nFig. 1(b) are given below\nhnxy\n1i=hnxy\n2i=hnxy\n3i=hnxy\n4i=1\n3+\u000enz (A1)\nhnyz\n1i=hnyz\n2i=hnzx\n3i=hnzx\n4i=1\n3+\u000enx (A2)\nwith the conditionP\n\u000bn\u000b\ni= 1 determining the other\nfour parameters. We obtain the single site mean \feld10\nHamiltonian for V.\nH0\nV=\u0000V\u0002\n(86\n3\u000enx+43\n3\u000enz)nyz+ (43\n3\u000enx+53\n3\u000enz)nxy\u0003\n(A3)\nSince above Tc, the high mean \feld Hamiltonian H0\nMF=\nH0\nV+HSOis time reversal invariant, we rotate into the\nbasis of total angular momentum Jwhich factors into\ntwo 3\u00023 blocks of doublets. The upper block may be\nchosen to be of the form below\n0\nBB@3\u0015\n2\u000043V(2\u000enx+\u000enz)\n3p\n6\u00007V\u000enzp\n2\n\u000043V(2\u000enx+\u000enz)\n3p\n67V\u000enz\n243V(2\u000enx+\u000enz)\n6p\n3\n\u00007V\u000enzp\n243V(2\u000enx+\u000enz)\n6p\n3\u00007V\u000enz\n21\nCCA\n(A4)\nwhere the basis jJ;mJiis given byj1=2;+1=2i,\nj3=2;\u00003=2i,j3=2;+1=2iin this order. Using \u0012=\narctan 43p\n3 (2\u000enx+\u000enz)=63\u000enz, we diagonalize the\nHamiltonian in the j= 3=2 block\n0\n@3\u0015\n2x y\nx\u0000\u0001 0\ny0 \u00011\nA (A5)\nwhere\n\u0001 =Vp\n1849\u000enx(\u000enx+\u000enz) + 793\u000en2z\n3p\n3(A6)\nx=\u0000V43p\n3(2\u000enx+\u000enz) cos\u0012\n2+ 63\u000enzsin\u0012\n2\n9p\n2(A7)\nandyis given byxwith sin\u0012!cos\u0012and cos\u0012!\u0000 sin\u0012\napplied. The lowest J= 3=2 doublet with energy \u0000\u0001 is\nmixed with the J= 1=2 doublet with amplitude \u00002x=3\u0015.\nWe project the magnetization operator M= 2S\u0000L\nonto this lowest doublet. Since nominally g= 0 for the\nj= 3=2 states, the \frst non-zero correction to the wave-\nfunction comes from mixing of the j= 3=2 andj= 1=2\nstates. From the projection, we obtain the gfactors for\nthis doublet in all three directions (ie. Mx=gx\u0016B\n2\u001bx,\netc) and compute the average gfactor obtained in a pow-\nder susceptibility measurement g2=1\n3\u0000\ng2\nx+g2\ny+g2\nz\u0001\nto\nobtain the powder average e\u000bective moment for the dou-\nblet. For the parameter regime we are interested in, \u000enz\nhas a negligible contribution to g, and thegfactor is\ngiven approximately by g= 344Vj\u000enxj=9p\n3\u0015so that the\nmoment is\u0016e\u000b= 172Vj\u000enxj\u0016B=9\u0015.\nNow we obtain the mean \feld orbital ordering temper-\natureTowhich occurs when the j= 3=2 states split. In\nthe limit that \u000enzis negligible, we self consistently solve\nfor the expectation value of the operator the projections\nof the operator \u000enx!nyz\u00001\n3within the 2\u00022 sub-\nspace of energies \u0000\u0001 and \u0001 (ie.jJ= 3=2;Jz=\u00003=2i\nandjJ= 3=2;Jz= +1=2i). The projection of the \u000enx\noperator to this subspace is\n\u000enx! \n\u00001\n2p\n3\u00001\n6\n\u00001\n61\n2p\n3!\n(A8)so that the mean \feld equations for \u000enxread\n\u000enx=1\n2p\n3tanh\f\u0001 (A9)\nwhere \u0001\u001943V\n3p\n3\u000enx. Then we \fnd kBTo= 43V=18\nwhich is consistent with Ref. 40. However, in contrast to\nRef. 40, our analysis shows that this orbital order is com-\npatible with both the FM and AFM phases and does not\ndisappear below Tcfor the AFM phase. We can relate\nthe ratios of these results as seen in Fig. 1(e) by\nkBTo=\u0015\n\u0016e\u000b=\u0016B=1\n8\u000enx: (A10)\nUsing the zeroth order approximation for \u000enxas 1=2p\n3,\nthis ratio becomes 0.43 which is close to that shown in\nFig. 1(e).11\n1W. Witczak-Krempa, G. Chen, Y. B. Kim, and L. Balents,\nAnnual Review of Condensed Matter Physics 5, 57 (2014).\n2J. G. Rau, E. K.-H. Lee, and H.-Y. Kee, Annual Review\nof Condensed Matter Physics 7, 195 (2016).\n3E. Kermarrec, C. A. Marjerrison, C. M. Thompson, D. D.\nMaharaj, K. Levin, S. Kroeker, G. E. Granroth, R. Flacau,\nZ. Yamani, J. E. Greedan, and B. D. Gaulin, Phys. Rev.\nB91, 075133 (2015).\n4A. E. Taylor, R. Morrow, R. S. Fishman, S. Calder, A. I.\nKolesnikov, M. D. Lumsden, P. M. Woodward, and A. D.\nChristianson, Phys. Rev. B 93, 220408 (2016).\n5A. M. Cook, S. Matern, C. Hickey, A. A. Aczel, and\nA. Paramekanti, Phys. Rev. B 92, 020417 (2015).\n6A. A. Aczel, A. M. Cook, T. J. Williams, S. Calder, A. D.\nChristianson, G.-X. Cao, D. Mandrus, Y.-B. Kim, and\nA. Paramekanti, Phys. Rev. B 93, 214426 (2016).\n7G. Khaliullin, Phys. Rev. Lett. 111, 197201 (2013).\n8A. Akbari and G. Khaliullin, Phys. Rev. B 90, 035137\n(2014).\n9K. Pajskr, P. Nov\u0013 ak, V. Pokorn\u0013 y, J. Koloren\u0014 c, R. Arita,\nand J. Kune\u0014 s, Phys. Rev. B 93, 035129 (2016).\n10O. N. Meetei, W. S. Cole, M. Randeria, and N. Trivedi,\nPhys. Rev. B 91, 054412 (2015).\n11T. Dey, A. Maljuk, D. V. Efremov, O. Kataeva, S. Gass,\nC. G. F. Blum, F. Steckel, D. Gruner, T. Ritschel, A. U. B.\nWolter, J. Geck, C. Hess, K. Koepernik, J. van den Brink,\nS. Wurmehl, and B. B uchner, Phys. Rev. B 93, 014434\n(2016).\n12B. Ranjbar, E. Reynolds, P. Kayser, B. J. Kennedy, J. R.\nHester, and J. A. Kimpton, Inorganic Chemistry 54, 10468\n(2015).\n13B. F. Phelan, E. M. Seibel, D. B. Jr., W. Xie, and R. Cava,\nSolid State Communications 236, 37 (2016).\n14K. I. Kugel' and D. I. Khomski, Soviet Physics Uspekhi\n25, 231 (1982).\n15G. Khaliullin, Progress of Theoretical Physics Supplement\n160, 155 (2005).\n16M. A. de Vries, A. C. Mclaughlin, and J.-W. G. Bos, Phys.\nRev. Lett. 104, 177202 (2010).\n17T. Aharen, J. E. Greedan, C. A. Bridges, A. A. Aczel,\nJ. Rodriguez, G. MacDougall, G. M. Luke, T. Imai, V. K.\nMichaelis, S. Kroeker, H. Zhou, C. R. Wiebe, and L. M. D.\nCranswick, Phys. Rev. B 81, 224409 (2010).\n18J. P. Carlo, J. P. Clancy, T. Aharen, Z. Yamani, J. P. C.\nRu\u000b, J. J. Wagman, G. J. Van Gastel, H. M. L. Noad,\nG. E. Granroth, J. E. Greedan, H. A. Dabkowska, and\nB. D. Gaulin, Phys. Rev. B 84, 100404 (2011).\n19M. A. d. Vries, J. O. Piatek, M. Misek, J. S. Lord, H. M.\nRnnow, and J.-W. G. Bos, New Journal of Physics 15,\n043024 (2013).\n20Z. Qu, Y. Zou, S. Zhang, L. Ling, L. Zhang, and Y. Zhang,\nJournal of Applied Physics 113, 17E137 (2013).\n21K. E. Stitzer, M. D. Smith, and H.-C. zur Loye, Solid\nState Sciences 4, 311 (2002).\n22A. S. Erickson, S. Misra, G. J. Miller, R. R. Gupta,\nZ. Schlesinger, W. A. Harrison, J. M. Kim, and I. R.\nFisher, Phys. Rev. Lett. 99, 016404 (2007).\n23A. J. Steele, P. J. Baker, T. Lancaster, F. L. Pratt,\nI. Franke, S. Ghannadzadeh, P. A. Goddard, W. Hayes,\nD. Prabhakaran, and S. J. Blundell, Phys. Rev. B 84,\n144416 (2011).24L. Lu, M. Song, W. Liu, A. P. Reyes, P. Kuhns, H. O. Lee,\nI. R. Fisher, and V. F. Mitrovi\u0013 c, ArXiv e-prints (2017),\narXiv:1701.06117 [cond-mat.str-el].\n25K. G. Bramnik, H. Ehrenberg, J. K. Dehn, and H. Fuess,\nSolid State Sciences 5, 235 (2003), dedicated to Sten An-\ndersson for his scienti\fc contribution to Solid State and\nStructural Chemistry.\n26C. A. Marjerrison, C. M. Thompson, G. Sala, D. D. Ma-\nharaj, E. Kermarrec, Y. Cai, A. M. Hallas, M. N. Wilson,\nT. J. S. Munsie, G. E. Granroth, R. Flacau, J. E. Greedan,\nB. D. Gaulin, and G. M. Luke, Inorganic Chemistry 55,\n10701 (2016).\n27H. L. Feng, Y. Shi, Y. Guo, J. Li, A. Sato, Y. Sun,\nX. Wang, S. Yu, C. I. Sathish, and K. Yamaura, Jour-\nnal of Solid State Chemistry 201, 186 (2013).\n28C. M. Thompson, J. P. Carlo, R. Flacau, T. Aharen, I. A.\nLeahy, J. R. Pollichemi, T. J. S. Munsie, T. Medina, G. M.\nLuke, J. Munevar, S. Cheung, T. Goko, Y. J. Uemura, and\nJ. E. Greedan, Journal of Physics: Condensed Matter 26,\n306003 (2014).\n29Y. Yuan, H. L. Feng, M. P. Ghimire, Y. Matsushita,\nY. Tsujimoto, J. He, M. Tanaka, Y. Katsuya, and\nK. Yamaura, Inorganic Chemistry 54, 3422 (2015), pMID:\n25751088, http://dx.doi.org/10.1021/ic503086a.\n30R. Morrow, A. E. Taylor, D. J. Singh, J. Xiong, S. Rodan,\nA. U. B. Wolter, S. Wurmehl, B. B uchner, M. B. Stone,\nA. I. Kolesnikov, A. A. Aczel, A. D. Christianson, and\nP. M. Woodward, Scienti\fc Reports 6, 32462 (2016).\n31T. Aharen, J. E. Greedan, C. A. Bridges, A. A. Aczel,\nJ. Rodriguez, G. MacDougall, G. M. Luke, V. K. Michaelis,\nS. Kroeker, C. R. Wiebe, H. Zhou, and L. M. D. Cran-\nswick, Phys. Rev. B 81, 064436 (2010).\n32A. A. Aczel, Z. Zhao, S. Calder, D. T. Adroja, P. J. Baker,\nand J.-Q. Yan, Phys. Rev. B 93, 214407 (2016).\n33C. M. Thompson, L. Chi, J. R. Hayes, A. M. Hallas, M. N.\nWilson, T. J. S. Munsie, I. P. Swainson, A. P. Grosvenor,\nG. M. Luke, and J. E. Greedan, Dalton Trans. 44, 10806\n(2015).\n34L. Xu, N. A. Bogdanov, A. Princep, P. Fulde, J. v. d. Brink,\nand L. Hozoi, npj Quantum Materials 1, 16029 (2016).\n35K.-H. Ahn, K. Pajskr, K.-W. Lee, and J. Kunes, ArXiv\ne-prints (2016), arXiv:1611.07749 [cond-mat.str-el].\n36H. J. Xiang and M.-H. Whangbo, Phys. Rev. B 75, 052407\n(2007).\n37K.-W. Lee and W. E. Pickett, EPL (Europhysics Letters)\n80, 37008 (2007).\n38S. Gangopadhyay and W. E. Pickett, Phys. Rev. B 91,\n045133 (2015).\n39J. Romh\u0013 anyi, L. Balents, and G. Jackeli, ArXiv e-prints\n(2016), arXiv:1611.00646 [cond-mat.str-el].\n40G. Chen, R. Pereira, and L. Balents, Phys. Rev. B 82,\n174440 (2010).\n41G. Chen and L. Balents, Phys. Rev. B 84, 094420 (2011).\n42T. Dodds, T.-P. Choy, and Y. B. Kim, Phys. Rev. B 84,\n104439 (2011).\n43H. Ishizuka and L. Balents, Phys. Rev. B 90, 184422\n(2014).\n44W. M. H. Natori, E. C. Andrade, E. Miranda, and R. G.\nPereira, Phys. Rev. Lett. 117, 017204 (2016).\n45Private communications.\n46A. Georges, L. de' Medici, and J. Mravlje, Annual Review12\nof Condensed Matter Physics 4, 137 (2013).\n47B. Yuan, J. P. Clancy, A. M. Cook, C. M. Thompson,\nJ. Greedan, G. Cao, B.-C. Jeon, T.-W. Noh, M. H. Upton,\nD. Casa, T. Gog, A. Paramekanti, and Y.-J. Kim, ArXive-prints (2017), arXiv:1701.06584 [cond-mat.str-el].\n48J. Goodenough, Magnetism and the Chemical Bond (In-\nterscience Publishers, 1963).\n49J. Kanamori, J. Phys. Chem. Solids 10, 87 (1959).\n50P. W. Anderson, Phys. Rev. 115, 2 (1959)."    },    {        "title": "1301.3596v1.Mechanical_generation_of_spin_current_by_spin_rotation_coupling.pdf",        "content": "Mechanical generation of spin current by spin-rotation coupling\nMamoru Matsuo1;2, Jun'ichi Ieda1;2, Kazuya Harii1;2, Eiji Saitoh1;2;3;4, and Sadamichi Maekawa1;2\n1Advanced Science Research Center, Japan Atomic Energy Agency, Tokai 319-1195, Japan\n2CREST, Japan Science and Technology Agency, Sanbancho, Tokyo 102-0075, Japan\n3Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan\n4WPI, Advanced Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan\n(Dated: October 13, 2018)\nSpin-rotation coupling, which is responsible for angular momentum conversion between the elec-\ntron spin and rotational deformations of elastic media, is exploited for generating spin current. This\nmethod requires neither magnetic moments nor spin-orbit interaction. The spin current generated\nin nonmagnets is calculated in presence of surface acoustic waves. We solve the spin di\u000busion equa-\ntion, extended to include spin-rotation coupling, and \fnd that larger spin currents can be obtained\nin materials with longer spin lifetimes. Spin accumulation induced on the surface is predicted to be\ndetectable by time-resolved Kerr spectroscopy.\nPACS numbers: 72.25.-b, 85.75.-d, 71.70.Ej, 62.25.-g\nIntroduction.| Spin current, a \row of spins, is a key\nconcept in the \feld of spintronics[1, 2]. It can be gen-\nerated from non-equilibrium spin states, i.e., spin accu-\nmulation and spin dynamics. The former is routinely\nproduced in nonlocal spin valves[3]. In ferromagnets,\nthe latter is excited by ferromagnetic resonance[4], tem-\nperature gradient[5], and sound waves in the magnetic\ninsulator[6]. Alternatively, spin currents in nonmagnets\nhave been generated by the spin Hall e\u000bect[7], in which a\nstrong spin-orbit interaction (SOI) is utilized. All these\nexisting methods rely on exchange coupling of spins with\nlocal magnetization or on SOI.\nIn this Letter, we pursue a new route for generating\nspin currents by considering spin-rotation coupling[8]:\nHS=\u0000\u0016h\n2\u001b\u0001\n; (1)\nwhere \u0016h\u001b=2 is the electron spin angular momentum and\n\nis the mechanical rotation frequency. The method re-\nquires neither magnetic moments nor SOI. In this sense,\nthe mechanism proposed here is particularly relevant in\nnonmagnets with longer spin lifetime.\nNonuniform rotational motion|. Here, we consider\nrotational motion of the lattice:\n\n=1\n2r\u0002 _u; (2)\nwhere uis the displacement vector of the lattice[9]. When\nthe lattice vibration has transverse modes, Eq. (2) does\nnot vanish. In such a case, the mechanical angular mo-\nmentum of the lattice can be converted into spin angular\nmomentum via HS[10]. However, as shown later, that\n\nis \fnite is insu\u000ecient to generate spin currents elas-\ntically. Both the time derivative and the gradient of ro-\ntational modes are necessary for the generation of spin\ncurrent. For this purpose, we focus on surface acous-\ntic waves (SAWs), which induce rotational deformations\nthat vary in space and time (Fig. 1).\nIn presence of SAW, a gradient of mechanical rotation\nis induced in the attenuation direction. We extend the\nFIG. 1. Snapshot of mechanical generation of spin current\ninduced by SAW. (a) In presence of a SAW propagating in\nthex-direction, a gradient of mechanical rotation around the\nz-axis is induced. The rotation couples to electron spins, and\nthen thez-polarized spin current \rows in the y-direction. (b)\nSpin accumulation induced on the surface. Because spins are\npolarized parallel to the rotation axis ( \u0006z), the striped pat-\ntern of spin accumulation arises at the surface.\nspin di\u000busion equation to include the coupling between\nelastic rotation and spin. By solving the equation in pres-\nence of SAW, we can evaluate the induced spin current\nfor metals and semiconductors.\nSpin di\u000busion equation with spin-rotation coupling.|\nFirst of all, we examine e\u000bects of spin-rotation couplingarXiv:1301.3596v1  [cond-mat.mes-hall]  16 Jan 20132\non spin density. When the mechanical rotation, \n, whose\naxis is in the z-direction, is applied, the electron spins\nalign parallel to the axis of rotation. This is known as the\nBarnett e\u000bect[11]. In this case, the bottom of the energy\nband of the electron is shifted by \u0016 h\n=2. The number\ndensity of up(down) spin electrons is then given by\nn\"(#)=Z\u0016\"(#)\n\u0006\u0016h\n=2d\"N 0(\"); (3)\nwhereN0is the density of states for electrons, and \u0016\"and\n\u0016#are chemical potentials for up and down spin electrons,\nrespectively. The z-direction is selected as the quantiza-\ntion axis. Then, spin density can be estimated as\nn\"\u0000n#\u0019N0(\u000e\u0016\u0000\u0016h\n); (4)\nwhere\u000e\u0016=\u0016\"\u0000\u0016#is spin accumulation. Here, a con-\nstant density of state is assumed for simplicity. Spin re-\nlaxation occurs in two processes: one is on-site spin \rip\nwith the spin lifetime, \u001csf, and the other is spin di\u000busion\nwith the di\u000busion constant, D. Equating these processes\nleads to@t(n\"\u0000n#) =\u001c\u00001\nsfN0\u000e\u0016+Dr2(N0\u000e\u0016). Next, we\nobtain the extended spin di\u000busion equation in presence\nof spin-rotation coupling:\n\u0000\n@t\u0000Dr2+\u001c\u00001\nsf\u0001\n\u000e\u0016= \u0016h@t\n; (5)\nThe R.H.S. of Eq. (5) is a source term originating from\nspin-rotation coupling. Z-polarized spin current can be\ncalculated from the solution of Eq. (5) as\nJz\ns=\u001b0\ner\u000e\u0016; (6)\nwith conductivity \u001b0. If the mechanical rotation is con-\nstant with time, the source term vanishes. Moreover,\neven if mechanical rotation depends on time, the uniform\nrotation in space cannot generate spin currents because\nspin accumulation is independent of space.\nSpin accumulation induced by SAW.| Let us con-\nsider generation of spin current due to spin-rotation cou-\npling of SAWs in nonmagnetic metals or semiconductors.\nOur setup is shown in Fig. 1 (a). SAWs are generated\nin thexz-plane and penetrates a nonmagnetic material\nalong they-direction. They then induce mechanical ro-\ntation around the z-axis, whose frequency \n= (0;0;\n)\nis given by[9]\n\n(x;y;t ) =!2u0\n2ctexpf\u0000kty+i(kx\u0000!t)g; (7)\nwhere!andu0are the frequency and amplitude of\nthe mechanical resonator, kis wave number, ctis the\ntransverse sound velocity, and ktis the transverse wave\nnumber. The frequency !is related to the wave num-\nber as!=ctk\u0018and the transverse wave number as\nkt=kp\n1\u0000\u00182, where\u0018satis\fes the equation \u00186\u00008\u00184+\n8\u00183(3\u00002c2\nt=c2\nl)\u000016(1\u0000c2\nt=c2\nl) = 0 andclis the longitu-\ndinal sound velocity. The Poisson ratio, \u0017, is related tothe ratio of velocities as ( ct=cl)2= (1\u00002\u0017)=2(1\u0000\u0017), and\n\u0017and\u0018are related as \u0018\u0019(0:875 + 1:12\u0017)=(1 +\u0017).\nSpin accumulation generated by the SAW can be eval-\nuated by solving Eq. (5). By inserting Eq. (7) and\n\u000e\u0016=\u000e\u0016y(y;t)eikxinto Eq. (5), the spin di\u000busion equa-\ntion can be rewritten as\n\u0000\n@t\u0000D@2\ny+ ~\u001c\u00001\nsf\u0001\n\u000e\u0016y(y;t) =\u0000i!\u0016h\n0e\u0000kty\u0000i!t;(8)\nwhere ~\u001csf=\u001csf(1+\u00152\nsk2)\u00001with the spin di\u000busion length\n\u0015s=pD\u001csfand \n 0=!2u0=2ct. With the boundary\ncondition@y\u000e\u0016= 0 on the surface y= 0, the solution is\ngiven by\n\u000e\u0016y(y;t) =\u0000i!\u0016h\n0Z1\n0dt0Z1\n0dy0\u0012(t\u0000t0)e\u0000(t\u0000t0)=~\u001csf\np\n4\u0019D(t\u0000t0)\n\u0002(e\u0000(y\u0000y0)2\n4D(t\u0000t0)+e\u0000(y+y0)2\n4D(t\u0000t0))e\u0000kty0\u0000i!t0:(9)\nHere, let us consider the time evolution of spin accu-\nmulation at the surface, y= 0. Because each spin aligns\nparallel to the rotation axis, i.e., the \u0006z-axis, a striped\npattern of spin accumulation [shown in Fig. 1 (b)] arises\nat the surface. The period of spatial pattern is the same\nas the wavelength of SAW, 2 \u0019=k.\nRecently, spin precession controlled by SAW was ob-\nserved by using the time-resolved polar megneto-optic\nKerr e\u000bect (MOKE)[12, 13]. In our case, in-plane spin\npolarization is induced. Therefore, transversal or longi-\ntudinal MOKE can be used to observe patterns shown in\nFig. 1 (b).\nSpin current from SAW.| From Eqs. (6) and (9)\nwe obtainz-polarized spin current in the y-direction. In\nFig. 2, the SAW-induced spin current is shown. The spin\ncurrent,Jz\ns, is plotted as a function of ktyand!tin Fig.\n2 (a). The spin current oscillates with the same frequency\nas that of the mechanical resonator, !. The maximum\namplitude, JMax\ns, is found near the surface, kty\u00191. In\nFig. 2 (b), maximum amplitude scaled by !3is plotted\nas a function of !\u001csf. The amplitude increases linearly\nwhen!\u001csf\u001c1, whereas it saturates when \u001csf\u001d!\u00001. In\nother words, JMax\ns/!4for!\u001csf\u001c1 whereasJMax\ns/!3\nfor!\u001csf\u001d1.\nTo clarify material dependence of spin current, we use\nan asymptotic solution of Eq. (5) for kty\u001d1:\n\u000e\u0016\u0019i!~\u001csf\ni!~\u001csf+\u00152sk2\nt\u00001\u0016h\n0e\u0000kty+i(kx\u0000!t);(10)\nwhich leads to\nJz\ns\u0019\u0000i!~\u001csf\ni!~\u001csf+\u00152sk2\nt\u00001\u0016h\u001b0\n2e!3u0\nc2\ntp\n1\u0000\u00182\n\u0018e\u0000kty+i(kx\u0000!t)\n(11)\nwhere \n 0=!2u0=2ct.\nWhen spin relaxation is absent, !~\u001csf\u001d1, one obtains\n\u000e\u0016\u0019\u0016h\n0e\u0000kty+i(kx\u0000!t)(12)3\nFIG. 2. (a) Spin current induced by SAW Jz\nsplotted as a\nfunction of ktyand!tfor \fxedxandz. The spin current\noscillates with time. Maximum amplitude is located near the\nsurface,kty\u00191. (b) Maximum amplitude of the spin current\nscaled by!3plotted as a function of !\u001csf. When!\u001csf\u001c1, the\nscaled amplitude, JMax\ns!\u00003, increases linearly. On the other\nhand, it saturates when !\u001csf\u001d1. Accordingly, the maximum\namplitude, JMax\ns, is proportional to !4in the former case,\nwhereasJMax\ns/!3in the latter case.\nand\nJz\ns\u0019\u0000(\u001b0=e)kt\u0016h\n0e\u0000kty+i(kx\u0000!t): (13)\nWhen!~\u001csf\u001c1 and\u0015skt\u001c1, the spin current be-\ncomes\nJz\ns\u0019!\u001csf\u0001\u0016h\u001b0\n2e!3u0\nc2\ntp\n1\u0000\u00182\n\u0018e\u0000kty+i(kx\u0000!t+\u0019=2):(14)\nAs seen in Eq. (14), the larger spin current can be\nobtained from materials with the longer spin lifetime,\nnamely, weaker SOI.Let us examine the SAW-induced spin current in typ-\nical nonmagnetic materials. Using Eqs. (6) and (9), the\nmaximum value of the spin current for Al, Cu, Ag, Au,\nand n-doped GaAs normalized by that of Pt, JMax;Pt\ns , is\ncomputed as listed in Table 1. The ratio of the maxi-\nmum amplitude of the spin current to that of Pt, \u0016Js, is\nde\fned as \u0016Js=JMax\ns=JMax;Pt\ns . The ratio depends mainly\non the conductivity, \u001b0, and spin lifetime, \u001csf. The or-\nder, \u0016JCu\ns>\u0016JAl\ns>\u0016JAg\ns>\u0016JAu\ns>\u0016JPt\ns= 1, is unchanged,\nsince these materials well satisfy !\u001csf\u001c1. For GaAs,\nthe ratio, \u0016JGaAs\ns, is greater than 1 for !<1GHz, whereas\nit becomes smaller than 1 for ! >1GHz. This happens\nbecause the spin lifetime of GaAs is much longer than\nthat of Pt; i,e., the dimensionless parameter, !\u001csf, be-\ncomes much greater than 1 when ! > 1GHz. In such a\ncase,JMax\ns=!3for GaAs saturates, whereas that for Pt\nlinearly increases, as shown in Fig. 2 (b).\nIt is worth noting that the spin current generated in a\nmetal with weak spin-orbit interaction such as those of Al\nand Cu is much larger than that in Pt. In addition, the\nspin current in n-doped GaAs is comparable to that in\nPt. Although conductivities of semiconductors are much\nsmaller than those of metals, the spin lifetime is much\nlonger. Hence, the amplitude of the induced spin current\nin GaAs is comparable to that in Pt for !\u001cGaAs\nsf<1.\nRecently, SAWs in the GHz frequency range have been\nused for spin manipulation[13, 14]. Here, we evaluate\nspin current at such high frequencies. In case of u0=\n10\u00009m,!=2\u0019= 10GHz, Pt has the maximum amplitude,\nJz;Pt\ns\u00194\u0002106A/m2.\nConventionally, generation of spin current in nonmag-\nnetic materials has required strong SOI because the spin\nHall e\u000bect has been utilized. In other words, nonmag-\nnetic materials with short spin lifetimes have been used.\nOn the contrary, the mechanism proposed here requires\nlonger spin lifetimes to generate larger spin currents.\nTherefore, more options are available for spin-current\ngeneration in nonmagnets than ever before.\nEnhancement of the SAW-induced spin current.|\nVery recently, it has been predicted that spin-rotation\ncoupling can be enhanced by an interband mixing of\nsolids[20]:\nH0\nS=\u0000(1 +\u000eg)\u0016h\n2\u001b\u0001\n: (15)\nHere,\u000egis given by \u000eg=g\u0000g0whereg0= 2 andg\nare electron gfactors in vacuum and solids, respectively.\nConsidering enhancement, the mechanical rotation, \n,\ninserted into the extended spin di\u000busion equation, Eq.\n(5), is replaced by (1 + \u000eg)\n. Consequently, the spin\naccumulation, \u000e\u0016, is modi\fed as \u000e\u0016!(1 +\u000eg)\u000e\u0016, and\naccordingly, the induced spin current as Jz\ns!(1+\u000eg)Jz\ns.\nFor lightly doped n-InSb at low temperature, g\u0019\u000049\nhas been employed in a recent experiment[21]. In this\ncase, one obtains \u000eg\u0019\u000051. Therefore, the amplitude of\nthe spin current can be 50 times larger.4\nTABLE I. SAW-induced spin current for Pt, Al, Cu, Ag, Au, and GaAs. The ratio is given by \u0016Js=JMax\ns=JMax;Pt\ns , where\nJMax;Pt\ns is the maximum amplitude of the spin current for Pt. The ratio \u0016Jsdepends on the Poisson's ratio, \u0017, the transverse\nvelocity,ct, conductivity, \u001b0, and spin lifetime, \u001csf.\n\u0017ct[m/s]\u001b0[107(\nm)\u00001]\u001csf[ps] \u0016Js(0.1GHz) \u0016Js(1GHz) \u0016Js(2.5GHz) \u0016Js(10GHz) Ref.\nPt 0.377 1730 0.96 0.3 1 1 1 1 [15]\nAl 0.345 3040 1.7 100 390 290 210 62 [16]\nCu 0.343 2270 7.0 42 950 700 650 330 [3]\nAg 0.367 1660 2.9 3.5 44 38 34 32 [17]\nAu 0.44 1220 2.5 2.8 42 35 33 30 [18]\nGaAs 0.31 2486 3.3\u000210\u000041051.6 0.13 0.050 0.013 [19]\nDiscussion and conclusion.| The method of spin-\ncurrent generation using spin-rotation coupling is purely\nof mechanical origin; i.e., it is independent of exchange\ncoupling and SOI. Lattice dynamics directly excites the\nnonequilibrium state of electron spins, and consequently,\nspin current can be generated in nonmagnets.\nAs an example, we have theoretically demonstrated\nthat SAW, a situation in which rotational motion of lat-\ntice couples with electron spins, can be exploited for spin\ncurrent generation. The spin di\u000busion equation is ex-\ntended to include e\u000bects of spin-rotation coupling. The\nsolution of the equation reveals that the spin current is\ngenerated parallel to the gradient of the rotation. More-\nover, it has been determined that larger spin current can\nbe generated in nonmagnetic materials with longer spin\nlifetimes. This means that Al and Cu, which have been\nconsidered as good materials for a spin conducting chan-\nnel, are favorable for generating spin current. Spin ac-\ncumulation induced by the SAW on the surface will be\nobserved by Kerr spectroscopy.\nThese results can be generalized for other lattice dy-\nnamics. SAW discussed above is the Rayleigh wave,\nwhich induces rotation with the axis parallel to the sur-\nface. For instance, the Love wave[22], horizontally polar-\nized shear wave, can be utilized to generate spin currents\nwhose spin polarization is perpendicular to the surface.\nThe use of spin rotation coupling, argued here, opens up\na new pathway for creating spin currents by elastic waves.\nThe authors thank S. Takahashi for valuable discus-\nsions. This study was supported by a Grant-in-Aid for\nScienti\fc Research from MEXT.\n[1] S. Maekawa, ed., Concepts in Spin Electronics (Oxford\nUniversity Press, Oxford, 2006).\n[2] S. Maekawa, S. Valenzuela, E. Saitoh, and T. Kimura,\ned., Spin Current (Oxford University Press, Oxford,\n2012).[3] F. J. Jedema, A. T. Filip, and B. J. van Wees, Nature\n(London), 410, 345 (2001).\n[4] E. Saitoh, M. Ueda, H. Miyajima, and G. Tatara. Appl.\nPhys. Lett. 88, 182509 (2006).\n[5] K. Uchida, S. Takahashi, K. Harii, J. Ieda, W. Koshibae,\nK. Ando, S. Maekawa, and E. Saitoh, Nature (London)\n455, 778 (2008).\n[6] K. Uchida, H. Adachi, T. An, T. Ota, M. Toda, B. Hille-\nbrands, S. Maekawa, and E. Saitoh, Nature Mater. 10,\n737 (2011); K. Uchida, T. An, Y. Kajiwara , M. Toda ,\nand E. Saitoh, Appl. Phys. Lett. 99, 212501 (2011); K.\nUchida, H. Adachi, T. An, H. Nakayama, M. Toda, B.\nHillebrands, S. Maekawa, and E. Saitoh, J. Appl. Phys.\n111, 053903 (2012).\n[7] Y. K. Kato, R. C. Myers, A. C. Gossard, and D. D.\nAwschalom, Science 306, 1910 (2004); J. Wunderlich, B.\nKaestner, J. Sinova, and T. Jungwirth, Phys. Rev. Lett.\n94, 047204 (2005); T. Kimura, Y. Otani, T. Sato, S.\nTakahashi, and S. Maekawa, Phys. Rev. Lett. 98, 156601\n(2007).\n[8] C. G. de Oliveira and J. Tiomno, Nuovo Cimento 24, 672\n(1962); B. Mashhoon, Phys. Rev. Lett. 61, 2639 (1988);\nJ. Anandan, Phys. Rev. Lett. 68, 3809 (1992); B. Mash-\nhoon, Phys. Rev. Lett. 68, 3812 (1992); F. W. Hehl and\nW.-T. Ni, Phys. Rev. D42, 2045 (1990).\n[9] L. D. Landau and E. M. Lifshitz, Theory of Elasticity\n(Pergamon, New York, 1959).\n[10] E. M. Chudnovsky, D. A. Garanin, and R. Schilling,\nPhys. Rev. B 72 , 094426 (2005); C. Calero, E. M. Chud-\nnovsky, and D. A. Garanin, Phys. Rev. Lett. 95, 166603\n(2005); C. Calero and E. M. Chudnovsky, Phys. Rev.\nLett. 99, 047201 (2007).\n[11] S. J. Barnett, Phys. Rev. 6, (1915) 239.\n[12] A. Hern\u0013 andez-M\u0013 \u0010nguez, K. Biermann, S. Lazi\u0013 c, R. Hey,\nand P. V. Santos, Appl. Phys. Lett. 97, 242110 (2010).\n[13] H. Sanada, T. Sogawa, H. Gotoh, K. Onomitsu, M. Ko-\nhda, J. Nitta, and P. V. Santos, Phys. Rev. Lett. 106,\n216602 (2011).\n[14] M. Weiler, L. Dreher, C. Heeg, H. Huebl, R. Gross, M.\nS. Brandt, and S. T. B. Goennenwein, Phys. Rev. Lett.\n106, 117601 (2011).\n[15] L. Vila, T. Kimura, and Y. C. Otani, Phys. Rev. Lett.\n99, 226604 (2007).\n[16] F. J. Jedema, H. B. Heersche, A. T. Filip, J. J. A. Basel-\nmans, and B. J. van Wees, Nature (London), 416, 7135\n(2002).\n[17] R. Godfrey and M. Johnson, Phys. Rev. Lett. 96, 136601\n(2006).\n[18] J.-H. Ku, J. Chang, H. Kim, and J. Eom, Appl. Phys.\nLett. 88, 172510 (2006).\n[19] J. M. Kikkawa and D. D. Awschalom, Phys. Rev. Lett.80, 4313 (1998).\n[20] M. Matsuo, J. Ieda, and S. Maekawa, arXiv:1211.0127.\n[21] C. M. Jaworski, R. C. Myers, E. Johnston-Halperin, and\nJ. P. Heremans, Nature (London) 484, 210 (2012).\n[22] A. E. H. Love, A Treatise on the Mathematical Theory of\nElasticity (Dover, New York, USA, 1967)."    },    {        "title": "1911.12180v2.How_spin_orbital_entanglement_depends_on_the_spin_orbit_coupling_in_a_Mott_insulator.pdf",        "content": "How spin-orbital entanglement depends on the spin-orbit coupling in a Mott insulator\nDorota Gotfryd,1, 2Ekaterina M. P arschke,3, 4Ji\u0014 r\u0013 \u0010 Chaloupka,5, 6Andrzej M. Ole\u0013 s,2, 7and Krzysztof Wohlfeld1\n1Institute of Theoretical Physics, Faculty of Physics, University of Warsaw, ul. Pasteura 5, PL-02093 Warsaw, Poland\n2Institute of Theoretical Physics, Jagiellonian University, Prof. S.  Lojasiewicza 11, PL-30348 Krak\u0013 ow, Poland\n3Department of Physics, University of Alabama at Birmingham, Birmingham, Alabama 35294, USA\n4Institute of Science and Technology Austria, Am Campus 1, A-3400 Klosterneuburg, Austria\n5Department of Condensed Matter Physics, Faculty of Science, Masaryk University,\nKotl\u0013 a\u0014 rsk\u0013 a 2, CZ-61137 Brno, Czech Republic\n6Central European Institute of Technology, Masaryk University, Kamenice 753/5, CZ-62500 Brno, Czech Republic\n7Max Planck Institute for Solid State Research, Heisenbergstrasse 1, D-70569 Stuttgart, Germany\n(Dated: March 3, 2020)\nThe concept of the entanglement between spin and orbital degrees of freedom plays a crucial role\nin understanding of various phases and exotic ground states in a broad class of materials, including\norbitally ordered materials and spin liquids. We investigate how the spin-orbital entanglement in\na Mott insulator depends on the value of the spin-orbit coupling of the relativistic origin. To this\nend, we numerically diagonalize a one-dimensional spin-orbital model with the `Kugel-Khomskii'\nexchange interactions between spins and orbitals on di\u000berent sites supplemented by the on-site\nspin-orbit coupling. In the regime of small spin-orbit coupling w.r.t. the spin-orbital exchange,\nthe ground state to a large extent resembles the one obtained in the limit of vanishing spin-orbit\ncoupling. On the other hand, for large spin-orbit coupling the ground state can, depending on\nthe model parameters, either still show negligible spin-orbital entanglement, or can evolve to a\nhighly spin-orbitally entangled phase with completely distinct properties that are described by an\ne\u000bective XXZ model. The presented results suggest that: (i) the spin-orbital entanglement may\nbe induced by large on-site spin-orbit coupling, as found in the 5 dtransition metal oxides, such as\nthe iridates; (ii) for Mott insulators with weak spin-orbit coupling of Ising-type, such as e.g. the\nalkali hyperoxides, the e\u000bects of the spin-orbit coupling on the ground state can, in the \frst order\nof perturbation theory, be neglected.\nI. INTRODUCTION\nA. Interacting Quantum Many-body Systems:\nCrucial Role of Entanglement\nOne of the main questions for a quantum interact-\ning many-body system concerns the nature of its ground\nstate. Perhaps the most fundamental question that can\nbe formulated here is as follows: Can the eigenstates of\nsuch a system be written in terms of a product of the `lo-\ncal' (e.g. single-site) basis states? If this is the case, then\nthe ground state can be understood using the classical\nphysics intuition. Moreover, the low lying excited states\ncan then be described as weakly interacting quasiparti-\ncles, carrying the quantum numbers of the constituents\nforming the system. Such physics is realized, for instance,\nby the ions in conventional crystals, spins in ordered mag-\nnets, or electrons in Fermi liquids [1].\nAn interesting situation occurs, however, when the an-\nswer to the above question is negative and we are left with\na `fully quantum' interacting many-body problem [2, 3].\nIn this case, the classical intuition fails, numerical de-\nscription of the ground state may become exponentially\ndi\u000ecult, and the low lying eigenstates cannot be de-\nscribed as weakly interacting quasiparticles. This can,\nfor example, be found in the spin liquids stabilized in the\none-dimensional (1D) or highly frustrated magnets [4] or\nincommensurate electronic systems with strong electron\ninteractions (the so-called non-Fermi liquids) [5]. In fact,a large number of condensed matter studies are nowadays\ndevoted to the understanding of such `exponentially dif-\n\fcult' problems.\nA way to characterize the fully quantum interacting\nmany-body problem is by introducing the concept of en-\ntanglement [6{8] and then by de\fning interesting aspects\nof its quantum structure through the entanglement en-\ntropy [9]. It is evident that studying entanglement always\nrequires \frst a de\fnition of what is entangled with what,\ni.e., specifying the division of the system into the sub-\nsystems that may become entangled. Perhaps the most\nwidely performed division has so far concerned splitting a\nlattice spin system into two subsystems in real space [10].\nSuch studies enabled to identify the relation between the\nentanglement of the spin system, its scaling with the sys-\ntem size, and the product nature of the ground state, cf.\nRefs. [10{16].\nB. Spin-Orbital Entanglement in\nTransition Metal Compounds\nTransition metal oxides involve often numerous com-\npeting degrees of freedom. Examples are the three-\ndimensional (3D) ground states of LaTiO 3, LaVO 3,\nYVO 3, Ba 3CuSb 2O9[17{22], where spin and orbital de-\ngrees of freedom are intertwined and entangled. A similar\nsituation occurs in MnP [23], the \frst Mn-based uncon-\nventional superconductor under pressure, and in some\ntwo-dimensional (2D) model systems [24{26]. In all thesearXiv:1911.12180v2  [cond-mat.str-el]  28 Feb 20202\ncases the ground state can only be explained by invok-\ning the joint spin-orbital \ructuations. Consequently, the\nmean \feld decoupling separating interactions into spin\nand orbital degrees of freedom fails and cannot be used.\nIn this paper we study the spin-orbital entanglement\nwhich manifests itself when a quantum many-body sys-\ntem with interacting spin and orbital degrees of freedom\nis split into the subsystems with separated degrees of\nfreedom, i.e., one attempts to write interacting spin and\norbital wave functions separately [27{29]. The concept of\nentanglement has been \frst introduced in these systems\nto understand the violation of the so-called Goodenough-\nKanamori rules [30, 31] in the ground states of several\ntransition metal oxides with partially \flled 3 dorbitals,\nstrong intersite spin-orbital (super)exchange interactions\nbut typically negligible value of the on-site spin-orbit cou-\npling. It was also realized that entanglement is impor-\ntant to understand the excited states where spin and or-\nbital variables are intertwined. Good examples are the\ntemperature evolution of the low energy excitations in\nLaVO 3[32, 33] and the renormalization of spin waves by\norbital tuning in spin-orbital systems due to the weak\ninteractions with the lattice [34]. Furthermore, the spin-\norbital entanglement is also crucial to understand the\n\frst unambiguous observations of the collective orbital\nexcitation (orbiton) in Sr 2CuO 3and CaCu 2O3[35, 36]\nand their interpretation in terms of the spin and orbital\nseparation in a 1D chain [37{39].\nCrucially, the spin-orbital entanglement is expected\nas well in the oxides with strong on-site spin-orbit\ncoupling|probably best exempli\fed by the partially\n\flled 4dand 5dorbitals as found in the ruthenium and\niridium oxides [40, 41], or in the recently discovered 5 dTa\nchlorides [42]. For instance, the concept of spin-orbital\nentanglement was recently invoked to understand the\ninelastic x-ray spectrum of Sr 3NiIrO 6[43], the ground\nstate of H 3LiIr2O6[44] or, in a di\u000berent physical set-\nting, the photoemission spectra of Sr 2RuO 4[45, 46].\nInterestingly, the peculiarities of the interplay between\nthe strong on-site spin-orbit coupling and the spin-\norbital (super)exchange interactions allowed for the on-\nset of several relatively exotic phenomena in this class of\ncompounds|such as a condensed matter analogue of a\nHiggs boson in Ca 2RuO 4[47, 48] or the strongly direc-\ntional, Kitaev-like, interactions between the low energy\ndegrees of freedom (pseudospins) in some of the iridates\nor ruthenates on a quasi-2D honeycomb lattice (Na 2IrO3,\nLi2IrO3,\u000b-RuCl 3, H3LiIr2O6) [44, 49, 50]. The latter\nmight be described to some extent by the exactly solv-\nable Kitaev model on the honeycomb lattice which, in-\nter alia , supports the onset of a novel spin-liquid ground\nstate with fractionalized `Majorana' excitations [51].\nC. Main question(s) and organization of the paper\nIt may come as a surprise that the concept of the spin-\norbital entanglement has so far been rigorously investi-gated only for the systems where the spins and orbitals at\nneighboring sites interact, as a result of the spin, orbital\nand spin-orbital (super)exchange processes in Mott insu-\nlators [27{29, 52{57]. This case is physically relevant to\nall Mott insulators with negligible spin-orbit coupling of\nrelativistic origin and with active orbital degrees of free-\ndom [58]|e.g. to the above mentioned case of transition\nmetal oxides with partially \flled 3 dorbitals.\nOn the other hand, to the best of our knowledge, such\nanalysis has not been done for the systems with strong on-\nsite coupling between spins and orbitals [59]|as in the\nabove-discussed case of the transition metal oxides with\npartially \flled 4 dand 5dorbitals and strong spin-orbit\ncoupling. We stress that in this case the spin and orbital\ndegrees of freedom can get entangled as a result of both\nthe nearest neighbor exchange interactions as well as on-\nsite spin-orbit coupling. In fact, one typically implic-\nitlyassumes that the spin-orbital entanglement should\nbe nonzero, since the spin Sand orbital Loperators cou-\nple at each site into a total angular momentum J=S+L\n[60]. The latter, `spin-orbital entangled' operators (also\ncalled pseudospins), then interact as a result of the ex-\nchange processes in the `relativistic' Mott insulators and\nare best described in terms of various e\u000bective pseudospin\nmodels, such as for example the Kitaev-like model dis-\ncussed above [61]. Finally, very few studies discuss the\nproblem of the evolution of a spin-orbital system between\nthe limit of weak and strong spin-orbit coupling [62{65].\nHere we intend to bridge the gap between the under-\nstanding of the spin-orbital physics in the above two lim-\nits. We ask the following questions: (i) what kind of evo-\nlution does the spin-orbital entanglement develop with\nincreasing spin-orbit coupling? (ii) can one always as-\nsume that in the limit of strong spin-orbit coupling the\nspin-orbital entanglement is indeed nonzero? (iii) how\ndoes the spin-orbital entanglement arise in the limit of\nthe strong spin-orbit coupling?\nTo answer the above questions we formulate a mini-\nmal 1D model with S=1=2spin andT=1=2orbital\n(pseudospin) degrees of freedom. The model has the\nSU(2)\nSU(2) intersite interactions between spins and or-\nbitals which are supplemented by the on-site spin-orbit\ncoupling of the Ising character|its detailed formulation\nas well as its relevance is discussed in Sec. II. Using exact\ndiagonalization (ED), the method of choice described in\nSec. III, we solve the 1D model and evaluate the vari-\nous correlation functions used to study the entanglement.\nNext, we present the evolution of the ground state prop-\nerties as a function of the model parameters: in Sec. IV A\nfor di\u000berent values of the three model parameters, and\nin Sec. IV B for a speci\fc choice of the relation between\nthe two out of the three model parameters. We then\nshow two distinct paths of ground state evolution in Secs.\nIV B 1 and IV B 2. The evolution of the exact spectra of\nthe periodic L= 4 chain is analyzed in Sec. IV B 3 for\nincreasing\u0015. We discuss obtained numerical results uti-\nlizing mapping of the model onto an e\u000bective XXZ model\nin Sec. V, which is valid in the limit of the strong on-site3\nspin-orbit coupling. We use the e\u000bective model to un-\nderstand: (i) how the spin-orbital entanglement sets in\nthe model system and (ii) how it depends on the value\nof the on-site spin-orbit coupling constant \u0015. The pa-\nper ends with the conclusions presented in Sec. VI and is\nsupplemented by an Appendix which discusses in detail\nthe mapping onto the e\u000bective XXZ model in the limit\nof large spin-orbit coupling \u0015!1 .\nD. Practical note on the organization of the paper\nWe note that, whereas the main results of the paper are\ngiven in the extensive Sec. IV, some of the main results\ncan be understood by using a mapping onto an e\u000bec-\ntive XXZ model in Sec. V. Thus, we refer the interested\naudience looking for the more physical and intuitive un-\nderstanding of (some of) the obtained numerical results\nto the latter section. Finally, we stress that allthe im-\nportant results of the paper are not only listed but also\ndiscussed in detail in Sec. VI A.\nII. MODEL\nIn this paper we study a spin-orbital model Hde\fned\nin the Hilbert space spanned by the eigenstates of the\nspinS=1=2and orbital (pseudospin) T=1=2operators\nat each lattice site of a 1D chain with periodic bound-\nary conditions. The model Hamiltonian consists of two\nqualitatively distinct terms,\nH=HSE+HSOC: (1)\nThe \frst term HSEdescribes the intersite (su-\nper)exchange interactions between spins and orbitals.\nThe spin-orbital (`Kugel-Khomskii') exchange reads,\nHSE=JX\ni[(Si\u0001Si+1+\u000b) (Ti\u0001Ti+1+\f)\u0000\u000b\f];(2)\nwhereJ > 0 is the exchange parameter and the con-\nstants\u000band\fare responsible for the relative strengths\nof the individual spin and orbital exchange interactions.\nThis 1D SU(2)\nSU(2) symmetric spin-orbital Hamilto-\nnian has been heavily studied in the literature|it is ex-\nactly solvable by Bethe Ansatz at the SU(4) point, i.e.,\nwhen\u000b=\f=1=4[66{68], has a doubly degenerate\nground state at the so-called Kolezhuk-Mikeska point\n\u000b=\f=3=4[69, 70], and was studied using various\nanalytical and numerical methods for several other rele-\nvant values off\u000b;\fgparameters [39, 71{75]. In partic-\nular, the entanglement between spin and orbital degrees\nof freedom in such a class of Hamiltonians is extremely\nwell-understood [28, 29, 52]. Last but not least, it was\nsuggested that this model may describe the low-energy\nphysics found in NaV 2O5and Na 2Ti2Sb2O [72], CsCuCl 3\nand BaCoO 3[76], as well as in the arti\fcial Mott insula-\ntors created in optical lattices [77, 78] and the so-called\nCoulomb impurity lattices [79].Altogether, this means that the spin-orbital exchange\ninteraction has the simplest possible form [58] that can,\nnevertheless, describe a realistic situation found in the\ntransition metal oxides. This, as already mentioned in In-\ntroduction, constitutes the main reason behind the choice\nof this form of spin-orbital intersite interaction. We note\nthat the spin-orbital exchange would often has a more\ncomplex form. For instance, this would be the case, if\ne.g. three instead of two active orbitals were taken into\naccount and the corrections from \fnite Hund's exchange\nwere included (as relevant for the 5 diridates, whose spin-\norbital exchange interactions are given by e.g. Eq. (3.11)\nof Ref. [19]).\nThe second term in the studied Hamiltonian (1) de-\nscribes the on-site interaction between the spin and or-\nbital degrees of freedom and reads\nHSOC= 2\u0015X\niSz\niTz\ni: (3)\nHere the parameter \u0015measures the strength of the on-\nsite spin-orbit coupling term (of relativistic origin). The\nabove Ising form of the spin-orbital coupling was chosen\nas the simplest possible and yet nontrivial one. Moreover,\nexactly such a form of the spin-orbit coupling is typically\nrealized in systems with twoactive orbitals. This is the\ncase of e.g. the active t2gdoublets in YVO 3[80] and\nSr2VO4[81], the molecular \u0019orbitals of KO 2[82], or on\noptical lattices. In fact, such a highly anisotropic form of\nspin-orbit coupling is valid for any system with an active\norbital doublet, either two p(pxandpy) or twot2g(xz\nandyz) orbitals.\nIII. METHODS AND CORRELATION\nFUNCTIONS\nAs we are interested in quantum entanglement,\nand moreover, the exchange Hamiltonian (2) itself\nbears a rather complex quantum many-body term\n/(SiSj)(TiTj), we opt for the exact diagonalization\n(ED) method which preserves the quantum \ructuations\nin the numerically found ground state. More speci\fcally,\nwe choose the ED calculations, since: (i) It allows us to\ninvestigate the system ground state in a numerically ex-\nact manner and in a completely unbiased way which for\nthe \frst study of its kind is usually selected as the method\nof choice; (ii) The analytically exact Bethe Ansatz ap-\nproach can only be applied to a few selected values of\nthe model parameters; (iii) The ED calculations can be\nrelatively easily repeated for a number of model parame-\nters and can typically address the qualitative properties\nof the ground state rather well; (iv) We are interested\nhere in rather local correlations which follow from the\nlocal spin-orbit and nearest neighbor exchange interac-\ntions.\nWe calculate the properties of the ground state of\nmodel (1) on \fnite chains with periodic boundary condi-\ntions. We utilize chains of length L= 4nsites, where nis4\nan integer number, in order to avoid a degenerate ground\nstate appearing in the case of a (4 n+ 2){site chain (see\nTable 1 of Ref. [67]). For chains L= 4 a standard full\nED procedure is performed, while for L= 8, 12, 16, and\n20 sites we restrict the ED calculations to the Lanczos\nmethod [83].\nTo capture the changes in the ground state of the spin-\norbital model at increasing spin-orbit coupling \u0015, we de-\n\fne and investigate the following correlation functions\nwhich will be used, besides the von Neumann spin-orbital\nentanglement entropy [84], to monitor the evolution of\nthe ground state with changing model parameters:\n(i) The intersite spin-orbital correlation function CSO:\nCSO=1\nLLX\ni=1h(Si\u0001Si+1)(Ti\u0001Ti+1)i\n\u00001\nLLX\ni=1hSi\u0001Si+1ihTi\u0001Ti+1i; (4)\nwhich measures the intersite (nearest neighbor) correla-\ntion between the spin and orbital degrees of freedom and\nhas already been used in the literature as a good qualita-\ntiveestimate for the von Neumann spin-orbital entangle-\nment entropy [55, 56]. This correlator can also be used to\nmonitor the failure of the mean \feld decoupling between\nthe spins and orbital pseudospins once CSO6= 0 [27].\n(ii) The intersite spin Sor orbitalTcorrelation func-\ntion,\nS=1\nLLX\ni=1hSi\u0001Si+1i; (5)\nT=1\nLLX\ni=1hTi\u0001Ti+1i; (6)\nwhich measures the intersite (nearest neighbor) correla-\ntion between the spin (orbital) degrees of freedom and\nis therefore sensitive to the changes in the ground state\nproperties taking place solely in the spin (orbital) sub-\nspace. We emphasize that these two functions are de\fned\non equal footing in the model with SU(2) \nSU(2) spin-\norbital superexchange.\n(iii) The\r{component S\r\rof spin scalar product:\nS\r\r=1\nLLX\ni=1\nS\r\niS\r\ni+1\u000b\n; (7)\nwhere\r=x;y;z . This function measures the component\n\rof the scalar product and thus allows one to investigate\npossible anisotropy of the intersite (nearest neighbor)\ncorrelations between the spin degrees of freedom. The\norbital scalar product component T\r\ris de\fned analo-\ngously to Eq. (7).\n(iv) Crucial for the systems with \fnite spin-orbit cou-\npling is the on-site spin-orbit correlation function OSO:\nOSO=1\nLLX\ni=1hSz\niTz\nii; (8)which measures the correlations between the zcompo-\nnents of the spin and orbital operators on the same site.\nThe precise form of this correlator is dictated by the Ising\nform (3) of the spin-orbit coupling present in Hamiltonian\n(1). Conveniently, the function (8) is one of the gener-\nators of the SU(4) group [68], which proved to be quite\nuseful for examining the range of the SU(4){symmetric\nground state.\nIV. NUMERICAL RESULTS\nA. von Neumann entropy in a general case\nThe main goal of this paper is to determine how\nthe spin-orbital entanglement changes in the spin-orbital\nmodel (1) upon increasing the value of the spin-orbit cou-\npling\u0015. To this end, we \frst de\fne the entanglement\nentropy calculated for a system that is bipartitioned into\ntwo subsystems: AandB. Typically such a subdivision\nrefers to two distinct parts of the real [10{13] or momen-\ntum [16] space. Here, however, it concerns spin ( A) and\norbital (B) degrees of freedom [27{29, 53].\nA standard measure of the entanglement entropy be-\ntween subsystems AandBin the ground state jGSiof\na system of size Lis due to von Neumann [84]. It is\nde\fned asSvN=\u0000TrAf\u001aAln\u001aAg=L, and is obtained by\nintegrating the density matrix, \u001aA= TrBjGSihGSjover\nsubsystem B. Consequently, in this paper we use the\nfollowing de\fnition of the von Neumann spin-orbital en-\ntanglement entropy:\nSvN=\u00001\nLTrSf\u001aSln\u001aSg; (9)\nwhere\n\u001aS= TrTjGSihGSj (10)\nis the reduced spin-only ( S) density matrix with the or-\nbital (T) degrees of freedom integrated out.\nThe spin-orbital von Neumann entropy is calculated\nusing ED on L{site chain for model (1) and is shown as\nfunction of the parameters f\u000b;\fgfor three representative\nvalues of the spin-orbit coupling \u0015in Fig. 1. In perfect\nagreement with Refs. [29, 52], the von Neumann entropy\nSvNis \fnite in a rather limited region of the f\u000b;\fgpa-\nrameters for \u0015= 0, i.e., in the entangled spin-orbital\nphase near the origin \u000b=\f= 0. The nonzero entan-\nglement in that case is well-understood and attributed\nto the onset of the dominant antiferromagnetic (AF) and\nalternating orbital (AO) \ructuations in the ground state\nwithout broken symmetry [29, 52], see discussion below\nin Sec. IV B 4.\nInterestingly, a \fnite but `small' spin-orbit coupling\n\u0015 < \u0015 CRIT (\u0015CRIT is discussed in more detail in\nSec. IV B) does not substantially increase the region in\nthef\u000b;\fg{parameter space for which the spin-orbital\nentropy is nonzero cf. Figs. 1(a) and 1(b). A drastic5\n-4.0 -2.0 0.0 2.0 4.0\nα-4.0-2.00.02.04.0β(a)\nλ/J=0\n-4.0 -2.0 0.0 2.0 4.0\nα(b)\nλ/J=0.1\n-4.0 -2.0 0.0 2.0 4.0\nα(c)\nλ/J→∞SvN\n0.00.10.20.30.40.5\nFIG. 1. The von Neumann spin-orbital entanglement entropy, SvN(9), calculated using ED on L= 12{site periodic chain\nfor the spin-orbital model Eq. (1) and for the increasing value of the spin-orbit coupling \u0015: (a)\u0015=J= 0, (b)\u0015=J= 0:1, and\n(c)\u0015=J!1 .\nchange in the behavior of the spin-orbital von Neumann\nentropy only happens for the dominant spin-orbit cou-\npling\u0015>\u0015 CRIT. In this case the region of nonzero spin-\norbital entanglement is not only much larger but also\ntakes place for di\u000berent values of the f\u000b;\fg{parameter\nspace. For instance, it is remarkable that the von Neu-\nmann entropy in the case of \u0015>\u0015 CRIT almost does not\ndepend on\u000balong the lines of constant \u000b+\f. Moreover,\n\fnite entanglement is activated when \u000b+\f >\u00001=2|\nhowever, the value of the von Neumann entropy strongly\ndecreases for \u000band\flocated `above' the stripe given\nby the inequalities \u00001=2\u0014\u000b+\f\u00142 and showing the\nhighest value of entropy. In fact, it will be shown later\n(see Sec. V) that the von Neumann entropy is expected\nto vanish in the limit of \u000b+\f!1 .\nAltogether, we observe that: (i) in the limit of small\n\u0015 < \u0015 CRIT the spin-orbital entanglement entropy does\nnot change substantially w.r.t. the case with vanishing\nspin-orbit coupling; (ii) in the limit of large \u0015 > \u0015 CRIT\nthe spin-orbital entanglement can become \fnite even if\nit vanishes for \u0015= 0; though it can also happen that\n(iii) in the limit of large \u0015 > \u0015 CRIT the spin-orbital en-\ntanglement vanishes when \u000b+\f <\u00001=2.\nB. von Neumann entropy for \f=\u0000\u000b:\nTwo distinct evolutions for increasing \u0015\nIn order to better understand the physics behind the\nobservations (i) and (ii) discussed in the end of the pre-\nvious subsection, here we study in great detail the onset\nof the spin-orbital entanglement once \f=\u0000\u000b. As shown\nin Fig. 1, for these values of the model parameters the\nregion of the nonzero spin-orbital entanglement increases\ndramatically with the increasing value of the spin-orbit\ncoupling\u0015.\nWe present in Fig. 2 the von Neumann spin-orbital en-tanglement entropy SvN(9) and the three spin-orbital\ncorrelation functions in the ground state of Hamilto-\nnian (1) with \u000b=\u0000\f, calculated using ED on an L= 12{\nsite chain. We begin the analysis by comparing the val-\nues of the three spin-orbital correlation functions (4), (8),\nand (5) against the von Neumann spin-orbital entangle-\nment entropy, see Figs. 2(b), 2(c), and 2(f). We observe\nthat only the intersite spin-orbital correlation function\nCSOcan be used as a qualitative measure for the von Neu-\nmann entropy, consistent with previous studies [55, 56].\nIn particular, the on-site spin-orbit correlation function\nOSOcannot be used to `monitor' the entanglement en-\ntropy, for it measures the correlations between spins and\norbitals locally and on the Ising level only. Nevertheless,\nbothOSOas well asSvNcan be used to identify various\nquantum phases obtained in the f\u000b;\u0015gparameter space\nof the Hamiltonian, as suggested before for system with\nnegligible spin-orbit coupling [56].\nNext, we study the evolution of the von Neumann\nspin-orbital entanglement entropy with increasing spin-\norbit coupling \u0015for various values of the parameter \u000b,\nsee Fig. 2(a). We observe that in the representative\n\u000b2[\u00001;1] interval there exist three distinct regimes of\nthe value of the von Neumann entropy: (i) two com-\npact areas in the f\u000b;\u0015gparameter space for which the\nvon Neumann entropy is vanishingly small, which exist\nin the large parameter range of j\u000bj&0:1 and\u0015=J.\n10\u00001\u0000100[the bottom left and bottom right parts of\npanel Fig. 2(a)]; (ii) one compact area in the f\u000b;\u0015gpa-\nrameter space for which the von Neumann entropy takes\nmaximal possible values, which exists in the large pa-\nrameter range \u0015=J&10\u00001\u0000101for all values of \u000b[the\ntop part of panel Fig. 2(a)]; (iii) the compact area in\nthef\u000b;\u0015gparameter space for which the von Neumann\nentropy is neither negligible nor takes maximal value,\nwhich exists in the relatively small parameter range be-\ntween cases (i) and (ii). In order to understand the on-6\n1.0\n 0.5\n0.0 0.5 1.0\nα10-310-210-1100101λ/JABA(a)SvN\n1.0\n 0.5\n0.0 0.5 1.0\nα(b)CSO\n1.0\n 0.5\n0.0 0.5 1.0\nα(c)OSO\n10-310-210-1100101102103\nλ/J0.00.10.20.30.40.5SvNA(d)\n10-310-210-1100101\nλ/JB(e)\n1.0\n 0.5\n0.0 0.5 1.0\nα10-310-210-1100101λ/J(f)S\n0.00.10.20.30.40.5\n0.25\n0.20\n0.15\n0.10\n0.05\n0.00\n0.25\n0.20\n0.15\n0.10\n0.05\n0.00\n0.50\n0.25\n0.000.250.50\nFIG. 2. Evolution of the von Neumann spin-orbital entanglement entropy and the three spin-orbital correlation functions in\nthe ground state of Hamiltonian (1) with \u000b=\u0000\f, calculated using ED with the periodic L= 12{site chain for logarithmically\nincreasing spin-orbit coupling \u0015: (a) the von Neumann spin-orbital entanglement entropy SvN(9) for\u000b2[\u00001:0;1:0]; (b) the\nintersite spin-orbital correlation function CSO(4) for\u000b2[\u00001:0;1:0]; (c) the on-site spin-orbit correlation function OSO(8) for\n\u000b2[\u00001:0;1:0]; (d) the von Neumann spin-orbital entanglement entropy SvN(9) obtained with j\u000bj= 0:5 [cut A in panel (a)]\nand \ftted with a logistic function (black thin line); (e) the von Neumann spin-orbital entanglement entropy SvN(9) obtained\nwith\u000b= 0 [cut B in panel (a)]; (f) the spin correlation function S(5) for\u000b2[\u00001:0;1:0].\nset of these three distinct regimes, we study below two\nqualitatively di\u000berent cases of the von Neumann entropy\nevolution with the increasing spin-orbit coupling: case\n`A' withj\u000bj= 0:5 and case `B' with \u000b= 0 [shown with\ndashed lines in Fig. 2(a)].\n1. From a product state to highly entangled state\nWhenj\u000bj= 0:5 (case A) the evolution of the von Neu-\nmann spin-orbital entanglement entropy SvN(9) with in-\ncreasing spin-orbit coupling can be well-approximated by\na logistic function, see Fig. 2(d). The von Neumann en-\ntropy has in\fnitesimally small values for \u0015=J.10\u00001,\nexperiences a rapid growth for \u0015=J2(10\u00001;101), and\nsaturates at ca. 0 :5 for\u0015=J&101. Comparable behav-\nior is observed for the intersite spin-orbital correlation\n(CSO), which, as already discussed, is a good and compu-\ntationally not expensive qualitative measure for the von\nNeumann entropy, see Figs. 3(a), 3(c), and 3(e). Cru-cially, the latter calculations are obtained for the spin-\norbital chains of di\u000berent length and [as well-visible in\nFigs. 3(a), 3(c), 3(e)] show relatively small \fnite-size ef-\nfects. This means that indeed the von Neumann entropy\nSvNdepends here mainly on short-range processes and\ncan remain negligibly small for a \fnite value of the spin-\norbit coupling even in the thermodynamic limit. Finally,\nFigs. 2 and 3 allow us to de\fne the critical value \u0015CRIT\nfor case A as being located in an interval of rapid growth\nof the spin-orbital entanglement: \u0015CRIT=J2(10\u00001;101).\nWhile the nature of the quantum phase for large\nspin-orbit coupling \u0015 > \u0015 CRIT is discussed in detail in\nSec. V B, here we merely mention that in this case the\nvalue of the spin-orbital entanglement entropy saturates\nat about 0:5 (0:504 forL= 12 site chain) per site. Hence,\nwe call this quantum phase a highly entangled state . Be-\nsides, in this case also the absolute value of the on-site\nspin-orbit correlation function OSOtakes its maximal\nvalue, while the spin (and orbital) correlation function\nS(T) is weakly AF (AO).7\n10-310-210-11001010.25\n0.20\n0.15\n0.10\n0.05\n0.00A\nL=8(a)\n10-210-1100B(b)\n10-310-210-11001010.25\n0.20\n0.15\n0.10\n0.05\n0.00\nL=12(c)\n10-210-1100(d)\nCSO OSO CSU(4)\nSO OSU(4)\nSO˜CSO˜OSO10-310-210-1100101\nλ/J0.25\n0.20\n0.15\n0.10\n0.05\n0.00\nL=16(e)\n10-210-1100\nλ/J(f)\nFIG. 3. The intersite spin-orbital correlation function CSO\n(green lines) and the on-site spin-orbit correlation function\nOSO(red lines) as functions of \u0015calculated for: case A (left),\ni.e.,\u000b= 0:5 [(a), (c), (e)] and case B (right), i.e., \u000b= 0 [(b),\n(d), (f)]. The ED results are shown for periodic chains of\nlengthL= 8 (top),L= 12 (middle), and L= 16 (bottom).\nThe dashed lines represent the asymptotic values of the above\ncorrelation functions: (i) the exact SU(4){point limit \u0015= 0,\n\u000b=\f= 0:25 is denoted byCSU(4)\nSO andOSU(4)\nSO (blue and light{\nred dashed lines); (ii) the \u0015=1,\f=\u0000\u000bXY limit|by ~CSO\nand ~OSO(dark{green and dark{red dashed lines). For further\ndetails see discussion in Secs. IV B 1-IV B 2 and Sec. V B).\nNext, we focus on the properties of the ground state\nobtained for small \u0015 < \u0015 CRIT. To this end we investi-\ngate the evolution of the two other correlation functions,\nthe on-site spin-orbit correlation function OSOand the\nspin correlation function S, for\u0015<\u0015 CRIT andj\u000bj= 0:5,\nsee Figs. 2(c), 2(f). We observe that whereas the on-site\nspin-orbit correlation function shows vanishingly small\nvalues in this limit, the spin correlation function S'0:25\n(S'\u00000:45) for\u000b= 0:5 (\u000b=\u00000:5), thus behaving sim-\nilarly to the 1D FM (AF) chain, respectively. We note\nthat the (unshown) analogous nearest neighbor orbital\n10-210-11001011020.2\n0.1\n0.00.10.2A (a)\nSδδ\nTδδSzz\nTzz/arrownorthwest\n/arrownorthwest /arrownortheast\n/arrownorthwest\n10-210-1100B (b)\nL=8\n10-210-11001011020.2\n0.1\n0.00.10.2(c)\nSδδ, Tδδ Szz, Tzz ˜Sδδ, ˜Tδδ ˜Szz, ˜Tzz10-210-1100(d)\nL=12\n10-210-1100101102\nλ/J0.2\n0.1\n0.00.10.2(e)\n10-210-1100\nλ/J(f)\nL=16FIG. 4. The anisotropic spin correlation function S\r\r(7) and\norbital correlation function T\r\rfor increasing \u0015. TheS\u000e\u000eand\nT\u000e\u000e(\u000e=x;y) components are marked by green color while\ntheSzzandTzzcomponents are marked by red color. The\ncorrelation functions are calculated for: case A (left), i.e.,\n\u000b= 0:5 [(a), (c), (e)] and case B (right), i.e., \u000b= 0 [(b),\n(d), (f)]. The ED results are shown for periodic chains of\nlengthL= 8 (top),L= 12 (middle), and L= 16 (bottom).\nThe asymptotic values of the correlation functions in the limit\n\u0015=1are shown for both \u000b= 0 and\u000b= 0:5 case and denoted\nas~S\u000e\u000e,~Szzand~T\u000e\u000e,~Tzz, see discussion in Sec. V B for further\ndetails.\ncorrelation function Tcalculated for \u000b=\u00060:5 takes\ncomplementary values to the spin correlation function for\n\u000b=\u00070:5, i.e.,T=\u0000S. Such behavior is again observed\nfor chains of various lengths, with S(T) better approx-\nimating the expected AF Bethe Ansatz value for larger\nchains, see Fig. 4. Altogether, this shows that the quan-\ntum phase that is observed for \u0015 < \u0015 CRIT qualitatively\nresembles the phases obtained in the limit of \u0015= 0: the\nFM\nAO (AF\nFO) for\u000b= 0:5 (\u000b=\u00000:5), respectively.\nThe above discussion contains just one caveat. Let us8\nlook at the evolution of the anisotropic spin (and orbital)\ncorrelation function S\r\r(andT\r\r) with the increasing\nspin-orbit coupling \u0015, see Figs. 4(a, c, e). We notice that\nwhenever\u0015=J > 0 for\u000b= 0:5 there exist an anisotropy\nbetween the zz(solid red lines) and the planar ( xx,yy,\nsolid green lines) correlation functions|which is absent\nfor\u0015= 0. However, for \u0015=J.3\u000110\u00001the anisotropy\nis only partial, being absent in the strongly AF T\r\rcor-\nrelations, in contrast to the S\r\rcorrelations. In fact,\nS\u000e\u000e(where\u000e=x;y), stay positive as in \u0015= 0 case\nwhileSzzbecomes negative. In this way the energy com-\ning from the \fnite spin-orbit coupling is `minimized' in\nthe ground state without qualitatively changing the na-\nture of the FM\nAO and AF\nFO ground states, allowing\nhowever for a very small value of the spin{orbital entan-\nglement. This is the reason why, in what follows, this\nquantum phase is called a perturbed FM\nAO product\nstate.\n2. From SU(4) singlet to a highly entangled state\nWe now investigate how the von Neumann spin-orbital\nentanglement entropy SvNevolves with the spin-orbit\ncoupling once \u000b= 0 (case B): i.e., from its \fnite value\nfor the SU(4){singlet ground state at \u0015= 0 [29, 52] to\nan even higher value obtained in the limit of large \u0015=Jin\nthe highly entangled state (i.e., the state already encoun-\ntered in case A). To this end, we \frst note that the von\nNeumann entropy SvNat\u000b= 0 changes with the spin-\norbit coupling in a qualitatively di\u000berent manner than\nin the case ofj\u000bj= 0:5, see Fig. 2(e). While we again\nencounter a monotonically growing function in \u0015, which\nsaturates at about 0 :5 for\u0015=J&0:2, this function seems\nto be discontinuous at three particular values of \u0015and\nthree `kinks' (for L= 12 sites) that can be easily identi-\n\fed in Fig. 2(e). A similar behavior is encountered in the\nqualitative measure for the von Neumann entropy|the\nspin-orbital correlation function CSO, see Fig. 2(b) and\nFig. 3(b, d, f).\nAs a side note let us mention that once \u0015= 0 and\n\u000b=\f= 0 the model (1) has an SU(4)-symmetric ground\nstate, as con\frmed by the remarkable convergence of\nthe functionsCSOandOSOto their asymptotic values\nCSU(4)\nSO andOSU(4)\nSO calculated at the exact SU(4) point\n\u000b=\f= 1=4, cf. Fig. 3(b, d, f). As the operator in the\nOSOfunction is one of the generators of the SU(4) group,\nits zero expectation value in the ground state is not only\nrelated to the absence of the spin-orbit coupling but also\nis a signature of the SU(4)-symmetric singlet [85].\nIt is clearly visible in Fig. 3(b, d, f) that the CSOand\nOSOcorrelations split from their SU(4)-singlet asymp-\ntotes in the subsequent kinks, which occur with the in-\ncreasing value of the spin-orbit coupling. Interestingly,\nthe number of kinks grows and their position changes\nwith the system size, see Fig. 3(b, d, f). In fact, L=4 kinks\nare observed for a chain of length L= 4;8;12;16;20, see\npanel (a) of Fig. 5. This naturally suggests that in the\n0.00 0.05 0.10 0.15 0.20 0.25\nλ/J0.25\n0.20\n0.15\n0.10\n0.05\n0.00OSO(a)\n4\n8\n1216\n20\n0.0 0.1 0.2 0.3\n1/L0.000.020.040.060.080.100.120.14λ/J(b)\n4\n8\n12\n1620\n0.00 0.01 0.02\n1/L20.160.170.180.190.200.21(c)\n812\n1620FIG. 5. Finite-size scaling of the boundaries of the interme-\ndiate entangled state for the case B in Fig. 3: (a) L=4 kinks\nforL= 4;8;12;16;20 shown on an example of OSO, (b) the\ndecreasing position of the `\frst' kink as a function of 1 =L,\n(c) the increasing position of the `\fnal' kink as a function of\n1=L2. These \fts use the ED numerical results obtained for\nL= 4;8;12;16;20 periodic chains presented as colorful dots\nin (b)&(c); the lines are the \fts (11) to the numerical data.\nin\fnite system the number of kinks will be in\fnite.\nBut what about the position of the `\frst' and the `last'\nkink in the thermodynamic limit? To answer this in-\ntriguing question with the available ED data we deduced\nqualitative values of \u0015=J which de\fne the regime where\ncorrelations take intermediate values and the entangled\nstate is not yet dominated by the large spin-orbit cou-\npling\u0015>\u0015 CRIT.\nThe \fnite size scaling performed here, shown in panels\n(b) and (c) of Fig. 5, uses a polynomial \ft similar as\nfor instance for the gap in the 1D half-\flled Hubbard\nmodel [86]. Here the positions of the `\frst' and the `last'\nkink scale di\u000berently with the increasing length of L= 4n\nchain. Namely, the position of the `\frst' kink k1is almost\nlinear in 1=Lwhile the `\fnal' kink's position kfscales\nalmost linearly with 1 =L2. By performing the \fts, we\nhave found that\nk1= 0:00004 + 0:69712x\u00000:50147x2;\nkf= 0:20132\u00001:13007x2+ 3:14415x4; (11)\nwherex\u00111=L. As a result, the position of the `\frst'\nkinkk1converges to \u0015=J = 0 whenL!1 , and it is\nindeed reasonable to expect that in\fnitesimal \u0015modi\fes\nweakly spin-orbital correlations in the thermodynamic\nlimit. In contrast, the `\fnal' kink kfwould then shift to\n\u0015=J'0:201. Therefore, for the case B we de\fne \u0015CRIT\nas a single number: \u0015CRIT=J'0:2. Altogether, this9\n10-1100\nλ/J-3.5-3.0-2.5-2.0E(a)\nA\n10-310-210-1100101\nλ/J-1.2-1.0-0.8-0.6(b)\nB\n-0.2 -0.1 0.0 0.1 0.2\nα-1.0-0.50.00.51.01.5˜E=E+2λ(c)\nλ/J=0.1\n-1.0 -0.5 0.0 0.5 1.0\nα0.010.020.030.040.0(d)\nλ/J=10\nFIG. 6. Top panels|the energy Eof the ground (blue) and\nlow lying excited states (gray) obtained for model (1) using\nED for periodic L= 4-site chain as a function of increasing\n\u0015=J[(a) case A for \u000b= 0:5, (b) case B for \u000b= 0].\nBottom panels|complete energy spectra for small and large\n\u0015[(c)\u0015=J = 0:1 and (d)\u0015=J = 10]. Note that we display\nhere ~E(12) to compare the spectra in similar energy range,\nindependently of the actual value of \u0015.\nmeans that the quantum phase encountered for 0 <\u0015<\n\u0015CRIT does not disappear in the thermodynamic limit\nand that its spin-orbital entanglement grows with the\nincreasing spin-orbit coupling in a continuous way. To\ncontrast this intermediate phase with the one showing\nthe maximal value of entanglement at \u0015 > \u0015 CRIT, we\ncall it an intermediate entangled state .\nTo better understand the properties of this phase,\nwe also consider the spin correlation function S, the\nanisotropic spin S\r\r, and the orbital T\r\rcorrelation\nfunctions, see Figs. 2(c, f), 3(b, d, f), and 4(b, d, f). Simi-\nlarly to the von Neumann entropy, also OSOorCSOcorre-\nlation functions show kinks due to \fnite-size e\u000bects which\nare expected to disappear in the thermodynamic limit.\nNoticeably, the behavior of S,S\r\r, andT\r\ris quite dis-\ntinct w.r.t. the one observed both for the highly entan-gled phase and seemingly for the perturbed FM \nAO or\nAF\nFO phases. This shows that the intermediate entan-\ngled phase observed at \u000b= 0 and for 0 < \u0015 < \u0015 CRIT is\nindeed qualitatively di\u000berent and constitutes a `genuine'\nquantum phase.\n3. Exact spectra for L= 4at increasing \u0015\nWe also note that the phase transition to the highly en-\ntangled phase with increasing \u0015is detected by level cross-\ning in Fig. 6(b) and by the discontinuity in the derivative\u0000@E\n@\u0015\u0001\n, which appears as the only kink for L= 4 { site\nchain, cf. Fig. 5(a). Other phase transitions occur by\nvarying\u000b|here at\u0015=J= 0:1 a phase transition is found\nfrom the FM (FO) phase with Stot= 2 andTtot= 0\n(Stot= 0 andTtot= 2) to an entangled SU(4) phase\n(with all 15 generators being equal to 0) at j\u000bj'0:08, see\nFig. 6(c). At this latter phase transition one \fnds also a\ndiscontinuous change of the von Neumann entropy [56].\nFor convenience, we introduce here the energy ~Ewhich\ndoes not decrease with increasing \u0015as in Fig. 6(b). For\na chain of length L= 4 it is de\fned as follows,\n~E\u0011E+ 2\u0015: (12)\nTo get more insight into the evolution of the spectra\nwith increasing spin-orbit coupling /\u0015, we consider in\nmore detail the exact spectra of the L= 4 periodic chain,\nsee Table I. At \u0015= 0 the ground state is the SU(4) sin-\nglet with energy E=\u00000:75. The degeneracies of the\nexcited states follow from the SandTquantum num-\nbers. Indeed, several states with higher values of Sand\nTexhibit huge degeneracies. Weak spin-orbit coupling\n\u0015=J= 0:1 perturbation of the superexchange introduces\nthe splittings of degenerate excited states and in fact the\nTABLE I. The energies of the ground state and eight \frst\nexcited states ~E(12), with their degeneracies d, obtained for\nthe periodic L= 4 chain at \u000b=\f= 0 describing the spin-\norbital model Eq. (1) at \u0015= 0 and for two representative\nvalues of\u0015(\u0015=J= 0:1 and 10), standing for weak and strong\nspin-orbit coupling.\n\u0015= 0 \u0015=J= 0:1 \u0015=J= 10\n~E d ~E d ~E d\n\u00000:75 1 \u00000:55 1 \u00000:45736 1\n\u00000:50 28 \u00000:46570 1 \u00000:25 2\n\u00000:45711 9 \u00000:37965 4 0 :24367 1\n\u00000:43301 12 \u00000:36944 8 0 :24684 2\n\u00000:40139 1 \u00000:30 15 0 :24688 4\n\u00000:25 48 \u00000:26229 2 0 :24691 1\n0:0 76 \u00000:25345 2 0 :25 2\n0:25 34 \u00000:25 2 0 :74369 2\n0:43301 12 \u00000:24561 2 0 :94779 110\nspectrum is quite dense, see Fig. 6(c) and the data in\nTable I. However, the SU(4) singlet ground state is still\nrobust as shown by the correlation functions CSOand\nOSOwhich do not change from their \u0015= 0 values, see\nthe dotted line in Fig. 5 (a). It indicates that the spin-\norbit term does not align here spin Szand orbital Tz\ncomponents. The energy of the ground state (12) is just\nmoved by 2 \u0015from\u00000:75 to ~E=\u00000:55 (Table I), and the\nspin-orbit coupling does not modify the ground state.\nAt\u0015CRIT = 0:14219Jthe energies of the ground state\nand of the lowest energy excited state cross and a com-\npletely di\u000berent situation arises|then the von Neumann\nentropy changes in a discontinuous way to the value cor-\nresponding to the strongly entangled state (for L= 4),\nand the spin-orbit correlation OSOdrops to\u00000:25, see\nFig. 5 (a). Since \u0015CRIT is de\fned based on the changes\nin the ground state , for\u0015 > \u0015 CRIT the full energy spec-\ntrum does change further and consists of several bands\nof states, separated by gaps of the order of \u0015, see Fig.\n6(d). The energies and their degeneracies within the low-\nest band of states are shown in the last two columns of\nTable I at large \u0015=J= 10 for the chain of length L= 4.\nWe have veri\fed that the 16 low energy states dis-\nplayed in Table I for \u0015=J= 10 collapse to the spectrum\nof the XY model in the limit of \u0015!1 , with the de-\ngeneracies 1, 2, 10, 2, 1, as expected and discussed in\nmore detail in Sec. V A. Thus, in general, the spectrum\nconsists of energy bands with the energies increasing in\nsteps of'\u0015, depending on the number of sites at which\nthe spin-orbit coupling aligns the expectation values of\nspin and orbital operators, hSz\niTz\nii, at each site i. In\nthis regime the spectra are dominated by the spin-orbit\ncoupling. Note that the highly entangled phase can in-\ndeed be regarded as a qualitatively unique phase, irre-\nspectively of the value of \u000b|provided that \f=\u0000\u000band\nthat\u0015>\u0015 CRIT.\n4. Summary\nWe have discussed in detail the evolution of the spin-\norbital entanglement, and its impact on the quantum\nphases, with the increasing value of the spin-orbit cou-\npling\u0015for two representative values of the parameter\n\u000b. We can now extend the above reasoning to the other\nvalues of\u000b, keeping\f=\u0000\u000b. However, in order to ob-\ntain a quantum phase diagram of the model we still need\nto investigate whether the transitions between the ob-\ntained ground states could be regarded as phase transi-\ntions or are rather just of the crossover type. Dependence\nof the ground states energy on the model parameters (see\nFig. 6) as well as the analytic characteristics of the von\nNeumann entropy [see Fig. 2(d-e); cf. Refs [87, 88]] sug-\ngest that the transitions along cuts A and B [Fig. 2(a)]\nare of distinct character. Whereas in case A the energy\n(as well as the von Neumann entropy) shows an analytic\nbehavior across the transition, [Fig. 6(a)], in case B such\nbehavior (both in energy as well as in von Neumann en-\nα βSU(4)FM\ndimers⊗AO AF⊗FOFM ⊗AO FM ⊗AO AF⊗F0\nlog scale/Jλ\nhighly entangled state\nperturbed perturbed IESFIG. 7. Schematic quantum phase diagram of Hamiltonian\n(1) in the here{discussed regime of the parameters. The limit\nof\u000b=\u0000\fis depicted by the colorful vertical plane and is\nbased on the results from Sec. IV B: whereas the four distinct\nphases are depicted with their names and separated by solid\nlines (IES stands for the intermediate entangled state), the\ntwo crossover regimes are denoted by yellow color and sepa-\nrated by the dashed lines. The limit \u0015= 0 is depicted by the\nhorizontal plane and is adopted from Fig. 1 of Ref. [52]|see\ntext for further details. We note that the shape of the phase\nboundaries depends on the logarithmic scale of \u0015, chosen here\nfor convenience. The schematic phase diagram is based on the\nED results on small clusters, see text for the validity of these\nresults in the thermodynamic limit.\ntropy) is clearly non-analytic [Fig. 6(b)]. This points to\na crossover (phase) transition in case A (B), respectively.\nAltogether, this allows us to draw, on a qualitative\nlevel, a quantum phase diagram in the f\u000b;\u0015gparame-\nter space (with \f=\u0000\u000b), see Fig. 7 (colorful vertical\nplane). As already discussed in Sec. IV, there are four\ndistinct ground states (\frst two shown in Fig. 7 in blue,\nand the other two in green and red, respectively): (i) the\nperturbed FM\nAO state for \u000b&0:08 and\u0015 < \u0015 CRIT,\n(ii) the perturbed AF \nFO state for \u000b.\u00000:08 and\n\u0015 < \u0015 CRIT, (iii) the intermediate entangled state for\nj\u000bj.0:08 and 0< \u0015 < \u0015 CRIT, and (iv) the highly\nentangled state for \u0015 > \u0015 CRIT and for all values of \u000b.\nThe latter state is discussed in more detailed in Sec. V.\nThe four clearly distinct states are supplemented by two\ncrossover regimes (shown in yellow in Fig. 7), which sep-\narate phases (i-ii) from phase (iv)|see also discussion\nabove.\nIt is instructive to place the above phase diagram in\nthe context of the one already known from the literature11\nand obtained for Hamiltonian (1) in the limit of the van-\nishing spin-orbit coupling \u0015but varying values of both \u000b\nand\f[29, 39, 52, 66{75]. As can be seen on the hori-\nzontal plane of Fig. 7, the \u0015= 0 phase diagram consists\nof three simple product phases (AF \nFO, FM\nAO and\nFM\nFO) as well as two spin-orbital entangled phases\n(cf. Fig. 1): a phase with previously mentioned `global'\nSU(4)-symmetric singlet ground state and gapless exci-\ntations [68] and a phase with the ground state breaking\ntheZ2symmetry and opening a \fnite gap by forming\nthe two nonequivalent patterns of the spin and orbital\ndimers [29, 52]. We would like to emphasize at this point\nthat the \fnite size e\u000bects for the spin-orbital model (at\n\u0015= 0) calculated on chains of length L= 16 (the maxi-\nmal size studied in Ref. [52]) and L= 20 (the maximal\nsize studied here) are already relatively small [52]. This\nmay suggest that the schematic phase diagram of Fig. 7\nisqualitatively correct also in the thermodynamic limit.\nV. DISCUSSION: THE LIMIT OF LARGE \u0015\nA. E\u000bective XXZ model\nTo better understand the numerical results obtained\nin Sec. IV for the Hamiltonian (1) in the limit of the\nlarge spin-orbit coupling, \u0015 > \u0015 CRIT, we derive an ef-\nfective low-energy description of the system. In fact, as\nalready discussed in Introduction, such an approach has\nbecome extremely popular in describing the physics of\nthe iridium oxides [61], for it has lead to the description\nof the latter in terms of e\u000bective Heisenberg or Kitaev-\nlike models. To obtain such an e\u000bective description for\nthe case of large spin-orbit coupling, \u0015>\u0015 CRIT, we \frst\nobtain the eigenstates of the spin-orbit coupling Hamil-\ntonian (3): these are two doublets, separated by the gap\n\u0001E=\u0015. Next, we restrict the Hilbert space to the lowest\ndoubletfj\"\u0000i;j#+ig, wherej#i(j\u0000i) denotes the state\nwithSz=\u00001=2(Tz=\u00001=2) quantum number. Lastly,\nwe project the intersite Hamiltonian (2) onto the lowest\ndoublet (see Appendix for details) and obtain the follow-\ning e\u000bective model:\nHe\u000b=J\n2X\ni\u0010\n~Jx\ni~Jx\ni+1+~Jy\ni~Jy\ni+1+ 2(\u000b+\f)~Jz\ni~Jz\ni+1\u0011\n;(13)\nwhere ~Jz\ni=\u00001\n2\u0000\nni;j\"\u0000i+ni;j#+i\u0001\nis an e\u000bective ~Jz=1=2\npseudospin operator.\nInterestingly, it turns out that this e\u000bective Hamilto-\nnian describes exactly a spin 1=2XXZ chain. Moreover,\nin the limit of \u000b=\u0000\fthe Ising interaction in Eq. (13)\ndisappears and we obtain an AF XY model. Thus, re-\nsembling the iridate case [61], the e\u000bective model in the\nlimit of large spin-orbit coupling has a surprisingly simple\nform.\nB. Validity of the e\u000bective XXZ model:benchmarking \u000b=\u0000\fcase\nFirst, let us show that the e\u000bective XXZ model indeed\ngives the correct description of the ground state of the full\nspin-orbital model (1) in the limit of \u0015>\u0015 CRIT. To this\nend, we compare the spin-orbital correlation functions\ncalculated using the e\u000bective and the full models.\nWe \frst express the spin-orbital correlation function\nCSO, the on-site spin-orbit correlation function OSO, and\nthe anisotropic spin (orbital) correlation functions S\r\r\n(T\r\r) in the basis spanned by the two lowest doublets\nper sitefj\"\u0000i;j#+ig|see the Appendix for the explicit\nformula. Next, we compare the values of the correlation\nfunctions in the two special \u000b=\u0000\fcases, already dis-\ncussed above: (i) case A with j\u000bj= 0:5, and (ii) case B\nwith\u000b= 0. As can be seen in Fig. 3 and Fig. 4, the cor-\nrelation functions calculated using the two distinct mod-\nels agree extremely well once \u0015=J&1{100 (\u0015=J&0:2)\nin case A (B), respectively. We note that calculations\nperformed for other values of the f\u000b;\fgparameters (un-\nshown) also show that the e\u000bective model describes the\nground state properties in the limit of \u0015 > \u0015 CRIT well.\nMoreover, once \u0015=J'106, the ground and lowest lying\nexcited states are quantitatively the same in the full and\nthe e\u000bective models.\nC. Why the spin-orbital entanglement can vanish\nHaving derived the e\u000bective model|and having shown\nits validity|we now discuss how it can help us with un-\nderstanding one of the crucial results of the paper: How\ncan the spin-orbital entanglement vanish in the limit of\nlarge spin-orbit coupling \u0015>\u0015 CRIT?\nWe start by expressing the measure for the spin-orbital\nentanglement for nearest neighbors, the spin-orbital cor-\nrelationCSO, in the basis of the e\u000bective model (see Ap-\npendix for details):\n~CSO=1\n2LLX\ni=1\u0014\nh~Jx\ni~Jx\ni+1+~Jy\ni~Jy\ni+1i\u00002h~Jz\ni~Jz\ni+1i2+1\n8\u0015\n;\n(14)\nwhere the averages are calculated in the ground state.\nTo evaluate Eq. (14), we calculate expectation values\nof the e\u000bective pseudospin operators using ED, which we\nshow in Fig. 8. (We note in passing that the presented\nED results for an XXZ L= 10 site chain agree well with\nthose which were published earlier, cf. Ref. [89].) The ob-\ntained ground state of the e\u000bective Hamiltonian (13) for\n\u000b+\f <\u00001=2 is described by a ferromagnetic Ising state,\nwhereh~Jz\ni~Jz\ni+1i=1=4and all other correlations vanish.\nSubstituting these into Eq. (14) explains why CSO= 0\nin the ground state of model (1) in the limit of large\n\u0015>\u0015 CRIT and when restricted to \u000b+\f <\u00001=2. In con-\nclusion, the spin-orbital entanglement for \u000b+\f <\u00001=2\nvanishes because not only the on-site interaction between\nspins and orbitals but also the intersite interactions in the12\n-2.0 -1.0 0.0 1.0 2.0\nα-2.0-1.00.01.02.0β/angbracketleftbig˜Ji˜Jj/angbracketrightbig\nXYAF Ising\nFM IsingAF H(a)\n-2.0 -1.0 0.0 1.0 2.0\nα/angbracketleftbig˜Jx\ni˜Jx\nj/angbracketrightbig\n=/angbracketleftbig˜Jy\ni˜Jy\nj/angbracketrightbig\nXYAF Ising\nFM IsingAF H(b)\n-2.0 -1.0 0.0 1.0 2.0\nα/angbracketleftbig˜Jz\ni˜Jz\nj/angbracketrightbig\nXYAF Ising\nFM IsingAF H(c)\n0.4\n0.2\n0.00.2\nFIG. 8. The zero-temperature phase diagram of the e\u000bective XXZ model (13) as a function of the model parameters \u000band\f\nobtained using ED on a L= 10 site chain. The panels present the correlations: (a) h~Ji~Jji; (b)h~J\u000e\ni~J\u000e\njiwith\u000e=x,y; (c)h~Jz\ni~Jz\nji.\nThe labels depict various ground states of the 1D XXZ model: AF H|the Heisenberg antiferromagnet, AF Ising|the Ising\nantiferromagnet, XY|the XY antiferromagnet, FM Ising|the Ising ferromagnet.\nground state are of purely Ising type, and e\u000bectively the\nground state is just a product state with no spin-orbital\nentanglement.\nD. Why the spin-orbital entanglement can be \fnite\nThe e\u000bective model (13) can also be used to explain the\npresence of \fnite spin-orbital entanglement in the limit of\nlarge spin-orbit coupling \u0015 > \u0015 CRIT while it vanishes in\nthe\u0015= 0 limit. Let us \frst look at the already discussed\nin detail\f=\u0000\u000bcase:\nIn this case and in the \u0015= 0 limit, the term in (2)\nwhich is explicitly responsible for the spin-orbital entan-\nglement,/(SiSj)(TiTj), can become relatively small\nfor large\u000bor\fdue to the presence of the \u000bTiTjand\n\fSiSjterms. Consequently, the region of signi\fcant\nspin-orbital entanglement is quite small without spin-\norbit coupling along the \f=\u0000\u000bline, see Fig. 1(a).\nThis situation, however, drastically changes in the limit\nof large\u0015>\u0015 CRIT, as discussed below.\nSpeci\fcally, downfolding the exchange Hamiltonian (2)\nterm by term onto the e\u000bective Hamiltonian (13) should\nreveal the origin of the spin-orbital entanglement in the\nlarge spin-orbit coupling limit. First, the \u000bTiTjand\n\fSiSjterms of Eq. (2) upon projecting onto spin-orbit\ncoupled basis produce \u000b~Jz\ni~Jz\njand\f~Jz\ni~Jz\nj, resulting in the\nIsing terms in the e\u000bective model (13). Note that in the\ncase that\f=\u0000\u000b, these Ising terms disappear. Second,\nthe term responsible for the spin-orbital entanglement,\ni.e., (SiSj)(TiTj) (cf. above), reduces exactly to the XY\nterms in the e\u000bective model. These terms do not vanish\nonce\f=\u0000\u000b. In fact, in this special limit the whole\ne\u000bective Hamiltonian is obtained from the term that is\nfully responsible for the spin-orbital entanglement in theoriginal Hamiltonian. Finally, as the ground state of the\nXY Hamiltonian carries `spatial entanglement' in pseu-\ndospins ~J, we expect the spin-orbital entanglement to be\n\fnite in the limit of large spin-orbit coupling \u0015>\u0015 CRIT\nand once\f=\u0000\u000b.\nThe above reasoning is con\frmed by calculating the\ntwo contributions to the intersite spin-orbital correla-\ntion function ~CSOin the e\u000bective model once \f=\u0000\u000b.\nThis can be done analytically for the XY model:\nh~Jz\ni~Jz\ni+1i=\u00001=\u00192andh~Jx\ni~Jx\ni+1i=h~Jy\ni~Jy\ni+1i=\u00001=(2\u0019).\n(These results agree with the correlations calculated us-\ning ED and presented in Fig. 8.)\nThe above discussion can now be extended to the case\nthat\f6=\u0000\u000band\u000b+\f >\u00001=2, for which \fnite, though\nincreasingly small for large and positive \u000b+\f, spin-\norbital entanglement can be observed, see Fig. 1(c). Such\nresult can be understood by using the e\u000bective model\nand by noting that the intersite spin-orbital correlation\n~CSOis always \fnite provided that \u000b+\fis \fnite and\n\u000b+\f >\u00001=2. This is because in this limit: (i) the cor-\nrelationsh~Jx\ni~Jx\ni+1i=h~Jy\ni~Jy\ni+1iare nonzero, see Fig. 8(b);\n(ii)h~Jz\ni~Jz\ni+1i6= 1=4, see Fig. 8(c). It is then only in the\nlimit\u000b+\f!1 that the spin-orbital entanglement can\nvanish, for the ground state of the XXZ model is `pure'\nIsing antiferromagnet. (A completely di\u000berent situation\noccurs once \u000b+\f <\u00001=2, i.e., for the FM ground state\nof the e\u000bective XXZ model, as already discussed in the\nprevious subsection|that explains why the spin-orbital\nentanglement can `sometimes' vanish even in the limit of\nlarge spin-orbit coupling, \u0015>\u0015 CRIT.)13\nVI. CONCLUSIONS\nA. Entanglement induced by spin-orbit coupling\nIn conclusion, in this paper we studied the spin-orbital\nentanglement in a Mott insulator with spin and orbital\ndegrees of freedom. We investigated how the spin-orbital\nentanglement gradually changes with the increasing value\nof the on-site spin-orbit coupling. The results, obtained\nby exactly diagonalizing a 1D model with the intersite\nSU(2)\nSU(2) spin-orbital superexchange /Jand the\non-site Ising-type spin-orbit coupling /\u0015, reveal that:\n1. For small \u0015<\u0015 CRIT [90]:\n(a) In general, the spin-orbital entanglement in\nthe ground state is not much more robust than\nin the\u0015= 0 case;\n(b) If the ground state had \fnite spin-orbital en-\ntanglement for \u0015= 0, it is driven into a novel\nspin-orbital strongly entangled phase upon in-\ncreasing\u0015;\n(c) If the ground state did notshow spin-orbital\nentanglement for \u0015= 0, it still shows none\nor negligible spin-orbital entanglement upon\nincreasing\u0015.\n2. In the limit of large \u0015>\u0015 CRIT:\n(a) In general, the spin-orbital entanglement in\nthe ground state is far more robust than in\nthe\u0015= 0 case;\n(b) The ground state may be driven into a novel\nspin-orbitally entangled phase even if it does\nnot show spin-orbital entanglement for \u0015= 0;\n(c) The ground state may still show vanishing\nspin-orbital entanglement, but only if the\nquantum \ructuations vanish in the ground\nstate of an e\u000bective model (as is the case of\nan Ising ferromagnet).\nThe statements mentioned under point 2. above, con-\ncerning\u0015>\u0015 CRIT, constitute, from the purely theoreti-\ncal perspective, the main results of this paper. In partic-\nular, they mean that: (i) the spin-orbital entanglement\nbetween spins and orbitals on di\u000berent sites can be trig-\ngered by a joint action of the on-site spin-orbit coupling\n(of relativistic origin) and the spin-orbital exchange (of\nthe `Kugel-Khomskii'{type); (ii) and yet, the onset of the\nspin-orbital entanglement in such a model does not have\nto be taken `for granted', for it can vanish even in the\nlarge spin-orbit coupling limit.\nCrucially, we have veri\fed that the spin-orbital entan-\nglement can be induced by the spin-orbit coupling, for\nthe latter interaction may enhance the role played by the\nspin-orbitally entangled ( SiSj)(TiTj) term by `quench-\ning' the bare spin ( SiSj) and orbital ( TiTj) exchange\nterms in an e\u000bective low-energy Hamiltonian valid in thislimit. Interestingly, such mechanism can be valid even if\nthe spin-orbit coupling has a purely `classical' Ising form\n(as for example in the case discussed in this paper). For\na more intuitive explanation of these results, in Sec. V we\npresented a detailed analysis of the e\u000bective low-energy\npseudospin XXZ model.\nB. Consequences for correlated materials\nThe results presented here may play an important role\nin the understanding of the correlated systems with non-\nnegligible spin-orbit coupling|such as e.g. the 5 diri-\ndates, 4druthenates, 3 dvanadates, the 2 palkali hyper-\noxides, and other to-be-synthesized materials. To this\nend, we argue that, even though obtained for a speci\fc\n1D model, some of the results presented here are to a\nlarge extent valid also for these 2D or 3D systems:\nFirst, this is partially the case for the results obtained\nin the limit of large \u0015>\u0015 CRIT. In particular, the map-\nping to the e\u000bective XXZ model is also valid in 2D and\n3D cases. Moreover, one can easily verify that the spin-\norbital correlation function [ ~CSO, Eq. (14)], which mea-\nsures spin-orbital entanglement never vanishes also in the\n2D and 3D cases, unless the quantum \ructuations com-\npletely disappear (as is the case of the 2D or 3D Ising\nferromagnet or antiferromagnet). Therefore, the main\nconclusions from Secs. IVC and IVD are also valid in\n2D and 3D cases and consequently also point 2 of the\nconcluding Section VI A holds. This means that, for ex-\nample, the results obtained here would apply to any Mott\ninsulator with two active t2gorbitals with small Hund's\ncoupling and with \u0015>\u0015 CRIT (such as e.g. Sr 2VO4[81]).\nNaturally, the question remains to what extent one\ncould use the reasoning discussed here to the understand-\ning of the spin-orbital ground state of the probably most\nfamous Mott insulators with active orbital degrees of free-\ndom and large spin-orbit coupling|the 5 diridates (such\nas e.g. Sr 2IrO4[41], Na 2IrO3, Li2IrO3, etc. [50]). Here\nwe suggest that, while the situation in the iridates might\nbe quite di\u000berent in detail and requires solving a dis-\ntinct spin-orbital model with three active t2gorbitals and\nan SU(2)-symmetric spin-orbit coupling (which is beyond\nthe scope of this work), we expect point 2(b) of the con-\ncluding Section VI A to hold also in this case: in fact, the\nquantum nature of the Heisenberg spin-orbit coupling of\nthe iridates (in contrast to the classical Ising spin-orbit\ncoupling studied in this paper), should only facilitate the\nonset of the spin-orbital entanglement. Thus, we suggest\nthat in principle also for the iridates the ground state may\nbe driven into a novel spin-orbitally entangled phase even\nif it does not show spin-orbital entanglement for \u0015= 0.\nSecond, we suggest that also the fact that the spin-\norbit coupling does not induce additional spin-orbital en-\ntanglement in the limit of small \u0015<\u0015 CRIT will carry on\nto higher dimensions and to spin-orbital models of lower\nsymmetry|for a priori there is no reason why the ten-\ndency observed in a 1D (and highly symmetric) model,14\ntowards a`more classical' behavior should fail in dimen-\nsions higher than one (and for more anisotropic mod-\nels). Thus, in general the spin-orbital entanglement of\nthe systems with weak spin-orbit coupling \u0015 < \u0015 CRIT\nand Ising-like spin-orbit coupling [80], such as e.g. the\nalkali hyperoxides with two active `molecular' 2 porbitals\n(e.g. KO 2[82]), should not qualitatively depend on the\nvalue of spin-orbit coupling. This means that, to simplify\nthe studies one may, in the \frst order of approximation,\nneglect the spin-orbit coupling in the e\u000bective models for\nthese materials.\nACKNOWLEDGMENTS\nWe thank Clio Agrapidis, Wojciech Brzezicki, Cheng-\nChien Chen, George Jackeli, Juraj Rusna\u0014 cko, and\nTakami Tohyama for insightful discussions. The cal-\nculations were performed partly at the Interdisci-\nplinary Centre for Mathematical and Computational\nModeling (ICM), University of Warsaw, under grant\nNo. G72-9. This research was supported in part\nby PLGrid Infrastructure (Academic Computer Cen-\nter Cyfronet AGH Krak\u0013 ow). We kindly acknowl-\nedge support by the Narodowe Centrum Nauki (NCN,\nPoland) under Projects Nos. 2016/22/E/ST3/00560 and\n2016/23/B/ST3/00839. E. M. P. acknowledges funding\nfrom the European Union's Horizon 2020 research and in-\nnovation programme under the Maria Sk lodowska-Curie\ngrant agreement No. 754411. J. Ch. acknowledges sup-\nport by M \u0014SMT \u0014CR under NPU II project CEITEC 2020\n(LQ1601). Computational resources were supplied by the\nproject \\e-Infrastruktura CZ\" (e-INFRA LM2018140)\nprovided within the program Projects of Large Re-\nsearch, Development and Innovations Infrastructures.\nA. M. Ole\u0013 s is grateful for an Alexander von Humboldt\nFoundation Fellowship (Humboldt-Forschungspreis).\nAPPENDIX: EFFECTIVE XXZ MODEL\nLet us consider the Hamiltonian (1) of the main text:\nH=HSE+HSOC; (15)\nwhere the intersite interaction HSEand on-site spin-orbit\ncoupling are described by\nHSE=JX\ni[(Si\u0001Si+1+\u000b)(Ti\u0001Ti+1+\f)\u0000\u000b\f];(16)\nHSOC= 2\u0015X\niSz\niTz\ni: (17)\nThe characteristic scales for HSEandHSOCare intersite\nexchange parameter Jand on-site SOC \u0015, respectively.\nIn the strong spin-orbit coupling limit, \u0015>\u0015 CRIT,HSE\ncan be considered as a perturbation to HSOC. The eigen-\nstates of the full Hamiltonian (15) in zeroth-order arethen obtained by the diagonalization of the on-site spin-\norbit partHSOC. In our simple case HSOCis already di-\nagonal with two doubly{degenerate energies \u0006\u0015=2. The\ncorresponding eigenstates de\fned by total momentum ~J\nform two doublets. The lower energy doublet consists of\n~J#=j+#i;\n~J\"=j\u0000\"i;\nwhile the higher doublet is given by:\n~J0\n\"=j+\"i;\n~J0\n#=j\u0000#i:\nHere,j\"i(j+i) denotes the state with Sz=1=2\n(Tz=1=2) quantum number. The on-site basis\ntransformation between the spin and orbital\nfjTz;Szig=fj+\"i;j+#i;j\u0000\"i;j\u0000#ig basis and spin-\norbit coupledf~J#;~J\";~J0\n\";~J0\n#gbasis consisting of two\ndoublets is described by a unitary matrix\nU=0\nBBB@0 0 1 0\n1 0 0 0\n0 1 0 0\n0 0 0 11\nCCCA: (18)\nWe then project the Hamiltonian (16) onto spin-orbit\ncoupled basisf~J,~J0g:HSOC\nSE=UyHSEU. As we are in-\nterested in the low-energy physics, we truncate Hilbert\nspace to the lowest doublet ~Jand obtain e\u000bective Hamil-\ntonian (13) from the main text:\nHe\u000b=J\n2X\ni\u0010\n~Jx\ni~Jx\ni+1+~Jy\ni~Jy\ni+1+ 2(\u000b+\f)~Jz\ni~Jz\ni+1\u0011\n:(19)\nTo analyze the e\u000bective model (19) and obtain impor-\ntant correlation functions, we \frst need to establish a\nlink between operators describing correlation functions\nin originalfjTz;Szigbasis and spin-orbit coupled f~J,~J0g\nbasis. To this end, we project each of the spin/orbital\noperators,Or=fS\r\nr;T\r\nrg,\r=fx;y;zg,r=fi;i+ 1g\nentering the original correlation functions (4) { (8) onto\nspin-orbit coupled basis: OSOC\nr =UyOrU. As most of\nthe correlation functions include intersite terms, the re-\nsult shall be written as a 16 \u000216 matrix, spanned by\nf~J,~J0gi\u0002f~J,~J0gjbasis.\nWe then once again drop out the high-energy doublet\non each site and obtain correlation functions as 4 \u00024\nmatrices de\fned in Hilbert space of f~Jgi\u0002f~Jgj:\n~S=~T=*0\nBBB@1\n40 0 0\n0\u00001\n40 0\n0 0\u00001\n40\n0 0 01\n41\nCCCA+\n=h~Jz\ni~Jz\nji; (20)\n~S\u000e\u000e=~T\u000e\u000e=*0\nBBB@0 0 0 0\n0 0 0 0\n0 0 0 0\n0 0 0 01\nCCCA+\n= 0; (21)15\nwhere\u000e=fx;yg,\n~Szz=~Tzz=*0\nBBB@1\n40 0 0\n0\u00001\n40 0\n0 0\u00001\n40\n0 0 01\n41\nCCCA+\n=D\n~Jz\ni~Jz\njE\n;\n~CSO=*0\nBBB@1\n160 0 0\n01\n161\n40\n01\n41\n160\n0 0 01\n161\nCCCA+\n\u00002\n6664*0\nBBB@1\n40 0 0\n0\u00001\n40 0\n0 0\u00001\n40\n0 0 01\n41\nCCCA+3\n77752\n=1\n2h~Jx\ni~Jx\nj+~Jy\ni~Jy\nji+1\n16\u0000h~Jz\ni~Jz\nji2:To express the on-site spin-orbit correlation function\nOSO, which does not include intersite terms, in the same\nbasis, we multiply it by a 2 \u00022 identity matrix represent-\ning the neighboring site:\n~OSO;i\nidj=*0\nBBB@\u00001\n40 0 0\n0\u00001\n40 0\n0 0\u00001\n40\n0 0 0\u00001\n41\nCCCA+\n=\u00001\n4:\n[1] D. I. Khomskii, Basic Aspects of the Quantum Theory of\nSolids: Order and Elementary Excitations (Cambridge\nUniversity Press, Cambridge, 2010).\n[2] T. Andrade, A. Krikun, K. Schalm, and J. Zaanen, Na-\nture Physics 14, 1049 (2018).\n[3] J. Zaanen, SciPost Phys. 6, 61 (2019).\n[4] L. Balents, Nature 464, 199 (2010).\n[5] C. Varma, Z. Nussinov, and W. van Saarloos, Physics\nReports 361, 267 (2002).\n[6] A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. 47,\n777 (1935).\n[7] H. Ollivier and W. H. Zurek, Phys. Rev. Lett. 88, 017901\n(2001).\n[8] I. Bentsson and K. Zyczkowski, Geometry of Quan-\ntum States: An Introduction to Quantum Entanglement\n(Cambridge University Press, Cambridge, 2006).\n[9] C. H. Bennett, H. J. Bernstein, S. Popescu, and B. Schu-\nmacher, Phys. Rev. A 53, 2046 (1996).\n[10] G. Vidal, J. I. Latorre, E. Rico, and A. Kitaev, Phys.\nRev. Lett. 90, 227902 (2003).\n[11] V. E. Korepin, Phys. Rev. Lett. 92, 096402 (2004).\n[12] M. M. Wolf, Phys. Rev. Lett. 96, 010404 (2006).\n[13] D. Gioev and I. Klich, Phys. Rev. Lett. 96, 100503\n(2006).\n[14] J. I. Lattore, E. Rico, and G. Vidal, Quantum Informa-\ntion & Computation 4, 48 (2004).\n[15] L. Cincio, J. Dziarmaga, and M. M. Rams, Phys. Rev.\nLett. 100, 240603 (2008).\n[16] R. Thomale, D. P. Arovas, and B. A. Bernevig, Phys.\nRev. Lett. 105, 116805 (2010).\n[17] G. Khaliullin and S. Maekawa, Phys. Rev. Lett. 85, 3950\n(2000).\n[18] G. Khaliullin, P. Horsch, and A. M. Ole\u0013 s, Phys. Rev.\nLett. 86, 3879 (2001).\n[19] G. Khaliullin, Prog. Theor. Phys. Suppl. 160, 155 (2005).\n[20] S. Nakatsuji, K. Kuga, K. Kimura, R. Satake,\nN. Katayama, E. Nishibori, H. Sawa, R. Ishii, M. Hagi-\nwara, F. Bridges, T. U. Ito, W. Higemoto, Y. Karaki,\nM. Halim, A. A. Nugroho, J. A. Rodriguez-Rivera, M. A.\nGreen, and C. Broholm, Science 336, 559 (2012).\n[21] Y. Ishiguro, K. Kimura, S. Nakatsuji, S. Tsutsui, A. Q. R.\nBaron, T. Kimura, and Y. Wakabayashi, Nature Com-\nmunications 4, 2022 (2013).\n[22] H. Man, M. Halim, H. Sawa, M. Hagiwara, Y. Wak-\nabayashi, and S. Nakatsuji, J. Phys.: Condens. Matter30, 443002 (2018).\n[23] B. Y. Pan, H. Jang, J.-S. Lee, R. Sutarto, F. He, J. F.\nZeng, Y. Liu, X. W. Zhang, Y. Feng, Y. Q. Hao, J. Zhao,\nH. C. Xu, Z. H. Chen, J. P. Hu, and D. L. Feng, Phys.\nRev. X 9, 021055 (2019).\n[24] B. Normand and A. M. Ole\u0013 s, Phys. Rev. B 78,\n094427 (2008); B. Normand, ibid.83, 064413 (2011);\nJ. Chaloupka and A. M. Ole\u0013 s, ibid.83, 094406 (2011).\n[25] P. Corboz, M. Lajk\u0013 o, A. M. L auchli, K. Penc, and\nF. Mila, Phys. Rev. X 2, 041013 (2012).\n[26] W. Brzezicki, J. Dziarmaga, and A. M. Ole\u0013 s, Phys. Rev.\nLett. 109, 237201 (2012).\n[27] A. M. Ole\u0013 s, P. Horsch, L. F. Feiner, and G. Khaliullin,\nPhys. Rev. Lett. 96, 147205 (2006).\n[28] A. M. Ole\u0013 s, J. Phys.: Condens. Matter 24, 313201 (2012).\n[29] Y. Chen, Z. D. Wang, Y. Q. Li, and F. C. Zhang, Phys.\nRev. B 75, 195113 (2007).\n[30] J. B. Goodenough, Magnetism and the Chemical Bond\n(Interscience, New York, 1963).\n[31] J. Kanamori, Journal of Physics and Chemistry of Solids\n10, 87 (1959).\n[32] S. Miyasaka, Y. Okimoto, and Y. Tokura, J. Phys. Soc.\nJpn.71, 2082 (2002).\n[33] G. Khaliullin, P. Horsch, and A. M. Ole\u0013 s, Phys. Rev. B\n70, 195103 (2004).\n[34] M. Snamina and A. M. Ole\u0013 s, New J. Phys. 21, 023018\n(2019).\n[35] J. Schlappa, K. Wohlfeld, K. J. Zhou, M. Mourigal,\nM. W. Haverkort, V. N. Strocov, L. Hozoi, C. Mon-\nney, S. Nishimoto, S. Singh, A. Revcolevschi, J. Caux,\nL. Patthey, H. M. Rnnow, J. van den Brink, and\nT. Schmitt, Nature 485, 82 (2012).\n[36] V. Bisogni, K. Wohlfeld, S. Nishimoto, C. Monney,\nJ. Trinckauf, K. Zhou, R. Kraus, K. Koepernik, C. Sekar,\nV. Strocov, B. B uchner, T. Schmitt, J. van den Brink,\nand J. Geck, Phys. Rev. Lett. 114, 096402 (2015).\n[37] K. Wohlfeld, M. Daghofer, S. Nishimoto, G. Khaliullin,\nand J. van den Brink, Phys. Rev. Lett. 107, 147201\n(2011).\n[38] K. Wohlfeld, S. Nishimoto, M. W. Haverkort, and\nJ. van den Brink, Phys. Rev. B 88, 195138 (2013).\n[39] C.-C. Chen, M. van Veenendaal, T. P. Devereaux, and\nK. Wohlfeld, Phys. Rev. B 91, 165102 (2015).\n[40] W. Witczak-Krempa, G. Chen, Y. B. Kim, and L. Ba-\nlents, Annual Review of Condensed Matter Physics 5, 5716\n(2014).\n[41] J. Bertinshaw, Y. K. Kim, G. Khaliullin, and B. J. Kim,\nAnnu. Rev. Condens. Matter Physics 10, 315 (2019).\n[42] H. Ishikawa, T. Takayama, R. K. Kremer, J. Nuss,\nR. Dinnebier, K. Kitagawa, K. Ishii, and H. Takagi,\nPhys. Rev. B 100, 045142 (2019).\n[43] E. Lefran\u0018 cois, A.-M. Pradipto, M. Moretti Sala, L. C.\nChapon, V. Simonet, S. Picozzi, P. Lejay, S. Petit, and\nR. Ballou, Phys. Rev. B 93, 224401 (2016).\n[44] K. Kitagawa, T. Takayama, Y. Matsumoto, A. Kato,\nR. Takano, Y. Kishimoto, S. Bette, R. Dinnebier,\nG. Jackeli, and H. Takagi, Nature 554, 341 (2018).\n[45] C. N. Veenstra, Z.-H. Zhu, M. Raichle, B. M. Lud-\nbrook, A. Nicolaou, B. Slomski, G. Landolt, S. Kittaka,\nY. Maeno, J. H. Dil, I. S. El\fmov, M. W. Haverkort, and\nA. Damascelli, Phys. Rev. Lett. 112, 127002 (2014).\n[46] G. Zhang, E. Gorelov, E. Sarvestani, and E. Pavarini,\nPhys. Rev. Lett. 116, 106402 (2016).\n[47] G. Khaliullin, Phys. Rev. Lett. 111, 197201 (2013).\n[48] A. Jain, M. Krautloher, J. Porras, G. H. Ryu, D. P. Chen,\nD. L. Abernathy, J. T. Park, A. Ivanov, J. Chaloupka,\nG. Khaliullin, B. Keimer, and B. J. Kim, Nature Physics\n13, 633 (2017).\n[49] J. Chaloupka, G. Jackeli, and G. Khaliullin, Phys. Rev.\nLett. 105, 027204 (2010).\n[50] S. M. Winter, A. A. Tsirlin, M. Daghofer, J. van den\nBrink, Y. Singh, P. Gegenwart, and R. Valent\u0013 \u0010, Journal\nof Physics: Condensed Matter 29, 493002 (2017).\n[51] A. Kitaev, Annals of Physics 321, 2 (2006).\n[52] R. Lundgren, V. Chua, and G. A. Fiete, Phys. Rev. B\n86, 224422 (2012).\n[53] W.-L. You, A. M. Ole\u0013 s, and P. Horsch, Phys. Rev. B 86,\n094412 (2012).\n[54] A. M. L auchli and J. Schliemann, Phys. Rev. B 85,\n054403 (2012).\n[55] W.-L. You, P. Horsch, and A. M. Ole\u0013 s, Phys. Rev. B 92,\n054423 (2015).\n[56] W.-L. You, A. M. Ole\u0013 s, and P. Horsch, New J. Phys. 17,\n083009 (2015).\n[57] V. E. Valiulin, A. V. Mikheyenkov, K. I. Kugel, and\nA. F. Barabanov, JETP Letters 109, 546 (2019).\n[58] K. I. Kugel and D. I. Khomskii, Sov. Phys. Usp. 25, 231\n(1982).\n[59] Note that in Ref. [52] the impact of relatively small\nspin-orbit coupling on the entanglement spectra was dis-\ncussed.\n[60] E. M. P arschke and R. Ray, Phys. Rev. B 98, 064422\n(2018).\n[61] G. Jackeli and G. Khaliullin, Phys. Rev. Lett. 102,\n017205 (2009).\n[62] M. V. Eremin, J. Deisenhofer, R. M. Eremina,\nJ. Teyssier, D. van der Marel, and A. Loidl, Phys. Rev.\nB84, 212407 (2011).\n[63] J. Choukroun, Phys. Rev. B 84, 014415 (2011).[64] C. Svoboda, M. Randeria, and N. Trivedi, Phys. Rev. B\n95, 014409 (2017).\n[65] A. Koga, S. Nakauchi, and J. Nasu, Phys. Rev. B 97,\n094427 (2018).\n[66] B. Sutherland, Phys. Rev. B 12, 3795 (1975).\n[67] Y. Yamashita, N. Shibata, and K. Ueda, Phys. Rev. B\n58, 9114 (1998).\n[68] Y.-Q. Li, M. Ma, D.-N. Shi, and F.-C. Zhang, Phys. Rev.\nB60, 12781 (1999).\n[69] A. K. Kolezhuk and H.-J. Mikeska, Phys. Rev. Lett. 80,\n2709 (1998).\n[70] A. K. Kolezhuk, H.-J. Mikeska, and U. Schollw ock, Phys.\nRev. B 63, 064418 (2001).\n[71] D. P. Arovas and A. Auerbach, Phys. Rev. B 52, 10114\n(1995).\n[72] S. K. Pati, R. R. P. Singh, and D. I. Khomskii, Phys.\nRev. Lett. 81, 5406 (1998).\n[73] C. Itoi, S. Qin, and I. A\u000feck, Phys. Rev. B 61, 6747\n(2000).\n[74] W. Zheng and J. Oitmaa, Phys. Rev. B 64, 014410\n(2001).\n[75] P. Li and S.-Q. Shen, Phys. Rev. B 72, 214439 (2005).\n[76] K. I. Kugel, D. I. Khomskii, A. O. Sboychakov, and S. V.\nStreltsov, Phys. Rev. B 91, 155125 (2015).\n[77] A. V. Gorshkov, M. Hermele, V. Gurarie, C. Xu, P. S.\nJulienne, J. Ye, P. Zoller, E. Demler, M. D. Lukin, and\nA. M. Rey, Nature Physics 6, 289 (2010).\n[78] X. Zhang, M. Bishof, S. L. Bromley, C. V. Kraus, M. S.\nSafronova, P. Zoller, A. M. Rey, and J. Ye, Science 345,\n1467 (2014).\n[79] X. Dou, V. N. Kotov, and B. Uchoa, Scienti\fc Reports\n6, 31737 (2016).\n[80] P. Horsch, G. Khaliullin, and A. M. Ole\u0013 s, Phys. Rev.\nLett. 91, 257203 (2003).\n[81] G. Jackeli and G. Khaliullin, Phys. Rev. Lett. 103,\n067205 (2009).\n[82] I. V. Solovyev, New J. Phys. 10, 013035 (2008).\n[83] E. Koch, The Lanczos Method, in: The LDA+DMFT\napproach to strongly correlated materials, edited by E.\nPavarini, E. Koch, D. Vollhardt, and A. Lichtenstein\n(Forschungszentrum J ulich, J ulich, 2011).\n[84] A. Kitaev and J. Preskill, Phys. Rev. Lett. 96, 110404\n(2006).\n[85] Y. Q. Li, M. Ma, D. N. Shi, and F. C. Zhang, Phys. Rev.\nLett. 81, 3527 (1998).\n[86] A. M. Ole\u0013 s, G. Tr\u0013 eglia, D. Spanjaard, and R. Jullien,\nPhys. Rev. B 32, 2167 (1985).\n[87] A. Osterloh, L. Amico, G. Falci, and R. Fazio, Nature\n416, 608 (2002).\n[88] X. Jia, A. R. Subramaniam, I. A. Gruzberg, and\nS. Chakravarty, Phys. Rev. B 77, 014208 (2008).\n[89] K. Hijii, A. Kitazawa, and K. Nomura, Phys. Rev. B 72,\n014449 (2005).\n[90]\u0015CRIT depends on the particular values of the model pa-\nrameters, see Sec. IV B."    },    {        "title": "1805.10328v2.Generation_of_Spin_Current_from_Lattice_Distortion_Dynamics__Spin_Orbit_Routes.pdf",        "content": "arXiv:1805.10328v2  [cond-mat.mes-hall]  20 Jul 2018Journal of the Physical Society of Japan LETTERS\nGeneration of Spin Current from Lattice Distortion Dynamic s: Spin-Orbit Routes\nTakumi Funato and Hiroshi Kohno\nDepartment of Physics, Nagoya University, Nagoya 464-8602 , Japan\nGeneration of spin current from lattice distortion dynamic s in metals is studied with special attention on the e ffect\nof spin-orbit coupling. Treating the lattice distortion by local coordinate transformation, we calculate spin curren t and\nspin accumulation with the linear response theory. It is fou nd that there are two routes to the spin-current generation:\none via the spin Hall e ffect and the other via the spin accumulation. The present e ffect due to spin-orbit coupling can\nbe comparable to, or even larger than, the one based on the spi n-vorticity coupling in systems with strong spin-orbit\ncoupling.\nIn the field of spintronics, spin current occupies a central\nposition for the development of new devices or the discov-\nery of novel physical phenomena. To date we know several\nmethods available to generate spin currents, which include\nspin pumping,1–3)spin Hall effect,4)spin accumulation at the\nferromagnet/nonmagnet interface,5)and spin Seebeck effect.6)\nThese are classified as magnetic, electrical, magnetoelect ric,\nand thermal means, respectively.\nRecently, there has also been interest in generating spin\ncurrents by mechanical means, namely, by converting angu-\nlar momentum associated with mechanical motion, such as\nthe rigid rotation of a solid or vorticity of a fluid, into spin an-\ngular momentum of electrons. In the experiments reported so\nfar, two mechanisms have been considered. One is the acous-\ntic spin pumping by Uchida et al. ,7, 8)which is based on the\nmagnon-phonon coupling. They succeeded in generating spin\ncurrent by injecting acoustic waves into yttrium iron garne t\n(YIG) from the attached piezoelectric element. Theoretica l\nanalyses were given by Adachi and Maekawa,9, 10)Keshtgar\net al. ,11)and Deymier et al.12)Another mechanism, proposed\nby Matsuo et al. ,13, 14)is based on the spin-rotation coupling\nor the spin-vorticity coupling (SVC). This is the coupling o f\nthe spin to the effective magnetic field that emerges in a ro-\ntating (non-inertial) frame of reference locally fixed on th e\nmaterial that is in motion. The first experiment for the SVC\nmechanism was conducted on liquid metals.15, 16)To realize\nthe SVC mechanism in solids, it was proposed to use sur-\nface acoustic waves.17, 18)Nozaki et al. used Py/Cu bilayer\nand injected surface acoustic waves into Cu from the attache d\nLiNbO 3(surface acoustic wave filter).19)The generated AC\nspin current was detected via the spin-torque ferromagneti c\nresonance.\nOne of the reasons that the mechanical generation of spin\ncurrent has attracted attention is that it does not rely on sp in-\norbit interaction (SOI). Therefore, previous works did not pay\nattention to the effects of SOI. However, it is well expected\nthat SOI plays certain roles in the mechanical processes of\nspin-current generation. For example, the previous experi -\nments7, 8)were conducted on systems with an interface, which\npotentially possesses Rashba SOI. Furthermore, the mechan i-\ncal generation method may be used in combination with other\n“conventional” mechanisms that utilize SOI, and thereby en -\nhance spin current.\nIn this paper, we study a mechanical generation of spin cur-rent by focusing on the e ffects of SOI. As a mechanical pro-\ncess, we consider dynamical lattice deformations of a solid\nwith metallic electrons and with SOI. To treat lattice defor -\nmations analytically, we use the method of Tsuneto develope d\nin the context of ultrasonic attenuation in superconductor s,20)\nwhich employs a local coordinate transformation. By calcu-\nlating spin current and spin accumulation induced by dynam-\nical lattice deformations, we found two routes to spin-curr ent\ngeneration: one via the spin accumulation and the other via\nthe spin Hall effect. As a related work, Wang et al.21)derived\nthe Hamiltonian that includes SOI in a general coordinate sy s-\ntem starting from the general relativistic Dirac equation, but\nthey did not give an explicit analysis of spin-current gener a-\ntion.\nModel: We consider a free-electron system in the presence\nof random impurities and the associated SOI. The Hamilto-\nnian is given by\nH=−∇′2\n2m+Vimp(r′)+iλso{[∇′Vimp(r′)]×σ}·∇′. (1)\nThe second term represents the impurity potential, Vimp(r′)=\nui/summationtext\njδ(r′−Rj), with strength uiand at position Rj(for jth\nimpurity), and the third term is the SOI associated with Vimp,\nwith strength λsoand the Pauli matrices σ=(σx,σy,σz).\nWhen the lattice is deformed, e.g., by sound waves, the\nHamiltonian becomes\nHlab=−∇′2\n2m+Vimp(r′−δR(r′,t))\n+iλso/braceleftBig/bracketleftBig\n∇′Vimp/parenleftbigr′−δR(r′,t)/parenrightbig/bracketrightBig\n×σ/bracerightBig\n·∇′,(2)\nwhereδR(r′,t) is the displacement vector of the lattice from\ntheir equilibrium position r′.\nFollowing Tsuneto,20)we make a local coordinate transfor-\nmation, r=r′−δR(r′,t), from the laboratory (Lab) frame\n(with coordinate r′) to a “material frame” (with coordinate r)\nwhich is fixed to the ‘atoms’ in a deformable lattice. At the\nsame time, the wave function needs to be redefined to keep\nthe normalization condition,\nψ(r,t)=[1+∇·δR]1/2ψ′(r′,t)+O(δR2), (3)\nwhereψ′(r′,t) is the wave function in the Lab frame, and\nψ(r,t) is the one in the material frame. Up to the first order\n1J. Phys. Soc. Jpn. LETTERS\ninδR, the Hamiltonian for ψ(r,t) is given by\nHmat=H+H′\nK+H′\nso, (4)\nwhere H=HK+Himp+Hsois the unperturbed Hamiltonian de-\nfined by HlabwithδR=0. Here, HK=/summationtext\nk(k2/2m)ψ†\nkψkis the\nkinetic energy, with ψk(ψ†\nk) being the electron annihilation\n(creation) operator. HimpandHsodescribe the impurity poten-\ntial and impurity SOI, respectively, Himp=/summationtext\nk,k′Vk′−kψ†\nk′ψk,\nHso=iλso/summationtext\nk,k′Vk′−k(k′×k)·ψ†\nk′σψk, where Vk′−kis the\nFourier component of Vimp(r). Assuming a uniformly ran-\ndom distribution, we average over the impurity positions as\n/an}b∇acketle{tVkVk′/an}b∇acket∇i}htav=niu2\niδk+k′,0, and/an}b∇acketle{tVkVk′Vk′′/an}b∇acket∇i}htav=niu3\niδk+k′+k′′,0,\nwhere niis the impurity concentration. The impurity-averaged\nretarded/advanced Green function is given by GR/A\nk(ε)=(ε+\nµ−k2/2m±iγ)−1, whereγ=πniu2\niN(µ)(1+2\n3λ2\nsok4\nF) is the\ndamping rate. Here, N(µ) is the Fermi-level density of states\n(per spin), and kFis the Fermi wave number. In this work, we\nconsider the effects of SOI up to the second order.\nThe effects of lattice distortion are contained in H′\nKandH′\nso,\nwhich come from HKandHso, respectively. In the first order\ninδR, they are given by\nH′\nK=/summationdisplay\nkWK\nn(k)un\nq,ωψ†\nk+q\n2ψk−q\n2, (5)\nH′\nso=/summationdisplay\nk,k′Vk′−kWso\nln(k,k′)un\nq,ωψ†\nk′+q\n2σlψk−q\n2. (6)\nHere, uq,ωis the Fourier component of the lattice velocity\nfield, u(r,t)=∂tδR(r,t), and we defined (see Fig. 1),\nWK\nn(k)=/bracketleftBigq·k\nmω−1/bracketrightBig\nkn, (7)\nWso\nln(k,k′)=λso\niω/bracketleftBig\n(k×q)lk′\nn−(k′×q)lkn/bracketrightBig\n. (8)\nThe first term in WK\nndescribes the coupling of the strain ∂iδRn\nto the stress tensor ∼/summationtext\nkkiknc†\nkckof electrons, and modifies\nthe effective mass tensor. Throughout this report, qrepresents\nthe wave vector of the lattice deformation and ωis its fre-\nquency. We assume that the spatial and temporal variations o f\nδRare slow and satisfy the conditions q≪ℓ−1andω≪γ,\nwhereℓis the mean free path.\nSpin and spin-current density operators are given by\nˆjα\ns,0(q)=ˆσα(q)=/summationdisplay\nkψ†\nk−q\n2σαv0ψk+q\n2, (9)\nˆjα\ns,i(q)=/summationdisplay\nkψ†\nk−q\n2σαviψk+q\n2+ˆja,α\ns,i(q), (10)\nwhereα=x,y,zspecifies the spin direction, i=x,y,zthe\ncurrent direction, and v0=1. Here,\nˆja,α\ns,i(q)=−iλsoǫαi j/summationdisplay\nk,k′Vk′−k(k′\nj−kj)ψ†\nk′−q\n2ψk+q\n2, (11)\nis the ‘anomalous’ part of the spin-current density, with ǫαi j\nbeing the Levi-Civita symbol. We calculate ˆjα\ns,µin (linear) re-\nsponse22)tou,\n/an}b∇acketle{tˆjα\ns,µ(q)/an}b∇acket∇i}htω=−/bracketleftBig\nKss,α\nµn+Ksj,α\nµn+Kso,α\nµn/bracketrightBig\nq,ωun, (12)\nwhere Kss,α\nµn(Ksj,α\nµn) is the skew-scattering (side-jump) type\ncontribution in response to H′\nK, and Kso,α\nµndescribes the re-\nsponse to H′\nso.\nFig. 1. Two types of vertices associated with the coupling to the lat tice\ndisplacement δR, or the velocity field u=dδR/dt.\nFig. 2. Skew-scattering type contributions to the spin current ( µ=x,y,z)\nand/or spin accumulation ( µ=0). The black (white) circles represent spin-\nflip (spin non-flip) vertices. The cross and the dashed line re present an impu-\nrity and the impurity potential, respectively. The shaded p art represents the\nimpurity ladder vertex corrections. The upside-down diagr ams are also con-\nsidered in the calculation.\nSkew-scattering process: The skew-scattering contribution\nwithout ladder vertex corrections, shown in Fig. 2, is given by\nKss,α\nµν(ω)=iλsoniu3\ni/summationdisplay\nk1,k2(k1×k2)αv1µWK\nν(k2)\n×ω\niπ/summationdisplay\np/bracketleftbigg\nGR\np/parenleftbiggω\n2/parenrightbigg\n−GA\np/parenleftbigg\n−ω\n2/parenrightbigg/bracketrightbigg\n×GR\nk1+GA\nk1−GR\nk2+GA\nk2−, (13)\nwith GR/A\nk±=GR/A\nk±q\n2(±ω\n2). By including ladder vertex correc-\ntions, spin accumulation and spin-current density are calc u-\nlated as23)\n/an}b∇acketle{tσα/an}b∇acket∇i}htss=αss\nSHneτ/parenleftBigg3\n5Dq2−iω/parenrightBigg(iq×u)α\nDq2−iω+τ−1\nsf, (14)\n/an}b∇acketle{tjα\ns,i/an}b∇acket∇i}htss=−Diq i/an}b∇acketle{tσα/an}b∇acket∇i}htss+αss\nSHǫαim/an}b∇acketle{tjm/an}b∇acket∇i}ht, (15)\nwhereαss\nSH=2π\n3k2\nFλsoN(µ)uiis the spin Hall angle due to skew\nscattering,24)ne=2\n3mk2\nFN(µ) is the electron number density,\nτ=(2γ)−1is the scattering time, D=1\n3v2\nFτis the diffusion\nconstant, and τ−1\nsf=(4λ2\nsok4\nF/3)τ−1is the spin relaxation rate\ndue to SOI. In Eq. (15), /an}b∇acketle{tjm/an}b∇acket∇i}htis the charge current,\n/an}b∇acketle{tjm/an}b∇acket∇i}ht=neτ/braceleftbigg\n−/parenleftBigg3\n5Dq2−iω/parenrightBigg\num\n+/parenleftBigg6\n5+1\nτ(Dq2−iω)/parenrightBigg\nDiq m(iq·u)/bracerightbigg\n, (16)\ngenerated by u.20)Here, in the first line, the term ∼Dq2um\n(the term∼iωum) is induced by the first (second) term in WK\nn,\nEq. (7), via the spatio-temporal variation of the strain ten sor\n∂iδRm(temporal variation of the velocity field um). The last\nterm is the diffusion current. We see that a spin accumulation\n(14) is induced by the vorticity of the lattice velocity field u.\nThe first term and the second terms in Eq. (15) are written\nwith the spin accumulation (Eq. (14)) and the charge current\n(Eq. (16)), respectively.\nSide-jump process: The side-jump contributions are ob-\n2J. Phys. Soc. Jpn. LETTERS\nFig. 3. Side-jump type contribution to the spin-current density ( i,µ=\nx,y,z) and/or spin accumulation ( µ=0). The diagrams in (a) come from the\nanomalous velocity, Eq. (11); hence they contribute only to the spin current.\nThe diagrams in (b) can be nonvanishing only when the lattice deformation is\nnonuniform. The upside-down diagrams are also included in t he calculation.\nFig. 4. Response to H′\nso, which turned out to vanish.\ntained from the two types of diagrams in Fig. 3. They give\nKsj (a),α\nin(ω)=iλsoniu2\niǫαi j/summationdisplay\nk1,k′\n1,k2(k′\n1,j−k1,j)WK\nn(k2)\n×ω\niπ/bracketleftBig\nδk′\n1k2GR\nk1++δk1k2GA\nk1′−/bracketrightBig\nGR\nk2+GA\nk2−, (17)\nKsj (b),α\nµν (ω)=λsoniu2\ni/summationdisplay\nk1,k2[(k1−k2)×iq]αv1µWK\nν(k2)\n×ω\niπGR\nk1+GA\nk1−GR\nk2+GA\nk2−, (18)\ncorresponding to the diagrams in Fig. 3 (a) and (b), respec-\ntively. With the ladder vertex corrections included, spin a ccu-\nmulation and spin-current density are calculated as23)\n/an}b∇acketle{tσα/an}b∇acket∇i}htsj=αsj\nSHneτ/parenleftBigg3\n5Dq2−iω/parenrightBigg(iq×u)α\nDq2−iω+τ−1\nsf, (19)\n/an}b∇acketle{tjα\ns,i/an}b∇acket∇i}htsj (a)=αsj\nSHǫαim/an}b∇acketle{tjm/an}b∇acket∇i}ht, (20)\n/an}b∇acketle{tjα\ns,i/an}b∇acket∇i}htsj (b)=αsj\nSHneτǫαimDiq m\nDq2−iωiq·u−Diq i/an}b∇acketle{tσα/an}b∇acket∇i}htsj. (21)\nwhereαsj\nSH=−λsom/τis the spin Hall angle due to side-jump\nprocesses.24)The diagrams in Fig. 3 (a) give only the spin cur-\nrent (Eq. (20)) since the left vertices come from the anoma-\nlous velocity. This contribution is also written with the ch arge\ncurrent/an}b∇acketle{tjm/an}b∇acket∇i}htgiven by Eq. (16). On the other hand, spin ac-\ncumulation coming from Fig. 3 (b) is again proportional to\nthe vorticity of the velocity field u. In Eq. (21), the first term\nis proportional to the di ffusion part of the charge current [the\nlast term in Eq (16)], and the second term is the di ffusion spin\ncurrent. We note that these contributions, coming from the d i-\nagrams of Fig. 3 (b), vanish when the external perturbation i s\nuniform, i.e., q=0.\nFinally, the response to H′\nso, shown in Fig. 4, turned out to\nvanish, Kso,α\nµn=0. This is also the case when the ladder vertex\ncorrections are included.\nResult: Taken together, the total spin accumulation and\nFig. 5. (Color online) Two routes to the generation of spin current f rom\nlattice distortion dynamics. The thick arrows indicate the processes governed\nby SOI. ‘AE’ means acousto-electric e ffect.\nspin-current density arising from the dynamical lattice di stor-\ntion via SOI have been obtained as\n/an}b∇acketle{tσα/an}b∇acket∇i}htSOI=αSHneτ/parenleftBigg3\n5Dq2−iω/parenrightBigg(iq×u)α\nDq2−iω+τ−1\nsf, (22)\n/an}b∇acketle{tjα\ns,i/an}b∇acket∇i}htSOI=−Diq i/an}b∇acketle{tσα/an}b∇acket∇i}htSOI+αSHǫαim/an}b∇acketle{tjm/an}b∇acket∇i}ht\n+αsj\nSHneτǫαimDiq m\nDq2−iωiq·u, (23)\nwhereαSH=αss\nSH+αsj\nSHis the ‘total’ spin Hall angle. As seen\nfrom Eq. (22), spin accumulation is induced by the vorticity of\nthe lattice velocity field via SOI. The resulting di ffusion spin\ncurrent contributes to Eq. (23) as the first term. In addition ,\ndynamical lattice distortion generates a charge current as well\n(known as the acousto-electric e ffect25)), which is then con-\nverted to a spin Hall current (in the transverse direction) v ia\nSOI, as expressed by the second and third terms in Eq. (23).\nTherefore, there are two routes to the spin-current generat ion\nin the present mechanism; one is the “di ffusion route” caused\nby the spin accumulation and the other is the “spin Hall route ”\nthat follows the acousto-electric e ffect.26)This is illustrated in\nFig. 5. In the latter (spin Hall) route, the longitudinal com -\nponent of ualso induces spin current via the generation of\ncharge current. Finally, we note that the induced spin accu-\nmulation (22) and the spin-current density (23) satisfy the spin\ncontinuity equation,\n∂t/an}b∇acketle{tσα/an}b∇acket∇i}htSOI+∇·/an}b∇acketle{tjα\ns/an}b∇acket∇i}htSOI=−/an}b∇acketle{tσα/an}b∇acket∇i}htSOI\nτsf. (24)\nThe term on the right-hand side represents spin relaxation d ue\nto SOI.\nThe above result does not include the e ffects of lattice dis-\ntortion on the spinorial character of the electron wave func -\ntion. Such effects are derived from the spin connection in the\ngeneral relativistic Dirac equation.13)The total spin current\nand spin accumulation are given by the sum of the contri-\nbutions from the SVC (previous work13)) and SOI (present\nwork). Next, we study the contribution from SVC, an e ffect\noriginating from the spin connection.\nSpin-rotation coupling: For comparing the present result\nwith the previous one that is based on the spin-vorticity cou -\npling (SVC),16)we also calculate the spin accumulation and\nspin-current density in response to the vorticity of the lat tice\nvelocity field,ω=∇× u.27)By treating the SVC Hamil-\ntonian HSV=−1\n4σ·ω(q,ω) as a perturbation, one has\n/an}b∇acketle{tjα\ns,µ/an}b∇acket∇i}htω=χα\nµβ(q,ω)ωβ(q,ω), whereχα\nµβ(q,ω) is the response\nfunction. The response function (without vertex correctio ns)\nis given as χα\nµβ(q,ω)=1\n2N(µ)δαβδµ0+iω\n4πδαβ/summationtext\nkvµGR\nk+GA\nk−.\nWith ladder vertex corrections, spin accumulation and spin -\n3J. Phys. Soc. Jpn. LETTERS\ncurrent density are obtained as\n/an}b∇acketle{tσα/an}b∇acket∇i}htSV=N(µ)\n2Dq2+τ−1\nsf\nDq2−iω+τ−1\nsfωα, (25)\n/an}b∇acketle{tjα\ns,i/an}b∇acket∇i}htSV=−iωN(µ)\n2Diq i\nDq2−iω+τ−1\nsfωα. (26)\nThey satisfy the spin continuity equation,\n(∂t+τ−1\nsf)/an}b∇acketle{tσα/an}b∇acket∇i}htSV+∇·/an}b∇acketle{tjα\ns/an}b∇acket∇i}htSV=N(µ)\n2τsfωα, (27)\nwith a source term ( ∼ω) on the right-hand side. Alterna-\ntively, one may define the “spin accumulation” δµα=µ↑−µ↓\nby/an}b∇acketle{tσα/an}b∇acket∇i}htSV=n↑−n↓=N(µ)(δµα+/planckover2pi1ωα/2),17)where the\nspin quantization axis has been taken along the ˆ αaxis. Then,\nEq. (25) leads to\n(∂t−D∇2+τ−1\nsf)δµα=−/planckover2pi1\n2˙ωα. (28)\nThis is the basic equation used in Ref. 16 to study spin-curre nt\ngeneration. Therefore, in the SVC mechanism, only the trans -\nverse acoustic waves generate spin current, and the generat ed\nspin current is purely of di ffusion origin. These are in stark\ncontrast with the SOI-induced mechanism.\nComparison: To see the magnitude of the present e ffect,\nwe estimate the di ffusion spin current generated via SOI,\nEq. (22), relative to the one due to SVC, Eq. (25),\nRdiff(f)≡/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/an}b∇acketle{tjα\ns,i/an}b∇acket∇i}htdiff\nSOI\n/an}b∇acketle{tjα\ns,i/an}b∇acket∇i}htSV/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle=8\n3αSHεFτ\n/planckover2pi1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1+6πi\n5D f\nv2a/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle, (29)\nwhere f=ω/2πis the frequency and va=ω/qis the (phase)\nvelocity of acoustic waves, εF=/planckover2pi12k2\nF/2mis the Fermi en-\nergy, and/planckover2pi1has been recovered. This ratio is larger for higher\nfrequency f, and for materials with stronger SOI.\nFor CuIr, the spin Hall angle is 2 αSH=2.1±0.6%, indepen-\ndent of impurity concentration, which is dominated by the ex -\ntrinsic, skew-scattering process.28)In the nearly free electron\napproximation with the Fermi wave number kF=1.36×1010\nm−1, Fermi velocity vF=1.57×106m/s, effective mass\nm∗=8.66×10−31kg,29)and resistivity ρimp=7.5µΩcm (for\n3% Ir), we estimate the scattering time as τimp=5.30×10−15\ns, and the diffusion constant as Dimp=4.35×10−3m2/s, due to\nimpurities. With the speed of the Rayleigh type surface acou s-\ntic wave, va=3.80×103m/s, on a single crystal of LiNbO 3,30)\nwe obtain\nRCuIr\ndiff(f)=1.51/radicalBig\n1+(1.14×f)2, (30)\nwhere fis expressed in GHz. Therefore the di ffusion spin\ncurrent/an}b∇acketle{tjα\ns,i/an}b∇acket∇i}htdiff\nSOIvia SOI is comparable to, or even larger than,\nthat from SVC in metals with strong SOI. It is thus expected\nthat the total contribution /an}b∇acketle{tjα\ns,i/an}b∇acket∇i}htSOI, which includes both the\ndiffusion spin current and the spin Hall current, can be larger\nthan/an}b∇acketle{tjα\ns,i/an}b∇acket∇i}htSV. The magnitude itself is, however, small; /an}b∇acketle{tjx\ns,z/an}b∇acket∇i}ht=\n1020∼1024m−2s−1=10∼105A/m2forτ−1\nsf=0∼5×\n1013s−1,δR=1Å, and f=3.8 GHz, as in the case of the\nSVC mechanism.14)\nTo summarize, we studied the generation of spin current\nand spin accumulation by dynamical lattice distortion in me t-\nals with SOI at the impurities. We identified two routes to\nthe spin-current generation, namely, the “spin Hall route” andthe “spin diffusion route.” In the former route, a charge cur-\nrent is first induced by dynamical lattice distortion, which is\nthen converted into a spin Hall current. In the latter route, a\nspin accumulation is first induced from the vorticity of the\nlattice velocity field, which then induces a di ffusion spin cur-\nrent. The result suggests that the spin accumulation (hence\nthe associated diffusion spin current) generated via SOI is\nlarger than that due to SVC for systems with strong SOI. Sim-\nilar effects are expected in systems with other types of SOI,\nsuch as Rashba, Weyl, etc., and such studies will be reported\nelsewhere. In this connection, we note that Xu et al. recently\nreported an experiment on the mechanical spin-current gen-\neration (due to magnon-phonon coupling) in a system with\nRashba SOI.31)\nAcknowledgment We would like to thank K. Kondou, J. Puebla, and M.\nXu for the valuable and informative discussion, and J. Ieda, S. Maekawa,\nM. Matsuo, M. Mori, and K. Yamamoto for their valuable advice and com-\nments. We also thank A. Yamakage, K. Nakazawa, T. Yamaguchi, Y . Imai,\nand J. Nakane for the daily discussions. This work is support ed by JSPS\nKAKENHI Grant Numbers 25400339, 15H05702 and 17H02929. TF i s sup-\nported by a Program for Leading Graduate Schools: “Integrat ive Graduate\nEducation and Research in Green Natural Sciences.”\n1) S. Mizukami, Y . Ando, and T. Miyazaki, J. Magn. Magn. Mater .226-\n230, 1640 (2001); Phys. Rev. B 66, 104413 (2002).\n2) Y . Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys. Rev. L ett.88,\n117601 (2002); Phys. Rev. B 66, 224403 (2002).\n3) E. Saitoh, M. Ueda, H. Miyajima, and G. Tatara, Appl. Phys. Lett. 88,\n182509 (2006).\n4) J. Sinova, S. O. Valenzuela, J. Wunderlich, C. H. Back and T . Jungwirth,\nRev. Mod. Phys. 87, 1213 (2015).\n5) M. Johnson and R. H. Silsbee, Phys. Rev. Lett. 55, 1709 (1985).\n6) K. Uchida, S. Takahashi, K. Harii, J. Ieda, W. Koshibae, K. Ando, S.\nMaekawa, and E. Saitoh Nature 455, 778 (2008).\n7) K. Uchida, H. Adachi, T. An, T. Ota, M. Toda, B. Hillebrands , S.\nMaekawa, and E. Saitoh, Nature Materials 10, 737 (2011).\n8) K. Uchida, T. An, Y . Kajiwara, M. Toda, and E. Saitoh, Appl. Phys. Lett.\n99, 212501 (2011).\n9) H. Adachi, K. Uchida, E. Saitoh, and S. Maekawa, Rep. Prog. Phys. 76,\n036501 (2013).\n10) H. Adachi, S. Maekawa, Solid State Comm. 198, 22 (2014).\n11) H. Keshtgar, M. Zareyan, and G. E. W. Bauer, Solid State Co mm. 198,\n30 (2014).\n12) P. A. Deymier, J. O. Vasseur, K. Runge, A. Manchon, and O. B ou-Matar,\nPhys. Rev. B 90, 224421 (2014).\n13) M. Matsuo, J. Ieda, E. Saitoh, and S. Maekawa, Phys. Rev. L ett.106,\n076601 (2011).\n14) M. Matsuo, J. Ieda, E. Saitoh, and S. Maekawa, Appl. Phys. Lett. 98,\n242501 (2011).\n15) R. Takahashi, M. Matsuo, M. Ono, K. Harii, H. Chudo, S. Oka yasu, J.\nIeda, S. Takahashi, S. Maekawa and E. Saitoh, Nature Physics 12, 52\n(2016).\n16) M. Matsuo, Y . Ohnuma, and S. Maekawa, Phys. Rev. B 96, 020401(R)\n(2017).\n17) M. Matsuo, J. Ieda, K. Harii, E. Saitoh, and S. Maekawa, Ph ys. Rev. B\n87, 180402(R) (2013).\n18) J. Ieda, M. Matsuo, and S. Maekawa, Solid State Comm. 198, 52-56\n(2014).\n19) D. Kobayashi, T. Yoshikawa, M. Matsuo, R. Iguchi, S. Maek awa, E.\nSaitoh, and Y . Nozaki, Phys. Rev. Lett. 119, 077202 (2017).\n20) T. Tsuneto, Phys. Rev. 121, 402 (1961).\n21) J. Wang, K. Ma, K. Li, and H. Fan, Ann. Phys. 362, 327 (2015).\n22) R. Kubo, J. Phys. Soc. Jpn. 12, 570 (1957).\n23) (Supplemental Material) Some details of the calculatio n of Eqs. (14),\n(15), (19), (20) and (21), together with all diagrams, are pr ovided on-\nline.\n24) We define the spin Hall angle as half of that of Ref.4)in order to be\nconsistent with the definition of spin-current operators.\n4J. Phys. Soc. Jpn. LETTERS\n25) R. H. Parmenter, Phys. Rev. 89, 990 (1953).\n26) These two routes were also identified in the electrical ge neration of spin\ncurrent in, K. Hosono, A. Yamaguchi, Y . Nozaki, and G. Tatara , Phys.\nRev. B 83, 144428 (2011).\n27) While the same Greek letter is used for the external frequ ency (ω) and\nthe vorticity (ωorωα), we hope no confusion will arise since the latter\nis denoted as a vector.\n28) Y . Niimi, M. Morota, D.H. Wei, C. Deranlot, M. Basletic, A . Hamzic,A. Fert, and Y . Otani, Phys. Rev. Lett. 106, 126601 (2011).\n29) N. W. Ashcroft and N. D. Mermin, Solid State Physics (Holt, Rinehart\nand Winston, New York, 1981).\n30) S. Ashida, Bulletin of the Japan Institute of Metals 17, 1004 (1978) (in\nJapanese).\n31) M. Xu, J. Puebla, F. Auvray, B. Rana, K. Kondou, and Y . Otan i, Phys.\nRev. B 97, 180301(R) (2018).\n5"    },    {        "title": "1707.02379v2.Interaction_induced_exotic_vortex_states_in_an_optical_lattice_clock_with_spin_orbit_coupling.pdf",        "content": "Interaction-induced exotic vortex states in an optical lattice clock with spin-orbit\ncoupling\nXiaofan Zhou,1, 2Jian-Song Pan,3, 4, 5Wei Yi,3, 4,\u0003Gang Chen,1, 2, †and Suotang Jia1, 2\n1State Key Laboratory of Quantum Optics and Quantum Optics Devices,\nInstitute of Laser spectroscopy, Shanxi University, Taiyuan 030006, China\n2Collaborative Innovation Center of Extreme Optics,\nShanxi University, Taiyuan, Shanxi 030006, China\n3Key Laboratory of Quantum Information, University of Science and Technology of China, CAS, Hefei, Anhui, 230026, China\n4Synergetic Innovation Center of Quantum Information and Quantum Physics,\nUniversity of Science and Technology of China, Hefei, Anhui 230026, China\n5Wilczek Quantum Center, School of Physics and Astronomy and T. D. Lee Institute,\nShanghai Jiao Tong University, Shanghai 200240, China\nMotivated by a recent experiment [L. F. Livi, et al. , Phys. Rev. Lett. 117, 220401(2016)],\nwe study the ground-state properties of interacting fermions in a one-dimensional optical lattice\nclock with spin-orbit coupling. As the electronic and the hyper\fne-spin states in the clock-state\nmanifolds can be treated as e\u000bective sites along distinct synthetic dimensions, the system can be\nconsidered as multiple two-leg ladders with uniform magnetic \rux penetrating the plaquettes of each\nladder. As the inter-orbital spin-exchange interactions in the clock-state manifolds couple individual\nladders together, we show that exotic interaction-induced vortex states emerge in the coupled-ladder\nsystem, which compete with existing phases of decoupled ladders and lead to a rich phase diagram.\nAdopting the density matrix renormalization group approach, we map out the phase diagram, and\ninvestigate in detail the currents and the density-density correlations of the various phases. Our\nresults reveal the impact of interactions on spin-orbit coupled systems, and are particularly relevant\nto the on-going exploration of spin-orbit coupled optical lattice clocks.\nI. INTRODUCTION\nSpin-orbit coupling (SOC) plays a key role in solid-\nstate topological materials such as topological insulators\nand quantum spin Hall systems [1{3]. The experimental\nrealization of synthetic SOC in cold atomic gases opens\nup the avenue of simulating synthetic topological mat-\nter on the versatile platform of cold atoms [4{18]. In\nmost of the previous studies, synthetic SOC is typically\nimplemented in alkali atoms using a two-photon Raman\nprocess, in which di\u000berent spin states in the ground-state\nhyper\fne manifold of the atoms are coupled. Thus, as the\natoms undergo Raman-assisted spin \rips, their center-\nof-mass momenta also change due to the photon recoil.\nAlternatively, by considering the atomic spin states as\ndiscrete lattice sites, the SOC can also be mapped to ef-\nfective tunneling in the so-called synthetic dimension [19{\n22]. Such an interpretation has led to the realization of\ntwo-leg ladder models with synthetic magnetic \rux, and\nto the subsequent experimental demonstration of chiral\nedge states using cold atoms [10, 23, 24].\nFor systems under the synthetic SOC generated by the\nRaman scheme, a key experimental di\u000eculty in reach-\ning the desired many-body ground states is the heat-\ning caused by high-lying excited states in the Raman\nprocess, whose single-photon detuning is limited by the\n\fne-structure splitting [4{6]. This problem can be over-\n\u0003Electronic address: wyiz@ustc.edu.cn\n†Electronic address: chengang971@163.comcome either by choosing atomic species with large \fne-\nstructure splitting [25, 26], or by using alkaline-earth-like\natoms [27], which feature long-lived excited states. In-\ndeed, in two recent experiments, synthetic SOCs with\nsigni\fcantly reduced heating have been experimentally\ndemonstrated by directly coupling the ground1S0(re-\nferred to asjgi) and the metastable3P0(referred to\nasjei) clock-state manifolds of87Sr or173Yb lattice\nclocks [10, 11]. In Ref. [10], the electronic states are fur-\nther mapped onto the e\u000bective lattice sites along a syn-\nthetic dimension, such that a two-leg ladder model with\nuniform magnetic \rux is realized and the resulting chiral\nedge currents are probed. Similar models for bosonic sys-\ntems have been extensively investigated in the past [28{\n35]. When the \rux is small, chiral edge currents emerge\nat the system boundary, where the currents along the\ntwo legs are opposite in direction. This is reminiscent of\nthe Meissner e\u000bects of superconductivity [36]. When the\n\rux becomes su\u000eciently large, the system undergoes a\nphase transition as the chiral edge currents are replaced\nby vortex lattices in the bulk, where currents exist on\nboth the rungs and the edges of the ladder.\nFurthermore, when taking the hyper\fne spin states\nin the clock-state manifolds into account, one can map\nboth the electronic and the spin degrees of freedom into\ndistinct synthetic dimensions, such that the system in\nRef. [10] can be extended to model multiple two-leg lad-\nders with synthetic magnetic \rux penetrating the pla-\nquettes of each ladder. Here, di\u000berent electronic states\nlabel the two legs of each ladder, and di\u000berent spin states\nlabel di\u000berent ladders. It would then be interesting to\nstudy the ground state of the system as the parame-arXiv:1707.02379v2  [cond-mat.quant-gas]  5 Sep 20172\nters such as the \rux or the interactions are tuned. In\nthe clock-state manifolds of alkaline-earth-like atoms, the\nnuclear and the electronic degrees of freedom are sep-\narated, and the short-range two-body interactions oc-\ncur either in the electronic spin-singlet channel j\u0000i=\n1\n2(jgei\u0000jegi)\n(j#\"i+j\"#i), or in the electronic spin-\ntriplet channelj+i=1\n2(jgei+jegi)\n(j#\"i\u0000j\"#i ) [37{\n39]. Here,j\"iandj#ilabel di\u000berent spin states in the\njgiorjeihyper\fne manifolds. As reported in previ-\nous studies, such an inter-orbital spin-exchange interac-\ntion would induce density-ordered states in either the\nspin or the charge channel, leading to spin-density wave\n(SDW), orbital-density order (ODW) or charge-density\nwave (CDW) phases [40{48]. More interestingly, these\ninteractions would couple the otherwise independent lad-\nders, which may induce new patterns of current \row in\nthe system.\nIn this work, adopting the concept of synthetic dimen-\nsions, we explicitly consider the hyper\fne spin states\nin the clock-state manifolds and map the system in\nRef. [10] to multiple two-leg ladders (see Fig. 1). Using\nthe density matrix renormalization group (DMRG) ap-\nproach [49, 50], we then numerically investigate the e\u000bect\nof interactions on the many-body ground-state properties\nsuch as the current \row and the density-density correla-\ntions. Our numerical results reveal a rich phase diagram,\nwhere the interactions drastically modify the Meissner\nand the vortex states in the non-interacting case. In par-\nticular, we show the existence of an exotic interaction-\ninduced vortex state, where spin currents emerge be-\ntween di\u000berent ladders together with SDW in the sys-\ntem. As the interactions in the clock-state manifolds can\nbe readily tuned by external magnetic \feld through the\norbital Feshbach resonance [51{53], or by transverse trap-\nping frequencies through the con\fnement-induced reso-\nnance [54, 55], our results have interesting implications\nfor future experiments.\nThe work is organized as follows. In Sec. II, we\npresent the system setup and the mode Hamiltonian.\nWe discuss the phase diagram of the non-interacting\ncase in Sec. III. We then study in detail the impact\nof interactions on the Meissner state and the vortex\nstate, respectively in Secs. IV and V. The detection\nand conclusion are given respectively in Secs. VI and VII.\nII. MODEL AND HAMILTONIAN\nWe consider a similar setup as in the recent experiment\non synthetic SOC in optical lattice clocks [10]. As shown\nin Fig. 1(a), a pair of counter-propagating laser with the\n\\magic\" wavelength \u0015L= 2\u0019=kLis used to generate a\none-dimensional (1D) optical lattice potential VL(x) =\nVxcos2(kLx), whereVxis the lattice depth. The synthetic\nSOC is implemented by an ultra-narrow \u0019-polarized clock\nlaser with a wavelength \u0015C, which drives a single-photon\ntransition between the clock states with the same nuclear\n(a)\n(b)\nVexΩe↓\ng↓ g↑e↑λLλC\nθπ\nλLFIG. 1: (a) Schematics of the experimental setup. The ul-\ntracold alkaline-earth-like atoms are trapped in a 1D optical\nlattice, which is generated by a pair of counter-propagating\nlasers with the \\magic\" wavelength \u0015L, such that states in the\nclock-state manifolds1S0and3P0are subject to the same lat-\ntice potential. An ultranarrow \u0019-polarized clock laser with a\nwavelength \u0015Cdrives a single-photon transition between the\nclock-state manifolds. By introducing an angle \u0012between the\nclock laser and that generating the optical lattice, the pho-\nton recoil momentum becomes kC= 2\u0019cos\u0012=\u0015Cand can be\ntuned experimentally. (b) Energy levels considering two nu-\nclear spin states j\"iandj#iin each manifold. While states\nin the1S0and3P0manifolds are coherently coupled by the\nspin-conserving clock laser (the red curves), the interaction\nin the clock-state manifolds can couple distinct spin states in\ndi\u000berent orbitals (the green curves).\nspins. Due to the existence of the angle \u0012between the\nwave vector of the clock laser and the alignment of the\n1D optical lattice [see Fig. 1(a)], the momentum transfer\nbecomeskC= 2\u0019cos\u0012=\u0015C[10]. A key ingredient in this\nsystem is the inter-orbital spin-exchange interaction [37{\n39], as shown by the green curves in Fig. 1(b). The full\nHamiltonian is written as ( ~= 1 hereafter)\n^HT=^HL+^HC+^HI (1)\nwith\n^HL=X\n\u000b\u001bZ\ndx^ †\n\u000b\u001b(x) [\u0000r2\n2m+VL(x)]^ \u000b\u001b(x);(2)\n^HC=\nR\n2X\n\u001bZ\ndx[^ †\ng\u001b(x)eikCx^ e\u001b(x) + H:c:];(3)\n^HI=g\u0006\n2Z\ndx[\t†\ng\"\t†\ne#\u0007\t†\ng#\t†\ne\"] [\te#\tg\"\u0007\te\"\tg#]\n+g\u0000Z\ndxh\n\t†\ng\"\t†\ne\"\te\"\tg\"+\t†\ng#\t†\ne#\te#\tg#i\n;(4)\nwhere\u000b=fg;egis the orbit index, \u001b=f\";#gis the\nspin index, \t \u000b\u001band \t†\n\u000b\u001bare the corresponding \feld op-\nerators, \n Ris the Rabi frequency of the clock laser, g\u00063\nare the 1D interaction strengths [51, 55], and H :c:is the\nHermitian conjugate. Note that in writing down Hamil-\ntonian (1), we only consider four nuclear spin states from\nthejgiandjeimanifolds. In principle, the other nu-\nclear spin states can be shifted away by imposing spin-\ndependent laser shifts [24].\nWhen the 1D optical lattice is deep enough and \n R\nis not too large [56{58], we may take the single-band\napproximation and write down the corresponding tight-\nbinding model\n^HTB=\u0000tX\n<i;j>;\u000b\u001b^c†\ni\u000b\u001b^cj\u000b\u001b+\n2X\nj;\u001b(ei\u001ej^c†\njg\u001b^cje\u001b+H:c:)\n+UX\nj(^njg\"^nje#+ ^njg#^nje\") +U0X\nj\u001b^njg\u001b^nje\u001b\n+VexX\nj(^c†\njg\"^c†\nje#^cje\"^cjg#+ H:c:); (5)\nwhere ^cj\u000b\u001b(^c†\nj\u000b\u001b) is the annihilation (creation) operator\nfor atoms on the ith site of the \u000borbital and the spin\n\u001b, ^nj\u000b\u001b= ^c†\nj\u000b\u001b^cj\u000b\u001b. The spin-conserving hopping rate\nt=\f\f\fR\ndxw(j)h\n\u0000r2\n2m+VL(x)i\nw(j+1)\f\f\f, withw(j)being the\nlowest-band Wannier function on the jth site of the lat-\ntice potential VL(x),. The spin-\ripping hopping rate \n= \n RR\ndxw(j)eikCxw(j).\u001e=1\n2kC\u0015L=\u0019\u0015Lcos\u0012=\u0015C\nis the synthetic magnetic \rux per plaquette induced\nby the SOC, U=1\n2(g++g\u0000)R\ndxw(j)w(j)w(j)w(j)\nandU0=g\u0000R\ndxw(j)w(j)w(j)w(j)are the inter-orbital\ndensity-density interaction strengths with the same\nand di\u000berent nuclear spins, respectively. Vex=\n1\n2(g\u0000\u0000g+)R\ndxw(j)w(j)w(j)w(j)is the inter-orbital\nspin-exchange interaction strength. Hamiltonian (5) has\nthe advantage that all parameters can be tuned indepen-\ndently. For example, tcan be controlled by the depth of\nthe optical lattice potential, \n and \u001ecan be controlled\nby the Rabi frequency and the angle of the clock laser,\nrespectively, and fVex,U,U0gcan be tuned through the\norbital Feshbach resonance [51{53] or the con\fnement\ninduced resonance [54, 55]. In the following, we take\nU0=Vex+U, which is dictated by the scattering param-\neters of173Yb atoms [54, 55].\nFrom the tight-binding Hamiltonian (5), it is clear\nthat if we drop the interaction terms and map the elec-\ntronic (\u000b) and the spin ( \u001b) states onto e\u000bective lattice\nsites along two di\u000berent synthetic dimensions, the non-\ninteracting tight-binding model describes a pair of two-\nleg ladders. We label the two synthetic dimensions as\nthe orbit\u000b- and the spin \u001b-directions, respectively, while\nthe optical lattice lies along the x-direction. We may\nthen denote the synthetic dimensions as SD \u000band SD\u001b,\nrespectively. As illustrated in Fig. 2, the pair of lad-\nders each lie within the ( x;\u000b) plane with the legs of\nboth ladders along the x-direction. The rung tunnel-\ning in each ladder is facilitated by the SOC, which also\ninduces uniform magnetic \rux in each plaquette of the\nladder. In the absence of interactions, the ladders are\nφ(b) φtSD αe↑\ng↑e↓\ng↓\n……        j-1         j         j+1       ……\n2ieφj\nSD σ\nReal dimension x\n φ(a)\nSD αt\n2ie φj\ne\ng\n……        j-1         j         j+1       ……\nReal dimension xFIG. 2: (a) A two-leg synthetic ladder with a synthetic mag-\nnetic \rux\u001e=\u0019\u0015L=\u0015Ccos\u0012in each plaquette. The orbital\nstates can be treated as an e\u000bective synthetic dimension de-\nnoted as SD \u000b. The two ladders along the synthetic dimen-\nsion SD\u001bare identical and decoupled. (b) A pair of two-leg\nsynthetic ladders (i.e., jg\";e\"iandjg#;e#i), with the same\n\rux, are coupled by the inter-orbital spin-exchange interac-\ntion (green curves). In this lattice, there are two-direction\nsynthetic dimensions, SD \u000band SD\u001b.\nnot coupled, as di\u000berent spin states are independent on\nthe single-body level. However, the inter-orbital spin-\nexchange interaction e\u000bectively couples the ladders to-\ngether [see Fig. 2(b)], which, as we will show later, induce\ninter-ladder currents along the \u001b-direction.\nAn important property here is the current along the\nlegs and the rungs of the ladder. Local and average cur-\nrents along the x-direction can be de\fne as [29, 30, 59],\nJk\nj;\u000b\u001b=i\u0010\n^c†\nj+1\u000b\u001b^cj\u000b\u001b\u0000^c†\nj\u000b\u001b^cj+1\u000b\u001b\u0011\n; (6)\nJk\n\u000b\u001b=1\nLX\njJk\nj;\u000b\u001b: (7)\nSimilarly, currents along the \u000b-direction can be de\fned\nas [30],\nJ?\nj;\u000b=i(ei\u001ej^c†\nje\u001b^cjg\u001b\u0000e\u0000i\u001ej^c†\njg\u001b^cje\u001b); (8)\nJ?\n\u000b=1\nLX\nj\f\fJ?\nj;\u000b\f\f; (9)\nFinally, we also de\fne currents along the \u001b-direction as\nJ?\nj;\u001b=i(^c†\nj\u000b#^cj\u000b\"\u0000^c†\nj\u000b\"^cj\u000b#); (10)\nJ?\n\u001b=1\nLX\nj\f\fJ?\nj;\u001b\f\f: (11)\nIn the following discussions, we adopt the DMRG\nformalism to calculate the ground state of the system,\nfrom which we characterize currents and density corre-\nlation functions. For the numerical calculation, we have\nconsidered length of chain Lup to 32 sites. We keep4\n0.00 .51 .0-0.10.00.10\n.00 .51 .00.000.010.020\n.00 .51 .0012345 \nJ  \nφ/π  \nJ   ||e\n/s61555  \nJ||g\n/s61555  J  ⊥(b)(\nc) φ\n/π J⊥σ\n \nJ⊥α\n  Ω/tφ\n/πVortex IMeissner(a)\nFIG. 3: (a) Phase diagram in the (\n ;\u001e) plane. The currents\n(b)Jkand (c)J?as functions of \u001e=\u0019. In all sub\fgures,\nU=U0=Vex= 0 andn= 1, and (b) and (c) have the other\nparameter \n =t= 2.\nthe maxstates m= 200 and achieve truncation errors of\n10\u000010. We mainly consider the case of half \flling, i.e.,\nn=N=(2L) = 1, where Lis the length of chain and N\nis the total number of atoms.\nIII. PHASES AND CURRENTS IN THE\nNON-INTERACTING CASE\nWe \frst discuss the ground-state phases of the system\nin the absence of interactions. In this case, di\u000berent spin\nstates are decoupled, and we may identify a pair of two-\nleg ladders, as illustrated in Fig. 2. From our numerical\ncalculations, we \fnd that only the Meissner and the\nvortex states appear in the ground-state phase diagram\nshown in Fig. 3(a). Typically, when the synthetic \rux is\nsmall, the ground state is the so-called Meissner state,\nwhere the edge currents Jk\ng\u001bandJk\ne\u001b\row in opposite\ndirections along the two legs of each ladder [10, 23, 29],\nas shown in Fig. 3(b). Upon increasing the \rux above\na critical value, the ground state features a vortex\nstate with rung currents and vortex lattice structures in\nthe bulk of each ladder, i.e., in the ( x;\u000b) plane. The\nexistence of this so-called Vortex I state is con\frmed\nin Fig. 3(c), where nonzero J?\n\u000bin the Vortex I state\nregime indicates a \fnite current along the rungs in the\n\u000b-direction for each ladder. The current J?\n\u001bremains\nzero in the non-interacting case, which is consistent with\nthe picture of two independent ladders of di\u000berent spins.\nWe also note that in the non-interacting case, there are\n-8-4048-8-4048-\n8-40480.00.51.0-\n8-4048-0.20.00.2-\n8-40480.00.10.2(a)M\neissnerCDW \n U/tV\n  /tVortex IIM\neissnere\nx(\nd)( c)(b) \nU/t \n Order S    \nC \nU/t \n J  \nJ||e\n/s61555  \nJ||g\n/s61555  J ⊥U\n/t J⊥σ\n    \nJ⊥α\nFIG. 4: (a) Phase diagram in the ( U;V ex) plane. (b) The SDW\norderSand the CDW order Cas well as the currents (c) Jk\nand (d)J?as functions of U=t. In all sub\fgures, \n =t= 4,\n\u001e=\u0019= 0:75, andn= 1, and (b)-(d) have the other parameter\nVex=t=\u00006.\nno density-ordered phases.\nIV. IMPACT OF INTERACTIONS ON THE\nMEISSNER STATE\nWe now study the impact of interactions on the Meiss-\nner state for \n =t= 4 and\u001e=\u0019= 0:75. As the interactions\nare turned on, the system can undergo phase transitions\ninto exotic vortex states or phases with density orders.\nWe map out the phase diagram in the ( U;V ex) plane,\nwhile \fxing other parameters. As shown in Fig. 4(a), the\nphase diagram consists of three di\u000berent phases: a simple\nMeissner state, a Meissner state with CDW, and an ex-\notic vortex state with SDW, which we label as Vortex II\nstate. The simple Meissner state resembles the Meissner\nstate in the non-interacting case with chiral edge currents\nand no density-orders in the bulk. The CDW Meissner\nstate features chiral edge currents as well as \fnite CDW\ndensity correlations in the bulk. The most interesting\nstate here is the Vortex II state, which features \fnite\nSDW correlations as well as currents and vortex lattice\nstructures in the ( x;\u001b) plane.\nThe phase boundaries between these phases can be de-\ntermined from the CDW and the SDW correlations, as\nwell as from the currents' calculations. As illustrated in\nFig. 4(b), for a \fxed Vex=t=\u00006, the CDW order\nC=1\n2LX\nj(\u00001)jnj (12)5\n08162432-0.10.00.10\n8162432-0.20.00.2(b)  J⊥j\n,σs\nite(  j)(a) \n nj,↑ − nj,↓s\nite(  j)\n  e↑\ng↑e↓\ng↓(c)SD αSD σ\nReal dimension x\nFIG. 5: (a) The currents J?and (b) density pro\fles nj;\"\u0000nj;#\nfor di\u000berent sites. (c) Sketch of currents of the Vortex II state.\nIn (a) and (b), \n =t= 4,\u001e=\u0019= 0:75,Vex=t=\u00006,U=t= 2,\nandn= 1.\nhas a \fnite value in the range \u00008< U=t < 0, while the\nSDW order\nS=1\nLX\nj(\u00001)j(nj;\"\u0000nj;#) (13)\nhas a \fnite value for 0 < U=t < 4:7. On the other\nhand, while the edge currents Jk\n\u000b\u001bare always \fnite\nand opposite in directions for di\u000berent sites in the \u000b\ndirection, the currents J?\n\u001bandJ?\n\u000bonly exist within a\nrange, as shown in Figs. 4(c) and 4(d). In particular,\nthe non-vanishing J?\n\u001bindicates inter-ladder currents\nand vortices in the ( x;\u001b) plane. To further characterize\nthe Vortex II state, in Figs. 5(a) and 5(b), we show\nthe spatial distribution of the inter-ladder currents as\nwell as the spin density. Apparently, the SDW and the\nvortex lattice structure in the ( x;\u001b) plane is due to the\ninterplay of the inter-orbital spin-exchange interaction\nand the synthetic magnetic \rux in the ( x;\u000b) plane, as\nshown in Fig. 5(c).\nV. IMPACT OF INTERACTIONS ON THE\nVORTEX STATE\nIn this section, we study the impact of interactions on\nthe vortex states. In Fig. 6, we map out the phase dia-\ngram in the ( U;V ex) plane for \n =t= 2 and\u001e=\u0019= 0:25.\nIn the absence of interactions, the system is in the vor-\ntex state. With interactions, the system can undergo\nphase transitions into various di\u000berent phases. As shown\nin Fig. 6(a), besides the Vortex II state and the CDW\nMeissner state, several other exotic phases emerge in the\n-8-4048-8-4048-\n8-40480.00.40.8-\n8-4048-0.20.00.2-\n8-40480.00.10.2exMeissnerODWVortex IODWV\nortex IM\neissnerCDW \n U/tV\n  /tVortex IIMeissnere\nx V  /t \n Order \nS \nOe\nx(d)( c)(b)V\n  /t \n J  \nJ||e\n/s61555  \nJ||g\n/s61555(a)e\nx  J⊥V\n  /t J⊥σ\n \nJ⊥α\nFIG. 6: (a) Phase diagram in the ( U;V ex) plane. (b) The\nSDW order Sand the ODW order Oas well as the currents\n(c)Jkand (d)J?as functions of Vex=t. In all sub\fgures,\n\n=t= 2,\u001e=\u0019= 0:25, andn= 1, and (b)-(d) have the other\nparameterU=t= 1.\nphase diagram. While the Vortex I resembles the vortex\nstate in the absence of interactions, interesting phases\nwith density correlations in the orbital channel appear,\nwhich can be further di\u000berentiated by their currents \rows\nas the ODW Meissner state and the ODW Vortex I state,\nwhere the vortex occurs in the ( x;\u000b) plane, together\nwith density-wave orders in the orbital channel. Here\nthe ODW order is de\fned as\nO=1\nLX\nj(\u00001)j(nj;g\u0000nj;e): (14)\nIn Figs. 6(b)-6(d), we plot the currents and the den-\nsity wave orders as functions of Vex=tforU=t = 1.\nForVex=t2[\u00008;\u00002:06], the SDW order Shas a \fnite\nvalue, the ODW order vanishes with O= 0, the cur-\nrentsJk\n\u000b\u001b>0 andJ?\n\u001b>0, the corresponding phase\nis the Vortex II state. When Vex=t2[\u00002:06;\u00001], the\nSDW order S= 0, the ODW order O= 0, the cur-\nrentsJk\n\u000b\u001b>0,J?\n\u000b>0, andJ?\n\u001b= 0, the corresponding\nphase is the Vortex I state. When Vex=t2[\u00001;0:77], the\nSDW order S= 0, the ODW order O > 0, the currents\nJk\n\u000b\u001b>0,J?\n\u000b>0, andJ?\n\u001b= 0, the corresponding phase\nis the ODW Vortex I state. When Vex=t2[0:77;8], the\nSDW order S= 0, the ODW order O= 0, the currents\nJk\n\u000b\u001b>0,J?\n\u000b= 0, andJ?\n\u001b= 0, the corresponding phase\nis the Meissner state.\nAccording to Hamiltonian (5), there are three dif-\nferent interaction parameters U,VexandU0. It is\nstraightforward to see that the CDW order is favored\nfor the attractive interactions ( U < 0,U0<0,Vex<0);6\nand that the SDW order is favored for the attractive\ninter-orbital spin-preserving interaction ( U0<0). This\nis because the CDW order can decrease the interaction\nenergy of all the attractive on-site interactions. While\nthe SDW order mainly decreases the energy of the at-\ntractive inter-orbital spin-preserving interactions. These\nare consistent with previous studies on the SU(2) ladder\nsystems [41, 43]. Note that for the phase diagrams in\nFigs. 4 and 6, we have \fxed U0=Vex+U. In the phase\ndiagrams, the CDW state becomes unstable against\nthe Vortex II state when Uis not su\u000eciently negative.\nApparently, the competition between the CDW and the\nSDW orders is driven by the interactions associated\nwithUandU0, which is eventually determined by the\nrelative values of UandVex. On the other hand, the\nemergence of the ODW order in Fig. 6 can be understood\nas a con\fguration in which the repulsive energies are\nminimized in the case of U > 0 andU0>0. A subtlety\nhere is the impact of the magnetic \rux in the x-\u000bplane\non the density ordered phases. From numerical analysis,\nwe see that an increase of magnetic \rux can lead to a\nstronger competition between di\u000berent density-ordered\nphases, which gives rise to a richer phase diagram. In\nany case, we emphasize that the exotic Vortex II state\nwith the SDW order is always robust in the negative U0\nlimit.\nVI. DETECTION\nGiven the rich phase diagram discussed above, a natu-\nral question is how to detect them experimentally. In gen-\neral, the di\u000berent phases of the system are characterized\nby their chiral edge currents as well as the density cor-\nrelations in di\u000berent channels such as CDW, ODW and\nSDW. Here, the density orders can be probed by state-\nselective measurements of density distributions. While\nthe CDW order can be identi\fed by oscillations of the\ntotal density distribution from site to site, the ODW and\nthe SDW orders can be identi\fed, respectively, by oscil-\nlations of the density distribution of a given orbital ( jgi\norjei) or of a given spin ( j\"iorj#i). For the detection\nof the Meissner and the vortex states, one can in princi-\nple follow the approach in Ref. [30], where, by projecting\nthe wave function into isolated double wells along each\nleg, the chiral currents can be calculated from the oscil-\nlatory density dynamics in the double wells. The vortex\nstate can be identi\fed either from the variation of the\nchiral currents, as the maximum of the chiral currentsappear at the phase boundary between the Meissner and\nthe vortex states [see Figs. 4(a) and 6(a)]. Alternatively,\none should also identify the Vortex I and the Vortex II\nstates, respectively, from the relative phase between the\noscillations of di\u000berent orbital and spin states. The rela-\ntive phase should be \u0019in the case of the Meissner state,\nand smaller than \u0019in the case of the vortex states [30].\nVII. CONCLUSION\nWe show that by implementing synthetic SOC in\nalkaline-earth-like atoms, one naturally realizes multiple\ntwo-leg ladders with uniform synthetic \rux. As inter-\nactions couple di\u000berent ladders together, the system\nfeatures a rich phase diagram. In particular, we demon-\nstrate the existence of an interaction-induced vortex\nstate, which possesses SDW in the spin channel. The\ndi\u000berent phases can be experimentally detected based\non their respective properties. As many phases in the\nphase diagram simultaneously feature density order and\nedge or bulk currents, a potential experimental challenge\nlies in the e\u000ecient detection of the various phases.\nThis is particularly so for the exotic interaction-induced\nVortex II state, whose SDW order as well as the bulk\ncurrents along SD \u001brequire spin-selective detections.\nNevertheless, with the state of the art quantum control\nover the clock states in alkaline-earth-like atoms, we\nexpect that these challenges can be overcome with\nexisting experimental techniques. Our results reveal the\nimpact of interactions on spin-orbit coupled systems,\nand are particularly relevant to the on-going exploration\nof spin-orbit coupled optical lattice clocks.\nAcknowledgments\nThis work is supported partly by the Na-\ntional Key R&D Program of China under Grants\nNo. 2017YFA0304203 and No. 2016YFA0301700; the\nNKBRP under Grant No. 2013CB922000; the NSFC un-\nder Grants No. 60921091, No. 11374283, No. 11434007,\nNo. 11422433, No. 11522545, and No. 11674200; \\Strate-\ngic Priority Research Program(B)\" of the Chinese\nAcademy of Sciences under Grant No. XDB01030200;\nthe PCSIRT under Grant No. IRT13076; the FANEDD\nunder Grant No. 201316; SFSSSP; OYTPSP; and\nSSCC. J.-S. P. acknowledges support from National\nPostdoctoral Program for Innovative Talents of China\nunder Grant No. BX201700156.\n[1] M. Z. Hasan and C. L. Kane, Topological insulators, Rev.\nMod. Phys. 82, 3045 (2010).\n[2] X.-L. Qi and S.-C. Zhang, Topological insulators and su-\nperconductors, Rev. Mod. Phys. 83, 1057 (2011).[3] J. Alicea, New directions in the pursuit of Majorana\nfermions in solid state systems, Rep. Prog. Phys. 75,\n076501 (2012).\n[4] Y.-J. Lin, K. Jim\u0013 enez-Garc\u0013 \u0010a, and I. B. Spielman, Spin-7\norbit-coupled Bose-Einstein condensates, Nature (Lon-\ndon)471, 83 (2011).\n[5] P. Wang, Z.-Q. Yu, Z. Fu, J. Miao, L. Huang, S. Chai,\nH. Zhai, and J. Zhang, Spin-Orbit Coupled Degenerate\nFermi Gases, Phys. Rev. Lett. 109, 095301 (2012).\n[6] L. W. Cheuk, A. T. Sommer, Z. Hadzibabic, T. Yefsah,\nW. S. Bakr, and M.W. Zwierlein, Spin-Injection Spec-\ntroscopy of a Spin-Orbit Coupled Fermi Gas, Phys. Rev.\nLett. 109, 095302 (2012).\n[7] Z. Meng, L. Huang, P. Peng, D. Li, L. Chen, Y. Xu,\nC. Zhang, P. Wang, and J. Zhang, Experimental Ob-\nservation of a Topological Band Gap Opening in Ultra-\ncold Fermi Gases with Two-Dimensional Spin-Orbit Cou-\npling, Phys. Rev. Lett. 117, 235304 (2016).\n[8] L. Huang, Z. Meng, P. Wang, P. Peng, S.-L. Zhang, L.\nChen, D. Li, Q. Zhou, and J. Zhang, Experimental real-\nization of two-dimensional synthetic spin{orbit coupling\nin ultracold Fermi gases, Nat. Phys. 12, 540 (2016).\n[9] Z. Wu, L. Zhang, W. Sun, X.-T. Xu, B.-Z. Wang, S.-C. Ji,\nY. Deng, S. Chen, X.-J. Liu, and J.-W. Pan, Realization\nof two-dimensional spin-orbit coupling for Bose-Einstein\ncondensates, Science 354, 83 (2016).\n[10] L. F. Livi, G. Cappellini, M. Diem, L. Franchi, C. Clivati,\nM. Frittelli, F. Levi, D. Calonico, J. Catani, M. Ingus-\ncio, and L. Fallani, Synthetic Dimensions and Spin-Orbit\nCoupling with an Optical Clock Transition, Phys. Rev.\nLett. 117, 220401 (2016).\n[11] S. Kolkowitz, S. L. Bromley, T. Bothwell, M. L. Wall,\nG. E. Marti, A. P. Koller, X. Zhang, A. M. Rey, J. Ye,\nSpin{orbit-coupled fermions in an optical lattice clock,\nNature (London) 542, 66 (2017).\n[12] V. Galitski and I. B. Spielman, Spin{orbit coupling in\nquantum gases, Nature (London) 494, 49 (2013).\n[13] N. Goldman, G. Juzeli\u0016 unas, P. Ohberg, and I. B. Spiel-\nman, Light-induced gauge \felds for ultracold atoms, Rep.\nProg. Phys. 77, 126401 (2014).\n[14] X. Zhou, Y. Li, Z. Cai, and C. Wu, Unconventional states\nof bosons with the synthetic spin{orbit coupling, J. Phys.\nB: At. Mol. Opt. Phys. 46, 134001 (2014).\n[15] H. Zhai, Degenerate quantum gases with spin-orbit cou-\npling: a review, Rep. Prog. Phys. 78, 026001 (2015).\n[16] W. Yi, W. Zhang, and X. Cui, Pairing super\ruidity\nin spin-orbit coupled ultracold Fermi gases, Sci. China:\nPhys. Mech. Astron. 58, 014201 (2015).\n[17] J. Zhang, H. Hu, X. J. Liu, and H. Pu, Fermi gases with\nsynthetic spin-orbit coupling, Ann. Rev. Cold At. Mol.\n2, 81 (2015).\n[18] Y. Xu and C. Zhang, Topological Fulde Ferrell super\ru-\nids of a spin orbit coupled Fermi gas, Int. J. Mod. Phys.\nB29, 1530001 (2015).\n[19] O. Boada, A. Celi, J. I. Latorre, and M. Lewenstein,\nQuantum Simulation of an Extra Dimension, Phys. Rev.\nLett. 108, 133001 (2012).\n[20] A. Celi, P. Massignan, J. Ruseckas, N. Goldman, I. B.\nSpielman, G. Juzeli\u0016 unas, and M. Lewenstein, Synthetic\nGauge Fields in Synthetic Dimensions, Phys. Rev. Lett.\n112, 043001 (2014).\n[21] O. Boada, A. Celi, J. Rodr\u0013 \u0010guez-Laguna, J. I. Latorre,\nand M. Lewenstein, Quantum simulation of non-trivial\ntopology, New J. Phys. 17, 045007 (2015).\n[22] E. Anisimovas, M. Ra\u0014 ci\u0016 unas, C. Str ater, A. Eckardt, I. B.\nSpielman, and G. Juzeli\u0016 unas, Semisynthetic zigzag opti-\ncal lattice for ultracold bosons, Phys. Rev. A 94, 063632(2016).\n[23] B. K. Stuhl, H.-I. Lu, L. M. Aycock, D. Genkina, and\nI. B. Spielman, Visualizing edge states with an atomic\nBose gas in the quantum Hall regime, Science 349, 1514\n(2015).\n[24] M. Mancini, G. Pagano, G. Cappellini, L. Livi, M. Rider,\nJ. Catani, C. Sias, P. Zoller, M. Inguscio, M. Dalmonte,\nand L. Fallani, Observation of chiral edge states with\nneutral fermions in synthetic Hall ribbons, Science 349,\n1510 (2015).\n[25] X. Cui, B. Lian, T.-L. Ho, B. L. Lev, and H. Zhai, Syn-\nthetic gauge \feld with highly magnetic lanthanide atoms,\nPhys. Rev. A 88, 011601 (2013).\n[26] N. Q. Burdick, Y. Tang, and B. L. Lev, Long-lived spin-\norbit-coupled degenerate dipolar Fermi gas, Phys. Rev.\nX6, 031022 (2016).\n[27] M. L. Wall, A. P. Koller, S. Li, X. Zhang, N. R. Cooper,\nJ. Ye, and A. M. Rey, Synthetic Spin-Orbit Coupling in\nan Optical Lattice Clock, Phys. Rev. Lett. 116, 035301\n(2016).\n[28] E. Orignac and T. Giamarchi, Meissner e\u000bect in a bosonic\nladder, Phys. Rev. B 64, 144515 (2001).\n[29] A. Petrescu and K. L. Hur, Bosonic Mott Insulator with\nMeissner Currents, Phys. Rev. Lett. 111, 150601 (2013).\n[30] M. Atala, M. Aidelsburger, M. Lohse, J. T. Barreiro, B.\nParedes, and I. Bloch, Observation of chiral currents with\nultracold atoms in bosonic ladders, Nat. Phys. 10, 588\n(2014).\n[31] M. Piraud, F. Heidrich-Meisner, I. P. McCulloch, S.\nGreschner, T. Vekua, and U. Schollw ock, Vortex and\nMeissner phases of strongly interacting bosons on a two-\nleg ladder, Phys. Rev. B 91, 140406(R) (2015).\n[32] S. S. Natu, Bosons with long-range interactions on two-\nleg ladders in arti\fcial magnetic \felds, Phys. Rev. A 92,\n053623 (2015).\n[33] F. Kolley, M. Piraud, I. P. McCulloch, U. Schollw ock,\nand F Heidrich-Meisner, Strongly interacting bosons on\na three-leg ladder in the presence of a homogeneous \rux,\nNew J. Phys. 17, 092001 (2015).\n[34] S. Greschner, M. Piraud, F. Heidrich-Meisner, I. P. Mc-\nCulloch, U. Schollw ock, and T. Vekua, Spontaneous In-\ncrease of Magnetic Flux and Chiral-Current Reversal in\nBosonic Ladders: Swimming against the Tide, Phys. Rev.\nLett. 115, 190402 (2015).\n[35] S. Greschner, M. Piraud, F. Heidrich-Meisner, I. P. Mc-\nCulloch, U. Schollw ock, and T. Vekua, Symmetry-broken\nstates in a system of interacting bosons on a two-leg lad-\nder with a uniform Abelian gauge \feld, Phys. Rev. A 94,\n063628 (2016).\n[36] A. K. Geim, S. V. Dubonos, J. G. S. Lok, M. Henini,\nand J. C. Maan, Paramagnetic Meissner e\u000bect in small\nsuperconductors, Nature (London) 396, 144 (1998).\n[37] X. Zhang, M. Bishof, S. L. Bromley, C. V. Kraus, M. S.\nSafronova, P. Zoller, A. M. Rey, and J. Ye, Spectroscopic\nobservation of SU(N)-symmetric interactions in Sr orbital\nmagnetism, Science 345, 1467 (2014).\n[38] G. Cappellini, M. Mancini, G. Pagano, P. Lombardi, L.\nLivi, M. Siciliani de Cumis, P. Cancio, M. Pizzocaro, D.\nCalonico, F. Levi, C. Sias, J. Catani, M. Inguscio, and\nL. Fallani, Direct Observation of Coherent Interorbital\nSpin-Exchange Dynamics, Phys. Rev. Lett. 113, 120402\n(2014); ibid114, 239903 (2015).\n[39] F. Scazza, C. Hofrichter, M. H ofer, P. C. De Groot, I.\nBloch, and S. F olling, Observation of two-orbital spin-8\nexchange interactions with ultracold SU(N)-symmetric\nfermions, Nat. Phys. 10, 779 (2014); ibid11, 514 (2015).\n[40] K. Kobayashi, M. Okumura, Y. Ota, S. Yamada, and\nM. Machida, Nontrivial Haldane phase of an atomic two-\ncomponent Fermi gas trapped in a 1D optical lattice,\nPhys. Rev. Lett. 109, 235302 (2012).\n[41] H. Nonne, M. Moliner, S. Capponi, P. Lecheminant, and\nK. Totsuka, Symmetry-protected topological phases of\nalkaline-earth cold fermionic atoms in one dimension, Eu-\nrophys. Lett. 102, 37008 (2013).\n[42] K. Duivenvoorden and T. Quella, Topological phases of\nspin chains, Phys. Rev. B 87, 125145 (2013).\n[43] V. Bois, S. Capponi, P. Lecheminant, M. Moliner, and K.\nTotsuka, Phase diagrams of one-dimensional half-\flled\ntwo-orbital SU(N) cold fermion systems, Phys. Rev. B\n91, 075121 (2015).\n[44] A. Roy and T. Quella, Chiral Haldane phases of SU(N)\nquantum spin chains in the adjoint representation,\narXiv: 1512.05229.\n[45] V. Bois, P. Fromholz, and P. Lecheminant, One-\ndimensional two-orbital SU(N) ultracold fermionic quan-\ntum gases at incommensurate \flling: A low-energy ap-\nproach, Phys. Rev. B 93, 134415 (2016).\n[46] S. Capponi, P. Lecheminant, and K. Totsuka, Phases of\none-dimensional SU(N) cold atomic Fermi gases|From\nmolecular Luttinger liquids to topological phases, Ann.\nPhys. 367, 50 (2016).\n[47] X. Zhou, J.-S. Pan, Z.-X. Liu, W. Zhang, W. Yi,\nG. Chen, and S. Jia, Symmetry-protected topologi-\ncal states for interacting fermions in alkaline-earth-like\natoms, arXiv: 1612.08880.\n[48] F. Iemini, L. Mazza, L. Fallani, P. Zoller, R. Fazio, and\nM. Dalmonte, Majorana Quasiparticles Protected by Z2\nAngular Momentum Conservation, Phys. Rev. Lett. 118,\n200404 (2017).\n[49] S. R. White, Density matrix formulation for quan-\ntum renormalization groups, Phys. Rev. Lett. 69, 2863(1992).\n[50] U. Schollw ok, The density-matrix renormalization group,\nRev. Mod. Phys. 77, 259 (2005).\n[51] R. Zhang, Y. Cheng, H. Zhai, and P. Zhang, Orbital\nFeshbach Resonance in Alkali-Earth Atoms, Phys. Rev.\nLett. 115, 135301 (2015).\n[52] M. H ofer, L. Riegger, F. Scazza, C. Hofrichter, D. R.\nFernandes, M. M. Parish, J. Levinsen, I. Bloch, and S.\nF olling, Observation of an Orbital Interaction-Induced\nFeshbach Resonance in173Yb, Phys. Rev. Lett. 115,\n265302 (2015).\n[53] G. Pagano, M. Mancini, G. Cappellini, L. Livi, C. Sias, J.\nCatani, M. Inguscio, and L. Fallani, Strongly Interacting\nGas of Two-Electron Fermions at an Orbital Feshbach\nResonance, Phys. Rev. Lett. 115, 265301 (2015).\n[54] M. Olshanii, Atomic Scattering in the Presence of an\nExternal Con\fnement and a Gas of Impenetrable Bosons,\nPhys. Rev. Lett. 81, 938 (1998).\n[55] R. Zhang, D. Zhang, Y. Cheng, W. Chen, P. Zhang,\nand H. Zhai, Kondo e\u000bect in alkaline-earth-metal atomic\ngases with con\fnement-induced resonances, Phys. Rev.\nA93, 043601 (2016).\n[56] J.-S. Pan, X.-J. Liu, W. Zhang, W. Yi, and G.-C. Guo,\nTopological Superradiant States in a Degenerate Fermi\nGas, Phys. Rev. Lett. 115, 045303 (2015).\n[57] L. Zhou and X. Cui, Spin-orbit coupled ultracold gases\nin optical lattices: High-band physics and insu\u000eciency\nof tight-binding models, Phys. Rev. B 92, 140502(R)\n(2015).\n[58] J.-S. Pan, W. Zhang, W. Yi, and G.-C. Guo, Bose-\nEinstein condensate in an optical lattice with Raman-\nassisted two-dimensional spin-orbit coupling, Phys. Rev.\nA94, 043619 (2016).\n[59] S. Barbarino, L. Taddia, D. Rossini, L. Mazza, and R.\nFazio, Magnetic crystals and helical liquids in alkaline-\nearth fermionic gases, Nat. Commun. 6, 8134 (2015)."    },    {        "title": "1503.06872v2.Spin_Orbit_Torques_in_Two_Dimensional_Rashba_Ferromagnets.pdf",        "content": "arXiv:1503.06872v2  [cond-mat.mtrl-sci]  13 Jul 2015Spin-Orbit Torques in Two-Dimensional Rashba Ferromagnet s\nA. Qaiumzadeh,1R.A. Duine,2and M. Titov1\n1Radboud University, Institute for Molecules and Materials , 6525 AJ Nijmegen, The Netherlands\n2Institute for Theoretical Physics and Centre for Extreme Ma tter and Emergent Phenomena,\nUtrecht University, 3584 CE Utrecht, The Netherlands\n(Dated: October 15, 2018)\nMagnetization dynamics in single-domain ferromagnets can be triggered by a charge current if the\nspin-orbit coupling is sufficiently strong. We apply functio nal Keldysh theory to investigate spin-\norbit torques in metallic two-dimensional Rashba ferromag nets in the presence of spin-dependent\ndisorders. A reactive, anti-damping-like spin-orbit torq ue as well as a dissipative, field-like torque\nis calculated microscopically, to leading order in the spin -orbit interaction strength. By calculating\nthe first vertex correction we show that the intrinsic anti-d amping-like torque vanishes unless the\nscattering rates are spin-dependent.\nPACS numbers: 72.15.Gd, 75.60.Jk, 75.70.Tj\nI. INTRODUCTION\nSpin-orbitronics1,2has attracted a lot of attention re-\ncently as a new subfield of spintronics3,4in which the\nrelativistic spin-orbit interaction (SOI) plays a central\nrole. Spin-orbitronics includes generation and detec-\ntion of spin-polarized currents through the spin Hall\neffect,5,6the induction of non-equilibrium spin accu-\nmulations in non-magnetic materials through the Edel-\nstein effect,7,8the triggering of magnetization dynam-\nics in single magnetic systems through spin-orbit torques\n(SOTs),9–11and magnonic charge pumping by means of\ninverseSOTs.12Spin-orbitronicsis believed to ultimately\nenable the faster and more efficient ways of magnetiza-\ntion switching needed for high density data storage and\ninformationprocessing,thereby providingnovelsolutions\nto address the essential challenges of spintronics. In this\npaper we investigate the microscopic origin of SOTs in\na two-dimensional (2D) metallic ferromagnet with spin-\norbit coupling.\nThe magnetization dynamics in ferromagnets is gov-\nerned by the seminal Landau-Lifshitz-Gilbert (LLG)\nequation,13–15\n∂m\n∂t=−γm×Heff+αGm×∂m\n∂t+T,(1)\nwheremis a unit vector along the magnetization di-\nrection|m|= 1,γis the gyromagnetic ratio, αGis the\nGilbert damping constant and Heffis an effective field\nwhich includes the effects of the external magnetic field,\nexchange interactions, and dipole and anisotropy fields.\nThe first term on the right-hand side of Eq. (1) describes\nthe precession of the magnetization vector maround\nthe effective field, while the second term describes the\nrelaxation of magnetization to its equilibrium orienta-\ntion. Furthermore, Tis a sum of different magnetization\ntorques not contained in the effectieve field or damping.\nThe spin-polarized current-induced magnetization dy-\nnamics in magnetic materials arises as a result of spin\ntransfer torque (STTs).13–15It is well known that STTmay induce magnetization dynamics in spin-valve struc-\ntures, and that the exchange interaction between the\nspin-polarized current and local spins leads, e.g., to\ndomain-wall motion. In uniformly magnetized single-\ndomain systems the transfer of spin angular momen-\ntum from the spin-current density jsto a local magne-\ntization is modelled by two different STT terms: i) an\nanti-damping-like (ADL) or Slonczewski in-plane torque\nT∝m×m×js, and ii) an out-of-plane field-like (FL)\ntorqueT∝m×js, which is typically negligible in con-\nventional metallic spin valves. On the other hand, in\nferromagnets with magnetic domains, in which spin tex-\ntures such as domain walls are necessarily present, the\nSTT also includes reactive, T∝(js·∇)m, and dissipa-\ntive,T∝m×(js·∇)m, torques.13–15\nRecently, it was demonstrated both theoretically and\nexperimentally that the current-induced nonequilibrium\nspin polarization7,8in (anti-)ferromagnets with inver-\nsion asymmetry may exert a so-called SOT on local-\nized spins and, consequently, may lead to a non-trivial\nmagnetization dynamics.16–32Unlike STT, the SOT phe-\nnomenon does not require an injection of spin current or\nthe presence of spatial inhomogeneities in the magneti-\nzation. The magnetization switching due to SOTs may\nbe achieved with current pulses as short as ∼180ps,\nwhile the critical charge current density can be as low as\n∼107Acm−2.29\nQuite generally Rashba SOTs can be classified as ei-\nther ADL or FL torques33. The first theoretical and ex-\nperimental studies of SOT have demonstrated that the\nADL SOT is proportional to the disorder strength and\ncan always be regarded as a small correction to the FL\nSOT.16–23Ontheotherhand, insomerecentexperiments\nthe opposite statement is made: the torques with ADL\nsymmetry are more likely to be the main source of the\nobservedmagnetization behavior.26–28,34–37These exper-\niments are performed with ferromagnetic metals grown\non top of a heavy metal with strong SOI and may, in\nprinciple, be explained by the spin Hall effect which in-\nduces a spin-polarized current. This spin current, in\nturn, exerts a torque on the magnetic layer via the STT2\nmechanism35,36,38so that the ADL symmetry term plays\nthe major role in the effect as discussed above.\nIt is, however, a serious experimental challenge to dis-\ntinguish between SOT and spin-Hall STT in bilayers,\nsince both torques have the same symmetry.35,36Very\nrecently Kurebayashi et al.39conducted an experiment\non the bulk of strained GaMnAs, which has an intrinsic\ncrystalline asymmetry. In these experiments, the contri-\nbution of a possible spin-Hall-effect STT was completely\neliminated, while sizable ADL torques were nevertheless\ndetected. This provides a strong argument in favour of\nthe ADL-SOT nature of the observed torque. The au-\nthorsofRef. 39attribute this torque toan intrinsic Berry\ncurvature, andestimate a scattering-independent, i.e. in-\ntrinsic, ADL-SOT.39–41This intrinsic ADL-SOT has also\nbeen reported by van der Bijl and Duine.33\nIn this paper we calculate both FL- and ADL-SOTs in\na 2D Rashba ferromagnetic metal microscopically by us-\ning a functional Keldysh theory approach.42By calculat-\ning the first vertex correction we show that the intrinsic\nADL-SOT vanishes unless the impurity scattering is spin\ndependent.\nThe rest of this paper is organized as follows. Section\nII introduces the model and method. In Sec. III we\ncalculate SOTs with and without vertex corrections. We\nconclude our work in Sec. IV.\nII. MODEL HAMILTONIAN AND METHOD\nWe start with the 2D mean-field Hamiltonian ( /planckover2pi1=c=\n1),\nH[ψ†,ψ] =/integraldisplay\nd2rψ†\nr,t/bracketleftBig\nH0+Vimp+ˆj·At/bracketrightBig\nψr,t.(2)\nwhereψ†= (ψ∗\n↑,ψ∗\n↓) is the Grassman coherent state\nspinor. Here, H0is the 2D conducting ferromagnet\nHamiltonian density in the presence of Rashba SOI,43\nH0=p2\n2me+αR(σ׈z)·p−1\n2∆σ·nr,t−1\n2∆Bσz,(3)\nwherepisthe2Dmomentumoperator, αRisthestrength\nof the SOI, ∆ and ∆ Bare the exchange energy and\nthe Zeeman splitting due to an external field in the z-\ndirection, respectively, nr,tis an arbitrary unit vector\nthat determines the quantizationaxis, and σis the three-\ndimensional vector of Pauli matrices.\nThe vector potential At=Ee−iΩt/iΩ is included in\nEq. (2) to model a dcelectric field in the limit Ω →0.\nIt is coupled to the current density operator, which is\ngiven by ˆj= (ie/2me)(← −∇−− →∇)−eαRσ׈z, whereeis\nthe electron charge and meis the electron effective mass.\nFinally, the impurity potential Vimpis of the form\nVimp(r) =/parenleftbigg\nV↑0\n0V↓/parenrightbigg/summationdisplay\niδ(r−Ri), (4)whereV↑(↓)is the strength of spin-up (down) disorder,\nand the index ilabels the impurity centers Ri. More\nspecifically, we restrict ourselves to the gaussian limit of\nthe disorder potential.\nThe impurity-averaged retarded Green’s function in\nthe Born approximation is given by44–46\nG+\nk,ε=/parenleftBig\ng−1\n↓σ↑+g−1\n↑σ↓+αR(σykx−σxky)/parenrightBig−1\n.(5)\nwhereg−1\ns=ε−εk+sM+iγs, fors=↑(+) or↓(−),\nσs= (σ0+sσz)/2,kandεarethewavevectorandenergy,\nrespectively, εk=k2/2me, andM= (∆ + ∆ B)/2. We\nhave also introduced the spin-dependent scattering rate\nγs=πνnimpV2\ns, whereν0=me/2πis the density of\nstates per spin for 2D electron gas, and nimpdenotes the\nimpurity concentration. Here wehaveassumedthat both\nspin-orbit split bands are occupied, i.e. the Fermi energy\nis larger than magnetization splitting, εF>M.\nFollowing Ref. 42 we minimize the effective action on\nthe Keldysh contour with respect to quantum fluctua-\ntions ofn. This procedure gives us directly the LLG\nequation which contains torque terms in linear response\nwith respect to the external field E. The effective action\nis given by S=/integraltext\nCKdtLF(t), whereCKstands for the\nKeldysh contour and LF(t) =/integraltext\nd2r(ˆψ†\nr,ti∂\n∂tˆψr,t−H) is\nthe mean-field Lagrangian.\nWe further assume that we are dealing with a ferro-\nmagnetic metal which is uniformly magnetized in the z-\ndirection. Thus, we can approximate the vector nas\nnr,t≃\nδnx\nr,t\nδny\nr,t\n1−1\n2(δnx\nr,t)2−1\n2(δny\nr,t)2)\n.(6)\nIn order to derive the LLG equation with torque terms\nit is sufficient to expand the effective action up to second\norder inδnand up to first order in the vector poten-\ntial:Seff=SSOT[O(δn),A] +Srest[O(δn2),A= 0]. A\nstraightforward calculation gives\nSSOT=/integraldisplay\nCKdt/integraldisplay\nCKdt′/integraldisplay\nd2r/integraldisplay\nd2r′χa;r−r′;t,t′δna\nr′,t′,(7)\nwhereχa(a={x,y}) is the response function,\nχa;r−r′;t,t′=i∆\n4/angbracketleftBig\njr,tψ†\nr′,t′σaψr′,t′/angbracketrightBig\n·At,(8)\nandj=ψ†ˆjψis the charge current density. Note that in\nthe absence of SOI the term SSOTis 0 and only second\norder terms, Srest42, remain. The field δncan be split\ninto the physical magnetization field δmand a quantum\nfluctuation field ξasδna\nr,t±=δma\nr,t±ξa\nr,t/2, where +\ncorresponds to the upper and −to the lower branch of\nthe Keldysh contour. At first order with respect to the\nquantum component we obtain\nSSOT=/integraldisplay\ndt/integraldisplay\ndt′/integraldisplay\nd2r′/integraldisplay\nd2rχ−\na;r−r′;t,t′ξa\nr′,t′,(9)3\n(b) aV(a) \nbb\ntcctctccc\nttc tAj\nAj\naVkk\n1k1k\n2k2k\nFIG. 1: Feynman diagrams related to the spin-torque re-\nsponse function Eq. (8): (a) undressed response function,\nand (b) the first vertex correction. The solid line correspon ds\ntoan electron propagator in theBorn approximation, the wig -\ngly line to the coupling to vector potential and current, and\nthe dashed represents a spin fluctuation. The vertical dotte d\nline describes the averaging over impurity positions.\nwhereχ−is the advanced component of the correla-\ntor, and the sum over repeated indices ais assumed.\nThe LLG equation is, then, derived by minimizing the\neffective action with respect to quantum fluctuations,\nδSeff/δξ= 0. Thus, the transverse components of the\nLLG equation in the Fourier space are given by,\nF/bracketleftbiggδSrest\nδξa/bracketrightbigg\nq=0,ε+χ−\na;q=0,ε=0= 0,(10)\nwhereF[...] represents the Fourier transformation op-\nerator. The functional derivative in Eq. (10) gives\nthe precession and Gilbert damping terms of the LLG\nequation,42while the second term describes the SOT.\nThe dependence of Gilbert damping on SOI is second\norder inαR,33and we focus below on SOT which is of\nfirst order in αR. The appearance of the zero-momentum\nresponse function χa;q=0,ε=0in the LLG equation shows\nthat the SOT is finite even for spatially uniform magne-\ntization, in contrast to the (non-)adiabatic STT which is\nof the first order in the gradient of magnetization.\nIII. CALCULATION OF SOTS\nIn what follows we evaluate the spin-torque response\nfunction of Eq. (8), shown diagrammatically in Fig. 1, to\nderive the SOT in the ballistic limit γs≪kBT, where\nkBTis the thermal energy. We calculate first the bare\n(undressed) part of the response function, χ(0), depicted\nin Fig. 1a. The final result for spin torque is, then,\nobtained by adding the first vertex correction, χ(1), de-\npicted in Fig. 1b. Throughout the calculation we assume\nthatγs≪kBT≪αRkF≪M, wherekFis the Fermi\nwavevector. The condition αRkF≪Mis normally ful-\nfilled in the metallic ferromagnets of interest. Whether\nor not the condition γs≪kBT≪αRkFis fulfilled de-\npends strongly on the sample quality. The analysis ofspin torques in diffusive regime kBT≪γswill require\ncalculation of the full vertex correction and will be done\nelsewhere.\nA. Undressed response function\nThe spin-torque response function of Eq. (8) without\nvertex corrections is given by\nχ(0)\na;t,t′=e∆\n4i/integraldisplayd2k\n(2π)2Tr[vkˇGk;t,t′σaˇGk;t′,t]·At.(11)\nwherevk=k/me−αRσ׈zis the velocity vector, and\nˇGis the Green’s function on the Keldysh contour. From\nEq. (11) we find retarded and advanced components of\nthe response function in the limit of zero frequency and\nmomentum as\nχ(0)±\na=e∆\n4ilim\nΩ→0/integraldisplayd2k\n(2π)2/integraldisplay/integraldisplay\ndωdω′fω′−fω\nΩ+ω′−ω±i0\n×1\nΩTr[(vk·E)Ak,ωσaAk,ω′], (12)\nwhereAk,ω=i(G+\nk,ω−G−\nk,ω)/2πis the spectral func-\ntion andfω= [e(ω−ǫF)/kBT+ 1]−1stands for the Fermi\ndistribution function.\nIn the limit of weak disorder, we can decompose the\nresponse function into two parts: the intrinsic part χin,\nwhich turns out not to depend on the scattering rate and\ndescribesinterbandtransitionsandtheextrinsicpart χex,\nwhich essentially depends on disorderand correspondsto\nintraband contributions. The intrinsic part corresponds\nto the principal value integration in Eq. (12), while\nthe extrinsic part is given by the corresponding delta-\nfunction contribution. To leading order in αRwe find\nχ(0)−\nin,a=eαR∆\n8Mν0Ea, (13a)\nχ(0)−\nex,a=eαR∆\n8Mν0/bracketleftbiggεF−M\nγ↓−εF+M\nγ↑/bracketrightbigg\n(ˆz×E)a.(13b)\nThe correspondingexpressionsforthe SOTsarethe ADL\nTADLand FLTFLcontributions, which do not take into\naccount vertex corrections,\nT(0)\nADL=−2eαRν0m×m×(ˆz×E), (14a)\nT(0)\nFL=−eαR∆ν0\nM/bracketleftbiggεF+M\nγ↑−εF−M\nγ↓/bracketrightbigg\nm×(ˆz×E).(14b)\nHence, we find that the ADL SOT in the absence of ver-\ntex corrections has an intrinsic origin, i.e., is disorder-\nindependent.\nB. Vertex correction\nLet us now turn to the calculation of the first vertex\ncorrection to the spin-torque response function depicted4\nin Fig. 1b. For the corresponding response function on\nthe Keldysh contour we find\nχ(1)\na;t,t′=e∆\n4i/integraldisplaydk1\n(2π)2/integraldisplaydk2\n(2π)2/integraldisplay\ncKdt1/integraldisplay\ncKdt2Tr[At·vk1\nסGk1;t,t1∝an}bracketle{tVimpˇGk2;t1,t′σaˇGk2;t′,t2Vimp∝an}bracketri}htˇGk1;t2,t].(15)\nThe advanced component of χ(1)at zero energy and mo-\nmentum is, then, given by\nχ(1)−\na=e∆\n4iηb/integraldisplayd2k1\n(2π)2/integraldisplayd2k2\n(2π)2/integraldisplay/integraldisplay\ndωdω′fω′−fω\nΩ+ω−ω′−i0\n×1\nΩTr/bracketleftbig\nE·vk1/parenleftbig\nG+\nk1,ωσbAk2,ωσaG+\nk2,ω′σbAk1,ω′\n+G+\nk1,ωσbAk2,ωσaAk2,ω′σbG−\nk1,ω′\n+Ak1,ωσbG−\nk2,ωσaG+\nk2,ω′σbAk1,ω′\n+Ak1,ωσbG−\nk2,ωσaAk2,ω′σbG−\nk1,ω′/parenrightbig/bracketrightbig\n,(16)\nwhere the summation over the index b={0,z}and the\nlimit Ω→0 are assumed. We have also used the nota-\ntionsη0=nimp(V↑+V↓)2/4 andηz=nimp(V↑−V↓)2/4.\nUsing the same approximationsas for the undressed part\nof the response function we obtain the intrinsic contribu-\ntion as\nχ(1)−\nin,a=−eαR∆\n8Mν0γ↑+γ↓\n2(γ↑γ↓)1\n2Ea, (17)\nwhile the corresponding extrinsic contribution is of the\nsecond order in scattering rates and can be neglected.\nThus, we obtain the FL and ADL torques in the limit\nγs≪αRkF≪Mto the leading order in the SOI as\nTFL=eαRν0∆\nM/bracketleftbiggεF−M\nγ↓−εF+M\nγ↑/bracketrightbigg\nm×(ˆz×E),(18)\nTADL=/bracketleftbiggγ↑+γ↓\n2√γ↑γ↓−1/bracketrightbigg\n2eαRν0m×(m×(ˆz×E)).(19)\nThese expressions provide the main result of this paper.\nIV. CONCLUSIONS\nThe SOTmechanismis basedonthe exchangeofangu-\nlar momentum between the crystal lattice and the local\nmagnetization via spin-orbit coupling. Here, we foundthe FL- and ADL-SOTs microscopically, Eqs. (18) and\n(19). The FL-SOT originates from the Fermi surface\ncontribution of the response function Eq. (8), while\nthe ADL-SOT is acquires contributions from the entire\nbands. Our main result in Eq. (19) immediately shows\nthattheintrinsiccontributiontoADL-SOTiscompletely\ncanceled in the presence of spin-independent scattering\nγ↑=γ↓. That is, the intrinsic component of the ADL\nSOT, which originates from virtual interbranch transi-\ntions, is canceled by the vertex correction when weak\nspin-independent impurity scattering is taken into ac-\ncount. Our result, therefore, explicitly elucidates the\ninterplay between intrinsic and extrinsic contributions\nto ADL SOT. This result resembles the suppression of\nboth spin Hall conductivity in nonmagnetic metals and\nanomalous Hall conductivity in magnetic metals, in the\npresence of spin-independent disorder.44–48In these ef-\nfects the cancelation is model dependent, and occurs for\nparabolic band dispersion and linear-in-momentum SOI.\nWe expect a similar scenario for intrinsic SOT.\nThe existence of a Rashba effect on the interface be-\ntween an ultrathin ferromagnet and a heavy metal is\nthe subject of intense discussion. Our results show\nthat the amplitudes of the FL and ADL SOTs can be\nof the same order of magnitude depending on the rel-\native strengths of the SOI, spin-dependent scattering\nrates, andexchangeinteraction. Ourresultsmayqualita-\ntively describe Co/Pt interfaces which are characterised\nby particularly large Rashba SOI of the magnitude of\n1eV˚A.26–28,49Relating the strength of the Rashba cou-\npling to the magnitude of the SOTs, however, would re-\nquire ab-initio modeling and additional experimental in-\nformation. Which of our results apply to more general\nmodels and band structures will be the subject of future\ninvestigation.\nACKNOWLEDGEMENT\nWe acknowledge Hiroshi Kohno and Dmitry Yudin for\nuseful discussions. The work was supported by Dutch\nScience Foundation NWO/FOM 13PR3118 and by EU\nNetwork FP7-PEOPLE-2013-IRSES Grant No 612624\n”InterNoM”. R.D. isamemberofthe D-ITPconsortium,\na program of the Netherlands Organisation for Scientific\nResearch(NWO), which is funded by the Dutch Ministry\nof Education, Culture and Science (OCW).\n1A. Manchon, Nature Phys. 10, 340 (2014).\n2T. Kuschel and G. Reiss, Nat. Nanotechnol. 10, 22 (2015).\n3J. Sinova and I. ˇZuti´ c, Nat. Mater. 11, 368 (2012).\n4I.ˇZuti´ c, J. Fabian, and S. Das Sarma, Rev. Mod. Phys.\n76, 323 (2004).\n5J. Sinova, S.O. Valenzuela, J. Wunderlich, C.H. Back,\nT. Jungwirth, arXiv:1411.3249 (2014).6N. Nagaosa, J. Sinova, S. Onoda, A.H. MacDonald, and\nN.P. Ong, Rev. Mod. Phys. 82, 1539 (2010).\n7V.M. Edelstein, Solid State Commun. 73, 233 (1990).\n8J.-i. Inoue, G.E.W. Bauer, and L.W. Molenkamp, Phys.\nRev. B67, 033104 (2003).\n9P. Gambardella and I.M. Miron, Phil. Trans. R. Soc. A\n369, 3175 (2011).5\n10A. Brataas and K.M.D. Hals, Nat. Nanotechnol. 9, 86\n(2014).\n11K.M.D. Hals and A. Brataas, Phys. Rev. B 88, 085423\n(2013).\n12C. Ciccarelli, K.M.D. Hals, A. Irvine, V. Novak,\nY. Tserkovnyak, H. Kurebayashi, A. Brataas, and A. Fer-\nguson, Nat. Nanotechnol. 10, 50 (2015).\n13N. Locatelli, V. Cros and J. Grollier, Nat. Mater. 13, 11\n(2013).\n14A. Brataas, A.D. Kent, and H. Ohno, Nat. Mater. 11, 372\n(2012).\n15D.C. Ralphand M.D. Stiles, J. Magn, Magn, Matter. 320,\n1190 (2008).\n16B.A. Bernevig and O. Vafek, Phys. Rev. B 72, 033203\n(2005).\n17A. Manchon and S. Zhang, Phys. Rev. B 78, 212405\n(2008);79, 094422 (2009).\n18A. Matos-Abiague and R.L. Rodr´ ıguez-Su´ arez, Phys. Rev.\nB80, 094424 (2009).\n19I. Garate, A.H. MacDonald, Phys. Rev. B 80, 134403\n(2009).\n20X. Wang and A. Manchon, Phys. Rev. Lett. 108, 117201\n(2012).\n21K.-W. Kim, S.-M. Seo, J. Ryu, K.-J. Lee, and H.-W. Lee,\nPhys. Rev. B 85, 180404(R) (2012).\n22D.A.Pesin andA.H.MacDonald, Phys.Rev.B 86, 014416\n(2012).\n23A. Chernyshov, M. Overby, X. Liu, J.K. Furdyna,\nY. Lyanda-Geller, and L.P. Rokhinson, Nature Phys. 5,\n656 (2009).\n24I.M. Miron, T. Moore, H. Szambolics, L.D. Buda-\nPrejbeanu, S. Auffret, B. Rodmacq, S. Pizzini, J. Vogel,\nM. Bonfim, A. Schuhl, and G. Gaudin, Nat. Mater. 10,\n419 (2011).\n25A.V. Khvalkovskiy, V. Cros, D. Apalkov, V. Nikitin,\nM. Krounbi, K. A. Zvezdin, A. Anane, J. Grollier, A. Fert,\nPhys. Rev. B 87, 020402(R) (2013).\n26I.M. Miron, G. Gaudin, S. Auffret, B. Rodmacq,\nA. Schuhl,S. Pizzini, J. Vogel, and P. Gambardella, Nat.\nMater.9, 230 (2010).\n27I.M. Miron, K. Garello, G. Gaudin, P.-J. Zermatten,\nM.V. Costache, S. Auffret, S. Bandiera, B. Rodmacq,\nA. Schuhl, and P. Gambardella, Nature 476, 189 (2011).\n28K. Garello, I.M. Miron, C.O. Avci, F.Freimuth,\nY. Mokrousov, S. Bl¨ ugel, S. Auffret, O. Boulle, G. Gaudin,\nand P. Gambardella, Nat. Nanotechnol. 8, 587 (2013).\n29K. Garello, C.O. Avci, I.M. Miron, M. Baumgartner,\nA. Ghosh, S. Auffret, O. Boulle, G. Gaudin, and P. Gam-\nbardella, Appl. Phys. Lett. 105, 212402 (2014).\n30J.ˇZelezn´ y, H. Gao, K. V´ yborn´ y, J. Zemen, J. Maˇ sek,\nA. Manchon, J. Wunderlich, J. Sinova, and T. Jungwirth,Phys. Rev. Lett. 113, 157201 (2014).\n31P. Wadley, B. Howells, J. ˇZelezn´ y, C. Andrews, V. Hills,\nR.P. Campion, V. Nov´ ak, F. Freimuth, Y. Mokrousov,\nA.W. Rushforth, K.W. Edmonds, B.L. Gallagher,\nT. Jungwirth, arXiv: 1503.03765 (2015).\n32J. Linder, Phys. Rev. B 87, 054434 (2013).\n33E. van der Bijl and R. A. Duine, Phys. Rev. B 86, 094406\n(2012).\n34L. Liu, C.-F. Pai, Y. Li, H.W. Tseng, D.C. Ralph, and\nR.A. Buhrman, Science 336, 555 (2012).\n35J. Kim, J. Sinha, M. Hayashi, M. Yamanouchi, S. Fukami,\nT. Suzuki, S. Mitani, and H. Ohno, Nat. Mater. 12, 240\n(2013).\n36X. Fan, H. Celik, J. Wu, C. Ni, K.-J. Lee, V.O. Lorenz,\nand J.Q. Xiao, Nat. Commun. 5, 3042 (2014).\n37G. Yu, P. Upadhyaya, Y. Fan, J.G. Alzate, W. Jiang,\nK.L. Wong, S. Takei, S.A. Bender, L.-T. Chang, Y. Jiang,\nM. Lang, J. Tang, Y. Wang, Y. Tserkovnyak, P.K. Amiri,\nand K.L. Wang, Nat. Nanotechnol. 9, 548 (2014); I.\nM. Miron, ibid.9, 502 (2014).\n38P.M. Haney, H.-W. Lee, K.-J. Lee, A. Manchon, and\nM.D. Stiles, Phys. Rev. B 87, 174411 (2013).\n39H. Kurebayashi, J. Sinova, D. Fang, A.C. Irvine,\nT.D. Skinner, J. Wunderlich, V. Nov´ ak, R.P. Campion,\nB.L. Gallagher, E.K. Vehstedt, L.P. Zˆ arbo, K. V´ yborn´ y,\nA.J. Ferguson, and T. Jungwirthn, Nat. Nanotechnol. 9,\n211 (2014).\n40H. Li, H. Gao, L.P. Zˆ arbo, K. V´ yborn´ y, X. Wang,\nI. Garate, F. Doˇ gan, A. ˇCejchan, J. Sinova, T. Jungwirth,\nA. Manchon, Phys. Rev. B 91, 134402 (2015).\n41F. Freimuth, S. Bl¨ ugel, and Y. Mokrousov, Phys. Rev. B,\n90, 174423 (2014).\n42R.A. Duine, A.S. N´ u˜ nez, J. Sinova, and A.H. MacDonald,\nPhys. Rev. B 75, 214420 (2007).\n43D. Bercioux and P. Lucignano, arXiv: 1502.00570 (2015).\n44T. Kato, Y. Ishikawa, H. Itoh, and J.-i. Inoue, New J.\nPhys.9, 350 (2007).\n45C.P. Moca and D.C. Marinescu, New J. Phys. 9, 343\n(2007).\n46T.S. Nunner, N.A. Sinitsyn, M.F. Borunda,\nV.K. Dugaev, A.A. Kovalev, Ar. Abanov, C. Timm,\nT. Jungwirth, J.-i. Inoue, A.H. MacDonald, and\nJ. Sinova, Phys. Rev. B 76, 235312 (2007).\n47J.-i. Inoue, T. Kato, Y. Ishikawa, H. Itoh, G.E.W. Bauer,\nand L.W. Molenkamp, Phys. Rev.Lett. 97, 046604 (2006).\n48A. Sakai, PhD Thesis (Osake University, 2014); A. Sakai\nand H. Kohno (unpublished).\n49J.-H. Park, C.H. Kim, H.-W. Lee, and J.H. Han, Phys.\nRev. B87, 041301(R) (2013)."    },    {        "title": "2103.04960v1.Spin_orbital_model_for_fullerides.pdf",        "content": "Spin-orbital model for fullerides\nRyuta Iwazaki and Shintaro Hoshino\nDepartment of Physics, Saitama University, Shimo-Okubo, Saitama 338-8570, Japan\n(Dated: March 9, 2021)\nThe multiorbital Hubbard model in the strong coupling limit is analyzed for the e\u000bectively antifer-\nromagnetic Hund's coupling relevant to fulleride superconductors with three orbitals per molecule.\nThe localized spin-orbital model describes the thermodynamics of the half-\flled (three-electron)\nstate with total spin-1/2, composed of singlon and doublon placed on the two of three orbitals.\nThe model is solved using the mean-\feld approximation and magnetic and electric ordered states\nare clari\fed through the temperature dependences of the order parameters. Combining the model\nwith the band structure from ab initio calculation, we also semi-quantitavely analyze the realistic\nmodel and the corresponding physical quantities. In the A15-structure fulleride model, there is an\nantiferromagnetic ordered state, and subsequently the two orbital ordered state appears at lower\ntemperatures. It is argued that the origin of these orbital orders is related to the Thpoint group\nsymmetry. As for the fcc-fulleride model, the time-reversal broken orbital ordered state is identi\fed.\nWhereas the spin degeneracy remains in our treatment for the geometrically frustrated lattice, it is\nexpected to be lifted by some magnetic ordering or quantum \ructuations, but not by the spin-orbital\ncoupling which is e\u000bectively zero for fullerides in the strong-coupling regime.\nI. INTRODUCTION\nStrongly correlated electron systems with multiple or-\nbital degrees of freedom show a variety of intriguing\nphenomena, and are realized in a wide range of materi-\nals such as iron-pnictides, heavy-electron materials, and\nmolecular-based organic materials. The alkali-doped ful-\nlerides are also the typical cases where the strong corre-\nlation e\u000bects with multiorbitals are relevant. This mate-\nrial has been attracting attention recent years for a lot of\nexperimental \fndings. The superconductivity with the\nhigh transition temperature \u001840K is one of the charac-\nteristic feature [1{8]. While the mechanism is identi\fed\nas the electron-phonon interaction [9{11], the supercon-\nducting dome in the temperature-pressure phase diagram\nis found to be located near the Mott insulator and an-\ntiferromagnetic phase, featuring the typical behaviors of\nthe strongly correlated superconductors [7, 12{14]. In the\nMott insulating phase, the localized electrons form a low-\nspin state and the imbalance of the occupancy in orbitals\nlead to the deformation of the fullerene molecule because\nof the coupling between electrons and anisotropic molec-\nular distortions (Jahn-Teller phonon). Interestingly, such\nbehavior can also be seen in the metallic phase near the\nMott insulator but is absent far away from it [15, 16].\nThis anomalous behavior is called the Jahn-Teller metal\nwhere the multiorbital degrees of freedom play an im-\nportant role. The fullerides are also crystallized on the\nsubstrate and the characteristic asymmetry between elec-\ntron and hole doping is identi\fed [17, 18]. Furthermore,\na possible superconducting state has been discussed un-\nder the excitation by light above the transition temper-\nature [19, 20]. Thus, the fulleride materials have been\nproviding the intriguing phenomena up until recently.\nThe alkali-doped fullerides are the systems with triply\ndegenerate t1umolecular orbitals which resembles atomic\np-electrons in nature. There, the Hund's coupling,\nwhich is usually acting ferromagnetically on the elec-trons located at the di\u000berent orbitals, is e\u000bectively an-\ntiferromagnetic due to the coupling to the anisotropic\nmolecular vibrations [10, 21, 22] and is crucial for the\nlow-temperature physics. The multiorbital Hubbard\nmodel with the antiferromagnetic Hund's coupling has\nbeen studied theoretically, and the various phase dia-\ngrams are clari\fed using the dynamical mean-\feld the-\nory suitable for the description of the electronically or-\ndered states [21{29]. The Jahn-Teller metal has been\ninterpreted as the spontaneous orbital selective Mott\nstate [26, 30] which is an unconventional type of orbital\norder. The orbital asymmetric feature has also been re-\nported in two-dimensional fullerides by using the many-\nvariable variational Monte Carlo method [31].\nWith the antiferromagnetic Hund's coupling, one of\nthe intra-molecular interaction, pair hopping, plays an\nimportant role: it activates the dynamics of the double\noccupancy in an orbital (doublon). In order to clarify the\ncharacters of the existing fulleride materials in detail, we\nfocus our attention on the Mott insulating phase, where\nthe doublon physics can be tackled with reasonable com-\nputational cost even in the realistic situation. As is well\nknown, for a single-orbital case, the electronic behav-\niors in the strong coupling regime are determined by the\nHeisenberg model of localized electrons. The extension\nof the Heisenberg model to the multiorbital system is\nknown as the Kugel-Khomskii model which has been de-\nrived for the ferromagnetic Hund's coupling [32, 33] and\ndescribes the degrees of freedom of the spin and orbital.\nThe spin-orbital models have been applied to the egor\nt2gorbital system [34{37]. On the other hand, the ful-\nlerides have antiferromagnetic Hund's coupling, so that\ntheir strongly correlated e\u000bective model di\u000bers from the\nusual Kugel-Khomskii model. While the localized model\nwith antiferromagnetic Hund's coupling have been con-\nstructed for a density-density type interaction [28], here\nwe deal with more complicated but realistic situations.\nIn this paper, we develop the localized spin-orbitalarXiv:2103.04960v1  [cond-mat.str-el]  8 Mar 20212\nmodel for the system with antiferromagnetic Hund's cou-\npling. We analyze both the symmetric model and the re-\nalistic model for fullerides, the former of which is easier\nto interpret the results and is useful as a reference. By\nusing the mean \feld theory, for the spherical model on\na bipartite lattice, we obtain the staggered magnetic or-\ndered state, and also the uniform orbital ordered state at\nlower temperature regime. This orbital ordered state is\nnot characterised by the ordinary orbital moment but by\nthe doublon's orbital moment. In the A15 fulleride e\u000bec-\ntive model, which is bipartite lattice, we reveal that there\nare two kinds of orbital ordered states below the antifer-\nromagnetic transition temperature. The obtained orbital\nordered states are interpreted as related to an e\u000bective\nrecovery of the four-fold symmetry at low temperatures\nin theThpoint group. We also analyze the geometri-\ncally frustrated fcc fulleride model seeking for a spatially\nuniform ordered state. We reveal that the fcc model has\nthe time-reversal symmetry broken orbital ordered state,\nwhere the spin ordered state is absent since the spin-orbit\ncoupling on the fullerene molecule is e\u000bectively zero.\nThis paper is organized as follows. We discuss the con-\nstruction of strongly correlated e\u000bective models and the\ntheoretical method in Sec. II. In Sec. III, we show numeri-\ncal results for the model with isotropic hopping (spherical\nmodel introduced in Sec. III A). Section IV provides nu-\nmerical results for the spin-orbital model combined with\nA15 and fcc fulleride band structure. We summarize the\nresults in Sec. V.\nII. CONSTRUCTION OF MODELS\nA. Three orbital Hubbard model in\nstrong-coupling limit\nLet us begin with the three-orbital Hubbard model\nH=Ht+HU; (1)\nHt=\u0000X\ni6=j;\r;\r0;\u001bt\r\r0\nijcy\ni;\r;\u001bcj;\r0;\u001b; (2)\nHU=U\n2X\ni;\r;\u001b;\u001b0cy\ni;\r;\u001bcy\ni;\r;\u001b0ci;\r;\u001b0ci;\r;\u001b\n+U0\n2X\ni;\r6=\r0;\u001b;\u001b0cy\ni;\r;\u001bcy\ni;\r0;\u001b0ci;\r0;\u001b0ci;\r;\u001b\n+J\n2X\ni;\r6=\r0;\u001b;\u001b0\u0010\ncy\ni;\r;\u001bcy\ni;\r0;\u001b0ci;\r;\u001b0ci;\r0;\u001b\n+cy\ni;\r;\u001bcy\ni;\r;\u001b0ci;\r0;\u001b0ci;\r;\u001b\u0011\n;(3)\nwhereci;\r;\u001b (cy\ni;\r;\u001b) is an annihilation (creation) opera-\ntor at siteiof fullerenes with the t1umolecular orbital\nindex\r=x;y;z and spin\u001b=\";#. We deal with the\nHilbert space with a \fxed number of electrons. We as-\nsume the condition U0=U\u00002Jfor the local interaction\npart in the following discussion, which is valid for the\n\t\u0001\u0001\u0001\u0001\u0001\n\f\n\t\u0001\u0001\u0001\u0001\u0001\n\f\t\u0001\u0001\u0001\u0001\u0001\n\f\n\u001bTJOHMPO\u001bEPVCMPO<latexit sha1_base64=\"anJOgOHXlpTDDY0YhXC+bSAHO3s=\">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</latexit>1p2\n<latexit sha1_base64=\"anJOgOHXlpTDDY0YhXC+bSAHO3s=\">AAAJGniclVbJbtRAEK0QFhOWLFyQuESMguAS9UQIEKco5AAXlIUsUmY0sj09xhq3bbo9Q4LjH+AHOICQQEKA+Au4gLjCIZ+AOAYJDhyorvEkIfGS2JJdXV3vdfXr7rKt0HNVxNjWwLHB4ydOnjJOD505e+788Mjo2LIKOtLmS3bgBXLVMhX3XJ8vRW7k8dVQclNYHl+x2nd0/0qXS+UG/oNoI+R1YTq+23JtM0JXY2Ss1pKmHVeTuKYeySieSpLGSIVNMrrGDxrV1KhAes0Fo4N/oAZNCMCGDgjg4EOEtgcmKLzXoAoMQvTVIUafRMulfg4JDCG2g1EcI0z0tvHpYGst9frY1pyK0OsYESAyQJYExmGCfWfv2Tb7zD6wH+xvLltMLDqbDXxbPSwPG8NPLy7+LkUJfEfwcBdVgLAwOn9Wul+i3UY7KowTpE8LoySq5eDoRfFarQijb5FKLqoWkkfrZ/fy7j55tr14e2EivsJes5+o2yu2xT6hcn73l/1mni88J3YfMY9JZUHz9nFlY/SrNPOE1rNOvn6GV5G/gt5x9O1mey3NN49RFDDWsJXFWcPoYlavgHUtk7NemmfW3EXm7MWR5p+lgMjRQBxJBZGpg8hUQuzTIp/VRE6Vw7iZw7hZkqlEv4ksScrXO2kSrRgWqEIU411qOZn4e3QWi/H6JPmZ6PKx75MiHqms2xbhalSz+v5ihtl9IyuMkeiZLUW2qYrF+G5k5D5D+GKGJmrXwjtr7s1D6d7JxLqH0ryH7NL3QLcV4nRd90uwDq1pzzLTvmLE7u7SVoh5liHUzn5StI/Lx+js7IEOVWaJd4CRZSvQRzUp2j8C0tlB6t2vvwGytDL2R/PI6+z5Auefzj5G7sXgP0J1/x/BQWN5arJ6Y5LNX69Mz6R/CwZcgstYMatwE6bhLszBEo66Di/hLbwzXhgfjS/G117osYEUcwH+u4xv/wDl3uPJ</latexit>1p2\n<latexit sha1_base64=\"anJOgOHXlpTDDY0YhXC+bSAHO3s=\">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</latexit>1p2<latexit sha1_base64=\"z7ibayWUbkOPH61LaKgFQQM9KO8=\">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</latexit>|\u0000,\u0000ii\n<latexit sha1_base64=\"UH9Wfb7wXhWadywsL8z8ujLLONA=\">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</latexit>|x,\"ii\n<latexit sha1_base64=\"3XX2eVu+Rh9TaIY01XXkIv79q1c=\">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</latexit>|y,\"ii\n<latexit sha1_base64=\"W0kxyk2VOCVIbX0jCAEyqs20SEk=\">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</latexit>|z,\"ii<latexit sha1_base64=\"Z2pgJTBhiNumLVAFTWJsBALRuPM=\">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</latexit>n=3<latexit sha1_base64=\"izRmVshPXfaqvICjoqW8/FO1zhM=\">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</latexit>n=1FIG. 1. Schematic pictures for the ground state wave func-\ntionsj\r;\u001b=\"iiof the local Hamiltonian for n= 3 andn= 1.\nspherical limit. In this paper, we consider a strong cou-\npling regime (HU\u001dHt). When we develop the e\u000bective\nmodel in this limit, the presence of the Hund's coupling\nJmakes theoretical treatment complicated since it real-\nizes quantum-mechanically superposed local wave func-\ntions. Especially for the negative (antiferromagnetic) J\nrelevant to fullerides, the pair hopping plays an impor-\ntant role which creates the dynamics of doubly occupied\nelectrons at an orbital (doublon). As shown in the follow-\ning, in order to diminish the di\u000eculty, we use a symbolic\nexpression without elaborating each intermediate process\nexplicitly.\nIn order to apply the perturbation theory from the\nstrong coupling limit, we \frst consider the ground state\nof the unperturbed Hamiltonian HU. Alkali-doped ful-\nlerides with half-\flled situation (three electrons per t1u\norbital) have six-fold degenerate ground states written as\nj\r;\u001bii=1p\n2cy\ni;\r;\u001bX\n\r06=\rby\ni;\r0j0i; (4)\nwhere we have de\fned an orbital-dependent doublon-\ncreation operator as\nby\ni;\r=cy\ni;\r;#cy\ni;\r;\": (5)\nThe vacuum has been expressed as j0i. These states are\nuniquely characterized by the spin and orbital of the elec-\ntron at the singly occupied orbital, which is called `sin-\nglon' to make contrast against doublons. The schematic\npicture of the three-electron state j\r;\u001b=\"iiis illustrated\nin Fig. 1.\nUsing the above Hamiltonian, the second-order e\u000bec-\ntive Hamiltonian is written as\nHe\u000b=PHt1\n\u0000HUQHtP; (6)3\nwherePis a projection operator to a model space de-\nscribed by Eq. (4) as\nP=X\ni;\r;\u001bj\r;\u001biiih\r;\u001bj; (7)\nandQ= 1\u0000P. We have used [P;HU] = 0. The energy\nis measured from the ground state of HU. The size of\nour model space is 6NwhereN=P\ni1 is the number of\nlattice sites.\nThe strategy for obtaining the concrete form of the ef-\nfective Hamiltonian is to consider the two-site problem.\nWe \frst prepare the 212\u0002212matrix expressions for the\nannihilation and creation operators for two-site problem\n(12 =P\ni;\r;\u001b1), and then de\fne all of the matrix ex-\npressions given in Eq. (6). Performing multiplications of\nsuch matrices, we obtain the two-site e\u000bective Hamilto-\nnian in the form of the 62\u000262matrix. We expand the\nabove e\u000bective hamiltonian by following local operators\nO\u0011\u0016\nide\fned as\nO\u0011\u0016\ni=X\n\r;\r0X\n\u001b;\u001b0j\r;\u001bii\u0015\u0011\n\r\r0\u001b\u0016\n\u001b\u001b0ih\r0;\u001b0j; (8)\nin the model Hilbert space. \u001b\u0016=0;x;y;zis Pauli matrix\n\u001b0=\u0012\n1 0\n0 1\u0013\n; \u001bx=\u0012\n0 1\n1 0\u0013\n;\n\u001by=\u0012\n0\u0000i\ni 0\u0013\n; \u001bz=\u0012\n1 0\n0\u00001\u0013\n; (9)\nwhich represents the degrees of freedom of the spin. An-\nother matrix \u0015\u0011=0;\u0001\u0001\u0001;8is given by\n\u00150=r\n2\n30\n@1 0 0\n0 1 0\n0 0 11\nA; \u00151=0\n@0\u00001 0\n\u00001 0 0\n0 0 01\nA;\n\u00152=0\n@0\u0000i 0\ni 0 0\n0 0 01\nA; \u00153=0\n@\u00001 0 0\n0 1 0\n0 0 01\nA;\n\u00154=0\n@0 0\u00001\n0 0 0\n\u00001 0 01\nA; \u00155=0\n@0 0 i\n0 0 0\n\u0000i 0 01\nA;\n\u00156=0\n@0 0 0\n0 0\u00001\n0\u00001 01\nA; \u00157=0\n@0 0 0\n0 0\u0000i\n0 i 01\nA;\n\u00158=r\n1\n30\n@1 0 0\n0 1 0\n0 0\u000021\nA; (10)\nwhere these matrices are slightly di\u000berent from ordinary\nde\fnition of the Gell-Mann matrices to make them suit-\nable forp-electron systems. We note that the above local\noperators satisfy the orthonormal relation\nTrh\nO\u0011\u0016\niO\u00110\u00160\nji\n= 4\u000eij\u000e\u0011\u00110\u000e\u0016\u00160: (11)Thus, the set of operators O\u0011\u0016\niis regarded as a basis set of\nthe extended Hilbert space (Liouville space). In contrast,\nthe statesj\r;\u001biiare the basis in the six-component model\nHilbert space. Extending the two-site problem to the full\nlattice, we obtain the e\u000bective Hamiltonian in the strong\ncoupling limit\nHe\u000b=X\ni;jX\n\u0011;\u00110X\n\u0016;\u00160I\u0011\u0016;\u00110\u00160\nijO\u0011\u0016\niO\u00110\u00160\nj: (12)\nThis model is to be analyzed in the rest of this paper.\nWe also comment on the orbital moments in the re-\nstricted Hilbert space. In terms of the original Hubbard\nmodel, the local orbital moment is de\fned by\nLi\u0011X\n\r;\r0;\u001bcy\ni;\r;\u001b`\r\r0ci;\r0;\u001b; (13)\nwhere the 3\u00023 matrices are given by `x=\u00157,`y=\u00155,\nand`z=\u00152. This angular momentum operator is, how-\never, zero for the restricted Hilbert space:\nPLiP=0: (14)\nThis anomalous disappearance of the angular momentum\nis due to the composite nature of the ground state [26]\nand is very di\u000berent from a singly occupied state. Then\nthe active orbital degrees of freedom are not of the origi-\nnal electrons but of the three-electron composite involv-\ning doublons. This feature also a\u000bects the spin-orbit cou-\npling which takes the form\nHSO=1\n2\u0015SOX\niX\n\r;\r0X\n\u001b;\u001b0cy\ni;\r;\u001b`\r\r0\u0001\u001b\u001b\u001b0ci;\r0;\u001b0;(15)\nin the language of the original multiorbital Hubbard\nmodel. The spin-orbit coupling for 2 p-electron in car-\nbon atom is nearly 2meV, and because of the extended\nnature of the fullerene molecular the spin-orbit coupling\n\u0015SOfort1uorbitals is one-hundred times smaller than the\natomic value ( \u0015SO\u001820\u0016eV) [38]. Furthermore, for the\nrestricted Hilbert space of n= 3 states, the e\u000bect of the\nspin-orbit coupling enters only through the second-order\nperturbation contribution as\nH(2)\nSO=PHSO1\n\u0000HUQHSOP (16)\n=1\n2\u0003SOX\niX\n\r;\r0X\n\u001b;\u001b0j\r;\u001bii`\r\r0\u0001\u001b\u001b\u001b0ih\r0;\u001b0j;(17)\nwhere \u0003 SO=11\u00152\nSO\n20JforJ < 0. Using the values\nfor the antiferromagnetic coupling J\u0018 \u0000 0:03eV for\nfullerdies [39], we obtain \u0003 SO\u00181neV which is so tiny.\nHence we can safely neglect the spin-orbit coupling in\nfullerides.\nIt is convenient to recognize that the above three elec-\ntron state is similar to the singly occupied state\njn= 1;\r;\u001bii=cy\ni;\r;\u001bj0i; (18)4\nwhich is the eigenstate with ni=P\n\r;\u001bcy\ni;\r;\u001bci;\r;\u001b = 1\nregardless of the sign of J(see the right column of Fig. 1).\nIn our paper, the number of electrons is \fxed at each site\nandniis sometimes simply written as n. In Eq. (18), we\nexplicitly write ` n= 1', and if it is dropped, the state\nrepresentsn= 3 state de\fned in Eq. (4). The ground\nstate forn= 3 is obtained by \flling the empty orbital in\nn= 1 state by the doublons as in Eq. (4).\nWe will consider the n= 1 case for reference to il-\nluminate the characteristics of n= 3 relevant to ful-\nlerides. When we deal with the second-order e\u000bective\nHamiltonian for the n= 1 states, we just replace j\r;\u001bii\nbyjn= 1;\r;\u001biide\fned in Eq. (18). We note that, in this\ncase, the angular momentum does not vanish as distinct\nfrom then= 3 multiplet. For the usual ferromagnetic\nHund's coupling ( J >0), the system corresponds to the\nspin-orbital model considered for the t2gorbitals [37].\nB. Mean \feld approximations\nIn this paper, we utilize the mean \feld approximation\n(MFA) for the obtained e\u000bective Hamiltonian. We apply\nthe external \feld for convenience and the full Hamilto-\nnian is written as\nHe\u000b=1\n2X\ni;j~OT\ni^Iij~Oj\u0000X\ni~HT\ni~Oi (19)\n\u0019\u0000X\ni;j\u0014\n~HT\ni\u000eij^1\u00001\n2~MT\ni\u0010\n^Iij+^IT\nji\u0011\u0015\n~Oj\n\u00001\n2X\ni;j~MT\ni^Iij~Mj\u0011HMF; (20)\nwhere the hat and arrow symbols represent the matrix\nand vector, respectively, with respect to the intra-site\ndegrees of freedom ( \u0011;\u0016). The vector ~Oiis the operator\nfor the order parameter at site i, whose matrix represen-\ntation is given in Eq. (8). Namely, it is a column vector\nhaving 35 components, each of which is a 6 \u00026 matrix\nwhere the identity is eliminated. The statistical aver-\nage~Mi=h~Oiiis the order parameter. In this paper,\nthe coupling constant ^Iijconnects only nearest-neighbor\n(NN) sites for the spherical model (Sec. III), and NN\nand next-nearest-neighbor (NNN) site for A15 and fcc\nfulleride model (Sec. IV). In the following of this section,\nwe concentrate on the bipartite lattice such as A15 struc-\nture. Then we introduce two kinds of AB-sublattice to\ndescribe staggered orders. For non-bipartite lattice (i.e.\nfcc), on the other hand, we consider only the uniform so-\nlution and the similar formula can easily be obtained by\nregarding the two sublattices as identical.The mean-\feld Hamiltonian is then rewritten as\nHMF=\u0000X\n\u000b\"\n~HT\n\u000b\u00001\n2X\n\u000e2NN~MT\n\u0016\u000b\u0010\n^I\u000e;0+^IT\n0;\u000e\u0011\n\u00001\n2X\n\u000e2NNN~MT\n\u000b\u0010\n^I\u000e;0+^IT\n0;\u000e\u0011#N=2X\ni2\u000b~Oi\n\u00001\n2N\n2X\n\u000b\"X\n\u000e2NN~MT\n\u0016\u000b^I\u000e;0~M\u000b+X\n\u000e2NNN~MT\n\u000b^I\u000e;0~M\u000b#\n;\n(21)\nwhere\u000b= A;B is the sub-lattice index and \u0016 \u000bis a com-\nplementary component of \u000b, i.e., \u0016A = B and \u0016B = A.N\nis the number of site. The number of \u000e2NN isz, 8 or\n12 respectively for the spherical, A15 or fcc model. As\nfor\u000e2NNN, both the A15 case (and fcc) has six sites.\nWe have used the fact that NN-connected sites belong to\nthe di\u000berent sub-lattices and the NNN-connected sites\nbelongs to the same sub-lattice. Since the coupling con-\nstants are dependent only on the direction of the vector\nconnecting two sites, we write the interaction parameter\nas^I\u000e;0, where the index 0 represents the site which we\nfocus on.\nFor the bipartite lattice, we introduce the uniform and\nstaggered moments as\n\u0012~Mu\n~Ms\u0013\n=1p\n2\u0012^1^1\n^1\u0000^1\u0013\u0012~MA\n~MB\u0013\n: (22)\nThis expression is useful in analyzing the mean-\feld so-\nlutions shown later.\nNow we explain the method of numerical calculation.\nThe solutions are obtained by renewing the order param-\neters iteratively using the self-consistent equation. The\nfree energy and the self-consistent equation are given by\nF=\u0000TlnZ; (23)\n~M\u000b=\u0000@F\n@~H\u000b; (24)\nwhereZ= Tr e\u0000\fHMFis the partition function made of\nthe mean-\feld Hamltonian. For the derivation of the self-\nconsistent equation, the parameters ~Hand~Mmust be\nregarded as independent variables.\nThe system with the present e\u000bective Hamiltonian has\n35 kinds of order parameters per site, and there may exist\nseveral solutions which take the same free energy as they\nare connected by symmetries. In the next sections, we\nshow the simplest form of the order parameters among\nthose energetically degenerate solutions.\nC. Response functions\nIn this subsection, we consider the response function to\nthe weak static \feld. We expand the mean-\feld Hamil-5\ntonian up to \frst order of the \feld\nHMF=H(0)+H(1)+O\u0000\nH2\u0001\n; (25)\nH(0)=X\ni;j\u0010\n~M(0)\ni\u0011T^Iij~Oj; (26)\nH(1)=\u0000X\ni;j\u0014\n~HT\ni\u000eij\u0000\u0010\n~M(1)\ni\u0011T^Iij\u0015\n~Oj;(27)\nwhere the superscript represents the perturbative or-\nder of the \feld and we have neglected the constant\nterm. When we de\fne the e\u000bective \feld as~~Hi=~Hi\u0000P\nj^IT\nji~M(1)\njand treatH(1)as perturbation, we obtain\nthe following linear response relation\n~M(1)\ni=X\nj^\u001f(0)\nij~~Hj=X\nj^\u001fij~Hj; (28)\nwhere ^\u001fis the full susceptibility for the bare external \feld\n~Hi. According to linear response theory, the zeroth-order\nsusceptibility is obtained by\n^\u001f(0)\nij=Z1=T\n0d\u001c\u0014D\nT\u001c~Oi~OT\nj(\u001c)E\n0\u0000~M(0)\ni\u0010\n~M(0)\nj\u0011T\u0015\n;\n(29)\nwhere\u001cis an imaginary time and T\u001cis imaginary time\nordering operator. The Heisenberg picture in an imagi-\nnary time is expressed as\n~Oi(\u001c) = e\u001cH(0)~Oie\u0000\u001cH(0): (30)\nh\u0001\u0001\u0001i0represents the statistical average with H(0). The\nsusceptibility matrix ^ \u001f(0)\nijhas only intra-site component\nsince each site is independent under MFA. Substituting\nthe concrete expression to the e\u000bective \feld in Eq. (28),\nwe obtain\nX\nj\"\n\u000eij^1 +X\nk^\u001f(0)\nik^IT\nkj#\n~M(1)\nj=X\nj^\u001f(0)\nij~Hj: (31)\nThen, taking matrix inverse of the left hand side and\ncombining it with Eq. (28), we obtain the susceptibil-\nity matrix ^\u001fij. For a bipartite lattice, we introduce the\nuniform and staggered susceptibilities by\n^\u001fu=1\nNX\ni;j^\u001fij; (32)\n^\u001fs=1\nNX\ni;jsisj^\u001fij; (33)\nwheresi= +1 fori2A andsi=\u00001 fori2B. This\nquantity will be shown in the next section. Although\nwe focus on the static response functions in this paper,\nthe above argument can easily be generalized for the dy-\nnamical susceptibility which captures the magnetic and\nelectric dynamics of the localized model.From the view point of Landau theory, we can also\ndiscuss the stability of the solution based on the suscep-\ntibilities. We write down the Landau free energy with an\norder parameter up to second order as\nFL=1\n2X\ni;j~MT\ni^aij~Mj\u0000X\ni~HT\ni~Mi; (34)\nwhere ^aijis a coe\u000ecient of the quadratic term. Note that,\nhere,~Mis de\fned as the deviation from its equilibrium\npoint. Then we obtain the following equation of states:\nX\nj^aij~Mj=~Hi: (35)\nComparing the linear response function, we \fnd that the\nHessian matrix is identical to the inverse susceptibility:\n@2FL\n@~Mi@~Mj= ^aij= (^\u001f\u00001)ij: (36)\nWe can consider the necessary and su\u000ecient condition for\nthe stable solution. Let \"nbe then-th eigenvalue of the\nmatrix ^aij. Each energy corresponds to the eigenenergy\nof the excitation modes. We must have the condition\n\"n\u00150; (37)\nfor alln, if the system is thermodynamically stable. If\n\"n= 0 is obtained, it indicates the presence of the\nNambu-Goldstone mode. With use of Eq. (36), in the\nactual calculations, we obtain \"nby diagonalizing the\ninverse susceptibility matrix.\nIII. NUMERICAL RESULTS FOR SPHERICAL\nMODELS\nIn the following of this paper, we will encounter the\nsuccessive phase transitions with decreasing temperature.\nThere, we denote each transition temperature as Tc1>\nTc2>\u0001\u0001\u0001. If there is only one transition temperature is\nidenti\fed, we use Tcto denote it. Note that we use the\nsame symbol for the transition temperatures in di\u000berent\nmodels.\nA. Spherical spin-orbital model\nFirst we consider the model in the spherical limit.\nNamely, we assume the hopping matrix given in Eq. (2)\nas\n^tij=0\n@t0 0\n0t0\n0 0t1\nA; (38)6\nfor a bipartite lattice with the coordination number z.\nUsing the spin-orbital operator O\u0011\u0016\nide\fned in the previ-\nous section, we obtain the spherical model as\nHe\u000b=\u0000X\nhijih\nISSi\u0001Sj+ILLi\u0001Lj+IQX\n\u0011Q\u0011\niQ\u0011\nj\n+IRX\n\u0016X\n\u0017R\u0017;\u0016\niR\u0017;\u0016\nj+ITX\n\u0016X\n\u0011T\u0011;\u0016\niT\u0011;\u0016\nj+I0i\n;\n(39)\nwhere the sum with hijiis taken over the pairs of\nthe NN sites. The superscript \u0016;\u0017 (=x;y;z ) and\u0011\n(=x2\u0000y2;z2;xy;yz;zx ) are the indices for the polyno-\nmials, which represents the component of the spin, rank 1\norbital and rank 2 orbital, respectively. We have rewrit-\nten the operators in accordance with their symmetries\nas\nS\u0016\ni=1\n2O0\u0016\ni; (40)\nLx\ni=1\n2O70\ni; Ly\ni=1\n2O50\ni; Lz\ni=1\n2O20\ni; (41)\nQx2\u0000y2\ni =1\n2O30\ni; Qz2\ni=1\n2O80\ni;\nQxy\ni=1\n2O10\ni; Qyz\ni=1\n2O60\ni; Qzx\ni=1\n2O40\ni; (42)\nRx;\u0016\ni=1\n2O7\u0016\ni; Ry;\u0016\ni=1\n2O5\u0016\ni; Rz;\u0016\ni=1\n2O2\u0016\ni; (43)\nTx2\u0000y2;\u0016\ni =1\n2O3\u0016\ni; Tz2;\u0016\ni=1\n2O8\u0016\ni;\nTxy;\u0016\ni=1\n2O1\u0016\ni; Tyz;\u0016\ni=1\n2O6\u0016\ni; Tzx;\u0016\ni=1\n2O4\u0016\ni:(44)\nThe physical meaning of each order parameter now be-\ncomes clearer with this notation. We call S\u0016\nia magnetic\nspin (MS or S),L\u0016\nia magnetic orbital (MO or L),Q\u0016\ni\na electric orbital (EO or Q),R\u0017;\u0016\nia electric spin-orbital\n(ESO orR) andT\u0011;\u0016\nia magnetic spin-orbital (MSO or\nT) moments. I0represents energy gain by the second\norder perturbation process. Obviously, Eq. (39) satis\fes\nSU(2)\u0002SO(3) symmetry in spin-orbital space.\nWe will show the numerical results of the n= 1 and\nn= 3 spherical models under MFA, both of which have\nthe six states per site in the model space as discussed in\nSec. II A. We beforehand introduce the following notation\nwith regard to the coupling constants de\fned in Eq. (39)\nas\nI\u0018=X\nnA\u0018nt2\n\u0001En; (45)\nfor\u0018=S;L;Q;R;T; 0, where \u0001 Enrepresents all pos-\nsible excitation energies. Its energy corresponds to the\ndenominator of Eq. (6). The coe\u000ecient Ais summarized\nin the tables in the following subsections (see Sec. III B\nor Sec. III C).\nBefore we show the mean-\feld results, we discuss the\nground state wave function for the two-site problem. Us-\ning the single site state de\fned in Eq. (4) or (18), weTABLE I. Coe\u000ecients Ade\fned in Eq. (45) for n= 1 spher-\nical model. The ground state energy is zero. We add the\ndetails for the intermediate state in the main text.\n\u0001EnU\u00003J U\u0000J U + 2J\nIS\u00002 10=3 2=3\nIL 3\u00005=3 2=3\nIQ 3\u00001=3\u00002=3\nIR 1 5=3\u00002=3\nIT 1 1=3 2=3\nI0\u00006\u000010=3\u00002=3\nobtain the two-site (i.e., sites at iandj) ground state as\njgsi=X\n\ri;\u001biX\n\rj;\u001bjC\ri\u001bi;\rj\u001bjj\ri;\u001biiij\rj;\u001bjij: (46)\nThe explicit form of the matrix Cis written as\n^C=\u00150\n(\u0000i\u001by): (47)\nThis shows that the ground-state wave function is spin-\nsinglet and symmetric on the orbital. This is valid for\nall the spherical cases considered in this section. For an\nin\fnite lattice, as in the single-orbital Hubbard model,\nthe inter-site spin-singlet state may favor the antiferro-\nmagnetic state in the ground state for a bipartite lattice.\nB.n= 1model\nFirst of all, we consider the results for the n= 1\nmodel. Although the results are not relevant to the alkali-\ndoped fullerides, the knowledge is useful in interpreting\nthe more complicated model for the spherical n= 3\nmodel (Sec. III C), the realistic A15- (Sec. IV A) and fcc-\nstructure fullerides (Sec. IV B).\n1. Coupling constant\nWe begin with the analysis of the intermediate states\nrelevant to the second-order perturbation theory. We\nshow the coe\u000ecients Ade\fned in Eq. (45) in Table I. We\nhave the three kinds of excited states, whose energy is de-\ntermined by the local Coulomb interaction. For \u0001 En=\nU\u00003J, the intermediate states are nine-fold degener-\nate spin-triplet states, as expressed, e.g., by cy\ni;y;\"cy\ni;x;\"j0i\nand1p\n2\u0010\ncy\ni;y;#cy\ni;x;\"+cy\ni;y;\"cy\ni;x;#\u0011\nj0i. For \u0001En=U\u0000J,\nthe intermediate states are the inter-orbital spin-singlet\nstates such as1p\n2\u0010\ncy\ni;y;#cy\ni;x;\"\u0000cy\ni;y;\"cy\ni;x;#\u0011\nj0i, and the\nintra-orbital spin-singlet states with anti-bonding or-\nbitals written asp\n2\n3\u0010\n2by\ni;z\u0000by\ni;x\u0000by\ni;y\u0011\nj0i. These two\nkinds of states take the same energy since there is the\nspherically symmetric condition U0=U\u00002J. For\n\u0001En=U+ 2J, there is only one intermediate state,7\n\u00000.4\u00000.20.00.2\nJ/U\u00006\u00004\u000020246I⇠/E0ni=1\nIS\nIL\nIQ\nIR\nIT\nFIG. 2. Hund's coupling ratio J=U dependence of the cou-\npling constants for n= 1 spherical model. The vertical axis\nis normalized by E0=t2=U.\nwhich is intra-orbital spin singlet and bonding state writ-\nten as1p\n3\u0010\nby\ni;x+by\ni;y+by\ni;z\u0011\nj0i.\nWe show the Hund's coupling dependence of the cou-\npling constants in Fig. 2. The perturbation theory is jus-\nti\fed for\u00001=2<J=U < 1=3 where the ground states are\nwritten in the form of Eq. (18). Taking J= 0, the cou-\npling constants become identical. This re\rects that the\nsystem has SU(6) symmetry and the degrees of freedom\nof the spin and orbital are equivalent in the absence of\nHund's coupling. The largest coupling constant is ISfor\nthe antiferromagnetic case ( J <0) andIQfor the ferro-\nmagnetic Hund's coupling ( J >0). This shows that the\nsystem tends to be antiferromagnetic (AFM) or antiferro-\norbital (AFO) order depending on the sign of the Hund's\ncoupling. This is understood from the intermediate state.\nIn the case of J > 0, which is relevant to the usual\nt2g-orbitald-electron systems with n= 1 per atom, the\nenergetically favorable intermediate two-electron state is\ninter-orbital spin triplet. To realize this intermediate\nstate, the initial state needs to occupy parallel spin con-\n\fguration with di\u000berent orbitals such as cy\ni;x;\"cy\nj;y;\"j0i.\nTherefore, the orbital order should be dominant for\nJ >0 as a leading-order ordering instability. If we take\nJ=U&0:2,IStakes a ferromagnetic coupling constant,\nwhich favors parallel spins at two sites.\nAs forJ <0, on the other hand, the intermediate state\ntends to be intra-orbital spin singlet and bonding state.\nThe corresponding initial state must be antiparallel spin\nwith the same orbital such as cy\ni;x;\"cy\nj;x;#j0i. Thus, the\nmagnetic order should be dominant for J <0.\n0.00.51.01.52.02.53.0T/E0\u00004\u000020Internal/Free energy densityni=1, J/U=\u00000.1\nUFUSULUQURUT(b)\n0.00.51.01.52.02.53.0T/E0\u00001.0\u00000.50.00.51.0Mu,sni=1, J/U=\u00000.1SzsQz2uTz2,zs\n<latexit sha1_base64=\"X52I/2YrE97X48H0XCX0tZyVuPU=\">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</latexit>Tc1\n<latexit sha1_base64=\"XlJmCfpEjjRXMdHUfDX7W/sBXjY=\">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</latexit>Tc2\n0.00.51.01.52.02.53.0T/E001234C/Nni=1, J/U=\u00000.1(d)\n0.00.51.01.52.02.53.0T/E00.00.51.01.5S/Nni=1, J/U=\u00000.1\n<latexit sha1_base64=\"iJITrNFThqM9eAjvFM/a/M71vKQ=\">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</latexit>ln 6<latexit sha1_base64=\"8y6Wk8tRec2BIDiGctZQySs6+4Q=\">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</latexit>ln 3(c)(a)\n0.8350.8401.01.1FIG. 3. Temperature dependence of (a) the order parameter,\n(b) the decomposed internal energy and total free energy den-\nsity, (c) entropy and (d) speci\fc heat for n= 1;J=U =\u00000:1\nbipartite spherical model. The inset in (c) is enlarged plot\naroundTc2. The energy unit of these plots are E0=t2=U.\n2. Mean-\feld solutions for antiferromagnetic Hund's\ncoupling (J <0)\nLet us turn our attention to the numerical results using\nMFA in the spherical model. We take the NN coordina-\ntion number z= 6 in the numerical calculation by assum-\ning a simple cubic lattice in three dimensions. Figure 3\nshows the temperature dependence of the physical quan-\ntities in the bipartite lattice model at J=U =\u00000:1 (an-\ntiferromagnetic Hund's coupling). We take E0\u0011t2=U\nas the unit of energy. The uniform and staggered order\nparameters are shown in Fig. 3(a), where the antiferro-\nmagnetic spin (AF- S) order appears \frst with decreasing\ntemperature from the high-temperature limit. This cor-\nresponds to the largest coupling constant ISin Fig. 2. At\nlower temperatures, the ferro (F)-orbital Qmoment ofz2\ntype appears together with the AF- T(MSO) moments.\nIn order to clarify which is the primary order parameter\nof the second phase transition at Tc2, we show in Fig. 3(b)\nthe internal energy and free energy per site, where the\ninternal energy is decomposed into each contribution as\nUS=IShSAi\u0001hSBi; (48)\nUL=ILhLAi\u0001hLBi; (49)\nUQ=IQX\n\u0011hQ\u0011\nAihQ\u0011\nBi; (50)\nUR=IRX\n\u0016X\n\u0017hR\u0017;\u0016\nAihR\u0017;\u0016\nBi; (51)\nUT=ITX\n\u0016X\n\u0011hT\u0011;\u0016\nAihT\u0011;\u0016\nBi: (52)\nThe total internal energy is given by U=P\n\u0018U\u0018for\n\u0018=S;L;Q;R;T , where the energy is measured from\nI0. We see from Fig. 3(b) that the energy UTis gained\nbelowTc2butUQis not. Hence, the AF- Tshould be the8\n0123T/E001020301/\u0000⌘µ;⌘µu[arb.units]ni=1, J/U=\u00000.1\n0123T/E001020301/\u0000⌘µ;⌘µs[arb.units]ni=1, J/U=\u00000.1(a)(b)\n\u00000.04\u00000.020.000.020.04\u00000.050.000.05\nSxSySzLxLyLzQx2\u0000y2Qz2QxyQyzQzxRx,xRy,xRz,xRx,yRy,yRz,yRx,zRy,zRz,zTx2\u0000y2,xTz2,xTxy,xTyz,xTzx,xTx2\u0000y2,yTz2,yTxy,yTyz,yTzx,yTx2\u0000y2,zTz2,zTxy,zTyz,zTzx,z\nFIG. 4. Temperature dependence of the inverse of (a) uni-\nform and (b) staggered component of the diagonal suscepti-\nbilities. The energy unit is E0=t2=U.\nprimary order parameter and F- Qis just induced by the\ncombination of AF- Splus AF-Tmoments. The results\nare consistent with the magnitude relation IT>IQseen\nin Fig. 2, where the larger energy gain is obtained from\nT-moment than the energy loss from Q.\nFigure 3(c) shows the temperature dependence of the\nentropy, where all the entropy is released in the ground\nstate. With increasing temperature, the entropy shows a\nkink atT=E 0'0:84, at which the value of the entropy is\nclose to ln 3 meaning that the orbital degeneracy is lifted\nbelow this transition temperature. The inset of (c) shows\nthe magni\fed picture of the entropy near Tc2, indicating\nthe \frst-order transition. The speci\fc heat C=@U=@T\nis also shown in Fig. 3(d). There are two discontinuity\ncorresponding to the spin and orbital orders.\nNext we show in Fig. 4 the inverse of the diagonal\nsusceptibilities \u001f\u0011\u0016;\u0011\u0016\nu (uniform) and \u001f\u0011\u0016;\u0011\u0016\ns (staggered)\nwhich are de\fned in Eqs. (32) and (33). First, we observe\nthat the susceptibilities shown here are all positive, in-\ndicating a stable solution. The AF- Ssusceptibility of\nx;y;z type diverges at T=E 0'2:3 signaling the on-\nset of the antiferromagnetic order. Below this transition\ntemperature, the longitudinal zcomponent is decreased\nwhile the perpendicular x;ycomponents remain diver-\ngent. This behavior indicates the presence of the Gold-\nstone mode, where the excitations are induced by rotat-\ning thezcomponent into xy-plane, as in the standard\nHeisenberg model. Inside this magnetic phase, the or-\nbital (F-Q) and spin-orbital (AF- T) susceptibility, which\narez2type in orbital part, continue to grow and tend\nto diverge at lower transition point ( Tc2). As shown in\nFig. 4(a), the `perpendicular' components, i.e. F- Qyz, F-\nQzx, remain divergent below Tc2, indicating the presence\nof the Goldstone mode even for the orbital order in the\nspherical model. Namely, because of the symmetry of the\nspin-orbital space, the energetically equivalent solutions\nexist and are obtained by rotating the order parameters.\nNext we discuss the ground state wave function, which\nincludes the information of order parameter at zero tem-\nperature limit. As is evident from the zero entropy at\nT= 0, we have the non-degenerate ground state. In\nthe present case, the ground state wave function is very\n01234T/E0\u00006\u00004\u000020Internal/Free energy densityni=1, J/U=0.1\nUFUSULUQURUT01234T/E0\u00001.0\u00000.50.00.51.0Mu,sni=1, J/U=0.1SzuQz2uQz2sTz2,zuTz2,zs\n<latexit sha1_base64=\"X52I/2YrE97X48H0XCX0tZyVuPU=\">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</latexit>Tc1\n<latexit sha1_base64=\"XlJmCfpEjjRXMdHUfDX7W/sBXjY=\">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</latexit>Tc2(a)(b)\n0.000.250.500.751.001.25C/N01234T/E00.00.51.01.5S/Nni=1, J/U=0.1SSASBC\n<latexit sha1_base64=\"iJITrNFThqM9eAjvFM/a/M71vKQ=\">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</latexit>ln 6\n<latexit sha1_base64=\"46zSoqOY4v2X8E4dXP5823PRQJc=\">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</latexit>ln 2\n<latexit sha1_base64=\"KOUbV8JEvMfIBzQnLzcUE4PwzHM=\">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</latexit>lnp2(c)\n01234T/E001020304050eigenvalues [arb.units](d)FIG. 5. Temperature dependence of (a) the order parame-\nter, (b) the decomposed internal energy and total free energy\ndensity, (c, left axis) the single site entropy, (c, right axis) the\nspeci\fc heat and (d) the eigenvalues of the Hessian matrix ^ a\nfor bipartite spherical model with n= 1;J=U = 0:1.\nsimple and is given using Eq. (4) by\nj Ai=jn= 1;z;#iA; (53)\nj Bi=jn= 1;z;\"iB; (54)\nfor each sublattice. This corresponds to the staggered\nspin ordered and uniform orbital ordered state, as is con-\nsistent with Fig. 3(a). More speci\fcally, we can construct\nthe order parameters from the direct product of the wave\nfunctions. In the present case, we obtain at sublattice \u000b\nas\nj \u000bih \u000bj=\u00071p\n6Sz\n\u000b\u00001p\n3Qz2\n\u000b\u00061p\n3Tz2;z\n\u000b+1\n6;(55)\nwhere the operators are de\fned in Eqs. (40){(44). The\nupper (lower) sign is chosen for \u000b= A (\u000b= B). The\nquantities that appear in the right-hand side are identical\nto the order parameters shown in Fig. 3(a).\n3. Mean-\feld solutions for ferromagnetic Hund's coupling\n(J >0)\nWe show the results for the J=U = 0:1 case, where the\nmodel is now relevant to materials with d-electrons, to\nmake contrast with behaviors of the systems with anti-\nferromagnetic Hund's coupling. Figure 5(a) shows the\ntemperature evolution of the order parameters. As seen\nin Fig. 2, the largest coupling constant is IQwhich is an-\ntiferro (IQ<0), and therefore the AF- Qorder ofz2-type\nappears at the highest transition temperature ( Tc1). The\nF-Qorder of the same z2-type is simultaneously induced.\nThe rise of the order parameters near the transition tem-\nperature behaves as \u0018pTc1\u0000Tfor AF-Qand\u0018Tc1\u0000T\nfor F-Q. Hence the AF- Qis the primary order. From the\nsymmetry argument, it can be shown that the F- Qorder9\narises from AF- Qorder since the coupling term in the\nLandau free energy has the form Qz2\nu(Qz2\ns)2. The exis-\ntence of such third-order term can be understood if one\nconsiders the symmetry in the plane of Qz2-Qx2\u0000y2[26].\nAt lower temperatures, the magnetic F- Sorder appears,\nwhereT-moments of Tz2;z-type are also \fnite. From the\ninternal-energy analysis shown in Fig. 5(b), the relevant\nordering at Tc2is induced from the interaction ITwhile\nISis energetically unfavorable. Thus, comparing with\ntheJ=U =\u00000:1 case, the roles of magnetic order and\nelectric (orbital) order are switched. This switching of\nthe magnetic and orbital ordered states depending on\nthe sign of Jhas also been reported in the two orbital\nmodel [25].\nWe next show the temperature dependence of the en-\ntropy and speci\fc heat in Fig. 5(c), where we have de\fned\nthe sublattice-dependent entropy (Shannon entropy) by\nS\u000b=\u0000X\nnp\u000b\nnlnp\u000b\nn; (56)\nwherep\u000b\nnis the probability for the n-th state as calculated\nfrom the local partition function Z\u000b=P\nnexp(\u0000\fE\u000b\nn) =P\nnp\u000b\nnZ\u000b. Since the entropy at zero temperature is zero\nat A sublattice and is \fnite at B sublattice, the two sub-\nlattices are inequivalent and are not simply connected by\nsymmetry operations. This is due to the presence of the\nboth uniform and staggered orbital order parameters in\nFig. 5(a). Indeed, the wave function in the ground state\nis written for each sublattice as\nj Ai=jn= 1;z;#iA; (57)\n\f\f\f~ BE\n= \njn= 1;x;#iB\njn= 1;y;#iB!\n: (58)\nThe remaining degeneracy at B sublattice is because the\nx- andy-orbital components are equivalent. Namely, the\ntriply degenerate state at each sublattice splits depending\non the sublattice: zorbital becomes energetically higher\nat A sublattice and lower at B sublattice. Thus the an-\ntiferro order of this type cannot lift the degeneracy com-\npletely.\nUsually, the degeneracy is lifted by the interaction ef-\nfects and the unique ground state is expected. Then, one\nmay suspect that the remaining degeneracy might indi-\ncate the instability of the solutions. In order to show\nthat our degenerate ground states are really stable, we\nshow the energy spectra of the Hessian matrix discussed\nin Sec. II C. As shown in Fig. 5(d), the excitation energies\nin terms of Landau theory are all positive or zero, and\nthe system is thus stable. The degeneracy at T= 0 is\ndue to the absence of the relevant interactions, and will\nbe resolved once the other types of the interaction are\nincluded in the more realistic situations.\nWe comment on the case where we allow only for the\nuniform solutions, by having the geometrical frustration\ne\u000bect in mind which does not favor a simple staggered or-\nders. Actually, the n= 1 uniform spherical model aroundTABLE II. Coe\u000ecients Ain Eq. (45) for n= 3 spherical\nmodel. The ground state is written as j\ri;\u001biiij\rj;\u001bjijand its\nenergy is 2(3 U\u00004J). We add the details for the intermediate\nstate in the main text.\n\u0001EnU\u00008J U\u00006J U\u00004J U\u00003J U\u0000J U + 2J\nIS 1=2\u00005=3 25=18\u00004=3 20=9 8=9\nIL 9=8\u00005=4 25=72 2\u000010=9 8=9\nIQ\u00009=8 1=4\u00001=72 2\u00002=9\u00008=9\nIR\u00001=8\u00005=12\u000025=72 2=3 10=9\u00008=9\nIT 1=8 1=12 1=72 2=3 2=9 8=9\nI0\u00009=2\u00005\u000025=18\u00004\u000020=9\u00008=9\nJ= 0 has no solution at any temperature because all of\nthe coupling constants are negative (antiferromagnetic)\nin the spherical model (see Fig. 2). On the other hand,\nfor relatively large jJjregion the uniform solutions can\nexist. However, since the typical value of Hund's cou-\npling isjJj=U\u00180:1 or less, we do not enter the regime\nwith largerjJjin this paper.\nC.n= 3model\nHere we consider the model with three electrons per\nmolecule and with the antiferromagnetic Hund's coupling\n(J < 0). This model is more relevant to the existing\nfullerides with half-\flled t1umolecular orbitals.\n1. Coupling constants\nWe show the coe\u000ecients A, which is de\fned by\nEq. (45), in Table II. Since we consider the half-\flled\nmodel, the initial and intermediate states for the two-\nsite problem at the sites iandjrelevant to Iijare\n(ni;nj) = (3;3) and (ni;nj) = (2;4), respectively. Here,\nni= 2 andni= 4 states are connected with each other by\nthe particle-hole (PH) transformation. The explicit form\nforni= 2 state is same as those given in Sec. III B, and\nthereby the n= 4 can also be constructed from n= 2\naccordingly. Below, we list the types of the intermedi-\nate states and their energies, speci\fcally focusing on the\nnj= 4 state.\nThe intermediate states with the excited energy\u0001 En=\nU\u00008Jare nine kinds of inter-orbital spin triplet\nstate forni= 2 and the PH transformed states\nfornj= 4 such as by\nj;zcy\nj;y;\"cy\nj;x;\"j0i. For \u0001En=\nU\u00006J, the intermediate states are the inter-orbital\nspin triplet states for ni= 2 and the PH trans-\nformed states which have inter-orbital spin singlet\nstates such as1p\n2by\nj;z\u0010\ncy\nj;y;#cy\nj;x;\"\u0000cy\nj;y;\"cy\nj;x;#\u0011\nj0ior\nintra-orbital spin singlet with anti-bonding such asp\n2\n3\u0010\n2by\nj;zby\nj;y\u0000by\nj;zby\nj;x\u0000by\nj;yby\nj;x\u0011\nj0i. For \u0001En=U\u00004J,\nthe intermediate states are the inter-orbital spin singlet\nor intra-orbital spin singlet with anti-bonding states for10\n\u00000.5\u00000.4\u00000.3\u00000.2\u00000.10.0\nJ/U\u00006\u00004\u000020246I⇠/E0ni=3\nIS\nIL\nIQ\nIR\nIT\nFIG. 6. Hund's coupling ratio J=U dependence of the cou-\npling constants for n= 3 spherical model.\nni= 2, and their PH transformed versions for the jsite.\nFor \u0001En=U\u00003J, the intermediate states are the intra-\norbital spin singlet and bonding states for ni= 2, and\nthe states which have inter-orbital spin triplet for nj= 4.\nFor \u0001En=U\u0000J, the intermediate states are the inter-\norbital spin singlet or intra-orbital spin singlet with anti-\nbonding states ( ni= 2), and intra-orbital spin singlet and\nbonding state such as1p\n3\u0010\nby\nj;zby\nj;y+by\nj;zby\nj;x+by\nj;yby\nj;x\u0011\nj0i\nfornj= 4. Finally, for \u0001 En=U+ 2J, which is the low-\nest among the excited states for J <0, the intermediate\nstate is non-degenerate and is written as the intra-orbital\nspin singlet with bonding state for ni= 2 and its PH\ntransformed states for nj= 4.\nFigure 6 shows the Hund's coupling dependence of the\ncoupling constants. The perturbation theory is justi\fed\nfor\u00001=2< J=U < 0 where any level cross for the un-\nperturbed Hamiltonian does not occur. If we consider\nJ >0, the ground state is a total spin S= 3=2 state (e.g.,\ncy\ni;z;\"cy\ni;y;\"cy\ni;x;\"j0ii) and is di\u000berent from J < 0. This\npoint is in contrast with n= 1 case where the ground\nstate of the local Hamiltonian is not dependent on the\nsign ofJas shown in Fig. 2. It is notable that the cou-\npling constants for n= 3 case are similar to those of the\nn= 1 spherical model in the region near J=U =\u00000:5,\nwhere the same physical behavior is expected.\n2. Mean-\feld solutions for bipartite lattice\nWe show in Fig. 7(a) the order parameters for the bi-\npartite lattice model with n= 3 andJ=U =\u00000:1. At\nTc1'2:3E0, the system shows the antiferromagnetic or-\nder, which is consistent with the largest coupling con-\nstant shown in Fig. 6. With decreasing temperature,\nthe second order at Tc2appears, where the F- Qz2and\nAF-Tz2;zorder parameters are additionally induced. We\nemphasize that this orbital order is not of the ordinaryorbital moment of electrons, but of the doublons relevant\nto the antiferromagnetic Hund's coupling as discussed in\nSec. II A.\nFigure 7(b) shows the temperature dependences of the\ninternal energies and free energy. We show the order-\nparameter-resolved energies and all the components de-\ncrease upon entering the ordered phase. While this is\nin contrast to n= 1 cases shown in the previous sub-\nsections, the largest energy gain arises from the AF- T\norder.\nWe show in Fig. 7(c) the entropy and speci\fc heat. The\nclear jump in the speci\fc heat at Tc1indicates the second-\norder phase transition, and the jump in the entropy at\nTc2is the \fngerprint of the \frst-order phase transition.\nThe wave function in the ground state is\nj Ai=jz;#iA; (59)\nj Bi=jz;\"iB: (60)\nThe ground state is thus non-degenerate as is consistent\nwith the zero entropy at T= 0.\n3. Single-sublattice solution\nHaving the geometrically frustrated lattice in mind,\nwe assume that the spatially modulated solutions are not\nrealized. Then we seek for the spatially uniform solutions\n(single-sublattice) only.\nFigure 8(a) shows the order parameter for the single-\nsublattice model with n= 3,J=U =\u00000:1. The system\nshows theQz2order atTc=E0'0:28, which is consis-\ntent with the magnitude of the coupling constant shown\nin Fig. 6. The entropy and speci\fc heat are shown in\nFig. 8(b) with left and right axis, respectively. The resid-\nual entropyS= ln 2 remains, which is in accordance with\nthe degeneracy of spin in the absence of the sublattice\ndegrees of freedom. Namely, the wave function of the\nground state is degenerated and is written as\n\f\f\f~ E\n= \njz;\"i\njz;#i!\n: (61)\nWe have con\frmed that the eigenvalues of ^ ain Eq. (36)\nare all non-negative (not shown) and thus the ordered\nstate is stable.\nWe also point out the other interesting possibilities.\nThe above orbital order is induced by the coupling con-\nstantIQ>0 in Fig. 6. In this \fgure, it is notable that the\nvalues ofIQandIRare very close with each other. Then\nwe try to search for another solutions by introducing the\nmodi\fed coupling constants de\fned as\n~IQ= (1 +r)IQ; (62)\n~IR= (1\u0000r)IR; (63)\nwhere the original spherical model corresponds to r= 0.11\n0.00.51.01.52.02.53.0T/E0\u00005\u00004\u00003\u00002\u000010Internal/Free energy densityni=3, J/U=\u00000.1UFUSULUQURUT(b)\n0.00.51.01.52.02.53.0T/E0\u00001.0\u00000.50.00.51.0Mu,sni=3, J/U=\u00000.1SzsQz2uTz2,zs\n<latexit sha1_base64=\"X52I/2YrE97X48H0XCX0tZyVuPU=\">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</latexit>Tc1\n<latexit sha1_base64=\"XlJmCfpEjjRXMdHUfDX7W/sBXjY=\">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</latexit>Tc2\n024681012C/N0123T/E00.00.51.01.5S/Nni=3, J/U=\u00000.1SC\n<latexit sha1_base64=\"iJITrNFThqM9eAjvFM/a/M71vKQ=\">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</latexit>ln 6(c)(a)\nFIG. 7. Temperature dependence of (a) the order parameter, (b) the decomposed internal energy and total free energy density,\n(c, left axis) entropy and (c, right axis) the speci\fc heat for n= 3 bipartite spherical model with J=U =\u00000:1. The horizontal\naxis are normalized by E0=t2=U.\n0.00.10.20.3T/E0\u00001.0\u00000.50.00.51.0Muni=3, J/U=\u00000.1Qz2u\n01234C/N0.00.10.20.3T/E00.00.51.01.5S/Nni=3, J/U=\u00000.1SC\n<latexit sha1_base64=\"iJITrNFThqM9eAjvFM/a/M71vKQ=\">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</latexit>ln 6<latexit sha1_base64=\"46zSoqOY4v2X8E4dXP5823PRQJc=\">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</latexit>ln 2(a)(b)\n<latexit sha1_base64=\"f5RI9QClJOPHM2dinkR4cFHy+hU=\">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</latexit>Tc<latexit sha1_base64=\"f5RI9QClJOPHM2dinkR4cFHy+hU=\">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</latexit>Tc0.00.10.20.3T/E0\u00001.0\u00000.50.00.51.0Muni=3, J/U=\u00000.1,r=\u00000.4Rx,xuRy,yuRz,zu\n<latexit sha1_base64=\"2hLODGMPDy74G5ldR05CV6wYc7g=\">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</latexit>r=\u00000.4\n0.00.10.20.3T/E0\u00001.0\u00000.50.00.51.0Muni=3, J/U=\u00000.1,r=\u00000.4Rx,xuRy,yuRz,zu\n<latexit sha1_base64=\"f5RI9QClJOPHM2dinkR4cFHy+hU=\">AAAJC3iclVZNa9RQFL1t/Yj1o61uBDfFoaKb8kZExVWpXehG2mmnLXSGIclkYpi8JCSZ0ZrOHxDEpYorBRfi0p24E8E/4KI/QVxW0IULT+5kprXNR5tAct9995x333nv3UTzbCsIhdgeGR07dvzESeXU+OkzZ89NTE6dXw3cjq8bVd21XX9dUwPDthyjGlqhbax7vqFKzTbWtPbduH+ta/iB5Tor4aZn1KVqOlbL0tUQrupKI9J7jcmSmBV8TR80yolRouRadKfG/lCNmuSSTh2SZJBDIWybVApwb1CZBHnw1SmCz4dlcb9BPRoHtoMoAxEqvG08TbQ2Eq+DdswZMPoxIlwgXbD0aJpmxHfxXuyIb+KD+CH+ZrJFzBJns4m31scaXmPi6cXl34UoiXdID3dROQgN0dmzivt92G3YYW6cZH1aiPKhlonR8+JjtUJE32aVLKjmsSfWT+/n3X3yYmf5TmUmuiLeip/Q7Y3YFl+gnNP9pb9bMiqvmd0B5hGrLHneDlY2gj9IMu/xetbZN8jwKvhL8E7Dt5vttSTfLEaZw1hDK42zhuh8VjuHdSOVs16YZ9rcZers5ZHmn6aAzNBAHkkFmaqDTFVC7tMim1UFZ5DBuJXBuFWQqQ+/CpZewtc/aT6siCpcIfLxFrfMVPx9Pov5+PgkOano4rEfsCI2qxy3NcbVuGYN/PkMC/tGDhDjw7NQiGxzFYvwbqTkPs/4fIYmtGvhTpt781C6d1Kx1qE07yO7/D2I2wFwcV13CrAmr2nfUpO+fMTu7ootD3kWIYLhfgp4HxeP0RnugQ5XZh+3i8iiFRigmhztHAFpDpHx7o+/AX5hZRyMZrPX3PMFzj6dA4y/F4N/hPL+P4KDxur12fLNWbF0ozQ3n/wtKHSJLqNilukWzdE9WqQqRrXoOb2kV8oz5aPySfncDx0dSTAX6L9L+foP1i3duw==</latexit>Tc(c)\nFIG. 8. Temperature dependence of (a) the order parameter, (b, left axis) the entropy and (b, right axis) the speci\fc heat for\nn= 3 uniform spherical model with J=U =\u00000:1. (c) Similar order-parameter plots for the n= 3 single-sublattice model with\ncoupling constant ratio r=\u00000:4. The energy unit is E0=t2=U.\nWe show the order parameters for n= 3,J=U =\u00000:1\nuniform model with the coupling constant ratio r=\u00000:4\nin Fig. 8(c). Since the magnitude of the modi\fed cou-\npling constants satis\fes ~IR>~IQin the present condition,\nwe obtain the solution for R\u0016;\u0016moments. Recalling the\nde\fnition of the Rmoment, we may rewrite the order\nparameter as R\u0016;\u0016\u0018L\u0016S\u0016symbolically. Therefore, it is\ninterpreted that the system has the e\u000bective spin-orbit\ncoupling spontaneously. The wave function is written as\n\f\f\f~ E\n=1p\n3 \njx;\"i\u0000 ijy;\"i\u0000jz;#i\n\u0000jx;#i\u0000 ijy;#i\u0000jz;\"i!\n; (64)\nwhich indicates that the ground state is entangled with\nrespect to spin and orbital. These doubly degenerate\nground states are connected with each other by the time-\nreversal symmetry.\nThis \\spontaneous spin-orbit coupling\" splits the six-\nfold degeneracy into two-fold and four-fold multiplets,\nand which is realized in the ground state is dependent\non the sign of the order parameters. Our solutions show\nthat the ground state is always doubly degenerate, and\nthis should be related to the minimization of the entropy\nat low temperatures.\nThus, although the system at the original parameter\nshows the doublon-orbital ordering ( Q), the system is\nlocated near the parameter range where the intriguing R\norder occurs. As discussed in Sec. II A the original spin-\norbit coupling \u0003 SOis tiny, but it might enter through the\nR-type ordering. Such situation is realized only for n= 3\nmodel with the antiferromagnetic Hund's coupling.\n020406080100T[K]\u00001.0\u00000.50.00.51.0Mµu,sJ/U=\u00000.1SzsQx2\u0000y2sQz2uQxyuTx2\u0000y2,zuTz2,zsTxy,zs\n<latexit sha1_base64=\"X52I/2YrE97X48H0XCX0tZyVuPU=\">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</latexit>Tc1\n<latexit sha1_base64=\"XlJmCfpEjjRXMdHUfDX7W/sBXjY=\">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</latexit>Tc2\n<latexit sha1_base64=\"U7nRNsthHSA2oaYSY1diN0tj1bk=\">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</latexit>Tc3\n\u00000.04\u00000.020.000.020.04\u00000.050.000.05\nSzsQx2\u0000y2sQz2uQxyuTx2\u0000y2,zuTz2,zsTxy,zs020406080100T[K]020040060080010001200eigenvalues [K](b)(a)FIG. 9. Temperature dependence of (a) the order parame-\nter and (b) the eigenvalues of the matrix ^ afor A15 fulleride\nmodel. The blue \flled symbol in (b) corresponds to the so-\nlution given in (a). The red circle represents the solution\nwithout the phase transition at Tc3.\nIV. NUMERICAL RESULTS FOR FULLERIDES\nWe show the numerical results for the fulleride in the\nstrong coupling regime by using the hopping parameters\nobtained by the \frst principles calculation [40]. We take\nthe intra-orbital Coulomb interaction U= 1eV and the\nHund's coupling J=U =\u00000:1 in the following.\nA. A15 structure\nFirst of all we show in Fig. 9(a) the temperature depen-\ndence of order parameters for the strong-coupling limit\nmodel of the realistic fulleride material with the A15\nstructure. The hopping parameters for Cs 3C60is cho-12\nsen (A15-Cs( Vopt\u0000P\nSC ) in Ref. [40]). The lattice struc-\nture is a bipartite lattice, and A and B sublattices are\nconnected with each other by screw transformation (i.e.,\ntranslation plus four-fold rotation). As shown in the \fg-\nure, atTc1'80K, the antiferromagnetic moment (AF- S)\nappears by the second-order phase transition. At lower\ntemperatures, we identify the two successive phase tran-\nsitions (Tc2;3) with orbital moment Qand spin-orbital\nmomentT. These two Q;T moments share the same\nsymmetry under the presence of AF- Szorder. We cannot\nsimply conclude which one is the primary order parame-\nter, because the interaction has complicated form for the\nrealistic model and cannot be decomposed to each con-\ntribution as in the spherical model. We also note that\nour choice of parameter is not \fne-tuned to reproduce\ncorrectly the transition temperature in the actual ma-\nterials, although our results can be compared with the\nexperiments semi-quantitatively.\nWe show in Fig. 9(b) the eigenvalues (\flled blue sym-\nbols) of the Hessian matrix de\fned in Eq. (36). All\nthe values are non-negative, and therefore the system\nis stable. On the other hand, we can also calculate the\nlow-temperature solutions by suppressing the ordering at\nTc3. The results are plotted as the open red symbols in\nFig. 9(b). In this case, the eigenvalues become partially\nnegative and hence the system is not stable although the\nentropy goes to zero even in this case. Thus, the emer-\ngence of the order at Tc3is essential in order to reach the\nstable ground state.\nWe discuss the origin of the second orbital order at Tc3\nin more detail. Below, we concentrate on the properties\nofQmoments to make the discussion simple, since the\nsymmetry of Qis same as that of Tbelow the transition\ntemperature Tc1. Figure 10(a) shows the orbital order pa-\nrameters for sublattice A (left panel) and B (right panel)\nslightly below the transition temperature Tc2(but above\nTc3). The three patterns are obtained depending on the\ninitial condition and hence are degenerate solutions. It\nis seen from Fig. 10(a) that the plane of X\u000b=Qz2\n\u000band\nY\u000b=Qx2\u0000y2\n\u000b has a three-fold rotational symmetry and\nthe equilateral triangle points, where the free energy min-\nima are located, are tilted from the Xaxis. This tilt angle\nremains \fnite at low temperatures below Tc3.\nThis result can be understood from the Landau theory:\nwe can show that, without four-fold rotational symmetry\nas inThpoint group symmetry in fulleride materials,\nthe Landau free energy is written in the restricted order-\nparameter space as\nFL=X\n\u000b=A;Bh\nc1X\u000b(X2\n\u000b\u00003Y2\n\u000b) +c2s\u000bY\u000b(3X2\n\u000b\u0000Y2\n\u000b)i\n;\n(65)\nwheres\u000b=A= +1 and s\u000b=B=\u00001. We have consid-\nered only the third-order term for our purpose. This is\nconsistent with the numerical results and the tilt of the\nangle is due to the presence of c2term. The tilt angle\nis estimated with the polar coordinates X=rcos\u0012and\n\u00000.250.000.25Qz2\u00000.4\u00000.20.00.20.4Qx2\u0000y2\u0000Qx2\nQy2↵=A,T.Tc2\n\u00000.250.000.25Qz2\u00000.4\u00000.20.00.20.4Qx2\u0000y2↵=B,T.Tc2\n\u00000.50.00.5Qz2\u00000.50\u00000.250.000.250.50Qx2\u0000y2↵=A,T!0\n\u00000.50.00.5Qz2\u00000.50\u00000.250.000.250.50Qx2\u0000y2↵=B,T!0(a)\n(b)FIG. 10. (a) Sublattice-dependent order parameters in the\nplane ofQz2-Qx2\u0000y2for A (left) an B (right) sublattices at\nT= 40:4K (< T c2). The similar plots at low-temperature\nlimit without the transition at Tc3is shown in (b). The dashed\ncircles in (a,b) correspond to the solutions in the system with\nfour-fold symmetries. Each color shows di\u000berent kind of solu-\ntions, which share the same free energy. The gray arrows with\nQx2orQy2are the guide for taking the other quantization\naxis. Speci\fcally, the solution given in Fig. 9(a) corresponds\nto the blue circle in the present \fgure (a). The angle \u001ein the\nleft panel of (a) is the deviation from the horizontal axis.\nY=rsin\u0012, leading to another expression of the free en-\nergyFL/cos(3\u0012+\u001e) with\u001e= tan\u00001c2=c1being the\ntilt angle. For example, one can estimate this angle from\nFig. 10(a) as \u001e= 6:76\u000e. The A15 structure has the screw\nsymmetry, i.e., the combination of the translation along\n[111] and four-fold rotation around x;y;z axes, which\nrelates the order parameters at A and B sublattices. In-\ndeed, the above Landau free energy is invariant under\nthe three-fold rotation and screw transformations.\nIf the four-fold symmetry is present, the condition c2=\n0 or\u001e= 0 is required. In Fig. 10(b), we show the order\nparameters at T!0without the second orbital ordering\nbelowTc3, where the four-fold symmetry seems to be\ne\u000bectively recovered since the tilt angle goes to zero when\nT!0. Hence, the origin of the second orbital order in\nFig. 9(a) below Tc3is interpreted as induced from this\nemergent symmetry at low temperatures which provides\nan additional free energy gain.13\n<latexit sha1_base64=\"f5RI9QClJOPHM2dinkR4cFHy+hU=\">AAAJC3iclVZNa9RQFL1t/Yj1o61uBDfFoaKb8kZExVWpXehG2mmnLXSGIclkYpi8JCSZ0ZrOHxDEpYorBRfi0p24E8E/4KI/QVxW0IULT+5kprXNR5tAct9995x333nv3UTzbCsIhdgeGR07dvzESeXU+OkzZ89NTE6dXw3cjq8bVd21XX9dUwPDthyjGlqhbax7vqFKzTbWtPbduH+ta/iB5Tor4aZn1KVqOlbL0tUQrupKI9J7jcmSmBV8TR80yolRouRadKfG/lCNmuSSTh2SZJBDIWybVApwb1CZBHnw1SmCz4dlcb9BPRoHtoMoAxEqvG08TbQ2Eq+DdswZMPoxIlwgXbD0aJpmxHfxXuyIb+KD+CH+ZrJFzBJns4m31scaXmPi6cXl34UoiXdID3dROQgN0dmzivt92G3YYW6cZH1aiPKhlonR8+JjtUJE32aVLKjmsSfWT+/n3X3yYmf5TmUmuiLeip/Q7Y3YFl+gnNP9pb9bMiqvmd0B5hGrLHneDlY2gj9IMu/xetbZN8jwKvhL8E7Dt5vttSTfLEaZw1hDK42zhuh8VjuHdSOVs16YZ9rcZers5ZHmn6aAzNBAHkkFmaqDTFVC7tMim1UFZ5DBuJXBuFWQqQ+/CpZewtc/aT6siCpcIfLxFrfMVPx9Pov5+PgkOano4rEfsCI2qxy3NcbVuGYN/PkMC/tGDhDjw7NQiGxzFYvwbqTkPs/4fIYmtGvhTpt781C6d1Kx1qE07yO7/D2I2wFwcV13CrAmr2nfUpO+fMTu7ootD3kWIYLhfgp4HxeP0RnugQ5XZh+3i8iiFRigmhztHAFpDpHx7o+/AX5hZRyMZrPX3PMFzj6dA4y/F4N/hPL+P4KDxur12fLNWbF0ozQ3n/wtKHSJLqNilukWzdE9WqQqRrXoOb2kV8oz5aPySfncDx0dSTAX6L9L+foP1i3duw==</latexit>Tc\n0510152025T[K]01002003004005006001/\u0000⌘µ;⌘µu[K]\n0510152025T[K]\u00001.0\u00000.50.00.51.0MµuJ/U=\u00000.1LzuQz2u(b)(a)\n\u00000.04\u00000.020.000.020.04\u00000.050.000.05\nSxSySzLxLyLzQx2\u0000y2Qz2QxyQyzQzxRx,xRy,xRz,xRx,yRy,yRz,yRx,zRy,zRz,zTx2\u0000y2,xTz2,xTxy,xTyz,xTzx,xTx2\u0000y2,yTz2,yTxy,yTyz,yTzx,yTx2\u0000y2,zTz2,zTxy,zTyz,zTzx,z\nFIG. 11. Temperature dependence of (a) the order parameter\nand (b) the inverse of the diagonal susceptibility for uniform\nfcc fulleride model with J=U =\u00000:1.\nB. fcc structure\nFinally, we consider the fulleride material with the fcc\nstructure. The spin-orbital model in the strong-coupling\nlimit is obtained by using the hopping parameters for\nRb3C60in Ref. [40]. Because of the geometrically frus-\ntrated nature of the fcc lattice, we here seek for only the\nspatially uniform ordered states.\nFigure 11(a) shows the temperature evolution of the\norder parameters. Here the primary order parameter is\nthe uniform Lzmoment which breaks the time-reversal\nsymmetry. The Lz-order arises as ( Tc\u0000T)1=2, and the\nQz2is also induced simultaneously with the linear tem-\nperature dependence /(Tc\u0000T). The latter Qmoment is\ninduced from the coupling term with the form ( Lz)2Qz2\nin the Landau free energy. We note that Lzis not in-\nduced when Qz2is a primary order parameter from that\ncoupling, since LzandQz2have di\u000berent time-reversal\nsymmetry [( Lz)2andQz2are same]. The ground-state\nwave function is written in a simple form as\n\f\f\f~ E\n=1p\n2 \njx;\"i\u0000 ijy;\"i\njx;#i\u0000 ijy;#i!\n; (66)\nwhere the complex wave function clearly shows the time-\nreversal symmetry breaking. We note that this or-\nbital moment is not a simple orbital motion around the\nfullerene molecule, but a complex motion of the three\nelectron state given in Eq. (4). In our calculations the\nspinSorder does not occur and the ground state is dou-\nbly degenerate at each cite. The stability of the solution\nis checked by the non-negative eigenvalues of the Hessian\nmatrix.\nThe results found here is di\u000berent from the spherical\ncase discussed in Sec. III C 3. The di\u000berence is due to the\nspeci\fc form of the tight-binding hopping parameters.\nWe show one of the coupling constant ILfor the nearestneighbour sites as\n0\nB@I1I20\nI2\u0000I30\n0 0I41\nCA;R=\u0010a\n2;a\n2;0\u0011\n; (67)\nwhereRis the direction of the NN molecules and ais\nthe lattice constant for the fcc fulleride. The information\nfor the other NN pairs is constructed from the symme-\ntry operations. The values of the matrix element are\nI1= 12:5;I2= 9:77;I3= 0:511 andI4= 20:1 in units of\nK in the present models. The coupling constant has the\nsame symmetry as the hopping parameters in Ref. [40]\nas required by the space group symmetry. The near-\nest neighbor coupling constant is largest and is positive,\nwhich favors the uniform magnetic orbital moment L.\nAs for the next nearest neighbour site, the coupling con-\nstant matrices are diagonal and every component of them\nis smaller than nearest neighbour ones.\nSince the spin Smoment has the same symmetry as\nL, it can in general be simultaneously induced under the\nsmall but \fnite spin-orbit coupling. However, as dis-\ncussed in Sec. II A, the magnitude of the e\u000bective spin-\norbit coupling for the doublon orbital is \u0003 SO\u001810\u00009eV,\nwhich can be regarded as zero in practice. Hence, the\nspin order can occur independently at low temperatures.\nThe absence of the spin Sorder is interpreted from the\npoint of view of the coupling constant. Figure 11(b)\nshows that the temperature dependence of the inverse\nof the diagonal susceptibilities. The blue lines represents\nthe magnetic susceptibility ( S), which indicates that the\ncoupling constants of Sare antiferromagnetic owing to\nthe negative Curie-Weiss temperature. In this case, the\ntransition temperature should be very low due to the ge-\nometrical frustration of fcc lattice, but \fnally the system\nshould show some magnetic ordering [41].\nC. Discussion\nThe models in this section are based on the band-\nstructure calculation results. Furthermore, the fulleride\nmaterials can be located in the Mott insulator regime\ndepending on the pressure. Hence, our results are poten-\ntially applied to the real materials. In fulleride materials,\nthe antiferromagnetic orders is experimentally identi\fed\nat low temperatures, while the orbital orders are not yet\nreported. Based on our results, we propose that at low\ntemperatures the orbital ordered moments Qare induced\nwith two successive transitions for A15 structures, and\nLmoments may appear for fcc structures. Such \fnger-\nprints of the orbital orders may be found in thermody-\nnamic quantities in principle. Here the orbital moment\nis not for a usual electron but for the doublons speci\fc\nto the systems with antiferromagnetic Hund's coupling\nas emphasized in the present paper. On the other hand,\nsince the real compounds are polycrystals and the disor-\nder e\u000bects are also present, the orbital orders might be14\nsmeared out in realistic situations. In this context, the\ne\u000bect of disorders on our spin-orbital model is interesting\nfuture issues which make it more direct to compare the\ntheoretical results with experimental observations. More-\nover, the antiferromagnetic Hund's coupling originates\nfrom the electron-phonon coupling. The resultant retar-\ndation e\u000bects are also the parts not included in this paper\nand an important issue for the more realistic arguments.\nV. SUMMARY AND OUTLOOK\nIn order to clarify the properties of strongly corre-\nlated electrons in fulleride superconductors, we have con-\nstructed the spin-orbital model in the strong coupling\nlimit. We begin with the three-orbital Hubbard model\nwith the antiferromagnetic Hund's coupling which is re-\nalized by the coupling between the electronic degrees\nof freedom and anisotropic Jahn-Teller molecular vibra-\ntions. In this case, the pair hopping e\u000bect among the\ndi\u000berent orbitals becomes relevant in strong contrast to\nthe multiorbital d-electron systems with the ferromag-\nnetic Hund's coupling. We have mainly considered the\nhalf-\flledn= 3 case relevant to real materials, where it\nis composed of the singly-occupied (singlon) plus doubly\noccupied orbitals (doublon) as illustrated in Fig. 1. The\ncorrelated ground state for an isolated fullerene molecule\nis six-fold degenerate and is characterized by the spin\nand orbital indices. This is the situation similar to the\nn= 1 ground states and the analogy between n= 3\nandn= 1 helps us for interpreting the results. The\nusual orbital moment, which is present for the n= 1\ncase, is absent for n= 3 because of the correlated na-\nture of the wave function, and instead the active orbital\nmoment characteristic for doublons exists. As the result,\nthe spin-orbit coupling, which is the order of 1meV for p-\nelectrons, becomes 1neV because of the extended nature\nof the molecular orbitals and the correlation e\u000bects.\nWe have applied the second-order perturbation the-\nory with respect to the inter-molecule hopping, and have\nobtained the localized spin-orbital model speci\fc to the\nfullerides. The obtained spin-orbital model is analyzed\nby employing the mean-\feld approximation. For refer-\nence, we have \frst solved the spherical n= 1 model for\nboth ferromagnetic and antiferromagnetic Hund's cou-\nplings with a spherical limit for the bipartite lattice. We\nthen apply our method to the n= 3 model where the\nmagnetic order is found at relatively high temperatures\nand the orbital order also occurs at lower temperatures.The temperature dependences of the physical quantities\nsuch as order parameters, internal and free energies, spe-\nci\fc heat, entropy, and susceptibilities are investigated\nin detail. The thermodynamic stability is also studied\nbased on the Hessian matrix derived from the inverse\nsusceptibilities, and are checked by con\frming that all\nthe eigenvalues are non-negative.\nWe have also considered the realistic situation in alkali-\ndoped fullerides, by using the tight-binding parameters\nderived from the \frst principles calculations. For the\nchoice of the lattice structure, we have taken both the\nbipartite A15 and fcc structures, whose hopping parame-\nters have been derived in Ref. [40]. For the A15 structure,\nthe antiferromagnetic order occurs at high temperatures,\nand the electric orbital orders arise at lower tempera-\ntures with two successive transitions. The \frst orbital\norder is already captured in the spherical model, but the\nsecond orbital order is characteristic for the Thsymmetry\nin fulleride materials where only the three-fold rotation\nsymmetry exists. This point has been discussed in de-\ntail based on the Landau theory. For the fcc model, we\nhave concentrated on the spatially uniform solutions due\nto the geometrically frustrated nature of the lattice. We\nhave found that the magnetic orbital order occurs. Al-\nthough this orbital moment has the same symmetry as\nthe electronic spin, the spin moment is not induced simul-\ntaneously in fulleride since the spin-orbit coupling is tiny\nas mentioned above. Thus the spin-moment can order in-\ndependently, and is expected to be antiferromagnetically\nordered in the ground state where the transition tem-\nperature is expected to be low owing to the geometrical\nfrustration of the fcc lattice.\nOur formalism itself is constructed in a very general\nway, and can be applied to any systems in the strong\ncoupling limit with integer \fllings per atom or molecule.\nIn this context, it would be desirable to develop the gen-\neral framework for the strong-coupling-limit spin-orbital\nmodel with the combination of the hopping parameters in\nthe Wannier functions obtained from the band-structure\ncalculations. This application is of interest speci\fcally\nin studying the ordered state of the multiorbital elec-\ntronic systems including transition metals and organic\nmaterials. This point remains to be explored and is an\nintriguing issue in the future.\nACKNOWLEDGEMENT\nThis work was supported by JSPS KAKENHI Grants\nNo. JP18K13490 and No. JP19H01842.\n[1] A. F. Hebard, M. J. Rosseinsky, R. C. Haddon, D. W.\nMurphy, S. H. Glarum, T. T. M. Palstra, A. P. Ramirez,\nand A. R. Kortan, Nature (London) 350, 600 (1991).[2] M. J. Rosseinsky, A. P. Ramirez, S. H. Glarum, D. W.\nMurphy, R. C. Haddon, A. F. Hebard, T. T. M. Palstra,\nA. R. Kortan, S. M. Zahurak, and A. V. Makhija, Phys.\nRev. Lett. 66, 2830 (1991).15\n[3] K. Holczer, O. Klein, S. mei Huang, R. B. Kaner, K. jian\nFu, R. L. Whetten, and F. Diederich, Science 252, 1154\n(1991).\n[4] K. Tanigaki, T. W. Ebbesen, S. Saito, J. Mizuki, J. S.\nTsai, Y. Kubo, and S. Kuroshima, Nature (London) 352,\n222 (1991).\n[5] R. M. Fleming, A. P. Ramirez, M. J. Rosseinsky, D. W.\nMurphy, R. C. Haddon, S. M. Zahurak, and A. V.\nMakhija, Nature (London) 352, 787 (1991).\n[6] A. Y. Ganin, Y. Takabayashi, Y. Z. Khimyak, S. Mar-\ngadonna, A. Tamai, M. J. Rosseinsky, and K. Prassides,\nNat. Mater. 7, 367 (2008).\n[7] Y. Takabayashi, A. Y. Ganin, P. Jegli\u0014 c, D. Ar\u0014 con,\nT. Takano, Y. Iwasa, Y. Ohishi, M. Takata, N. Takeshita,\nK. Prassides, and M. J. Rosseinsky, Science 323, 1585\n(2009).\n[8] A. Y. Ganin, Y. Takabayashi, P. Jegli\u0014 c, D. Ar\u0014 con,\nA. Poto\u0014 cnik, P. J. Baker, Y. Ohishi, M. T. McDonald,\nM. D. Tzirakis, A. McLennan, G. R. Darling, M. Takata,\nM. J. Rosseinsky, and K. Prassides, Nature (London)\n466, 221 (2010).\n[9] O. Gunnarsson, Rev. Mod. Phys. 69, 575 (1997).\n[10] M. Fabrizio and E. Tosatti, Phys. Rev. B 55, 13465\n(1997).\n[11] M. Capone, M. Fabrizio, C. Castellani, and E. Tosatti,\nScience 296, 2364 (2002).\n[12] M. Capone, M. Fabrizio, C. Castellani, and E. Tosatti,\nRev. Mod. Phys. 81, 943 (2009).\n[13] Y. Nomura, S. Sakai, M. Capone, and R. Arita, J. Phys.\nCondens. Matter 28, 153001 (2016).\n[14] Y. Takabayashi and K. Prassides, Phil. Trans. R. Soc. A\n374, 20150320 (2016).\n[15] R. H. Zadik, Y. Takabayashi, G. Klupp, R. H. Colman,\nA. Y. Ganin, A. Poto\u0014 cnik, P. Jegli\u0014 c, D. Ar\u0014 con, P. Ma-\ntus, K. Kamar\u0013 as, Y. Kasahara, Y. Iwasa, A. N. Fitch,\nY. Ohishi, G. Garbarino, K. Kato, M. J. Rosseinsky, and\nK. Prassides, Sci. Adv. 1, e1500059 (2015).\n[16] Y. Kasahara, Y. Takeuchi, R. H. Zadik, Y. Takabayashi,\nR. H. Colman, R. D. McDonald, M. J. Rosseinsky,\nK. Prassides, and Y. Iwasa, Nat. Commun. 8, 14467\n(2017).\n[17] S. Han, M.-X. Guan, C.-L. Song, Y.-L. Wang, M.-Q. Ren,\nS. Meng, X.-C. Ma, and Q.-K. Xue, Phys. Rev. B 101,\n85413 (2020).\n[18] M.-Q. Ren, S. Han, S.-Z. Wang, J.-Q. Fan, C.-L. Song,\nX.-C. Ma, and Q.-K. Xue, Phys. Rev. Lett. 124, 187001\n(2020).\n[19] M. Mitrano, A. Cantaluppi, D. Nicoletti, S. Kaiser,\nA. Perucchi, S. Lupi, P. D. Pietro, D. Pontiroli, M. Ricc\u0012 o,S. R. Clark, D. Jaksch, and A. Cavalleri, Nature (Lon-\ndon)530, 461 (2016).\n[20] A. Cantaluppi, M. Buzzi, G. Jotzu, D. Nicoletti, M. Mi-\ntrano, D. Pontiroli, M. Ricc\u0012 o, A. Perucchi, P. D. Pietro,\nand A. Cavalleri, Nat. Phys. 14, 837 (2018).\n[21] M. Capone, M. Fabrizio, P. Giannozzi, and E. Tosatti,\nPhys. Rev. B 62, 7619 (2000).\n[22] Y. Nomura, S. Sakai, M. Capone, and R. Arita, Sci. Adv.\n1, e1500568 (2015).\n[23] A. Koga and P. Werner, Phys. Rev. B 91, 085108 (2015).\n[24] S. Hoshino and P. Werner, Phys. Rev. B 93, 155161\n(2016).\n[25] K. Steiner, S. Hoshino, Y. Nomura, and P. Werner, Phys.\nRev. B 94, 075107 (2016).\n[26] S. Hoshino and P. Werner, Phys. Rev. Lett. 118, 177002\n(2017).\n[27] K. Ishigaki, J. Nasu, A. Koga, S. Hoshino, and P. Werner,\nPhys. Rev. B 98, 235120 (2018).\n[28] K. Ishigaki, J. Nasu, A. Koga, S. Hoshino, and P. Werner,\nPhys. Rev. B 99, 085131 (2019).\n[29] C. Yue, S. Hoshino, and P. Werner, Phys. Rev. B 102,\n195103 (2020).\n[30] S. Hoshino, P. Werner, and R. Arita, Phys. Rev. B 99,\n235133 (2019).\n[31] T. Misawa and M. Imada, arXiv:1711.10205 (2017).\n[32] K. I. Kugel and D. I. Khomskii, ZhETF Pis. Red. 15,\n629 (1972), [Sov. Phys. JETP Lett. 15, 446 (1972)].\n[33] K. I. Kugel and D. I. Khomskii, Zh. Eksp. Teor. Fiz. 64,\n1429 (1973), [Sov. Phys. JETP 37, 725 (1973)].\n[34] S. Inagaki, J. Phys. Soc. Jpn. 39, 596 (1975).\n[35] S. Ishihara, J. Inoue, and S. Maekawa, Phys. Rev. B 55,\n8280 (1997).\n[36] L. F. Feiner and A. M. Ole\u0013 s, Phys. Rev. B 59, 3295\n(1999).\n[37] B. Normand and A. M. Ole\u0013 s, Phys. Rev. B 78, 094427\n(2008).\n[38] E. Tosatti, N. Manini, and O. Gunnarsson, Phys. Rev. B\n54, 17184 (1996).\n[39] Y. Nomura and R. Arita, Phys. Rev. B 92, 245108 (2015).\n[40] Y. Nomura, K. Nakamura, and R. Arita, Phys. Rev. B\n85, 155452 (2012).\n[41] Y. Kasahara, Y. Takeuchi, T. Itou, R. H. Zadik, Y. Tak-\nabayashi, A. Y. Ganin, D. Ar\u0014 con, M. J. Rosseinsky,\nK. Prassides, and Y. Iwasa, Phys. Rev. B 90, 014413\n(2014)."    },    {        "title": "2208.09409v1.Spin_triplet_Superconductivity_in_Nonsymmorphic_crystals.pdf",        "content": "Spin-triplet Superconductivity in Nonsymmorphic crystals\nShengshan Qin,1,\u0003Chen Fang,2, 1Fu-chun Zhang,1, 3and Jiangping Hu2, 1, 4,y\n1Kavli Institute for Theoretical Sciences and CAS Center for Excellence in Topological Quantum Computation,\nUniversity of Chinese Academy of Sciences, Beijing 100190, China\n2Beijing National Research Center for Condensed Matter Physics,\nand Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China\n3Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, China\n4South Bay Interdisciplinary Science Center, Dongguan, Guangdong Province, China\n(Dated: August 22, 2022)\nSpin-triplet superconductivity is known to be a rare quantum phenomenon. Here we show that nonsymmor-\nphic crystalline symmetries can dramatically assist spin-triplet superconductivity in the presence of spin-orbit\ncoupling. Even with a weak spin-orbit coupling, the spin-triplet pairing can be the leading pairing instability in a\nlattice with a nonsymmorphic symmetry. The underlining mechanism is the spin-sublattice-momentum lock on\nelectronic bands that are protected by the nonsymmorphic symmetry. We use the nonsymmorphic space group\nP4=nmm to demonstrate these results and discuss related experimental observables. Our work paves a new\nway in searching for spin-triplet superconductivity.\nPACS numbers:\nIntroduction. The spin-triplet superconductors, which are\nthe superconducting analogy of the3He superfluid1, have\nbeen long-pursued. They have been proposed to be natu-\nral candidates for the topological superconductors2–8, host-\ning the Majorana modes which are expected to play an es-\nsential role in the fault-tolerant quantum computations9–11.\nIn the past decades, great efforts have been made in pursu-\ning the spin-triplet superconductors12–14. Theoretically, var-\nious mechanisms have been proposed in favor of the spin-\ntriplet superconductivity. For instance, the spin-triplet su-\nperconductivity may arise at ultra low temperature through\nthe Kohn-Luttinger mechanism15; and it can also be induced\nfrom the ferromagnetic spin fluctuations or the ferromag-\nnetic exchange coupling13. Experimentally, Sr 2RuO 4has\nbeen suggested to be a promising candidate for the spin-\ntriplet superconductors16,17. However, recent experiments\nraise doubts on this issue18. In many heavy fermion sys-\ntem such as UPt 319, UTe 220, and the recently synthesized\nK2Cr3As321,22, signatures for the spin-triplet superconductiv-\nity have been observed.\nDuring the past decades, the spin-orbit coupled systems\nhave attracted more and more research attentions. The spin-\norbit coupling (SOC) has been revealed to play an impor-\ntant role in various exotic condensed matter systems such\nas the topological materials23–27. Recent studies suggest\nthat the SOC can help the spin-triplet superconductivity.\nFor instance, it has been predicted the spin-triplet super-\nconductivity may exist in doped superconducting topological\ninsulators28–32and semimetals33,34. Especially, in the doped\ntopological insulator Bi 2Se3nematic superconductivity has\nbeen confirmed experimentally35–40, indicating possible odd-\nparity spin-triplet superconductivity in the system28,29. Be-\nsides the topological materials, in the two-dimensional (2D)\nelectron gas formed at the interface between LaAlO 3and\nSrTiO 341,42, the spin-triplet superconductivity is also pro-\nposed based on large Rashba SOC43,44. However, all these\nstudies rely on a strong SOC, in which pairing forces may be\nsignificantly weakened by the SOC as well.In this Letter, we show that the spin-triplet superconduc-\ntivity can be stabilized by nonsymmorphic symmetries in the\nspin-orbit coupled systems. Even with a weak SOC, the spin-\ntriplet pairing can be the leading pairing instability in a lattice\nwith a nonsymmorphic symmetry. We specify our study with\nthe nonsymmorphic space group P4=nmm (#:129). Due to\nthe nonsymmorphic symmetries, the sublattice degree always\nexists in the system. In the presence of the SOC, the sub-\nlattice degree intertwines with the spin degree. Correspond-\ningly, the spin, sublattice and momentum are locked with each\nother, forming a spin-sublattice-momentum lock texture on the\nnormal-state energy bands. The spin-triplet pairing state is\nalways favored due to spin-sublattice-momentum lock when\nthere is a pairing force between the two sublattices.\nWe first briefly review the space group G=P4=nmm ,\nwhich is nonsymmorphic. There are 16 symmetry operations\nin its quotient group G=TwithTbeing the translation group.\nMore specially,G=Tcan be written in the following concise\ndirect product form45\nG=T=D2d\nZ2; (1)\nin a sense that symmetry operations are equivalent if they dif-\nfer by a lattice translation. We specify the symmetry group\nwith a quasi-2D lattice shown in Fig.1(a), which is similar to\nthe structure of the monolayer FeSe. As shown in the lattice,\nthe fixed point of point group D2din Eq.(1) is at the lattice\nsites, andZ2is a two-element group including the inversion\nsymmetry which is defined at the bond center between two\nnearest lattice sites. According to Eq.(1), we can choose the\ngenerators of the quotient group G=Tas the inversion sym-\nmetryfIj\u001c0g, the mirror symmetry fMyj0gand the rotoin-\nversion symmetry fS4zj0g46, where the symmetry operators\nhave been expressed in the form of the Seitz operators and\n\u001c0=a1=2 +a2=2witha1anda2being the primitive lattice\ntranslations along the xandydirections in Fig.1(a).\nLow-energy theory near (\u0019;\u0019).A standard group theory\nanalysis shows that the space group P4=nmm merely has\none single 4D irreducible representation at the Brillouin zonearXiv:2208.09409v1  [cond-mat.supr-con]  19 Aug 20222\ncorner (\u0019;\u0019), i.e. the M point, in the spinful condition47.\nThe above conclusion straightforwardly leads to three impor-\ntant implications. (i) For systems respecting the space group\nP4=nmm , in the presence of SOC all the energy bands are\nfourfold degenerate at the M point. (ii) All the fourfold de-\ngenerate bands respect the same low-energy effective model.\n(iii) One can use arbitrary orbital to construct the low-energy\neffective theory near M, and for simplicity we consider one s\norbital at each lattice site in Fig.1(a) in the following.\nWith the above preparation, we can construct the low-\nenergy effective theory near M. As all the symmetry opera-\ntions inG=Tpreserve at (\u0019;\u0019), we need to derive the matrix\nform of the symmetry generators of G=T. By a careful analy-\nsis, we obtain the matrix form of the symmetry operators as,\nI=s0\u001b1,My=is2\u001b3andS4z=eis3\u0019=4\u001b3, whereI,My\nandS4zstand forfIj\u001c0g,fMyj0gandfS4zj0grespectively\n(details in SM). In the matrix form, siand\u001bi(i= 1;2;3)\nare the Pauli matrices for the spin and the two sublattices\nrespectively, and s0and\u001b0the corresponding identity ma-\ntrices. The above matrices are actually a set of irreducible\nrepresentation matrices for space group P4=nmm at M. Be-\nsides the crystalline symmetries, the time reversal symmetry\nisT=is2\u001b0KwithKthe complex conjugation operation.\nThe low-energy effective theory near M for group\nP4=nmm is generally depicted by the sixteen \u0000 =si\u001bjma-\ntrices. In deriving the effective model, it is convenient to\nfirst constrain the system by the time reversal symmetry and\nthe inversion symmetry, and then consider the constraints of\nother crystalline symmetries. After some algebra, we classify\nthe symmetry allowed \u0000matrices along with the k-dependent\nfunctions, and obtain the low-energy effective Hamiltonian as\nfollows48(details in SM)\nHeff(k) =m(k)s0\u001b0+\u0015kxs2\u001b3+\u0015kys1\u001b3\n+t0kxkys0\u001b1; (2)\nwherem(k) =t(k2\nx+k2\ny). Notice that kx=yis defined accord-\ning to the M point here. To have a more intuitive impression\non the effective theory in Eq.(2), one can understand the pa-\nrameters in the lattice shown in Fig.1(a). Specifically, t(t0)\ndescribes the hopping between the intrasublattice (intersub-\nlattice) nearest neighbours, and \u0015is the inversion-symmetric\nRashba SOC arising from the mismatch between the lattice\nsites and the inversion center49, i.e. the local inversion-\nsymmetry breaking50. We show the band structures calculated\nfrom the effective Hamiltonian in Eq.(2) in Fig.1(b). Due to\nthe presence of both the time reversal and inversion symme-\ntries, all the energy bands are twofold degenerate.\nSpin-sublattice-momentum lock. In centrosymmetric sys-\ntems, the local inversion-symmetry breaking can intertwine\nthe different degrees of freedom51–53. Here, for systems re-\nspecting the space group P4=nmm , symmetries enforce the\nspin degree locked to the sublattice degree on the energy\nbands and the sublattice-distinguished spin is nearly fully po-\nlarized for small Fermi surfaces near (\u0019;\u0019). Before the de-\ntailed calculations, we first consider the symmetry constraints.\nAs shown in Fig.1(a), the inversion symmetry exchanges the\ntwo sublattices, while the time reversal symmetry does not\nFIG. 1: (color online) (a) A sketched quasi-2D lattice structure re-\nspecting the P4=nmm space group: A and B indicate the sublat-\ntices related by the nonsymmorphic symmetries, the shadow region\nindicates the unit cell, and the red point is the inversion center lo-\ncated at the bond center between two nearest neighbouring sites. (b)\nThe band structure near the M point, plotted from the Hamiltonian\nin Eq.(2) with parameters ft;t0;\u0015g=f1:0;0:8;0:12g. (c) and (d)\nshow the spin polarizations on the lower and upper energy bands in\n(b) respectively: the spin polarization contributed by the A (B) sub-\nlattice is labeled by the red (blue) arrowed line, with the length of\nthe line indicating the strength of the polarization. The gray lines in\n(c)(d) show the Fermi surfaces for chemical potential \u0016= 0:2.\nchange the spacial position. On the other hand, the inver-\nsion symmetry preserves the spin but the time reversal sym-\nmetry flips the spin. Therefore, considering the combina-\ntion of the time reversal symmetry and inversion symmetry,\nat each kpoint the spin polarizations from the two sublattices\nare always opposite. Moreover, due to the mirror symmetry\nfMx=yj0g, the spin is polarized perpendicular to the mirror\nplane along kx=y= 0, i.e. the Brillouin zone boundary.\nBased on the low-energy effective theory in Eq.(2), the\nspin polarizations contributed by the different sublattices,\ni.e.hsAiandhsBi, can be calculated analytically (details\nin SM). A direct calculation shows that hsBi=\u0000hsAi=\n(sin\u0012;cos\u0012;0)=(q\n1 +t02k2sin22\u0012=4\u00152)at point k, with k\nwritten as (kx;ky) = (kcos\u0012;ksin\u0012). We sketch the results\nin Fig.1(c)(d). As shown, both hsAiandhsBilie in thexy\nplane and wind around the M point anticlockwise. Moreover,\nthe spin polarization reaches its minimum along kx=ky, and\nis nearly fully polarized on the bands near the Brillouin zone\nboundary. It is worth pointing out that, the fully polarized spin\nalong the Brillouin zone boundary satisfying hsAi=\u0000hsBi\nis consistent with the matrix form of the mirror symmetries at\nM, i.e.Mx=is1\u001b3andMy=is2\u001b354.\nSuperconductivity. For the effective theory in Eq.(2), we3\nconsider the possible superconductivity induced by the phe-\nnomenological density-density interactions\nHint=Z\ndq[U2X\ni=1ni(q)ni(\u0000q) + 2Vn1(q)n2(\u0000q)];(3)\nwheren1(q) =P\n\u0014=\";#1p\nNR\ndkcy\nk;\u0014ck+q;\u0014andn2(q) =\nP\n\u0014=\";#1p\nNR\ndkdy\nk;\u0014dk+q;\u0014are the density operators for the\nA and B sublattices in Fig.1(a) respectively, and UandVare\nthe intrasublattice and intersublattice interactions respectively.\nIn Eq.(3), we focus on the momentum-independent interac-\ntions in the weak-coupling condition. Obviously, the negative\nU(V) correspond to the attractive interaction. Actually, the\nphenomenological interactions in Eq.(3) arise from the short-\nrange density-density interactions in the real space. Specifi-\ncally,Uis the onsite interaction, and Vis the leading-order\nterm, i.e. the momentum-independent part, in the intersublat-\ntice interaction between nearest neighbours (details in SM).\nFrom the interactions in Eq.(3), in the mean-field level\nonly the momentum-independent superconducting orders are\nexpected55. Due to the fermionic statistics of electrons,\nthe pairing orders are required to satisfy ^\u0001(\u0000k)\u0001is2\u001b0=\n\u0000(^\u0001(k)\u0001is2\u001b0)T, where the pairing term is  y(k)^\u0001(k)\u0001\nis2\u001b0 y(\u0000k)in the basis  y(k) = (cy\nk;\";cy\nk;#;dy\nk;\";dy\nk;#).\nThe pairing orders can be further classified in accordance\nwith the symmetry group of the system, and we classify the\nmomentum-independent pairing orders and present the results\nin Table.I. As shown, the pairing orders belong to five differ-\nent pairing symmetries in the A1g,B2g,A2u,B2uandEu\nrepresentations of the D4hgroup, with the AandBrepresen-\ntations being 1D and the Erepresentation 2D. To show the\nmeaning of the pairing orders clear, we list the explicit form\nof the superconducting pairing as follows\n^\u0001A1g:cy\nk;\"cy\n\u0000k;#+dy\nk;\"dy\n\u0000k;#; (4)\n^\u0001B2g:cy\nk;\"dy\n\u0000k;#+dy\nk;\"cy\n\u0000k;#;\n^\u0001A2u:\u0000icy\nk;\"dy\n\u0000k;#+idy\nk;\"cy\n\u0000k;#;\n^\u0001B2u:cy\nk;\"cy\n\u0000k;#\u0000dy\nk;\"dy\n\u0000k;#;\n^\u0001Eu: (icy\nk;\"dy\n\u0000k;\"\u0000icy\nk;#dy\n\u0000k;#;cy\nk;\"dy\n\u0000k;\"+cy\nk;#dy\n\u0000k;#):\nAs mentioned, the inversion symmetry in space group\nP4=nmm exchanges the two sublattices. ^\u0001A1gand^\u0001B2gare\nspin-singlet pairings with even parity, and ^\u0001A1goccurs in the\nsame sublattice while ^\u0001B2gis between the different sublat-\ntices. ^\u0001B2uis the intrasublattice spin-singlet pairing with odd\nparity, whereas ^\u0001A2uand^\u0001Euare the odd-parity spin-triplet\npairings between the different sublattices.\nTo find out the superconducting ground state, we solve the\nfollowing linearized gap equations (details in SM)\n^\u0001A1g;B2u:\u0000U\u001fA1g;B2u(Tc) = 1; (5)\n^\u0001B2g;A2u;Eu:\u0000V\u001fB2g;A2u;Eu(Tc) = 1;\nwhere we have used the fact that ^\u0001A1gand^\u0001B2ucan only\nresult from the intrasublattice interaction U, and ^\u0001B2g,^\u0001A2uTABLE I: Classification of the possible momentum-independent\npairing potentials corresponding to the interactions in Eq.(3), accord-\ning to the irreducible representations of the D4hpoint group. Here,\nthe pairing potentials are in the form  y(k)^\u0001\u0001is2\u001b0 y(\u0000k), with\nthe basis being  y(k) = (cy\nk;\";cy\nk;#;dy\nk;\";dy\nk;#).\nES4zI M y^\u0001\nA1g 1 1 1 1 s0\u001b0\nB2g 1 -1 1 -1 s0\u001b1\nA2u 1 -1 -1 1 s3\u001b2\nB2u 1 1 -1 1 s0\u001b3\nEu 2 0 -2 0 (s1\u001b2;s2\u001b2)\nand^\u0001Euonly arise from the intersublattice interaction V. In\nEq.(5),\u001fis the finite-temperature superconducting suscepti-\nbility for each irreducible representation pairing channel in\nTable.I, which can be calculated as\n\u001f(Tc) =F(Tc)X\nsZ\nd\u0012D(\u0012)X\ns0=s;\u0016sjhus;kj^\u0001jus0;kij2:(6)\nIn the above equation, jus;kiis the wavefunction for the state\non the Fermi surface contributed by band s, andju\u0016s;ki=\nITjus;kiis the state degenerate with jus;kidue to the pres-\nence of both the inversion symmetry Iand the time re-\nversal symmetryT.F(Tc) =1\n2NR!0\n\u0000!01\n2\u0018tanh\f\u0018\n2d\u0018is a\ntemperature-dependent constant with \f= 1=kBTcand!0the\nenergy cutoff near the Fermi energy. D(\u0012) = 2dk0=d\u0018s;k0\nis the density of states on the Fermi surface. By solving\nEq.(5), we can get the superconducting transition temperature\nfor each pairing channel, and the state with the highest Tcis\nthe ground state.\nFIG. 2: (color online) Superconducting phase diagram versus the\nchemical potential \u0016, the SOC\u0015, and the interaction U=V , assuming\nUandVboth attractive. The colored surface in the figure is the phase\nboundary between the different superconducting ground states. In\nthe calculation, the other parameters are ft;t0g=f1:0;0:8g. Here,\nonly the condition for 2jUj>jVjis shown. For the following two\nconditions, (i) 2jUj<jVjand (ii)U > 0andV < 0, only theB2g\nandA2ustates can appear in the phase diagram, with their phase\nboundary always the same with that at 2U=V.\nAccording to Eqs.(5)(6), the superconducting instability\ncan merely arise from the attractive interactions. Moreover,4\na direct calculation shows that the superconducting suscep-\ntibilities always satisfies \u001fB2u< \u001fA1gand\u001fA2u= 2\u001fEu\n(details in SM), meaning that the B2uandEupairing states\ncan never be the ground states. Consequently, in the condi-\ntion withU < 0andV > 0, i.e. the intrasublattice attractive\nand intersublattice repulsive interactions, the ground state is\nalways theA1gstate; and in the condition with U > 0and\nV < 0, theB2gandA2ustates can be the superconducting\nground states. If both of the interactions are attractive, the\nA1g,B2gandA2upairing states can appear in different re-\ngions in the parameter space, and the corresponding phase di-\nagram is presented in Fig.2. In the phase diagram, we merely\nshow the condition for U=V > 0:5. Whereas, the phase dia-\ngram forU=V < 0:5is independent with U=V , and only the\nB2gandA2ustates can be the ground states with their phase\nboundary always the same with that at 2U=V. The phase\nboundary between the B2gandA2ustates at 2U=Valso\napplies to the condition with U > 0andV < 0. It is worth\npointing out that, all the states in the phase diagram are fully\ngapped. Especially, the B2gstate is actually similar to the\nnodelessd-wave state in the iron-based superconductors56–58.\nA remarkable feature in the phase diagram in Fig.2 is that,\nthe spin-triplet A2ustate occupies a large area and it can\nbe the ground state even in the weak SOC limit. The phe-\nnomenon is closely related to the symmetry-enforced spin-\nsublattice-momentum lock on the normal-state energy bands\nshown in Fig.1. As analyzed, near the M point the sublattice-\ndistinguished spin polarization lies in the xyplane and the\nstrength is proportional to 1=q\n1 +t02k2sin22\u0012=4\u00152satisfy-\ninghsB(k)i=hsA(\u0000k)i. In the small chemical potential\ncondition, the spin on the Fermi surface is nearly fully polar-\nized. When Cooper pair forms between two electrons with op-\nposite momenta, the spin-sublattice-momentum lock in Fig.1\nenforces the equal-spin pairing state with the spin polarized\nin thexyplane in the intersublattice channel, which is exactly\ntheA2ustate in Fig.2. Moreover, since the fully polarized\nspin near M is enforced by symmetries which is regardless\nof the strength of the SOC, the A2ustate can appear as the\nground state in the weak SOC condition as long as the chem-\nical potential is small. In the large chemical potential con-\ndition, the average spin polarization on the large Fermi sur-\nface is weak. Correspondingly, the spin-singlet A1gandB2g\nstates become more favorable. It is worth mentioning that in\nthe limit\u0016!0and\u0015!0, the spin-singlet states compete\nwith the spin triplet state, due to the concentric Fermi surface\nstructure arising from the fourfold band degeneracy at M as\nindicated in Fig.1.\nExperimental signatures. The different states in the phase\ndiagram in Fig.2 can be distinguished in experiments. In nu-\nclear magnetic resonance measurements, the temperature de-\npendence of the Knight shift Kssand the spin relaxation rate\n1=T1can provide essential information on the superconduct-\ning orders13,19,59,60. We calculate Kssand1=T1for the dif-\nferent superconducting ground states in Fig.2. With a strong\nSOC, as shown in Fig.3(a) \u0018(c) theA2ustate has a distinguish-\ning feature in the Knight shift, i.e. the constant Kzzcorre-\nsponding to magnetic fields applied along the zdirection, andtheA1gstate is characterized by the Hebel Slichter coherence\npeak in the spin relaxation rate as presented in Fig.3(d) \u0018(f),\nas the temperature cools down below Tc. For theB2gstate,\nthe Knight shift is always suppressed and the Hebel Slichter\ncoherence peak in the spin relaxation rate is absent. At an\nultra low temperature, all the three states show similar expo-\nnential scaling behavior in the spin relaxation rate as indicated\nin Fig.3(d)\u0018(f), due to their nodeless gap structures. We want\nto note that the Knight shift results can change if the strength\nof the SOC is comparable to the pairing order (more details in\nSM), while the features in the spin relaxation rate always hold\nfor the different states.\nFIG. 3: (color online) (a) \u0018(c) show the Knight shift and (d) \u0018(f)\nshow the spin relaxation rate versus the reduced temperature T=Tc,\nfor the three possible superconducting ground states obtained in the\nphase diagram in Fig.2. The red and blue lines in (a) \u0018(c) correspond\nto the out-of-plane and in-plane Knight shift respectively. In the cal-\nculations, we set \u0015= 0:2and the superconducting order \u0001 = 0:05\nfor theA1gandA2ustates, andf\u0015;\u0001g=f0:1;0:1gfor theB2g\nstate, in accordance with the phase diagram in Fig.2. The other pa-\nrameters areft;t0;\u0016g=f1:0;0:8;0:3g.\nAnother characteristic feature for the A2ustate is its in-\nplane upper critical field exceeding the Pauli limit, which is\nclosely related to the following facts. (i) In the A2ustate,\nthe Cooper pair forms between an electron and its inversion\npartner, and the magnetic field preserves the inversion sym-\nmetry. (ii) The in-plane magnetic field only modifies the spin\npolarization in Fig.1 which is vital for the A2ustate as ana-\nlyzed in the above. For the A1gandB2gstates, due to the\nspin-singlet nature, their in-plane upper critical fields obey the\nPauli limit. In the SM, we roughly estimate the in-plane upper\ncritical fields numerically. Notice that, here we omit the possi-\nble superconducting phase transitions, i.e. the phase transition\nfrom the even parity state to the odd parity state61,62and the\ntransition to the Fulde-Ferrell-Larkin-Ovchinnikov state63,64,\ndriven by the magnetic field; and we also ignore the symme-\ntry breaking effect arising from the in-plane magnetic field.\nIn summary, we find that the nonsymmorphic lattice sym-\nmetries can greatly assist the spin-triplet superconductivity in\nthe presence of SOC. In a system respecting the space group\nP4=nmm , the nonsymmorphic symmetry makes the spin-\ntripletA2ustate be the leading pairing instability because\nof the spin-sublattice-momentum lock on electronic bands.\nTopologically, the spin-triplet A2ustate is trivial. The triv-5\niality can be easily understood from the concentric Fermi\nsurface structure arising from the fourfold band degeneracy\nat M according to the parity criterion for centrosymmetric\nsuperconductors28,65. Our work unveils a new way in search-\ning for the spin-triplet superconductors.\nThe authors are grateful to Xianxin Wu for fruitful dis-\ncussions. This work is supported by the Ministry of Sci-ence and Technology of China 973 program (Grant No.\n2017YFA0303100), National Science Foundation of China\n(Grant No. NSFC-12174428, NSFC-11888101 and NSFC-\n11920101005), and the Strategic Priority Research Program\nof Chinese Academy of Sciences (Grant No. XDB28000000\nand No. XDB33000000).\n\u0003Electronic address: qinshengshan@ucas.ac.cn\nyElectronic address: jphu@iphy.ac.cn\n1G. E. V olovik, The universe in a helium droplet (Oxford Univer-\nsity Press, 2003).\n2M. Z. Hasan and C. L. Kane, Colloquium: Topological insulators ,\nRev. Mod. Phys. 82, 3045 (2010).\n3X.-L. Qi and S.-C. Zhang, Topological insulators and supercon-\nductors , Rev. Mod. Phys. 83, 1057 (2011).\n4C.-K. Chiu, J. C. Y . Teo, A. P. Schnyder, and S. Ryu, Classifica-\ntion of topological quantum matter with symmetries , Rev. Mod.\nPhys. 88, 035005 (2016).\n5J. Alicea, New directions in the pursuit of Majorana fermions in\nsolid state systems , Rep. Prog. Phys. 75, 076501 (2012).\n6A. Y . Kitaev, Unpaired Majorana fermions in quantum wires ,\nPhys. Usp. 44, 131 (2001).\n7N. Hao and J. P. Hu, Topological quantum states of matter in\niron-based superconductors: from concept to material realization ,\nNatl. Sci. Rev. 6, 213 (2019).\n8X. X. Wu, R.-X. Zhang, G. Xu, J. P. Hu, and C.-X. Liu, In the Pur-\nsuit of Majorana Modes in Iron-based High-T cSuperconductors ,\narXiv:2005.03603.\n9C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. Das Sarma,\nNon-Abelian anyons and topological quantum computation , Rev.\nMod. Phys. 80, 1083 (2008).\n10S. Vijay, T. H. Hsieh, and L. Fu, Majorana Fermion Surface Code\nfor Universal Quantum Computation , Phys. Rev. X 5, 041038\n(2015).\n11B. Lian, X.-Q. Sun, A. Vaezi, X.-L. Qi, and S.-C. Zhang, Topo-\nlogical quantum computation based on chiral Majorana fermions ,\nPNAS 115, 10938 (2018).\n12M. Sigrist and K. Ueda, Phenomenological theory of unconven-\ntional superconductivity , Rev. Mod. Phys. 63, 239 (1991).\n13A. P. Mackenzie and Y . Maeno, The superconductivity of Sr 2RuO 4\nand the physics of spin-triplet pairing , Rev. Mod. Phys. 75, 657\n(2003).\n14M. Smidman, M. B. Salamon, H. Q. Yuan, and D. F.\nAgterberg, Superconductivity and spin-orbit coupling in non-\ncentrosymmetric materials: a review , Rep. Prog. Phys. 80, 036501\n(2017).\n15W. Kohn and J. M. Luttinger, New Mechanism for Superconduc-\ntivity , Phys. Rev. Lett. 15, 524 (1965).\n16Y . Maeno, H. Hashimoto, K. Yoshida, S. Nishizaki, T. Fujita, J.\nG. Bedonorz, and F. Lichtenberg, Superconductivity in a layered\nperovskite without copper , Nature 372, 532 (1994).\n17T. M. Rice and M. Sigrist, Sr2RuO 4: An electronic analogue of\n3He? J. Phys. Condens. Matter 7, L643 (1995).\n18A. Pustogow, Y . K. Luo, A. Chronister, Y .-S. Su, D. A. Sokolov,\nF. Jerzembeck, A. P. Mackenzie, C. W. Hicks, N. Kikugawa, S.\nRaghu, E. D. Bauer, and S. E. Brown, Constraints on the super-\nconducting order parameter in Sr 2RuO 4from oxygen-17 nuclear\nmagnetic resonance , Nature 574, 72 (2019).\n19R. Joynt and L. Taillefer, The superconducting phases of UPt 3,Rev. Mod. Phys. 74, 235 (2002).\n20S. Ran, C. Eckberg, Q.-P. Ding, Y . Furukawa, T. Metz, S. R.\nSaha, I.-L. Liu, M. Zic, H. Kim, J. Paglione, and N. P. Butch,\nNearly ferromagnetic spin-triplet superconductivity , Science 365,\n684 (2019).\n21J.-K. Bao, J.-Y . Liu, C.-W. Ma, Z.-H. Meng, Z.-T. Tang, Y .-L. Sun,\nH.-F. Zhai, H. Jiang, H. Bai, C.-M. Feng, Z.-A. Xu, and G.-H.\nCao, Superconductivity in quasi-one-dimensional K 2Cr3As3with\nsignificant electron correlations , Phys. Rev. X 5, 011013 (2015).\n22J. Yang, J. Luo, C. J. Yi, Y . G. Shi, Y . Zhou, and G.-Q.\nZheng, Spin-Triplet Superconductivity in K 2Cr3As3, Sci. Adv. 7,\neabl4432 (2021).\n23A. Bansil, H. Lin, and T. Das, Colloquium: Topological band the-\nory, Rev. Mod. Phys. 88, 021004 (2016).\n24Y . Ando, Topological Insulator Materials , J. Phys. Soc. Jpn. 82,\n102001 (2013).\n25Y . Ando and L. Fu, Topological Crystalline Insulators and Topo-\nlogical Superconductors: From Concepts to Materials , Annu.\nRev. Condens. Matter Phys. 6, 361 (2015).\n26N. P. Armitage, E. J. Mele, and A. Vishwanath, Weyl and Dirac\nsemimetals in three-dimensional solids , Rev. Mod. Phys. 90,\n015001 (2018).\n27B. H. Yan and C. Felser, Topological Materials: Weyl Semimetals ,\nAnnu. Rev. Condens. Matter Phys. 8, 337 (2017).\n28L. Fu and E. Berg, Odd-Parity Topological Superconductors: The-\nory and Application to Cu xBi2Se3, Phys. Rev. Lett. 105, 097001\n(2010).\n29L. Fu, Odd-parity topological superconductor with nematic order:\nApplication to Cu xBi2Se3, Phys. Rev. B 90, 100509 (R) (2014).\n30T. Mizushima, A. Yamakage, M. Sato, and Y . Tanaka, Dirac-\nfermion-induced parity mixing in superconducting topological in-\nsulators , Phys. Rev. B 90, 184516 (R) (2014).\n31J. W. F. Venderbos, V . Kozii, and L. Fu, Odd-parity superconduc-\ntors with two-component order parameters: Nematic and chiral,\nfull gap, and Majorana node , Phys. Rev. B 94, 180504 (R) (2016).\n32T. Kawakami, T. Okamura, S. Kobayashi, and M. Sato, Topologi-\ncal Crystalline Materials of J= 3=2Electrons: Antiperovskites,\nDirac Points, and High Winding Topological Superconductivity ,\nPhys. Rev. X 8, 041026 (2018).\n33S. Kobayashi and M. Sato, Topological Superconductivity in\nDirac Semimetals , Phys. Rev. Lett. 115, 187001 (2015).\n34T. Hashimoto, S. Kobayashi, Y . Tanaka, and M. Sato, Dirac-\nfermion-induced parity mixing in superconducting topological in-\nsulators , Phys. Rev. B 94, 014510 (2016).\n35K. Matano, M. Kriener, K. Segawa, Y . Ando, and G.-Q. Zheng,\nSpin-rotation symmetry breaking in the superconducting state of\nCuxBi2Se3, Nat. Phys. 12, 852 (2016).\n36S. Yonezawa, K. Tajiri, S. Nakata, Y . Nagai, Z. W. Wang, K.\nSegawa, Y . Ando, and Y . Maeno, Thermodynamic evidence for ne-\nmatic superconductivity in Cu xBi2Se3, Nat. Phys. 13, 123 (2017).\n37Y . Pan, A. M. Nikitin, G. K. Araizi, Y . K. Huang, Y . Matsushita,\nT. Naka, and A. de Visser, Rotational symmetry breaking in the6\ntopological superconductor Sr xBi2Se3probed by upper-critical\nfield experiments , Sci. Rep. 6, 28632 (2016).\n38G. Du, Y . F. Li, J. Schneeloch, R. D. Zhong, G. D. Gu, H. Yang,\nH. Lin, and H.-H. Wen, Superconductivity with two-fold symmetry\nin topological superconductor Sr xBi2Se3, Sci. China Phys. Mech.\nAstron. 60, 037411 (2017).\n39T. Asaba, B. J. Lawson, C. Tinsman, L. Chen, P. Corbae, G. Li,\nY . Qiu, Y . S. Hor, L. Fu, and L. Li, Rotational Symmetry Breaking\nin a Trigonal Superconductor Nb-doped Bi 2Se3, Phys. Rev. X 7,\n011009 (2017).\n40R. Tao, Y .-J. Yan, X. Liu, Z.-W. Wang, Y . Ando, Q.-H. Wang,\nT. Zhang, and D.-L. Feng, Direct Visualization of the Nematic\nSuperconductivity in Cu xBi2Se3, Phys. Rev. X 8, 041024 (2018).\n41N. Reyren, S. Thiel, A. D. Caviglia, L. Fitting Kourkoutis, G.\nHammerl, C. Richter, C. W. Schneider, T. Kopp, A.-S. R ¨uetschi,\nD. Jaccard, M. Gabay, D. A. Muller, J.-M. Triscone, and J.\nMannhart, Superconducting Interfaces Between Insulating Ox-\nides, Science 317, 1196 (2007).\n42A. D. Caviglia, M. Gabay, S. Gariglio, N. Reyren, C. Cancellieri,\nand J.-M. Triscone, Tunable Rashba Spin-Orbit Interaction at Ox-\nide Interfaces , Phys. Rev. Lett. 104, 126803 (2010).\n43S. Nakosai, Y . Tanaka, and N. Nagaosa, Topological Supercon-\nductivity in Bilayer Rashba System , Phys. Rev. Lett. 108, 147003\n(2012).\n44K. Yada, S. Onari, Y . Tanaka, and J. Inoue, Electrically controlled\nsuperconducting states at the heterointerface SrTiO 3/LaAlO 3,\nPhys. Rev. B 80, 140509 (R) (2009).\n45J. P. Hu, Iron-Based Superconductors as Odd-Parity Supercon-\nductors , Phys. Rev. X 3, 031004 (2013).\n46The point group parts in the three symmetries act on the coor-\ndinates in Fig.1(a) as, I: (x;y;z )7!(\u0000x;\u0000y;\u0000z),My:\n(x;y;z )7!(x;\u0000y;z), andS4z: (x;y;z )7!(y;\u0000x;\u0000z).\n47S. S. Qin, et al., to appear\n48Notice that the effetive theory in Eq.(2) can be transformed to that\nin Ref.58with a similarity transformation. In constructing the ef-\nfective theory, though the symmetry group P4=nmm is nonsym-\nmorphic, the \u0000matrices and f(k)can be classified according to\nthe point group D4hdue to the bilinear form of the single-partile\nHamiltonian.\n49S. S. Qin, C. Fang, F.-C. Zhang, and J. P. Hu, Topological Super-\nconductivity in an Extended s-Wave Superconductor and Its Impli-\ncation to Iron-Based Superconductors , Phys. Rev. X 12, 011030\n(2022).\n50M. H. Fischer, M. Sigrist, D. F. Agterberg, and Y . Yanase,\nSuperconductivity and Local Inversion-Symmetry Breaking ,\narXiv:2204.02449.\n51X. W. Zhang, Q. H. Liu, J.-W. Luo, A. J. Freeman, and A. Zunger,\nHidden spin polarization in inversion-symmetric bulk crystals ,\nNat. Phys. 10, 387 (2014).\n52S.-L. Wu, K. Sumida, K. Miyamoto, K. Taguchi, T. Yoshikawa,\nA. Kimura, Y . Ueda, M. Arita, M. Nagao, S. Watauchi, I. Tanaka,and T. Okuda, Direct evidence of hidden local spin polarization in\na centrosymmetric superconductor LaO 0:55F0:45BiS2, Nat. Com-\nmun. 8, 1919 (2017).\n53Y . J. Zhang, P. F. Liu, H. Y . Sun, S. X. Zhao, H. Xu, and Q. H. Liu,\nSymmetry-Assisted Protection and Compensation of Hidden Spin\nPolarization in Centrosymmetric Systems , Chinese Phys. Lett. 37,\n087105 (2020).\n54The matrix form of the mirror symmetry and the nearly fully po-\nlarized spin near the Brillouin zone boundary, is closely related\nto the nonsymmorphic symmetry. To show this, in the SM we\nanalyze the condition near the Brillouin zone center where the\nspin polarization turns out to be vanishing samll. This is because\nthe nonsymmorphic symmetry has nontrivial effect merely on the\nBrillouin boundary, and it is simply equivalent to the point group\nsymmetry at the Brillouin zone center.\n55Strictly speaking, the superconducting orders refer to leading or-\nder approximation of the momentum-dependent superconducting\norders. More detailed analysis is shown in the SM.\n56I. I. Mazin, Symmetry analysis of possible superconducting states\nin KxFeySe2superconductors , Phys. Rev. B 84, 024529 (2011).\n57P. J. Hirschfeld, M. M. Korshunov, and I. I. Mazin, Gap symmetry\nand structure of Fe-based superconductors , Rep. Prog. Phys. 74,\n124508 (2011).\n58D. F. Agterberg, T. Shishidou, J. O’Halloran, P. M. R. Brydon,\nand M. Weinert, Resilient Nodeless d-Wave Superconductivity in\nMonolayer FeSe , Phys. Rev. Lett. 119, 267001 (2017).\n59L. C. Hebel and C. P. Slichter, Nuclear Relaxation in Supercon-\nducting Aluminum , Phys. Rev. 107, 901 (1957).\n60L. C. Hebel and C. P. Slichter, Nuclear Spin Relaxation in Normal\nand Superconducting Aluminum , Phys. Rev. 113, 1504 (1959).\n61S. Khim, J. F. Landaeta, J. Banda, N. Bannor, M. Brando, P. M.\nR. Brydon, D. Hafner, R. K ¨uchler, R. Cardoso-Gil, U. Stockert,\nA. P. Mackenzie, D. F. Agterberg, C. Geibel, and E. Hassinger,\nNonsymmorphic symmetry and field-driven odd-parity pairing in\nCeRh 2As2, Science 373, 1012 (2021).\n62D. C. Cavanagh, T. Shishidou, M. Weinert, P. M. R. Brydon, and\nD. F. Agterberg, Nonsymmorphic symmetry and field-driven odd-\nparity pairing in CeRh 2As2, Phys. Rev. B 105, L020505 (2022).\n63P. Fulde and R. A. Ferrell, Superconductivity in a Strong Spin-\nExchange Field , Phys. Rev. 135, A550 (1964).\n64A. I.Larkin and Y . N. Ovchinnikov, Nonuniform state of super-\nconductors , Sov. Phys. JETP 20, 762 (1965).\n65L. Fu and C. L. Kane, Topological insulators with inversion sym-\nmetry , Phys. Rev. B 76, 045302 (2007).\n66L. P. Gorkov, On the energy spectrum of superconductors , J. Ex-\nptl. Theoret. Phys. (U. S. S. R.) 34, 735 (1958), Sov. Phys. JETP\n7, 505 (1958).\n67B. Mhlschlegel, Die thermodynamischen Funktionen des\nSupraleiters , Z. Phys. 155, 313 (1959).\nAppendix A: Matrix form for the symmetry operators at G and M\nThe nonsymmorphic symmetry must lead to multiple sublattices in the system, as indicated in the lattice structure respecting\nthe space group P4=nmm in the main text. Due to the sublattice degree of freedom, the matrix form for the symmetry operations\nisk-dependent. In the following, we take the fS4zj0gsymmetry for instance and construct its matrix form at G and M. To do\nthis, we need to figure out how the symmetry operations act on the basis (j\u001eA(k)i;j\u001eB(k)i) = (cy\nk;dy\nk)j0i, wherej0iis the\nvacuum and the spin index has been omitted for convenience. The bases in the reciprocal space and in the real space are related7\nTABLE A1: Matrix form for the symmetry operations at G and M. Here, I,My,Mxy,S4zandTstanding forfIj\u001c0g,fMyj0g,fMxyj\u001c0g,\nfS4zj0gand the time reversal symmetry respectively.\nI M yMxy S4zT\nGs0\u001b1is2\u001b0i(s1+s2)\u001b1=p\n2eis3\u0019=4\u001b0is2\u001b0K\nMs0\u001b1is2\u001b3i(s1+s2)\u001b1=p\n2eis3\u0019=4\u001b3is2\u001b0K\nby the Fourier transform\nj\u001eA(k)i=X\njeik\u0001Rj\nAj\u001eA(Rj\nA)i;j\u001eB(k)i=X\njeik\u0001Rj\nBj\u001eB(Rj\nB)i; (A1)\nwhere Rj\nA(Rj\nB) labels the position of the A (B) site in the jthunit cell and Rj\nA\u0000Rj\nB=\u001c0=a1=2 +a2=2as shown in in\nthe lattice in the main text. Under fS4zj0g, the G point is left unchanged, i.e. gk=k; however, the M point is transfromed as\nk!k\u0000b1. Accordingly, when fS4zj0gacts on the basis function, we have\nG:fS4zj0g(j\u001eA(k)i;j\u001eB(k)i) =eis3\u0019=4(j\u001eA(k)i;j\u001eB(k)i) (A2a)\nM:fS4zj0g(j\u001eA(k)i;j\u001eB(k)i) =eis3\u0019=4(j\u001eA(k\u0000b1)i;j\u001eB(k\u0000b1)i)\n=X\njeis3\u0019=4(ei(k\u0000b1)\u0001Rj\nAj\u001eA(Rj\nA)i;ei(k\u0000b1)\u0001Rj\nAe\u0000i(k\u0000b1)\u0001\u001c0j\u001eB(Rj\nB)i)\n=X\njeis3\u0019=4(eik\u0001Rj\nAj\u001eA(Rj\nA)i;eik\u0001Rj\nAe\u0000ik\u0001\u001c0eib1\u0001\u001c0j\u001eB(Rj\nB)i)\n=X\njeis3\u0019=4(eik\u0001Rj\nAj\u001eA(Rj\nA)i;\u0000eik\u0001Rj\nBj\u001eB(Rj\nB)i)\n=eis3\u0019=4(j\u001eA(k)i;\u0000j\u001eB(k)i); (A2b)\nwhere we have taken use of the fact that Rj\nA\u0001b1mod 2\u0019equals 0 and b1\u0001\u001c0=\u0019. Therefore, in the normal state fS4zj0ghas\nthe matrix form eis3\u0019=4\u001b0at G andeis3\u0019=4\u001b3at M. Similar analysis can be applied to other symmetry operations, and we get\nthe results in Table.A1.\nAs pointed out in the main text, for the space group P4=nmm in the spinful condition it merely has one 4D irreducible\nrepresentation at M, which is contributed by states with angular momenta Jz=\u00061=2andJz=\u00063=2defined according to\nfS4zj0g(or the fourfold rotation fC4zj\u001c0g). However, in constructing the low-energy effective model in the main text we only\nconsider the sorbital. At first glance, the sorbital can not contribute states beyond Jz=\u00061=2. According to Eq.(A2b), the\nadditional angular momentum origins from the plane wave part of the Bloch wave function.\nAppendix B: Effective model at G\nIn this section, we present the detailed construction of the low-energy effective model Heff;G(k)near the Brillouin zone\ncenter, i.e. the G point. The effective model near the M point shown in the main text can be constructed in a similar way.\nWe first consider the time reversal symmetry and the inversion symmetry, which constrain the system as\nTH eff;G(k)T\u00001=Heff;G(\u0000k);IH eff;G(k)I\u00001=Heff;G(\u0000k): (B1)\nThe four-band model Heff;G(k)can be generally expressed in the form of the sixteen \u0000 =si\u001bjmatrices. The constraints in\nEq.(B1) merely allow six \u0000matrices, i.e. s0\u001b0,s0\u001b1,s0\u001b2,s1\u001b3,s2\u001b3ands3\u001b3, to appear inHeff;G(k). Then, we consider the\nconstraints of the crystalline symmetries. Based on the matrix form of the symmetry operations in Table.A1, one can classify\nthe above six matrices as shown in Table.B1. Therefore, Heff;G(k)must take the following form\nHeff;G(k) =m(k)s0\u001b0+t0s0\u001b1+\u0015kzs2\u001b3+\u0015kys1\u001b3; (B2)\nwithm(k) =t(k2\nx+k2\ny)andt;t0;\u0015all the coefficients.\nIn fact, the effective model at G shown in Eq.(B2) has similar form with the model Hamiltonian considered in Ref.43, but\ntheir physical meanings are different. However, if we consider the phenomenological density-density interactions similar to that\nin the main text, we can expect similar conclusions with that in Ref.43and the spin-triplet superconductivity can appear in the\nstrong SOC condition.8\nTABLE B1: Classification of the si\u001bjmatrices which are allowed by the time reversal symmetry and the inversion symmetry, and the\nfunctionsf(k)at G. The classification is according to the D4hpoint group.\nES4zI M yMxy space si\u001bj\nA1g 1 1 1 1 1 x2+y2s0\u001b0,s0\u001b1\nB1u 1 1 -1 -1 1 s3\u001b3\nB2u 1 1 -1 1 -1 s0\u001b2\nEu 2 0 -2 0 0 (x;y) ( s2\u001b3;s1\u001b3)\nAppendix C: Spin polarization on the energy bands\nIn this part, we present the detailed calculations for the spin polarization on the energy bands based on the low-energy effective\ntheory near the M point in the main text. The effective Hamiltonian near M can be solved as\nj'1(k)i=1p\n20\nB@\u00001\ncos\u0010\nisin\u0010e\u0000i\u0012\n01\nCA;j'2(k)i=\u00001p\n20\nB@0\nisin\u0010ei\u0012\ncos\u0010\n\u000011\nCA;\nj'3(k)i=1p\n20\nB@1\ncos\u0010\nisin\u0010e\u0000i\u0012\n01\nCA;j'4(k)i=\u00001p\n20\nB@0\nisin\u0010ei\u0012\ncos\u0010\n11\nCA; (C1)\nwhere sin\u0010=\u0015kp\n\u00152k2+t02k4sin22\u0012=4andcos\u0010=t0k2sin 2\u0012\n2p\n\u00152k2+t02k4sin22\u0012=4withkwritten in the polar coordinates (kx;ky) =\n(ksin\u0012;kcos\u0012). In Eq.(C1),j'1(k)iandj'2(k)iare the two degenerate eigenstates corresponding to eigenvalue E\u0000=\ntk2\u0000q\n\u00152k2+t02k4sin22\u0012=4, whilej'3(k)iandj'4(k)iare the two degenerate eigenstates with eigenvalue E+=\ntk2+q\n\u00152k2+t02k4sin22\u0012=4.\nThe sublattice-distinguished spin operators are sA=B;i =si(\u001b0\u0006\u001b3)=2. Straightforwardly, the spin polarization on\ntheE\u0000bands can be calculated as hsA=Bi=h'1jsA=Bj'1i+h'2jsA=Bj'2i, which turns out to be hsBi=\u0000hsAi=\n(sin\u0010sin\u0012;sin\u0010cos\u0012;0). Obviously,jhsA=Bijatkis proportional to1p\n1+t02k2sin22\u0012=4\u00152. The spin polarization on the E+\nbands can be calculated similarly.\nFor the Fermi surfaces near the G point, according to Eq.(B2), the spin polarization can be obtained as jhsA=Bij=\n1p\n1+t02=4\u00152k2. Comparing the results near G and M, one immediately comes to the conclusion, the spin polarization is van-\nishing small for small Fermi surfaces near G, while it is nearly fully polarized for small Fermi surfaces near M.\nAppendix D: Derivation of the superconducting ground state\nIn the main text, we get the superconducting ground states by solving the linearized gap equations. Here, we present the\ndetails on the derivation of the linearized gap equation, and present more analysis on the calculations of the superconducting\nsusceptibility.\n1. Linearized gap equation\nIn the superconducting state, the Green functions can be defined as\nGij(k;\u001c) =\u0000hT\u001cci(k;\u001c)cy\nj(k;0)i; (D1)\nFij(k;\u001c) =hT\u001cci(k;\u001c)cj(\u0000k;0)i;\nFy\nij(k;\u001c) =hT\u001ccy\ni(\u0000k;\u001c)cy\nj(k;0)i:\nIn the mean-field level, the superconducting order can be calculated as\n\u0001(k) =\u00001\nNX\nk0U(k;k0)F(k0;\u001c= 0) =1\nN\fX\nk0;nU(k;k0)F(k0;i!n): (D2)9\nNotice that the Fourier transformation for Eq.(D1) is as follows: g(\u001c) =1\n\fP+1\nn=\u00001e\u0000i!n\u001cg(i!n)andg(i!n) =R\f\n0ei!n\u001cg(\u001c)\nwith!n=2n\u0019\n\ffor boson and !n=(2n+1)\u0019\n\ffor fermion. According to Eq.(D2), we must calculate F(k0;\u001c= 0) firstly. To do\nthis, we consider the Gor’kov equations66,i:e:the equation of motion in the superconducting state, which read as\nG\u00001\n0(k;i!)G(k;i!) +\u0001(k)Fy(k;i!) = 1; (D3)\nG\u00001\n0(k;i!)F(k;i!)\u0000\u0001(k)GT(\u0000k;\u0000i!) = 0;\n\u0000(G\u00001\n0)T(\u0000k;\u0000i!)Fy(k;i!) +\u0001y(k)G(k;i!) = 0:\nAccording to the Gor’kov equations, we can derive the following equations\nFy(k;i!) =GT\n0(\u0000k;\u0000i!)\u0001y(k)G(k;i!); (D4)\nF(k;i!) =G0(k;i!)\u0001(k)GT(\u0000k;\u0000i!);\nG\u00001(k;i!) =G\u00001\n0(k;i!) +\u0001(k)GT\n0(\u0000k;\u0000i!)\u0001y(k);\nwhereG0(k;i!)is the normal-state Green function. In the weak-coupling condition, \u0001is small and we have\nF(k;i!)'G0(k;i!)\u0001(k)GT\n0(\u0000k;\u0000i!) =G0(k;i!)\u0001(k)G\u0003\n0(\u0000k;i!); (D5)\nwhere we have used the identity G0(\u0000k;\u0000i!) =Gy\n0(\u0000k;i!). The normal-state Green function can be written in the band basis\nG0(k;i!) =1\ni!\u0000h0(k)=X\nsjus;kihus;kj\ni!\u0000\u0018s;k; (D6)\nwheresis the band index, and jus;ki(\u0018s;k) is the eigenfunction (eigenenergy) for band s. Accordingly, the anomalous super-\nconducting Green function is\nF(k;i!) =X\ns;s0jus;kihus;kj\u0001(k)ju\u0003\ns0;\u0000kihu\u0003\ns0;\u0000kj\n(i!\u0000\u0018s;k)(\u0000i!\u0000\u0018s0;\u0000k); (D7)\nwhere we useju\u0003\ns0;\u0000kito lable (jus0;\u0000ki)\u0003. In the weak-coupling condition, the superconductivity is mainly contributed by\nelectrons on the Fermi surfaces. Moreover, since the system in our consideration possesses both the time reversal symmetry and\ninversion symmetry, the electron on the Fermi surfaces can always form Cooper pair with its time reversal or inversion partner,\nnamelys0= \u0016sors0=s, and\nF(k;i!) =X\ns;s0jus;kihus;kj\u0001(k)ju\u0003\ns0;\u0000kihu\u0003\ns0;\u0000kj\n!2+\u00182\ns;k: (D8)\nCorrespondingly, the superconducting order parameter in Eq.(D2) is\n\u0001(k) =\u00001\nN\fX\nk0;nU(k;k0)F(k0;i!n) =\u00001\nNX\nk0;s;s0U(k;k0)jus;k0ihus;k0j\u0001(k0)ju\u0003\ns0;\u0000k0ihu\u0003\ns0;\u0000k0j1\n\fX\nn1\n!2n+\u00182\ns;k0\n=\u00001\nNX\nk0;sU(k;k0)\n2\u0018s;k0tanh\f\u0018s;k0\n2X\ns0=s;\u0016sjus;k0ihus;k0j\u0001(k0)ju\u0003\ns0;\u0000k0ihu\u0003\ns0;\u0000k0j; (D9)\nwhere only the electronic states on the Fermi surfaces are taken into account in the weak-coupling condition. In calculating the\nfrequency summation in Eq.(D9), we have used the relationH\njzj!1dz\n2\u0019i1\n\u00182\u0000z21\ne\fz+1= 0.\nFor each irreducible representation channel, the superconducting order parameter \u0001(k)can be expanded according to the\ncorresponding bases. Considering the orthonormality of the basis functions, we have\nX\nktr[\u0001(k)\u0001y\n\u00170(k)] =X\nk;\u0017\u0014\u0017tr[\u0001\u0017(k)\u0001y\n\u00170(k)] =\u0014\u0017 (D10)\n=\u0000X\nktr[1\nNX\nk0;sU(k;k0)\n2\u0018s;k0tanh\f\u0018s;k0\n2X\ns0=s;\u0016sjus;k0ihus;k0j\u0001(k0)ju\u0003\ns0;\u0000k0ihu\u0003\ns0;\u0000k0j\u0001y\n\u00170(k)]\n=\u0000X\n\u00171\nNX\nk;k0;sU(k;k0)\n2\u0018s;k0tanh\f\u0018s;k0\n2X\ns0=s;\u0016shus;k0j\u0001\u0017(k0)ju\u0003\ns0;\u0000k0ihu\u0003\ns0;\u0000k0j\u0001y\n\u00170(k)jus;k0i\u0014\u0017\n=\u0000X\n\u00171\nNX\nk;k0;sU(k;k0)\n2\u0018s;k0tanh\f\u0018s;k0\n2X\ns0=s;\u0016shus;k0j^\u0001\u0017(k0)jus0;k0ihus0;k0j^\u0001y\n\u00170(k)jus;k0i\u0014\u0017;10\nwhere \u0001\u0017(k) =^\u0001\u0017(k)\u0001is2. Eq.(D10) can be intuitively expressed in the form \u0014\u0001X=\u0014with\nX\u0017;\u00170(\f) =\u00001\nNX\nk;k0;sU(k;k0)\n2\u0018s;k0tanh\f\u0018s;k0\n2X\ns0=s;\u0016shus;k0j^\u0001\u0017(k0)jus0;k0ihus0;k0j^\u0001y\n\u00170(k)jus;k0i: (D11)\nIn our consideration the interaction is k-independent, and in the continuum condition we have\nX\u0017;\u00170(\f) =\u00001\nNX\nsZ\ndk0U\n2\u0018s;k0tanh\f\u0018s;k0\n2X\ns0=s;\u0016shus;k0j^\u0001\u0017jus0;k0ihus0;k0j^\u0001y\n\u00170jus;k0i; (D12)\n=\u00001\nNX\nsZdk0\nd\u0018s;k0d\u0018s;k0d\u0012U\n2\u0018s;k0tanh\f\u0018s;k0\n2X\ns0=s;\u0016shus;k0j^\u0001\u0017jus0;k0ihus0;k0j^\u0001y\n\u00170jus;k0i;\n=\u0000U\nNX\nsZ!0\n\u0000!0d\u0018s;k01\n2\u0018s;k0tanh\f\u0018s;k0\n2Z\nd\u0012D(\u0012)X\ns0=s;\u0016shus;k0j^\u0001\u0017jus0;k0ihus0;k0j^\u0001y\n\u00170jus;k0i;\n=\u0000UF (\f)X\nsZ\nd\u0012D(\u0012)X\ns0=s;\u0016shus;k0j^\u0001\u0017jus0;k0ihus0;k0j^\u0001y\n\u00170jus;k0i=\u0000U\u001f\u0017;\u00170(Tc):\nIn Eq.(D12),F(\f) =1\n2NR!0\n\u0000!01\n2\u0018s;k0tanh\f\u0018s;k0\n2d\u0018s;k0is a temperature-dependent constant with !0the energy cutoff near the\nFermi energy, and D(\u0012) = 2dk0=d\u0018s;k0is the density of states on the Fermi surface. By solving the characteristic equation\n\u0000U\u001f(Tc) =I, we can get the superconducting ground state.\n2. Superconductivity from density-density interactions\nIn the main text, we consider superconductivity induced from the phenomenological density-density interactions\nHint=Z\ndq[U2X\ni=1ni(q)ni(\u0000q) + 2Vn1(q)n2(\u0000q)]; (D13)\n=1\nNZ\ndkdk0dq(Ucy\nk0+q;\u001bck0;\u001bcy\nk\u0000q;\u0016\u001bck;\u0016\u001b+Udy\nk0+q;\u001bdk0;\u001bdy\nk\u0000q;\u0016\u001bdk;\u0016\u001b+ 2Vcy\nk0+q;\u001bck0;\u001bdy\nk\u0000q;\u001b0dk;\u001b0):\nIn the superconducting channel, i.e. k=\u0000k0, we have\nHint=1\nNZ\ndk0dq(Ucy\nk0+q;\u001bck0;\u001bcy\n\u0000k0\u0000q;\u0016\u001bc\u0000k0;\u0016\u001b+Udy\nk0+q;\u001bdk0;\u001bdy\n\u0000k0\u0000q;\u0016\u001bd\u0000k0;\u0016\u001b+ 2Vcy\nk0+q;\u001bck0;\u001bdy\n\u0000k0\u0000q;\u001b0d\u0000k0;\u001b0)\n=1\nNZ\ndkdk0(Ucy\nk;\u001bcy\n\u0000k;\u0016\u001bc\u0000k0;\u0016\u001bck0;\u001b+Udy\nk;\u001bdy\n\u0000k;\u0016\u001bd\u0000k0;\u0016\u001bdk0;\u001b+ 2Vcy\nk;\u001bdy\n\u0000k;\u001b0d\u0000k0;\u001b0ck0;\u001b): (D14)\nThe interactions in Eq.(D14) can be expanded according to the superconducting orders. When we consider Fermi surfaces near\nthe M point, we need to expand the interactions according to the pairing orders classified at M which is shown in the main text\nHint;M =1\nNZ\ndkdk0[U(cy\nk;\u001bcy\n\u0000k;\u0016\u001bc\u0000k0;\u0016\u001bck0;\u001b+dy\nk;\u001bdy\n\u0000k;\u0016\u001bd\u0000k0;\u0016\u001bdk0;\u001b) + 2Vcy\nk;\u001bdy\n\u0000k;\u001b0d\u0000k0;\u001b0ck0;\u001b] (D15)\n=1\n4NZ\ndkdk0[U(^\u0001A1g^\u0001y\nA1g+^\u0001B2u^\u0001y\nB2u) +V(^\u0001A2u^\u0001y\nA2u+^\u0001B2g^\u0001y\nB2g+^\u0001(1)\nEu^\u0001(1)y\nEu+^\u0001(2)\nEu^\u0001(2)y\nEu)]:\nBased on Eqs.(D12)(D15), we can ge the linearized gap equations for each irreducible representation channel shown in the main\ntext.11\n3. Calculations of the superconducting susceptibility\nBased on the wave functions in Eq.(C1), we can calculate the superconducting susceptibility for each irreducible representation\nchannel at M shown in the main text straightforwardly\n\u001fA1g=F(\f)X\nsX\ns0=s;\u0016sZ\nd\u0012[D\u0000(\u0012)jhu\u0000\ns;k0js0\u001b0ju\u0000\ns0;k0ij2+D+(\u0012)jhu+\ns;k0js0\u001b0ju+\ns0;k0ij2] =F(\f)Z\nd\u0012[D\u0000(\u0012) +D+(\u0012)]2;\n\u001fB2g=F(\f)Z\nd\u0012[D\u0000(\u0012) +D+(\u0012)]2 cos2\u0010; \u001f A2u=F(\f)Z\nd\u0012[D\u0000(\u0012) +D+(\u0012)]2 sin2\u0010;\n\u001fB2u=F(\f)Z\nd\u0012[D\u0000(\u0012) +D+(\u0012)]2 sin2\u0010; \u001f Eu=F(\f)Z\nd\u0012[D\u0000(\u0012) +D+(\u0012)]2 sin2\u0010sin2\u0012: (D16)\nIn the above equations, D\u0006andju\u0006\ns0iare the density of states and the eigenstates on the Fermi surface contributed by the energy\nbandE\u0006respectively. According to the results in Eq.(D16) it is obvious to notice that, in the intrasublattice pairing channels\n\u001fA1g>\u001fB2uand in the intersublattice channels \u001fA2u>\u001fEu. Moreover, considering that \u001fA2u+\u001fB2g=\u001fA1g, if we compare\ntheA1gstate with a special case in the intersublattice pairing channels where \u001fA2u=\u001fB2g, i.e. the phase boundary between the\nA2uandB2gstate, one can find that the A1gstate can never be the ground state for 2jUj<jVjassumingVattractive, which is\nconsistent with the phase diagram in the main text.\nAppendix E: Lattice Model\nIn the main text, we analyze the k-independent superconducting ground states based on a low-energy effective model. Here,\nwe show that these k-independent states are the leading order approximation, i.e. the superconducting orders preserved to the\n0thorder of k, in the minimal lattice model.\nTaking the lattice structure in the main text into consideration and substituting kx=ywith appropriate trigonometric functions,\nwe can get the corresponding lattice model\nH(k) = 2t(coskx+ cosky+ 2)s0\u001b0\u0000\u0015sinkxs2\u001b3\u0000\u0015sinkys1\u001b3+ 4t0coskx\n2cosky\n2s0\u001b1: (E1)\nThe above lattice model respects the space group P4=nmm with the lattice sites located at the D2dinvariant points, and at each\nlattice site an sorbital is considered as pointed out in the main text. In Eq.(E1), tis the intrasublattice nearest-neighbour hopping,\nt0is the intersublattice nearest-neighbour hopping, and \u0015=2is the inversion-symmetric Rashba SOC. Based on the lattice model,\nwe calculate the sublattice-distinguished spin polarization on the energy bands and plot the results in Fig.E1. As shown, the\nlattice model captures the essential features analyzed in the main text and in the above sections: for a system respecting the\nspace group P4=nmm , it has fully polarized sublattice-distinguished spin polarization on the energy bands near the Brillouin\nzone boundary, i.e. kx=y=\u0019; while the spin polarization is vanishing small near the Brillouin zone center.\nFIG. E1: (color online) Sketch for the sublattice-distinguished spin polarizations on the lower energy bands (a) and upper energy bands (b),\nplotted from the lattice model in Eq.(E1). The parameters are set to be ft;t0;\u0015g=f\u00001:0;0:8;0:8g. The symbols in the figures are the same\nwith that in the main text.\nFor the interacting part, we consider the following electron density-density interaction\nHint(k) =UX\ni;\u001bni\u001bni\u0016\u001b+VX\nhiji;\u001b;\u001b0ni\u001bnj\u001b0; (E2)12\nwhereUis the onsite interaction and Vis the intersublattice nearest-neighbour interaction. By doing a Fourier transformation,\nci=1p\nNP\nkckeik\u0001riand1\nNP\niei(k\u0000k0)\u0001ri=\u000ek;k0, we obtain\nHint(k) =U\nNX\nkk0q(cy\nk0+q;\u001bck0;\u001bcy\nk\u0000q;\u0016\u001bck;\u0016\u001b+dy\nk0+q;\u001bdk0;\u001bdy\nk\u0000q;\u0016\u001bdk;\u0016\u001b) +V\nNX\nkk0qcy\nk0+q;\u001bck0;\u001bdy\nk\u0000q;\u001b0dk;\u001b0X\n\u000e=hijieiq\u0001(ri\u0000rj)\n=U\nNX\nkk0q(cy\nk0+q;\u001bck0;\u001bcy\nk\u0000q;\u0016\u001bck;\u0016\u001b+dy\nk0+q;\u001bdk0;\u001bdy\nk\u0000q;\u0016\u001bdk;\u0016\u001b) +V\nNX\nkk0qcy\nk0+q;\u001bck0;\u001bdy\nk\u0000q;\u001b0dk;\u001b04 cosqx\n2cosqy\n2:\n(E3)\nIn the superconducting channel, we set k=\u0000k0and get\nHint(k) =U\nNX\nk0q(cy\nk0+q;\u001bck0;\u001bcy\n\u0000k0\u0000q;\u0016\u001bc\u0000k0;\u0016\u001b+dy\nk0+q;\u001bdk0;\u001bdy\n\u0000k0\u0000q;\u0016\u001bd\u0000k0;\u0016\u001b)\n+V\nNX\nk0qcy\nk0+q;\u001bck0;\u001bdy\n\u0000k0\u0000q;\u001b0d\u0000k0;\u001b04 cosqx\n2cosqy\n2\n=U\nNX\nkk0(cy\nk;\u001bcy\n\u0000k;\u0016\u001bc\u0000k0;\u0016\u001bck0;\u001b+dy\nk;\u001bdy\n\u0000k;\u0016\u001bd\u0000k0;\u0016\u001bdk0;\u001b)\n+V\nNX\nkk0cy\nk;\u001bdy\n\u0000k;\u001b0d\u0000k0;\u001b0ck0;\u001b4 coskx\u0000k0\nx\n2cosky\u0000k0\ny\n2: (E4)\nApparently, the onsite interaction can only contribute the constant intrasublattice pairing order. For the intersublattice part, we\nhave\nHint;NN (k) =V\nNX\nkk0cy\nk;\u001bdy\n\u0000k;\u001b0d\u0000k0;\u001b0ck0;\u001b(coskx\n2cosky\n2cosk0\nx\n2cosk0\ny\n2+ sinkx\n2sinky\n2sink0\nx\n2sink0\ny\n2\n+ coskx\n2sinky\n2cosk0\nx\n2sink0\ny\n2+ sinkx\n2cosky\n2sink0\nx\n2cosk0\ny\n2): (E5)\nAccording to the equation, it can be noticed that the term with form factor sinkx\n2sinky\n2sink0\nx\n2sink0\ny\n2dominates the other\nterms for small Fermi surfaces near the M point ( kx=y\u0018\u0019), since sinkx=y\n2= 1\u0000k2\nM;x=y\n2andcoskx=y\n2=\u0000kM;x=y\n2with\nkM=k\u0000KMandKMbeing the M point ( kMis samll). Therefore, for small Fermi surfaces near M, it is reasonable we only\nconsider the constant pairing orders between different sublattices, i.e. the 0thorder of k-dependent pairing orders contributed\nby the sinkx\n2sinky\n2sink0\nx\n2sink0\ny\n2term in Eq.(E5), which is exactly the consideration in the main text.\nAppendix F: Magnetic response\nIn this part, we provide more details on the numerical simulation for the Knight shift and the spin relaxation rate, and present\na rough numerical estimation for the in-plane upper critical field, for the superconducting ground states in the phase diagram in\nthe main text.\n1. Knight shift and spin relaxation rate\nIn the nuclear magnetic resonance, the Knight shift Kssand the spin relaxation rate 1=T1are measured through the static spin\nsusceptibility. In the general condition, the spin susceptibility is defined as\n\u001fst(q;i!) =Z\f\n0d\u001c\u001fst(q;\u001c) =Z\f\n0d\u001chT\u001cSs(q)St(\u0000q)iei!\u001c: (F1)\nIn our consideration, the Knight shift in the nuclear magnetic resonance reads\nKss(T)/X\n\u000b\u001f\u000b\u000b\nss(0;0)/\u0000X\nk;m;n;\u000bjh\u001em(k)jS\u000b\nsj\u001en(k)ij2n(Ekm)\u0000n(Ekn)\nEkm\u0000Ekn: (F2)13\nThe spin relaxation rate is\n1\nT1(T)/lim\n!!0X\nq;\u000b;sjA(q)j2Im\u001f\u000b\u000b\nss(q;!+i0+)\n!(F3)\n/ \u0000X\nk;k0;m;n;s;\u000bjA(k\u0000k0)j2jh\u001em(k)jS\u000b\nsj\u001en(k0)ij2@n(E)\n@E\f\f\f\f\nE=Ekm\u000e(Ekm\u0000Ek0n):\nIn the above equations, S\u000b\ns(s= 1;2;3and\u000b=A;B ) is the spin operator for sublattice \u000bin the Nambu space\n( y(k);is2 (\u0000k)), with y(k)being the basis for the normal-state Hamiltonian shown in the main text. Specifically,\nSA\ns=ss\n\u001b0+\u001b3\n2\n\u001c0andSB\ns=ss\n\u001b0\u0000\u001b3\n2\n\u001c0, with\u001c0=I2\u00022in the Nambu space. Ekmand\u001em(k)are the energy\nand wavefunction for the mth eigenstate for the superconducting Hamiltonian respectively, and A(q)in Eq.(F3) is the structure\nfactor which is set to be 1. In the numerical calculations, we set the superconducting transition temperature kbTc= \u0001 0=3:53\nwith \u00010the zero temperature pairing order, and consider the T-dependent superconducting order \u0001(T) = \u0001 0f(T=Tc)with\nf(T=Tc)being the BCS-type normalized gap presented in Ref.67.\nFIG. F1: (color online) The Knight shift for the A1gandA2ustates in different conditions. In the calculations, the other parameters are\nchosen asft;t0;\u0016g=f1:0;0:8;0:3g. The symbols in the figures are the same with that in the main text.\nIn the main text, we claim that for the system in our consideration, the Knight shift can be affected by the SOC and the\ninterband pairing. Here, we present more numerical results. We mainly consider the A1gandA2ustates, since the B2gstate can\nonly appear in the weak SOC condition (if the SOC is strong, the B2gstate is nodal and cannot be the ground state) where the\nKnight shift is qualitatively the same with that in the main text. As indicated in Fig.F1 and the results shown in the main text,\nin the weak pairing condition, i.e. \u0001<<\u0015 , the Knight shift for the two states merely changes quantitatively as the SOC varies.\nTheA2ustate can be distinguished from the other states through the unsuppressed Kzz. However, if the pairing is strong, i.e.\n\u0001\u0018\u0015, the interband pairing changes the results qualitatively and it suppresses Kzzin the superconducting state. Therefore, in\nthe strong pairing condition, it is not a good choice to use the Knight shift to distinguish the different states.\n2. In-plane upper critical field\nFor the superconducting ground states in the phase diagram in the main text, the in-plane upper critical field of spin-triplet\nA2ustate is larger than the Pauli limit. Here, we show this by carrying out rough numerical simulations for the pairing orders in\npresence of the in-plane magnetic field. We consider the following Hamiltonian\nHtotal=Heff+Hint+Hmag; (F4)\nwhereHmag=Bs1\u001b0is the in-plane Zeeman field, and HeffandHintare shown in the main text. In the calculations, we set\nthe parameters as,ft;t0;\u0016\u0015g=f1:0;0:8;0:3;0:2gfor theA1gandA2ustates, andft;t0;\u0016\u0015g=f1:0;0:8;0:3;0:1gfor theA1g\nfor theB2gstate. For the Hamiltonian in Eq.(F4), we solve the superconducting gap equation \u0001(k) =\u00001\nNP\nk0UF(k0;\u001c= 0)\nitinerantly. In solving the gap equation, for the A1gchannel we choose U=\u00000:8which leads to pairing order in the absence\nof the Zeeman field \u00010;A1g= 0:06; and for the B2gandA2uchannels, we set V=\u00001:0corresponding to the pairing orders in14\nthe absence of the Zeeman field \u00010;B2g= 0:11and\u00010;A2u= 0:07. Turning on the in-plane magnetic field, we get the pairing\norders in Fig.F2. Obviously, for the parameters chosen in the above, the pairing orders for the A1gandB2gstates vanish at\nB\u0018\u00010, while theA2ustate has the upper critical field B\u00185\u00010.\nFIG. F2: (color online) By solving the Hamiltonian in Eq.(F4), we get the pairing orders in the presence of the in-plane magnetic field."    },    {        "title": "1403.4265v1.Single_parameter_spin_pumping_in_driven_metallic_rings_with_spin_orbit_coupling.pdf",        "content": "arXiv:1403.4265v1  [cond-mat.mes-hall]  17 Mar 2014Single-parameter spin-pumping in driven metallic rings wi th\nspin-orbit coupling\nJ. P. Ramos,1L. E. F. Foa Torres,2P. A. Orellana,3and V. M. Apel1\n1Departamento de F´ ısica, Universidad Cat´ olica del Norte,\nAngamos 0610, Casilla 1280, Antofagasta, Chile\n2Instituto de F´ ısica Enrique Gaviola (CONICET) and FaMAF,\nUniversidad Nacional de C´ ordoba, Ciudad Universitaria 50 00, C´ ordoba, Argentina\n3Departamento de F´ ısica, Universidad Federico Santa Mar´ ı a,\nAvenida Vicu˜ na Mackenna 3939, San Joaquin, Santiago, Chil e\n(Dated: May 8, 2019)\nAbstract\nWe consider the generation of a pure spin-current at zero bia s voltage with a single time-\ndependent potential. To such end we study a device made of a me soscopic ring connected to\nelectrodes and clarify the interplay between a magnetic flux , spin-orbit coupling and non-adiabatic\ndriving in the production of a spin and electrical current. B y using Floquet theory, we show that\nthe generated spin to charge current ratio can be controlled by tuning the spin-orbit coupling.\n1I. INTRODUCTION\nLargelyforgottenduringtheearlydecadesofnanoelectronics, t hespindegreeoffreedomis\nbecoming ever closer to the center of the research stage1,2. Indeed, generating and detecting\nspin-currents is now a fascinating field of research3,4with applications in future electronics5,\nquantum computing6and information storage7. Among the many ways of harnessing the\nelectron spin, spin orbit interaction (SOI) in two-dimensional electr on gases is a promising\none since the spin transport properties can be controlled simply by a pplying an electric\nfield8,9.\nMost of the proposals aiming at the control of the spin degree of fr eedom use static elec-\ntric or magnetic fields3,4. Here we follow a different path and use alternating fields (ac) as\nin10,11. The time-dependence introduced by the alternating fields12provide an avenue for\nexploring new phenomena including the opening of a laser-induced ban dgap13,14or chiral\nedge states15and, more generally, the tuning of its topological properties15–18. Another\nstriking phenomena is the coherent generation of a current at zer o bias voltage (termed\nquantum charge pumping )19–21and, as shown below, the generation of a pure spin-currents\nthroughspinpumping. Quantumpumping isusuallyachieved bydrivinga sampleconnected\nto electrodes through ac gate voltages. Within the adiabatic appro ximation22, pumping a\nnon-vanishing charge requires the presence of at least two time-d ependent parameters (typ-\nically constituted by gate voltages) and has been widely studied in man y systems including\npristine23,24and disordered graphene25. But beyond the adiabatic approximation single-\nparameter pumping is also possible as predicted theoretically26–28(similar to the mesoscopic\nphotovoltaic effect predicted earlier in29) and achieved in careful experiments30–32. Besides\nreducing the burden of adding more contacts in a nanoscale sample, a single parameter\nsetup could also prove advantageous (as a compared to a two-par ameter one) in reducing\ncapacitive effects and crosstalk between time-dependent gates.\nHere we address the effect of spin-orbit coupling and its interplay wit h a single time-\ndependent field in the generation of non-adiabatic spin current at z ero bias voltage. To such\nend we consider a setup as the one represented in Fig.1, where a nan oscale ring is connected\nto electrodes and has a quantum dot embeded in one of its arms. The time dependence is\nintroduced as an alternating gate applied to the quantum dot and do es not break neither\ntime-reversal nor inversion symmetry (parity). Crucial to the ge neration of pumped current\n2is the addition of a magnetic flux threading the ring as shown in Fig. (1) . The spin-orbit\ncoupling is introduced as an additional spin-dependent flux. In this p aper we show how this\nsimple setup is able to provide a minimal model where a pure spin-curre nt can be achieved.\nFIG. 1. Scheme of setup considered in the text, a quantum ring with a magnetic field cross them\nand a quantum dot embedded in one of its arms, driven by an ac vo ltage source.\nII. HAMILTONIAN MODEL AND ITS SOLUTION THROUGH FLOQUET THE-\nORY\nLet us start our discussion by presenting our model Hamiltonian for the situation repre-\nsented in Fig.(1). The total Hamiltonian H(⊔) is written as:\nH(t) =HC+HQD(t)+HT, (1)\nwhereHCrepresents the left and right contacts and the lower arm of the rin g (represented\nby sitej= 0 in the notation below), HQD(t) the quantum dot in the upper arm of the\nring (which for simplicity is taken to be a single level), and HTthe tunneling Hamiltonian\nbetween the quantum-ring and the contacts, which are given by\nHC=∞/summationdisplay\nj=−∞,σ(εjc†\nj,σcj,σ+γc†\nj,σcj+1,σ)+h.c., (2)\nHQD(t) =εd(t)/summationdisplay\nσd†\nσdσ (3)\nHT=/summationdisplay\nσ(Vσ\nLc†\n−1,σdσ+VRc†\n1,σdσ)+h.c.. (4)\nThe time dependence is introduced as a modulation of the energy leve ls of the quantum\ndot. For a single-level quantum-dot, this is is achieved through ε0(t) =ε0+vcos(Ω0t). We\n3consider a magnetic and electric fields in the system, their contribut ions to the Hamiltonian\nare embedded in the hopping matrix elements Vσ\nL=V0exp[i2π(φAB+σφSO)/φ0], where\nφABandφSOare the phases due to the Aharonov-Bohm effect and spin-orbit int eraction\nrespectively, σis the spin index ( σ=↑,↓orσ= 1,−1) andφ0is the flux quantum.\nSince we are interested in a single-parameter pumping configuration as in28,30,33,34, the\ncalculation of the electrical response requires going beyond the ad iabatic theory. Floquet\ntheory offers a suitable framework12,21. Here we use it in combination with Green’s func-\ntions, then we have a Floquet-Green function denoted by GFdefined from the Floquet’s\nHamiltonian HFas28,35GF= [EI −H F]−1.\nIf the spin-orbit coupling does not couple different spin channels, as in our case, the dc\ncomponent of the current is given by:\n¯Iσ=1\nτ/integraldisplayτ\n0dtIσ(t) (5)\n¯Iσ=e\nh× (6)\n/summationdisplay\nn/integraldisplay/bracketleftBig\nT(n)\n(R,σ),(L,σ)(ε)fL(ε)−T(n)\n(L,σ),(R,σ)(ε)fR(ε)/bracketrightBig\ndε,\nwhereT(n)\n(R,σ),(L,σ)(ε) is the probability for an electron on the left ( L) with spin σand energy\nεto be transmitted to the right ( R) reservoir while exchanging nphotons and τ= 2π/Ω0.\nThese probabilities are weighted by the usual Fermi-Dirac distributio n functions fR(L)for\neach electrode and are given, in terms of Floquet-Green function, by\nT(n)\n(R,σ),(L,σ)(ε) = 4Γ R(ε+n/planckover2pi1Ω0)|G(n)\nLR,σ(ε)|2ΓL(ε), (7)\nwhere the probability in opposite direction is described exchanging th e indexLwithR, and\nΓL(R)is the matrix coupling with left (right) electrode, defined as the imagin ary part of the\nelectrode’s self energies, i. e. Γ L(R)=−Im(ΣL(R)).\nThe associated spin-current is ¯Is=¯I↑−¯I↓while the charge current is ¯I=¯I↑+¯I↓.\nIII. RESULTS AND DISCUSSION\nUsing the model introduced before we now turn to our results for t he pumped electric and\nspin currents. To start with we consider the system in the absence of spin-orbit coupling.\n4We consider the leads in thermodynamic equilibrium ( i. e.fL(ε) =fR(ε) =f(ε)) as a semi-\ninfinite 1d system with nearest neighbor coupling γ, which is used as energy parameter. The\nac field frequency is set to Ω 0such that /planckover2pi1Ω0=γ/5, and the field magnitude is v= 0.07γ/e.\nThe hopping between the contacts and QD is V0=γ/4.\nFIG. 2. (Color online) (a) Transmission probability from le ft to right as a function of the applied\nmagnetic flux and the Fermi energy (for vanishing spin-orbit interaction). (b) Same as (a) for the\npumped current. Note the emergence of local maxima/minima c lose to the parameters where a\ntransmission zero is observed.\nFigure 2a shows a contour plot of the transmission probability as a fu nction of the Fermi\nlevel position and the magnetic flux. There we can observe the pres ence of a region where\nthe transmission is very close to zero (close to the intersection of t he dashed lines). This is\ndue to a destructive quantum interference known asFanoresona nce or antiresonance36–38(for\na recent review39). Interestingly, the pumped current shown in Fig. 2b achieves a ma ximum\n5intensity whenever the parameters are tuned close to the transm ission zero. Besides, we can\nsee that the sign of the observed maxima is reversed when travers ing the transmission zero.\nWe note that a single-time dependent harmonic potential does not b reak time reversal\nsymmetry (being defined as the existence of a time t0such that the Hamiltonian which\nis a function of the time tsatisfiesH(t0+t) =H(t0−t)). It is the magnetic field that\nbreaks TRS and allows for pumping to occur. Note, however that th is is true only whenever\nmagnetic flux is different fromthe half integer multiples of theflux qua ntum. For a magnetic\nflux ofπfor example, the Hamiltonian does not change upon time-reversal ( the phase in\nthe hopping term VLchanges from exp( iπ) to exp( −iπ) and therefore there is no pumped\ncurrent as observed in Fig. 2-b. On the other hand one should note that the magnetic\nfield alone would not produce pumping. This is the point where the time- dependent field\nenters into the game. Its role in this setup is to provide for additiona l effective channels\nfor transport, thereby circumventing the constraint of phase- rigidity40and allowing for the\ndirectional asymmetry in the transmission probabilities.\nThe addition of spin-orbit interaction breaks the spin degeneracy a nd at zero bias both\nchargeandspincurrentsaregenerated. ByexaminingFig. (2)onecanimaginet hatthespin-\norbit phase may be used to tune the working point of our pump for ea ch spinindependently .\nIndeed, the term φAB+σφSOenters as an effective spin-dependent flux φeff\nσ. In particular,\nwe could choose this spin-orbit phase so that it cancels out for one s pin direction (leading\nto a vanishing pumped charge for this spin) and adds up for the othe r, or in such a way as\nto cancel the charge current while summing up towards the spin cur rent.\nFigures 3 (a) and (b) show the charge (solid lines) and spin (dashed lin es) currents as\na function of the Fermi energy for different values of the spin-orb it and Aharanov-Bohm\nphases, while Figures 3 (c) and (d) show the pumped current for ea ch spin, spin up (black\nsolid lines) and down (red dashed lines). As anticipated, the paramet ers can be chosen so\nthat the currents for each spin direction have opposite signs (Fig. 3 (d)), thereby leading to\na pure spin-current as on Fig.3 (b). In this situation, the charge cu rrent cancels out whether\nthe spin-current is maximal.\nA point that needs to be emphasized in this proposal is that the pump ed current is in-\ntrinsically non-adiabatic (this contrasts for example with Ref.10using a similar setup but\nwith two time-dependent parameters). An adiabatic calculation wou ld actually give a van-\nishing response. Going beyond this adiabatic (low-frequency) limit is t herefore mandatory\n6\u0000\nFIG. 3. (Color online) a-b Pumpedcharge ( ¯I, dashed line) and spin currents ( ¯Is, solid line), dashed\nand solid for different values of the applied static flux and spi n-orbit interaction: (a) φAB= 0.2φ0,\nφSO= 0.1φ0and b)φAB= 0.5φ0,φSO= 0.4φ0. Onecan see that in (b) the charge current vanishes\nbut the spin current is enhanced. The spin-resolved contrib utions to the current for the same cases\nare shown in (c) and (d).\njustifying the use of Floquet theory. On the other hand, the pump ed currents in this\ncase emerges as an interplay between photon-assisted processe s and the interference in the\nAharanov-Bohm ring28. A similar setup but without contacts to electrodes were considere d\nin41,42. The spin-orbit coupling allows to obtain spin polarized pumped curren ts and the key\nrole of the time-dependent field is to provide for additional paths fo r interference breaking\nphase-rigidity40, although it does not break time-reversal symmetry.\n7The setup discussed here can be realized by using the present tech nologies. A quantum\ndot inserted in a mesoscopic ring has been fabricated by several lab oratories in the last\ndecades43,44. A particularly interesting case would be an InGAsquantum-dot inserted in\na mesoscopic quantum-ring since InGAshas a strong spin-orbit coupling and this coupling\ncan be controlled by an electric field45.\nIV. FINAL REMARKS\nIn summary, we study quantum spin-pumping with a single parameter in a configuration\nwhere the effect of the time-dependent field is reduced to the esse ntial one: providing for\nadditional channels for transport. The spin-orbit interaction bre aks the spin degeneracy and\nwe exploit it to generate a pure pumping spin-current through the in dependent tuning of\nthe phases dues to Aharonov-Bohm effect and spin-orbit coupling.\nV. ACKNOWLEDGMENTS\nWe acknowledge support of the SPU-Argentina (Project PPCP-02 5), SeCyT-UNC,\nANPCyT-FonCyT, FONDECYT project number 1100560, CONICYT ( Project PSD-65),\nand the scholarship CONICYT academic year 2012 number 22121816 .\n1S. A. Wolf et al., Science 294, 1488 (2001).\n2F. Pulizzi, Nature 11, 367 (2012).\n3I. Zutic, I., J. Fabian, and S. Das Sarma, Rev. Mod. Phys. 76, 323 (2004).\n4J. Sinova and I. Zutic, Nat Mater 11, 368 (2012).\n5R. Jansen, Nature Materials 11, 400 (2012).\n6D. D. Awschalom et al., Science 339, 1174 (2013).\n7D. R. McCamey, J. Van Tol, G. W. Morley, and C. Boehme, Science 330, 1652 (2010).\n8S. Datta and B. Das, Applied Physics Letters 56, 665 (1990).\n9J. Nitta, F. E. Meijer, and H. Takayanagi, Applied Physics Le tters75, 695 (1999).\n10R. Citro and F. Romeo, Phys. Rev. B 73, 233304 (2006).\n811B. H. Wu and J. C. Cao, Phys. Rev. B 75, 113303 (2007).\n12S. Kohler, J. Lehmann, and P. H¨ anggi, Physics Reports 406, 379 (2005).\n13Y. H. Wang, H. Steinberg, P. Jarillo-Herrero, N. Gedik, Scie nce342, 453 (2013).\n14T. Oka and H. Aoki, Phys. Rev. B 79, 081406 (2009); H. L. Calvo, H. M. Pastawski, S. Roche,\nL. E. F. Foa Torres, Appl. Phys. Lett. 98, 232103 (2011).\n15P. M. Perez-Piskunow, G. Usaj, C. A. Balseiro, L. E. F. Foa Tor res, Phys. Rev. B 89, 121401(R)\n(2014).\n16N. H. Lindner, G. Refael, V. Galitski, Nature Physics 7, 490 (2011).\n17Y. Tenenbaum Katan and D. Podolsky, Phys. Rev. B 88, 224106 (2013).\n18M. Thakurathi, K. Sengupta, D. Sen, Majorana edge modes in th e Kitaev model, arxiv:\narXiv:1310.4701 [cond-mat.mes-hall] (unpublished).\n19D. J. Thouless, Phys. Rev. B 27, 6083 (1983).\n20M. B¨ uttiker and M. Moskalets, in Lecture Notes in Physics , edited by J. Asch and A. Joye\n(Springer Berlin Heidelberg , 2006), Vol. 690, pp. 33–44–.\n21M. Moskalets and M. B¨ uttiker, Phys. Rev. B 66, 205320 (2002).\n22P. W. Brouwer, Phys. Rev. B 58, R10135 (1998).\n23E. Prada, P. San-Jose, and H. Schomerus, Phys. Rev. B 80, 245414 (2009).\n24R. Zhu and H. Chen, Applied Physics Letters 95, 122111 (2009).\n25L. Ingaramo and L. E. F. Foa Torres, Appl. Phys. Lett. 103, 123508 (2013).\n26T. A. Shutenko, I. L. Aleiner, and B. L. Altshuler, Phys. Rev. B61, 10366 (2000).\n27L. Arrachea, Phys. Rev. B 72, 121306 (2005).\n28L. E. F. Foa Torres, Phys. Rev. B 72, 245339 (2005).\n29V. Falko and D. Khmelnitskii, Sov. Phys. JETP 68, 186 (1989).\n30B. Kaestner et al., Phys. Rev. B 77, 153301 (2008).\n31A. Fujiwara, K. Nishiguchi, and Y. Ono, Appl. Phys. Lett. 92, 042102 (2008).\n32B. Kaestner et al., Appl. Phys. Lett. 94, 012106 (2009).\n33A. Agarwal and D. Sen, Phys. Rev. B 76, 235316 (2007).\n34L. E. F. Foa Torres, H. L. Calvo, C. G. Rocha, and G. Cuniberti, Appl. Phys. Lett. 99, 092102\n(2011).\n35L. E. F. Foa Torres and G. Cuniberti, Appl. Phys. Lett. 94, 222103 (2009).\n36F. Guinea and J. A. Verg´ es, Phys. Rev. B 35, 979 (1987).\n937J. L. D’Amato, H. M. Pastawski, and J. F. Weisz, Phys. Rev. B 39, 3554 (1989).\n38M. L. Ladron de Guevara and P. A. Orellana, Phys. Rev. B 73, 205303 (2006).\n39A. E. Miroshnichenko, S. Flach, and Y. S. Kivshar, Rev. Mod. P hys.82, 2257 (2010).\n40M. B¨ uttiker, IBM J. Res. Dev. 32, 317 (1988).\n41V. E. Kravtsov and V. I. Yudson, Phys. Rev. Lett. 70, 210 (1993).\n42V. I. Yudson and V. E. Kravtsov, Phys. Rev. B 67, 155310 (2003).\n43E. Buks et al., Phys. Rev. Lett. 77, 4664 (1996).\n44K. Kobayashi, H. Aikawa, S. Katsumoto, and Y. Iye, Phys. Rev. B68, 235304 (2003).\n45S. Amaha et al., Appl. Phys. Lett. 92, 202109 (2008).\n10"    },    {        "title": "1208.5841v1.Radio_frequency_spectroscopy_of_weakly_bound_molecules_in_spin_orbit_coupled_atomic_Fermi_gases.pdf",        "content": "arXiv:1208.5841v1  [cond-mat.quant-gas]  29 Aug 2012Radio-frequency spectroscopy of weakly bound molecules in spin-orbit coupled atomic\nFermi gases\nHui Hu1, Han Pu2, Jing Zhang3, Shi-Guo Peng4, and Xia-Ji Liu1∗\n1ARC Centre of Excellence for Quantum-Atom Optics and Centre for Atom Optics and Ultrafast Spectroscopy,\nSwinburne University of Technology, Melbourne 3122, Austr alia\n2Department of Physics and Astronomy and Rice Quantum Instit ute, Rice University, Houston, TX 77251, USA\n3State Key Laboratory of Quantum Optics and Quantum Optics De vices,\nInstitute of Opto-Electronics, Shanxi University, Taiyua n 030006, P. R. China\n4State Key Laboratory of Magnetic Resonance and Atomic and Mo lecular Physics,\nWuhan Institute of Physics and Mathematics, Chinese Academ y of Sciences, Wuhan 430071, P. R. China\n(Dated: October 29, 2018)\nWe investigate theoretically radio-frequency spectrosco py of weakly bound molecules in an ul-\ntracold spin-orbit-coupled atomic Fermi gas. We consider t wo cases with either equal Rashba and\nDresselhaus coupling or pure Rashba coupling. The former sy stem has been realized very recently\nat Shanxi University [Wang et al., arXiv:1204.1887] and MIT [Cheuk et al., arXiv:1205.3483]. We\npredict realistic radio-frequency signals for revealing t he unique properties of anisotropic molecules\nformed by spin-orbit coupling.\nPACS numbers: 03.75.Ss, 03.75.Hh, 05.30.Fk, 67.85.-d\nI. INTRODUCTION\nThe coupling between the spin of electrons to their\norbital motion, the so-called spin-orbit coupling, lies at\nthe heart of a variety of intriguing phenomena in di-\nverse fields of physics. It is responsible for the well-\nknown fine structure of atomic spectra in atomic physics,\nas well as the recently discovered topological state of\nmatter in solid-state physics, such as topological insu-\nlators and superconductors [1, 2]. For electrons, spin-\norbit coupling is a relativistic effect and in general not\nstrong. Most recently, in a milestone experiment at the\nNational Institute of Standards and Technology (NIST),\nsynthetic spin-orbit coupling was created and detected\nin an atomic Bose-Einstein condensate (BEC) of87Rb\natoms [3]. Using the same experimental technique, non-\ninteracting spin-orbit-coupled Fermi gases of40K atoms\nand6Li atoms have also been realized, respectively, at\nShanxi University [4] and at Massachusetts Institute of\nTechnology (MIT) [5]. These experiments have paved an\nentirely new way to investigate the celebrated effects of\nspin-orbit coupling.\nOwing to the high-controllability of ultracold atoms in\natomic species, interactions, confining geometry and pu-\nrity, the advantage of using synthetic spin-orbit coupling\nis apparent: (i) The strength of spin-orbit coupling be-\ntween ultracold atoms can be made very strong, much\nstronger than that in solids; (ii) New bosonic topological\nstates that have no analogy in solid-state systems may\nbe created; (iii) Ultracold atoms are able to realize topo-\nlogical superfluids that are yet to be observed in the solid\nstate; (iv) Strongly correlated topological states can be\nreadily realized, whose understanding remains a grand\n∗Electronic address: xiajiliu@swin.edu.auchallenge. At the moment, there has been a flood of the-\noretical work on synthetic spin-orbit coupling in BECs\n[6–18] and atomic Fermi gases [19–31], addressing par-\nticularly new exotic superfluid phases arising from spin-\norbit coupling [8, 9, 13, 23].\nIn this paper, we investigate theoretically momentum-\nresolved radio-frequency (rf) spectroscopy of an inter-\nacting two-component atomic Fermi gas with spin-orbit\ncoupling. The interatomic interactions can be easily ma-\nnipulated using a Feshbach resonance in40K or6Li atoms\n[32]. It is known that weakly bound molecules with\nanisotropic mass and anisotropic wave function (in mo-\nmentum space) may be formed due to spin-orbit coupling\n[19, 22, 23]. Here, we aim to predict observable rf signals\nof these anisotropic molecules. Our calculation is based\non the Fermi’s golden rule for the bound-free transition of\na stationary molecule [33]. We consider two kinds of spin-\norbit coupling: (1) the equal Rashba and Dresselhaus\ncoupling,λkxσy, which has been realized experimentally\nat Shanxi University [4] and MIT [5], and (2) the pure\nRashba coupling, λ(kyσx−kxσy),which is yet to be real-\nized. Here σxandσyare the Pauli matrices, kxandky\nare momenta and λis the strength of spin-orbit coupling.\nThe latter case with pure Rashba spin-orbit coupling is\nof particular theoretical interest, since molecules induc ed\nby spin-orbit coupling exist even for a negative s-wave\nscattering length above Feshbach resonances [19, 22, 23].\nThe paper is structured as follows. In the next section,\nwe discuss briefly the rf spectroscopy and the Fermi’s\ngolden rule for the calculation of rf transfer strength.\nIn Sec. III, we consider the experimental case of equal\nRashba and Dresselhaus spin-orbit coupling. We intro-\nduce first the model Hamiltonian and the single-particle\nand two-particle wave functions. We then derive an\nanalytic expression for momentum-resolved rf transfer\nstrength following the Fermi’s golden rule. It can be\nwritten explicitly in terms of the two-body wave function.2\nWe discuss in detail the distinct features of momentum-\nresolved rf spectroscopy in the presence of spin-orbit cou-\npling, with the use of realistic experimental parameters.\nIn Sec. IV, we consider an atomic Fermi gas with pure\nRashba spin-orbit coupling, a system anticipated to be\nrealized in the near future. Finally, in Sec. V, we con-\nclude and make some final remarks.\nII. RADIO-FREQUENCY SPECTROSCOPY\nAND THE FERMI’S GOLDEN RULE\nRadio-frequency spectroscopy, including momentum-\nresolved rf-spectroscopy, is a powerful tool to characteri ze\ninteracting many-body systems. It has been widely used\nto study fermionic pairing in a two-component atomic\nFermi gas near Feshbach resonances when it crosses from\na Bardeen-Cooper-Schrieffer (BCS) superfluid of weakly\ninteracting Cooper pairs into a BEC of tightly bound\nmolecules [34–36]. Most recently, it has also been used\nto detect new quasiparticles known as repulsive polarons\n[37, 38], which occur when “impurity” fermionic particles\ninteract repulsively with a fermionic environment.\nThe underlying mechanics of rf-spectroscopy is sim-\nple. For an atomic Fermi gas with two hyperfine states,\ndenoted as |1/angb∇acket∇ight=|↑/angb∇acket∇ightand|2/angb∇acket∇ight=|↓/angb∇acket∇ight, the rf field drives\ntransitions between one of the hyperfine states (i.e., |↓/angb∇acket∇ight)\nand an empty hyperfine state |3/angb∇acket∇ightwhich lies above it by\nan energy /planckover2pi1ω3↓due to the magnetic field splitting in bare\natomic hyperfine levels. The Hamiltonian for rf-coupling\nmay be written as,\nVrf=V0ˆ\ndr/bracketleftBig\nψ†\n3(r)ψ↓(r)+ψ†\n↓(r)ψ3(r)/bracketrightBig\n,(1)\nwhereψ†\n3(r)(ψ†\n↓(r)) is the field operator which creates an\natom in|3/angb∇acket∇ight(|↓/angb∇acket∇ight) at the position rand,V0is the strength\nof the rf drive and is related to the Rabi frequency ωR\nwithV0=/planckover2pi1ωR/2.\nFor the rf-spectroscopy of weakly bound molecules that\nis of interest in this work, a molecule is initially at rest in\nthe bound state |Φ2B/angb∇acket∇ightwith energy E0=−ǫB. HereǫBis\nthe binding energy of the molecules. A radio-frequency\nphoton with energy /planckover2pi1ωwill break the molecule and trans-\nfer one of the atoms to the third state |3/angb∇acket∇ight. In the case\nthat there is no interaction between the state |3/angb∇acket∇ightand the\nspin-up and spin-down states, the final state |Φf/angb∇acket∇ightinvolves\na free atom in the third state and a remaining atom in\nthe system. According to the Fermi’s golden rule, the rf\nstrength of breaking molecules and transferring atoms is\nproportional to the Franck-Condon factor [33],\nF(ω) =|/angb∇acketleftΦf|Vrf|Φ2B/angb∇acket∇ight|2δ/bracketleftbigg\nω−ω3↓−Ef−E0\n/planckover2pi1/bracketrightbigg\n,(2)\nwhere the Dirac delta function ensures energy conserva-\ntion andEfis the energy of the final state. The inte-\ngrated Franck-Condon factor over frequency should be\nunity,´+∞\n−∞F(ω)dω= 1, if we can find a complete setof final states for rf transition. Hereafter, without any\nconfusion we shall ignore the energy splitting in the bare\natomic hyperfine levels and set ω3↓= 0. To calculate the\nFranck-Condon factor Eq. (2), it is crucial to understand\nthe initial two-particle bound state |Φ2B/angb∇acket∇ightand the final\ntwo-particle state |Φf/angb∇acket∇ight.\nIII. EQUAL RASHBA AND DRESSELHAUS\nSPIN-ORBIT COUPLING\nLet us first consider a spin-orbit-coupled atomic Fermi\ngas realized recently at Shanxi University [4] and at MIT\n[5]. In these two experiments, the spin-orbit coupling is\ninduced by the spatial dependence of two counter propa-\ngating Raman laser beams that couple the two spin states\nof the system. Near Feshbach resonances, the system may\nbe described by a model Hamiltonian H=H0+Hint,\nwhere\nH0=/summationdisplay\nσˆ\ndrψ†\nσ(r)/planckover2pi12ˆk2\n2mψσ(r)+\nˆ\ndr/bracketleftbigg\nψ†\n↑(r)/parenleftbiggΩR\n2ei2kRx/parenrightbigg\nψ↓(r)+H.c./bracketrightbigg\n(3)\nis the single-particle Hamiltonian and\nHint=U0ˆ\ndrψ†\n↑(r)ψ†\n↓(r)ψ↓(r)ψ↑(r) (4)\nis the interaction Hamiltonian describing the contact in-\nteraction between two spin states. Here, ψ†\nσ(r)is the\ncreation field operator for atoms in the spin-state σ,\n/planckover2pi1ˆk≡ −i/planckover2pi1∇is the momentum operator, ΩRis the cou-\npling strength of Raman beams, kR= 2π/λRis deter-\nmined by the wave length λRof two Raman lasers and\ntherefore 2/planckover2pi1kRis the momentum transfer during the two-\nphoton Raman process. The interaction strength is de-\nnoted by the bare interaction parameter U0. It should be\nregularized in terms of the s-wave scattering length as,\ni.e.,1/U0=m/(4π/planckover2pi12as)−/summationtext\nkm/(/planckover2pi12k2).\nTo solve the many-body Hamiltonian Eq. (3), it is\nuseful to remove the spatial dependence of the Raman\ncoupling term, by introducing the following new field op-\nerators˜ψσvia\nψ↑(r) =e+ikRx˜ψ↑(r), (5)\nψ↓(r) =e−ikRx˜ψ↓(r). (6)\nWith the new field operators ˜ψσ, the single-particle\nHamiltonian then becomes,\nH0=/summationdisplay\nσˆ\ndr˜ψ†\nσ(r)/planckover2pi12/parenleftBig\nˆk±kRex/parenrightBig2\n2m˜ψσ(r)\n+ΩR\n2ˆ\ndr/bracketleftBig\n˜ψ†\n↑(r)˜ψ↓(r)+H.c./bracketrightBig\n, (7)3\nwhere in the first term on the right hand side of the\nequation we take “ +” for spin-up atoms and “ −” for spin-\ndown atoms. The form of the interaction Hamiltonian is\ninvariant,\nHint=U0ˆ\ndr˜ψ†\n↑(r)˜ψ†\n↓(r)˜ψ↓(r)˜ψ↑(r).(8)\nHowever, the rf Hamiltonian acquires an effective mo-\nmentum transfer kRex,\nVrf=V0ˆ\ndr/bracketleftBig\ne−ikRxψ†\n3(r)˜ψ↓(r)+H.c./bracketrightBig\n.(9)\nFor later reference, we shall rewrite the rf Hamiltonian\nin terms of field operators in the momentum space,\nVrf=V0/summationdisplay\nq/parenleftBig\nc†\nq−kRex,3cq↓+H.c./parenrightBig\n, (10)whereψ†\n3(r)≡/summationtext\nqc†\nq3eiq·rand˜ψ↓(r)≡/summationtext\nqcq↓eiq·r.\nHereafter, we shall denote ck3andckσas the field op-\nerators (in the momentum space) for atoms in the third\nstate and in the spin-state σ, respectively.\nA. Single-particle solution\nUsing the Pauli matrices, the single-particle Hamilto-\nnian takes the form,\nH0=ˆ\ndr[˜ψ†\n↑(r),˜ψ†\n↓(r)]/bracketleftBigg\n/planckover2pi12/parenleftbig\nk2\nR+k2/parenrightbig\n2m+hσx+λkxσz/bracketrightBigg/bracketleftbigg˜ψ↑(r)\n˜ψ↓(r)/bracketrightbigg\n, (11)\nwhere for convenience we have defined the spin-orbit couplin g constant denoted as λ≡/planckover2pi12kR/mand an “effective”\nZeeman field h≡ΩR/2. This Hamiltonian is equivalent to the one with equal Rashba and Dresselhaus spin-orbit\ncoupling,λkxσy. To see this, let us take the second transformation and intro duce new field operators Ψσ(r)via\n˜ψ↑(r) =1√\n2[Ψ↑(r)−iΨ↓(r)], (12)\n˜ψ↓(r) =1√\n2[Ψ↑(r)+iΨ↓(r)]. (13)\nUsing these new field operators, the single-particle Hamilt onian now takes the form,\nH0=ˆ\ndr[Ψ†\n↑(r),Ψ†\n↓(r)]/bracketleftBigg\n/planckover2pi12/parenleftbig\nk2\nR+k2/parenrightbig\n2m+λkxσy+hσz/bracketrightBigg/bracketleftbigg\nΨ↑(r)\nΨ↓(r)/bracketrightbigg\n, (14)\nwhich is precisely the Hamiltonian with equal Rashba\nand Dresselhaus spin-orbit coupling.\nThe single-particle Hamiltonian Eq. (11) can be diag-\nonalized to yield two eigenvalues\nEk±=/planckover2pi12k2\nR\n2m+/planckover2pi12k2\n2m±/radicalbig\nh2+λ2k2x. (15)\nHere “±” stands for the two helicity branches. The single-\nparticle eigenstates or the field operators in the helicity\nbasis take the form\nck+= +ck↑cosθk+ck↓sinθk, (16)\nck−=−ck↑sinθk+ck↓cosθk, (17)\nwhere\nθk= arctan[(/radicalbig\nh2+λ2k2x−λkx)/h]>0(18)is an angle determined by handkx. Note that,\ncos2θk=1\n2/parenleftBigg\n1+λkx/radicalbig\nh2+λ2k2x/parenrightBigg\n, (19)\nsin2θk=1\n2/parenleftBigg\n1−λkx/radicalbig\nh2+λ2k2x/parenrightBigg\n. (20)\nNote also that the minimum energy of the single-particle\nenergy dispersion is given by [24]\nEmin=/planckover2pi12k2\nR\n2m−mλ2\n2/planckover2pi12−/planckover2pi12h2\n2mλ2=−/planckover2pi12h2\n2mλ2, (21)\nifh<mλ2//planckover2pi12.4\nB. The initial two-particle bound state |Φ2B/angbracketright\nIn the presence of spin-orbit coupling, the wave func-\ntion of the initial two-body bound state has both spinsinglet and triplet components [19, 22, 23]. The wave\nfunction at zero center-of-mass momentum, |Φ2B/angb∇acket∇ight, may\nbe written as [22],\n|Φ2B/angb∇acket∇ight=1√\n2C/summationdisplay\nk/bracketleftBig\nψ↑↓(k)c†\nk↑c†\n−k↓+ψ↓↑(k)c†\nk↓c†\n−k↑+ψ↑↑(k)c†\nk↑c†\n−k↑+ψ↓↓(k)c†\nk↓c†\n−k↓/bracketrightBig\n|vac/angb∇acket∇ight, (22)\nwherec†\nk↑andc†\nk↓are creation field operators of spin-up and spin-down atoms w ith momentum kandC ≡/summationtext\nk[|ψ↑↓(k)|2+|ψ↓↑(k)|2+|ψ↑↑(k)|2+|ψ↓↓(k)|2]is the normalization factor. From the Schrödinger equation\n(H0+Hint)|Φ2B/angb∇acket∇ight=E0|Φ2B/angb∇acket∇ight, we can straightforwardly derive the following equations f or coefficients ψσσ′appearing\nin the above two-body wave function [22]:\n/bracketleftbigg\nE0−/parenleftbigg/planckover2pi12k2\nR\nm+/planckover2pi12k2\nm+2λkx/parenrightbigg/bracketrightbigg\nψ↑↓(k) = +U0\n2/summationdisplay\nk′[ψ↑↓(k′)−ψ↓↑(k′)]+hψ↑↑(k)+hψ↓↓(k), (23)\n/bracketleftbigg\nE0−/parenleftbigg/planckover2pi12k2\nR\nm+/planckover2pi12k2\nm−2λkx/parenrightbigg/bracketrightbigg\nψ↓↑(k) =−U0\n2/summationdisplay\nk′[ψ↑↓(k′)−ψ↓↑(k′)]+hψ↑↑(k)+hψ↓↓(k), (24)\n/bracketleftbigg\nE0−/parenleftbigg/planckover2pi12k2\nR\nm+/planckover2pi12k2\nm/parenrightbigg/bracketrightbigg\nψ↑↑(k) =hψ↑↓(k)+hψ↓↑(k), (25)\n/bracketleftbigg\nE0−/parenleftbigg/planckover2pi12k2\nR\nm+/planckover2pi12k2\nm/parenrightbigg/bracketrightbigg\nψ↓↓(k) =hψ↑↓(k)+hψ↓↑(k), (26)\nwhereE0=−ǫB<0is the energy of the two-body\nbound state. Let us introduce Ak≡ −ǫB−(/planckover2pi12k2\nR/m+\n/planckover2pi12k2/m)<0and different spin components of the wave-\nfunctions,\nψs(k) =1√\n2[ψ↑↓(k)−ψ↓↑(k)], (27)\nψa(k) =1√\n2[ψ↑↓(k)+ψ↓↑(k)]. (28)\nIt is easy to see that,\nψ↑↑(k) =√\n2h\nAkψa(k), (29)\nψ↓↓(k) =√\n2h\nAkψa(k), (30)\nψa(k) =λkx/bracketleftbigg1\nAk−2h+1\nAk+2h/bracketrightbigg\nψs(k),(31)\nand\n/bracketleftbigg\nAk−4λ2k2\nx\nAk−4h2/Ak/bracketrightbigg\nψs(k) =U0/summationdisplay\nk′ψs(k′).(32)\nAs required by the symmetry of fermionic system, the\nspin-singlet wave function ψs(k)is an even function\nof the momentum k, i.e.,ψs(−k) =ψs(k),and the\nspin-triplet wave functions are odd functions, satisfy-\ningψa(−k) =−ψa(k),ψ↑↑(−k) =−ψ↑↑(k)andψ↓↓(−k) =−ψ↓↓(k). The un-normalized wavefunction\nψs(k) = [Ak−4λ2k2\nx/(Ak−4h2/Ak)]−1is given by,\nψs(k) =1\nh2+λ2k2x/bracketleftbiggh2\nAk+λ2k2\nxAk\nA2\nk−4(h2+λ2k2x)/bracketrightbigg\n.(33)\nUsing Eq. (32) and un-normalized wave function ψs(k),\nthe bound-state energy E0or the binding energy ǫBis\ndetermined by U0/summationtext\nkψs(k) = 1, or more explicitly,\nm\n4π/planckover2pi12as−/summationdisplay\nk/bracketleftBig\nψs(k)+m\n/planckover2pi12k2/bracketrightBig\n= 0. (34)\nHere we have replaced the bare interaction strength U0\nby the s-wave scattering length asusing the standard\nregularization scheme mentioned earlier. The normaliza-\ntion factor of the total two-body wave function is given\nby,\nC=/summationdisplay\nk|ψs(k)|2/bracketleftBigg\n1+2λ2k2\nx\n(Ak−2h)2+2λ2k2\nx\n(Ak+2h)2/bracketrightBigg\n.\n(35)\nC. The final two-particle state |Φf/angbracketright\nLet us consider now the final state |Φf/angb∇acket∇ight. For this pur-\npose, it is useful to calculate\nVrf|Φ2B/angb∇acket∇ight=V0/summationdisplay\nqc+\n−q−kRex,3c−q↓|Φ2B/angb∇acket∇ight (36)5\nand then determine possible final states. It can be readily\nseen that,\nVrf|Φ2B/angb∇acket∇ight=−V0√\n2C/summationdisplay\nqc+\n−q−kRex,3/braceleftBig\n[ψ↑↓(q)−ψ↓↑(−q)]c†\nq↑+[ψ↓↓(q)−ψ↓↓(−q)]c†\nq↓/bracerightBig\n|vac/angb∇acket∇ight. (37)\nRewritingψ↑↓andψ↓↑in terms of ψsandψaas shown in Eqs. (27) and (28), and exploiting the parity of th e wave\nfunctions, we obtain a general result valid for any type of sp in-orbit coupling,\nVrf|Φ2B/angb∇acket∇ight=−/radicalbigg\n1\nCV0/summationdisplay\nqc+\n−q−kRex,3/braceleftBig\n[ψs(q)+ψa(q)]c†\nq↑+√\n2ψ↓↓(q)c†\nq↓/bracerightBig\n|vac/angb∇acket∇ight. (38)\nTo proceed, we need to rewrite the field operators c†\nq↑andc†\nq↓in terms of creation operators in the helicity basis. For\nthe case of equal Rashba and Dresselhaus spin-orbit couplin g, using Eqs. (16) and (17), we find that\nc†\nq↑= cosθqc†\nq+−sinθqc†\nq−, (39)\nc†\nq↓= sinθqc†\nq++cosθqc†\nq−. (40)\nThus, we obtain\nVrf|Φ2B/angb∇acket∇ight=−/radicalbigg\n1\nCV0/summationdisplay\nqc+\n−q−kRex,3/bracketleftBig\nsq+c†\nq+−sq−c†\nq−/bracketrightBig\n|vac/angb∇acket∇ight, (41)\nwhere\nsq+= [ψs(q)+ψa(q)]cosθq+√\n2ψ↓↓(q)sinθq, (42)\nsq−= [ψs(q)+ψa(q)]sinθq−√\n2ψ↓↓(q)cosθq. (43)\nEq. (41) can be interpreted as follows. The rf photon breaks a stationary molecule and transfers a spin-down atom to\nthe third state. We have two possible final states: (1) we may h ave two atoms in the third state and the upper helicity\nstate, respectively, with a possibility of |sq+|2/C; and (2) we may also have a possibility of |sq−|2/Cfor having two\natoms in the third state and the lower helicity state, respec tively.\nD. Momentum-resolved rf spectroscopy\nTaking into account these two final states and using the Fermi ’s golden rule, we end up with the following expression\nfor the Franck-Condon factor,\nF(ω) =1\nC/summationdisplay\nq/bracketleftbigg\ns2\nq+δ/parenleftbigg\nω−Eq+\n/planckover2pi1/parenrightbigg\n+s2\nq−δ/parenleftbigg\nω−Eq−\n/planckover2pi1/parenrightbigg/bracketrightbigg\n, (44)\nwhere\nEq±≡ǫB+/planckover2pi12/parenleftbig\nk2\nR+q2/parenrightbig\n2m±/radicalbig\nh2+λ2q2x+/planckover2pi12(q+kRex)2\n2m. (45)\nThe two Dirac delta functions in Eq. (44) are due to energy con servation. For example, the energy of the initial state\n(of the stationary molecule) is E0=−ǫB, while the energy of the final state is /planckover2pi12(q+kRex)2/(2m)for the free atom\nin the third state and /planckover2pi12(k2\nR+q2)/(2m)+/radicalbig\nh2+λ2q2xfor the remaining atom in the upper branch. Therefore, the\nrf energy /planckover2pi1ωrequired to have such a transfer is given by Eq+, as shown by the first Dirac delta function. It is easy to\ncheck that the Franck-Condon factor is integrated to unity,´+∞\n−∞F(ω) = 1.\nExperimentally, in addition to measuring the total number o f atoms transferred to the third state, which is propor-\ntional toF(ω), we may also resolve the transferred number of atoms for a giv en momentum or wave-vector kx. Such6\na momentum-resolved rf spectroscopy has already been imple mented for a non-interacting spin-orbit coupled atomic\nFermi gas at Shanxi University and at MIT. Accordingly, we ma y define a momentum-resolved Franck-Condon factor,\nF(kx,ω) =1\nC/summationdisplay\nq⊥/bracketleftbigg\ns2\nq+δ/parenleftbigg\nω−Eq+\n/planckover2pi1/parenrightbigg\n+s2\nq−δ/parenleftbigg\nω−Eq−\n/planckover2pi1/parenrightbigg/bracketrightbigg\n, (46)\nwhere the summation are now over the wave-vector q⊥≡(qy,qz)and we have defined kx≡qx+kRby shifting the\nwave-vector qxby an amount kR. This shift is due to the gauge transformation used in Eqs. (5 ) and (6). With the\nhelp of the two Dirac delta functions, the summation over q⊥may be done analytically. We finally arrive at,\nF(kx,ω) =m\n8π2/planckover2pi1C/bracketleftbig\ns2\nq+Θ/parenleftbig\nq2\n⊥,+/parenrightbig\n+s2\nq−Θ/parenleftbig\nq2\n⊥,−/parenrightbig/bracketrightbig\n, (47)\nwhereΘ(x)is the step function and\nq2\n⊥,±=m\n/planckover2pi1/parenleftBig\nω−ǫB\n/planckover2pi1/parenrightBig\n−/parenleftBig\nk2\nR+q2\nx+qxkR±m\n/planckover2pi12/radicalbig\nh2+λ2q2x/parenrightBig\n. (48)\nIt is understood that we will use q= (qx,q⊥,+)in the\ncalculation of sq+andq= (qx,q⊥,−)insq−.\nWe may immediately realize from the above expres-\nsion that the momentum-resolved Franck-Condon factor\nis an asymmetric function of kx, due to the coexistence of\nspin-singlet and spin-triplet wave functions in the initia l\ntwo-body bound state. Moreover, the contribution from\ntwo final states or two branches should manifest them-\nselves in the different frequency domain in rf spectra. As\nwe shall see below, these features give us clear signals of\nanisotropic bound molecules formed by spin-orbit cou-\npling. On the other hand, from Eq. (47), it is readily\nseen that once the momentum-resolved rf spectroscopy\nis measured with high resolution, it is possible to deter-\nmine precisely s2\nq+ands2\nq−and then re-construct the\ntwo-body wave function of spin-orbit bound molecules.\nE. Numerical results and discussions\nFor equal Rashba and Dresselhaus spin-orbit coupling,\nthe bound molecular state exists only on the BEC side\nof Feshbach resonances with a positive s-wave scattering\nlength,as>0[24]. Thus, it is convenient to take the\ncharacteristic binding energy EB=/planckover2pi12/(ma2\ns)as the unit\nfor energy and frequency. For wave-vector, we use kR=\nmλ//planckover2pi12as the unit. The strength of spin-orbit coupling\nmay be measured by the ratio\nEλ\nEB=/bracketleftbigg/planckover2pi12\nmλas/bracketrightbigg−2\n, (49)\nwhere we have defined the characteristic spin-orbit en-\nergyEλ≡mλ2//planckover2pi12=/planckover2pi12k2\nR/m. Note that, the spin-orbit\ncoupling is also controlled by the effective Zeeman field\nh= ΩR/2. In particular, in the limit of zero Zeeman field\nΩR= 0, there is no spin-orbit coupling term as shown\nin the original Hamiltonian Eq. (3), although the char-\nacteristic spin-orbit energy Eλ/negationslash= 0. UsingkRandEB\nas the units for wave-vector and energy, we can writea set of dimensionless equations for the binding energy\nǫB=−E0, normalization factor C, Franck-Condon factor\nF(ω)and the momentum-resolved Franck-Condon fac-\ntorF(ω,kx). We then solve them for given parameters\nEλ/EBandh/Eλ. In accord with the normalization con-\ndition´+∞\n−∞F(ω) = 1, the units for F(ω)andF(kx,ω)\nare taken to be 1/EBand1/(EBkR), respectively.\n/s48 /s49 /s50 /s51 /s52 /s53 /s54/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51\n/s48 /s50 /s52 /s54/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51\n/s32/s32\n/s32/s32\n/s69 /s32/s61/s32 /s69\n/s66\n/s48/s46/s53/s69 /s47/s69\n/s66/s61/s50/s70 /s40 /s41\n/s47/s69\n/s66/s104 /s32/s61/s32/s48/s46/s53 /s69/s49\nFigure 1: (color online) Franck-Condon factor of weakly\nbound molecules formed by equal Rashba and Dresselhaus\nspin-orbit coupling, in units of E−1\nB. Here we take h=Eλ/2\norΩR=/planckover2pi12k2\nR/mand set Eλ/EB= 0.5,1, and2. The re-\nsult without spin-orbit coupling is plotted by the thin dash ed\nline. The Inset shows the different contribution from the two\nfinal states at Eλ/EB= 1. The one with a remaining atom\nin the lower (upper) helicity branch is plotted by the dashed\n(dot-dashed) line.\nFig. 1 displays the Franck-Condon factor as a func-\ntion of the rf frequency at h/Eλ= 0.5and at several\nratios ofEλ/EBas indicated. For comparison, we show\nalso the rf line-shape without spin-orbit coupling [33],\nF(ω) = (2/π)√ω−EB/ω2, by the thin dashed line. In\nthe presence of spin-orbit coupling, the existence of two\npossible final states is clearly revealed by the two peaks\nin the rf spectra. This is highlighted in the inset for7\nEλ/EB= 1, where the contribution from the two possi-\nble final states is plotted separately. The main rf response\nis from the final state with the remaining atom staying\nin the lower helicity branch, i.e., the second term in the\nFranck-Condon factor Eq. (44). The two peak positions\nmay be roughly estimated from Eq. (48) for the threshold\nfrequencyωc±of two branches,\n/planckover2pi1ωc±=ǫB+/bracketleftBigg\n/planckover2pi12/parenleftbig\nk2\nR+q2\nx+qxkR/parenrightbig\nm±/radicalbig\nh2+λ2q2x/bracketrightBigg\nmin.\n(50)\nWith increasing spin-orbit coupling, the low-frequency\npeak becomes more and more pronounced and shifts\nslightly towards lower energy. In contrast, the high-\nfrequency peak has a rapid blue-shift.\n/s45/s51 /s45/s50 /s45/s49 /s48 /s49 /s50/s48/s49/s50/s51/s52/s53/s54/s32\n/s32\n/s32/s47/s69\n/s66/s40/s97/s41/s32 /s69 /s47/s69\n/s66/s32/s61/s32/s48/s46/s49\n/s45/s51 /s45/s50 /s45/s49 /s48 /s49 /s50 /s51\n/s32\n/s32/s40/s98/s41/s32 /s69 /s47/s69\n/s66/s32/s61/s32/s48/s46/s53\n/s45/s51 /s45/s50 /s45/s49 /s48 /s49 /s50 /s51\n/s32/s32\n/s40/s100/s41/s32 /s69 /s47/s69\n/s66/s32/s61/s32/s50/s46/s48\n/s107\n/s120/s47/s107\n/s82/s48\n/s48/s46/s49\n/s45/s51 /s45/s50 /s45/s49 /s48 /s49 /s50/s48/s49/s50/s51/s52/s53/s54/s32\n/s32\n/s107\n/s120/s47/s107\n/s82/s47/s69\n/s66\n/s40/s99/s41/s32 /s69 /s47/s69\n/s66/s32/s61/s32/s49/s46/s48\nFigure 2: (color online) Linear contour plot of momentum-\nresolved Franck-Condon factor, in units of (EBkR)−1. Here\nwe takeh=Eλ/2and consider Eλ/EB= 0.1,0.5,1, and2.\n/s45/s51 /s45/s50 /s45/s49 /s48 /s49 /s50 /s51/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s104 /s32/s61/s32/s48/s46/s53 /s69 /s44/s32 /s69 /s32/s61/s32 /s69\n/s66\n/s47/s69 /s32/s61/s32/s49/s46/s48/s70 /s40/s107\n/s120/s44/s32 /s41\n/s107\n/s120/s47/s107\n/s82/s51/s46/s48/s50/s46/s53/s50/s46/s48/s49/s46/s53\nFigure 3: (color online) Energy distribution curve of\nthe momentum-resolved Franck-Condon factor, in units of\n(EBkR)−1. We consider several values of the rf frequency ωas\nindicated, under given parameters h=Eλ/2andEλ=EB.\nFig. 2 presents the corresponding momentum-resolved\nFranck-Condon factor. We find a strong asymmetric dis-tribution as a function of the momentum kx. In par-\nticular, the contribution from two final states are well\nseparated in different frequency domains and therefore\nshould be easily observed experimentally. The asymmet-\nric distribution of F(kx,ω)is mostly evident in energy\ndistribution curve, as shown in Fig. 3, where we plot\nF(kx,ω)as a function of kxat several given frequencies\nω. In the experiment, each of these energy distribution\ncurves can be obtained by a single-shot measurement.\n/s48 /s49 /s50 /s51 /s52/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56\n/s45/s50 /s48 /s50/s48/s50/s52/s54/s47/s69\n/s66\n/s32/s32\n/s107\n/s120/s47/s107\n/s82/s104 /s61 /s69\n/s48/s46/s52/s48/s46/s54/s48/s46/s56/s104 /s47/s69 /s61/s32/s49/s46/s48/s70 /s40 /s41\n/s47/s69\n/s66\nFigure 4: (color online) Zeeman-field dependence of the\nFranck-Condon factor at Eλ=EB. Here we vary the effective\nZeeman fields, h/Eλ= 0.4,0.6,0.8and1.0. The inset shows\nthe momentum-resolved Franck-Condon factor at h=Eλ.\nWe finally discuss the effect of the effective Zeeman\nfieldh= ΩR/2. Fig. 4 shows how the line-shape of\nFranck-Condon factor evolves as a function of the Zee-\nman field at Eλ/EB= 1. In general, the larger Zeeman\nfield the stronger spin-orbit coupling. Therefore, as the\nsame as in Fig. 1, the increase in Zeeman field leads to\na pronounced peak at about the binding energy. There\nis a red-shift in the peak position as the binding energy\nbecomes smaller as the Zeeman field increases. As antic-\nipated, the larger the Zeeman field, the more asymmetric\nF(kx,ω)becomes. In the inset, we show as an example\nthe contour plot of F(kx,ω)ath/Eλ= 1.\nIV. RASHBA SPIN-ORBIT COUPLING\nWe now turn to the case with pure Rashba spin-orbit\ncoupling,λ(kyσx−kxσy), which may be realized exper-\nimentally in the near future. The single-particle Hamil-\ntonian may be written as [23],\nH0=ˆ\ndr/bracketleftBig\nψ†\n↑(r),ψ†\n↓(r)/bracketrightBig\nS/bracketleftbigg\nψ↑(r)\nψ↓(r)/bracketrightbigg\n,(51)\nwhere the matrix\nS=/bracketleftbigg\n/planckover2pi12/parenleftbig\nk2\nR+k2/parenrightbig\n/(2m)iλ(kx−iky)\n−iλ(kx+iky)/planckover2pi12/parenleftbig\nk2\nR+k2/parenrightbig\n/(2m)/bracketrightbigg\n.(52)\nHereλis the coupling strength of Rashba spin-orbit cou-\npling,kR≡mλ//planckover2pi12, and we have added a constant term8\n/planckover2pi12k2\nR/(2m)to make the minimum single-particle energy\nzero [24], i.e., Emin= 0.\nA. Single-particle solution\nWe diagonalize the matrix Sto obtain two helicity\neigenvalues [23],\nEk±=/planckover2pi12/parenleftbig\nk2\nR+k2/parenrightbig\n2m±λk⊥, (53)\nwherek⊥≡/radicalBig\nk2x+k2yand “±”stands for the two helicity\nbranches. For later reference, the field operators in the\noriginal spin basis and in the helicity basis are related by,\nc†\nk↑=1√\n2/parenleftBig\nc†\nk++ieiϕkc†\nk−/parenrightBig\n, (54)\nc†\nk↓=1√\n2/parenleftBig\nie−iϕkc†\nk++c†\nk−/parenrightBig\n. (55)Hereϕk≡arg(kx,ky)is the azimuthal angle of the wave-\nvectork⊥in thex−yplane.\nB. The initial two-particle bound state |Φ2B/angbracketright\nIn the case of Rashba spin-orbit coupling, the two-body\nwave function can still be written in the same form as\nin the previous case, i.e., Eq. (22), and the Schrödinger\nequation leads to [22],\nAkψ↑↓(k) = +U0\n2/summationdisplay\nk′[ψ↑↓(k′)−ψ↓↑(k′)]−λ(ky−ikx)ψ↑↑(k)+λ(ky+ikx)ψ↓↓(k), (56)\nAkψ↓↑(k) =−U0\n2/summationdisplay\nk′[ψ↑↓(k′)−ψ↓↑(k′)]+λ(ky−ikx)ψ↑↑(k)−λ(ky+ikx)ψ↓↓(k), (57)\nAkψ↑↑(k) =−λ(ky+ikx)ψ↑↓(k)+λ(ky+ikx)ψ↓↑(k), (58)\nAkψ↓↓(k) = +λ(ky−ikx)ψ↑↓(k)−λ(ky−ikx)ψ↓↑(k), (59)\nwhereAk≡E0−(/planckover2pi12k2\nR/m+/planckover2pi12k2/m)<0. It is easy to\nshow thatψa(k) = 0 and\n/bracketleftbigg\nAk−4λ2k2\n⊥\nAk/bracketrightbigg\nψs(k) =U0/summationdisplay\nk′ψs(k′).(60)\nThus, we obtain the (un-normalized) wavefunction:\nψs(k) =1\n2/bracketleftbigg1\nE0−2Ek++1\nE0−2Ek−/bracketrightbigg\n, (61)\nand the equation for the energy E0,\nm\n4π/planckover2pi12as=/summationdisplay\nk/bracketleftbigg1/2\nE0−2Ek++1/2\nE0−2Ek−+m\n/planckover2pi12k2/bracketrightbigg\n.\n(62)\nThe spin-triplet wave functions ψ↑↑(k)andψ↓↓(k)are\ngiven by,\nψ↑↑(k) =/bracketleftBigg\n−ie−iϕk√\n2λk⊥\nE0−2ǫk/bracketrightBigg\nψs(k),(63)\nψ↓↓(k) =/bracketleftBigg\n−ie+iϕk√\n2λk⊥\nE0−2ǫk/bracketrightBigg\nψs(k),(64)whereǫk≡/planckover2pi12k2/(2m). The normalization factor for the\ntwo-body wave function is therefore,\nC=/summationdisplay\nk|ψs(k)|2/bracketleftBigg\n1+4λ2k2\n⊥\n(E0−2ǫk)2/bracketrightBigg\n. (65)\nC. The final two-particle state |Φf/angbracketright\nTo obtain the final state, we consider again Vrf|Φ2B/angb∇acket∇ight.\nIn the present case, we assume that the rf Hamiltonian\nis given by,\nVrf=V0/summationdisplay\nq/parenleftBig\nc†\nq3cq↓+c†\nq↓cq3/parenrightBig\n. (66)\nFollowing the same procedure as in the case of equal\nRashba and Dresselhaus coupling, it is straightforward\nto show that,9\nVrf|Φ2B/angb∇acket∇ight=−/radicalbigg\n1\nCV0/summationdisplay\nqc+\n−q3/bracketleftBig\nψs(q)c†\nq↑+√\n2ψ↓↓(q)c†\nq↓/bracketrightBig\n|vac/angb∇acket∇ight. (67)\nUsing Eqs. (54) and (55) to rewrite c†\nq↑andc†\nq↓in terms of c†\nq+andc†\nq−, we obtain,\nVrf|Φ2B/angb∇acket∇ight=−/radicalbigg\n1\n2CV0/summationdisplay\nq/bracketleftBigg\nc+\n−q3c†\nq+\nE0−2Eq++ieiϕqc+\n−q3c†\nq−\nE0−2Eq−/bracketrightBigg\n|vac/angb∇acket∇ight. (68)\nTherefore, we have again two final states, differing in the hel icity branch that the remaining atom stays. The\nremaining atom stays in the upper branch with probability (2C)−1(E0−2Eq+)−2, and in the lower branch with\nprobability (2C)−1(E0−2Eq−)−2.\nD. Momentum-resolved rf spectroscopy\nUsing the Fermi’s golden rule, we have immediately the Franc k-Condon factor,\nF(ω) =1\nC/summationdisplay\nk/bracketleftBigg\nδ(ω−Ek+//planckover2pi1)\n2(ǫB+2Ek+)2+δ(ω−Ek−//planckover2pi1)\n2(ǫB+2Ek−)2/bracketrightBigg\n, (69)\nwhere\nEk±≡ǫB+/planckover2pi12k2\nR\n2m+/planckover2pi12k2\nm±λk⊥. (70)\nFor Rashba spin-orbit coupling, it is reasonable to define th e following momentum-resolved Franck-Condon factor,\nF(k⊥,ω) =1\nC/summationdisplay\nkz/bracketleftBigg\nδ(ω−Ek+//planckover2pi1)\n2(ǫB+2Ek+)2+δ(ω−Ek−//planckover2pi1)\n2(ǫB+2Ek−)2/bracketrightBigg\n, (71)\nwhere we have summed over the momentum kz. Integrating over kzwith the help of the two Dirac delta functions,\nwe find that,\nF(k⊥,ω) =m\n16π3/planckover2pi1C/bracketleftBigg\nΘ/parenleftbig\nk2\nz+/parenrightbig\n(/planckover2pi1ω+/planckover2pi12k2\nR/2m+λk⊥)2kz++Θ/parenleftbig\nk2\nz−/parenrightbig\n(/planckover2pi1ω+/planckover2pi12k2\nR/2m−λk⊥)2kz−/bracketrightBigg\n, (72)\nwhere\nk2\nz±=m\n/planckover2pi1/parenleftBig\nω−ǫB\n/planckover2pi1/parenrightBig\n−/parenleftbiggk2\nR\n2+k2\n⊥±kRk⊥/parenrightbigg\n. (73)\nIt is easy to see that the threshold frequencies for the two\nfinal states are given by,\n/planckover2pi1ωc+=ǫB+/planckover2pi12k2\nR\n2m, (74)\n/planckover2pi1ωc−=ǫB+/planckover2pi12k2\nR\n4m, (75)\nwhich differ by an amount of /planckover2pi12k2\nR/(4m) =Eλ/4. Near\nωc−, we find approximately that F(ω)∝Θ(ω−ωc−)/ω2.\nThus, the lineshape near the threshold is similar to that\nof a two-dimensional (2D) Ferm gas [36]. This similarity\nis related to the fact that at low energy a 3D Fermi gaswith Rashba spin-orbit coupling has exactly the same\ndensity of states as a 2D Fermi gas [14].\nE. Numerical results and discussions\nFor the pure Rashba spin-orbit coupling, the molec-\nular bound state exists for arbitrary s-wave scattering\nlengthas[19, 22, 23]. We shall take kR=mλ//planckover2pi12and\nEλ≡mλ2//planckover2pi12as the units for wave-vector and energy,\nrespectively. With these units, the dimensionless inter-\naction strength is given by /planckover2pi12/(mλas). The spin-orbit10\neffect should be mostly significant on the BCS side with\n/planckover2pi12/(mλas)<0, where the bound state cannot exist with-\nout spin-orbit coupling.\n/s48 /s49 /s50 /s51 /s52/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48\n/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s50\n/s47/s40 /s109 /s97\n/s115/s41\n/s32/s32\n/s32/s32\n/s32/s32/s70 /s40 /s41\n/s47/s69\nFigure 5: (color online) Franck-Condon factor of weakly\nbound molecules formed by Rashba spin-orbit coupling, in\nunits ofE−1\nλ. Here we take /planckover2pi12/(mλas) =−1(dashed line), 0\n(solid line), and 1(dot-dashed line). In the deep BCS limit,\n/planckover2pi12/(mλas)→ −∞ , the Franck-Condon factor peaks sharply\nat/planckover2pi1ω≃Eλ/2and becomes a delta-like distribution.\nFig. 5 shows the Franck-Condon factor at three dif-\nferent interaction strengths /planckover2pi12/(mλas) =−1,0, and+1.\nThe strong response in the BCS regime ( as<0) or in\nthe unitary limit ( as→ ±∞ ) is an unambiguous signal\nof the existence of Rashba molecules. In particular, the\nrf line-shape in the BCS regime shows a sharp peak at\nabout/planckover2pi1ω≃Eλ/2and decays very fast at high frequency.\nIn Fig. 6, we present the corresponding momentum-\nresolved Franck-Condon factor F(k⊥,ω), in the form of\ncontour plots. We can see clearly the different response\nfrom the two final states. The momentum-resolved rf\nspectroscopy is particularly useful to identify the contri -\nbution from the final state that has a remaining atom in\nthe upper branch, which, being integrated over k⊥, be-\ncomes too weak to be resolved in the total rf spectroscopy.\nFinally, we report in Fig. 7 energy distribution curves of\nF(k⊥,ω)in the unitary limit /planckover2pi12/(mλas) = 0. We find\ntwo sharp peaks in each energy distribution curve, aris-\ning from the two final states. When measured experi-\nmentally, these sharp peaks would become much broader\nowing to the finite experimental energy resolution.\nV. CONCLUSIONS\nIn conclusions, we have investigated theoretically the\nradio-frequency spectroscopy of weakly bound molecules\nin a spin-orbit coupled atomic Fermi gas. The wave\nfunction of these molecules is greatly affected by spin-\norbit coupling and has both spin-singlet and spin-triplet\ncomponents. As a result, the line-shape of the to-\ntal radio-frequency spectroscopy is qualitatively differ-\nent from that of the conventional molecules at the BEC-/s48 /s49 /s50 /s51/s48/s49/s50/s51/s52/s47/s69/s47/s69/s32\n/s32/s47/s69\n/s40/s97/s41/s32/s50\n/s47/s40 /s109 /s97\n/s115/s41/s32/s61/s32\n/s48 /s49 /s50 /s51/s48/s49/s50/s51/s52\n/s32\n/s40/s98/s41/s32/s50\n/s47/s40 /s109 /s97\n/s115/s41/s32/s61/s32/s48/s46/s48/s48/s49\n/s48/s46/s49\n/s48 /s49 /s50 /s51/s48/s49/s50/s51/s52\n/s32/s32\n/s40/s99/s41/s32/s50\n/s47/s40 /s109 /s97\n/s115/s41/s32/s61/s32\n/s107 /s47/s107\n/s82\nFigure 6: (color online) Contour plot of momentum-resolved\nFranck-Condon factor of weakly bound molecules formed by\nRashba spin-orbit coupling, in units of (EλkR)−1. The in-\ntensity increases from blue to red in a logarithmic scale. We\nconsider /planckover2pi12/(mλas) =−1(a),0(b), and 1(c).\nBCS crossover without spin-orbit coupling. In addi-\ntion, the momentum-resolved radio-frequency becomes\nhighly asymmetric as a function of the momentum.\nThese features are easily observable in current experi-\nments with spin-orbit coupled Fermi gases of40K atoms\nand6Li atoms. On the other hand, from the high-\nresolution momentum-resolved radio-frequency, we may\nre-construct the two-body wave function of the bound\nmolecules.\nWe consider so far the molecular response in the radio-\nfrequency spectroscopy. Our results should be quantia-\ntively reliable in the deep BEC limit with negligible num-\nber of atoms, i.e., in the interaction parameter regime\nwith1/(kFas)>2. However, in a real experiment, in or-\nder to maximize the spin-orbit effect, it is better to work\ncloser to Feshbach resonances, i.e., 1/(kFas)∼0.5. Un-\nder this situation, the spin-orbit coupled Fermi gas con-\nsists of both atoms and weakly bound molecules, which\nmay strongly interact with each other. Our prediction\nfor the molecular response is still qualitatively valid, wi th\nthe understanding that there would be an additional pro-\nnounced atomic response in the rf spectra. A more in-\ndepth investigation of radio-frequency spectroscopy re-11\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48 /s50/s46/s53 /s51/s46/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56\n/s32/s32 /s47/s69 /s32/s61/s32\n/s32/s32/s32/s32/s32/s32/s32/s32/s32/s52/s46/s48/s70 /s40/s107 /s44/s32 /s41\n/s107 /s47/s107\n/s82/s51/s46/s53\n/s51/s46/s48\n/s50/s46/s53\n/s50/s46/s48\n/s49/s46/s53\n/s49/s46/s48\nFigure 7: (color online) Momentum-resolved Franck-Condon\nfactor of Rashba molecules at /planckover2pi12/(mλas) = 0, shown in the\nform of energy distribution curves at several rf frequencie s as\nindicated.quires complicated many-body calculations beyond our\nsimple two-body picture pursued in the present work.\nAcknowledgments\nWe would like to thank Zeng-Qiang Yu and Hui Zhai\nfor useful discussions. HH and XJL are supported by the\nARC Discovery Projects (Grant No. DP0984522 and No.\nDP0984637) and the National Basic Research Program of\nChina (NFRP-China, Grant No. 2011CB921502). HP is\nsupported by the NSF and the Welch Foundation (Grant\nNo. C-1669). JZ is supported by the NSFC Project\nfor Excellent Research Team (Grant No. 61121064) and\nthe NFRP-China (Grant No. 2011CB921601). SGP is\nsupported by the NSFC-China (Grant No. 11004224)\nand the NFRP-China (Grant No. 2011CB921601).\n[1] X.-L. Qi and S.-C. Zhang, Physics Today 63, 33 (2010).\n[2] M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045\n(2010).\n[3] Y.-J. Lin, K. Jiménez-García, and I. B. Spielman, Nature\n(London) 471, 83 (2011).\n[4] P. Wang, Z.-Q. Yu, Z. Fu, J. Miao, L. Huang, S. Chai, H.\nZhai, and J. Zhang, Phys. Rev. Lett. 109, 095301 (2012).\n[5] L. W. Cheuk, A. T. Sommer, Z. Hadzibabic, T. Yefsah,\nW. S. Bakr, and M. W. Zwierlein, Phys. Rev. Lett. 109,\n095302 (2012).\n[6] T. D. Stanescu, B. Anderson, and V. Galitski, Phys. Rev.\nA78, 023616 (2008).\n[7] J. Larson and E. Sjöqvist, Phys. Rev. A 79, 043627\n(2009).\n[8] C. Wang, C. Gao, G.-M. Jian, and H. Zhai, Phys. Rev.\nLett.105, 160403 (2010).\n[9] C. Wu, I. Mondragon-Shem, and X.-F. Zhou, Chin. Phys.\nLett.28, 097102 (2011).\n[10] T.-L. Ho and S. Zhang, Phys. Rev. Lett. 107, 150403\n(2011).\n[11] X. Q. Xu and J. H. Han, Phys. Rev. Lett. 107, 200401\n(2011).\n[12] S. Sinha, R. Nath, and L. Santos, Phys. Rev. Lett. 107,\n270401 (2011).\n[13] H. Hu, B. Ramachandhran, H. Pu, and X.-J. Liu, Phys.\nRev. Lett. 108, 010402 (2012).\n[14] H. Hu and X.-J. Liu, Phys. Rev. A 85, 013619 (2012).\n[15] B. Ramachandhran, B. Opanchuk, X.-J. Liu, H. Pu, P. D.\nDrummond, and H. Hu, Phys. Rev. A 85, 023606 (2012).\n[16] R. Barnett, S. Powell, T. Gra, M. Lewenstein, and S. Das\nSarma, Phys. Rev. A 85, 023615 (2012).\n[17] Y. Deng, J. Cheng, H. Jing, C.-P. Sun, and S. Yi, Phys.\nRev. Lett. 108, 125301 (2012).\n[18] Q. Zhu, C. Zhang, and B. Wu, eprint arXiv:1109.5811.\n[19] J. P. Vyasanakere, V. B. Shenoy, Phys. Rev. B 83, 094515\n(2011).\n[20] M. Iskin and A. L. Subas, Phys. Rev. Lett. 107, 050402(2011).\n[21] S. L. Zhu, L. B. Shao, Z. D. Wang, and L. M. Duan,\nPhys. Rev. Lett. 106, 100404 (2011).\n[22] Z. Q. Yu and H. Zhai, Phys. Rev. Lett. 107, 195305\n(2011).\n[23] H. Hu, L. Jiang, X.-J. Liu, and H. Pu, Phys. Rev. Lett.\n107, 195304 (2011).\n[24] L. Jiang, X.-J. Liu, H. Hu, and H. Pu, Phys. Rev. A 84,\n063618 (2011).\n[25] M. Gong, S. Tewari, and C. Zhang, Phys. Rev. Lett. 107,\n195303 (2011).\n[26] L. Han and C. A. R. Sá de Melo, Phys. Rev. A 85,\n011606(R) (2012).\n[27] X.-J. Liu, L. Jiang, H. Pu, and H. Hu, Phys. Rev. A 85,\n021603 (2012).\n[28] X. Cui, Phys. Rev. A 85, 022705 (2012).\n[29] X.-J. Liu and H. Hu, Phys. Rev. A 85, 033622 (2012).\n[30] L. He and X.-G. Huang, Phys. Rev. Lett. 108, 145302\n(2012).\n[31] W. Yi and W. Zhang, arXiv:1204.6476.\n[32] C. Chin, R. Grimm, P. S. Julienne, and E. Tiesinga, Rev.\nMod. Phys. 82, 1225 (2010).\n[33] C. Chin and P. S. Julienne, Phys. Rev. A 71, 012713\n(2005).\n[34] C. Chin, M. Bartenstein, A. Altmeyer, S. Riedl, S.\nJochim, J. Hecker Denschlag, and R. Grimm, Science\n305, 1128 (2004).\n[35] C. H. Schunck, Y. Shin, A. Schirotzek, and W. Ketterle,\nNature (London) 454, 739 (2008).\n[36] Y. Zhang, W. Ong, I. Arakelyan, J. E. Thomas, Phys.\nRev. Lett. 108, 235302 (2012).\n[37] C. Kohstall, M. Zaccanti, M. Jag, A. Trenkwalder, P.\nMassignan, G. M. Bruun, F. Schreck and R. Grimm, Na-\nture (London) 485, 615 (2012).\n[38] M. Koschorreck, D. Pertot, E. Vogt, B. Fröhlich, M. Feld\nand M. Köhl, Nature (London) 485, 619 (2012)."    },    {        "title": "1807.07653v1.Three_boson_spectrum_in_the_presence_of_1D_spin_orbit_coupling__Efimov_s_generalized_radial_scaling_law.pdf",        "content": "arXiv:1807.07653v1  [cond-mat.quant-gas]  19 Jul 2018Three-boson spectrum in the presence of 1D spin-orbit coupl ing: Efimov’s generalized\nradial scaling law\nQ. Guan1and D. Blume1\n1Homer L. Dodge Department of Physics and Astronomy,\nThe University of Oklahoma, 440 W. Brooks Street, Norman, Ok lahoma 73019, USA\n(Dated: May 2, 2019)\nSpin-orbit coupled cold atom systems, governed by Hamilton ians that contain quadratic kinetic\nenergy terms typical for a particle’s motion in the usual Sch r¨ odinger equation and linear kinetic\nenergy terms typical for a particle’s motion in the usual Dir ac equation, have attracted a great\ndeal of attention recently since they provide an alternativ e route for realizing fractional quantum\nHall physics, topological insulators, and spintronics phy sics. The present work focuses on the three-\nboson system in the presence of 1D spin-orbit coupling, whic h is most relevant to ongoing cold\natom experiments. In the absence of spin-orbit coupling ter ms, the three-boson system exibits the\nEfimov effect: the entire energy spectrum is uniquely determi ned by the s-wave scattering length\nand a single three-body parameter, i.e., using one of the ene rgy levels as input, the other energy\nlevels can be obtained via Efimov’s radial scaling law, which is intimately tied to a discrete scaling\nsymmetry. It is demonstrated that the discrete scaling symm etry persists in the presence of 1D spin-\norbit coupling, implying the validity of a generalized radi al scaling law in five-dimensional space.\nThe dependence of the energy levels on the scattering length , spin-orbit coupling parameters, and\ncenter-of-mass momentum is discussed. It is conjectured th at three-body systems with other types\nof spin-orbit coupling terms are also governed bygeneraliz ed radial scaling laws, provided the system\nexhibits the Efimov effect in the absence of spin-orbit coupli ng.\nPACS numbers:\nI. INTRODUCTION\nUnder which conditions do two, three, or more par-\nticles form weakly-bound states, i.e., bound states that\nare larger than the range of the two-, three-, and higher-\nbody forces that bind the particles together? And under\nwhichconditionsarethecharacteristicsofthesefew-body\nbound states governed by underlying symmetries? These\nquestions are of utmost importance across physics. For\nexample, the existence of bound tetra-quark systems [1],\nfirst proposed in 1964 by Gell-Mann [2], has been chal-\nlenging our understanding of QCD. The existence of the\nextremely weakly-bound triton has a profound effect on\nthe nuclear chart, including the existence of larger exotic\nhalo nuclei [3, 4]. Historically, the triton has played an\nimportant role in the context of the Thomas collapse [5]\nand the Efimov effect [6, 7], which is intimately tied to a\ndiscrete scaling symmetry of the three-body Schr¨ odinger\nequation.\nThe three-boson system with two-body short-range\ninteractions is considered the holy grail of few-body\nphysics. It has captured physicists’ attention since Efi-\nmov’s bizarre and counterintuitive predictions in the\nearly 70ies [6, 7] and has spurred a flurry of theoreti-\ncal and experimental works from nuclear to atomic to\ncondensed matter to particle physics [8–21]. The unique\nscaling laws exhibited by Efimov trimers can be traced\nback to the existence of just one large length scale in the\nproblem, namely the two-body s-wave scattering length.\nThe main focus of the present work is on investigating\nwhat happens to the three-boson Efimov states in the\npresence of 1D spin-orbit coupling. Similar to few-bodysystems on the lattice [22], the 1D spin-orbit coupling in-\ntroduces a parametric dependence of the relative Hamil-\ntonian on the center-of-mass momentum. This center-of-\nmass momentum dependence leads, as we will show, to\na modification of the lowest break-up threshold and has\na profound effect on the binding energy. Despite this de-\npendence on the center-of-mass momentum and despite\nthe fact that the spin-orbit coupling terms depend on\nthree additional parameters (namely, kso, Ω andδ; see\nbelow), it is argued that the three-boson system in the\npresence of 1D spin-orbit coupling possesses, in the zero-\nrange limit, a discrete scaling symmetry and it is shown\nthat the energy spectrum is described by a generalized\nradial scaling law.\nThe 1D spin-orbit coupling terms, which break the ro-\ntational symmetry, introduce an unusual single-particle\ndispersion. The Hamiltonian ˆHjof thej-th particle with\nmassmand momentum operator ˆ/vector pj(with components\nˆpj,x, ˆpj,y, and ˆpj,z) is not simply given by ˆ/vector p2\nj/(2m) but\nincludes a term that emulates a spin-1/2 particle inter-\nacting with a momentum-dependent “magnetic field” of\ninfinite range [23–27],\nˆHj=ˆ/vector p2\nj\n2mIj+ˆ/vectorB(ˆpj,z)·ˆ/vector σj. (1)\nHere,Ijdenotes the 2x2 identity matrix that spans the\nspin degrees of freedom of the j-th particle, the vector ˆ/vector σj\ncontains the three Pauli matrices ˆ σj,x, ˆσj,y, and ˆσj,zof\nthej-th particle, andˆ/vectorBrepresents the effective magnetic\nfield,ˆ/vectorB= (Ω/2,0,/planckover2pi1ksoˆpj,z/m+δ/2), felt by the j-th par-\nticle. The Raman coupling Ω, detuning δ, and spin-orbit2\ncoupling strength kso, which characterize the two-photon\nRaman transitionthat couples (effectively) two hyperfine\nstates of an ultracold atom, describe the deviations from\nthe “normal” quadratic single-particle dispersion curves,\nEj,±=/vector p2\nj\n2m±/radicaligg/parenleftbigg/planckover2pi1ksopj,z\nm+δ\n2/parenrightbigg2\n+Ω2\n4,(2)\nwhere/vector pjandpj,z(both without “hat”) are expecta-\ntion values of the corresponding operators. For large\n|/vector pj|, the dispersion curves Ej,±approach/vector p2\nj/(2m). For\nsmall|/vector pj|, in contrast, the Ej,±curves deviate appre-\nciably from /vector p2\nj/(2m). The momenta /vector pjare generalized\nmomenta (sometimes also referred to as quasi-momenta)\nand not mechanical momenta (sometimes also referred\nto as kinetic momenta) [28]. Throughout this article, we\nfrequently drop the prefix “generalized” and refer to /vector pj\nas momentum vector of the j-th atom. The Hamiltonian\ngiven in Eq. (1) can also be realized by lattice shaking\ntechniques as well as in photonic crystals and mechanical\nsetups [27, 29–31].\nIf two-body short-range interactions are added, the\nmodified single-particle dispersion curves can signifi-\ncantly alter the properties of weakly-bound two- and\nthree-body states. This has been demonstrated exten-\nsively for two identical fermions for 1D, 2D, and 3D\nspin-orbit coupling [32–41] and for two identical bosons\nfor 2D and 3D spin-orbit coupling [41–45] but not for\nthe 1D spin-orbit coupling considered in this work. The\npresent work presents the first study of how the experi-\nmentally most frequently realized 1D spin-orbit coupling\ntermsmodify the three-bosonenergyspectrum. Wenote,\nhowever, that several three-body studies for bosonic and\nfermionic systems with other types of spin-orbit coupling\nexist [46–49]. All of these earlier studies limited them-\nselvestovanishingcenter-of-massmomentum. Ourwork,\nin contrast, allows for finite center-of-mass momenta.\nThe key objective of the present work is to show that\nthe three-boson system in the presence of 1D spin-orbit\ncoupling obeys a generalized radial scaling law, which re-\nflects the existence of a discrete scaling symmetry in the\nlimit of zero-range interactions. The scaling parameter\nλ0,λ0≈22.694, is the same as in the absence of the\nspin-orbit coupling terms. The generalized radial scaling\nlaw relates the energy for a given 1 /as,kso, Ω, and ˜δ[˜δ\nis a generalized detuning that is defined in terms of the\ndetuningδand thez-component of the center-of-mass\nmomentum, see Eq. (21)] to the energy for a scaled set\nof parameters, namely λ0/as,λ0kso, (λ0)2Ω, and (λ0)2˜δ.\nCorrespondingly, the term “radial” does not refer to the\nradius in a two-dimensional space as in the usual Efi-\nmov scenario but to the radius in a five-dimensional\nspace. The fact that the discrete scaling symmetry “sur-\nvives” when the spin-orbit coupling terms are added to\nthethree-bosonHamiltonianwithzero-rangeinteractions\ncan be intuitively understood from the observation that\nkso, Ωand˜δcanbethoughtofasintroducingfinitelength\nscales into the system. In the standard Efimov scenario,asintroduces a finite length scale and the radial scaling\nlaw holds regardless of whether |as|is larger or smaller\nthan the size of the trimer, provided |as|is much larger\nthan the intrinsic scalesof the underlying two- and three-\nbody interactions. In the generalized Efimov scenario\nconsidered here, the parameters as,kso, Ω, and ˜δeach\nintroduce a finite length scale. Correspondingly, the gen-\neralized radial scaling law holds regardless of whether\nthese length scales are larger or smaller than the size\nof the trimer, provided the length scales are much larger\nthan the intrinsic scalesof the underlying two- and three-\nbody interactions.\nOur findings for the experimentally most frequently\nrealized 1D spin-orbit coupling are consistent with\nRef. [48]. References [46, 48] considered an impurity\nwith 3D spin-orbit coupling that interacts with two iden-\ntical fermions that do not feel any spin-orbit coupling\ntermsandinteractwiththeimpuritythroughshort-range\ntwo-body potentials. Restricting themselves to vanish-\ning center-of-mass momenta, Ref. [46] stated that the\ntrimersformassratio /greaterorsimilar13.6“nolongerobey the discrete\nscaling symmetry even at resonance” because the spin-\norbitcoupling“introducesanadditionallength scale”. In\nRef. [48], the same authors arrive at a seemingly differ-\nentconclusion, namely“in the presenceofSO[spin-orbit]\ncoupling, the system exhibits a discrete scaling behav-\nior” and “the scaling ratio is identical to that without\nSO [spin-orbit] coupling”. The two statements can be\nreconciled by noting that the discrete scaling symmetry\nrequires an enlarged parameter space, an aspect that was\nrecognizedin Ref. [48]but notin Ref. [46]. We conjecture\nthat the discrete scaling symmetry holds for any type of\nspin-obit coupling and all center-of-mass momenta. De-\npending on the type of the spin-orbit coupling, the gen-\neralized Efimov plot is four- or five-dimensional and the\ngeneralized radial scaling law applies to the entire low-\nenergy spectrum. The dependence of the energy levels\non the system parameters has to be calculated explicitly\nonce for each type of spin-obit coupling.\nThe remainder of this article is organized as follows.\nTo set the stage, Sec. II reviews the standard Efimov\nscenariofor three identical bosons. Section III introduces\nthe system Hamiltonian in the presence of 1D spin-orbit\ncoupling and discussesthe associatedcontinuousand dis-\ncrete scaling symmetries. The generalized radial scaling\nlaw for the three-boson system in the presence of 1D\nspin-orbit coupling is confirmed numerically in Sec. IV.\nSection V highlights the role of the center-of-mass mo-\nmentum and discusses possible experimental signatures\nof this dependence. Finally, Sec. VI presents an outlook.\nTechnical details are relegated to several appendices.\nII. REVIEW OF STANDARD EFIMOV\nSCENARIO\nThe relative Hamiltonian for two identical bosons of\nmassminteracting through the zero-range contact inter-3\nactionV2b,zr(/vector r),\nV2b,zr(/vector r) =4π/planckover2pi12as\nmδ(3)(/vector r)∂\n∂rr, (3)\nwhereasdenotes the two-body s-wave scattering length\nand/vector rthe internucleardistance vector ( r=|/vector r|), possesses\na continuous scaling symmetry [8]. Performingthe trans-\nformation\nas→λas, /vector r→λ/vector r,andt→λ2t, (4)\nwheretdenotes the time and λa real number (scal-\ning parameter), the relative two-body time-dependent\nSchr¨ odinger equation remains unchanged.\nImportantly, the continuous scaling symmetry extends\nto three identical mass mbosons with position vectors /vector rj\nthat interact through pairwise s-wave zero-rangeinterac-\ntionsV2b,zr(/vector rjk) [8]. To see this, we consider the time-\ndependent Schr¨ odinger equation for the relative three-\nbody Hamiltonian ˆHrel,\nˆHrel=/summationdisplay\nj=1,2−/planckover2pi12\n2µj∇2\n/vector ρj+2/summationdisplay\nj=13/summationdisplay\nk=j+1V2b,zr(/vector rjk)+V3b,zr(R),\n(5)\nwhere/vector ρjdenotes the j-th relative Jacobi vector and µj\nthe associated Jacobi mass. We use a “K-tree” (see Ap-\npendix A) in which µ1for the two-body system is given\nbym/2 andµ1andµ2for the three-body system are\ngiven bym/2 and 2m/3. The zero-range three-body po-\ntentialV3b,zr(R),\nV3b,zr(R) =g3/planckover2pi12\nmδ(6)(R), (6)\nis written in terms of a six-dimensional delta-function\nin the three-body hyperradius R,R2=r2\n12+r2\n13+r2\n23.\nSince thecouplingconstant g3hasunits of length4, it can\nbe rewritten as g3=Cκ−4\n∗, whereCis a real constant\nandκ∗the three-body binding momentum of one of the\nthree-bosonboundstatesatunitarity(infinite as). While\nV3b,zr(R) has to be regularized in practice, the explicit\nregularization is irrelevant for our purpose. Performing\nthe transformation\nas→λas, /vector rjk→λ/vector rjk, t→λ2t,andκ∗→λ−1κ∗,(7)\nthe Schr¨ odinger equation for the relative Hamiltonian\ngiven in Eq. (5) remains unchanged, i.e., the three-body\nsystem possesses a continuous scaling symmetry.\nIntriguingly, the three-body systemwith zero-rangein-\nteractions additionally exhibits an exact discrete scaling\nsymmetry [8]. The discrete transformation is given by\nas→(λ0)nas, /vector rjk→(λ0)n/vector rjk, t→(λ0)2nt,\nandκ∗→κ∗,(8)\nwheren=±1,±2,···,±∞andλ0≈22.694. The dis-\ncrete scaling transformation, which underlies the three-\nbody Efimov effect, is illustrated in Fig. 1(a). Fixing thethree-body parameter κ∗[see Eq. (8)], the Efimov plot\ndepictsKas a function of 1 /as, where\nK=−/radicalbig\nm|E|//planckover2pi12 (9)\nandEdenotes the eigen energy of the Hamiltonian ˆHrel\ngivenin Eq. (5). The thick solidline in Fig. 1(a) shows K\nfor one of the three-body eigen energies. The thick solid\nline merges with the three-atom threshold on the nega-\ntiveas-side and with the atom-dimer threshold (dashed\nline) on the positive as-side. The thick solid line is\nobtained by solving the time-independent Schr¨ odinger\nequation for the three-body Hamiltonian ˆHrel. Provided\nthe thick solid line is known (a parametrization can be\nfound in Refs. [8, 11]), the thin solid lines—which cor-\nrespond to other three-body eigen energies—can be ob-\ntained using the discrete scaling symmetry without hav-\ning to explicitly solve the Schr¨ odinger equation again.\nFor the construction, it is convenient to switch from the\nvector/vector y= (1/as,K)Tto a radius y=|/vector y|and an angle ξ,\nK=−ysinξ (10)\nand\n(as)−1=ycosξ, (11)\nwhereξgoes fromπ/4 toπ. The limits π/4 andπare\nsetbythe atom-dimerandthree-atomthresholds, respec-\ntively. To obtain the thin solid lines in Fig. 1(a) from the\nthick solid line, one fixes the angle ξand reads off the\nvalues of the pair (1 /as,K) corresponding to the solid\nline. Using\ny=/radicalbig\n(as)−2+K2, (12)\nit can be seen that the discrete scaling transformation\nas→(λ0)nasandE→(λ0)−2nEimpliesy→(λ0)−ny.\nThus, dividing the radius yof the thick solid line by\n(λ0)±1,(λ0)±2,···and using the scaled value of yin\nEqs. (10) and (11), one obtains the values of the vec-\ntors/vector y= (1/as,K)Tcorresponding to the thin solid lines.\nThis construction, referred to as Efimov’s radial scaling\nlaw, is a direct consequence of the discrete scaling sym-\nmetry. If the three-bosonsystemis characterizedby κnew\n∗\ninstead ofκ∗, the entire energy spectrum is scaled, i.e.,\nif/vector y= (1/as,K)Tdescribes a point on the Efimov plot\nforκ∗, then (κnew\n∗/κ∗)/vector ydescribes a point on the Efimov\nplot forκnew\n∗.\nIII. SYMMETRIES IN THE PRESENCE OF 1D\nSPIN-ORBIT COUPLING\nThis section generalizes the symmetry discussion pre-\nsentedintheprevioussectiontothetwo-andthree-boson\nsystems in the presence of 1D spin-obit coupling. As\na first step, we derive the relative two- and three-body\nHamiltonian with zero-range interactions in the presence4\n-0.2 -0.1 0 0.1 0.2\nSign(as)|r0/as|1/2-0.2-0.10-|E/Esr|1/41/asKξλ0\nλ0(a)\n(b)(0,0)\nFIG. 1: (Color online) Radial scaling law for the standard\nEfimov scenario. (a) The solid lines show the quantity Kas a\nfunction of 1 /asfor the zero-range three-boson Hamiltonian.\nTo make this plot, λ0has been artificially set to 2 instead\nof 22.694. The dashed line shows the atom-dimer threshold.\nThe thin radially outgoing solid lines and arrows illustrat e the\nscaling law. Circles and squares mark the critical scatteri ng\nlengthsa−at which the trimer energy is degenerate with the\nthree-atom threshold and the critical scattering lengths a∗at\nwhich the trimer energy is degenerate with the atom-dimer\nthreshold, respectively. (b) Collapse of neighboring ener gy\nlevels for the finite-range interaction model [ ˆHrelin Eq. (5)\nwithV2b,zrandV3b,zrreplaced by V2b,GandV3b,G, respec-\ntively;R0=√\n8r0and (κ∗)−1≈66.05r0]. The solid line\nshows the fourth-root of the energy of the lowest three-boso n\nstate as a function of the square-root of the inverse of the\ns-wave scattering length. The dashed line shows the asso-\nciated atom-dimer threshold. The dots show the energy of\nthe second-lowest three-boson state, with the radial scali ng\nlaw applied in reverse so as to collapse the second-lowest le vel\n(dots) onto the lowest level (solid line). For clarity, the s caled\natom-dimer threshold for the second-lowest three-boson st ate\nis not shown.\nof 1D spin-orbit coupling. In a second step, it is shown\nthat these systems possess a continuous scaling symme-\ntry. In a third step, it is argued that the three-boson\nsystem additionally exhibits a discrete scaling symmetry,\nsuggesting the existence of a generalized radial scaling\nlaw. Numerical evidence that supports our claim that\nthe three-boson system with 1D spin-orbit coupling is\ngoverned by a generalized radial scaling law is presented\nin Sec. IV.\nWestartwiththe firststep. The N-bosonHamiltonian\nin the presence of 1D spin-orbit coupling reads\nˆH=ˆHni+ˆVint, (13)where the non-interacting and interacting pieces are\ngiven by\nˆHni=N/summationdisplay\nj=1ˆ/vector p2\nj\n2mI1,···,N+\nN/summationdisplay\nj=1/parenleftbigg/planckover2pi1kso\nmˆpj,z+δ\n2/parenrightbigg\nI1,···,j−1ˆσj,zIj+1,···,N+\nN/summationdisplay\nj=1Ω\n2I1,···,j−1ˆσj,xIj+1,···,N (14)\nand\nˆVint=\nN/summationdisplay\nj=1,j<kV2b(rjk)+N/summationdisplay\nj=1,j<k<lV3b(rjkl)\nI1,···,N.\n(15)\nHere,Ij,···,kwithj < kspans the spin degrees of free-\ndom of particles jthroughk,Ij,···,k=Ij⊗···⊗Ik. For\nN= 2, onlyV2bcontributes. For N= 3,rjklis equal\nto the three-body hyperradius R. The interaction model\nconsidered throughout this work assumes that the inter-\nactions are the same in all spin channels.\nIt is, just as in the case without spin-orbit coupling,\nconvenient to use Jacobi coordinates /vector ρjand associated\nmomentum operators ˆ/vector qjinstead of the single-particle\nquantities/vector rjandˆ/vector pj. Importantly, the N-th Jacobi\n“quantities” /vector ρNandˆ/vector qNcorrespond to the center-of-\nmass vector and center-of-mass momentum operator. It\ncan be shown straightforwardly that the Hamiltonian ˆH\ncommutes with the center-of-mass momentum operator\nˆ/vector qN[50], i.e., the Schr¨ odinger equation ˆHΨ =EΨ can be\nsolved for each fixed /vector qN. Using this and Jacobi coordi-\nnates, thenon-interactingfixed- /vector qNHamiltonian, denoted\nbyˆ¯Hni, reads\nˆ¯Hni=ˆ¯Hni,rel+/vector q2\nN\n2µNI1,···,N, (16)\nwhere\nˆ¯Hni,rel=N−1/summationdisplay\nj=1ˆ/vector q2\nj\n2µjI1,···,N+\nN−1/summationdisplay\nj=1/planckover2pi1kso\nmˆqj,zˆΣj,z+\nN/summationdisplay\nj=1Ω\n2I1,···,j−1ˆσj,xIj+1,···,N+\n/parenleftbigg/planckover2pi1kso\nµNqN,z+δ\n2/parenrightbigg\n×\n\nN/summationdisplay\nj=1I1,···,j−1⊗ˆσj,z⊗Ij+1,···,N\n.(17)5\nThe explicit form of the operators ˆΣj,zwithj=\n1,···,N−1 is given in Appendix A. Note that the first\nandsecondlinesofEq.(17)containmomentumoperators\nwhile the fourth line of Eq. (17) and the second term on\nthe right hand side of Eq. (16) contain expectation val-\nues of the center-of-mass momentum operators (and not\noperators). As “usual”, the interaction ˆVintdepends on\n/vector ρ1,···,/vector ρN−1but not on the center-of-mass vector /vector ρN.\nThis implies that the eigen states Ψ can be written as\nΨ = Φ cmΦrel, (18)\nwhere [51]\nΦcm= exp/parenleftbiggı/vector qN·/vector ρN\n/planckover2pi1/parenrightbigg\n(19)\nand where the Φ rel, which are eigen states ofˆ¯Hrel,\nˆ¯Hrel=ˆ¯Hni,rel+ˆVint, (20)\ndepend on the Jacobi vectors /vector ρ1,···,/vector ρN−1and the spin\ndegrees of freedom.\nEquation (17) shows that the eigen energies of ˆHde-\npend on the generalized detuning ˜δ,\n˜δ\n2=/planckover2pi1kso\nµNqN,z+δ\n2, (21)\ni.e.,qN,zandδenter as a combination and not as in-\ndependent parameters. This observation suggests that\nthe center-of-mass momentum may play a decisive role\nin determining the characteristics of the weakly-bound\ntwo- and three-body states (see also Refs. [35, 37]). The\nparametric dependence of the Hamiltonianˆ¯Hrelon the\nz-component qN,zof the center-of-mass momentum is a\ndirect consequence of the fact that the presence of the\nspin-orbit coupling breaks the Galilean invariance [26].\nOne immediate consequence of the broken Galilean in-\nvariance is that knowing the energy of an eigen state\nwithqN,z= 0 does not, in general, suffice for predicting\nthe energy of an eigen state with qN,z/negationslash= 0. Importantly,\nthe eigen states Ψ depend, in general, explicitly on qN,z\nandδand not just on ˜δ.\nWe are now ready to address the second step.\nParametrizingthe two-body interactions V2bby the zero-\nrange potential V2b,zr, theN= 2 relative Hamiltonian\ndepends on four parameters, namely as,kso, Ω, and ˜δ.\nIt can be readily checked that the corresponding time-\ndependent Schr¨ odinger equation is invariant under the\ntransformation\nas→λas, kso→λ−1kso,Ω→λ−2Ω,˜δ→λ−2˜δ,\n/vector r→λ/vector r,andt→λ2t,(22)\ni.e., theN= 2 system with zero-range interactions pos-\nsesses a continuous scaling symmetry. The continuous\nscalingsymmetryextendstothethree-bosonsystemwithzero-range interactions [ V2b=V2b,zrandV3b=V3b,zrin\nEq. (15)] in the presence of spin-orbit coupling since the\ncorresponding time-dependent N= 3 Schr¨ odinger equa-\ntion is invariant under the transformation\nas→λas, kso→λ−1kso,Ω→λ−2Ω,˜δ→λ−2˜δ,\n/vector r→λ/vector r, t→λ2t,andκ∗→λ−1κ∗.(23)\nEquations (22) and (23) generalize Eqs. (4) and (7) from\nSec. II.\nParalleling the discussion of Sec. II, step three poses\nthe question whether or not the three-boson system in\nthe presence of spin-orbit coupling additionally possesses\na discrete scaling symmetry in the zero-range interaction\nlimit. Our claim is that it does and that the discrete\ntransformation is given by\nas→(λ0)nas, kso→(λ0)−nkso,Ω→(λ0)−2nΩ,\n˜δ→(λ0)−2n˜δ, /vector r→(λ0)n/vector r, t→(λ0)2nt,andκ∗→κ∗,\n(24)\nwhereλ0is identical to the scaling factor of the stan-\ndard Efimov scenario, i.e., λ0≈22.694. Since no general\nanalytical solutions exist to the three-boson Schr¨ odinger\nequation in the presence of spin-orbit coupling, we rely\non numerics to support our claim. The claim that the\ndiscrete scaling symmetry survives in the presence of the\nspin-orbit coupling terms can be understood intuitively\nby realizing that the spin-orbit coupling terms modify\nthe low- but not the high-energy portions of the single-\nparticledispersioncurves. Tosetthestageforthenumer-\nical calculations presented in the next section, we discuss\na number of consequences of the discrete scaling symme-\ntry.\nThe discrete scaling symmetry suggests a generalized\nradial scaling law for the three-boson system in the pres-\nence of 1D spin-orbit coupling in which the Efimov plot\nfor/vector y= (1/as,K)Tdiscussed in the previous section is\nreplaced by a generalized Efimov plot for\n/vector y= (1/as,kso,Sign(Ω)/radicalbig\nm|Ω|//planckover2pi12,Sign(˜δ)/radicalig\nm|˜δ|//planckover2pi12,K)T.\n(25)\nIn the limit that the second, third, and fourth param-\neters vanish, each of the usual Efimov energies is four-\nfold degenerate due to the fact that the spin degrees\nof freedom enlarge the three-boson Hilbert space by a\nfactor of four (from the 23= 8 independent spin con-\nfigurations, one can construct four fully symmetric spin\nfunctions). For non-vanishing kso, Ω, and ˜δ, we expect\nthat the three-boson system supports four “unique” en-\nergy levels. Each of the four energies, collectively re-\nferred to as a manifold, is characterized by a vector /vector y.\nKnowing the dependence of each of these energy curves\non 1/as,kso, Sign(Ω)/radicalbig\nm|Ω|//planckover2pi12, and Sign( ˜δ)/radicalig\nm|˜δ|//planckover2pi12,\nthere should exist other energy manifolds for the same\nκ∗that can be obtained from the manifold that has been6\nmapped out without explicitly solving the three-boson\nSchr¨ odinger equation again.\nTo see how, we switch from the five parameters given\ninEq.(25)tothelength y=|/vector y|andfourangles ξ1,···,ξ4\nfor each of the four energy levels in the “reference mani-\nfold”,\nK=−ysinξ1sinξ2sinξ3sinξ4,(26)\nSign(˜δ)/radicalig\nm|˜δ|//planckover2pi12=ycosξ1sinξ2sinξ3sinξ4,(27)\nSign(Ω)/radicalbig\nm|Ω|//planckover2pi12=ycosξ2sinξ3sinξ4, (28)\nkso=ycosξ3sinξ4, (29)\nand\n1/as=ycosξ4. (30)\nThe full range of possible as,kso, Ω, and ˜δis covered if\nξ1,ξ2,ξ3,ξ4∈[0,π]. The range of the angles is further\nconstrained by the energy surfaces of the three-atom and\natom-dimer thresholds (see Secs. IV and V). To obtain\ntheKfor other manifolds, one chooses a direction of the\nvector/vector yby fixing the angles ξ1toξ4and reads off the\nvalues of the components of /vector yfor each of the four known\nenergy levels. Using\ny=/radicaligg\n(as)−2+(kso)2+m|Ω|\n/planckover2pi12+m|˜δ|\n/planckover2pi12+K2,(31)\nit can be seen that the discrete transformation as→\n(λ0)nas,kso→(λ0)−nkso, Ω→(λ0)−2nΩ,˜δ→(λ0)−2n˜δ,\nE→(λ0)−2nEimpliesy→(λ0)−ny. Thus, dividing the\n“hyperradius” ycorresponding to the j-th energy in the\nreference manifold by ( λ0)±1, (λ0)±2,···and using the\nscaled value of yin Eqs. (26)-(30), one obtains the values\nof the components of /vector yfor thej-th energy level in the\nother manifolds. The generalized scaling law is tested\nin the next section by considering two neighboring en-\nergy manifolds and confirming that the energy manifolds\ncollapse onto each other if the discrete scaling transfor-\nmation isapplied to the energylevelsin the moreweakly-\nbound manifold.\nIV. NUMERICAL TEST OF THE\nGENERALIZED RADIAL SCALING LAW\nTo facilitate the numerical calculations, we replace\nthe two-body zero-range potential V2b,zrby an attractive\nGaussianV2b,Gwith range r0and depthv0,\nV2b,G(rjk) =v0exp/parenleftigg\n−r2\njk\n2r2\n0/parenrightigg\n, (32)\nwherev0is negative and adjusted such that V2b,G(rjk)\nsupports at most one two-body s-wave bound state. To\nreduce finite-range effects, we aim to work in the regimewhere the absolute value of the free-space s-wavescatter-\ning lengthasis notably larger than r0. Parameter com-\nbinations where the absolute value of the free-space p-\nwave scattering volume is large are excluded. The three-\nbodyzero-rangepotential V3b,zrisreplacedbyarepulsive\nGaussianV3b,Gwith range R0and height V0,\nV3b,G(rjkl) =V0exp/parenleftigg\n−r2\njkl\n2R2\n0/parenrightigg\n. (33)\nIn our numerical calculations, R0is fixed at√\n8r0and\nV0(V0≥0) is varied to dial in the desired three-body\nparameterκ∗. Specifically, we define κ∗to be the bind-\ning momentum of the energetically lowest-lying universal\nthree-body state at unitarity (infinite as) forkso= Ω =\n˜δ= 0. Without the repulsive three-body potential, the\nlowest three-body state is not universal [52]. The repul-\nsive three-body potential pushes the lowest three-body\nenergyupandweadjust V0, forfixedv0(infiniteas), such\nthattheenergyofthelowestthree-bodystateforfinite V0\nis identical to the energy of the first excited three-body\nstate forV0= 0. This corresponds to ( κ∗)−1≈66.05r0,\ni.e., the trimer is much larger than the intrinsic scales of\nthe two- and three-body interactions. With the repulsive\nthree-bodypotentialturnedon, theradialscalinglawcan\nbe tested using the two lowest-lying energy manifolds.\nWe start the discussion of our numerical results by\nlooking at the three-body spectrum for the standard Efi-\nmov scenario ( kso= Ω =˜δ= 0). The reason for dis-\ncussing this “reference system” is two-fold: it illustrates\nhow to check the validity of the radial scaling law for a\ncase where it is known to hold and it gives us a sense\nfor the finite-range corrections expected in the presence\nof spin-orbit coupling. The solid line in Fig. 1(b) shows\ntherelativethree-bodyenergyoftheenergeticallylowest-\nlying state as a function of the inverse of the s-wave\nscattering length for the Hamiltonian given in Eq. (5)\nwithV2b,zrandV3b,zrreplaced by V2b,GandV3b,G, re-\nspectively. To “compress” the data, the horizontal and\nvertical axis employ a square-root and fourth-root repre-\nsentation. The scattering length is scaled by r0and the\nenergy byEsr,\nEsr=/planckover2pi12\nmr2\n0. (34)\nThe trimer energy merges with the three-atom threshold\non the negative scattering length side at r0/|as| ≈0.01.\nTo get a feeling for the finite-range effects, we assume\nthat the radial scaling law holds and apply it “in re-\nverse”. Specifically, using numerically determined pairs\n(1/as,K) corresponding to the excited state, the dots in\nFig.1(b) showthe points( λ0r0/as,λ0r0K), using—asfor\nthe lowest state—the square-root and fourth-root depic-\ntion. In the zero-range limit ( r0→0 andR0→0), the\ndots would lie on top of the solid line. The nearly perfect\nagreementbetweenthesolidlineandthe dotsin Fig.1(b)\nindicates that the finite-range effects are negligibly small\nfor the parameter combinations considered.7\nTo test the generalized radial scaling law proposed in\nSec. III, we calculate the eigen energies of states in the\nlowest and second-lowest manifolds ofˆ¯Hrel(there are at\nmost four states in each manifold) and scale the energies\nin the second-lowest manifold assuming that the gener-\nalized radial scaling law holds. If the energy curves col-\nlapse, the generalized radial scaling law is validated.\nIn the presence of the 1D spin-orbit coupling, the gen-\neralized Efimov plot has five axes. Clearly, visualizing\nenergy surfaces that depend on four parameters is im-\npossible and fully mapping out these high-dimensional\ndependences is computationally demanding. Thus, we\nconsider selected cuts in the five-dimensional space. Our\nfirst cut uses ( kso)−1= 50r0, Ω = 2Eso= 0.04Esr, where\nEso=(/planckover2pi1kso)2\n2m, (35)\nand˜δ= 0. For these parameters, we calculate the rela-\ntive energy Eof the states in the lowest energy man-\nifold. The solid lines in Fig. 2(a) show the quantity\n−|(E−Eaaa\nth)/Esr|1/4as a function of Sign( as)|r0/as|1/2,\nwhereEaaa\nthdenotes the energy of the lowest three-atom\nthreshold whose wave function has the same total mo-\nmentumq3,zalong thez-axis as the three-body system.\nThe determination of Eaaa\nthis discussed in Appendix C.\nThe energy Eaaa\nthis independent of κ∗and referencing E\nrelative to the lowest three-atom threshold does not alter\nthe generalizedradialscalinglaw. Figure2(a) showsthat\nthe lowest energy manifold consists of, depending on the\nvalue ofr0/as, zero, one, or two energy levels [the second\nand third excited states of the lowest manifold exist at\nlargerr0/asthan those shown in Fig. 2(a)].\nThe lowest three-body energy merges with the three-\natom threshold on the negative s-wave scattering length\nside and with the atom-dimer threshold [dashed line in\nFig. 2(a)] on the positive scattering length side. The de-\ntermination of the atom-dimer threshold energy Ead\nthis\ndiscussed in Appendix D. Just as the three-atom thresh-\nold, the atom-dimer threshold is independent of κ∗and\ndetermined such that the momentum q3,zof the atom-\ndimer system is the same as that of the three-body sys-\ntem. The second lowest state does not merge with the\nthree-atom threshold on the negative asside but with\nthe atom-dimer threshold on the positive asside.\nHaving determined the energies of the states in the\nlowest energy manifold, the next step is to calculate the\nenergies of the states in the second-lowest energy man-\nifold. To map the energies of the states in the second-\nlowest manifold onto the energies of the states in the\nlowest energy manifold, we use the same r0,R0, andκ∗\nand calculate the energies of the states in the second-\nlowest energy manifold for a ksothat isλ0times smaller\nthan theksoused to calculate the energiesof the states in\nthe lowest energy manifold [i.e., for ( kso)−1≈1,135r0],\nfor a Ω that is ( λ0)2times smaller than the Ω used\nto calculate the energies in the lowest energy manifold\n(i.e., for Ω ≈7.77×10−7Esr), and for ˜δ= 0 (the scal-\ning does not change zero) as a function of r0/as. Hav--0.2-0.10-|(E-Ethaaa)/Esr|1/4\n-0.2-0.10-|(E-Ethaaa)/Esr|1/4\n-0.2 -0.1 0 0.1 0.2\nSign(as)(r0/as)1/2-0.2-0.10-|(E-Ethaaa)/Esr|1/4(a)\n(b)\n(c)\nFIG. 2: (Color online) Testing the generalized radial scal-\ning law in the presence of 1D spin-orbit coupling. Pan-\nels (a)-(c) demonstrate the collapse of two neighboring en-\nergy manifolds for the finite-range interaction model [ˆ¯Hrelin\nEq. (20) with V2b=V2b,GandV3b=V3b,G;R0=√\n8r0and\n(κ∗)−1≈66.05r0]. The energies of states in the lowest man-\nifold (solid lines) are obtained for Ω = 2 Eso,˜δ= 0, and (a)\n(kso)−1= 50r0, (b) (kso)−1= 25r0, and (c) ( kso)−1= 100r0.\nIn all three panels, the dashed lines show the atom-dimer\nthreshold. The dots show the energies of states in the second -\nlowest manifold, with the generalized radial scaling law ap -\nplied in reverse so as to collapse the three-body energies of\nstates in the second-lowest manifold (dots) onto the three-\nbody energies of the states in the lowest manifold (solid lin es).\nFor clarity, the scaled atom-dimer thresholds for the secon d-\nlowest energy manifold are not shown in any of the panels.\ning calculated the energies of the states in the second-\nlowest manifold for the scaled kso, Ω, and ˜δ, the pairs\n(1/as,E−Eaaa\nth) are scaled (note that Eaaa\nthfor the ex-\ncited state manifold is calculated using the scaled kso,\nΩ, and˜δvalues). The dots in Fig. 2(a) show the scaled\npairs (λ0r0/as,−(λ0)2|E−Eaaa\nth|/Esr), using—as for the\nlowest energy manifold—the square-root and fourth-root\ndepiction. It can be seen that the solid lines and dots\nagree very well. Note that the atom-dimer threshold for\nthe second-lowest energy manifold also needs to be re-\ncalculated using the scaled kso, Ω, and ˜δ[the resulting\nenergies lie essentially on top of the dashed line and are\nnot shown in Fig. 2(a)]. The deviation between the solid\nline and dots is 0 .025% for (r0/as)1/2= 0 and 0.80 % for8\n(r0/as)1/2= 0.19. These deviations are comparable to\nthose between the corresponding atom-dimer thresholds\n[inthiscase, thedeviationsare0 .023%for(r0/as)1/2= 0\nand 0.82 % for (r0/as)1/2= 0.19]. We conclude that our\nnumerical results are consistent with the generalized ra-\ndial scaling law.\nWe emphasize that the scaling law has to be applied to\nall five axes of the generalized Efimov plot, i.e., to obtain\nthe dots in Fig. 2(a) it is imperative to not only scale the\ntwo axes depicted but also the parameters corresponding\nto the three axes that are not depicted. The ratio of the\nlowest energy in neighboring manifolds at unitarity, e.g.,\nis only equal to 22 .6942if the direction of ˆ yis the same\nfor the two energy levels under consideration.\nThe energy scales Eso,|Ω|, and|˜δ|are much smaller\nthan|E−Eaaa\nth|for a large portion of Fig. 2(a). The\nregion close to the three-atom threshold is an exception.\nAs such it might be argued that the spin-orbit coupling\nterms are too weak to notably influence the energy spec-\ntrum, possibly suggesting that the applicability of the\ngeneralized radial scaling law is trivial. One fact that\nspeaks against this argumentation is that the shape of\nthe energy levels is notably influenced by the spin-orbit\ncoupling terms. This is, e.g., reflected by the fact that\nthe energy levels in a given energy manifold are not de-\ngenerate. To more explicitly demonstrate that the gen-\neralized radial scaling law holds when one or more of\nthe energy scales associated with the spin-orbit coupling\ntermsis/arelargerthanthebindingenergy,werepeatthe\ncalculations for larger ksothan those used in Fig. 2(a).\nSpecifically, to determine the energy of the lowest state\nin the lowest energy manifold [solid line in Fig. 2(b)], we\nuse (kso)−1= 25r0while keeping r0,R0,κ∗, and˜δun-\nchanged. The Raman coupling strength Ω is set to be\nequal to 2Eso. To demonstrate the collapse of the ener-\ngies of the lowest states in the second-lowest and lowest\nmanifolds, we apply the generalized radial scaling law in\nthe same way as in Fig. 2(a). The energy of the lowest\nstate in the second-lowest manifold is shown by dots in\nFig. 2(b). The agreement with the solid line is excellent,\nsupporting our claim that the generalized radial scaling\nlaw is not limited to the case where the energy scales\nassociated with the spin-orbit coupling are smaller than\nthe binding energy of the trimer, provided these energies\nare notably smaller than Esr.\nTo show the characteristics of the excited states in the\nlowest manifold in more detail, we consider a smaller kso,\n(kso)−1= 100r0, and as before ˜δ= 0 and Ω = 2 Eso. The\nuse of a smaller kso[solid lines in Fig. 2(c)] moves the\nmerging points of the three-body energies corresponding\nto the excited states with the atom-dimer threshold to\nthe left compared to Fig. 2(a). Again, scaling the pa-\nrameters appropriately, the dots in Fig. 2(c) show the\nenergies of the states in the second-lowest energy mani-\nfold. The dots agree nearly perfectly with the solid lines\nnot onlyfor the loweststatein the twomanifolds but also\nforthe excited statesin the twomanifolds, lending strong\nnumerical support for the validity of the generalized ra-dialscalinglawandhence fortheexistence ofthe discrete\nscaling symmetry in the presence of 1D spin-orbit cou-\npling terms in the zero-range limit.\nAsalreadymentioned, thethree-bodyparameter κ∗for\nV0= 0, defined using the energy of the first excited state\nat unitarity in the absence of spin-orbit coupling, is iden-\ntical to the κ∗for the three-body interaction with finite\nV0used throughout this section. Turning on the spin-\norbit coupling, we checked that the energies of states in\nthe second-lowestmanifold for V0= 0 agreewell with the\nenergies of states in the lowest manifold for the finite V0.\nThis providesevidence that the generalizedradialscaling\nlaw is, just as the standard radial scaling law, indepen-\ndent of the details of the underlying microscopic interac-\ntion model. To confirm the continuous scaling symmetry\nofthethree-bodyHamiltonianinthepresenceof1Dspin-\norbit coupling, we checked that the energies for different\nκ∗can be mapped onto each other: If /vector ydescribes a point\non the Efimov plot for κ∗, then (κnew\n∗/κ∗)/vector ydescribes a\npoint on the Efimov plot for the new κnew\n∗.\nV. EXPERIMENTAL IMPLICATIONS: ROLE\nOF CENTER-OF-MASS MOMENTUM\nMeasuring signatures associated with two consecutive\ntrimer energy levels is challenging, especially for equal-\nmass bosons, due to the relatively large discrete scaling\nfactor of 22 .694. The reason is that the absolute value\nof the scattering length should, on the one hand, be no-\ntably larger than the van der Waals length rvdWand,\non the other hand, be smaller than the de Broglie wave\nlengthλdB[8, 11]. Despite these challenges, the discrete\nscaling symmetry underlying the standard Efimov sce-\nnario has been confirmed experimentally by monitoring\nthe atom losses of an ultracold thermal gas of Cs atoms\nas a function of the s-wave scattering length [20] (for\nunequal mass mixtures, see Refs. [57, 58]). When the\ntrimer energy is degenerate with the three-atom thresh-\nold [dots in Fig. 1(a); the corresponding critical scatter-\ning lengths are denoted by a(n)\n−] or with the atom-dimer\nthreshold [squares in Fig. 1(a); the corresponding criti-\ncal scattering lengths are denoted by a(n)\n∗], the losses are\nenhanced. Since the critical scattering lengths for con-\nsecutive trimer states are related to the scaling factor λ0,\nthese atom-loss measurements provide a direct confirma-\ntion of the discrete scaling symmetry. In addition, other\ncharacteristicsofthe standardEfimovscenariohavebeen\nmeasured [8, 11–13]. For example, the critical scattering\nlengthsa(n)\n−anda(n)\n∗for a given trimer level nare re-\nlated to each other by a universal number. Correspond-\ningly, the experimentally determined ratio a(n)\n−/a(n)\n∗can\nbe viewed as a test of the functional form of the en-\nergy levels shown in Fig. 1(a). Other experimental tests\nof the standard Efimov scenario include the determina-\ntion of the binding energy of an Efimov trimer via radio-\nfrequency spectroscopy [17, 18], the imaging of the quan-9\n-\n-1.2-0.9-0.6-0.3 0 0.300.10.20.30.40.50.6\n(a0/as)×104(δ˜/h)/kHz\nFIG. 3: (Color online) The contours show the negative of\nthe three-boson binding energy, in kHz, of the lowest state i n\nthe second lowest manifold as functions of ( as)−1and˜δfor\nkso/κ∗≈1.32 and Ω = 2 Eso, whereκ∗denotes the binding\nmomentum of the first excited Efimov trimer at unitarity in\nthe absence of spin-orbit coupling. The conversion to a0and\nkHz is done using the experimentally determined value of a(1)\n−\nfor Cs (see text). The calculations are performed forˆ¯Hrelwith\nV2b=V2b,GandV3b= 0 [(κ∗)−1≈66.05r0].\ntum mechanical density of the helium Efimov trimer via\nCoulomb explosion [21], and the observation of four- and\nfive-body loss features that are universally linked to the\ncritical scattering lengths of the Efimov trimer [53–56].\nDirectly measuring the discrete scaling symmetry in\nthe presence of spin-orbit coupling requires varying the\ninverse of the s-wave scattering length by the scaling fac-\ntorλ0, as in the standard Efimov scenario, as well as\nvarying the spin-orbit coupling parameters Ω, Eso, and\n˜δby (λ0)2. Covering such a wide range of parameters is\nexpected to be very challenging experimentally. In what\nfollows we instead focus on the situation where the spin-\norbit coupling parameters ksoand Ω are held fixed while\nthes-wave scattering length asand generalized detuning\n˜δare varied. An analogous study for the standard Efi-\nmov scenario would look at the three-boson system for\na fixed finite s-wave scattering length. In this case, the\nenergy spacing would not be ( λ0)2; however, the energies\nof neighboring states would still be uniquely related to\neach other.\nFor concreteness, we consider the133Cs system [59],\nfor which the three-atom resonances in the absence of\nspin-orbitcouplingoccuratthecriticalscatteringlengths\na(0)\n−≈ −936a0anda(1)\n−≈ −20,190a0[20]. Here, a0\ndenotes the Bohr radius and the superscripts “(0)” and\n“(1)”indicatethatthesecriticalscatteringlengthsarefor\nthe ground and first excited Efimov trimers, respectively.\nApplying ournumerical result κ∗a−=−1.505to the first\nexcited state, the Cs system is characterized by ( κ∗)−1≈\n13,416a0.\nFigure 3 shows the negative of the binding energy of\nthe lowest state in the first excited manifold for kso/κ∗≈1.32 and Ω = 2 Esoas functions of the inverse of the s-\nwave scattering length and the generalized detuning ˜δ\nusing, as in the previous sections, that the scattering\nlengths are the same for all spin channels. Using Cs’s\na(1)\n−, these parameterscorrespondto ( kso)−1≈10,156a0,\nEso/h≈0.132kHz, and Ω /h≈0.264kHz. Compari-\nson with the87Rb experiment at NIST [28], which uses\n(kso)−1≈3,410a0(corresponding to Eso/h≈1.786kHz)\nand Ω/hvalues ranging from zero to about 10kHz, sug-\ngests that the parameter regime covered in Fig. 3 is rea-\nsonable. Figure 3 shows that the three-boson binding\nenergy for a fixed scattering length is largest for ˜δ= 0\n(this is where the three-atom threshold has a degener-\nacy of six; see Appendix C). In addition, there exists an\nenhancement of the binding for ˜δ/h≈0.301kHz (this\nis where the three-atom threshold has a degeneracy of\nfour; see Appendix C). As ˜δgoes to infinity, the trimer\nin the presenceofthe 1Dspin-orbit couplingbecomesun-\nbound at the same scattering length as the correspond-\ning trimer in the absence of spin-orbit coupling (i.e., at\nas≈ −20,190a0).\nThe three-boson binding energy shown in Fig. 3 is\ncalculated by enforcing that the three-boson threshold\nhas the same center-of-mass momentum as the trimers\n(see Appendices C-E). If the detuning δis equal to zero,\nthe generalized detuning ˜δis directly proportional to the\nz-component q3,zof the center-of-mass momentum [see\nEq. (21)]. In this case, the trimer is bound maximally\nforq3,z= 0. However, for finite detuning δ, the most\nstrongly bound trimer has a finite center-of-mass mo-\nmentum. A similar dependence on the center-of-mass\nmomentum was pointed out in Refs. [35, 37, 60] for the\ntwo-fermion system.\nThe dependence ofˆ¯HrelonqN=3,zis a key character-\nistic of systems with 1D spin-orbit coupling. A similar\ndependence exists for three-body systems in the pres-\nence of 2D or 3D spin-orbit coupling (in these cases, the\nrelative Hamiltonian depends on two or all three compo-\nnentsof/vector qN=3) andforthree-bodysystemsonalattice (in\nthis case,/vector qN=3is a lattice or quasi-momentum vector).\nIn all works known to us [16, 46–48], the assumption\n/vector qN=3= 0 is made prior to obtaining concrete results.\nTable I contrasts studies for systems, which possess a\ncenter-of-mass momentum dependence, with the “stan-\ndard” three-boson Efimov system (first row), for which\nthe relative Hamiltonian is independent of /vector qN=3. In the\nstandard Efimov case, the lowest atom-dimer threshold\nof the relative Hamiltonian is given by the energy Eaof\nan atom with vanishing atom momentum vector /vector qa(Eais\nequal to zero) plus the energy Ed(/vector qd= 0) of a dimer with\nvanishing dimer momentum vector /vector qd. When the relative\nHamiltonian depends on /vector qN=3, the atom-dimer threshold\nneeds to be determined carefully, since the trimer with\nfixed/vector qN=3can break up into an atom with finite momen-\ntum and into a dimer with finite momentum in such a\nway that the generalized three-body center-of-mass mo-\nmentum is conserved. Of the many break-up configu-10\nsystemˆ¯Hrel=ˆ¯Hrel(/vector qN=3)?restriction? atom-dimer threshold (rel. Ham.)\n3-spinless bosons; “standard” Efimov scenario [6] no no Ed(/vector qd= 0)+Ea(/vector qa= 0)\nFFX; X feels 2D SOC; Borromean binding [47] yes /vector qN=3= 0 min/vector qdEd(/vector qd)+min /vector qaEa(/vector qa)\nFFX; X feels 3D SOC; universal/Efimov trimers [46, 48] yes /vector qN=3= 0min/vector qd+/vector qa=/vector qN=3[Ed(/vector qd)+Ea(/vector qa)]\nBBB quasi-particles on lattice; Efimov trimers [16] yes /vector qN=3= 0 Ed(/vector qd= 0)+Ea(/vector qa= 0)\nBBB; B’s feel 1D SOC (this work) yes no min/vector qd+/vector qa=/vector qN=3[Ed(/vector qd)+Ea(/vector qa)]\nTABLE I: Summary of three-particle studies. The standard Efi mov scenario (first row) is contrasted with three-particle s ystems\nfor which the relative Hamiltonianˆ¯Hreldepends parametrically on the generalized three-body cent er-of-mass momentum vector\n/vector qN=3. The last column lists the atom-dimer threshold of the relat ive Hamiltonian. The symbols Ed,Ea,/vector qd, and/vector qadenote\nthe energy of the dimer, energy of the atom, generalized mome ntum of the dimer, and generalized momentum of the atom,\nrespectively. “SOC” stands for “spin-orbit coupling”, “F” for “fermion”, “X” for a particle different from “F”, and “B” f or\n“boson”.\nrations that conserve the three-body center-of-mass mo-\nmentum, theonewiththelowestenergydefinestheatom-\ndimer threshold. Table I shows that the definition of\nthe lowest atom-dimer threshold of the relative Hamil-\ntonian varies in the literature. The definitions employedin Refs. [16, 47] disagree with the definition used in the\npresent work (last row of Table I). While the definition\nof Ref. [47] may be meaningful in a many-body context\n(see also comment [60]), we fail to see how the definition\nof Ref. [16] can, in general, be correct.\nIt is proposed that the center-of-mass momentum de-\npendence can be observed experimentally by performing\natom-loss measurements for fixed kso, Ω, andδon a cold\nthermalatomic gas. Tuning the s-wavescatteringlength,\none expects—just asin the casewhere the spin-orbitcou-\npling is absent—enhanced losses when the trimer energy\nis degenerate with the three-atomthreshold. However, in\ncontrast to the standard Efimov scenario, such a degen-\neracy exists for a rangeof scattering lengths provided the\ntrimersembedded inthethermalgashavedifferentthree-\nbody center-of-mass momenta (the exact distribution of\ncenter-of-mass momenta is set by the temperature of the\ngas sample). Figure 3 shows that the critical scattering\nlengtha(1)\n−changes, for the Cs example, from −20,190a0\nfor large ˜δto−7,791a0for˜δ= 0. Provided the three-\nbody center-of-mass momenta are spread over the range\ncovered on the vertical axis in Fig. 3, one expects en-\nhanced losses over the entire scattering length window.\nThe difference between the losses in the presence and ab-\nsence of the spin-orbit coupling terms can be interpreted\nas a few-body probe of the breaking of the Galilean in-\nvariance in the presence of spin-orbit coupling.\nAn important question is whether the changes of the\nloss features related to the lowest state in the second-\nlowest manifold will be washed out by finite tempera-\nture effects. A comprehensive answer to this question\nwill require performing three-body recombination calcu-\nlations, which include thermal averaging, in the pres-\nence of spin-orbit coupling. Such calculations are beyond\nthe scope of this work. Given that the energy scales\nassociated with the spin-orbit coupling are, for the ex-\nample considered in Fig. 3, comparable to /planckover2pi12κ2\n∗/mand\nthat the binding energy of the trimer near the three-atom threshold is much smaller than /planckover2pi12κ2\n∗/m, we are\nhopeful that the temperatures realized in previous Cs\nexperiments ( T≥7.7nK) [20] are low enough to ob-\nserve the impact of the spin-orbit coupling terms on the\nloss features. For example, without spin-orbit coupling,\nthe loss coefficient L3is maximal around −20,000a0\nand reaches about half of its maximum value at around\n−10,000a0[see Fig. 1(a) of Ref. [20]]. In the pres-\nence of spin-orbit coupling, the loss feature is expected\nto be centered over the range −20,190a0to−7,790a0,\nleading to an observable modification of the shoulder\non the less negative scattering length side. The shape\nof the shoulder is expected to carry a signature of\nthe non-monotonic dependence of the critical scatter-\ning length a(1)\n−on the center-of-mass momentum. For\nexample, three different ˜δcorrespond to the same a(1)\n−\nfora(1)\n−∈[−10,330a0,−9,160a0] but each ˜δcorresponds\nto a unique a(1)\n−fora(1)\n−∈[−20,190a0,−10,330a0] and\na(1)\n−∈[−9,160a0,−7,790a0].\nFor the same spin-orbit coupling parameters, the criti-\ncalscatteringlength a(0)\n−, associatedwiththe loweststate\nin the lowest manifold, displays essentially no depen-\ndence on ˜δ, i.e., the associated three-atom loss feature\nshould only be minimally affected by the spin-orbit cou-\npling terms. Intuitively, this can be understood byrealiz-\ning that the energy scales associated with the spin-orbit\ncoupling parameters are much smaller than the binding\nenergy of the lowest-lying trimer state. The fact that\nthe loss features for the lowest state in the lowest and\nsecond-lowest manifolds are expected to be very differ-\nent can also be understood from the generalized radial\nscaling law. Fixing the spin-orbit coupling parameters11\ncorresponds to looking at particular cuts in the five-\ndimensional parameterspace as opposed to looking along\na specific radial direction. As a consequence, the loss fea-\ntures for the two manifolds can be very different even if\nthe scattering lengths at which the loss features occur\nare, roughly, spaced by λ0.\nVI. CONCLUDING REMARKS\nThis work analyzed what happens to the three-boson\nEfimov spectrum if 1D spin-orbit coupling terms, realiz-\nable in cold atoms as well as in photonic crystalsand me-\nchanical setups, are added to the Hamiltonian. The spin-\norbit coupling terms introduce a parametric dependence\noftherelativeHamiltonianonthecenter-of-massmomen-\ntum vector. A similar center-of-mass momentum vector\ndependence exists for few-body systems with short-range\ninteractions on a lattice. The present work mapped out,\nfor the first time, the three-boson spectrum as a function\nof the center-of-mass momentum vector. It was found\nthat the three-boson system in the presence of 1D spin-\norbit coupling obeys a generalized radial scaling law in\na five-dimensional parameter space, which is associated\nwith a discrete scaling symmetry. Within the framework\nof effective field theory, the existence of the discrete scal-\ning symmetry can be rationalized by scale separation:\nThe discrete scaling symmetry of the standard Efimov\nscenario “survives” provided the additional length scales\nare much larger than the ranges of the intrinsic interac-\ntions. While our work focused on 1D spin-orbit coupling,\nthe discrete scaling symmetry should persist for other\ntypes of spin-obit coupling schemes as well.\nThe spin degrees of freedom lead—for the type of spin-\norbit coupling considered in this work—to a quadru-\npling of each Efimov trimer (manifold of four states).\nThe three-body states in a given manifold are tied to\none of the three two-boson states [61]. The point\n(1/as,kso,Ω,˜δ) = (0,0,0,0) serves as an accumulation\npoint for all four states of the manifold, i.e., in its vicin-\nity, there exist infinitely many three-body bound states.\nThe rich structure of two- and three-boson states should\nbe amenable to experimental verification. Due to the\ndependence of the trimers on the center-of-mass momen-\ntum, the scattering length at which the lowest trimer in\nthe second-lowest manifold merges with the atom-atom-\natom threshold is, in fact, a scattering length window.\nSimilar scattering length windows exist for the excited\nstates in the second-lowest manifold. It was argued that\nthese scattering length windows should be observable in\ncold atom loss experiments, providing a direct few-body\nsignature of the breaking of the Galilean invariance of\nsystems with spin-orbit coupling.\nIf one considers a cut in the generalized five-\ndimensional Efimov plot, energy levels are not spaced\nby the scaling factor ( λ0)2. Let us consider the situa-\ntion where asis infinitely large and where kso, Ω, and ˜δ\nare finite. In this case, the low-energy scales associatedwith the 1D spin-orbit coupling terms lead to a cut-off\nof the hyperradial −1/R2Efimov potential curve ( Rde-\nnotes the three-body hyperradius). As a consequence,\nthe number of three-body bound states at unitarity is\nnot infinitely large. Albeit due to a different mechanism,\nthis is similar in spirit to the disappearance of Efimov\nstates if an Efimov trimer is placed into a gas of bosons\nor fermions [62, 63]. This is also similar in spirit to a\nrather different system, namely the H−ion. Taking only\nCoulomb interactions into account, one obtains a −1/r2\nattraction [64], where ris the distance between the ex-\ntra electron and the atom. Relativistic effects introduce\nan additional length scale, which renders the number of\nbound states finite [64].\nThe calculationsin this workwereperformedassuming\nthat the interactions between the different spin-channels\nare all equal. If one of the scattering lengths is large and\ntunable while the others are close to zero, the discrete\nscaling symmetry should still hold (approximately). To\nfind the functional form of the energies for this scenario,\nthe spectrum has to be recalculated.\nThestudypresentedshouldbeviewedasafirststepto-\nward uncovering the rich three- and higher-body physics\nthat emerges as a consequence of the unique coupling be-\ntween the relative and center-of-mass degrees of freedom\nin cold atom systems in the presence of artificial Gauge\nfields. While somewhat different in nature, the coupling\nof these degrees of freedom in the relativistic Klein Gor-\ndon and Dirac equations and quantum field theories has\ncaptured physicists’ imagination for many decades.\nVII. ACKNOWLEDGEMENT\nSupport by the National Science Foundation through\ngrant numbers PHY-1509892and PHY-1745142is grate-\nfully acknowledged. This work used the Extreme Sci-\nence and Engineering Discovery Environment (XSEDE),\nwhich is supported by NSF Grant No. OCI-1053575.\nSome of the computing for this project was performed at\nthe OU Supercomputing Center for Education and Re-\nsearch (OSCER) at the University of Oklahoma (OU).\nAppendix A: Jacobi coordinates\nThe single-particleand Jacobicoordinatesemployedin\nthis work are related through the matrix U[65],\n(/vector ρ1,···,/vector ρN)T=U(/vector r1,···,/vector rN)T, (A1)\nwhere\nU=/parenleftigg\n1−1\n1/2 1/2/parenrightigg\n(A2)12\nfor the equal-mass two-particle system and\nU=\n1−1 0\n1/2 1/2−1\n1/3 1/3 1/3\n (A3)\nfor the equal-mass three-particle system. The transfor-\nmation matrix Ualso defines the matrices ˆΣj,z(j=\n1,···,N−1). For the two-body system, we have\nˆΣ1,z= ˆσ1,z⊗I2−I1⊗ˆσ2,z. (A4)\nFor the three-body system, we have\nˆΣ1,z= ˆσ1,z⊗I2⊗I3−I1⊗ˆσ2,z⊗I3(A5)\nand\nˆΣ2,z=1\n2(ˆσ1,z⊗I2⊗I3+I1⊗ˆσ2,z⊗I3)−\nI1⊗I2⊗ˆσ3,z. (A6)\nAppendix B: Explicitly correlated basis set\nexpansion approach\nThis appendix discusses our approach to solving the\nfew-particle time-independent Schr¨ odinger equation us-\ning a basis set expansion in terms of explicitly correlated\nGaussian basis functions, which contain non-linear varia-\ntional parameters that are optimized semi-stochastically.\nAs discussed in the main text, we are interested in\nboundstates, i.e., eigenstatesthatapproachzeroatlarge\ninterparticle distances. To solve the time-independent\nSchr¨ odingerequation,weexpandtherelativeportionΦ rel\nof the eigen state Ψ sought in terms of a set of non-\northogonal eigen functions ψj[65, 66],\nΦrel=Nb/summationdisplay\nj=1cjψj. (B1)\nThe basis functions ψjdepend on the relative spatial and\nthe spin degrees of freedom,\nψj=ˆS(φj(/vector ρ1,···,/vector ρN−1)χj), (B2)\nwhereχjdenotes anN-particle spin function that is cho-\nsen from the complete set of 2Npossible spin functions.\nThe spatial parts φjare written in terms of a total of\n(N−1)(N/2+3) non-linear variational parameters d(j)\nkl\nand/vector s(j)\nk,\nφj= exp/parenleftigg\n−N/summationdisplay\nk<lr2\nkl\n2d(j)\nkl+N−1/summationdisplay\nk=1ı/vector s(j)\nk·/vector ρk/parenrightigg\n.(B3)\nHere, the superscript “( j)” serves to remind us that each\nbasis function is characterized by a set of non-linear vari-\national parameters. The non-linear variational parame-\ntersd(j)\nkldetermine the widths of the Gaussian factors ofthe basis functions. These widths are governed, roughly,\nby the two-body interaction terms in the Hamiltonian.\nThe non-linear parameters /vector s(j)\nkdetermine the spatial os-\ncillations due to the kso-dependent one-body terms. We\ndo find that the values of /vector s(j)\nkcan depend quite strongly\non the spin basis function considered. This shows that\nthe parameters /vector s(j)\nkgovern, to leading order, the inter-\nplay between the spatial and spin degrees of freedom.\nOf course, strictly speaking, the influence of the single-\nparticle and two-particle interaction terms in the Hamil-\ntonian on the eigen states cannot be separated; rather,\nthe eigen states are the result of the relative importance\nof each of these terms and the kinetic energy terms. The\nN(N−1)/2 non-linear parameters d(j)\nkland the 3(N−1)\nnon-linear parameters contained in /vector s(j)\nkare optimized\nsemi-stochastically, i.e., the basis set is constructed so\nas to minimize the energy of the eigen state under study.\nThe symmetrizer ˆSin Eq. (B2) ensures that the basis\nfunctions, and hence the eigen state sought, are fully\nsymmetrized. For the two-boson system, e.g., the sym-\nmetrizer reads ˆS= (1 +ˆP12)/√\n2, where ˆP12exchanges\nthe spatial and spin degrees of freedom of particles 1 and\n2. ForNidentical particles, the symmetrizer contains N!\nterms.\nThe linear parameters cjare determined by solving\nthe generalized eigen value problem that depends on the\nHamiltonian matrix and, due to the non-orthogonalityof\nthe basis functions, the overlap matrix [65]. The results\npresented in this paper utilize basis sets consisting of up\ntoNb= 1,800 basis functions. Key strengths of the nu-\nmericalapproachemployedare that the Hamiltonian and\noverlap matrix elements have compact analytical expres-\nsions and that the non-linear variational parameters can\nbe adjusted so as to capture correlations that occur on\nscales smaller than r0and larger than ( kso)−1.\nThe energies for the N= 2 system depend on the pa-\nrametersas,kso, Ω, and ˜δ. ForN= 3, they additionally\ndependonκ∗. Inpractice,weset /vector qN= 0andscanΩ /Eso,\nδ/Eso, andaskso(forN= 3, we fix κ∗). To obtain the\nrelative eigen energies Eand eigen states Φ relfor finite\nqN,z, we do not need to redo the numerical calculations.\nInstead, we use the “conversion” implied by Eq. (21),\ni.e., the relative eigen energy and eigen states are ob-\ntained by changing from δto˜δ. The full eigen states Ψ\nare obtained by multiplying the /vector qN= 0 eigen states by\nthe center-of-mass piece Φ cm.\nAs in the case without spin-orbit coupling [67], it tends\nto be more efficient to describe each relative eigen state\nΦrelby its own basis set as opposed to constructing one\nbasis set that describes multiple eigen states well. We\nrefer to the eigen state sought—this could be the ground\nstate or one of the excited states—as target state. To\nconstruct the basis, we add one basis function at a time.\nLet us assume that we have a basis set of size Ninitialand\nthat we want to enlarge the basis set by one basis func-\ntion. To do this, we generate Ntrialtrial basis functions\n(Ntrialis of the order of a few hundred to a few thou-13\nsand) and calculate the energy Ek(k= 1,···,Ntrial) of\nthe target state for each of these enlarged basis sets. As-\nsuming that the generalized eigen value problem for the\nbasis set of size Ninitialhas been solved, the trial ener-\ngies can be calculated via a root-finding procedure [65],\nwhich is, generally, computationally significantly faster\nthan solving the generalized eigen value problem. The\nbasis function to be added is determined by which of the\nNtrialtrial energies is lowest, i.e., by looking for the trial\nbasis function that lowers the energy of the target state\nthe most. After the “best” trial function has been added,\nthe generalizedeigen value problemis solvedfor the basis\nset of sizeNinitial+ 1 and the procedure is repeated to\ngenerate a basis set of size Ninitial+2.\nThe convergence of the energy with increasing basis\nset size can be improved significantly by choosing “good”\nbasisfunctions, i.e., bygeneratingbasisfunctions that ef-\nficiently cover the entire Hilbert space. Conversely, if the\nbasis set is not constructed carefully, the energy may not\neven converge. In our implementation, the parameters\nd(j)\nkland/vector s(j)\nkthat characterizethespatialpartofthebasis\nfunctions are choosen from carefully adjusted parameter\nwindows. For example, the d(j)\nklare chosen so as to cover\nlength scales ranging from less than r0to a few times the\nlengthsetbythebindingenergyofthetargetstate. Since\nthe target energy is, in general, not known a priori, the\nparameter windows are typically refined based on results\nobtained in preliminary calculations. To choose parame-\nter windows for the x-,y- andz-components of /vector s(j)\nk, we\nare guided by the non-interacting two- and three-particle\ndispersion curves. For example, if the non-interacting\ntwo-particle dispersion along the z-coordinate exhibits\ntwo minima, we choose s(j)\n1,zuniformly from the windows\n[−smax,z,−smin,z] and [smin,z,smax,z], with the windows\nincluding the momenta at which the dispersion curve is\nminimal. The parameters s(j)\n1,xands(j)\n1,yare selected from\nwindows that include zero. The widths of the parame-\nter windows are adjusted through an educated trial and\nerror procedure. Among other things, we check if the pa-\nrametersselected by the code areclumped in a particular\nregion of the parameter space.\nAppendix C: Three-atom threshold\nThe three-atom system with center-of-mass momen-\ntum/vector qN,N= 3,isboundifitsenergyislowerthanthatof\nthree infinitely farseparatedatoms with the samecenter-\nof-mass momentum and, if a two-body bound state ex-\nists, lower than that of an infinitely far separated dimer\nand atom with the same center-of-mass momentum. To\ndetermine the lowest three-body scattering threshold,\none thus needs to know the lowest dimer binding energy\nfor all two-body center-of-mass momenta. As a conse-\nquence, the three-body scattering threshold depends on\n˜δ/Esoand Ω/Esoas well as on askso. We define the\nscattering threshold Ethusingˆ¯Hreland denote the eigen\nFIG. 4: (Color online) Lowest non-interacting relative thr ee-\natom dispersion curve. The contours show the lowest non-\ninteracting relative three-atom dispersion curve, in unit s of\nEso, as functions of q1,zandq2,z. Panels (a)-(c) are for\nΩ/Eso= 0 and ˜δ/Eso= 0, 8/3, and 3 .5, respectively. Pan-\nels (d)-(f) are for Ω /Eso= 2 and ˜δ/Eso= 0, 2.278, and 4,\nrespectively.\nenergies ofˆ¯HrelbyE.\nWe start by determining the lowestrelative three-body\nscattering threshold in the absence of two-body bound\nstates. In this case, the lowest relative scattering thresh-\nold is determined by the minimum energy of the non-\ninteracting relative dispersion curves for fixed ˜δ/Esoand\nΩ/Eso. The dispersion curves depend on two relative\nmomenta (namely q1,zandq2,z) and the total number of\ndispersion curves is eight. Figures 4(a)-4(c) are for the\nuncoupled case (Ω = 0) and ˜δ/Eso= 0, 8/3, and 3.5,\nrespectively. The number of global minima changes from\nsix for˜δ= 0 [see Fig. 4(a)] to three for 0 <˜δ/Eso<8/3\n(not shown) to four for ˜δ/Eso= 8/3 [see Fig. 4(b)] to\none for˜δ/Eso>8/3 [see Fig. 4(c)]. A finite Raman cou-\npling strength Ω introduces a coupling between the dif-14\nFIG. 5: (Color online) Lowest relative three-atom scatteri ng\nthreshold. Thecontoursshowtheenergy Eaaa\nth, inunitsof Eso,\nof the lowest three-atom scattering threshold as functions of\nΩ/Esoand˜δ/Eso. The thickopencircles andthickdashedline\nindicate the parameter combinations ( ˜δ/Eso,Ω/Eso) at which\nthe degeneracy of the lowest three-atom scattering thresho ld\nis six and four, respectively.\nferent spin channels. As anexample, Figs. 4(d)-4(f) show\nthe lowest relative non-interacting dispersion curves for\nΩ/Eso= 2 and ˜δ/Eso= 0, 2.278, and 4, respectively.\nAs for vanishing Raman coupling, the number of global\nminima changes from six to three (not shown) to four\nto one with increasing ˜δ. However, the critical general-\nized detuning ˜δat which these changes occur differs for\nΩ/Eso= 2 and Ω = 0.\nThe minimum of the non-interacting relative three-\natom dispersion curves defines, assuming two-body\nbound states are absent, the lowest three-atom scatter-\ning threshold. Figure 5 shows the lowest three-atom\nscattering threshold energy Eaaa\nthas functions of ˜δ/Eso\nand Ω/Eso. The thick open circles and thick dashed\nline indicate the parameter combinations at which the\nnumber of global minima is six and four, respectively.\nFor parameter combinations above the thick open circles\nand below the thick dashed line the number of global\nminima of the lowest non-interacting relative dispersion\ncurve is equal to three. Above the thick dashed line\nthe number of global minima is equal to one. If we\nassume that the three-body binding energy is, approx-\nimately, largest when the degeneracy of the lowest non-\ninteractingrelativedispersioncurveislargest,then Fig.5\nsuggeststhatthe three-bodysystemonthenegativescat-\ntering length side, provided two-body bound states are\nabsent, is enhanced the most compared to the energy of\nthe system without spin-orbit coupling when ˜δ= 0 or\nq3,z=−3mδ/(2/planckover2pi1kso). The main text shows that this\nreasoning provides an intuitive understanding for the be-\nhavior of the lowest three-boson state in each manifold.\nHowever, the situation for the excited states in a mani-\nfold is more intricate [61].Appendix D: Atom-dimer threshold\nAs already mentioned, the determination of the low-\nest atom-dimer scattering threshold requires knowledge\nof the dimer binding energy and the single-particle dis-\npersion curve. Since the z-component of the center-of-\nmass momentum q1,zof the dimer, formed by particles\n1 and 2, can be written as a linear combination of q2,z\nandq3,z,q1,zis not a free parameter. As a consequence,\ntheatom-dimerdispersioncurvesdepend onlyon q2,zbut\nnot onq1,z. Physically, this makes sense since the three-\nbody system breaks up into two units (a dimer and an\natom), with the momentum between the two units deter-\nmining the division of the three-body momentum among\nthe dimer and the atom.\nTo quantify this, we rewrite the Hamiltonianˆ¯Hrelby\narbitrarily singling out the third atom and treating the\nexpectation value q2,zof ˆq2,zas a parameter,\nˆ¯Hrel/vextendsingle/vextendsingle\n/angbracketleftˆq2,z/angbracketright=q2,z=ˆH12(q2,z)⊗I3+\nI1⊗I2⊗ˆH3(q2,z)+\nVcoupling. (D1)\nHere, the “dimer Hamiltonian” ˆH12(q2,z) reads\nˆH12(q2,z) =/parenleftiggˆ/vector q2\n1\n2µ1+V2b(r12)/parenrightigg\nI1⊗I2+\n/planckover2pi1ksoq1,z\nm(ˆσ1,z⊗I2−I1⊗ˆσ2,z)+\n/parenleftigg\n/planckover2pi1ksoq2,z\n2m+˜δ\n2/parenrightigg\n(ˆσ1,z⊗I2+I1⊗ˆσ2,z)+\nΩ\n2(ˆσ1,x⊗I2+I1⊗ˆσ2,x). (D2)\nIdentifying ˜δ12,eff,\n˜δ12,eff\n2=/planckover2pi1ksoq2,z\n2m+˜δ\n2, (D3)\nas a new effective dimer detuning, the eigen energies of\nˆH12(q2,z) are the same as those of the two-body Hamil-\ntonian. The “atom Hamiltonian” ˆH3(q2,z),\nˆH3(q2,z) =\n/vector q2\n2\n2µ2⊗I3+/parenleftigg\n−/planckover2pi1ksoq2,z\nm+˜δ\n2/parenrightigg\nˆσ3,z+Ω\n2ˆσ3,x,(D4)\ndescribes the Jacobi particle with mass µ2and effective\natom detuning ˜δ3,eff, where\n˜δ3,eff\n2=−/planckover2pi1ksoq2,z\nm+˜δ\n2. (D5)\nNote that the effective dimer detuning ˜δ12,effand the\neffective atom detuning ˜δ3,effdepend on the “true de-\ntuning”δ, which is fixed by the experimental set-up, on15\nthez-component q3,zof the three-body center-of-mass\nmomentum, which is a conserved quantity, and on q2,z,\nwhich is treated as a parameter. Assuming that the dis-\ntance between the center-of-mass of the dimer and the\natom is large compared to the size of the dimer and com-\npared to the ranges r0andR0of the two- and three-body\ninteractions, the coupling term Vcoupling,\nVcoupling= [V2b(r13)+V2b(r23)+V3b(r123)]I1⊗I2⊗I3,\n(D6)\ncan be set to zero. Thus, the q2,z-dependent relative\natom-dimer dispersion curves are obtained by adding the\neigen energies of ˆH12andˆH3, which depend parametri-\ncally onq2,z.\nEquations(D1)-(D6) assumedthatthe dimerisformed\nby atoms 1 and 2. Alternatively, the dimer could be\nformed by atoms 1 and 3 or by atoms 2 and 3. These\nalternative divisions yield atom-dimer dispersion curves\nthat depend on the z-component of the momentum that\nis associated with the distance vector between particle\n2 and the center-of-mass of the 13-dimer and the z-\ncomponent of the momentum that is associated with the\ndistance vector between particle 1 and the center-of-mass\nof the 23-dimer, respectively. Since we are considering\nthree identical bosons, the three divisions are equivalent.\nIn what follows, we use qad,zto reflect that we could\nsingle out any of the three atoms. The corresponding\natom-dimer energy is denoted by Ead\nth.\nSince there exist up to three two-boson bound\nstates [61], the three-boson system supports up to six\natom-dimer dispersion curves (there could be four or\ntwo). As an example, Fig. 6 shows the energy Ead\nthof the\nlowest relative atom-dimer dispersion curve as a func-\ntion ofqad,zfor (askso)−1= 0.01128, Ω/Eso= 2, and\nvarious˜δ, i.e.,˜δ= 0,2.287, and 3.5. The system sup-\nports, for this Raman coupling strength and scattering\nlength, one weakly-bound two-boson state for all two-\nbody center-of-mass momenta. For ˜δ= 0 (solid line in\nFig. 6), the atom-dimer dispersion is symmetric with re-\nspect toqad,z= 0 and supports two global minima at\nfiniteqad,z. The break-up into a dimer and an atom is\nenergetically most favorable when qad,z/(/planckover2pi1kso) is equal\nto±0.76. This translates, using Eqs. (D3) and (D5),\ninto˜δ12,eff/Eso=±1.52 and˜δ3,eff/Eso=±3.04. For\n˜δ >0 (the dashed and dotted lines in Fig. 6 are for\n˜δ/Eso= 2.287 and 3.5, respectively), the atom-dimer\ndispersions are asymmetric with respect to qad,z= 0 and\nexhibit a global minimum at negative qad,z, which ap-\nproachesqad,z= 0 in the ˜δ→ ∞limit. Intuitively, the\nasymmetry can be understood by realizing that the atom\nand the dimer already see a detuning. Thus, moving\nin the positive momentum direction is not equivalent to\nmoving in the negative momentum direction. The mini-\nmum of the lowest relative atom-dimer dispersion curve\ndecreases with increasing ˜δ.-1.5 -1 -0.5 0 0.5 1 1.5\nqad,z/(h_kso)-6-5-4-3Ead\nth/Eso\nFIG. 6: (Color online) Relative atom-dimer dispersion curv es\nfor (askso)−1= 0.01128 and Ω /Eso= 2. The solid, dashed,\nand dotted lines show the energy Ead\nthas a function of the\nz-component qad,zof the atom-dimer momentum for ˜δ/Eso=\n0,2.287, and 3 .5, respectively.\nAppendix E: Three-body threshold\nThe three-boson threshold is given by the minimum of\nthe lowest three-atom threshold and the lowest atom-\ndimer threshold. It depends on the values of Ω, kso,\n˜δ, and thes-wave scattering length. Using ksoandEso\nas units, Fig. 7 shows a contour plot of the lowest rela-\ntive three-boson threshold as functions of the generalized\ndetuning ˜δ/Esoand the inverse ( askso)−1of thes-wave\nscattering length for Ω /Eso= 2. As already discussed,\nthe parameter regime in which two-boson bound states\nexist depends on the value of as. Correspondingly, the\nthick dotted line, which marks the separation of the re-\ngion in which the three-atom threshold has the lowest\nenergy (to the left of the thick dotted line) and that in\nwhich the atom-dimer threshold has the lowest energy\n(to the right of the thick dotted line), shows a distinct\ndependence on the s-wave scattering length. For large ˜δ,\nthe thick dotted line approaches the ( askso)−1= 0 line.\nFor a fixed ˜δ, the energy Ead\nthof the lowest atom-dimer\nthreshold decreases with increasing ( askso)−1. This can\nbe traced back to the increase of the binding energy of\nthe two-boson ground state with increasing ( askso)−1.\nThe parameter combinations with the largest degener-\nacy of the scattering threshold are shown by the thick\nopen circles (three-atom threshold; the degeneracyis six)\nand the thick dash-dotted line (atom-dimer threshold;\nthe degeneracy is two). For all as, the largest degener-\nacy of the scattering threshold is found for ˜δ= 0. As\ndiscussed in the main text, our numerical three-boson\ncalculations show that the binding energy of the most\nstrongly-bound state in each manifold, determined as\nfunctions of ( askso)−1and˜δ/Eso, is largest for vanishing\n˜δ, i.e., where the degeneracy of the lowest three-boson\nscattering threshold is maximal.16\nFIG. 7: (Color online) Lowest relative three-boson scatter ing\nthreshold for Ω /Eso= 2. The contours show the energy Eth,\nin units of Eso, of the lowest three-boson scattering threshold\nas functions of ( askso)−1and˜δ/Eso. To the left of the thick\ndotted line, bound dimers are not supported and the lowest\nthreshold is given by Eaaa\nth. In this regime, the lowest scatter-\ning threshold is independent of ( askso)−1. To the right of the\nthick dotted line, a weakly-bound bosonic dimer exists and\nthe lowest threshold is given by Ead\nth. In this regime, the low-\nest scatteringthresholddependson( askso)−1. The thickopen\ncircles and thick dashed line mark the (( askso)−1,˜δ/Eso) com-\nbinations for which the lowest three-atom threshold has a de -\ngeneracy of six and four, respectively. The thick dash-dott ed\nline marks the (( askso)−1,˜δ/Eso) combinations for which the\nlowest atom-dimer threshold has a degeneracy of two. In the\nregion encircled by the thick open circles, the thick dotted\nline, the thick dashed line, and the left edge of the figure the\ndegeneracy of the three-atom threshold is equal to three. In\nthe region encircled by the thick dashed line, the thick dott ed\nline, the upper edge of the figure, and the left edge of the\nfigure, the degeneracy of the three-atom threshold is equal t o\none. In the region encircled by the thick dash-dotted line, t he\nright edge of the figure, the upper edge of the figure, and the\nthick dotted line the degeneracy of the atom-dimer threshol d\nis equal to one. The range of scattering lengths shown on the\nhorizontal axis is, even though different units are used, the\nsame as in Fig. 3.\n[1] H.-X.Chen, W.Chen, X.Liu, andS.-L.Zhu, Thehidden-\ncharm pentaquark and tetraquark states, Phys. Rep.\n639, 1 (2016).\n[2] M. Gell-Mann, A Schematic Model of Baryons and\nMesons, Phys. Lett. 8, 214 (1964).\n[3] E. Epelbaum, H.-W. Hammer, and U.-G. Meißner, Mod-\nern theory of nuclear forces, Rev. Mod. Phys. 81, 1773\n(2009).\n[4] I. Tanihata, Neutron halo nuclei, J. Phys. G: Nucl. Part.\nPhys.22, 157 (1996).\n[5] L. H. Thomas, The Interaction Between a Neutron and\na Proton and the Structure of H3, Phys. Rev. 47, 903\n(1935).\n[6] V. Efimov, Energy levels arising from resonant two-bodyforces in a three-body system, Phys. Lett. B 33, 563\n(1970).\n[7] V. Efimov, Weakly-bound states of three resonantly-\ninteracting particles, Yad. Fiz. 12, 1080 (1970) [Sov. J.\nNucl. Phys. 12, 589 (1971)].\n[8] E. Braaten and H.-W. Hammer, Universality in few-body\nsystems with large scatteringlength, Phys.Rep. 428, 259\n(2006).\n[9] E. Braaten and H.-W. Hammer, Efimov physics in cold\natoms, Annals of Physics 322, 120 (2007).\n[10] H.-W. Hammer and L. Platter, Efimov States in Nuclear\nand Particle Physics, Ann. Rev. Nucl. Part. Sci. 60, 207\n(2010).\n[11] P. Naidon and S. Endo, Efimov Physics: a review, Rep.17\nProg. Phys. 80, 056001 (2017).\n[12] F. Ferlaino and R. Grimm, Trend: Forty years of Efimov\nphysics: Howa bizarre prediction turnedintoa hot topic,\nPhysics3, 9 (2010).\n[13] C. H. Greene, Universal insights from few-body land,\nPhysics Today 63, 40 (2010).\n[14] T. Kraemer, M. Mark, P. Waldburger, J. G. Danzl, C.\nChin, B. Engeser, A. D. Lange, K. Pilch, A. Jaakkola, H.-\nC. N¨ agerl, and R. Grimm, Evidence for Efimov quantum\nstates in an ultracold gas of caesium atoms, Nature 440,\n315 (2006).\n[15] M. Zaccanti, B. Deissler, C. D’Errico, M. Fattori, M.\nJona-Lasinio, S. M¨ uller, G. Roati, M. Inguscio, and G.\nModugno, Observation of an Efimov spectrum in an\natomic system, Nat. Phys. 5, 586 (2009).\n[16] Y. Nishida, Y. Kato, and C. D. Batista, Efimov effect in\nquantum magnets, Nat. Phys. 9, 93 (2013).\n[17] T. Lompe, T. B. Ottenstein, F. Serwane, A. N. Wenz,\nG. Z¨ urn, and S. Jochim, Radio-Frequency Association of\nEfimov Trimers, Science 330, 940 (2010).\n[18] S. Nakajima, M. Horikoshi, T. Mukaiyama, P. Naidon,\nand M. Ueda, Measurement of an Efimov Trimer Binding\nEnergyinaThree-ComponentMixtureof6Li, Phys.Rev.\nLett.106, 143201 (2011).\n[19] M. Berninger, A. Zenesini, B. Huang, W. Harm, H.-C.\nN¨ agerl, F. Ferlaino, R. Grimm, P. S. Julienne, and J.\nM. Hutson, Universality of the Three-Body Parameter\nfor Efimov States in Ultracold Cesium, Phys. Rev. Lett.\n107, 120401 (2011).\n[20] B. Huang, L. A. Sidorenkov, R. Grimm, and J. M. Hut-\nson, Observation of the Second Triatomic Resonance in\nEfimov’s Scenario, Phys. Rev. Lett. 112, 190401 (2014).\n[21] M. Kunitski, S. Zeller, J. Voigtsberger, A. Kalinin, L.\nP. H. Schmidt, M. Sch¨ offler, A. Czasch, W. Sch¨ ollkopf,\nR. E. Grisenti, T. Jahnke, D. Blume, and R. D¨ orner,\nObservation of the Efimov state of the helium trimer,\nScience348, 551 (2015).\n[22] D. C. Mattis, The few-body problem on a lattice, Rev.\nMod. Phys. 58, 361 (1986).\n[23] J. Dalibard, F. Gerbier, G. Juzeli¯ unas, and P. ¨Ohberg,\nColloquium: Artificial gauge potentials for neutral\natoms, Rev. Mod. Phys. 83, 1523 (2011).\n[24] V. Galitski and I. B. Spielman, Spin-orbit coupling in\nquantum gases, Nature 494, 49 (2013).\n[25] N. Goldman, G. Juzeli¯ unas, P. ¨Ohberg, and I. B. Spiel-\nman, Light-inducedgauge fieldsfor ultracold atoms, Rep.\nProg. Phys. 77, 126401 (2014).\n[26] H. Zhai, Degenerate quantum gases with spin-orbit cou-\npling: a review, Rep. Prog. Phys. 78, 026001 (2015).\n[27] M. Aidelsburger, S. Nascimbene, and N. Goldman, Ar-\ntificial gauge fields in materials and engineered systems,\narXiv:1710.00851.\n[28] Y.-J. Lin, K. Jim´ enez-Garc´ ıa, and I. B. Spielman, Spi n-\norbit-coupled Bose-Einstein condensates, Nature (Lon-\ndon)471, 83 (2011).\n[29] J. Struck, C. ¨Olschl¨ ager, M. Weinberg, P. Hauke, J. Si-\nmonet, A. Eckardt, M. Lewenstein, K. Sengstock, and\nP. Windpassinger, Tunable Gauge Potential for Neutral\nand Spinless Particles in Driven Optical Lattices, Phys.\nRev. Lett. 108, 225304 (2012).\n[30] C. V. Parker, L.-C. Ha, and C. Chin, Direct observa-\ntion of effective ferromagnetic domains of cold atoms in\na shaken optical lattice, Nat. Phys. 9, 769 (2013).[31] T. Ozawa, H. M. Price, A. Amo, N. Goldman, M.\nHafezi, L. Lu, M. Rechtsman, D. Schuster, J. Simon,\nO. Zilberberg, and I. Carusotto, Topological Photonics,\narXiv:1802.04173 (2018).\n[32] J. P. Vyasanakere and V. B. Shenoy, Bound states of two\nspin-1/2 fermions in a synthetic non-Abelian gauge field,\nPhys. Rev. B 83, 094515 (2011).\n[33] J. P. Vyasanakere, S. Zhang, and V. B. Shenoy, BCS-\nBEC crossover induced by a synthetic non-Abelian gauge\nfield, Phys. Rev. B 84, 014512 (2011).\n[34] J. P. Vyasanakere and V. B. Shenoy, Rashbons: prop-\nerties and their significance, New. J. Phys. 14, 043041\n(2012).\n[35] J. P. VyasanakereandV. B. Shenoy, Flow-enhanced pair-\ning and other unusual effects in Fermi gases in synthetic\ngauge fields, Phys. Rev. A 88, 033609 (2013).\n[36] X. Cui, Mixed-partial-wave scattering with spin-orbi t\ncoupling and validity of pseudopotentials, Phys. Rev. A\n85, 022705 (2012).\n[37] L. Dong, L. Jiang, H. Hui, and H. Pu, Finite-momentum\ndimer bound state in a spin-orbit-coupled Fermi gas,\nPhys. Rev. A 87, 043616 (2013).\n[38] Z. Yu, Short-range correlations in dilute atomic Fermi\ngases with spin-orbit coupling, Phys. Rev. A. 85, 042711\n(2012).\n[39] Y. WuandZ. Yu, Short-range asymptotic behavior ofthe\nwave functions of interacting spin-1/2 fermionic atoms\nwith spin-orbit coupling: A model study, Phys. Rev. A\n87, 032703 (2013).\n[40] X. Cui, Multichannel molecular state and rectified shor t-\nrange boundary condition for spin-orbit-coupled ultra-\ncold fermions near p-wave resonances, Phys. Rev. A 95,\n030701(R) (2017).\n[41] Q. Guan and D. Blume, Scattering framework for two\nparticles with isotropic spin-orbit coupling applicable t o\nall energies, Phys. Rev. A 94, 022706 (2016).\n[42] S.-J. Wang and C. H. Greene, General formalism for\nultracold scattering with isotropic spin-orbit coupling,\nPhys. Rev. A 91, 022706 (2015).\n[43] D. Luo and L. Yin, BCS-pairing state of a dilute Bose\ngas with spin-orbit coupling, Phys. Rev. A 96, 013609\n(2017).\n[44] R. Li and L. Yin, Pair condensation in a dilute Bose\ngas with Rashba spin-orbit coupling, New J. Phys. 16,\n053013 (2014).\n[45] Z.Xu, Z.Yu,andS.Zhang, Evidencefor correlatedstate s\nin a cluster of bosons with Rashba spin-orbit coupling,\nNew J. Phys. 18, 025002 (2016).\n[46] Z.-Y. Shi, X. Cui, and H. Zhai, Universal Trimers In-\nduced by Spin-Orbit Coupling in Ultracold Fermi Gases,\nPhys. Rev. Lett. 112, 013201 (2014).\n[47] X. Cui and W. Yi, Universal Borromean Binding in Spin-\nOrbit-Coupled Ultracold Fermi Gases, Phys. Rev. X 4,\n031026 (2014).\n[48] Z.-Y. Shi, H. Zhai, and X. Cui, Efimov physics and uni-\nversal trimers inspin-orbit-coupledultracold atomic mix -\ntures, Phys. Rev. A 91, 023618 (2015).\n[49] X. Qiu, X. Cui, and W. Yi, Universal trimers emerging\nfrom a spin-orbit-coupled Fermi sea, Phys. Rev. A 94,\n051604(R) (2016).\n[50] Q. Guan, One-, two-, and three-body systems with spin-\norbit coupling, Ph.D. thesis, Washington State Univer-\nsity (2017).\n[51] The normalization factor of the plane wave center-of-18\nmass wave function is not written out for notational con-\nvenience.\n[52] Y. Yan and D. Blume, Energy and structural proper-\nties ofN-boson clusters attached to three-body Efimov\nstates: Two-body zero-range interactions and the role\nof the three-body regulator, Phys. Rev. A 92, 033626\n(2015).\n[53] F. Ferlaino, S. Knoop, M. Mark, M. Berninger, H.\nSch¨ obel, H.-C. N¨ agerl, andR. Grimm, Collisions between\nTunable Halo Dimers: Exploring and Elementary Four-\nBody Process with Identical Bosons, Phys. Rev. Lett.\n101, 023201 (2008).\n[54] F. Ferlaino, S. Knoop, M. Berninger, W. Harm, J. P.\nD’Incao, H.-C. N¨ agerl, and R. Grimm, Evidence for Uni-\nversal Four-Body States Tied to an Efimov Trimer, Phys.\nRev. Lett. 102, 140401 (2009).\n[55] S. E. Pollack, D. Dries, and R. G. Hulet, Universality in\nThree- and Four-Body Bound States of Ultracold Atoms,\nScience326, 1683 (2009).\n[56] A. Zenesini, B. Huang, M. Berninger, S. Besler, H.-C.\nN¨ agerl, F. Ferlaino, R. Grimm, C. H. Greene, and J.\nvon Stecher, Resonant five-body recombination in an ul-\ntracold gas of bosonic atoms, New J. Phys. 15, 043040\n(2013).\n[57] R. Pires, J. Ulmanis, S. H¨ afner, M. Repp, A. Arias,\nE. D. Kuhnle, and M. Weidem¨ uller, Observation of Efi-\nmov Resonances in a Mixture with Extreme Mass Imbal-\nance, Phys. Rev. Lett. 112, 250404 (2014).\n[58] S.-K. Tung, K. Jim´ enez-Garc´ ıa, J. Johansen, C. V.\nParker, and C. Chin, Geometric Scaling of Efimov States\nin a6Li-133Cs Mixture, Phys. Rev. Lett. 113, 240402\n(2014).\n[59] Spin-orbit coupled experiments with Cs, which has tun-\nable Feshbach resonances, have not yet been conducted.\nOn the other hand, while many experiments utilize Rb\nto realize spin-orbit coupling, the known Feshbach res-\nonances in Rb are accompanied by large losses, making\nthe resonances challenging to use. Proposals for imple-\nmenting spin-orbit coupling schemes in Cs exist [68].\n[60] When we apply our formalism to the two-identical\nfermion system, our binding energy agrees with the re-\nsults obtained by Shenoy [35] but disagrees with the re-\nsults presented in Ref. [37]. Reference [37] calculated the\ntwo-fermion binding energy by allowing the z-component\nof the center-of-mass momentum of the eigen state of the\ninteracting system and that of the two-atom state associ-\nated with thethreshold with respect towhich the binding\nenergy is being measured to be different. The definition\nemployed [see Eq. (14) of Ref. [37]] implies that the z-\ncomponent of the center-of-mass momentum is not a con-served quantity. A straightforward calculation, however,\nestablishes that the three components of the center-of-\nmass momentum are individually conserved (see, e.g.,\nRef. [50]). Correspondingly, the quantity considered in\nRef. [37] is inconsistent with the accepted definition of\nthe term “two-body binding energy”. The quantity de-\nfined in Ref. [37] does, however, become meaningful in a\nmany-body context. To see this, we consider a two-body\nbound state with total energy E2(/vector q2) =E+/vector q2\n2/(4m) em-\nbedded into a many-body background (here, Eis the\neigen energy of the relative two-particle Hamiltonian;\nEdepends on q2,z). Assuming that the dimer can re-\nduce its energy via center-of-mass changing inelastic col-\nlisions with the background atoms, the lowest energy\nof the dimer is given by min /vector q2E2(/vector q2). The dimer can\nthen be considered stable if the global energy Eglobal,2,\nEglobal,2= min /vector q2E2(/vector q2)−min/vector q2[Eth+/vector q2\n2/(4m)] (Ethde-\nnotes the lowest relative atom-atom scattering threshold\nfor fixed q2,z) is negative. For the global energy Eglobal,2,\nreferred to as binding energy in Ref. [37], to be meaning-\nful, the dimer needs to be embedded into a background\nthat can, via inelastic collisions, absorb energy on a time\nscale that is fast compared to the time scales associated\nwith other loss processes.\n[61] Q. Guan and D. Blume, unpublished.\n[62] N. T. Zinner, Efimov states of heavyimpurities inaBose-\nEinstein condensate, EPL 101, 60009 (2013).\n[63] D. J. MacNeill and F. Zhou, Pauli Blocking Effect on\nEfimov States near a Feshbach Resonance, Phys. Rev.\nLett.106, 145301 (2011).\n[64] M. Ga ˘ilitis and R. Damburg, Some features of the thresh-\nold behavior of the cross sections for excitation of hydro-\ngen by electrons due to the existence of a linear Stark\neffect in hydrogen, Sov. Phys. JETP 17, 1107 (1963).\n[65] Y. Suzuki and K. Varga, Stochastic Variational Ap-\nproach to Quantum-Mechanical Few-Body Problems , 1st\ned. (Springer-Verlag Berlin Heidelberg, 1998).\n[66] J. Mitroy, S. Bubin, W. Horiuchi, Y.Suzuki, L. Adamow-\nicz, W. Cencek, K. Szalewicz, J. Komasa, D. Blume, and\nK. Varga, Theory and application of explicitly correlated\nGaussians, Rev. Mod. Phys. 85, 693 (2013).\n[67] D. Rakshit, K. M. Daily, and D. Blume, Natural and un-\nnatural parity states of small trapped equal-mass two-\ncomponent Fermi gases at unitarity and fourth-order\nvirial coefficient, Phys. Rev. A 85, 033634 (2012).\n[68] B.-Z. Wang, Y.-H. Lu, W. Sun, S. Chen, Y. Deng,\nand X.-J. Liu, Dirac-, Rashba, and Weyl-type spin-orbit\ncouplings: Toward experimental realization in ultracold\natoms, Phys. Rev. A 97, 011605(R) (2018)."    },    {        "title": "1911.09244v1.Evidence_for_orbital_ordering_in_Ba__2_NaOsO__6___a_Mott_insulator_with_strong_spin_orbit_coupling__from_First_Principles.pdf",        "content": "Evidence for orbital ordering in Ba 2NaOsO 6, a Mott insulator with strong spin orbit\ncoupling, from First Principles\nR. Cong1, Ravindra Nanguneri2, Brenda Rubenstein2;y, and V. F. Mitrovi\u0013 c1;y\n1Department of Physics, Brown University, Providence, Rhode Island 02912, USA and\n2Department of Chemistry, Brown University, Providence, Rhode Island 02912, USA\n(Dated: March 7, 2022)\nWe present \frst principles calculations of the magnetic and orbital properties of Ba 2NaOsO 6\n(BNOO), a 5 d1Mott insulator with strong spin orbit coupling (SOC) in its low temperature emergent\nquantum phases. Our computational method takes into direct consideration recent NMR results that\nestablished that BNOO develops a local octahedral distortion preceding the formation of long range\nmagnetic order. We found that the two-sublattice canted ferromagnetic ground state identi\fed in Lu\net al. , Nature Comm. 8, 14407 (2017) is accompanied by a two-sublattice staggered orbital ordering\npattern in which the t2gorbitals are selectively occupied as a result of strong spin orbit coupling. The\nstaggered orbital order found here using \frst principles calculations asserts the previous proposal\nof Chen et al. , Phys. Rev. B 82, 174440 (2010) and Lu et al. , Nature Comm. 8, 14407 (2017),\nthat two-sublattice magnetic structure is the very manifestation of staggered quadrupolar order.\nTherefore, our results a\u000erm the essential role of multipolar spin interactions in the microscopic\ndescription of magnetism in systems with locally entangled spin and orbital degrees of freedom.\nI. INTRODUCTION\nThe competition between electron correlation and spin\norbit coupling (SOC) present in materials containing 4-\nand 5dtransition metals is an especially fruitful ten-\nsion predicted to lead to the emergence of a plethora\nof exotic quantum phases, including quantum spin liq-\nuids, Weyl semimetals, Axion insulators, and phases with\nexotic magnetic orders1{11. There has been an active\nquest to develop microscopic theoretical models to de-\nscribe such systems with comparably strong correlations\nand SOC to enable the prediction of their emergent quan-\ntum properties1{5. In strong Mott insulators, mean \feld\ntheories predict strong SOC to partially lift the degen-\neracy of total angular momentum eigenstates by entan-\ngling orbital and spin degrees of freedom to produce\nhighly nontrivial anisotropic exchange interactions2,3,5,7.\nThese unusual interactions are anticipated to promote\nquantum \ructuations that generate such novel quantum\nphases as an unconventional antiferromagnet with dom-\ninant magnetic octuple and quadrupole moments and\na noncollinear ferromagnet whose magnetization points\nalong the [110] axis and possesses a two-sublattice struc-\nture.\nBecause their SOC and electron correlations are\nof comparable magnitude6, 5ddouble perovskites\n(A2BB0O6) are ideal materials for testing these predic-\ntions. Indeed, recent NMR measurements on a repre-\nsentative material of this class, Ba 2NaOsO 6(BNOO),\nrevealed that it possesses a form of exotic ferromag-\nnetic order: a two-sublattice canted ferromagnetic (cFM)\nstate, reminiscent of theoretical predictions12. Speci\f-\ncally, upon lowering its temperature, BNOO evolves from\na paramagnetic (PM) state with perfect fcccubic sym-\nmetry into a broken local symmetry (BLPS) state. This\nBLPS phase precedes the formation of long-range mag-\nnetic order, which at su\u000eciently low temperatures, co-\nexists with the two-sublattice cFM order, with a netmagnetic moment of \u00190:2\u0016Bper osmium atom along\nthe [110] direction. One key question that remains is\nwhether such cFM order implies the existence of com-\nplex orbital/quadrupolar order.\nIn this paper, we report a two-sublattice orbital order-\ning pattern that coexists with cFM order in BNOO, as\nrevealed by DFT+U calculations. Evidence for this or-\nder is apparent in BNOO's selective occupancy of the t2g\norbitals and spin density distribution. More speci\fcally,\nthe staggered orbital pattern is manifest in BNOO's par-\ntial density of states and band structure, which possesses\na distinctt2gorbital contribution along high symmetry\nlines. This staggered orbital pattern is not found in the\nFM[110] phase. The results of our \frst principles cal-\nculations paint a coherent picture of the coexistence of\ncFM order with staggered orbital ordering in the ground\nstate of BNOO. Therefore, the staggered orbital order\ndiscovered here validates the previous proposal that the\ntwo-sublattice magnetic structure, which de\fnes the cFM\norder in BNOO, is the very manifestation of staggered\nquadrupolar order with distinct orbital polarization on\nthe two-sublattices2,12. Furthermore, our results a\u000erm\nthat multipolar spin interactions are an essential ingredi-\nent of quantum theories of magnetism in SOC materials.\nThis paper is organized as follows. The details of our\n\frst principles (DFT) simulations calculations of NMR\nobservables is \frst described in Section II. In Section III,\nwe present our numerical results for the nature of the or-\nbital order for a given imposed magnetic state. Lastly,\nin Section IV, we conclude with a summary of our cur-\nrent \fndings and their bearing on the physics of related\nmaterials.\nII. COMPUTATIONAL APPROACH\nAll of the following computations were performed using\nthe Vienna Ab initio Simulation Package (VASP), com-arXiv:1911.09244v1  [cond-mat.str-el]  21 Nov 20192\nplex version 5.4.1/.4, plane-wave basis DFT code13{16\nwith the Generalized-Gradient Approximation (GGA)\nPW9117functional and two-component spin orbit cou-\npling. We used 500 eV as the plane wave basis cuto\u000b en-\nergy and we sampled the Brillouin zone using a 10 \u000210\u00025\nk-point grid. The criterion for stopping the DFT self-\nconsistency cycle was a 10\u00005eV di\u000berence between suc-\ncessive total energies. Two tunable parameters, Uand\nJ, were employed. Udescribes the screened-Coulomb\ndensity-density interaction acting on the Os 5 dorbitals\nandJis the Hund's interaction that favors maximizing\nSz\ntotal18. In this work, we set U= 3:3 eV andJ= 0:5 eV\nbased upon measurements from Ref. 12 and calculations\nin Ref. 21. We note that these parameters are similar in\nmagnitude to those of the SOC contributions we observe\nin the simulations presented below, which are between\n1-2 eV. This is in line with previous assertions that the\nSOC and Coulomb interactions in 5 dperovskites are sim-\nilar in magnitude. Projector augmented wave (PAW)19,20\npseudopotentials (PPs) that include the psemi-core or-\nbitals of the Os atom, which are essential for obtaining\nthe observed electric \feld gradient (EFG) parameters21,\nwere employed to increase the computational e\u000eciency.\nA monoclinic unit cell with P2 symmetry is required to\nrealize cFM order (see the Supplemental Information).\nThe lattice structure with BLPS characterized by the or-\nthorhombic Q2 distortion mode that was identi\fed as\nbeing in the best agreement with NMR \fndings and re-\nferred to as Model A.3 in Ref.21was imposed.\nThe general outline of the calculations we performed is\ndescribed in the following. We \frst carried out single self-\nconsistent or `static' calculations with GGA+SOC+U\nwith a \fxed BLPS structure for Model A, representing\nthe orthorhombic Q2 distortion mode. Then, a mag-\nnetic order with [110] easy axes, as dictated by experi-\nmental \fndings12,22, is imposed on the osmium lattices.\nTypically, we found that the \fnal moments converged to\nnearly the same directions as the initial ones. Speci\f-\ncally, two types of such initial order are considered: (a)\nsimple FM order with spins pointing along the [110] di-\nrection; and, (b) non-collinear, cFM oder in which ini-\ntial magnetic moments are imposed on the two osmium\nj~Sj\u001e(S) j~Lj\u001e(L)M\u001e(M)\ncFM\nOs1 0.55 -41.56 0.44 90+46.29 0.12 -34\nOs2 0.55 90+28.73 0.43 -31.07 0.11 110\nFM110\nOs 0.83 45 0.52 225 0.31 45\nTABLE I. cFM and FM[110] magnetic moments for the im-\nposed representative BLPS structure using GGA+SOC+U.\nThe angles, \u001e, are in degrees and measured anti-clockwise\nwith respect to the + xaxis. The magnitudes of spin, orbital,\nand total moments are denoted by j~Sj,j~Lj, andj~Mj, in units\nof\u0016B, respectively. The small net magnetic moment is due\nto the anti-aligment of ~Sand~Le\u000bin theJe\u000b=3\n2state. As of\nnow, the FM110 state has not been experimentally identi\fed\nin a 5ddouble perovskite.\nSx\nSy\nSzSublattice 1 Sublattice 2FIG. 1. Contour plots of the spin density on two distinct\nsublattices of the BLPS structure (Model A.3 in Ref.21)\nfrom GGA+SOC+U calculations. The S x(top row), Sy\n(center row), and Sz(bottom row) components of the spin\ndensity on a single Os octahedron from sublattice 1 (left\ncolumn) and sublattice 2 (right column) are plotted. The\ndi\u000berent colors denote the signs of the Sx;y;z projections.\nThe isovalues are blue for positive Sx;y;z, 0:001, and yellow\nfor negative Sx;y;z,\u00000:001. On the top left, the negative Sx\ndensity is sandwiched between the lobes of the positive Sx\ndensities on the Os atom, and vice-versa for the Os atom\non the top right. On the top left, four of the O atoms have\na cloverleaf spin density pattern with alternating positive\nand negative Sxdensities, while on the top right, only the\ntwo axial O atoms have this pattern. The other O atoms\nin the top two OsO 6octahedra have spin densities that are\nuniformly polarized.\nsublattices in the directions determined in Ref. 12. We\nused the Methfessel-Paxton (MP) smearing technique23\nto facilitate charge density convergence. For the density\nof states and band structure calculations, we employed\nthe tetrahedron smearing with Bl ochl corrections24and\nGaussian methods, respectively.\nIII. ORBITAL ORDERING WITH IMPOSED\nMAGNETIC CFM AND FM110 ORDERS\nIn the following subsections, we report our results for\nthe orbital order, band structure, and density of states\nof BNOO when we impose magnetic order with [110]\neasy axes and the local orthorhombic distortion that\nbest matched experiments21. In Table I, we summarize\nthe converged orbital and spin magnetic moments. In3\n total on Sublattice 1 & 2\nc\naSx\nb\nc\nb\nab\nb\na\nc\nb\na\nc\nb\nac\nb\naccFM\nFM [110]\nb\na\nc\nFIG. 2. Two views of the Sx-component of the spin density\nfor imposed FM order and an orthorhombic Q2 distortion\n(Model A.3 in Ref. 21 on both sublattices as viewed along\nthe -aand -cdirections. This component shows only the\nSx-projection of the spin vector \feld. The isovalues are\nblue, positive S x: 0:001, and yellow, negative S x: -0:001.\nBNOO,M= 2S+Le\u000b= 0, since the t 2glevel can be re-\ngarded as a pseudospin with Le\u000b=\u00001. The magnitude\nof the spin moment, j~Sj, is in the vicinity of \u00190:5\u0016B,\nwhile the orbital moment, j~Lj, is\u00190:4\u0016B. These values\nare reduced from their purely local moment limit due to\nhybridization with neighboring atoms, and, in the case\nof~L, by quenching caused by the distorted crystal \feld.\nFor imposed cFM order, we \fnd that the relative angle\n\u001ewithin the two sublattices is in agreement with our\nNMR \fndings in Ref. 12. Indeed, \frst principles cal-\nculations, performed outside of our group, taking into\naccount multipolar spin interactions found that the re-\nported canted angle of \u001967\u000ecorresponds to the global\nenergy minimum25.\nNext, we will explore the nature of the orbital order-\ning. Previous \frst principles works hinted at the presence\nof orbital order in BNOO, but did not fully elucidate its\nnature26. Since we imposed cFM order and SOC, we\nwere able to obtain a more exotic orbital order than uni-\nform ferro-order. We report below evidence for a type of\nlayered, anti-ferro-orbital-order (AFOO) that has been\nshown to arise in the mean \feld treatment of multipolar\nHeisenberg models with SOC5.\nFirst, we analyze the nature of the orbital order by\ncomputing the spin density. The spin density is a contin-\nuous vector \feld of the electronic spin, and can point in\nnon-collinear directions. Its operators are the product of\nthe electrons' density and their spin-projection operators,\nsuch as \u0001z(~ r) =P\ni\u000e(~ ri\u0000~ r)Sz\ni:\nThe spin densities are given by the expectation value,\nh\u0001z(~ r)i=Tr[\u001ad\u0001z(~ r)]; (1)\nwhere\u001adis the 5d-shell single-particle density matrix\nobtained from DFT+U calculations. In Fig. 1, theh\u0001x;y;z(~ r)i, obtained via GGA+SOC+U calculations,\nare displayed for two distinct Os sublattices. The re-\nsults illustrate that the spins are indeed localized about\nthe Os atoms, and that there is a noticeable imbalance\nin the distribution of the n\"andn#spin densities, which\nmanifests in their di\u000berence, h\u0001z(~ r)i\u0011n\"(~ r)\u0000n#(~ r).\nThe di\u000berence in the spatial distribution between the two\nsublattice spin densities is indicative of the orbital order-\ning. The net spin moments are obtained by integrating\nthe spin density over the volume of a sphere enclosing\nthe Os atoms.\nIn Fig. 1, it is visually clear that: I.TheSx(top)\nandSy(center) spin density components are overwhelm-\ningly of a single sign, which gives rise to net moments\nin the (a;b) plane; and II.The signs of SxandSybe-\ntween the two sublattices are reversed, indicating that\nthe sublattice spins are canted symmetrically about the\n[110] direction and the angle between them exceeds 90\u000e.\nIn contrast, for Sz(bottom), both signs of Szcontribute\nequally, so that the net Sz\u00190 after integrating over the\nsphere.\nIn Fig. 2, we plot the total Sx-component of the spin\ndensity over two sublattices for both types of imposed\nmagnetic order. It is evident that the staggered orbital\nE (eV)-1 1.5 -0.5 0.0 0.50 1.0dxy\ndyzdz\ndzx\ndx   - y2\n2 2My\nMy\nTotal Total\nE (eV)-1 1.5 -0.5 0.0 0.50 1.0\nE (eV)-1 1.5 -0.5 0.0 0.50 1.0E (eV)-1 1.5 -0.5 0.0 0.50 1.0E (eV)-1 1.5 -0.5 0.0 0.50 1.0\nE (eV)-1 1.5 -0.5 0.0 0.50 1.0Mx\nMx\nDOS (states/eV/cell)-2-1012\n-3dxy\ndyz\ndz\ndzx\ndx   - y2\n2 2\ndxy\ndyz\ndz\ndzx\ndx   - y2\n2 2dxy\ndyz\ndz\ndzx\ndx   - y2\n2 2dxy\ndyzdz\ndzx\ndx   - y2\n2 2DOS (states/eV/cell)-2-1012\n-3\nDOS (states/eV/cell)\n00.51.01.52.02.53.03.5DOS (states/eV/cell)\n00.51.01.52.02.53.03.5\nDOS (states/eV/cell)-2-1012\n-3DOS (states/eV/cell)-2-1012\n-3\ndxy\ndyzdz\ndzx\ndx   - y2\n2 2a) b)\nc) d)\ne) f)Os1 Os1\nOs1 Os2Os2 Os2\nFIG. 3. The partial density of states (PDOS) for spin de-\ncomposed parts of the Q2 orthorhombic distortion (Model\nA.3 in Ref. 21) for the Os atom in each sublattice, Os1 and\nOs2.4\npattern only arises when cFM order is imposed. There-\nfore, we demonstrate that the staggered orbital order can\nsolely coexist with cFM order.\nWe note also that in Figs. 1 and 2 there is non-\nnegligible spin density on the O atoms of the OsO 6oc-\ntahedra. It is usually thought that atoms with closed\nshells, like O in stoichiometric compounds, possess neg-\nligible spin densities. This is an unexpected feature in\nBNOO that has been previously noted in Ref. 26, and\nis due to the stronger 5 d-2phybridization, which results\nin OsO 6cluster orbitals. The spin imbalance is a quan-\ntity associated with the cluster rather than the individual\natoms, which is why we see the spin densities on the O\natoms.\nFor non-collinear systems, the orbital character of each\nosmium's 5 dmanifold can be further decomposed into\nthe Cartesian components of the spin magnetization:\nhSii\u0011Mi,i=x;y;z . Since the spins lie in the ( xy)\nplane and the Mzcomponent is zero for both sublattices,\nwe only plotted Mx,My, and the total PDOS for the two\nsublattices. We see in Fig. 3 that, \frstly, for both sub-\nlattices, only the t2gorbitals have an appreciable density\nof states consistent with the fact that the calculated d\noccupation at the Os sites is hndi<6. Secondly, below\nthe band gap, the dyzorbital has the same occupation\non both sublattices, while the dxyorbital is occupied on\none sublattice and the dzxorbital on the other. This\npattern in which certain dorbitals are preferentially oc-cupied at di\u000berent sites deviates from the case without\norbital ordering, in which each of the dxy,dyz, anddzx\norbitals have the same occupancies on both Os sites, as\nshown in Ref. 27. These orbital occupations are con-\nsistent with mean \feld predictions of the occupancy of\nthe Osdorbitals at zero temperature, which also pre-\ndict a staggered pattern5. This staggered pattern arises\nfrom BNOO's distinctive blend of cFM order with strong\nSOC.\nTo study this ordering in greater depth, we can com-\npute the occupation matrices, which after diagonaliza-\ntion, yield the occupation number (ON) eigenvalues and\ncorresponding natural orbital (NO) eigenvectors. For a\ngiven Os atom, the 5 dspin-orbitals have unequal ampli-\ntudes in each NO, as expected for the AFOO. The NOs\nalso all have di\u000berent occupation numbers. Regardless of\ntheir precise occupations, the unequal spin-orbital super-\npositions in the NOs endow the Os with a net non-zero\nspin and orbital moment. We moreover note that, due to\n5d-2phybridization, there is signi\fcant charge transfer\nfrom O to Os, such that the charge on the 5 dshell of Os-\nmium ishndi\u00195-6, which is very di\u000berent from the nom-\ninal heptavalent 5 d1\flling from simple valence counting.\nFurthermore, the ten NOs are fractionally occupied with\nthe largest ON close to hn1i\u00191:0 and the other nine\nNOs having occupations ranging from 0 :37\u00000:56. For\nthe NO,j1i, with the occupation hn1i\u00191:0, the coe\u000e-\ncients of the egorbitals are an order of magnitude smaller\nthan those for the t2gorbitals.\n0.000.050.100.150.200.250.300.35\n0.00.10.20.30.4\nX1RΓ Γ X2M2M1X1RΓ Γ X2M2M1X1RΓ Γ X2M2M1X1RΓ Γ X2M2M1X1RΓ X2M2M1X1RΓ Γ X2M2M1X1RΓ Γ X2M2M1X1RΓ Γ X2M2M1total\ntotalE (eV)\n-0.4-0.40.00.20.40.6E (eV)\n-0.4-0.40.00.20.40.6E (eV)\n-0.4-0.40.00.20.40.6E (eV)\n-0.4-0.40.00.20.40.6E (eV)\n-0.4-0.40.00.20.40.6E (eV)\n-0.4-0.40.00.20.40.6E (eV)\n-0.4-0.40.00.20.40.6E (eV)\n-0.4-0.40.00.20.40.6\nOs1 Os1 Os1 Os1\nOs2 Os2 Os2 Os2dxy\ndxydyz\ndyzdzx\ndzxΓ\nFIG. 4. The band structures of the two sublattice Os atoms near the Fermi level with 5 dpartial characters for the Q2\northorhombic distortion (Model A.3 in Ref.21): Os1 sublattice (top), Os2 sublattice (bottom). The projection of each 5 d\norbital onto the Kohn-Sham bands is represented by the color shading. The color bar on the left shows the color scaling\nfor the partial characters of the t2gorbitals, while the color bar on the right shows the scaling for all of the orbitals. The\nchosen high symmetry points are \u0000=(0,0,0), X 1=(1\n2,0,0), M 1=(0,1\n2,0), M 2=(1\n2,0,1\n2), X 2=(0,0,1\n2), and R=(0,1\n2,1\n2).\nIn Fig. 4, we plot the band structures of the two\nsublattice Os atoms along the high symmetry direc-\ntions of the monoclinic cell, with the total partial char-acters of the Os 5 dbands color-coded proportional to\ntheir squared-amplitude contributions to the Kohn-Sham\neigenvectors, the so-called fat-bands. The total par-5\ntial character is the root of the sum of the squares of\nthe partial characters of the Cartesian spin projections,q\nM2x+M2y+M2z. We plot the partial characters of the\nspin projections of MxandMyin the Supplemental In-\nformation Figs. 3 and 4, but not Mzbecause it is two\norders of magnitude smaller than the other two. For both\nOs atoms along these high symmetry directions, only t2g\norbitals are occupied, consistent with hndi<6. Thet2g\nandegare irreducible representations of perfect cubic,\noctahedral, or tetrahedral symmetry. Because these sym-\nmetries are broken in the structure with Q2 distortion,\nthere are no pure t2goregorbitals, nor a \u0001( eg\u0000t2g) en-\nergy splitting, and there will be a small mixing between\nthe two sets of orbitals.\nWe see that for both sublattices, below the band gap,\nthe d yzorbital is most heavily occupied (as denoted by\nthe brighter green color), especially along the X 1-M2di-\nrection, while the d xyand d zxorbitals are less occupied.\nHowever, around the \u0000 point, the dxyorbital obviously\nhas the largest occupancy. We point out that here the\ndorbital character contribution is only for the selected\nhigh symmetry directions. Thus, it can not be directly\ncompared with the PDOS result. Nevertheless, the di\u000ber-\nent orbital character contributions re\rected in the color\ncan also be observed for all three t2gorbitals, especially\nalong the X 2-\u0000 line. We can also see that the dispersions\nare largest along the X 1-M2and X 1-\u0000 paths, while the\nbands are \ratter from M 2to R. The band gap is indirect\nand\u00190:06 eV in magnitude.\nFinally, we have computed the gaps for the imposed\ncFM phase. We found that the gaps for the cFM phase\nwith the DFT+U parameters of U= 3:3 eV andJ= 0:5\neV are \fnite, but too small to be considered Mott insu-\nlating gaps. However, we \fnd that the gap opens dra-\nmatically as we raise Uto 5.0 eV, as shown in detail in\nthe Supplemental Information. In fact, even a \\small\"\nincrease to U= 4:0 eV is su\u000ecient to open the gap to\nEgap= 0:244 eV. This indicates that the true value\nofUfor the osmium 5 dshell in BNOO could plausi-\nbly approach 4 :0 eV, but not exceed it. Previously, it\nwas found that LDA+U with U= 4 eV> W was in-\nsu\u000ecient to open a gap27. Here, we demonstrated that\nGGA+SOC+U is su\u000ecient to open a gap for U\u00194:0 eV.\nIV. CONCLUSIONS\nIn this work, we carried out DFT+U calculations on\nthe magnetic Mott insulator Ba 2NaOsO 6, which hasstrong spin orbit coupling. Our numerical work is in-\nspired by our recent NMR results revealing that this ma-\nterial exhibited a broken local point symmetry (BLPS)\nphase followed by a two-sublattice exotic canted ferro-\nmagnetic order (cFM). The local symmetry is broken by\nthe orthorhombic Q2 distortion mode21. The question\nwe addressed here is whether this distortion is accom-\npanied by the emergence of orbital order. It was pre-\nviously proposed that the two-sublattice magnetic struc-\nture, revealed by NMR, is the very manifestation of stag-\ngered quadrupolar order with distinct orbital polariza-\ntion on the two sublattices arising from multipolar ex-\nchange interactions2,12. Moreover, it was indicated via a\ndi\u000berent mean \feld formalism that the anisotropic inter-\nactions result in orbital order that stabilizes exotic mag-\nnetic order5. Therefore, distinct mean \feld approaches2,5\nwith a common ingredient of anisotropic exchange inter-\nactions imply that exotic magnetic order, such as the\ncFM reported in Ref. 12, is accompanied/driven by an\norbital order.\nMotivated by the cFM order detected in NMR exper-\niments, here we investigated BNOO's orbital ordering\npattern from \frst principles. We found two-sublattice\norbital ordering, illustrated by the spin density plots,\nwithin the alternating planes in which the total magnetic\nmoment resides. An auxiliary signature of the orbital or-\ndering is revealed by the occupancies of the t2gorbitals\nin the density of states and band structures. Our \frst\nprinciples work demonstrates that this two-sublattice or-\nbital ordering mainly arises from cFM order and strong\nSOC. Moving forward, it would be worthwhile to more\nthoroughly investigate the cFM order observed in this\nwork using other functionals or methods more adept at\nhandling strong correlation to eliminate any ambiguities\nthat stem from our speci\fc computational treatment.\nV. ACKNOWLEDGMENTS\nThe authors thank Jeong-Pil Song and Yiou Zhang\nfor enlightening discussions. We are especially grate-\nful to Ian Fisher for his long term collaboration on the\nphysics of Ba 2NaOsO 6. This work was supported in\npart by U.S. National Science Foundation grants DMR-\n1608760 (V.F.M.) and DMR-1726213 (B.M.R.). The cal-\nculations presented here were performed using resources\nat the Center for Computation and Visualization, Brown\nUniversity, which is supported by NSF Grant No. ACI-\n1548562.\nyCorresponding authors V. F. M. (vemi@brown.edu)\n& B. R. (brenda rubenstein@brown.edu)\n1B. J. Kim, H. Jin, S. J. Moon, J.-Y. Kim, B.-G. Park,\nC. S. Leem, J. Yu, T. W. Noh, C. Kim, S.-J. Oh, J.-H.\nPark, V. Durairaj, G. Cao, and E. Rotenberg, Phys. Rev.\nLett. 101, 076402 (2008).2G. Chen, R. Pereira, and L. Balents, Phys. Rev. B 82,\n174440 (2010).\n3G. Chen and L. Balents, Physical Review B 84, 094420\n(2011).6\n4H. Ishizuka and L. Balents, Physical Review B 90, 184422\n(2014).\n5C. Svoboda, M. Randeria, and N. Trivedi,\narXiv:1702.03199v1 (unpublished) (2017).\n6W. Witczak-Krempa, G. Chen, Y. B. Kim, and L. Balents,\nAnnual Review of Condensed Matter Physics 5, 57 (2014).\n7J. Romh\u0013 anyi, L. Balents, and G. Jackeli, Phys. Rev. Lett.\n118, 217202 (2017).\n8B. J. Kim, H. Ohsumi, T. Komesu, S. Sakai, T. Morita,\nH. Takagi, and T. Arima, Science 323, 1329 (2009).\n9H. Zhang, K. Haule, and D. Vanderbilt, Phys. Rev. Lett.\n111, 246402 (2013).\n10S. J. Moon, H. Jin, K. W. Kim, W. S. Choi, Y. S. Lee,\nJ. Yu, G. Cao, A. Sumi, H. Funakubo, C. Bernhard, and\nT. W. Noh, Phys. Rev. Lett. 101, 226402 (2008).\n11X. Wan, A. M. Turner, A. Vishwanath, and S. Y.\nSavrasov, Phys. Rev. B 83, 205101 (2011).\n12L. Lu, M. Song, W. Liu, A. P. Reyes, P. Kuhns, H. O. Lee,\nI. R. Fisher, and V. F. Mitrovi\u0013 c, Nature Communications\n8, 14407 EP (2017).\n13G. Kresse and J. Hafner, Phys. Rev. B 47, 558 (1993).\n14G. Kresse and J. Hafner, Phys. Rev. B 49, 251 (1994).\n15G. Kresse and J. Furthmuller, Comput. Mat. Sci. 6, 15\n(1996).16G. Kresse and J. Furthmuller, Phys. Rev. B 54, 11169\n(1996).\n17J. P. Perdew and Y. Wang, Phys. Rev. B 45, 13244 (1992).\n18F. Z. Hund, Zeitschrift f ur Physik 33, 345.\n19P. Bl ochl, Phys. Rev. B 50, 17953 (1994).\n20G. Kresse and J. Joubert, Phys. Rev. B 59, 1758 (1999).\n21R. Cong, R. Nanguneri, B. M. Rubenstein, and\nV. F. Mitrovic, arXiv e-prints , arXiv:1908.09014 (2019),\narXiv:1908.09014.\n22A. Erickson, S. Misra, G. J. Miller, R. Gupta,\nZ. Schlesinger, W. Harrison, J. Kim, and I. Fisher, Phys-\nical Review Letters 99, 016404 (2007).\n23M. Methfessel and A. T. Paxton, Phys. Rev. B 40, 3616\n(1989).\n24P. E. Bl ochl, O. Jepsen, and O. K. Andersen, Phys. Rev.\nB49, 16223 (1994).\n25D. F. Mosca, \\Master Thesis - Universit\u0012 a di Bologna:\nQuantum Magnetism in Relativistic Osmates from First\nPrinciples,\" (2019).\n26S. Gangopadhyay and W. E. Pickett, Phys. Rev. B 91,\n045133 (2015).\n27K.-W. Lee and W. E. Pickett, EPL (Europhysics Letters)\n80, 37008 (2007)."    },    {        "title": "1608.02230v1.Spin_orbit_coupling_in_Fe_based_superconductors.pdf",        "content": "Noname manuscript No.\n(will be inserted by the editor)\nSpin-orbit coupling in Fe-based superconductors\nM.M. Korshunov \u0001Yu.N. Togushova \u0001I. Eremin \u0001P.J. Hirschfeld\nReceived: date / Accepted: date\nAbstract We study the spin resonance peak in re-\ncently discovered iron-based superconductors. The res-\nonance peak observed in inelastic neutron scattering\nexperiments agrees well with predicted results for the\nextendeds-wave (s\u0006) gap symmetry. Recent neutron\nscattering measurements show that there is a dispar-\nity between longitudinal and transverse components of\nthe dynamical spin susceptibility. Such breaking of the\nspin-rotational invariance in the spin-liquid phase can\noccur due to spin-orbit coupling. We study the role of\nthe spin-orbit interaction in the multiorbital model for\nFe-pnictides and show how it a\u000bects the spin resonance\nfeature.\nKeywords Fe-based superconductors \u0001Spin-resonance\npeak\u0001Spin-orbit coupling\nThe nature of the superconductivity and gap sym-\nmetry and structure in the recently discovered Fe-based\nsuperconductors (FeBS) are the most debated topics\nin condensed matter community [1]. These quasi two-\ndimensional systems shows a maximal Tcof 55 K plac-\ning them right after high- Tccuprates. Fe d-orbitals form\nM.M. Korshunov\nE-mail: korshunov@phys.u\r.edu\nL.V. Kirensky Institute of Physics, Krasnoyarsk 660036, Rus-\nsia\nM.M. Korshunov and Yu.N. Togushova\nSiberian Federal University, Svobodny Prospect 79, Krasno-\nyarsk 660041, Russia\nP.J. Hirschfeld\nDepartment of Physics, University of Florida, Gainesville,\nFlorida 32611, USA\nI. Eremin\nInstitut f ur Theoretische Physik III, Ruhr-Universitat\nBochum, D-44801 Bochum, Germany\nKazan Federal University, Kazan 420008, Russiathe Fermi surface (FS) which in the undoped systems\nconsists of two hole and two electron sheets. Nesting\nbetween these two groups of sheets is the driving force\nfor the spin-density wave (SDW) long-range magnetism\nin the undoped FeBS and the scattering with the wave\nvector Qconnecting hole and electron pockets is the\nmost probable candidate for superconducting pairing in\nthe doped systems. In the spin-\ructuation studies [2,3,\n4], the leading instability is the extended s-wave gap\nwhich changes sign between hole and electron sheets\n(s\u0006state) [5].\nNeutron scattering is a powerful tool to measure\ndynamical spin susceptibility \u001f(q;!). It carries infor-\nmation about the order parameter symmetry and gap\nstructure. For the local interactions (Hubbard and Hund's\nexchange),\u001fcan be obtained in the RPA from the bare\nelectron-hole bubble \u001f0(q;!) by summing up a series of\nladder diagrams to give \u001f(q;!) = [I\u0000Us\u001f0(q;!)]\u00001\u001f0(q;!),\nwhereUsandIare interaction and unit matrices in or-\nbital space, and all other quantities are matrices as well.\nScattering between nearly nested hole and electron\nFermi surfaces in FeBS produce a peak in the normal\nstate magnetic susceptibility at or near q=Q= (\u0019;0).\nFor the uniform s-wave gap, sign \u0001k= sign\u0001k+Qand\nthere is no resonance peak. For the s\u0006order parameter\nas well as for an extended non-uniform s-wave symme-\ntry,Qconnects Fermi sheets with the di\u000berent signs\nof gaps. This ful\flls the resonance condition for the in-\nterband susceptibility, and the spin resonance peak is\nformed at a frequency below \nc= min (j\u0001kj+j\u0001k+qj)\n(compare normal and s\u0006superconductor's response in\nFig. 1) [6,7,8]. The existence of the spin resonance in\nFeBS was predicted theoretically [6,7] and subsequently\ndiscovered experimentally with many reports of well-\nde\fned spin resonances in 1111, 122, and 11 systems\n[9,10,11].arXiv:1608.02230v1  [cond-mat.supr-con]  7 Aug 20162 M.M. Korshunov et al.\n 0 1 2 3 4 5 6 7 8 9\n 0  1  2  3  4  5Im χ(q=[π,0],ω)\nω/∆0non-SC, χ+-\nnon-SC, 2 ×χzz\ns++, χ+-\ns++, 2×χzz\ns±, χ+-\ns±, 2×χzz\nFig. 1 Fig. 1. Calculated Im \u001f(Q;!) in the normal state, and\nfor thes++ands\u0006pairing symmetries. In the latter case, the\nresonance is clearly seen below != 2\u00010. Spin-orbit coupling\nconstant\u0015= 100 meV, intraorbital Hubbard U= 0:9 eV,\nHund'sJ= 0:1 eV, interorbital U0=U\u00002J, and pair-hopping\ntermJ0=J.\nOne of the recent puzzles in FeBS is the discov-\nered anisotropy of the spin resonance peak in Ni-doped\nBa-122 [12]. It was found that \u001f+\u0000and 2\u001fzzare dif-\nferent. This contradicts the spin-rotational invariance\n(SRI)hS+S\u0000i= 2hSzSziwhich have to be obeyed in\nthe disordered system. One of the solution to the puzzle\nis the spin-orbit (SO) interaction which can break the\nSRI like it does in Sr 2RuO 4[13]. Here we incorporate\nthe e\u000bect of the SO coupling in the susceptibility cal-\nculation for FeBS to shed light on the spin resonance\nanisotropy.\nThe simplest model for pnictides in the 1-Fe per\nunit cell Brillouin zone comes from the three t2gd-\norbitals. The xzandyzcomponents are hybridized and\nform two electron-like FS pockets around ( \u0019;0) and\n(0;\u0019) points, and one hole-like pocket around \u0000= (0;0)\npoint. Thexyorbital is considered to be decoupled from\nthem and form an outer hole pocket around \u0000point.\nThe one-electron part of the Hamiltonian is given by\nH0=P\nk;\u001b;l;m\"lm\nkcy\nkl\u001bckm\u001b, wherelandmare orbital\nindices,ckm\u001bis the annihilation operator of a particle\nwith momentum kand spin\u001b. This model for pnic-\ntides is similar to the one for Sr 2RuO 4and, in particu-\nlar, thexyband does not hybridize with the xzandyz\nbands. Keeping in mind the similarity to the Sr 2RuO 4\ncase, for simplicity we consider only the Lz-component\nof the SO interaction [13]. Due to the structure of the\nLz-component, the interaction a\u000bects xzandyzbands\nonly.\nFollowing Ref. [14], we write the SO coupling term,\nHSO=\u0015P\nfLf\u0001Sf, in the second-quantized form asHSO= i\u0015\n2P\nl;m;n\u000flmnP\nk;\u001b;\u001b0cy\nkl\u001bckm\u001b0^\u001bn\n\u001b\u001b0, where\u000flmnis\nthe completely antisymmetric tensor, indices fl;m;ng\ntake valuesfx;y;zg$fdyz;dzx;dxyg$f 2;3;1g, and\n^\u001bn\n\u001b\u001b0are the Pauli spin matrices.\nThe matrix of the Hamiltonian H=H0+HSOis\nthen\n^\"k\u001b=0\n@\"1k 0 0\n0\"2k\"4k+ i\u0015\n2sign\u001b\n0\"4k\u0000i\u0015\n2sign\u001b \" 3k1\nA (1)\nAs for Sr 2RuO 4, eigenvalues of ^ \"k\u001bdo not depend on\nspin\u001b, therefore, spin-up and spin-down states are still\ndegenerate in spite of the SO interaction.\nWe calculated both + \u0000(longitudinal) and zz(trans-\nverse) components of the spin susceptibility and found\nthat in the normal state \u001f+\u0000>2\u001fzzat small frequen-\ncies, see Fig. 1. As expected, for the s++supercon-\nductor (conventional isotropic s-wave) there is no res-\nonance peak and the disparity between \u001f+\u0000and 2\u001fzz\nis very small. For the s\u0006superconductor, however, the\nsituation is opposite { we observe a well de\fned spin\nresonance and \u001f+\u0000is larger than 2 \u001fzzby about 15%\nnear the peak position (Fig. 1).\nIn summary , we have shown that the spin resonance\npeak in FeBS gains anisotropy in the spin space due\nto the spin-orbit coupling. This result is in qualitative\nagreement with experimental \fndings. We do not ob-\nserve changes in the peak position but this may be due\nto the simple model that we studied.\nAcknowledgements Partial support was provided by DOE\nDE-FG02-05ER46236 (P.J.H. and M.M.K.) and NSF-DMR-\n1005625 (P.J.H.). M.M.K. acknowledge support from RFBR\n(grants 09-02-00127, 12-02-31534 and 13-02-01395), Presid-\nium of RAS program \\Quantum mesoscopical and disordered\nstructures\" N20.7, FCP Scienti\fc and Research-and-Educational\nPersonnel of Innovative Russia for 2009-2013 (GK 16.740.12.0731\nand GK P891), and President of Russia (grant MK-1683.2010.2),\nSiberian Federal University (Theme N F-11), Program of SB\nRAS #44, and The Dynasty Foundation and ICFPM. I.E. ac-\nknowledges support of the SFB Transregio 12, Merkur Foun-\ndation, and German Academic Exchange Service (DAAD PPP\nUSA No. 50750339).\nReferences\n1. P. J. Hirschfeld, M. M. Korshunov, and I. I. Mazin, Rep.\nProg. Phys. 74, 124508 (2011).\n2. S. Graser et al. , New. J. Phys. 11, 025016 (2009).\n3. K. Kuroki et al. , Phys. Rev. Lett. 101, 087004 (2008).\n4. S. Maiti et al. , Phys. Rev. Lett. 107, 147002 (2011).\n5. I. I. Mazin et al. , Phys. Rev. Lett. 101, 057003 (2008).\n6. M. M. Korshunov and I. Eremin, Phys. Rev. B 78,\n140509(R) (2008).\n7. T. A. Maier and D. J. Scalapino, Phys. Rev. B 78,\n020514(R) (2008).\n8. T. A. Maier et al. , Phys. Rev. B 79, 134520 (200).Spin-orbit coupling in Fe-based superconductors 3\n9. A. D. Christianson et al. , Nature 456, 930 (2008).\n10. D. S. Inosov et al. , Nature Physics 6, 178 (2010).\n11. D. N. Argyriou et al., Phys. Rev. B 81, 220503(R) (2010).\n12. O. J. Lipscombe et al. , Phys. Rev. B 82, 064515 (2010).\n13. I. Eremin, D. Manske, and K. H. Bennemann, Phys. Rev.\nB65, 220502(R) (2002).\n14. K. K. Ng and M. Sigrist, Europhys. Lett. 49, 473 (2000)."    },    {        "title": "2311.16268v2.Gilbert_damping_in_two_dimensional_metallic_anti_ferromagnets.pdf",        "content": "Gilbert damping in two-dimensional metallic anti-ferromagnets\nR. J. Sokolewicz,1, 2M. Baglai,3I. A. Ado,1M. I. Katsnelson,1and M. Titov1\n1Radboud University, Institute for Molecules and Materials, 6525 AJ Nijmegen, the Netherlands\n2Qblox, Delftechpark 22, 2628 XH Delft, the Netherlands\n3Department of Physics and Astronomy, Uppsala University, Box 516, SE-751 20, Uppsala, Sweden\n(Dated: March 29, 2024)\nA finite spin life-time of conduction electrons may dominate Gilbert damping of two-dimensional\nmetallic anti-ferromagnets or anti-ferromagnet/metal heterostructures. We investigate the Gilbert\ndamping tensor for a typical low-energy model of a metallic anti-ferromagnet system with honeycomb\nmagnetic lattice and Rashba spin-orbit coupling for conduction electrons. We distinguish three\nregimes of spin relaxation: exchange-dominated relaxation for weak spin-orbit coupling strength,\nElliot-Yafet relaxation for moderate spin-orbit coupling, and Dyakonov-Perel relaxation for strong\nspin-orbit coupling. We show, however, that the latter regime takes place only for the in-plane\nGilbert damping component. We also show that anisotropy of Gilbert damping persists for any\nfinite spin-orbit interaction strength provided we consider no spatial variation of the N´ eel vector.\nIsotropic Gilbert damping is restored only if the electron spin-orbit length is larger than the magnon\nwavelength. Our theory applies to MnPS 3monolayer on Pt or to similar systems.\nI. INTRODUCTION\nMagnetization dynamics in anti-ferromagnets con-\ntinue to attract a lot of attention in the context\nof possible applications1–4. Various proposals utilize\nthe possibility of THz frequency switching of anti-\nferromagnetic domains for ultrafast information storage\nand computation5,6. The rise of van der Waals magnets\nhas had a further impact on the field due to the pos-\nsibility of creating tunable heterostructures that involve\nanti-ferromagnet and semiconducting layers7.\nUnderstanding relaxation of both the N´ eel vector and\nnon-equilibrium magnetization in anti-ferromagnets is\nrecognized to be of great importance for the function-\nality of spintronic devices8–13. On one hand, low Gilbert\ndamping must generally lead to better electric control of\nmagnetic order via domain wall motion or ultrafast do-\nmain switching14–16. On the other hand, an efficient con-\ntrol of magnetic domains must generally require a strong\ncoupling between charge and spin degrees of freedom due\nto a strong spin-orbit interaction, that is widely thought\nto be equivalent to strong Gilbert damping.\nIn this paper, we focus on a microscopic analysis of\nGilbert damping due to Dyakonov-Perel and Elliot-Yafet\nmechanisms. We apply the theory to a model of a two-\ndimensional N´ eel anti-ferromagnet with a honeycomb\nmagnetic lattice.\nTwo-dimensional magnets typically exhibit either\neasy-plane or easy-axis anisotropy, and play crucial\nroles in stabilizing magnetism at finite temperatures17,18.\nEasy-axis anisotropy selects a specific direction for mag-\nnetization, thereby defining an axis for the magnetic or-\nder. In contrast, easy-plane anisotropy does not select a\nparticular in-plane direction for the N´ eel vector, allowing\nit to freely rotate within the plane. This situation is anal-\nogous to the XY model, where the system’s continuous\nsymmetry leads to the suppression of out-of-plane fluc-\ntuations rather than fixing the magnetization in a spe-\ncific in-plane direction19,20. Without this anisotropy, themagnonic fluctuations in a two-dimensional crystal can\ngrow uncontrollably large to destroy any long-range mag-\nnetic order, according to the Mermin-Wagner theorem21.\nRecent density-functional-theory calculations for\nsingle-layer transition metal trichalgenides22, predict the\nexistence of a large number of metallic anti-ferromagnets\nwith honeycomb lattice and different types of magnetic\norder as shown in Fig. 1. Many of these crystals may\nhave the N´ eel magnetic order as shown in Fig. 1a and are\nmetallic: FeSiSe 3, FeSiTe 3, VGeTe 3, MnGeS 3, FeGeSe 3,\nFeGeTe 3, NiGeSe 3, MnSnS 3, MnSnS 3, MnSnSe 3,\nFeSnSe 3, NiSnS 3. Apart from that it has been predicted\nthat anti-ferromagnetism can be induced in graphene by\nbringing it in proximity to MnPSe 323or by bringing it\nin double proximity between a layer of Cr 2Ge2Te6and\nWS224.\nPartly inspired by these predictions and recent\ntechnological advances in producing single-layer anti-\nferromagnet crystals, we propose an effective model to\nstudy spin relaxation in 2D honeycomb anti-ferromagnet\nwith N´ eel magnetic order. The same system was studied\nby us in Ref. 25, where we found that spin-orbit cou-\npling introduces a weak anisotropy in spin-orbit torque\nand electric conductivity. Strong spin-orbit coupling was\nshown to lead to a giant anisotropy of Gilbert damping.\nOur analysis below is built upon the results of Ref. 25,\nand we investigate and identify three separate regimes\nof spin-orbit strength. Each regime is characterized by\nqualitatively different dependence of Gilbert damping on\nspin-orbit interaction and conduction electron transport\ntime. The regime of weak spin-orbit interaction is dom-\ninated by exchange field relaxation of electron spin, and\nthe regime of moderate spin-orbit strength is dominated\nby Elliot-Yafet spin relaxation. These two regimes are\ncharacterized also by a universal factor of 2 anisotropy\nof Gilbert damping. The regime of strong spin-orbit\nstrength, which leads to substantial splitting of electron\nFermi surfaces, is characterized by Dyakonov-Perel relax-\nation of the in-plane spin component and Elliot-Yafet re-arXiv:2311.16268v2  [cond-mat.dis-nn]  28 Mar 20242\nFIG. 1. Three anti-ferromagnetic phases commonly found\namong van-der-Waals magnets. Left-to-right: N´ eel, zig-zag,\nand stripy.\nlaxation of the perpendicular-to-the-plane Gilbert damp-\ning which leads to a giant damping anisotropy. Isotropic\nGilbert damping is restored only for finite magnon wave\nvectors such that the magnon wavelength is smaller than\nthe spin-orbit length.\nGilbert damping in a metallic anti-ferromagnet can be\nqualitatively understood in terms of the Fermi surface\nbreathing26. A change in the magnetization direction\ngives rise to a change in the Fermi surface to which the\nconduction electrons have to adjust. This electronic re-\nconfiguration is achieved through the scattering of elec-\ntrons off impurities, during which angular momentum is\ntransferred to the lattice. Gilbert damping, then, should\nbe proportional to both (i) the ratio of the spin life-time\nand momentum life-time of conduction electrons, and (ii)\nthe electric conductivity. Keeping in mind that the con-\nductivity itself is proportional to momentum life-time,\none may conclude that the Gilbert damping is linearly\nproportional to the spin life-time of conduction electrons.\nAt the same time, the spin life-time of localized spins is\ninversely proportional to the spin life-time of conduc-\ntion electrons. A similar relation between the spin life-\ntimes of conduction and localized electrons also holds\nfor relaxation mechanisms that involve electron-magnon\nscattering27.\nOur approach formally decomposes the magnetic sys-\ntem into a classical sub-system of localized magnetic mo-\nments and a quasi-classical subsystem of conduction elec-\ntrons. A local magnetic exchange couples these sub-\nsystems. Localized magnetic moments in transition-\nmetal chalcogenides and halides form a hexagonal lat-\ntice. Here we focus on the N´ eel type anti-ferromagnet\nthat is illustrated in Fig. 1a. In this case, one can de-\nfine two sub-lattices A and B that host local magnetic\nmoments SAandSB, respectively. For the discussion of\nGilbert damping, we ignore the weak dependence of both\nfields on atomic positions and assume that the modulus\nS=|SA(B)|is time-independent.\nUnder these assumptions, the magnetization dynamics\nof localized moments may be described in terms of two\nfields\nm=1\n2S\u0000\nSA+SB\u0001\n,n=1\n2S\u0000\nSA−SB\u0001\n, (1)\nwhich are referred to as the magnetization and staggeredmagnetization (or N´ eel vector), respectively. Within the\nmean-field approach, the vector fields yield the equations\nof motion\n˙n=−Jn×m+n×δs++m×δs−, (2a)\n˙m=m×δs++n×δs−, (2b)\nwhere dot stands for the time derivative, while δs+and\nδs−stand for the mean staggered and non-staggered non-\nequilibrium fields that are proportional to the variation of\nthe corresponding spin-densities of conduction electrons\ncaused by the time dynamics of nandmfields. The en-\nergy Jis proportional to the anti-ferromagnet exchange\nenergy for localized momenta.\nIn Eqs. (2) we have omitted terms that are propor-\ntional to easy axis anisotropy for the sake of compact-\nness. These terms are, however, important and will be\nintroduced later in the text.\nIn the framework of Eqs. (2) the Gilbert damping can\nbe computed as the linear response of the electron spin-\ndensity variation to a time change in both the magneti-\nzation and the N´ eel vector (see e. g. Refs.25,28,29).\nIn this definition, Gilbert damping describes the re-\nlaxation of localized spins by transferring both total and\nstaggered angular momenta to the lattice by means of\nconduction electron scattering off impurities. Such a\ntransfer is facilitated by spin-orbit interaction.\nThe structure of the full Gilbert damping tensor can be\nrather complicated as discussed in Ref. 25. However, by\ntaking into account easy axis or easy plane anisotropy we\nmay reduce the complexity of relevant spin configurations\nto parameterize\nδs+=α∥\nm˙m∥+α⊥\nm˙m⊥+αmn∥×(n∥×˙m∥),(3a)\nδs−=α∥\nn˙n∥+α⊥\nn˙n⊥+αnn∥×(n∥×˙n∥), (3b)\nwhere the superscripts ∥and⊥refer to the in-plane\nand perpendicular-to-the-plane projections of the corre-\nsponding vectors, respectively. The six coefficients α∥\nm,\nα⊥\nm,αm,α∥\nn,α⊥\nn, and αnparameterize the Gilbert damp-\ning.\nInserting Eqs. (3) into the equations of motion of\nEqs. (2) produces familiar Gilbert damping terms. The\ndamping proportional to time-derivatives of the N´ eel vec-\ntornis in general many orders of magnitude smaller than\nthat proportional to the time-derivatives of the magneti-\nzation vector m25,30. Due to the same reason, the higher\nharmonics term αmn∥×(n∥×∂tm∥) can often be ne-\nglected.\nThus, in the discussion below we may focus mostly on\nthe coefficients α∥\nmandα⊥\nmthat play the most important\nrole in the magnetization dynamics of our system. The\nterms proportional to the time-derivative of ncorrespond\nto the transfer of angular momentum between the sub-\nlattices and are usually less relevant. We refer to the\nresults of Ref. 25 when discussing these terms.\nAll Gilbert damping coefficients are intimately related\nto the electron spin relaxation time. The latter is rel-\natively well understood in non-magnetic semiconductors3\nwith spin-orbital coupling. When a conducting electron\nmoves in a steep potential it feels an effective magnetic\nfield caused by relativistic effects. Thus, in a disordered\nsystem, the electron spin is subject to a random magnetic\nfield each time it scatters off an impurity. At the same\ntime, an electron also experiences precession around an\neffective spin-orbit field when it moves in between the\ncollisions. Changes in spin direction between collisions\nare referred to as Dyakonov-Perel relaxation31,32, while\nchanges in spin-direction during collisions are referred to\nas Elliot-Yafet relaxation33,34.\nThe spin-orbit field in semiconductors induces a char-\nacteristic frequency of spin precession Ω s, while scalar\ndisorder leads to a finite transport time τof the con-\nducting electrons. One may, then, distinguish two limits:\n(i) Ω sτ≪1 in which case the electron does not have\nsufficient time to change its direction between consec-\nutive scattering events (Elliot-Yafet relaxation), and (ii)\nΩsτ≫1 in which case the electron spin has multiple pre-\ncession cycles in between the collisions (Dyakonov-Perel\nrelaxation).\nThe corresponding processes define the so-called spin\nrelaxation time, τs. In a 2D system the spin life-time\nτ∥\ns, for the in-plane spin components, appears to be dou-\nble the size of the life-time of the spin component that\nis perpendicular to the plane, τ⊥\ns32. This geometric ef-\nfect has largely been overlooked. For non-magnetic 2D\nsemiconductor one can estimate35,36\n1\nτ∥\ns∼(\nΩ2\nsτ,Ωsτ≪1\n1/τ, Ωsτ≫1, τ∥\ns= 2τ⊥\ns. (4)\nA pedagogical derivation and discussion of Eq. 4 can\nbe found in Refs. 35 and 36. Because electrons are con-\nfined in two dimensions the random spin-orbit field is\nalways directed in-plane, which leads to a decrease in the\nin-plane spin-relaxation rate by a factor of two compared\nto the out-of-plane spin-relaxation rate as demonstrated\nfirst in Ref. 32 (see Refs. 36–40 as well). The reason is\nthat the perpendicular-to-the-plane component of spin is\ninfluenced by two components of the randomly changing\nmagnetic field, i. e. xandy, whereas the parallel-to-the-\nplane spin components are only influenced by a single\ncomponent of the fluctuating fields, i. e. the xspin pro-\njection is influenced only by the ycomponent of the field\nand vice-versa. The argument has been further general-\nized in Ref. 25 to the case of strongly separated spin-orbit\nsplit Fermi surfaces. In this limit, the perpendicular-to-\nthe-plane spin-flip processes on scalar disorder potential\nbecome fully suppressed. As a result, the perpendicular-\nto-the-plane spin component becomes nearly conserved,\nwhich results in a giant anisotropy of Gilbert damping in\nthis regime.\nIn magnetic systems that are, at the same time, con-\nducting there appears to be at least one additional energy\nscale, ∆ sd, that characterizes exchange coupling of con-\nduction electron spin to the average magnetic moment of\nlocalized electrons. (In the case of s-d model descriptionit is the magnetic exchange between the spin of conduc-\ntionselectron and the localized magnetic moment of d\norfelectron on an atom.) This additional energy scale\ncomplicates the simple picture of Eq. (4) especially in the\ncase of an anti-ferromagnet. The electron spin precession\nis now defined not only by spin-orbit field but also by\n∆sd. As the result the conditions Ω sτ≪1 and ∆ sdτ≫1\nmay easily coexist. This dissolves the distinction between\nElliot-Yafet and Dyakonov-Perel mechanisms of spin re-\nlaxation. One may, therefore, say that both Elliot-Yafet\nand Dyakonov-Perel mechanisms may act simultaneously\nin a typical 2D metallic magnet with spin-orbit coupling.\nThe Gilbert damping computed from the microscopic\nmodel that we formulate below will always contain both\ncontributions to spin-relaxation.\nII. MICROSCOPIC MODEL AND RESULTS\nThe microscopic model that we employ to calculate\nGilbert damping is the so-called s–dmodel that couples\nlocalized magnetic momenta SAandSBand conducting\nelectron spins via the local magnetic exchange ∆ sd. Our\neffective low-energy Hamiltonian for conduction electrons\nreads\nH=vfp·Σ+λ\n2\u0002\nσ×Σ\u0003\nz−∆sdn·σΣzΛz+V(r),(5)\nwhere the vectors Σ,σandΛdenote the vectors of Pauli\nmatrices acting on sub-lattice, spin and valley space,\nrespectively. We also introduce the Fermi velocity vf,\nRashba-type spin-orbit interaction λ, and a random im-\npurity potential V(r).\nThe Hamiltonian of Eq. (5) can be viewed as the\ngraphene electronic model where conduction electrons\nhave 2D Rashba spin-orbit coupling and are also cou-\npled to anti-ferromagnetically ordered classical spins on\nthe honeycomb lattice.\nThe coefficients α∥\nmandα⊥\nmare obtained using linear\nresponse theory for the response of spin-density δs+to\nthe time-derivative of magnetization vector ∂tm. Impu-\nrity potential V(r) is important for describing momen-\ntum relaxation to the lattice. This is related to the an-\ngular momentum relaxation due to spin-orbit coupling.\nThe effect of random impurity potential is treated pertur-\nbatively in the (diffusive) ladder approximation that in-\nvolves a summation over diffusion ladder diagrams. The\ndetails of the microscopic calculation can be found in the\nAppendices.\nBefore presenting the disorder-averaged quantities\nα∥,⊥\nm, it is instructive to consider first the contribution\nto Gilbert damping originating from a small number of\nelectron-impurity collisions. This clarifies how the num-\nber of impurity scattering effects will affect the final re-\nsult.\nLet us annotate the Gilbert damping coefficients with\nan additional superscript ( l) that denotes the number\nof scattering events that are taken into account. This4\n01234\u0016\u000b(i)\n?[\"\u001c]\n\u0016\u000b(0)\n?\u0016\u000b(1)\n?\u0016\u000b(2)\n? \u0016\u000b(1)\n?\n10\u0000210\u00001100101\n\u0015\u001c01234\u0016\u000b(i)\nk[\"\u001c]\n\u0016\u000b(0)\nk\u0016\u000b(1)\nk\u0016\u000b(2)\nk\u0016\u000b(1)\nk\nFIG. 2. Gilbert damping in the limit ∆ sd= 0. Dotted (green)\nlines correspond to the results of the numerical evaluation of\n¯α(l)\nm,⊥,∥forl= 0,1,2 as a function of the parameter λτ. The\ndashed (orange) line corresponds to the diffusive (fully vertex\ncorrected) results for ¯ α⊥,∥.\nm.\nmeans, in the diagrammatic language, that the corre-\nsponding quantity is obtained by summing up the ladder\ndiagrams with ≤ldisorder lines. Each disorder line cor-\nresponds to a quasi-classical scattering event from a sin-\ngle impurity. The corresponding Gilbert damping coeffi-\ncient is, therefore, obtained in the approximation where\nconduction electrons have scattered at most lnumber\nof times before releasing their non-equilibrium magnetic\nmoment into a lattice.\nTo make final expressions compact we define the di-\nmensionless Gilbert damping coefficients ¯ α∥,⊥\nmby extract-\ning the scaling factor\nα∥,⊥\nm=A∆2\nsd\nπℏ2v2\nfS¯α∥,⊥\nm, (6)\nwhere Ais the area of the unit cell, vfis the Fermi ve-\nlocity of the conducting electrons and ℏ=h/2πis the\nPlanck’s constant. We also express the momentum scat-\ntering time τin inverse energy units, τ→ℏτ.\nLet us start by computing the coefficients ¯ α∥,⊥(l)\nm in the\nformal limit ∆ sd→0. We can start with the “bare bub-\nble” contribution which describes spin relaxation without\na single scattering event. The corresponding results read\n¯α(0)\nm,⊥=ετ1−λ2/4ε2\n1 +λ2τ2, (7a)\n¯α(0)\nm,∥=ετ\u00121 +λ2τ2/2\n1 +λ2τ2−λ2\n8ε2\u0013\n, (7b)\nwhere εdenotes the Fermi energy which we consider pos-\nitive (electron-doped system).In all realistic cases, we have to consider λ/ε≪1,\nwhile the parameter λτmay in principle be arbitrary. For\nλτ≪1 the disorder-induced broadening of the electron\nFermi surfaces exceeds the spin-orbit induced splitting.\nIn this case one basically finds no anisotropy of “bare”\ndamping: ¯ α(0)\nm,⊥= ¯α(0)\nm,∥. In the opposite limit of substan-\ntial spin-orbit splitting one gets an ultimately anisotropic\ndamping ¯ α(0)\nm,⊥≪¯α(0)\nm,∥. This asymptotic behavior can be\nsummarized as\n¯α(0)\nm,⊥=ετ(\n1 λτ≪1,\n(λτ)−2λτ≫1,(8a)\n¯α(0)\nm,∥=ετ(\n1 λτ≪1,\n1\n2\u0000\n1 + (λτ)−2\u0001\nλτ≫1,(8b)\nwhere we have used that ε≫λ.\nThe results of Eq. (8) modify by electron diffusion. By\ntaking into account up to lscattering events we obtain\n¯α(l)\nm,⊥=ετ(\nl+O(λ2τ2) λτ≪1,\n(1 +δl0)/(λτ)2λτ≫1,(9a)\n¯α(l)\nm,∥=ετ(\nl+O(λ2τ2) λτ≪1,\n1−(1/2)l+1+O((λτ)−2)λτ≫1,(9b)\nwhere we have used ε≫λagain.\nFrom Eqs. (9) we see that the Gilbert damping for\nλτ≪1 gets an additional contribution of ετfrom each\nscattering event as illustrated numerically in Fig. 2. This\nleads to a formal divergence of Gilbert damping in the\nlimit λτ≪1. While, at first glance, the divergence looks\nlike a strong sensitivity of damping to impurity scatter-\ning, in reality, it simply reflects a diverging spin life-time.\nOnce a non-equilibrium magnetization mis created it\nbecomes almost impossible to relax it to the lattice in\nthe limit of weak spin-orbit coupling. The formal diver-\ngence of α⊥\nm=α∥\nmsimply reflects the conservation law\nfor electron spin polarization in the absence of spin-orbit\ncoupling such that the corresponding spin life-time be-\ncomes arbitrarily large as compared to the momentum\nscattering time τ.\nBy taking the limit l→ ∞ (i. e. by summing up the\nentire diffusion ladder) we obtain compact expressions\n¯α⊥\nm≡¯α(∞)\nm,⊥=ετ1\n2λ2τ2, (10a)\n¯α∥\nm≡¯α(∞)\nm,∥=ετ1 +λ2τ2\nλ2τ2, (10b)\nwhich assume ¯ α⊥\nm≪¯α∥\nmforλτ≫1 and ¯ α⊥\nm= ¯α∥\nm/2\nforλτ≪1. The factor of 2 difference that we observe\nwhen λτ≪1, corresponds to a difference in the elec-\ntron spin life-times τ⊥\ns=τ∥\ns/2 that was discussed in the\nintroduction32.\nStrong spin-orbit coupling causes a strong out-of-plane\nanisotropy of damping, ¯ α⊥\nm≪¯α∥\nmwhich corresponds to5\na suppression of the perpendicular-to-the-plane damping\ncomponent. As a result, the spin-orbit interaction makes\nit much easier to relax the magnitude of the mzcompo-\nnent of magnetization than that of in-plane components.\nLet us now turn to the dependence of ¯ αmcoefficients on\n∆sdthat is illustrated numerically in Fig. 3. We consider\nfirst the case of absent spin-orbit coupling λ= 0. In\nthis case, the combination of spin-rotational and sub-\nlattice symmetry (the equivalence of A and B sub-lattice)\nmust make Gilbert damping isotropic (see e. g.25,41). The\ndirect calculation for λ= 0 does, indeed, give rise to the\nisotropic result ¯ α⊥\nm= ¯α∥\nm=ετ(ε2+∆2\nsd)/2∆2\nsd, which is,\nhowever, in contradiction to the limit λ→0 in Eq. (10).\nAt first glance, this contradiction suggests the exis-\ntence of a certain energy scale for λover which the\nanisotropy emerges. The numerical analysis illustrated\nin Fig. 4 reveals that this scale does not depend on the\nvalues of 1 /τ, ∆sd, orε. Instead, it is defined solely by\nnumerical precision. In other words, an isotropic Gilbert\ndamping is obtained only when the spin-orbit strength\nλis set below the numerical precision in our model.\nWe should, therefore, conclude that the transition from\nisotropic to anisotropic (factor of 2) damping occurs ex-\nactly at λ= 0. Interestingly, the factor of 2 anisotropy is\nabsent in Eqs. (8) and emerges only in the diffusive limit.\nWe will see below that this paradox can only be re-\nsolved by analyzing the Gilbert damping beyond the in-\nfinite wave-length limit.\nOne can see from Fig. 3 that the main effect of finite\n∆sdis the regularization of the Gilbert damping diver-\ngency ( λτ)−2in the limit λτ≪1. Indeed, the limit of\nweak spin-orbit coupling is non-perturbative for ∆ sd/ε≪\nλτ≪1, while, in the opposite limit, λτ≪∆sd/ε≪1,\nthe results of Eqs. (10) are no longer valid. Assuming\n∆sd/ε≪1 we obtain the asymptotic expressions for the\nresults presented in Fig. 3 as\n¯α⊥\nm=1\n2ετ(2\n3ε2+∆2\nsd\n∆2\nsdλτ≪∆sd/ε,\n1\nλ2τ2 λτ≫∆sd/ε,(11a)\n¯α∥\nm=ετ(2\n3ε2+∆2\nsd\n∆2\nsdλτ≪∆sd/ε,\n1 +1\nλ2τ2λτ≫∆sd/ε,(11b)\nwhich suggest that ¯ α⊥\nm/¯α∥\nm= 2 for λτ≪1. In the op-\nposite limit, λτ≫1, the anisotropy of Gilbert damping\ngrows as ¯ α∥\nm/¯α⊥\nm= 2λ2τ2.\nThe results of Eqs. (11) can also be discussed in terms\nof the electron spin life-time, τ⊥(∥)\ns = ¯α⊥(∥)\nm/ε. For the\ninverse in-plane spin life-time we find\n1\nτ∥\ns=\n\n3∆2\nsd/2ε2τ λτ ≪∆sd/ε,\nλ2τ ∆sd/ε≪λτ≪1,\n1/τ 1≪λτ,(12)\nthat, for ∆ sd= 0, is equivalent to the known result of\nEq. (4). Indeed, for ∆ sd= 0, the magnetic exchange\n10\u0000310\u0000210\u00001100101\n\u0015\u001c10\u00001101103105\u0016\u000bm;k;?[\"\u001c]\n\u0001sd=\"= 0:1\u0001sd=\"= 0\u0016\u000bm;k\n\u0016\u000bm;?FIG. 3. Numerical results for the Gilbert damping compo-\nnents in the diffusive limit (vertex corrected)as the function\nof the spin-orbit coupling strength λ. The results correspond\ntoετ= 50 and ∆ sdτ= 0.1 and agree with the asymptotic\nexpressions of Eq. (11). Three different regimes can be dis-\ntinguished for ¯ α∥\nm: i) spin-orbit independent damping ¯ α∥\nm∝\nε3τ/∆2\nsdfor the exchange dominated regime, λτ≪∆sd/ε, ii)\nthe damping ¯ α∥\nm∝ε/λ2τfor Elliot-Yafet relaxation regime,\n∆sd/ε≪λτ≪1, and iii) the damping ¯ α∥\nm∝ετfor the\nDyakonov-Perel relaxation regime, λτ≫1. The latter regime\nis manifestly absent for ¯ α⊥\nmin accordance with Eqs. (12,13).\nplays no role and one observes the cross-over from Elliot-\nYafet ( λτ≪1) to Dyakonov-Perel ( λτ≫1) spin relax-\nation.\nThis cross-over is, however, absent in the relaxation of\nthe perpendicular spin component\n1\nτ⊥s= 2(\n3∆2\nsd/2ε2τ λτ ≪∆sd/ε,\nλ2τ ∆sd/ε≪λτ,(13)\nwhere Elliot-Yafet-like relaxation extends to the regime\nλτ≫1.\nAs mentioned above, the factor of two anisotropy in\nspin-relaxation of 2 Dsystems, τ∥\ns= 2τ⊥\ns, is known in the\nliterature32(see Refs.36–38as well). Unlimited growth of\nspin life-time anisotropy, τ∥\ns/τ⊥\ns= 2λ2τ2, in the regime\nλτ≪1 has been described first in Ref. 25. It can be qual-\nitatively explained by a strong suppression of spin-flip\nprocesses for zspin component due to spin-orbit induced\nsplitting of Fermi surfaces. The mechanism is effective\nonly for scalar (non-magnetic) disorder. Even though\nsuch a mechanism is general for any magnetic or non-\nmagnetic 2D material with Rashba-type spin-orbit cou-\npling, the effect of the spin life-time anisotropy on Gilbert\ndamping is much more relevant for anti-ferromagnets. In-\ndeed, in an anti-ferromagnetic system the modulus of m\nis, by no means, conserved, hence the variations of per-\npendicular and parallel components of the magnetization\nvector are no longer related.\nIn the regime, λτ≪∆sd/εthe spin life-time is de-\nfined by exchange interaction and the distinction between\nDyakonov-Perel and Elliot-Yafet mechanisms of spin re-\nlaxation is no longer relevant. In this regime, the spin-\nrelaxation time is by a factor ( ε/∆sd)2larger than the\nmomentum relaxation time.\nLet us now return to the problem of emergency of the6\n10\u00006410\u00005410\u00004410\u00003410\u00002410\u000014\n\u0015\u001c12\u0016\u000bk=\u0016\u000b?n= 32\nn= 64n= 96\nn= 128\nFIG. 4. Numerical evaluation of Gilbert damping anisotropy\nin the limit λ→0. Isotropic damping tensor is restored only\nifλ= 0 with ultimate numerical precision. The factor of 2\nanisotropy emerges at any finite λ, no matter how small it\nis, and only depends on the numerical precision n, i.e. the\nnumber of digits contained in each variable during computa-\ntion. The crossover from isotropic to anisotropic damping can\nbe understood only by considering finite, though vanishingly\nsmall, magnon qvectors.\nfactor of 2 anisotropy of Gilbert damping at λ= 0. We\nhave seen above (see Fig. 4) that, surprisingly, there ex-\nists no energy scale for the anisotropy to emerge. The\ntransition from the isotropic limit ( λ= 0) to a finite\nanisotropy appeared to take place exactly at λ= 0. We\ncan, however, generalize the concept of Gilbert damping\nby considering the spin density response function at a\nfinite wave vector q.\nTo generalize the Gilbert damping, we are seeking a\nresponse of spin density at a point r,δs+(r) to a time\nderivative of magnetization vectors ˙m∥and ˙m⊥at the\npoint r′. The Fourier transform with respect to r−r′\ngives the Gilbert damping for a magnon with the wave-\nvector q.\nThe generalization to a finite q-vector shows that the\nlimits λ→0 and q→0 cannot be interchanged. When\nthe limit λ→0 is taken before the limit q→0 one\nfinds an isotropic Gilbert damping, while for the oppo-\nsite order of limits, it becomes a factor of 2 anisotropic.\nIn a realistic situation, the value of qis limited from\nbelow by an inverse size of a typical magnetic domain\n1/Lm, while the spin-orbit coupling is effective on the\nlength scale Lλ= 2πℏvf/λ. In this picture, the isotropic\nGilbert damping is characteristic for the case of suffi-\nciently small domain size Lm≪Lλ, while the anisotropic\nGilbert damping corresponds to the case Lλ≪Lm.\nIn the limit qℓ≪1, where ℓ=vfτis the electron mean\n\u00002 0 2\nk[a.u.]\u00002:50:02:5energy [a.u.]\u0015=\u0001sd= 4\n\u00002 0 2\nk[a.u.]\u0015=\u0001sd= 2\n\u00002 0 2\nk[a.u.]\u0015=\u0001sd= 1FIG. 5. Band-structure for the effective model of Eq. (5)\nin a vicinity of Kvalley assuming nz= 1. Electron bands\ntouch for λ= 2∆ sd. The regime λ≤2∆sdcorresponds to\nspin-orbit band inversion. The band structure in the valley\nK′is inverted. Our microscopic analysis is performed in the\nelectron-doped regime for the Fermi energy above the gap as\nillustrated by the top dashed line. The bottom dashed line\ndenotes zero energy (half-filling).\nfree path, we can summarize our results as\n¯α⊥\nm=ετ\n\nε2+∆2\nsd\n2∆2\nsdλτ≪qℓ≪∆sd/ε,\n1\n3ε2+∆2\nsd\n∆2\nsdqℓ≪λτ≪∆sd/ε,\n1\n2λ2τ2 λτ≫∆sd/ε,, (14a)\n¯α∥\nm=ετ\n\nε2+∆2\nsd\n2∆2\nsdλτ≪qℓ≪∆sd/ε,\n2\n3ε2+∆2\nsd\n∆2\nsdqℓ≪λτ≪∆sd/ε,\n1 +1\nλ2τ2λτ≫∆sd/ε,(14b)\nwhich represent a simple generalization of Eqs. (11).\nThe results of Eqs. (14) correspond to a simple behav-\nior of Gilbert damping anisotropy,\n¯α∥\nm/¯α⊥\nm=(\n1 λτ≪qℓ,\n2\u0000\n1 +λ2τ2\u0001\nqℓ≪λτ,(15)\nwhere we still assume qℓ≪1.\nIII. ANTI-FERROMAGNETIC RESONANCE\nThe broadening of the anti-ferromagnet resonance\npeak is one obvious quantity that is sensitive to Gilbert\ndamping. The broadening is however not solely defined\nby a particular Gilbert damping component but depends\nalso on both magnetic anisotropy and anti-ferromagnetic\nexchange.\nTo be more consistent we can use the model of Eq. (5)\nto analyze the contribution of conduction electrons to an\neasy axis anisotropy. The latter is obtained by expanding\nthe free energy for electrons in the value of nz, which has\na form E=−Kn2\nz/2. With the conditions ε/λ≫1 and\nε/∆sd≫1 we obtain the anisotropy constant as\nK=A\n2πℏ2v2(\n∆2\nsdλ 2∆sd/λ≤1,\n∆sdλ2/2 2∆ sd/λ≥1,(16)7\nwhere Ais the area of the unit cell. Here we assume\nboth λand ∆ sdpositive, therefore, the model natu-\nrally gives rise to an easy axis anisotropy with K > 0.\nIn real materials, there exist other sources of easy axis\nor easy plane anisotropy. In-plane magneto-crystalline\nanisotropy also plays an important role. For example,\nN´ eel-type anti-ferromagnets with easy-axis anisotropy\nare FePS 3, FePSe 3or MnPS 3, whereas those with easy\nplane and in-plane magneto-crystalline anisotropy are\nNiPS 3and MnPSe 3. Many of those materials are, how-\never, Mott insulators. Our qualitative theory may still\napply to materials like MnPS 3monolayers at strong elec-\ntron doping.\nThe transition from 2∆ sd/λ≥1 to 2∆ sd/λ≤1 in\nEq. (16) corresponds to the touching of two bands in the\nmodel of Eq. (5) as illustrated in Fig. 5.\nAnti-ferromagnetic magnon frequency and life-time in\nthe limit q→0 are readily obtained by linearizing the\nequations of motion\n˙n=−Jn×m+Km×n⊥+n×(ˆαm˙m), (17a)\n˙m=Kn×n⊥+n×(ˆαn˙n), (17b)\nwhere we took into account easy axis anisotropy Kand\ndisregarded irrelevant terms m×(ˆαn˙n) and m×(ˆαm˙m).\nWe have also defined Gilbert damping tensors such as\nˆαm˙m=α∥\nm˙m∥+α⊥\nm˙m⊥, ˆαn˙n=α∥\nn˙n∥+α⊥\nn˙n⊥.\nIn the case of easy axis anisotropy we can use the lin-\nearized modes n=ˆz+δn∥eiωt,m=δm∥eiωt, hence we\nget the energy of q= 0 magnon as\nω=ω0−iΓ/2, (18)\nω0=√\nJK, Γ =Jα∥\nn+Kα∥\nm (19)\nwhere we took into account that K≪J. The expression\nforω0is well known due to Kittel and Keffer42,43.\nUsing Ref. 25 we find out that α∥\nn≃α⊥\nm(λ/ε)2and\nα⊥\nn≃α∥\nm(λ/ε)2, hence\nΓ≃α∥\nm\u0012\nK+J/2\nε2/λ2+ε2τ2\u0013\n, (20)\nwhere we have simply used Eqs. (10). Thus, one may\noften ignore the contribution Jα∥\nnas compared to Kα∥\nm\ndespite the fact that K≪J.\nIn the context of anti-ferromagnets, spin-pumping\nterms are usually associated with the coefficients α∥\nnin\nEq. (3b) that are not in the focus of the present study.\nThose coefficients have been analyzed for example in Ref.\n25. In this manuscript we simply use the known results\nforαnin Eqs. (17-19), where we illustrate the effect of\nboth spin-pumping coefficient αnand the direct Gilbert\ndamping αmon the magnon life time. One can see from\nEqs. (19,20) that the spin-pumping contributions do also\ncontribute, though indirectly, to the magnon decay. The\nspin pumping contributions become more important in\nmagnetic materials with small magnetic anisotropy. The\nprocesses characterized by the coefficients αnmay also be\n10\u0000310\u0000210\u00001100101\n\u0015\u001c0:000:010:021=\u0016\u000bk\nm\u0015=\"= 0:04\n\u0015=\"= 0:02\n\u0015=\"= 0:01FIG. 6. Numerical evaluation of the inverse Gilbert damping\n1/¯α∥\nmas a function of the momentum relaxation time τ. The\ninverse damping is peaked at τ∝1/λwhich also corresponds\nto the maximum of the anti-ferromagnetic resonance quality\nfactor in accordance with Eq. (21).\ninterpreted in terms of angular momentum transfer from\none AFM sub-lattice to another. In that respect, the spin\npumping is specific to AFM, and is qualitatively differ-\nent from the direct Gilbert damping processes ( αm) that\ndescribe the direct momentum relaxation to the lattice.\nAs illustrated in Fig. 6 the quality factor of the anti-\nferromagnetic resonance (for a metallic anti-ferromagnet\nwith easy-axis anisotropy) is given by\nQ=ω0\nΓ≃1\nα∥\nmr\nJ\nK. (21)\nInterestingly, the quality factor defined by Eq. (21) is\nmaximized for λτ≃1, i. e. for the electron spin-orbit\nlength being of the order of the scattering mean free path.\nThe quantities 1 /√\nKand 1 /¯α∥\nmare illustrated in\nFig. 6 from the numerical analysis. As one would ex-\npect, the quality factor vanishes in both limits λ→0\nandλ→ ∞ . The former limit corresponds to an over-\ndamped regime hence no resonance can be observed. The\nlatter limit corresponds to a constant α∥\nm, but the reso-\nnance width Γ grows faster with λthan ω0does, hence\nthe vanishing quality factor.\nIt is straightforward to check that the results of\nEqs. (20,21) remain consistent when considering systems\nwith either easy-plane or in-plane magneto-crystalline\nanisotropy. Thus, the coefficient α⊥\nmnormally does not\nenter the magnon damping, unless the system is brought\ninto a vicinity of spin-flop transition by a strong external\nfield.\nIV. CONCLUSION\nIn conclusion, we have analyzed the Gilbert damping\ntensor in a model of a two-dimensional anti-ferromagnet\non a honeycomb lattice. We consider the damping mech-\nanism that is dominated by a finite electron spin life-time8\ndue to a combination of spin-orbit coupling and impu-\nrity scattering of conduction electrons. In the case of a\n2D electron system with Rashba spin-orbit coupling λ,\nthe Gilbert damping tensor is characterized by two com-\nponents α∥\nmandα⊥\nm. We show that the anisotropy of\nGilbert damping depends crucially on the parameter λτ,\nwhere τis the transport scattering time for conduction\nelectrons. For λτ≪1 the anisotropy is set by a geo-\nmetric factor of 2, α∥\nm= 2α⊥\nm, while it becomes infinitely\nlarge in the opposite limit, α∥\nm= (λτ)2α⊥\nmforλτ≫1.\nGilbert damping becomes isotropic exactly for λ= 0, or,\nstrictly speaking, for the case λ≪ℏvfq, where qis the\nmagnon wave vector.\nThis factor of 2 is essentially universal, and is a geomet-\nric effect: the z-component relaxation results from fluctu-\nations in two in-plane spin components, whereas in-plane\nrelaxation stems from fluctuations of the z-component\nalone. This reflects the subtleties of our microscopic\nmodel, where the mechanism for damping is activated\nby the decay of conduction electron momenta, linked to\nspin-relaxation through spin-orbit interactions.\nWe find that Gilbert damping is insensitive to mag-\nnetic order for λ≫∆sd/ετ, where ∆ sdis an effective\nexchange coupling between spins of conduction and local-\nized electrons. In this case, the electron spin relaxation\ncan be either dominated by scattering (Dyakonov-Perel\nrelaxation) or by spin-orbit precession (Elliot-Yafet re-\nlaxation). We find that the Gilbert damping component\nα⊥\nm≃ε/λ2τis dominated by Elliot-Yafet relaxation irre-\nspective of the value of the parameter λτ, while the other\ncomponent crosses over from α∥\nm≃ε/λ2τ(Elliot-Yafet\nrelaxation) for λτ≪1, to α∥\nm≃ετ(Dyakonov-Perel re-\nlaxation) for λτ≫1. For the case λ≪∆sd/ετthe spin\nrelaxation is dominated by interaction with the exchange\nfield.\nCrucially, our results are not confined solely to the N´ eel\norder on the honeycomb lattice: we anticipate a broader\napplicability across various magnetic orders, including\nthe zigzag order. This universality stems from our focus\non the large magnon wavelength limit. The choice of the\nhoneycomb lattice arises from its unique ability to main-\ntain isotropic electronic spectra within the plane, coupled\nwith the ability to suppress anisotropy concerning in-\nplane spin rotations. Strong anisotropic electronic spec-\ntra would naturally induce strong anisotropic in-plane\nGilbert damping, which are absent in our results.\nFinally, we show that the anti-ferromagnetic resonance\nwidth is mostly defined by α∥\nmand demonstrate that the\nresonance quality factor is maximized for λτ≈1. Our\nmicroscopic theory predictions may be tested for systems\nsuch as MnPS 3monolayer on Pt and similar heterostruc-\ntures.ACKNOWLEDGMENTS\nWe are grateful to O. Gomonay, R. Duine, J. Sinova,\nand A. Mauri for helpful discussions. This project has\nreceived funding from the European Union’s Horizon\n2020 research and innovation program under the Marie\nSklodowska-Curie grant agreement No 873028.\nAppendix A: Microscopic framework\nThe microscopic model that we employ to calculate\nGilbert damping belongs to a class of so-called s–dmod-\nels that describe the physical system in the form of a\nHeisenberg model for localized spins and a tight-binding\nmodel for conduction electrons that are weakly coupled\nby a local magnetic exchange interaction of the strength\n∆sd.\nOur effective electron Hamiltonian for a metallic\nhexagonal anti-ferromagnet is given by25\nH0=vfp·Σ+λ\n2[σ×Σ]z−∆sdn·σΣzΛz,(A1)\nwhere the vectors Σ,σandΛdenote the vectors of Pauli-\nmatrices acting on sub-lattice, spin and valley space re-\nspectively. We also introduce the Fermi velocity vf,\nRashba-type spin-orbit interaction λ.\nTo describe Gilbert damping of the localized field n\nwe have to add the relaxation mechanism. This is pro-\nvided in our model by adding a weak impurity potential\nH=H0+V(r). The momentum relaxation due to scat-\ntering on impurities leads indirectly to the relaxation of\nHeisenberg spins due to the presence of spin-orbit cou-\npling and exchange couplings.\nFor modeling the impurity potential, we adopt a delta-\ncorrelated random potential that corresponds to the\npoint scatter approximation, where the range of the im-\npurity potential is much shorter than that of the mean\nfree path (see e.g. section 3.8 of Ref. 44), i.e.\n⟨V(r)V(r′)⟩= 2πα(ℏvf)2δ(r−r′), (A2)\nwhere the dimensionless coefficient α≪1 characterizes\nthe disorder strength. The corresponding scattering time\nfor electrons is obtained as τ=ℏ/παϵ , which is again\nsimilar to the case of graphene.\nThe response of symmetric spin-polarization δs+to the\ntime-derivative of non-staggered magnetization, ∂tm, is\ndefined by the linear relation\nδs+\nα=X\nβRαβ|ω=0˙mβ, (A3)\nwhere the response tensor is taken at zero frequency25,45.\nThe linear response is defined generally by the tensor\nRαβ=A∆2\nsd\n2πSZdp\n(2πℏ)2\nTr\u0002\nGR\nε,pσαGA\nε+ℏω,pσβ\u0003\u000b\n,(A4)9\nwhere GR(A)\nε,pare standing for retarded(advanced) Green\nfunctions and the angular brackets denote averaging over\ndisorder fluctuations.\nThe standard recipe for disorder averaging is the diffu-\nsive approximation46,47that is realized by replacing the\nbare Green functions in Eq. (A4) with disorder-averaged\nGreen functions and by replacing one of the vertex op-\nerators σxorσywith the corresponding vertex-corrected\noperator that is formally obtained by summing up ladder\nimpurity diagrams (diffusons).\nIn models with spin-orbit coupling, the controllable dif-\nfusive approximation for non-dissipative quantities may\nbecome, however, more involved as was noted first in\nRef. 48. For Gilbert damping it is, however, sufficient to\nconsider the ladder diagram contributions only.\nThe disorder-averaged Green function is obtained by\nincluding an imaginary part of the self-energy ΣR(not\nto be confused here with the Pauli matrix Σ 0,x,y,z) that\nis evaluated in the first Born approximation\nIm ΣR= 2παv2\nfZdp\n(2π)2Im1\nε−H0+i0. (A5)\nThe real part of the self-energy leads to the renormaliza-\ntion of the energy scales ε,λand ∆ sd.\nIn the first Born approximation, the disorder-averaged\nGreen function is given by\nGR\nε,p=1\nε−H0−iIm ΣR. (A6)\nThe vertex corrections are computed in the diffusive\napproximation. The latter involves replacing the vertex\nσαwith the vertex-corrected operator,\nσvc\nα=∞X\nl=0σ(l)\nα, (A7)\nwhere the index lcorresponds to the number of disorder\nlines in the ladder.\nThe operators σ(l)\nαcan be defined recursively as\nσ(l)\nα=2ℏv2\nf\nετZdp\n(2π)2GR\nε,pσ(l−1)\nαGA\nε+ℏω,p, (A8)\nwhere σ(0)\nα=σα.\nThe summation in Eq. (A7) can be computed in the\nfull operator basis, Bi={α,β,γ}=σαΣβΛγ, where each\nindex α,βandγtakes on 4 possible values (with zero\nstanding for the unity matrix). We may always normalize\nTrBiBj= 2δijin an analogy to the Pauli matrices. The\noperators Biare, then, forming a finite-dimensional space\nfor the recursion of Eq. (A8).\nThe vertex-corrected operators Bvc\niare obtained by\nsumming up the matrix geometric series\nBvc\ni=X\nj\u00121\n1− F\u0013\nijBj, (A9)where the entities of the matrix Fare given by\nFij=ℏv2\nf\nετZdp\n(2π)2Tr\u0002\nGR\nε,pBiGA\nε+ℏω,pBj\u0003\n.(A10)\nOur operators of interest σxandσycan always be de-\ncomposed in the operator basis as\nσα=1\n2X\niBiTr (σαBi), (A11)\nhence the vertex-corrected spin operator is given by\nσvc\nα=1\n2X\nijBvc\niTr(σαBi). (A12)\nMoreover, the computation of the entire response tensor\nof Eq. (A4) in the diffusive approximation can also be\nexpressed via the matrix Fas\nRαβ=α0ετ\n8ℏX\nij[TrσαBi]\u0014F\n1− F\u0015\nij[TrσβBj],(A13)\nwhere α0=A∆2\nsd/πℏ2v2\nfSis the coefficient used in\nEq. (6) to define the unit of the Gilbert damping.\nIt appears that one can always choose the basis of\nBioperators such that the computation of Eq. (A13)\nis closed in a subspace of just three Bioperators with\ni= 1,2,3. This enables us to make analytical computa-\ntions of Eq. (A13).\nAppendix B: Magnetization dynamics\nThe representation of the results can be made some-\nwhat simpler by choosing xaxis in the direction of the\nin-plane projection n∥of the N´ eel vector, hence ny= 0.\nIn this case, one can represent the result as\nδs+=c1n∥×(n∥×∂tm∥) +c2∂tm∥+c3∂tm⊥+c4n,\nwhere ndependence of the coefficients cimay be param-\neterized as\nc1=r11−r22−r31(1−n2\nz)/(nxnz)\n1−n2z, (B1a)\nc2=r11−r31(1−n2\nz)/(nxnz), (B1b)\nc3=r33, (B1c)\nc4= (r31/nz)∂tmz+ζ(∂tm)·n. (B1d)\nThe analytical results in the paper correspond to the\nevaluation of δs±up to the second order in ∆ sdusing\nperturbative analysis. Thus, zero approximation corre-\nsponds to setting ∆ sd= 0 in Eqs. (A1,A5).\nThe equations of motion on nandmare given by\nEqs. (2),\n∂tn=−Jn×m+n×δs++m×δs−, (B2a)\n∂tm=m×δs++n×δs−, (B2b)10\nIt is easy to see that the following transformation leaves\nthe above equations invariant,\nδs+→δs+−ξn, δ s−→δs−−ξm, (B3)\nfor an arbitrary value of ξ.\nSuch a gauge transformation can be used to prove that\nthe coefficient c4is irrelevant in Eqs. (B2).\nIn this paper, we compute δs±to the zeroth order in\n|m|– the approximation which is justified by the sub-\nlattice symmetry in the anti-ferromagnet. A somewhat\nmore general model has been analyzed also in Ref. 25 to\nwhich we refer the interested reader for more technical\ndetails.\nAppendix C: Anisotropy constant\nThe anisotropy constant is obtained from the grand po-\ntential energy Ω for conducting electrons. For the model\nof Eq. (A1) the latter can be expressed as\nΩ =−X\nς=±1\nβZ\ndε g(ε)νς(ε), (C1)\nwhere β= 1/kBTis the inverse temperature, ς=±is\nthe valley index (for the valleys KandK′),GR\nς,pis the\nbare retarded Green function with momentum pand in\nthe valley ς. We have also defined the function\ng(ε) = ln (1 + exp[ β(µ−ε)]), (C2)\nwhere µis the electron potential, and the electron density\nof states in each of the valleys is given by,\nνς(ε) =1\nπZdp\n(2πℏ)2Im Tr GR\nς,p, (C3)\nwhere the trace is taken only over spin and sub-lattice\nspace,\nIn the metal regime considered, the chemical potential\nis assumed to be placed in the upper electronic band.\nIn this case, the energy integration can be taken only for\npositive energies. The two valence bands are always filled\nand can only add a constant shift to the grand potential\nΩ that we disregard.\nThe evaluation of Eq. (C1) yields the following density\nof states\nντ(ε) =1\n2πℏ2v2\nf\n\n0 0 < ε < ε 2\nε/2 +λ/4ε2< ε < ε 1,\nε ε > ε 1,(C4)where the energies ε1,2correspond to the extremum\npoints (zero velocity) for the electronic bands. These\nenergies, for each of the valleys, are given by\nε1,ς=1\n2\u0000\n+λ+p\n4∆2+λ2−4ς∆λnz\u0001\n, (C5a)\nε2,ς=1\n2\u0000\n−λ+p\n4∆2+λ2+ 4ς∆λnz\u0001\n(C5b)\nwhere ς=±is the valley index.\nIn the limit of zero temperature we can approximate\nEq. (C1) as\nΩ =−X\nς=±1\nβZ∞\n0dε(µ−ε)νς(ε). (C6)\nThen, with the help of Eq. (C1) we find,\nΩ =−1\n24πℏ2v2\nfX\nς=±\u0002\n(ε1,ς−µ)2(4ε1,ς−3λ+ 2µ)\n+(ε2,ς−µ)2(4ε2,ς+ 3λ+ 2µ)\u0003\n. (C7)\nBy substituting the results of Eqs. (C5) into the above\nequation we obtain\nΩ =−1\n24πℏ2v2\nfh\n(4∆2−4nz∆λ+λ2)2/3\n+(4∆2+ 4nz∆λ+λ2)2/3−24∆µ+ 8µ3i\n.(C8)\nA careful analysis shows that the minimal energy cor-\nresponds to nz=±1 so that the conducting electrons\nprefer an easy-axis magnetic anisotropy. By expanding\nin powers of n2\nzaround nz=±1 we obtain Ω = −Kn2\nz/2,\nwhere\nK=1\n2πℏ2v2(\n|∆2λ| | λ/2∆| ≥1,\n|∆λ2|/2|λ/2∆| ≤1.(C9)\nThis provides us with the easy axis anisotropy of Eq. (16).\n1S. A. Siddiqui, J. Sklenar, K. Kang, M. J.\nGilbert, A. Schleife, N. Mason, and A. Hoff-\nmann, Journal of Applied Physics 128, 040904\n(2020), https://pubs.aip.org/aip/jap/article-pdf/doi/10.1063/5.0009445/15249168/040904 1online.pdf.\n2V. V. Mazurenko, Y. O. Kvashnin, A. I. Lichtenstein, and\nM. I. Katsnelson, Journal of Experimental and Theoretical\nPhysics 132, 506 (2021).11\n3L.ˇSmejkal, J. Sinova, and T. Jungwirth, Phys. Rev. X\n12, 040501 (2022).\n4B. A. Bernevig, C. Felser, and H. Beidenkopf, Nature 603,\n41 (2022).\n5T. G. H. Blank, K. A. Grishunin, B. A. Ivanov, E. A.\nMashkovich, D. Afanasiev, and A. V. Kimel, Phys. Rev.\nLett. 131, 096701 (2023).\n6W. Wu, C. Yaw Ameyaw, M. F. Doty, and\nM. B. Jungfleisch, Journal of Applied Physics 130,\n091101 (2021), https://pubs.aip.org/aip/jap/article-\npdf/doi/10.1063/5.0057536/13478272/091101 1online.pdf.\n7M. Gibertini, M. Koperski, A. F. Morpurgo, and K. S.\nNovoselov, Nature Nanotechnology 14, 408 (2019).\n8K. M. D. Hals, Y. Tserkovnyak, and A. Brataas, Physical\nReview Letters 106, 107206 (2011), publisher: American\nPhysical Society.\n9R. Cheng, D. Xiao, and A. Brataas, Physical Review Let-\nters116, 207603 (2016), publisher: American Physical So-\nciety.\n10S. Urazhdin and N. Anthony, Physical Review Letters 99,\n046602 (2007), publisher: American Physical Society.\n11R. Cheng, M. W. Daniels, J.-G. Zhu, and D. Xiao, Phys-\nical Review B 91, 064423 (2015), publisher: American\nPhysical Society.\n12R. Khymyn, I. Lisenkov, V. Tiberkevich, B. A. Ivanov,\nand A. Slavin, Scientific Reports 7, 43705 (2017), number:\n1 Publisher: Nature Publishing Group.\n13R. Cheng, J. Xiao, Q. Niu, and A. Brataas, Physical\nReview Letters 113, 057601 (2014), publisher: American\nPhysical Society.\n14A. Mougin, M. Cormier, J. P. Adam, P. J. Metaxas, and\nJ. Ferr´ e, Europhysics Letters (EPL) 78, 57007 (2007), pub-\nlisher: IOP Publishing.\n15A. A. Thiele, Physical Review Letters 30, 230 (1973), pub-\nlisher: American Physical Society.\n16R. Weber, D.-S. Han, I. Boventer, S. Jaiswal, R. Lebrun,\nG. Jakob, and M. Kl¨ aui, Journal of Physics D: Applied\nPhysics 52, 325001 (2019), publisher: IOP Publishing.\n17V. Y. Irkhin, A. A. Katanin, and M. I. Katsnelson, Phys.\nRev. B 60, 1082 (1999).\n18D. V. Spirin, Journal of Magnetism and Magnetic Materi-\nals264, 121 (2003).\n19J. Kosterlitz, Journal of Physics C: Solid State Physics 7,\n1046 (1974).\n20J. M. Kosterlitz and D. J. Thouless, in Basic Notions Of\nCondensed Matter Physics (CRC Press, 2018) pp. 493–515.\n21N. Mermin and H. Wagner, Phys. Rev. Lett. 17, 1133\n(1966).\n22B. L. Chittari, D. Lee, N. Banerjee, A. H. MacDonald,\nE. Hwang, and J. Jung, Phys. Rev. B 101, 085415 (2020).\n23P. H¨ ogl, T. Frank, K. Zollner, D. Kochan, M. Gmitra, and\nJ. Fabian, Phys. Rev. Lett. 124, 136403 (2020).24K. Dolui, M. D. Petrovi´ c, K. Zollner, P. Plech´ aˇ c, J. Fabian,\nand B. K. Nikoli´ c, Nano Lett. 20, 2288 (2020).\n25M. Baglai, R. J. Sokolewicz, A. Pervishko, M. I. Katsnel-\nson, O. Eriksson, D. Yudin, and M. Titov, Physical Re-\nview B 101, 104403 (2020), publisher: American Physical\nSociety.\n26M. F¨ ahnle and D. Steiauf, Physical Review B 73, 184427\n(2006).\n27H. T. Simensen, A. Kamra, R. E. Troncoso, and\nA. Brataas, Physical Review B 101, 020403 (2020), pub-\nlisher: American Physical Society.\n28A. Brataas, Y. Tserkovnyak, and G. E. W. Bauer, Physical\nReview Letters 101, 037207 (2008), publisher: American\nPhysical Society.\n29H. Ebert, S. Mankovsky, D. K¨ odderitzsch, and P. J. Kelly,\nPhysical Review Letters 107, 066603 (2011), publisher:\nAmerican Physical Society.\n30Q. Liu, H. Y. Yuan, K. Xia, and Z. Yuan, Physical Review\nMaterials 1, 061401 (2017), publisher: American Physical\nSociety.\n31M. Dyakonov and V. Perel, Sov. Phys. Solid State, Ussr\n13, 3023 (1972).\n32M. Dyakonov and V. Kachorovskij, Fizika i tehnika\npoluprovodnikov 20, 178 (1986).\n33R. Elliott, Phys. Rev. 96, 266 (1954).\n34Y. Yafet, in Solid State Physics , Vol. 14, edited by F. Seitz\nand D. Turnbull (Elsevier, 1963) pp. 1–98.\n35M. Dyakonov, arXiv:cond-mat/0401369 (2004).\n36M. I. Dyakonov, ed., Spin Physics in Semiconductors , 2nd\ned., Springer Series in Solid-State Sciences (Springer Inter-\nnational Publishing, 2017).\n37N. Averkiev, L. Golub, and M. Willander, Semiconductors\n36, 91 (2002).\n38A. Burkov, A. S. N´ u˜ nez, and A. MacDonald, Phys. Rev.\nB70, 155308 (2004).\n39A. A. Burkov and L. Balents, Physical Review B 69,\n245312 (2004), publisher: American Physical Society.\n40N. A. Sinitsyn and Y. V. Pershin, Reports on Progress in\nPhysics 79, 106501 (2016), publisher: IOP Publishing.\n41A. Kamra, R. E. Troncoso, W. Belzig, and A. Brataas,\nPhys. Rev. B 98, 184402 (2018).\n42C. Kittel, Phys. Rev. 82, 565 (1951).\n43F. Keffer and C. Kittel, Phys. Rev. 85, 329 (1952).\n44J. Rammer and H. Smith, Rev. Mod. Phys. 58, 323 (1986).\n45I. Ado, O. A. Tretiakov, and M. Titov, Phys. Rev. B 95,\n094401 (2017).\n46J. Rammer, Quantum Transport Theory (CRC Press, New\nYork, 2018).\n47G. D. Mahan, Many-particle physics (Springer Science &\nBusiness Media, 2013).\n48I. Ado, I. Dmitriev, P. Ostrovsky, and M. Titov, EPL 111,\n37004 (2015)."    },    {        "title": "1705.09659v1.Role_of_the_spin_orbit_coupling_in_the_Kugel_Khomskii_model_on_the_honeycomb_lattice.pdf",        "content": "arXiv:1705.09659v1  [cond-mat.str-el]  26 May 2017Role of the spin-orbit coupling in the Kugel-Khomskii model on the honeycomb lattice\nAkihisa Koga, Shiryu Nakauchi, and Joji Nasu\nDepartment of Physics, Tokyo Institute of Technology, Megu ro, Tokyo 152-8551, Japan\n(Dated: May 30, 2017)\nWe study the effective spin-orbital model for honeycomb-lay ered transition metal compounds, ap-\nplyingthe second-order perturbation theory tothethree-o rbital Hubbardmodel with the anisotropic\nhoppings. This model is reduced to the Kitaev model in the str ong spin-orbit coupling limit. Com-\nbining the cluster mean-field approximations with the exact diagonalization, we treat the Kugel-\nKhomskii type superexchange interaction and spin-orbit co upling on an equal footing to discuss\nground-state properties. We find that a zigzag ordered state is realized in the model within nearest-\nneighbor interactions. We clarify how the ordered state com petes with the nonmagnetic state, which\nis adiabatically connected to the quantum spin liquid state realized in a strong spin-orbit coupling\nlimit. Thermodynamic properties are also addressed. The pr esent work should provide another\nroute to account for the Kitaev-based magnetic properties i n candidate materials.\nOrbital degrees of freedom have been studied as a cen-\ntral topic of strongly correlated electron systems as they\npossess own quantum dynamics and are strongly entan-\ngled with other degrees of freedom such as charge and\nspin [1]. Recently, multiorbital systems with strong spin-\norbit (SO) couplings have attracted considerable atten-\ntion [2, 3]. One of the intriguing examples is the series\nof the Mott insulators with honeycomb-based structures\nsuch asA2IrO3(A= Na,Li) [4–6], and β-Li2IrO3[7]. In\nthese compounds, a strong SO coupling for 5 delectrons\nlifts the triply degenerate t2glevels and the low-energy\nKramersdoublet, which is referred to as an isospin, plays\nan important role at low temperatures. Furthermore,\nanisotropic electronic clouds intrinsic in the t2gorbitals\nresult in peculiar exchange couplings and the system is\nwell described by the Kitaev model for the isospins [8, 9].\nThe ground state of this model is a quantum spin liquid\n(QSL), and hence a lot of experimental and theoretical\nworks have been devoted to the iridium oxides in this\ncontext [10–18]. Very recently, the ruthenium compound\nα-RuCl 3with 4delectrons has been studied actively as\nanother Kitaev candidate material [19–27]. In general,\nthe SO coupling in 4 dorbitals is weaker than that in\n5dorbitals and is comparable with the exchange energy.\nTherefore, it is highly desired to deal with SO and ex-\nchange couplings on an equal footing although the mag-\nnetic properties for honeycomb-layered compounds have\nbeen mainly discussed within the isospin model with the\nKitaev and other exchange couplings including longer-\nrange interactions [10, 28–32].\nIn this Letter, we study the role of the SO cou-\npling in the Mott insulator with orbital degrees of free-\ndom. We examine the localized spin-orbital model with\nthe Kugel-Khomskii type superexchange interactions be-\ntween nearest-neighbor sites and onsite SO couplings on\nthe two-dimensional honeycomb lattice. In the strong\nSO coupling limit, this model is reduced to the Kitaev\nmodel and the QSL state is realized. On the other hand,\na conventional spin-orbital ordered state may be stabi-\nlized in the small SO coupling case. To examine thecompetition between the magnetically disordered and or-\ndered states in the intermediate SO coupling region, we\nfirst use the cluster mean-field (CMF) theory [33] with\ntheexactdiagonalization(ED).Wedeterminetheground\nstatephasediagraminthemodelandclarifythatazigzag\nmagneticallyorderedstate is realizeddue to the competi-\ntion between distinct exchanges. Calculating the specific\nheat and entropy in terms of the thermal pure quantum\n(TPQ) state [34], we discuss how thermodynamic prop-\nerties characteristic of the Kitaev model appear in the\nintermediate SO coupling region.\nWe start with the three-orbital Hubbard model on the\nhoneycomb lattice. This should be appropriate to de-\nscribe the electronic state of the t2gorbitals in the com-\npoundsA2IrO3andα-RuCl 3since there exists a large\ncrystalline electric field for the dorbitals. The transfer\nintegraltbetweenthe t2gorbitalsvialigand porbitalsare\nevaluated from the Slater-Koster parameters, where the\nneighboringoctahedra consisting of six ligands surround-\ning transition metal ions share their edges. Note that the\ntransfer integrals involving one of the three t2gorbitals\nvanish due tothe anisotropicelectronicclouds[9]. We re-\nfer to this as an inactive orbital and the other orbitals as\nactive ones. These depend on three inequivalent bonds,\nwhich are schematically shown as the distinct colored\nlines in Fig. 1. Moreover, we consider the onsite intra-\nand inter-orbital Coulomb interactions, UandU′, Hund\ncoupling K, and pair hopping K′in the conventional\nmanner. In the following, we restrict our discussions to\nthe conditions U=U′+2KandK′=K, which are lead\nby the symmetry argument of the degenerate orbitals.\nWe use the second-order perturbation theory in the\nstrong coupling limit since the Mott insulating state is\nrealized in the honeycomb-layered compounds. We then\nobtainthe Kugel-Khomskii-typeexchangemodel, assum-\ning that five electrons occupy the t2gorbitals in each site.\nBy taking the SO coupling into account, the effective\nHamiltonian is explicitly given as\nH=/summationdisplay\n/angbracketleftij/angbracketrightγHex(γ)\nij−λ/summationdisplay\niLi·Si, (1)2\nwhereλis the SO coupling, and SiandLiare spin and\norbital angular-momentum operators at the ith site, re-\nspectively. The exchange Hamiltonian Hex(γ)\nij, which de-\npends on the bond γ(=x,y,z) of the honeycomb lattice(see Fig. 1), is given as\nHex(γ)\nij=H(γ)\n1;ij+H(γ)\n2;ij+H(γ)\n2;ij, (2)\nwith\nH(γ)\n1;ij= 2J1/parenleftbigg\nSi·Sj+3\n4/parenrightbigg/bracketleftbigg\nτ(γ)\nixτ(γ)\njx−τ(γ)\niyτ(γ)\njy−τ(γ)\nizτ(γ)\njz+1\n4τ(γ)\ni0τ(γ)\nj0−1\n4/parenleftBig\nτ(γ)\ni0+τ(γ)\nj0/parenrightBig/bracketrightbigg\n, (3)\nH(γ)\n2;ij= 2J2/parenleftbigg\nSi·Sj−1\n4/parenrightbigg/bracketleftbigg\nτ(γ)\nixτ(γ)\njx−τ(γ)\niyτ(γ)\njy−τ(γ)\nizτ(γ)\njz+1\n4τ(γ)\ni0τ(γ)\nj0+1\n4/parenleftBig\nτ(γ)\ni0+τ(γ)\nj0/parenrightBig/bracketrightbigg\n, (4)\nH(γ)\n3;ij=−4\n3(J2−J3)/parenleftbigg\nSi·Sj−1\n4/parenrightbigg/bracketleftbigg\nτ(γ)\nixτ(γ)\njx+τ(γ)\niyτ(γ)\njy−τ(γ)\nizτ(γ)\njz+1\n4τ(γ)\ni0τ(γ)\nj0/bracketrightbigg\n, (5)\n(a)\n(b)\nFIG. 1. Honeycomb lattice. (a) Effective cluster model with\nten sites, which are treated in the framework of the CMF\nmethod. (b) Twelve-site cluster for the TPQ states.\nwhere we follow the notation of Ref. [35], and J1=\n2t2/U[1−3K/U]−1,J2= 2t2/U[1−K/U]−1,J3=\n2t2/U[1+2K/U]−1are the exchange couplings between\nnearest neighbor spins. Here, we have newly introduced\nthe orbital pseudospin operators τ(γ)\nlwithl=x,y,z,0.\nNote that its definition depends on the direction of the\nbond (γ-bond) between the nearest neighbor pair /angbracketleftij/angbracketright.\nτ(γ)\nlis represented by the 3 ×3 matrix based on the\nthree orbitals: the 2 ×2 submatrix on the two active\norbitals is given by σl/2 forl=x,y,zand the identity\nmatrix for l= 0, and the other components for one in-\nactive orbital are zero, where σlis the Pauli matrix. We\nhere note that Hamiltonian H1enhances ferromagnetic\ncorrelations, while H2andH3lead to antiferromagnetic\ncorrelations. Therefore, spin frustration should play an\nimportant role for the ground state in the small K/U\nregion, where J1∼J2∼J3.\nWhat is the most distinct from ordinary spin-orbital\nmodelsisthatthe presentsystemdescribesnotonlyspin-\norbital orders but also the QSL state realized in the Ki-\ntaev model. When the SO coupling is absent, the system\nis reduced to the standard Kugel-Khomskii type Hamil-tonian. In the large Hund coupling case, the Hamilto-\nnianH(γ)\n1;ijis dominant. Then, the ferromagnetically or-\ndered ground state should be realized despite the pres-\nence of orbital frustration. In the smaller case of the\nHund coupling, the ground state is not trivial due to\nthe existence of spin frustration, discussed above. On\nthe other hand, in the case λ→ ∞, the SO coupling\nlifts the degeneracy at each site and the lowest Kramers\ndoublet, |˜σ/angbracketright= (|xy,σ/angbracketright ∓ |yz,¯σ/angbracketright+i|zx,¯σ/angbracketright)/√\n3, plays a\ncrucial role for low temperature properties. Then, the\nmodel Hamiltonian Eq. (1) is reduced to the exactly\nsolvable Kitaev model with the spin-1/2 isospin opera-\ntor˜S, asHeff=−˜J/summationtext\n/angbracketleftij/angbracketrightγ˜Siγ˜Sjγ(γ=x,y,z), where\n˜J[= 2(J1−J2)/3] is the effective exchange coupling [8].\nIt is known that, in this effective spin model, the QSL\nground state is realized with the spin gap. At finite\ntemperatures, a fermionic fractionalization appears to-\ngether with double peaks in the specific heat [15, 16].\nIn the following, we set the exchange coupling J1as a\nunit of energy. We then study ground-state and finite-\ntemperature properties in the spin-orbital system with\nparameters K/Uandλ/J1.\nFirst, we discuss ground state properties in the spin-\norbital model by means of the CMF method [33]. In\nthe method, the original lattice model is mapped to an\neffective cluster model, where spin and orbital corre-\nlations in the cluster can be taken into account prop-\nerly. Intercluster correlations are treated through sev-\neral mean-fields at ith site,/angbracketleftSik/angbracketright,/angbracketleftτ(γ)\nil/angbracketrightand/angbracketleftSikτ(γ)\nil/angbracketright,\nwherek=x,y,zandl=x,y,z,0. These mean-fields\naredetermined via the self-consistent conditions imposed\non the effective cluster problem. The method is compa-\nrable with the numerically exact methods if the cluster\nsize is large, and has successfully been applied to quan-\ntum spin [33, 36–38] and hard-core bosonic systems [39–\n41]. To describe some possible ordered states such as the\nzigzag and stripy states [10], we introduce two kinds of\nclustersinthehoneycomblattice, whichareshownasdis-\ntinct colors in Fig. 1(a). Using the ED method, we self-3\n 0.4 0.42 0.44 0.46 0.48 0.5\n 0.1  0.15  0.2  0.25  0.3mS\nK/U-1.43-1.42-1.41-1.4-1.39\n 0.1  0.2  0.3Eg/J1Nλ/J1=0.0\nλ/J1=0.2\nFIG.2. ThespinmomentsasafunctionoftheHundcoupling\nK/U. Solidandopencircles (squares)representtheresultsfor\nthe ferromagnetically and zigzag ordered states in the syst em\nwithλ/J1= 0.0 (0.2). The ground state energy is shown in\nthe inset.\nconsistently solve two effective cluster problems. To dis-\ncuss magnetic properties at zero temperature, we calcu-\nlate spin and orbital moments, mα\nS=|/summationtext\ni(−1)δα\ni/angbracketleftSi/angbracketright|/N\nandmα\nL=|/summationtext\ni(−1)δα\ni/angbracketleftLi/angbracketright|/N, whereNis the number of\nsites and δα\niis the phase factor for an ordered state α.\nWhenλ= 0, the spin and orbital degrees of freedom\nare decoupled. Here, we show in Fig. 2 the spin mo-\nmentsmf\nSandmz\nSfor the ferromagnetically and zigzag\norderedstates, respectively, whichareobtainedby means\nof the ten-site CMF method (CMF-10). Namely, we have\nconfirmedthatotherorderedstatessuchasantiferromag-\nnetic and stripy states are never stabilized in the present\ncalculations, and thereby we do not show them in Fig. 2.\nMeanwhile, the local orbital moment disappears in the\ncaseλ= 0. In the system with the large Hund coupling,\nthe exchange coupling J1is dominant, and the ferromag-\nnetically ordered ground state is realized with the fully-\npolarized moment mf\nS= 0.5, as shown in Fig. 2. On the\nother hand, in the smaller Kregion, the exchange cou-\nplingsJ2andJ3are comparable with J1. SinceH2and\nH3should enhance antiferromagnetic correlations, the\nferromagnetically ordered state becomes unstable. We\nfind that a zigzag magnetically ordered state is realized\nwith finite mz\nSaroundK/U∼0.12. To study the compe-\ntition between these ordered states, we show the ground\nstate energies in the inset of Fig. 2. We clearly find the\nhysteresis in the curves, which indicates the existence of\nthe first-order phase transition. By examining the cross-\ning point, we clarify that the quantum phase transition\nbetween ferromagnetically and zigzag ordered states oc-\ncurs atK/U∼0.15. In the case with K/U <0.1, due\nto strong frustration, it is hard to obtain the converged\nsolutions. This will be interesting to clarify this point in\na future investigation. 0 0.5 1 1.5\n 0  0.5  1(a) K/U=0.12\nmz\nµ\nmz\nLmz\nSmf\nµ\nmf\nL\nmf\nSm\nλ/J1 0 0.5  1 1.5  2(b) K/U=0.3\nmf\nµ\nmf\nL\nmf\nS\nλ/J1\nFIG. 3. Total magnetic moment mµ, spin moment mS,\nand orbital moment mLin the spin-orbital systems with (a)\nK/U= 0.12 and (b) K/U= 0.3.\nThe introduction of λcouples the spin and orbital de-\ngrees of freedom. The spin moments slightly decrease in\nboth states, as shown in Fig. 2. The zigzag and ferro-\nmagnetically ordered states are stable against the small\nSO coupling and the first-order transition point has little\neffectontheSOcoupling. Todiscussthestabilityofthese\nstates against the strong SO coupling, we calculate the\nspinand orbitalmomentsin the systemwith K/U= 0.12\nand 0.3, as shown in Fig. 3. The introduction of the\nSO coupling slightly decreases the spin moment, as dis-\ncussedabove. Bycontrast,theorbitalmomentisinduced\nparallel to the spin moment. Therefore, the total mag-\nnetic moment mα\nµ=|/summationtext\ni(−1)δα\ni/angbracketleft2Si+Li/angbracketright|/Nincreases.\nWhenK/U= 0.12, the zigzag ordered state becomes\nunstable and the first-order phase transition occurs to\nthe ferromagnetically ordered state at λ/J1∼0.4. Fur-\nther increase of the SO coupling decreases the total mo-\nmentmf\nµ. Finally, a jump singularity appears around\nλ/J1∼0.8(1.8) in the system with K/U= 0.12(0.3). It\nis also found that the magnetic moment is almost zero\nand each orbital is equally occupied as in the isospin\nstates|˜σ/angbracketrightin the larger SO coupling region. Therefore,\nwe believe that this state is essentially the same as the\nQSL state realized in the Kitaev model.\nBy performing similar calculations, we obtain the\nground state phase diagram, as shown in Fig. 4. The\ndisordered (QSL) state is realized in the region with\nlargeλ/J1. The ferromagnetically ordered state is re-\nalized in the region with small λ/J1and large K/U. The\ndecrease of the Hund coupling induces spin frustration,\nwhich destabilizes the ferromagnetically ordered state.\nWe wish to note that the zigzag ordered state is stable in\nthe small SO coupling region, which is not directly taken\ninto account in the Kitaev model.\nNext, we discuss thermodynamic properties in the sys-4\n 0  0.5 1  1.5 2 \n 0.1  0.15  0.2  0.25  0.3 Ferro \nZigzagDisorder λ/J 1\nK/U \nFIG. 4. The ground state phase diagram of the spin-orbital\nmodel. Transition points are obtained by the CMF-10.\ntem. It is known that, in the Kitaev limit ( λ→ ∞), the\nexcitations are characterized by two energy scales, which\ncorrespond to localized and itinerant Majoranafermions.\nThis clearly appears in the specific heat as two peaks at\nT/˜J= 0.012 and 0 .38 [16]. To clarify how the double\npeak structure appears in the intermediate SO coupling\nregion, we make use of the TPQ state for the twelve-site\ncluster with the periodic boundary condition [see Fig.\n1(b)]. According to the previous study [30], the dou-\nble peak structure appears in the spin-1/2 Kitaev model\neven with the twelve-site cluster. Therefore, we believe\nthat thermodynamic properties in the system can be dis-\ncussed, at least, qualitatively in our calculations.\nHere, we fix the Hund coupling as K/U= 0.3 to dis-\ncuss finite temperature properties in the system with the\nintermediate SO coupling. Figure 5 shows the specific\nheat and entropy in the system with λ/J1= 0,1,2,4 and\n10. In this calculation, the quantities are deduced by the\nstatistical average of the results obtained from, at least,\ntwenty independent TPQ states. When λ= 0, we find a\nbroad peak around T/J1= 0.4 in the curve of the spe-\ncific heat. In addition, most of the entropy is released\natT/J1∼0.1, as shown in Fig. 5(b). This can be ex-\nplained by the fact that ferromagnetic correlations are\nenhanced and spin degrees of freedom are almost frozen.\nThe appearance of the large residual entropy should be\nan artifact in the small cluster with the orbital frustra-\ntion. The introduction of the SO coupling leads to in-\nteresting behavior. It is clearly found that the broad\npeak shifts to higher temperatures. This indicates the\nformation of the Kramers doublet and a part of the en-\ntropyS= log(6) −log(2) is almost released, as shown in\nFig. 5(b). In addition, we find in the case λ/J1≥2, two\npeaksinthespecificheatatlowertemperatures. Thecor-\nresponding temperatures are little changed by the mag-\nnitude of the SO coupling and the curves are quantita-\ntively consistent with the results for the isospin Kitaev\nmodel on the twelve sites, which are shown as dashed 0 0.5 1 1.5\n 0.01  0.1  1  10  100log 2log 6(b)\nS\nT/J1 0 0.5 1(a)\nCλ/J1=0\nλ/J1=1\nλ/J1=2\nλ/J1=4\nλ/J1=10\nFIG. 5. The specific heat (a) and entropy (b) as a function\nof the temperature for the system with λ/J1= 0,1,2,4, and\n10. Dashed lines represent the results for the isospin Kitae v\nmodel with twelve sites.\nlines. Therefore, we believe that the Kitaev physics ap-\npears in the region. On the other hand, when λ/J1= 1,\na single peak structure appears in the specific heat, indi-\ncating that the Kitaev physics is hidden by the formation\nof the Kramers doublet due to the competition between\nthe exchange interaction and SO coupling. We have used\nthe TPQ states to clarify how the double peak structure\ninherent in the Kitaev physics appears, in addition to the\nbroad peak for the formation of the Kramers doublet at\nhigher temperatures.\nTo conclude, we have studied the effective spin-orbital\nmodel obtained by the second-order perturbation the-\nory. Combining the CMF theory with the ED method,\nwe have treated the Kugel-Khomskii type superexchange\ninteraction and SO coupling on an equal footing to de-\ntermine the ground-state phase diagram. We have clar-\nified how the magnetically ordered state competes with\nthe nonmagnetic state, which is adiabatically connected\nto the QSL state realized in a strong SO coupling limit.\nParticularly, we have revealed that a zigzag orderedstate\nis realized in this effective spin-orbital model with finite\nSO couplings. The present study suggests another mech-\nanism to stabilize the zigzag ordered phase close to the\nQSL in the plausible situation, and also will stimulate\nfurther experimental studies in the viewpoint of the SO\ncoupling effect on magnetic properties in Kitaev candi-\ndate materials.5\nThe authors would like to thank S. Suga for valuable\ndiscussions. Parts of the numerical calculations are per-\nformed in the supercomputing systems in ISSP, the Uni-\nversity of Tokyo. This work was partly supported by the\nGrant-in-Aid for Scientific Research from JSPS, KAK-\nENHI Grant Number JP17K05536, JP16H01066 (A.K.),\nand JP16K17747, JP16H00987 (J.N.).\n[1] Y. Tokura and N. Nagaosa, Science 288, 462 (2000).\n[2] W. Witczak-Krempa, G. Chen, Y. Baek Kim, and L.\nBalents, Ann. Rev. Condens. Matter Phys. 5, 57 (2014).\n[3] H. Watanabe, T. Shirakawa, and S. Yunoki, Phys. Rev.\nLett.105, 216410 (2010).\n[4] Y. Singh and P. Gegenwart, Phys. Rev. B 82, 064412\n(2010).\n[5] Y. Singh, S. Manni, J. Reuther, T. Berlijn, R. Thomale,\nW. Ku, S. Trebst, and P. Gegenwart, Phys. Rev. Lett.\n108, 127203 (2012).\n[6] R. Comin, G. Levy, B. Ludbrook, Z.-H. Zhu, C. N. Veen-\nstra, J. A. Rosen, Y. Singh, P. Gegenwart, D. Stricker,\nJ. N. Hancock, D. van der Marel, I. S. Elfimov, and A.\nDamascelli, Phys. Rev. Lett. 109, 266406 (2012).\n[7] T. Takayama, A. Kato, R. Dinnebier, J. Nuss, H. Kono,\nL. S. I. Veiga, G. Fabbris, D. Haskel, and H. Takagi,\nPhys. Rev. Lett. 114, 077202 (2015).\n[8] A. Kitaev, Ann. Phys. 321, 2 (2006).\n[9] G. Jackeli and G. Khaliullin, Phys. Rev. Lett. 102,\n017205 (2009).\n[10] J. Chaloupka, G. Jackeli, and G. Khaliullin, Phys. Rev.\nLett.110, 097204 (2013).\n[11] V. M. Katukuri, S. Nishimoto, V. Yushankhai, A. Stoy-\nanova, H.Kandpal, S.Choi, R.Coldea, IRousochatzakis,\nL. Hozoi, and J. van den Brink, New J. Phys. 16, 013056\n(2014).\n[12] J. Knolle, D. L. Kovrizhin, J. T. Chalker, and R. Moess-\nner, Phys. Rev. Lett. 112, 207203 (2014).\n[13] Y. Yamaji, Y. Nomura, M. Kurita, R. Arita, and M.\nImada, Phys. Rev. Lett. 113, 107201 (2014).\n[14] J. Knolle, G.-W. Chern, D. L. Kovrizhin, R. Moessner,\nand N. B. Perkins, Phys. Rev. Lett. 113, 187201 (2014).\n[15] J. Nasu, M. Udagawa, and Y. Motome, Phys. Rev. Lett.\n113, 197205 (2014).\n[16] J. Nasu, M. Udagawa, and Y. Motome, Phys. Rev. B 92,\n115122 (2015).\n[17] T. Suzuki, T. Yamada, Y. Yamaji, and S. I. Suga, Phys.\nRev. B92, 184411 (2015).\n[18] J. Yoshitake, J. Nasu, and Y. Motome, Phys. Rev. Lett.\n117, 157203 (2016).\n[19] K. W. Plumb, J. P. Clancy, L. J. Sandilands, V. V.\nShankar, Y. F. Hu, K. S. Burch, H.-Y. Kee, and Y.-J.\nKim, Phys. Rev. B 90, 041112 (2014).[20] Y. Kubota, H. Tanaka, T. Ono, Y. Narumi, and K.\nKindo, Phys. Rev. B 91, 094422 (2015).\n[21] J. A. Sears, M. Songvilay, K. W. Plumb, J. P. Clancy, Y.\nQiu, Y. Zhao, D. Parshall, and Y.-J. Kim, Phys. Rev. B\n91, 144420 (2015).\n[22] M. Majumder, M. Schmidt, H. Rosner, A. A. Tsirlin,\nH. Yasuoka, and M. Baenitz, Phys. Rev. B 91, 180401\n(2015).\n[23] L. J. Sandilands, Y. Tian, K. W. Plumb, Y.-J. Kim, and\nK. S. Burch, Phys. Rev. Lett. 114, 147201 (2015).\n[24] R. D. Johnson, S. C. Williams, A. A. Haghighirad, J.\nSingleton, V. Zapf, P. Manuel, I. I. Mazin, Y. Li, H. O.\nJeschke, R. Valent´ ı, and R. Coldea, Phys. Rev. B 92,\n235119 (2015).\n[25] J. Nasu, J. Knolle, D. L. Kovrizhin, Y. Motome, and R.\nMoessner, Nat. Phys. 12, 912 (2016).\n[26] A. Koitzsch, C. Habenicht, E. M¨ uller, M. Knupfer, B.\nB¨ uchner, H. C. Kandpal, J. van den Brink, D. Nowak,\nA. Isaeva, and T. Doert, Phys. Rev. Lett. 117, 126403\n(2016).\n[27] L. J. Sandilands, Y. Tian, A. A. Reijnders, H.-S. Kim,\nK. W. Plumb, Y.-J. Kim, H.-Y. Kee, and K. S. Burch,\nPhys. Rev. B 93, 075144 (2016).\n[28] H.-S. Kim, Vilay Shankar V., A. Catuneanu, and H.-Y.\nKee, Phys. Rev. B 91, 241110 (2015).\n[29] A. Banerjee, C. A. Bridges, J.-Q. Yan, A. A. Aczel, L. Li,\nM. B. Stone, G. E. Granroth, M. D. Lumsden, Y. Yiu, J.\nKnolle, S. Bhattacharjee, D. L. Kovrizhin, R. Moessner,\nD. A. Tennant, D. G. Mandrus, and S. E. Nagler, Nat.\nMater.15, 733 (2016).\n[30] Y. Yamaji, T. Suzuki, T. Yamada, S. I. Suga, N.\nKawashima, and M. Imada, Phys. Rev. B 93, 174425\n(2016).\n[31] S.-H. Do, S.-Y. Park, J. Yoshitake, J. Nasu, Y. Motome,\nY. S. Kwon, D. T. Adroja, D. J. Voneshen, K. Kim,\nT.-H. Jang, J.-H. Park, K.-Y. Choi, and S. Ji, preprint\narXiv:1703.01081.\n[32] S. M. Winter, K. Riedl, A. Honecker, and R. Valenti,\npreprint arXiv:1702.08466.\n[33] T. Oguchi, Prog. Theor. Phys. 13, 148 (1955).\n[34] S. Sugiura and A. Shimizu, Phys. Rev. Lett. 108, 240401\n(2012); Phys. Rev. Lett. 111, 010401 (2013).\n[35] G. Khaliullin, Prog. Theor. Phys. 160155 (2005).\n[36] T. Sakai and M. Takahashi, J. Phys. Soc. Jpn. 58, 3131\n(1989).\n[37] A. Kawaguchi, A. Koga, K. Okunishi, and N. Kawakami,\nPhys. Rev. B 65, 214405 (2002).\n[38] D. Yamamoto, Phys. Rev. B 79, 144427 (2009).\n[39] S. R. Hassan, L. de Medici, and A.-M. S. Tremblay, Phys.\nRev. B76, 144420 (2007).\n[40] D. Yamamoto, I. Danshita, and C. A. R. S´ a de Melo,\nPhys. Rev. A 85, 021601 (2012).\n[41] R. Suzuki and A. Koga, JPS Conf. Proc. 3, 016005\n(1014); J. Phys. Soc. Jpn. 83, 064003 (2014)."    },    {        "title": "1803.05158v2.Inter_orbital_topological_superconductivity_in_spin_orbit_coupled_superconductors_with_inversion_symmetry_breaking.pdf",        "content": "arXiv:1803.05158v2  [cond-mat.supr-con]  31 May 2018Inter-orbital topological superconductivity in spin-orb it coupled superconductors with\ninversion symmetry breaking\nYuri Fukaya,1Shun Tamura,1Keiji Yada,1Yukio Tanaka,1Paola Gentile,2and Mario Cuoco2\n1Department of Applied Physics, Nagoya University, Nagoya 4 64-8603, Japan\n2SPIN-CNR, I-84084 Fisciano (Salerno), Italy and Dipartime nto di Fisica ”E. R. Caianiello”,\nUniversitá di Salerno, I-84084 Fisciano (Salerno), Italy\nWe study the superconducting state of multi-orbital spin-o rbit coupled systems in the presence of\nan orbitally driven inversion asymmetry assuming that the i nter-orbital attraction is the dominant\npairing channel. Although the inversion symmetry is absent , we show that superconducting states\nthat avoid mixing of spin-triplet and spin-singlet configur ations are allowed, and remarkably, spin-\ntriplet states that are topologically nontrivial can be sta bilized in a large portion of the phase\ndiagram. The orbital-dependent spin-triplet pairing gene rally leads to topological superconductivity\nwith point nodes that are protected by a nonvanishing windin g number. We demonstrate that the\ndisclosed topological phase can exhibit Lifshitz-type tra nsitions upon different driving mechanisms\nand interactions, e.g., by tuning the strength of the atomic spin-orbit and inversion asymmetry\ncouplings or by varying the doping and the amplitude of order parameter. Such distinctive signatures\nof the nodal phase manifest through an extraordinary recons truction of the low-energy excitation\nspectra both in the bulk and at the edge of the superconductor .\nI. INTRODUCTION\nSpin-triplet pairing is at the core of intense investiga-\ntion especially because of its foundational aspect in un-\nconventional superconductivity1–4and owing to its tight\nconnection with the occurrence of topological phases with\nzero-energy surface Andreev bound states5–10marked by\nMajorana edge modes11–19. Some of the fundamental\nessences of topological spin-triplet superconductivity a re\nbasically captured by the Kitaev model20and its gener-\nalized versions where non-Abelian states of matter and\ntheir employment for topological quantum computation\ncan be demonstrated20–24. Another remarkable element\nof odd-parity superconductivity is given by the poten-\ntial of having active spin degrees of freedom making\nsuch states of matter also appealing for superconduct-\ning spintronics applications based on spin control and\ncoherent spin manipulation of Cooper pairs25–31. The\ninterplay of magnetism and spin-triplet superconductiv-\nity can manifest within different unconventional physical\nscenarios, such as the case of the emergent spin-orbital\ninteraction between the superconducting order parame-\nter and interface magnetization32,33, the breakdown of\nthe bulk-boundary correspondence34, and the anomalous\nmagnetic35,36and spin-charge current37effects occurring\nin the proximity between chiral or helical p-wave and\nspin-singlet superconductors. Achieving spin-triplet ma -\nterials platforms, thus, sets the stage for the development\nof emergent technologies both in nondissipative spintron-\nics and in the expanding area of quantum devices.\nAlthough embracing strong promises, spin-triplet su-\nperconductivity is quite rare in nature and the mecha-\nnisms for electron pairs gluing are not completely set-\ntled. The search for spin-triplet superconductivity has\nbeen performed along different routes. For instance, sci-\nentific exploration has been focused on the regions of the\nmaterials phase diagram that are in proximity to ferro-magnetic quantum phase transitions38,39, as in the case\nof heavy fermions superconductivity, i.e., UGe2, URhGe,\nandUIr2, or in materials on the verge of a magnetic in-\nstability, e.g., ruthenates2,40.\nAnother remarkable route to achieve spin-triplet pair-\ning relies on the presence of a source of inversion symme-\ntry breaking, both at the surface/interface and in the\nbulk, or alternatively, in connection with noncollinear\nmagnetic ordering41–51. Paradigmatic examples along\nthese directions are provided by quasi-one-dimensional\nheterostructures whose interplay of inversion and time-\nreversal symmetry breaking or noncollinear magnetism\nhave been shown to convert spin-singlet pairs into spin-\ntriplet ones and in turn to topological phases48,52–55.\nSimilar mechanisms and physical scenarios are also en-\ncountered at the interface between spin-singlet super-\nconductors and inhomogeneous ferromagnets with even\nand odd-in time spin-triplet pairing that are generally\ngenerated25. Semimetals have also been indicated as fun-\ndamental building blocks to generate spin-triplet pairing\nas theoretically proposed and demonstrated in topologi-\ncal insulators interfaced with conventional superconduc-\ntors or by doping Dirac/Weyl phases56, e.g., in the case\nof Cu-doped Bi2Se357–63in anti-perovskites materials64,\nas well as Cd 3As265,66.\nGenerally, there are two fundamental interactions to\ntake into account in inversion asymmetric microscopic\nenvironments: i) the Rashba spin-orbit coupling67due\nto inversion symmetry breaking at the surface or inter-\nface in heterostructures, and ii) the Dresselhaus coupling\narising from the inversion asymmetry in the bulk of the\nhost material68. For the present analysis, it is worth\nnoting that typically in multi-orbital materials, it is the\ncombination of the atomic spin-orbit interaction with the\ninversion symmetry-breaking sources that effectively gen-\nerates both Rashba and Dresselhaus emergent interac-\ntions within the electronic manifold close to the Fermi\nlevel. Another general observation is that the lack of in-2\nversion symmetry is expected to lead to a parity mixing\nof spin-singlet and spin-triplet configurations69–71with\nan ensuing series of unexpected features ranging from\nanomalous magneto-electric72effects to unconventional\nsurface states73, topological phases74–76, and non-trivial\nspatial textures of the spin-triplet pairs77. Such symme-\ntry conditions in intrinsic materials are, however, funda-\nmentally linked to the momentum dependent structure\nof the superconducting order parameter. In contrast,\nwhen considering multi-orbital systems, more channels\nare possible with emergent unconventional paths for elec-\ntron pairing that are expected to be strongly tied to the\norbital character of the electron-electron attraction and\nof the electronic states close to the Fermi level.\nOrbital degrees of freedom are key players in quantum\nmaterials when considering the degeneracy of d-bands of\nthe transition elements not being completely removed by\nthe crystal distortions or due to the intrinsic spin-orbita l\nentanglement78triggered by the atomic spin-orbit cou-\npling. In this context, a competition of different and\ncomplex types of order is ubiquitous in realistic mate-\nrials, such as transition metal oxides, mainly owing to\nthe frustrated exchange arising from the active orbital\ndegrees of freedom. Such scenarios are commonly en-\ncountered in materials where the atomic physics plays a\nsignificant role in setting the character of the electronic\nstructure close to the Fermi level. As the d-orbitals have\nan anisotropic spatial distribution, the nature of the elec -\ntronic states is also strongly dependent on the system’s\ndimensionality. Indeed, two-dimensional (2D) confined\nelectron liquids originating at the interface or surface of\nmaterials generally manifest a rich variety of spin-orbita l\nphenomena79. Along this line, understanding how elec-\ntron pairing is settled in quantum systems exhibiting a\nstrong interplay between orbital degrees of freedom and\ninversion symmetry breaking represents a fundamental\nproblem in unconventional superconductivity, and it can\nbe of great relevance for a large class of materials.\nIn this study, we investigate the nature of the super-\nconducting phase in spin-orbit coupled systems in the\nabsence of inversion symmetry assuming that the inter-\norbital attractive channel is dominant and sets the elec-\ntrons pairing. We demonstrate that the underlying in-\nversion symmetry breaking leads to exotic spin-triplet\nsuperconductivity. Isotropic spin-triplet pairing config -\nurations, without any mixing with spin-singlet, gener-\nally occur among the symmetry allowed solutions and are\nshown to be the ground-state in a large part of the pa-\nrameters space. We then realize an isotropic spin-triplet\nsuperconductor whose orbital character can make it topo-\nlogically non trivial. Remarkably, the topological phase\nexhibits an unconventional nodal structure with unique\ntunable features. An exotic fingerprint of the topologi-\ncal phases is that the number and k-position of nodes can\nbe controlled by doping, orbital polarization, through the\ncompetition between spin-orbit coupling and lattice dis-\ntortions, and temperature (or equivalently, the amplitude\nof the order parameter).The paper is organized as follows. In Sec. II, we in-\ntroduce the model Hamiltonian and present the classi-\nfication of the inter-orbital pairing configurations with\nrespect to the point-group and time-reversal symmetries.\nSection III is devoted to an analysis of the stability of the\nvarious orbital entangled superconducting states and the\nenergetics of the isotropic superconducting states. Sec-\ntion IV focuses on the electronic spectra of the energet-\nically most favorable phases and the ensuing topological\nconfigurations both in the bulk and at the boundary. Fi-\nnally, in Sec. V, we provide a discussion of the results\nand few concluding remarks.\nII. MODEL AND SYMMETRY\nCLASSIFICATION OF SUPERCONDUCTING\nPHASES WITH INTER-ORBITAL PAIRING\nOne of the most common crystal structures of transi-\ntion metal oxides is the perovskite structure, with tran-\nsition metal (TM) elements surrounded by oxygen (O)\nin an octahedral environment. For cubic symmetry, ow-\ning to the crystal field potential generated by the oxy-\ngen around the TM, the fivefold orbital degeneracy is\nremoved and dorbitals split into two sectors: t2g, i.e.,\nyz,zx, andxy, andeg, i.e.,x2−y2and3z2−r2. In the\npresent study, the analysis is focused on two-dimensional\n(2D) electronic systems with broken out-of-plane inver-\nsion symmetry and having only the t2gorbitals (Fig. 1)\nclose to the Fermi level to set the low energy excitations.\nFor highly symmetric TM-O bonds, the three t2gbands\nare directional and basically decoupled, e.g., an electron\nin thedxyorbital can only hop along the yorxdirection\nthrough the intermediate pxorpyorbitals. Similarly, the\ndyzanddzxbands are quasi-one-dimensional when con-\nsidering a 2D TM-O bonding network. Furthermore, the\natomic spin-orbit interaction (SO) mixes the t2gorbitals\nthus competing with the quenching of the orbital angu-\nlar momentum due to the crystal potential. Concerning\nthe inversion asymmetry, we consider microscopic cou-\nplings that arise from the out-of-plane oxygen displace-\nments around the TM. Indeed, by breaking the reflection\nsymmetry with respect to the plane placed in between\nthe TM-O bond80, a mixing of orbitals that are even and\nodd under such a transformation is generated. Such crys-\ntal distortions are much more relevant and pronounced in\n2D electron gas forming at the interface of insulating po-\nlar and nonpolar oxide materials or on their surface and\nthey result in the activation of an effective hybridization,\nwhich is odd in space, of dxyanddyzordzxorbitals\nalong theyorxdirections, respectively. Although the\npolar environment tends to amplify the out-of-plane oxy-\ngen displacements with respect to the position of the TM\nion, such types of distortions can also occur at the inter-\nface of nonpolar oxides and in superlattices81.\nThus, the model Hamiltonian, including the t2ghop-\nping connectivity, the atomic spin-orbit coupling, and the3\ninversion symmetry breaking term, reads as\nH=/summationdisplay\nkˆC(k)†H(k)ˆC(k), (1)\nH(k) =H0(k)+HSO(k)+His(k), (2)\nwhereˆC†(k) =/bracketleftBig\nc†\nyz↑k,c†\nzx↑k,c†\nxy↑k,c†\nyz↓k,c†\nzx↓k,c†\nxy↓k/bracketrightBig\nis\na vector whose components are associated with the elec-\ntron creation operators for a given spin σ[σ= (↑,↓)],\norbitalα[α= (xy,yz,zx )], and momentum kin the\nBrillouin zone. In Fig. 1(a), we report a schematic illus-\nFIG. 1. (a) dyz,dzx, anddxy-orbitals with L= 2orbital angu-\nlar momentum. (b) Schematic image of the orbital dependent\nhopping amplitudes for εyz,εxy, and the orbital connectivity\nassociated with the inversion asymmetry term ∆is. Here, we\ndo not explicitly indicate the intermediate p-orbitals of the\noxygen ions surrounding the transition metal element that e n-\nter the effective d−dhopping processes. εzxis obtained from\nεyzby rotating π/2aroundz-axis.∆iscorresponds to the\nodd-in-space hopping amplitude from dxytodzxalong they-\ndirection. Similarly, the odd-in-space hopping amplitude from\ndxytodyzalong thex-direction is obtained by π/2rotation\naround the z-axis. (c) Sketch of the orbital mixing through\nthe spin-orbit coupling term in the Hamiltonian. σdenotes\nthe spin state, and ¯σis the opposite spin of σ.∆tgives the\nlevel splitting between dxy-orbital and dyz/dzx-orbitals. (d)\nSchematic illustration of inter-orbital interaction.\ntration of the local orbital basis for the t2gstates.H0(k),\nHSO(k), andHis(k)indicate the kinetic term, the spin-\norbit interaction, and the inversion symmetry breaking\nterm, respectively. In the spin-orbital basis, H0(k)isgiven by\nH0(k) =−µ/bracketleftBig\nˆl0⊗ˆσ0/bracketrightBig\n+ ˆεk⊗ˆσ0, (3)\nˆεk=\nεyz0 0\n0εzx0\n0 0εxy\n,\nεyz= 2t1(1−cosky)+2t3(1−coskx),\nεzx= 2t1(1−coskx)+2t3(1−cosky),\nεxy= 4t2−2t2coskx−2t2cosky+∆t,\nwhereˆl0andˆσ0are the unit matrices in orbital and spin\nspace, respectively. Here, µis the chemical potential, and\nt1,t2, andt3are the orbital dependent hopping ampli-\ntudes as schematically shown in Fig. 1(b). ∆tdenotes\nthe crystal field potential owing to the symmetry lower-\ning from cubic to tetragonal symmetry. The symmetry\nreduction yields a level splitting between dxyorbital and\ndyz/dzxorbitals.HSO(k)denotes the atomic L·Sspin-\norbit coupling,\nHSO(k) =λSO/bracketleftBig\nˆlx⊗ˆσx+ˆly⊗ˆσy+ˆlz⊗ˆσz/bracketrightBig\n,(4)\nwithˆσi(i=x,y,z)being the Pauli matrix in spin space.\nIn order to write down the L·Sinteraction, it is conve-\nnient to introduce the matrices ˆlx,ˆlyandˆlz, which are\nthe projections of the L= 2angular momentum operator\nonto thet2gsubspace, i.e.,\nˆlx=\n0 0 0\n0 0i\n0−i0\n, (5)\nˆly=\n0 0−i\n0 0 0\ni0 0\n, (6)\nˆlz=\n0i0\n−i0 0\n0 0 0\n, (7)\nassuming {(dyz,dzx,dxy)}as orbital basis. Finally, as\nmentioned above, the breaking of the mirror plane in be-\ntween the TM-O bond, due to the oxygen displacements,\nleads to an inversion symmetry breaking term His(k)of\nthe type\nHis(k) = ∆is/bracketleftBig\nˆly⊗ˆσ0sinkx−ˆlx⊗ˆσ0sinky/bracketrightBig\n.(8)\nThis contribution gives an inter-orbital process, due to\nthe broken inversion symmetry, that mixes dxyanddyz\nordzxalongxoryspatial directions [Fig. 1(b)]. Hisre-\nsembles a Rashba-type Hamiltonian that, however, cou-\nples the momentum to the orbital angular momentum\nrather than the spin. Its origin is due to distortions or\nother sources of inversion symmetry breaking that lead\nto local asymmetries deforming the orbital lobes and in\nturn antisymmetric hopping terms within the orbitals in4\nthet2gsector. In this respect, it is worth pointing out\nthat it is the combination of the local spin-orbit cou-\npling and the antisymmetric inversion symmetry inter-\naction that leads to a nontrivial momentum dependent\nspin-orbital splitting. While the original Rashba effect67\nfor the single-band system describes a linear spin split-\nting and is typically very small, the multi-band char-\nacter of the model Hamiltonian yields a more complex\nspin-orbit coupled structure with significant splitting82.\nIndeed, near the Γpoint of the Brillouin zone, one can\nhave a linear spin splitting with respect to the momentum\nfor the lowest energy bands, but a cubiclike splitting in\nmomentum83,84for the intermediate ones with enhanced\nanomalies when the filling is close to the transition from\ntwo to four Fermi surfaces. The Rashba-like effects due\nto the combined atomic spin-orbit coupling and the or-\nbitally driven inversion-symmetry term can be influenced\nby the application of an external electric field (e.g., via\ngating) in a dual way. On one hand, the gating directly\nmodifies the filling concentration and, on the other, it\ncan affect the deformation of the orbital lobes by chang-\ning the amplitude of the polar distortion85,86.\nIn this paper, we set t1=t2≡tas a unit of energy\nfor convenience and clarity of presentation. The analysis\nis performed for a representative set of hopping param-\neters, i.e.,t3/t= 0.10and∆t/t=−0.50. The primary\nreason for the choice of the electronic parameters is that\nwe aim to model superconductivity in transition-metal\nbased layered materials with low electron concentration\nin thet2gsector at the Fermi level both in the presence of\natomic spin-orbit and inversion symmetry breaking cou-\nplings. In this framework, the set of selected parameters\nis representative of a general physical regime where the\nhierarchy of the energy scales is such that ∆t>∆is>λSO\nand∆t∼t. The choice of this regime is also motivated\nby the fact that this relation can be generally encoun-\ntered in 3d(or4d) layered oxides or superlattices in the\npresence of tetragonal distortions with flat octahedra and\ninterface driven inversion-symmetry breaking potential.\nFor instance, in the case of the two-dimensional elec-\ntron gas (2DEG) forming at the interface of two band\ninsulators [e.g., the n-type 2DEG in LaAlO 3/SrTiO 387\n(LAO/STO)] or the 2DEG at the surface of a band in-\nsulator [e.g., in SrTiO 3(STO)], the energy scales for\nthe electronic parameters, as given by abinitio80,83,88or\nspectroscopic studies89, are such that the bare ∆t∼50-\n100meV,∆is∼20meV, andλSO∼10meV, while the\neffective main hopping amplitudes (i.e., t) can be in the\n200-300meV range. Similar electronic energies can be\nalso encountered in 4dlayered oxides. Slight variations\nof these parameters are expected; however, they do not\nalter the qualitative aspects of the achieved results. We\nalso point out that our analysis is not intended for a spe-\ncific material case and that variations in the amplitude\nof the electronic parameters that keep the indicated hi-\nerarchy do not alter the qualitative outcome and do not\nlead to significant changes in the results.\nThe electronic structure of the examined model sys-−0.5−0.2500.250.5\nΓ Μ XE / t\nΜ0 ππ\n−π0 ΓΜ\nXY\nkxky\n0 ππ\n−π0 ΓΜ\nXY\nkxky\n0 ππ\n−π0 ΓΜ\nXY\nkxky(a) (b)\n(c) (d)(b)(c)(d)\nFIG. 2. (a) Band structure close to the Fermi energy in the\nnormal state at λSO/t= 0.10and∆is/t= 0.20. (b)-(d) Fermi\nsurfaces at (b) µ/t=−0.25, (c)µ/t= 0.0, and (d)µ/t= 0.35.\ntem can be accessed by direct diagonalization of the ma-\ntrix Hamiltonian. Representative dispersions for λSO/t=\n0.10and∆is/t= 0.20are shown in Fig. 2(a). We ob-\nserve six non degenerate bands due to the presence of\nbothHSO(k)andHis(k). Once the dispersions are de-\ntermined, one can immediately notice that the number\nof Fermi surfaces and the structure can be varied by tun-\ning the chemical potential µ. Indeed, for µ/t=−0.25,\nµ/t= 0.0, andµ/t= 0.35one can single out all the main\npossible cases with two, four, and six Fermi sheets, as\ngiven in Figs. 2(b), (c), and (d), respectively. For the\nexplored regimes of low doping, all the Fermi surfaces\nare made of electron-like pockets centered around origin\nof the Brillouin zone ( Γ). The dispersion of the lowest\noccupied band has weak anisotropy as it has a dominant\ndxycharacter (Fig. 2(a)); moreover, moving to higher\nelectron concentrations, the outer Fermi sheets exhibit a\nhighly anisotropic profile that becomes more pronounced\nwhen the chemical potential crosses the bands mainly\narising from the dyzanddzx-orbitals.\nAfter having considered the normal state properties,\nwe concentrate on the possible superconducting states\nthat can be realized, their energetics and their topolog-\nical behavior. The analysis is based on the assumption\nthat the inter-orbital local attractive interaction is the\nonly relevant pairing channel that contributes to the for-\nmation of Cooper pairs. Then, the intra-orbital pair-\ning coupling is negligible. Such a hypothesis can be\nphysically applicable in multi-orbital systems because th e\nintra-band Coulomb interaction is typically larger than\nthe inter-band one. Indeed, in the t2grestricted sector\nthe Coulomb interaction matrix elements of low-energy5\nlattice Hamiltonian can be evaluated by employing the\nHubbard-Kanamori parametrization90in terms of U,U′\nandJH, after symmetrizing the Slater-integrals91within\nthet2gshell assuming a cubic splitting of the t2gand\negorbitals.Ucorresponds to the intra-orbital Coulomb\nrepulsion, whereas U′(withU′=U−2JHin a cu-\nbic symmetry) is the inter-orbital interaction which is\nreduced by Hund exchange, JH. Hence, one has that\nthe inter-orbital Coulomb repulsion is generally always\nsmaller than the intra-orbital one. Estimates for transi-\ntion metal oxide materials in d1,d2ord3configurations,\nbeing relevant for the t2gshell and thus for our work,\nindicate that U∼3.5eV andU′∼2.5eV92. Thus, it is\nplausible to expect that the Coulomb repulsion tends to\nfurther suppress the electron pairing that occurs within\nthe same band. In addition, in the case of having the\nelectron-phonon coupling as a source of electrons attrac-\ntion, it is shown that the effective inter- and intra-orbital\nattractive interaction can be of the same magnitude (see\nAppendix for more details).\nIn this framework, we point out that topological super-\nconductivity is proposed to occur, owing to inter-orbital\npairing, in Cu-doped Bi2Se3for an inversion symmetric\ncrystal structure60. Here, although similar inter-orbital\npairing conditions are considered, we pursue the super-\nconductivity in low-dimensional configurations, e.g., at\nthe interface of oxides, with the important constraint of\nhaving a broken inversion symmetry. Concerning the or-\nbital structure of the pairing interaction, owing to the\ntetragonal crystalline symmetry, the coupling between\nthedxy-orbital and dyz/dzx-orbital is equivalent, and\nthus one can assume that only two independent chan-\nnels of attraction are allowed, as shown in Fig. 1(d).\nIndeed,Vxydenotes the interaction between the dxyand\ndyz/dzx-orbitals, while Vzrefers to the coupling between\nthedyzanddzx-orbitals. Then, the pairing interaction is\ngiven by\nHI=Vxy/summationdisplay\ni[nxy,inyz,i+nxy,inzx,i]\n+Vz/summationdisplay\ninyz,inzx,i, (9)\nnα,i=c†\nα↑icα↑i+c†\nα↓icα↓i, (10)\nwhereidenotes the lattice site.\nA. Irreducible representation and symmetry\nclassification\nIn this subsection, we classify the inter-orbital super-\nconducting states according to the point group symme-\ntry. The system upon examination has a tetragonal sym-\nmetry associated with the point group C4v, marked by\nfour-fold rotational symmetry C4and mirror symmetries\nMyzandMzx. Based on the rotational and reflection\nsymmetry transformations, all the possible inter-orbital\nisotropic pairings can be classified into five irreduciblerepresentations of the C4vpoint group as summarized\nin Table I. For our purposes, only solutions that do not\nTABLE I. Irreducible representation of the inter-orbital\nisotropic superconducting states for the tetragonal group C4v.\nIn the columns, we report the sign of the order parameter\nupon a four-fold rotational symmetry transformation, C4, and\nthe reflection mirror symmetry Myz, as well as the explicit\nspin and orbital structure of the gap function. In the E repre -\nsentation, +and−of the subscript mean the doubly degen-\nerate mirror-even ( +) and mirror-odd ( −) solutions, respec-\ntively.\nC4vC4Myzorbital basis function\n(dxy,dyz)d(xy,yz)\ny\nA1+ + (dxy,dzx)d(xy,zx)\nx=−d(xy,yz)\ny\n(dyz,dzx)d(yz,zx)\nz\nA2+−(dxy,dyz)d(xy,yz)\nx\n(dxy,dzx)d(xy,zx)\ny=d(xy,yz)\nx\nB1−+(dxy,dyz)d(xy,yz)\ny\n(dxy,dzx)d(xy,zx)\nx=d(xy,yz)\ny\n(dxy,dyz)d(xy,yz)\nx\nB2− − (dxy,dzx)d(xy,zx)\ny=−d(xy,yz)\nx\n(dyz,dzx)ψ(yz,zx)\nE±i±(dxy,dyz)ψ(xy,yz),d(xy,yz)\nz\n(dxy,dzx)ψ(xy,zx)\n+=∓id(xy,yz)\nz+\nd(xy,zx)\nz−=∓iψ(xy,yz)\n−\n(dyz,dzx)d(yz,zx)\nx ,d(yz,zx)\ny\nbreak the time-reversal symmetry are considered and are\nreported in Table I. Then, the superconducting order\nparameter associated to bands αandβcan be classi-\nfied as an isotropic ( s-wave) spin-triplet/orbital-singlet\nd(α,β)-vector and s-wave spin-singlet/orbital-triplet with\namplitudeψ(α,β)or as a mixing of both configurations.\nWith these assumptions, one can generally describe the\nisotropic order parameter with spin-singlet and triplet\ncomponents as\nˆ∆α,β=iˆσy/bracketleftBig\nψ(α,β)+ˆσ·d(α,β)/bracketrightBig\n, (11)\nwithαandβstanding for the orbital index, and having\nfor each channel three possible orbital flavors. Further-\nmore, owing to the selected tetragonal crystal symmetry,\none can achieve three different types of inter-orbital pair-\nings. The spin-singlet configurations are orbital triplets\nand can be described by a symmetric superposition of op-\nposite spin states in different orbitals. On the other hand,\nspin-triplet components can be expressed by means of the\nfollowing d-vectors:\nd(xy,yz)=/parenleftBig\nd(xy,yz)\nx,d(xy,yz)\ny,d(xy,yz)\nz/parenrightBig\n,\nd(xy,zx)=/parenleftBig\nd(xy,zx)\nx,d(xy,zx)\ny,d(xy,zx)\nz/parenrightBig\n,\nd(yz,zx)=/parenleftBig\nd(yz,zx)\nx,d(yz,zx)\ny,d(yz,zx)\nz/parenrightBig\n,6\nwithd(α,β)indicating the spin-triplet configuration built\nwithαandβ-orbitals. In general, independently of the\norbital mixing, spin-triplet pairing can be expressed in a\nmatrix form as\n∆T=/parenleftBigg\n∆↑↑∆↑↓\n∆↓↑∆↓↓/parenrightBigg\n=/parenleftBigg\n−dx+idydz\ndzdx+idy/parenrightBigg\n,(12)\nwhere the d-vector components are related to the pair-\ning order parameter with zero spin projection along the\ncorresponding symmetry axis. The three components\ndx=1\n2(−∆↑↑+∆↓↓),dy=1\n2i(∆↑↑+∆↓↓)anddz= ∆↑↓\nare expressed in terms of the equal spin ∆↑↑and∆ ↓↓, and\nthe anti-aligned spin ∆↑↓gap functions. As the compo-\nnents of the d-vector are associated with the zero spin\nprojection of spin-triplet configuration, if the d-vector\npoints along a given direction, the parallel spin config-\nurations lie in the plane perpendicular to the d-vector\norientation. In the presence of time-reversal symmetry,\nthe superconducting order parameter should satisfy the\nfollowing relations:\n∆↓↓\nα,β=/bracketleftBig\n∆↑↑\nα,β/bracketrightBig∗\n, (13)\n∆↑↓\nα,β=−/bracketleftBig\n∆↓↑\nα,β/bracketrightBig∗\n, (14)\nwith the appropriate choice of the U(1) gauge. In addi-\ntion, the pairing order parameter has four-fold rotational\nsymmetry and mirror reflection symmetry with respect\nto theyzandzxplanes as dictated by the point group\nC4v. Thus, it has to be transformed according to the\nfollowing relations:\nC4ˆ∆Ct\n4=einπ\n2ˆ∆,\nMyzˆ∆Mt\nyz=±ˆ∆,\nwherenequals to 0for A representation, 2for B repre-\nsentation, 1, and3for E representation. Such properties\nare very important to distinguish the symmetry of the so-\nlutions obtained by the Bogoliubov-de Gennes equation.\nThe energy gap functions are then explicitly constructed\nby taking into account the corresponding irreducible rep-\nresentations. For the one-dimensional representations,\ntheA1state is given by\nd(xy,zx)\nx=−d(xy,yz)\ny,\n∆↑↑\nxy,yz= ∆↓↓\nxy,yz=id(xy,yz)\ny,\n∆↑↑\nxy,zx=−∆↓↓\nxy,zx=−d(xy,zx)\nx,\n∆↑↓\nyz,zx= ∆↓↑\nyz,zx=d(yz,zx)\nz,\nwhile for the A2representation,\nd(xy,zx)\ny=d(xy,yz)\nx,\n∆↑↑\nxy,yz=−∆↓↓\nxy,yz=−d(xy,yz)\nx,\n∆↑↑\nxy,zx= ∆↓↓\nxy,zx=id(xy,zx)\ny,theB1representation,\nd(xy,zx)\nx=d(xy,yz)\ny,\n∆↑↑\nxy,yz= ∆↓↓\nxy,yz=id(xy,yz)\ny,\n∆↑↑\nxy,zx=−∆↓↓\nxy,zx=−d(xy,zx)\nx,\nand theB2representation,\nd(xy,zx)\ny=−d(xy,yz)\nx,\n∆↑↑\nxy,yz=−∆↓↓\nxy,yz=−d(xy,yz)\nx,\n∆↑↑\nxy,zx= ∆↓↓\nxy,zx=id(xy,zx)\ny,\n∆↑↓\nyz,zx=−∆↓↑\nyz,zx=ψ(yz,zx).\nFinally, for the E representation, there are doubly degen-\nerate mirror-even (+)and mirror-odd (−)solutions:\nψ(xy,zx)\n+=∓id(xy,yz)\nz+,\nd(xy,zx)\nz−=∓iψ(xy,yz)\n−,\n∆↑↓\nxy,yz=α−ψ(xy,yz)\n−+α+d(xy,yz)\nz+,\n∆↓↑\nxy,yz=−α−ψ(xy,yz)\n−+α+d(xy,yz)\nz+,\n∆↑↓\nxy,zx=α+ψ(xy,zx)\n++α−d(xy,zx)\nz−,\n∆↓↑\nxy,zx=−α+ψ(xy,zx)\n++α−d(xy,zx)\nz−,\n∆↑↑\nyz,zx=−α−d(yz,zx)\nx−+iα+d(yz,zx)\ny+,\n∆↓↓\nyz,zx=α−d(yz,zx)\nx−+iα+d(yz,zx)\ny+,\nwhereα+andα−denote arbitrary constants for the lin-\near superposition. As a consequence of the symmetry\nconstraint and of the inter-orbital structure of the or-\nder parameter, different types of isotropic spin-triplet\nand singlet-triplet mixed configurations can be obtained.\nEqual spin-triplet and opposite spin-triplet pairings are\nmixed in the A1representation. On the other hand, in\ntheB2representation, equal spin-triplet and spin-singlet\npairings are mixed. For the A2andB1representations,\nonly equal spin-triplet pairings are allowed, and all types\nof pairings can be realized in the E representation. It\nis worth noting that A1,B2, and E representations have\npairings between all the orbitals in the yz-zxandxy-\nyz/zx channels, while A2andB1can make electron pair-\nings only in the xy-yz/zx channel, that is, by mixing\nthedxyanddyz/dzx-orbitals as shown in Table I. This\nsymmetry constraint is important when searching for the\nground-state configuration.\nIII. ENERGY GAP EQUATION AND PHASE\nDIAGRAM\nIn order to investigate which of the possible symmetry-\nallowed solutions is more stable energetically, we solve th e\nEliashberg equation within the mean field approximation\nby taking into account the multi-orbital effects near the7\ntransition temperature. The linearized Eliashberg equa-\ntion within the weak coupling approximation is given by\nΛ∆στ\nα,γ=−kBT\nNVα,γ/summationdisplay\nk′,iεmFασ,γτ(k′,iεm), (15)\nVxy,yz=Vxy,zx=Vyz,xy=Vzx,xy≡Vxy,\nVyz,zx=Vzx,yz≡Vz,\nFασ,γτ(k′,iεm) (16)\n=/summationdisplay\nβ,δ/summationdisplay\nσ′,τ′∆σ′τ′\nβ,δGσσ′\nα,β(k′,iεm)Gττ′\nγ,δ(−k′,−iεm),\nwhereΛis the eigenvalue of the linearized Eliashberg\nequation. Here, σ,τ,σ′, andτ′denote the spin\nstates and α,β,γ, andδstand for the orbital in-\ndices.Fασ,γτ(k′,iεm)is the anomalous Green’s func-\ntion. As we assume an isotropic Cooper pairing, which\nisk-independent, the summation over momentum and\nMatsubara frequency in Eq. (16) gets simplified. Fi-\nnally, the problem is reduced to the diagonalization of\nthe24×24matrix. We then study the relative stabil-\nity of the irreducible representations as listed in Table\nI. An analysis of the energetically most favorable super-\nconducting states is performed as a function of Vz/Vxy,\nassuming that Vxy/t=−1.0and for a given temperature\nT/t= 5.0×10−5. When we keep the ratio Vz/Vxy, the\neigenvalue Λis proportional to Vxywithin the mean-field\napproximation. The choice of the representative coupling\nVxy/t=−1.0is guided by the fact that one aims to ac-\ncess a physical regime for the superconducting phase that\nin principle can be compared to realistic superconduct-\ning materials in the weak coupling limit. For instance, if\none chooses t∼200−300meV, which is common in ox-\nides, and considering that the superconducting transition\ntemperature Tcis obtained when the magnitude of the\ngreatest eigenvalue gets close to 1, then one would find\nTcto be of the order of 100−300mK, which is reasonable\nfor the 2DEG superconductivity at the oxide interface.\nFigure 3 shows the superconducting phase diagram\nfor representative amplitudes of the spin-orbit coupling,\nλSO/t= 0.10, and inversion asymmetry interaction,\n∆is/t= 0.20, while varying both the chemical potential\nand the ratio of the pairing couplings Vz/Vxy. Owing to\nthe inequivalent mixing of the orbitals in the paired con-\nfigurations, it is plausible to expect a significant compe-\ntition between the various symmetry allowed states and\nthat such an interplay is sensitive not only to the pairing\norbital anisotropy, but also to the structure and the num-\nber of Fermi surfaces. A direct observation is that for Vz\nlarger than Vxy, theA1phase is stabilized with respect\nto theB1phase because it contains a d(yz,zx)\nz channel of\na spin-triplet pairing in the yz-zxsector that is absent\ninB1phase. However, such a simple deduction does not\ndirectly explain why the A1phase wins the competition\nwith other superconducting phases, e.g., the B2and E\nphases, which also can gain condensation energy by pair-\ning electrons in the yz-zxsector. As a different type0.5 0.75 1 1.25−0.3−0.2−0.100.10.20.30.4\nµ / t\nVz / Vxy# of FS = 2# of FS = 4# of FS = 6B1\nB1A1\nA1\nA1B1\nFIG. 3. Phase diagram as a function of Vz/VxyatλSO/t=\n0.10,∆is/t= 0.20,T/t= 5.0×10−5, andVxy/t=−1.0.\nThe brown solid line is the border between A1andB1states.\nThe black solid line indicates the value of the chemical po-\ntential for which the number of Fermi surfaces changes. The\nblack dotted lines correspond to the values of the chemical\npotentials used in Fig. 2 for the normal state Fermi surfaces .\nofd-vector orientation enters into the A1and E config-\nurations, yet in the B2state, theyz-zxchannel has a\nspin-singlet pairing, one can deduce that the interplay\nbetween the spin of the Copper pairs and that of the\nsingle-electron states close to the Fermi level is relevant\nto single out the most favorable superconducting phase.\nThe boundary between the A1andB1phases exhibits\na sudden variation when one tunes the chemical potential\nacross the value for which the number of Fermi surfaces\nchanges. Such an abrupt transition is, however, plau-\nsible when passing through a Lifshitz point in the elec-\ntronic structure of the normal state because other pairing\nchannels get activated at the Fermi level. The relation\nbetween the modification of the superconducting state\nand electronic topological or Lifshitz transition93that the\nFermi surface can undergo is a subject of general interest.\nIndeed, there are many theoretical studies and experi-\nmental signatures pointing to a subtle interplay of Lif-\nshitz transitions and superconductivity in cuprates94,95,\nheavy-fermion superconductors96and more recently in\niron-based superconductors97–100. In those cases, major\nchanges of the superconducting state seem to occur when\ngoing through a Lifshitz transition because Fermi pockets\ncan appear or disappear at the Fermi level and in turn\nlead to different physical effects.\nHere, along this line of investigation, the role of the\nelectron filling is also quite important and sets the com-\npetition between the energetically most stable phases.\nIndeed, one can notice that the A1(B1) phase is sta-\nbilized for higher (lower) Vz/Vxyand lower (higher) µ.\nFurthermore, we find that, in the case of two Fermi sur-\nfaces, the A1state is further stabilized by decreasing the\nchemical potential and moving to a regime of extremely\nlow concentration. On the other hand, a transition to8\ntheB1phase is achieved by electron doping. In the dop-\ning regime of four bands at the Fermi level, the A1-B1\nboundary evolves approximately as a linear function of\nVz/Vxy. This implies that the A1configuration tends\nto be less stable and a higher ratio Vz/Vxyis needed to\nachieve such a configuration at a given chemical poten-\ntial. Finally, approaching the doping regime of six Fermi\nsurfaces, the A1-B1boundary becomes independent of\nthe amplitude of µ. It is remarkable that the doping can\nsubstantially alter the competition between the A1and\nB1phases, thus manifesting the intricate consequences\nof the spin-orbital character of the electronic structure\nclose to the Fermi level.\nTo explicitly and quantitatively demonstrate the en-\nergy competition among all the symmetry allowed\nphases, one can follow the behavior of the eigenvalues\nof the linearized Eliashberg equations as a function of\nthe ratioVz/Vxy(Fig. 4). Figures. 4(a)-(c) show the\n0 0.5 1 1.5 2 2.500.511.522.5\n0 0.5 1 1.5 2 2.500.511.52\n0 0.5 1 1.5 2 2.500.20.40.60.8\nVz / VxyThe eigenvalueA1 representation\nB1 representation\n(a) (b) (c)\nThe eigenvalueB2 representationA2 representation\nE representation\nThe eigenvalue\nVz / Vxy Vz / Vxy\nFIG. 4. Evolution of the eigenvalue of the Eliashberg matrix\nequation as a function of Vz/Vxyat (a)µ/t=−0.25, (b)\nµ/t= 0.0, and (c)µ/t= 0.35atλSO/t= 0.10,∆is/t= 0.20,\nT/t= 5.0×10−5, andVxy/t=−1.0. (a)A1representation is\ndominant. (b)and(c) B1is dominant for small amplitude of\nthe ratioVz/Vxy.\neigenvalues of the Eliashberg matrix equation for all the\nirreducible representations as a function of Vz/Vxywhen\nthe number of Fermi surfaces is (a) two ( µ/t=−0.25),\n(b) four (µ/t= 0.0), and (c) six ( µ/t= 0.35) as indicated\nby the dotted lines in Fig. 3. With the increase in Vz, the\nmagnitude of the eigenvalues of the irreducible represen-\ntations including the yz-zxchannel, i.e., A1,B2, and E\nrepresentations, increases in all the cases with two, four,\nand six Fermi surfaces. On the other hand, the eigen-\nvalues of the A 2and B 1representations are independent\nofVz, asVzis irrelevant for this pairing channel. When\nthe number of Fermi surfaces is two, the A1representa-\ntion is the most dominant pairing for all Vz. Although\nthe magnitude of the eigenvalues for the B2andErep-\nresentations also increases with Vz, these solutions never\nbecome dominant as compared with the A1state. When\nthe number of Fermi surfaces is four or six, the eigenvalueof theB1phase is larger than that of the A1representa-\ntion for lower Vz.\nFinally, we have investigated the phase diagram by\nscanning a larger range of temperatures for few repre-\nsentative cases of pairing interaction and filling concen-\ntration (see Appendix). The results are not significantly\nchanged except in a region of extremely high tempera-\nture, corresponding to an unphysically large amplitude\nof the pairing interaction. There, although B 1keeps be-\ning the most stable state, the largest eigenvalues indicate\na competition between the B 1and B2rather than the B 1\nand A 1configurations.\nIV. TOPOLOGICAL PROPERTIES AND\nENERGY EXCITATION SPECTRUM IN THE\nBULK AND AT THE EDGE\nIn the previous section, we confirmed that both the\nA1andB1pairings can be energetically stabilized in a\nlarge region of the parameter space. Thus, it is relevant\nto further consider the nature of the electronic structure\nof these superconducting phases in order to provide key\nelements and indications that can be employed for the\ndetection of the most favorable inter-orbital supercon-\nductivity. The analysis is based on the solution of the\nBogoliubov-de Gennes (BdG) equation for the evalua-\ntion of the low-energy spectral excitations both in the\nbulk and at the edge of the superconductor for both the\nA1andB1phases. The matrix Hamiltonian in momen-\ntum space is given by\nHBdG(k) =/parenleftBigg\nH(k)ˆ∆\nˆ∆†−H∗(−k)/parenrightBigg\n. (17)\nwithH(k)being the normal state Hamiltonian.\nA. Bulk energy spectrum and topological\nsuperconductivity\nIn order to determine the excitation spectrum, we solve\nthe BdG equations for both the A1andB1configurations.\nFor convenience, we introduce the gap amplitude |∆0|,\nand we set the components of the d-vectors to be\nd(xy,yz)\ny=−d(xy,zx)\nx=d(yz,zx)\nz=|∆0|, (18)\nforA1and\nd(xy,yz)\ny=d(xy,zx)\nx=|∆0|, (19)\nforB1state. Here, the parameter |∆0|/t= 1.0×10−3is\nset as a scale of energy.\nWe start focusing on the doping regime of four bands\nat the Fermi level. In this case, the A1state has a fully\ngapped electronic structure for all the bands at the Fermi\nlevel as demonstrated by the inspection of the in-plane9\n10−1100101\n10−1100101\n10−1100101\n10−1100101kxkyΓπ/2\n−π/2\n−π/2 π/2Fermi surface\n00\nπ0 −π\nθπ0−π\nθgap / | ∆0|\ngap / | ∆0|\nπ0 −π\nθgap / | ∆0|\nπ0 −π\nθgap / | ∆0|(a)\n(b) (c)\n(d)(e)θ(b)\n(c)\n(d)\n(e)\nFIG. 5. (a) Fermi surfaces at µ/t= 0.0in the normal state.\n[(b)-(e)] A1quasi-particle energy gap along the Fermi surface\nas a function of the polar angle θas shown in (a) for λSO/t=\n0.10,∆is/t= 0.20, and|∆0|/t= 1.0×10−3corresponding to\nthe Fermi surfaces in (a).\nangular dependence of the gap magnitude [Figs. 5(b)-\n(e)]. In particular, we notice that the gap amplitude is\nnot isotropic and orbital dependent when moving from\nthe outer to the inner Fermi surface [Figs. 5(b)-(e)]. The\nnodal state (Fig. 6), on the other hand, exhibits a more\nregular behavior of the gap amplitude which is basically\norbital independent and point nodes occurring only along\nthe diagonal of the Brillouin zone on the various Fermi\nsurfaces.\nIt is interesting to further investigate the nature of the\nnodalB1phase by determining whether the existence of\nthe nodes is related to a non-vanishing topological invari-\nant. As the model Hamiltonian owes particle-hole and\ntime-reversal symmetry, one can define a chiral operator\nˆΓas a product of the particle-hole ˆCand time-reversal\nˆΘoperators. As the chiral symmetry operator anticom-\nmutes with HBdG(k), by employing a unitary transforma-\ntion rotating the basis in the eigenbasis of ˆΓ, the Hamilto-\nnian can be put in an off-diagonal form with antidiagonal\nblocks. Hence, the determinant of each block can be put\nin a complex polar form and, as long as the eigenvalues\nare non-zero, it can be used to obtain a winding number\nby evaluating its trajectory in the complex plane. On a10−1010−5100\n10−1010−5100\n10−1010−5100\n10−1010−5100kxky\nFermi surface1D winding number = +1π/2\n−π/2\n−π/2 π/21D winding number = −1\n00\n0−π πgap / |∆0|\nθ0 −π πgap / |∆0|\nθ\n0−π πgap / |∆0|\nθ0−π πgap / |∆0|\nθ(a)\n(b) (c)\n(d) (e)θΓ(b)\n(c)\n(d)\n(e)\nFIG. 6. (a) Fermi surfaces and position of the nodes at µ/t=\n0.0. We indicate the winding numbers defined at each node.\n(b)-(e) indicate the quasi-particle energy spectra for the B1\nstate withλSO/t= 0.10, ∆is/t= 0.20, and|∆0|/t= 1.0×\n10−3at the corresponding Fermi surfaces shown in (a).\ngeneral ground, we point out that the number of singu-\nlarities in the phase of the determinant is a topological\ninvariant101because it cannot change without the am-\nplitude going to zero, thus implying a gap closing and\na topological phase transition. For this symmetry class,\nthen, one can associate and determine the winding num-\nber around each node by following, for instance, the ap-\nproach already applied successfully in Refs. [102–104].\nThe chiral, particle-hole, and time-reversal operators ar e\nexpressed as\nˆΓ =−iˆCˆΘ, (20)\nˆC=/parenleftBigg\n0ˆI6×6\nˆI6×60/parenrightBigg\n=ˆl0⊗ˆσx⊗ˆτ0, (21)\nˆΘ =ˆl0⊗iˆσy⊗ˆτ0. (22)\nHere,ˆI6×6andˆτ0denote the 6×6unit matrix and the\nidentity matrix in the particle-hole space, respectively.\nAs we consider time-reversal symmetric pairings, the chi-\nral operator anticommutes with the Hamiltonian:\n{HBdG(k),ˆΓ}= 0. (23)\nOne can then introduce a unitary matrix ˆUΓthat diago-10\n0.5 0.6 0.70.50.60.7\n0π / 2\nπ / 2 kxky\nkxky C(a) (b)\n+1 : \n−1 : \nFIG. 7. (a) Fermi surfaces at λSO/t= 0.10,∆is/t= 0.20,\nµ/t= 0.0, and point nodes position (winding number) at\n∆0/t= 1.0×10−3. (b) Zoomed view of the plot in (a) and a\ncontour of the integral C.\nnalizes the chiral operator ˆΓ:\nˆU†\nΓˆΓˆUΓ=/parenleftBigg\nˆI6×60\n0−ˆI6×6/parenrightBigg\n, (24)\nˆU†\nΓ=ˆUΓ=1√\n2/parenleftBigg\nˆI6×6ˆl0⊗ˆσy\nˆl0⊗ˆσy−ˆI6×6/parenrightBigg\n.(25)\nIn this basis the BdG Hamiltonian is block antidiagonal-\nized byˆUΓ,\nˆU†\nΓHBdG(k)ˆUΓ=/parenleftBigg\n0 ˆq(k)\nˆq†(k) 0/parenrightBigg\n, (26)\nˆq=H(k)/bracketleftBig\nˆl0⊗ˆσy/bracketrightBig\n−ˆ∆. (27)\nThen, the determinant of the ˆq(k)matrix block can be\nput in a complex polar form, and as long as the eigen-\nvalues are nonzero, it can be used to obtain the winding\nnumberWby evaluating its trajectory in the complex\nplane as\nW=1\n2π/contintegraldisplay\nCdθ(k), (28)\nθ(k)≡arg[det ˆq(k)].\nCin Eq. (28) is a closed line contour that encloses a\ngiven node as schematically shown in Fig. 7(b). From\nthe explicit calculation, we find that the amplitude of\nWis±1[see Figs. 6(a) and 7(a)]. If the nodes have a\nnonzero winding number, edge states appear due to the\nbulk-edge correspondence. It is known103that the follow-\ning index theorem is satisfied: for any one-dimensional\ncut in the Brillouin zone that is indicated by a given\nmomentum k/bardblthat is parallel to the edge, one has that\nw(k/bardbl) =n+−n−, withn+andn−being the number of\nthe eigenstates associated to the eigenvalues +1and−1\nof the chiral operator ˆΓ, respectively. The number of edge\nstates is equal to |w(k/bardbl)|when considering a boundary\nconfiguration with a conserved k/bardbl. We can easily show\nthatWwhich is given in Eq. (28) and w(k/bardbl)are deeply10−310−210−1−0.3−0.2−0.100.10.20.30.410−310−210−1−0.3−0.2−0.100.10.20.30.4\n10−310−210−1−0.3−0.2−0.100.10.20.30.410−310−210−1\n10−310−210−1\n0.05 0.1 0.150.10.20.3µ / t\n| ∆ 0 | / tFS = 6\nFS = 4\nFS = 2\nFS = 2FS = 4FS = 6\nn = 6(a)\n(b)\n(c) FS = # of FS\nn = # of point nodes ( Γ−M)(d)\n(e)\nn = 6 n = 4\nFS = 2FS = 4FS = 6FS = 6\nFS = 4\nFS = 2\nFS = 6\nFS = 4\nFS = 2n = 2n = 6\nn = 4µ / t\nµ / tn = 4\nn = 2\nn = 2n = 0\nn = 4 n = 2\nn = 2n = 0n = 4n = 4 n = 6n = 4\nn = 4n = 2\nn = 2\nn = 2n = 0\nn = 0n = 2n = 2\nn = 0\nn = 0\nn = 0n = 0n = 2\nn = 2n = 2n = 2n = 4n = 6 n = 4n = 2\nn = 0\nn = 0\nλSO / t∆is / t\n(a)(b)(c)\n(d) (e)(f)| ∆ 0 | / t\n| ∆ 0 | / t| ∆ 0 | / t| ∆ 0 | / t\nFIG. 8. Phase diagram of the Lifshitz transitions for the\nnodalB1phase at (a) λSO/t= 0.10and∆is/t= 0.10,\n(b)λSO/t= 0.10and∆is/t= 0.20, (c)λSO/t= 0.10and\n∆is/t= 0.30, (d)λSO/t= 0.05and∆is/t= 0.20, and (c)\nλSO/t= 0.15and∆is/t= 0.20. (f) Schematic plot of the cor-\nrespondence between the spin-orbit and the inversion asym-\nmetry couplings and the panels (a)-(e). Black and red labels\ndenote the number of Fermi surfaces in the normal state and\nthe point nodes in the superconducting phase that are locate d\nalong the diagonal of the Brillouin zone from Γto M, respec-\ntively. The black dotted line and the orange solid line indi-\ncate the two-to-four Fermi surface separation and the four- to-\nsix Fermi surface boundary in the normal state. Circles and\nsquares set the transition lines for the nodal superconduct or\nbetween configurations having different number of nodes in\nthe excitation spectrum.\nlinked:w(k1)−w(k2) =−Wsgn(k1−k2)whereWis the\ntotal winding number around the nodes between k/bardbl=k1\nandk/bardbl=k2. Thus, nonzero Won the nodes means\nnonzerow(k/bardbl)and the existence of the zero energy edge\nstate with appropriate choice of the crystal plane. Such\nrelation sets the main physical connection between the\nwinding number and the properties of the topological su-\nperconductor.\nWe generally find that two to six point nodes can oc-11\ncur along the Γ-M direction, and their number is related\nto that of the Fermi surfaces. Interestingly, the position\nof the point nodes is not fixed and pinned to the lines\nof the Fermi surface in the normal state. In general,\ntheir position along the diagonal of the Brillouin zone\ndepends on the amplitude |∆0|and indirectly on the val-\nues of the spin-orbit and inversion asymmetry couplings.\nThus, two adjacent point nodes with opposite winding\nnumbers can, in principle, be moved until they merge\nand then disappear by opening a gap in the excitation\nspectrum. This behavior is generally demonstrated in\nFig. 8. A phase diagram can be determined in terms of\nthe amplitude |∆0|and the chemical potential µ. The\nnodal superconductor can undergo different types of Lif-\nshitz transitions, and in general, those occurring in the\nnormal state are not linked to the nodal merging in the\nsuperconducting phase. Indeed, one of the characteristic\nfeatures of the nodal superconductor is that, by changing\nthe filling, through µ, one can drive a transition from two\nto four and six point nodes independently of the number\nof bands crossing the Fermi level in the normal state. It\nis rather the strength of |∆0|that plays an important\nrole in tuning the nodal superconductor. An increase in\n|∆0|tends to reduce the number of nodes until a fully\ngapped phase appears. As the critical lines are sensitive\nto the spin-orbit λSOand inversion asymmetry ∆iscou-\nplings, one can get line crossings that allow for multiple\nmerging of nodes such that the superconductor can un-\ndergo a direct transition from six to two at µ/t∼0.40\n[Figs. 8(e) and (d)] or from four to zero point nodes,\nas for instance nearby the crossing between the blue and\norange lines at µ/t∼0.10in the Fig. 8. As the positions\nof the point nodes are fixed, each Fermi surface in the\nlimit of small |∆0|and its distance in the Brillouin zone\nincreases with level splitting by ∆isandλSO, a larger\n|∆0|is required to annihilate the point nodes when both\n∆isandλSOgrow in amplitude as demonstrated by the\nshift of the green and blue critical lines in Figs. 8(a)-8(c)\nfor different values of ∆is, and Figs. 8(d), 8(b), and 8(e)\nin terms of λSO. When considering these results in the\ncontext of two-dimensional superconductors that emerge\nat the surface or interface of band insulators we observe\nthat the achieved topological transitions can be driven\nby gate voltage and temperature, as µand∆isare tun-\nable by electric fields, and the amplitude of |∆0|can be\ncontrolled by the temperature and the electric field as\nwell.\nB. Local density of states at the edge of the\nsuperconductor\nHaving established that the nodes in the B1config-\nuration are protected by a nonvanishing winding num-\nber, one can expect that flat zero-energy surface Andreev\nbound states (SABS) occur at the boundary of the su-\nperconductor.\nIn this section, we investigate the SABS and the localdensity of states (LDOS) for two different terminations of\nthe two-dimensional superconductor, i.e., the (100) and\n(110) oriented edges. We start by discussing the LDOS\nfor the (100) and (110) edges at representative values of\nλSO/t= 0.10,∆is/t= 0.20, and|∆0|/t= 1.0×10−3, and\nby varying the chemical potential in order to compare\nthe cases with a different number of point nodes in the\nbulk energy spectrum at µ/t=−0.25,µ/t= 0.0, and\nµ/t= 0.35as shown in Figs. 9(a)-9(c), respectively.\nAs expected, the momentum-resolved LDOS indicates\nthat zero-energy SABS can be observed but only for spe-\ncific orientations of the edge. Indeed, as reported in Figs.\n9(d)-9(i), one has zero-energy SABS (ZESABS) for the\n(110) boundary while they are absent for the (100) edge.\nThe reason for having inequivalent SABS edge modes is\ndirectly related to the presence of a nontrivial winding\nnumber that is protecting the point nodes. For the (110)\nedge, isolated point nodes exist in the surface Brillouin\nzone, and they have winding numbers with opposite sign.\nThus, the ZESABS, which connects the nodes with a pos-\nitive and negative winding number, emerge in the gap.\nOn the other hand, when considering the (100) oriented\ntermination, the winding numbers for positive kxand\nnegativekxare completely opposite in sign, and they\ncancel each other when projected on the (100) surface\nBrillouin zone. Thus, flat zero-energy states cannot oc-\ncur for the (100) edge. Nevertheless, helical edge modes\nare observed inside the energy gap as demonstrated in\nFig. 9(d). This is because the Majorana edge modes\nwith positive and negative chirality can couple, get split,\nand acquire a dispersion. The differences in the edge\nABS also manifest in the momentum integrated LDOS.\nFor the (110) edge, owing to the presence of the ZESABS,\nthe LDOS normalized by its normal state value at E= 0\nshows pronounced zero-energy peaks (see dash-dotted\nline in Figs. 9(j), (k), and (l)). On the other hand, for\nthe (100) boundary, they lead to a broad peak or exhibit\nmany narrow spectral structures reflecting the complex\ndispersion of the edge states.\nFinally, we discuss the |∆0|dependence of LDOS at\nzero energy, i.e., E= 0. For the (110) edge, the zero-\nenergy peak mainly originates from the zero-energy flat\nband. The height of the zero-energy peak can then be\ncharacterized by (i) the strength of the localization of the\nedge state and (ii) the total length of the ZESABS within\nthe surface Brillouin zone. The strength of the localiza-\ntion is defined by the inverse of the localization length 1/ξ\nand1/ξ∝ |∆0|. In other words, the peak height gener-\nally increases with |∆0|. On the other hand, as shown in\nFig. 8, the extension in the momentum space of the zero-\nenergy flat states becomes shorter with increasing |∆0|.\nFor simplicity, one can focus on the two Fermi surface\nconfiguration. In this case, the total length of the zero-\nenergy flat band is roughly estimated as δk(1−|∆0|/|∆c\n0|)\nfor|∆0|<|∆c\n0|and zero for |∆0|>|∆c\n0|, whereδkis the\nFermi surface splitting along the Γ-M direction and |∆c\n0|\nis a critical value above which the point-nodes disappear.\nThen, the height of the zero-energy peak is proportional12\nFIG. 9. Momentum-resolved and angular averaged LDOS for B 1representation. The Fermi surfaces and the position of the\npoint nodes are shown for (a) µ/t=−0.25, (b)µ/t= 0.0, and (c)µ/t= 0.35. The momentum ( k/bardbl) resolved LDOS at the\n(100) oriented surface for (d) µ/t=−0.25, (e)µ/t= 0.0, and (f)µ/t= 0.35. The momentum ( k/bardbl) resolved LDOS at the (110)\noriented surface for (g) µ/t=−0.25, (h)µ/t= 0.0, and (i)µ/t= 0.35. LDOS normalized by its normal state value at E= 0\n[(DOS N(E= 0)] at the (100) and (110) oriented surfaces, and in the bulk for (j)µ/t=−0.25, (k)µ/t= 0.0, and (l)µ/t= 0.35.\nThe red solid line, blue dash-dotted line, and black dashed l ine denote the LDOS at the (100) oriented surface, (110) orie nted\nsurface, and in the bulk, respectively. Other parameters ar eλSO/t= 0.10,∆is/t= 0.20, and|∆0|/t= 1.0×10−3.13\n 0 50 100 150 200 250 300 350\n10-310-210-1(110) surface\nµ/tDOSSC(E=0)/DOSN(E=0)\n|∆0|/t0.35\n0\n-0.25\nFIG. 10. The LDOS at E= 0for the (110) oriented surface for\ntheB1representation as a function of |∆0|/tatλSO/t= 0.10\nand∆is/t= 0.20. The red solid line, blue dotted line, and\ngreen dashdotted line correspond to µ/t=−0.25,µ/t= 0.0,\nandµ/t= 0.35.\nto|∆0|(1− |∆0|/|∆c\n0|)for|∆0|<|∆c\n0|and vanishes for\n|∆0|>|∆c\n0|. This is a nonmonotonic dome-shaped be-\nhavior of the ZELDOS as a function of |∆0|. The ex-\nplicit profile can be seen in Fig. 10 at µ/t=−0.25and\nµ/t= 0.0. Forµ/t= 0.35, the point nodes still exist\nin this parameter regime, and the height of the zero en-\nergy peak develops with |∆0|. Thus, we have that the\nZESABS get strongly renormalized and are tunable by a\nvariation in the electron filling ( µ) and amplitude of the\norder parameter |∆0|as shown in Fig. 8.\nV. DISCUSSION AND SUMMARY\nWe investigated and determined the possible super-\nconducting phases arising from inter-orbital pairing in\nan electronic environment marked by spin-orbit coupling\nand inversion symmetry breaking while focusing on mo-\nmentum independent paired configurations. One remark-\nable aspect is that, although the inversion symmetry is\nabsent, one can have symmetry-allowed solutions that\navoid mixing of spin-triplet and spin-singlet configura-\ntions. Importantly, states with only spin-triplet pairing s\ncan be stabilized in a large portion of the phase diagram.\nWithin those spin-triplet superconducting states, we\nunveiled an unconventional type of topological phase in\ntwo-dimensional superconductors that arises from the\ninterplay of spin-orbit coupling and orbitally driven\ninversion-symmetry breaking. For this kind of a model\nsystem, atomic physics plays a relevant role and in-\nevitably tends to yield orbital entanglement close to the\nFermi level. Thus we assumed that local inter-orbital\npairing is the dominant attractive interaction. As already\nmentioned, this type of pairing in the presence of inver-\nsion symmetry breaking allows for solutions that do not\nmix spin-singlet and triplet configurations. The orbital-\nsinglet/spin-triplet superconducting phase can have atopological nature with distinctive spin-orbital finger-\nprints in the low-energy excitations spectra that make\nit fundamentally different from the topological configu-\nration that is usually obtained in single band noncen-\ntrosymmetric superconductors. Here, a remarkable find-\ning is that, contrary to the common view that an isotropic\npairing structure leads to a fully gapped spectrum, a\nnodal superconductivity can be achieved when consid-\nering an isotropic spin-triplet pairing. Although in a dif-\nferent context, we noticed that akin paths for the genera-\ntion of an anomalous nodal-line superconductor can also\nbe encountered when local spin-singlet pairing occur in\nantiferromagnetic semimetals105.\nIn the present study, for a given symmetry, the super-\nconducting phase can exhibit point nodes that are pro-\ntected by a nonvanishing winding number. The most\nstriking feature of the disclosed topological superconduc -\ntivity is expressed by its being prone to both topological\nand Lifshitz-type transitions upon different driving mech-\nanisms and interactions, e.g., when tuning the strength\nof intrinsic spin-orbit and orbital-momentum couplings\nor by varying doping and the amplitude of order pa-\nrameter by, for example, varying the temperature. The\nessence of such a topologically and electronically tunable\nsuperconductivity phase is encoded in the fundamental\nobservation of having control of the nodes position in the\nBrillouin zone. Indeed, the location of the point nodes\nis not determined by the symmetry of the order parame-\nter in the momentum space, as occurs in the single band\nnoncentrosymmetric system, but rather it is a nontrivial\nconsequence of the interplay between spin-triplet pair-\ning and the spin-orbital character of the electronic struc-\nture. In particular, their position and existence in the\nBrillouin zone can be manipulated through various types\nof Lifshitz transitions, if one varies the chemical poten-\ntial, the amplitude of the spin-triplet order parameter,\nthe inversion symmetry breaking term, and the atomic\nspin-orbit coupling. While electron doping can induce a\nchange in the number of Fermi surfaces, such electronic\ntransition is not always accompanied by a variation in\nthe number of nodes within the superconducting state.\nThis behavior allows one to explore different physical sce-\nnarios that single out notable experimental paths for the\ndetection of the targeted topological phase. Owing to the\nstrong sensitivity of the topological and Lifshitz transi-\ntions with respect to the strength of the superconducting\norder parameter, one can foresee the possibility of observ-\ning an extraordinary reconstruction of the superconduct-\ning state both in the bulk and at the edge by employing\nthe temperature to drive the pairing order parameter to a\nvanishing value, i.e., at the critical temperature, starti ng\nfrom a given strength at zero temperature. Then, a sub-\nstantial thermal reorganization of the superconducting\nphase can be obtained. While a variation in the number\nof nodes in the low energy excitations spectra cannot be\neasily extracted by thermodynamic bulk measurements,\nwe find that the electronic structure at the edge of the su-\nperconductor generally undergoes a dramatic reconstruc-14\ntion that manifests into a non-monotonous behavior of\nthe zero bias conductance or in an unconventional ther-\nmal dependence of the in-gap states. Another impor-\ntant detection scheme of the examined spin-triplet su-\nperconductivity emerges when considering its sensitivity\nto the doping or to the strength of the inversion symme-\ntry breaking coupling, which can be accessed by applying\nan electrostatic gating or pressure. Such gate/distortive\ncontrol can find interesting applications, especially when\nconsidering two-dimensional electron gas systems.\nAnother interesting feature of the multiple-nodes topo-\nlogical superconducting phase is given by the strong sen-\nsitivity of the edge states to the geometric termination, as\ndemonstrated in Fig. 9. This is indeed a consequence of\nthe presence of nodes with an opposite sign winding num-\nber within the Brillouin zone. Hence, when considering\nthe electronic transport along a profile that is averaging\ndifferent terminations, it is natural to expect multiple\nin-gap features.\nOwing to the multi-orbital character of the supercon-\nducting state, we expect that non-trivial odd-in-time\npair amplitudes are also generated106–110. In particu-\nlar, we predict that both local odd-in-time spin-singlet\nand triplet states can be obtained in the bulk and at the\nedge. The local spin-singlet odd-in-time pair correlation s\nare an exquisite consequence of the multi-orbital super-\nconducting phase. Accessing the nature of their competi-\ntion/cooperation and its connection to the nodal super-\nconducting phase is a general and relevant problem in\nrelation to the generation, manipulation, and control of\nodd-in-time pair amplitudes.\nIt is also relevant to comment on the impact of an\nintra-orbital pairing on the achieved results. Here, there\nare few fundamental observations to make. Firstly, one\nmay ask whether the topological B 1phase is robust\nto the adding of an extra pairing component which in\nthe intra-orbital channel is most likely to have a spin-\nsinglet symmetry. For this circumstance, one can start\nby pointing out that for any intra-orbital pairing com-\nponent that does not break the chiral symmetry protect-\ning the nodal structure of the superconducting state, the\nB1configuration can only undergo a Lifshitz-type tran-\nsition associated with the merging of nodes having oppo-\nsite sign in the winding number. Moreover, specifically\nfor the B 1irreducible representation, the intra-orbital\nspin-singlet component would have a dx2−y2-wave sym-\nmetry (∼coskx−cosky) and thus its amplitude would be\nvanishing along the Γ-M direction of the Brillouin zone\nwhere the nodes of the B 1phase are placed. Hence, the\nintra-orbital component cannot affect at all the nodal\nstructure of the B 1phase. From this perspective, the\nB1phase is remarkably robust to the inclusion of spin-\nsinglet intra-orbital pairing components. In Appendix,\nthe intra-orbital spin-singlet pairings other than B 1rep-\nresentation ( dx2−y2-wave) are discussed.\nConcerning the experimental consequences of the topo-\nlogical superconducting phase, one can observe that,\napart from the direct spectroscopic access to the tem-perature dependence of the edge states, the use of\na superconductor-normal metal-superconductor (S-N-S)\njunction can also contribute to design of experiments to\ndirectly probe the peculiar behavior of the B 1phase. In\nparticular, by scanning its temperature dependent prop-\nerties, since the B 1state can undergo a series of Lifshitz\ntransitions within the superconducting phase by gapping\nout part of the nodes, a dramatic modification of the\nAndreev spectrum at the S-N boundary is expected to\noccur. Hence, upon the application of a phase difference\nbetween the superconductors in the S-N-S junction, the\nJosephson current is expected to exhibit an anomalous\ntemperature behavior. In particular, the abrupt changes\nin the Andreev bound states will drive a rapid variation in\nthe Josephson current through the S-N-S junction when\nthe superconductor undergoes transitions in the number\nof nodes.\nFinally, we point out that the examined model Hamil-\ntonian is generally applicable to two-dimensional lay-\nered materials, in the low/intermediate doping regime,\nhavingt2gd-bands at the Fermi level and subjected to\nboth atomic spin-orbit coupling and inversion symme-\ntry breaking, for instance owing to lattice distortions\nand bond bending. Many candidate material cases can\nbe encountered in the family of transition metal ox-\nides. There, unconventional low-dimensional quantum\nliquids with low electron density can be obtained by\nengineering a 2DEG at polar/nonpolar interfaces be-\ntween two band insulators, on the surface of band insu-\nlators (i.e., STO) or by designing single monolayer het-\nerostructures, ultrathin films or superlattices. A paradig -\nmatic case of superconducting 2DEG is provided by the\nLAO/STO heterostructure111–114. Recent experimental\nobservations by tunneling spectroscopy have pointed out\nthat the superconducting state can be unconventional\nowing to the occurrence of in-gap states with peaks at\nzero and finite energies115. Although these peaks may\nbe associated with a variety of concomitant physical\nmechanisms, e.g., surface Andreev bound states5–10, the\nanomalous proximity effect by odd-frequency spin-triplet\npairing15,116–128, and bound states owing to the presence\nof magnetic impurities129, their nature can provide key\ninformation about the pairing symmetry of the super-\nconductor. Furthermore, the observation of Josephson\ncurrents130across a constriction in the 2DEG confirms a\nfundamental unconventional nature of the superconduct-\ning state131–133. A common aspect emerging from the\ntwo different spectroscopic probes is that the supercon-\nducting state seems to have a multi-component character.\nAlthough it is not easy to disentangle the various contri-\nbutions that may affect the superconducting phase in the\n2DEG, we speculate that the proposed topological phase\ncan be also included within the possible candidates for\naddressing the puzzling properties of the superconduc-\ntivity of the oxide interface.15\nACKNOWLEDGMENTS\nThis work was supported by a JSPS KAKENHI\n(Grants No. JP15H05853, No. JPH06136, and No.\nJP15H03686), and the JSPS Core-to-Core program \"Ox-\nide Superspin\", and the project Quantox of QuantERA\nERA-NET Cofund in Quantum Technologies, imple-\nmented within the EU H2020 Programme.\nVI. APPENDIX\nIn this section we address three different issues related\nto the presented results. Firstly, we investigate how a\nmodification of the pairing interaction affects the phase\ndiagram and the relative competition between the var-\nious configurations by scanning a larger range of tem-\nperatures at representative cases of filling concentration .\nThen, we consider the classification of the irreducible rep-\nresentations of the superconducting phases in the pres-\nence of an intra-orbital attractive interaction. Moreover ,\nwe demonstrate that the intra-orbital and inter-orbital\npairing interactions mediated by phonons have the same\namplitude.\nStarting from the impact of the pairing interaction\non the phase diagram, in Fig. 11 we show that at a\ngiven temperature the maximal eigenvalue in the various\nirreducible representations scales with the values of Vz\nandVxyatλSO/t= 0.10,∆is/t= 0.20andµ/t= 0.0.\nWhen we keep the ratio Vz/Vxy, the eigenvalue Λis pro-\nportional to Vxywithin the mean field approximation.\nHence, the phase diagram is basically determined by the\nratioVz/Vxy. In addition, since the transition tempera-\ntureTcis achieved when the magnitude of the greatest\neigenvalue gets close to 1, then, according to this relation ,\none can identify the regime of temperatures which is close\nto the superconducting transition by suitably scaling the\npairing interactions. In this way, the corresponding irre-\nducible representation with the largest eigenvalue is the\nmost stable according to the solution of the gap equation.\nIn order to understand how a change in the criti-\ncal temperature can affect the relative stability, in Fig.\n12 we report the eigenvalues for the various irreducible\nrepresentations at λSO/t= 0.10,∆is/t= 0.20and\nµ/t= 0.0as a function of T/tat two different ratio (a)\nVz/Vxy= 1.0and (b)Vz/Vxy= 0.70. We notice that\nthe most stable configuration is not affected by a change\nin temperature or the strength of the pairing coupling.\nHowever, the eigenvalues of the B 2and E representation\nbecome larger than that of A 1aboveT/t∼1.0×10−2\n(see Fig. 12(b)), thus affecting the competition between\nthe A1and B1configurations. Otherwise, the analysis at\ndifferent temperatures demonstrate that even for larger\nvalues of the pairing interaction the phase diagram is not\nmuch affected.\nConcerning the role of the intra-orbital spin-singlet\npairing, in Ref. [1], we can classify the possible ir-\nreducible representations for the tetragonal group C4v0 1 200.050.10.150.2\n0 1 200.511.52\n0 1 200.250.50.751\nVz / Vxy Vz / Vxy(a) (c)the eigenvalueVxy / t = −0.10 Vxy / t = −1.0 A1 representation\nB1 representationA2 representation\nB2 representation\nE representation\nVz / Vxy(b)Vxy / t = −0.5\nFIG. 11. The eigenvalues for various irreducible represen-\ntations as a function of Vz/Vxyat (a)Vxy/t=−0.10, (b)\nVxy/t=−0.50and (c)Vxy/t=−1.0, assuming that T/t=\n1.0×10−5,λSO/t= 0.10,∆is/t= 0.20, andµ/t= 0.0.\n10−510−410−310−210−110000.20.40.60.81\n10−510−410−310−210−1100\nT / tthe eigenvalueA1 representation\nB1 representationA2 representation\nB2 representation\nE representation\n(a) (b)Vz / Vxy = 1.00 Vz / Vxy = 0.70\nT / t\nFIG. 12. The eigenvalues for various irreducible represent a-\ntions as a function of T/tat (a)Vz/Vxy= 1.0and (b)Vz/Vxy=\n0.70, assuming that Vxy/t=−1.0,λSO/t= 0.10,∆is/t=\n0.20, andµ/t= 0.0.\nassuming both isotropic inter-orbital pairing and intra-\norbital ones with isotropic and anisotropic structures\ncompatible with the symmetry configuration as shown\nin Table II.\nFinally, we consider the relative strength of the attrac-\ntive interaction in the inter- and intra-orbital channel as\ndue to electron-phonon coupling in a t2gmulti-orbital\nsystem.\nConsider the electron phonon coupling in t2gsystem,\nHep=1√\nN/summationdisplay\nk,q,m,l,l′σαm\nll′(q)c†\nk+q,l,σck,l′,σ,(29)\nwheremdenotes the phonon mode, αm\nll′(q)is the\nelectron-phonon coupling constant, and landl′stands\nfor orbital indices in the basis of yz,zx, andxy. Here,16\nTABLE II. Irreducible representation of isotropic inter-o rbital\nsuperconducting states and intra-orbital spin-singlet on es\nwith isotropic and anisotropic structures for the tetragon al\ngroupC4v. In the columns, we report the sign of the order\nparameter upon a four-fold rotational symmetry transforma -\ntion,C4, and the reflection mirror symmetry Myz, as well as\nthe explicit spin and orbital structure of the gap function. In\nthe E representation, +and−of the subscript mean the dou-\nbly degenerate mirror-even ( +) and mirror-odd ( −) solutions,\nrespectively.\nC4vC4Myzorbital basis function\nA1+ +(dyz,dyz) ψ(yz,yz)=const.\n(dzx,dzx) ψ(zx,zx)=ψ(yz,yz)\n(dxy,dxy) ψ(xy,xy)=const.\n(dxy,dyz) d(xy,yz)\ny\n(dxy,dzx) d(xy,zx)\nx=−d(xy,yz)\ny\n(dyz,dzx) d(yz,zx)\nz\nA2+−(dyz,dyz)ψ(yz,yz)= sinkxsinky(coskx−cosky)\n(dzx,dzx)ψ(zx,zx)(kx,ky) =ψ(yz,yz)(ky,−kx)\n(dxy,dxy)ψ(xy,xy)= sinkxsinky(coskx−cosky)\n(dxy,dyz) d(xy,yz)\nx\n(dxy,dzx) d(xy,zx)\ny=d(xy,yz)\nx\nB1−+(dyz,dyz)ψ(yz,yz)∝coskx−cosky\n(dzx,dzx)ψ(zx,zx)(kx,ky) =−ψ(yz,yz)(ky,−kx)\n(dxy,dxy)ψ(xy,xy)∝coskx−cosky\n(dxy,dyz) d(xy,yz)\ny\n(dxy,dzx) d(xy,zx)\nx=d(xy,yz)\ny\nB2− −(dyz,dyz)ψ(yz,yz)∝sinkxsinky\n(dzx,dzx)ψ(zx,zx)(kx,ky)=−ψ(yz,yz)(ky,−kx)\n(dxy,dxy)ψ(xy,xy)∝sinkxsinky\n(dxy,dyz) d(xy,yz)\nx\n(dxy,dzx) d(xy,zx)\ny=−d(xy,yz)\nx\n(dyz,dzx) ψ(yz,zx)\nE±i±(dxy,dyz) ψ(xy,yz),d(xy,yz)\nz\n(dxy,dzx)ψ(xy,zx)\n+=∓id(xy,yz)\nz+\nd(xy,zx)\nz−=∓iψ(xy,yz)\n−\n(dyz,dzx) d(yz,zx)\nx ,d(yz,zx)\nywe consider only the diagonal elements, which are rele-\nvant to the attractive interaction. Off-diagonal ones are\nrelevant to the pair hopping, which enhance the transi-\ntion temperature.\nHep=1√\nN/summationdisplay\nk,q,m,l,σαm\nll(q)c†\nk+q,l,σck,l,σ.(30)\nThe effective interaction in the Eliashberg equation due\nto this electron-phonon coupling is given by\nVm\nll′(q,ωn) =−αm\nll(q)αm\nl′l′(q)Dm(q,ωn),(31)\nwhereDm(q,ωn)is the Green’s function of phonon\nDm(q,ωn) =2ωm(q)\nω2m(q)+ω2n, (32)\nwith phonon’s frequency ωm(q)and bosonic Matsubara\nfrequencyωn= 2nπkBT. Here, we suppose the A 1modes\nare the most relevant to the interaction. In A 1modes,\nαm\nyz,yz(q) =αm\nzx,zx(q) (33)\nin the tetragonal symmetry. Then, we have the relation\nVm\nyz,yz(q,ωn) =Vm\nzx,zx(q,ωn) =Vm\nyz,zx(q,ωn) =Vm\nzx,yz(q,ωn).\nThis means that the effective inter and intraorbital at-\ntractive interaction in these two orbitals are the same.\nWe can also include xy-orbitals into this relation when\n∆tand∆isis small and the symmetry goes toward the\ncubic.\n1M. Sigrist and K. Ueda, Rev. Mod. Phys. 63, 239 (1991).\n2Y. Maeno, H. Hashimoto, K. Yoshida, S. Nishizaki, T. Fu-\njita, J. G. Bednorz, and F. Lichtenberg, Nature 372, 532\n(1994).\n3H. Tou, Y. Kitaoka, K. Ishida, K. Asayama, N. Kimura,\nY. Onuki, E. Yamamoto, Y. Haga, and K. Maezawa,\nPhys. Rev. Lett. 80, 3129 (1998).\n4S. Kashiwaya, H. Kashiwaya, H. Kambara, T. Furuta,\nH. Yaguchi, Y. Tanaka, and Y. Maeno, Phys. Rev. Lett.\n107, 077003 (2011).\n5L. J. Buchholtz and G. Zwicknagl, Phys. Rev. B 23, 5788\n(1981).6J. Hara and K. Nagai, Prog. Theor. Phys. 76, 1237 (1986).\n7C. R. Hu, Phys. Rev. Lett. 72, 1526 (1994).\n8Y. Tanaka and S. Kashiwaya, Phys. Rev. Lett. 74, 3451\n(1995).\n9S. Kashiwaya and Y. Tanaka, Rep. Prog. Phys. 63, 1641\n(2000).\n10T. Löfwander, V. S. Shumeiko, and G. Wendin, Super-\ncond. Sci. Technol. 14, R53 (2001).\n11H. Kwon, K. Sengupta, and V. Yakovenko, Eur. Phys. J.\nB37, 349 (2004).\n12A. P. Schnyder, S. Ryu, A. Furusaki, and A. W. W.\nLudwig, Phys. Rev. B 78, 195125 (2008).17\n13S. Ryu, A. P. Schnyder, A. Furusaki, and A. Ludwig,\nNew J. Phys. 12, 065010 (2010).\n14X.-L. Qi and S.-C. Zhang, Rev. Mod. Phys. 83, 1057\n(2011).\n15Y. Tanaka, M. Sato, and N. Nagaosa, J. Phys. Soc. Jpn.\n81, 011013 (2012).\n16M. Leijnse and K. Flensberg, Semiconductor Science and\nTechnology 27, 124003 (2012).\n17C. W. J. Beenakker, Annu. Rev. Condens. Matter Phys.\n4, 113 (2013).\n18M. Sato and S. Fujimoto, J. Phys. Soc. Jpn. 85, 072001\n(2016).\n19M. Sato and Y. Ando, Rep. Prog. Phys. 80, 076501 (2017).\n20A. Y. Kitaev, Usp. Fiz. Nauk (Suppl.) 171, 131 (2001).\n21C. Nayak, S. H. Simon, A. Stern, M. Freedman, and\nS. Das Sarma, Rev. Mod. Phys. 80, 1083 (2008).\n22F. Wilczek, Nat. Phys. 5, 614 (2009).\n23J. Alicea, Rep. Prog. Phys. 75, 076501 (2012).\n24R. Aguado, La Rivista del Nuovo Cimento 40, 523 (2017).\n25F. S. Bergeret, A. F. Volkov, and K. B. Efetov,\nRev. Mod. Phys. 77, 1321 (2005).\n26A. I. Buzdin, Rev. Mod. Phys. 77, 935 (2005).\n27T. S. Khaire, M. A. Khasawneh, W. P. Pratt, and N. O.\nBirge, Phys. Rev. Lett. 104, 137002 (2010).\n28W. A. Robinson, J. D. S. Witt, and M. G. Blamire, Sci-\nence329, 59 (2010).\n29M. Eschrig, J. Kopu, J. C. Cuevas, and G. Schön, Phys.\nRev. Lett. 90, 137003 (2003).\n30M. Eschrig and T. Löfwander, Nature Physics 4, 138\n(2008).\n31J. Linder and J. W. A. Robinson, Nat. Phys. 11, 307\n(2015).\n32P. Gentile, M. Cuoco, A. Romano, C. Noce, D. Manske,\nand P. M. R. Brydon, Phys. Rev. Lett. 111, 097003\n(2013).\n33D. Terrade, D. Manske, and M. Cuoco, Phys. Rev. B 93,\n104523 (2016).\n34M. T. Mercaldo, M. Cuoco, and P. Kotetes, Phys. Rev.\nB94, 140503(R) (2016).\n35A. Romano, P. Gentile, C. Noce, I. Vekhter, and\nM. Cuoco, Phys. Rev. Lett. 110, 267002 (2013).\n36A. Romano, P. Gentile, C. Noce, I. Vekhter, and\nM. Cuoco, Phys. Rev. B 93, 014510 (2016).\n37A. Romano, C. Noce, I. Vekhter, and M. Cuoco, Phys.\nRev. B 96, 054512 (2017).\n38D. Fay and J. Appel, Phys. Rev. B 22, 3173 (1980).\n39C. Pfleiderer, Rev. Mod. Phys. 81, 1551 (2009).\n40A. P. Mackenzie and Y. Maeno, Rev. Mod. Phys. 75, 657\n(2003).\n41I. Martin and A. F. Morpurgo, Phys. Rev. B 85, 144505\n(2012).\n42K. Pöyhönen, A. Westström, J. Röntynen, and T. Oja-\nnen, Phys. Rev. B 89, 115109 (2014).\n43F. Pientka, L. I. Glazman, and F. von Oppen, Phys. Rev.\nB88, 155420 (2013).\n44B. Braunecker and P. Simon, Phys. Rev. Lett. 111, 147202\n(2013).\n45J. Klinovaja, P. Stano, A. Yazdani, and D. Loss, Phys.\nRev. Lett. 111, 186805 (2013).\n46Y. Kim, M. Cheng, B. Bauer, R. M. Lutchyn, and\nS. Das Sarma, Phys. Rev. B 90, 060401 (2014).\n47F. Pientka, L. I. Glazman, and F. von Oppen, Phys. Rev.\nB89, 180505 (2014).\n48S. Nadj-Perge, I. K. Drozdov, B. A. Bernevig, and A. Yaz-dani, Phys. Rev. B 88, 020407 (2013).\n49S. Nakosai, Y. Tanaka, and N. Nagaosa, Phys. Rev. B\n88, 180503 (2013).\n50A. Heimes, P. Kotetes, and G. Schön, Phys. Rev. B 90,\n060507 (2014).\n51D. Mendler, P. Kotetes, and G. Schön,\nPhys. Rev. B 91, 155405 (2015).\n52J. D. Sau, R. M. Lutchyn, S. Tewari, and S. DasSarma,\nPhys. Rev. Lett. 104, 040502 (2010).\n53J. Alicea, Phys. Rev. B 81, 125318 (2010).\n54R. M. Lutchyn, J. D. Sau, and S. DasSarma, Phys. Rev.\nLett.105, 077001 (2010).\n55Y. Oreg, G. Refael, and F. von Oppen, Phys. Rev. Lett.\n105, 177002 (2010).\n56G. Volovik, The Universe in a Helium Droplet (Oxford\nScience Publications, New York, 2003).\n57Y. S. Hor, A. J. Williams, J. G. Checkelsky, P. Roushan,\nJ. Seo, Q. Xu, H. W. Zandbergen, A. Yazdani, N. P. Ong,\nand R. J. Cava, Phys. Rev. Lett 104, 057001 (2010).\n58L. A. Wray, S. Y. Xu, Y. Xia, Y. S. Hor, D. Qian, A. V.\nFedorov, H. Lin, A. Bansil, R. J. Cava, and M. Z. Hasan,\nNature Phys. 6, 855 (2010).\n59M. Kriener, K. Segawa, Z. Ren, S. Sasaki, and Y. Ando,\nPhys.Rev. Lett. 106, 127004 (2011).\n60L. Fu and E. Berg, Phys. Rev. Lett. 105, 097001 (2010).\n61S. Sasaki, M. Kriener, K. Segawa, K. Yada, Y. Tanaka,\nM. Sato, and Y. Ando, Phys. Rev. Lett. 107, 217001\n(2011).\n62L. Hao and T. K. Lee, Phys. Rev. B 83, 134516 (2011).\n63A. Yamakage, K. Yada, M. Sato, and Y. Tanaka, Phys.\nRev. B 85, 180509 (2012).\n64M. Oudah, A. Ikeda, J. N. Hausmann, S. Yonezawa,\nT. Fukumoto, S. Kobayashi, M. Sato, and Y. Maeno,\nNat. Commun. 7, 13617 (2016).\n65L. Aggarwal, A. Gaurav, G. S. Thakur, Z. Haque, A. K.\nGanguli, and G. Sheet, Nat. Matter 15, 32 (2015).\n66H. Wang, H. Wang, H. Liu, H. Lu, W. Yang, S. Jia, X.-J.\nLiu, X. C. Xie, J. Wei, and J. Wang, Nat. Matter 15, 38\n(2015).\n67E. I. Rashba, Sov. Phys. Solid State 2, 1109 (1960).\n68G. Dresselhaus, Phys. Rev. 100, 580 (1955).\n69L. P. Gorkov and E. I. Rashba, Phys. Rev. Lett. 87,\n037004 (2001).\n70P. A. Frigeri, D. F. Agterberg, A. Koga, and M. Sigrist,\nPhys. Rev. Lett. 92, 097001 (2004).\n71K. Yada, S. Onari, Y. Tanaka, and J.-i. Inoue,\nPhys. Rev. B 80, 140509 (2009).\n72C. K. Lu and S. Yip, Phys. Rev. B 78, 132502 (2008).\n73A. B. Vorontsov, I. Vekhter, and M. Eschrig, Phys. Rev.\nLett.101, 127003 (2008).\n74Y. Tanaka, T. Yokoyama, A. V. Balatsky, and N. Na-\ngaosa, Phys. Rev. B 79, 060505(R) (2009).\n75M. Sato and S. Fujimoto, Phys. Rev. B 79, 094504 (2009).\n76A. P. Schnyder, P. M. R. Brydon, D. Manske, and\nC. Timm, Phys. Rev. B 82, 184508 (2010).\n77Z.-J. Ying, M. Cuoco, C. Ortix, and P. Gentile, Phys.\nRev. B 96, 100506 (R) (2017).\n78A. M. Oleś, J. Phys.: Condens. Matter 24, 313201 (2012).\n79H. Y. Hwang, Y. Iwasa, M. Kawasaki, B. Keimer, N. Na-\ngaosa, and Y. Tokura, Nat. Mater. 11, 103 (2012).\n80G. Khalsa, B. Lee, and A. H. MacDonald, Phys. Rev. B\n88, 041302 (2013).\n81C. Autieri, M. Cuoco, and C. Noce, Phys. Rev. B 89,\n075102 (2014).18\n82R. Winkler, Spin-Orbit Coupling Effects in Two-\nDimensional Electron and Hole Systems (Springer Berlin\nHeidelberg, 2003).\n83Z. Zhong, A. Tóth, and K. Held, Phys. Rev. B 87,\n161102(R) (2013).\n84Y. Kim, R. M. Lutchyn, and C. Nayak, Phys. Rev. B 87,\n245121 (2013).\n85A. Joshua, S. Pecker, J. Ruhman, E. Altman, and S. Ilani,\nNat. Comm. 3, 1129 (2012).\n86K. Steffen, F. Loder, and T. Kopp, Phys. Rev. B 91,\n075415 (2015).\n87A. Ohtomo and H. Y. Hwang, Nature (London) 427, 423\n(2004).\n88J. Zabaleta, V. S. Borisov, R. Wanke, H. O. Jeschke,\nS. C. Parks, B. Baum, A. Teker, T. Harada, K. Syassen,\nT. Kopp, N. Pavlenko, R. Valentí, and J. Mannhart,\nPhys. Rev. B 93, 235117 (2016).\n89M. Salluzzo, J. C. Cezar, N. B. Brookes, V. Bisogni,\nG. M. D. Luca, C. Richter, S. Thiel, J. Mannhart, M. Hui-\njben, A. Brinkman, G. Rijnders, and G. Ghiringhelli,\nPhys. Rev. Lett. 102, 166804 (2009).\n90Sugano, Y. Tanabe, and H. Kamimura, Multiplets of\ntransition-metal ions in crystal (Academic Press, New\nYork London, 1970).\n91J. Slater, Quantum Theory of Atomic Structure (McGraw-\nHill, New York, 1960).\n92L. Vaugier, H. Jiang, and S. Biermann, Phys. Rev. B 86,\n165105 (2012).\n93I. M. Lifshitz, Sov. Phys. JETP 11, 1130 (1960).\n94S. Benhabib, A. Sacuto, M. Civelli, I. Paul, M. Cazayous,\nY. Gallais, M.-A. Méasson, R. D. Zhong, J. Schneeloch,\nG. D. Gu, D. Colson, and A. Forget, Phys. Rev. Lett.\n114, 147001 (2015).\n95M. R. Norman, J. Lin, and A. J. Millis, Phys. Rev. B 81,\n180513(R) (2010).\n96E. A. Yelland, J. M. Barraclough, W. Wang, K. V.\nKamenev, and A. D. Huxley, Nat. Phys. 7, 890 (2011).\n97X. Shi, Z.-Q. Han, X.-L. Peng, P. Richard, T. Qian, X.-\nX. Wu, M.-W. Qiu, S. Wang, J. P. Hu, Y.-J. Sun, and\nH. Ding, Nat. Comm. 8, 14988 (2017).\n98M. Ren, Y. Yan, X. Niu, R. Tao, D. Hu, R. Peng, B. Xie,\nJ. Zhao, T. Zhang, and D.-L. Feng, Sci. Adv. 3, 1603238\n(2017).\n99S. N. Khan and D. D. Johnson, Phys. Rev. Lett. 112,\n156401 (2014).\n100C. Liu, A. D. Palczewski, R. S. Dhaka, T. Kondo, R. M.\nFernandes, E. D. Mun, H. Hodovanets, A. N. Thaler,\nJ. Schmalian, S. L. Bud’ko, P. C. Canfield, and A. Kamin-\nski, Phys. Rev. B 84, 020509(R) (2011).\n101S. Tewari and J. D. Sau, Phys. Rev. Lett. 109, 150408\n(2012).\n102K. Yada, M. Sato, Y. Tanaka, and T. Yokoyama, Phys.\nRev. B. 83, 064505 (2011).\n103M. Sato, Y. Tanaka, K. Yada, and T. Yokoyama, Phys.\nRev. B 83, 224511 (2011).\n104P. M. R. Brydon, A. P. Schnyder, and C. Timm,\nPhys. Rev. B 84, 020501 (2011).\n105W. Brzezicki and M. Cuoco, Phys. Rev. B 97, 064513\n(2018).\n106A. M. Black-Schaffer and A. V. Balatsky,\nPhys. Rev. B 88, 104514 (2013).107A. M. Black-Schaffer and A. V. Balatsky,\nPhys. Rev. B 87, 220506 (2013).\n108L. Komendová, A. V. Balatsky, and A. M. Black-Schaffer,\nPhys. Rev. B 92, 094517 (2015).\n109Y. Asano and A. Sasaki, Phys. Rev. B 92, 224508 (2015).\n110L. Komendová and A. M. Black-Schaffer,\nPhys. Rev. Lett. 119, 087001 (2017).\n111N. Reyren, S. Thiel, A. D. Caviglia, L. F. Kourkoutis,\nG. Hammerl, C. Richter, C. W. Schneider, T. Kopp, A.-\nS. Ruetschi, D. Jaccard, M. Gabay, D. A. Muller, J.-M.\nTriscone, J. Mannhart, F. L. Kourkoutis, and G. Ham-\nmerl, Science 317, 1196 (2007).\n112C. Cen, S. Thiel, G. Hammerl, C. W. Schneider, K. E.\nAndersen, C. S. Hellberg, J. Mannhart, and J. Levy, Nat.\nMater. 7, 298 (2008).\n113A. D. Caviglia, S. Gariglio, N. Reyren, D. Jac-\ncard, T. Schneider, M. Gabay, S. Thiel, G. Hammerl,\nJ. Mannhart, and J.-M. Triscone, Nature 456, 624 (2008).\n114T. Schneider, A. D. Caviglia, S. Gariglio, N. Reyren, and\nJ.-M. Triscone, Phys. Rev. B 79, 184502 (2009).\n115L. Kuerten, C. Richter, N. Mohanta, T. Kopp,\nA. Kampf, J. Mannhart, and H. Boschker,\nPhys. Rev. B 96, 014513 (2017).\n116X. Liu, J. D. Sau, and S. Das Sarma, Phys. Rev. B 92,\n014513 (2015).\n117Z. Huang, P. Woelfle, and A. V. Balatsky, Phys. Rev. B\n92, 121404(R) (2015).\n118P. Burset, B. Lu, G. Tkachov, Y. Tanaka, E. Hankiewicz,\nand B. Trauzettel, Phys. Rev. B 92, 205424 (2015).\n119S.-P. Lee, R. M. Lutchyn, and J. Maciejko,\nPhys. Rev. B 95, 184506 (2017).\n120O. Kashuba, B. Sothmann, P. Burset, and B. Trauzettel,\nPhys. Rev. B 95, 174516 (2017).\n121J. Cayao and A. M. Black-Schaffer, Phys. Rev. B 96,\n155426 (2017).\n122Y. Tanaka and S. Kashiwaya, Phys. Rev. B 70, 012507\n(2004).\n123Y. Tanaka, S. Kashiwaya, and T. Yokoyama, Phys. Rev.\nB71, 094513 (2005).\n124Y. Asano, Y. Tanaka, and S. Kashiwaya, Phys. Rev. Lett.\n96, 097007 (2006).\n125Y. Tanaka, Y. Asano, A. A. Golubov, and S. Kashiwaya,\nPhys. Rev. B 72, 140503(R) (2005).\n126Y. Tanaka and A. A. Golubov, Phys. Rev. Lett. 98,\n037003 (2007).\n127Y. Tanaka, A. A. Golubov, S. Kashiwaya, and M. Ueda,\nPhys. Rev. Lett. 99, 037005 (2007).\n128Y. Tanaka, Y. Tanuma, and A. A. Golubov, Phys. Rev.\nB76, 054522 (2007).\n129H. Ebisu, K. Yada, H. Kasai, and Y. Tanaka, Phys. Rev.\nB91, 054518 (2015).\n130D. Stornaiuolo, D. Massarotti, R. Di Capua, P. Lu-\ncignano, G. P. Pepe, M. Salluzzo, and F. Tafuri,\nPhys. Rev. B 95, 140502 (2017).\n131Y. Tanaka and S. Kashiwaya, Phys. Rev. B 53, R11957\n(1996).\n132Y. S. Barash, H. Burkhardt, and D. Rainer, Phys. Rev.\nLett.77, 4070 (1996).\n133Y. Tanaka and S. Kashiwaya, Phys. Rev. B 56, 892 (1997)."    },    {        "title": "1703.08405v2.Gate_control_of_the_spin_mobility_through_the_modification_of_the_spin_orbit_interaction_in_two_dimensional_systems.pdf",        "content": "Gate control of the spin mobility through the modi\fcation of\nthe spin-orbit interaction in two-dimensional systems\nM. Luengo-Kovac,1F. C. D. Moraes,2G. J. Ferreira,3A. S. L. Ribeiro,2\nG. M. Gusev,2A. K. Bakarov,4V. Sih,1and F. G. G. Hernandez2,\u0003\n1Department of Physics, University of Michigan, Ann Arbor, Michigan 48109, United States\n2Instituto de F\u0013 \u0010sica, Universidade de S~ ao Paulo, S~ ao Paulo, SP 05508-090, Brazil\n3Instituto de F\u0013 \u0010sica, Universidade Federal de Uberl^ andia, Uberl^ andia, MG 38400-902, Brazil\n4Institute of Semiconductor Physics and Novosibirsk State University, Novosibirsk 630090, Russia\n(Dated: November 14, 2021)\nSpin drag measurements were performed in a two-dimensional electron system set close to the\ncrossed spin helix regime and coupled by strong intersubband scattering. In a sample with uncom-\nmon combination of long spin lifetime and high charge mobility, the drift transport allows us to\ndetermine the spin-orbit \feld and the spin mobility anisotropies. We used a random walk model\nto describe the system dynamics and found excellent agreement for the Rashba and Dresselhaus\ncouplings. The proposed two-subband system displays a large tuning lever arm for the Rashba\nconstant with gate voltage, which provides a new path towards a spin transistor. Furthermore,\nthe data shows large spin mobility controlled by the spin-orbit constants setting the \feld along the\ndirection perpendicular to the drift velocity. This work directly reveals the resistance experienced\nin the transport of a spin-polarized packet as a function of the strength of anisotropic spin-orbit\n\felds.\nThe pursuit for a new active electronic component\nbased on \row of spin, rather than that of charge, strongly\nmotivates research in semiconductor spintronics [1{5].\nSince the Datta-Das proposal for a ballistic spin tran-\nsistor, full electrical control of the spin state was sug-\ngested using the gate-tunable Rashba spin-orbit interac-\ntion (SOI) [6{10]. Further studies, including the Dressel-\nhaus SOI [11], were made to assure a nonballistic tran-\nsistor robust against spin-independent scattering [12{14].\nFor example, it has been demonstrated that SU(2) spin\nrotation symmetry, preserving the spin polarization, can\nbe obtained in the persistent spin helix (PSH) formed\nwhen the strengths of the Rashba and Dresselhaus SOI\nare equal (\u000b=\f) [15{19]. This is possible because the\nuniaxial alignment of the spin-orbit \feld suppresses the\nrelaxation mechanism when the spins precess about this\n\feld while experiencing momentum scattering [20]. Gate\ncontrol of this symmetry point was experimentally ob-\nserved [21{23] and allowed to produce a transition to the\nPSH\u0000(\u000b=\u0000\f) in the same subband [24]. Drift in those\nsystems showed surprising properties [25, 26] such as the\ncurrent-control of the temporal spin-precession frequency\n[27]. Although the helical spin-density texture could be\neven transported without dissipation under certain con-\nditions [15], the spin transport su\u000bers additional resis-\ntance from the spin Coulomb drag [28{32]. These fric-\ntional forces appear as a lower mobility for spins than for\ncharge and studies in new systems are still necessary to\nunderstand this important constraint for future devices.\nA two-dimensional electron gas (2DEG) hosted in a\nquantum well (QW) with two occupied subbands o\u000bers\nunexplored opportunities for the study of spin transport\n[33, 34]. Theoretically, the inter- and intra-subband spin-\norbit couplings (SOCs) have been extensively studied[35{38]. In terms of a random walk model (RWM) [39],\nthe spin drift and di\u000busion was recently developed for\nthese systems displaying two possible scenarios regarding\nthe intersubband scattering (ISS) rate [40]. The interplay\nbetween the two subbands may introduce new features to\nthe PSH dynamics, for example, a crossed persistent spin\nhelix [41] may arise when the subbands are set to orthog-\nonal PSHs (i.e., \u000b1=\f1and\u000b2=\u0000\f2) in the weak ISS\nlimit. In this report, we experimentally study spin drag\nin a system with the two-subbands individually set close\nto the PSH+and PSH\u0000, but with strong ISS, where the\ndynamics is given by the averaged SOCs of both sub-\nbands. The combination of long spin lifetime and high\ncharge mobility allows us to determine the spin mobility\nand the spin-orbit \feld anisotropies with the application\nof an accelerating in-plane voltage. We are able to con-\ntrol the SOCs in both subbands and to show a linear\ndependence for the sum of the Rashba constants with\ngate voltage. Finally, we determine an inverse relation\nfor the spin mobility dependence on the SOCs directly\nrevealing the resistance experienced in the transport of\na spin-polarized packet as a function of the strength of\nanisotropic spin-orbit \felds.\nThe sample consists of a single 45 nm wide GaAs QW\ngrown in the [001] ( z) direction and symmetrically doped.\nDue to the Coulomb repulsion of the electrons, the charge\ndistribution experiences a soft barrier inside the well.\nFigure 1(a) shows the calculated QW band pro\fle and\ncharge density for both subbands. The electronic system\nhas a con\fguration with symmetric and antisymmetric\nwave functions for the two lowest subbands with sub-\nband separation of \u0001 SAS = 2 meV. The subband den-\nsity (n 1= 3.7, n 2= 3.3\u00021011cm\u00002) was obtained from\nthe Shubnikov-de Hass (SdH) oscillations as shown inarXiv:1703.08405v2  [cond-mat.mes-hall]  18 May 20172\n0102030405060\n00.511.522.53\n00.511.522.53Rxx (:)Rxy (k:)\nBext (T)6789FFT f (T)n2n1246-15-10-50-0.100.10.2n (x1011 cm-2)Vg (V)Ez (V/Pm)n1n2n1+n2-20-100-0.100.10.2E (meV)Ez (V/Pm)H1H2H3-20020z (nm)(a)(b)\n(d)(c)\nFIG. 1. (a) Longitudinal ( Rxx) and Hall ( Rxy) magnetoresis-\ntance of the two-subband QW. From the SdH periodicity, one\ncan obtain the subbands density n\u0017in the lower inset. The\ntop inset shows the potential pro\fle and subbands charge den-\nsity calculated from the self-consistent solution of Schr odinger\nand Poisson equations for Ez=0. (b) Subband energy levels\nand (c) electron concentration dependence on V gandEz. (d)\nGeometry of the device and contacts con\fguration.\nFig. 1(a) and the low-temperature charge mobility was\n2.2\u0002106cm2/Vs [42]. A device was fabricated in a cross-\nshaped con\fguration with width of w=270\u0016m and chan-\nnels along the [1 \u001610] (x) and [110] (y) directions. Lateral\nOhmic contacts deposited l=500\u0016m apart were used to\napply an in-plane voltage ( Vip) in order to induce drift\ntransport. For the \fne tuning of the subband SOCs,\na semitransparent contact on top of the mesa structure\n(Vg) was used to modify structural symmetry and sub-\nband occupation. The e\u000bect of Vgon the subband energy\nlevels (\u000f\u0017) and densities ( n\u0017) is shown in Fig. 1(b) and\n(c) as a function of the out-of-plane electric \feld ( Ez).\nNote that the total density changes linearly with Vgand\nthatVg=0 corresponds to a built-in electric \feld of 0.15\nV/\u0016m. Figure 1(d) displays the experimental scheme\nwith the connection of VipandVg[43].\nTo describe the magnetization dynamics and the mea-\nsured SO \felds for our two-subband system, we combine\nthe calculated SOCs with RWM [39, 40, 44]. For a [001]\nGaAs 2DEG, the x and y components of the SO \felds\nfor each subband \u0017=f1;2gare\nBSO;\u0017(k) =2\ng\u0016B0\nB@\u0000\n+\u000b\u0017+\f1;\u0017+ 2\f3;\u0017k2\nx\u0000k2\ny\nk2\u0001\nky\n\u0000\n\u0000\u000b\u0017+\f1;\u0017\u00002\f3;\u0017k2\nx\u0000k2\ny\nk2\u0001\nkx1\nCA;\n(1)\nplus corrections due to the intersubband SOCs [23, 35{\n38, 41]. Above, g=\u00000:44 is the electron g-factor for\nGaAs and\u0016Bis the Bohr magneton. The SOCs are the\nusual Rashba \u000b\u0017and linear\f1;\u0017and cubic\f3;\u0017Dressel-\nhaus terms. Considering the strong intersubband scat-\ntering (ISS) regime of the RWM [40], the randomization\nof the momenta k(within the Fermi circle k=kF) and\nsubband\u0017is much faster than the spin precession. Con-\nsequently, the dynamics is governed by an averaged SOC\n-3-2-10123\n-0.3-0.2-0.100.10.20.3DX/EX\nEz (V/Pm)2ndsubband\n1stsubbandPSH+PSH--40-2002040-40-2002040x (Pm)y (Pm)-40-2002040-40-2002040x (Pm)y (Pm)\n-40-2002040-40-2002040\nx (Pm)y (Pm)-40-2002040-40-2002040\nx (Pm)y (Pm)-40-2002040-40-2002040\nx (Pm)y (Pm)-2-10123\n-0.3-0.1500.150.3SOC (meVA)D1D2K6DXEz (V/Pm)-1012\n-0.3-0.1500.150.3E1,2*E1,1Ez (V/Pm)0.40.60.81\n-0.3-0.1500.150.3Ez (V/Pm)E3,1E3,26E*X(a)(b)\n(d)(c)\n(e)1stPSH+2ndPSH-\nEz=0Ez=0.15Ez=0.3FIG. 2. (a)-(c) Calculated SOCs for the Rashba ( \u000b\u0017),\nlinear (\f1;\u0017) and cubic ( \f3;\u0017) Dresselhaus for each subband\n\u0017=f1;2g, as well as intersubband SOCs \u0011and \u0000 as a func-\ntion ofEz. The purple lines give the sum of \u000b\u0017and\f\u0003\n\u0017. (d)\nThe ratio\u000b\u0017=\f\u0017=\u00061 when the subband \u0017is set to the\nPSH\u0006regime. The insets show the single-subband magne-\ntization maps on the xyplane for the PSH\u0006regimes, and\nthe self-consistent potentials and subband densities for the\nrespectiveEz. (e) Two-subband magnetization maps in the\nstrong ISS regime for di\u000berent Ez. AtEz= 0 the well is sym-\nmetric (\u000b\u0017= 0) and the magnetization shows an isotropic\nBessel pattern. For \fnite Ezthe broken symmetry leads\nto the stripped PSH pattern in accordance with the pos-\nitive ratioP\u000b\u0017=P\f\u0017[purple line in panel (d)]. The ar-\nrows in the Fermi circle show the \frst harmonic component\nofPBSO;\u0017(k), illustrating the transition from isotropic to\nuniaxial \feld with increasing Ez. All the xy maps are frames\nof the spin pattern at t = 13 ns.\n\feldhBSOi= (hBx\nSOi;hBy\nSOi) transverse to the drift ve-\nlocity vdr= (vx\ndr;vy\ndr) Namely, the \feld components read\nhBx\nSOi=hm\n~g\u0016B2X\n\u0017=1(+\u000b\u0017+\f\u0003\n\u0017)i\nvy\ndr; (2)\nhBy\nSOi=hm\n~g\u0016B2X\n\u0017=1(\u0000\u000b\u0017+\f\u0003\n\u0017)i\nvx\ndr; (3)\nwhere\f\u0003\n\u0017=\f1;\u0017\u00002\f3;\u0017, andm= 0:067m0is the e\u000bec-\ntive electron mass for GaAs and ~is Planck's constant.\nSince Bx(y)\nSO/vy(x)\ndr, it is convenient to analyze the linear\ncoe\u000ecients bx(y)= By(x)\nSO=vx(y)\ndr, which are given by the3\nterms between square brackets above.\nThe intra- and intersubband SOCs are calculated\nwithin the self-consistent Hartree approximation [35{38]\nfor GaAs quantum wells tilted by Ez. The chemical\npotential is set to return the density n=n1+n2=\n7\u00021011cm\u00002forEz= 0, while it varies linearly for\n\fniteEzin Fig. 1(c). The SOCs are de\fned from\nthe matrix elements \u0011\u0017;\u00170=h\u0017j\u0011wV0+\u0011HV0\nHj\u00170iand\n\u0000\u0017;\u00170=\rh\u0017jk2\nzj\u00170i, wherej\u0017iis the eigenket for subband\n\u0017,\u0011w= 3:47\u0017A2and\u0011H= 5:28\u0017A2are bulk coe\u000ecients\n[35{38, 45], V0=@zV(z) andV0\nH=@zVH(z) are the\nderivatives of the heterostructure and Hartree potentials\nalongz,\r= 11 eV \u0017A3is the bulk Dresselhaus constant,\nandkzis the z-component of the momentum. The usual\nintrasubband Rashba and linear Dresselhaus SOCs are\n\u000b\u0017=\u0011\u0017;\u0017and\f1;\u0017= \u0000\u0017;\u0017. The non-diagonal terms are\nthe intersubband SOCs \u0011=\u001112and \u0000 = \u0000 12. The calcu-\nlated SOCs, plotted in Fig. 2(a)-(c) as a function of Ez,\nshow agreement with previous studies [46, 47]. The high-\ndensitynmakes the cubic Dresselhaus \f3;\u0017\u0019\r\u0019n\u0017=2\ncomparable with \f1;\u0017, strongly a\u000becting the PSH tuning\n[17]\u000b\u0017=\f\u0017, with\f\u0017=\f1;\u0017\u0000\f3;\u0017.\nNearEz\u00190:04 V/\u0016m, the SOCs reach almost simul-\ntaneously the balanced condition for the PSH+in the\n\frst subband ( \u000b1=\f1= +1) and for the PSH\u0000in the\nsecond subband ( \u000b2=\f2=\u00001), as shown by the ratio\n\u000b\u0017=\f\u0017in Fig. 2(d). The expected magnetization pat-\nterns for the single-subband PSH is shown in the inset\nof Fig. 2(d). The PSH\u0000shows more stripes than the\nPSH+due to the higher value of \u000b, which grows quickly\nwithin the Ezrange. However, the ratio of the aver-\naged SOCs (P\u000b\u0017)=(P\f\u0017) approaches the PSH regimes\nonly forjEzj>0:3 V/\u0016m. As we will see next, the\nexperimental data matches well the strong ISS regime\nof the RWM, therefore the dynamics is governed by the\naveraged SOCs. In this case, the expected magnetiza-\ntion patterns are shown in Fig. 2(e). With increasing Ez\nthe system transitions from isotropic ( Ez= 0) to uniax-\nial (Ez>0:3 V/\u0016m), as indicated by the formation of\nstripes and the orientation of the \frst harmonic compo-\nnent of the total \feldPBSO;\u0017(k) [arrows in Fig. 2(e)].\nWe are interested in the determination of the\nanisotropy for the coe\u000ecients bx(y), estimated in one or-\nder of magnitude in Fig. 3(a) and (b). We measured\nthe spin polarization using time-resolved Kerr rotation\nas function of the space and time separation of pump\nand probe beams. All optical measurements were per-\nformed at 10 K. A mode-locked Ti:Sapphire laser with a\nrepetition rate of 76 MHz tuned to 816.73 nm was split\ninto pump and probe pulses. The polarization of the\npump beam was controlled by a photoelastic modulator\nand the intensity of the probe beam was modulated by\nan optical chopper for cascaded lock-in detection. An\nelectromagnet was used to apply an external magnetic\n\feld in the plane of the QW. The spatial positioning of\nthe pump relative to the probe (d) was controlled using\n00.511.52\n-50-250255075A(d) (a.u.)d (Pm)dc = 27.1 Pm0255075vdr || yVip (mV)\n0123\n-40-30-20-10010203040Bext (mT)0IK (a.u.)255075vdr || yVip (mV)0246810\n0246810\n00.511.522.53i (mA)BSO (mT)vdr (Pm/ns) by=(3.0  0.3) mTns/Pmbx=(0.4  0.2) mTns/Pmvdr || yvdr || x00.511.522.53\n01020304050607080vdr (Pm/ns)Pys=(1.4  0.1) x105cm2/VsPxs=(2.2  0.1)x105cm2/VsEip (V/m)020406080100120140160\nVip (mV)(a)(b)\n(f)(e)±\n±\n±±(d)(c)bxbyFIG. 3. Calculated coe\u000ecients b with vdrparallel to (a) x\nand (b) y for each subband (colored) and total \feld (black).\n(c) Amplitude of the drifting spin polarization in space show-\ning, for example, the center of the packet d cfor 75 mV. (d)\nLinear dependence of v drwith the channel V ip. The slope\ngives the spin mobility along v drin x or y. (e) Field scan\nof\u001eKfor several V ipmeasured at d c. (f) By(x)\nSOas function\nof vx(y)\ndrand the current \rowing in that channel. The slopes\nbx(y)give the strength of the SOCs that generate the \feld\nalong y(x) for drift in x(y). The solid lines are gaussian (c)\nand linear (d and f) \fttings. Scans taken at t=13 ns.\na scanning mirror. We de\fned the spin injection point\nto be x=y=0 at t=0. The application of an in-plane\nelectric \feld (E ip=Vip/l), in the x or y-oriented channel,\nadds a drift velocity to the 2DEG electrons and allows\nus to determine the spin mobility and the spin-orbit \feld\ncomponents [48{50].\nThe sample was rotated such that each channel un-\nder study was oriented parallel to the external magnetic\n\feldBextkvdrfor all measurements reported here. From\nthe SOI form in k-space, we expected BSO?vdrimply-\ning that the observable BSOdirection will be BSO?Bext.\nConsidering this orientation, we can model the Kerr rota-\ntion signal as \u001eK(Bext;d) = A(d) cos ( !t) with the pre-\ncession frequency given by != (g\u0016B=~)p\nB2\next+B2\nSO,\nwhere A(d) is the amplitude at a given pump-probe spa-\ntial separation and B SOis the internal SO \feld compo-\nnent perpendicular to Bext(and to vdr).\nFigure 3 shows the results of the spin drag experiment\nwith the gate contact open. Scanning the pump-probe\nseparation in space at \fxed long time delay (13 ns), we\ndetermined the central position d cof the spin packet4\namplitude for several V ipin a given crystal orientation.\nFrom the values of d cin Fig. 3(c), we calculated the drift\nvelocity as v dr=dc/t and plotted it as a function of V ip\nin Fig. 3(d). The slope of the linear \ft give us spin mo-\nbilities (\u0016x;y\ns) in the range of 105cm2/Vs. Values in the\nsame order of magnitude have been measured by Doppler\nvelocimetry for the transport in single subband samples\n[32]. Nevertheless, in those systems the spin lifetimes\nwere restricted to the picosecond range and the trans-\nport was limited to the nanometer scale.\nFollowing the drifting spin packet in space, Fig. 3(e)\ndisplays a B extscan from where changes in the amplitude\nof zeroth resonance determined B SOstrength at d c. As\nexplained above, the data con\frmed the perpendicular\norientation between BSOandvdrand did not show a\ncomponent parallel to Bextwithin the experimental res-\nolution [51]. From the Lorenztian shape of the B extscan\n[52, 53], we evaluated a spin lifetime of 7 ns at V ip=0.\nThis experiment was only possible due to the nanosecond\nspin lifetime in our sample that extends the spin trans-\nport to several tens of micrometers [54, 55].\nFigure 3(f) shows the \ftted values of B SOfor sev-\neral Vipapplied along x and y. We observed highly\nanisotropic spin-orbit \felds in the range of several mT\nas expected from Fig 3(a) and (b). The B SOorientation\nwas aligned primary with the x axis in agreement with\nthe simulation in Fig 2(e). The slopes bx(y)=By(x)\nSO/vx(y)\ndr\ngive the strength of the SOCs that generate the \feld ac-\ncording to Eqs. 2 and 3. For this condition of the sample\nas-grown, we foundP\u000b\u0017= 0.57 meV \u0017A andP\f\u0003\n\u0017= 0.75\nmeV\u0017A.\nNote the inverse behaviour on V ipfor the mobility and\nfor BSOstrength in perpendicular directions. In Fig.\n3(c) and (d), the axis with the largest mobility is also\nthe axis with smaller spin-orbit \feld in the perpendic-\nular direction. This result may be related to the spin\nCoulomb drag observed previously in the transport of\nspin-polarized electrons [31, 32]. Next, we demonstrate\nthe direct control of the spin mobility through the gate\nmodi\fcation of the subband SOCs.\nFigure 4(a) shows that the magnitude and the orienta-\ntion with the largest \u0016scan be tuned by E z. BSOdisplays\nanisotropic components with Bx\nSObeing larger in all the\nstudied range, which con\frms the preferential alignment\ntowards the PSH+in Fig. 2(e). The variation of Bx\nSO\nhas a minimum (indicated by an arrow) close to position\nwhen the second subband attains the PSH\u0000(with BSO\nalong y). Dividing Fig. 4(a) panels, the values for b are\nplotted in Fig. 4(b). The lines plotted together with\nthe data are the expected values using Eq. 2 and 3 with\nthe SOCs from Fig. 2(a)-(c). When the QW approaches\nthe symmetric condition (E z=0), bx(y)decreases remov-\ning the anisotropy of B SOas simulated in Fig. 2(e).\nThe addition and subtraction of bxandbygive the sum\nof the Rashba and Dressselhaus SOCs displayed in Fig.\n4(c). Dashed lines corresponding to the purple curves in\n-0.200.20.40.60.81\n00.050.10.15SOC (meVA)6DXEz (V/Pm)6E*X00.511.522.53\n-0.0500.050.10.15b (mT/Pm/ns)vdr || yvdr || xEz (V/Pm)-4.4 eÅ235.0 eÅ20123456-0.0500.050.10.15BSO (mT)Ez (V/Pm)BSO || xBSO || y00.511.522.53-15-10-50Ps (x105cm2/Vs)Vg (V)vdr || yvdr || x(a)(b)\n(d)(c)\n01234\n00.20.40.60.811.26\u000bDX+E*X), 6\u000b\u0010DX+E*X) (meVA)Ps (x105cm2/Vs)vdr || yvdr || xP0s-1.1x105𝑐𝑚2𝑉𝑠𝑚𝑒𝑉Å-3.4x105 𝑐𝑚2𝑉𝑠𝑚𝑒𝑉ÅFIG. 4. (a) Spin mobility and B SOas a function of the\ngate-tunable E z. (b) Ratio bx(y)from (a), showing a crossing\nat Ez=0. (c) SOCs obtained from the addition and subtrac-\ntion ofbxandbyin (b). (d) Spin mobility as function of the\nSOCs that de\fne the B SOstrength along the direction per-\npendicular to v dr. The solid lines are linear \fttings and the\ndashed lines (b,c) are the theoretical results from the RWM\ncombined with the self-consistent calculation of the SOCs.\nFig.2(a) and (c) are plotted together displaying excellent\nagreement. The slope for the Rashba SOI indicates a tun-\ning lever arm of 35 e \u0017A2. This value is considerably larger\nthan those reported in recent studies for single subband\nsamples, typically below 10 e \u0017A2[17, 23]. Finally, Fig.\n4(d) presents \u0016x(y)\ns[from (a)] against the SOCs de\fn-\ning By(x)\nSO:P(\u0000\u000b\u0017+\f\u0003\n\u0017) andP(\u000b\u0017+\f\u0003\n\u0017), respectively.\nThis last plot illustrates the inverse dependence, with\nnegative slope, for the spin mobility and strength of the\nSOCs perpendicular to the drift direction. The di\u000berent\nslopes for x and y channels give us a hint that this e\u000bect\ndepends not only on how B SOchanges with v dr(given\nby the SOCs) but also in the magnitude of the \felds. A\ncommon maximum value \u00160\ns=3\u0002105cm2/Vs was found\nindependent of vdrorientation.\nIn conclusion, we have studied a 2DEG system with\ntwo subbands set close to the crossed PSH regime under\nstrong intersubband scattering and successfully described\nit using a random walk model. In the spin transport\nwith nanosecond lifetimes over micrometer distances, we\ndemonstrate the control of the subbands spin-orbit cou-\nplings with gate voltage and observed spin mobilities in\nthe range of 105cm2/Vs. Speci\fcally, the sum of the\nRashba SOCs presents a linear behaviour with remark-\nably large tunability lever arm with gate voltage. We5\ntailored the spin mobility by controlling the strength of\nthe spin-orbit interaction in the direction perpendicular\nto the drift velocity. Our \fndings provided evidence of\nthe rich physical phenomena behind multisubband sys-\ntems and experimentally demonstrated relevant proper-\nties required for the implementation of a nonballistic spin\ntransistor.\nThis work is a result of the collaboration initiative\nSPRINT No. 2016/50018-1 of the S~ ao Paulo Research\nFoundation (FAPESP) and the University of Michi-\ngan. F.G.G.H also acknowledges \fnancial support from\nFAPESP Grants No. 2009/15007-5, No. 2013/03450-\n7, and No. 2014/25981-7 and 2015/16191-5. G.J.F.\nacknowledges the \fnancial support from CNPq and\nFAPEMIG. The work at the University of Michigan is\nsupported by the National Science Foundation under\nGrant No. DMR-1607779.\n\u0003Corresponding author.\nElectronic address: felixggh@if.usp.br\n[1] S. A. Wolf, D. D. Awschalom, R. A. Buhrman, J.\nM. Daughton, S. von Moln\u0013 ar, M. L. Roukes, A. Y.\nChtchelkanova, and D. M. Treger, Science 294, 1488\n(2001).\n[2] I. ^Zuti\u0013 c, J. Fabian, and S. Das Sarma, Rev. Mod. Phys.\n76, 323 (2004).\n[3] D. D. Awschalom and M. E. Flatt\u0013 e, Nat. Phys. 3, 153\n(2007).\n[4] J. Wunderlich, B.-G. Park, A. C. Irvine, L. P. Z^ arbo,\nE. Rozkotov\u0013 a, P. Nemec, V. Nov\u0013 ak, J. Sinova, and T.\nJungwirth, Science 330, 1801 (2010).\n[5] J. C. Egues, G. Burkard, and D. Loss, Appl. Phys. Lett.\n82, 2658 (2003).\n[6] E. I. Rashba, Sov. Phys. Solid State 2, 1109 (1960); Y. A.\nBychkov and E. I. Rashba, J. Phys. C 17, 6039 (1984);\nJETP Lett. 39, 78 (1984).\n[7] S. Datta and B. Das, Appl. Phys. Lett. 56, 665 (1990).\n[8] J. Nitta, T. Akazaki, H. Takayanagi, and T. Enoki, Phys.\nRev. Lett. 78, 1335 (1997).\n[9] V. Lechner, L. E. Golub, P. Olbrich, S. Stachel, D. Schuh,\nW. Wegscheider, V. V. Bel'kov and S. D. Ganichev, Appl.\nPhys. Lett. 94, 242109 (2009).\n[10] P. Chuang, S.-C. Ho, L. W. Smith, F. S\fgakis, M. Pep-\nper, C.-H. Chen, J.-C. Fan, J. P. Gri\u000eths, I. Farrer, H.\nE. Beere, G. A. C. Jones, D. A. Ritchie, and T.-M. Chen,\nNat. Nanotechnol. 10, 35 (2014).\n[11] G. Dresselhaus, Phys. Rev. 100, 580 (1955).\n[12] J. Schliemann, J. C. Egues, and D. Loss, Phys. Rev. Lett .\n90, 146801 (2003).\n[13] M. Ohno and K. Yoh, Phys. Rev. B 77, 045323 (2008).\n[14] Y. Kunihashi, M. Kohda, H. Sanada, H. Gotoh, T. So-\ngawa, and J. Nitta, Appl. Phys. Lett. 100, 113502 (2012).\n[15] B. A. Bernevig, J. Orenstein, and S.-C. Zhang, Phys.\nRev. Lett. 97, 236601 (2006).\n[16] J. D. Koralek, C. Weber, J. Orenstein, B. Bernevig, S.-\nC. Zhang, S. Mack, and D. Awschalom, Nature 458, 610\n(2009).\n[17] M. P. Walser, C. Reichl, W. Wegscheider, and G. Salis,Nat. Phys. 8, 757 (2012).\n[18] J. Schliemann, Rev. Mod. Phys. 89, 011001 (2017).\n[19] M. Kohda and G. Salis, Semiconductor Science and Tech-\nnology (2017). at press: https://doi.org/10.1088/1361-\n6641/aa5dd6\n[20] M. I. D'yakonov and V. I. Perel', Sov. Phys. Solid State\n13, 3023 (1972).\n[21] M. Kohda, V. Lechner, Y. Kunihashi, T. Dollinger, P.\nOlbrich, C. Sch onhuber, I. Caspers, V. V. Bel'kov, L.\nE. Golub, D. Weiss, K. Richter, J. Nitta, and S. D.\nGanichev, Phys. Rev. B 86, 081306(R) (2012).\n[22] J. Ishihara, Y. Ohno, and H. Ohno, Appl. Phys. Exp. 7,\n013001 (2014).\n[23] F. Dettwiler, J. Fu, S. Mack, P. J. Weigele, J. C. Egues,\nD. D. Awschalom, and D. M. Zumb uhl, arXiv:1702.05190\n[cond-mat.mes-hall].\n[24] K. Yoshizumi, A. Sasaki, M. Kohda, and J. Nitta, Appl.\nPhys. Lett. 108, 132402 (2016). Here, the PSH\u0000is la-\nbeled as inverse PSH (iPSH).\n[25] L. Yang, J. D. Koralek, J. Orenstein, D. R. Tibbetts, J.\nL. Reno, and M. P. Lilly Phys. Rev. Lett. 109, 246603\n(2012).\n[26] Y. Kunihashi, H. Sanada, H. Gotoh, K. Onomitsu, M.\nKohda, J. Nitta, and T. Sogawa, Nat. Commun. 7, 10722\n(2016).\n[27] P. Altmann, F. G. G. Hernandez, G. J. Ferreira, M. Ko-\nhda, C. Reichl, W. Wegscheider, and G. Salis, Phys. Rev.\nLett.116, 196802 (2016).\n[28] I. D'Amico and G. Vignale, Europhys. Lett. 55, 566\n(2001).\n[29] I. D'Amico and G. Vignale, Phys. Rev. B 68, 045307\n(2003).\n[30] W.-K. Tse and S. Das Sarma, Phys. Rev. B 75, 045333\n(2007).\n[31] C. P. Weber, N. Gedik, J. E. Moore, J. Orenstein,\nJ. Stephens, and D. D. Awschalom, Nature 437, 1330\n(2005).\n[32] L. Yang, J. D. Koralek, J. Orenstein, D. R. Tibbetts, J.\nL. Reno, and M. P. Lilly Nat. Phys. 8, 153 (2012).\n[33] F. G. G. Hernandez, L. M. Nunes, G. M. Gusev, and A.\nK. Bakarov, Phys. Rev. B 88, 161305(R) (2013).\n[34] F. G. G. Hernandez, G. M. Gusev, and A. K. Bakarov,\nPhys. Rev. B 90, 041302(R) (2014).\n[35] E. Bernardes, J. Schliemann, M. Lee, J. C. Egues, and\nD. Loss, Phys. Rev. Lett. 99, 076603 (2007).\n[36] E. Bernardes, J. Schliemann, C. Egues, and D. Loss,\nPhys. Status Solidi C 3, 4330 (2006).\n[37] R. S. Calsaverini, E. Bernardes, J. C. Egues, and D. Loss,\nPhys. Rev. B 78, 155313 (2008).\n[38] J. Fu and J. C. Egues, Phys. Rev. B 91, 075408 (2015).\n[39] L. Yang, J. Orenstein, and D.-H. Lee, Phys. Rev. B 82,\n155324 (2010).\n[40] G. J. Ferreira, F. G. G. Hernandez, P. Altmann, G. Salis,\nPhys. Rev. B 95, 125119 (2017).\n[41] J. Fu, P. H. Penteado, M. O. Hachiya, D. Loss, and J. C.\nEgues, Phys. Rev. Lett. 117, 226401 (2016).\n[42] The barriers were made of short-period AlAs/GaAs su-\nperlattices in order to shield the doping ionized impuri-\nties and e\u000eciently enhance the mobility, more details in\nK.-J. Friedland, R. Hey, H. Kostial, R. Klann, and K.\nPloog, Phys. Rev. Lett. 77, 4616 (1996).\n[43] The measured low-temperature resistance was 10 \n for\nthe sample and 100 \n for the lateral Ohmic contacts.\nThus, the e\u000bective V ipin the channel is about 10 times6\nsmaller than the applied voltage. The values of V ipused\nfor the plots and calculations are the e\u000bective ones.\n[44] The model neglects changes in the intersubband\nscattering-time that depends on the QW symmetry as\nstudied in: N. C. Mamani, G. M. Gusev, T. E. Lamas,\nand A. K. Bakarov, O. E. Raichev, Phys. Rev. B 77,\n205327 (2008); N. C. Mamani, G. M. Gusev, E. C. F. da\nSilva, O. E. Raichev, A. A. Quivy, and A. K. Bakarov,\nPhys. Rev. B 80, 085304 (2009).\n[45] R. Winkler, Spin-Orbit Coupling E\u000bects in Two-\nDimensional Electron and Hole Systems , Springer Tracts\nin Modern Physics (Springer, Berlin, 2003).\n[46] M. Studer, M. P. Walser, S. Baer, H. Rusterholz, S.\nSch on, D. Schuh, W. Wegscheider, K. Ensslin, and G.\nSalis Phys. Rev. B 82, 235320 (2010).\n[47] L. Meier, G. Salis, I. Shorubalko, E. Gini, S. Sch on and\nK. Ensslin, Nat. Phys. 3, 650 (2007).\n[48] J. M. Kikkawa and D. D. Awschalom, Nature 397, 139\n(1999).[49] Y. Kato, R. C. Myers, A. C. Gossard and D. D.\nAwschalom, Nature 427, 50 (2004).\n[50] B. M. Norman, C. J. Trowbridge, J. Stephens, A. C.\nGossard, D. D. Awschalom, and V. Sih, Phys. Rev. B\n82, 081304(R) (2010).\n[51] V. K. Kalevich and V. L. Korenev, Zh. Eksp. Teor. Fiz .\n52, 859 (1990) [JETP Lett. 52, 230 (1990)].\n[52] R. I. Dzhioev, K. V. Kavokin, V. L. Korenev, M. V.\nLazarev, B. Ya. Meltser, M. N. Stepanova, B. P. Za-\nkharchenya, D. Gammon, and D. S. Katzer, Phys. Rev.\nB66, 245204 (2002).\n[53] S. A. Crooker, M. Furis, X. Lou, P. A. Crowell, D. L.\nSmith, C. Adelmann, and C. J. Palmstr\u001cm, J. Appl.\nPhys. 101, 081716 (2007).\n[54] S. Ullah, G. M. Gusev, A. K. Bakarov, and F. G. G.\nHernandez, J. Appl. Phys. 119, 215701 (2016).\n[55] F. G. G. Hernandez, S. Ullah, G. J. Ferreira, N. M. Kawa-\nhala, G. M. Gusev, and A. K. Bakarov, Phys. Rev. B 94,\n045305 (2016)."    },    {        "title": "1908.07766v2.Spin_orbit_coupled_quantum_memory_of_a_double_quantum_dot.pdf",        "content": "Spin-orbit-coupled quantum memory of a double quantum dot\nL. Chotorlishvili,1A. Gudyma,2J. Wätzel,1A. Ernst,2,3and J. Berakdar1\n1Institut für Physik, Martin-Luther Universität Halle-Wittenberg, D-06120 Halle/Saale, Germany\n2Max Planck Institute of Microstructure Physics, Weinberg 2, 06120 Halle/Saale, Germany\n3Institute for Theoretical Physics, Johannes Kepler University, Altenberger Strasse 69, 4040 Linz, Austria\n(Dated: October 29, 2019)\nThe concept of quantum memory plays an incisive role in the quantum information theory. As\nconfirmed by several recent rigorous mathematical studies, the quantum memory inmate in the\nbipartite system ρABcan reduce uncertainty about the part B, after measurements done on the\npartA. In the present work, we extend this concept to the systems with a spin-orbit coupling and\nintroducea notionof spin-orbitquantum memory. We self-consistently exploreUhlmann fidelity, pre\nand post measurement entanglement entropy and post measurement conditional quantum entropy\nof the system with spin-orbit coupling and show that measurement performed on the spin subsystem\ndecreases the uncertainty of the orbital part. The uncovered effect enhances with the strength of\nthe spin-orbit coupling. We explored the concept of macroscopic realism introduced by Leggett and\nGarg and observed that POVM measurements done on the system under the particular protocol are\nnon-noninvasive. For the extended system, we performed the quantum Monte Carlo calculations\nand explored reshuffling of the electron densities due to the external electric field.\nI. INTRODUCTION\nLet us consider a typical setting of a bipartite quantum\nsystem1, described by the density matrix ˆρABshared by\ntwo parties Alice ( A) and Bob ( B). Suppose Aperforms\ntwo consecutive measurements of Hermitian observables\nXandY. The uncertainty relation in the Robertson’s\nform2,3states that the product of the standard devia-\ntions is larger or equal to the expectation value of com-\nmutator/triangleX/triangleY≥1\n2/vextendsingle/vextendsingle/angbracketleftψ/vextendsingle/vextendsingle/bracketleftbig\nX,Y/bracketrightbig/vextendsingle/vextendsingleψ/angbracketright/vextendsingle/vextendsinglewith respect to the\nshared quantum state |ψ/angbracketright. From Bob’s perspective, the\nuncertainty in Alice measurement results depends how-\never on the nature of the quantum state/vextendsingle/vextendsingleψ/angbracketright, meaning\non whether/vextendsingle/vextendsingleψ/angbracketrightis entangled or separable. Berta et al.4\nshowed that, for Bob, entanglement may decrease the\nlower bound for the uncertainty in Alice measurements\noutcome. That means Bob may become more certain\nabout the results of Alice measurements done on her part\nA, if Bob subsystem Bis entangled with A. More specif-\nically, Bob uncertainty concerning Alice measurements\nis determined by the quantum conditional entropy de-\nfined as follows S/parenleftbig\nA|B/parenrightbig\n=S/parenleftbig\nˆ/rho1AB/parenrightbig\n−S/parenleftbig\ntrA/parenleftbig\nˆ/rho1AB/parenrightbig/parenrightbig\n. Here\nˆ/rho1ABis the post-measurement density matrix of the bi-\npartite system and S/parenleftbig\nˆ/rho1AB/parenrightbig\nis the von Neumann entropy\nS/parenleftbig\nˆ/rho1AB/parenrightbig\n=−tr/parenleftbig\nˆ/rho1ABln/parenleftbig\nˆ/rho1AB/parenrightbig/parenrightbig\n. A negative quantum con-\nditional entropy is in contrast to conventional wisdom\nregarding entropy of a classical system, as classically, en-\ntropy is an extensive quantity and hence the entropy of\nthe whole system should not be lower than the entropy\nof the subsystem.\nPhysical realizations of the subsystems AandBare\ndiverse1. For instance, AandBcould be two electrons\nin a double quantum dot, each hosting spin and orbital\ndegrees of freedom. The spin and orbital degrees of free-\ndom of electrons may be entangled. However, due to the\nSOinteraction, spindegreesoffreedommaybeentangled\nwith orbital degrees as well. Thus, in a double quantumdot the quantum state of the system may hold spin, or-\nbitalandspin-orbitentanglement. Suchsolid-state-based\nsystems are very attractive due to their scalability and\nthe various tools at hand to control, read and write in-\nformation. When the two dots are in close proximity\ntunneling sets in, as well as orbital correlation mediated\nby the Coulomb interaction. In the presence of a spin or-\nbital (SO) interaction, of the Rashba type5for instance,\nthe spin becomes affected by the orbital motion. Our in-\nterest in this work is devoted to the information obtain-\nable on the orbital subsystem through a measurements\ndone on the spin subsystem and how the quality of this\ninformation is affected by the SO interaction. We note\nin this context, that the strength of SO interaction in a\nsemiconductor-based quantum structures can be tuned\nto certain extent by a static electric field. The orbital\npart can be assessed for example by exploiting the differ-\nent relaxation times of the electrons pair to a reservoir\ndepending on their spin state6,7.\nIn what follows, we show that measurement done on\nthe spin subsystem reduces the uncertainty about the\norbital part, meaning that information about one sub-\nsystem can be extracted indirectly through the measure-\nment done on another subsystem. We also study the\nuncertainty of two incompatible measurements done on\nthe spin subsystem and explore factor of quantum mem-\nory. Namely, we prove that when the system is in a pure\nstate, quantum memory reduces the uncertainty of two\nincompatible spin measurements.\nOur focus here is on the case when Alice does two in-\ncompatible quantum measurements on one of the parts\nof the bipartite system. Say, Alice measures two non-\ncommuting spin components of the qubit at her hand.\nThe concept of quantum memory states that the entan-\nglement between qubits of Alice and Bob permits Bob\nto reduce the upper limit of the uncertainty bound of\nthe measurements done by Alice. In what follows we\nhighlight and illustrate by direct numerical examples thearXiv:1908.07766v2  [quant-ph]  28 Oct 20192\nsubtle effects of spin-orbital coupling on quantum mem-\nory. In particular, spin-orbit-coupled systems may store\nthree different types of entanglements related to spin-\nspin, spin-orbit, and orbit-orbit parts. We prove that\nonly the entire entanglement allows a reduction of the\nupper bound for the uncertainty. After the elimination\nof the spin-orbit and the orbit-orbit parts, the residual\nspin-spin entanglement is not enough to reduce the un-\ncertainty. Our result is generic and is expected to apply\nto a broad class of materials with spin-orbital coupling.\nThe paper is structured as follows: In section IIwe re-\nview the experimental studies relevant to our work. Sec-\ntionIIIpresents the theoretical model, in section IVwe\ndescribe measurement procedures and explore the post-\nmeasurement states, in section Vwe study the Uhlmann\nfidelity between pre and post measurement states of the\nspin subsystem and evaluate post-measurement quantum\nconditional entropy, in the section VIwe study effect of\nthe SO interaction on the quantum discord, in the sec-\ntionVIIwestudynon-invasivemeasurements, inthesec-\ntionVIIIwe explore the impact of Coulomb interaction,\nin the section IXwe present results of quantum Monte\nCarlo calculations for the electron density obtained for\nthe extended system, in the section Xwe discuss the\nproblem of quantum memory and conclude the work.\nII. EXPERIMENTALLY FEASIBLE POVM\nPROTOCOLS IN QUANTUM DOTS\nQuantum dots are assumed as an experimental real-\nization to the theory below, similar to the first quan-\ntum computing scheme based on spins in isolated quan-\ntum dots which was proposed by D. Loss and D. Di-\nVincenzo8, see also9–11and references therein. An ulti-\nmate goal of a quantum gate and a quantum information\nprotocol is to read out and record the outcome state.\nSeveral types of local spin measurements were realized\nexperimentally12–15. In quantum dots, the spin can be\nmeasured selectively through the spin-to-charge conver-\nsion16–19. Our focus is on the experimentally feasible\nspin POVM (positive operator-valued measure) measure-\nments see20, the only measurement considered through-\noutthepresentwork. Fundamentallimitsfornondestruc-\ntive measurement of a single spin in a quantum dot was\nstudied recently21.\nHere we briefly look back to experimental and concep-\ntual aspects of the POVM spin measurement in quan-\ntum dots. The spin-resolved filter (barrier) permits to\npass through the gate only electrons with particular spin\norientation, i.e., transmits |1/angbracketrightand bans the|0/angbracketright. Thus if\nparticlepasses,forsureweknowtheprojectionofitsspin.\nHowever, what is detected in the experiment is not a spin\nprojection but a charge. Through the change in the elec-\ntric charge recognized by the electrometer, we infer the\ninformation that electron has passed through the filter.\nThe beauty of this scheme is simpleness that allows in-\ntroducing POVM projectors ΠA\n0=|0/angbracketright/angbracketleft0|A,ΠA\n1=|1/angbracketright/angbracketleft1|Afor a quantum dot in the formal theoretical discussion.\nOf interest is also the single-shot measurement scheme\nthat can selectively access the singlet or the triplet\ntwo-electron states in a quantum dot7. The scheme\nexploits the different coupling strengths of the triplet\nand singlet states to the reservoir. Therefore, charge\nrelaxation times are different too 1/ΓT<1/ΓS. A\nnondestructive measurement is achieved by an electric\npulse of duration τthat shifts temporally the chemical\npotential of the dot with respect to the Fermi level\nof the reservoir, where 1/ΓT< τ < 1/ΓSis chosen.\nFor the dot in the singlet electron state, the time\nis too short for tunneling, but the triplet state may\ntunnel. If two consecutive measurements are done\nwithin a time interval shorter than relaxation time T1,\nthe measurement procedure is invasive, meaning that\nthe outcome of the second measurement depends on\nthe first measurement. The measurement procedure is\nnoninvasive if the time interval between measurements\nexceeds ∆τ1,2> T 1. In the experiment7values of the\nparameters for GaAs/Al xGa1−xheterostructure read:\n1/ΓT= 5µs,τ= 20µs, and 1/ΓS= 100µs.\nIII. MODEL OF THE SYSTEM\nThe issue of quantum memory has already been ad-\ndressed for a number of model systems22–29. Here, we\nfocus particularly on the interacting two-electron dou-\nble quantum dots5,30–40. We self-consistently explore the\nUhlmann fidelity, pre and post measurement entangle-\nment entropy, and post measurement conditional quan-\ntum entropy of the system and show that a measure-\nment performed on the spin subsystem decreases the un-\ncertainty of the orbital part. This effect becomes more\nprominent with increasing the strength of SO coupling.\nWe consider a double quantum dot characterized by\na rather strong quantum confinement potential in the\nyandzdirections, see pictorial Fig.(1). For sin-\ngle particle we use the orbitals Ψnx,ny,nz(x,y,z ) =\nNφnx(x)Yny(y)Znz(z)where/angbracketleftφnx|φn/primex/angbracketright=δnx,n/primex,\n/angbracketleftYny|Yn/primey/angbracketright=δny,n/primeyand/angbracketleftZnz|Zn/primez/angbracketright=δnz,n/primez. We consider\na situation with a strong confinement in yandzdirec-\ntions such that only the lowest subbands with ny= 0\nandnz= 0are occupied. The relevant dynamics takes\nplace in the xdirection only, subject to the effective one-\ndimensionalpotential V(x). TheHamiltonianofconfined\nelectrons reads\nˆH0=−¯h2\n2m∗N/summationdisplay\nn=1∂2\n∂x2n+N/summationdisplay\nn<mVC(rrrn,rrrm) +ˆHSO\n+N/summationdisplay\nn=1(V(xn) +eExn). (1)\nHere,V(x) =m∗ω2min/bracketleftbig\n(x−∆/2)2,(x+ ∆/2)2/bracketrightbig\n/2is3\nFIG. 1. Schematic representation of the considered double\nquantum dot system in the presence of an external electric\nfield and spin-orbit coupling. In a quasi-one-dimensional con-\nductive channel, two quantum dots are created and controlled\nby two local gates: ”gate 1” and ”gate 2”. The quantum con-\nfinement in y-direction is strong enough such that the only\nlowest subband state ny= 0is occupied. The ”gate 0” is\nused to control tunnel junction between the dots (that mimics\nthe changing of the interdot distance). The applied constant\nelectric field, polarized in x-direction, is represented by the\nsky blue arrow.\nthe double-dot confinement potential,\nˆHSO=−iαN/summationdisplay\nn=1∂\n∂xnˆσy\nn+BN/summationdisplay\nn=1ˆσz\nn,(2)\nis the Rashba SO term with the magnetic field, ∆ =/lscriptd0\nis the inter-dot distance with the dimensionless scaling\nfactor/lscript,m∗is the electron effective mass, eis the ab-\nsolute value of the electron charge, and the strength of\nthe constant static electric field is E. The trial magnetic\nfieldBapplied along the z-axis has no particular effect\non the phase of the wave function in 1D case but specifies\nthe quantization axis and shifts energy levels. Note that\nthe Coulomb potential VC(rrr1,rrr2) =e2/κ|rrr1−rrr2|, where\nκis the dielectric constant, still depends on the six co-\nordinatesrrr1= (x1,y1,z1)andrrr2= (x2,y2,z2)and will\nbe reduced to the (x1,x2)-variables later in the text. In\nwhat follows we introduce dimensionless units by setting\nx1,2→x1,2/d0,β=m∗ωd2\n0/¯h,ˆH0→ˆH0/(¯h2/(m∗d2\n0)),\nE0=m∗ed3\n0E/¯h2. We adopt parameters of the semicon-\nductormaterialGaAs, β= 1inthiscasecorrespondstoa\nconfinement energy ¯hω= 11.4meV,d0= 10nm. Dimen-\nsionless electric field E0= 1is equivalent to the applied\nexternal field of the strength E= 1.1V/µm. The single\nparticleenergylevelsofasingledot( /lscript= 0)aregiventhen\nbyεn=β(n+ 1/2). For the sake of simplicity to start,\nwe neglect the Coulomb term and treat SO coupling per-\nturbatively. The antisymmetric total wave-functions are\npresented as direct products of the orbital and spin parts\nΨ(1)\nn=ψS\nn⊗χA/parenleftbig\n1,2/parenrightbig\nandΨ(2)\nn=ψA\nn⊗χS/parenleftbig\n1,2/parenrightbig\n, where/vextendsingle/vextendsingleχT+\nS/parenleftbig\n1,2/parenrightbig/angbracketrightbig\n=|1↑/angbracketrightbig\n|2↑/angbracketrightbig\n,/vextendsingle/vextendsingleχT−\nS/parenleftbig\n1,2/parenrightbig/angbracketrightbig\n=|1↓/angbracketrightbig\n|2↓/angbracketrightbig\n,\n|χT0\nS/parenleftbig\n1,2/parenrightbig/angbracketrightbig\n=1√\n2/parenleftbig\n|1↑/angbracketright|2↓/angbracketright+|1↓/angbracketright|2↑/angbracketright/parenrightbig\nand the asym-metric spin function read/vextendsingle/vextendsingleχA/parenleftbig\n1,2/parenrightbig/angbracketrightbig\n=1√\n2/parenleftbig\n|1↑/angbracketright|2↓/angbracketright−\n|1↓/angbracketright|2↑/angbracketright/parenrightbig\n. We define the two-electron symmetric and\nantisymmetric coordinate wave functions of a double\nquantum well as follows:\n/vextendsingle/vextendsingleψS,A\nn,n/prime/angbracketrightbig\n=1/radicalbig\n2(1±S2)/bracketleftbig\nψL,n/parenleftbig\nx1/parenrightbig\nψR,n/prime/parenleftbig\nx2/parenrightbig\n±ψL,n/parenleftbig\nx2/parenrightbig\nψR,n/prime/parenleftbig\nx1/parenrightbig/bracketrightbig\n,(3)\nwhereψL,n/parenleftbig\nx/parenrightbig\nis the single particle wave function cor-\nresponding to the left dot and quantum state n, while\nψR,n/prime/parenleftbig\nx/parenrightbig\nis associated with the right dot and quantum\nstaten/primeandS=/angbracketleftψL,n|ψR,n/prime/angbracketrightistheoverlapintegral. The\nresults of the exact numerical calculations (not shown)\nhave confirmed that for the large values of the parameter\nβ/greatermuch1overlap integral is zero S= 0and tunneling pro-\ncesses are not activated. As will be shown bellow effect of\nthe Coulomb term in this case is less relevant and can be\nneglected safely. In the double quantum dot, the equilib-\nrium positions of electrons shifts along the x-axis by the\ndistance±d0/2. The harmonic oscillator eigenfunctions\nfollow Heitler-London ansatz41,42and readψL(R),n/parenleftbig\nx/parenrightbig\n=\nφL(R),n/parenleftbig\nx/parenrightbig\n, whereφL(R),n/parenleftbig\nx/parenrightbig\n=1√\n2nn!/parenleftbigg\nβ\nπ/parenrightbigg1/4\n×\nexp/parenleftBig\n−β(x±1/2−d)2\n2/parenrightBig\nHn/parenleftbig\nx√β/parenrightbig\n,Hn/parenleftbig\nx√β/parenrightbig\nis Hermite\npolynomial and d=eE/d 0m∗ω2.\nThe energy spectrum of unperturbed system is de-\nscribed by the sum of energies of non-interacting oscil-\nlatorsEN=β/parenleftbig\nn+ 1/2/parenrightbig\n+β/parenleftbig\nn/prime+ 1/2/parenrightbig\n, N = (n,n/prime)\nand we introduced the following notations for brevity./vextendsingle/vextendsingleΦN/angbracketrightbig\n=/vextendsingle/vextendsingleψS,A\nn,n/prime/angbracketrightbig\n⊗/vextendsingle/vextendsingleχA,S/angbracketrightbig\n, see Eq. (3).\nThe presence of the SO term mixes different spin sec-\ntors and spin and orbital states. Considering/vextendsingle/vextendsingleΨM/angbracketrightbig\n=/vextendsingle/vextendsingleψA\n0,1/angbracketrightbig\n⊗/vextendsingle/vextendsingleχT+\nS/angbracketrightbig\nas an unperturbed wave function we ob-\ntain:\n/vextendsingle/vextendsingleΦM/angbracketrightbig\n=/vextendsingle/vextendsingleψA\n0,1/angbracketrightbig\n⊗/vextendsingle/vextendsingleχT+\nS/angbracketrightbig\n+α\n2√β/parenleftbigg1\n2/vextendsingle/vextendsingleψS\n1,1/angbracketrightbig\n−/vextendsingle/vextendsingleψS\n0,0/angbracketrightbig/parenrightbigg\n⊗/vextendsingle/vextendsingleχA/angbracketrightbig\n. (4)\nUsing Eq. (4) and tracing out orbital (spin) parts\nwe construct reduced density matrix of the spin (or-\nbital) subsystem respectively: ˆρS=1\nZ/braceleftbig/vextendsingle/vextendsingleχT+\nS/angbracketrightbig/angbracketleftbig\nχT+\nS/vextendsingle/vextendsingle+\n5α2\n16β/vextendsingle/vextendsingleχA/angbracketrightbig/angbracketleftbig\nχA/vextendsingle/vextendsingle/bracerightbig\n,ˆρor=1\nZ/parenleftbigg/vextendsingle/vextendsingleψA\n0,1/angbracketrightbig/angbracketleftbig\nψA\n0,1/vextendsingle/vextendsingle+α2\n16β/vextendsingle/vextendsingleψS\n1,1/angbracketrightbig/angbracketleftbig\nψS\n1,1/vextendsingle/vextendsingle+\nα2\n4β/vextendsingle/vextendsingleψS\n0,0/angbracketrightbig/angbracketleftbig\nψS\n0,0/vextendsingle/vextendsingle/parenrightbigg\n, whereZ= 1 +5α2\n16β.\nIV. POVM MEASUREMENTS AND\nPOST-MEASUREMENT STATES\nThe generic state of two non-interacting particles is a\nproduct state. Therefore a density matrix of a system\ncan be factorized as a direct product of density matrices\nof individual particles. After tracing out states of one4\nparticle, product state leaves a system in a pure state\nwith a zero entropy. However, in the case of fermions,\nthe Pauli principle imposes quantum correlation even in\nthe absence of interaction. As for an interacting bipartite\nsystem in most of the cases, the state is entangled43–50.\nTracing out part of bipartite entangled state results in\na mixed state and finite entropy. Quantum correlation\nmanifests in continuous variables systems as well51–58.\nTherefore under certain conditions, we expect the or-\nbital part to be entangled. After setting theoretical ma-\nchinery of the problem, we proceed with the information\nmeasures of uncertainty and quantum correlations in the\nsystem. In particular, we specify pre-measurement von\nNeumann entropy of the orbital and spin subsystems:\nS/parenleftbig\nˆρor/parenrightbig\n=−tr/parenleftbig\nˆρorln/parenleftbig\nˆρor/parenrightbig/parenrightbig\n=−α2\n16βln/parenleftbigα2\n16β/parenrightbig\n−α2\n4βln/parenleftbigα2\n4β/parenrightbig\nandS/parenleftbig\nˆρs/parenrightbig\n=−tr/parenleftbig\nˆρsln/parenleftbig\nˆρs/parenrightbig/parenrightbig\n=−5α2\n16βln/parenleftbig5α2\n16β/parenrightbig\nrespec-\ntively, where we assumed that Z≈1. Spin and or-\nbital von Neumann entropies increase with the Rashba\nSO coupling constant β. Let us assume that Alice per-\nforms POVM measurement59on the first qubit at her\nhand (in what follows we use notations A= 1andB= 2\nfor the first and second qubit).\nAfter measurement the initial sate collapses either to\nthe post-measurement state\n|Ψ(1)\nAB/angbracketrightbig\n=/parenleftbig\nΠA\n0/circlemultiplytextIB/parenrightbig/vextendsingle/vextendsingleΦM/angbracketrightbig\n/radicalBig/angbracketleftbig\nΦM/vextendsingle/vextendsingle/parenleftbig\nΠA\n0/circlemultiplytextIB/parenrightbig/vextendsingle/vextendsingleΦM/angbracketrightbig,(5)\nwith probability\nΓA\n0=/angbracketleftΦM/vextendsingle/vextendsingle/parenleftbig\nΠA\n0/circlemultiplytextIB/parenrightbig/vextendsingle/vextendsingleΦM/angbracketrightbig\n/angbracketleftΦM/vextendsingle/vextendsingleΦM/angbracketrightbig, (6)\nor to the post-measurement state\n|Ψ(2)\nAB/angbracketrightbig\n=/parenleftbig\nΠA\n1/circlemultiplytextIB/parenrightbig/vextendsingle/vextendsingleΦM/angbracketrightbig\n/radicalBig/angbracketleftbig\nΦM/vextendsingle/vextendsingle/parenleftbig\nΠA\n1/circlemultiplytextIB/parenrightbig/vextendsingle/vextendsingleΦM/angbracketrightbig,(7)\nwith probability\nΓA\n1=/angbracketleftΦM/vextendsingle/vextendsingle/parenleftbig\nΠA\n1/circlemultiplytextIB/parenrightbig/vextendsingle/vextendsingleΦM/angbracketrightbig\n/angbracketleftΦM/vextendsingle/vextendsingleΦM/angbracketrightbig. (8)\nPOVM operators have form: ΠA\n0=|0/angbracketright/angbracketleft0|A,ΠA\n1=\n|1/angbracketright/angbracketleft1|A,IBis the identity operator acting on the\nqubitB. Easy to see that ΓA\n0= 5α2//parenleftbig\n10α2+\n32β/parenrightbig\n;ΓA\n1=/parenleftbig\n5α2+ 32β/parenrightbig\n//parenleftbig\n10α2+ 32β/parenrightbig\nand ΓA\n1>\nΓA\n0. After involved calculations we derive explicit\nexpressions for the post-measurement reduced or-\nbital ˆ/rho1(1,2)\nAB = trs/parenleftbig\n|Ψ(1,2)\nAB/angbracketrightbig/angbracketleftbig\nΨ(1,2)\nAB|/parenrightbig\nand spin ˆσ(1,2)\nAB =\ntror/parenleftbig\n|Ψ(1,2)\nAB/angbracketrightbig/angbracketleftbig\nΨ(1,2)\nAB|/parenrightbig\ndensity matrices:\nˆσ(1)\nAB=/vextendsingle/vextendsingle1↓/angbracketright|2↑/angbracketright/angbracketleft2↑|/angbracketleft1↓|,\nˆσ(2)\nAB=1\n1 + 5α2/32β/parenleftbigg\n|1↑/angbracketright|2↑/angbracketright/angbracketleft2↑|/angbracketleft1↑|\n+5α2\n32β|1↑/angbracketright|2↓/angbracketright/angbracketleft2↓|/angbracketleft1↑|/parenrightbigg\n, (9)and\nˆ/rho1(1)\nAB=4\n5/parenleftbigg1\n4|ψs\n1,1/angbracketright/angbracketleftψs\n1,1|+|ψs\n0,0/angbracketright/angbracketleftψs\n0,0|\n−1\n2|ψs\n1,1/angbracketright/angbracketleftψs\n0,0|−1\n2|ψs\n0,0/angbracketright/angbracketleftψs\n1,1|/parenrightbigg\n,\nˆ/rho1(2)\nAB=1\n1 + 5α2/32β/parenleftbigg\n|ψA\n0,1/angbracketright/angbracketleftψA\n0,1|+α2\n8β/parenleftbigg1\n4|ψs\n1,1/angbracketright/angbracketleftψs\n1,1|\n+|ψs\n0,0/angbracketright/angbracketleftψs\n0,0|−1\n2|ψs\n1,1/angbracketright/angbracketleftψs\n0,0|−1\n2|ψs\n0,0/angbracketright/angbracketleftψs\n1,1|/parenrightbigg/parenrightbigg\n.(10)\nSinceα/√βis the small parameter, with high accuracy\nwe set 1 + 5α2/32β≈1.\nV. THE UHLMANN FIDELITY AND THE\nPOST-MEASUREMENT QUANTUM\nCONDITIONAL ENTROPY\nBefore study the entropy of the system we explore\nthe fidelity between pre and post measurement states\nof the spin subsystem. In its most general form, the\nfidelity problem was formulated by Uhlmann. For de-\ntails about the Uhlmann fidelity, we refer to59. At\nfirst, let us perform the standard purification proce-\ndure of the pre ˆρsand post measurement spin den-\nsity matrices ˆσAB. We adopt spectral decompositions\nˆρs=/summationtext\nxPX/parenleftbig\nx/parenrightbig\n|x/angbracketright/angbracketleftx|AB,ˆσAB=/summationtext\nxQY/parenleftbig\ny/parenrightbig\n|y/angbracketright/angbracketlefty|AB, as-\nsociated with the ensembles {PX,|x/angbracketright},{QY,|Y/angbracketright}where\nrandom variables x,ybelong to the different alphabets.\nA purification with respect to the reference system R\nwe define as follows: |φρ/angbracketrightR,AB=/summationtext\nx/radicalBig\nPX/parenleftbig\nx/parenrightbig\n|x/angbracketrightR|x/angbracketrightAB,\n|φσ/angbracketrightR,B= trA/parenleftbigg/summationtext\ny/radicalBig\nQY/parenleftbig\ny/parenrightbig\n|y/angbracketrightR|y/angbracketrightAB/parenrightbigg\n. The Uhlmann\nfidelity between two mixed states read:\nF/parenleftbig\ntrA/parenleftbig\nˆσAB/parenrightbig\n,ˆρs/parenrightbig\n=\nmax (Uσ,Uρ)/vextendsingle/vextendsingle/angbracketleftφσ|/parenleftbig\nU†\nρUσ/parenrightbigR⊗IAB|φσ/angbracketrightR,AB|2.(11)\nThe Uhlmann theorem59facilitates calculation of\nUhlmann fidelity and finally, we deduce:\nF/parenleftbig\ntrA/parenleftbig\nˆσ(2)\nAB/parenrightbig\n,ˆρs/parenrightbig\n=/parenleftbigg\n1 +5α2\n16β√\n2/parenrightbigg\n×1\n1 + 5α2/32β×1\n1 + 5α2/16β. (12)\nFor the small SO coupling we obtain asymptotic esti-\nmation:F/parenleftbig\ntrA/parenleftbig\nˆσ(2)\nAB/parenrightbig\n,ˆρs/parenrightbig\n≈1−5/parenleftbig\n3−√\n2/parenrightbig\nα2/32β. As\nwe see the distance between pre and post-measurement\nstates decays with SO constant α.\nTaking into account Eq. (9), Eq. (10) and probabili-\nties Eq. (6), Eq. (8) we deduce the expression of the post\nmeasurement von Neumann entropy of the spin subsys-\ntemS/parenleftbig\nˆσ(2)\nAB/parenrightbig\n=−ΓA\n15α2\n32βln/parenleftbig5α2\n32β/parenrightbig\n. The difference between\npre and post measurement entropies of the spin subsys-\ntemS/parenleftbig\nˆσS/parenrightbig\n−S/parenleftbig\nˆσAB/parenrightbig\n=−5α2\n16βln/parenleftbig5α2\n16β/parenrightbig\n+ ΓA\n15α2\n32βln/parenleftbig5α2\n32β/parenrightbig5\nFIG. 2. Dependence of the von Neumann entropy on the system’s and field parameters: (a) Planes describe the pre (green)\nand post (orange) measurement entropies as a function of the spin-orbit coupling strength αand the applied external electric\nfieldE0. The effective inter-dot distance is ∆ = 0.8d0. (b) The difference between pre and post measurement entropies of the\norbital subsystem S/parenleftbig\nˆρor/parenrightbig\n−S/parenleftbig\nˆ/rho1AB/parenrightbig\nas a function of the spin-orbit coupling α, plotted for different inter-dot distances.\nis positive for any ΓA\n1<1and that means POVM\nmeasurement decreases the entropy of the spin subsys-\ntem. The post measurement von Neumann entropy\nof the orbital subsystem S/parenleftbig\nˆ/rho1AB/parenrightbig\n=−ΓA\n15α2\n32βln/parenleftbig5α2\n32β/parenrightbig\n.\nThe difference between pre and post measurement en-\ntropies of the orbital subsystem S/parenleftbig\nˆρor/parenrightbig\n−S/parenleftbig\nˆ/rho1AB/parenrightbig\n=\n−α2\n16βln/parenleftbigα2\n16β/parenrightbig\n−α2\n4βln/parenleftbigα2\n4β/parenrightbig\n+ ΓA\n15α2\n32βln/parenleftbig5α2\n32β/parenrightbig\n. Easy to see\nthat−α2\n4βln/parenleftbigg\nα2\n4β/parenrightbigg\n>5α2\n32βln/parenleftbig5α2\n32β/parenrightbig\nand the entropy after\nmeasurement decreases S/parenleftbig\nˆρor/parenrightbig\n−S/parenleftbig\nˆ/rho1AB/parenrightbig\n>0. An inter-\nesting observation is that POVM measurement done on\nthe spin subsystem through the SO interaction decreases\nthe von Neumann entropy of the orbital part. As larger\nis SO coupling constant α, larger is a decrement of the\norbital entropy. Even more surprising is that measure-\nment equates post-measurement von Neumann entropies\nof the spin and orbital subsystems S/parenleftbig\nˆσ(2)\nAB/parenrightbig\n=S/parenleftbig\nˆ/rho1AB/parenrightbig\n.\nThe pair concurrence of the spin subsystem is defined\nas follows:C=max/parenleftbig\n0,√R1−√R2−√R3−√R4/parenrightbig\n, with\nthe eigenvalues Rn, n= 1,...4of the following matrix\nR= ˆρS/parenleftbig\nˆσy\n1⊗ˆσy\n2/parenrightbig/parenleftbig\nˆρS/parenrightbig∗/parenleftbig\nˆσy\n1⊗ˆσy\n2/parenrightbig\n. For the pre and post\nmeasurement concurrence we obtain: C/parenleftbig\nˆρS/parenrightbig\n= 5α2/16β,\nC/parenleftbig\nˆσAB/parenrightbig\n= 0. The measurement disentangles the system.\nTaking into account Eq. (9), for the von Neumann en-\ntropy of the subsystem Bwe deduce S/parenleftbig\ntrA/parenleftbig\nˆσ(2)\nAB/parenrightbig/parenrightbig\n=\n−ΓA\n15α2\n32βln/parenleftbig5α2\n32β/parenrightbig\n. Therefore for the post-measurement\nconditional quantum entropy we obtain: S/parenleftbig\nA|B/parenrightbig\n=\nS/parenleftbig\nˆσAB/parenrightbig\n−S/parenleftbig\ntrA/parenleftbig\nˆσAB/parenrightbig/parenrightbig\n= 0.\nNotethattheconditionalquantumentropyofthepost-measurement state quantifies the uncertainty that Bob\nhas about the outcome of Alice’s measurement. The zero\nvalue ofS/parenleftbig\nA|B/parenrightbig\nmeans that Bob has precise information\nabout the measurement result. The same effect we see in\nthe post-measurement entropy of the orbital subsystem.\nDue to the SO coupling, the measurement done on the\nqubitAreduces the post-measurement entropy of the or-\nbital subsystem. The effect of the electric field, Coulomb\ninteraction and tunneling processes activated in case of\nsmall inter-dot distance may modify this picture. In case\nof a short inter-dot distance (i.e., parameter βis an order\nof1<β < 10) effect of the quantum tunneling processes\nassisted by the Coulomb interaction becomes important.\nWe explore this problem using numeric methods.\nVI. QUANTUM GENERALIZATION OF\nCONDITIONAL ENTROPY\nThe quantum mutual information quantifies all cor-\nrelations in the quantum bipartite system, and at least\npart of these correlations can be classical. Vedral, Zurek,\nand others asked the question: whether it is possible to\nhave a more subtle notion of quantum correlations rather\nthan the entanglement60–63. For pure states, the quan-\ntum discord is equivalent to the quantum entanglement\nbut is distinct when the state is mixed. The central issue\nfor the quantum discord is a quantum generalization of\nconditional entropy (the quantity that is distinct from\nthe conditional quantum entropy). Quantum discord is\nquantified as follows:\nDA/parenleftbig\nˆρs/parenrightbig\n= min/braceleftbig\nΠB\nj/bracerightbig/braceleftbigg\nS(A)−S(A,B)−S/parenleftbigg/summationdisplay\njpjtr/braceleftbigg/parenleftbig\nΠB\njˆρsΠB\nj/parenrightbig\n/pj/bracerightbigg\nln/braceleftbigg/parenleftbig\nΠB\njˆρsΠB\nj/parenrightbig\n/pj/bracerightbigg/parenrightbigg/bracerightbigg\n. (13)6\nWe omit details of calculations and present result for the\ndifference between pre and post-measurement quantum\ndiscordsDA/parenleftbig\nˆρs/parenrightbig\n−DA/parenleftbig\nˆσAB/parenrightbig\n=5α2\n32βln 4. From this re-\nsult we see that similar to the pre and post-measurement\nvon Neumann entropy, quantum discord decreases after\nmeasurement.\nVII. QUANTUM WITNESS AND\nNON-INVASIVE MEASUREMENTS\nThe concept of macroscopic realism introduced by\nLeggett and Garg64postulates criteria of noninvasive\nmeasurability. In the sequence of two measurements, the\nfirst blind measurement has no consequences on the out-\ncome of the second measurement if a system is classi-\ncal. However, in the case of quantum systems, any mea-\nsurement alters the state of the system independently\nfrom the fact was the first measurement either blind (i.e.,\nthe measurement result is not recorded) or not. Simi-\nlar to the Bell’s inequalities, quantumness (i.e., entangle-\nment) may violate the macroscopic realism and Leggett-\nGarg inequalities. This effect is widely discussed in the\nliterature65–67. Quantum witness introduced in68is the\ncentral characteristic of invasive measurements. In this\nsection we discus particular type of non-invasive mea-\nsurement protocol.\nThe directly measured probability we define in terms\nof the following expression PB/parenleftbig\n1/parenrightbig\n= tr/braceleftbig\nΠB\n1N/parenleftbig\nˆρs/parenrightbig/bracerightbig\n. Here\nΠB\n1=|1/angbracketright/angbracketleft1|Bis the operator of the projective measure-\nment done on the second qubit, N/parenleftbig\nˆρs/parenrightbig\n=/summationtext\ni=1,2ˆLiˆρsˆL†\ni\nis the trace preserving quantum channel with Kraus\noperatorsL1=|0/angbracketright/angbracketleft1|A,L2=|1/angbracketright/angbracketleft0|A. The blind-\nmeasurement probability we define as follows: GB/parenleftbig\n1/parenrightbig\n=\ntr/braceleftbig\nΠB\n1N/parenleftbigˆΞs/parenrightbig/bracerightbig\n, where density matrix of the system after\nblind measurement is given by ˆΞs=/summationtext\ni=0,1ΠA\niˆρsΠA\ni.\nThe quantum witness that quantifies invasiveness of the\nquantum measurements is given by the formula:\nW=|tr/braceleftbig\nΠB\n1/parenleftbig\nN/parenleftbig\nˆρs/parenrightbig\n−N/parenleftbigˆΞs/parenrightbig/parenrightbig/bracerightbig\n|. (14)\nDirect calculations for our system shows that\nGB/parenleftbig\n1/parenrightbig\n=PB/parenleftbig\n1/parenrightbig\n=1\nZ/parenleftbig\n1 + 5α2/32β/parenrightbig\n.(15)\nThe quantum witness is zero W= 0indicating that mea-\nsurements done on the system within this particular pro-\ncedure are noninvasive.\nVIII. THE EFFECT OF THE COULOMB\nINTERACTION\nWe study the case of a short inter-dot distance and\nthe effect of the Coulomb interaction. We utilize the\nconfigurational interaction (CI) ansatz and perform ex-\ntensive numerical calculations. Utilizing the single par-\nticle orbitals we solve the stationary one-dimensionalSchrödinger equation in absence of the Coulomb term.\nBy means of numerical diagonalization of the single par-\nticle Hamiltonian ˆHSP=−∂2\nx/2+V(x)+xE0discretized\non a fine space grid we obtain the single-particle orbitals\nφi(x) =ci,Lφi,L(x)+ci,Rφi,R(x)andenergies εi. Wecon-\nstructed the symmetric and anti-symmetric two-electron\nwave functions labeled as (+,−)and evaluate matrix el-\nements of ˆH0including the Coulomb term:\n/angbracketleftΥn/prime\n0|ˆH0|Υn\n0/angbracketright=/epsilon10\nnδn,n/prime+/angbracketleftΥn/prime\n0|VC|Υn\n0/angbracketrightδb,b/prime,(16)\nwherebis a part of the index n={i,j,b = (+,−)}. Note\nthat two-electron wave-functions |Υn\n0/angbracketrightaccounts the ef-\nfect of doubly-occupied states as well. We diagonalize\nthe matrix Eq. (16) and obtain the fully correlated two-\nelectron eigenstates and eigenvalues {|Ψn/angbracketright,/epsilon1n}. For a\ngoodconvergenceandreliabilityofthespectrum, weused\n80 single-particle orbitals |φi/angbracketright. In the last step we add\nthe Rashba SOC term to Eq. (1). The matrix elements\nof the total Hamiltonian including the SO term read\n/angbracketleftΨn/primeχ/prime|ˆH0+ˆHSO|Ψnχ/angbracketright=/epsilon1nδn,n/primeδχχ/prime\n−iα2/summationdisplay\ni=1/angbracketleftΨ/prime\nn|∂xi|Ψn/angbracketright/angbracketleftχ/prime|σy\ni|χ/angbracketright.\n(17)\nHere the last term corresponds to the Rashba SO interac-\ntion in the matrix form. The spin-resolved two-electron\neigenstates|Φn/angbracketrightand the corresponding energies Enwe\nobtain by means of numerical diagonalization of Eq. (17).\nIn Fig. 2 (a) pre and post measurement von Neu-\nmann entropies are plotted for the fixed inter-dote dis-\ntance ∆ = 0.8d0. The values of the applied electric\nfield and SO coupling are in the range of 0< E 0<8\nand 0< α < 1, i.e.E0= 1corresponds to a static\nelectric field≈1.1V/µm,β= 1is equivalent to the\nrealistic parameters adopted for GaAs ¯hω= 11,4meV,\nm∗= 0,067me,d0= 10nm. The post-measurement\nvon Neumann entropy S(ˆρAB)is always smaller than\nthe pre-measurement entropy S(ˆρor). Electric field en-\nhances both pre and post-measurement entropies and\nforE0>2we see the saturation effect. The difference\nbetween pre and post measurement entropies of the or-\nbital subsystem S(ˆρor)−S(ˆρAB)at different inter-dot\ndistances is plotted in Fig. 2 (b). As we see measure-\nment done on the spin subsystem reduces the entropy\nof the orbital part. Reduction of entropy increases with\nthe strength of SO coupling term α. On the other hand\nat small inter-dot distances the differences between pre-\nand post-measurement entropies of the orbital subsystem\nS(ˆρor)−S(ˆρAB)is smaller due to the Coulomb term. We\nnote that when SO coupling is zero, the reduced density\nmatrix of the orbital subsystem corresponds to the pure\nstate, and therefore von Neumann entropy is zero see\nFig. 2 (a) and Fig. 2 (b). The maximum value of the von\nNeumann entropy depends on the number of the quan-\ntum states involved in the process and reaches the peak7\nfor the maximally mixed state. Strong electric field in-\ncreases the amount of the involved quantum states, and\nvon Neumann entropy reaches its saturation value. Nu-\nmerical calculations frankly confirm the validity and cor-\nrectness of analytical results.\nIX. QUANTUM MONTE CARLO\nCALCULATIONS ELECTRON DENSITY\nHere we consider the extended system 1 of four elec-\ntrons in the four-dot confinement potential\nV(x) =m∗ω2\n2min/bracketleftbigg\n(x−3∆\n2)2,(x−∆\n2)2,(x+∆\n2)2,\n(x+3∆\n2)2/bracketrightbigg\n. (18)\nWe perform numerical simulations with the modi-\nfied continuous spin Variational Monte Carlo (CSVMC)\nalgorithm69,70. We introduce auxiliary spinor vector\nχ†(s) =N/productdisplay\nn=1⊗[eisn,e−isn], (19)\nwheresnare auxiliary variables defined on [0,2π)with\nthe periodic boundary conditions. We construct effec-\ntive scalar wave-function as a scalar product of the wave-\nfunctions and vectors χ†(s)follows\nψ(x,s) =χ†(s)·Ψ(x) (20)\nThe inverse transformation is done through the integra-\ntion over the auxiliary variables\nΨ(x) =1\n(2π)N/integraldisplayN/productdisplay\nn=1dsnψ(x,s)χ(s).(21)\nWe write the effective Schrödinger equation for the scalar\nwave-function\ni¯h∂\n∂tψ(x,s) =ˆHeffψ(x,s), (22)\nwhere ˆHeffis the effective Hamiltonian. We construct\nthe effective Hamiltonian replacing the spinor operators\nby the following operators:\nˆσx= cos(2s)−sin(2s)∂\n∂s, (23a)\nˆσy= sin(2s) + cos(2s)∂\n∂s. (23b)\nˆσz=−i∂\n∂s, (23c)This transformation expands Hilbert space of the prob-\nlem from the particular spin sector s=1\n2to arbitrary\nspin. Toselectthedesiredsolutionfromthesetofallpos-\nsible solutions we introduce equality constraints s2=3\n4\nands2\nz=1\n4. First of these constraints fulfils automati-\ncally while second one in introduced directly into the La-\ngrange function L=/angbracketleftBig\nˆH/angbracketrightBig\n+λ/parenleftbig/angbracketleftbig\nˆσ2\nz/angbracketrightbig\n−1/parenrightbig\n. The Lagrange\nfunction is constructed through minimization of the ef-\nfective Hamiltonian with the additional spin-variable ki-\nnetic energy term. doing an importance sampling with a\nguiding wave function ψT. We use trial wave-function in\nthe Slater-Jastrow form\nψT=DeJ, (24)\nwhereJis the Jastrow factor which takes into ac-\ncount correlations introduced through the many-body\ninteraction71. The none-interacting part is chosen to be\na Slater determinant spanned in the lowest lying single-\nparticle orbitals. Single particle orbitals are approxi-\nmated with product of Heitler-London functions41,42and\nphase calculated from the homogeneous system.\nIn Fig. 3 the pair distribution function is shown for dif-\nferent values of the trapping parameter β= 1,3and10.\nThe Rashba constant is equal to α= 0.4. In the regime,\nβ/greatermuch1the electronic density is localized in the vicin-\nity of minimums of the trapping potential and the over-\nlap between neighboring trapping gaps is small (Fig. 3c).\nWith the decrease of trapping barrier, electrons delocal-\nize (Figs. 3a-b). The effect of the electric field is pre-\nsented in Fig. 4. Pair distribution function for β= 1,3\nand10andE0= 1is plotted in Fig. 4. Coordinates x1\nandx2are centered at the minimums of the V(x) +eEx.\nAt the finite electric field minimums in the direction of\nthe field are energetically preferable and total density\nshifts towards the direction of the applied field.\nX. QUANTUM MEMORY\nWe already showed that measurement done on the spin\nsubsystemreducestheorbitalentanglement. Nowwedis-\ncuss a different scheme when Alice does two incompatible\nquantum measurements on one of the parts of the bipar-\ntite system, and we try to answer the question: whether\nthe spin-orbit interaction can reduce Bob’s total uncer-\ntainty about measurements done by Alice?\nWe consider two cases: in the first case Alice and Bob\nshare the total density matrix of a bipartite SO system\nEq. (4)8\nFIG. 3. The pair distribution function ρ(x1,x2)at zero magnetic and electric fields for various values of the trapping parameter\nβ. The Rashba constant α= 0.4. Parameter βdefines the inverse localization length of wave-function. When the localization\nlength exceeds the distance between minimums of trapping potential V(x)the electronic wave-function is delocalized. (3a-b)\nDelocalized pair distribution function for β= 1andβ= 3. With the increase of βpotential barrier between minimums of the\npotential increases and electrons become localized in the minimums of the potential. (3c) Localized pair distribution function\nforβ= 10.\nFIG. 4. The pair distribution function. The applied electric field steers the electronic density to the edge of the sample in\nthe direction of the field. The pair distribution function of the first two particles ρ(x1,x2)is centered at the minimums of\nV(x) +eExforE0= 1. Various values of the trapping parameter are considered β= 1,3,10. The Rashba constant is equal to\nα= 0.4.\nˆρAB=1\nZ/braceleftbigg/vextendsingle/vextendsingleψA\n0,1/angbracketrightbig/angbracketleftbig\nψA\n0,1/vextendsingle/vextendsingle⊗/vextendsingle/vextendsingleχT+\nS/angbracketrightbig/angbracketleftbig\nχT+\nS/vextendsingle/vextendsingle+5α2\n16β/parenleftbigg1√\n5/vextendsingle/vextendsingleψS\n1,1/angbracketrightbig\n−2√\n5/vextendsingle/vextendsingleψS\n0,0/angbracketrightbig/parenrightbigg/parenleftbigg1√\n5/angbracketleftbig\nψS\n1,1/vextendsingle/vextendsingle−2√\n5/angbracketleftbig\nψS\n0,0/vextendsingle/vextendsingle/parenrightbigg\n⊗/vextendsingle/vextendsingleχA/angbracketrightbig/angbracketleftbig\nχA/vextendsingle/vextendsingle\n+√\n5α\n4√β/parenleftbigg/vextendsingle/vextendsingleψA\n0,1/angbracketrightbig/parenleftbigg1√\n5/angbracketleftbig\nψS\n1,1/vextendsingle/vextendsingle−2√\n5/angbracketleftbig\nψS\n0,0/vextendsingle/vextendsingle/parenrightbigg\n⊗/vextendsingle/vextendsingleχT+\nS/angbracketrightbig/angbracketleftbig\nχA/vextendsingle/vextendsingle+/parenleftbigg1√\n5/vextendsingle/vextendsingleψS\n1,1/angbracketrightbig\n−2√\n5/vextendsingle/vextendsingleψS\n0,0/angbracketrightbig/parenrightbigg/angbracketleftbig\nψA\n0,1/vextendsingle/vextendsingle⊗/vextendsingle/vextendsingleχA/angbracketrightbig/angbracketleftbig\nχT+\nS/vextendsingle/vextendsingle/parenrightbigg/bracerightbigg\n,(25)\nor they share the mixed state formed after trac-\ning the orbital subsystem ˆρS\nAB=1\nZ/braceleftbig/vextendsingle/vextendsingleχT+\nS/angbracketrightbig/angbracketleftbig\nχT+\nS/vextendsingle/vextendsingle+\n5α2\n16β/vextendsingle/vextendsingleχA/angbracketrightbig/angbracketleftbig\nχA/vextendsingle/vextendsingle/bracerightbig\n, where/vextendsingle/vextendsingleχT+\nS/parenleftbig\n1,2/parenrightbig/angbracketrightbig\n=|1↑/angbracketrightbig\n|2↑/angbracketrightbig\n,/vextendsingle/vextendsingleχA/parenleftbig\n1,2/parenrightbig/angbracketrightbig\n=1√\n2/parenleftbig\n|1↑/angbracketright|2↓/angbracketright−| 1↓/angbracketrightbig\n|2↑/angbracketrightbig/parenrightbig\n,Z= 1 +5α2\n16β\nand the functions/vextendsingle/vextendsingleψA\n0,1/angbracketrightbig\n,/vextendsingle/vextendsingleψS\n1,1/angbracketrightbig\n,/vextendsingle/vextendsingleψS\n0,0/angbracketrightbig\nare defined in the\nsection III. Bob sends Alice subsystem Aand Alice does\ntwo incompatible measurements (she measures σz\nAand\nσx\nA). The post-measurement states are given by4:\nˆρRB=/summationdisplay\nn|ψn/angbracketright/angbracketleftψn|⊗IBˆρS\nAB|ψn/angbracketright/angbracketleftψn|⊗IB,\nˆρQB=/summationdisplay\nn|φn/angbracketright/angbracketleftφn|⊗IBˆρS\nAB|φn/angbracketright/angbracketleftφn|⊗IB.(26)HereIBis the identity operator acting on the subsystem\nB, and|ψ1/angbracketright=|1/angbracketright,|ψ2/angbracketright=|0/angbracketright,|φ1,2/angbracketright=1√\n2(|0/angbracketright±|1/angbracketright)\nare the eigenfunctions of σz\nA,σx\nA. Bob has not precise\ninformation about the measurements of Alice. The un-\ncertainty about outcomes of measurements is quantified\nthrough the entropy measure:\nS/parenleftbig\nR|B/parenrightbig\n+S/parenleftbig\nQ|B/parenrightbig\n≥ln/parenleftbigg1\nc/parenrightbigg\n+S(A|B/parenrightbig\n.(27)\nHerec=maxn,m|/angbracketleftψm|φn/angbracketright|2,S/parenleftbig\nR|B/parenrightbig\n=−ˆρRBln ˆρRB+\ntrR(ˆρRB) ln trR(ˆρRB)is the conditional quantum infor-\nmation, and the last term S(A|B/parenrightbig\ndescribes the effect\nof the quantum memory, meaning that for a negative\nS(A|B/parenrightbig\n<0quantum memory reduces the uncertainty.9\nNote that negative conditional quantum entropy points\nto entanglement in the system. The inverse statement\nis not always true, i.e., not for all entangled states, con-\nditional quantum entropy is negative. Nevertheless, for\na pure state Eq. (25) shared by Alice and Bob ˆρAB, the\nconditionalquantumentropycanbecalculatedexplicitly,\nand it reads:\nS(A|B/parenrightbig\nˆρAB=5α2\n32βZln/parenleftbigg5α2\n32βZ/parenrightbigg\n+5α2+ 32β\n32βZln/parenleftbigg5α2+ 32β\n32βZ/parenrightbigg\n. (28)Easy to see that for any 0< α <√βconditional quan-\ntum entropy is negative for a pure state S(A|B/parenrightbig\nˆρAB<0.\nThis fact means that correlations stored in the spin-orbit\nsystem work as quantum memory and reduce the uncer-\ntainties of measurements. However in case of the mixed\nstate ˆρS\nABsituation is different. All entropy measures can\nbe calculated analytically, and we deduce:\nS(A|B)ˆρS\nAB=−1\nZln/parenleftbigg1\nZ/parenrightbigg\n+5α2\n32βZln/parenleftbigg5α2\n32βZ/parenrightbigg\n−5α2\n16βZln/parenleftbigg5α2\n16βZ/parenrightbigg\n+5α2+ 32β\n32βZln/parenleftbigg5α2+ 32β\n32βZ/parenrightbigg\n,(29)\nS(R|B) =−1\nZln/parenleftbigg1\nZ/parenrightbigg\n−5α2\n32βZln/parenleftbigg5α2\n32βZ/parenrightbigg\n+5α2+ 32β\n32βZln/parenleftbigg5α2+ 32β\n32βZ/parenrightbigg\n, (30)\nS(Q|B) =5α2\n32βZln/parenleftbigg5α2\n32βZ/parenrightbigg\n+5α2+ 32β\n32βZln/parenleftbigg5α2+ 32β\n32βZ/parenrightbigg\n−5α2+ 16β+/radicalbig\n25α4+ 256β2\n32βZln/parenleftBigg\n5α2+ 16β+/radicalbig\n25α4+ 256β2\n32βZ/parenrightBigg\n−5α2+ 16β−/radicalbig\n25α4+ 256β2\n32βZln/parenleftBigg\n5α2+ 16β−/radicalbig\n25α4+ 256β2\n32βZ/parenrightBigg\n. (31)\nFor strong confinement potential and realistic SO\ncouplingα/√β < 1,Z= 1 +5α2\n16β≈1. Apparently\nS(A|B)ˆρS\nAB>0meaning that spin orbit coupling\nin case of a mixed states enhances uncertainties of\nmeasurements. The reason for this nontrivial effect\nis the following. The total entanglement between\nsubsystems AandBstored in the state ˆρABconsists\nof spin-spin, spin-orbit, and orbit-orbit contributions.Averaging over the orbital states eliminates part of\nentanglement. The residual spin-spin entanglement is\nnot enough to reduce the uncertainty of measurements\ndone by Alice. To support this statement, we compare\nthe entanglement stored in the states ˆρABand ˆρS\nAB.\nThe reduced density matrix ˆρA=trB(ˆρAB)has theform:\nˆρA=/parenleftBig1\n2Zα2\n32β/vextendsingle/vextendsingleψL,1/angbracketrightbig/angbracketleftbig\nψL,1/vextendsingle/vextendsingle+1\n2Zα2\n32β/vextendsingle/vextendsingleψR,1/angbracketrightbig/angbracketleftbig\nψR,1/vextendsingle/vextendsingle+1\n2Z4α2\n32β/vextendsingle/vextendsingleψL,0/angbracketrightbig/angbracketleftbig\nψL,0/vextendsingle/vextendsingle+1\n2Z4α2\n32β/vextendsingle/vextendsingleψR,0/angbracketrightbig/angbracketleftbig\nψR,0/vextendsingle/vextendsingle/parenrightBig\n⊗/vextendsingle/vextendsingle1↓/angbracketrightbig/angbracketleftbig\n1↓/vextendsingle/vextendsingle\n+/parenleftbigg1\n2Zα2\n32β/vextendsingle/vextendsingleψL,1/angbracketrightbig/angbracketleftbig\nψL,1/vextendsingle/vextendsingle+1\n2Z/parenleftbigg\n1 +α2\n32β/parenrightbigg/vextendsingle/vextendsingleψR,1/angbracketrightbig/angbracketleftbig\nψR,1/vextendsingle/vextendsingle+1\n2Z/parenleftbigg\n1 +4α2\n32β/parenrightbigg/vextendsingle/vextendsingleψL,0/angbracketrightbig/angbracketleftbig\nψL,0/vextendsingle/vextendsingle+1\n2Z4α2\n32β/vextendsingle/vextendsingleψR,0/angbracketrightbig/angbracketleftbig\nψR,0/vextendsingle/vextendsingle/parenrightbigg\n⊗/vextendsingle/vextendsingle1↑/angbracketrightbig/angbracketleftbig\n1↑/vextendsingle/vextendsingle (32)\nThe corresponding von Neumann entropy:\nS(ˆρA) =−3\n2Zα2\n32βln/parenleftbigg1\n2Zα2\n32β/parenrightbigg\n−3\n2Z4α2\n32βln/parenleftbigg1\n2Z4α2\n32β/parenrightbigg\n−1\n2Z/parenleftbigg\n1 +α2\n32β/parenrightbigg\nln/parenleftbigg1\n2Z/parenleftbigg\n1 +α2\n32β/parenrightbigg/parenrightbigg\n−1\n2Z/parenleftbigg\n1 +4α2\n32β/parenrightbigg\nln/parenleftbigg1\n2Z/parenleftbigg\n1 +4α2\n32β/parenrightbigg/parenrightbigg\n. (33)10\nThe von Neumann entropy for the state ˆρs\nA=trB(ˆρs\nAB):\nS(ˆρS\nA) =−1\nZ/parenleftbigg\n1 +5α2\n32β/parenrightbigg\nln/parenleftbigg1\nZ/parenleftbigg\n1 +5α2\n32β/parenrightbigg/parenrightbigg\n−5α2\n32βZln/parenleftbigg5α2\n32βZ/parenrightbigg\n. (34)\nApparently S(ˆρA)> S(ˆρs\nA)and part of entanglement is\nlost after averaging over the orbital states.\nXI. CONCLUSIONS\nCombining the analytical method with extensive nu-\nmeric calculations, in the present work, we studied the\ninfluence of the spin-orbit interaction on the effect of\nquantum memory. We observed that measurement done\non the spin subsystem through the spin-orbit channel al-\nlows to extract information about the orbital subsystem\nand reduce the entropy of the orbital part. On the handresult of two incompatible measurements done on the\nspin subsystem, depends on the fact whether the den-\nsity matrix of the system is pure or mixed. In the case\nof pure states, the spin-orbit coupling works as quantum\nmemory and reduces the uncertainty about the measure-\nment results, whereas, in the case of mixed states, the\nspin-orbit coupling enhances uncertainty.\nXII. ACKNOWLEDGMENT\nWe acknowledge financial support from DFG through\nSFB 762.\n1G. Alber, T. Beth, M. Horodecki, P. Horodecki,\nR. Horodecki, M. Rötteler, H. Weinfurter, R. Werner,\nand A. Zeilinger, Quantum information: An introduction\nto basic theoretical concepts and experiments , Vol. 173\n(Springer, 2003).\n2W. Heisenberg, Zeitschrift für Physik 43, 172 (1927).\n3H. P. Robertson, Phys. Rev. 34, 163 (1929).\n4M. Berta, M. Christandl, R. Colbeck, J. M. Renes, and\nR. Renner, Nature Physics 6, 659 (2010).\n5A. Brataas and E. I. Rashba, Phys. Rev. B 84, 045301\n(2011).\n6J.R.Petta, A.C.Johnson, A.Yacoby, C.M.Marcus, M.P.\nHanson, and A. C. Gossard, Phys. Rev. B 72, 161301\n(2005).\n7T.Meunier,I.T.Vink,L.H.WillemsvanBeveren, F.H.L.\nKoppens, H. P. Tranitz, W. Wegscheider, L. P. Kouwen-\nhoven, and L. M. K. Vandersypen, Phys. Rev. B 74,\n195303 (2006).\n8D. Loss and D. P. DiVincenzo, Phys. Rev. A 57, 120\n(1998).\n9B. E. Kane, Nature 393, 133 (1998).\n10P. Recher, E. V. Sukhorukov, and D. Loss, Phys. Rev.\nLett.85, 1962 (2000).\n11R. Hanson and G. Burkard, Phys. Rev. Lett. 98, 050502\n(2007).\n12J. Elzerman, R. Hanson, L. W. Van Beveren, B. Witkamp,\nL. Vandersypen, and L. P. Kouwenhoven, nature 430, 431\n(2004).\n13R. Hanson, L. H. W. van Beveren, I. T. Vink, J. M. Elz-\nerman, W. J. M. Naber, F. H. L. Koppens, L. P. Kouwen-\nhoven, and L. M. K. Vandersypen, Phys. Rev. Lett. 94,\n196802 (2005).\n14R.HansonandD.D.Awschalom,Nature 453,1043(2008).\n15R. Hanson, L. P. Kouwenhoven, J. R. Petta, S. Tarucha,\nand L. M. K. Vandersypen, Rev. Mod. Phys. 79, 1217(2007).\n16T. Fujisawa, D. G. Austing, Y. Tokura, Y. Hirayama, and\nS. Tarucha, Nature 419, 278 (2002).\n17R. Hanson, B. Witkamp, L. M. K. Vandersypen, L. H. W.\nvan Beveren, J. M. Elzerman, and L. P. Kouwenhoven,\nPhys. Rev. Lett. 91, 196802 (2003).\n18J. A. Folk, R. M. Potok, C. M. Marcus,\nand V. Umansky, Science 299, 679 (2003),\nhttps://science.sciencemag.org/content/299/5607/679.full.pdf.\n19W. Lu, Z. Ji, L. Pfeiffer, K. West, and A. Rimberg, Nature\n423, 422 (2003).\n20D. D. Awschalom, D. Loss, and N. Samarth, Semicon-\nductor Spintronics and Quantum Computation (Springer\nScience & Business Media, 2002).\n21D. Scalbert, Phys. Rev. B 99, 205305 (2019).\n22F. Adabi, S. Haseli, and S. Salimi, EPL (Europhysics Let-\nters)115, 60004 (2016).\n23G. Adesso, T. R. Bromley, and M. Cianciaruso, Journal\nof Physics A: Mathematical and Theoretical 49, 473001\n(2016).\n24A. K. Pati, M. M. Wilde, A. R. U. Devi, A. K. Rajagopal,\nand Sudha, Phys. Rev. A 86, 042105 (2012).\n25P. J. Coles and M. Piani, Phys. Rev. A 89, 022112 (2014).\n26T.Nemoto, F.Bouchet, R.L.Jack, andV.Lecomte,Phys.\nRev. E 93, 062123 (2016).\n27P. J. Coles, M. Berta, M. Tomamichel, and S. Wehner,\nRev. Mod. Phys. 89, 015002 (2017).\n28S. Wehner and A. Winter, New Journal of Physics 12,\n025009 (2010).\n29R. Kumar and S. Ghosh, Journal of Physics B: Atomic,\nMolecular and Optical Physics 51, 165301 (2018).\n30E. I. Rashba and A. L. Efros, Phys. Rev. Lett. 91, 126405\n(2003).\n31E. I. Rashba and A. L. Efros, Applied Physics Letters 83,\n5295 (2003), https://doi.org/10.1063/1.1635987.11\n32D. V. Khomitsky and E. Y. Sherman, Phys. Rev. B 79,\n245321 (2009).\n33D. V. Khomitsky, L. V. Gulyaev, and E. Y. Sherman,\nPhys. Rev. B 85, 125312 (2012).\n34E. Y. Sherman and D. Sokolovski, New Journal of Physics\n16, 015013 (2014).\n35S. Faniel, T. Matsuura, S. Mineshige, Y. Sekine, and\nT. Koga, Phys. Rev. B 83, 115309 (2011).\n36P. I. Tamborenea and H. Metiu, Phys. Rev. Lett. 83, 3912\n(1999).\n37D. Stepanenko, M. Rudner, B. I. Halperin, and D. Loss,\nPhys. Rev. B 85, 075416 (2012).\n38J. Pawłowski, P. Szumniak, A. Skubis, and S. Bednarek,\nJournal of Physics: Condensed Matter 26, 345302 (2014).\n39Q. Li, L. Cywiński, D. Culcer, X. Hu, and S. Das Sarma,\nPhys. Rev. B 81, 085313 (2010).\n40X. Hu and S. Das Sarma, Phys. Rev. A 61, 062301 (2000).\n41W. Heitler and F. London, Zeitschrift für Physik 44, 455\n(1927).\n42P. Fazekas, Series in Modern Condensed Matter Physics:\nLecture Notes on Electron Correlation and Magnetism\n(World Scientific Publishing Co. Pte. Ltd.: River Edge,\nNJ, 1999).\n43J. Stolze and D. Suter, Quantum Computing: a short\ncourse from Theory to Experiment (John Wiley & Sons,\n2008).\n44K. Życzkowski, P. Horodecki, A. Sanpera, and M. Lewen-\nstein, Phys. Rev. A 58, 883 (1998).\n45W. K. Wootters, Phys. Rev. Lett. 80, 2245 (1998).\n46H. Ollivier and W. H. Zurek, Phys. Rev. Lett. 88, 017901\n(2001).\n47R. Horodecki, P. Horodecki, M. Horodecki, and\nK. Horodecki, Rev. Mod. Phys. 81, 865 (2009).\n48L.-M. Duan, G. Giedke, J. I. Cirac, and P. Zoller, Phys.\nRev. Lett. 84, 2722 (2000).\n49L.Amico, R.Fazio, A.Osterloh, andV.Vedral,Rev.Mod.\nPhys. 80, 517 (2008).\n50S. L. Braunstein and P. van Loock, Rev. Mod. Phys. 77,\n513 (2005).51R. Simon, Phys. Rev. Lett. 84, 2726 (2000).\n52C. Weedbrook, S. Pirandola, R. García-Patrón, N. J. Cerf,\nT. C. Ralph, J. H. Shapiro, and S. Lloyd, Rev. Mod. Phys.\n84, 621 (2012).\n53C.Silberhorn, P.K.Lam, O.Weiß, F.König, N.Korolkova,\nand G. Leuchs, Phys. Rev. Lett. 86, 4267 (2001).\n54G. Adesso and F. Illuminati, Journal of Physics A: Math-\nematical and Theoretical 40, 7821 (2007).\n55F. Toscano, A. Saboia, A. T. Avelar, and S. P. Walborn,\nPhys. Rev. A 92, 052316 (2015).\n56G. Adesso, A. Serafini, and F. Illuminati, Phys. Rev. A\n70, 022318 (2004).\n57E. Shchukin and W. Vogel, Phys. Rev. Lett. 95, 230502\n(2005).\n58R. F. Werner and M. M. Wolf, Phys. Rev. Lett. 86, 3658\n(2001).\n59M. M. Wilde, Quantum information theory (Cambridge\nUniversity Press, 2013).\n60V. Vedral, Rev. Mod. Phys. 74, 197 (2002).\n61L. Henderson and V. Vedral, Journal of Physics A: Math-\nematical and General 34, 6899 (2001).\n62H. Ollivier and W. H. Zurek, Phys. Rev. Lett. 88, 017901\n(2001).\n63W. H. Zurek, Phys. Rev. A 67, 012320 (2003).\n64A.J.LeggettandA.Garg,Phys.Rev.Lett. 54,857(1985).\n65G. Schild and C. Emary, Phys. Rev. A 92, 032101 (2015).\n66K. Wang, G. C. Knee, X. Zhan, Z. Bian, J. Li, and P. Xue,\nPhys. Rev. A 95, 032122 (2017).\n67D. Avis, P. Hayden, and M. M. Wilde, Phys. Rev. A 82,\n030102 (2010).\n68C.-M. Li, N. Lambert, Y.-N. Chen, G.-Y. Chen, and\nF. Nori, Scientific reports 2, 885 (2012).\n69C. A. Melton, M. Zhu, S. Guo, A. Ambrosetti, F. Pederiva,\nand L. Mitas, Phys. Rev. A 93, 042502 (2016).\n70C. A. Melton and L. Mitas, Phys. Rev. E 96, 043305\n(2017).\n71N. D. Drummond, M. D. Towler, and R. J. Needs, Phys.\nRev. B 70, 235119 (2004)."    },    {        "title": "1812.08884v1.Non_local_Spin_charge_Conversion_via_Rashba_Spin_Orbit_Interaction.pdf",        "content": "Non-local Spin-charge Conversion via Rashba Spin-Orbit Interaction\nJunji Fujimoto\u0003\nInstitute for Chemical Research, Kyoto University, Uji, Kyoto 611-0011, Japan and\nRIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan\nGen Tatara\nRIKEN Center for Emergent Matter Science (CEMS) and RIKEN\nCluster for Pioneering Research (CPR), Wako, Saitama 351-0198, Japan\n(Dated: December 24, 2018)\nAbstract\nWe show theoretically that conversion between spin and charge by spin-orbit interaction in metals occurs\neven in a non-local setup where magnetization and spin-orbit interaction are spatially separated if electron\ndi\u000busion is taken into account. Calculation is carried out for the Rashba spin-orbit interaction treating the\ncoupling with a ferromagnet perturbatively. The results indicate the validity of the concept of e\u000bective spin\ngauge \feld (spin motive force) in the non-local con\fguration. The inverse Rashba-Edelstein e\u000bect observed\nfor a trilayer of a ferromagnet, a normal metal and a heavy metal can be explained in terms of the non-local\ne\u000bective spin gauge \feld.\n1arXiv:1812.08884v1  [cond-mat.mes-hall]  20 Dec 2018(F)Ferromagnetmagnetization\nRasbha spin-orbit interaction\nat the NM-HM interface\nNonmagnetic Metal\n(NM)\nHeavy Metal\n(HM)FIG. 1. Schematic \fgure of a trilayer of a ferromagnet (F), a normal metal (NM) and a heavy metal (HM).\nThe Rashba spin-orbit interaction is localized at the NM-HM interface.\nI. INTRODUCTION\nThe objective of spintronics is to manipulate spins by electric means and vice versa. For gen-\nerating spin accumulation and spin current, several methods have been experimentally established\nin the last two decades, including the spin pumping e\u000bect1{4, where magnetization precession of a\nferromagnet (F) is used to generate spin current into a normal metal (NM) in a F-NM junction.\nFor electric detection of spin current, so called the inverse spin Hall e\u000bect induced by spin-orbit\ninteraction of heavy metal is widely used5.\nAnother electric detection of spin is by interfacial Rashba spin-orbit interaction, called the\ninverse Rashba-Edelstein e\u000bect. The e\u000bect, the reciprocal e\u000bect of the current-induced spin po-\nlarization studied theoretically by Edelstein6, has been experimentally demonstrated in a trilayer\nof a ferromagnet, a normal metal and a heavy metal (HM) (Fig. 1)7. The Rashba interaction is\nexpected to be localized at the interface of NM and HM, and the normal metal works as a spacer to\nseparate the magnetization and the Rashba interaction. The current observed was argued to sup-\nport the spin current picture, in which a spin current generated by spin pumping e\u000bect propagates\nthrough the normal metal, forming spin accumulation at the NM-HM interface, \fnally resulting in\na current as a result of inverse Rashba-Edelstein e\u000bect.\nTheoretically, current generation indicates existence of an e\u000bective electric \feld (motive force).\nIn the present magnetic systems, it is the one driving electron spin, namely, e\u000bective spin gauge \feld\nor spin motive force. E\u000bective spin gauge \feld has been known to arise for a slowly-varying magnetic\ntexture in ferromagnets, as the texture gives rise to a phase (spin Berry's phase) for electron spin\nwave function as a result of sdexchange interaction8,9. The spin Berry's phase generates a current\nwhen magnetization has dynamics besides spatial texture. The concept of e\u000bective spin gauge \feld\nwas shown to be generalized to include spin-orbit interactions that is linear in the wave vector,\nlike the Rashba interaction10{12. It was demonstrated that magnetization Mand the Rashba\n\feld\u000bgive rise to an e\u000bective spin gauge \feld proportional to \u000b\u0002M, leading to an e\u000bective\n2electric \feld and current proportional to \u000b\u0002_M. It was also pointed out that spin relaxation leads\nto a perpendicular e\u000bective spin electric \feld, \u000b\u0002(M\u0002_M)13. The latter component leads to\na direct current (DC) for a precessing magnetization, while the former induces only alternating\ncurrent (AC). The DC is in the direction perpendicular to both \u000band the average of M, which\nagrees with the geometry of the experimentally-observed inverse Rashba-Edelstein e\u000bect. The\nabove theories, however, do not directly apply to the experimental situations with a spacer layer,\nas the coexistence of magnetization and Rashba interaction is assumed in the theories.\nThe objective of the paper is to demonstrate theoretically that the concept of e\u000bective electric\n\feld can be generalized to describe non-local con\fgurations where the magnetization and the\nRashba interaction are spatially separated by a nonmagnetic metal. For charge transport in metals,\nlonger distance than the electron mean free path is possible due to electron di\u000busion; Current\ngeneration in disordered metals can therefore occur non-locally. As far as the di\u000busion is induced\nby elastic scattering conserving spin, the spin information is expected to be equally transported\nlong distance. In fact, long-range di\u000busive component of spin current induced by magnetization\ndynamics was studied in Ref.14. The spin-charge conversion e\u000bect was brie\ry mentioned there,\nbut assuming uniform Rashba interaction. It was also pointed out that long-range spin chirality\ncontributes to the anomalous Hall e\u000bect in disordered ferromagnet if the spatial size is less than\nspin di\u000busion length15, indicating that spin Berry's phase has long-range components. Moreover,\nspin Hall and inverse spin Hall e\u000bects were recently formulated in terms of non-local conversion of\nspin and electric current including electron di\u000busion16. In this representation, the observed spin\ndensity (or current) in the direct (inverse) spin Hall e\u000bect is directly related to the driving electric\n\feld (or spin pumping \feld) via a non-local response function of spin and electric current.\nIn this paper, we calculate electric current generated in the system of conduction electron with\nsdexchange interaction with a dynamic magnetization and the Rashba interaction having spatial\ndistributions. Although experiments are carried out in trilayers with rather sharp interfaces, we\nhere describe the slowly-varying case, which can be accessible straightforwardly by an expansion\nwith respect to the wave vectors of the two interactions. The vertex corrections (VCs) to the\ncorrelation function representing the e\u000bect turns out to contain a singular pole at slowly-varying\nlimit, indicating the di\u000busive nature. The di\u000busion propagator arising from this pole is shown\nto connect the information of the magnetization and the Rashba interaction even when they are\nspatially separated, resulting in a non-local current generation. The e\u000bect is interpreted in terms\nof the non-local component of spin electric \feld.\n3II. MODEL AND GREEN FUNCTIONS\nWe consider the following model Hamiltonian, H=H0+HR+Hsd(t) with\nH0=Z\ndr y(r) \n\u0000~2r2\n2me+uVNiX\ni=1\u000e(r\u0000Ri)!\n (r); (1a)\nHR=\u0000i~\n2\u000fijkZ\ndr\u000bi(r)\u0010\n y(r)\u001bk\u0000\nrj (r)\u0001\n\u0000\u0000\nrj y(r)\u0001\n\u001bk (r)\u0011\n; (1b)\nHsd(t) =\u0000Z\ndr y(r)\u0000\nM(r;t)\u0001\u001b\u0001\n (r); (1c)\nwhere (r) =t( \"(r);  #(r)) is the spinor form of the annihilation operator of electron with the\nmass being me, andH0consists of the kinetic term and non-magnetic impurity potential with the\nstrength being u.HRis the Rashba spin-orbit interaction with the spatial-dependent Rashba \feld\ndenoted by\u000b(r) = (\u000bx(r);\u000by(r);\u000bz(r)). Thesdexchange interaction is given by Hsd(t), where\nM(r;t) is the magnetization vector including the sdinteraction strength. We deal with H0as\nthe unperturbed Hamiltonian and treat HRandHsdperturbatively. Here, Vis the volume of the\nsystem,\u001b= (\u001bx;\u001by;\u001bz) is the vector form of the Pauli matrices, ~is the Planck constant divided\nby 2\u0019, and\u000fijkis the Levi-Civita symbol. We consider slowly-varying case with weak spatial\ndependencies of M(r) and\u000b(r). Our particular interest is the case where M(r) and\u000b(r) do not\ncoexist, such as a tri-layer structure composed of F, NM and HM.\nThe charge current density operator of the system is given by\nj(r) =\u0000e~\n2mei\u0010\n y(r)\u0000\nr (r)\u0001\n\u0000\u0000\nr y(r)\u0001\n (r)\u0011\n+e\u000b(r)\u0002s(r); (2)\nthe \frst two terms of which we call the current density for the normal velocity and the last term\nis called that for the anomalous velocity, where e(>0) is the elementary charge, and s(r) =\n y(r)\u001b (r) is the spin density. In the Fourier forms, the Hamiltonians of Eqs. (1) are given as\nH0=X\nk\u000fkcy\nkck+uX\nk;qNiX\ni=1eiq\u0001Ricy\nk+q\n2ck\u0000q\n2; (3a)\nHsd(t) =\u0000Zd!\n2\u0019e\u0000i!tX\nqS(\u0000q)\u0001M(q;!); (3b)\nHR=X\nk;q(\u000bq\u0002~k)\u0001cy\nk+q\n2\u001bck\u0000q\n2; (3c)\nwhere\u000fk=~2k2=2me, ands(q) is the Fourier component of the spin operator given by\ns(q) =1\nVX\nkcy\nk\u0000q\n2\u001bck+q\n2(4)\n4The current density in the Fourier form is given as\nj(q) =\u0000e\nVX\nk~k\nmecy\nk\u0000q\n2ck+q\n2+eX\nq0\u000bq0\u0002s(q\u0000q0); (5)\nwhere the \frst and second terms correspond to the currents of the normal and anomalous velocities,\nrespectively.\nWe denote the thermal Green function for the Hamiltonian H0+HRasGk;k0(i\u000fn), which is\nevaluated up to the \frst order with respect to HRas\nGk;k0(i\u000fn)'gk(i\u000fn)\u000ek;k0+~\n2gk(i\u000fn)\u001b\u0001\u0000\n\u000bk\u0000k0\u0002(k+k0)\u0001\ngk0(i\u000fn); (6)\nwhere\u000fn= (2n+ 1)\u0019kBTis the Matsubara frequency of fermion, gk(i\u000fn) is the thermal Green\nfunction for the Hamiltonian H0given by\ngk(i\u000fn) =1\ni\u000fn\u0000\u000fk+isgn(\u000fn)~=(2\u001c)(7)\nwith the signum function sgn( x). The lifetime of electron evaluated within the Born approximation\nis given as\n~\n2\u001c=\u0019niu2\u0017 (8)\nwith\u0017=\u0017(\u000fF) being the density of states (DOS) at the Fermi energy \u000fFof NM.\nIII. NON-LOCAL EFFECTIVE ELECTRIC FIELDS\nBy evaluating the non-local charge current induced by the magnetization dynamics, and using\nthe Drude conductivity, we show that the charge current is driven by the non-local e\u000bective electric\n\felds. We consider the exchange interaction up to the second order and the Rashba interaction in\nthe \frst order in this section.\nA. Linear response to exchange interaction\nFor the linear response of the charge current hj(r;t)i(1)to the external \feld n(r0;t0), where the\nexternal Hamiltonian is given by Hsd(t0), the current is calculated based on the Kubo formula17\n(see Appendix A 2) as\nhji(r;t)i(1)=i\n~Zt\n\u00001dt0\n[ji(r;t);Hsd(t0)]\u000b\n=Z\ndr0Z1\n\u00001dt0\u001f(1)\nij(r;r0;t\u0000t0)Mj(r0;t0); (9)\n5whereji(r;t) is the Heisenberg representation of Eq. (2), [ A;B] =AB\u0000BAis the communicator,\nh\u0001\u0001\u0001i is the thermal average for H0+HR, and the linear response coe\u000ecient \u001f(1)\nij(r;r0;t\u0000t0) is the\nretarded correction function between the charge current density and the spin density,\n\u001f(1)\nij(r;r0;t\u0000t0) =\u0000i\n~\u0012(t\u0000t0)\n[ji(r;t);sj(r0;t0)]\u000b\n(10)\nwith\u0012(t) being the Heaviside step function and sj(r0;t0) being the Heisenberg representation of\nthe spin density. Note that, since the Rashba \feld in the system has the spatial dependence, the\nlinear response coe\u000ecient cannot be expressed as \u001f(1)\nij(r\u0000r0;t\u0000t0), which also means that the\nspace translational symmetry is not assumed in the system.\nIn the Fourier form, the charge current is given as\nhji(q;!)i(1)=X\nq0\u001fR;(1)\nij(q;q0;!)Mj(q0;!) (11)\nHere,\u001fR;(1)\nij(q;q0;!) can be calculated from the following correlation function in the Matsubara\nformalism,\n\u001f(1)\nij(q;q0;i!\u0015) =\u0000VZ\f\n0d\u001cei!\u0015\u001c\nT\u001cji(q;\u001c)sj(\u0000q0;0)\u000b\n; (12)\nby taking the analytic continuation, i!\u0015!~!+i0 as\n\u001fR;(1)\nij(q;q0;!) =\u001f(1)\nij(q;q0;!+i0); (13)\nwhere\f= 1=kBTis the inverse temperature with the Boltzmann constant kB, and!\u0015= 2\u0019\u0015=\f\n(\u0015= 0;\u00061;\u0001\u0001\u0001) is the Matsubara frequency of boson. Note that the Matsubara frequencies are\nde\fned as in unit of energy instead of frequency. By means of the thermal Green function for\n(1, n)\ninin+iωλ\nk+q/2 k+q/2\nk−q/2 k−q/2ki\nmσj\n(1, a)\ninin+iωλ\nk−q k+q−qσjσmk kilmαq,l\nFIG. 2. The Feynman diagrams of \u001f(1;n)\nij and\u001f(1;a)\nij. The solid lines with two arrows denote the Green\nfunctions including the Rashba interaction given by Eq. (6), the \flled circle represents the spin vertex, the\nun\flled triangle describes the normal velocity vertex, and the dashed wavy line indicates the anomalous\nvelocity vertex without the Pauli matrix.\n6H0+HR, Eq. (12) is expressed as\n\u001f(1)\nij=\u001f(1;n)\nij+\u001f(1;a)\nij+\u001f(1;a)(df)\nij +\u001f(1;n)(df)\nij; (14)\nwhere\u001f(1;n)\nij and\u001f(1;a)\nijare the contributions from the normal and anomalous velocities without\nvertex corrections, which are given as\n\u001f(1;n)\nij(q;q0;i!\u0015) =\u0000e\n\fVX\nnX\nk;k0~ki\nmetrh\nGk+q\n2;k0+q0\n2(i\u000fn+i!\u0015)\u001bjGk0\u0000q0\n2;k\u0000q\n2(i\u000fn)i\n; (15a)\n\u001f(1;a)\nij(q;q0;i!\u0015) =e\n\fVX\nnX\nq00\u000film\u000bq00;lX\nk;k0tr\u0002\n\u001bmGk;k0(i\u000fn+i!\u0015)\u001bjGk0\u0000q0;k+q00\u0000q(i\u000fn)\u0003\n(15b)\nFigure 2 depicts \u001f(1;n)\nij and\u001f(1;a)\nij. Contributions \u001f(1;n)(df)\nij and\u001f(1;a)(df)\nij contain the di\u000busion\nladder VCs, whose diagrams and expressions are given in Appendix B (Fig. 4d-j). In order to\nevaluate them up to the \frst order of the Rashba interaction, we expand the Green functions in\nEq. (15a) by using Eq. (6). As Eq. (15b) is already the \frst order of the Rashba interaction, the\nGreen functions there can be approximated as Gk;k0(i\u000fn) =gk(i\u000fn)\u000ek;k0. We take the analytic\ncontinuation, i!\u0015!~!+i0, and calculate the !-linear contribution, which leads to a contribution\nproportional to _M. We also expand them up to the second order with respect to qandq0. The\ndetails of the calculations are shown in Appendix C. Finally, we obtain\n\u001fR;(1)\nij(q;q0;!) =e\u000fmlj\u000bq\u0000q0;l\u0010\n\u0011(1)\nq;q0;im+i!'(1)\nq;q0;im+\u0001\u0001\u0001\u0011\n; (16)\n'(1)\nq;q0;im=2\u0017\u001c\nq02\u0000\n\u000eim(q\u0000q0)\u0001q0\u0000(qi\u0000q0\ni)q0\nm\u0001\n; (17)\nwhere\u0011(1)\nq;q0;imis the static response to the magnetization, and '(1)\nq;q0;imis the dynamical response of\nour interest. In the real space, using the Drude conductivity \u001bD= 2e2\u0017\u000fF\u001c=(3me) =e2\u0017D0with\n\u0017=\u0017(\u000fF) being DOS at the Fermi energy of NM and D0being the di\u000busion constant, we \fnd the\nlinear-order current as\nhj(r;t)i(1)=\u0000el2\u0017\u001c\n3Z\ndr0D(r\u0000r0)\u0010\u0000\nrr0\u0001rr\u0001\n[\u000b(r)\u0002_M(r0;t)]\u0000rr\u0000\nrr0\u0001[\u000b(r)\u0002_M(r0;t)]\u0001\u0011\n(18)\n=\u0000el2\u0017\u001c\n3Z\ndr0D(r\u0000r0)\u0014\nrr0\u0002\u0010\nrr\u0002[\u000b(r)\u0002_M(r0;t)]\u0011\u0015\n; (19)\nwhere\nD(r)\u00111\nVX\nqeiq\u0001r\nD0q2\u001c=3\n4\u0019l21\nr; (20)\nis the di\u000busion propagator, l\u0011p3D0\u001cbeing the elastic mean free path.\n7B. Second order response to exchange interaction\nFor the second order response to the exchange Hamiltonian, the charge current is given by\nhji(r;t)i(2)=\u0012i\n~\u00132Zt\n\u00001dt0Zt0\n\u00001dt00Dh\n[ji(r;t);Hsd(t0)];Hsd(t00)iE\n=ZZ\ndr0dr00ZZ1\n\u00001dt0dt00\u001fR;(2)\nijk(r;r0;r00;t\u0000t0;t0\u0000t00)Mj(r0;t0)Mk(r00;t00);(21)\nwhere the second order response coe\u000ecient \u001fR;(2)\nijk(r;r0;r00;t\u0000t0;t0\u0000t00) is expressed as\n\u001fR;(2)\nijk(r;r0;r00;t\u0000t0;t0\u0000t00) =1\n2\u0010\nQijk(r;r0;r00;t\u0000t0;t0\u0000t00) +Qikj(r;r00;r0;t\u0000t00;t00\u0000t0)\u0011\n;\n(22)\nQijk(r;r0;r00;t\u0000t0;t0\u0000t00) =\u0012i\n~\u00132\n\u0012(t\u0000t0)\u0012(t0\u0000t00)Dh\n[ji(r;t);sj(r0;t0)];sk(r00;t00)iE\n(23)\nHere,\u001fR;(2)\nikj(r;r00;r0;t\u0000t00;t00\u0000t0) =\u001fR;(2)\nijk(r;r0;r00;t\u0000t0;t0\u0000t00). In the Fourier form, the current\nis given as\nhji(q;!)i(2)=X\nq0;q00Z1\n\u00001d!0\n2\u0019\u001fR;(2)\nijk(q;q0;q00;!;!0)Mj(q0;!\u0000!0)Mk(q00;!0) (24)\nFrom Appendix A 3, the non-linear response coe\u000ecient \u001fR;(2)\nijk(q;q0;q00;!;!0) is evaluated from\n\u001f(2)\nijk(q;q0;q00;i!\u0015;i!\u00150) =V2\n2ZZ\f\n0d\u001cd\u001c0ei!\u0015\u001c+i!\u00150\u001c0\nT\u001cji(q;\u001c+\u001c0)sj(\u0000q0;\u001c0)sk(\u0000q00;0)\u000b\n(25)\nby taking the analytic continuations as\ni!\u0015!~!+ 2i0; i!\u00150!~!0+i0 (26)\nNote that, in order to obtain the precise response coe\u000ecient, the order of the analytic continuations\nfori!\u0015andi!\u00150must be speci\fed; taking i!\u00150to the real frequency ~!0and then taking i!\u0015to\n~!from the upper plane ( !\u0015(0)>0). Hence, we have set Eq. (26). We should emphasise that\nthe Matsubara Green function method can apply to the non-linear responses as demonstrated by\nJujo18and by Kohno and Shibata19. (See Appendix A for general cases.)\nHere, we separate Eq. (25) into three components in the similar way of the calculation of \u001f(1)\nij\n[Eq. (14)],\n\u001f(2)\nijk=\u001f(2;n)\nijk+\u001f(2;a)\nijk+\u001f(2;n)(df)\nijk+\u001f(2;a)(df)\nijk; (27)\n8+(2, n)\nσj\nσkki\nmk+q\n2k+q\n2\nk−q\n2\nk+q\n2\nk−q\n2k−q\n2in+iωλ\nin+iωλ\nin σjσk\nki\nm\nk+q\n2\nk−q\n2k+q\n2\nk−q\n2k+q\n2\nk−q\n2\nin−iωλin−iωλin\n(2, a)\n+σj\nσm\nσkin+iωλ\nin+iωλ\ninkk\nk−q\nk\nk−qk−q+qilmαq,lσmk\nk\nk−qk−q+q\nσjσk\nin−iωλin−iωλink\nk−q\nilmαq,lFIG. 3. The Feynman diagrams of \u001f(2;n)\nijkand\u001f(2;a)\nijk. The lines and symbols are de\fned in the caption of\nFig. 2.\nwhere the \frst two terms are the contributions from the normal and anomalous velocities without\nVCs, respectively, given by\n\u001f(2;n)\nijk(q;q0;q00;i!\u0015;i!\u00150) =\u0000e\n2\fVX\nnX\nk;k0;k00~ki\nm\n\u0002trh\nGk+q\n2;k0+q0\n2(i\u000fn+i!\u0015)\u001bjGk0\u0000q0\n2;k00+q00\n2(i\u000fn+i!\u00150)\u001bkGk00\u0000q00\n2;k\u0000q\n2(i\u000fn)\n+Gk+q\n2;k00+q00\n2(i\u000fn)\u001bkGk00\u0000q00\n2;k0+q0\n2(i\u000fn\u0000i!\u00150)\u001bjGk0\u0000q0\n2;k\u0000q\n2(i\u000fn\u0000i!\u0015)i\n;\n(28)\n\u001f(2;a)\nijk(q;q0;q00;i!\u0015;i!\u00150) =e\n2\fVX\nnX\nk;k0;k00\u000filmX\nq000\u000bq000;l\n\u0002trh\n\u001bmGk;k0(i\u000fn+i!\u0015)\u001bjGk0\u0000q0;k00(i\u000fn+i!\u00150)\u001bkGk00\u0000q00;k\u0000q+q000(i\u000fn)\n+\u001bmGk;k00(i\u000fn)\u001bkGk00\u0000q00;k0(i\u000fn\u0000i!\u00150)\u001bjGk0\u0000q0;k\u0000q+q000(i\u000fn\u0000i!\u0015)i\n(29)\nThe last two terms in Eq. (27) include the ladder type VCs of \u001f(2;n)\nijkand\u001f(2;a)\nijk, which are given\nin Appendix B. We evaluate them up to the \frst order of the Rashba interaction. For \u001f(2;n)\nijk, we\nexpand the Green functions in the \frst term of Eq. (15) by using Eq. (6). As \u001f(2;a)\nijkis already the\n\frst order of the Rashba interaction, the Green functions in the second term can be approximated\n9asGk;k0(i\u000fn) =gk(i\u000fn)\u000ek;k0. We take the analytic continuation as shown by Eq. (26) and evaluate\nthe!0-linear contribution, which leads to a contribution proportional to M\u0002_M. Then, we expand\nthem up to the second order with respect to q,q0andq00. The details of the calculations are shown\nin Appendix C. After all, we have the followings:\n\u001fR;(2)\nijk(q;q0;q00;!;!0) = 2ie\u000folm\u000fmjk\u000bq\u0000q0\u0000q00;l\u0010\n\u0011(2)\nq;q0;q00;io+i!#(2)\nq;q0;q00;io+i!0'(2)\nq;q0;q00;io+\u0001\u0001\u0001\u0011\n;\n(30)\n'(2)\nq;q0;q00;io=\u00002i\u0017\u001c2\n~\u000eio(q\u0000q0\u0000q00)\u0001(q0+q00)\u0000(qi\u0000q0\ni\u0000q00\ni)(q0\no+q00\no)\n(q0+q00)2; (31)\nwhere\u0011(2)\nq;q0;q00;iois static response to the magnetization, #(2)\nq;q0;q00;iois the dynamical response pro-\nportional to d( MjMk)=dt, which are negligible. '(2)\nq;q0;q00;iois the dynamical response of our interest,\nwhich is proportional to such as Mjd(Mk)=dt. In the real space, dynamically-induced current is\nhj(r;t)i(2)=4el2\u0017\u001c2\n3~Z\ndr0D(r\u0000r0)\u0014\nrr0\u0002\u0012\nrr\u0002\u0014\n\u000b(r)\u0002\u0012\nM(r0;t)\u0002_M(r0;t)\u0013\u0015\u0013\u0015\n:(32)\nIV. RESULTS AND DISCUSSION\nThe generated charge current to the second order responses to the exchange interaction is\nthereforej(r;t) =hj(r;t)i(1)+hj(r;t)i(2). The current is expressed as a response to a non-local\ne\u000bective electric \feld as j(r;t) =\u001bDEe\u000b(r;t), where\nEe\u000b(r;t) =mel2\n2e\u000fFZ\ndr0D(r\u0000r0)\u0014\nrr0\u0002\u0012\nrr\u0002\u0014\n\u000b(r)\u0002\u0012\n_M(r0;t) +4\u001c\n~[M(r0;t)\u0002_M(r0;t)]\u0013\u0015\u0013\u0015\n:\n(33)\nNote that the magnetization M(r;t) is de\fned as including the sdinteraction strength. The linear\nresponse term, E(1)\ne\u000b, is written as E(1)\ne\u000b=\u0000_A(1)\ne\u000b, where\nA(1)\ne\u000b(r;t) =\u0000mel2\n2e\u000fFZ\ndr0D(r\u0000r0)\u0014\nrr0\u0002\u0012\nrr\u0002\u0014\n\u000b(r)\u0002M(r0;t)\u0015\u0013\u0015\n(34)\nis a non-local extension of e\u000bective gauge \feld discussed in Refs.10{12. In contrast, the second-order\ncontribution, proportional to spin damping, M\u0002_M, does not have the corresponding gauge \feld\nlike in the local case13.\nFor junctions like a trilayer homogeneous in the xy-plane, the spatial derivative is \fnite only in\nthez-direction. The in-plane current, which is of experimental interest, in this case reads\njk(r;t) =mel2\n2e\u000fF\u001bDZ\ndr0D(r\u0000r0)(rz\nr\u000b(r))\u0002rz\nr0\u0014\n_M(r0;t) +4\u001c\n~[M(r0;t)\u0002_M(r0;t)]\u0015\n:(35)\n10This result indicates that the spatially-inhomogeneity of precessing spin at the F-NM interface\ndrives an in-plane e\u000bective motive force at the NM-HM interface as a result of electron di\u000busion.\nThis motive force is an alternative and direct interpretation of inverse Rashba-Edelstein e\u000bect.\nFor describing the case of a spacer thicker than spin di\u000busion length, spin relaxation e\u000bect needs\nto be included in the di\u000busion. As was discussed in Ref.16, the result in this case becomes Eq. (33)\nwith di\u000busion D(r) replaced by the one including spin di\u000busion length,\nDs(r)\u00111\nVX\nqeiq\u0001r\nD0q2\u001c+\rs; (36)\nwhere\rs, proportional to spin relaxation rate, is related to spin di\u000busion length lsasls=l=p3\rs.\nThe non-local e\u000bective electric \feld found in the present study is an electric counterpart of the\nnon-local e\u000bective magnetic \feld (non-local spin Berry's phase) discussed in the context of the\nanomalous Hall e\u000bect15. Although the spin Berry's phase itself arises from static magnetization\ntextures, calculation of the non-local contribution in the present formalism requires including an\nexternal \feld with a \fnite or in\fnitesimal frequency, as the electron di\u000busion applies to non-\nequilibrium situations only.\nACKNOWLEDGMENTS\nJF would like to thank A. Shitade and S.C. Furuya for giving informative comments. JF is\nsupported by a Grant-in-Aid for Specially Promoted Research (No. 15H05702). GT thanks a\nGrant-in-Aid for Exploratory Research (No.16K13853) and a Grant-in-Aid for Scienti\fc Research\n(B) (No. 17H02929) from the Japan Society for the Promotion of Science, a Grant-in-Aid for\nScienti\fc Research on Innovative Areas (No.26103006) from The Ministry of Education, Culture,\nSports, Science and Technology (MEXT), Japan, and the Graduate School Materials Science in\nMainz (MAINZ) (DFG GSC 266) for \fnantial support.\nAppendix A: Matsubara formalism for non-linear responses\nHere, we show the formulation of the non-linear response theory based on the Matsubara for-\nmalism. Some textbooks20,21explain that it is not possible. However, by a careful treatment of\nanalytic continuations, this formulation can be done and leads to the exactly same result from the\nKeldysh formalism. In this Appendix, we show the way to evaluate the responses up to the second\norder with respect to the external force. We also discuss brie\ry the evaluations for the higher order\nresponses.\n111. Setup\nIn this Appendix, we assume that the system we consider is expressed by the Hamiltonian H,\nand the mechanical external force is F\u0016(t) (\u0016is index), which couples to the physical quantity ^A\u0016,\nhence the external Hamiltonian given by\n^H0(t) =\u0000^A\u0016F\u0016(t) (A1)\n(One should presume that the dummy index \u0016sums over all the external forces.) We introduce\n\u0011 > 0 as an in\fnitesimal quantity to ensure that the system is in thermal equilibrium and the\nexternal force is zero at the time t!\u00001 , and the external force is turned on adiabatically from\nthe time:\nF\u0016(t) =e\u0011tZ1\n\u00001d!\n2\u0019e\u0000i!tF\u0016(!) =Z1\n\u00001d!\n2\u0019e\u0000i(!+i\u0011)tF\u0016(!) (A2)\nFollowing the paper by Kubo17, the response of the physical quantity ^Bto the external force F\u0016\nis given byh^Bi(t) =h^Bi0+\u00011B(t) +\u00012B(t) +\u0001\u0001\u0001+\u0001kB(t) +\u0001\u0001\u0001with thek-th order response\n\u0001kB(t) =\u0012\u00001\ni~\u0013kZt\n\u00001dt1Zt1\n\u00001dt2\u0001\u0001\u0001Ztk\u00001\n\u00001dtk\n\u0002Tr\u001a\u0014\n^A\u00161(t1);h\n^A\u00162(t2);\u0002\n\u0001\u0001\u0001;[^A\u0016k(tk);^\u001a]\u0003\n\u0001\u0001\u0001i\u0015\n^B(t)\u001b\nF\u00161(t1)F\u00162(t2)\u0001\u0001\u0001F\u0016k(tk);\n(A3)\nwhereh^Bi0is the expectation value without any external \felds, ^A(t) =ei^Ht=~^Ae\u0000i^Ht=~is the\nHeisenberg representation of ^A, [^A;^B] = ^A^B\u0000^B^Ais communicator, ^ \u001ais the density matrix\noperator for ^H, and\n^\u001a=e\u0000\f^H=Tre\u0000\f^H=e\f(\n\u0000^H)(A4)\nwith\f= 1=kBTand with \n =\u0000kBTln Trfe\u0000\f^Hgbeing the thermodynamic potential. Introducing\njnias the eigenstates of the Hamiltonian, ^Hjni=Enjni, Trf\u0001\u0001\u0001g is given as\nTrf^Ag=X\nnhnj^Ajni (A5)\nThe thermal average h\u0001\u0001\u0001i for the systemHin the temperature Tis de\fned by\nD\n^AE\n= Trf^\u001a^Ag=X\nne\f(\n\u0000En)hnj^Ajni (A6)\nWe also note that the time translational symmetry is held in thermal equilibrium.\n122. Linear response\nWe \frst look at the linear response ( k= 1). Using the cyclic relation, Tr f^A^B^Cg= Trf^B^C^Ag=\nTrf^C^A^Bg, we \fnd\n\u00011B(t) =\u00001\ni~Zt\n\u00001dt0Trn\n[^A\u0016(t0);^\u001a]^B(t)o\nF\u0016(t0)\n=Z1\n\u00001dt0QR\n\u0016(t\u0000t0)F\u0016(t0); (A7)\nwhereQR\n\u0016(t) is the retarded two-point Green function,\nQR\n\u0016(t) =i\n~\u0012(t)D\n[^B(t);^A\u0016(0)]E\n(A8)\nBy using the Fourier transformation22,\u00011B(!) =QR\n\u0016(!)F\u0016(!), where\nQR\n\u0016(!) =i\n~Z1\n0dtei(!+i0)tD\n[^B(t);^A\u0016(0)]E\n(A9)\nHere,!+i0 stands for lim \u0011!0+!+i\u0011.\nThe Matsubara Green function corresponding to QR\n\u0016(!) is\nQ\u0016(i!\u0015) =1\n~Z~\f\n0d\u001cei!\u0015\u001cD\nT\u001cf^B(\u001c)^A\u0016(0)gE\n; (A10)\nwhere!\u0015= 2\u0019\u0015=~\f(\u0015= 0;\u00061;\u00062;\u0001\u0001\u0001) is the Matsubara frequency of bosons, \u001cis the imaginary\ntime, and ^A(\u001c) =e^H\u001c=~^Ae\u0000^H\u001c=~is the so-called Heisenberg representation of ^Ain the imaginary\ntime (t=\u0000i\u001c) and de\fned in the region \u0000~\f\u0014\u001c\u0014~\f. T\u001cf\u0001\u0001\u0001g is the time ordering operator of\n\u001c. Note that as one shows the periodicity hT\u001cf^B(\u001c\u0000~\f)^A\u0016(0)gi=hT\u001cf^B(\u001c)^A\u0016(0)gifor\u001c\u00150\nusing Eq. (A6), it can be expressed by means of the Fourier series of ei!\u0015\u001c.\nThe correspondence between QR\n\u0016(!) andQ\u0016(i!\u0015) is proven easily by representing them in the\nLehmann representation and taking the analytic continuation, i!\u0015!!+i0, resulting to\nQR\n\u0016(!) =Q\u0016(!+i0) (A11)\nFrom these, we can evaluate the linear response coe\u000ecient QR\n\u0016(!) from the corresponding Matsub-\nara Green function Q\u0016(i!\u0015) by taking the analytic continuation, i!l!!+i0.\n3. Second order response\nNext, we show the way to evaluate the second order response precisely. This procedure is\nsimilar to the evaluation of the linear response; (1) \fnd the correlation function in the Matsubara\n13formalism corresponding to the response coe\u000ecient, (2) calculate the correlation function, and (3)\ntake the precise analytic continuation. The procedures (1) and (3) are of the central theme in this\nAppendix since the procedure (2) is same as the well-known procedure.\nFork= 2 in Eq. (A3), the second order response is given as\n\u00012B(t) =\u0012\u00001\ni~\u00132Zt\n\u00001dt1Zt1\n\u00001dt2Trnh\n^A\u0016(t1);\u0002^A\u0017(t2);^\u001a\u0003i\n^B(t)o\nF\u0016(t1)F\u0017(t2)\n=Z1\n\u00001dt1Z1\n\u00001dt2QR\n\u0016\u0017(t;t1;t2)F\u0016(t1)F\u0017(t2)\n=1\n2!Z1\n\u00001dt1Z1\n\u00001dt2\u0010\nQR\n\u0016\u0017(t;t1;t2) +QR\n\u0017\u0016(t;t2;t1)\u0011\nF\u0016(t1)F\u0017(t2); (A12)\nwhereQR\n\u0016\u0017(t;t0;t00) is the retarded three-points correlation function given by\nQR\n\u0016\u0017(t;t0;t00) =\u00001\n~2\u0012(t\u0000t0)\u0012(t0\u0000t00)D\u0002\n[^B(t);^A\u0016(t0)];^A\u0017(t00)\u0003E\n(A13)\nHere, we should point out that we treated the external forces F\u0016(t1) andF\u0017(t2) symmetrically; we\nadded the term interchanging \u0016and\u0017as well ast1andt2and divided them by 2! as in the last\nequal of Eq. (A12). From the following relation by using Eq. (A5),\nQR\n\u0016\u0017(t;t1;t2) =\u00001\n~2\u0012(t\u0000t1)\u0012(t1\u0000t2)X\nn;m;le\f(\n\u0000En)(1\u0000e\f(En\u0000El))\n\u0002n\nei(En\u0000Em)(t\u0000t1)=~+i(En\u0000El)(t1\u0000t2)=~hnj^Bjmihmj^A\u0016jlihlj^A\u0017jni+ (c:c:)o\n;\n(A14)\none \fndQR\n\u0016\u0017(t;t1;t2) =QR\n\u0016\u0017(t\u0000t1;t1\u0000t2) with\nQR\n\u0016\u0017(t;t0) =\u00001\n~2\u0012(t)\u0012(t0)D\u0002\n[^B(t+t0);^A\u0016(t0)];^A\u0017(0)\u0003E\n(A15)\nBy means of QR\n\u0016\u0017(t;t0), Eq. (A12) is rewritten as\n\u00012B(t) =Z1\n\u00001dt1Z1\n\u00001dt2\u001eR\n\u0016\u0017(t\u0000t1;t1\u0000t2)F\u0016(t1)F\u0017(t2); (A16)\n\u001eR\n\u0016\u0017(t;t0) =1\n2!\u0010\nQR\n\u0016\u0017(t;t0) +QR\n\u0017\u0016(t+t0;\u0000t0)\u0011\n(A17)\nThen, the second order response [Eq. (A12)] is expressed in the Fourier space23as\n\u00012B(!) =Z1\n\u00001d!0\n2\u0019\u001eR\n\u0016\u0017(!;!0)F\u0016(!\u0000!0)F\u0017(!0); (A18)\n\u001eR\n\u0016\u0017(!;!0) = (QR\n\u0016\u0017(!;!0) +QR\n\u0017\u0016(!;!\u0000!0))=2!; (A19)\nQR\n\u0016\u0017(!;!0) =\u00001\n~2Z1\n0dtZ1\n0dt0ei(!+i\u0011)t+i(!0+i\u00110)t0D\u0002\n[^B(t+t0);^A\u0016(t0)];^A\u0017(0)\u0003E\n(A20)\n14We introduced the convergence factor \u0011and\u00110, but these two must have the relation \u0011>\u00110because\nofQR\n\u0017\u0016(!;!\u0000!0). Hence, we use 2 i0 as the convergence factor for !andi0 as that for !0.\nAs we show in Appendix A 4, the corresponding correlation function in the Matsubara formu-\nlation to\u001eR\n\u0016\u0017(!;!0) (not toQR\n\u0016\u0017(!;!0)) is given as\n'\u0016\u0017(i!\u0015;i!\u00150) =1\n2!~2Z~\f\n0d\u001cZ~\f\n0d\u001c0ei!\u0015(\u001c\u0000\u001c0)+i!\u00150\u001c0D\nT\u001c;\u001c0f^B(\u001c)^A\u0016(\u001c0)^A\u0017(0)gE\n(A21)\nTaking the analytic continuation i!\u0015!!+ 2i0 andi!\u00150!!0+i0, the following relation is held\n\u001eR\n\u0016\u0017(!;!0) ='\u0016\u0017(!+ 2i0;!0+i0) (A22)\nHence, we can evaluate the second order response [Eq. (A18)] from the corresponding correlation\nfunction in the Matsubara formalism, '\u0016\u0017(i!\u0015;i!\u00150), by taking the analytic continuations.\n4. Correspondence between '\u0016\u0017(i!\u0015;i!\u00150)and\u001eR\n\u0016\u0017(!;!0)\nHere, we show Eq. (A22). First, we perform the integrals of tandt0inQR\n\u0016\u0017(!;!0). Introducing\n!+=!+ 2i0 and!0\n+=!0+i0,\nQR\n\u0016\u0017(!;!0) =\u00001\n~2X\nn;m;le\f(\n\u0000En)(1\u0000e\f(En\u0000El))Z1\n0dtZ1\n0dt0ei!+t+i!0\n+t0\n\u0002n\nei(En\u0000Em)t=~+i(En\u0000El)t0=~hnj^Bjmihmj^A\u0016jlihlj^A\u0017jni\n+e\u0000i(En\u0000Em)t=~\u0000i(En\u0000El)t0=~hmj^Bjnihnj^A\u0017jlihlj^A\u0016jmio\n=X\nn;m;le\f(\n\u0000En)(1\u0000e\f(En\u0000El))\n\u0002(\nhnj^Bjmihmj^A\u0016jlihlj^A\u0017jni\n(~!++En\u0000Em)(~!0\n++En\u0000El)+hmj^Bjnihnj^A\u0017jlihlj^A\u0016jmi\n(~!+\u0000En+Em)(~!0\n+\u0000En+El))\n;\n(A23)\nand forQR\n\u0017\u0016(!;!\u0000!0), by interchanging n$min the above equation, we obtain\nQR\n\u0017\u0016(!;!\u0000!0) =X\nn;m;le\f(\n\u0000En)e\f(En\u0000Em)(1\u0000e\f(Em\u0000El))\n\u0002(\nhmj^Bjnihnj^A\u0017jlihlj^A\u0016jmi\n(~!+\u0000En+Em)(~!+\u0000~!0\n++Em\u0000El)+hnj^Bjmihmj^A\u0016jlihlj^A\u0017jni\n(~!++En\u0000Em)(~!+\u0000~!0\n+\u0000Em+El))\n15Here,!+\u0000!0\n+=!\u0000!0+i(\u0011\u0000\u00110), and\u0011>\u00110is needed for the convergence in the limit t!1 ,\nhence putting \u0011= 2\u00110. From these, \u001eR\n\u0016\u0017(!;!0) =QR\n\u0016\u0017(!;!0) +QR\n\u0017\u0016(!;!\u0000!0) is given as\n\u001eR\n\u0016\u0017(!;!0) =1\n2X\nn;m;le\f(\n\u0000En) \nhnj^Bjmihmj^A\u0016jlihlj^A\u0017jni\n~!++En\u0000Ema(!+;!0\n+)\n+hmj^Bjnihnj^A\u0017jlihlj^A\u0016jmi\n\u0000~!++En\u0000Ema(\u0000!+;\u0000!0\n+)!\n; (A24)\nwhere\na(!+;!0\n+)\u00111\u0000e\f(En\u0000El)\n~!0\n++En\u0000El+e\f(En\u0000Em)(1\u0000e\f(Em\u0000El))\n~!+\u0000~!0\n+\u0000Em+El(A25)\nNext, we perform the integrals of the correlation function in the Matsubara formalism [Eq. (A21)].\nFrom the time-ordering operator, for \u001c >\u001c0>0,\nD\nT\u001c;\u001c0f^B(\u001c)^A\u0016(\u001c0)^A\u0017(0)gE\n=X\nm;n;le\f(\n\u0000En)e(Em\u0000El)\u001c0=~hnj^Bjmihmj^A\u0016jlihlj^A\u0017jnie(En\u0000Em)\u001c=~;\n(A26)\nhence we \fnd\n1\n~2Z~\f\n0d\u001c0Z~\f\n\u001c0d\u001cei!\u0015(\u001c\u0000\u001c0)+i!\u00150\u001c0D\nT\u001c;\u001c0f^B(\u001c)^A\u0016(\u001c0)^A\u0017(0)gE\n=X\nm;n;le\f(\n\u0000En)hnj^Bjmihmj^A\u0016jlihlj^A\u0017jni1\n~2Z~\f\n0d\u001c0ei(!\u00150\u0000!\u0015)\u001c0e(Em\u0000El)\u001c0=~Z~\f\n\u001c0d\u001ce(i~!\u0015+En\u0000Em)\u001c=~\n=X\nm;n;le\f(\n\u0000En)hnj^Bjmihmj^A\u0016jlihlj^A\u0017jni\ni~!\u0015+En\u0000Ema(i!\u0015;i!\u00150) (A27)\nAlso for\u001c0>\u001c > 0,\nD\nT\u001c;\u001c0f^B(\u001c)^A\u0016(\u001c0)^A\u0017(0)gE\n=X\nm;n;le\f(\n\u0000El)e(El\u0000Em)\u001c0=~hmj^Bjnihnj^A\u0017jlihlj^A\u0016jmie(Em\u0000En)\u001c=~;\n(A28)\nand then, we obtain\n1\n~2Z~\f\n0d\u001c0Z\u001c0\n0d\u001cei!\u0015(\u001c\u0000\u001c0)+i!\u00150\u001c0D\nT\u001c;\u001c0f^B(\u001c)^A\u0016(\u001c0)^A\u0017(0)gE\n=X\nm;n;le\f(\n\u0000El)hmj^Bjnihnj^A\u0017jlihlj^A\u0016jmi1\n~2Z~\f\n0d\u001c0ei(!\u00150\u0000!\u0015)\u001c0e(El\u0000Em)\u001c0=~Z\u001c0\n0d\u001ce(i~!\u0015+Em\u0000En)\u001c=~\n=X\nm;n;le\f(\n\u0000En)hmj^Bjnihnj^A\u0017jlihlj^A\u0016jmi\n\u0000i~!\u0015\u0000Em+Ena(\u0000i!\u0015;\u0000i!\u00150) (A29)\n16Therefore, Eq. (A21) is rewritten as\n'\u0016\u0017(i!\u0015;i!\u00150) =1\n2X\nn;m;le\f(\n\u0000En) \nhnj^Bjmihmj^A\u0016jlihlj^A\u0017jni\ni~!\u0015+En\u0000Ema(i!\u0015;i!\u00150)\n+hmj^Bjnihnj^A\u0017jlihlj^A\u0016jmi\n\u0000i~!\u0015+En\u0000Ema(\u0000i!\u0015;\u0000i!\u00150)!\n; (A30)\nand as compared with Eq. (A24), it is obvious that Eq. (A22) is held.\n5. Third and higher order responses\nThe third order response, k= 3 for Eq. (A3), reads\n\u00013B(t) =Z1\n\u00001dt1Z1\n\u00001dt2Z1\n\u00001dt3\u001eR\n\u0016\u0017\u0018(t;t1;t2;t3)F\u0016(t1)F\u0017(t2)F\u0018(t3); (A31)\nwhere\u001eR\n\u0016\u0017\u0018(t;t1;t2;t3) is a symmetrized response coe\u000ecient given as\n\u001eR\n\u0016\u0017\u0018(t;t1;t2;t3) =1\n3!\u0010\nQR\n\u0016\u0017\u0018(t;t1;t2;t3) +QR\n\u0016\u0018\u0017(t;t1;t3;t2) +QR\n\u0017\u0016\u0018(t;t2;t1;t3)\n+QR\n\u0017\u0018\u0016(t;t2;t3;t1) +QR\n\u0018\u0017\u0016(t;t3;t1;t2) +QR\n\u0018\u0016\u0017(t;t3;t2;t1)\u0011\n; (A32)\nQR\n\u0016\u0017\u0018(t;t1;t2;t3) =\u0012\u00001\ni~\u00133\n\u0012(t\u0000t1)\u0012(t1\u0000t2)\u0012(t2\u0000t3)Dh\u0002^B(t);^A\u0016(t1)];^A\u0017(t2)\u0003\n;^A\u0018(t3)iE\n:\n(A33)\nOne can see QR\n\u0016\u0017\u0018(t;t1;t2;t3) =QR\n\u0016\u0017\u0018(t\u0000t1;t1\u0000t2;t2\u0000t3) by using Eq. (A5), and the Fourier form\nis shown as\nQR\n\u0016\u0017\u0018(!;!0;!00) =\u0012\u00001\ni~\u00133ZZZ1\n0dtdt0dt00ei(!+i\u0011)t+i(!0+i\u00110)t0+i(!00+i\u001100)t00\n\u0002Dh\u0002^B(t+t0+t00);^A\u0016(t0+t00)];^A\u0017(t00)\u0003\n;^A\u0018(0)iE\n: (A34)\nHence, the third order response in the Fourier space is given as\n\u00013B(!) =1\n3!Z1\n\u00001d!0\n2\u0019Z1\n\u00001d!00\n2\u0019\u001eR\n\u0016\u0017\u0018(!;!0;!00)F\u0016(!\u0000!0)F\u0017(!0\u0000!00)F\u0018(!00) (A35)\nwith\n\u001eR\n\u0016\u0017\u0018(!;!0;!00) =QR\n\u0016\u0017\u0018(!;!0;!00) +QR\n\u0016\u0018\u0017(!;!0;!0\u0000!00) +QR\n\u0017\u0016\u0018(!;!+!00\u0000!0;!00)\n+QR\n\u0017\u0016\u0018(!;!+!00\u0000!0;!\u0000!00) +QR\n\u0018\u0016\u0017(!;!\u0000!00;!0\u0000!00) +QR\n\u0018\u0017\u0016(!;!\u0000!0;!\u0000!00):\n(A36)\nEquation (A36) leads that the convergence factors need to have the relation \u0011 >\u00110>\u001100. Hence,\nwe put\u0011= 3\u001100,\u00110= 2\u001100.\n17There is the corresponding correlation function in the Matsubara formalism to \u001eR\n\u0016\u0017\u0018(!;!0;!00)\ngiven by\n'\u0016\u0017\u0018(i!\u0015;i!\u00150;i!\u001500) =1\n3!~3ZZZ~\f\n0d\u001cd\u001c0d\u001c00ei!\u0015(\u001c\u0000\u001c0)+i!\u00150(\u001c0\u0000\u001c00)+i!\u001500\u001c00D\nTf^B(\u001c)^A\u0016(\u001c0)^A\u0017(\u001c00)^A\u0018(0)gE\n:\n(A37)\nBy taking the analytic continuations, i!\u0015!!+ 3i0,i!\u00150!!0+ 2i0,i!\u001500!!00+i0, we have\n\u001eR\n\u0016\u0017\u0018(!;!0;!00) ='\u0016\u0017\u0018(!+ 3i0;!0+ 2i0;!00+i0): (A38)\nFrom thek= 1;2;3-th order responses, it is expected that the k-th order response is evaluated\nas follows: the response of ^Bto the external forces is expressed as\n\u0001kB(t) =ZZ\n\u0001\u0001\u0001Z1\n\u00001dt1dt2\u0001\u0001\u0001dtk\u001eR\n\u00161\u00162\u0001\u0001\u0001\u0016k(t\u0000t1;t1\u0000t2;\u0001\u0001\u0001;tk\u00001\u0000tk)F\u00161(t1)F\u00162(t2)\u0001\u0001\u0001F\u0016k(tk);\n(A39)\nwhere\u001eR\n\u00161\u00162\u0001\u0001\u0001\u0016k(t\u0000t1;t1\u0000t2;\u0001\u0001\u0001;tk\u00001\u0000tk) is the response coe\u000ecient already symmetrized, whose\nFourier component \u001eR\n\u00161\u00162\u0001\u0001\u0001\u0016k(!;! 1;!2;\u0001\u0001\u0001;!k\u00001) is evaluated from the corresponding correlation\nfunction in the Matsubara formalism\n'\u00161\u00162\u0001\u0001\u0001\u0016k(i!\u0015;i!\u00151;i!\u00152;\u0001\u0001\u0001;i!\u0015k\u00001)\n=1\nk!~kZZ\n\u0001\u0001\u0001ZZ~\f\n0d\u001cd\u001c1\u0001\u0001\u0001d\u001ck\u00002d\u001ck\u00001ei!\u0015(\u001c\u0000\u001c1)+i!\u00151(\u001c1\u0000\u001c2)+\u0001\u0001\u0001+i!\u0015k\u00002(\u001ck\u00002\u0000\u001ck\u00001)+i!\u0015k\u00001\u001ck\u00001\n\u0002D\nTf^B(\u001c)^A\u00161(\u001c1)^A\u00162(\u001c2)\u0001\u0001\u0001^A\u0016k\u00002(\u001ck\u00002)^A\u0016k\u00001(\u001ck\u00001)^A\u0016k(0)gE\n(A40)\nby taking the analytic continuations, i!\u0015!!+ki0,i!\u00151!!1+(k\u00001)i0,\u0001\u0001\u0001,i!\u0015k\u00002!!k\u00002+2i0,\ni!\u0015k\u00001!!k\u00001+i0;\n\u001eR\n\u00161\u00162\u0001\u0001\u0001\u0016k(!;! 1;!2;\u0001\u0001\u0001;!k\u00001) ='\u00161\u00162\u0001\u0001\u0001\u0016k(!+ki0;!1+ (k\u00001)i0;!2+ (k\u00002)i0;\u0001\u0001\u0001;!k\u00001+i0):\n(A41)\nAppendix B: Expressions of diagrams\nIn this Appendix, we show the expressions of all the diagrams contributing the non-local emer-\ngent electric \felds shown in Fig. 4 for the linear response and in Fig. 6 for the second order\nresponse.\n18(a)\nkk+q k+qmloαq−q,lk+q+q\n2 m\nσo(b)\nk−q k−qk\nσo\nmloαq−q,lk−q+q\n2 m(c)\nkk+q\nσmilmαq−q,l\n(d)\nkk+qk+q\nk+q\nk(e)\nk−q\nk−qk\nk−qk\n(f)\nkk+qk+q\nk(g)\nk−qkk k+q\nk−q k−q\n(h) (i)\n(j)\nk kk+q k+qFIG. 4. The Feynman diagrams of \u001f(1)\nij(i!\u0015) in the \frst order of the Rashba spin-orbit interaction; (a)-\n(c) without the ladder type VCs and (d)-(j) with the VCs. The solid lines with arrows denote the Green\nfunctions without the Rashba interaction, given by Eq. (6), the \flled circle represents the spin vertex, the\nun\flled triangle describes the normal velocity vertex, the dashed wavy line indicates the anomalous velocity\nvertex, and the solid wavy line depicts the Rashba-interaction vertex without the spin component.\nniu\nu\nkkk+qk+q\ninin+iωλ\nkk+q\nFIG. 5. The diagrammatic description for the four-point vertex of the di\u000busion ladder. The dotted lines\ndenote the impurity potential, and the cross symbol represents the impurity concentration. The solid lines\nwithout arrows are for the external momentums.\nEquations (15a) and (15b) in the \frst order of the Rashba interaction are given respectively by\n19Fig. 4 (a)-(c), which reads\n\u001f(1;n)\nij(q;q0;i!\u0015) =e\u000fmlj\u000bq\u0000q0;l1\n\fX\nnX\n\u001b=\u0006\u0010\n\u000ej;z\u0005im;\u001b\u001b\nq;q0(i\u000f+\nn;i\u000fn) +\u000ej;?\u0005im;\u001b \u0016\u001b\nq;q0(i\u000f+\nn;i\u000fn) + (i\u000f+\nn$i\u000fn)\u0011\n;\n(B1a)\n\u001f(1;a)\nij(q;q0;i!\u0015) =e\u000filj\u000bq\u0000q0;l1\n\fX\nnX\n\u001b=\u0006\u0000\n\u000ej;z\u0003\u001b\u001b\nq0(i\u000f+\nn;i\u000fn) +\u000ej;?\u0003\u001b\u0016\u001b\nq0(i\u000f+\nn;i\u000fn)\u0001\n; (B1b)\nwherei\u000f+\nn=i\u000fn+i!\u0015and\u000ej;?= (1\u0000\u000ej;z), and\n\u0005im;\u001b\u001b0\nq;q0(i\u000fm;i\u000fn) =1\nVX\nk~2\nme\u0010\nk+q\n2\u0011\ni\u0012\nk+q+q0\n2\u0013\nmgk+q;\u001b(i\u000fm)gk+q0;\u001b0(i\u000fm)gk;\u001b(i\u000fn);(B2)\n\u0003\u001b\u001b0\nq0(i\u000fm;i\u000fn) =1\nVX\nkgk+q0;\u001b(i\u000fm)gk;\u001b0(i\u000fn) (B3)\nHere, we used g\u0000k;\u001b(i\u000fn) =gk;\u001b(i\u000fn) for calculating the diagram of Fig.4 (b), resulting in the term\nwhich is interchanged i\u000f+\nnandi\u000fnfor the \frst two terms in Eq. (B1a). The Green function gk;\u001b(i\u000fn)\nis here expressed depending spin \u001b=\u0006, but we will evaluate it as gk;\u001b(i\u000fn) =gk(i\u000fn) in the next\nsection. The four-point vertex of the di\u000busion ladder is given by Fig. 5,\n\u0000\u001b\u001b0\nq(i\u000f+\nn;i\u000fn) =niu2+(niu2)2\nVX\nkgk+q;\u001b(i\u000f+\nn)gk;\u001b0(i\u000fn) +(niu2)3\nV X\nkgk+q;\u001b(i\u000f+\nn)gk;\u001b0(i\u000fn)!2\n+\u0001\u0001\u0001\n=niu2\n1\u0000niu2\u0003\u001b\u001b0\nq(i\u000f+n;i\u000fn): (B4)\nThe di\u000busion ladder VCs of \u001f(1;n)\nijare given by Fig. 4 (d)-(i), which read\n\u001f(1;n)(df)\nij (q;q0;i!\u0015) =\u001f(d)+(e)\nij +\u001f(f)+(g)\nij +\u001f(h)+(i)\nij (B5)\nwith\n\u001f(d)+(e)\nij =e\u000fmlj\u000bq\u0000q0;l1\n\fX\nnX\n\u001b=\u0006h\n\u000ej;z\u0005im;\u001b\u001b\nq;q0(i\u000f+\nn;i\u000fn)\u0000\u001b\u001b\nq0(i\u000f+\nn;i\u000fn)\u0003\u001b\u001b\nq0(i\u000f+\nn;i\u000fn)\n+\u000ej;?\u0005im;\u001b \u0016\u001b\nq;q0(i\u000f+\nn;i\u000fn)\u0000\u0016\u001b\u001b\nq0(i\u000f+\nn;i\u000fn)\u0003\u0016\u001b\u001b\nq0(i\u000f+\nn;i\u000fn) + (i\u000f+\nn$i\u000fn)i\n;\n(B6a)\n\u001f(f)+(g)\nij =e\u000fmlj\u000bq\u0000q0;l1\n\fX\nnX\n\u001b=\u0006h\n\u000ej;z\u0003i;\u001b\nq(i\u000f+\nn;i\u000fn)\u0000\u001b\u001b\nq(i\u000f+\nn;i\u000fn)\u0005m;\u001b\u001b\nq;q0(i\u000f+\nn;i\u000fn)\n+\u000ej;?\u0003i;\u001b\nq(i\u000f+\nn;i\u000fn)\u0000\u001b\u001b\nq(i\u000f+\nn;i\u000fn)\u0005m;\u001b\u0016\u001b\nq;q0(i\u000f+\nn;i\u000fn) + (i\u000f+\nn$i\u000fn)i\n;\n(B6b)\n\u001f(h)+(i)\nij =e\u000fmlj\u000bq\u0000q0;l1\n\fX\nnX\n\u001b=\u0006h\n\u000ej;z\u0003i;\u001b\nq(i\u000f+\nn;i\u000fn)\u0000\u001b\u001b\nq(i\u000f+\nn;i\u000fn)\u0005m;\u001b\u001b\nq;q0(i\u000f+\nn;i\u000fn)\u0000\u001b\u001b\nq0(i\u000f+\nn;i\u000fn)\u0003\u001b\u001b\nq0(i\u000f+\nn;i\u000fn)\n+\u000ej;?\u0003i;\u001b\nq(i\u000f+\nn;i\u000fn)\u0000\u001b\u001b\nq(i\u000f+\nn;i\u000fn)\u0005m;\u001b\u0016\u001b\nq;q0(i\u000f+\nn;i\u000fn)\u0000\u0016\u001b\u001b\nq0(i\u000f+\nn;i\u000fn)\u0003\u0016\u001b\u001b\nq0(i\u000f+\nn;i\u000fn)\n+ (i\u000f+\nn$i\u000fn)i\n; (B6c)\n20where\n\u0005i;\u001b\u001b0\nq;q0(i\u000fm;i\u000fn) =~\nVX\nk\u0012\nk+q+q0\n2\u0013\nigk+q;\u001b(i\u000fm)gk+q0;\u001b0(i\u000fm)gk;\u001b(i\u000fn); (B7)\n\u0003i;\u001b\nq(i\u000fm;i\u000fn) =1\nVX\nk~\nme\u0010\nk+q\n2\u0011\nigk+q;\u001b(i\u000fm)gk;\u001b(i\u000fn); (B8)\nand the di\u000busion ladder VCs of \u001f(1;a)\nijis given by Fig. 4 (j), which reads\n\u001f(1;a)(df)\nij (q;q0;i!\u0015) =e\u000filj\u000bq\u0000q0;l1\n\fX\nnX\n\u001b=\u0006h\n\u000ej;z\u0003\u001b\u001b\nq0(i\u000f+\nn;i\u000fn)\u0000\u001b\u001b\nq0(i\u000f+\nn;i\u000fn)\u0003\u001b\u001b\nq0(i\u000f+\nn;i\u000fn)\n+\u000ej;?\u0003\u001b\u0016\u001b\nq0(i\u000f+\nn;i\u000fn)\u0000\u0016\u001b\u001b\nq0(i\u000f+\nn;i\u000fn)\u0003\u0016\u001b\u001b\nq0(i\u000f+\nn;i\u000fn)i\n:(B9)\nWe calculate all the above quantities in Appendix C.\ni+\nn,k+q\nin,ki+\nn,k+q+q\ni+\nn,k+qin,k+qin,k+q+q\nin,k+q\ni−\nn,ki+\nn,k\nin,k−q−qin,k−qi+\nn,k−qin,k\ni−\nn,k−qin,k−q\ni−\nn,k−q−q\nin,ki+\nn,k+q+q\ni+\nn,k+qin,k+q+q\nin,k+q\ni−\nn,ki+\nn,k+q−q\ni+\nn,k+qi+\nn,k+q\nin,kin,k+q\ni−\nn,kin,k+qin,k+q−q(a) (b) (c) (d)\n(h) (g) (f) (e)\nFIG. 6. The Feynman diagrams of \u001f(2)\nijk(i!\u0015;i!\u00150) in the \frst order of the Rashba interaction without the\ndi\u000busion ladder VCs. The lines and symbols are de\fned in the caption of Fig. 4. (a)-(f): The contributions of\nthe normal velocity term \u001f(2;n)\nijk(i!\u0015;i!\u00150) and (g)-(h): that of the anomalous velocity term \u001f(2;a)\nijk(i!\u0015;i!\u00150).\nNote that (b), (d), (f) and (h) are same contributions as that which are obtained by replacing j$k,\nq0$q00,\u000fn!\u000f\u0000\nn,\u000f+\nn!\u000fn, and\u000f0+\nn!\u000f0\u0000\nnin (a), (c), (e), and (g), respectively.\nFor the second order response discussed in Sec. III B, expanding Eqs. (28) and (29) in the\n\frst order of the Rashba interaction, we have the diagrams in Fig. 6. The diagrams shown in\nFig. 6 (a)-(f) and (g)-(h) are obtained from Eqs. (28) and (29), respectively, which reads\n\u001f(2;n)\nijk(i!\u0015;i!\u00150) =ie\u000folm\u000fmjk\u000bq\u0000q0\u0000q00;l1\n\fX\nnX\n\u001b=\u0006\n\u0002h\n\u000em;z\u0010\n\u0004io;\u001b\u001b \u0016\u001b\nq;q0;q00(i\u000f+\nn;i\u000f0+\nn;i\u000fn) + \u0004io;\u001b\u001b \u0016\u001b\nq;q00;q0(i\u000fn;i\u000f0+\nn;i\u000f+\nn)\u0000\u0002io;\u001b\u0016\u001b\u0016\u001b\nq;q0;q00(i\u000f+\nn;i\u000f0+\nn;i\u000fn)\u0011\n+\u000ej;z\u0010\n\u0004io;\u001b\u0016\u001b\u0016\u001b\nq;q0;q00(i\u000f+\nn;i\u000f0+\nn;i\u000fn) + \u0004io;\u001b\u0016\u001b\u001b\nq;q00;q0(i\u000fn;i\u000f0+\nn;i\u000f+\nn)\u0000\u0002io;\u001b\u001b \u0016\u001b\nq;q0;q00(i\u000f+\nn;i\u000f0+\nn;i\u000fn)\u0011\n21(i) (j) (k)\n(n) (o) (p)(l) (m)\nin,k,σim,k+q,σim,k+q,σ\nin,k,σ(q) (r)FIG. 7. All the Feynman diagrams of \u001f(2)\nijk(i!\u0015;i!\u00150) in the \frst order of the Rashba interaction; (i), (j), (k)\nand (n) include the diagrams shown in Fig. 6 (a)-(b), (c)-(d), (g)-(h) and (e)-(f), respectively. The three\ndiagrams surrounded by a thick line include main contributions to the non-local emergent electric \felds. In\nthe diagrams (i)-(p), the momentums and Matsubara frequencies are not displayed for readability; they are\nsame as in the diagrams of Fig. 6 for (i), (j), (k), and (n). The momentums and Matsubara frequencies in\n(l), (m), (o) and (p) are expected from (i), (j) and (n) by using Fig. 5. The \flled triangle and double circle\nare the full vertexes of the normal velocity and the spin, respectively, given by (q) and (r). The momentums\nand Matsubara frequencies of the arrowed lines are shared with the corresponding diagrams in Fig. 6.\n+\u000ek;z\u0010\n\u0004io;\u001b\u0016\u001b\u001b\nq;q0;q00(i\u000f+\nn;i\u000f0+\nn;i\u000fn) + \u0004io;\u001b\u0016\u001b\u0016\u001b\nq;q00;q0(i\u000fn;i\u000f0+\nn;i\u000f+\nn)\u0000\u0002io;\u001b\u0016\u001b\u001b\nq;q0;q00(i\u000f+\nn;i\u000f0+\nn;i\u000fn)\u0011\n+ (j$k;q0$q00;\u000fn!\u000f\u0000\nn;\u000f+\nn!\u000fn;\u000f0+\nn!\u000f0\u0000\nn)i\n; (B10a)\n\u001f(2;a)\nijk(i!\u0015;i!\u00150) =ie\u000film\u000fmjk\u000bq\u0000q0\u0000q00;l1\n\fX\nnX\n\u001b=\u0006h\n\u000em;z\u0003\u001b\u0016\u001b\u001b\nq0;q00(i\u000f+\nn;i\u000f0+\nn;i\u000fn) +\u000ej;z\u0003\u001b\u001b\u0016\u001b\nq0;q00(i\u000f+\nn;i\u000f0+\nn;i\u000fn)\n+\u000ek;z\u0003\u001b\u0016\u001b\u0016\u001b\nq0;q00(i\u000f+\nn;i\u000f0+\nn;i\u000fn) + (j$k;q0$q00;\u000fn!\u000f\u0000\nn;\u000f+\nn!\u000fn;\u000f0+\nn!\u000f0\u0000\nn)i\n;\n(B10b)\nwherei\u000f\u0006\nn=i\u000fn\u0006i!\u0015,i\u000f0\u0006\nn=i\u000fn\u0006i!\u00150, and\n\u0004ij;\u001b\u001b0\u001b00\np;q;r(i\u000fl;i\u000fm;i\u000fn) =1\nVX\nk~2\nme\u0010\nk+p\n2\u0011\ni\u0012\nk+p+q+r\n2\u0013\nj\n\u0002gk+p;\u001b(i\u000fl)gk+q+r;\u001b0(i\u000fl)gk+r;\u001b00(i\u000fm)gk;\u001b(i\u000fn); (B11a)\n\u0002ij;\u001b\u001b0\u001b00\np;q;r(i\u000fl;i\u000fm;i\u000fn) =1\nVX\nk~2\nme\u0010\nk+p\n2\u0011\ni\u0012\nk+p\u0000q+r\n2\u0013\nj\n\u0002gk+p;\u001b(i\u000fl)gk+p\u0000q;\u001b0(i\u000fm)gk+r;\u001b00(i\u000fm)gk;\u001b(i\u000fn); (B11b)\n22\u0003\u001b\u001b0\u001b00\nq;r(i\u000fl;i\u000fm;i\u000fn) =1\nVX\nkgk+q+r;\u001b(i\u000fl)gk+r;\u001b0(i\u000fm)gk;\u001b00(i\u000fn) (B11c)\nThe normal velocity terms containing di\u000busion ladder VCs are shown in Fig. 6 (i)-(j), (l)-(p) and\ngiven as\n\u001f(2;n)(df)\nijk(i!\u0015;i!\u00150) =\u001f(i)+(j)\nijk+\u001f(l)+(m)\nijk+\u001f(n)+(o)+(p)\nijk; (B12)\n\u001f(2;a)(df)\nijk(i!\u0015;i!\u00150) =\u001f(k)\nijk(B13)\nwhere\n\u001f(l)+(m)\nijk=ie\u000folm\u000fmjk\u000bq\u0000q0\u0000q00;l1\n\fX\nnX\n\u001b=\u0006\n\u0002h\n\u000em;z\u0010\n\u0005io;\u001b\u001b\nq;q0+q00(i\u000f+\nn;i\u000fn)\u0000\u001b\u001b\nq0+q00(i\u000f+\nn;i\u000fn)\u0003\u001b\u0016\u001b\u001b\nq0;q00(i\u000f+\nn;i\u000f0+\nn;i\u000fn) + (i\u000f+\nn$i\u000fn)\u0011\n+\u000ej;z\u0010\n\u0005io;\u001b\u0016\u001b\nq;q0+q00(i\u000f+\nn;i\u000fn)\u0000\u0016\u001b\u001b\nq0+q00(i\u000f+\nn;i\u000fn)\u0003\u0016\u001b\u0016\u001b\u001b\nq0;q00(i\u000f+\nn;i\u000f0+\nn;i\u000fn)\n+ \u0005io;\u001b\u0016\u001b\nq;q0+q00(i\u000fn;i\u000f+\nn)\u0000\u001b\u0016\u001b\nq0+q00(i\u000f+\nn;i\u000fn)\u0003\u0016\u001b\u001b\u001b\nq0;q00(i\u000fn;i\u000f0+\nn;i\u000f+\nn)\u0011\n+\u000ek;z\u0010\n\u0005io;\u001b\u0016\u001b\nq;q0+q00(i\u000f+\nn;i\u000fn)\u0000\u0016\u001b\u001b\nq0+q00(i\u000f+\nn;i\u000fn)\u0003\u0016\u001b\u001b\u001b\nq0;q00(i\u000f+\nn;i\u000f0+\nn;i\u000fn)\n+ \u0005io;\u001b\u0016\u001b\nq;q0+q00(i\u000fn;i\u000f+\nn)\u0000\u001b\u0016\u001b\nq0+q00(i\u000f+\nn;i\u000fn)\u0003\u0016\u001b\u0016\u001b\u001b\nq0;q00(i\u000fn;i\u000f0+\nn;i\u000f+\nn)\u0011\n+ (j$k;q0$q00;\u000fn!\u000f\u0000\nn;\u000f+\nn!\u000fn;\u000f0+\nn!\u000f0\u0000\nn)i\n; (B14)\n\u001f(k)\nijk=ie\u000film\u000fmjk\u000bq\u0000q0\u0000q00;l1\n\fX\nnX\n\u001b=\u0006h\n\u000em;z\u0003\u001b\u001b\nq0+q00(i\u000f+\nn;i\u000fn)\u0000\u001b\u001b\nq0+q00(i\u000f+\nn;i\u000fn)\u0003\u001b\u0016\u001b\u001b\nq0;q00(i\u000f+\nn;i\u000f0+\nn;i\u000fn)\n+\u000ej;z\u0003\u001b\u0016\u001b\nq0+q00(i\u000f+\nn;i\u000fn)\u0000\u001b\u0016\u001b\nq0+q00(i\u000f+\nn;i\u000fn)\u0003\u001b\u001b\u0016\u001b\nq0;q00(i\u000f+\nn;i\u000f0+\nn;i\u000fn)\n+\u000ek;z\u0003\u001b\u0016\u001b\nq0+q00(i\u000f+\nn;i\u000fn)\u0000\u001b\u0016\u001b\nq0+q00(i\u000f+\nn;i\u000fn)\u0003\u001b\u0016\u001b\u0016\u001b\nq0;q00(i\u000f+\nn;i\u000f0+\nn;i\u000fn)\n+ (j$k;q0$q00;\u000fn!\u000f\u0000\nn;\u000f+\nn!\u000fn;\u000f0+\nn!\u000f0\u0000\nn)i\n; (B15)\nand\u001f(i)+(j)\nijkand\u001f(n)+(o)+(p)\nijkcontains di\u000berent types of di\u000busion from \u001f(k)+(l)+(m)\nijk.\nAppendix C: Calculation details\nIn this Appendix, we show the details of the calculations of the response coe\u000ecients at absolute\nzero,T= 0. In the present perturbative approach, the free Green functions are spin unpolarized,\nwhich means gk;\u001b(i\u000fn) is equivalent to gk(i\u000fn) de\fned as in Eq. (7). First, we calculate the liner\nresponse coe\u000ecient. Equations. (B1a) and (B1b) are reduced to the following simple forms,\n\u001f(1;n)\nij(q;q0;i!\u0015) = 2e\u000fmlj\u000bq\u0000q0;l1\n\fX\nn\u0000\n\u0005im\nq;q0(i\u000f+\nn;i\u000fn) + \u0005im\nq;q0(i\u000fn;i\u000f+\nn)\u0001\n; (C1a)\n\u001f(1;a)\nij(q;q0;i!\u0015) = 2e\u000filj\u000bq\u0000q0;l1\n\fX\nn\u0003q0(i\u000f+\nn;i\u000fn); (C1b)\n23and their di\u000busion VCs which mainly contribute to the non-local emergent electric \felds are given\nas\n\u001f(1;n)(df)\nij (q;q0;i!\u0015)'\u001f(d)+(e)\nij\n= 2e\u000fmlj\u000bq\u0000q0;l1\n\fX\nnh\n\u0005im\nq;q0(i\u000f+\nn;i\u000fn)\u0000q0(i\u000f+\nn;i\u000fn)\u0003q0(i\u000f+\nn;i\u000fn) + (i\u000f+\nn$i\u000fn)i\n;\n(C1c)\n\u001f(1;a)(df)\nij (q;q0;i!\u0015) = 2e\u000filj\u000bq\u0000q0;l1\n\fX\nnh\n\u0003q0(i\u000f+\nn;i\u000fn)\u0000q0(i\u000f+\nn;i\u000fn)\u0003q0(i\u000f+\nn;i\u000fn)i\n; (C1d)\nwhere \u0005im\nq;q0(i\u000f+\nn;i\u000fn),\u0003q0(i\u000f+\nn;i\u000fn), and\u0000q0(i\u000f+\nn;i\u000fn) are given respectively by that dropped the\nspin dependences in Eqs. (B2), (B3), and (B4). Using the following relations\n\u0000q0(i\u000f+\nn;i\u000fn)\u0003q0(i\u000f+\nn;i\u000fn) =\u00001 +1\n1\u0000niu2\u0003q0(i\u000f+n;i\u000fn); (C2)\nand\u0003q0(i\u000f+\nn;i\u000fn) =\u0003q0(i\u000fn;i\u000f+\nn) due tog\u0000k(i\u000fn) =gk(i\u000fn), we \fnd\n\u001f(1)\nij(q;q0;i!\u0015)'2e\u000fmlj\u000bq\u0000q0;l1\n\fX\nn\u0005im\nq;q0(i\u000f+\nn;i\u000fn) + \u0005im\nq;q0(i\u000fn;i\u000f+\nn) +\u000eim\u0003q0(i\u000f+\nn;i\u000fn)\n1\u0000niu2\u0003q0(i\u000f+n;i\u000fn)(C3)\nWe rewrite the Matsubara summation of i\u000fnto the contour integral, and then change the contour\npath as the two path of [ \u00001\u0006i0;+1\u0006i0] and [\u00001\u0000i!\u0015\u0006i0;+1\u0000i!\u0015\u0006i0]. After taking the\nanalytic continuation i!\u0015!!+i0, we obtain\n\u001fR;(1)\nij(q;q0;!) = 2e\u000fmlj\u000bq\u0000q0;l\u0010\n\u0011(1)\nq;q0;im+i!'(1)\nq;q0;im+\u0001\u0001\u0001\u0011\n; (C4)\nwhere\u0011(1)\nq;q0;imis the zeroth order term of !. The!-linear term '(1)\nq;q0;imis obtained as\n'(1)\nq;q0;im=~\n2\u00192Re\u0002\n\u0005im\nq;q0(+i0;\u0000i0)\u0003\n+\u000eim\u0003q0(+i0;\u0000i0)\n1\u0000niu2\u0003q0(+i0;\u0000i0); (C5)\n\u0005im\nq;q0(+i0;\u0000i0) =1\nVX\nk~2\nme\u0010\nk+q\n2\u0011\ni\u0012\nk+q+q0\n2\u0013\nmgR\nk+qgR\nk+q0gA\nk; (C6)\n\u0003q0(+i0;\u0000i0) =1\nVX\nkgR\nk+q0gA\nk; (C7)\nwheregR\nk= (\u0000\u000fk+\u0016+i~=2\u001c)\u00001andgA\nk= (gR\nk)\u0003. Here, we remained the terms which contains\nboth the retarded and advanced Green functions in Eq. (C5). Expanding \u0005im\nq;q0(+i0;\u0000i0) and\n24\u0003q0(+i0;\u0000i0) up toq2andq02and performing the k-summations, we obtain\nRe\u0002\n\u0005im\nq;q0(+i0;\u0000i0)\u0003\n'\u000eim\u0012\n\u00001\n2I011+~2(q2+q02)\n3meRe\u0014\nI131+4\n5I241\u0015\n+4~2q\u0001q0\n15meRe\u0002\nI241\u0003\u0013\n+~2\n3me(q+q0)i(q+q0)mRe\u0014\nI131+4\n5I241\u0015\n+4~2\n15me(qiqm+q0\niq0\nm)Re\u0002\nI241\u0003\n+~2\n4meqi(qm+q0\nm)Re\u0014\nI021+4\n3I131\u0015\n(C8)\n'\u0019\u0017\u001c\n~\u0002\n\u000eim\u0000\n\u00001 +D0\u001cq\u0001q0\u0001\n\u0000D0\u001c(qi\u0000q0\ni)q0\nm\u0003\n; (C9)\n\u0003q0(+i0;\u0000i0)'I011+~2q02\n2me\u0012\nI021+4\n3I131\u0013\n'2\u0019\u0017\u001c\n~\u0000\n1\u0000D0q02\u001c\u0001\n; (C10)\nwhereD0= 2\u000fF\u001c=3meis the di\u000busion constant, and we used Eqs. (D8) and neglected the higher\norder contributions of ~=\u000fF\u001c(\u001c1). Hence, using Eq. (8), we \fnd\n'(1)\nq;q0;im=\u0017\u001c\nq02\u0000\n\u000eim(q\u0000q0)\u0001q0\u0000(qi\u0000q0\ni)q0\nm\u0001\n; (C11)\nwhich leads to Eq. (16). It should be noted that '(1)\nq;q0;ijmdoes not contain any terms proportional\nto 1=D0q02, which means that there is no contribution such as\nhj(r;t)i(1)/Z\u000b(r)\u0002_M(r0;t)\njr\u0000r0jdr0: (C12)\nNext, we calculate the second order response coe\u000ecient. The coe\u000ecient is also simpli\fed in the\ncase of\u0016+=\u0016\u0000. Equations (B10a) and (B10b) are given as\n\u001f(2;n)\nijk(i!\u0015;i!\u00150) = 2ie\u000folm\u000fmjk\u000bq\u0000q0\u0000q00;l1\n\fX\nnh\n\u0004io\nq;q0;q00(i\u000f+\nn;i\u000f0+\nn;i\u000fn) + \u0004io\nq;q00;q0(i\u000fn;i\u000f0+\nn;i\u000f+\nn)\n\u0000\u0004io\nq;q00;q0(i\u000fn;i\u000f0\u0000\nn;i\u000f\u0000\nn)\u0000\u0004io\nq;q0;q00(i\u000f\u0000\nn;i\u000f0\u0000\nn;i\u000fn)\n+ \u0002io\nq;q0;q00(i\u000f+\nn;i\u000f0+\nn;i\u000fn)\u0000\u0002io\nq;q00;q0(i\u000fn;i\u000f0\u0000\nn;i\u000f\u0000\nn)i\n;\n(C13)\n\u001f(2;a)\nijk(i!\u0015;i!\u00150) = 2ie\u000film\u000fmjk\u000bq\u0000q0\u0000q00;l1\n\fX\nnh\n\u0003q0;q00(i\u000f+\nn;i\u000f0+\nn;i\u000fn)\u0000\u0003q00;q0(i\u000fn;i\u000f0\u0000\nn;i\u000f\u0000\nn)i\n;(C14)\nwhere \u0004io\nq;q0;q00(i\u000f+\nn;i\u000f0+\nn;i\u000fn), \u0002io\nq;q0;q00(i\u000f+\nn;i\u000f0+\nn;i\u000fn), and \u0003q0;q00(i\u000f+\nn;i\u000f0+\nn;i\u000fn) are same respectively\nas that dropped the spin indexes in Eqs. (B11a), (B11b), and (B11c). However, Eqs. (C13) does\nnot contribute the non-local emergent electric \felds of our interest because it is canceled by the\ndi\u000busion ladder VCs shown in Fig. 6 (i), (j), and (n). We also \fnd that Eq. (C14) is canceled by\nthe di\u000busion ladder VCs of the spin vertex \u001bjor\u001bk, which gives rise to 1 =D0q002or 1=D0q02and\ndo not contribute the emergent electric \feld we focus in this paper.\n25The main contributions of the di\u000busion VCs [Eqs. (B12) and (B13)] to the non-local emergent\nelectric \feld are given as\n\u001f(2;n)(df)\nijk(i!\u0015;i!\u00150) = 2ie\u000folm\u000fmjk\u000bq\u0000q0\u0000q00;l1\n\fX\nnh\n\u0005io\nq;q0+q00(i\u000f+\nn;i\u000fn)\u0000q0+q00(i\u000f+\nn;i\u000fn)\u0003q0;q00(i\u000f+\nn;i\u000f0+\nn;i\u000fn)\n\u0000\u0005io\nq;q0+q00(i\u000fn;i\u000f\u0000\nn)\u0000q0+q00(i\u000fn;i\u000f\u0000\nn)\u0003q00;q0(i\u000fn;i\u000f0\u0000\nn;i\u000f\u0000\nn)\n+ \u0005io\nq;q0+q00(i\u000fn;i\u000f+\nn)\u0000q0+q00(i\u000fn;i\u000f+\nn)\u0003q0;q00(i\u000fn;i\u000f0+\nn;i\u000f+\nn)\n\u0000\u0005io\nq;q0+q00(i\u000f\u0000\nn;i\u000fn)\u0000q0+q00(i\u000f\u0000\nn;i\u000fn)\u0003q00;q0(i\u000f\u0000\nn;i\u000f0\u0000\nn;i\u000fn)i\n;\n(C15)\n\u001f(2;a)(df)\nijk(i!\u0015;i!\u00150) = 2ie\u000film\u000fmjk\u000bq\u0000q0\u0000q00;l1\n\fX\nnh\n\u0003q0+q00(i\u000f+\nn;i\u000fn)\u0000q0+q00(i\u000f+\nn;i\u000fn)\u0003q0;q00(i\u000f+\nn;i\u000f0+\nn;i\u000fn)\n\u0000\u0003q0+q00(i\u000fn;i\u000f\u0000\nn)\u0000q0+q00(i\u000fn;i\u000f\u0000\nn)\u0003q00;q0(i\u000fn;i\u000f0\u0000\nn;i\u000f\u0000\nn)i\n;\n(C16)\nwhere \u0003q(i\u000f+\nn;i\u000fn) and \u0000q(i\u000f+\nn;i\u000fn) are de\fned respectively by that dropped the spin indexes in\nEqs. (B3) and (B4). For Eqs. (C15) and (C16), we perform the similar procedures as in the\ncalculations of the linear response coe\u000ecient; rewriting the Matsubara summation of i\u000fnto the\ncontour integral, changing the integral path into the three paths [ \u00001\u0000iPi\u0006i0;+1\u0000iPi\u0006i0] with\nP1=!\u0015,P2=!\u00150andP3= 0 for the terms which depend on the frequencies set ( i\u000f+\nn;i\u000f0+\nn;i\u000fn),\nand withP1= 0,P2=\u0000!\u00150andP3=\u0000!\u0015for the terms which depend on the frequencies set\n(i\u000fn;i\u000f0\u0000\nn;i\u000f\u0000\nn). Then, taking the analytic continuations as Eq. (26), we obtain\n\u001fR;(2)\nijk(q;q0;q00;!;!0) = 2ie\u000folm\u000fmjk\u000bq\u0000q0\u0000q00;l\u0010\n\u0011(2)\nq;q0;q00;io+i!#(2)\nq;q0;q00;io+i!0'(2)\nq;q0;q00;io+\u0001\u0001\u0001\u0011\n;\n(C17)\nwhere the \frst term gives rise to the current depending on the magnetization \u0018Mj(t)Mk(t),\nnot on its dynamics, and the second term leads to the current due to the total derivative of the\nmagnetizations\u0018d(Mj(t)Mk(t))=dt. The third term of Eq. (C17) is the component that we are\nfocusing, which is obtained as\n'(2)\nq;q0;q00;io='(1)\nq;q0+q00;ioniu2\n2\bq0;q00; (C18)\n\bq0;q00=X\ns;t=\u0006s\u0003q0;q00(it0;is0;\u0000it0) =2i\nVX\nkIm\u0002\n(gR\nk\u0000gA\nk)gR\nk+q0gA\nk\u0000q00\u0003\n(C19)\nwhere'(1)\nq;q0+q00;iois given by Eq. (C5), we neglected the terms which contains only retarded/advanced\nGreen functions because they are just higher order contributions with respect to ~=\u000fF\u001c, and we\n26used \u0003 q00;q0(\u0000i0;\u0006i0;i0) = \u0003 q0;q00(i0;\u0006i0;\u0000i0). Here, expanding \b q0;q00with respect to q0andq00\nup to the second order, we have\n\bq0;q00=\u00002iIm\u0014\n2I012+~2(3q02+ 4q0\u0001q00+ 3q002)\n2meI013+4~2(2q02+ 3q0\u0001q00+ 2q002)\n3meI114\u0015\n'\u00008i\u0019\u0017\u001c2\n~2\b\n1\u0000D0\u001c(2q02+ 3q0\u0001q00+ 2q002)\t\n; (C20)\nwhereIlmnis given by Eq. (D1) and left in the leading order of \u000fF\u001c=~in the second equal by means\nof Eqs. (D8). Hence, using Eqs. (C9), (C10), and (8), we \fnally \fnd\n'(2)\nq;q0;q00;io=\u00002i\u0017\u001c2\n~\u000eio(q\u0000q0\u0000q00)\u0001(q0+q00)\u0000(qi\u0000q0\ni\u0000q00\ni)(q0\no+q00\no)\n(q0+q00)2+O(q2;q02;q002);(C21)\nwhich leads to Eq. (32).\nAppendix D: Integrals\nIn this Appendix, we show k-integralsIlmnthat we use in this paper;\nIlmn=1\nVX\nk\u0012~2k2\n2me\u0013l\u0000\ngR\nk\u0001m\u0000\ngA\nk\u0001n; (D1)\nwhere we set m+n\u0015l+ 3=2 to its convergence. We rewrite the summation over kinto the energy\nintegral,\nIlmn=Z1\n\u0000\u000fFd\u0018\u0017(\u0018+\u000fF)(\u0018+\u000fF)l\n(\u0000\u0018+i~=2\u001c)m(\u0000\u0018\u0000i~=2\u001c)n; (D2)\nwhere\u0017(\u000f)/p\u000fis DOS. For evaluating Ilmnwith respect to the leading order of ~=\u000fF\u001c, it is valid\nto approximate \u0018+\u000fF'\u000fF, which means that DOS and the energy approximates to the values at\nthe Fermi level, and to regard the lower limit of the integral as \u00001. However, we need to evaluate\nthe higher order contributions precisely such as in Eq. (16). Hence, we calculate Ilmnwithout any\napproximations.\nConsidering \u0017(x)/x1=2, the analyticity is as follows: Supposed that z=x+iyandw=pz=\nX+iY, and in the polar coordinate, z=rei(\u0012+2n\u0019)with\u0000\u0019\u0014\u0012<\u0019 andn= 0;\u00061;\u00062;\u0001\u0001\u0001,\nw=e1\n2logz=e1\n2logr+i\u0012\n2+n\u0019i=8\n<\n:prei\u0012\n2(n= 0);\n\u0000prei\u0012\n2(n= 1);(D3)\nwheren= 0;1 means the n-th Riemann surface: [ \u0000\u0019;\u0019) forn= 0, and [\u0019;3\u0019) forn= 1. For the\nRiemann surface of n= 0,\nX=prcos\u0012\n2; Y =prsin\u0012\n2; (D4)\n27and, the condition \u0000\u0019\u0014\u0012 < \u0019 , usingx=rcos\u0012=r(2 cos2(\u0012=2)\u00001),y=rsin\u0012=\n2rsin(\u0012=2) cos(\u0012=2),\ncos\u0012\n2=r\n1\n2\u0010\n1 +x\nr\u0011\n;sin\u0012\n2= sign(y)r\n1\n2\u0010\n1\u0000x\nr\u0011\nSimilarly, for the Riemann surface of n= 1,\ncos\u0012\n2=\u0000r\n1\n2\u0010\n1 +x\nr\u0011\n;sin\u0012\n2=\u0000sign(y)r\n1\n2\u0010\n1\u0000x\nr\u0011\nThese are collectively expressed as\nw=8\n<\n:px(c+(y=x) +isign(y)c\u0000(y=x)) (n= 0);\n\u0000px(c+(y=x) +isign(y)c\u0000(y=x)) (n= 1);(D5)\nwhere\nc\u0006(\u000e) =p\n1 +\u000e2 p\n1 +\u000e2\u00061\n2!1\n2\n(D6)\nFrom Eq. (D5), we can rewrite the path of the integral in Eq. (D2) as (see Fig. 8)\n1\n2\u0019Z1\n\u0000\u000fFd\u0018\u0017(\u0018+\u000fF)\u0017(\u0018+\u000fF)(\u0018+\u000fF)l\n(\u0000\u0018+i~=2\u001c)m(\u0000\u0018\u0000i~=2\u001c)n\n=1\n4\u0019\u0012Z1+i0\n\u0000\u000fF+i0+Z\nCR+Z\u0000\u000fF\u0000i0\n1\u0000i0+Z\nC0\u0013\nd\u0018\u0017(\u0018+\u000fF)(\u0018+\u000fF)l\n(\u0000\u0018+i~=2\u001c)m(\u0000\u0018\u0000i~=2\u001c)n\n=i\n2X\n\u0011=\u00061Res\u0018=i\u0011~=2\u001c\u0014\u0017(\u0018+\u000fF)(\u0018+\u000fF)l\n(\u0000\u0018+i~=2\u001c)m(\u0000\u0018\u0000i~=2\u001c)n\u0015\n; (D7)\nwhere the path CRis given by \u0018=Rei\u0012, 0\u0014\u0012\u00142\u0019, (changing the Riemann surfaces at \u0012=\u0019),\nC0is given by \u0018=\u000eei\u0012, 0\u0014\u0012 <2\u0019, andc\u0006=c\u0006(~=2\u000fF\u001c). Noted that the sign of the Residue at\n\u0018=\u0000i~=2\u001cis minus, because we gather it of the Riemann surface of n= 1.\nHere, we show the results of the integrals Ilmnwithm= 1.\nI011=2\u0019\u0017\u001c\n~c+; (D8a)\nI012=2\u0019\u0017\u001c2\n~2\u0014\u0012\ni\u0000\u000e\n21\n1\u0000i\u000e\u0013\nc++\u000e\n2ic\u0000\n1\u0000i\u000e\u0015\n; (D8b)\nI013=\u00002\u0019\u0017\u001c3\n~3\u0014\u0012\n1 +1\n2i\u000e\n1\u0000i\u000e+1\n4\u000e2\n(1\u0000i\u000e)2\u0013\nc++\u000e\n2c\u0000\n1\u0000i\u000e\u0012\n1\u00001\n2i\u000e\n1\u0000i\u000e\u0013\u0015\n; (D8c)\nI112=2\u0019\u0017\u000fF\u001c2\n~2\u0014\u0012\ni\u00003\u000e\n2\u0013\nc++i\u000e\n2c\u0000\u0015\n; (D8d)\nI113=\u00002\u0019\u0017\u000fF\u001c3\n~3\u0014\u0012\n1 +3i\u000e\n2\u00003\n4\u000e2\n1\u0000i\u000e\u0013\nc++\u000e\n2\u0012\n1 +3\n2i\u000e\n1\u0000i\u000e\u0013\nc\u0000\u0015\n; (D8e)\n28FIG. 8. The path of the integral is described. The solid and dashed line of the path CRandC0denote the\npathes on the Riemann surfaces n= 0 andn= 1, respectively. The contributions from C0andCRvanish\nin the limit of R!1 .\nI114=2\u0019\u0017\u000fF\u001c4\n~4\u0014\u0012\n\u0000i+3\u000e\n2+3\n4i\u000e2\n1\u0000i\u000e+1\n4\u000e3\n(1\u0000i\u000e)2\u0013\nc+\u0000\u000e\n2\u0012\ni\u00003\n2\u000e\n1\u0000i\u000e+1\n2i\u000e2\n(1\u0000i\u000e)2\u0013\nc\u0000\u0015\n;\n(D8f)\nI214=2\u0019\u0017\u000f2\nF\u001c4\n~4\u0014\u0012\n\u0000i+5\u000e\n2+9i\u000e2\n4\u00005\n4\u000e3\n1\u0000i\u000e\u0013\nc+\u0000\u000e\n2\u0012\ni\u00005\u000e\n2\u00005i\u000e2\n21\n1\u0000i\u000e\u0013\nc\u0000\u0015\n; (D8g)\nI215=2\u0019\u0017\u000f2\nF\u001c5\n~5\u0014\u0012\n1 +5i\u000e\n2\u00009\u000e2\n4\u00005\n4i\u000e3\n1\u0000i\u000e\u00005\n16\u000e4\n(1\u0000i\u000e)2\u0013\nc++\u000e\n2\u0012\n1 +5i\u000e\n2\u00005\n2\u000e2\n1\u0000i\u000e+5\n8i\u000e3\n(1\u0000i\u000e)2\u0013\nc\u0000\u0015\n;\n(D8h)\nwhere\u000e=~=2\u000fF\u001c(\u001c1). For small \u000e(>0),\nc+(\u000e) = 1 + (5=8)\u000e2\u0000(13=128)\u000e4+\u0001\u0001\u0001; (D9a)\nc\u0000(\u000e) =\u000e=2 + (3=16)\u000e3\u0000(17=256)\u000e5\u0000\u0001\u0001\u0001 (D9b)\n\u0003E-mail address: fujimoto.junji.s8@kyoto-u.ac.jp\n1R. H. Silsbee, A. Janossy, and P. Monod, Phys. Rev. B 19, 4382 (1979).\n2S. Mizukami, Y. Ando, and T. Miyazaki, J. Magn. Magn. Mater. 226-230 , 1640 (2001).\n3Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys. Rev. Lett. 88, 117601 (2002).\n4M. V. Costache, M. Sladkov, S. M. Watts, C. H. van der Wal, and B. J. van Wees, Phys. Rev. Lett. 97,\n216603 (2006).\n5E. Saitoh, M. Ueda, H. Miyajima, and G. Tatara, Appl. Phys. Lett. 88, 182509 (2006).\n6V. Edelstein, Solid State Communications 73, 233 (1990).\n297J. C. R. S\u0013 anchez, L. Vila, G. Desfonds, S. Gambarelli, J. P. Attan\u0013 e, J. M. De Teresa, C. Mag\u0013 en, and\nA. Fert, Nat. Commun. 4, 2944 (2013).\n8G. E. Volovik, J. Phys. C Solid State Phys. 20, L83 (3 1987).\n9G. Tatara, Physica E: Low-dimensional Systems and Nanostructures 106, 208 (2019).\n10K.-W. Kim, J.-H. Moon, K.-J. Lee, and H.-W. Lee, Phys. Rev. Lett. 108, 217202 (2012).\n11A. Takeuchi and G. Tatara, J. Phys. Soc. Jpn. 81, 033705 (2 2012).\n12N. Nakabayashi and G. Tatara, New J. Phys. 16, 015016 (2014).\n13G. Tatara, N. Nakabayashi, and K.-J. Lee, Phys. Rev. B 87, 054403 (2013).\n14A. Takeuchi, K. Hosono, and G. Tatara, Phys. Rev. B 81, 144405 (2010).\n15K. Nakazawa and H. Kohno, J. Phys. Soc. Jpn. 83, 073707 (2014).\n16G. Tatara, Phys. Rev. B 98, 174422 (2018).\n17P. D. R. Kubo, P. D. M. Toda, and P. D. N. Hashitsume, in Statistical Physics II , Springer Series in\nSolid-State Sciences No. 31 (Springer Berlin Heidelberg, 1991) pp. 146{202.\n18T. Jujo, J. Phys. Soc. Jpn. 75, 104709 (2006).\n19H. Kohno and J. Shibata, J. Phys. Soc. Jpn. 76, 063710 (2007).\n20A. M. Zagoskin, Quantum Theory of Many-Body Systems: Techniques and Applications (Springer- Verlag,\n1998).\n21G. Rickayzen, Green's Functions and Condensed Matter , reprint ed. (Dover Publications, New York,\n2013).\n22As we have introduced the convergence factor \u0011as in Eq. (A2), \u00011B(t) is also assumed to be expressed\nas\u00011B(t) =e\u0011tR\n\u00011B(!)e\u0000i!td!=2\u0019for the time-translational symmetry.\n23Considering the convergence factors, we have assumed \u00012B(t) =e(\u0011+\u00110)tR\n\u00012B(!)e\u0000i!tdt=2\u0019.\n30"    },    {        "title": "1111.5466v1.Rashba_spin_torque_in_an_ultrathin_ferromagnetic_metal_layer.pdf",        "content": "arXiv:1111.5466v1  [cond-mat.mtrl-sci]  23 Nov 2011Rashba spin torque in an ultrathin ferromagnetic metal laye r\nXuhui Wang∗and Aurelien Manchon†\nPhysical Science & Engineering Division, KAUST, Thuwal 239 55-6900, Kingdom of Saudi Arabia\n(Dated: August 24, 2018)\nIn a two-dimensional ferromagnetic metal layer lacking inv ersion symmetry, the itinerant electrons\nmediate the interaction between the Rashba spin-orbit inte raction and the ferromagnetic order\nparameter, leading to a Rashba spin torque exerted on the mag netization. Using Keldysh technique,\nin the presence of both magnetism and a spin-orbit coupling, we derive a spin diffusion equation\nthat provides a coherent description to the diffusive spin dy namics. The characteristics of the spin\ntorque and its implication on magnetization dynamics are di scussed in the limits of large and weak\nspin-orbit coupling.\nPACS numbers: 75.60.Jk,75.70.Tj,72.25.-b,72.10.-d\nI. INTRODUCTION\nBy transferring angular momentum between the elec-\ntronic spin and the orbital, spin-orbit coupling fills the\nneedforelectricalmanipulationofspindegreeoffreedom.\nOutstanding examples are the electrically generated bulk\nspin polarization1,2and the well-known spin Hall effect\n(SHE)3–5in a two dimensional electron gas where the\nspin-orbit interaction, particularly of the Rashba-type,6\nplays the leading role. Rashba spin-orbit interaction not\nonly introduces an effective field perpendicular to the lin-\near momentum but also provides the backbone to the\nspin-relaxation through the so-called D’yakonov-Perel\nmechanism,7which is dominant in a two-dimensional\nsystem. Besides its prominent role in semiconductors,\nRashba spin-orbit coupling is believed to exist at ferro-\nmagnetic/heavy metal as well as ferromagnetic/metal-\noxide interfaces, in which the inversion symmetry break-\ning offers a potential gradient empowering the spin-orbit\ncoupling.\nMeanwhile, magnetism continuously stimulates the in-\ndustrial and academic appetite. In the pursuit of fast\nmagnetization switching, Slonczewski-Berger spin trans-\nfer torque8employs a polarized spin current instead of\na cumbersome magnetic field. This celebrated scheme\ndemands non-collinear magnetic textures in forms of, for\nexample, spin valves or domain wall structures.9\nIn the presence of inversion symmetry breaking (such\nasasymmetricinterfaces),aferromagneticmetallayeras-\nsembles both magnetism and spin-orbit coupling, hence\noffering an alternative switching mechanism:10,11Spin-\norbit coupling transfers the orbital angular momentum\ncarried by an electric current to the electronic spin, thus\ncreating an effective magnetic field (Rashba field). As\nlong as the effective field is mis-aligned with the magne-\ntization direction, the so-called Rashba torque emerges,\nthus exciting the magnetization.\nCurrent-driven magnetization dynamics by spin-orbit\ntorque has been demonstrated by several experiments\non metal-oxide based systems.12–14In fact, the Rashba\ntorque can be categorized into to a broader family\nof spin-orbit interaction induced torque that has been\nobserved in diluted magnetic semiconductors.16–18Re-cently, Miron et al.,15has demonstrated the current-\ninducedmagnetizationswitchingusinga singleferromag-\nnet in Pt/Co/AlO xtrilayers, which further consolidates\nthe feasibility of the Rashba torque. The same type of\nspin-orbit coupling induced torque is predicted to im-\nprove current-driven domain wall motion,11,19which is\nsupported by experimental observations.20At this stage,\nwe are aware of an alternative explanation, as pointed\nout by Liu et al.,21in terms of the spin Hall effect (SHE)\noccurring in the underlying heavy metal layer, such as\nPt or Ta. The distinction between the spin Hall induced\neffect and the Rashba one is discussed in the last section\nof this article.\nIn searching for a general form of the Rashba torque\nin ferromagnetic metal layers,10we found an expression\nthat consists of two components:22An in-plane torque\n(∝m×(ˆy×m)) and an out-of-plane one ( ∝ˆy×m),\ngivenˆyisthein-planedirectiontransversetotheinjected\ncurrent and mis the magnetization direction. Numerical\nsolution on a two-dimensional nano-wire with one open\ntransport direction has been carried out to appreciate\nthe significance of diffusive motion on the spin torque.\nWe found that the in-plane component of the torque in-\ncreases when narrowing the magnetic wire22.\nIn the present article, we give a full theoretical deriva-\ntion of the coupled diffusive equation for spin dynam-\nics in a ferromagnetic metal layer and describe the form\nof the Rashba torque in both weak and strong Rashba\nlimits. In Sec. II, we combine the Keldysh formalism\nand the gradient expansion technique to derive a cou-\npled diffusion equation for charge and non-equilibrium\nspin densities. To demonstrate that the diffusion equa-\ntion provides a coherent framework to describe the spin\ndynamics, we dedicate Sec. III to the spin diffusion in\na ferromagnetic metal, which shows an excellent agree-\nment to early result on the same system. In Sec.IV, we\nillustrate that the absence of magnetism (in our diffusion\nequation) describes the well-know phenomenon of elec-\ntrically induced spin polarization. The cases of a weak\nand a strong spin-orbit coupling are discussed in Sec.V\nand Sec. VI, respectively, where we provide an analytical\nform of the Rashba torque in an infinite medium. In Sec.\nVII, we discuss the implication of the Rashba torque on2\nmagnetization dynamics as well as its distinction from\nspin Hall effect induced torque.\nII. DIFFUSION EQUATIONS\nThe system of interest is defined as a quasi-two-\ndimensional ferromagnetic metal layer rolled out in the\nxy-plane. Two asymmetric interfaces provide a confine-\nment in z-direction, along which the potential gradi-\nent generates a Rashba spin-orbit coupling. Therefore\na single-particle Hamiltonian for an electron of momen-\ntumˆkis (/planckover2pi1= 1 is assumed throughout)\nˆH=ˆk2\n2m+αˆσ·(ˆk׈z)+1\n2∆xcˆσ·m+Hi(1)\nwhereˆσis the Pauli matrix, mthe effective mass,\nandmthe magnetization direction. The ferromag-\nnetic exchange splitting is given by ∆ xcandαrepre-\nsents the Rashba constant (parameter). Hamiltonian\nˆHi=/summationtextN\nj=1V(r−rj) sums the contribution of the non-\nmagnetic impurity scattering potential V(r) localized at\nrj.\nTo derive a diffusion equation for the non-\nequilibriumchargeandspindensities, weemployKeldysh\nformalism.23Using Dyson equation, in a 2 ×2 spin space,\nwe obtain a kinetic equation that assembles the retarded\n(advanced) Green’s function ˆGR(ˆGA), the Keldysh com-\nponent of the Green function ˆGK, and the self-energy\nˆΣK, i.e.,\n[ˆGR]−1ˆGK−ˆGK[ˆGA]−1=ˆΣKˆGA−ˆGRˆΣK,(2)\nwhere all Green’s functions are full functions with inter-\nactions taken care of by the self-energies ˆΣR,A,K. The\nretarded (advanced) Green’s function in momentum and\nenergy space is\nˆGR(A)(k,ǫ) =1\nǫ−ǫk−ˆσ·b(k)−ˆΣR(A)(k,ǫ),(3)\nwhereǫk=k2/(2m) is the single-particle energy. We\nhave introduced a k-dependent effective field b(k) =\n∆xcm/2+α(k×z) of the magnitude bk=|∆xcm/2+\nα(k×z)|and the direction ˆb=b(k)/bk.\nNeglecting localization effect and electron-electron in-\nteractions, we assume a short-range δ-function type im-\npurity scattering potential. At a low concentration and\na weak coupling to electrons, the second-order Born ap-\nproximation is justified,23i.e., the self-energy is24\nˆΣR,A,K(r,r′) =δ(r,r′)\nmτˆGR,A,K(r,r) (4)\nwhere the momentum relaxation time reads\n1\nτ≈2π/integraldisplayd2k′\n(2π)2|V(k−k′)|2δ(ǫk−ǫk′),(5)whereV(k) is the Fourier transform of the scattering\npotential and the magnitude of momentum kandk′is\nevaluated at Fermi vector kF.\nThe quasi-classical distribution function ˆ g≡\nˆgk,ǫ(T,R), defined as the Wigner transform of the\nKeldysh function ˆGK(r,t;r′,t′), is obtained by inte-\ngrating out the relative spatial-temporal coordinates\nwhile retaining the center-of-mass ones R= (r+r′)/2\nandT= (t+t′)/2. As long as the spatial profile\nof the quasi-classical distribution function is smooth\nat the scale of Fermi wave length, we may apply the\ngradient expansion technique on Eq.(2),25which gives\nus a transport equation associated with macroscopic\nquantities. The left-hand side of the kinetic equation in\ngradient expansion becomes\n[ˆGR]−1ˆGK−ˆGK[ˆGA]−1\n≈[ˆg,ˆσ·b(k)]+i\nτˆg+i∂ˆg\n∂T\n+i\n2/braceleftbiggk\nm+α(ˆz׈σ),∇Rˆg/bracerightbigg\n,(6)\nwhere{·,·}denotes the anti-commutator. The relax-\nation time approximation indulges the right-hand side\nof Eq.(2) as\nˆΣKˆGA−ˆGRˆΣK\n≈1\nτ/bracketleftBig\nˆρ(ǫ,T,R)ˆGA(k,ǫ)−ˆGR(k,ǫ)ˆρ(ǫ,T,R)/bracketrightBig\n(7)\nwherewe haveintroducedthe densitymatrix by integrat-\ning out the momentum kin ˆg, i.e.,\nˆρ(E,T,R) =1\n2πN0/integraldisplayd2k′\n(2π)2ˆgk′,ǫ(T,R).(8)\nFor the convenience of discussion, time variable is\nchanged from Ttot. At this stage, we have a kinetic\nequation depending on ˆ ρas well as on ˆ g\ni[ˆσ·b(k),ˆg]+1\nτˆg+∂ˆg\n∂t+1\n2/braceleftbiggk\nm+α(ˆz׈σ),∇Rˆg/bracerightbigg\n=i\nτ/bracketleftBig\nˆGR(k,ǫ)ˆρ(ǫ)−ˆρ(ǫ)ˆGA(k,ǫ)/bracketrightBig\n. (9)\nA Fouriertransformationon temporal variableto the fre-\nquency domain ωleads to\nΩˆg−bk[ˆUk,ˆg] =iˆK, (10)\nwhere Ω = ω+i/τand the operator ˆUk≡ˆσ·ˆbsatisfies\nˆUkˆUk= 1. The right hand side of Eq.(10) is partitioned\naccording to\nˆK=−1\n2/braceleftbiggk\nm+α(ˆz׈σ),∇Rˆg/bracerightbigg\n/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright\nˆK(1)\n+i\nτ/bracketleftBig\nˆGR(k,ǫ)ˆρ(ǫ)−ˆρ(ǫ)ˆGA(k,ǫ)/bracketrightBig\n/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright\nˆK(0).(11)3\nThe equilibrium part is denoted by ˆK(0)while the gradi-\nent term ˆK(1)is regarded as perturbation. Functions ˆ g\nand ˆρare both in frequency domain. We solve Eq. (10)\nformally to find a solution to ˆ g\nˆg=i(2b2\nk−Ω2)ˆK+2b2\nkˆUkˆKˆUk−Ωbk[ˆUk,ˆK]\nΩ(4b2\nk−Ω2)≡L[ˆK].\n(12)\nAn iterationprocedureto solveEq.(12) hasbeenoutlined\nby Mishchenko et al.,in Ref.[24]. We follow this proce-\ndure here: According to the partition scheme on ˆK, we\nuseˆK(0)to obtain the zero-th order approximation as\nˆg(0)≡L[ˆK(0)(ˆρ)], which replaces ˆ ginˆK(1)to generate\na correction due to the gradient term, i.e., ˆK(1)(ˆg(0)).\nWe further insert ˆK(1)(ˆg(0)) back to Eq.(12) to obtain\na correction given by L[ˆK(1)(ˆg(0))], then we obtain the\nfirst order approximation to the quasi-classical distribu-tion function,\nˆg(1)= ˆg(0)+L[ˆK(1)(ˆg(0))]. (13)\nThe above procedure is repeated to any desired order,\ni.e.,\nˆg(n)= ˆg(n−1)+L[ˆK(1)(ˆg(n−1))].(14)\nIn this paper, the second order approximation is suffi-\ncient. The full expression of the second orderapproxima-\ntion istedious thusnot included in the following. Adiffu-\nsion equation is derived by an angle averagingin momen-\ntum space, which allowsalltermsthat areoddorderin ki\n(i=x,y) to vanish while the combinations such as kikj\ncontribute to the averaging by a factor k2\nFδij, givenkF\nthe Fermi wave vector.25Further more, a Fourier trans-\nform from frequency domain back to the real time brings\na diffusion type equation for the density matrix,\n∂\n∂tˆρ(t) =D∇2ˆρ−1\nτxcˆρ+1\n2τxc(ˆz׈σ)·ˆρ(ˆz׈σ)+iC[ˆz׈σ,∇ˆρ]−B{ˆz׈σ,∇ˆρ}\n+Γ[(m×∇)zˆρ−ˆσ·m∇ˆρ·(ˆz׈σ)−(ˆz׈σ)·∇ˆρˆσ·m]\n+1\n2Txc(ˆσ·mˆρˆσ·m−ˆρ)−i˜∆xc[ˆσ·m,ˆρ]−2R{ˆσ·m,(m×∇)zˆρ}, (15)\nwhere all quantities are evaluated at Fermi energy ǫF.\nIn a two-dimensional system, the diffusion constant D=\nτv2\nF/2 is given in terms of Fermi velocity vFand mo-\nmentum relaxation time τ. The renormalized exchange\nsplitting reads ˜∆xc= (∆xc/2)/(4ξ2+ 1) where ξ2=\n(∆2\nxc/4+α2k2\nF)τ2. The other parameters are\nC=αkFvFτ\n(4ξ2+1)2,Γ =α∆xcvFkFτ2\n2(4ξ2+1)2, R=α∆2\nxcτ2\n2(4ξ2+1),\n1\nτxc=2α2k2\nFτ\n4ξ2+1, B=2α3k2\nFτ2\n4ξ2+1,1\nTxc=∆2\nxcτ\n4ξ2+1.\nτxcis the relaxation time due to the so-called D’yakonov-\nPerel mechanism.1Equation (15) is valid in the dirty\nlimitξ≪1, whichenablestheapproximation1+4 ξ2≈1.\nCharge density nand the non-equilibrium spin density S\nare introduced by the vector decomposition on the den-\nsity matrix ˆ ρ=n/2 +S·ˆσ. In a real experimental\nsetup,12,15,20spin transport in ferromagnetic layers suf-\nfersfromrandommagneticscatterers, forwhichweintro-\nduce an isotropic spin-flip relaxation S/τsfphenomeno-\nlogically.\nEventually, we obtain a set of diffusion equations for\nthe charge and spin densities, i.e.,\n∂n\n∂t=D∇2n+B∇z·S\n+Γ∇z·mn+R∇z·m(S·m),(16)and\n∂S\n∂t=D∇2S−1\nτ/bardblS/bardbl−1\nτ⊥S⊥\n−∆xcS×m−1\nTxcm×(S×m)\n+B∇zn+2C∇z×S+2R(m·∇zn)m\n+Γ[m×(∇z×S)+∇z×(m×S)],(17)\nwhere∇z≡ˆz×∇. The spin density S/bardbl≡Sxˆx+Syˆyis\nrelaxed at a rate 1 /τ/bardbl≡1/τxc+ 1/τsfwhileS⊥≡Szˆz\nhas a rate 1 /τ⊥≡2/τxc+1/τsf.\nForabroadrangeoftherelativestrengthbetweenspin-\norbit coupling and the exchange splitting, i.e., αkF/∆xc,\nEq.(16) and Eq.(17) describe the spin dynamics in a fer-\nromagnetic layer. When the magnetism vanishes (∆ xc=\n0), theB-term provides a source that generates spin den-\nsity electrically.2,24On the other hand, when the spin-\norbit coupling is absent ( α= 0), the first two lines in\nEq.(17) describe a diffusive motion of spin density in\na ferromagnetic metal, which, to be shown in the next\nsection, agrees excellently with early results in the cor-\nresponding limit.26C-term describes the coherent pre-\ncession of the spin density around the effective Rashba\nfield. The precession of the spin density (induced by the\nRashba field) around the exchange field is described by\nthe Γ-term, thus a higher order (compared to C) in the4\ndirty limit for Γ = ∆ xcτC/2. TheR-term contributes to\nthe magnetization renormalization.\nIII. SPIN DIFFUSION IN A FERROMAGNET\nSpin diffusion in a ferromagnet has been discussed ac-\ntively in the field of spintronics.26–29In this section we\nshow explicitly that, by suppressing the spin-orbit cou-\npling, Eq.(17) describes the spin diffusion equation in the\ncorresponding limits.\nIn the present model, vanishing Rashba spin-orbit cou-\npling means α= 0, then Eq.(17) reduces to\n∂\n∂tS=D∇2S+1\nτ∆m×S\n−1\nτsfS−1\nTxcm×(S×m),(18)\nwhereτ∆≡1/∆xcis the time scale of the coherent pre-\ncession of the spin density around the magnetization.\nThis equation differs from the result of Zhang et al.,27\nonly by a dephasing of the transverse component of the\nspin density that is set by the time scale Txc. In a fer-\nromagnetic metal, we may divide the spin density into\nalongitudinal component that follows the magnetization\ndirection adiabatically, and a deviation that is perpendic-\nularto the magnetization, i.e., S=s0m+δSwheres0\nis the local equilibrium spin density. Such a partition,\nafter restoring the electric field by ∇→∇+eE∂ǫ, gives\nrise to\n∂\n∂tδS+∂\n∂ts0m\n=s0D∇2m+D∇2δS+DePFNFE·∇m\n−δS\nτsf−s0m\nτsf−δS\nTxc+∆xcm×δS,(19)\nwhere the magnetic order parameteris allowedto be spa-\ntial dependent, i.e., m=m(r,t). The energy derivative\nis treated as ∂ǫS≈PFNFmgivenPFthe spin polariza-\ntion and NFthe density of state, both at Fermi energy.\nFor a smooth magnetic texture in which the character-\nistic length scale of the magnetic profile is much larger\nthan the length scale for electron transport, we discard\nthe contribution D∇2δS.26The diffusion of the equilib-\nrium spin density follows s0D∇2m≈s0m/τsf. In this\npaper, we retain only terms that are first order in tem-\nporal derivative, which simplifies Eq.(19) to\n−1\nτ∆m×δS+/parenleftbigg1\nτsf+1\nTxc/parenrightbigg\nδS=\n−s0∂\n∂tm+DePFNFE·∇m.(20)The last equation can be solved exactly\nδS=τ∆\n1+ς2/bracketleftbiggPF\nem×(je·∇)m+ςPF\ne(je·∇)m\n−s0m×∂m\n∂t−ςs0∂m\n∂t/bracketrightbigg\n(21)\nwhereς=τ∆(1/τsf+ 1/Txc) and the electric current\nje=e2nτE/mis given in terms of electron density\nn. Apart from the inclusion of the dephasing of trans-\nversecomponentasimplementedinparameter ς, thenon-\nequilibrium spin density Eq.(21) agrees excellently with\nEq.(8) in Ref.[26].\nGiven the knowledge of the spin density, the spin\ntorque, defined as\nT=−1\nτ∆m×δS+1\nTxcδS, (22)\nis given by\nT=1\n1+ς2/bracketleftbigg\n−ηs0∂m\n∂t+βs0m×∂m\n∂t\n+ηPF\ne(je·∇)m−βPF\nem×(je·∇)m/bracketrightbigg\n(23)\nwhereη= 1 +ςτ∆/Txcandβ=τ∆/τsf. Assum-\ning a long dephasing time of the transverse component\n(i.e.,Txc→ ∞), thenη≈1 and Eq. (23) reproduces the\nEq.(9) in Ref.[26]. On the other hand, a short dephasing\ntime (of the transverse component) enhances parameter\nηtherefore increases the temporal spin torque (i.e., the\nfirst term in Eq.(23)).\nIV. ELECTRICALLY GENERATED SPIN\nDENSITY\nThe effect of an electrically generated non-equilibrium\nspin density due to spin-orbit coupling2can be ex-\ntractedfrom Eq.(17) bysetting exchangeinteractionzero\n(i.e., ∆ xc= 0). Retaining D’yakonov-Perel as the only\nspin relaxation mechanism and letting τsf=∞, Eq.(17)\nends up in\nD∇2S−1\nτxc(S+Szˆz)\n+2C(ˆz×∇)×S+B(ˆz×∇)n= 0 (24)\nwhich reduces to the results in the well-known spin Hall\neffect.24,30,31In the case of an infinite medium along\ntransport direction, i.e., ˆx-direction, Eq.(24) gives rise\nto a solution to the spin density\nS=τxcBeE1\nǫFnˆy=eEζ\nπvFˆy,\nwhere only the linear term in electric field has been re-\ntained. On the right hand side, we have used the charge5\ndensity in a 2D system n=k2\nF/(2π) and introduced the\nparameter ζ=αkFτas used in Ref. [24].\nIn the following sections, we explore the spin torque in\nthe presence of both exchange and Rashba field in an in-\nfinite medium. The primary focus is on two cases: Weak\nand a strong spin-orbit coupling, when comparing to the\nmagnitude of exchange splitting. In general, Eq.(17) is\napplicable through a broad range of relative strength be-\ntween spin-orbit coupling and exchange splitting. A full\nscale numerical simulation on the diffusion equation is\nbeyond the scope of this paper, we refer the readers to\nRef.[22] for further interests.\nV. WEAK SPIN-ORBIT COUPLING\nA weak Rashba spin-orbit coupling implies a small\nD’yakonov-Perel relaxation rate 1 /τxc∝α2, such that\nτxc≫τsf,τ∆, which allows spin relaxation to be dom-\ninated by random magnetic impurities. In this regime,\nwhen comparing to the magnitudes of Cand Γ, the con-\ntribution from BandRare at a higher order in α, thus\nto be disregarded. We consider a stationary state where\n∂S/∂t= 0. An electric field applied along ˆx-direction,\ni.e.,E=Eˆx. In an infinite medium,10all the spatial\nderivatives vanishes ( ∇→0) and the dynamic equation\nreads\n−1\nτ∆m×S+1\nTxcm×(S×m)+1\nτsfS\n=2eECˆy×∂ǫS\n+eEΓ[ˆy×(m×∂ǫS)+m×(ˆy×∂ǫS)].(25)\nIn addition to the spin density induced by exchange\nsplitting, a weak spin-orbit interaction leads to a devi-\nation in spin density that can be considered as a per-\nturbation. Therefore, we may well apply the partition\nS=S⊥+S/bardblmto separate the longitudinal and the\ntransverse components. Eq.(25) is thus reduced to\n1\nτ∆m×S⊥−1\nT⊥S⊥−1\nτsfS/bardblm\n=−2eECP FNFˆy×m\n−eEΓPFNFm×(ˆy×m) (26)\nwhere 1/T⊥≡1/Txc+ 1/τsfand we have again em-\nployedthe approximationon the energy derivative ∂ǫS≈\nPFNFmand replaced the energy derivative of the\ncharge density by the density of states at Fermi energy\n(i.e.,∂ǫn≈n/ǫF=NF). We solve Eq.(26) to obtain a\nsolution to the non-equilibrium spin density\nS⊥=τ∆\n1+ς2eEPFNF[(2C+ςΓ)m×(ˆy×m)\n−(Γ−2ςC)(ˆy×m)]. (27)\nandS/bardbl= 0. In Eq.(27), the second component, oriented\nalong the direction ˆy×m, is actually perpendicular totheplanespannedbythemagnetizationdirectionandthe\neffective Rashba field (along ˆy), which, as to be shown\nbelow, contributes to a Rashba torque that fulfils the\nsymmetry described in a recent experiment.15The defi-\nnition Eq.(22) leads to a general expression for the spin\ntorque\nT=T⊥ˆy×m+T/bardblm×(ˆy×m),(28)\nwhich consists of an out-of-plane and anin-plane com-\nponents with magnitudes determined by\nT⊥=eEPFNF\n1+ς2(2ηC+βΓ), (29)\nT/bardbl=eEPFNF\n1+ς2(ηΓ−2βC). (30)\nTheplaneis defined by the magnetization direction m\nand the direction of the effective Rashba field that in the\npresent setting is aligned along ˆy-direction. Note that\nthe sign of the in-plane torque, Eq. (30), can change\ndepending on the interplay between spin relaxation and\nprecession.\nTo compare directly with the results in Ref.[10], we al-\nlow the spin relaxation time τsf→ ∞, therefore β≈0.\nWe also consider the transverse dephasing time to be\ninfinite.26,27Under these assumptions, η≈1 andς≈0\nand we have T⊥≈2eEPFNFCandT/bardbl≈eEPFNFΓ. In\nthe dirty limit, Γ ≪Cdue to ∆ xcτ≪1, therefore mak-\ning use of the relation for the polarization PF= ∆xc/ǫF\nand the Drude relation je=e2nτE/m, we obtain an\nout-of-plane torque\nT= 2αm∆xc\neǫFjeˆy×m, (31)\nwhich agrees excellently with the spin torque in an in-\nfinite system in the corresponding limit as derived in\nRef.[10].\nVI. STRONG SPIN-ORBIT COUPLING\nThe opposite limit to Sec.V is a strong spin-orbit\ncoupling. In this case, we consider the scenario that\nαkF≫∆xcand the D’yakonov-Perel relaxation mech-\nanism is dominating, i.e., 1 /τxc≫1/τsf, due to the\nfact 1/τxc∝α2. Therefore, it is not physical to sim-\nply assume that the direction of spin density is domi-\nnantlyalignedalongthemagnetizationdirection, aswhat\nis treated in the case of a weak spin-orbit coupling. A\nself-consistent solution from Eq.(17) to the spin density\nis more justified.\nAgain, as in Sec.V, we consider an infinite system\nwhere an electric field Eis applied at ˆx-direction. The\nmagnetization direction is left arbitrary. We approxi-\nmate the energy derivative by ∂ǫ≈1/ǫF. The above6\nassumptions simplify Eq.(17) to\n1\nτ∆S×m+1\nTxcm×(S×m)−2eEC\nǫFˆy×S\n+1\nτxc(S+Szˆz) =eE\nǫFnBˆy,(32)\nwherea strongspin-orbitcouplingrendersΓ and Rterms\nnegligible. By considering Txc≫τ∆,τxc, Eq. (32) re-\nduces to\n1\nτ∆S׈m+1\nτxc(S+Szˆz)\n−2eEC\nǫFˆy×S=eE\nǫFnBˆy,(33)\nwhich is a set of linear equations for the non-equilibrium\nspin density. We are interested in the linear response\nregime, which implies that at the distance as defined by\nthe Fermi wave length 1 /kF, we have eE/kF≪αkF.\nTherefore up to the first order in exchange splitting, we\nextract the spin density from the above equation to be\nS=eE\nǫFnτxcB/parenleftBig\nˆy−χˆy×m−χ\n2mxˆz/parenrightBig\n(34)\nwhereχ≡τxc/τ∆we have used the identity ˆy×m=\nmzˆx−mxˆz. This yields a spin torque\nT=αm∆xc\neǫFje(ˆy×m\n+χm×(ˆy×m)−χ\n2mxˆz×m/parenrightBig\n.(35)\nThis torque is slightly different from the weak Rashba\nlimit and has a strong implication in terms of magneti-\nzation dynamics. The torque is dominated by a field-like\ntorque along ˆy, similarly to the weak Rashba case. First,\nin contrast to the weak Rashba case [see Eq. (30)], the\nsign of the in-plane torque remains positive. Secondly,\nthe anisotropic spin relaxation coming from D’yakonov-\nPerel mechanism yields an additional component of spin\naccumulation that is oriented along ˆz. The implication\nof this torque on the current-driven magnetization dy-\nnamics is discussed in the next section.\nVII. DISCUSSION\nCurrent-induced magnetization dynamics in a sin-\ngle ferromagnetic layer has been observed in vari-\nous structures that involve interfaces between transi-\ntion metal ferromagnets, heavy metals and/or metal-\noxide insulators. Existing experimental systems are\nPt/Co/AlO x,12,13,15,20Ta/CoFeB/MgO,14Pt/NiFe and\nPt/Co bilayers.21Besides the structural complexity in\nsuch systems, an unclear form of spin-orbit coupling in\nthe bulk and interfaces places a challenge to understand\nthe nature of the torque.A. Validity of Rashba model\nThe celebrated Rashba-type effective interfacial spin-\norbit Hamiltonian was pioneered by E. I. Rashba to\nmodel the influence of asymmetric interfaces in semicon-\nducting 2DEG:6A sharp potential drop, emerging at the\ninterface (say, in the xy-plane) between two materials,\ngivesrisetoapotentialgradient ∇Vthatisnormaltothe\ninterface, i.e., ∇V≈ξ(r)ˆz. In case a rotational symme-\ntry exists in the interface plane, a spherical Fermi surface\nassumptionallowsthe spin-orbitinteractionHamiltonian\ntohavetheform ˆHR=αˆσ·(p׈z), whereα≈ ∝an}b∇acketle{tξ∝an}b∇acket∇i}ht/4m2c2.\nAs a matter of fact, in semiconducting interfaces where\nthe transport is described by a limited number of bands\naround a high symmetry point, the Rashba form can be\nrecovered through k·ptheory.32\nAs far as metallic interfaces areconcerned, a spin-orbit\nsplitting of the Rashba-type in the conduction band has\nbeen observedat Au surfaces,33Gd/GdO interfaces,34Bi\nsurfacesandcompounds,35andmetallicquantumwells.36\nThe presence of a Rashba interaction in graphene37and\nat oxide hetero-interfaces38has also been reported re-\ncently. It is quite interesting to notice that the sym-\nmetry breaking-induced spin splitting of the conduction\nband seems rather general and might not be restricted to\nheavy metal interfaces36.\nIn the case of transition metals, however, the free elec-\ntron approximation fails to characterize the band struc-\nture accuratelydue to both alargenumberofband cross-\ningattheFermienergyandastronghybridizationamong\ns,panddorbitals. Density functional theory (DFT) is a\nsuccessful tool to investigate the nature of spin-orbit in-\nteraction at metallic surfaces. For example, in Refs.[39],\nthe authors observe a band splitting that possesses simi-\nlarpropertiesasRashbaspin-orbitinteractionanddecays\nexponentially away from the surface.39Alternatively, the\nspin-orbit interaction at metallic surfaces has been ad-\ndressed using tight-binding models for the porbitals.40,41\nAt such sharp interfaces, the magnitude of the orbital\nangular momentum (OAM) is considered to play a dom-\ninant role at the onset of a Rashba-type spin splitting.\nThis finding is consistent with the long stand-\ning work on interfacial magnetic anisotropy at\na ferromagnet/heavy metal,42and more recently,\nferromagnetic/metal-oxide interface.43In such systems,\na perpendicular magnetic anisotropy arises from the\norbital overlap between the 3 dstates of the ferromagnets\nand the spin-orbit coupled states of the normal metal.\nThe observation of perpendicular magnetic anisotropy\nat Co/metal-oxide interfaces tends to support the major\nrole of large interfacial OAM in the onset of interfacial\nspin-orbit effects.41,43The presence of interfacial Rashba\nspin-orbit coupling has also been shown to produce\ninterfacial perpendicular magnetic anisotropy.44\nAll these previoustheoreticaland experimental studies\nstrongly suggest that the interfacial spin splitting exists\nin the presence of a large OAM and potential gradient.\nHowever, a microscopic description of realistic interfaces7\nis still missing. Although the Rashba spin-orbit interac-\ntion is a convenient Hamiltonian to extract qualitative\nbehaviors, its applicability to realistic metallic interfaces\nwith complex band structures remains to be tested.\nB. Spin Hall effect versus Rashba torque\nRecently, Liu et al.,21proposedtomanipulatethemag-\nnetization of a Pt/Co or Pt/NiFe bilayer using the spin\ncurrent generatedby spin Hall effect in the underlying Pt\nlayer. When injecting a charge current jeinto a normal\nmetal accommodating a strong spin-orbit coupling, the\nasymmetricspinscatteringinducesatransversepurespin\ncurrent that has the form J= (αH/e)je׈µ⊗ˆµ, where\nαHis the spin Hall angle and ˆµis the spin direction.45\nWhen impinging on the ferromagnetic layer deposited on\ntop of the Pt layer, the spin current transverse to the\nlocal magnetization is absorbed and generates a torque\nTSHE= (bH/e)(1−βm×)m×(ˆµ×m) (to be called\nSHE torque thereafter). Here, bH=αHjeµB/eis the\nspintorqueamplitudewheretheregularspinpolarization\nPis replaced by the spin Hall angle αH.βis the non-\nadiabaticity parameter proposed by Zhang and Li26and\nit stems from the presence of spin-flip scattering in the\nsystem. In the configuration adopted by Liu et al.,the\ncharge current is injected along ˆxand the torque is given\nby\nTSHE=αHµBje\ne(m×(ˆy×m)+βˆy×m).(36)\nNote that a more realistic model should account for spin\ndiffusion in Co and Pt, as discussed in Ref. [46]. An\nimportant conclusion is that, besides the correction in\nthe case of a strong Rashba coupling, both Rashba and\nSHE produce the same type of torque, see Eq.(28) and\nEq.(35) in this article.\nNevertheless, distinctions can be made. First, in the\nabsence of the corrections due to spin-flip and spin pre-\ncession, the Rashba torque reduces to the field-like term,\nˆy×m, whereas the SHE torque reduces to the (anti-\n)dampingterm m×(ˆy×m). Thisassertionmustbescru-\ntinizede carefully since the actual relative magnitude be-\ntween the field-like and the damping torques depends on\nthe width of the magnetic wire as well as on the detailed\nspin dynamics in presence of spin-flip and precession.22\nFurthermore, for such an ultra-small system the spin-flip\nscattering giving rise to the non-adiabaticity parameter\n(β) might be significantly different from the one mea-\nsured in a more conventional thin film.\nA second important difference arises from the fact that\nthe Rashba torque arises from spin-orbit fields generated\nbyinterfacial currents, whereas the SHE torque is due to\nthe current flowing in the bulkof the Pt layer. Therefore,\nforaconstantexternalelectricfield, varyingthethickness\nof Pt layer shall enhance the SHE torque, while keeping\nthe Rashba torque unchanged.The torques as a function of the Co layer thickness is\nmore difficult to foresee. Although one could claim that\nRashba spin-orbit interaction is expected to be localized\nat the interface, where the potential gradient is large,\nnumerical simulations show that the Rashba-type inter-\nactionsurvivesafew monolayers39(which istypically the\nthickness of the Co layer under consideration). In addi-\ntion, the presenceofquantumwellstatesmightalsomod-\nifythenatureofthespin-orbitinteractionintheultrathin\nmagnetic layer in a system such as Pt/Co/AlO x.36\nThe same is true for the SHE torque. The injection\nof spin current into a Co layer is accompanied by spin\nprecession that takes place over a very short decoherence\nlength. This decoherence length has been studied experi-\nmentally and theoretically in spin valves and found to be\nof the order of a few monolayers.47In the typical case of\n3 or 4 monolayer-thickferromagnets, the SHE torque can\nnot be considered as a purely interfacial phenomenon.\nC. Magnetization Dynamics\nIn Pt/Co/AlO xtrilayers, Miron et alhave observed\na current-driven domain wall nucleation,12an enhanced\ncurrent-driven domain wall velocity20and a current-\ndriven magnetization switching.15The symmetry of the\nspin torque required to explain the experimental findings\nagree well with Rashba torque proposed in Ref. 10. On\na similar structure, Pi et al13and Suzuki et al14also ob-\nserved an effective field torque that could be interpreted\nintermsoftheRashbatorque. Recently,Liu et al21inter-\npreted their experiments on Pt/NiFe and Pt/Co bilayers\nusing SHE in the underlying Pt layer.\n1. Magnetization switching\nAccording to our previous discussions, both Rashba\ntorque and SHE torque have a general form T=T⊥ˆy×\nm+T/bardblm×(ˆy×m). The first term acts like a field\noriented along the direction transverse current direction\nwhereas the second term acts like an (anti-)damping\nterm, mimicking a conventional spin transfer torque that\nwould arise from a polarizer pointing to ˆy.\nAsaconsequence,bothRashbatorqueandSHEtorque\npossess the appropriate symmetry to excite the magne-\ntization of a single ferromagnet and induce switching, as\nobservedby Miron et al15and Liuet al.21In the case of a\nlarge Rashba spin-orbit coupling, the torque acquires an\nadditional component that acts like an effective magnetic\nfield along ˆz, vanishing as the magnetization component\nmxis zero (see Section VI), which provides an additional\ntorque that helps destabilize the magnetization.8\n2. Current-driven domain wall motion\nThe influence of Rashba/SHE torque on a domain wall\ncan be illustrated within the rigid Bloch wall approxima-\ntion. The perpendicularly magnetized Bloch wall is pa-\nrameterized by m= (cosφsinθ,sinφsinθ,cosθ) where\nφ=φ(t) andθ(x,t) = 2tan−1[e(x−xc(t))/∆], where xc\nrefers to the center of the domain wall and ∆ is defined\nas the domain wall width. To describe the dynamics of\na Bloch wall, Landau-Lifshitz-Gilbert (LLG) equation\n∂tm=−γm×Heff+αG∂tm×m+τ(37)\nhas to be augmented by the current induced torque τ\nτ=bJ∇m−βbJm×∇m\n+bJ(τ⊥ˆy×m+τ/bardblm×(ˆy×m)\n+τzmxˆz×m).(38)\nThe torque τis written in the most general form, where\nthe first two terms are the regular adiabatic and the so-\ncalled non-adiabatic torques; the next two terms ( τ/bardbland\nτ⊥) emerge from the presenceof Rashbaand/orspin Hall\neffect and the last term τzappears only in large Rashba\nlimit (see Sec. VI). The magnitude of the adiabatic\ntorque is bJ=µBPje/e. The effective field is given by\nHeff=2A\nMs∇2m+HKmxˆx+H⊥mzˆz.(39)\nParameter γin LLG is the gyromagnetic ratio, αGis the\nGilbert damping, Ais the exchange constant, Msis the\nsaturation magnetization, HKis the in-plane magnetic\nanisotropy and H⊥is the combination of an out-of-plane\nanisotropyand ademagnetizingfield. Themagnetization\ndynamics can be obtained readily from Eqs. (37)-(39) by\nintegrating over the magnetic volume\n∂tφ+αG∂txc\n∆=/bracketleftbigg∆π\n2(τ/bardbl−τz\n2)cosφ−β/bracketrightbiggbJ\n∆(40)\nαG∂tφ−∂txc\n∆=−γHK\n2sin2φ+/parenleftbigg\n1+∆π\n2τ⊥cosφ/parenrightbiggbJ\n∆.\n(41)\nWe observe that the in-plane torque τ/bardbldistorts the do-\nmain wall texture, while the perpendicular torque τ⊥\ndrives the domain wall motion. The additional torque\nτz, arising in the large Rashba limit, only contributes to\nthe in-plane torque. Therefore, in the following, we will\nrefer to the in-plane torque as τ∗\n/bardbl=τ/bardbl−τz/2. Below the\nWalker breakdown ( ∂tφ= 0), the velocity is given by\n∂txc=−/parenleftbigg\nβ−∆π\n2τ∗\n/bardblcosφ/parenrightbiggbJ\nαG(42)\nγHK\n2sin2φ=/bracketleftbigg\nαG−β+∆π\n2(αGτ⊥+τ∗\n/bardbl)cosφ/bracketrightbiggbJ\nαG∆,\n(43)where the tilting angle φis given by the competition be-\ntween the magnetic anisotropy, the non-adiabatictorque,\nand the Rashba/SHEtorque. In the case ofweak Rashba\n(τz= 0), assuming τ/bardbl=βτ⊥and omitting the correction\nto the spin precession, we recover the results of Ref. [48].\nWhen neglecting the in-plane torque and accounting for\nthe perpendicular Rashba torque ( τ∗\n/bardbl= 0), the Rashba\ntorque only acts like an effective transverse field and en-\nhances the Walker breakdown limit20[see Eq. (43)].\nAccounting for the in-plane component τ/bardblarising ei-\nther from corrections to Rashba torque or from the SHE,\nthis torque appears to modify the domain wall veloc-\nity. Therefore, depending on the strength and the sign\nof Rashba/SHE torque as well as on the resulting tilt-\ning angle φ, it is possible to obtain a vanishing or even\na reversed domain wall velocity, as has been shown nu-\nmerically in Ref. [48] and illustrated in Eq. (42). A full\nscale numerical investigation is beyond the scope of this\narticle, but it will help understand the profound effect of\nRashba and SHE torque on the domain wall structures.\nVIII. CONCLUSION\nUsing Keldysh technique, in the presence of both mag-\nnetism and a Rashba spin-orbit coupling, we derive a\nspin diffusion equation that provides a coherent descrip-\ntion to the diffusive spin dynamics. In particular, we\nhave derived a general expression for the Rashba torque\nin the bulk of a ferromagnetic metal layer, at both weak\nand strong Rashba limits. We find that the torque is in\ngeneral composed of two components, a field-like torque\nand the other (anti-)damping one. Being aware of the\nrecent alternative interpretation on the current-induced\nmagnetization switching in a single ferromagnet, we have\ndiscussedthe differencebetweentheRashbaandtheSHE\ntorques. While exploring the common features, we found\nthat the magnetization dynamics driven by the Rashba\ntorque presents several interesting similarities to that in-\nduced by SHE torque. Nevertheless, further investiga-\ntion involving structural modification of the system is\nexpected to provide a deeper knowledge on the nature\nof the interfacial spin-orbit interaction as well as the\ncurrent-induced magnetization switching in a single fer-\nromagnet.\nAcknowledgments\nWe thank G. E. W. Bauer, J. Sinova, M. D. Stiles, X.\nWaintal, and S. Zhang for stimulating discussions. We\nare specially grateful to K. -J. Lee and H. -W. Lee for\ninspiring discussion about the magnetization dynamics.9\n∗Electronic address: xuhui.wang@kaust.edu.sa\n†Electronic address: aurelien.manchon@kaust.edu.sa\n1M. I. D’yakonov and V. I. Perel, Sov. Phys. JETP Lett.\n13, 467 (1971); Phys. Lett. A 35, 459 (1971).\n2V. M. Edelstein, Solid. Stat. Comm. 73, 233 (1989).\n3S. Murakami et al.,Science301, 1348 (2004).\n4J. Sinova et al., Phys. Rev. Lett. 92, 126603 (2004).\n5Y. K. Kato et al.,Science306, 1910 (2004).\n6Yu. A. Bychkov and E. I. Rashba, J. Phys. C: Solid. Stat.\nPhys.17, 6039 (1984).\n7M. I. D’yakonov and V. I. Perel, Sov. Phys. Solid State 13,\n3023 (1971).\n8J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996);\nL. Berger, Phys. Rev. B. 54, 9353 (1996).\n9M. D. Stiles and J. Miltat, Top. Appl. Phys. 101, 225\n(2006); D. C. Ralph and M. D. Stiles, J. Magn. Magn.\nMater.320, 1190 (2008); J. Z. Sun and D. C. Ralph, ibid.\n320, 1227 (2008).\n10A. Manchon and S. Zhang, Phys. Rev. B. 78, 212405\n(2008);ibid.79, 094422 (2009).\n11A. Matos-Abiague and R. L. Rodriguez-Suarez, Phys. Rev.\nB80, 094424 (2009); I. Garate and A. H. MacDonald, ibid.\n80, 134403 (2009); P. M. Haney, and M. D. Stiles, Phys.\nRev. Lett. 105, 126602 (2010).\n12I. M. Miron et al.,Nature Materials 9, 230 (2010).\n13U. H. Pi et al.,Appl. Phys. Lett. 97, 162507 (2010).\n14T. Suzuki et al.,Appl. Phys. Lett. 98, 142505 (2011).\n15I. M. Miron et al.,Nature (London) 476, 189 (2011).\n16A. Chernyshov et al.,Nature Physics 5, 656 (2010).\n17D. Fang et al.,Nature Nanotechnology 6, 413 (2011).\n18M. Endo et al.,Appl. Phys. Lett. 97, 222501 (2010).\n19K. Obata, and G. Tatara, Phys Rev. B 77, 214429 (2008).\n20I. M. Miron et al.,Nature Materials 10, 419 (2011).\n21L. Liu et al.,Phys. Rev. Lett. 106, 036601 (2011);\narXiv:1110.6846 (2011).\n22X. Wang and A. Manchon, arXiv:1111.1216 (2011).\n23J. Rammer and H. Smith, Rev. Mod. Phys. 58, 323 (1986).\n24E. G. Mishchenko et al.,Phys. Rev. Lett. 93, 226602\n(2004).\n25J. Rammer, Quantum Field Theory of Non-equilibrium\nStates(Cambridge University Press, Cambridge, 2007).\n26S. Zhang and Z. Li, Phys. Rev. Lett. 93, 127204 (2004).\n27S. Zhang et al.,Phys. Rev. Lett. 88, 236601 (2002).28Y. Tserkovnyak et al.,Rev. Mod. Phys. 77, 1375 (2005).\n29Y. Tserkovnyak et al.,Phys. Rev. B 79, 094415 (2009).\n30A. A. Burkov et al.,Phys. Rev. B 70, 155308 (2004).\n31˙I. Adagideli and G. E. W. Bauer, Phys. Rev. Lett. 95,\n256602 (2005).\n32R. Winkler, Spin-Orbit Coupling Effects in Two-\nDimensional Electron and Hole Systems , Springer, Berlin,\n(2003).\n33S. LaShell et al.,Phys. Rev. Lett. 77, 3419 (1996); F.\nReinertet al.,Phys. Rev. B 63, 115415 (2001); G. Nicolay\net al., ibid .65, 033407 (2001); M. Hoesch et al., ibid .69,\n241401(R) (2004).\n34O. Krupin et al.,Phys. Rev. B 71, 201403(R) (2005); O\nKrupinet al.,New. J. Phys. 11, 013035 (2009).\n35Ph. Hofmann, Prog. Surf. Sci. 81, 191 (2006); T. Hirahara\net al.,Phys. Rev. B 76, 153305 (2007); H. Mirhosseini\net al., ibid .79, 245428 (2009); A. Takayama et al.,Phys.\nRev. Lett. 106, 166401 (2011); K. Ishizaka et al.,Nature\nMaterials 10, 521 (2011).\n36A. Varykhalov et al.,Phys. Rev. Lett. 101, 256601 (2008);\nJ. Hugo Dil et al., ibid .101, 266802 (2008); A. G. Rybkin\net al.,Phys. Rev. B 82, 233403 (2010).\n37Yu. S. Dedkov et al.,Phys. Rev. Lett. 100, 107602 (2008);\nO. Rader et al., ibid .102, 057602 (2009).\n38A. D. Caviglia et al.,Phys. Rev. Lett. 104, 126803 (2010).\n39G. Bihlmayer et al.,Phys. Rev. B 75, 195414 (2007); G.\nBihlmayer et al.,Surf. Sci. 600, 3888 (2006); M. Nagano\net al.,J. Phys.: Cond. Mater. 21, 064239 (2009).\n40L. Petersen and P. Hedeg˚ ard, Surf. Sci. 459, 49 (2000).\n41S. R. Park et al.,Phys. Rev. Lett. 107, 156803 (2011).\n42K. Kyuno et al.,J. Phys. Soc. Jpn. 61, 2099 (1992); D.\nWelleret al.,Phys. Rev. B 49, 12888 (1994); G. H. O.\nDaalderop et al., ibid .50, 9989 (1994).\n43H. X. Yang et al.,Phys. Rev. B 84, 054401 (2011); C.\nNistoret al., ibid .84, 054464 (2011).\n44A. Manchon, Phys. Rev. B 83, 172403 (2011).\n45J. E. Hirsch, Phys. Rev. Lett. 83, 1834 (1999); S. Zhang,\nibid.85, 393 (2000).\n46A. Vedyayev et al.,arXiv:1108.2589 (2011).\n47S. Wang et al.,Phys. Rev. B 77, 184430 (2008).\n48K.-W. Kim et al.,arXiv:1111.3422 (2011)."    },    {        "title": "1908.01236v2.Experimental_evidence_for_Zeeman_spin_orbit_coupling_in_layered_antiferromagnetic_conductors.pdf",        "content": "Experimental evidence for Zeeman spin-orbit coupling\nin layered antiferromagnetic conductors\nR. Ramazashvili,1,∗P. D. Grigoriev,2, 3, 4,†T. Helm,5, 6, 7,‡F. Kollmannsberger,5, 6M. Kunz,5, 6,§W.\nBiberacher,5E. Kampert,7H. Fujiwara,8A. Erb,5, 6J. Wosnitza,7, 9R. Gross,5, 6, 10and M. V. Kartsovnik5,¶\n1Laboratoire de Physique Th´ eorique, Universit´ e de Toulouse, CNRS, UPS, France\n2L. D. Landau Institute for Theoretical Physics, 142432 Chernogolovka, Russia\n3National University of Science and Technology MISiS, 119049 Moscow, Russia\n4P. N. Lebedev Physical Institute, 119991 Moscow, Russia\n5Walther-Meißner-Institut, Bayerische Akademie der Wissenschaften,\nWalther-Meißner-Strasse 8, D-85748 Garching, Germany\n6Physik-Department, Technische Universit¨ at M¨ unchen, D- 85748 Garching, Germany\n7Hochfeld-Magnetlabor Dresden (HLD-EMFL) and W¨ urzburg-Dresden Cluster of Excellence ct.qmat,\nHelmholtz-Zentrum Dresden-Rossendorf, 01328 Dresden, Germany\n8Department of Chemistry, Graduate School of Science,\nOsaka Prefecture University, Osaka 599-8531, Japan\n9Institut f¨ ur Festk¨ orper- und Materialphysik, TU Dresden, 01062 Dresden, Germany\n10Munich Center for Quantum Science and Technology (MCQST), D-80799 Munich, Germany\n(Dated: December 23, 2020)\nMost of solid-state spin physics arising from spin-orbit coupling, from fundamental phenomena\nto industrial applications, relies on symmetry-protected degeneracies. So does the Zeeman spin-\norbit coupling, expected to manifest itself in a wide range of antiferromagnetic conductors. Yet,\nexperimental proof of this phenomenon has been lacking. Here, we demonstrate that the N´ eel state\nof the layered organic superconductor κ-(BETS) 2FeBr 4shows no spin modulation of the Shubnikov-\nde Haas oscillations, contrary to its paramagnetic state. This is unambiguous evidence for the spin\ndegeneracy of Landau levels, a direct manifestation of the Zeeman spin-orbit coupling. Likewise,\nwe show that spin modulation is absent in electron-doped Nd 1.85Ce0.15CuO 4, which evidences the\npresence of N´ eel order in this cuprate superconductor even at optimal doping. Obtained on two very\ndifferent materials, our results demonstrate the generic character of the Zeeman spin-orbit coupling.\nINTRODUCTION\nSpin-orbit coupling (SOC) in solids intertwines elec-\ntron orbital motion with its spin, generating a variety of\nfundamental effects1,2. Commonly, SOC originates from\nthe Pauli termHP=¯h\n4m2\neσ·p×∇V(r) in the electron\nHamiltonian3,4, where ¯his the Planck constant, methe\nfree electron mass, pthe electron momentum, σits spin\nandV(r) its potential energy depending on the position.\nRemarkably, N´ eel order may give rise to SOC of an en-\ntirely different nature, via the Zeeman effect5,6:\nHso\nZ=−µB\n2/bracketleftbig\ng/bardbl(B/bardbl·σ) +g⊥(k)(B⊥·σ)/bracketrightbig\n,(1)\nwhereµBis the Bohr magneton, Bthe magnetic field,\nwhileg/bardblandg⊥define the g-tensor components with\nrespect to the N´ eel axis. In a purely transverse field\nB⊥, a hidden symmetry of a N´ eel antiferromagnet pro-\ntects double degeneracy of Bloch eigenstates at a spe-\ncial set of momenta in the Brillouin zone(BZ)5,6: at such\nmomenta,g⊥must vanish. The scale of g⊥is set by\ng/bardbl, which renders g⊥(k) substantially momentum depen-\ndent, and turnsHso\nZinto a veritable SOC5–8. This cou-\npling was predicted to produce unusual effects, such as\nspin degeneracy of Landau levels in a purely transverse\nfieldB⊥9,10and spin-flip transitions, induced by an AC\nelectric rather than magnetic field10. Contrary to the\ntextbook Pauli spin-orbit coupling, this mechanism doesnot require heavy elements. Being proportional to the\napplied magnetic field (and thus tunable!), it is bound\nonly by the N´ eel temperature of the given material. In\naddition to its fundamental importance as a novel spin-\norbit coupling mechanism, this phenomenon opens new\npossibilities for spin manipulation, much sought after in\nthe current effort11–13to harness electron spin for future\nspintronic applications. While the novel SOC mechanism\nmay be relevant to a vast variety of antiferromagnetic\n(AF) conductors such as chromium, cuprates, iron pnic-\ntides, hexaborides, borocarbides, as well as organic and\nheavy-fermion compounds6, it has not received an exper-\nimental confirmation yet.\nHere, we present experimental evidence for the spin\ndegeneracy of Landau levels in two very different lay-\nered conductors, using Shubnikov–de Haas (SdH) os-\ncillations as a sensitive tool for quantifying the Zee-\nman effect14. First, the organic superconductor κ-\n(BETS) 2FeBr 4(hereafterκ-BETS)15is employed for\ntesting the theoretical predictions. The key features mak-\ning this material a perfect model system for our purposes\nare (i) a simple quasi-two-dimensional (quasi-2D) Fermi\nsurface and (ii) the possibility of tuning between the AF\nand paramagnetic (PM) metallic states, both showing\nSdH oscillations, by a moderate magnetic field15,16. We\nfind that, contrary to what happens in the PM state, the\nangular dependence of the SdH oscillations in the AF\nstate of this compound is notmodulated by the ZeemanarXiv:1908.01236v2  [cond-mat.str-el]  21 Dec 20202\nsplitting. We show that such a behavior is a natural con-\nsequence of commensurate N´ eel order giving rise to the\nZeeman SOC in the form of Eq. (1).\nHaving established the presence of the Zeeman SOC\nin an AF metal, we utilize this effect for probing\nthe electronic state of Nd 2−xCexCuO 4(NCCO), a pro-\ntotypical example of electron-doped high- Tccuprate\nsuperconductors17. In these materials, superconduc-\ntivity coexists with another symmetry-breaking phe-\nnomenon manifested in a Fermi-surface reconstruction\nas detected by angle-resolved photoemission spectroscopy\n(ARPES)18–21and SdH experiments22–25. The involve-\nment of magnetism in this Fermi-surface reconstruction\nhas been broadly debated26–40. Here, we present detailed\ndata on the SdH amplitude in optimally doped NCCO,\ntracing its variation over more than two orders of mag-\nnitude with changing the field orientation. The oscil-\nlation behavior is found to be very similar to that in\nκ-BETS. Given the crystal symmetry and the position\nof the relevant Fermi-surface pockets, this result is firm\nevidence for antiferromagnetism in NCCO. Our finding\nnot only settles the controversy in electron-doped cuprate\nsuperconductors—but also clearly demonstrates the gen-\nerality of the novel SOC mechanism.\nTheoretical background — Before presenting the ex-\nperimental results, we recapitulate the effect of Zee-\nman splitting on quantum oscillations. The superposi-\ntion of the oscillations coming from the conduction sub-\nbands with opposite spins results in the well-known spin-\nreduction factor in the oscillation amplitude14:Rs=\ncos/parenleftBig\nπg\n2m\nme/parenrightBig\n; heremeandmare, respectively, the free\nelectron mass and the effective cyclotron mass of the\nrelevant carriers. We restrict our consideration to the\nfirst-harmonic oscillations, which is fully sufficient for the\ndescription of our experimental results. In most three-\ndimensional (3D) metals, the dependence of Rson the\nfield orientation is governed by the anisotropy of the cy-\nclotron mass. At some field orientations Rsmay vanish,\nand the oscillation amplitude becomes zero. This spin-\nzeroeffect carries information about the renormalization\nof the product gmrelative to its free-electron value 2 me.\nFor 3D systems, this effect is obviously not universal. For\nexample, in the simplest case of a spherical Fermi surface,\nRspossesses no angular dependence whatsoever, hence\nno spin zeros. By contrast, in quasi-2D metals with their\nvanishingly weak interlayer dispersion such as in layered\norganic and cuprate conductors, spin zeros are41–46a ro-\nbust consequence of the monotonic increase of the cy-\nclotron mass, m∝1/cosθ, with tilting the field by an\nangleθaway from the normal to the conducting layers.\nIn an AF metal, the g-factor may acquire a kdepen-\ndence through the Zeeman SOC mechanism. It becomes\nparticularly pronounced in the purely transverse geome-\ntry, i.e., for a magnetic field normal to the N´ eel axis. In\nthis case,Rscontains the factor ¯ g⊥averaged over the cy-\nclotron orbit, see Supplementary Information (hereafter\nSI) for details. As a result, the spin-reduction factor ina layered AF metal takes the form:\nRs= cos/bracketleftbiggπ\ncosθ¯g⊥m0\n2me/bracketrightbigg\n, (2)\nwherem0≡m(θ= 0◦). Often, the Fermi surface is\ncentered at a point k∗, where the equality g⊥(k∗) = 0\nis protected by symmetry6– as it is for κ-BETS (see\nSI). Such a k∗belongs to a line node g⊥(k) = 0 crossing\nthe Fermi surface. Hence, g⊥(k) changes sign along the\nFermi surface, and ¯ g⊥in Eq. (2) vanishes by symmetry of\ng⊥(k). Consequently, the quantum-oscillation amplitude\nis predicted to have no spin zeros9. For pockets with\nFermi wave vector kFwell below the inverse AF coheren-\nce length 1/ξ,g⊥(k) can be described by the leading term\nof its expansion in k. For such pockets, the present result\nwas obtained in Refs.9,10,47. According to our estimates\nin the SI, both in κ-BETS and in optimally doped NCCO\n(x= 0.15), the product kFξconsiderably exceeds unity.\nYet, the quasi-classical consideration above shows that\nforkFξ >1 the conclusion remains the same: ¯ g⊥= 0,\nsee SI for the explicit theory.\nWe emphasize that centering of the Fermi surface at\na point k∗withg⊥(k∗) = 0 – such as a high-symmetry\npoint of the magnetic BZ boundary6– is crucial for a\nvanishing of ¯ g⊥. Otherwise, Zeeman SOC remains inert,\nas it does in AF CeIn 3, whosedFermi surface is centered\nat the Γ point (see SI and Refs.48–50), and in quasi-2D\nEuMnBi 2, with its quartet of Dirac cones centered away\nfrom the magnetic BZ boundary51,52. With this, we turn\nto the experiment.\nRESULTS\nAF Organic Superconductor κ-(BETS) 2FeBr 4—This\nis a quasi-2D metal with conducting layers of BETS\ndonor molecules, sandwiched between insulating FeBr−\n4-\nanion layers15. The material has a centrosymmetric or-\nthorhombic crystal structure (space group Pnma ), with\ntheacplane along the layers. The Fermi surface consists\nof a weakly warped cylinder and two open sheets, sep-\narated from the cylinder by a small gap ∆ 0at the BZ\nboundary, as shown in Fig. 115,16,53.\nThe magnetic properties of the compound are mainly\ngoverned by five localized 3 d-electron spins per Fe3+ion\nin the insulating layers. Below TN≈2.5 K, theseS= 5/2\nspins are ordered antiferromagnetically, with the unit cell\ndoubling along the caxis and the staggered magnetiza-\ntion pointing along the aaxis15,54. Above a critical mag-\nnetic fieldBc∼2−5 T, dependent on the field orien-\ntation, antiferromagnetism gives way to a saturated PM\nstate55.\nThe SdH oscillations in the high-field PM state and\nin the N´ eel state are markedly different (see Fig. 1b).\nIn the former, two dominant frequencies corresponding\nto a classical orbit αon the Fermi cylinder and to a\nlarge magnetic-breakdown (MB) orbit βare found, in\nagreement with the predicted Fermi surface16,53. The3\n0 2 4 6 8 10 12 140.130.140.150.16R (W)\nB (T)Bc\n0 2 40.010.1110\nb \nF (kT)a \n0.05 0.100.010.1110\nFFT ampl. (Arb. units) F (kT)d \ngꓕ(k)\nkcka\nkc0AF\nd\nQAF\n/2c /c /2c /c 0\nba\n/c /c 0ka\nkc\n0(a) (b) (c)Paramagnetic\nphaseAntiferromagnetic\nphase\nAF PM\nFIG. 1. 2D Fermi surface of κ-BETS in the para- and antiferromagnetic phases. (a) Fermi surface of κ-BETS in\nthe PM state15,53(blue lines). The blue arrows show the classical cyclotron orbits αand the red arrows the large MB orbit β,\nwhich involves tunneling through four MB gaps ∆ 0in a strong magnetic field. (b)Interlayer magnetoresistance of the κ-BETS\nsample, recorded at T= 0.5 K with field applied nearly perpendicularly to the layers ( θ= 2◦). The vertical dash indicates the\ntransition between the low-field AF and high-field PM states. The insets show the fast Fourier transforms (FFT) of the SdH\noscillations for field windows [2 −5] T and [12 −14] T in the AF and PM state, respectively. (c)The BZ boundaries in the AF\nstate with the wave vector QAF= (π/c,0) and in the PM state are shown by solid-black and dashed-black lines, respectively.\nThe dotted-blue and solid-orange lines show, respectively, the original and reconstructed Fermi surfaces16. The shaded area in\nthe corner of the magnetic BZ, separated from the rest of the Fermi surface by gaps ∆ 0and ∆ AF, is theδpocket responsible\nfor the SdH oscillations in the AF state. The inset shows the function g⊥(k).\nAntiferromagnetic\nphase\n0 1 2 3 40510\n5\n5\n0°16°34° 60°65.4°  Rosc/Rbackg (%)\nB cosq  (T)52° \n3\n65.4°  0°\n16° \n34° \n52° \n03060901200246AFFT (arb. unit)\nF cos (q) (T)2\n10\n0 1 2 3 4 5 6051015202530\n0°R (W)\nB (T)16°34°52°q : 65.4°\nk-BETSBc(b) (a)\nFIG. 2. SdH oscillations in the antiferromagnetic phase of κ-BETS .(a)Examples of the field-dependent interlayer\nresistance at different field orientations, at T= 0.42 K. The AF – PM transition field Bcis marked by vertical dashes. Inset: the\norientation of the current Jand magnetic field Brelative to the crystal axes and the N´ eel axis N.(b)Oscillating component,\nnormalized to the non-oscillating B-dependent resistance background, plotted as a function of the out-of-plane field component\nB⊥=Bcosθ. The curves corresponding to different tilt angles θare vertically shifted for clarity. For θ≥52◦the ratio\nRosc/Rbackg is multiplied by a constant factor, as indicated. The vertical dashed lines are drawn to emphasize the constant\noscillation phase in these coordinates; Inset: FFT spectra of the SdH oscillations taken in the field window [3 −4.2] T. The\nFFT amplitudes at θ= 52◦and 65.4◦are multiplied by a factor of 2 and 10, respectively.\noscillation amplitude exhibits spin zeros as a function\nof the field strength and orientation, which is fairly\nwell described by a field-dependent spin-reduction factor\nRs(θ,B), with theg-factorg= 2.0±0.2 in the presence of\nan exchange field BJ≈−13 T, imposed by PM Fe3+ionson the conduction electrons45,56. In the SI, we provide\nfurther details of the SdH oscillation studies on κ-BETS.\nBelowBc, in the AF state, new, slow oscillations at the\nfrequencyFδ≈62 T emerge, indicating a Fermi-surface\nreconstruction16. The latter is associated with the fold-4\ning of the original Fermi surface into the magnetic BZ,\nandFδis attributed to the new orbit δ, see Fig. 1c. This\norbit emerges due to the gap ∆ AFat the Fermi-surface\npoints, separated by the N´ eel wave vector ( π/c,0)57.\nFigure 2 shows examples of the field-dependent inter-\nlayer resistance of κ-BETS, recorded at T= 0.42 K, at\ndifferent tilt angles θ. The field was rotated in the plane\nnormal to the N´ eel axis (crystallographic aaxis). In ex-\ncellent agreement with previous reports16,58, slow oscil-\nlations with frequency Fδ= 61.2 T/cosθare observed\nbelowBc, see inset in Fig. 2b. Thanks to the high crys-\ntal quality, even in this low-field region the oscillations\ncan be traced over a wide angular range |θ|≤70◦.\nThe angular dependence of the δ-oscillation amplitude\nAδis shown in Fig. 3. The amplitude was determined\nby fast Fourier transform (FFT) of the zero-mean oscil-\nlating magnetoresistance component normalized to the\nmonotonicB-dependent background, in the field window\nbetween 3.0 and 4.2 T, so as to stay below Bc(θ) for all\nfield orientations. The lines in Fig. 3 are fits using the\nLifshitz-Kosevich formula for the SdH amplitude14:\nAδ=A0m2\n√\nBRMBexp(−KmT D/B)\nsinh(KmT/B )Rs(θ), (3)\nwhereA0is a field-independent prefactor, B= 3.5 T\n(the midpoint of the FFT window in 1 /Bscale),m\nthe effective cyclotron mass ( m= 1.1meatθ=\n0◦16, growing as 1 /cosθwith tilting the field as in\nother quasi-2D metals59,60),K= 2π2kB/¯he,T=\n0.42 K,TDthe Dingle temperature, and RMBthe MB\nfactor. For κ-BETS,RMBtakes the form RMB =/bracketleftbig\n1−exp/parenleftbig\n−B0\nBcosθ/parenrightbig/bracketrightbig /bracketleftbig\n1−exp/parenleftbig\n−BAF\nBcosθ/parenrightbig/bracketrightbig\n, with two cha-\nracteristic MB fields B0andBAFassociated with the\ngaps ∆ 0and ∆ AF, respectively. The Zeeman splitting\neffect is encapsulated in the spin factor Rs(θ). In Eq. (1),\nthe geometry of our experiment implies B/bardbl= 0, thus in\nthe N´ eel state Rs(θ) takes the form of Eq. (2).\nExcludingRs(θ), the other factors in Eq. (3) decrease\nmonotonically with increasing θ. By contrast, Rs(θ) in\nEq. (2), generally, has an oscillating angular dependence.\nFor ¯g⊥=g= 2.0 found in the PM state45, Eq. (2) yields\ntwo spin zeros, at θ≈43◦and 64◦. Contrary to this, we\nobserve nospin zeros, but rather a monotonic decrease of\nAδby over two orders of magnitude as the field is tilted\naway from θ= 0◦to±70◦, i.e., in the entire angular\nrange where we observe the oscillations. The different\ncurves in Fig. 3 are our fits using Eq. (3) with A0and\nTDas fit parameters, and different values of the g-factor.\nWe used the MB field values B0= 20 T and BAF= 5 T.\nWhile the exact values of B0andBAFare unknown, they\nhave virtually no effect on the fit quality, as we demon-\nstrate in the SI. The best fit is achieved with g= 0, i.e.\nwith an angle-independent spin factor Rs= 1. The ex-\ncellent agreement between the fit and the experimental\ndata confirms the quasi-2D character of the electron con-\nduction, with the 1 /cosθdependence of the cyclotron\nmass.\nComparison of the curves in Fig. 3 with the data rules\n/s45/s54/s48 /s45/s52/s48 /s45/s50/s48 /s48 /s50/s48 /s52/s48 /s54/s48/s48/s46/s48/s49/s48/s46/s49/s49/s49/s48\n/s69/s120/s112/s101/s114/s105/s109/s101/s110/s116/s124/s65\n/s100/s124/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41\n/s113 /s32/s40/s100/s101/s103/s114/s101/s101/s115/s41/s32/s32/s32/s32/s70/s105/s116/s115/s32/s119/s105/s116/s104/s32/s69/s113/s46/s32/s40/s51/s41/s58\n/s32/s32/s32/s32 /s32/s32/s32 /s103 /s32/s61/s32/s50/s46/s48\n/s32/s32/s32/s32 /s32/s32/s32 /s103 /s32/s61/s32/s49/s46/s48\n/s32/s32/s32/s32 /s32/s32/s32 /s103 /s32/s61/s32/s48/s46/s53\n/s32/s32/s32/s32 /s32/s32/s32 /s103 /s32/s61/s32/s48/s46/s50\n/s32/s32/s32/s32 /s32/s32/s32 /s103 /s32/s61/s32/s48\n/s107 /s45/s66/s69/s84/s83FIG. 3. Angular dependence of the SdH amplitude Aδ\nin the AF state of κ-BETS. The lines are fits using Eq. (3)\nwith different values of the g-factor.\nout ¯g⊥>0.2. Given the experimental error bars, we\ncannot exclude a nonzero ¯ g⊥<∼0.2, yet even such a small\nfinite value would be in stark contrast with the textbook\ng= 2.0, found from the SdH oscillations in the high-field,\nPM state45. Below we argue that, in fact, ¯ g⊥in the N´ eel\nstate is exactly zero.\nOptimally doped NCCO — This material has a\nbody-centered tetragonal crystal structure (space group\nI4/mmm ), where (001) conducting CuO 2layers alter-\nnate with their insulating (Nd,Ce)O 2counterparts17.\nBand-structure calculations61,62predict a holelike cylin-\ndrical Fermi surface, centered at the corner of the\nBZ. However, angle-resolved photoemission spectroscopy\n(ARPES)18,19,21,63reveals a reconstruction of this Fermi\nsurface by a ( π/a,π/a ) order. Moreover, magnetic quan-\ntum oscillations23–25show that the Fermi surface remains\nreconstructed even in the overdoped regime, up to the\ncritical doping xc(≈0.175 for NCCO), where the super-\nconductivity vanishes64. The origin of this reconstruc-\ntion remains unclear: while the ( π/a,π/a ) periodicity is\ncompatible with the N´ eel order observed in strongly un-\nderdoped NCCO, coexistence of antiferromagnetism and\nsuperconductivity in electron-doped cuprates remains\ncontroversial. A number of neutron-scattering and muon-\nspin rotation studies31–34have detected short-range N´ eel\nfluctuations, but no static order within the superconduct-\ning doping range. However, other neutron scattering35,36\nand magnetotransport37–39experiments have produced\nevidence of static or quasi-static AF order in supercon-\nducting samples at least up to optimal doping xopt. Al-\nternative mechanisms of the Fermi-surface reconstruc-\ntion have been proposed, including a d-density wave28,\na charge-density wave29, or coexistent topological and\nfluctuating short-range AF orders30.5\n(a) (b)\n20 30 40 50 60 700123\n0.0 0.3 0.6 0.90.00.20.40.60.8Rosc/Rbackg (%)\nB cosq  (T)0°37°55°60°64°\n ´ 2´ 5\n67°\nAFFT (arb. unit)\nF cos q (kT)´ 10\n´ 15\n0 20 40 60 800.00.20.4\n 67° 55°  37°R (W)\nB (T)q :  0°\nNCCOc\n b\nFIG. 4. Examples of MR and angle-dependent SdH oscillations in optimally doped NCCO. (a) Examples of the\nB-dependent interlayer resistance at different field orientations, at T= 2.5 K;(b)oscillating component, normalized to the\nnon-oscillating B-dependent resistance background, plotted as a function of the out-of-plane field component B⊥=Bcosθ.\nThe curves corresponding to different tilt angles θare vertically shifted for clarity. For θ= 64◦and 67◦the ratioRosc/Rbackg is\nmultiplied by a factor of 2 and 5, respectively. The vertical dashed lines are drawn to emphasize the constant oscillation phase\nin these coordinates; Inset: FFT spectra of the SdH oscillations taken in the field window 45 to 64 T.\nTo shed light on the relevance of antiferromagnetism\nto the electronic ground state of superconducting NCCO,\nwe have studied the field-orientation dependence of the\nSdH oscillations of the interlayer resistance in an opti-\nmally doped, xopt= 0.15, NCCO crystal. The overall\nmagnetoresistance behavior is illustrated in Fig. 4(a).\nAt low fields the sample is superconducting. Imme-\ndiately above the θ-dependent superconducting critical\nfield the magnetoresistance displays a non-monotonic\nfeature, which has already been reported for optimally\ndoped NCCO in a magnetic field normal to the lay-\ners22,65. This anomaly correlates with an anomaly in\nthe Hall resistance and has been associated with mag-\nnetic breakdown through the energy gap, created by the\n(π/a,π/a )-superlattice potential64. With increasing θ,\nthe anomaly shifts to higher fields, consistently with the\nexpected increase of the breakdown gap with tilting the\nfield.\nSdH oscillations develop above about 30 T. Figure 4(b)\nshows examples of the oscillatory component of the mag-\nnetoresistance, normalized to the field-dependent non-\noscillatory background resistance Rbackg, determined by\na low-order polynomial fit to the as-measured R(B) de-\npendence. In our conditions, B<∼65 T,T= 2.5 K, the\nonly discernible contribution to the oscillations comes\nfrom the hole-like pocket αof the reconstructed Fermi\nsurface22. This pocket is centered at the reduced BZ\nboundary, as shown in the inset of Fig. 5. While mag-\nnetic breakdown creates large cyclotron orbits βwith the\narea equal to that of the unreconstructed Fermi surface,\neven in fields of 60-65 T the fast βoscillations are more\nthan two orders of magnitude weaker than the αoscilla-\ntions24,64.The oscillatory signal is plotted in Fig. 4(b) as a func-\ntion of the out-of-plane field component B⊥=Bcosθ. In\nthese coordinates the oscillation frequency remains con-\nstant, indicating that F(θ) =F(0◦)/cosθand thus con-\nfirming the quasi-2D character of the conduction. In the\ninset we show the respective FFTs plotted against the\ncosθ-scaled frequency. They exhibit a peak at Fcosθ=\n294 T, in line with previous reports. The relatively large\nwidth of the FFT peaks is caused by the small number\nof oscillations in the field window [45 −64] T. This re-\nstrictive choice is dictated by the requirement that the\nSdH oscillations be resolved over the whole field window\nat all tilt angles up to θ≈72◦. In the SI we provide an\nadditional analysis of the amplitude at fixed field values\nin the SI confirming the FFT results. Furthermore, in\nFig. 4(b) one can see that the phase of the oscillations is\nnot inverted, and stays constant in the studied angular\nrange. This is fully in line with the absence of spin zeros,\nsee Eq. (2).\nThe main panel of Fig. 5 presents the angular depen-\ndence of the oscillation amplitude (symbols), in a field\nrotated in the ( ac) plane. The amplitude was determined\nby FFT of the data taken at T= 2.5 K in the field win-\ndow 45 T≤B≤64 T. The lines in the figure are fits using\nEq. (3), for different g-factors. The fits were performed\nusing the MB factor RMB= [1−exp(−B0/B)]24,64, the\nreported values for the MB field B0= 12.5 T, and the ef-\nfective cyclotron mass m(θ= 0◦) = 1.05m064, while tak-\ning into account the 1 /cosθangular dependence of both\nB0andm. The prefactor A0and Dingle temperature TD\nwere used as fit parameters, yielding TD= (12.6±1) K,\nclose to the value found in the earlier experiment64. Note\nthat, contrary to the hole-doped cuprate YBa 2Cu3O7−x,6\n/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48 /s54/s48 /s55/s48/s48/s46/s48/s49/s48/s46/s49/s49/s49/s48\n/s32/s32\n/s32/s32 /s103 /s32/s61/s32/s50/s46/s48\n/s32/s32 /s103 /s32/s61/s32/s49/s46/s48\n/s32/s32 /s103 /s32/s61/s32/s48/s46/s50\n/s32/s32 /s103 /s32/s61/s32/s48/s124/s65\n/s97/s124/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41\n/s113 /s32/s40/s100/s101/s103/s114/s101/s101/s115/s41/s32/s69/s120/s112/s101/s114/s105/s109/s101/s110/s116 /s70/s105/s116/s115/s32/s119/s105/s116/s104/s32/s69/s113/s46/s32/s40/s51/s41/s58\n/s78/s67/s67/s79/s97\nFIG. 5. Angular dependence of the SdH amplitude in\noptimally doped NCCO. The lines are fits using Eq. (3)\nwith different g-factor values. Inset: The first quadrant of\nthe BZ with the Fermi-surface reconstructed by a superlattice\npotential with wave vector Q= (π/a,π/a ). If this potential\ninvolves N´ eel order, the function g⊥(k) (red line in the inset)\nvanishes at the reduced BZ boundary (dashed line). The SdH\noscillations are associated with the oval hole pocket αcentered\nat (π/2a,π/2a)22.\nwhere the analysis of earlier experiments46,66was compli-\ncated by the bilayer splitting of the Fermi cylinder46, the\nsingle-layer structure of NCCO poses no such difficulty.\nSimilar toκ-BETS, the oscillation amplitude in NCCO\ndecreases by a factor of about 300, with no sign of spin\nzeros as the field is tilted from θ= 0◦to 72.5◦. Again,\nthis behavior is incompatible with the textbook value\ng= 2, which would have produced two spin zeros in the\ninterval 0◦≤θ≤70◦, see the green dash-dotted line in\nFig. 5. A reduction of the g-factor to 1.0 would shift\nthe first spin zero to about 72◦, near the edge of our\nrange (blue dotted line in Fig. 5). However, this would\nsimultaneously suppress the amplitude at small θby a\nfactor of ten, contrary to our observations. All in all, our\ndata rule out a constant g>0.2.\nDISCUSSION\nIn both materials, our data impose on the effective g-\nfactor an upper bound of 0 .2. At first sight, one could\nsimply view this as a suppression of the effective gto\na small nonzero value. However, below we argue that,\nin fact, our findings imply ¯ g⊥= 0 and point to the im-\nportance of the Zeeman SOC in both materials. The\nquasi-2D character of electron transport is crucial for this\nconclusion: as mentioned above, in three dimensions, the\nmere absence of spin zeros imposes no bounds on the g-\nfactor.Inκ-BETS the interplay between the crystal symmetry\nand the periodicity of the N´ eel state5,6,47guarantees that\ng⊥(k) vanishes on the entire line kc=π/2cand is an\nodd function of kc−π/2c, see the inset of Fig. 1c and\nFig. S2 in the SI. The δorbit is centered on the line\nkc=π/2c; hence ¯g⊥in Eq. (2) vanishes, implying the\nabsence of spin zeros, in agreement with our data. At the\nsame time, quantum oscillations in the PM phase clearly\nreveal the Zeeman splitting of Landau levels with g=\n2.0. Therefore, we conclude that ¯ g⊥= 0 is an intrinsic\nproperty of the N´ eel state.\nIn optimally doped NCCO, as already mentioned, the\npresence of a (quasi)static N´ eel order has been a sub-\nject of debate. However, if indeed present, such an order\nleads tog⊥(k) = 0 at the entire magnetic BZ boundary\n(see SI). For the hole pockets, producing the observed\nFα/similarequal300 T oscillations, ¯ g⊥= 0 by symmetry of g⊥(k)\n(see inset of Fig. 5 and Fig. S3 in SI). Such an inter-\npretation requires that the relevant AF fluctuations have\nfrequencies below the cyclotron frequency in our experi-\nment,νc∼1012Hz at 50 T.\nFinally, we address mechanisms – other than Zeeman\nSOC of Eq. (1) – which may also lead to the absence\nof spin zeros. While such mechanisms do exist, we will\nshow that none of them is relevant to the materials of\nour interest.\nWhen looking for alternative explanations to our ex-\nperimental findings, let us recall that, generally, the\neffectiveg-factor may depend on the field orientation.\nThis dependence may happen to compensate that of\nthe quasi-2D cyclotron mass, m/cosθ, in the expres-\nsion (2) for the spin-reduction factor Rs, and render the\nlatter nearly isotropic, with no spin zeros. Obviously,\nsuch a compensation requires a strong Ising anisotropy\n[g(θ= 0◦)/greatermuchg(θ= 90◦)] – as found, for instance, in\nthe heavy-fermion compound URu 2Si2, with the values\ngc= 2.65±0.05 andgab= 0.0±0.1 for the field along\nand normal to the caxis, respectively67,68. However, this\nscenario is irrelevant to both materials of our interest: In\nκ-BETS, a nearly isotropic g-factor, close to the free-\nelectron value 2.0, was revealed by a study of spin zeros\nin the paramagnetic state45. In NCCO, the conduction\nelectrong-factor may acquire anisotropy via an exchange\ncoupling to Nd3+local moments. However, the low-\ntemperature magnetic susceptibility of Nd3+in the basal\nplane is some 5 times larger than along the caxis69,70.\nTherefore, the coupling to Nd3+may only increase gab\nrelative togc, and thereby only enhance the angular de-\npendence of Rsrather than cancel it out. Thus, we are\nlead to rule out a g-factor anisotropy of crystal-field ori-\ngin as a possible reason behind the absence of spin zeros\nin our experiments.\nAs follows from Eq. (2), another possible reason for the\nabsence of spin zeros is a strong reduction of the ratio\ngm/2me. However, while some renormalization of this\nratio in metals is commonplace, its dramatic suppression\n(let alone nullification) is, in fact, exceptional. Firstly, a\nvanishing mass would contradict m/m e≈1, experimen-7\ntally found in both materials at hand. On the other hand,\na Landau Fermi-liquid renormalization14g→g/(1+G0)\nwould require a colossal Fermi-liquid parameter G0≥10,\nfor which there is no evidence in NCCO, let alone κ-\nBETS with its already mentioned g≈2 in the paramag-\nnetic state45.\nA sufficient difference of the quantum-oscillation am-\nplitudes and/or cyclotron masses for spin-up and spin-\ndown Fermi surfaces might also lead to the absence of\nspin zeros. Some heavy-fermion compounds show strong\nspin polarization in magnetic field, concomitant with\na substantial field-induced difference of the cyclotron\nmasses of the two spin-split subbands71,72. As a result,\nfor quantum oscillations in such materials, one spin am-\nplitude considerably exceeds the other, and no spin zeros\nare expected. Note that this physics requires the presence\nof a very narrow conduction band, in addition to a broad\none. In heavy-fermion compounds, such a band arises\nfrom thefelectrons, but is absent in both materials of\nour interest.\nAnother extreme example is given by the single fully\npolarized band in a ferromagnetic metal, where only one\nspin orientation is present, and spin zeros are obviously\nabsent. Yet, no sign of ferro- or metamagnetism has\nbeen seen in either NCCO or κ-BETS. Moreover, in κ-\nBETS,the spin-zero effect has been observed in the para-\nmagnetic state45, indicating that the quantum-oscillation\namplitudes of the two spin-split subbands are compara-\nble. However, for NCCO one may inquire whether spin\npolarization could render interlayer tunneling amplitudes\nfor spin-up and spin-down different enough to lose spin\nzeros, especially in view of an extra contribution of Nd3+\nspins in the insulating layers to spin polarization. In the\nSI, we show that this is notthe case.\nThus, we are lead to conclude that the absence of spin\nzeros in the AF κ-BETS and in optimally doped NCCO\nis indeed a manifestation of the Zeeman SOC. Our ex-\nplanation relies only on the symmetry of the N´ eel state\nand the location of the carrier pockets, while being in-\nsensitive to the mechanism of the antiferromagnetism or\nto the orbital makeup of the relevant bands.\nMETHODS\nCrystals of κ-(BETS) 2FeBr 4were grown electrochem-\nically and prepared for transport measurements as re-\nported previously45. The interlayer ( I||b) resistance was\nmeasured by the standard four-terminal a.c. technique\nusing a low-frequency lock-in amplifier. Magnetoresis-\ntance measurements were performed in a superconduct-\ning magnet system at fields of up to 14 T. The samples\nwere mounted on a holder allowing in-situ rotation of the\nsample around an axis perpendicular to the external field\ndirection. The orientation of the crystal was defined by a\npolar angle θbetween the field and the crystallographic\nbaxis (normal to the conducting layers).\nOptimally doped single crystals of Nd 1.85Ce0.15CuO 4,grown by the traveling solvent floating zone method, were\nprepared for transport measurements as reported previ-\nously23. Measurements of the interlayer ( I||c) resistance\nwere performed on a rotatable platform using a standard\nfour-terminal a.c. technique at frequencies of 30 −70 kHz\nin a 70 T pulse-magnet system, with a pulse duration of\n150 ms, at the Dresden High Magnetic Field Laboratory.\nThe raw data were collected by a fast digitizing oscillo-\nscope and processed afterwards by a digital lock-in pro-\ncedure22. The orientation of the crystal was defined by a\npolar angle θbetween the field and the crystallographic\ncaxis (normal to the conducting layers).\nACKNOWLEDGMENTS\nIt is our pleasure to thank G. Knebel, H. Sakai,\nI. Sheikin, and A. Yaresko for illuminating discus-\nsions. This work was supported by HLD at HZDR,\na member of the European Magnetic Field Labora-\ntory (EMFL). P. D. G. acknowledges State Assignment\n#0033-2019-0001 for financial support and the Labo-\nratoire de Physique Th´ eorique, Toulouse, for the hos-\npitality during his visit. R. G. acknowledges finan-\ncial support from the German Research Foundation\n(Deutsche Forschungsgemeinschaft, DFG) via Germany’s\nExcellence Strategy (EXC-2111-390814868). J. W. ac-\nknowledges financial support from the DFG through the\nW¨ urzburg-Dresden Cluster of Excellence on Complexity\nand Topology in Quantum Matter - ct.qmat (EXC 2147,\nproject-id 390858490).\nAUTHOR CONTRIBUTIONS\nT.H., F.K., M.K., E.K., W.B., and M.V.K performed\nthe experiments and analyzed the data. R.R., P.D.G.,\nand M.V.K. initiated the exploration and performed the\ntheoretical analysis and interpretation of the experimen-\ntal results. H.F. and A.E. provided high-quality single\ncrystals. R.R., P.D.G., T.H., W.B., E.K., J.W., R.G.,\nand M.V.K. contributed to the writing of the manuscript.\nDATA AVAILABILITY\nThe authors declare that all essential data supporting\nthe findings of this study are available within the paper\nand its supplementary information. The complete data\nset in ASCII format is available from the corresponding\nauthors upon reasonable request.\nADDITIONAL INFORMATION\nSupplementary Information: Supplementary In-\nformation is available at [URL to be provided by the\npublisher].8\nCompeting interests: The authors declare no com-\npeting interests.\nREFERENCES\n∗revaz@irsamc.ups-tlse.fr\n†grigorev@itp.ac.ru\n‡t.helm@hzdr.de\n§Present address: TNG Technology Consulting GmbH,\n85774 Unterf¨ ohring, Germany\n¶mark.kartsovnik@wmi.badw.de\n1R. Winkler, Spin-orbit Coupling Effects in Two-\ndimensional Electron and Hole Systems, Springer Tracts\nin Modern Physics, Vol. 191, Berlin – Heidelberg, 2003.\n2M. I. Dyakonov (Editor), Spin Physics in Semiconductors,\nSpringer Series in Solid-State Sciences, Vol. 157, Berlin –\nHeidelberg, 2017.\n3V. B. Berestetski˘ ı, E. M. Lifshitz and L. P. Pitaevski˘ ı,\n“Course of Theoretical Physics”, Vol. 4, Quantum Elec-\ntrodynamics , Pergamon Press (1982).\n4C. Kittel, Quantum Theory of Solids , John Wiley & Sons,\nInc., New York – London (1963).\n5R. Ramazashvili, Kramers Degeneracy in a Magnetic Field\nand Zeeman Spin-Orbit Coupling in Antiferromagnetic\nConductors , Phys. Rev. Lett. 101, 137202 (2008).\n6R. Ramazashvili, Kramers degeneracy in a magnetic field\nand Zeeman spin-orbit coupling in antiferromagnetic con-\nductors , Phys. Rev. B 79, 184432 (2009).\n7S. A. Brazovskii and I. A. Luk’yanchuk, Symmetry of elec-\ntron states in antiferromagnets , Sov. Phys. JETP 69, 1180-\n1184 (1989).\n8S. A. Brazovskii, I. A. Luk’yanchuk, and R. R. Ramaza-\nshvili, Electron paramagnetism in antiferromagnets , JETP\nLett. 49, 644-646 (1989).\n9V.V. Kabanov and A. S. Alexandrov, Magnetic quantum\noscillations in doped antiferromagnetic insulators , Phys.\nRev. B 77, 132403 (2008); 81, 099907(E) (2010).\n10R. Ramazashvili, Electric excitation of spin resonance in\nantiferromagnetic conductors , Phys. Rev. B 80, 054405\n(2009).\n11E. I. Rashba and V. I. Sheka, ”Electric-Dipole Spin Res-\nonances”, in Landau Level Spectroscopy , edited by G.\nLandwehr and E. I. Rashba (Elsevier, New York, (1991).\n12M. Shafiei et al. ,Resolving Spin-Orbit- and Hyperfine-\nMediated Electric Dipole Spin Resonance in a Quantum\nDot, Phys. Rev. Let. 110, 107601 (2013).\n13J. W. G. van den Berg et al. ,Fast Spin-Orbit Qubit in\nan Indium Antimonide Nanowire , Phys. Rev. Let. 110,\n066806 (2013).\n14D. Shoenberg, Magnetic Oscillations in Metals , Cambridge\nUniversity Press (1984).\n15H. Fujiwara et al. ,A Novel Antiferromagnetic Or-\nganic Superconductor κ-(BETS) 2FeBr 4[Where BETS\n= Bis(ethylenedithio)tetraselenafulvalene] , J. Am. Chem.\nSoc.123, 306-314 (2001).\n16T. Konoike et al. ,Fermi surface reconstruction in the\nmagnetic-field-induced superconductor κ-(BETS) 2FeBr 4,\nPhys. Rev. B 72, 094517 (2005).17N. P. Armitage, P. Fournier, and R. L. Greene, Progress\nand perspectives on electron-doped cuprates , Rev. Mod.\nPhys. 82, 2421-2487 (2010).\n18A. F. Santander-Syro et al. ,Two-Fermi-surface supercon-\nducting state and a nodal d-wave energy gap of the electron-\ndoped Sm 1.85Ce0.15CuO 4−δcuprate superconductor , Phys.\nRev. Lett. 106, 197002 (2011).\n19D. Song et al. ,Electron Number-Based Phase Diagram of\nPr1−xLaCexCuO 4−δand Possible Absence of Disparity be-\ntween Electron- and Hole-Doped Cuprate Phase Diagrams ,\nPhys. Rev. Lett. 118, 137001 (2017).\n20J.-F. He et al. ,Fermi surface reconstruction in electron-\ndoped cuprates without antiferromagnetic long-range order ,\nProc. Natl. Acad. Sci. USA 116, 3449-3453 (2019).\n21H. Matsui et al. ,Evolution of the pseudogap\nacross the magnet-superconductor phase boundary of\nNd2−xCexCuO 4, Phys. Rev. B 75, 224514 (2007).\n22T. Helm et al. ,Evolution of the Fermi surface\nof the electron-doped high-temperature superconductor\nNd2−xCexCuO 4revealed by Shubnikov - de Haas oscilla-\ntions , Phys. Rev. Lett. 103, 157002 (2009).\n23T. Helm et al. ,Magnetic breakdown in the electron-\ndoped cuprate superconductor Nd 2−xCexCuO 4: The recon-\nstructed Fermi surface survives in the strongly overdoped\nregime , Phys. Rev. Lett. 105, 247002 (2010).\n24M. V. Kartsovnik et al. ,Fermi surface of the electron-doped\ncuprate superconductor Nd 2−xCexCuO 4probed by high-\nfield magnetotransport , New J. Phys. 13, 015001 (2011).\n25J. S. Higgins et al. ,Quantum oscillations from the recon-\nstructed Fermi surface in electron-doped cuprate supercon-\nductors , New J. Phys. 20, 043019 (2018).\n26T. Das, R. S. Markiewicz, and A. Bansil, Superconductivity\nand topological Fermi surface transitions in electron-doped\ncuprates near optimal doping , J. Phys. Chem. Solids 69,\n2963-2966 (2008).\n27S. Sachdev, Where is the quantum critical point in the\ncuprate superconductors? , Phys. Status Solidi B 247, 537-\n543 (2010).\n28S. Chakravarty, R. B. Laughlin, D. K. Morr, and C. Nayak,\nHidden order in cuprates , Phys. Rev. B 63, 094503 (2001).\n29E. H. da Silva Neto et al. ,Charge ordering in the electron-\ndoped superconductor Nd 2–xCexCuO 4, Science 347, 282-\n285 (2015).\n30S. Sachdev, Topological order, emergent gauge fields, and\nFermi surface reconstruction , Rep. Prog. Phys. 82, 014001\n(2019); S. Sachdev, H. D. Scammell, M. S. Scheurer, and G.\nTarnopolsky, Gauge theory for the cuprates near optimal\ndoping , Phys. Rev. B 99, 054516 (2019).\n31G. M. Luke et al. ,Magnetic order and electronic phase\ndiagrams of electron-doped copper oxide materials , Phys.\nRev. B 42, 7981-7988 (1990).\n32E. M. Motoyama et al. ,Spin correlations in the\nelectron-doped high-transition-temperature superconductor9\nNd2−xCexCuO 4−δ, Nature 445, 186-189 (2007).\n33P. K. Mang et al. ,Phase decomposition and chemical inho-\nmogeneity in Nd 2−xCexCuO 4±δ, Phys. Rev. B 70, 094507\n(2004).\n34H. Saadaoui et al. ,The phase diagram of electron-doped\nLa2−xCexCuO 4−δ, Nat. Commun. 6, 6041 (2015).\n35K. Yamada et al. ,Commensurate Spin Dynamics in the\nSuperconducting State of an Electron-Doped Cuprate Su-\nperconductor , Phys. Rev. Lett. 90, 137004 (2003).\n36H. J. Kang et al. ,Electronically competing phases and their\nmagnetic field dependence in electron-doped nonsupercon-\nducting and superconducting Pr 0.88LaCe 0.12CuO 4±δ, Phys.\nRev. B 71, 214512 (2005).\n37Y. Dagan et al. ,Origin of the Anomalous Low Temperature\nUpturn in the Resistivity of the Electron-Doped Cuprate\nSuperconductors , Phys. Rev. Lett. 94, 057005 (2005).\n38W. Yu, J. S. Higgins, P. Bach, and R. L. Greene, Trans-\nport evidence of a magnetic quantum phase transition\nin electron-doped high-temperature superconductors , Phys.\nRev. B 76, 020503(R) (2007).\n39A. Dorantes et al. ,Magnetotransport evidence for irre-\nversible spin reorientation in the collinear antiferromag-\nnetic state of underdoped Nd2−xCexCuO 4, Phys. Rev. B\n97, 054430 (2018).\n40R. L. Greene, P. R. Mandal, N. R. Poniatowski, and T.\nSarkar, The Strange Metal State of the Electron-Doped\nCuprates , Annu. Rev. Condens. Matter Phys. 11, 213-229\n(2020).\n41J. Wosnitza et al. ,de Haas–van Alphen studies\nof the organic superconductors α-(ET) 2NH4Hg(SCN) 4\nandκ-(ET) 2Cu(NCS) 2with ET = bis(ethelenedithio)-\ntetrathiafulvalene , Phys. Rev. B 45, 3018-3025 (1992).\n42A. E. Kovalev, M. V. Kartsovnik, and N. D. Kushch, Quan-\ntum and quasi-classical magnetoresistance oscillations in\na new organic metal (BEDT-TTF) 2TlHg(SeCN) 4, Solid\nState Commun. 87, 705-708 (1993).\n43F. A. Meyer et al. ,High-Field de Haas-Van Alphen Studies\nofκ-(BEDT-TTF) 2Cu(NCS) 2, Europhys. Lett. 32, 681-\n686 (1995).\n44S. Uji et al. ,Fermi surface and internal magnetic field\nof the organic conductors λ-(BETS) 2FexGa1−xCl4, Phys.\nRev. B 65, 113101 (2002).\n45M. V. Kartsovnik et al. ,Interplay Between Conducting and\nMagnetic Systems in the Antiferromagnetic Organic Super-\nconductorκ-(BETS) 2FeBr 4, J. Supercond. Nov. Magn. 29,\n3075-3080 (2016).\n46B. J. Ramshaw et al. ,Angle dependence of quantum os-\ncillations in YBa 2Cu3O6:59 shows free-spin behaviour of\nquasiparticles , Nat. Phys. 7, 234-238 (2011).\n47R. Ramazashvili, Quantum Oscillations in Antiferromag-\nnetic Conductors with Small Carrier Pockets , Phys. Rev.\nLett. 105, 216404 (2010).\n48R. Settai et al. ,De Haas – van Alphen studies of rare earth\ncompounds , J. Magn. Magn. Mater. 140–144 , 1153-1154\n(1995).\n49T. Ebihara, N. Harrison, M. Jaime, S. Uji, and J. C. Lash-\nley,Emergent Fluctuation Hot Spots on the Fermi Surface\nof CeIn 3in Strong Magnetic Fields , Phys. Rev. Lett. 93,\n246401 (2004).\n50L. P. Gor’kov and P. D. Grigoriev, Antiferromagnetism and\nhot spots in CeIn 3, Phys. Rev. B 73, 060401(R) (2006).\n51H. Masuda et al. ,Impact of antiferromagnetic order\non Landau-level splitting of quasi-two-dimensional Dirac\nfermions in EuMnBi 2, Phys. Rev. B 98, 161108(R) (2018).52S. Borisenko et al. ,Time-reversal symmetry breaking type-\nII Weyl state in YbMnBi 2, Nat. Commun. 10, 3424 (2019).\n53S. Uji et al. ,Two-dimensional Fermi surface for the or-\nganic conductor κ-(BETS) 2FeBr 4, Physica B 298, 557-561\n(2001).\n54T. Mori and M. Katsuhara, Estimations of πd-interactions\nin organic conductors including magnetic anions , J. Phys.\nSoc. Jpn. 71, 826-844 (2002).\n55T. Konoike et al. ,Magnetic-field-induced superconductiv-\nity in the antiferromagnetic organic superconductor κ-\n(BETS) 2FeBr 4, Phys. Rev. B 70, 094514 (2004).\n56O. C´ epas, R. H. McKenzie, and J. Merino, Magnetic-\nfield-induced superconductivity in layered organic molecu-\nlar crystals with localized magnetic moments , Phys. Rev.\nB65, 100502(R) (2002).\n57R. E. Peierls, Quantum Theory of Solids , Oxford Univer-\nsity Press (2001).\n58T. Konoike et al. ,Anomalous Magnetic-Field-Hysteresis of\nQuantum Oscillations in κ-(BETS) 2FeBr 4, J. Low Temp.\nPhys. 142, 531-534 (2006).\n59J. Wosnitza, Fermi Surfaces of Low-Dimensional Organic\nMetals and Superconductors , Springer Verlag, Berlin – Hei-\ndelberg (1996).\n60M. V. Kartsovnik, High Magnetic Fields: A Tool for\nStudying Electronic Properties of Layered Organic Metals ,\nChem. Rev. 104, 5737-5782 (2004).\n61S. Massidda, N. Hamada, J. Yu, and A. J. Freeman, Elec-\ntronic structure of Nd-Ce-Cu-O, a Fermi liquid supercon-\nductor , Physica C 157, 571-574 (1989).\n62O. K. Andersen, A. I. Liechtenstein, O. Jepsen, and F.\nPaulsen, LDA energy bands, low-energy Hamiltonians, t/prime,\nt/prime/prime,t⊥(k), andJ⊥, J. Phys. Chem. Solids 56, 1573-1591\n(1995).\n63N. P. Armitage et al. ,Doping dependence of an n-type\ncuprate superconductor investigated by angle-resolved pho-\ntoemission spectroscopy , Phys. Rev. Lett. 88, 257001\n(2002).\n64T. Helm et al. ,Correlation between Fermi surface trans-\nformations and superconductivity in the electron-doped\nhigh-Tcsuperconductor Nd 2−xCexCuO 4, Phys. Rev. B 92,\n094501 (2015).\n65P. Li, F. F. Balakirev, and R. L. Greene, High-Field\nHall Resistivity and Magnetoresistance of Electron-Doped\nPr2−xCexCuO 4−δ, Phys. Rev. Lett. 99, 047003 (2007).\n66S. E. Sebastian et al. ,Spin-Order Driven Fermi Surface\nReconstruction Revealed by Quantum Oscillations in an\nUnderdoped High TcSuperconductor , Phys. Rev. Lett. 103,\n256405 (2009).\n67M. M. Altarawneh et al. ,Superconducting Pairs with Ex-\ntreme Uniaxial Anisotropy in URu 2Si2, Phys. Rev. Lett.\n108, 066407 (2012).\n68G. Bastien et al. ,Fermi-surface selective determination\nof theg-factor anisotropy in URu 2Si2, Phys. Rev. B 99,\n165138 (2019).\n69M. F. Hundley, J. D. Thompson, S-W. Cheong, Z. Fisk,\nand S. B. Oseroff, Specific heat and anisotropic magnetic\nsusceptibility of Pr 2CuO 4, Nd 2CuO 4and Sm 2CuO4 crys-\ntals, Physica C 158, 102-108 (1989).\n70Y. Dalichaouch, M. C. de Andrade, and M. B.\nMaple, Synthesis, transport, and magnetic properties of\nLn2−xCexCu0 4−ysingle crystals (Ln = Nd, Pr, Sm) ,\nPhysica C 218, 309-315 (1993).\n71N. Harrison, D. W. Hall, R. G. Goodrich, J. J. Vuillemin,\nand Z. Fisk, Quantum Interference in the Spin-Polarized10\nHeavy Fermion Compound CeB 6: Evidence for Topologi-\ncal Deformation of the Fermi Surface in Strong Magnetic\nFields , Phys. Rev. Lett. 81, 870 (1998).\n72A. McCollam et al. ,Spin-dependent masses and field-induced quantum critical points , Physica B 359-361 , 1-8\n(2005).1\nSUPPLEMENTARY INFORMATION\nI. SPIN-REDUCTION FACTOR Rs(θ)IN A\nMAGNETIC FIELD PERPENDICULAR TO THE\nN´EEL AXIS\nFor a quasi-2D metal, the phase ϕof the first quantum-\noscillation harmonic (i.e., of the fundamental frequency\noscillations) is, up to a constant, ϕ=F\nB=¯h\neF\nBcosθ,\nwhereθis the tilt angle between the field and the nor-\nmal to the conducting plane, Fstands for the oscil-\nlation frequency at θ= 0◦, andFthe corresponding\nFermi-surface area. The Zeeman effect splits the spin-\ndegenerate Fermi surface, breaking up FintoF+and\nF−, withδF=F+−F−∝B. Adding the two harmonic\noscillations at close frequencies F+andF−is equivalent\nto a single oscillation at frequency F, with an amplitude\nmodulated by the spin-reduction factor\nRs(θ) = cos/bracketleftbigg¯h\n2eδF\nBcosθ/bracketrightbigg\n. (1)\nFor the field perpendicular to the N´ eel axis, B=B⊥,\nEq. (1) of the main text yields the single-particle Hamil-\ntonian:H=E(k)−1\n2µBg⊥(k)(B⊥·σ), whereE(k)\nis the zero-field dispersion near the Fermi surface, and\nσ= (σx,σy,σz) is a vector composed of the three Pauli\nmatrices. Upon turning on the field, a given point kof\nthe Fermi surface undergoes a small shift δksuch that\nδk·∇kE(k) =±1\n2µBg⊥(k)B⊥, where the±signs cor-\nrespond to the ‘up’ and ‘down’ spin projections on B⊥\nand to the subscript of the resulting Fermi-surface areas\nF±. As shown in Fig. S1, upon the shift by δkan ele-\nmentdkof the Fermi surface contributes the shaded area\ndk(δk·ˆlk) =±1\n2µBB⊥dkg⊥(k)/|∇kE(k)|to the variation\nof the total area, enclosed by the Fermi surface. Here,\nˆlk=∇kE(k)/|∇kE(k)|is the local unit vector, normal\nto the Fermi surface. Therefore, to linear order in B⊥,\nthe areasF±of the two spin-split Fermi surfaces differ\nkδ\ndkl\nFIG. S1. Fermi surface of a two-dimensional conductor\nin zero field (dashed line) and in a transverse field B ⊥\n(solid line). Upon turning on B⊥, a Fermi-surface element\ndkshifts by a small momentum δk(see the main text), adding\nthe shaded trapezoid of the area dF=dk(δk·ˆlk) to the area\nenclosed by the Fermi surface, where ˆlk=∇kE(k)/|∇kE(k)|\nis the local unit vector, normal to the Fermi surface. The total\nvariation of the Fermi-surface area is given by integrating dF\nover the Fermi surface, as explained in the main text.by\nδF=F+−F−=µBB⊥/contintegraldisplay\nFSdkg⊥(k)\n|∇kE(k)|, (2)\nwhere the line integral is taken along the zero-field Fermi\nsurface. It is convenient to introduce the transverse g-\nfactor, averaged over the Fermi surface:\n¯g⊥=/contintegraldisplay\nFS¯h2dk\n2πmg⊥(k)\n|∇kE(k)|, (3)\nwhere\nm=1\n2π/contintegraldisplay\nFS¯h2dk\n|∇kE(k)|(4)\nis the (θ= 0◦) cyclotron mass. Substituting µB=e¯h\n2me\ninto Eq. (2), we combine it with Eq. (1) to arrive at the\nexpression for the spin-reduction factor:\nRs(θ) = cos/bracketleftbiggπ\ncosθ¯g⊥m\n2me/bracketrightbigg\n, (5)\nthat is Eq. (2) of the main text. For a momentum-\nindependent g⊥, Eq. (5) matches the textbook expres-\nsion1for the spin-reduction factor in two dimensions.\nII. SYMMETRY ANALYSIS OF THE N ´EEL\nSTATES OF κ-(BETS) AND NCCO\nA. Symmetry analysis of the N´ eel state of\nκ-(BETS) 2FeBr 4in a transverse field\nThe existence of a special set of momenta in the Bril-\nlouin zone (BZ), where Bloch eigenstates of a N´ eel an-\ntiferromagnet remain degenerate in transverse magnetic\nfield, is a general phenomenon. However, the precise geo-\nmetry of this set depends on the interplay between the\nperiodicity of the N´ eel order and the symmetry of the\nunderlying crystal lattice2,3. Here, we describe this set\nforκ-(BETS) 2FeBr 4, hereafter referred to as κ-BETS.\nUpon transition from the paramagnetic to N´ eel state,\nthe lattice period of κ-BETS along the caxis doubles,\nand the symmetry of the paramagnetic state with respect\nto both the time reversal ˆθand the elementary transla-\ntion ˆTcalong thecaxis is broken. Yet, the product ˆθˆTc\nremains a symmetry operation, along with the spin rota-\ntion ˆUn(φ) around the N´ eel axis nby an arbitrary angle\nφ.\nApplied transversely to n, a magnetic field breaks the\nsymmetry with respect to both ˆθˆTcandˆUn(φ); however,\nˆUn(π)ˆθˆTcremains a symmetry operation2,3. It maps a\nBloch eigenstate |k/angbracketrightat wave vector konto a degenerate\northogonal eigenstate ˆUn(π)ˆθˆTc|k/angbracketrightat wave vector−k,\nas shown in Fig. S2. Upon combination with reflection\nˆRa: (kc,ka)→(kc,−ka), the resulting symmetry opera-\ntion ˆRaˆUn(π)ˆθˆTcmaps|k/angbracketrightat wave vector k= (kc,ka)arXiv:1908.01236v2  [cond-mat.str-el]  21 Dec 20202\nka\n/aπ\nkc\n/cπ π/2c\nπ/2cδ\nk\nQg (    ) ck\nkc U  (  )   T  nπθckR  U  (  )   T  a n πθck\nFIG. S2. Schematic view of the BZ of κ-BETS. The BZ\nin its paramagnetic state (dashed line) and in the N´ eel state\nwith wave vector Q= (π/c,0) (solid line). The δpocket is\ncentered at the corner k= (±π/2c,±π/a) of the magnetic\nBZ. The arrows show an exact Bloch eigenstate |k/angbracketrightat a wave\nvector kon the vertical segment kc=π/2cof the magnetic\nBZ boundary – and its symmetry partners Un(π)θTc|k/angbracketrightand\nRaUn(π)θTc|k/angbracketright. The orthogonality /angbracketleftk|RaUn(π)θTc|k/angbracketright= 0\nimplies that g⊥(k) vanishes on the segment kc=π/2c. The\ninset illustrates the g⊥(k) being an odd function of kc−π/2c.\nonto a degenerate orthogonal eigenstate ˆRaˆUn(π)ˆθˆTc|k/angbracketright\nat wave vector (−kc,ka)2,3.\nFor an arbitrary k= (kc,ka) at the vertical segment\nkc=π/2cof the magnetic BZ boundary, the wave vectors\n(−kc,ka) and (kc,ka) differ by the reciprocal wave vector\nQ= (π/c,0) of the N´ eel state; in the nomenclature of\nthe magnetic BZ, they are one and the same vector. The\ndegeneracy of such a |k/angbracketrightwith ˆθˆTcˆRaˆUn(π)|k/angbracketrightmeans\nthatg⊥(k) vanishes at the entire segment kc=π/2c.\nTheδpocket is centered at ( ±π/2c,±π/a) and is sym-\nmetric with respect to reflection around the line kc=\n±π/2c, as shown in Figs. 1 and S2. At the same time,\nas shown in Supplemental Material IV and illustrated in\nthe insets of Figs. 1 and S2, g⊥(k) is odd under reflection\naround the same line. As a result, for the δpocket ¯g⊥in\nEq. (3) vanishes, as stated in the main text.\nB. Symmetry analysis of the N´ eel state of\nNd2−xCexCuO 4in a transverse field\nIn the antiferromagnetic state of Nd 2−xCexCuO 4\n(hereafter NCCO), the Cu2+spins point along the lay-\ners. At zero field, they form a so-called non-collinear\nstructure: the staggered magnetization vectors of adja-\ncent layers are normal to each other, pointing along the\ncrystallographic directions [100] and [010], respectively\n(see Ref.4for a review). However, an in-plane field above\n5 T transforms this spin structure into a collinear one,\nwith the staggered magnetization in all the layers aligned\ntransversely to the field. Therefore, in our experiment,\nπ/a\nπ/akQ\nky kx\nkg (    )kxkx\nR  U  (  )   T  ynθa π\nU  (  )   T  nπ θ akFIG. S3. Schematic view of the BZ of NCCO. The BZ\nin its paramagnetic state (dashed line) and in the N´ eel state\nwith wave vector Q= (π/a,π/a ) (solid line). The carrier\npocketsαare centered at the midpoints k= (±π/2a,±π/2a)\nof the magnetic BZ boundary. The blue arrows show an ex-\nact Bloch eigenstate |k/angbracketrightat a wave vector kon the magnetic\nBZ boundary – and its symmetry partners Un(π)θTa|k/angbracketrightand\nRyUn(π)θTa|k/angbracketright. The orthogonality /angbracketleftk|RyUn(π)θTa|k/angbracketright= 0\nimplies that g⊥(k) vanishes on the segment kxa=±π/√\n2.\nThe inset illustrates g⊥(k) being an odd function of kxa−\nπ/√\n2.\nwith the field B > 45 T rotated around the [100] axis,\nthe staggered magnetization is normal to the field at all\ntilt angles except for a narrow interval 0◦<|θ|<∼5◦.\nThus, we can restrict ourselves to the purely\ntransverse-field geometry, with the field normal to\nthe N´ eel axis, which makes the analysis similar to\nthat forκ-BETS. The only difference is that, given\nthe tetragonal symmetry of NCCO, the triple product\nˆUn(π)ˆθˆTacan now be combined with reflections ˆRx:\n(kx,ky)→(−kx,ky) and ˆRy: (kx,ky)→(kx,−ky).\nAs a result, for any wave vector kat the magnetic\nBrillouin-zone boundary, one finds /angbracketleftk|ˆRyˆUn(π)ˆθˆTa|k/angbracketright=\n/angbracketleftk|ˆRxˆUn(π)ˆθˆTa|k/angbracketright= 0. This guarantees double degen-\neracy of Bloch eigenstates, hence the equality g⊥(k) = 0\nat the entire boundary of the magnetic BZ, as shown in\nFig. S3.\nThe charge-carrier pockets of our interest are believed\nto be centered at ( ±π/2a,±π/2a), and are symmetric\nwith respect to reflections RxandRyaround the kxand\nkyaxes, as shown in Figs. 4 and S3. At the same time, as\nshown in Section IVand illustrated in the inset of Figs. 4\nand S3,g⊥(k) is odd under the very same reflections. As\na result, ¯g⊥in Eq. (3) vanishes for these pockets, as stated\nin the main text.3\nIII. ESTIMATING THE PRODUCT kFξ\nA.δpocket in κ-BETS\nLooking only for a crude estimate, we assume a\nparabolic energy dispersion and treat the δpocket as\ncircular of radius kFand areaFδ= 2πeFδ/¯h. Defining\nthe antiferromagnetic coherence length as ξ= ¯hvF/∆AF,\nwe find:\nkFξ= ¯hkFvF/∆AF= 2εF/∆AF. (6)\nThe Fermi energy εFin Eq. (6) can be expressed via the\nShubnikov-de Haas (SdH) frequency Fδ= 61 T:\nεF=¯h2k2\nF\n2m=¯h2Fδ\n2πm=¯heFδ\nm≈6 meV. (7)\nAssuming a BSC-like relation between the N´ eel tem-\nperature,TN≈2.5 K, and the antiferromagnetic gap\n∆AFin the electron spectrum, we evaluate the latter\nas ∆ AF/similarequal1.8kBTN≈0.4 meV. A similar estimate is\nobtained from the critical field, Bc≈5 T, required to\nsuppress the N´ eel state: ∆ AF∼µBBc≈0.3 meV.\nThus, we find kFξ/similarequal2εF\n∆AF∼30−40/greatermuch1, which means\nthatg⊥(k) is nearly constant over most of the Fermi sur-\nface, except in a small vicinity of kc=π/2c, where it\nchanges sign, cf. Figs. S2 and S4.\nB. Small hole pocket of the reconstructed Fermi\nsurface in NCCO\nIn NCCO, the small Fermi-surface pocket α, responsi-\nble for the observed oscillations, is far from being circu-\nlar. Therefore, we can no longer estimate kFξthe same\nway as we did for the δpocket inκ-BETS. Instead, we\nevaluate the relevant Fermi wave vector and the antifer-\nromagnetic coherence length separately.\nThe value of the Fermi wave vector in the direction\nnormal to the magnetic BZ boundary can be found from\nARPES maps of the Fermi surface5–7:kF= 0.4±\n0.1 nm−1.\nThe coherence length can be estimated using the MB\ngap value, ∆ AF≈16 meV, and parameters of the (ap-\nproximately circular) large parent Fermi surface obtained\nfrom the analysis of MB quantum oscillations8. Using\nthe corresponding SdH frequency F= 11.25 kT and cy-\nclotron mass mc= 3.0me, we estimate the Fermi velocity,\nvF∼¯hkF/mc≈√\n2¯heF/mc≈2.2×105m/s, which leads\nto the coherence length ξ∼¯hvF/∆AF≈9 nm.\nThis yields the product kFξ∼3−5, which implies\ng⊥(k) being piecewise nearly constant over most of the\nFermi surface, except in a small vicinity of the magnetic\nBZ boundary, where g⊥(k) changes sign, cf. Figs. S3 and\nS4.IV. SYMMETRY PROPERTIES OF g⊥(k)\nIn Section II A, we have shown that in κ-BETS the\nfactorg⊥(k) vanishes at the entire kc=±π/2csegment\nof the magnetic BZ boundary. Here, we establish an im-\nportant general symmetry property of g⊥(k). In the case\nofκ-BETS, this property implies that g⊥(k) is an odd\nfunction of kc−π/2c. Theδpocket, responsible for the\nobserved SdH oscillations, is centered on this segment,\nat the corner of the magnetic BZ (see Fig. 1). As a re-\nsult, for this pocket the “effective g-factor” ¯g⊥in Eq. (3)\nvanishes by symmetry.\nWithout loss of generality, we consider the simplest\ncase of double commensurability, relevant to both ma-\nterials of our interest. In both of them, the underlying\nnon-magnetic state is centrosymmetric, with the relevant\nelectron band having the spectrum ε(k). Spontaneous\nN´ eel magnetization with wave vector Qinteracts with\nthe conduction-electron spin σvia the exchange term\n(∆AF·σ), coupling the states at wave vectors kand\nk+Q. In the N´ eel phase, subjected to magnetic field B,\nthe electron Hamiltonian takes the form9\nHk=/bracketleftbigg\nε(k)−g(B·σ) ( ∆AF·σ)\n(∆AF·σ)ε(k+Q)−g(B·σ)/bracketrightbigg\n,(8)\nwhere the factor µB/2 has been absorbed into the defi-\nnition of B, and ∆AF=JSis the product of the an-\ntiferromagnetically ordered moment Sand its exchange\ncouplingJto the conduction electrons. In a purely trans-\nverse field B⊥⊥∆AF, the Hamiltonian (8) can be easily\ndiagonalized2,10to yield the spectrum\nE(k) =ε+(k)±/radicalBig\n∆2\nAF+ [ε−(k)−g(B⊥·σ)]2,(9)\n/s45/s49\n/s107/s103 /s40/s107 /s41\nFIG. S4. Theg-factor for magnetic field normal to\nthe N´ eel axis. Schematic plot of g⊥(k) as a function of the\nmomentum component k, normal to the line g⊥(k) = 0. At\nsmallk < 1/ξ, the function g⊥(k) is linear: g⊥(k)≈gξk.\nBeyondk≈1/ξ,g⊥(k) is nearly constant: g⊥(k)≈g. Here,\nξ= ¯hvF/∆AFis the antiferromagnetic coherence length, and\n∆AFis the energy gap in the electron spectrum (9) of the\nN´ eel state.4\nwhereε±(k) =1\n2[ε(k)±ε(k+Q)]. Equation (9) shows\nthat ∆ AFis the energy gap in the electron spectrum of\nthe N´ eel state. From Eq. (9), one easily finds the effective\ntransverseg-factorg⊥(k)2,11\ng⊥(k) =gε−(k)/radicalBig\n∆2\nAF+ε2\n−(k), (10)\nplotted in Fig. S4 as a function of momentum component\nk, normal to the line g⊥(k) = 0.\nThe parent paramagnetic state is invariant under\ntime reversal, thus ε(k) =ε(−k). Also, in a doubly-\ncommensurate antiferromagnet with N´ eel wave vector Q,\nthe wave vector 2 Qis a reciprocal lattice vector of the\nunderlying non-magnetic state; thus, ε(k+ 2Q) =ε(k).\nFrom these properties, it follows that E(k) =E(−k+Q)\nandg⊥(k) =−g⊥(−k+Q)2. We will now show how this\nsymmetry property leads to ¯ g⊥= 0. In NCCO as well as\nin the N´ eel state of κ-BETS, the relevant Fermi surface\nconsists of two symmetric parts, which map onto each\nother under transformation k→−k+Q. Contributions\nof these two parts to the integral in the right-hand side\nof Eq. (2) cancel each other exactly; hence, Eq. (3) yields\n¯g⊥= 0. In other words, Eq. (2) yields δF= 0, and thus,\nin Eqs. (2) and (3) one finds Rs(θ) = 1: in a transverse\nfield, the amplitude of magnetic quantum oscillations has\nno spin zeros.\nThe arguments above rely on a quasi-classical descrip-\ntion. Note that the key conclusion, the absence of spin\nzeros in a transverse field, holds regardless of how the\nFermi wave vector kFcompares with the inverse antifer-\nromagnetic coherence length 1 /ξ∼∆AF/¯hvF, where the\nbehavior of g⊥(k) crosses over from linear to constant as\nillustrated in Fig. S4.\nIn the limit of kFξ<∼1, the problem can be analyzed by\nreducing the Hamiltonian to the leading terms of its mo-\nmentum expansion around the band extremum. The con-\nclusion remains intact: in a purely transverse field, the\nZeeman term of Eq. (1) does notlift the spin degeneracy\nof Landau levels12; hence, the quantum-oscillation am-\nplitude has no spin zeros. The present work extends the\nvalidity range of this result from a small Fermi-surface\npocket to an arbitrarily large Fermi surface.\nV. QUANTUM OSCILLATIONS IN CeIn3\nAs we pointed out in the main text, for certain Fermi\nsurfaces Zeeman spin-orbit coupling may not manifest it-\nself in quantum oscillations: this is sensitive to where\nthe Fermi surface is centered. An illustrative exam-\nple is provided by CeIn 3, a heavy-fermion compound\nof simple-cubic Cu 3Au structure, with a moderately en-\nhanced Sommerfeld coefficient, γ= 130 mJ/K2mol. Be-\nlowTN≈10 K, it develops a type-II antiferromagnetic\nstructure with wave vector Q= (π\na,π\na,π\na) and an ordered\nmoment of about (0.65 ±0.1)µBper Ce atom13, shown\nin Fig. S5(a). The material remains a metallic down\nCe\n(a)\nL\n/CapSigma\nWX\n/LParen1b/RParen1FIG. S5. Geometry of CeIn 3in real and reciprocal\nspace. (a) Cubic unit cell of CeIn 3in its N´ eel state, show-\ning Ce atoms and their magnetic moments13. Indium atoms\n(not shown) are located at the face centers of the unit cell.\n(b) Cubic BZ of paramagnetic CeIn 3shown by dashed lines\nand, inside, the magnetic BZ. Hexagonal faces (dark gray)\nform the reciprocal-space surface of the symmetry-protected\ndegeneracy g⊥(k) = 0 [2].\n[H]\nFIG. S6. Schematic view of the dsheet of CeIn 3in the\nN´ eel state. We only show a single pair of necks for clarity.\nThe outer sheet illustrates the Fermi-surface reconstruction\nwith broadened necks, the inner sheet shows a reconstruction\nwith the necks truncated. The red lines represent two cy-\nclotron trajectories: a smaller cyclotron-mass trajectory in a\nmagnetic field pointing along ΓL, and a larger cyclotron-mass\ntrajectory, approaching the neck of the Fermi surface.\nto the lowest temperatures. The BZs in the paramag-\nnetic and in the N´ eel state are shown in Fig. S5(b). In\na transverse magnetic field, anti-unitary symmetry pro-\ntects double degeneracy (and thus the equality g⊥(k) =\n0) on hexagonal faces of the magnetic BZ in Fig. S5(b)2.\nWe will focus on the dbranch of the Fermi surface\nof CeIn 3, a nearly spherical sheet centered at the zone\ncenter Γ, with a radius close to√\n3\n2π\na, whereais the\nlattice constant. In the paramagnetic state, this sheet\nhas necks protruding out near the Lpoints in Fig. S5(b),\nconnecting it to another sheet centered at the corner of5\nthe paramagnetic BZ14,15.\nThedsheet has been studied in detail by quantum\noscillations, which revealed a large enhancement of the\ncyclotron mass upon tilting the field, from m≈2m0for\ncyclotron trajectories passing far from the necks (e.g.,\nfor magnetic field B/bardbl/angbracketleft100/angbracketright) tom> 12m0for trajectories\napproaching the necks (such as B/bardbl/angbracketleft110/angbracketright)16,17. This mass\nenhancement can be explained18by the Fermi-surface ge-\nometry, illustrated in Fig. S6: approaching a neck, the\ncyclotron trajectory inevitably runs into a saddle point\nwith its concomitant logarithmic enhancement of the cy-\nclotron mass.\nUpon transition to the N´ eel state, the Fermi surface\nundergoes a reconstruction by folding into the magnetic\nBZ. Depending on the neck size and on the value of\nthe N´ eel gap in the electron spectrum, the reconstructed\nsheets may have their necks broadened or truncated al-\ntogether18, as shown in Fig. S6.\nThe cyclotron-mass enhancement for those field orien-\ntations, for which the quasiclassical trajectories approach\na neck16,17is consistent with the Fermi sheet having\nnecks in the N´ eel state. So is the observation of spin\nzeros16: the enhancement of m/m 0alone produces spin\nzeros as the cyclotron trajectory approaches a neck with\ntilting the field. Crucially for the Zeeman effect that we\nare interested in, the average of g⊥(k) over a cyclotron\ntrajectory on the dsheet does notvanish – simply be-\ncause most (if not all) of the cyclotron trajectory in ques-\ntion lies within the first magnetic BZ, and thus g⊥(k)\nkeeps its sign over most (or all) of the trajectory. Indeed,\nbeing centered at the Γ point, the dsheet cannot possi-\nbly be symmetric with respect to the surface g⊥(k) = 0\nin Fig. S5(b), and thus our symmetry argument for the\nvanishing ¯g⊥inevitably breaks down. The observation\nof spin zeros16in CeIn 3is thus perfectly consistent with\nthe physics of the Zeeman spin-orbit coupling described\nin the main text.\nVI. SdH OSCILLATIONS IN THE HIGH-FIELD\nPARAMAGNETIC STATE OF κ-BETS\nThe SdH oscillations in the high-field, paramagnetic\n(PM) state of κ-BETS have been described in detail in\na number of publications19–23, revealing a Fermi sur-\nface largely consistent with that obtained from band-\nstructure calculations24. Here, we give a brief overview,\nreferring to our own data, which show perfect agreement\nwith the previous reports. The measurements were per-\nformed on the same crystal as discussed in the main text.\nFigure 1(b) in the main text shows an example of the\nlow-temperature interlayer resistance measured in a mag-\nnetic field up to 14 T, directed nearly perpendicularly to\nthe conducting layers. The kink around Bc≈5.2 T re-\nflects the transition from the low-field AF to the high-\nfield PM state. The slow SdH oscillations associated\nwith the small pocket δ(shown in Fig. 1c) of the re-\nconstructed FS in the AF state collapse at Bc. In thePM state, the oscillation spectrum is composed of two\nfundamental frequencies Fα≈840 T andFβ≈4200 T\nand their combinations, as shown in the inset in Fig.1(b).\nThe dominant contribution comes from the αoscillations\nassociated with the closed portion of the Fermi surface\ncentered on the BZ boundary shown in Fig. 1(a). The β\noscillations, which are considerably weaker than the αos-\ncillations, originate from the large magnetic-breakdown\n(MB) orbit enveloping the whole Fermi surface [dashed\norange line in Fig. 1(b)]. Additionally, there are sizable\ncontributions of the second harmonic of the αoscillations\nand combination frequencies, β−pαwithp= 1,2, in the\nSdH spectrum, which are often observed in clean, highly\ntwo-dimensional organic metals in the intermediate MB\nregime25. The oscillation amplitude shows spin-zeros not\nonly in the angular dependence but also with changing\nthe field strength at a fixed orientation. This is due to\nthe presence of an exchange field BJimposed on the con-\nduction electrons by saturated paramagnetic Fe3+ions,\nalthough some details of its behavior are still to be under-\nstood1,23,26. The analysis of the spin-zero positions in the\nPM state yields the g-factor very close to that of free elec-\ntrons,g= 2.0±0.2 and the exchange field BJ≈−13 T\npointing against the external magnetic field23.\nNote that both αandβoscillations are rapidly sup-\npressed with lowering the field and cannot be resolved\nnear the transition field Bc. This rapid suppression is\ndue to the relatively high cyclotron masses: compared to\ntheδ-oscillation mass, the αandβmasses are some 5 and\n7 times higher, respectively. The exponential dependence\nof the SdH amplitude on the cyclotron mass, see Eq. (3)\nof the main text, renders the amplitude factor for the α\noscillations approximately 2 orders of magnitude smaller\nthan that for the slow δoscillations at fields near Bc. In\naddition to the higher cyclotron mass, the MB gap ∆ 0\ncontributes to the damping of the βoscillations via the\nMB damping factor. This is why the βoscillations are\nonly seen at highest fields, ≥10 T, as weak distortions\nof theα-oscillation wave form and as a small peak in the\nFFT spectrum.\nVII. DETAILS OF THE SdH FIT FOR κ-BETS\nIN THE AF STATE\nIn the main text, we noted a large uncertainty of B0\nandBAF. Here, we show that the quality of our SdH\namplitude fits is insensitive to the exact values of B0and\nBAF. Equation (3) for the amplitude Aδcontains the\nMB factor\nR[δ]\nMB=/bracketleftbigg\n1−exp/parenleftbigg\n−B0\nBcosθ/parenrightbigg/bracketrightbigg/bracketleftbigg\n1−exp/parenleftbigg\n−BAF\nBcosθ/parenrightbigg/bracketrightbigg\n,\n(11)\nwhich must be taken into account when analyzing the\nangular dependence Aδ(θ).\nA rough estimate of B0can be obtained from the\nratio between the α- andβ-oscillation amplitudes at\na certain field and temperature, using the LK formula6\n[Eq. (3) of the main text] with the MB factors R[α]\nMB=\n[1−exp (−B0/B)] andR[β]\nMB= exp (−2B0/B) for the\nαandβoscillations, respectively. From the data in\nFig. 1(b) we estimate the ratio between the FFT am-\nplitudes of the αandβoscillations in the field interval\n10 to 14 T, at T= 0.5 K, asAβ/Aα≈0.03. To com-\nplete the estimation, we also need to know the cyclotron\nmasses, which have been determined in earlier experi-\nments:mα= 5.2m0andmβ= 7.9m020, and the Dingle\ntemperature TD. The latter arises from scattering on\ncrystal imperfections1and, therefore, varies from sample\nto sample. As shown below, for the present crystal a\nreasonable estimate of the Dingle temperature in the AF\nstate,TD/similarequal0.7 K, can be obtained from the angular de-\npendence of the δ-oscillation amplitude. Assuming that\nTDis momentum-independent and remains the same in\nthe PM state, we substitute this value in the LK formula,\narriving at the MB field value B0≈12 T. This is three\ntimes higher than the upper end of the field interval used\nfor the SdH oscillation analysis in the AF state. There-\nfore, the first factor in the right-hand side of Eq. (11)\nis close to unity and does not contribute significantly to\nthe angular dependence Aδ(θ)27. To confirm this, we\nhave checked how our fits are affected by varying B0in\nthe range between 5 T and 50 T, as will be presented\nbelow.\nThe MB field BAFis due to magnetic ordering and\ncan be estimated from the gap ∆ AFwith the help of the\nBlount criterion1,28,\nBAF∼mc\n¯he·∆2\nAF/εF/similarequal0.15 T. (12)\nHere, we estimated the Fermi energy εFfrom SdH os-\ncillations in the paramagnetic state: εF∼¯h2k2\nF/2m∼\n¯heFβ/mc,β, with the SdH frequency Fβ= 4280 T and\ncorresponding cyclotron mass mc,β= 7.9m020.\nOf course, these are only rough estimates. Moreover,\nthe observation of the δoscillations in fields up to Bc/similarequal\n5 T implies that the relevant MB field must be in the\nrange of a few tesla, to provide a non-vanishing second\nfactor in the right-hand side of Eq. (11). On the other\nhand, the MB field cannot be much higher than the fields\nwe applied ( B<∼Bc). We have tentatively set the upper\nestimate for BAFto about 5 T and checked how our fits\nare affected by varying BAFfrom 0.15 T to 5 T.\nThe results are summarized in Fig. S7, which presents\nseveral fits to the experimental data for κ-BETS (the\nsame as in Fig. 3 of the main text), with values of B0\nandBAFvarying in a broad range: 5 T ≤B0≤50 T\nand 0.15 T≤BAF≤5 T. All the fits assume g⊥= 0;\nas shown in the main text, a finite value for g⊥would\nsimply lead to sharp spin zeros, insensitive to the\nmonotonic θdependence of RMB. One can clearly see\nthat the fits are nearly indistinguishable and virtually\ninsensitive to variation of B0within the given range,\nwhereas the variation of BAFbarely results in a 15%\nchange of the Dingle temperature: TD= (0.69±0.05) K.\nThe parameter A0in Eq. (3) changes roughly in inverse\n/s45/s56/s48 /s45/s54/s48 /s45/s52/s48 /s45/s50/s48 /s48 /s50/s48 /s52/s48 /s54/s48 /s56/s48/s48/s46/s48/s49/s48/s46/s49/s49/s49/s48\n/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s70/s105/s116/s115/s32/s119/s105/s116/s104/s32/s103 /s32/s61/s32/s48/s58\n/s32/s32 /s66\n/s65/s70/s32/s61/s32/s53/s32/s84/s44/s32 /s32/s32/s32/s32/s32/s66\n/s48/s32/s61/s32/s32/s53/s32/s84/s59/s32/s32/s32 /s84\n/s68/s32/s61/s32/s48/s46/s54/s57/s50/s32 /s32/s48/s46/s48/s49/s56/s32/s75\n/s32/s32 /s66\n/s65/s70/s32/s61/s32/s53/s32/s84/s44/s32/s32/s32/s32/s32/s32 /s66\n/s48/s32/s61/s32/s50/s48/s32/s84/s59/s32/s32 /s84\n/s68/s32/s61/s32/s48/s46/s54/s51/s56/s32 /s32/s48/s46/s48/s49/s54/s32/s75\n/s32/s32 /s66\n/s65/s70/s32/s61/s32/s53/s32/s84/s44/s32/s32/s32/s32/s32/s32 /s66\n/s48/s32/s61/s32/s53/s48/s32/s84/s59/s32/s32 /s84\n/s68/s32/s61/s32/s48/s46/s54/s51/s55/s32 /s32/s48/s46/s48/s49/s54/s32/s75\n/s32/s32 /s66\n/s65/s70/s32/s61/s32/s48/s46/s49/s53/s32/s84/s44/s32 /s66\n/s48/s32/s61/s32/s50/s48/s32/s84/s59/s32/s32 /s84\n/s68/s32/s61/s32/s48/s46/s55/s52/s48/s32 /s32/s48/s46/s48/s49/s55/s32/s75\n/s32/s32 /s66\n/s65/s70/s32/s61/s32/s48/s46/s49/s53/s32/s84/s44/s32 /s66\n/s48/s32/s61/s32/s53/s48/s32/s84/s59/s32/s32 /s84\n/s68/s32/s61/s32/s48/s46/s55/s51/s57/s32 /s32/s48/s46/s48/s49/s55/s32/s75/s65 /s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41\n/s32/s40/s100/s101/s103/s46/s41/s32/s69/s120/s112/s101/s114/s105/s109/s101/s110/s116/s97/s108/s32/s100/s97/s116/s97FIG. S7. Angle-dependent δoscillation amplitude in κ-\nBETS compared to theoretical fits. Black dots are the\nexperimental data and lines are fits using Eq. (3) of the main\ntext with different fixed values of the MB fields B0andBAF.\nThe angle-independent Dingle temperature TDand amplitude\nprefactorA0are the fitting parameters and g⊥is set to zero.\nThe plot demonstrates insensitivity of the angular dependence\non the concrete choice of B0andBAF.\nproportion to BAF. However, A0is largely an empirical\nparameter, irrelevant to our study. Thus, we conclude\nthat the mentioned uncertainty of the MB fields has no\neffect on the quality of our fits, as stated in the main text.\nVIII. AMPLITUDE ANALYSIS FOR NCCO\nIn our experiment on NCCO we were restricted to a rel-\natively narrow field interval, between 45 and 64 T where\nthe oscillations were observable over the whole angular\nrange. Due to the low frequency, this interval contains\nonly a few oscillation periods, see Fig. S8 and Fig. 4 in\nthe main text. Under these conditions the FFT spectrum\nis sensitive to the choice of the field window and details\nof the background subtraction. The choice of the upper\nbound of the field window is most important here: this af-\nfects the largest-amplitude oscillation and thus gives the\ndominant contribution to the overall error. In the course\nof our analysis we always ensured that the order of the\npolynomial fit (≤4) to the non-oscillating B-dependent\nbackground did not affect the height of the FFT peak cor-\nresponding to the SdH oscillations, within our resolution.\nThe main source of error is the variation of the oscillation\nphase at the (fixed) upper end of the field window caused\nby the changing frequency F(θ): this disturbs the shape\nand amplitude of the largest oscillation. In order to re-\nduce this parasitic effect, we kept the FFT window more\nnarrow than that of the background fit; the upper Bbor-\nder of the FFT window was at least 1 T lower than that\nof the background fit. In our analysis we found that the7\n30 40 50 60 70-0.010.000.010.020.030.04Rosc/Rbackg\nB (T)0°50°\n37°\n20°54°54.5°55°(a) (b) (c)\n40 50 60 70-20246810Rosc/Rbackg (´10-3)\nB (T)x10\nx3\n57.8°58.2°60.1°67.2°70.7°\n64.2°\n62.8°\n-10 010 20 30 40 50 60 70 800.010.1110|Aa| (arb. unit)\nq (degree) 52.8 T\n 55 T\n 60 T\n AFFT [45 - 64] T\n LK Fit\nFIG. S8. Amplitude analysis of the angle-dependent αoscillations in NCCO. (a) and (b) oscillating resistance\ncomponent, normalized to the non-oscillating B-dependent background, plotted as a function of magnetic field. The curves\ncorresponding to different tilt angles θare vertically shifted for clarity. For θ= 67.2◦and 70.7◦the ratioRosc/Rbackg is\nmultiplied by a factor of 3 and 10, respectively. The vertical dashed lines indicate the field values at which we determine A(θ)\nshown in (c). (c) Normalized peak-to-peak oscillation amplitude determined for B= 52.8 T, 55 T, and 60 T in comparison to\nthe amplitude determined by FFT for B= [45−64] T. Red dashed line is the fit with g= 0, see Fig. 5 in the main text.\nassociated error bar for the height of the FFT peak did\nnot exceed±10 %. While this uncertainty may be impor-\ntant, for example, for an exact evaluation of the effective\ncyclotron mass, in our case, when the oscillation ampli-\ntude is expected to change an order of magnitude near a\nspin-zero, it is not significant. Only at highest angles, the\nerror becomes comparable to the signal, which is related\nto the low signal-to-noise ratio at these conditions.\nTo crosscheck our FFT analysis we present an alterna-\ntive amplitude analysis in Fig. S8. For each Rosc/Rbackg\ncurve, we determine the peak-to-peak amplitude from\nthe linearly interpolated envelopes (dashed black lines\nin Fig. S8a,b) at a fixed field value. We chose the\nmiddle of the FFT window in 1 /Bscale, i.e., B=\n[(1/45 + 1/64)/2]−1T= 52.8 T, as well as the values 55 T\nand 60 T, with oscillations discernible up to 67 °. Fig. S8c\npresents these points plotted versus the tilt angle θand\ncompares them to the FFT values and LK fit from Fig. 3shown in the main text. The extracted data from both\napproaches match very well. Clearly, our main conclusion\nholds, that is, the overall angular dependence exhibits no\nindication of a spin-zero effect for angles of up to at least\n70°.\nIX. ESTIMATING THE DISPARITY OF\nSPIN-UP AND SPIN-DOWN TUNNELING\nAMPLITUDES IN NCCO\nThe Nd3+spins in the insulating layers of NCCO are\npolarized by strong magnetic field. Consequently one\nmay wonder whether this spin polarization could ren-\nder interlayer tunneling amplitudes for spin-up and spin-\ndown different enough to lose spin zeros in the c-axis\nmagnetoresistance oscillations. In the following we will\nshow that this possibility can be ruled out. The ratio\nof interlayer the conductivities σ↑andσ↓can be crudely\nestimated via the tunneling amplitudes w↑andw↓as\nσ↑\nσ↓≈w2\n↑\nw2\n↓= exp/parenleftbigg\n−2\n¯h/integraldisplay\ndz/bracketleftBig/radicalbig\n2m(U(z)−EF−EZ/2)−/radicalbig\n2m(U(z)−EF+EZ/2)/bracketrightBig/parenrightbigg\n,\nwhereU(z) is the tunneling potential and EZthe Zee-\nman splitting, responsible for the difference between\nσ↑andσ↓. For a rough estimate, it suffices to re-\nplaceU(z) by a rectangular potential barrier of spa-\ntial widthd, the unperturbed tunneling amplitude beingw= exp/parenleftBig\n−d/radicalbig\n2m(U(z)−EF)/¯h/parenrightBig\n. The expression in\nthe exponent above can then be expanded in small EZ\nto yield\nσ↑\nσ↓≈exp/parenleftBigg\nEZ\nU−EF/radicalbig\n2m(U−EF)d\n¯h/parenrightBigg\n= exp/bracketleftbigg\n−EZ\nU−EFlnw/bracketrightbigg\n.8\nTo put in the numbers, notice that the deviation of the\nexperimental data in Fig. 5 of the main text from the\ntheoretical fit with Rs= 1 does not exceed 20%. Assum-\ning this deviation to be entirely due to an interference\nof unequal spin-up and spin-down oscillation amplitudes\nwould imply σ↑/σ↓∼20. Let us estimate the EZre-\nquired for such a behavior. Recall that wis essentially the\nratio of a typical interlayer hopping amplitude tz∼10−2eV to the in-plane Fermi scale U−EF∼1 eV, and thus\nlnw≈−4.6. The ratio σ↑/σ↓∼20 would then mean\nEZ>∼0.6 eV: an unrealistic bound, which allows us to\nrule out this scenario.\nREFERENCES\n1D. Shoenberg, Magnetic Oscillations in Metals (Cambridge\nUniversity Press, Cambridge, 1984).\n2R. Ramazashvili, Phys. Rev. B 79, 184432 (2009).\n3R. Ramazashvili, Phys. Rev. Lett. 101, 137202 (2008).\n4N. P. Armitage, P. Fournier, and R. L. Greene, Rev. Mod.\nPhys. 82, 2421-2487 (2010).\n5J.-F. He, C. R. Rotundu, M. S. Scheurer, Y. He,\nM. Hashimoto, K. Xu, Y. Wang, E. W. Huang, T. Jia,\nS.-D. Chen, B. Moritz, D.-H. Lu, Y. S. Lee, T. P. Dev-\nereaux, and Z.-X. Shen, Proc. Natl. Acad. Sci. USA 116,\n3449-3453 (2019).\n6N. P. Armitage, F. Ronning, D. H. Lu, C. Kim, A. Dam-\nascelli, K. M. Shen, D. L. Feng, H. Eisaki, Z.-X. Shen,\nP. K. Mang, N. Kaneko, M. Greven, Y. Onose, Y. Taguchi,\nand Y. Tokura, Phys. Rev. Lett. 88, 257001 (2002).\n7H. Matsui, T. Takahashi, T. Sato, K. Terashima, H. Ding,\nT. Uefuji, and K. Yamada, Phys. Rev. B 75, 224514 (2007).\n8T. Helm, M. V. Kartsovnik, C. Proust, B. Vignolle,\nC. Putzke, E. Kampert, I. Sheikin, E.-S. Choi, J. S. Brooks,\nN. Bittner, W. Biberacher, A. Erb, J. Wosnitza, and\nR. Gross, Phys. Rev. B 92, 094501 (2015).\n9N. I. Kulikov and V. V. Tugushev, Sov. Phys. Usp. 27,\n954-976 (1984).\n10S. A. Brazovskii, I. A. Luk’yanchuk, and R. R. Ramaza-\nshvili, JETP Lett. 49, 644-646 (1989).\n11V. V. Kabanov and A. S. Alexandrov, Phys. Rev. B 77,\n132403 (2008); 81, 099907(E) (2010)\n12R. Ramazashvili, Phys. Rev. B 80, 054405 (2009).\n13J. M. Lawrence and S. M. Shapiro, Phys. Rev. B 22, 4379-\n4388 (1980).\n14M. Biasini, G. Ferro, and A. Czopnik, Phys. Rev. B 68,\n094513 (2003).\n15J. Rusz and M. Biasini, Phys. Rev. B 71, 233103 (2005).\n16R. Settai, T. Ebihara, M. Takashita, H. Sugawara,\nN. Kimura, K. Motoki, Y. Onuki, S. Uji, and H. Aokii,J. Magn. Magn. Mat. 140-144 , 1153-1154 (1995).\n17T. Ebihara, N. Harrison, M. Jaime, S. Uji, and J. C. Lash-\nley, Phys. Rev. Lett. 93, 246401 (2004).\n18L. P. Gor’kov and P. D. Grigoriev, Phys. Rev. B 73, 060401\n(R) (2006).\n19L. Balicas, J. S. Brooks, K. Storr, D. Graf, S. Uji, H. Shina-\ngawa, E. Ojima, H. Fujiwara, H. Kobayashi, A. Kobayashi,\nand M. Tokumoto, Solid State Commun. 116, 557-562\n(2000).\n20S. Uji, H. Shinagawa, Y. Terai, T. Yakabe, C. Terakura,\nT. Terashima, L. Balicas, J. S. Brooks, E. Ojima, H. Fu-\njiwara, H. Kobayashi, A. Kobayashi, and M. Tokumoto,\nPhysica B 298, 557-561 (2001).\n21T. Konoike, S. Uji, T. Terashima, M. Nishimura, S. Ya-\nsuzuka, K. Enomoto, H. Fujiwara, E. Fujiwara, B. Zhang,\nand H. Kobayashi, Phys. Rev. B 72, 094517 (2005).\n22T. Konoike, S. Uji, T. Terashima, M. Nishimura, T. Ya-\nmaguchi, K. Enomoto, H. Fujiwara, B. Zhang, and\nH. Kobayashi, J. Low Temp. Phys. 142, 531-534 (2006).\n23M. V. Kartsovnik, M. Kunz, L. Schaidhammer, F. Koll-\nmannsberger, W. Biberacher, N. D. Kushch, A. Miyazaki,\nand H. Fujiwara, J. Supercond. Nov. Magn. 29, 3075-3080\n(2016).\n24H. Fujiwara, E. Fujiwara, Y. Nakazawa, B. Zh. Narymbe-\ntov, K. Kato, H. Kobayashi, A. Kobayashi, M. Tokumoto,\nand P. Cassoux, J. Am. Chem. Soc. 123, 306-314 (2001).\n25M. V. Kartsovnik, Chem. Rev. 104, 5737-5782 (2004).\n26O. C´ epas, R. H. McKenzie, and J. Merino, Phys. Rev. B\n65, 100502 (2002).\n27One may expect that TDin the PM state is somewhat lower\nthan the value TD≈0.7 K obtained for the AF state, due\nto the absence of scattering on defects of the AF order (e.g.,\ndomain walls). If so, the actual MB field is even higher than\n12 T, hence the corresponding MB factor is even closer to\nunity.\n28E. I. Blount, Phys. Rev. 126, 1636-1653 (1962)."    },    {        "title": "1409.5600v1.Angular_dependence_of_spin_orbit_spin_transfer_torques.pdf",        "content": "arXiv:1409.5600v1  [cond-mat.mtrl-sci]  19 Sep 2014Angular dependence of spin-orbit spin transfer torques\nKi-Seung Lee1,∗∗, Dongwook Go2,∗∗, Aur´ elien Manchon3, Paul M.\nHaney4, M. D. Stiles4, Hyun-Woo Lee2,∗and Kyung-Jin Lee1,5†\n1Department of Materials Science and Engineering, Korea Uni versity, Seoul 136-701, Korea\n2PCTP and Department of Physics, Pohang University of Scienc e and Technology, Pohang 790-784, Korea\n3Physical Science and Engineering Division, King Abdullah U niversity\nof Science and Technology (KAUST), Thuwal 23955-6900, Saud i Arabia\n4Center for Nanoscale Science and Technology, National Inst itute\nof Standards and Technology, Gaithersburg, Maryland 20899 , USA\n5KU-KIST Graduate School of Converging Science and Technolo gy, Korea University, Seoul 136-713, Korea\nIn ferromagnet/heavy metal bilayers, an in-plane current g ives rise to spin-orbit spin transfer\ntorque which is usually decomposed into field-like and dampi ng-like torques. For two-dimensional\nfree-electron and tight-bindingmodels with Rashba spin-o rbit coupling, the field-like torque acquires\nnontrivial dependence on the magnetization direction when the Rashba spin-orbit coupling becomes\ncomparable to the exchange interaction. This nontrivial an gular dependence of the field-like torque\nis related to the Fermi surface distortion, determined by th e ratio of the Rashba spin-orbit coupling\nto the exchange interaction. On the other hand, the damping- like torque acquires nontrivial angular\ndependence when the Rashba spin-orbit coupling is comparab le to or stronger than the exchange\ninteraction. It is related to the combined effects of the Ferm i surface distortion and the Fermi\nsea contribution. The angular dependence is consistent wit h experimental observations and can be\nimportant to understand magnetization dynamics induced by spin-orbit spin transfer torques.\nI. INTRODUCTION\nIn-plane current-induced spin-orbit spin transfer\ntorques in ferromagnet/heavy metal bilayers provide an\nefficient way of inducing magnetization dynamics and\nmay play a role in future magnetoelectronic devices.1–14\nTwo mechanisms for spin-orbit torques have been pro-\nposed to date; the bulk spin Hall effect in the heavy\nmetal layer15–19and interfacial spin-orbit coupling effect\nat the ferromagnet/heavymetal interface20–30frequently\nreferred to as the Rashba effect. Substantial efforts have\nbeen expended in identifying the dominant mechanism\nforthespin-orbittorque.29–34Forthispurpose, oneneeds\nto go beyond qualitative analysis since both the mecha-\nnismsresultinqualitativelyidenticalpredictions,i.e. two\nvector components of spin-orbit torque (see Eq. (1)). For\nquantitative analysis, we adopt the commonly used de-\ncomposition of the spin-orbit torque T,\nT=τfˆM׈y+τdˆM×(ˆM׈y), (1)\nwhere the first term is commonly called the field-like\nspin-orbit torque, the second term the damping-like spin-\norbit torque or the Slonczewski-like spin-orbit torque,\nˆM= (cosφsinθ,sinφsinθ,cosθ) is the unit vector along\nthe magnetization direction, ˆyis the unit vector perpen-\ndicular to both current direction ( ˆx) and the direction\nin which the inversion symmetry is broken ( ˆz),τfand\nτddescribe the magnitude of field-like and damping-like\nspin-orbit torque terms, respectively. Since Tshould be\northogonal to ˆM, the two terms in Eq. (1), which are or-\nthogonal to ˆMand also to each other, provide a perfectly\ngeneral description of the spin-orbit torque regardless of\nthe detailed mechanism of T. The quantitative analysis\nofTthen amounts to the examination of the propertiesofτfandτd.\nOneofintriguingfeaturesofspin-orbittorqueobserved\nin some experiments is the strong dependence of τfand\nτdon the magnetization direction.35,36Comparing the\nmeasured and calculated angular dependence will pro-\nvidecluestothemechanismofthespin-orbittorque. The\ndetailed angular dependence also determines the magne-\ntization dynamics and hence is important for device ap-\nplications based on magnetization switching1,3,10–13, do-\nmain wall dynamics2,5–9,14, and magnetic skyrmion mo-\ntion37.\nTheories based on the bulk spin Hall effect com-\nbined with a drift-diffusion model or Boltzmann trans-\nport equation29predict no angular dependence of τfand\nτd, which is not consistent with the experimental re-\nsults.35,36For theories based on the interfacial spin-orbit\ncoupling, the angular dependence is subtle. Based on\nthe Rashba model including D’yakonov-Perel spin relax-\nation, Pauyac et al.38studied the angular dependence of\nspin-orbit torque perturbatively in weak Rashba regime\n(r≡αRkF/J≪1) and strong Rashba regime ( r≫1)\nwhereαRis the strength of the Rashba spin-orbit cou-\npling,kFis the Fermi wave vector, and Jis the exchange\ncoupling. They found that both τfandτdare almost\nindependent of the angular direction of ˆMin the weak\nRashba regime. In the strong Rashba regime, on the\nother hand, they found that τdexhibits strong angular\ndependence. Theoriginoftheangulardependencewithin\nthis model is the anisotropy of the spin relaxation, which\narises naturally since the Rashba spin-orbit interaction\nis responsible for the anisotropic D’yakonov-Perel spin\nrelaxation mechanism. For τf, in contrast, they found\nit to be almost constant in the strong Rashba regime\neven when the spin relaxationis anisotropic. Experimen-\ntally,35,36both the damping-like and the field-like contri-2\nbutions depend strongly on the magnetization direction.\nHerewe reexaminethe angulardependence ofthe spin-\norbit torque based on the Rashba interaction motivated\nby the following two observations. The first motivation\ncomes from a first-principles calculation39of Co/Pt bi-\nlayers, according to which both the spin-orbit potential\nand the exchange splitting are large near the interface\nbetween the heavy metal and the ferromagnet. This im-\nplies that the problem of interest is not in the analyti-\ncally tractableweak Rashbaor strongRashbaregime but\nin the intermediate Rashba regime ( r≈1). We exam-\nine this intermediate regime numerically and find that in\ncontrast to both the strong and weak Rashba regimes, τf\nhas a strong angular dependence. The second motivation\ncomes from a recent calculation28,30showing that the\ninterfacial spin-orbit coupling can generate τdthrough\na Berry phase contribution40. In contrast, earlier the-\nories25–27of the interfacial spin-orbit coupling found a\nseparate contribution to τdfrom spin relaxation. More-\nover those calculations30suggest that the Berry phase\ncontribution to τdis much largerthan the spin relaxation\ncontribution. Here, we examine the angular dependence\nof the Berry phase contribution.\nTo be specific, we examine the angular dependence of\nthe spin-orbit torques for a free-electron model of two-\ndimensionalferromagneticsystemswiththeRashbaspin-\norbit coupling. When an electric field is applied to gener-\nate an in-plane current, the spin-orbit torque arises from\nthe two types of changes caused by the electric field. One\nis the electron occupation change. For a small electric\nfield, the net occupation change is limited to the Fermi\nsurface so that the spin-orbit torque caused by the oc-\ncupation change comes from the Fermi surface. For this\nreason, this contribution is referred to as the Fermi sur-\nface contribution. The other is the state change. The\nelectric field modifies the potential energy of the system,\nwhich in turn modifies wavefunctions of all single parti-\ncle states. Thus the spin-orbit torque caused by the state\nchangecomesnotonlyfromthestatesneartheFermisur-\nface but also from all states in the entire Fermi sea. This\ncontribution is referred to as the Fermi sea contribution\nand often closely related to the momentum-space Berry\nphase30.\nWe find that in the absence of spin relaxation, the\nFermi surface contribution to τdis vanishingly small,\nwhileτfremains finite. The τfhas a substantial angu-\nlar dependence in the intermediate Rashba regime. This\nnontrivial angular dependence of τfis related to Fermi\nsurface distortion, which becomes significant when the\nRashba spin-orbit coupling energy ( ∼αRkF) is compa-\nrabletotheexchangecoupling( ∼J). Ontheotherhand,\nthe Fermi sea contribution generates primarily τdwhich\nexhibits strong angular dependence in both the interme-\ndiate and strong Rashba regimes. The nontrivial angular\ndependence of τdis caused by the combined effects of the\nFermi surface distortion and the Fermi sea contribution.\nWe also compute the angular dependence of the spin-\norbit torques for a tight-binding model and find that theresults are qualitatively consistent with those for a free-\nelectron model.\nII. SEMICLASSICAL MODELS\nIn this section, we use subscripts (1) and (2) to de-\nnote the Fermi surface and the Fermi sea contributions,\nrespectively. The model Hamiltonian for an electron in\nthe absence of an external electric field is\nH0=p2\n2m+αRσ·(k׈ z)+Jσ·ˆM,(2)\nwherek= (kx,ky) is the two-dimensionalwavevector, m\nis the electronmass, J(>0) is the exchangeparameter, p\nis the momentum, and σis the vector of Pauli matrices,\nandMx,My, andMzare thex-,y-, andz-components\nofˆM, respectively. When ˆMis position-independent,\nwhich will be assumed all throughout this paper, kis a\ngood quantum number. For each k, there are two energy\neigenvalues since the spin may point in two different di-\nrections. Thus the energy eigenvalues of H0form two\nenergy bands, called majority and minority bands. The\none-electron eigenenergy of H0is\nE±\nk=¯h2k2\n2m∓ǫk, (3)\nwhere the upper (lower) sign corresponds to the majority\n(minority)band, k2=k2\nx+k2\ny, andǫk=|JˆM+αR(k׈ z)|.\nTo determine the spin state of the majorityand minor-\nity bands, it is useful to combine the last two terms of H0\ninto an effective Zeeman energy term (= −µBBeff,k·σ),\nwhere the effective magnetic field is k-dependent and\ngiven by\nBeff,k=−J\nµBˆM−αR\nµB(k׈ z). (4)\nHereµBis the Bohr magneton. Beff,kfixes the spin\ndirection of the majority and minority bands. For\nthe eigenstate |ψk,±∝angb∇acket∇ightof an eigenstate in the major-\nity/minority band, its spin expectation value s±\nk(1)≡\n(¯h/2)∝angb∇acketleftψk,±|σ|ψk,±∝angb∇acket∇ightis given by\ns±\nk(1)=±¯h\n2ˆBeff,k, (5)\nwhereˆBeff,kis the unit vector along Beff,k. In terms\nof thek-dependent angle θkandφk, which are defined\nbyˆBeff,k= (sinθkcosφk,sinθksinφk,cosθk), the eigen-\nstate|ψk,±∝angb∇acket∇ightis given by\n|ψk,+∝angb∇acket∇ight= eik·r/parenleftbiggcos(θk/2)\nsin(θk/2)eiφk/parenrightbigg\n(6)\n|ψk,−∝angb∇acket∇ight= eik·r/parenleftbiggsin(θk/2)\n−cos(θk/2)eiφk/parenrightbigg\n(7)3\nTogether with the energy eigenvalue E±\nkin Eq. (3), the\neigenstate |ψk,±∝angb∇acket∇ightcompletely specifies properties of the\nequilibriumHamiltonian. Thegroundstateofthe system\nisthenachievedbyfillingupallsingleparticleeigenstates\n|ψk,±∝angb∇acket∇ight, below the Fermi energy EF.\nWhen an electric field E=Eˆ xis applied, one of the\neffects is the modification of the state occupation. This\neffect generates the non-equilibrium spin density s±\n(1)as\ns±\n(1)=/integraldisplaydk2\n(2π)2/bracketleftbigg\nf±/parenleftbigg\nk−eEτ\n¯hˆx/parenrightbigg\n−f±(k)/bracketrightbigg\ns±\nk(1),(8)\nwhere−eis the electron charge, τis the relaxation time,\nandf±(k) = Θ(EF−E±\nk) is the zero-temperature elec-\ntron occupation function where Θ( x) is the Heaviside\nstep function. Note that the net contribution to s±\n(1)\narises entirely from the states near EFdue to the can-\ncellation effect between the two occupation functions in\nEq. (8). Thus s±\n(1)is aFermi surface contribution. The\ntotal spin density generated by the occupation change\nbecomes s(1)=s+\n(1)+s−\n(1). This is related to the spin-\norbit torque T(1)generatedby the occupation change via\nT(1)= (J/¯h)s(1)׈M. In Eq. (8), we use the relaxation\ntime approximation with the assumption that the scat-\ntering probability is isotropic and spin-independent.\nThe other important effect of the electric field other\nchanging than the occupation is that it modifies the po-\ntential energy that the electrons feel, and hence modifies\ntheir wave functions, generating in turn a correction to\ns±\nk(1). We call this correction s±\nk(2). It is calculated in\nAppendix A and given by\ns±\nk(2)=±¯h\n2αReE/bracketleftbiggJ\n2ǫ3\nk(ˆM׈y)+αR\n2ǫ3\nk(ˆx×k)/bracketrightbigg\n.(9)\nSumming over all occupied states in the major-\nity/minority band, gives the total spin density s±\n(2)gen-\nerated by the state change in that band, and is given\nby\ns±\n(2)=/integraldisplaydk2\n(2π)2f±(k)s±\nk(2). (10)\nNote that the equilibrium occupation function fappears\nin Eq. (10) rather than the difference between the two\noccupation functions. The occupation change effect is\nignoredin Eq.(10)sinceweareinterestedin lineareffects\nof the electric field Eands±\nk(2)is already first order in\nE. Note that all the occupied single particle states in\nthe Fermi sea contribute to s±\n(2). Thuss±\n(2)amounts is a\nFermi sea contribution. The total spin density generated\nby the state change becomes s(2)=s+\n(2)+s−\n(2). This is\nrelated to the spin-orbit torque T(2)generated by the\nstate change via T(2)= (J/¯h)s(2)׈M.\nAfewremarksareinorder. In Eq.(8), the twooccupa-\ntion functions cancel each other for most kvalues. They\ndo not cancel for kpoints that correspond to electronexcitation slightly above the Fermi surface or the hole\nexcitation slightly below the Fermi surface. Thus the di-\nrection of s±\n(1)can be estimated simply by evaluating the\ndifference of Beff,kbetween two k’s of electron-like and\nhole-like excitations. This shows that s±\n(1)points along\nEˆx׈z=−Eˆy. Thus the spin-orbit torque T(1)should\nbe proportional to Eˆy׈M, which is nothing but the\nfield-like spin-orbit torque. Thus the Fermi surface con-\ntribution T(1)contributes mostly to τf. To be precise,\nhowever, this statement is not valid for spin-dependent\nscattering, which we neglect in deriving Eq. (8). If the\nscattering is spin-dependent, T(1)producesτdas well as\nτfas demonstrated in Refs. 25–27. In this paper, we\nneglect the contribution to the angular dependence of τd\nfromT(1)andspin-dependent scatteringsinceithasbeen\nalready treated in Ref. 38. The contribution to τdin our\nstudy comes from the Fermi sea contribution T(2). One\ncan easily verify that the first term in Eq. (9) generates\nthe spin-orbit torque proportionalto ( ˆM׈y)׈M, which\nhas the form of the damping-like spin-orbit torque. The\nsecond term in Eq. (9) on the other hand almost vanishes\nuponkintegration in Eq. (10). This demonstrates that\nthe Fermi sea contribution T(2)contributes mostly to τd.\nWe also compute the spin-orbit torques based on a\ntight-binding model because free electron models with\nlinear Rashba coupling, like we use here, can exhibit\npathological behavior when accounting for vertex correc-\ntions to the impurity scattering. For example, the intrin-\nsic spin Hall effect40, that has the same physical origin of\nthe Fermi sea contribution to spin-orbit torque, gives a\nuniversalresult that vanishes when vertexcorrectionsare\nincluded.41–45However, the intrinsic spin Hall effect does\nnot vanish when the electron dispersion deviates from\nfree electron behavior or the spin-orbit coupling is not\nlinear in momentum.46–49Since we neglect vertex correc-\ntions in the calculations presented in this paper, it is nec-\nessary to check whether or not the angular dependence\nof spin-orbit torque obtained in a free-electron model is\nqualitatively reproduced in a tight-binding model, where\nthe electron dispersion deviates from free electron be-\nhavior and the spin-orbit coupling is not strictly linear\nin momentum. To compute spin-orbit torque in the two-\nband (majority and minority spin bands) tight-binding\nmodel on a square lattice with the lattice constant a, we\nreplacekxandkyby sin(kxa)/aand sin(kya)/a, respec-\ntively. The corresponding spin density is then calculated\nby integrating the electric field-induced spin expectation\nvalue up to the point of band filling. For most cases in\na tight-binding model, the result converges for a k-point\nmesh with mesh spacing dk=0.052 nm−1and 80,000 k-\npoints, wherethe convergencecriteriais 1percentchange\nofresults with a finer mesh by factor of 2. All results pre-\nsented in this paper are converged to this criteria.4\n/s45/s49/s46/s56/s45/s49/s46/s50/s45/s48/s46/s54/s48/s46/s48/s48/s46/s54/s49/s46/s50/s49/s46/s56\n/s45/s49/s46/s56 /s45/s49/s46/s50 /s45/s48/s46/s54 /s48/s46/s48 /s48/s46/s54 /s49/s46/s50 /s49/s46/s56 /s45/s49/s46/s56 /s45/s49/s46/s50 /s45/s48/s46/s54 /s48/s46/s48 /s48/s46/s54 /s49/s46/s50 /s49/s46/s56/s45/s49/s46/s56/s45/s49/s46/s50/s45/s48/s46/s54/s48/s46/s48/s48/s46/s54/s49/s46/s50/s49/s46/s56\n/s45/s49/s46/s56 /s45/s49/s46/s50 /s45/s48/s46/s54 /s48/s46/s48 /s48/s46/s54 /s49/s46/s50 /s49/s46/s56/s69/s108/s101/s99/s116/s114/s105/s99/s32/s102/s105/s101/s108/s100/s32/s100/s105/s114/s101/s99/s116/s105/s111/s110/s122\n/s72/s101/s97/s118/s121/s32/s109/s101/s116/s97/s108/s32/s32\n/s40/s97/s41\n/s72/s101/s97/s118/s121/s32/s109/s101/s116/s97/s108/s70/s101/s114/s114/s111/s109/s97/s103/s110/s101/s116/s120\n/s50/s45/s100/s105/s109/s101/s110/s115/s105/s111/s110/s97/s108/s32/s82/s97/s115/s104/s98/s97/s32/s105/s110/s116/s101/s114/s102/s97/s99/s101/s40/s102/s41\n/s32/s32/s40/s98/s41\n/s32/s32\n/s40/s100/s41/s32\n/s32/s107\n/s121/s32/s40/s120/s49/s48/s49/s48\n/s32/s109/s45/s49\n/s41\n/s32/s107\n/s120/s32/s40/s120/s49/s48/s49/s48\n/s32/s109/s45/s49\n/s41/s40/s101/s41\n/s32/s32/s77/s97/s106/s111/s114/s105/s116/s121 /s32/s98/s97/s110/s100\n/s32/s114/s32/s61/s32/s48\n/s32/s114/s32/s61/s32/s49/s46/s48/s40/s99/s41\n/s32\nFIG. 1: (color online) Fermi surface and spin direction for a free-electron model. (a) r= 0 (only exchange splitting), (b)\nr=∞(only Rashba spin-orbit coupling and non-magnetic), (c) co mparison of Fermi surfaces (majority band) for r= 0 and\nr= 1.0, (d)r= 0.8, (e)r= 1.0, and (f) r= 1.2. We assume ˆM= (0,1,0),EF= 10 eV, m=m0, andJ= 1 eV. Here m0\nis the free electron mass. The outer red (inner blue) Fermi su rface corresponds to majority (minority) band. Arrows are t he\neigendirections of spins on the Fermi surface. The coordina te system is shown on the right.\nIII. RESULTS AND DISCUSSION\nWe first discuss Fermi surface distortion as a function\nofr(=αRkF/J). Figure 1 shows the Fermi surface and\nthe spin direction at each k-point for various values of\nthe ratior. Without Rashba spin-orbit coupling ( r= 0),\nthe spin directiondoesnot depend on kforferromagnetic\nsystems (Fig. 1(a)). Without exchange coupling (non-\nmagnetic Rashba system ( r=∞)), on the other hand,\nthe spins point in the azimuthal direction (Fig. 1(b)).\nFor these extreme cases, the Fermi surfaces of two bands\nare concentric circles.\nThe Fermi surfaces distort significantly when r≈1.\nFigure 1(c) comparestwo Fermi surfaces(majorityband)\nforr= 0andr= 1.0whenˆM= (0,1,0). Whenthemag-\nnetizationhasanin-planecomponentasinthiscase,each\nsheet of the Fermi surface shifts in a different direction\nand distorts from perfect circularity(Fig. 1(c): note that\nthe dotted Fermi surfaceis for r= 0 andis a circle). This\ndistortion arises because the k-dependent effective mag-\nnetic field (Eq. (4)) contains contributions both from the\nexchange and Rashba spin-orbit couplings. An effective\nfield from the exchange is aligned along ˆMand uniform\nregardless of k, whereas that from the Rashba spin-orbit\ncoupling lies in the x−yplane and is k-dependent. For\nexample, for ˆM= (0,1,0) and majority band, an effec-\ntive field from the Rashba spin-orbit coupling is parallel\n(anti-parallel) to that from the exchange at k= (kF,1,0)\n(k= (kF,2,0)), wherekF,1(>0) andkF,2(<0) are the\nFermi wave vectors corresponding to the electric field-induced electron-like and hole-like excitations, respec-\ntively. The k-dependent effective field distorts the Fermi\nsurface distortion as demonstrated in Fig. 1(c)-(f).\nThis Fermi surface distortion also affects the spin di-\nrection at each k-point because the spin eigendirection is\nk-dependent due to the Rashba spin-orbit coupling. In\nthe weak (strong) Rashba regime, the spin landscape is\nsimilar with that in Fig. 1(a) (Fig. 1(b)). In these ex-\ntreme cases, the spin landscape is not significantly mod-\nified by the change in the magnetization direction as one\nof the effective fields (either from the exchange or from\nthe Rashba spin-orbit coupling) is much stronger than\nthe other. As a result, τfhas almost no angular distor-\ntionintheseregimes. Thespinlandscapefor r≈1onthe\nother hand becomes highly complicated (Fig. 1(d)-(f))\nas the Fermi surface distortion is maximized. One can\neasily verify that the spin landscape for r≈1 varies sig-\nnificantly with the magnetization direction because the\nFermi surface distortion is closely related to the in-plane\ncomponent of the magnetization as explained above.\nAs the non-equilibrium spin density corresponding to\nτf(i.e.s(1)) is obtained from the integration of the\nspins on the Fermi surface, this magnetization-angle-\ndependent change in the spin landscape generates a non-\ntrivial angular dependence of τf. A similar argument\nis valid for τd(i.e.s(2)) which comes from the Fermi\nsea contribution because the Fermi surface distortion af-\nfects the interval of the integration. Therefore, the re-\nsults shown in Fig. 1 suggest that the spin-orbit torque\noriginatingfromtheinterfacialspin-orbitcouplingshould\nhave a strong dependence on the magnetization angles θ5\n/s48/s49/s50\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48/s48/s49/s50/s48/s51/s54/s57\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48/s48/s51/s54/s57/s45/s50/s46/s48\n/s45/s49/s46/s54\n/s45/s49/s46/s50\n/s45/s48/s46/s56\n/s32/s32/s102/s32/s40/s120/s49/s48/s50/s51\n/s32/s101/s69 /s41\n/s70/s114/s101/s101/s45/s101/s108/s101/s99/s116/s114/s111/s110\n/s32/s61/s32/s48/s114/s32/s61/s32\n/s82/s107\n/s70/s47/s74/s32/s61/s32/s45/s48/s46/s52\n/s40/s100/s41\n/s45/s50/s46/s48\n/s45/s49/s46/s54\n/s45/s49/s46/s50\n/s45/s48/s46/s56\n/s114/s32/s61/s32\n/s82/s107\n/s70/s47/s74/s32/s61/s32/s45/s48/s46/s52/s102/s32/s40/s120/s49/s48/s50/s51\n/s32/s101/s69 /s41\n/s80/s111/s108/s97/s114/s32/s97/s110/s103/s108/s101/s32 /s47/s70/s114/s101/s101/s45/s101/s108/s101/s99/s116/s114/s111/s110\n/s32/s61/s32 /s47/s50/s40/s97/s41\n/s84/s105/s103/s104/s116/s45/s98/s105/s110/s100/s105/s110/s103\n/s32/s61/s32/s48/s82/s32/s40/s101/s86/s46/s110/s109/s41\n/s32/s45/s48/s46/s48/s50/s32/s32/s32 /s32/s45/s48/s46/s48/s52\n/s32/s45/s48/s46/s48/s54/s32/s32/s32 /s32/s45/s48/s46/s48/s56\n/s32/s45/s48/s46/s50/s56/s102/s32/s40/s120/s49/s48/s50/s51\n/s32/s101/s69 /s41\n/s82/s32/s40/s101/s86/s46/s110/s109/s41\n/s32/s45/s48/s46/s48/s50/s32/s32/s32 /s32/s45/s48/s46/s48/s52\n/s32/s45/s48/s46/s48/s54/s32/s32/s32 /s32/s45/s48/s46/s48/s56\n/s32/s45/s48/s46/s50/s56/s40/s99/s41\n/s84/s105/s103/s104/s116/s45/s98/s105/s110/s100/s105/s110/s103\n/s32/s61/s32 /s47/s50/s40/s98/s41\n/s102/s32/s40/s120/s49/s48/s50/s51\n/s32/s101/s69 /s41\n/s80/s111/s108/s97/s114/s32/s97/s110/s103/s108/s101/s32 /s47\nFIG. 2: (color online) Polar angle ( θ) dependence of field-\nlike spin-orbit torque coefficient τf. Free-electron model (a\nand b): (a) azimuthal angle of magnetization φ= 0 and (b)\nφ=π/2. Tight-binding model (c and d): (c) φ= 0 and\n(d)φ=π/2. For a free-electron model, we use EF= 10 eV,\nm=m0, andJ= 1 eV. For a tight-binding model, we use\nm=m0,J= 1 eV, a= 0.3 nm, and normalized electron\ndensityn=N/Nmax= 0.5 where Nis the electron density of\nfilled bands and Nmax(= 2.2×1019m−2) is the maximum\nelectron density. For a free-electron model, the results fo r\nr= 1 are excluded because it is singular. For a tight-binding\nmodel, the results for -0.28 eV ·nm< αR<-0.08 eV ·nm are\nnot included because of bad convergence.\nandφwhenris close to 1.\nSeveral additional remarks for the Fermi surface dis-\ntortionareasfollows. First, the twoFermi surfacestouch\nexactly for r= 1 (Fig. 1(e)) and they anticross for\nr>1 (Fig. 1(f)). As a result, the spin landscape rapidly\nchanges when rvaries around 1 so that a similar dras-\ntic change in the angular dependence of the spin-orbit\ntorqueisexpected. Second, alleffectsfromtheFermisur-\nface distortion, described for a free-electronmodel above,\nshould also affect the results obtained for a tight-binding\nmodel. However, as the shape of the Fermi surface is\ndifferent for the two models (i.e. for J= 0 andαR= 0,\nthe Fermi surface for a free-electron model is a circle,\nwhereas that for a tight-binding model with half band-\nfilling is a rhombus), the results for the two models are\nquantitatively different.\nWe next show the angular dependence of τfandτd\nfor the two models. Here we do not attempt to analyze\nthe detailed angular dependence quantitatively, because\nit is very parameter sensitive. In contrast, our intention\nis to identify the general trends that emerge from these\nnumerical calculations. Figure 2 shows the angular de-\npendence of τffor a free-electron model ((a) and (b)) and\na tight-binding model ((c) and (d)). In both models, we\nobtain nontrivial angular dependence of τfin certain pa-\nrameter regimes. In the free-electron model, we find τf\nis almost constant in weak ( r≪1) and strong ( r≫1)/s48/s46/s48 /s48/s46/s53 /s49/s46/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s48/s46/s48 /s48/s46/s53 /s49/s46/s48/s48/s50/s52/s54\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48/s48/s53/s49/s48/s49/s53/s50/s48/s80/s111/s108/s97/s114/s32/s97/s110/s103/s108/s101/s32 /s47/s114/s32/s61/s32\n/s82/s107\n/s70/s47/s74\n/s32/s45/s48/s46/s52\n/s32/s45/s48/s46/s56\n/s32/s45/s49/s46/s50\n/s32/s45/s49/s46/s54\n/s32/s45/s50/s46/s48\n/s32/s32/s100/s32/s40/s120/s49/s48/s56\n/s32/s101/s69/s41/s70/s114/s101/s101/s45/s101/s108/s101/s99/s116/s114/s111/s110\n/s32/s61/s32/s48\n/s114/s32/s61/s32\n/s82/s107\n/s70/s47/s74\n/s32/s45/s48/s46/s52/s32/s32 /s32/s45/s48/s46/s56\n/s32/s45/s49/s46/s50/s32/s32 /s32/s45/s49/s46/s54\n/s32/s45/s50/s46/s48/s40/s100/s41/s100/s32/s40/s120/s49/s48/s56\n/s32/s101/s69/s41\n/s65/s122/s105/s109/s117/s116/s104/s97/s108/s32/s97/s110/s103/s108/s101/s32 /s47/s70/s114/s101/s101/s45/s101/s108/s101/s99/s116/s114/s111/s110\n/s32/s61/s32 /s47/s50/s40/s97/s41\n/s80/s111/s108/s97/s114/s32/s97/s110/s103/s108/s101/s32 /s47/s84/s105/s103/s104/s116/s45/s98/s105/s110/s100/s105/s110/s103\n/s32/s61/s32/s48/s82/s32/s40/s101/s86/s46/s110/s109/s41\n/s32/s45/s48/s46/s48/s50/s32/s32/s32 /s32/s45/s48/s46/s48/s54\n/s32/s45/s48/s46/s49/s48/s32/s32/s32 /s32/s45/s48/s46/s49/s52\n/s32/s45/s48/s46/s49/s56/s100/s32/s40/s120/s49/s48/s56\n/s32/s101/s69/s41\n/s82/s32/s40/s101/s86/s46/s110/s109/s41\n/s32/s45/s48/s46/s48/s50/s32/s32/s32 /s32/s45/s48/s46/s48/s54\n/s32/s45/s48/s46/s49/s48/s32/s32/s32 /s32/s45/s48/s46/s49/s52\n/s32/s45/s48/s46/s49/s56/s40/s99/s41\n/s84/s105/s103/s104/s116/s45/s98/s105/s110/s100/s105/s110/s103\n/s32/s61/s32 /s47/s50/s40/s98/s41\n/s100/s32/s40/s120/s49/s48/s56\n/s32/s101/s69/s41\n/s65/s122/s105/s109/s117/s116/s104/s97/s108/s32/s97/s110/s103/s108/s101/s32 /s47\nFIG. 3: (color online) Angular dependence of damping-like\nspin-orbit torque coefficient τd. Free-electron model (a and\nb): (a) polar angle dependence at φ= 0 and (b) azimuthal\nangle dependence at θ=π/2. Tight-binding model (c and d):\n(c) polar angle dependence at φ= 0 and (d) azimuthal angle\ndependence at θ=π/2. Same parameters are used as in Fig.\n2.\nRashba regimes, consistent with earlierworks.22,23In the\nintermediate Rashba regimes, however, τfis not a con-\nstant. We find that τfdepends not only on the polar\nangleθbut also the azimuthal angle φ, as expected from\nthe Fermi surface distortion (Fig. 1). In Fig. 2(b), τf\nforr <1 (r >1) is maximal (minimal) at θ=π/2,\nwhich is caused by the anticrossing of the two Fermi sur-\nfaces (Fig. 1(d)-(f)). Despite the strong angular depen-\ndence, the sign of τfis preserved since the spin direction\nof nonequilibrium spin density is unambiguously deter-\nmined once the direction of electric field and the sign of\nαRarefixed. Theseoveralltrendsarequalitativelyrepro-\nduced in a tight-binding model (Fig. 2(c) and (d)). The\nmagnitude and angular dependence of τfdiffer quantita-\ntively from those of the free electron model, due to the\ndifferent shape of the Fermi surfaces for the two models.\nFigure 3 shows the angular dependence of τdfor a free-\nelectron model ((a) and (b)) and a tight-binding model\n((c) and (d)). This τdresults from the Fermi sea contri-\nbution (Eqs. (9) and (10)). In both models, we obtain\nnontrivial angular dependence of τdboth in the inter-\nmediate and strong Rashba regimes (Fig. 3(a) and (c)).\nThis is in contrast to τfwhich exhibits nontrivial angular\ndependence only in the intermediate Rashba regime. To\nunderstand this difference, we derive an approximate τd\nby expanding up to third order inαRkF\nJand assuming no\nFermi surface distortion (i.e. the Fermi wave vector kF\ndoes not depend on the direction of k), which is analyt-\nically tractable. By integrating Eq. (10) with these as-\nsumptions, wefind τd∝(16J2−3α2\nRk2\nF−9α2\nRk2\nFcos(2θ)).\nTherefore, the Fermi sea contribution induces an intrin-\nsicangulardependence in τd, which increaseswith|αR|kF\nJ6\nirrespective of the Fermi surface distortion. The results\nin Fig. 3, which are obtained numerically, include the\nFermi surface distortion, so that the nontrivial angular\ndependence of τdresults from the combined effects of the\nintrinsic Fermi sea contribution and the Fermi surface\ndistortion. For example, Fig. 3(a) shows a sharp differ-\nence in the angular dependence of τdforr>1 andr<1.\nThis is qualitatively similar to the results of τfshown in\nFig. 2(b), showing that the Fermi surface distortion also\nhas a role in the angular dependence of τd.\nThe sign of τddoes not change with the magnetization\nangle despite the strong angular dependence, similar to\nthe behavior of τf. Whenθ=π/2 (Fig. 3(d)), a steep\nincrease of τdis obtained at φ=π/4 and 3π/4, origi-\nnating from the shape of the Fermi surface. We expect\nthat this strong dependence of τdonφcan be observed\nin epitaxial bilayers but may be absent in sputtered bi-layers as sputtered thin films consist of small grains with\ndifferent lattice orientation in the film plane. However,\nthe dependence of τdon the polar angle θ(Fig. 3(a) and\n(c)) is irrelevant to this in-plane crystallographic issue\nso we expect that it will be observable in experiments\nwhen the interfacial spin-orbit coupling is comparable to\nor stronger than the exchange coupling. We note that a\nstrong dependence of τdonθ(but a very weak depen-\ndence onφ) was experimentally observed in sputtered\nbilayers.35\nWe finally illustrate the connection between τd(i.e.\nT(2)) and the Berry phase. This examination is mo-\ntivated by Ref. 30, which called T(2)the Berry phase\ncontribution. To clarify the connection, it is useful to ex-\npressthe Fermi seacontribution s(2)in the Kubo formula\nform,\ns(2)=1\n2eE¯h2AIm/summationdisplay\nab/integraldisplayd2k\n(2π)2[fa(k)−fb(k)]∝angb∇acketleftk,a|σ|k,b∝angb∇acket∇ight∝angb∇acketleftk,b|vx|k,a∝angb∇acket∇ight\n[Ea(k)−Eb(k)+2iδ]2, (11)\nwherea,bare band indices, and δis an infinitesimally\nsmall positive constant. In the present case, |k,a∝angb∇acket∇ightis ei-\nther|ψk,+∝angb∇acket∇ightor|ψk,−∝angb∇acket∇ight. One then uses the relations\nvx=1\n¯h∂H0(k,M)\n∂kx, σα=1\nJ∂H0(k,M)\n∂Mα,(12)\nwhere the notation H0(k,M) emphasizes that the un-perturbed Hamiltonian H0is a function of the momen-\ntumkand the magnetization M. Note that here we use\nMinstead of ˆMsince one needs to relax the constraint\n|ˆM|= 1toestablishtheconnectionwiththeBerryphase.\nEquation (12) allows one to convert the numerator of\nEq. (11) as follows,\n∝angb∇acketleftk,a|σ|k,b∝angb∇acket∇ight=−1\nJ[Ea(k)−Eb(k)]∝angb∇acketleftk,a|∇M|k,b∝angb∇acket∇ight,∝angb∇acketleftk,b|vx|k,a∝angb∇acket∇ight=−1\n¯h[Eb(k)−Ea(k)]∝angb∇acketleftk,b|∂\n∂kx|k,a∝angb∇acket∇ight.(13)\nThus the numerator of Eq. (11) acquires the factor\n[Ea(k)−Eb(k)]2, which cancels the denominator in thelimitδ→0. Then one of the two summations for the\nband indices aandbcan be performed to produce\n[s(2)]α=1\n2eE¯hA\nJ/summationdisplay\na/integraldisplayd2k\n(2π)2fa(k)/bracketleftbigg∂\n∂kxAa\nMα(k)−∂\n∂MαAa\nkx(k)/bracketrightbigg\n, (14)\nwhere the spin-space Berry phase Aa\nMα(k) and the momentum-space Berry phase Aa\nkx(k) are defined by\nAa\nMα(k) =i∝angb∇acketleftk,a|∂\n∂Mα|k,a∝angb∇acket∇ight (15)\nAa\nkx(k) =i∝angb∇acketleftk,a|∂\n∂kx|k,a∝angb∇acket∇ight.7\nHere these Berry phases are manifestly real. Equa-\ntion (14) establishes the connection between T(2)and\nthe spin-momentum-space Berry phase.\nA few remarks are in order. First, through an explicit\nevaluation of the Berry phases, one can verify that the\nintegrand of Eq. (14) generates s±\nk(2)in Eq. (9) precisely.\nSecond, Eq. (14) contains the occupation function fa(k)\nitself rather than difference between the occupation func-tions or derivatives of the occupation function. Thus s(2)\nmay be classified as a Fermi sea contribution. We note,\nhowever, that the Fermi sea contribution Eq. (14) may\nbe converted to a different form50, where the net contri-\nbution is evaluated only at the Fermi surface. To demon-\nstrate this point, we integrate Eq. (14) by parts, which\ngenerates\n[s(2)]α=1\n2eE¯hA\nJ/summationdisplay\na/integraldisplayd2k\n(2π)2/bracketleftbigg\n−∂fa(k)\n∂kxAa\nMα(k)+∂fa(k)\n∂MαAa\nkx(k)/bracketrightbigg\n. (16)\nNotethatinthezerotemperaturelimit, both ∂fa(k)/∂kx\nand∂fa(k)/∂Mαare nonzero only at the Fermi surface,\nand thus the net contribution to s(2)depends only on\nproperties evaluated at the Fermi surface. In this sense,\nthis Fermi surface contribution is analogous to Friedel\noscillations. Friedel oscillations form near surfaces when\nelectrons reflect and the incoming and outgoing waves\ninterfere. Then, each electron below the Fermi energy\nmakes an oscillatory contribution to the density with a\nwavelength that depends on the energy. However, inte-\ngrating up from the bottom of the band to the Fermi\nenergy gives a result that only depends on the proper-\nties of the electrons at the Fermi energy where there is a\nsharp cut-off in the integration.\nIV. SUMMARY\nWe use simple models to examine the angular depen-\ndenceofspin-orbittorquesasafunctionoftheratioofthe\nspin-orbit interaction to the exchange interaction. We\nfind that both the field-like and damping-like torques\nare angle independent when the spin-orbit coupling is\nweak but become angle-dependent when the spin-orbit\ncoupling becomes comparable to the exchange coupling.\nWhen the spin-orbit coupling becomes much stronger\nthan the exchange coupling, the angular dependence of\nthe field-like torque goes away, but that of the damping-\nlike torque remains. The angulardependence of the field-\nlike torque becomes significant when the spin-orbit cou-\npling becomes strong enough to distort the Fermi surface\nso that it changes when the direction of the magnetiza-\ntion changes. On the other hand, the angular depen-\ndence of the damping-like torque is caused by the com-\nbined effects of the intrinsic Fermi sea contribution and\nthe Fermi surface distortion. We expect that these qual-\nitative conclusions will hold for more realistic treatments\nof the interface. The strong angular dependence of the\nspin-orbit torques will significantly impact their role in\nlargeamplitudemagnetizationdynamicslikeswitchingor\ndomain wall motion. This suggests caution when com-paring measurement of the strength of torques with the\nmagnetizations in different directions.\nAcknowledgments\nK.-J.L. acknowledges support from the NRF (2011-\n028163, NRF-2013R1A2A2A01013188) and under the\nCooperative Research Agreement between the Univer-\nsity of Maryland and the National Institute of Stan-\ndards and Technology Center for Nanoscale Science\nand Technology, Award 70NANB10H193, through the\nUniversity of Maryland. H.-W.L. was supported by\nNRF (2013R1A2A2A05006237) and MOTIE (Grant No.\n10044723). A.M. acknowledges support by the King Ab-\ndullah University of Science and Technology. D.G. ac-\nknowledges support from the Global Ph.D. Fellowship\nProgram funded by NRF (2014H1A2A101).\nAppendix A: Derivation of Eq. (9)\nHere, we derive the Fermi sea contribution of the spin-\norbit torque. First, we calculate the change of the eigen-\nstatesinthepresenceofanexternalelectricfield. Second,\nwe calculate the resulting spin accumulation. We use\ntime-dependent perturbation theory, adiabatically turn-\ning on the electric field, which is treated as the pertur-\nbation. Weadopttime-dependent perturbationapproach\ninsteadofthe Kuboformulaforpedagogicalreasonssince\nit directly shows how the states change due to the per-\nturbation. One can show that both approaches give the\nsame result.\nLet us consider an in-plane electric field E′(t) =\nEexp(δt) , where exp( δt) gives the adiabatic turning-on\nprocess. The electric field starts toincreasefrom t=−∞\nuntilt= 0, for very small δwhich will be set to be zero\nat the end of the calculation. This is represented by the\nvectorpotential A=−texp(δt)EsinceE=−∂A/∂t. In\nthe presence of a vector potential, the momentum opera-\ntorpis replaced by p+eA. Thus, the total Hamiltonian8\nbecomes\nH=(p+eA)2\n2m+αR\n¯hσ·[(p+eA)׈z]+Jσ·ˆM\n=H0+H1(t)+O(E2) (A1)\nwhere\nH1(t) =−e(E·p)\nmtexp(δt) (A2)\n+αR\n¯h[(eE×σ)·ˆz]texp(δt).\nHere, the first term comes from the kinetic energy and\nthe the second term from the Rashbaspin-orbit coupling.\nIn the interaction picture, the propagator of the order of\nO(E1) is\nU(I)\n1=−i\n¯h/integraldisplay0\n−∞dtH(I)\n1(t) (A3)\nwhere\nH(I)\n1(t) =eiH0t/¯hH1(t)e−iH0t/¯h\n=−e(E·p)\nmtexp(δt)\n+αR\n¯h[(eE×σ(I)(t))·ˆz]texp(δt),(A4)and\nσ(I)(t) =eiH0t/¯hσe−iH0t/¯h\n=σcos/parenleftbigg2ǫkt\n¯h/parenrightbigg\n+(ˆn×σ)sin/parenleftbigg2ǫkt\n¯h/parenrightbigg\n+ˆn(ˆn·σ)/bracketleftbigg\n1−cos/parenleftbigg2ǫkt\n¯h/parenrightbigg/bracketrightbigg\n.(A5)\nHere we define\nǫk=/vextendsingle/vextendsingle/vextendsingleJˆM+αRk׈z/vextendsingle/vextendsingle/vextendsingle, (A6)\nand\nˆn=1\nǫk/parenleftBig\nJˆM+αRk׈z/parenrightBig\n. (A7)\nThus,\nU(I)\n1(k) =U(I)\n1(a)(k)+U(I)\n1(b)(k), (A8)\nwhere\nU(I)\n1(a)(k) =−ie(E·k)\nm1\nδ2, (A9)\nand\nU(I)\n1(b)(k) =−i\n¯hαR\n¯h/braceleftbigg1\n2[(eE×σ)·ˆz]/parenleftbigg1\n(2ǫk/¯h+iδ)2+1\n(2ǫk/¯h−iδ)2/parenrightbigg\n+1\n2i[(eE×(ˆn×σ))·ˆz]/parenleftbigg1\n(2ǫk/¯h+iδ)2−1\n(2ǫk/¯h−iδ)2/parenrightbigg\n+[(eE׈n)·ˆz](ˆn·σ)/bracketleftbigg\n−1\nδ2−1\n2/parenleftbigg1\n(2ǫk/¯h+iδ)2+1\n(2ǫk/¯h−iδ)2/parenrightbigg/bracketrightbigg/bracerightbigg\n. (A10)\nThus, the change in the state due to the adiabatically\nturned on electric field is given by\nδ|ψk,±∝angb∇acket∇ight=U(I)\n1(k)|ψk,±∝angb∇acket∇ight. (A11)\nNow, the spin accumulation arising from the changes\nin the occupied states is\ns±\nk(2)=¯h\n2[(δ∝angb∇acketleftψk,±|)σ|ψk,±∝angb∇acket∇ight+∝angb∇acketleftψk,±|σ(δ|ψk,±∝angb∇acket∇ight)]δ→0\n=¯h\n2×2Re/bracketleftBig\n∝angb∇acketleftψk,±|σU(1)\n1(k)|ψk,±∝angb∇acket∇ight/bracketrightBig\nδ→0\n=±¯h\n2αReE/braceleftbiggJ\n2ǫ3\nk[ˆM×(ˆz׈E)]+αR\n2ǫ3\nk(ˆE×k)/bracerightbigg\n,\n(A12)where±indicates majority/minoritybands, respectively.\nWhen the electric field is applied along the ˆxdirection,\nwe arrive at Eq. (6). Note that U(I)\n1(a)does not contribute\nto the spin expectation value since it is purely imaginary.\n(∗∗) These authors equally contributed to this work.9\n∗Electronic address: hwl@postech.ac.kr\n†Electronic address: kj˙lee@korea.ac.kr\n1I.M. Miron, K.Garello, G.Gaudin, P.-J. Zermatten, M.V.\nCostache, S. Auffret, S. Bandiera, B. Rodmacq, A. Schuhl,\nand P. Gambadella, Nature (London) 476, 189 (2011).\n2I. M. Miron, T. Moore, H. Szambolics, L. D. Buda-\nPrejbeanu, S. Auffret, B. Rodmacq, S. Pizzini, J. Vogel,\nM. Bonfim, A. Schuhl, and G. Gaudin, Nat. Mater. 10,\n419 (2011).\n3L. Liu, C.-F. Pai, Y. Li, H. W. Tseng, D. C. Ralph and R.\nA. Buhrman, Science 4, 555 (2012).\n4L. Liu, C.-F. Pai, D. C. Ralph and R. A. Buhrman, Phys.\nRev. Lett. 109, 186602 (2012).\n5S.-M. Seo, K.-W. Kim, J. Ryu, H.-W. Lee, and K.-J. Lee,\nAppl. Phys. Lett. 101, 022405 (2012).\n6A. Thiaville, S. Rohart, ´E. Ju´ e, V. Cros, and A. Fert, Eu-\nrophys. Lett. 100, 57002 (2012).\n7P. P. J. Haazen, E. Mur´ e, J. H. Franken, R. Lavrijsen, H.\nJ. M. Swagten, and B. Koopmans, Nat. Mater. 12, 299\n(2013).\n8S. Emori, U. Bauer, S.-M. Ahn, E. Martinez, and G. S. D.\nBeach, Nat. Mater. 12, 611 (2013).\n9K.-S. Ryu, L. Thomas, S.-H. Yang, and S. S. P. Parkin,\nNat. Nanotech. 8, 527 (2013).\n10K.-S. Lee, S.-W. Lee, B.-C. Min, and K.-J. Lee, Appl.\nPhys. Lett. 102, 112410 (2013).\n11A. van den Brink, S. Cosemans, S. Cornelissen, M. Man-\nfrini, A. Vaysset, W. van Roy, T. Min, H. J. M. Swagten,\nand B. Koopmans, Appl. Phys. Lett. 102, 112410 (2013).\n12K. Garello, C. O. Avci, I. M. Miron, O. Boulle, S. Auffret,\nP. Gambadella, and G. Gaudin, arXiv:1310.5586.\n13K.-S. Lee, S.-W. Lee, B.-C. Min, and K.-J. Lee, Appl.\nPhys. Lett. 104, 072413 (2014).\n14Y. Yoshimura, T. Koyama, D. Chiba, Y. Nakatani,\nS. Fukami, M. Yamanouchi, H. Ohno, K.-J. Kim, T.\nMoriyama, and T. Ono, Appl. Phys. Express 7, 033005\n(2014).\n15M. I. D’yakonov and V. I. Perel, JETP Lett. 13, 467\n(1971).\n16J. E. Hirsch, Phys. Rev. Lett. 83, 1834 (1999).\n17S. F. Zhang, Phys. Rev. Lett. 85, 393 (2000).\n18K. Ando, S. Takahashi, K. Harii, K. Sasage, J. Ieda, S.\nMaekawa, and E. Saitoh, Phys. Rev. Lett. 101, 036601\n(2008).\n19L. Liu, T. Moriyama, D. C. Ralph and R. A. Buhrman,\nPhys. Rev. Lett. 106, 036601 (2011).\n20Yu. A. Bychkov and E. I. Rashba, JETP. Lett. 39, 78\n(1984).\n21V. M. Edelstein, Solid State Commun. 73, 233 (1990).\n22K. Obata, and G. Tatara, Phys Rev. B 77, 214429 (2008).\n23A. Manchon and S. Zhang, Phys. Rev. B 78, 212405\n(2008).\n24A. Matos-Abiague and R. L. Rodriguez-Suarez, Phys. Rev.\nB80, 094424 (2009).25X. Wang and A. Manchon, Phys. Rev. Lett. 108, 117201\n(2012).\n26K.-W. Kim, S.-M. Seo, J. Ryu, K.-J. Lee, and H.-W. Lee,\nPhys. Rev. B 85, 180404(R) (2012).\n27D. A. Pesin and A. H. MacDonald, Phys. Rev. B 86,\n014416 (2012).\n28E. van der Bijl and R. A. Duine, Phys. Rev. B 86, 094406\n(2012).\n29P. M. Haney, H.-W. Lee, K.-J. Lee, A. Manchon, and M.\nD. Stiles, Phys. Rev. B 87, 174411 (2013).\n30H. Kurebayashi, J. Sinova, D. Fang, A. C. Irvine, T. D.\nSkinner, J. Wunderlich, V. Nov´ ak, R. P. Campion, B. L.\nGallagher, E. K. Vehstedt, L. P. Zarobo, K. Vyborny, A. J.\nFerguson, and T. Jungwirth, Nat. Nanotech. 9, 211 (2014).\n31J. Kim, J. Sinha, M. Hayashi, M. Yamanouchi, S. Fukami,\nT. Suzuki, S. Mitani, and H. Ohno, Nat. Mater. 12, 240\n(2013).\n32X. Qiu, K. Narayanapillai, Y. Wu, P. Deorani, X.\nYin, A. Rusydi, K.-J. Lee, H.-W. Lee, and H. Yang,\narXiv:1311.3032.\n33X. Fan, H. Celik, J. Wu, C. Ni, K.-J. Lee, V. O. Lorenz,\nand J. Q. Xiao, Nat. Commun. 53042 (2014).\n34R. H. Liu, W. L. Lim, and S. Urazhdin, Phys. Rev. B 89,\n220409(R) (2014).\n35K. Garello, I. M. Miron, C. O. Avci, F. Freimuth, Y.\nMokrousov, S. Bl¨ ugel, S. Auffret, O. Boulle, G. Gaudin,\nand P. Gambardella, Nat. Nanotech. 8, 587 (2013).\n36X. Qiu, P. Deorani, K. Narayanapillai, K.-S. Lee, K.-J.\nLee, H.-W. Lee, and H. Yang, Sci. Rep. 4, 4491 (2014).\n37J. Sampaio, V. Cros, S. Rohart, A. Thiaville, and A. Fert,\nNat. Nanotech. 8, 839 (2013).\n38C. O. Pauyac, X. Wang, M. Chshiev, and A. Manchon,\nAppl. Phys. Lett. 102, 252403 (2013).\n39P. M. Haney, H.-W. Lee, K.-J. Lee, A. Manchon, and M.\nD. Stiles, Phys. Rev. B 88, 214417 (2013).\n40J. Sinova, D. Culcer, Q. Niu, N. A. Sinitsyn, T. Jungwirth,\nand A. H. MacDonald, Phys. Rev. Lett. 92, 126603 (2004).\n41J.-i. Inoue, G. E. W. Bauer, and L. W. Molenkamp, Phys.\nRev. B67, 033104 (2003).\n42J. Schliemann and D. Loss, Phys. Rev. B 69, 165315\n(2004).\n43A. A. Burkov, A. S. N´ u˜ nez and A. H. MacDonald, Phys.\nRev. B70, 155308 (2004).\n44E. G. Mishchenko, A. V. Shytov, and B. I. Halperin, Phys.\nRev. Lett. 93, 226602 (2004).\n45O. V. Dimitrova, Phys. Rev. B 71, 245327 (2005).\n46S. Murakami, Phys. Rev. B 69, 241202(R) (2004).\n47B. A. Bernevig and S.-C. Zhang, Phys. Rev. Lett. 95,\n016801 (2005).\n48A. V. Shytov, E. G. Mishchenko, H.-A. Engel, and B. I.\nHalperin, Phys. Rev. B 73, 075316 (2006).\n49A. Khaetskii, Phys. Rev. B 73, 115323 (2006).\n50F. D. M. Haldane, Phys. Rev. Lett. 93, 206602 (2004)"    },    {        "title": "1501.04022v3.Entanglement_Driven_Phase_Transitions_in_Spin_Orbital_Models.pdf",        "content": "arXiv:1501.04022v3  [cond-mat.str-el]  23 Jun 2015Entanglement driven phase transitions in\nspin-orbital models\nWen-Long You1,2, Andrzej M. Ole´ s1,3and Peter Horsch1\n1Max-Planck-Institut f¨ ur Festk¨ orperforschung,\nHeisenbergstrasse 1, D-70569 Stuttgart, Germany\n2College of Physics, Optoelectronics and Energy, Soochow Universit y,\nSuzhou, Jiangsu 215006, People’s Republic of China\n3Marian Smoluchowski Institute of Physics, Jagiellonian University,\nprof. S. /suppress Lojasiewicza 11, PL-30348 Krak´ ow, Poland\nE-mail:a.m.oles@fkf.mpg.de\nAbstract. To demonstrate the role played by the von Neumann entropy spect ra\nin quantum phase transitions we investigate the one-dimensional an isotropic\nSU(2)⊗XXZ spin-orbital model with negative exchange parameter. In the cas e\nof classical Ising orbital interactions we discover an unexpected n ovel phase\nwith Majumdar-Ghosh-like spin-singlet dimer correlations triggered by spin-orbital\nentanglement and having k=π/2 orbital correlations, while all the other phases are\ndisentangled. For anisotropic XXZ orbital interactions both spin-orbital entanglement\nand spin-dimer correlations extend to the antiferro-spin/alterna ting-orbital phase.\nThis quantum phase provides a unique example of two coupled order p arameters which\nchange the character of the phase transition from first-order t o continuous. Hereby we\nhave established the von Neumann entropy spectral function as a valuable tool to\nidentify the change of ground state degeneracies and of the spin- orbital entanglement\nof elementary excitations in quantum phase transitions.\nPACS numbers: 75.25.Dk, 03.67.Mn, 05.30.Rt, 75.10.Jm\nSubmitted to: New J. Phys.Entanglement driven phase transitions in spin-orbital mod els 2\n1. Spin-orbital physics and von Neumann entropy spectra\nIn the Mott-insulating limit of a transition metal oxide the low-energy physics can be\ndescribed by Kugel-Khomskii-type models [1], where both spin and orb ital degrees of\nfreedom undergo joint quantum fluctuations and novel types of s pin-orbital order [2]\nor disorder [3] may emerge. Following the microscopic derivation from the multiorbital\nHubbard model, the generic structure of spin-orbital superexch ange takes the form of a\ngeneralized Heisenberg model [4,5],\nH=/summationdisplay\n/angbracketleftij/angbracketright/bardblγ/braceleftBig\nJ(γ)\nij(/vectorTi,/vectorTj)/vectorSi·/vectorSj+K(γ)\nij(/vectorTi,/vectorTj)/bracerightBig\n, (1)\nas indeed found not only for the simplest systems with S= 1/2 spins: KCuF 3[1], the\nRTiO3perovskites [6], LiNiO 2and NaNiO 2[7], Sr 2CuO3[8], or alkali RO 2hyperoxides\n[9], but also for larger spins as e.g. for S= 2 in LaMnO 3[10]. In such models\nthe parameters that determine the spin- SHeisenberg interactions stem from orbital\noperatorsJ(γ)\nijandK(γ)\nij— they depend on the bond direction and are controlled by\nthe orbital degree of freedom which is described by pseudospin ope rators{/vectorTi}. That\nis, these parameters are not necessarily fixed by rigid orbital orde r [3,11], but quantum\nfluctuations of orbital occupation [12,13] may strongly influence t he form of the orbital\noperators, particularly in states with spin-orbital entanglement ( SOE) [14,15]. As a\nconsequence, amplitudes and even the signs of the effective excha nge can fluctuate in\ntime. Such entangled spin-orbital degrees of freedom can form ne w states of matter,\nas for instance the orbital-Peierls state observed at finite temper ature in YVO 3[16,17].\nAnother example are the collective spin and orbital excitations in a on e-dimensional\n(1D) spin-orbital chain under a crystal field which can be universally described by\nfractionalized fermions [18]. It is challenging to ask which measure of S OE would be\nthe most appropriate one to investigate quantum phase transition s in such systems.\nThe subject is rather general and it has become clear that entang lement and other\nconcepts from quantum information provide a useful perspective for the understanding\nof electronic matter [19–23]. Other examples of entangled system s are: topologically\nnontrivial states [24], relativistic Mott insulators with 5 dions [25], ultracold alkaline-\nearth atoms [26], and skyrmion lattices in the chiral metal MnSi [27].\nOne well-known characterization of a quantum system is the entang lement\nentropy (EE) determined by bipartitioning a system into AandBsubsystems. This\nsubdivision can refer for example to space [19,28], momentum [28,29 ], or different\ndegrees of freedom such as spin and orbital [30]. A standard measu re is the von\nNeumann entropy (vNE), S0\nvN≡ −TrA{ρ0\nAlog2ρ0\nA}, for the ground state |Ψ0∝an}bracketri}htwhich\nis obtained by integrating the density matrix, ρ0\nA= TrB|Ψ0∝an}bracketri}ht∝an}bracketle{tΨ0|, over subsystem B.\nAnother important measure is the entanglement spectrum (ES) int roduced by Li and\nHaldane [31], which has been explored for gapped 1D spin systems [32 ], quantum\nHeisenbergladders[33], topologicalinsulators[34], bilayersandspin- orbitalsystems[28].\nThe ES is a property of the ground state and basically represents the eigenvalues piof\nthe reduced density matrix ρ0\nAobtained by bipartitioning of the system. InterestinglyEntanglement driven phase transitions in spin-orbital mod els 3\na correspondence of the ES and the tower of excitations relevant for SU(2) symmetry\nbreaking has been pointed out recently [35]. It was also noted that t he ES can exhibit\nsingular changes, although the system remains in the same phase [36 ]. This suggests\nthat the ES has less universal character than initially assumed [37].\nIn this paper we explore a different entanglement measure, namely t he vNE\nspectrum which monitors the vNE of ground and excited states of the system, for\ninstance of a spin-orbital system as defined in equation (1). In this case we consider\nthe entanglement obtained from the bipartitioning into spin and orbit al degrees of\nfreedom in the entire system [30]. Here the vNE is obtained from the d ensity matrix,\nρ(n)\ns= Tro|Ψn∝an}bracketri}ht∝an}bracketle{tΨn|, by taking the trace over the orbital degrees of freedom (Tr o) for\neach eigenstate |Ψn∝an}bracketri}ht. We show below that the vNE spectrum,\nSvN(ω) =−/summationdisplay\nnTrs{ρ(n)\nslog2ρ(n)\ns}δ{ω−ωn}, (2)\nreflects the changes of SOE entropy for the different states at p hase transitions. The\nexcitation energies, ωn=En−E0, of eigenstates |Ψn∝an}bracketri}htare measured with respect to\nthe ground state energy E0. It has already been shown that the vNE spectra uncover\na surprisingly large variation of entanglement within elementary excit ations [30]. Also\ncertain spectral functions have been proposed, that can be det ermined by resonant\ninelastic x-ray scattering [38], and provide a measure of the vNE sp ectral function.\nHere we generalize this function to arbitrary excitations |Ψn∝an}bracketri}ht, i.e.,beyondelementary\nexcitations which refer to a particular ground state. We demonstr ate that focusing on\ngeneral excited states opens up a new perspective that sheds ligh t on quantum phase\ntransitions and the entanglement in spin-orbital systems.\nThe paper is organized as follows: In section 2 we introduce the 1D sp in-orbital\nmodel with ferromagnetic exchange, and in section 3 we present its phase diagrams\nfor the Ising limit of orbital interactions and for the anisotropic SU( 2)⊗XXZmodel\nwith enhanced Ising component. SOE is analyzed in section 4 using bot h the spin-\norbital correlation function and the entanglement entropy and we show that these two\nmeasures are equivalent. In section 5 we present the entanglemen t spectra and discuss\ntheir relation to the quantum phase transitions. The main conclusion s and summary are\ngiven in section 6. The distance dependence ofspin correlations in th eantiferromagnetic\nphase is explored in the Appendix.\n2. Ferromagnetic SU(2) ⊗XXZspin-orbital model\nThe motivation for our theoretical discussion of spin-orbital phys ics comes from t2g\nelectron systems in which orbital quantum fluctuations are enhanc ed by an intrinsic\nreduction of the dimensionality of the electronic structure [12]. Exa mples of strongly\nentangled quasi-1D t2gspin-orbital systems due to dimensional reduction arewell known\nand we mention here just LaTiO 3[6], LaVO 3and YVO 3[13], where the latter two\ninvolve{yz,zx}orbitals along the ccubic axis; as well as pxandpyorbital systems in\n1D fermionic optical lattices [39–41]. This motivates us to consider th e 1D spin-orbitalEntanglement driven phase transitions in spin-orbital mod els 4\nmodel forS= 1/2 spins and T= 1/2 orbitals with anisotropic XXZinteraction, i.e.,\nwith reduced quantum fluctuation part in orbital interactions. The |+∝an}bracketri}htand|−∝an}bracketri}htorbital\nstates are a local basis at each site and play a role of yzandzxstates int2gsystems,\nH(x,y,∆) =−JL/summationdisplay\nj=1/parenleftBig\n/vectorSj·/vectorSj+1+x/parenrightBig/parenleftBig\n[/vectorTj·/vectorTj+1]∆+y/parenrightBig\n, (3)\n[/vectorTj·/vectorTj+1]∆≡∆/parenleftbig\nTx\njTx\nj+1+Ty\njTy\nj+1/parenrightbig\n+Tz\njTz\nj+1, (4)\nwhereJ >0 andweuse periodicboundaryconditions fora ringof Lsites, i.e.,L+1≡1.\nThe parameters of this model are {x,y}and ∆. At x=y= 1/4 and ∆ = 1 the model\nhas SU(4) symmetry. Hund’s exchange coupling does not only modify xandybut also\nleads to the XXZanisotropy (∆ <1), a typical feature of the orbital sector in real\nmaterials [5,12]. The antiferromagnetic model ( J=−1) is Bethe-Ansatz integrable at\nthe SU(4) symmetric point [42] and its phase diagram is well establish ed by numerical\nstudies [43,44]. It includes two phases with dimer correlations [45] wh ich arise near the\nSU(4) point. Some of its ground states could be even determined ex actly at selected\n(x,y,∆) points [46–50].\nHere we are interested in the complementary and less explored mode l with negative\n(ferromagnetic) coupling ( J= 1), possibly realized in multi-well optical lattices [51],\nwhich has been studied so far only for SU(2) orbital interaction (∆ = 1) [30]. This\nmodel is physically distinct from the antiferromagnetic ( J=−1) model, except for the\nIsing limit (∆ = 0) where the two models can be mapped onto each other , but none\nwas investigated so far. The phase diagrams for J= 1, see figure 1, determined using\nthe fidelity susceptibility [52] display a simple rule that the vNE (2) vanis hes for exact\nground states of rings of length Lwhich can be written as products of spin ( |ψs∝an}bracketri}ht) and\norbital (|ψo∝an}bracketri}ht) part,|Ψ0∝an}bracketri}ht=|ψs∝an}bracketri}ht⊗|ψo∝an}bracketri}ht.\n3. Phase transitions in the spin-orbital model\nTo understand the role played by the SOE in the 1D spin-orbital mode l (3) and (4) we\nconsider the phase diagrams for ∆ = 0 and ∆ = 0 .5, see figure 1. In the case ∆ = 0\nall trivial combinations of ferro (F) and antiferro (A) spin-orbital phases labeled I-IV\nhaveS0\nvN= 0, i.e., spins and orbitals disentangle in all these ground states: FS/ FO,\nAS/FO, AS/AO, FS/AO. If both subsystems exhibit quantum fluctu ations, the ground\nstate|Ψ0∝an}bracketri}htcan no longer be written in the product form. This occurs for the AS /AO\nphase III at ∆ >0.\nMost remarkable is the strongly entangled phase V at ∆ = 0 and y <0, see figure\n1(a). This phase occurs near x≃ −∝an}bracketle{t/vectorSj·/vectorSj+1∝an}bracketri}htAF≡ln2−1/4, i.e., when the uniform\nantiferromagnetic spin correlations in phase III are compensated by the parameter x, so\nthat the energy associated with Hamiltonian (3) de facto disappears . This triggers state\nV with strong SOE (see below) as the only option for the system to ga in substantial\nenergy in this parameter range by nonuniform spin-orbital correla tions. The analysis ofEntanglement driven phase transitions in spin-orbital mod els 5\n/s45/s49 /s48 /s49/s45/s49/s48/s49/s40/s97/s41\n/s120/s121/s73/s73/s86\n/s73/s73/s73/s73/s73/s86\n/s45/s49 /s48 /s49/s45/s49/s48/s49/s40/s98/s41\n/s86\n/s120/s121/s73/s86 /s73\n/s73/s73/s73/s73/s73/s86/s73\nFigure 1. Phase diagrams of the spin-orbital model [equation (3)] obtained b y two\nmethods, fidelity susceptibility or an exact diagonalization of an L= 8 site model, for:\n(a) ∆ = 0, and (b) ∆ = 0 .5. The spin-orbital correlations in phases I-IV correspond to\nFS/FO, AS/FO, AS/AO, FS/AO order (see text). At ∆ = 0 only the gr ound state of\na novel phase V has finite EE, S0\nvN>0 (shaded), whereas at ∆ >0 the EE in phases\nIII and VI is also finite.\nphase V in terms of the longitudinal equal-time spin (orbital) structu re factors\nOzz(k) =1\nL2L/summationdisplay\nm,n=1e−ik(m−n)∝an}bracketle{tOz\nmOz\nn∝an}bracketri}ht, (5)\nwhereO=SorT, reveals in figure 2(a) at ∆ = 0 and y=−1/4 for the spin structure\nfactorSzz(k)∝(1−cosk). This is a manifestation of nearest neighbour correlations,\nwhile further neighbour spin correlations vanish and moreover we fin d a quadrupling in\ntheorbitalsector, seefigure2(b). Thusthespincorrelationsind icateeitherashort-range\nspin liquid or a translational invariant dimer state.\nThehidden spin-dimer order [53] can be detected by the four-spin correlator (we\nuse periodic boundary conditions),\nD(r) =1\nLL/summationdisplay\nj=1/bracketleftbigg/angbracketleftBig\n(/vectorSj·/vectorSj+1)(/vectorSj+r·/vectorSj+1+r)/angbracketrightBig\n−/angbracketleftBig\n/vectorSj·/vectorSj+1/angbracketrightBig2/bracketrightbigg\n. (6)\nAt ∆ = 0 we find |D(r)|with long-range dimer correlations in phase V, but not in III.\nPhase III is a state with alternating ( k=π) spin (orbital) correlations in the range\nx <0.17 shown in figures 2(a,b). Interestingly for ∆ >0 the dimer spin correlations\n|D(r)|are not only present in phase V but also appear in phase III. Moreov er a phase\nVI emerges, complementary to phase V, with interchanged role of s pins and orbitals, see\nfigure 1(b). The order parameters for phase VI follow from the fo rm of structure factors\nwhich develop similar but complementary momentum dependence to th at for phase V\nseen in figure 2(a), i.e., maxima at π/2 forSzz(k) and atπforTzz(k). We remark that\nphases V and VI are unexpected and they were overlooked before for the SU(2) ⊗SU(2)\nmodel at ∆ = 1 [30]. From the size dependence of |D(r)|in figures 2(c,d) we concludeEntanglement driven phase transitions in spin-orbital mod els 6\n/s48 /s49 /s50/s48/s46/s48/s48/s46/s53/s49/s46/s48\n/s49 /s50/s48/s49/s50/s40/s97/s41\n/s32/s32/s83/s122/s122\n/s40/s107/s41\n/s107/s47/s32 /s120/s60/s48/s46/s49/s55\n/s32 /s32/s120 /s48/s46/s49/s55\n/s32/s40/s49/s45/s99/s111/s115/s32/s107/s41/s47/s50/s40/s98/s41/s84/s122/s122\n/s40/s107/s41/s32/s32\n/s107/s47/s32 /s32/s120/s60/s48/s46/s49/s55\n/s32 /s32/s120 /s48/s46/s49/s55\n/s48/s46/s49 /s48/s46/s51 /s48/s46/s53/s48/s46/s48/s48/s46/s49\n/s48/s46/s49 /s48/s46/s51 /s48/s46/s53/s40/s100/s41/s40/s99/s41\n/s120/s62 /s48/s46/s55/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s53\n/s48/s46/s51/s32/s32/s32/s32/s32 /s48/s46/s52\n/s32/s32/s32/s32/s32\n/s32/s32/s32/s32/s32/s32/s124/s68/s40/s114/s41/s124\n/s49/s47/s114/s32/s120/s61 /s48/s46/s54\n/s32/s32/s32/s32/s32\n/s32\n/s32/s32\nFigure 2. Top— Spin [ Szz(k)] and orbital [ Tzz(k)] structure factors (5) for the 1D\nspin-orbital model (3) of L= 8 sites at ∆ = 0 and y=−1/4: (a)Szz(k) and (b)\nTzz(k). Bottom— Spin dimer correlations D(r) equation (6) found at ∆ = 0 .5 for\ndecreasing 1 /rfor clusters of (c) L= 12 and (d) L= 16 sites.\nthat the dimer correlations are long-ranged at ∆ = 0 .5 in phase V, but also in III, as\nseen from the data for x∈[0.0,0.4), where they coexist with the AS correlations.\nThese results suggest thatthegroundstateVinfigure1(a)isfor medby spin-singlet\nproduct states\n|ΦD\n1∝an}bracketri}ht= [1,2][3,4][5,6]···[L−1,L],\n|ΦD\n2∝an}bracketri}ht= [2,3][4,5][6,7]···[L,1], (7)\nwhere [l,l+1] = (|↑↓∝an}bracketri}ht−|↓↑∝an}bracketri}ht )/√\n2 denotes a spin singlet. They arenot coupled to orbital\nsinglets on alternating bonds as it happens for the AFantiferromag netic SU(2) ⊗SU(2)\nspin-orbital chain in a different parameter regime [46], but to Ising co nfigurations in theEntanglement driven phase transitions in spin-orbital mod els 7\norbital sector. The four-fold ( k=π/2) periodicity of orbital correlations is consistent\nwith four orbital states:\n|Ψz\n1∝an}bracketri}ht=|++−−++···−−∝an}bracketri}ht,\n|Ψz\n2∝an}bracketri}ht=|−++−−+···+−∝an}bracketri}ht,\n|Ψz\n3∝an}bracketri}ht=|−−++−−···++∝an}bracketri}ht,\n|Ψz\n4∝an}bracketri}ht=|+−−++−···−+∝an}bracketri}ht. (8)\nThe decoupling of singlets is complete for y=−1/4 and ∆ = 0, where (++) and ( −−)\nbonds yield vanishing coupling in equation (3), and the phase boundar ies of region V\narexc\nIII,V= 3/4 + 2∝an}bracketle{t/vectorSj·/vectorSj+1∝an}bracketri}htAF≃0.136 andxc\nV,II= 3/4 in the thermodynamic limit;\nmoreover we find perfect long-range order of spin singlets, i.e., D(r) = (3/8)2(−1)r.\nThe dimerized spin-singlet state at ∆ = 0 has the same spin structure as the\nMajumdar-Ghosh (MG) state [54], however its origin is different. While the MG state\nin aJ1-J2Heisenberg chain results from frustration of antiferromagnetic e xchange (at\nJ2=J1/2), here the spin singlets are induced by the SOE. At ∆ = 0 the only pha se with\nfinite SOE S0\nvN= 1 is phase V, see figure 3. In contrast, for ∆ >0 one finds finite EE\nalso in phase III, when the original product ground state changes into a more complex\nsuperposition ofstatesandjointspin-orbital fluctuations[14]a ppear. These correlations\ncontrol the SOE and give equivalent information to S0\nvN, see section 4. Furthermore, EE\nincreases with xtowards phase V where it is further amplified andexceeds S0\nvN= 1. The\nrelated softening of orbital order will be discussed below. Interes tingly we find a one-\nto-one correspondence of finite EE and long-range order in the sp in dimer correlations\n/s45/s49/s46/s48 /s45/s48/s46/s53 /s48/s46/s48 /s48/s46/s53 /s49/s46/s48/s48/s49/s50/s32\n/s32/s83\n/s118/s78\n/s120/s32\n/s32\n/s32\n/s32\nFigure 3. Spin-orbital entanglement entropy S0\nvNin the ground state of the spin-\norbital model (3) for the three phases III, V and II as a function ofxfor various\n∆. Solid line for ∆ = 0 stands for the k= 0 ground state in the limit of ∆ →0.\nParameters: y=−0.5 andL= 8.Entanglement driven phase transitions in spin-orbital mod els 8\n|D(r)|.\nThe superstructure of phase V emerges from the interplay of spin and orbitals,\nwhere orbitals modulate the interaction of spins in equation (3), and vice versa . It is\nimportant to distinguish this from the Peierls effect, where the coup ling to the lattice\nis an essential mechanism. The orbital Peierls effect observed in van adates [16,17] or\nthe orbital-selective Peierls transition studied recently [55] fall into the former category,\nyet, as they involve orbital singlets — they are distinct from the cas e discussed here.\n4. Spin-orbital entanglement\nThedescriptionofspin-orbitalentanglement intermsofthevNEen tropy, asdiscussed in\nsection 3, is a very convenient measure of entanglement. But it is als o a highly abstract\nmeasure. To capture its meaning, one has to refer to mathematica l intuition, namely to\nthe fact that any product state, |Ψ∝an}bracketri}ht=|ψs∝an}bracketri}ht⊗|ψo∝an}bracketri}ht, has zero vNE. That is, an entangled\nstate is a state that cannot be written as a single product. A more p hysical measure\nare obviously spin-orbital correlation functions relative to their me an-field value [14].\nSuch correlation functions vanish for product states where mean -field factorization of\nthe relevant product is exact, i.e., spins and orbitals are disentangle d.\nTo detect spin-orbital entanglement in the ground state we evalua te here the joint\nspin-orbital bond correlation function C1for the SU(2) ⊗XXZmodel (3), defined as\nfollows for a nearest neighbour bond ∝an}bracketle{ti,i+1∝an}bracketri}htin the ring of length L[14],\nC1≡1\nLL/summationdisplay\ni=1/braceleftBig/angbracketleftBig\n(/vectorSi·/vectorSi+1)(/vectorTi·/vectorTi+1)/angbracketrightBig\n−/angbracketleftBig\n/vectorSi·/vectorSi+1/angbracketrightBig/angbracketleftBig\n/vectorTi·/vectorTi+1/angbracketrightBig/bracerightBig\n.(9)\nThe conventional intersite spin- and orbital correlation functions are:\nSr≡1\nLL/summationdisplay\ni=1/angbracketleftBig\n/vectorSi·/vectorSi+r/angbracketrightBig\n, (10)\nTr≡1\nLL/summationdisplay\ni=1/angbracketleftBig\n/vectorTi·/vectorTi+r/angbracketrightBig\n. (11)\nThe above general expressions imply averaging over the exact (tr anslational invariant)\nground state found from Lanczos diagonalization of a ring. While SrandTrcorrelations\nindicate the tendency towards particular spin and orbital order, C1quantifies the spin-\norbital entanglement — if C1∝ne}ationslash= 0 spin and orbital degrees of freedom are entangled\nand the mean-field decoupling in equation (3) cannot be applied as it ge nerates\nuncontrollable errors.\nFigures 4(a) and 4(b) show the nearest neighbour correlation fun ctionsS1,T1and\nC1aty=−0.5, for ∆ = 0 and ∆ = 0 .5, respectively, as functions of x. The nearest\nneighbour spin correlation function S1is antiferromagnetic (negative) in all phases III,\nV and II shown in figure 4, while ( negative)T1indicates AO correlations in phase III\nandferro-orbital ( positive) inphase II. Finite ∆ = 0 .5 triggers orbital fluctuations whichEntanglement driven phase transitions in spin-orbital mod els 9\n/s45/s48/s46/s52/s48/s46/s48/s48/s46/s52\n/s45/s49/s46/s48 /s45/s48/s46/s53 /s48/s46/s48 /s48/s46/s53 /s49/s46/s48/s45/s48/s46/s52/s48/s46/s48/s48/s46/s52/s40/s97/s41\n/s84\n/s122/s61/s48/s84\n/s122/s61/s54/s73/s73/s73 /s86/s73/s73\n/s32/s32/s73/s73/s86\n/s84\n/s122/s61/s54\n/s120/s32/s67\n/s49\n/s32/s84\n/s49\n/s32/s83\n/s49/s84\n/s122/s61/s48/s73/s73/s73 /s40/s98/s41\nFigure 4. Nearest neighbour spin S1(10), orbital T1(11), and joint spin-orbital\nC1(9) correlations as obtained for a spin-orbital ring (3) with L= 12 sites and\ny=−0.5, as functions of xfor: (a) ∆ = 0, and (b) ∆ = 0 .5. The III-V phase\nboundary (dotted vertical line) in (b) has been determined by the m aximum of the\nfidelity susceptibility [52].\nlowerT1below the classical value of 0.25 found at ∆ = 0. In the intermediate sp in dimer\nphaseT1is negative for all ∆ >0, while it is zero for ∆ = 0.\nIt is surprising that C1is positive in phase V at ∆ = 0 in spite of the classical Ising\norbital interactions, see figure 4(a). It is also positive in phases II I and V at ∆ = 0 .5 [see\nfigure 4(b)]. Note that positiveC1is found in the present spin-orbital chain with J >0,\nwhileC1isnegative whenJ <0 [42]. In phase II C1vanishes in the entire parameter\nrange as then the ground state can be written as a product. The s ame is true for phase\nIII at ∆ = 0. We emphasize that the dependence of C1onxis completely analogous to\nthat of the von Neumann entropy in figure 3, which also displays a bro ad maximum in\nthe vicinity of the III-Vphase transition at ∆ = 0 .5, and a step-like structure in phase VEntanglement driven phase transitions in spin-orbital mod els 10\nat ∆ = 0. Thus we conclude here that the vNE yields a faithful measur e of SOE in the\nground state that is qualitatively equivalent to the more direct entanglement measure\nvia the spin-orbital correlation function C1[14].\n5. Entanglement spectra and quantum phase transitions\nFigure 3 stimulates the question about the origin and the understan ding of the sudden\nor gradual EE changes at phase transitions. This can be resolved b y exploring the\nvNE spectral function defined in equation (2) and shown in figures 5 (a) and 5(b)\nfor ∆ = 0 and 0.5, where colors encode the vNE of states. The excita tion energies\nωn(x) =En(x)−E0(x) are plotted here as function of the parameter x. Only the lowest\nexcitationsareshownthatarerelevantforthephasetransitions andthelow-temperature\nphysics. They include: ( i) the elementary excitations of the respective ground state,\nand (ii) the many-body excited states that are relevant for the phase t ransition(s) and\nmay become ground states or elementary excitations in neighbourin g phases when the\nparameterxis varied.\nThe AS/FO ground state of phase II in figure 5(a) obtained for a rin g ofL= 8\nsites is an AS singlet ( S= 0) with a maximal orbital quantum number, T=L/2 = 4,\nand a twofold ( k= 0,π) degeneracy at ∆ = 0. The spin excitation spectrum appears\nas horizontal (red) lines and consists of gapless triplet S= 1 excitations. The low-\nlying excitations of the Bethe-Ansatz-solvable antiferromagnetic Heisenberg chain form\na two-spinon ( s-¯s) continuum, whose lower bound is given by ε(k) =π|sink|/2 in the\nthermodynamic limit [56]. For the L= 8 ring the spectrum is discrete with a ∆ k=π/4\nspacing, and it is known that the energy of triplet excitations εS(π) will scale to zero\nas 1/L[57–59]. Red lines in phase II with finite slope are orbital excitations. T he\nx-dependence is due to the spin part of H(3) which determines both the orbital energy\nscale and the dispersion, JT≡(x+∝an}bracketle{t/vectorSj·/vectorSj+1∝an}bracketri}htAF)(1−∆cosk). This energy changes with\nxand at finite ∆ also with momentum k, see figure 5(b). While the orbitons are gapped,\nthe low-lying excitations are either magnons or x-dependent spin-orbital excitations. It\nis remarkable that the latter are entangled in general, although the ground state II is\ndisentangled.\nWithdecreasing xa first-orderphase transitionfromIItoVoccursby level crossin g\nof disentangled (red) and entangled (green) ground states. The spin-singlet ( S= 0)\nground state of phase V has degeneracy 4 at ∆ = 0, and its compone nts are labeled\nby the momenta k= 0,±π/2,π. This is reflected by finite ϕTorder parameter in\nfigure 6(a). Note that at ∆ >0 this four-fold ground state degeneracy is lifted. In the\nspin-dimer phase a gap opens in the spectrum of elementary spin exc itations [60,61].\nThe one-magnon triplet gap ∆ S(δ)∝δ3/4depends on yvia the dimerization parameter\nδ≡1/|4y|. In phase V even the pure magnetic excitations are entangled [see h orizontal\ngreen lines in figure 5(a)]. The lowest excitations in the vicinity of the p hase transitions\nhave orbital character. From finite EE in figure 5(a) one recognize s that these states\nare inseparable spin-orbital excitations.Entanglement driven phase transitions in spin-orbital mod els 11\n/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52\n/s83/s79/s83/s79\n/s107/s61/s48/s47/s107/s61/s48/s47\n/s32/s32\n/s122\n/s40 /s41/s32/s115\n/s40 /s41\n/s115\n/s40 /s41\n/s115\n/s40 /s41/s69\n/s110/s45/s69\n/s48\n/s120/s73/s73/s73/s86/s73/s73/s115\n/s40 /s41\n/s107/s61/s48/s47/s83/s61/s48/s32/s84/s61/s48/s40/s97/s41\n/s48/s46/s48/s48/s48/s46/s50/s53/s48/s48/s46/s53/s48/s48/s48/s46/s55/s53/s48/s49/s46/s48/s48/s49/s46/s50/s53/s49/s46/s53/s48/s49/s46/s55/s53/s50/s46/s48/s48\n/s83/s61/s48/s32/s84/s61/s48\n/s83/s61/s48/s32/s84/s61/s52\n/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s48/s50/s52/s122\n/s40 /s41\n/s83/s79\n/s40 /s41/s32/s84\n/s40 /s41\n/s83/s61/s48/s32/s32/s32 /s32/s32/s84\n/s40 /s41\n/s83/s61/s48/s32/s84/s61/s52/s32 /s107/s61/s84\n/s40 /s41 /s32/s32\n/s122\n/s40 /s41\n/s122\n/s40 /s41/s115\n/s40 /s41/s69\n/s110/s45/s69\n/s48\n/s120/s73/s73/s73 /s86 /s73/s73/s48/s46/s48/s48/s48/s46/s51/s50/s48/s48/s46/s54/s52/s48/s48/s46/s57/s54/s48/s49/s46/s50/s56/s49/s46/s54/s48/s49/s46/s57/s50/s50/s46/s50/s52/s50/s46/s53/s54/s50/s46/s56/s56\n/s83/s61/s48/s32/s107/s61/s48/s40/s98/s41\nFigure 5. vNE-spectrum of lowest energies En(x) (relative to the ground state energy\nE0(x)) versusxwith colors representing the size of the vNE of individual states. Da ta\nfor the three phases III, V and II is shown for y=−0.5,L= 8 and: (a) ∆ = 0, and\n(b) ∆ = 0.5. HereεS(k) [εT(k)] denotes spin (orbital) excitation, εz(k) corresponds\nto an elementary excitation having the same SandTas the ground state, and εSO(k)\nstands for the spin excitation under simultaneous flipping of orbitals .\nThe phase transition from the dimer phase V to the AS phase III ( S= 0) appears\nsingular in the sense that it is first order at ∆ = 0 and continuous othe rwise [figures\n5(a,b)]. Tolocatethecenter ofthecontinuous phasetransitionbe tween phases IIIandVEntanglement driven phase transitions in spin-orbital mod els 12\n/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48\n/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s45/s49/s46/s53/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48/s69\n/s48/s45/s69/s86 /s48/s48/s46/s48/s48/s48/s46/s50/s53/s48/s46/s53/s48\n/s32/s67/s83\n/s49\n/s32/s84\n/s32/s111/s114/s100/s101/s114/s32/s112/s97/s114/s97/s109/s101/s116/s101/s114/s40/s97/s41\n/s32\n/s120/s48/s46/s48/s48/s48/s46/s50/s53/s48/s46/s53/s48/s40/s98/s41\n/s32/s67/s83\n/s49\n/s32/s32/s84\n/s32/s32\nFigure 6. Ground state energy relative to phase V, E0(x)−EV\n0(x) (dots), orbital order\nparameters (dashed), ψT= [Tzz(π)]1\n2,ϕT= [Tzz(π/2)]1\n2, and bond spin correlations\n|S1|=|∝an}bracketle{t/vectorS1·/vectorS2∝an}bracketri}ht|(solid line), for phases III, V and II (from left to right), for: (a) ∆ = 0.0\nand (b) ∆ = 0 .5. Parameters: y=−0.5 andL= 12.\nat ∆>0, we have selected the peak of the first derivative of the entangle ment entropy,\nsee figure 3. Yet also the peaks in the derivatives of the fidelity susc eptibility, the orbital\ncorrelation function T1[see figure 4(b)] and the orbital order parameters ψTandϕTin\nfigure 6(b) may be used. Finally, we note that the scaling of entangle ment with system\nsize has quite different behaviour in phases III and V, indicating that a phase transition\nseparates them.\nFurthermore, the peculiar feature of the AS/AO phase III manife sts itself in a\ntwofold degeneracy and zero SOE at ∆ = 0 in contrast to the nondeg enerate ground\nstate and finite SOE at finite ∆. The entanglement has two sources, namely: (i) the\ninterplayofquantumfluctuationsinthespinandorbitalsectorsan d(ii)thedimerization\norder which coexists with antiferromagnetic spin correlations in pha se III at finite ∆.\nThe latter is the origin of the nondegenerate ground state as it yield s a coupling to\ntheεSO(π) excitation (nearly horizontal in x), and the emergence of the spin-dimer\ncorrelations D(r) leads to a faster decay of the spin correlations in phase III than in the\n1D antiferromagnetic Heisenberg chain, see the Appendix. The orb ital order parameters\nψTandϕTcompete in phases III and V, see figure 6(b), near the phase boun dary inEntanglement driven phase transitions in spin-orbital mod els 13\nfigure 1(b). This also explains why the transition from phase V to III is smooth at finite\n∆ in terms of both the vNE (figure 3) and the nearest neighbour spin correlations |S1|.\n6. Conclusions and summary\nSummarizing, we have studied the quantum phases and the spin-orb ital entanglement of\nthe 1D ferromagnetic SU(2) ⊗XXZmodel by means of the Lanczos method. We have\ndiscovered a previously unknown translational invariant phase V wit h long-range spin\nsinglet order and four-fold periodicity in the orbital sector. Its me chanism is distinct\nfrom the dimer phases found in the 1D antiferromagnetic spin-orbit al model near the\nSU(4) symmetric point [45]. Both III-V and II-V phase transitions arise from the\nspin-orbital entanglement in the case of Ising orbital interactions . When the orbital\ninteractions change from Ising to anisotropic XXZ-type, the entanglement develops\nin phase III, where antiferromagnetic spin correlations and long-r ange spin dimer\norder coexist, changing the quantum phase transition from first- order to continuous.\nFurthermore in the regime of finite orbital fluctuations (∆ >0) another phase VI\nemerges, which is complementary in many aspects to phase V, but wit h the important\ndifference that phase VI disappears in the limit ∆ = 0.\nWe have shown that the von Neumann entropy spectral function SvN(ω) (2) is\na valuable tool that captures the spin-orbital entanglement SOE o f excitations and\nexplains the origin of the entanglement entropy change at a phase t ransition. From the\nperspective of spin-orbital entanglement we encounter ( i) first-order transitions between\ndisentangled (II) and entangled (V) phases, ( ii) a continuous transition involving two\ncompeting order parameters between two entangled phases, III and V, and ( iii) trivial\nfirst-order transitions between two disentangled phases. Case ( ii) goes beyond the\ncommonly accepted paradigm of a single order parameter to charac terize a quantum\nphase.\nMoreover, we have presented two simple measures of entanglemen t in the ground\nstateandshownthattheyarebasicallyequivalent —thedirectmeas ureviathe(quartic)\nspin-orbital bond correlation function C1(9) and the von Neumann entropy S0\nvN. The\nlatterisdefinedbyseparating globallyspinfromorbitaldegreesoff reedomintheground\nstate.\nAcknowledgments\nWe thank Bruce Normand and Krzysztof Wohlfeld for insightful disc ussions. W-L You\nacknowledges support by the Natural Science Foundation of Jiang su Province of China\nunder Grant No. BK20141190 and the NSFC under Grant No. 11474 211. A M Ole´ s\nkindly acknowledges support by Narodowe Centrum Nauki (NCN, Na tional Science\nCenter) under Project No. 2012/04/A/ST3/00331.Entanglement driven phase transitions in spin-orbital mod els 14\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48/s48/s50/s52/s54/s56\n/s48/s46/s53 /s49/s46/s48/s124/s60/s83\n/s48/s83\n/s114/s62/s124\n/s49/s47/s114/s32/s120/s61/s45/s49/s46/s48\n/s32/s120/s61/s45/s48/s46/s53\n/s32/s120/s61/s48/s46/s48\n/s32/s120/s61/s48/s46/s49\n/s32/s120/s61/s48/s46/s50\n/s32/s120/s61/s48/s46/s51\n/s32/s120/s61/s48/s46/s52\n/s32/s120/s61/s48/s46/s53\n/s32/s120/s61/s48/s46/s54\n/s32/s120/s40/s97/s41/s40/s98/s41\n/s49/s47/s114\nFigure 7. Modulus of spin correlations Srequation (10) versus the inverse distance\n1/ras obtained for a spin-orbital ring (3) with L= 16 sites, for: (a) ∆ = 0, and (b)\n∆ = 0.5. Parameter: y=−0.5.\nAppendix: Distance dependence of the antiferromagnetic sp in correlations\nHere we explore in more detail the competition of the antiferromagn etic (AF) spin\ncorrelations of the spin-orbital chain in the AS/AO phase III and th e Majumdar-Ghosh\nlike spin-singlet dimer correlations that coexist at finite ∆, as we foun d in our work.\nFor ∆ = 0 the spin correlations in phase III are those of an AF Heisenb erg spin chain,\n/angbracketleftBig\n/vectorSi·/vectorSi+r/angbracketrightBig\n∼(−1)r/radicalbig\nln|r|\n|r|, (12)\nwhichreveal thetypical1 /r-powerlawdecaycombined withlogarithmiccorrectionsthat\nwere first predicted by conformal field theory [62,63] as well as by renormalization group\nmethods [64], and subsequently confirmed [65] by numerical density matrix method [66].\nIn figure 7(a) we present our numerical data for the spin-correla tion function Sr\nequation (10) (i.e., for translational invariant ground states) for several values of x,\nand for ∆ = 0 and y=−0.5. In the ∆ = 0 case there are only two distinct types of\nbehaviour of Sr, namely exponential decay in phase V and the power law decay of the\n1D quantum N´ eel spin liquid state, which are the same in phases II an d III.\nFigure 7(b) displays Srat ∆ = 0 .5 for different x-values. Here again the\nunperturbed AF correlations of the 1D N´ eel spin-liquid state appe ar in phase II\n(x≥0.7). It is evident that in phase III the AF spin correlations are stron gly reduced,\ndue to the competition with the coexisting long-range ordered spin- singlet correlations.\nThe spin singlet order increases with xin phase III, and as a consequence we observe\nhere that the decay of Srbecomes stronger as xapproaches the III/V transition.\nIn figure 8 we present a logarithmic plot which highlights the different d ecays ofSrEntanglement driven phase transitions in spin-orbital mod els 15\n/s48 /s50 /s52 /s54 /s56/s45/s52/s45/s50/s48/s50/s108/s110/s40/s124/s60/s83\n/s48/s83\n/s114/s62/s124/s41\n/s114/s32/s120/s61/s48/s46/s55\n/s32/s108/s105/s110/s101/s97/s114/s32/s102/s105/s116\n/s32/s120/s61/s48/s46/s51\n/s32/s108/s105/s110/s101/s97/s114/s32/s102/s105/s116\n/s32/s120/s61/s48/s46/s54\n/s32/s108/s105/s110/s101/s97/s114/s32/s102/s105/s116\nFigure 8. Logarithm of modulus of spin correlations Srequation (10) for increasing\ndistanceras obtained for the spin-orbital model equation (3) on a ring of L= 16 sites\nfor ∆ = 0.5,y=−0.5, and three values of x. Exponential decay of Srwith increasing\nris obtained for phase V ( x= 0.6).\nfor ∆ = 0.5 in the three different phases: III, V, and II. We have selected th e values for\nx= 0.3, 0.6 and 0.7, respectively, for greater transparency. The log-plot shows c learly\nthe exponential decay of Srin phase V. It also shows that the L= 16 system reveals\nstrong finite size effects in phase II where Srhas power law decay. Nevertheless it is\nclear already from the L= 16 data that the AF spin correlations in phase III (here\nshown forx= 0.3) are strongly suppressed and approach the exponential decay ofSrin\nphase V (x= 0.5 and 0.6) when xapproaches the III-V phase boundary from the left.\nSummarizing, we find that in phase III the AF spin correlations of the 1D N´ eel\nspin liquid state decay much more rapidly as the competing spin-singlet order emerges.\nThis effect is particularly strong near the boundary of phase III to the spin-singlet\ndimer phase V. Whether in the thermodynamic limit the correlations Sralso decay\nexponentially in phase III as in V cannot be decided here, and this que stion is beyond\nthe scope of the present work.\nReferences\n[1] Kugel K I and Khomskii D I 1973 JETP37725\nKugel K I and Khomskii D I 1982 Sov. Phys. Usp. 25231\n[2] Brzezicki W, Ole´ s A M and Cuoco M 2015 Phys. Rev. X 5011037\n[3] Corboz P, Lajk´ o M, L¨ auchli A M, Penc K and Mila F 2012 Phys. Rev. X 2041013\n[4] Tokura Y and Nagaosa N 2000 Science288462\n[5] Ole´ s A M, Khaliullin G, Horsch P and Feiner L F 2005 Phys. Rev. B 72214431\n[6] Khaliullin G and Maekawa S 2000 Phys. Rev. Lett. 853950Entanglement driven phase transitions in spin-orbital mod els 16\nMochizuki M and Imada M 2004 New J. Phys. 6154\nPavarini E, Yamasaki A, Nuss J and Andersen O K 2005 New J. Phys. 7188\n[7] Reitsma A, Feiner L F and Ole´ s A M 2005 New J. Phys. 7121\n[8] Wohlfeld K, Nishimoto S, Haverkort M W and van den Brink J 2013 Phys. Rev. B 88195138\n[9] Solovyev I V 2008 New J. Phys. 10013035\nWohlfeld K, Daghofer M and Ole´ s A M 2011 Europhys. Lett. 9627001\n[10] Feiner L F and Ole´ s A M 1999 Phys. Rev. B 593295\n[11] Zhou J-S, Ren Y, Yan J-Q, Mitchell J F and Goodenough J B 2008 Phys. Rev. Lett. 100046401\n[12] Khaliullin G 2005 Prog. Theor. Phys. Suppl. 160155\n[13] Horsch P, Ole´ s A M, Feiner L F and Khaliullin G 2008 Phys. Rev. Lett. 100167205\n[14] Ole´ s A M, Horsch P, Feiner L F and Khaliullin G 2006 Phys. Rev. Lett. 96147205\n[15] Ole´ s A M 2012 J. Phys.: Condens. Matter 24313201\n[16] Ulrich U, Khaliullin G, Sirker J, Reehuis M, Ohl M, Miyasaka S, Tokura Y and Keimer B 2003\nPhys. Rev. Lett. 91257202\nHorsch P, Khaliullin and Ole´ s A M 2003 Phys. Rev. Lett. 91257203\n[17] Sirker J, Herzog A, Ole´ s A M and Horsch P 2008 Phys. Rev. Lett. 101157204\nHerzog A, Horsch P, Ole´ s A M and Sirker J 2011 Phys. Rev. B 83245130\n[18] Chen C-C, van Veenendaal M, Devereaux T P and Wohlfeld K 2015 Phys. Rev. B 91165102\n[19] Amico L, Fazio R, Osterloh A and Vedral V 2008 Rev. Mod. Phys. 80517\n[20] Zhou H-Q, Barthel T, Fjaerestad J O and Schollw¨ ock U 2006 Phys. Rev. A 74050305(R)\n[21] Byczuk K, Kunes J, Hofstetter W and Vollhardt D 2012 Phys. Rev. Lett. 108087004\n[22] Held K and Mauser N J 2013 Eur. Phys. J. B 86328\n[23] Veenstra C N, Zhu Z-H, Raichle M, Ludbrook B M, Nicolaou A, Sloms ki B, Landolt G, Kittaka S,\nMaeno Y, Dil J H, Elfimov I S, Haverkort M W and Damascelli A 2014 Phys. Rev. Lett. 112\n127002\n[24] Hasan M Z and Kane C L 2010 Rev. Mod. Phys. 823045\n[25] Jackeli G and Khaliullin G 2009 Phys. Rev. Lett. 102017205\n[26] Gorshkov A V, Hermele M, Gurarie V, Xu C, Julienne P S, Ye J, Zoller P, Demler E, Lukin M D\nand Rey A M 2010 Nature Phys. 6289\n[27] M¨ uhlbauer S, Binz B, Jonietz F, Pfleiderer C, Rosch A, Neubaue r A, Georgii R and B¨ oni P 2009\nScience323915\n[28] Lundgren R, Chua V and Fiete G A 2012 Phys. Rev. B 86224422\n[29] Thomale R, Arovas D P and Bernevig B A 2010 Phys. Rev. Lett. 105116805\n[30] You W-L, Ole´ s A M and Horsch P 2012 Phys. Rev. B 86094412\n[31] Li H and Haldane F D M 2008 Phys. Rev. Lett. 101010504\n[32] Pollmann F, Turner A M, Berg E and Oshikawa M 2010 Phys. Rev. B 81064439\nAlba V, Haque M and L¨ auchli A M 2012 Phys. Rev. Lett. 108227201\n[33] Poilblanc D 2010 Phys. Rev. Lett. 105077202\n[34] Fidkowski L 2010 Phys. Rev. Lett. 104130502\nRegnault N and Bernevig B A 2011 Phys. Rev. X 1021014\n[35] Kolley F, Depenbrock S, McCulloch I P, Schollw¨ ock U and Alba V 201 3Phys. Rev. B 88144426\n[36] Lundgren R, Blair J, Greiter M, L¨ auchli A, Fiete G A and Thomale R 2014Phys. Rev. Lett. 113\n256404\n[37] Chandran A, Khemani V and Sondhi S L 2014 Phys. Rev. Lett. 113060501\n[38] Ament L J P, van Veenendaal M, Devereaux T P, Hill J P and van de n Brink J 2011 Rev. Mod.\nPhys.83705\n[39] Zhao E and Liu W V 2008 Phys. Rev. Lett. 100160403\nWu C 2008 Phys. Rev. Lett. 100200406\nWu C and Das Sarma 2008 Phys. Rev. B 77235107\n[40] Sun G, Jackeli G, Santos L and Vekua T 2012 Phys. Rev. B 86155159\n[41] Zhou Z, Zhao E and Liu W V 2015 Phys. Rev. Lett. 114100406Entanglement driven phase transitions in spin-orbital mod els 17\n[42] Li Y-Q, Ma M, Shi D-N and Zhang F-C 1998 Phys. Rev. Lett. 813527\nFrischmuth B, Mila F and Troyer M 1999 Phys. Rev. Lett. 82835\n[43] Itoi C, Qin S and Affleck I 2000 Phys. Rev. B 616747\nOrignac E, Citro R and Andrei N 2000 Phys. Rev. B 61, 11533\n[44] Yamashita Y, Shibata N and Ueda K 2000 J. Phys. Soc. Jpn. 69242\n[45] Li Peng and Shen Shun-Qing 2005 Phys. Rev. B 72214439\n[46] Kolezhuk A K and Mikeska H-J 1998 Phys. Rev. Lett. 802709\n[47] Kolezhuk A K, Mikeska H-J and Schollw¨ ock U 2001 Phys. Rev. B 63064418\n[48] Martins M J and Nienhuis B 2000 Phys. Rev. Lett. 854956\n[49] Kumar B 2013 Phys. Rev. B 87195105\n[50] Brzezicki W, Dziarmaga J and Ole´ s A M 2014 Phys. Rev. Lett. 112117204\n[51] Belemuk A M, Chtchelkatchev N M and Mikheyenkov A V 2014 Phys. Rev. A 90023625\n[52] You W-L, Li Y-W, and Gu S-J 2007 Phys. Rev. E 76022101\n[53] Yu Y, M¨ uller G and Viswanath V S 1996 Phys. Rev. B 549242\n[54] Majumdar C K and Ghosh D 1969 J. Math. Phys. 101388\n[55] Streltsov S V and Khomskii D I 2014 Phys. Rev. B 89161112(R)\n[56] des Cloizeaux J and Pearson J J 1962 Phys. Rev. 1282131\n[57] Horsch P and von der Linden W 1988 Z. Phys. B 72181\n[58] Affleck I 1993 Rev. Math. Phys. 6887\n[59] Koma T and Tasaki H 1994 J. Stat. Phys. 76745\n[60] Spronken G, Fourcade B and L´ epine Y 1986 Phys. Rev. B 331886\n[61] Uhrig G S and Schulz H J 1996 Phys. Rev. B 54R9624\n[62] Affleck I, Gepner D, Schulz H and Ziman T 1989 J. Phys. A: Math. and Theor. 22511\n[63] Lukyanov S 1998 Nuclear Physics B 522533\n[64] Giamarchi T and Schulz H J 1989 Phys. Rev. B 394620\nSingh R R, Fisher M E and Shankar R 1989 Phys. Rev. B 392562\n[65] Hallberg K A, Horsch P and Mart´ ınez G 1995 Phys. Rev. B 52R719\n[66] Schollw¨ ock U 2005 Rev. Mod. Phys. 77259"    },    {        "title": "1610.02927v2.The_Role_of_Interaction_in_the_Pairing_of_Two_Spin_orbit_Coupled_Fermions.pdf",        "content": "Role of interaction in the binding of two Spin-orbit Coupled Fermions\nChong Ye,1, 2Jie Liu,2, 3Li-Bin Fu1,\u0003\n1Graduate School, China Academy of Engineering Physics, Beijing 100193, China\n2National Laboratory of Science and Technology on Computational Physics,\nInstitute of Applied Physics and Computational Mathematics, Beijing 100088, China\n3HEDPS, CAPT, and CICIFSA MoE, Peking University, Beijing 100871, China\nWe investigate role of an attractive s-wave interaction with positive scattering length in the\nbinding of two spin-orbit coupled fermions in the vacuum and on the top of a Fermi sea in the\nsingle impurity system, motivated by current interests in exploring exotic binding properties in the\nappearance of spin-orbit couplings. For weak spin-orbit couplings where the density of states is\nnot signi\fcantly altered, we analytically show that the high-energy states become more important\nin determining the binding energy when the scattering length decreases. Consequently, tuning the\ninteraction gives rise to a rich behavior, including a zigzag of the momentum of the bound state\nor inducing transitions among the meta-stable states. By exactly solving the two-body quantum\nmechanics for a spin-orbit coupled Fermi mixture of40K-40K-6Li, we demonstrate that our analysis\ncan also apply to the case when the density of states is signi\fcantly modi\fed by the spin-orbit\ncoupling. Our \fndings pave a way for understanding and controlling the binding of fermions in the\npresence of spin orbit couplings.\nPACS numbers: 03.65.Ge, 71.70.Ej, 67.85.Lm\nI. INTRODUCTION\nIn ultracold physics, many schemes have been proposed\nto generated various types of synthetic spin-orbit cou-\nplings (SOC) by controlling atom-light interaction [1].\nIn 2011, I. B. Spielman's group in NIST had gener-\nated an equal weight combination of Rashba-type and\nDresselhaus-type SOC in87Rb [2]. Afterwards, SOC has\ntriggered a great amount of experimental interest [3{5].\nIn the appearance of SOC, the ultracold atomic gases\nhave been altered dramatically [6{8].\nOne basic issue is the binding of two spin-orbit cou-\npled fermions in the vacuum [9{16] where SOC has given\nrise to the change of binding energy and the appearance\nof \fnite-momentum dimer bound states. Another rele-\nvant issue is the binding of two fermions on the top of\na Fermi sea (the molecular state) for the case where a\nsingle impurity is immersed in a noninteracting Fermi\ngas [16{21]. In the appearance of SOC, the center-of-\nmass (c.m.) momentum of the molecular state becomes\n\fnite [16, 21]. All of these can be understood from the\nperspective of two-body quantum mechanics. It general\ncontains three components: the threshold energy associ-\nated with the c.m. momentum, the density of states, and\nthe interacting strength. For extremely weak attractive\ninteraction, changes of two-body properties under SOC\ncame from the di\u000berent threshold behavior of the density\nof states [9, 10, 14]. However, in the strong interacting\nregime, the binding of two fermions presents a rich behav-\nior [13, 16] such as the variation of the c.m. momentum\nand the competition between two meta-stable states with\nthe tuning of interacting strength. These phenomena can\n\u0003lbfu@gscaep.ac.cnnot simply owe to the threshold behavior of the density\nof states. Therefore, the mechanism as to how all these\nthree components cooperate with each other in deter-\nmining the novel two-body properties is pressing needed.\nThe establishment of such a comprehensive picture will\nshed light on ongoing explorations of the intriguing be-\nhavior of spin-orbit coupled Fermi gases [9{16]. Below,\nwe report a theoretical contribution to address this issue,\nwhich also allows predictions of new phenomena.\nWe investigate the two-body quantum mechanisms of\nthe binding of two spin-orbit coupled fermions in the vac-\nuum and on the top of a Fermi sea in the single impurity\nFermi gas. We consider an attractive s-wave interaction\nwith positive scattering length, the strength of which can\nbe tuned in a wide range via a Feshbach resonance [22].\nFrom Sec. II to Sec. IV, we give analyses which do\nnot dependent on the concrete type of SOC. In Sec. II,\nby decomposing the two-body energy (molecular energy)\ninto the threshold energy and the binding energy, both\nof which depend on the c.m. momentum of two fermions,\nwe establish a direct relation between the interaction, the\ndensity of states, and the binding energy. In Sec. III,\nwith the \frst-order perturbation analysis in the weak\nSOC limit, we reveals that the low-energy states play\na decisive role in determining the binding energy when\nthe scattering length is large, in contrast to the small\nscattering length case where the high-energy states can\ndominate. This allows us to elucidate the mechanism\nunderlying interesting phenomena such as a zigzag be-\nhavior of the two-body ground state momentum and the\ncompetition between two meta-stable states in Sec. IV.\nIn Sec. V, we illustrate our analysis with an interact-\ning Fermi mixture of40K-40K-6Li with40K containing\nan (\u000bkx\u001bz+h\u001bx)-type SOC, which can be realized by\nthe state of the art experimental techniques using cold\natoms [4, 23]. Remarkably, by exactly solving the two-arXiv:1610.02927v2  [cond-mat.quant-gas]  24 Feb 20182\nbody problem for this system, we show that our analysis\na\u000bords insights into the main properties of the binding of\ntwo spin-orbit coupled fermions, even when the density\nof states is signi\fcantly altered by SOC. Our \fndings re-\nveal the role of interaction in the binding of two spin-orbit\ncoupled fermions and allow deep physical understandings\nof the rich two-body properties in the presence of SOC.\nII. BINDING OF TWO FERMIONS WITH SOC\nWe consider two di\u000berent spin-orbit coupled fermionic\nspecies in three dimensions (3D) at zero temperature:\natom A (B) has Na(Nb) components, with the corre-\nsponding non-interacting Hamiltonian Ha(Hb). We con-\nsider an attractive s-wave contact interaction with posi-\ntive scattering length between the two fermionic species\nas described by\nHint=U\nVX\nQ;k;k0ay\nk;l0by\nQ\u0000k;m0ak0;l0bQ\u0000k0;m0;(1)\nwithQthe c.m. momentum of two scattering fermions.\nHereay\nk;l0(by\nk;m0) denotes the creation operator of a\nSOC-free atom A (B) in the l0-th (m0-th) spin compo-\nnent with momentum k,Uis the bare interaction, and\nVis the quantization volume. The total Hamiltonian is\nthusH=Ha+Hb+Hint.\nWith this Hamiltonian, we address to the binding of\ntwo fermionic atoms A and B (i) in the vacuum [9{16] and\n(ii) on the top of a non-interacting Fermi sea of atoms\nAin the situation where a single impurity of Bis im-\nmersed in a non-interacting Fermi gas of A[17{21]. The\nansatz wave function of the two-body bound state and\nthe molecular state can be expressed in a general form\nj\tQi=X\ni;j0X\nk i;j\nQ;k\u000by\nk;i\fy\nQ\u0000k;jj?i; (2)\nwhere Qis the c.m. momentum of two particles. For (i),\nj?iis the vacuum state and the summationP0\nkincludes\nall the states. For (ii), j?iis the non-interacting spin-\norbit coupled Fermi sea of Aand the summationP0\nk\nexcludes the states below the Fermi surfaces, re\recting\nthe e\u000bect of Pauli blocking. Here, \u000by\nk;i=P\ni0\u0015i;i0\nkay\nk;i0\n(\fy\nk;i=P\ni0\u0011i;i0\nkby\nk;i0) is the creation operator of an atom\nA (B) in the i-th eigen-state of Hamiltonian Ha(Hb) with\nmomentum kand energy \"a\nk;i(\"b\nk;i) and i;j\nQ;kdenotes\nthe variational coe\u000ecient. The coe\u000ecients \u0015i;i0\nkand\u0011i;i0\nk\nare \fxed by SOC. Solving the eigen-equation Hj\tQi=\nEQj\tQigives\n i;j\nQ;k=(\u0015i;l0\nk\u0011j;m0\nQ\u0000k)\u0003\nEQ\u0000Eij\nQ;kU\nV0X\nk0;i0;j0 i0;j0\nQ;k0\u0015i0;l0\nk0\u0011j0;m0\nQ\u0000k0;(3)\nwithEij\nQ;k=\"a\nk;i+\"b\nQ\u0000k;j. Rearranging Eq. (3), we\nobtain a self-consistent equation for two-body energy(molecular energy) EQin the momentum-space repre-\nsentation, i.e.,\n1\nU=1\nVX\ni;jX\nk0j\u0015i;l0\nkj2j\u0011j;m0\nQ\u0000kj2\nEQ\u0000Eij\nQ;k: (4)\nA key step of our treatment next constitutes a decom-\nposition ofEQ: De\fning the threshold energy associated\nwith the c.m. momentum QbyEQ\nth\u0011mini;j;kfEij\nQ;kg,\nwe writeEij\nQ=EQ\nth+\". The rest of the two-body energy\n(molecular energy) is therefore Esc\nQ\u0011EQ\u0000EQ\nth. While\nEQ\nthis only a\u000bected by SOC, Esc\nQencodes the e\u000bect of\ninteraction. Such decomposition of EQin terms of Esc\nQ\nandEQ\nth, as we shall see, allows a transparent correspon-\ndence to the SOC-free counterpart. Following a standard\nprocedure, we obtain the self-consistent equation for Esc\nQ\nin the energy domain of \"as in Ref.[24]\nZ1\n0\r\"\nQd\"\nEsc\nQ\u0000\"=1\nU: (5)\nHere,\r\"\nQis de\fned by\n\r\"\nQ=X\niX\njZ0\nj\u0015i;l0\nkj2j\u0011j;m0\nQ\u0000kj2jJjd\u0017d\u0016; (6)\nwhich describes the density of states in 3D[25]. For (i),\nthe integrationR0d\u0017d\u0016 includes all the states. For (ii),\nthe integrationR0d\u0017d\u0016 excludes the states below the\nFermi surfaces. In Eq. (6), \u0016and\u0017label the degrees\nof freedom other than \", andJdenotes the standard Ja-\ncobian. These formulas can be also easily adapted to de-\nscribing the binding of two homo-nuclear fermions where\nAandBare the same fermionic species.\nEquation (5) establishes a direct relation between the\ninteraction U, the density of states \r\"\nQ, andEsc\nQ. Intu-\nition behind it can be gained in the limit of vanishing\nSOC in case (i), where EQ\nth=Q2=(2m\u0016) withm\u0016the\nreduced mass of two fermions, and \r\"\nQ=\r\"\n0= 2p2m\u0016\".\nThen,Esc\nQis independent of Qas ensured by Eq. (5),\nand can be identi\fed as Esc\nQ\u0011\"b=\u00001=(2m\u0016a2\ns) [~\u00111]\nwithas>0 the s-wave scattering length, i.e., the binding\nenergy at rest. In this case, Eq. (5) reduces to, in the\nmomentum space representation, the well known renor-\nmalization equation for two scattering particles, i.e.,\n1\nU=m\u0016\n2\u0019as\u00001\nVX\nk2m\u0016\nk2: (7)\nEquation (5) thus extends the standard prescription for\ntwo interacting fermions to the presence of SOC, where\nEsc\nQis the counterpart of the binding energy \"b.\nIII. ROLE OF INTERACTION\nBased on above treatment, below we elucidate how the\ninteraction cooperates with the e\u000bect of SOC in deter-3\nmining the behavior of Esc\nQ, when the interaction strength\na\u00001\nsis tuned in a wide range via Feshbach resonance [22].\nTo compare to the SOC-free case, we introduce the quan-\ntity\u0018Q\u0011Esc\nQ\u0000\"b. For weak SOC that does not signi\f-\ncantly alter the density of states, the leading term of \u0018Q\ncan be derived from Eq. (5) as [26]:\n\u0018Q=\u0000hZ1\n0\r\"\nQ\n(\"\u0000\"b)2d\"i\u00001Z1\n0\r\"\nQ\u0000\r\"\n0\n\"\u0000\"bd\": (8)\nHere we have ignored the modi\fcation of the renormal-\nization relation by SOC [27{31]. In discussing the e\u000bect\nof interaction on \u0018Q, we will be interested in (i)@\u0018Q\n@a\u00001\ns\nand (ii) \u0001 QQ0\u0011\u0018Q0\u0000\u0018Q: The sign of the former re\rects\nhow\u0018Qfor \fxed Qchanges with interaction, while that\nof the latter tells whether a large or small Qis energet-\nically favored for a given interaction. Using Eq. (8), we\n\fnd \u0001 QQ0'\u0000[R1\n0\r\"\nQ\n(\"\u0000\"b)2d\"]\u00001R1\n0\r\"\nQ0\u0000\r\"\nQ\n\"\u0000\"bd\". Both of\n\u0018Qand \u0001 QQ0rely crucially on \r\"\nQ. Thus, while the form\nof\r\"\nQvaries with speci\fc setups [see Eq. (6)], its qualita-\ntive analysis a\u000bords insights into generic behavior of \u0018Q,\nas we elaborate next. In order to give some analyses, we\napply the further approximation\n\u0018Q'\u0000hZ1\n0\r\"\n0\n(\"\u0000\"b)2d\"i\u00001Z1\n0\r\"\nQ\u0000\r\"\n0\n\"\u0000\"bd\"\n/p\u0000\"bZ1\n0\r\"\nQ\u0000\r\"\n0\n\"\u0000\"bd\": (9)\nConsider \frst the simplest case where \r\"\nQ\u0000\r\"\n0>0 for\nall energy levels \"[32], i.e., SOC induces an increase in\nthe number of available scattering states at all energies.\nFrom Eq. (9), we see \u0018Q<0, hence binding with \fnite Q\nleads to an energy decrease as compared to the SOC-free\ncase, irrespective of the interacting strength. Such energy\ndrop, following from@\u0018Q\n@a\u00001\ns>0, can be further enhanced\nby increasing a\u00001\ns. If, moreover, \r\"\nQincreases monotoni-\ncally with Q, we have \u0001 QQ0<0, i.e.,\u0018Qdecreases with\nincreasing Qfor \fxed scattering length. The amplitude\nof this decrease can be controlled by tuning the scattering\nlength, which enhances with increased a\u00001\ns.\nIn contrast, if the e\u000bect of SOC is such that \r\"\nQ\u0000\n\r\"\n0alters sign depending on the energy \"of the state,\n\u0018Qcan exhibit a very rich behavior. To demonstrate it,\nconsider\r\"\nQ\u0000\r\"\n0has opposite sign in the low- and high-\nenergy regimes, with a sign \rip occurring at the energy\n\"0. Applying the mean value theorem to Eq. (9), we \fnd\nZ1\n0\r\"\nQ\u0000\r\"\n0\n\"\u0000\"bd\"=fl=(\"1\u0000\"b) +fh=(\"2\u0000\"b);(10)\nwith\"12(0;\"0), and\"22(\"0;1). Herefl=R\"0\n0(\r\"\nQ\u0000\n\r\"\n0)d\"andfh=R1\n\"0(\r\"\nQ\u0000\r\"\n0)d\"are the number of scat-\ntering states in the low- and high-energy regimes, respec-\ntively. Since flandfhhave opposite signs, the contribu-\ntion from the high-energy states to \u0018Qis suppressed by\nthe smaller pre-factor compared to the low-energy states.Yet, such suppression becomes less signi\fcant when a\u00001\ns\nincreases, following similar reasoning as before. We thus\nexpect the sign of \u0018Qto be mainly determined by the\nlow-energy states for large as, whereas the high-energy\nstates can become decisive for small as. This has in-\nteresting physical implications: by tuning the scattering\nlength and hence the sign of \u0018Qand \u0001 QQ0, we can con-\ntrol whether a bound pair favors nonzero Q, and even\nthe speci\fc choice of Q.\nIV. TYPICAL BEHAVIORS OF TWO-BODY\nGROUND STATES\nWe now show that, combining EQ\nth, above insights into\nthe cooperative e\u000bects of interaction and SOC on Esc\nQ\nallows predictions on generic features of the dispersion\nEQ. This can be best illustrated in two following cases.\n(i) IfEQ\nthhas only one minimum, without interaction,\nthe two-body (molecular) ground state c.m. momentum\nQgwill locate at Q1whereEQ\nthis minimized. By con-\ntrast, adding interaction can strongly modify Esc\nQand\nthusEQ, according to previous analysis, which renders\nQgto deviate from Q1. Such deviation intimately de-\npends on the behavior of Esc\nQ: IfEsc\nQvaries monotonically\nwithQfor a \fxed scattering length, Qgshifts from Q1\nin such a way that a smaller Esc\nQcan be reached. Such\nshift can be further enhanced by increasing a\u00001\ns, provided\nit does not qualitatively alter the behavior of Esc\nQ, i.e,.\nEsc\nQstays increasing (or decreasing) with Qwhen varying\na\u00001\ns[c.f. inset of Fig. 1(d)]. If, instead, the behavior of\nEsc\nQundergoes a qualitative change when a\u00001\nsincreases,\ne.g. from increasing to decreasing with Q[see inset of\nFig. 2(c)], Qgwill \frst exhibit a zigzag away from Q1\nbefore increasing above Q1monotonically [see inset of\nFig. 2(d)].\n(ii) In general EQ\nthcan have multiple local minima,\neach corresponding to a meta-stable state. For individ-\nual meta-stable state, the associated c.m. momentum ex-\nhibits similar behavior as in (i). An interesting question\nthen concerns how two-body (molecular) ground state\ntransits among multiple meta-stable states when the in-\nteraction is tuned. To address it, suppose for simplicity\nthatEQ\nthhas two degenerate local minima at Q1and\nQ2respectively, and Esc\nQvaries monotonically with Q\nfor a \fxed scattering length. The two-body (molecular)\nground state c.m. momentum Qis expected to be close to\nQ1orQ2, depending on which corresponds to a smaller\nEsc\nQ. If the behavior of Esc\nQcan be changed qualitatively\nby tuninga\u00001\ns, say from increase to decrease with Q,\na transition of the system between the two meta-stable\nstates can be induced. This phenomenon also occurs\nwhen the two local minima EQ\nthbecome non-degenerate,\ndue to the competition between EQ\nthandEsc\nQ, which is the\norigin of the transition discussed in Ref. [16]. In addition,\nwith the increasing of a\u00001\ns,Esc\nQwill dominate over EQ\nthin4\n� ���\n(c)EEth,+Q\nEth,-Q- - 1(a)-\nQ\n(b)0kas(           )-1\n(d) - - - - - 1\n- - - - - - - - - - - - - - - -\n1\n0kas(           )-1\n0kas(           )-1\nFigure 1. Binding of spin-orbit coupled fermions in the vacuum. (a) The distribution of \r\"\nQ\u0000\r\"\n0. (b)\u0018Qas a function of Q\nwith di\u000berent ( k0as)\u00001according to Eq.(8). (c) The helicity-dependent threshold energy EQ\nth;+(EQ\nth;\u0000) is the minimum energy\nof two particles with A in the upper (lower) helicity branch and a c.m. momentum Q. (d) The two-body energy with di\u000berent\ninteracting strengths by exactly solving Eq. (4). The inset shows the variation of the ground state c.m. momentum. Here\nQ0=\u00001:5k0ex.\ndetermining the dispersion of EQ. This may qualitative\nchange the dispersion of two-body (molecular) energy,\nsay from a double-well type with two meta-stable states\nto a single-well type with one meta-stable state, which\nmay cause the disappear of the transition.\nV. SPIN-ORBIT COUPLED\nTHREE-COMPONENT FERMI MIXTURE\nPrevious discussions from Sec. II to Sec. IV are not\ndependent on the concrete type of the SOC. To give an\nexample, below we present concrete calculations by solv-\ning Eq. (4) for a system of interacting Fermi mixture\nof40K-40K-6Li (A-A-B), where the atom40K couples to\nSOC and the atom6Li is spinless. Here, we choose an\n(\u000bkx\u001bz+h\u001bx)-type SOC which can be readily realized\nexperimentally in40K [4]. In this three-component mix-\nture, the6Li fermions are tuned close to a wide Feshbach\nresonance with spin up species of40K [23]. The Hamil-tonian for the system reads\nH=X\nk;\u001b\"a\nkay\nk;\u001bak;\u001b+X\nk(hay\nk;\"ak;#+hay\nk;#ak;\")\n+X\nk\"b\nkby\nkbk+U\nVX\nk;k0;qay\nq\n2+k;\"by\nq\n2\u0000kbq\n2\u0000k0aq\n2+k0;\"\n+X\nk(\u000bkxay\nk;\"ak;\"\u0000\u000bkxay\nk;#ak;#): (11)\nHereak;\u001b(\u001b=\";#) denotes the annihilation operator of\na SOC-free particle Awith spin\u001band momenta k, while\nthe operator bkannihilates a particle Bwith momenta k.\nIn addition, \"a(b)\nk=k2=(2ma(b)) is the kinetic energy of\nparticleA(B). The SOC parameters hand\u000bare respec-\ntively proportional to the Raman coupling strength and\nthe momentum transfer in the Raman process generating\nthe SOC [4]. We also note that via a global pseudo-spin\nrotation such SOC can be transformed to an equal weight\ncombination of Rashba-type and Dresselhaus-type SOC\n(\u000bkx\u001by+h\u001bz) which is the \frst SOC generated in ultra-\ncold atomic gases [2]. Therefore, the ( \u000bkx\u001bz+h\u001bx)-type\nSOC can be interpreted as an equal weight combination5\nQEth,+Q\nEth,-QEQ\nQ\n(c)3\n1 2(b)Q1 Q3Q20.8\n0.6\n-1.72 -1.68 -1.64Q1\n0.3\n0.4\n0.5\n0.60kas(           )-1\n(d)0.3\n0.4\n0.5\n0.60kas(           )-1\n0kas(           )-1Q1Q\n0.2 0.4 0.6-1.7-1.5-1.3\nQ1Q2 Q3(a)Q1 Q2 Q3-\nFigure 2. Binding of spin-orbit coupled fermions on top of a Fermi sea in single impurity system. (a) The distribution of\n\r\"\nQ\u0000\r\"\n0. (b)\u0018Qas a function of Qwith di\u000berent ( k0as)\u00001according to Eq.(8). The inset shows \u0018Qin the region near Q1.\n(c) The threshold EQ\nth= minfEQ\nth;\u0000;EQ\nth;+g. The helicity-dependent threshold energy EQ\nth;+(EQ\nth;\u0000) is the minimum energy of\ntwo particles with A in the upper (lower) helicity branch and a c.m. momentum Q. (d) The two-body energy with di\u000berent\ninteracting strengths by exactly solving Eq. (4). The inset shows the variation of the ground state c.m. momentum. Here\nQ0=\u00002k0ex.\nof Rashba-type and Dresselhaus-type SOC [4].\nIn the presence of SOC, the single-particle eigenstates\nofAin the helicity basis are created by operators ay\nk;\u0006=\n\u0015\u0006;\"\nkay\nk;\"+\u0015\u0006;#\nkay\nk;#, with\u0015\u0006;\"\nk=\u0006\u0010\u0006\nk,\u0015\u0006;#\nk=\u0010\u0007\nk, and\n\u0010\u0006\nk= [p\nh2+\u000b2k2x\u0006\u000bkx]1=2=p\n2[h2+\u000b2k2\nx]1=4, with\n+(\u0000) labelling the upper (lower) helicity branch. The\nsingle particle dispersions of two helicity branches are\n\"a\nk;\u0006=\"a\nk\u0006p\nh2+\u000b2k2x. Here we have measured the\nenergy in the unit of E0= 2\u000b2ma=~2, the momentum in\nthe unit of k0= 2\u000bma=~2, andh= 0:4E0.\nWe \frst present our results for the binding of AandB\nin the vacuum, as summarized in Fig. 1. The density of\nstates [see Fig. 1(a)] exhibits a monotonic decrease with\nboth Qand\". As expected, Esc\nQwill change monoton-\nically with respect to both Qanda\u00001\ns[see Fig. 1(b)].\nTogether with EQ\nth[see Fig. 1(c)], we see that the actual\nground state c.m. momenta will be pulled to the direc-\ntion with a smaller magnitude than Q1and the increase\nofa\u00001\nswill enhance this tendency [see Fig. 1(d)].\nWe now turn to the binding of AandBon the top of\nthe Fermi sea of Ain the situation where a single impu-rity ofBimmerses in a non-interacting Fermi sea of spin-\norbit coupled Awith the Fermi energy Eh=\u00001:5E0, as\nillustrated in Fig. 2. There, both the density of states\n[see Fig. 2(a)] and Esc\nQ[see Fig. 2(b)] exhibit a rich be-\nhavior. In addition, from EQ\nthin Fig. 2(c), we see that\nthere exist two meta-stable states near Q1andQ2, re-\nspectively. Let us \frst analyze the c.m. momenta asso-\nciated with the meta-stable states, e.g., the one formed\nnearQ1. Seen from Fig. 2(a), \r\"\nQfor c.m. momentum\nnearQ1decreases with Qin the low energy region (e.g.\n0< \" < 2E0), but increases in the high energy region\n(e.g. 6E0< \" < 10E0). In addition, near Q1,EQ\nsc[see\nthe inset of Fig. 2(b)] shows a qualitative change with in-\ncreasing of a\u00001\ns. We thus expect from earlier discussions\na zigzag behavior of c.m. momenta of the meta-stable\nstate, as con\frmed by our results plotted in the inset of\nFig. 2(d). Next, we discuss which of the two meta-stable\nstates is energetically favored. Due to the degeneracy of\nthe two local minima of EQ\nth, this is determined by the\ndensity of states, which is larger near Q1than that near\nQ2[see Fig. 2(a)]. Hence the meta-stable state near Q16\n(b)-Q Q0Q1Q2\nQ30.8\n1.0\n1.2\n1.40kas(           )-1\n(c)EEth,+Q\nEth,-Q\nQ1Q2Q3\n(d)Q1Q2Q30.8\n1.0\n1.2\n1.40kas(           )-1(a)- Q1 Q2 Q3\nFigure 3. (a) The distribution of \r\"\nQ\u0000\r\"\n0. (b)\u0018Qas a function of Qwith di\u000berent ( k0as)\u00001according to Eq.(8). (c) The\nthreshold energy EQ\nth= minfEQ\nth;\u0000;EQ\nth;+g. The helicity-dependent threshold energy EQ\nth;+(EQ\nth;\u0000) is the minimum energy of\ntwo particles with A in the upper (lower) helicity branch and a c.m. momentum Q. (d) The two-body energy with di\u000berent\ninteracting strengths by exactly solving Eq. (4). Here Q0=\u00003k0ex.\nis energetically favored by EQ\nsc[see Fig. 2(b)]. We thus\nexpect the molecular ground state c.m. momentum to be\nnearQ1, well agreeing with Fig. 2(d).\nComparing the binding of AandBin the vacuum and\non top of the \flled Fermi sea, we observe that the pres-\nence of Fermi sea not only elevates EQ\nthin the regime\nQ1<Q<Q2, giving rise to two minima, but also en-\nhances the density of states there. Consequently, the\nminimum of EQ\nscoccurs at Q3, and the two meta-stable\nstates merge together [see Fig. 2(d)] following from pre-\nvious analysis. We remark that, while the SOC here has\ndramatically modi\fed the density of states compared to\nthe SOC-free case, our analyses based on perturbation\ntreatment agree remarkably well with the exact numeri-\ncal results.\nVI. CONCLUDING DISCUSSIONS AND\nSUMMARY\nWhen the Fermi sea has only one Fermi surface, the\ntwo meta-stable states formed near Q1andQ2are fa-\nvored by the threshold energy and the density of states,\nrespectively, see Fig. 3. In Ref. [16] with high Fermienergy, tuning the interaction can induce a transition\nbetween the two meta-stable states. In contrast, as il-\nlustrated in Fig. 3 where the Fermi energy Eh= 0, such\ntransition is missing and the increase of a\u00001\nswill eventu-\nally cause a merge of the two meta-stable states. With an\nincrease of the Fermi energy, our case crossovers to that\ndiscussed in Ref. [16]. In addition, we note that for the\nsingle impurity Fermi system we only consider the low-\nest energy state within our ansatz, the ground state of\nthe system should be given by connecting the molecular\nground state to the polaron ground state which describes\nthe particle-hole excitations above the Fermi sea.\nSummarizing, we have investigated how the tuning of\ninteracting strength of an attractive s-wave interaction\na\u000bects two-body energy under certain distribution of the\ndensity of states. Combining with the dispersion of the\nthreshold energy, we can predict typical behavior of the\ntwo-body bound state when tuning the scattering length\nand hence the interaction, including the change of the\nc.m. momentum of the two-body ground state and the\ncompetition between multiple meta-stable states. Our\nperturbation analyses are not dependent on the concrete\ntype of SOC and corroborated by the exact numerical\nsolution of the two-body problem for a spin-orbit coupled7\nFermi mixture of40K-40K-6Li, even though the density\nof states is signi\fcantly altered by the e\u000bect of SOC.VII. ACKNOWLEDGMENTS\nWe thanks Ying Hu for helpful discussion. The work\nis supported by the National Basic Research Program of\nChina (973 Program) (Grants No. 2013CBA01502 and\nNo. 2013CB834100), the National Natural Science Foun-\ndation of China (Grants No. 11374040, No. 11475027,\nNo. 11575027, and No. 11274051).\n[1] Jean Dalibard, Fabrice Gerbier, Gediminas Juzeli\u0016 unas,\nand Patrik Ohberg, Rev. Mod. Phys. 83, 1523 (2011).\n[2] Y.-J. Lin, K. Jimen\u0013 ez-Garc\u0013 \u0010a, and I. B. Spielman, Nature\n(London) 471, 83 (2011).\n[3] Jin-Yi Zhang, Si-Cong Ji, Zhu Chen, Long Zhang, Zhi-\nDong Du, Bo Yan, Ge-Sheng Pan, Bo Zhao, YouJin\nDeng, Hui Zhai, Shuai Chen, and Jian-Wei Pan, Phys.\nRev. Lett. 109, 115301 (2012).\n[4] Pengjun Wang, Zeng-Qiang Yu, Zhengkun Fu, Jiao Miao,\nLianghui Huang, Shijie Chai, Hui Zhai, and Jing Zhang,\nPhys. Rev. Lett. 109, 095301 (2012).\n[5] Lawrence W. Cheuk, Ariel T. Sommer, Zoran Hadz-\nibabic, Tarik Yefsah, Waseem S. Bakr, and Martin W.\nZwierlein, Phys. Rev. Lett. 109, 095302 (2012); Lianghui\nHuang, Zengming Meng, Pengjun Wang, Peng Peng,\nShao-Liang Zhang, Liangchao Chen, Donghao Li, Qi\nZhou and Jing Zhang, Nat. Phys. 10, 1038 (2016).\n[6] H. Zhai, Int. J. Mod. Phys. B 26, 1230001 (2012).\n[7] V. Galitski and I. B. Spielman, Nature, 494, 49 (2013).\n[8] H. Zhai, Rep. Prog. Phys., 78, 026001 (2015).\n[9] Jayantha P. Vyasanakere, and Vijay B. Shenoy, Phys.\nRev. B 83, 094515 (2011).\n[10] Jayantha P. Vyasanakere, Shizhong Zhang, and Vijay B.\nShenoy, Phys. Rev. B 84, 014512 (2011).\n[11] Hui Hu, Lei Jiang, Xia-Ji Liu, and Han Pu, Phys. Rev.\nLett.107, 195304 (2011).\n[12] Zeng-Qiang Yu and Hui Zhai, Phys. Rev. Lett. 107,\n195305 (2011).\n[13] Ren Zhang, Fan Wu, Jun-Rong Tang, Guang-Can Guo,\nWei Yi, and Wei Zhang, Phys. Rev. A. 87. 033629 (2013).\n[14] Vijay B. Shenoy, Phys. Rev. A. 88. 033609 (2013).\n[15] Fan Wu, Ren Zhang, Tian-Shu Deng, Wei Zhang, Wei Yi,\nand Guang-Can Guo, Phys. Rev. A. 89.063610 (2014).\n[16] Lihong Zhou, Xiaoling Cui, and Wei Yi, Phys. Rev. Lett.\n112, 195301 (2014).\n[17] F. Chevy, Phys. Rev. A 74, 063628 (2006).\n[18] R. Combescot, A. Recati, C. Lobo and F. Chevy, Phys.\nRev. Lett, 98, 180402 (2007).[19] Sascha Zollner, G. M. Bruun, and C. J. Pethick, Phys.\nRev. A 83, 021603(R) (2011).\n[20] Marco Koschorreck, Daniel Pertot, Enrico Vogt, Bernd\nFrohlich, Michael Feld and Michael Kohl, Nature 485,\n619 (2012).\n[21] Wei Yi and Wei Zhang, Phys. Rev. Lett 109, 140402\n(2012).\n[22] Cheng Chin, Rudolf Grimm, Paul Julienne, and Eite\nTiesinga, Rev. Mod. Phys. 82, 1225 (2010).\n[23] Andr\u0013 e Schirotzek, Cheng-Hsun Wu, Ariel Sommer, and\nMartin W. Zwierlein, Phys. Rev. Lett. 102, 230402\n(2009); M. Koschorreck, D. Pertot, E. Vogt, B. Fr ohlich,\nM. Feld, and M. K ohl, Nature (London) 485, 619 (2012).\n[24] W. Ketterle and M. W. Zwierlein, Rivista del Nuovo Ci-\nmento 31, 247-422 (2008).\n[25] We note that for homo-nuclear systems, our derivation\nfor\r\"\nQreduces to the singlet density of states as discussed\nin Ref. [14].\n[26] With Eq.(5) and renomolization eqaution, we haveR1\n0\r\"\n0\n\"b\u0000\"d\"=R1\n0\r\"\nQ\n\u0018Q+\"b\u0000\"d\". The right hand of the equa-\ntion can be writen as followR1\n0\r\"\nQ\n\u0018Q+\"b\u0000\"d\"=R1\n0[\r\"\nQ\n\"b\u0000\"\u0000\n\u0018Q\r\"\nQ\n(\"b\u0000\")2+\u00182\nQ\r\"\nQ\n(\"b\u0000\")3+:::]d\". To \frst order of \u0018Q(j\"Q\n\"bj<<1),\nwe arrive Eq.(8).\n[27] Xiaoling Cui, Phys. Rev. A 85, 022705 (2012).\n[28] Peng Zhang, Long Zhang, and Wei Zhang, Phys. Rev. A\n86, 042707 (2012).\n[29] Peng Zhang, Long Zhang, and Youjin Deng, Phys. Rev.\nA86, 053608 (2012).\n[30] Yuxiao Wu and Zhenhua Yu, Phys. Rev. A, 87, 032703\n(2013).\n[31] Hao Duan, Li You, and Bo Gao, Phys. Rev. A, 87, 052708\n(2013).\n[32] the analysis for the case with \r\"\nQ\u0000\r\"\n0>0 is similar."    },    {        "title": "0908.2961v2.Theory_of_anisotropic_exchange_in_laterally_coupled_quantum_dots.pdf",        "content": "arXiv:0908.2961v2  [cond-mat.mes-hall]  23 Feb 2010Theory of anisotropic exchange in laterally coupled quantu m dots\nFabio Baruffa1, Peter Stano2,3and Jaroslav Fabian1\n1Institute for Theoretical Physics, University of Regensbu rg, 93040 Regensburg, Germany\n2Institute of Physics, Slovak Academy of Sciences, 84511 Bra tislava, Slovak Republic\n3Physics Department, University of Arizona, 1118 E 4thStreet, Tucson, AZ 85721, USA\nThe effects of spin-orbit coupling on the two-electron spect ra in lateral coupled quantum dots are\ninvestigated analytically and numerically. It is demonstr ated that in the absence of magnetic field\nthe exchange interaction is practically unaffected by spin- orbit coupling, for any interdot coupling,\nboosting prospects for spin-based quantum computing. The a nisotropic exchange appears at finite\nmagnetic fields. A numerically accurate effective spin Hamil tonian for modeling spin-orbit-induced\ntwo-electron spin dynamics in the presence of magnetic field is proposed.\nPACS numbers: 71.70.Gm, 71.70.Ej, 73.21.La, 75.30.Et\nThe electron spins in quantum dots are natural and\nviable qubits for quantum computing,[1] as evidenced by\nthe impressive recent experimental progress [2, 3] in spin\ndetection and spin relaxation,[4, 5] as well as in coherent\nspin manipulation.[6, 7] In coupled dots, the two-qubit\nquantumgatesarerealizedbymanipulatingtheexchange\ncouplingwhichoriginatesin theCoulombinteractionand\nthe Pauliprinciple.[1, 8] How is the exchangemodified by\nthe presence of the spin-orbit coupling? In general, the\nusual (isotropic) exchange changes its magnitude while\na new, functionally different form of exchange, called\nanisotropic, appears, breaking the spin-rotational sym-\nmetry. Such changes are a nuisance from the perspective\nof the error correction,[9] although the anisotropic ex-\nchange could also induce quantum gating.[10, 11]\nTheanisotropicexchangeofcoupledlocalizedelectrons\nhasaconvolutedhistory[12–18]. Thequestionboilsdown\nto determining the leading order in which the spin-orbit\ncoupling affects both the isotropic and anisotropic ex-\nchange. At zero magnetic field, the second order was\nsuggested,[19] with later revisions showing the effects are\nabsent in the second order.[12, 20] The analytical com-\nplexities make a numerical analysis particularly useful.\nHere we perform numerically exact calculations of the\nisotropic and anisotropic exchange in realistic GaAs cou-\npled quantum dots in the presence of both the Dres-\nselhaus and Bychkov-Rashba spin-orbit interactions.[21]\nThe numerics allowsus to make authoritative statements\nabout the exchange. We establish that in zero magnetic\nfield the second-order spin-orbit effects are absent at all\ninterdot couplings. Neither is the isotropic exchange af-\nfected, nor is the anisotropic exchange present. At finite\nmagnetic fields the anisotropic coupling appears. We de-\nrive a spin-exchange Hamiltonian describing this behav-\nior, generalizing the existing descriptions; we do not rely\non weak coupling approximations such as the Heitler-\nLondon one. The model is proven highly accurate by\ncomparison with our numerics and we propose it as a re-\nalistic effective model for the two-spin dynamics in cou-\npled quantum dots.\nOur microscopic description is the single band effec-tive mass envelope function approximation; we neglect\nmultiband effects.[22, 23] We consider a two electron\ndouble dot whose lateral confinement is defined electro-\nstatically by metallic gates on the top of a semiconduc-\ntor heterostructure. The heterostructure, grown along\n[001] direction, provides strong perpendicular confine-\nment, such that electrons are strictly two dimensional,\nwith the Hamiltonian (subscript ilabels the electrons)\nH=/summationdisplay\ni=1,2(Ti+Vi+HZ,i+Hso,i)+HC.(1)\nThe single electron terms are the kinetic energy, model\nconfinement potential, and the Zeeman term,\nT=P2/2m= (−i¯h∇+eA)2/2m, (2)\nV= (1/2)mω2[min{(x−d)2,(x+d)2}+y2],(3)\nHZ= (g/2)(e¯h/2me)B·σ=µB·σ, (4)\nand spin-orbit interactions—linear and cubic Dressel-\nhaus, and Bychkov-Rashba[21],\nHd= (¯h/mld)(−σxPx+σyPy), (5)\nHd3= (γc/2¯h3)(σxPxP2\ny−σyPyP2\nx)+Herm. conj. ,(6)\nHbr= (¯h/mlbr)(σxPy−σyPx), (7)\nwhich we lump together as Hso=w·σ. The posi-\ntionrand momentum Pvectors are two dimensional\n(in-plane); m/meis the effective/electron mass, eis the\nproton charge, A=Bz(−y,x)/2 is the in-plane vector\npotential to magnetic field B= (Bx,By,Bz),gis the\nelectrong-factor, σarePaulimatrices,and µistherenor-\nmalized magnetic moment. The double dot confinement\nis modeled by two equal single dots displaced along [100]\nby±d, each with a harmonic potential with confinement\nenergy ¯hω. The spin-orbit interactions are parametrized\nby the bulk material constant γcand the heterostruc-\nture dependent spin-orbit lengths lbr,ld. Finally, the\nCoulomb interaction is HC= (e2/4πǫ)|r1−r2|−1, with\nthe dielectric constant ǫ.\nThe numerical results are obtained by exact diagonal-\nization(configurationinteractionmethod). Thetwoelec-\ntron Hamiltonian is diagonalized in the basis of Slater2\n500 250 100 50 25 10 5tunneling energy [ µeV]\n0246energy [meV]\n0 25 50 75\ninterdot distance [nm]JΨ-Ψ+\n2T∆\nFIG. 1: Calculated double dot spectrum as a function of the\ninterdot distance/tunneling energy. Spin is not considere d\nand the magnetic field is zero. Solid lines show the two elec-\ntron energies. The two lowest states are explicitly labeled ,\nsplit by the isotropic exchange Jand displaced from the near-\nest higher excited state by ∆. For comparison, the two lowest\nsingle electron states are shown (dashed), split by twice th e\ntunneling energy T. State spatial symmetry is denoted by\ndarker (symmetric) and lighter (antisymmetric) lines.\ndeterminants constructed from numerical single electron\nstates in the double dot potential. Typically we use 21\nsingle electron states, resulting in the relative error for\nenergies of order 10−5. We use material parameters of\nGaAs:m= 0.067me,g=−0.44,γc= 27.5 meV˚A3, a\ntypical single dot confinement energy ¯ hω= 1.1 meV, and\nspin-orbit lengths ld= 1.26µm andlbr= 1.72µm from a\nfit to a spin relaxation experiment.[24, 25]\nLet us first neglect the spin and look at the spectrum\nin zero magnetic field as a function of the interdot dis-\ntance (2d)/tunneling energy, Fig. 1. At d= 0 our model\ndescribes a single dot. The interdot coupling gets weaker\nas one moves to the right; both the isotropic exchange\nJand the tunneling energy Tdecay exponentially. The\nsymmetry of the confinement potential assures the elec-\ntronwavefunctionsaresymmetricorantisymmetricupon\ninversion. The two lowest states, Ψ ±, are separated from\nthe higher excited states by an appreciable gap ∆, what\njustifies the restriction to the two lowest orbital wave-\nfunctions for the spin qubit pair at a weak coupling. Our\nfurther derivations are based on the observation\nPΨ±=±Ψ±, I 1I2Ψ±=±Ψ±,(8)\nwhereIf(x,y) =f(−x,−y) is the inversion operator\nandPf1g2=f2g1is the particle exchange operator.\nFunctions Ψ ±in the Heitler-Londonapproximationfulfill\nEq. (8). However, unlike Heitler-London, Eq. (8) is valid\ngenerally in symmetric double dots, as we learn from nu-\nmerics (we saw it valid in all cases we studied).Let us reinstate the spin. The restricted two qubit\nsubspace amounts to the following four states ( Sstands\nfor singlet, Tfor triplet),\n{Φi}i=1,...,4={Ψ+S,Ψ−T+,Ψ−T0,Ψ−T−},(9)\nWithin this basis, the system is described by a 4 by 4\nHamiltonian with matrix elements ( H4)ij=/an}bracketle{tΦi|H|Φj/an}bracketri}ht.\nWithout spin-orbit interactions, this Hamiltonian is di-\nagonal, with the singlet and triplets split by the isotropic\nexchange J,[1, 8] and the triplets split by the Zeeman en-\nergyµB. It is customary to refer only to the spinor part\nof the basis states, using the sigma matrices, resulting in\nthe isotropic exchange Hamiltonian,\nHiso= (J/4)σ1·σ2+µB·(σ1+σ2).(10)\nA naive approach to include the spin-orbit interaction\nis to consider it within the basis of Eq. (9). This gives\nthe Hamiltonian H′\nex=Hiso+H′\naniso, where\nH′\naniso=a′·(σ1−σ2)+b′·(σ1×σ2),(11)\nwith the six real parameters given by spin-orbit vectors\na′= Re/an}bracketle{tΨ+|w1|Ψ−/an}bracketri}ht,b′= Im/an}bracketle{tΨ+|w1|Ψ−/an}bracketri}ht.(12)\nThe form of the Hamiltonian follows solely from the in-\nversion symmetry Iw=−wand Eq. (8). The spin-orbit\ncoupling appears in the first order.\nThe Hamiltonian H′\nexfares badly with numerics. Fig-\nure 2 shows the energy shifts caused by the spin-orbit\ncoupling for selected states, at different interdot cou-\nplings and perpendicular magnetic fields. The model is\ncompletely off even though we use numerical wavefunc-\ntions Ψ ±in Eq. (12) without further approximations.\nTo improve the analytical model, we remove the\nlinear spin-orbit terms from the Hamiltonian using\ntransformation[20, 26, 27]\nU= exp[−(i/2)n1·σ1−(i/2)n2·σ2],(13)\nwheren= (x/ld−y/lbr,x/lbr−y/ld,0).\nUp to the second order in small quantities (the spin-\norbit and Zeeman interactions), the transformed Hamil-\ntonianH=UHU†is the same as the original, Eq. (1),\nexcept for the linear spin-orbit interactions:\nHso=−(µB×n)·σ+(K−/¯h)Lzσz−K+,(14)\nwhereK±= (¯h2/4ml2\nd)±(¯h2/4ml2\nbr). In the unitarily\ntransformed basis, we again restrict the Hilbert space to\nthe lowest four states, getting the effective Hamiltonian\nHex= (J/4)σ1·σ2+µ(B+Bso)·(σ1+σ2)\n+a·(σ1−σ2)+b·(σ1×σ2)−2K+.(15)\nThe operational form is the same as for H′\nex. The qual-\nitative difference is in the way the spin-orbit enters the\nparameters. First, a contribution to the Zeeman term,\nµBso=ˆ z(K−/¯h)/an}bracketle{tΨ−|Lz,1|Ψ−/an}bracketri}ht, (16)3\n-0.6-0.4-0.200.2\n0 50 100\ninterdot distance [nm]-0.6-0.4-0.200.2                                 energy shift [ µeV]\n0 0.5 1\nmagnetic field [T]ab\ncdHex\nnumericalHex'\nFIG. 2: The spin-orbit induced energy shift as a function of\nthe interdot distance (left) and perpendicular magnetic fie ld\n(right). a) Singlet in zero magnetic field, c) singlet at 1 Tes la\nfield, b) and d) singlet and triplet T+at the interdot distance\n55 nm corresponding to the zero field isotropic exchange of\n1µeV. The exchange models H′\nex(dashed) and Hex(dot-\ndashed) are compared with the numerics (solid).\nappears due to the inversion symmetric part of Eq. (14).\nSecond, the spin-orbit vectors are linearly proportional\nto both the spin-orbit coupling and magnetic field,\na=−µB×Re/an}bracketle{tΨ+|n1|Ψ−/an}bracketri}ht, (17a)\nb=−µB×Im/an}bracketle{tΨ+|n1|Ψ−/an}bracketri}ht. (17b)\nThe effective model and the exact data agree very well\nfor all interdot couplings, as seen in Fig. 2.\nAt zero magnetic field, only the first and the last term\nin Eq. (15) survive. This is the result of Ref. [20], where\nprimed operators were used to refer to the fact that the\nHamiltonian Hexrefers to the transformed basis, {UΦi}.\nNote that if a basis separable in orbital and spin part is\nrequired,undoing UnecessarilyyieldstheoriginalHamil-\ntonian Eq. (1), and the restriction to the four lowest\nstates gives H′\nex. Replacing the coordinates ( x,y) by\nmean values ( ±d,0)[12] visualizes the Hamiltonian Hex\nas an interaction through rotated sigma matrices, but\nthis is just an approximation, valid if d,lso≫l0.\nOne of our main numerical results is establishing the\nvalidity of the Hamiltonian in Eq. (15) for B= 0, con-\nfirming recent analytic predictions and extending their\napplicability beyond the weak coupling limit. In the\ntransformed basis, the spin-orbit interactions do not lead\nto any anisotropic exchange, nor do they modify the\nisotropic one. In fact, this result could have been antic-\nipated from its single-electron analog: at zero magnetic\nfield there is no spin-orbit contribution to the tunneling\nenergy,[28] going opposite to the intuitive notion of the\nspin-orbit coupling induced coherent spin rotation and\nspin-flip tunneling amplitudes. Figure 3a summarizes\nthis case, with the isotropic exchangeas the only nonzeroFIG. 3: a) The isotropic and anisotropic exchange as func-\ntions of the interdot distance at zero magnetic field. b) The\nisotropic exchange J, anisotropic exchange c/c′, the Zeeman\nsplitting µB, and its spin-orbit part µBsoat perpendicular\nmagnetic field of 1 Tesla. c) Schematics of the exchange-spli t\nfour lowest states for the three models, Hiso,H′\nex, andHex,\nwhich include the spin-orbit coupling in no, first, and secon d\norder, respectively, at zero magnetic field (top). The latte r\ntwo models are compared in perpendicular and in-plane mag-\nnetic fields as well. The eigenenergies are indicated by the\nsolid lines. The dashed lines show which states are coupled\nby the spin-orbit coupling. The arrows indicate the redistr i-\nbution of the couplings as the in-plane field direction chang es\nwith respect to the crystallographic axes (see the main text ).\nparameterof model Hex. In contrast, model H′\nexpredicts\na finite anisotropic exchange.[36]\nFrom the concept of dressed qubits[29] it follows that\nthe main consequence of the spin-orbit interaction, the\ntransformation Uof the basis, is not a nuisance for quan-\ntum computation. We expect this property to hold also\nfor aqubit array, since the electrons areat fixed positions\nwithoutthepossibilityofalongdistancetunneling. How-\never, a rigorous analysis of this point is beyond the scope\nof this article. If electrons are allowed to move, Uresults\nin the spin relaxation.[30]\nFigure 3b shows model parameters in 1 Tesla perpen-\ndicular magnetic field. The isotropic exchange again\ndecays exponentially. As it becomes smaller than the\nZeeman energy, the singlet state anticrosses one of the\npolarized triplets (seen as cusps on Fig. 2). Here it is\nT+, due to the negative sign of both the isotropic ex-\nchange and the g-factor. Because the Zeeman energy\nalways dominates the spin-dependent terms and the sin-\nglet and triplet T0are never coupled (see below), the\nanisotropic exchange influences the energy in the sec-\nond order.[12] Note the difference in the strengths. In\nH′\nextheanisotropicexchangefallsoffexponentially, while\nHexpredicts non-exponential behavior, resulting in spin-\norbit effects larger by orders of magnitude. The effective\nmagnetic field Bsois always much smaller than the real\nmagnetic field and can be neglected in most cases.4\nFigure3ccomparesanalyticalmodels. Inzerofieldand\nno spin-orbit interactions, the isotropic exchange Hamil-\ntonianHisodescribes the system. Including the spin-\norbit coupling in the first order, H′\nex, gives a nonzero\ncoupling between the singlet and triplet T0. Going to the\nsecond order, the effective model Hexshows there are no\nspin-orbit effects (other than the basis redefinition).\nThe Zeeman interaction splits the three triplets in a fi-\nnite magnetic field. Both H′\nexandHexpredict the same\ntype of coupling in a perpendicular field, between the\nsinglet and the two polarized triplets. Interestingly, in\nin-plane fields the two models differ qualitatively. In\nH′\nexthe spin-orbit vectors are fixed in the plane. Rota-\ntion of the magnetic field “redistributes” the couplings\namong the triplets. (This anisotropy with respect to\nthe crystallographic axis is due to the C2vsymmetry of\nthe two-dimensional electron gas in GaAs, imprinted in\nthe Bychkov-Rashba and Dresselhaus interactions.[21])\nIn contrast, the spin-orbit vectors of Hexare always per-\npendicular to the magnetic field. Remarkably, aligning\nthe magnetic field along a special direction (here we al-\nlow anarbitrarypositioned dot, with δthe anglebetween\nthe main dot axis and the crystallographic xaxis),\n[lbr−ldtanδ,ld−lbrtanδ,0], (18)\nall the spin-orbit effects disappear once again , as ifB\nwere zero. (An analogous angle was reported for a sin-\ngle dot in Ref. [31]). This has strong implications for\nthe spin-orbit induced singlet-triplet relaxation. Indeed,\nS↔T0transitions are ineffective at any magnetic field ,\nas these two states are never coupled in our model. Sec-\nond,S↔T±transitions will show strong (orders ofmag-\nnitude) anisotropy with respect to the field direction,\nreaching minimum at the direction given by Eq. (18).\nThis prediction is straightforwardly testable in experi-\nments on two electron spin relaxation.\nOur derivation was based on the inversion symmetry\nof the potential only. What are the limits of our model?\nWe neglected third order terms in Hsoand, restricting\nthe Hilbert space, corrections from higher excited orbital\nstates. (Among the latter is the non-exponential spin-\nspin coupling[12]). Compared to the second order terms\nwe keep, these are smaller by (at least) d/lsoandc/∆,\nrespectively.[35] Apart from the analytical estimates, the\nnumerics, which includes all terms, assures us that both\nof these are negligible. Based on numerics we also con-\nclude our analytical model stays quantitatively faithful\neven at the strong coupling limit, where ∆ →0. More\ninvolved is the influence of the cubic Dresselhaus term,\nwhich is not removed by the unitary transformation.\nThis term is the main source for the discrepancy of the\nmodel and the numerical data in finite fields. Most im-\nportantly, it does not change our results for B= 0.\nConcluding, we studied the effects of spin-orbit cou-\npling on the exchange in lateral coupled GaAs quantumdots. Wederiveandsupportbyprecisenumericsaneffec-\ntive Hamiltonian for two spin qubits, generalizing the ex-\nisting models. The effective anisotropic exchange model\nshouldbeusefulinpreciseanalysisofthephysicalrealiza-\ntions of quantum computing schemes based on quantum\ndot spin qubits, as well as in the physics of electron spins\nin quantum dots in general. Our analysis should also\nimprove the current understanding of the singlet-triplet\nspin relaxation [32–35].\nThisworkwassupportedbyDFGGRK638,SPP1285,\nNSF grant DMR-0706319, RPEU-0014-06, ERDF OP\nR&D “QUTE”, CE SAS QUTE and DAAD.\n[1] D. Loss and D. P. DiVincenzo, Phys. Rev. A 57, 120\n(1998).\n[2] R. Hanson, L. P. Kouwenhoven, J. R. Petta, S. Tarucha,\nand L. M. K. Vandersypen, Rev. Mod. Phys. 79, 1217\n(2007).\n[3] J. M. Taylor, J. R. Petta, A. C. Johnson, A. Yacoby,\nC. M. Marcus, andM. D.Lukin, Phys. Rev.B 76, 035315\n(2007).\n[4] J. M. Elzerman, R. Hanson, L. H. Willems van Beveren,\nB. Witkamp, L. M. K. Vandersypen, and L. P. Kouwen-\nhoven, Nature 430, 431 (2004).\n[5] F. H. L. Koppens, K. C. Nowack, and L. M. K. Vander-\nsypen, Phys. Rev. Lett. 100, 236802 (2008).\n[6] J. R. Petta, A. C. Johnson, J. M. Taylor, E. A. Laird,\nA. Yacoby, M. D. Lukin, C. M. Marcus, M. P. Hanson,\nand A. C. Gossard, Science 309, 2180 (2005).\n[7] K. C. Nowack, F. H. L. Koppens, Y. V. Nazarov, and\nL. M. K. Vandersypen, Science 318, 1430 (2007).\n[8] X. Hu and S. Das Sarma, Phys. Rev. A 61, 062301\n(2000).\n[9] D. Stepanenko, N. E. Bonesteel, D. P. DiVincenzo,\nG. Burkard, and D. Loss, Phys. Rev. B 68, 115306\n(2003).\n[10] D. Stepanenko and N. E. Bonesteel, Phys. Rev. Lett. 93,\n140501 (2004).\n[11] N. Zhao, L. Zhong, J.-L. Zhu, and C. P. Sun, Phys. Rev.\nB74, 075307 (2006).\n[12] S. Gangadharaiah, J. Sun, and O. A.Starykh, Phys. Rev.\nLett.100, 156402 (2008).\n[13] L. Shekhtman, O. Entin-Wohlman, and A. Aharony,\nPhys. Rev. Lett. 69, 836 (1992).\n[14] A. Zheludev, S. Maslov, G. Shirane, I. Tsukada, T. Ma-\nsuda, K. Uchinokura, I. Zaliznyak, R. Erwin, and L. P.\nRegnault, Phys. Rev. B 59, 11432 (1999).\n[15] Y. Tserkovnyak and M. Kindermann, Phys. Rev. Lett.\n102, 126801 (2009).\n[16] S. Chutia, M. Friesen, and R. Joynt, Phys. Rev. B 73,\n241304(R) (2006).\n[17] L. P. Gorkov and P. L. Krotkov, Phys. Rev. B 67, 033203\n(2003).\n[18] S. D. Kunikeevand D. A. Lidar, Phys. Rev. B 77, 045320\n(2008).\n[19] K. V. Kavokin, Phys. Rev. B 64, 075305 (2001).\n[20] K. V. Kavokin, Phys. Rev. B 69, 075302 (2004).\n[21] J. Fabian, A. Matos-Abiagus, C. Ertler, P. Stano, and\nI.ˇZuti´ c, Acta Phys. Slov. 57, 565 (2007).5\n[22] S. C. Badescu, Y. B. Lyanda-Geller, and T. L. Reinecke,\nPhys. Rev. B 72, 161304(R) (2005).\n[23] M. M. Glazov and V. D. Kulakovskii, Phys. Rev. B 79,\n195305 (2009).\n[24] P. Stano and J. Fabian, Phys. Rev. Lett. 96, 186602\n(2006).\n[25] P. Stano and J. Fabian, Phys. Rev. B 74, 045320 (2006).\n[26] I. L. Aleiner and V. I. Fa ˇlko, Phys. Rev. Lett. 87, 256801\n(2001).\n[27] L. S. Levitov and E. I. Rashba, Phys. Rev. B 67, 115324\n(2003).\n[28] P. Stano and J. Fabian, Phys. Rev. B 72, 155410 (2005).\n[29] L.-A. Wu and D. A. Lidar, Phys. Rev. Lett. 91, 097904\n(2003).\n[30] K. V. Kavokin, Semicond. Sci. Technol. 23, 114009(2008).\n[31] V. N. Golovach, A. Khaetskii, and D. Loss, Phys. Rev.\nB77, 045328 (2008).\n[32] K. Shen and M. W. Wu, Phys. Rev. B 76, 235313 (2007).\n[33] E. Y. Sherman and D. J. Lockwood, Phys. Rev. B 72,\n125340 (2005).\n[34] J. I. Climente, A. Bertoni, G. Goldoni, M. Rontani, and\nE. Molinari, Phys. Rev. B 75, 081303(R) (2007).\n[35] O. Olendski and T. V. Shahbazyan, Phys. Rev. B 75,\n041306(R) (2007).\n[36] The spin-orbit vectors are determined up to the relativ e\nphase of states Ψ +and Ψ −. The observable quantity is\nc=√a2+b2and analogously for c′=√a′2+b′2."    },    {        "title": "1709.07948v1.Quantification_of_spin_accumulation_causing_spin_orbit_torque_in_Pt_Co_Ta_stack.pdf",        "content": "1 \n Quantification of  spin accumulation causing  spin-orbit torque in \nPt/Co/Ta  stack  \nFeilong Luo, Sarjoosing Goolaup, Christian Engel, and Wen Siang Lew* \nSchool of Physical and Mathematical Sciences, Nanyang Technological University,  \n21 Nanyang Link, Singapore 637371  \n \nAbstract  \nSpin accumulation induced by spin-orbit coupling  is experimentally quantified in \nstack with in -plane magnetic anisotropy via the contribution of spin accumulation to Hall \nresistances.  Using a biasing direct current the spin accumulation within the structure can \nbe tuned, enabling quantification.  Quantification shows the spin accumulation can be more \nthan ten percentage of local magnetization , when the electric current is 1011 Am−2. The spin \naccumulation is dependent  of the thickness of Ta  layer , the trend  agrees with that of  spin \nHall angle indicating the capability of Ta and  Pt in generating spins .  \n \n \n \n \n \n \n                                                           \n*Corresponding author: wensiang@ntu.edu.sg  2 \n Introduction  \nCurrent -induced spin accumulation  causes  spin-orbit torque (SOT) on the \nmagnetization of a ferromagnetic metal (FM) layer sandwiched by two heavy metal (HM)  \nlayers , via exchange interaction  [1]. The spin accumulation originates from t wo spin-orbit \ncoupling  effects: Rashba effect and spin Hall effect  [2-10]. The SOT is reflected  in the \nrevised Landau –Lifshitz –Gilbert equation by  the term  \n0 JMs  , where \n0  is the \ngyromagnetic coefficient, M is the magnetization of the FM layer, s is the spin \naccumulation , and J is a coefficient related to spin diffusion length of accumulated spin s \nin the FM layer. The term \n0 JMs  can be decomposed into a fieldlike torque \nFF H τ Mp\n and a damping like torque  \nDD H  τ M m p , where p represents  the \nspin orientation  of the electrons diffusing into the FM layer , and m is the unit vector of M \n[1, 10 -14]. The corresponding effective field s arising from  SOT  can be written as the \nfieldlike term  \nFFHHp  and damping like term  \nDDHH m p , alternatively, \nFD J H H  s p m p\n [6, 8, 10, 12, 15 -20]. The effective field , Js, which is a combination \nof spin accumulation and a spin-diffusion related coefficient, has been widely characterized  \nvia current -induced domain wall motion [8, 21, 28 -30], ferromagnetic resonance (FMR) \ntechniques [31 -38], and SOT -assisted magnetization switching [6, 20, 22, 35, 39] . \nQuantification of the spin accumulation, which plays a crucial role in the origins of the \nSOT, has remained e lusive . \nIn this letter , we provide  a concise solution to quantify the  spin accumulation  in the \nsandwich ed structure with in -plane magnetic anisotropy (IMA) . We propose the spin \naccumulation s contribute s to the second harmonic Hall resistance  in the harmonic Hall 3 \n voltage scheme , in addition to  the SOT effective field Js as expected. Applying a  biasing \ndirect current (DC) enables the extraction of the contribution  of the spin accumulation  from  \nthe second harmonic Hall resistances . Analogized  to first harmonic Hall resistance  which \nis induced by  the magnetization,  modulation of the second Hall resistance via DC current \ncan be used to compute the spin accumulation . Results of the computation show the spin \naccumulation is dependent of the thickness of HM layers . This quantification allows us to \nunderstand the anatomy of Js and distinguish the role s of J and s. \n \nMain body  \nFollowing the transfer of momentum  to the local magnetization, the accumulated \nspins s adopt similar polarization as the magne tization  orientation of the FM layer . The \nstructure comprises of Ta/Co/Pt multilayer, where the FM layer exhibits IMA . The initial \npolarization of s is induced by R ashba effect due to the asymmetric HM/FM interface and \nspin Hall effect within the  Ta and Pt layers [2-10, 8, 15, 18, 19, 21 -27]. The Rashba effect \nre-orientates the spin with in the  conduction electrons  of FM layer  to provide a net resultant \nspin in the FM layer  [5]. Additionally, t he spin Hall effect induces a spin -selective \nseparation of electron s in the HM layer; the spin polarized electrons then diffuses into the \nFM layer  [10]. In the Co layer, the transfer of spin torque  from the spin polarized electron \nto the FM layer occurs on the nanosecond time scale [ 5]. A schematic of the spin transfer \nprocess is depicted in Fig 1(a) . At the end of the spin transfer, s is in relaxation  state, hence \nit adopt s the same orientation as the local magnetic moment  Mm as depicted in Fig. 1(b). \nIn experiment, w ithin the low frequency regime of hundreds of Hertz, corresponding to \nperiod of oscillation of alternating current (AC)  in millisecond scale, it is reasonable t o 4 \n consider that the accumulated spins s follow the orientation of Mm. Similarly, e xtending \nto direct current ( DC) bias regime , an identical  approximation can be made  and the spins s \nsimilarly aligns along  m. Therefore , after the electron spins have transferred the \nmomentum to the local magnetization , the resultant polarization direction of the electron \nis along the magnetization orientation of  the FM layer. Thus, the spin accumulation can be \nwritten as sm. Consequently , the total magnetization of the stack becomes to  Mm+sm from \nMm. \nWe propose that t he spin accumulation sm results in additional planar Hall \nresistance, analogized to the local magnetization Mm. The magnitude of planar Hall \nresistance,  RPHE due to the local magnetic moment Mm is parabolic with respect to  \nmagnitude of the local magnetization  M via a coefficient k, \n2\nPHER kM  [40-42]. The Hall \nresistance induced by sm as an ext ra magnetization should present  the same behavior of \nthat induced by Mm. The planar Hall resistance  due to Mm is expressed as  \nP PHE sin 2 RR  \n, where φ is the azimuthal angle of magnetization Mm [7, 12, 43, 44 ]. \nAnalogically,  the planar Hall resistance,  𝑟p due to the extra magnetization, sm, should \nfollow a similar trend  as \nP PHE sin 2 rr   and \n2\nPHEr ks . Obtaining k from the expression \nof RPHE, we derive the expression of  𝑟PHE as \n2 PHE\nPHE 2=RrsM  . As such, 𝑟PHE can be used to \ncalculate the magnitude of spin accumulation.  \nApplying a biasing  DC increases the magnitude of 𝑟PHE to measureable  levels . \nWhen AC and DC are applied in the wire concurrently , the harmonic Hall voltage induced \nby sm can be written as \n 2 PHE\ns,Hall AC DC 2sin 2 sinRv s j t jM , where jAC and ω are 5 \n the amplitude and frequency of AC density  respectively , and jDC is the magnitude of DC \ndensity.  At steady -state which is  the rate of spin decay equaling to that of spin generating , \ns is proportional to the current density and can be written as \n AC DCsin s j t j , where \nζ is the coefficient constant.  Substituting s by \n AC DCsinj t j   in vs,Hall gives   \n 2\n3 2 2 2 3 3\ns,Hall PHE DC DC AC AC DC AC 2sin 2 3 sin 3 sin sin v R j j j t j j t j tM       \n.    (1) \nIn Eq. (1), we substitute \n2sin t  with \n1 cos2\n22t , as such, eliminate the constant 1\n2 item \nto obtain a  second harmonic Hall voltage as \n2\n2\ns,2ndHall PHE AC DC 23sin 2 cos22v R j j tM  . \nConsequently, sm induces a second harmonic Hall resistance 𝑟𝛼±𝛽 as \ns,2ndHall\nACsin 2vrzj\n \n,                                             (2) \nWhere \n2\nPHE DC AC 23\n2z R j jM\n , α and β correspond to the factors α× 1010 Am−2 for jAC and \nβ× 1010 Am−2 for jDC, ±  indicat es the sign of DC.  Compared with the expression of  \n2 PHE\nPHE 2=RrsM\n, the expression \n2\nPHE DC AC 23\n2z R j jM\n  is the other expression of rPHE \nwhich includes the electric current. Similarly, Eq. (2) is the other expression of rP. For a  \nconstant amplitude of jAC, the resistance 𝑧𝛼±𝛽 is proportional to the amplitude of applied \nDC. As such,  jDC provides a way to modulate the resistance from the baseline provided by \njAC. 6 \n This resistance  𝑟𝛼±𝛽 can be obtained , by subtracting the  second harmonic Hall \nresistance ℜ𝛼±𝛽 which is measured in experiment by DC -biased AC from  that of ℜ𝛼0 which \nis solely due to AC only. The measured second harmonic Hall resistance is induced by both \nsm and Mm concurrently . As such, ℜ𝛼±𝛽 is the sum of 𝑟𝛼±𝛽 due to  sm and 𝑅𝛼±𝛽 due to  Mm, \nThe second harmonic Hall resistance 𝑅𝛼±𝛽 due to Mm is \n,, 42\nAHE PHE\nextcos 2cos cos2D AC F AC\nxHHR R RHH\n   \n   \n,                    (3) \nwhere HꞱ is the effective perpendicular anisotropy field, RAHE is the amplitude of \nanomalous Hall effect (AHE) resistance  [7, 12, 43, 44, 45 ]. DC has no effect  on 𝑅𝛼±𝛽, \nalternatively, \n0RR\n  [Appendix ]. Both HD,AC and HF,AC are only determined by the AC \ncomponent of electric current , HD,AC is along z-axis while HF,AC is along y-axis, for the \nwires with IMA  [45]. Hence, according to Eq. (2), the measured second harmonic Hall \nresistance by AC only,  is \n000R   . While , the measured second harmonic Hall \nresistance by DC -biased AC is  \n0Rr\n     , where β is not equal to 0 . Therefore, \nsubtract ion of ℜ𝛼0 from ℜ𝛼±𝛽, 𝛥ℜ𝛼±𝛽(ℜ𝛼±𝛽−ℜ𝛼0), equals to  𝑟𝛼±𝛽. Based on Eq. (2), we \nconclude  \n2\nPHE DC AC 23sin 22R j jM\n  . As such, theoretically , the coefficient \nindicating the magnitude of spin accumulation can be quantified by 𝛥ℜ𝛼±𝛽. \nMeasurements of the second harmonic Hall resistances ℜ𝛼±𝛽with respect to the \nazimuthal angle  of magnetization were carried out  in magnetic wire with stack s of Ta( 8 \nnm)/Co(2 nm)/Pt(5 nm) . The wire has IMA as evidenced by hysteresis loop measurements \nusing  both Kerr and anomalous Hall effects  [45]. For all measurements, a lock -in amplifier 7 \n was used to obtain the harmonic Hall voltage signals. T he second harmonic Hall resistance \nℜ𝛼±𝛽 is calculated by dividing  the second harmonic Hall voltage  with the magnitude of the \nAC. Only the Hall resistance mo dulation has been considered removing the offset \nresistance for each measurement.  Each measured  ℜ𝛼±𝛽 as well as the following Δℜ𝛼±𝛽 has \nbeen moved to be around 0 Ω by eliminating  a constant  offset  for easy comparison . A \nschematic of the measurement setup  is shown in Fig. 1(c) . The azimuthal angle of \nmagnetization of the wire depends on the applied field s as \next\nextarctany\nxH\nH\n , where \ntransvers Hy-ext sweep s from  −1800 Oe to +1 800 Oe  along y-axis while  ±Hx-ext keeps ± 560 \nOe to orientate  Mm along ±x-axis. In the following, we do not distinguish Hy-ext from  φ as \nφ is equivalent to Hy-ext, since the φ corresponds to unique Hy-ext for constant Hx-ext. The \nlongitudinal magnetic field  Hx-ext was used to ensure a uniform  magnetization along  the \nwire axis [45]. \nThe second harmonic Hall resistances , ℜ40,±6, with respect to the azimuthal angle  \nof magnetization, measured at Hx-ext = −560 Oe are presented  in Fig. 1( d). For AC-Js only, \nthe derived equation to represent  the second Hall resistance is given in Eq. 3 . Through \nsubstituting HꞱ, RAHE and RPHE in Eq. 3 with experimental values of 5790 Oe,  26 mΩ and \n6 mΩ , respectively,  the measured 𝑅40 is in good agreement with Eq. 3. A damping like term  \nHD,AC of 29 Oe and fieldlike term HF,AC of 4 Oe  are obtained , as shown in Fig. 1( d). This \ngood agreement  suggests that AC-Js \n  ,,F AC D ACHHp m p  result s in the symmetric \nbehavior of ℜ40 with respect to Hy-ext. For the second harmonic Hall resistance obtained  \nwith using both DC and AC concurrently , an asymmetric behavior around Hy-ext = 0 Oe  is \nobserved  for ℜ4±6. ℜ4+6 and ℜ4−6 are mirror symmetric to each other at Hy-ext = 0 Oe. These 8 \n variations indicate that both magnitude and sign of the DC bias in the wire contributes to \nthe corresponding  signals , ℜ4±6. The derived equation for the  second harmonic Hall \nresistance with DC bias is  still Eq. 3 [45]. As such, w e may expect to include DC induced \nHD,DC and HF,DC in Eq. 3 as offset s of HꞱ and Hy-ext to explain  the behavior of  ℜ4±6. However, \nthe revised Eq. 3 fails to fit ℜ4±6 with fitting root-mean -square error ( RMSE ) reaches \nminimum  as shown in Fig. 1(d) , as the measured  ℜ4±6 and the fitted  ℜ4±6 by Eq. 3  do not \noverlap . The failure clarifies the negligible role of DC -Js in the behavior of  ℜ4±6. \nThe differences, Δℜ4±6, are explored to investigate the behavior of ℜ4±6 with \nrespect to Hy-ext. Δℜ𝛼±𝛽 is computed by subtracting the second harmonic Hall resistance \nobtained without DC bias ( 𝑅𝛼0) from that with  DC bias  as Δℜ𝛼±𝛽=ℜ𝛼±𝛽−ℜ𝛼0, \nconsequently,  Δℜ4±6=ℜ4±6−ℜ40. As shown in Fig. 1(d), Δℜ4±6 adapts  the behavior \nsimilar to  the first harmonic Hall resistance  with respect to Hy-ext as shown in Fig. 1( f). The \nanalytical  expression of the first harmonic Hall resistance , which is mainly from PHE,  is \n1stHall PHE sin 2 RR  \n. We use \n66\n44 sin 2Z   to fit  Δℜ4±6, where  2𝑍4±6 equals to  the \ndifference between maximum and minimum values of Δℜ4±6, 60 µΩ . The experimental \nΔℜ4±6 are in good agreement  with \n6\n4sin 2Z  as shown in Fig. 1( d). R1stHall  is due to the \nmagnetization of the wire. Analogically, Δℜ𝛼±𝛽 is due to an extra magnetization  and \nsin 2Z\n  \n,                                                 (4) \nwhere 𝑍𝛼±𝛽 is the amplitude of Δℜ𝛼±𝛽. ℜ𝛼0 is given by Eq . 3. Respective of the expression \nof Δℜ𝛼±𝛽=ℜ𝛼±𝛽−ℜ𝛼0, ℜ𝛼±𝛽 is determined by both the pre -known AC -Js and the extra \nmagnetization.  9 \n In the following, w e confirm  the extra magnetization is  sm, as we further show  that \nΔℜ𝛼±𝛽 follows 𝑟𝛼±𝛽 with respect to the orientation of Mm,  𝑍𝛼±𝛽 is equal to the derived  𝑧𝛼±𝛽 \nof Eq. (2) , 𝑍𝛼±𝛽 with respect to JDC and JAC follows the predict ed behavior of or \n2\nPHE DC AC 23\n2z R j jM\n\n in experiments .  \nThe proposal  is validated  through analyzing  𝛥ℜ𝛼±𝛽 obtained with varying  magnetic \nvector of Hx-ext. Figure 2(a) shows 𝛥ℜ4±6 measured with Hx-ext = +560 Oe. As Hy-ext varies \nfrom −1800 Oe to +1800, the magnitude of 𝛥ℜ4+6 changes from −20 µΩ to +20 µΩ . In Fig. \n1(d) which represents 𝛥ℜ4+6 with Hx-ext = −560 Oe, 𝛥ℜ4+6 changes fr om +20 µΩ to −20 µΩ. \nThe change  of sign for 𝛥ℜ𝛼±𝛽 is present ed in the 𝛥ℜ4+6, as well as in 𝛥ℜ4−6. 𝛥ℜ𝛼±𝛽 follow \nR1stHall  or 𝑟𝛼±𝛽 to change their signs when Hx-ext is orientated along opposite directions. An \nalternative approach to substantiate the proposal is that  𝛥ℜ𝛼±𝛽 should follow  R1stHall  or 𝑟4±6 \nto present extremum at Hy-ext = ±Hx-ext. In experiments, the extremum s of R1stHall , ±6 mΩ, \nare at Hy-ext = ± 360 Oe and ± 1000 Oe, when the applied constant field Hx-ext equals to +360 \nOe and +1000 Oe, respectively, as shown in Fig. 2(b). Similarly, the extremum values of \n𝛥ℜ4+6, which are ± 30 µΩ, are at Hy-ext = ± 360 Oe as shown in Fig. 2(c) where Hx-ext is \napplied as +360 Oe and at ± 1000 Oe as shown in Fig. 2(d) where Hx-ext is applied as +1000 \nOe. \nAs such, experimentally on condition that 𝑍𝛼±𝛽 follows  a linear function of jDC and \njAC as predicted  by \n2\nPHE DC AC 23\n2R j jM , it is allowed to conclude th e extra magnetization is \nfrom the spin accumulation  sm. For different DC biases, ℜ4±[1 to 5] were measured to 10 \n compute Δℜ4±[1 to 5]. ℜ4±3 and Δℜ4±3 are exhibited  in Fig. 3(a) with Hx-ext = −560 Oe. ℜ4±3 \nand Δℜ4±3 show the behavior similar to ℜ4±6 and Δℜ4±6, respectively. 2𝑍4±3 is calculated to \nbe 30 µΩ less than 2𝑍4±6. Figure 3(b) shows all the computed 2𝑍4±[1 𝑡𝑜 5]. We find that  2𝑍4 \nis as a linear function of the DC density. For different AC biases, ℜ6,5,4,3,2,1±4 were measured \nto compute 𝛥ℜ6,5,4,3,2,1±4. ℜ4±4 and Δℜ4±4 as examples are presented, as shown in Fig. 3(c) \nwith Hx-ext = −560 Oe. ℜ4±4 and Δℜ4±4 show the behavior similar to ℜ4±6 and Δℜ4±6, \nrespectively. Figure 3(d) shows the computed resistances 2𝑍6,5,4,3,2,1±4. 2𝑍4±4 is ca lculated \nto be 4 4 µΩ lower than 2𝑍6±4 of 62  µΩ. 2𝑍±4 is as a linear function of the AC density. \nHence , we conclude th e extra magnetization is sm, and  \nr\n   as well as  \n  \n2\nPHE DC AC 23\n2Z z R j jM\n .                                    ( 5) \nThe coefficient, ζ, which indicates the capability of electric current inducing spin \naccumulation can be extracted from the measurement . ζ2 is proportional to 𝑍𝛼±𝛽 as shown  \nin Eq. ( 5). 𝑍𝛼±𝛽 is extracted from Hall resistances measured at various combinations of  AC \nand DC current densities. For each AC current density, the extracted Z shows a linear \nbehavior with respect to the DC current density. Through  carry ing out partial  derivative of  \n𝑍𝛼±𝛽 over jDC for each AC density , \nDCZ\nj\n\n  is obtained as shown in the inset of Fig. 4( a). \nDCZ\nj\n\n\n is a linear function to jAC. The slope  of \nDCZ\nj\n\n  with respect to or as a function of  jAC, \nAC DCZ\njj\n\n\n is calculated to be  1.3 µΩ∙[1010 Am−2]−2. We obtain  11 \n \n2\n2\nAC DC3\n2PHEZRj j M\n  from Eq. (5) . By substituting RPHE and M with 5.7 mΩ and 458 \nemu∙cc [45], respectively, ζ is computed to be 56 emu/cc per 1011 Am−2. For the sample \nstack with t=4, \nDCZ\nj\n\n  is shown in Fig. 4( b).  \nAC DCZ\njj\n\n  is obtained as 0.8 µΩ∙[1010 \nAm−2]−2, similarly, ζ is computed to be 45 emu/cc per 1011 Am−2, with  RPHE=5.6 mΩ and \nM=466 emu /cc [45]. For the sample stacks with t=2, 6, 10, \nDCZ\nj\n\n  is obtained at jAC=4× 1010 \nAm−2 as shown in Fig. 4(c). The expression of \nDCZ\nj\n\n  is \n2\nPHE AC 2\nDC3\n2ZRjjM\n  . Therefore, \nsubstituting jAC, RPHE and M with the ex perimental values in Fig. 4(c), we obtain ζ for the \nsample stacks with t=2, 6, 10, as shown in Fig. 4(d ) where  ζ of t=4 and 8 are also included. \nFor samples with t≤6, ζ is ~47 emu/cc per 1011 Am−2. For samples with  t>6, ζ increases  \nfrom 56 emu/cc per 1011 Am−2, reaching a maximum of 107 emu/cc per 1011 Am−2 at t = \n10. \nThe critical current density for SOT induced magnetization switching and domain \nwall driving is in the order of  1×1011 Am−2. The total current density in our experiment is \nin the same order for wires investigated.  For films with t=2, 4, 6, 8, and 10, the measured \nspin accumulation is 54 emu/cc, 45 emu/cc, 44 emu/cc,  56 emu/cc  and 107 emu/cc , \nrespectively. The corresponding local magnetization is 581  emu/cc,  446 emu/cc,  436 \nemu/cc,  458 emu/cc,  and 592 emu/cc. Therefore, t he ratio of the spin accumulation s to the \nlocal magnetization M, is ~10% for the films with t=2, 4 and 6, 12% for t=8, and 18% for \n10. The p ercentage indicates that the spin density induced by a current density of 1×1011 \nAm−2 is in the comparable order  to the local magnetization  in all the stacks , as we firstly 12 \n claim. As the initial orientation of spin accumulation is along y-axis, when the local \nmagnetization orientates along y-axis, the spin accumulation can vary the magnetization \nby as much  as 13% in the wire with t=10. Therefore, to switch magnetization and drive \ndomain wall motion by a current density of  1×1011 Am−2 is within expectation. If current \ndensity is increased to 1×1012 Am−2 which is the critical current for spin -transfer torque to \nswitch magnetization and drive domain wall  motion,  the spin accumulation can be 540 \nemu/cc, 450 emu/cc, 440 emu/cc,  560 emu/cc  and 1070 emu/cc  for films with t=2, 4, 6, 8, \nand 10, respectively. The magnitudes of spin accumulation are close to  the magnitudes of \nthe corresponding local magnetization.  In this case, the magnetic moments can be \nreorganized in the wires. Hence, the orientation of magnetization could be determined by \nthe spin accumulation .  \nAs shown in Fig. 4(d), ζ follows the spin Hall angles of the samples which have \nbeen reported in our previous work  with respect to the thickness of Ta layer  [45], \nirrespective of their magnitudes. ζ represents the extra magnetization generated by an \nelectric current, while spin Hall angle of Ta/Pt indicates the perc entage of spin current \nconverted by Ta/Pt in an electric current. As such, the same trend of ζ and spin Hall angle \nwith respect to the thickness of Ta is expected , since the extra magnetization should be as \na linear function to magnitude of spin current.  Therefore , the same trend  confirms the spin \naccumulation in Co laye r is from the Ta and Pt layers.  \n \nConclusion  13 \n We have experimentally quantified the s pin accumulation induced by electric \ncurrent in series  stack s of Ta/Co/Pt. We find that spin accumulation is around dozens of \npercentage of local magnetization when the current density is  1011 Am−2. As our results \ndemonstrate for the first time, when the spins from Ta and Pt are in relaxation state, they \nstill contribute to secon d harmonic Hall resistance, instead when they are in initial state \nonly as expected . The coefficient of spin accumulation over electric current is consistent \nwith spin Hall angle . This consistency  suggests that the coefficient can be used to evaluate  \nthe c apability of a heavy metal converting electric current to spin current. As the \nmeasurements are easily carried out, w e offer a concise solution  to estimate  spin \naccumulation.  \n \nAcknowledgements  \nThis work was supported by the Singapore National Research Foundation, Prime Minister's \nOffice, under a Competitive Research Programme (Non -volatile Magnetic Logic and \nMemory Integrated Circuit Devices, NRF -CRP9 -2011 -01), and an Industry -IHL \nPartnership Program (NRF2015 -IIP001 -001). The work was also supported by a MOE -\nAcRF Tier 2 Grant (MOE 2013 -T2-2-017). WSL is a member of the Singapore Spintronics \nConsortium (SG -SPIN).  \n \nReferences  \n[1] A. Manchon and S. Zhang, Phys. Rev. B 78, 212405 (2008); 79, 094422 (2009).  \n[2] E.I. Rashba, Sov. Phys. Solid State 2, 1224 (1960).  14 \n [3] M. I. Dyakonov and V. I. Perel, Phys. Lett. A 35, 459 (1971).  \n[4] Y. A. Bychkov and E. I. Rashba, J. Phys. C 17, 6039 (1984).  \n[5] J. Stohr and H.C. Siegmann, Magnetism from Fundamentals to Nanoscale Dynamics  \n(Springer -Verlag Berlin, Heidelberg,  2006), p. 107.  \n[6] I. M. Miron, G. Gaudin, S. Auffret, B. Rodmacq, A. Schuhl, S. Pizzini,   J. Vogel, and \nP. Gambardella, Nat. Mater. 9, 230 (2010).  \n[7] I. M. Miron, K. Garello, G. Gaudin, P. -J. Zermatten, M. V. Costache, S. Auffret, S. \nBandiera, B. Rodmac q, A. Schuhl, and P. Gambardella, Nature 476, 189 (2011).  \n[8] S. Emori, U. Bauer, S -M, Ahn, E. Martinez, and G. S. D. Beach, Nat. Mater. 12, 611 \n(2013).  \n[9] X. Fan, H. Celik, J. Wu, C. Ni, K. -J. Lee, V. O. Lorenz, and J. Q. Xiao, Nat. Commun. \n5, 3042 (2014 ). \n[10] T. D. Skinner, K. Olejní k, L. K. Cunningham, H. Kurebayashi, R. P. Campion, B. L. \nGallagher, T. Jungwirth, and A. J. Ferguson, Nat. Commun. 6, 6730 (2015).  \n[11] L. Berger, Phys. Rev. B 54, 9353 (1996).  \n[12] J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996); 247, 324 (2002).  \n[13] S. Zhang, P. M. Levy, and A. Fert, Phys. Rev. Lett. 88, 236601 (2002).  \n[14] S. Zhang and Z. Li, Phys. Rev. Lett. 93, 127204 (2004).  \n[15] K. Garello, I. M. Miron, C. O. Avci, F. Freimuth, Y. Mokrousov, S. Blu ¨ gel, S. A uffret, \nO. Boulle, G. Gaudin, and P. Gambardella, Nat. Nanotechnol. 8, 587 (2013).  15 \n [16] X. Wang and A. Manchon, Phys. Rev. Lett. 108, 117201 (2012).  \n[17] S. -M. Seo, K. -W. Kim, J. Ryu, H. -W. Lee, and K. -J. Lee, Appl. Phys. Lett. 101, \n022405 (2012).  \n[18] J. Kim, J. Sinha, M. Hayashi, M. Yamanouchi, S. Fukami, T. Suzuki, S. Mitani, and \nH. Ohno, Nat. Mater. 12, 240 (2013).  \n[19] M. Jamali, K. Narayanapillai, X. Qiu, L. M. Loong, A. Manchon, and H. Yang, Phys. \nRev. Lett. 111, 246602 (2013).  \n[20] M. Hayashi, J. Ki m, M. Yamanouchi, and H. Ohno, Phys. Rev. B 89, 144425, (2014).  \n[21] Satoru Emori, David C. Bono, and Geoffrey S. D. Beach, Appl. Phys. Lett. 101, \n042405 (2012).  \n[22] L. Q. Liu, O. J. Lee, T. J. Gudmundsen, D. C. Ralph, and R. A. Buhrman, Phys. Rev. \nLett. 109, 096602 (2012).  \n[23] S. Woo, M. Mann, A. J. Tan, L. Caretta, and G. S. D. Beach, Appl. Phys. Lett. 105, \n212404 (2014).  \n[24] H. R. Lee, K. Lee, J. Cho, Y. H. Choi, C. Y. You, M. H. Jung, F. Bonell, Y. Shiota, S. \nMiwa, and Y. Suzuki, Sci. Rep. 4, 6548 (2 014).  \n [25] X. Qiu, P. Deorani, K. Narayanapillai, K. -S. Lee, K. -J. Lee, H. -W. Lee, and H. Yang, \nSci. Rep. 4, 4491 (2014).  \n[26] M. Montazeri, P. Upadhyaya, M. C. Onbasli, G. Yu, K. L. Wong, M. Lang, Y. Fan, \nX. Li, P. K. Amiri, R. N. Schwartz, C. A. Ross, a nd K. L. Wang, Nat. Commun. 6, 8958 \n(2015).  16 \n [27] X. Qiu, K. Narayanapillai, Y. Wu, P. Deorani, D. H. Yang, W. S. Noh, J. H. Park, K. \nJ. Lee, H. W. Lee, and H. Yang, Nat. Nanotechnol. 10, 333 (2015).  \n[28] D. Chiba, M. Kawaguchi, S. Fukami, N. Ishiwata, K. Shimamura, K. Kobayashi, and  \nT. Ono, Nat. Commun. 3, 888 (2012).  \n[29] P. P. J. Haazen, E. Mure, J. H. Franken, R. Lavrijsen, H. J. M. Swagten, and B. \nKoopmans, Nat. Mater. 12, 299 (2013).  \n[30] K. S. Ryu, L. Thomas, S. H. Yang, and S. Parkin, Nat. Nanotechn ol. 8, 527 (2013).  \n[31] T. Kimura, Y. Otani, T. Sato, S. Takahashi, and S. Maekawa, Phys. Rev. Lett. 98, \n156601 (2007).  \n[32] K. Ando, S. Takahashi, K. Harii, K. Sasage, J. Ieda, S. Maekawa, and E. Saitoh, Phys. \nRev. Lett. 101, 036601 (2008).  \n[33] O. Mosend z, J. E. Pearson, F. Y. Fradin, G. E. W. Bauer, S. D. Bader, and A. \nHoffmann, Phys. Rev. Lett. 104, 046601 (2010).  \n[34] L. Liu, T. Moriyama, D. C. Ralph, and R. A. Buhrman, Phys. Rev. Lett. 106, 036601 \n(2011).  \n[35] L. Liu, C. F Pai, Y. Li, H. W. Tseng, D. C. Ralph, and R. A. Buhrman, Science 336, \n6081   (2012).  \n[36] G. Allen, S. Manipatruni, D. E. Nikonov, M. Doczy, and I. A. Young, Phys. Rev. B \n91, 144412 (2015).  \n[37] K. Ueda, C. F. Pai, A. J. Tan, M. Mann, and G. S. D. Beach, Appl. Phys. Lett. 108, \n232405 (2016).  17 \n [38] X. Qiu, W. Legrand, P. He, Y. Wu, J. Yu, R. Ramaswamy, A. Manchon, and H. Yang, \nPhys. Rev. Lett. 117, 217206 (2016).  \n[39] T. Suzuki , S. Fukami, N. Ishiwata, M. Yamanouchi, S. Ikeda, N. Kasai, and H. Ohno, \nAppl. Phys. Lett. 98, 142505 (2011).  \n[40] V. D. Ky, and I. A. N SSR, Ser. Fiz. 29(4), 576 (1965).  \n[41] M. L. Yu, and J. T. H. Chang, J. Phys. Chem. Solids 31, 1997 (1970).  \n[42] V. D. Ky,  Phys. Stat. Sol. 26, 565 (1968).  \n[43] T. R. McGuire and R. I. Potter, IEEE Trans. Magn. 11, 1018 (1975).  \n[44] H. X. Tang, R. K. Kawakami, D. D. Awschalom, and M. L. Roukes, Phys. Rev. Lett. \n90, 107201 (2003).  \n[45] F. Luo, S. Goolaup, W. C. Law, S. Li, F. Tan, C. Engel, T. Zhou, W. S . Lew, Phys. \nRev. B. 95,174415 (2017).  \n \n \nFigures and captions  \n    \n  18 \n \n  \n  \n  \n  \nFig. 1 (a) the transfer of spin torque from s to M in dozens of nanoseconds; (b) s relaxes to be along \nM after SOT transferring; (c) schematic of measurement setup; ( d) measured ℜ4±6 and obtained  \nΔℜ4±6 with respect to Hy-ext, fit of Δℜ4±6 indicate that Δℜ4±6 is fitted by Eq. (2) where Hx-ext is 560 \nOe; ( e) measured ℜ4±6 with respect to Hy-ext, fit of Δℜ4+6 indicate that Δℜ4+6 is fitted to reaching \nminimum RMSE by Eq. (1) with changing Hx-ext to 560+69 Oe and Hy-ext to 560+38 Oe, fit of Δℜ4−6 \nindicate that Δℜ4−6 is fitted to reaching minimum RMSE by Eq. (1) with changing Hx-ext to 560+60 \nOe and Hy-ext to 560−39 Oe ; (f) measured first harmonic Hall resistances with respect to Hy-ext when \nobtaining ℜ4+6 with Hx-ext =± 560 Oe.  In (d), the magenta and violet lines show the fit to the \nexperimental data .  19 \n \n  \n  \n \nFig. 2 ( a) measured ℜ4±6 and obtained Δℜ4±6 with respect to Hy-ext when Hx-ext=−560 Oe; (b) \nobtained first harmonic Hall resistance  with respect to Hy-ext when measuring \ncorresponding ℜ4±6 with Hx-ext equating to +360 Oe  and +1000 Oe; (c) measured \nℜ4±6 and obtained  Δℜ4±6 with respect to Hy-ext when Hx-ext=+360 Oe; (d) measured \nℜ4±6 and obtained  Δℜ4±6 with respect to Hy-ext when Hx-ext=+1000 Oe.  In (a), (c) and (d), the \nmagenta and violet lines show the fit to the experimental data . \n \n 20 \n \n \n \nFig. 3 (a) measured ℜ4±3 and obtained  Δℜ4±3 with respect to Hy-ext when Hx-ext=−560 Oe; \n(b) calculated 𝑍4±𝛽 with respect to the DC offset β when the AC density is fixed to be \n4× 1010 Am−2; (c) measured ℜ4±4 and obtained  Δℜ4±4 with respect to Hy-ext when Hx-\next=−560 Oe; (d) calculated 𝑍𝛼±4 with respect to the AC α when the DC densities are fixed \nto be ± 4× 1010 Am−2. In (a) and (c), the magenta and violet lines show the fit to the experimental \ndata. \n \n 21 \n \n \n \nFig. 4 calculated 𝑍𝛼±𝛽 with respect to the DC offset β under different AC densities, inset is \nthe slope of  𝑍𝛼±𝛽 to β with respect to α for sample (a) Ta(8 nm)/Co(2 nm)/Pt(5 nm) and \nsample (b) Ta(4 nm)/Co(2 nm)/Pt(5 nm); (c) calculated 𝑍4±𝛽 with respect to the DC offset \nβ for the samples of Ta( t nm)/Co(2 nm)/Pt(5 nm) with t=2, 6 and 10; (d) calculated ζ  (black \nline) and reported spin Hall angle (red line) with respect to the thickness of Ta.  \n "    },    {        "title": "2002.02415v1.Correlated_motion_of_particle_hole_excitations_across_the_renormalized_spin_orbit_gap_in___rm_Sr_2_Ir_O_4_.pdf",        "content": "arXiv:2002.02415v1  [cond-mat.str-el]  5 Feb 2020Correlated motion of particle-hole excitations\nacross the renormalized spin-orbit gap in Sr2IrO4\nShubhajyoti Mohapatra1and Avinash Singh1,∗\n1Department of Physics, Indian Institute of Technology, Kanpu r - 208016, India\n(Dated: February 7, 2020)\nThe high-energy collective modes of particle-hole excitat ions across the spin-orbit\ngap in Sr 2IrO4are investigated using the transformed Coulomb interactio n terms\nin the pseudo-spin-orbital basis constituted by the J= 1/2 and 3/2 states arising\nfrom spin-orbit coupling. With appropriate interaction st rengths and renormalized\nspin-orbit gap, these collective modes yield two well-defin ed propagating spin-orbit\nexciton modes, with energy scale anddispersioninexcellen t agreement with resonant\ninelastic X-ray scattering (RIXS) measurements.\nPACS numbers: 75.30.Ds, 71.27.+a, 75.10.Lp, 71.10.Fd2\nI. INTRODUCTION\nThe iridium based transition-metal oxides exhibiting novel J=1/2 Mott insulating states\nhave attracted considerable interest in recent years in view of the ir potential for host-\ning collective quantum states such as quantum spin liquids, topologica l orders, and high-\ntemperature superconductors.1The effective J=1/2 antiferromagnetic (AFM) insulating\nstateiniridates arises fromanovel interplay between crystal field , spin-orbit coupling (SOC)\nand intermediate Coulomb correlations. Exploration of the emerging quantum states in the\niridate compounds therefore involves investigation of the correlat ed spin-orbital entangled\nelectronic states and related magnetic properties.\nAmongtheiridiumcompounds, thequasi-two-dimensional (2D)squa re-latticeperovskite-\nstructured iridate Sr 2IrO4is of special interest as the first spin-orbit Mott insulator to be\nidentified and because of its structural and physical similarity with L a2CuO4.2,3It exhibits\ncanted AFM ordering of the pseudospins below N´ eel temperature TN≈240 K. The canting\nof the in-plane magnetic moments tracks the staggered IrO 6octahedral rotations about the\ncaxis. The effectively single (pseudo) orbital ( J=1/2) nature of this Mott insulator has\nmotivated intensive finite doping studies aimed at inducing the superc onducting state as in\nthe cuprates.4–10\nTechnological advancements and improved energy resolution in res onant inelastic X-ray\nscattering (RIXS) have been instrumental in the elucidation of the pseudospin dynamics in\nSr2IrO4. Recent measurements point to a partially resolved ∼30 meV magnon gap at the\nΓ point,11which has been further resolved via high-resolution RIXS and inelast ic neutron\nscattering (INS), bothof which indicate another magnongapbetw een 2 to 3 meV at ( π,π).12\nThese low-energyfeaturescorrespondtodifferent magnonmode sassociatedwithbasal-plane\nand out-of-plane fluctuations, indicating the presence of anisotr opic spin interactions. In\naddition to magnon modes, RIXS experiments have also revealed a hig h-energy dispersive\nfeature in the energy range 0.4-0.8 eV. Attributed to electron-ho le pair excitations across\nthe spin-orbit gap between the J=1/2 and 3/2 bands, this distinctive mode is referred to as\nthe spin-orbit exciton.13–17\nAmong the theoretical approaches, the spin-orbit exciton was ide ntified as a bound state\ninthespectral functionofthetwo-particleGreen’sfunctionwithin themulti-orbitalitinerant\nelectron picture.16However, the full dispersion was not obtained, and the original t2gbasis3\nwas employed instead of the more natural SOC-split Jstates with intrinsic spin-orbit gap.\nIn another approach, the exciton dispersion was obtained in analog y with hole motion in an\nAFM background.13,15However, the bare exciton dispersion was neglected, and an appro ach\nwhich allows for a unified description of both magnon and spin-orbit ex citon on the same\nfooting will be desirable as both excitations are observed in the same RIXS measurements.\nIn this paper, we therefore plan to investigate the correlated mot ion of inter-orbital\nparticle-hole excitations across the renormalized spin-orbit gap (b etweenJ=1/2 andJ=3/2\nsectors), along with detailed comparison with RIXS data for the spin -orbit exciton modes\nin Sr2IrO4. Similar comparison for the magnon dispersion involving intra-orbital (J=1/2)\nparticle-hole excitations has provided experimental evidence of se veral distinctive features\nassociated with the rich interplay of spin-orbit coupling, Coulomb inte raction, and realistic\nmulti-orbital electronicbandstructure, such as(i)finite- Uandfinite-SOCeffects, (ii)mixing\nand coupling between the J=1/2 and 3/2 sectors, and (iii) Hund’s-coupling-induced true\nmagnetic anisotropy and magnon gap.18–20\nThestructure ofthepaper isasfollows. After a briefaccount oft hetransformedCoulomb\ninteraction terms in the pseudo-spin-orbital basis in Sec. II, the A FM state of the three\norbital model is discussed in Sec. III. The spin-orbit gap renormaliz ation due to the relative\nenergy shift between the J=1/2 and 3/2 sectors arising from the density interaction terms\nis discussed in Sec. IV. The spin-orbit exciton as a resonant state f ormed by the correlated\npropagation of the inter-orbital, spin-flip, particle-hole excitation across the renormalized\nspin-orbit gap is investigated in Sec. V. Finally, conclusions are prese nted in Sec. VI.\nII. COULOMB INTERACTION IN THE PSEUDO-SPIN-ORBITAL BASIS\nDue to large crystal-field splitting ( ∼3 eV) in the IrO 6octahedra, the low-energy physics\nind5iridates is effectively described by projecting out the empty e glevels which are well\nabove thet2glevels. Spin-orbit coupling (SOC) further splits the t 2gstates into (upper)\nJ=1/2 doublet and (lower) J=3/2 quartet with an energy gap of 3 λ/2. Four of the five\nelectrons fill the J=3/2 states, leaving one electron for the J=1/2 sector, rendering it mag-\nnetically active in the ground state.\nThe three Kramers pairs above correspond to pseudo orbitals (l= 1,2,3) withpseudo4\nFIG. 1: The pseudo-spin-orbital energy level scheme for the three Kramers pairs along with their\norbital shapes. The colors represent the weights of real spi n↑(red) and ↓(blue) in each pair.\nspins(τ=↑,↓) each, with the |J,mj/angbracketrightand corresponding |l,τ/angbracketrightstates having the form:\n|l= 1,τ=σ/angbracketright=/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\n2,±1\n2/angbracketrightbigg\n= [|yz,¯σ/angbracketright±i|xz,¯σ/angbracketright±|xy,σ/angbracketright]/√\n3\n|l= 2,τ=σ/angbracketright=/vextendsingle/vextendsingle/vextendsingle/vextendsingle3\n2,±1\n2/angbracketrightbigg\n= [|yz,¯σ/angbracketright±i|xz,¯σ/angbracketright∓2|xy,σ/angbracketright]/√\n6\n|l= 3,τ= ¯σ/angbracketright=/vextendsingle/vextendsingle/vextendsingle/vextendsingle3\n2,±3\n2/angbracketrightbigg\n= [|yz,σ/angbracketright±i|xz,σ/angbracketright]/√\n2 (1)\nwhere|yz,σ/angbracketright,|xz,σ/angbracketright,|xy,σ/angbracketrightare the t 2gstates and the signs ±correspond to spins σ=↑/↓.\nThe coherent superposition of different-symmetry t2gorbitals, with opposite spin polariza-\ntion between xz/yzandxylevels implies spin-orbital entanglement, and also imparts unique\nextended 3D shape to the pseudo-orbitals l= 1,2,3, as shown in Fig 1.\nInverting the above transformation, the three real-spin-orbita l basis states can be repre-\nsented in terms of the pseudo-spin-orbital basis states, given be low in terms of the corre-\nsponding creation operators:\n\na†\nyzσ\na†\nxzσ\na†\nxyσ\n=\n1√\n31√\n61√\n2\niσ√\n3iσ√\n6−iσ√\n2\n−σ√\n3√\n2σ√\n30\n\na†\n1τ\na†\n2τ\na†\n3τ\n(2)\nwhere,σ=↑/↓andτ=σ.\nWe consider the on-site Coulomb interaction terms:\nHint=U/summationdisplay\ni,µniµ↑niµ↓+U′/summationdisplay\ni,µ<ν,σniµσniνσ+(U′−JH)/summationdisplay\ni,µ<ν,σniµσniνσ\n+JH/summationdisplay\ni,µ/negationslash=ν(a†\niµ↑a†\niν↓aiµ↓aiν↑+a†\niµ↑a†\niµ↓aiν↓aiν↑) (3)5\nin the real-spin-orbital basis ( µ,ν=yz,xz,xy ), including the intra-orbital ( U) and inter-\norbital (U′) density interaction terms, the Hund’s coupling term ( JH), and the pair hopping\nterm (JH). Herea†\niµσandaiµσare the creation and annihilation operators for site i, orbital\nµ, spinσ=↑,↓, and the density operator niµσ=a†\niµσaiµσ.\nUsing the transformation from the t2gbasis to the pseudo-spin-orbital basis given above,\nand keeping the Hubbard, density, and Hund’s coupling like interactio n terms which are\nrelevant for the present study, we obtain (for site i):\nHint(i) =1\n2/summationdisplay\nm,m′,τ,τ′Uττ′\nmm′nmτnm′τ′+/parenleftbiggU−U′\n3/parenrightbigg/summationdisplay\nτa†\n1τa†\n2τa1τa2τ\n+/parenleftbiggU−2JH−U′\n6/parenrightbigg/summationdisplay\nτ/parenleftig\na†\n2τa†\n3τa2τa3τ+2a†\n3τa†\n1τa3τa1τ/parenrightig\n(4)\nwhere the transformed interaction matrices Uττ′\nmm′in the new basis ( m,m′= 1,2,3):\nUττ\nmm′=\n0U′U′−2\n3JH\nU′0U′−1\n3JH\nU′−2\n3JHU′−1\n3JH0\n,\nUττ\nmm′=\n1\n3(U+2U′)1\n3(U+2U′−3JH)1\n3(U+2U′−JH)\n1\n3(U+2U′−3JH)1\n2(U+U′)1\n6(U+5U′−4JH)\n1\n3(U+2U′−JH)1\n6(U+5U′−4JH)1\n2(U+U′)\n(5)\nfor pseudo-spins τ′=τandτ′=τ, whereτ=↑,↓. Similar transformation to the Jbasis\nhas been discussed recently, focussing only on the density interac tion terms.21\nUsing the spherical symmetry condition ( U′=U-2JH), the transformed interaction Hamil-\ntonian (4) simplifies to:\nHint(i) =/parenleftbigg\nU−4\n3JH/parenrightbigg\nn1↑n1↓+(U−JH)[n2↑n2↓+n3↑n3↓]\n−4\n3JHS1.S2+2JH[Sz\n1Sz\n2−Sz\n1Sz\n3]\n+/parenleftbigg\nU−13\n6JH/parenrightbigg\n[n1n2+n1n3]+/parenleftbigg\nU−7\n3JH/parenrightbigg\nn2n3. (6)\nThe symmetry features of the interaction terms above are consis tent with a general pseudo-\nspin rotation symmetry analysis which shows that the Hund’s coupling (JH) and pair-\nhopping (JH) interaction terms in Eq. (3) explicitly break this symmetry systema tically,\nwhile the Hubbard ( U) and density ( U′) interaction terms do not.226\nIII. ANTIFERROMAGNETIC STATE OF THE THREE-ORBITAL MODEL\nWe consider the various interaction terms in Eq. (6) in the Hartree- Fock (HF) approxi-\nmation, focussing on the staggered field terms corresponding to ( π,π) ordered AF state on\nthe square lattice. The charge terms corresponding to density co ndensates will be discussed\nin the next section. For general ordering direction with component s∆l= (∆x\nl,∆y\nl,∆z\nl), the\nstaggered field term for sector lin the pseudo-orbital basis is given by:\nHsf(l) =/summationdisplay\nksψ†\nkls/parenleftig\n−sτ.∆l/parenrightig\nψkls=/summationdisplay\nks−sψ†\nkls\n∆z\nl∆x\nl−i∆y\nl\n∆x\nl+i∆y\nl−∆z\nl\nψkls (7)\nwhereψ†\nkls= (a†\nkls↑a†\nkls↓),s=±1 for the two sublattices A/B, and the staggered field\ncomponents ∆α=x,y,z\nl=1,2,3are self-consistently determined from:\n2∆α\n1=U1mα\n1+2JH\n3mα\n2+JH(mα\n3−mα\n2)δαz\n2∆α\n2=U2mα\n2+2JH\n3mα\n1−JHmα\n1δαz\n2∆α\n3=U3mα\n3+JHmα\n1δαz (8)\nin terms of the staggered pseudo-spin magnetization components mα=x,y,z\nl=1,2,3. In practice, it is\neasier to choose set of ∆l=1,2,3and self-consistently determine the Hubbard-like interaction\nstrengths Ul=1,2,3such that U1=U−4\n3JHandU2=U3=U−JHusing Eq. (8).\nTransforming the staggered-field term back to the three-orbita l basis (yzσ,xzσ,xy ¯σ),\nand including the SOC and band terms,23the full HF Hamiltonian considered in our band\nstructure and spin fluctuation analysis is given by HHF=HSO+Hband+Hsf, where,\nHband=/summationdisplay\nkσsψ†\nkσs\n\nǫyz\nk′0 0\n0ǫxz\nk′0\n0 0ǫxy\nk′\nδss′+\nǫyz\nkǫyz|xz\nk0\n−ǫyz|xz\nkǫxz\nk0\n0 0 ǫxy\nk\nδ¯ss′\nψkσs′ (9)\nin the composite three-orbital, two-sublattice basis, showing the d ifferent hopping terms\nconnecting the same and opposite sublattice(s). Corresponding t o the hopping terms in the7\ntight-binding model, the various band dispersion terms in Eq. (9) are given by:\nǫxy\nk=−2t1(coskx+cosky)\nǫxy\nk′=−4t2coskxcosky−2t3(cos2kx+cos2ky)+µxy\nǫyz\nk=−2t5coskx−2t4cosky\nǫxz\nk=−2t4coskx−2t5cosky\nǫyz|xz\nk=−2tm(coskx+cosky). (10)\nHeret1,t2,t3are respectively the first, second, and third neighbor hopping ter ms for the\nxyorbital, which has energy offset µxyfrom the degenerate yz/xzorbitals induced by the\ntetragonal splitting. For the yz(xz) orbital,t4andt5are the NN hopping terms in y(x)\nandx(y) directions, respectively. Mixing between xzandyzorbitals is represented by the\nNN hopping term tm. We have taken values of the tight-binding parameters ( t1,t2,t3,t4,\nt5,tm,µxy,λ) = (1.0, 0.5, 0.25, 1.028, 0.167, 0.0, -0.7, 1.35) in units of t1, where the energy\nscalet1= 280 meV. Using above parameters, the calculated electronic band structure shows\nAFM insulating state and mixing between pseudo-orbital sectors.18,23As the pseudo-spin\ncanting is not relevant for the following discussion, we have set tmto zero by going to the\nlocally rotated coordinate frame.\nTo illustrate the AF state calculation, we have taken staggered field values ∆x\nl=1,2,3=\n(0.92,0.08,−0.06) in units of t1, which ensures self-consistency for all three orbitals, with\nthe given relations U2=U3=U1+JH/3. Using the calculated sublattice magnetization val-\nuesmx\nl=1,2,3=(0.65,0.005,-0.038), we obtain Ul=1,2,3=(0.80,0.83,0.83) eV, which finally yields\nU=U1+4\n3JH=0.93 eV for JH=0.1 eV. For these parameter values, the calculated magnon\ndispersion and energy gap are in very good agreement with RIXS mea surements.11–13,15The\neasyx-yplane anisotropy arising from Hund’s coupling results in energy gap ≈40 meV for\nthe out-of-plane ( z) magnon mode.19\nThe electron fillings are obtained as nl=1,2,3≈(1.064,1.99,1.946) in the three pseudo\norbitals. Finitemixing betweenthe J=1/2and3/2sectorsisreflectedinthesmalldeviations\nfrom ideal fillings and also in the very small magnetic moment values for l= 2,3 as given\nabove, which play a crucial role in the expression of true magnetic an isotropy and magnon\ngap in view of the Hund’s coupling induced anisotropic interactions in Eq . (6). The values\nλ=0.38 eV,U=0.93 eV, and JH=0.1 eV taken above lie well within the estimated parameter\nrange for Sr 2IrO4.16,248\nIV. RENORMALIZED SPIN-ORBIT GAP\nIn this section, we obtain the relative energy shift between the J=1/2 and 3/2 states\narising from the transformed density interaction terms in Eq. (6. T his relative shift effec-\ntively renormalizes the spin-orbit gap and plays an important role in de termining the energy\nscale of the spin-orbit exciton, as discussed in the next section. Co rresponding to the total\ndensity condensate /angbracketleftnl↑+nl↓/angbracketrightin the HF approximation of the density interaction terms, the\nspin-independent self-energy contributions for the three pseud o orbitals are obtained as:\nΣl=1\ndens=U/angbracketleftbigg1\n2n1+n2+n3/angbracketrightbigg\n−JH/angbracketleftbigg2\n3n1+13\n6n2+13\n6n3/angbracketrightbigg\nΣl=2\ndens=U/angbracketleftbigg\nn1+1\n2n2+n3/angbracketrightbigg\n−JH/angbracketleftbigg13\n6n1+1\n2n2+7\n3n3/angbracketrightbigg\nΣl=3\ndens=U/angbracketleftbigg\nn1+n2+1\n2n3/angbracketrightbigg\n−JH/angbracketleftbigg13\n6n1+7\n3n2+1\n2n3/angbracketrightbigg\n(11)\nThe formally unequal contributions will result in relative energy shift s between the three\norbitals depending on the electron filling. With /angbracketleftn1/angbracketright=1 and/angbracketleftn2/angbracketright=/angbracketleftn3/angbracketright=2 for thed5system\nhaving nominally half-filled and filled orbitals, the relative energy shift:\n∆dens= Σl=1\ndens−Σl=2,3\ndens=U−3JH\n2(12)\nbetweenl=1 and (degenerate) l=2,3 orbitals.\nForU >3JH, the relative energy shift enhances the energy gap between J=1/2 and\n3/2 sectors, effectively resulting in a correlation-induced renormaliza tion of the spin-orbit\ngap and the spin-orbit coupling. The SOC strength is renormalized as ˜λ=λ+ 2∆dens/3\nby the relative energy shift. With ∆ dens= (U−3JH)/2≈0.3 eV for the parameter values\nconsideredearlier, weobtain ˜λ≈0.6eV,whichisinagreementwiththecorrelation-enhanced\nSOC strength obtained in a recent DFT study of Sr 2IrO4.24Ford4systems with nominally\n/angbracketleftn1/angbracketright=0, the relative energy shift increases to U−3JH. This enhancement of the spin-orbit\ngap renormalization is seen in recent DFT study of the hexagonal irid ates Sr 3LiIrO6and\nSr4IrO6with Ir5+(5d4) and Ir4+(5d5) ions, respectively.25\nV. SPIN-ORBIT EXCITON\nMagnon excitations modes in Sr 2IrO4essentially involve collective modes of intra-orbital,\nspin-flip, particle-hole excitations within the magnetically active J=1/2 sector.18,19In anal-9\nFIG. 2: Propagator of inter-orbital, spin-flip, particle-h ole excitations across the renormalized\nspin-orbit gap between the nominally filled J=3/2 sector and the half-filled J=1/2 sector.\nogy, we will investigate here the collective modes of inter-orbital, pa rticle-hole excitations\nacross the renormalized spin-orbit gap between the nominally filled J=3/2 sector and the\nhalf-filledJ=1/2 sector. We will consider both pseudo-spin-flip and non-pseud o-spin-flip\ncases for these spin-orbit exciton modes. Starting first with the s pin-flip case, we consider\nthe composite pseudo-spin-orbital fluctuation propagator in the z-ordered AFM state:\nχ−+\nso(q,ω) =/integraldisplay\ndt/summationdisplay\nieiω(t−t′)e−iq.(ri−rj)/angbracketleftΨ0|T[S−\ni,m,n(t)S+\nj,m,n(t′)]|Ψ0/angbracketright (13)\ninvolving the inter-orbital spin-raising and -lowering operators S+\nj,m,n=a†\njm↑ajn↓and\nS−\ni,m,n=a†\nin↓aim↑at lattice sites jandi, describing the propagation of a spin-flip, particle-hole\nexcitation between different pseudo orbitals mandn. Althoughthe most general propagator\nwould involve S−\ni,m,nandS+\nj,m′,n′, the above simplified propagator is a good approximation\nin view of the orbital restrictions on the particle-hole states as disc ussed below. Also, we\nhave considered the z-ordered AFM state for simplicity as the JH-induced weak easy-plane\nanisotropy has negligible effect on the spin-orbit exciton.\nIn the ladder-sum approximation, the spin-orbital propagator is o btained as:\n[χ−+\nso(q,ω)] =[χ0\nso(q,ω)]\n1−U[χ0\nso(q,ω)](14)\nwhere the relevant interactions U=Uττ\nmnfor the spin-flip particle-hole pair are given in\nEq. (5). The ladder-sum approximation with repeated (attractive ) interactions (as shown\nin Fig. (2) for the retarded case) represents resonant scatter ing of the particle-hole pair,\nresulting in a resonant state split-off from the particle-hole continu um, which we identify as\nthe spin-orbit exciton modes.10\nThe bare particle-hole propagator in the above equation:\n[χ0\nso(q,ω)]mn\nss′=/summationdisplay\nk/bracketleftigg\n/angbracketleftϕn\nk−q|τ−|ϕm\nk/angbracketrights/angbracketleftϕm\nk|τ+|ϕn\nk−q/angbracketrights′\nE+\nk−q−E−\nk+ω−iη+/angbracketleftϕn\nk−q|τ−|ϕm\nk/angbracketrights/angbracketleftϕm\nk|τ+|ϕn\nk−q/angbracketrights′\nE+\nk−E−\nk−q−ω−iη/bracketrightigg\n(15)\nwasevaluatedinthetwo-sublattice basisbyintegratingoutthefer mionsinthe( π,π)ordered\nstate. Here Ekandϕkare the eigenvalues and eigenvectors of the Hamiltonian matrix in\nthe pseudo-spin-orbital basis, and the Eksuperscript +( −) refers to particle (hole) energies\nabove (below) the Fermi energy. The projected amplitudes ϕm\nkτabove were obtained by\nprojecting the kstates in the three-orbital basis |µ,σ/angbracketrighton to the pseudo-orbital basis |m,τ/angbracketright\ncorresponding to the J= 1/2 and 3/2 sector states, as given below:\nϕ1\nk↑=1√\n3/parenleftbig\nφyz\nk↓−iφxz\nk↓+φxy\nk↑/parenrightbig\nϕ1\nk↓=1√\n3/parenleftbig\nφyz\nk↑+iφxz\nk↑−φxy\nk↓/parenrightbig\nϕ2\nk↑=1√\n6/parenleftbig\nφyz\nk↓−iφxz\nk↓−2φxy\nk↑/parenrightbig\nϕ2\nk↓=1√\n6/parenleftbig\nφyz\nk↑+iφxz\nk↑+2φxy\nk↓/parenrightbig\nϕ3\nk↑=1√\n2/parenleftbig\nφyz\nk↓+iφxz\nk↓/parenrightbig\nϕ3\nk↓=1√\n2/parenleftbig\nφyz\nk↑−iφxz\nk↑/parenrightbig\n(16)\nin terms of the amplitudes φµ\nkσin the three-orbital basis ( µ=yz,xz,xy ). The [χ0(q,ω)]\nmatrix was evaluated by performing the ksum over the 2D Brillouin zone divided into a\n300×300 mesh.\nThe dominant contribution to [ χ0\nso(q,ω)] above will correspond to particle (+) states in\nthe nominally half-filled pseudo-orbital m=1 (J=1/2 sector) and hole ( −) states in the nom-\ninally filled pseudo-orbitals n=2,3 (J=3/2 sector). Due to these restrictions, the bare propa-\ngatoressentially becomes diagonalinthecomposite particle-holeor bitalbasis( m′=m,n′=n),\nwhich justifies the simplified propagator considered above. In orde r to focus exclusively on\nthe high-energy spin-orbit exciton modes, particle-hole excitation s within the J=1/2 sector\n(which yield the low-energy magnon modes) have been excluded.\nFig. 3 shows the spin-orbit exciton spectral function:\nAq(ω) =1\nπIm Tr/bracketleftbig\nχ−+\nso(q,ω)/bracketrightbig\n(17)\nas an intensity plot for qalong the high symmetry directions of the BZ. For clarity, we have\nconsidered here the particle-hole propagator in Eq. (15) separat ely for (m,n)=(1,3) and\n(1,2). The relevant interaction terms for these two cases are: Uττ\n13=U-5JH/3 andUττ\n12=U-\n7JH/3. Here, we have taken U=0.93 eV and JH=0.1 eV as obtained in Sec. III, and the\nrenormalized spin-orbit gap (Sec. IV) has been incorporated.11\n(π/2,π/2)(π,0) (π,π) (π/2,π/2) (0,0) ( π,0) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9ω  (eV)\n 0.001 0.01 0.1 1 10\nn = 3 (a)\n(π/2,π/2)(π,0) (π,π) (π/2,π/2) (0,0) ( π,0) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9ω  (eV)\n 0.001 0.01 0.1 1 10\nn = 2 (b)\nFIG. 3: The spin-orbit exciton spectral function Aq(ω) for the two cases: (a) ( m,n)=(1,3) and (b)\n(m,n)=(1,2), showing well defined dispersive modes near the lowe r edge of the continuum. The\nexciton represents collective spin-orbital excitations a cross the renormalized spin-orbit gap.\nThespin-orbitexcitonspectralfunctioninFig. 3(a)clearlyshowsa welldefinedpropagat-\ning mode near the lower edge of the continuum with significantly higher intensity compared\nto the continuum background. With increasing interaction strengt h, this mode progressively\nshifts to lower energy further away from the continuum, and beco mes less dispersive and\nmore prominent in intensity, indicating enhanced localization of the sp in-orbit exciton.\nFig. 3(b) shows a similar exciton mode for the other case ( m,n)=(1,2), with slightly\nhigher energy and reduced dispersion as well as significant damping. The relatively reduced\ninteraction strength Uττ\n12for this mode accounts for the slightly higher energy. We have\nsimilarly obtained the spectral functions for the non-spin-flip case s by considering operators\nnτ\nj,m,n=a†\njmτajnτandnτ†\ni,m,n=a†\ninτaimτinstead ofS+\nj,m,nandS−\ni,m,nin Eq. (13) with appropriate\ninteractions Uττ\nmn. The spectral functions for these cases are nearly identical, as e xpected\nfrom the non-magnetic character of the filled J=3/2 sector.\nThe calculated dispersion and energy scale of the two spin-orbit exc iton modes are in\nexcellent agreement with RIXS measurements in Sr 2IrO4.11,13Comparison of the calculated\nAq(ω) with the observed RIXS intensity and its momentum dependence is b eyond the scope\nof this work. The basic RIXS mechanism involved in the creation of the spin-orbit exciton,\nwhose propagation is considered in Eq. (14), is explained below.\nTheL3-edge RIXS essentially involves second-order dipole-allowed transit ions between\n2p3/2core level and t2glevels. The incoming photon resonantly excites a 2 p3/2electron\nto the unfilled t2gstates (upper Hubbard band of the nominally J= 1/2 sector). In the12\nFIG. 4: The optical (i) excitation ( i→2p3/2) and (ii) de-excitation (2 p3/2→f) processes (in the\nhole picture) involved in the RIXS mechanism for the particl e-hole excitation across the renormal-\nized spin-orbit gap. The real spin is conserved in optical tr ansitions.\nsubsequent radiative de-excitation, an electron from the filled t2gstates fills the 2 p3/2core\nhole, the loss in photon energy thereby corresponding to the over all particle-hole excitation\nin thet2gmanifold. The magnon and spin-orbit exciton cases correspond to t he final-state\nt2ghole created in the J= 1/2 and 3/2 sectors, respectively.\nIn the magnon case, with both initial and final hole states in the J= 1/2 sector (in the\nhole picture), the dipole matrix elements /angbracketleft2p3/2|Dǫ|i/angbracketrightand/angbracketleftf|D†\nǫ′|2p3/2/angbracketrightinvolving pseudo-\nspin-flip have been shown to be finite,26,27implying that RIXS is fully allowed, and the\nobserved low-energy RIXS spectrum corresponds to the magnon excitation. In the spin-\norbit exciton case, with final hole state in the J=3/2 sector, the optical excitation and\nde-excitation processes are shown in Fig. 4. These processes invo lve no change in real\nspin which is conserved in optical transitions.27However, due to the spin-orbital entangled\nnature of the Jstates, both pseudo-spin-flip and non-pseudo-spin-flip cases ar e allowed with\nrespect to the initial and final hole states. For example, the pseud o-spin-flip case is realized\nifi→2p3/2involves excitation of ( xy,σ=↑) hole from |l= 1,τ=↑/angbracketrightstate and 2 p3/2→f\ninvolves de-excitation of hole to the ( yz,σ=↑) component of |l= 3,τ=↓/angbracketrightstate.13\nVI. CONCLUSIONS\nWell-defined propagating spin-orbit exciton modes were obtained re presenting collective\nmodes of inter-orbital, particle-hole excitations across the renor malized spin-orbit gap, with\nboth dispersion and energy scale in excellent agreement with RIXS st udies. The relevant\ninteraction terms for the two exciton modes as well as the renorma lized spin-orbit gap,\nwhich play an important role in the spin-orbit exciton energy scale, we re obtained from the\ntransformation of the various Coulomb interaction terms to the ps eudo-spin-orbital basis\nformed by the J=1/2 and 3/2 states. The approach presented here allows for a un ified\ndescription of magnons and spin-orbit excitons in spin-orbit coupled systems.\n∗Electronic address: avinas@iitk.ac.in\n1W. Witczak-Krempa, G. Chen, Y. B. Kim, and L. Balents, Annu. R ev. Condens. Matter Phys.\n5, 57-82 (2014).\n2J. G. Rau, E. Kin-Ho Lee, and H.-Y. Kee, Annu. Rev. Condens. Ma tter Phys. 7, 195-221 (2016).\n3J. Bertinshaw, Y. K. Kim, G. Khaliullin, and B. J. Kim, Annu. R ev. Condens. Matter Phys.\n(in press).\n4F. Wang and T. Senthil, Phys. Rev. Lett. 106, 136402 (2011).\n5Y. K. Kim, O. Krupin, J. D. Denlinger, A. Bostwick, E. Rotenbe rg, Q. Zhao, J. F. Mitchell, J.\nW. Allen, and B. J. Kim, Science 345, 187190 (2014).\n6A. de la Torre, S. McKeown Walker, F. Y. Bruno, S. Ricc´ o, Z. Wa ng, I. Gutierrez Lezama, G.\nScheerer, G. Giriat, D. Jaccard, C. Berthod, T. K. Kim, M. Hoe sch, E. C. Hunter, R. S. Perry,\nA. Tamai, and F. Baumberger, Phys. Rev. Lett. 115, 176402 (2015).\n7Y. K. Kim, N. H. Sung, J. D. Denlinger, and B. J. Kim, Nature Phy sics12, 3741 (2016).\n8H. Gretarsson, N. Sung, J. Porras, J. Bertinshaw, C. Dietl, J an A. N. Bruin, A. F. Bangura,\nY. K. Kim, R. Dinnebier, J. Kim, A. Al-Zein, M. Moretti Sala, M . Krisch, M. Le Tacon, B.\nKeimer, and B. J. Kim, Phys. Rev. Lett. 117, 107001 (2016).\n9X. Chen, J. L. Schmehr, Z. Islam, Z. Porter, E. Zoghlin, K. Fin kelstein, J. P. C. Ruff, and S.\nD. Wilson, Nat. Commun. 9, 103 (2018).\n10S. Bhowal, J. M. Kurdestany and S. Satpathy, J. Phys.: Conden s. Matter 30235601 (2018).14\n11D. Pincini, J. G. Vale, C. Donnerer, A. de la Torre, E. C. Hunte r, R. Perry, M. Moretti Sala,\nF. Baumberger, and D. F. McMorrow, Phys. Rev. B 96, 075162 (2017).\n12J. Porras, J. Bertinshaw, H. Liu, G. Khaliullin, N. H. Sung, J .-W. Kim, S. Francoual, P.\nSteffens, G. Deng, M. Moretti Sala, A. Effimenko, A. Said, D. Casa , X. Huang, T. Gog, J. Kim,\nB. Keimer, and B. J. Kim, Phys. Rev. B 99, 085125 (2019).\n13J. Kim, D. Casa, M. H. Upton, T. Gog, Y.-J. Kim, J. F. Mitchell, M. van Veenendaal, M.\nDaghofer, J. van den Brink, G. Khaliullin, and B. J. Kim, Phys . Rev. Lett. 108, 177003 (2012).\n14B. H. Kim, G. Khaliullin, and B. I. Min, Phys. Rev. Lett. 109, 167205 (2012).\n15J. Kim, M. Daghofer, A. H. Said, T. Gog, J. van den Brink, G. Kha liullin, and B. J. Kim, Nat.\nCommun. 5, 4453 (2014).\n16J-i Igarashi and T. Nagao, Phys. Rev. B 90, 064402 (2014).\n17X. Lu, P. Olalde-Velasco, Y. Huang, V. Bisogni, J. Pelliciar i, S. Fatale, M. Dantz, J. G. Vale, E.\nC. Hunter, J. Chang, V. N. Strocov, R. S. Perry, M. Grioni, D. F . McMorrow, H. M. Rønnow,\nand T. Schmitt, Phys. Rev. B 97, 041102(R) (2018).\n18S. Mohapatra, J. van den Brink, and A. Singh, Phys. Rev. B 95, 094435 (2017).\n19S. Mohapatra and A. Singh, arXiv: 1903.03360 (2019).\n20J-i Igarashi and T. Nagao, J. Phys. Soc. Jpn. 83, 053709 (2014).\n21C. Martins, M. Aichhorn, and S. Biermann, J. Phys.: Condens. Matter29, 263001 (2017).\n22S. Mohapatra and A. Singh, arXiv: 2001.00190 (2020).\n23H. Watanabe, T. Shirakawa, and S. Yunoki, Phys. Rev. Lett. 105, 216410 (2010).\n24S. Zhou, K. Jiang, H. Chen, and Z. Wang, Phys. Rev. X 7, 041018 (2017).\n25X. Ming, X. Wan, C. Autieri, J. Wen, and X. Zheng, Phys. Rev. B 98, 245123 (2018).\n26L. J. P. Ament, G. Khaliullin, and J. van den Brink, Phys. Rev. B,84, 020403 (2011).\n27S. Boseggia, Ph.D. Thesis, Magnetic order and excitations in perovskite iridates stud ied with\nresonant X-ray scattering techniques (2014)."    },    {        "title": "1612.03447v4.The_Spinon_Fermi_Surface_U_1__Spin_Liquid_in_a_Spin_Orbit_Coupled_Triangular_Lattice_Mott_Insulator_YbMgGaO4.pdf",        "content": "The spinon Fermi surface U(1) spin liquid in a spin-orbit-coupled\ntriangular lattice Mott insulator YbMgGaO 4\nYao-Dong Li1, Yuan-Ming Lu2, and Gang Chen1;3\u0003\n1State Key Laboratory of Surface Physics, Department of Physics,\nCenter for Field Theory & Particle Physics, Fudan University, Shanghai, 200433, China\n2Department of Physics, The Ohio State University, Columbus, OH, 43210, United States and\n3Collaborative Innovation Center of Advanced Microstructures, Nanjing, 210093, China\n(Dated: August 24, 2017)\nMotivated by the recent progress on the spin-orbit-coupled triangular lattice spin liquid candi-\ndate YbMgGaO 4, we carry out a systematic projective symmetry group analysis and mean-\feld\nstudy of candidate U(1) spin liquid ground states. Due to the spin-orbital entanglement of the Yb\nmoments, the space group symmetry operation transforms both the position and the orientation of\nthe local moments, and hence brings di\u000berent features for the projective realization of the lattice\nsymmetries from the cases with spin-only moments. Among the eight U(1) spin liquids that we \fnd\nwith the fermionic parton construction, only one spin liquid state, that was proposed and analyzed\nin Yao Shen, et. al. , Nature 540, 559-562 (2016) and labeled as U1A00 in the present work, stands\nout and gives a large spinon Fermi surface and provides a consistent explanation for the spectro-\nscopic results in YbMgGaO 4. Further connection of this spinon Fermi surface U(1) spin liquid with\nYbMgGaO 4and the future directions are discussed. Finally, our results may apply to other spin-\norbit-coupled triangular lattice spin liquid candidates, and more broadly, our general approach can\nbe well extended to spin-orbit-coupled spin liquid candidate materials.\nI. INTRODUCTION\nThe interplay between strong spin-orbit coupling\n(SOC) and strong electron correlation has attracted a\nsigni\fcant attention in recent years1. At the mean time,\nthe abundance of strongly correlated materials with 5 d\nand 4felectrons, such as iridates and rare-earth mate-\nrials1,2, brings a fertile arena to explore various emer-\ngent and exotic phases that arise from such an inter-\nplay3{32. The recently discovered quantum spin liquid\n(QSL) candidate YbMgGaO 433, where the rare-earth Yb\natoms form a perfect triangular lattice, is an ideal sys-\ntem that involves strong spin-orbital entanglement in the\nstrong Mott insulating regime of the Yb electrons34{41.\nIn YbMgGaO 4, the thirteen 4 felectrons of the Yb3+\nions are well localized and form a spin-orbit-entangled\ntotal moment JwithJ= 7=234,35. The eight-fold de-\ngeneracy of the J= 7=2 moment is further split by the\nD3dcrystal electric \felds. The resulting ground state\nKramers doublet of the Yb3+ion, whose two-fold de-\ngeneracy is protected by the time-reversal symmetry,\nis well separated from the excited doublets and is re-\nsponsible for the low-temperature magnetic properties\nof YbMgGaO 4. No signature of time-reversal symmetry\nbreaking is observed for YbMgGaO 4down to the lowest\nmeasured temperature36{38. Applying the recent theoret-\nical result on spin-orbit-coupled Mott insulators42, two of\nus and collaborators have proposed YbMgGaO 4to be the\n\frst QSL candidate in the spin-orbit-coupled Mott in-\nsulator with odd electron \fllings34{36,39. More broadly,\nYbMgGaO 4represents a new class of rare-earth materi-\nals where the strong spin-orbit entanglement of the local\nmoments meets with the geometrical frustration of the\ntriangular lattice such that exotic quantum phases may\nbe stabilized.Apart from the absence of magnetic ordering, the heat\ncapacity was found to be Cv/T0:7at low tempera-\ntures33,34,37,43, and is close to the well-known T2=3heat\ncapacity44{46. The latter was the one obtained within a\nrandom phase approximation for the spinon-gauge cou-\npling in a spinon Fermi surface U(1) QSL44{46. More\nsubstantially, the broad continuum36,37of the magnetic\nexcitation with a clear dispersion for the upper excitation\nedge agrees reasonably with the particle-hole continuum\nof the spinon Fermi surface36. However, due to the scat-\ntering with the phonon degrees of freedom, the thermal\ntransport measurement in YbMgGaO 4was unable to ex-\ntract the intrinsic magnetic contribution to the thermal\nconductivity43. Partly motivated by the spin liquid be-\nhaviors in YbMgGaO 4and more broadly by the families\nof rare-earth magnets with identical structures, in this\npaper, we carry out a systematic projective symmetry\ngroup (PSG) analysis for a triangular lattice Mott insu-\nlator with spin-orbital-entangled local moments. Unlike\nthe cases for the spin-only moments in the pioneering\nwork by X.-G. Wen47, the space group symmetry opera-\nFIG. 1. (a) The intralayer symmetries of the R \u00163m space group\nfor YbMgGaO 435. (b) The same lattice symmetry group with\na di\u000berent complete set of elementary transformations. Here\nS6\u0011C\u00001\n3I. The bold arrow is the axis for the C2rotation\n(see Appendix).arXiv:1612.03447v4  [cond-mat.str-el]  23 Aug 20172\ntion, in particular, the rotation, transforms both the po-\nsition and the orientation of the Yb local moments35,39.\nWe \fnd that, among the eight U(1) QSL states, the\nspinon mean-\feld state that was introduced in Ref. 36\nand labeled as the U1A00 state in our PSG classi\fcation,\ncontains a large spinon Fermi surface and gives a large\nspinon scattering density of states that is consistent with\nthe inelastic neutron scattering (INS) results.\nThe following part of the paper is organized as follows.\nIn Sec. II, we describe the space group symmetry and the\nthe multiplication rules for the symmetry transformation.\nIn Sec. III, we introduce the fermionic spinon construc-\ntion and the fermionic spinon mean-\feld Hamiltonian. In\nSec. IV, we explain the scheme for the projective sym-\nmetry group classi\fcation when the spin-orbit coupling\nis present. In Sec. V, we explain the relationship between\nthe spinon band structure and the projective symmetry\ngroup of the spinon mean-\feld states. In Sec. VI, we\nfocus on the U1A00 state and study the spectroscopic\nproperties of this state. Finally in Sec. VII, we discuss\nthe experimental relevance and remark on the thermal\ntransport result and the competing scenarios and pro-\nposals. The details of the calculation are presented in\nthe Appendices.\nII. SPACE GROUP SYMMETRY\nIt was pointed out that the intralayer symmetries in-\nvolves two translations, T1andT2, one two-fold rota-\ntion,C2, one three-fold rotation, C3, and one spatial in-\nversionI(see Fig. 1(a))35,39. Here we use a di\u000berent\ncomplete set of elementary transformations for the space\ngroup symmetries that involve two translations, T1and\nT2, one two-fold rotation, C2, and one more operation,\nS6(see the de\fnition in Fig. 1(b)). It is ready to con-\n\frmI=S3\n6;C3=S2\n6with the de\fnition S6\u0011C\u00001\n3I. The\nmultiplication rules of this symmetry group is given as\nT\u00001\n1T2T1T\u00001\n2=T\u00001\n1T\u00001\n2T1T2= 1; (1)\nC\u00001\n2T1C2T\u00001\n2=C\u00001\n2T2C2T\u00001\n1= 1; (2)\nS\u00001\n6T1S6T2=S\u00001\n6T2S6T\u00001\n2T\u00001\n1= 1; (3)\n(C2)2= (S6)6= (S6C2)2= 1: (4)\nDue to the presence of time reversal in\nYbMgGaO 434,36{38, we further supplement the symmetry\ngroup with the time reversal Tsuch thatO\u00001TOT = 1\nandT2= 1, whereOis a lattice symmetry operation.\nIII. FERMIONIC PARTON CONSTRUCTION\nTo describe the U(1) QSL that we propose for\nYbMgGaO 4, we introduce the fermionic spinon opera-\ntorfr\u000b(\u000b=\";#) that carries spin-1/2, and express the\nYb local moment as\nSr=1\n2X\n\u000b;\ffy\nr\u000b\u001b\u000b\ffr\f; (5)U(1) QSLWT1rWT2rWC2rWS6r\nU1A00I2\u00022I2\u00022I2\u00022I2\u00022\nU1A10I2\u00022I2\u00022i\u001byI2\u00022\nU1A01I2\u00022I2\u00022I2\u00022i\u001by\nU1A11I2\u00022I2\u00022i\u001byi\u001by\nTABLE I. List of the gauge transformations for the four U1A\nPSGs. For the time reversal, all PSGs here have WT\nr=I2\u00022.\nThe last two letters in the labels of the U(1) QSLs are extra\nquantum numbers in the PSG classi\fcation48.\nwhere\u001b= (\u001bx;\u001by;\u001bz) is a vector of Pauli matrices. We\nfurther impose a constraintP\n\u000bfy\nr\u000bfr\u000b= 1 on each site\nto project back to the physical Hilbert space of the spins.\nThe choice of fermionic spinons allows a local SU(2)\ngauge freedom47.\nAs a direct consequence of the spin-orbital entangle-\nment, the spinon mean-\feld Hamiltonian for the U(1)\nQSL should generically involve both spin-preserving and\nspin-\ripping hoppings, and has the following form\nHMF=\u0000X\n(rr0)X\n\u000b\f\u0002\ntrr0;\u000b\ffy\nr\u000bfr0\f+h:c:\u0003\n; (6)\nwheretrr0;\u000b\fis the spin-dependent hopping. The choice\nof the mean-\feld ansatz in Eq. (6) breaks the local SU(2)\ngauge freedom down to U(1). Here, to get a more com-\npact form for Eq. (6), we follow Ref. 49 and introduce the\nextended Nambu spinor representation for the spinons\nsuch that \tr= (fr\";fy\nr#;fr#;\u0000fy\nr\")Tand\nHMF=\u00001\n2X\n(r;r0)\u0002\n\ty\nrurr0\tr0+h:c:\u0003\n; (7)\nwhereurr0is a hopping matrix that is related to trr0;\u000b\f.\nWith the extended Nambu spinor, the spin operator Sr\nand the generator Grfor the SU(2) gauge transformation\nare given by47,50{53\nSr=1\n4\ty\nr(\u001b\nI2\u00022)\tr; (8)\nGr=1\n4\ty\nr(I2\u00022\n\u001b)\tr; (9)\nwhereI2\u00022is a 2\u00022 identity matrix. Under the symme-\ntry operationO, \trtransforms as\n\tr!UOGO\nO(r)\tO(r)=GO\nO(r)UO\tO(r); (10)\nwhereGO\nO(r)is the local gauge transformation that cor-\nresponds to the symmetry operation O, and we add a\nspin rotationUObecause the spin components are trans-\nformed whenOinvolves a rotation. In Eq. (10), the\ngauge transformation and the spin rotation are commu-\ntative54simply because [ S\u0016\nr;G\u0017\nr] = 0. Moreover, from\nEq. (9), the gauge transformation GO\nris block diagonal\nwithGO\nr=I2\u00022\nWO\nr, whereWO\nris a 2\u00022 matrix (see\nAppendix).3\nIV. PROJECTIVE SYMMETRY GROUP\nCLASSIFICATION\nFor the spinon mean-\feld Hamiltonian in Eq. (6), the\nlattice symmetries are realized projectively and form the\nprojective symmetry group (PSG). To respect the lat-\ntice symmetry transformation O, the mean-\feld ansatz\nshould satisfy\nurr0=GOy\nO(r)Uy\nOuO(r)O(r0)UOGO\nO(r0): (11)\nThe ansatz itself is invariant under the so-called invariant\ngauge group (IGG) with urr0=G1y\nrurr0G1\nr0. The IGG\ncan be regarded as a set of gauge transformations that\ncorrespond to the identity transformation. For an U(1)\nQSL, IGG = U(1).\nA general group relation O1O2O3O4= 1 for the lattice\nsymmetry turns into the following group relation for the\nPSG\nUO1GO1\nrUO2GO2\nO2O3O4(r)UO3GO3\nO3O4(r)UO4GO4\nO4(r)\n=UO1UO2UO3UO4GO1\nrGO2\nO2O3O4(r)GO3\nO3O4(r)GO4\nO4(r)(12)\n2IGG; (13)\nwhere we used the fact that the gauge transformation\ncommutes with the spin rotation. As the series of rota-\ntionsO1O2O3O4either rotate the spinons by 0 or 2 \u0019,\nUO1UO2UO3UO4=\u0006I4\u00024; (14)\nwhereI4\u00024is a 4\u00024 identity matrix. Since\nf\u0006I4\u00024g\u001aIGG, then\nGO1\nrGO2\nO2O3O4(r)GO3\nO3O4(r)GO4\nO4(r)2IGG: (15)\nThis immediately indicates that, to classify the PSGs for\na spin-orbit-coupled Mott insulator, we only need to fo-\ncus on the gauge part, \frst \fnd the gauge transformation\nwith the same procedures as those for the conventional\nMott insulators with spin-only moments47, and then ac-\ncount for the spin rotation.\nFor the mean-\feld ansatz in HMF, we choose the\n\\canonical gauge\" for the IGG with\nIGG =fI2\u00022\nei\u001e\u001bzj\u001e2[0;2\u0019)g: (16)\nUnder the canonical gauge, the gauge transformation as-\nsociated with the symmetry operation Otakes the form\nof\nGO\nr=I2\u00022\nWO\nr\n\u0011I2\u00022\n\u0002\n(i\u001bx)nOei\u001eO[r]\u001bz\u0003\n; (17)\nwherenO= 0;1. For translations, one can always choose\na gauge such that\nWT1\nr= (i\u001bx)n1; (18)\nWT2\nr= (i\u001bx)n2ei\u001e2[x;y]\u001bz(19)\nwithn1;n2= 0;1 and\u001e2[0;y] = 0. The group rela-\ntion in Eq. (3) further demands n1=n2= 0. Thus the\nFIG. 2. (a,b,c) The mean-\feld spinon bands along the high-\nsymmetry momentum lines (see (d)) of the U1A00, U1A01\nand U1A11 states, where t1;t0\n1andt2are hoppings in their\nspinon mean-\feld Hamiltonians (see Appendix). The Dirac\ncones are highlighted in dashed circles. The dashed line refers\nto the Fermi level. (d) The Brioullin zone of the triangular\nlattice.\ngroup relation in Eq. (1) gives WT1r= 1;WT2r=eix\u001e1\u001bz,\nwhere\u001e1is the \rux through each unit cell of the triangu-\nlar lattice and takes the value of 0 or \u0019(see Appendix).\nThe PSGs with \u001e1= 0 (\u0019) are labeled by U1A (U1B).\nAmong the sixteen algebraic PSGs that we \fnd, eight un-\nphysical solutions have T2= 1 for the spinons and give\nvanishing spinon hoppings everywhere. In Tab. I and the\nAppendix, we list the remaining eight PSGs that have\nT2=\u00001 consistent with the fact that fermionic spinons\nare Kramers doublets (see Appendix).\nV. MEAN-FIELD STATES\nHere we obtain the spinon mean-\feld Hamiltonian\nfrom Tab. I and explain why the U1A00 state stands\nout as the candidate ground state for YbMgGaO 4. We\nstart with the U1A states. Among the four U1A states,\nthe U1A10 state gives a vanishing mean-\feld Hamilto-\nnian for the spinon hoppings between the \frst and the\nsecond neighbors, the remaining ones except the U1A00\nstate all have symmetry protected band touchings at the\nspinon Fermi level (see Fig. 2). To illustrate the idea55,\nwe consider the U1A01 state where the spinon Hamilto-\nnian has the form HU1A01\nMF =P\nkh\u000b\f(k)fy\nk\u000bfk\fin the\nmomentum space and h(k) is a 2\u00022 matrix with\nh(k) =d0(k)I2\u00022+3X\n\u0016=1d\u0016(k)\u001b\u0016: (20)\nFor this band structure there are nondegenerate band\ntouchings at \u0000, M and K points that are protected by\nthe PSG of the U1A01 state. Under the operation S6,4\nthe PSG demands that spinons to transform as\nfk\"!\u0000e\u0000i\u0019=3fy\n\u0000S\u00001\n6k;#; (21)\nfk#!ei\u0019=3fy\n\u0000S\u00001\n6k;\": (22)\nApplyingS6three times and keeping HMFinvariant, we\nrequire\nh(k) =\u0000[\u001byh(k)\u001by]T(23)\nwhich forces d0(k) = 0. The time reversal sym-\nmetry (T=i\u001by\nI2\u00022K) further requires that\nd\u0016(k) =\u0000d\u0016(\u0000k). Thus we have symmetry pro-\ntected band touchings with h(k) = 0 at the time reversal\ninvariant momenta \u0000 and M. The K points are invariant\nunderC2andS6because the spinon partile-hole trans-\nformation is involved for S6(see Appendix). Using those\ntwo symmetries, we further establish the band touching\nat the K points. Likewise, for the U1A11 state, the\nPSG demands the band touchings at \u0000 and M points.\nBecause there are only two spinon bands for the U1A\nstates, these band touchings generically occur at the\nspinon Fermi level.\nDue to the Dirac band touchings at the Fermi level, the\nlow-energy dynamic spin structure factor, that measures\nthe spinon particle-hole continuum, is concentrated at a\nfew discrete momenta that correspond to the intra-Dirac-\ncone and the inter-Dirac-cone scatterings36. Clearly, this\nis inconsistent with the recent INS result that observes a\nbroad continuum covering a rather large portion of the\nBrillouin zone36,37.\nFor the U1B states, the spinons experience a \u0019back-\nground \rux in each unit cell. The direct consequence of\nthe\u0019background \rux is that the U1B states support an\nenhanced periodicty of the dynamic spin structure in the\nBrillouin zone47,56,57. Such an enhanced periodicity is\nabsent in the INS result36,37. In particular, unlike what\none would expect for an enhanced periodicity, the spec-\ntral intensity at the \u0000 point is drastically di\u000berent from\nthe one at the M point in the existing experiments36,37.\nThe above analysis leads to the conclusion that the\nU1A00 state is the most promising candidate U(1) QSL\nfor YbMgGaO 4, and this conclusion is independent from\nany microscopic model. The spinon mean \feld Hamilto-\nnian, allowed by the U1A00 PSG, is remarkably simple\nand is given as58\nHU1A00\nMF =\u0000t1X\nhrr0i;\u000bfy\nr\u000bfr\u000b\u0000t2X\nhhrr0ii;\u000bfy\nr\u000bfr\u000b;(24)\nwhere the spinon hopping is isotropic for the \frst and\nthe second neighbors. This mean-\feld state only has a\nsingle band that is 1/2-\flled, so it has a large spinon\nFermi surface. From HU1A00\nMF , we construct the mean-\n\feld ground state by \flling the spinon Fermi sea,\nj\tU1A00\nMFi=Y\n\u000fk<\u000fFfy\nk\"fy\nk#j0i (25)where\u000fkis the spinon dispersion and \u000fFis the spinon\nFermi energy. The mean-\feld variational energy is\nEvar=h\tU1A00\nMFjHspinj\tU1A00\nMFi; (26)\nwhere\nHspin=X\nhrr0iJzzSz\nrSz\nr0+J\u0006(S+\nrS\u0000\nr0+S\u0000\nrS+\nr0)\n+J\u0006\u0006(\rrr0S+\nrS+\nr0+\r\u0003\nrr0S\u0000\nrS\u0000\nr0)\n\u0000i\n2Jz\u0006\u0002\n(\r\u0003\nrr0S+\nr\u0000\rrr0S\u0000\nr)Sz\nr0\n+Sz\nr(\r\u0003\nrr0S+\nr0\u0000\rrr0S\u0000\nr0)\u0003\n(27)\nis the microscopic spin model that was introduced in\nRefs. 34 and 35, and \rrr0is a bond-dependent phase fac-\ntor due to the spin-orbit-entangled nature of the Yb mo-\nments35. The anisotropic nature of the spin interaction\nhas been clearly supported by the recent polarized neu-\ntron scattering measurement59. For the speci\fc choice\nwithJ\u0006= 0:915Jzz, we \fnd the minimum variational en-\nergyEvar=\u00000:39Jzzand occurs at t2= 0:2t1(see Ap-\npendix). Here, the expectation values of the J\u0006\u0006andJz\u0006\ninteractions simply vanish, and this is an artifact of the\nfree spinon mean-\feld theory with the isotropic hoppings\nin Eq. (24). We here establish that the U1A00 state is a\nspinon Fermi surface U(1) QSL.\nVI. SPECTROSCOPIC PROPERTIES\nFor the U1A00 state, the dynamic spin structure essen-\ntially detects the spinon particle-hole excitation across\nthe Fermi surface. The information about the Fermi\nsurface is encoded in the pro\fle of the dynamic spin\nstructure factor. We evaluate the dynamic spin struc-\nture factor within the free spinon mean-\feld theory (see\nAppendix) (see Fig. 3(a)). Qualitatively similar to the\nmean-\feld theory with only \frst neighbor spinon hop-\npings, the improved free-spinon mean-\feld theory of\nHU1A00\nMF captures the crucial features of the INS re-\nsults36,37. The spinon particle-hole continuum covers a\nlarge portion of the Brillouin zone, and vanishes beyond\nthe spinon bandwidth. More importantly, the \\V-shape\"\nupper excitation edge near the \u0000 point in Fig. 3(a) was\nclearly observed in the experiments36,37, and the slope of\nthe \\V-shape\" is the Fermi velocity.\nDue to the isotropic spinon hoppings, HU1A00\nMF does not\nexplicitly re\rect the absence of spin-rotational symmetry\nthat is brought by the J\u0006\u0006andJz\u0006interactions. To\nincorporate the J\u0006\u0006andJz\u0006interactions, we follow the\nphenomenological RPA treatment for the \\ t-J\" model in\nthe context of cuprate superconductors60and consider\nH=HU1A00\nMF +H0\nspin; (28)\nwhereH0\nspinare theJ\u0006\u0006andJz\u0006interactions (see Ap-\npendix). While the free spinon results from HU1A00\nMF al-\nready capture the main features of the neutron scatter-\ning data36,37, the anisotropic spin interaction H0\nspin, in-\ncluded by RPA, merely redistributes the spectral weight5\nFIG. 3. (a)S(q;!) along the high-symmetry momentum\nlines from HU1A00\nMF witht2= 0:2t1. The spinon bandwidth\nB= 9:6t1. (b) The RPA corrected SRPA(q;!) along the high\nsymmetry momentum lines. We have set the parameters in\nthe spin model to be J\u0006=Jzz= 0:915,J\u0006\u0006=Jzz= 0:35, and\nJz\u0006=Jzz= 0:2. The ratio Jzz=t1is obtained from Refs. 34\nand 36 and \fxed to be 1 :0 for concreteness.\nin the momentum space. We \fnd in Fig. 3(b) that, the\nlow-energy spectral weight at M is slightly enhanced, a\nfeature observed in Refs. 36 and 37. From our choice\nof the parameters, it is plausible that this peak results\nfrom the proximity to a phase with a stripe-like magnetic\norder35,36,39.\nVII. DISCUSSION\nWe have demonstrated that the spinon Fermi surface\nU(1) QSL gives a consistent explanation of the INS re-\nsult in YbMgGaO 4. Moreover, the anisotropic spin in-\nteraction, slightly enhances the spectral weight at the M\npoints. The U(1) gauge \ructuation in the spinon Fermi\nsurface U(1) QSL44,45was suggested to be the cause for\nthe sublinear temperature dependence of the heat capac-\nity in YbMgGaO 435,36,39,46.\nIn YbMgGaO 4, the coupling between the Yb moments\nis relatively weak34. It is feasible to fully polarize the spin\nwith experimentally accessible magnetic \felds35,37,39,61\nand to study the evolution of the magnetic properties un-\nder the magnetic \feld. Recently, two of us have predicted\nthe spectral weight shift of the INS for YbMgGaO 4under\na weak magnetic \feld41, and the predicted spectral cross-\ning at the \u0000 point and the dispersion of the spinon con-\ntinuum have actually been con\frmed in the recent INS\nmeasurement62. Numerically, it is useful to perform nu-\nmerical calculation with \fxed J\u0006andJzzthat are close\nto the ones for YbMgGaO 4, and obtain the phase dia-\ngram of our spin model by varying J\u0006\u0006andJz\u000635,39,63.\nMore care needs to be paid to the disordered region of the\nmean-\feld phase diagram35where quantum \ructuation\nis found to be strong35. The \\2kF\" oscillation in the spin\ncorrelation would be the strong indication of the spinon\nFermi surface. Noteworthily recent DMRG works64,65\nhave actually provided some useful information about\nthe ground states of the system, in particular, Ref. 65\nsuggested the scenario of exchange disorders. Certain\namount of exchange disorder may be created by the crys-\ntal electric \feld disorder that stems from the Mg/Gamixing in the non-magnetic layers37,61, but recent polar-\nized neutron scattering measurement did not \fnd strong\nexchange disorder59. Regardless of the possibilities of\nexchange disorders, the spin quantum number fraction-\nalization, that is one of the key properties of the QSLs,\ncould survive even with weak disorders. The approach\nand results in our present work are phenomenologically\nbased and are independent of the microscopic mechanism\nfor the possible QSL ground state in YbMgGaO 4.\nRef. 43 claimed the absence of the magnetic thermal\nconductivity in YbMgGaO 4by extrapolating the low-\ntemperature thermal conductitivity data in the zero mag-\nnetic \feld. Here, we provide an alternative understand-\ning for this thermal transport result. The hint lies in\nthe \feld dependence of the thermal conductivity. It was\nfound that, when strong magnetic \felds are applied to\nYbMgGaO 4, the thermal conductivity \u0014xx=Tat 0.2K is\nincreased compared with the one at zero \feld43. If one ig-\nnores the disorder e\u000bect and assumes the zero-\feld ther-\nmal conductivity is a simple addition of the magnetic\ncontribution and the phonon contribution with\n\u0014xx=\u0014spin;xx+\u0014phonon;xx; (29)\nthe strong magnetic \feld almost polarizes the spins com-\npletely and creates a spin gap for the magnon excitation,\nhence suppress the magnetic contribution. The high-\n\feld thermal conductivity would be purely given by the\nphonon contribution, and we would expect a decreasing\nof the thermal conductivity in the strong \feld compared\nto the zero \feld result. This is clearly inconsistent with\nthe experimental result. Therefore, the zero-\feld ther-\nmal conductivity is not a simple addition of the magnetic\ncontribution and the phonon contribution, i.e.,\n\u0014xx6=\u0014spin;xx+\u0014phonon;xx: (30)\nThis also strongly suggests the presence rather than the\nabsence of magnetic excitations in the thermal conductiv-\nity result at zero magnetic \feld. If there is no magnetic\nexcitation in the system at low temperatures, the low-\ntemperature thermal conductivity at zero \feld should\njust be the phonon contribution, and we would expect\nthe zero-\feld thermal conductivity to be the same as the\none in the strong \feld limit, (although the intermediate\n\feld regime could be di\u000berent). This is again inconsis-\ntent with the experiments. This means that the magnetic\nexcitation certainly does not have a large gap and could\njust be gapless as we propose from the spinon Fermi sur-\nface state. In fact, the gapless nature of the magnetic\nexcitation is consistent with the power-law heat capac-\nity results in YbMgGaO 4. What suppresses \u0014xxcould\narise from the mutual scattering between the magnetic\nexcitations and the the phonons. In fact, similar \feld de-\npendence of thermal conductivity \u0014xxhas been observed\nin other rare-earth systems such as Tb 2Ti2O766{68and\nPr2Zr2O769. It was suggested there67{69that the spin-\nphonon scattering is the cause. The Yb local moment,\nthat is a spin-orbit-entangled object, involves the orbital6\ndegree of freedom. The orbital degree of freedom is sen-\nsitive to the ion position, and thus couples to the phonon\nstrongly. This is probably the microscopic origin for the\nstrong coupling between the magnetic moments and the\nphonons in the rare-earth magnets. This is quite dif-\nferent from the organic spin liquid candidates and the\nherbertsmithite kagome system where the orbital degree\nof freedom does not seem to be involved70{73.\nIf the ground state of YbMgGaO 4is a QSL with the\nspinon Fermi surface, the \feld-driven transition from the\nQSL ground state to the fully polarized state is neces-\nsarily a unconventional transition beyond the traditional\nLandau's paradigm and has not been studied in the pre-\nvious spin liquid candidates70{73. The smooth growth of\nthe magnetization with varying external \felds indicates\na continuous transition34. Since we propose YbMgGaO 4\nto be a spinon Fermi surface U(1) QSL and gapless, the\ntransition would be associated with the openning of the\nspin gap at the critical \feld. The continuous nature of\nthe transition suggests the spin gap to open in a contin-\nuous manner. Moreover, the spinon con\fnement would\nbe concomitant with the spin gap that suppresses the\nspinon density of states and allows the instanton events\nof the U(1) gauge \feld to proliferate. Therefore, it might\nbe interest to identify the critical \feld and obtain the\ncritical properties of the \feld-driven transition. Thermo-\ndynamic, spectroscopic, and thermal transport measure-\nments with \fner \feld variation would be helpful.\nFinally, several families of rare-earth triangular lattice\nmagnets have been discovered recently35,39,74{79. Their\nproperties have not been studied carefully. Our general\nclassi\fcation results and the prediction of the spectro-\nscopic properties would apply to the QSL candidates that\nmay emerge in these families of materials. It is certainly\nexciting if one \fnds the new QSL candidates in these\nfamilies behave like YbMgGaO 435.\nVIII. ACKNOWLEDGEMENTS\nWe thank one anonymous referee for the suggestion for\nimprovement to this paper, and Zhu-Xi Luo for point-\ning out some typos. G.C. acknowledges the discussion\nwith Xuefeng Sun from USTC and Yuji Matsuda about\nthermal transports in rare-earth magnets, and the dis-\ncussion with Professor Sasha Chernyshev about the re-\nlated matters. This work is supported by the Min-\nistry of Science and Technology of China with the Grant\nNo.2016YFA0301001 (G.C.), the Start-Up Funds of OSU\n(Y.M.L.) and Fudan University (G.C.), the National Sci-\nence Foundation under Grant No. NSF PHY-1125915\n(Y.M.L and G.C.), the Thousand-Youth-Talent Program\n(G.C.) of China, and the \frst-class university construc-\ntion program of Fudan University.Appendix A: The coordinate System and space\ngroup symmetry\nFollowing our convention in Fig. 1 in the main text, we\nchoose the coordinate system of the triangular lattice to\nbe\na1= (1;0); (A1)\na2= (\u00001\n2;p\n3\n2): (A2)\nWe label the triangular lattice sites by r=xa1+ya2.\nRestricted to the triangular layer, the space group con-\ntains two translations T1along thea1direction,T2along\nthea2direction, a counterclockwise three-fold rotation\nC3around the lattice site, a two-fold rotation C2around\na1+a2, and the inversion Iat the lattice site. Their\nactions on the lattice indices are\nT1: (x;y)!(x+ 1;y); (A3)\nT2: (x;y)!(x;y+ 1); (A4)\nC3: (x;y)!(\u0000y;x\u0000y); (A5)\nC2: (x;y)!(y;x); (A6)\nI: (x;y)!(\u0000x;\u0000y): (A7)\nIn the formulation introduced in the main text, we\nconsider an equivalent set of generators, fT1;T2;C2;S6g,\nwhere the operation S6isde\fned asS6\u0011C\u00001\n3Iand acts\non the lattice indices as\nS6: (x;y)!(x\u0000y;x): (A8)\nIt is evident that these two sets of generators are equiva-\nlent, since we merely rede\fne the symmetry rather than\nintroducing any new symmetry.\nThe multiplication rule of this symmetry group is given\nin the main text. For the convenience of the presentation\nbelow, we also list these rules here,\nT\u00001\n1T2T1T\u00001\n2=T\u00001\n1T\u00001\n2T1T2= 1; (A9)\nC\u00001\n2T1C2T\u00001\n2=C\u00001\n2T2C2T\u00001\n1= 1; (A10)\nS\u00001\n6T1C6T2=S\u00001\n6T2C6T\u00001\n2T\u00001\n1= 1;(A11)\n(C2)2= (C6)6= (S6C2)2= 1: (A12)\nIncluding the time reversal symmetry, we further have\nT\u00001\n1TT1T=T\u00001\n2TT2T= 1; (A13)\nC\u00001\n2TC2T=S\u00001\n6TS6T= 1; (A14)\nT2= 1: (A15)\nAppendix B: Projective symmetry group\nclassi\fcation\nAs we describe in the main text, we consider the U(1)\nQSL. The spinon mean-\feld Hamiltonian has the follow-\ning form\nHMF=\u0000X\n(rr0)X\n\u000b\f\u0002\ntrr0;\u000b\ffy\nr\u000bfr0\f+h:c:\u0003\n;(B1)7\nU(1) QSLWT1rWT2rWC2r WS6r\nU1A00I2\u00022I2\u00022I2\u00022 I2\u00022\nU1A10I2\u00022I2\u00022i\u001byI2\u00022\nU1A01I2\u00022I2\u00022I2\u00022 i\u001by\nU1A11I2\u00022I2\u00022i\u001byi\u001by\nU1B00I2\u00022(\u00001)xI2\u00022(\u00001)xyI2\u00022(\u00001)xy\u0000y(y\u00001)\n2I2\u00022\nU1B10I2\u00022(\u00001)xI2\u00022i\u001by(\u00001)xy(\u00001)xy\u0000y(y\u00001)\n2I2\u00022\nU1B01I2\u00022(\u00001)xI2\u00022(\u00001)xyI2\u00022i\u001by(\u00001)xy\u0000y(y\u00001)\n2\nU1B11I2\u00022(\u00001)xI2\u00022i\u001by(\u00001)xyi\u001by(\u00001)xy\u0000y(y\u00001)\n2\nTABLE II. List of the gauge transformations for the sym-\nmetry operations of the eight U(1) PSGs, where ( x;y) is the\ncoordinate in the oblique coordinate system. For time rever-\nsal symmetry, all PSGs have the same gauge transformation\nWT\nr=I2\u00022.\nwheretrr0;\u000b\fis the spin-dependent hopping. With\nthe extended Nambu spinor representation49\tr=\n(fr\";fy\nr#;fr#;\u0000fy\nr\")T,HMFhas a more compact form\nHMF=\u00001\n2X\n(r;r0)\u0002\n\ty\nrurr0\tr0+h:c:\u0003\n; (B2)\nwhereurr0is a hopping matrix that is related to trr0;\u000b\f,\nurr0=0\nBBBBB@trr0;\"\" 0trr0;\"# 0\n0\u0000t\u0003\nrr0;## 0t\u0003\nrr0;#\"\ntrr0;#\" 0trr0;## 0\n0t\u0003\nrr0;\"# 0\u0000t\u0003\nrr0;\"\"1\nCCCCCA:(B3)\n1. Spatial symmetry\nFirst of all, the gauge transformation and spin rotation\nare commutative. So in the PSG classi\fcation, we only\nneed to focus on the gauge part of the PSG transforma-\ntion. In the canonical gauge IGG = fI2\u00022\nei\u001e\u001bzj\u001e2\n[0;2\u0019)g, the gauge transformation associated with a given\nsymmetry operation Otakes the form\nGO\nr=I2\u00022\nWO\nr\u0011I2\u00022\n\u0002\n(i\u001bx)nOei\u001eO[r]\u001bz\u0003\n;(B4)\nwherenO= 0;1. For the symmetry multiplication rule\nO1O2O3O4= 1 whereOiis an unitary transformation,\nthe corresponding PSG relation becomes\nGO1\nrGO2\nO2O3O4(r)GO3\nO3O4(r)GO4\nO4(r)2IGG (B5)\nor equivalently,\nWO1\nrWO2\nO2O3O4(r)WO3\nO3O4(r)WO4\nO4(r)\n2fei\u001e\u001bzj\u001e2[0;2\u0019)g: (B6)We start with T1andT2, where\nWT1\nr= (i\u001bx)nT1; (B7)\nWT2\nr= (i\u001bx)nT2ei\u001eT2[r]\u001bz: (B8)\nThrough Eq. (A10) that connects T1andT2, one imme-\ndiately has nT1=nT2. From Eq. (A11) where the to-\ntal number of T1andT2is odd, one immediately has\nnT1=nT2= 0. So we have\nWT1\nr= 1; (B9)\nWT2\nr=ei\u001eT2[x;y]\u001bz: (B10)\nUsing Eq. (A9), we have\n[WT1T1]\u00001[WT2T2][WT1T1][WT2T2]\u00001\n=T\u00001\n1(WT1)\u00001WT2T2WT1T1T\u00001\n2W\u00001\nT2\n2fei\u001e\u001bzj\u001e2[0;2\u0019)g;(B11)\nwhich leads to the result\n\u001eT2[x+ 1;y]\u0000\u001eT2[x;y]\u0011\u001e1 (B12)\nwith\u001e1to be determined. Since it is always possible\nto choose a gauge such that \u001eT2[0;y] = 0, then we have\n\u001eT2[x;y] =\u001e1x.\nSimilarly,T\u00001\n1T\u00001\n2T1T2= 1 leads to\n\u001eT2[x+ 1;y+ 1]\u0000\u001eT2[x;y+ 1] =\u001e2:(B13)\nIt is ready to \fnd \u001e2=\u001e1.\nWe continue to \fnd WS6randWC2r. For the operation\nS6withWS6r= (i\u001bx)nS6ei\u001eS6[x;y]\u001bz, Eq. (A11) leads to\n\u0000\u001eS6[T1(r)] +\u001eS6[r] =\u0000\u001e1y+\u001e3; (B14)\n\u0000\u001eS6[T2(r)] +\u001eS6[r] =\u001e4\u0000\u001e1x+\u001e1y;(B15)\nfornS6= 0, and\n\u0000\u001eS6[T1(r)] +\u001eS6[r] =\u0000\u001e1y+\u001e3 (B16)\n\u0000\u001eS6[T2(r)] +\u001eS6[r] =\u001e4+\u001e1x+\u001e1y:(B17)\nfornS6= 1. So we obtain\nwhennS6= 0;\n\u001eS6[r] =\u001e1xy\u0000\u001e3x\u0000\u001e4y\u0000\u001e1y(y\u00001)\n2(B18)\nwhennS6= 1;\n\u001eS6[r] =\u001e1xy\u0000\u001e3x\u0000\u001e4y\u0000\u001e1y(y\u00001)\n2:(B19)\nFornS6= 1, we further require \u001e1= 0;\u0019.S6\n6= 1 is\nautomatically satis\fed with the above relations for both\nnS6= 0 andnS6= 1.\nForWC2rwithWC2r= (i\u001bx)nC2ei\u001eC2[x;y]\u001bz, we need to\nconsider two separate cases with nc2= 0;1, respectively.\nIfnC2= 0, Eq. (A10) leads to\n\u0000\u001eT2[C\u00001\n2T1(r)]\u0000\u001eC2[T1(r)] +\u001eC2[r] =\u001e5;(B20)\n\u0000\u001eC2[T2(r)] +\u001eT2[T2(r)] +\u001eC2[r] =\u001e6:(B21)8\nSo we obtain \u001eC2[x;y] =\u0000\u001e5x\u0000\u001e6y\u0000xy\u001e 1and\u001e1= 0;\u0019\nfornC2= 0. Similary, for nC2= 1, we obtain \u001eC2[x;y] =\n\u0000\u001e5x\u0000\u001e6y\u0000xy\u001e 1.\nUsingC2\n2= 1, we further have \u001e6=\u0000\u001e5fornC2= 0,\nand\u001e6=\u001e6fornC2= 1. So we arrive at the result\nnC2= 0; \u001eC2[x;y] =\u0000\u001e5(x\u0000y)\u0000xy\u001e 1;(B22)\nnC2= 1; \u001eC2[x;y] =\u0000\u001e5(x+y)\u0000xy\u001e 1:(B23)\nHere, to simplify the above expression, we choose a pure\ngauge tranformation ~Wa\nr=eix\u001bz\u001e5. Under the pure\ngauge transformation, the gauge part of the PSG trans-\nforms as\nWO\nr!~Wa\nrWO\nr~Way\nO\u00001(r): (B24)\nClearly ~Wa\nronly modi\fes WT1andWT2by an overall\nphase shift, but WC2rbecomes\nWC2\nr= (i\u001bx)nC2e\u0000ixy\u001e 1\u001bz(B25)\nfor bothnC2= 0;1, except that we require \u001e1= 0;\u0019for\nnC2= 0.\nFor the relation ( S6C2)2= 1, we need to consider the\nfour cases with nS6= 0;1 andnC2= 0;1.\nFornS6=nC2= 0, we have \u001e1=\u0019, and (S6C2)2= 1\ngives\u001e3+ 2\u001e4= 0. We then introduce a pure gauge\ntransformation ~Wb\nr,\n~Wb\nr=e\u0000i(x+y)\u001e4\u001bz: (B26)\nAfter applying ~Wb\nr, we have\n\u001eC2=\u0000xy\u001e 1; (B27)\n\u001eS6=xy\u001e 1\u0000\u001e1y(y\u00001)\n2(B28)\nwith\u001e1= 0;\u0019.\nFornS6= 0 andnC2= 1, we obtain \u001e3= 0. We\nintroduce a pure gauge transformation ~Wc\nr,\n~Wc\nr=e\u0000i(x\u0000y)\u001e4\u001bz: (B29)\nAfter applying ~Wb\nr, we have\n\u001eC2=\u0000xy\u001e 1; (B30)\n\u001eS6=xy\u001e 1\u0000\u001e1y(y\u00001)\n2: (B31)\nFornS6= 1 andnC2= 0, we obtain \u001e3= 0. We apply\na pure gauge transformation ~Wb\nrand obtain\n\u001eC2=\u0000xy\u001e 1; (B32)\n\u001eS6=xy\u001e 1\u0000\u001e1y(y\u00001)\n2: (B33)\nFornS6= 1 andnC2= 1, we obtain \u001e3+ 2\u001e4= 0. We\napply a pure gauge transformation ~Wc\nrand obtain\n\u001eC2=\u0000xy\u001e 1; (B34)\n\u001eS6=xy\u001e 1\u0000\u001e1y(y\u00001)\n2: (B35)In summary, we have\nWT1\nr= 1; WT2\nr=ei\u001e1x: (B36)\nand\nWC2\nr= (i\u001bx)nC2e\u0000i\u001e1xy\u001bz; (B37)\nWS6\nr= (i\u001bx)nS6ei\u001e1[xy\u0000y(y\u00001)\n2]\u001bz; (B38)\nwhere\u001e1= 0;\u0019fornC2= 0 ornS6= 1.\n2. Time reversal symmetry\nBecause time reversal is an antiunitary symmetry, the\nproductO\u00001T\u00001OTbecomes\n(WO\nr)y[(WT\nr)yWO\nrWT\nO\u00001(r)]\u0003(B39)\nfor the PSGs, where WTis the gauge transformation\nassociated with the time reversal. We here rede\fne\nWT\nr=\u0016WT\nr(i\u001by); (B40)\nso that\nO\u00001T\u00001OT ! (WO\nr)y(\u0016WT\nr)yWO\nr\u0016WT\nO\u00001(r):(B41)\n\u0016WT\nrhas the general form \u0016WT\nr= (i\u001bx)nTei\u001eT[r]\u001bz.\nWe start with nT= 0. The relation in Eq. (A13) leads\nto\n\u001eT[x;y]\u0000\u001eT[x\u00001;y] =\u0000\u001e7; (B42)\n\u001eT[x;y+ 1]\u0000\u001eT[x;y] =\u0000\u001e8; (B43)\nso we have \u001eT[x;y] =\u0000\u001e7x\u0000\u001e8y. Applying this result\nto Eq. (A14), we have\n\u0000\u001eC2[y;x]\u0000\u001eT[y;x] +\u001eC2[y;x]\n+\u001eT[x;y] =\u001e9;\n\u0000\u001eS6[x;y]\u0000\u001eT[x;y] +\u001eS6[x;y]\n+\u001eT[y;\u0000x+y] =\u001e10;(B44)\nfornC2=nS6= 0. The above equations give\n\u001e7=\u001e8= 0, so we have \u0016WT\nr= 1. Other cases can be\nobtained likewise. We \fnd that for both nT= 0 and\nnT= 1, there is \u001eT[x;y] = 0 and\u001e1= 0;\u0019. So we have\n\u0016WT\nr= 1;i\u001by; (B45)\nwhere we have used a global and uniform rotation ei\u0019\n4\u001bz\nto rotate\u001bxto the basis of \u001by.\nIncluding the time reversal, there are 16 PSG solutions.\nBut for \u0016WT\nr= 1, the mean-\feld ansatz is found to vanish\neverythere. This makes sense as these PSGs have T2= 1\nfor the fermionic spinons that are expected to Kramers\ndoublets. So only 8 of them with T2=\u00001 for the spinons\nsurvive. Replacing ei\u001e1\u001bzwith\u00061, we present the PSG\nsolutions in the table of the main text.9\nAppendix C: Spinon band structures and mean-\feld\nHamiltonians\nAs we establish in the previous section and the main\ntext, there are four U1A PSGs and four U1B PSGs. In\nthe main text, we have argued that the experimental re-\nsuls in YbMgGaO 4is against the U1B states. So here\nwe focus on the U1A states. From the U1A PSGs, it is\nstraight to obtain the spinon transformations. We list\nthe results in Tab. III.\n1. Spinon band structures\nUsing Tab. III, we obtain the spinon mean-\feld Hamil-\ntonian. In particular, the U1A10 state gives vanishing\nspinon hoppings on the \frst and second neighbors, and\nthe U1A01 state gives an isotropic spinon hopping on\nboth \frst and second neighbors. The U1A10 state, as\nwe described in the main text, has symmetry protected\nband touchings at the \u0000, M and K points. The U1A11\nstate has symmetry protected band touchings at the \u0000\nand M points.\nFor the U1A10 state, the spinon mean-\feld Hamilto-\nnian has the form\nHU1A01\nMF =X\nkh\u000b\f(k)fy\nk\u000bfk\f; (C1)\nwhereh\u000b\f(k) is given by\nh(k) =d0(k)I2\u00022+3X\n\u0016=1d\u0016(k)\u001b\u0016: (C2)\nIn the main text, we have used ( S6)3andTto showd0(k) = 0 and the band touchings at \u0000 and M. To account\nfor the band touching at the K point, we need to use S6\nandC2. UnderS6,\nS6HS\u00001\n6=X\nk\u0002\nei2\u0019\n3h(\u0000S\u00001\n6(k))\"#fy\nk\"fk#+h:c:\u0003\n=H; (C3)\nwhereh(k)\"#=dx(k)\u0000idy(k). Since K is invariant un-\nderS6,\ndx(K)\u0000idy(K) =ei2\u0019\n3[dx(K)\u0000idy(K)];(C4)\nhencedx(K) =dy(K) = 0.\nTheC2symmetry constraints the dzterm, we have\nC2HC\u00001\n2=X\nkdz(C\u00001\n2(k))fy\nk#fk#\u0000dz(C\u00001\n2(k))fy\nk\"fk\"\n=H: (C5)\nSince K is also invariant under C2, we obtain dz(K) =\n\u0000dz(K). Hence dz(K) = 0. We conclude that h(K) = 0\nand there exists a band touching at K.\nFor the U1A11 state, TandS6are implemented in the\nsame way as the U1A01 state, and we arrive at the same\nconclusion that there are band touchings at the \u0000 and M\npoints. At the K point, however, the band structure is\ngenerally gapped due to a nonzero dz.\n2. Spinon mean-\feld Hamiltonians\nThe U1A00 state has the isotropic spinon hoppings on\n\frst and second neighboring bonds, and the mean-\feld\nHamiltonain HU1A00\nMF has already been given in the main\ntext. This states gives a large spinon Fermi surface in\nthe Brioullin zone. The spinon mean-\feld states of the\nU1A01 state and the U1A11 state are given by\nHU1A01\nMF =X\nx;yt1h\n\u0000ify\n(x+1;y);\"f(x;y);#\u0000ify\n(x+1;y);#f(x;y);\"\u0000e\u0000i\u0019\n6fy\n(x;y+1);\"f(x;y);#\n+ei\u0019\n6fy\n(x;y+1);#f(x;y);\"\u0000ei\u0019\n6fy\n(x+1;y+1);\"f(x;y);#+e\u0000i\u0019\n6fy\n(x+1;y+1);#f(x;y);\"+h:c:i\n+t2h\nei2\u0019\n3fy\n(x+1;y\u00001);\"f(x;y);#+ei\u0019\n3fy\n(x+1;y\u00001);#f(x;y);\"+fy\n(x+1;y+2);\"f(x;y);#\n\u0000fy\n(x+1;y+2);#f(x;y);\"+ei\u0019\n3fy\n(x+2;y+1);\"f(x;y);#+ei2\u0019\n3fy\n(x+2;y+1);#f(x;y);\"+h:c:i\n; (C6)10\nTABLE III. The transformation for the spinons under four U1A PSGs that are labeled by U1A nC2nS6.\nU(1) PSGs T1 T2 C2 S6\nU1A00f(x;y);\"!f(x+1;y);\"\nf(x;y);#!f(x+1;y);#f(x;y);\"!f(x;y+1);\"\nf(x;y);#!f(x;y+1);#f(x;y);\"!ei\u0019\n6f(y;x);#\nf(x;y);#!ei5\u0019\n6f(y;x);\"f(x;y);\"!e\u0000i\u0019\n3f(x\u0000y;x);\"\nf(x;y);#!e+i\u0019\n3f(x\u0000y;x);#\nU1A10f(x;y);\"!f(x+1;y);\"\nf(x;y);#!f(x+1;y);#f(x;y);\"!f(x;y+1);\"\nf(x;y);#!f(x;y+1);#f(x;y);\"!ei\u0019\n6fy\n(y;x);\"\nf(x;y);#!e\u0000i\u0019\n6fy\n(y;x);#f(x;y);\"!e\u0000i\u0019\n3f(x\u0000y;x);\"\nf(x;y);#!e+i\u0019\n3f(x\u0000y;x);#\nU1A01f(x;y);\"!f(x+1;y);\"\nf(x;y);#!f(x+1;y);#f(x;y);\"!f(x;y+1);\"\nf(x;y);#!f(x;y+1);#f(x;y);\"!ei\u0019\n6f(y;x);#\nf(x;y);#!ei5\u0019\n6f(y;x);\"f(x;y);\"!\u0000e\u0000i\u0019\n3fy\n(x\u0000y;x);#\nf(x;y);#!e+i\u0019\n3fy\n(x\u0000y;x);\"\nU1A11f(x;y);\"!f(x+1;y);\"\nf(x;y);#!f(x+1;y);#f(x;y);\"!f(x;y+1);\"\nf(x;y);#!f(x;y+1);#f(x;y);\"!ei\u0019\n6fy\n(y;x);\"\nf(x;y);#!e\u0000i\u0019\n6fy\n(y;x);#f(x;y);\"!\u0000e\u0000i\u0019\n3fy\n(x\u0000y;x);#\nf(x;y);#!e+i\u0019\n3fy\n(x\u0000y;x);\"\nand\nHU1A11\nMF =X\nx;yt1h\nify\n(x+1;y);\"f(x;y);\"\u0000ify\n(x+1;y);#f(x;y);#+ify\n(x;y+1);\"f(x;y);\"\n\u0000ify\n(x;y+1);#f(x;y);#\u0000ify\n(x+1;y+1);\"f(x;y);\"+ify\n(x+1;y+1);#f(x;y);#+h:c:i\n+t0\n1h\n\u0000fy\n(x+1;y);\"f(x;y);#+fy\n(x+1;y);#f(x;y);\"+ei\u0019\n3fy\n(x;y+1);\"f(x;y);#\n+ei2\u0019\n3fy\n(x;y+1);#f(x;y);\"+ei2\u0019\n3fy\n(x+1;y+1);\"f(x;y);#+ei\u0019\n3fy\n(x+1;y+1);#f(x;y);\"+h:c:i\n+t2h\nei\u0019\n6fy\n(x+1;y\u00001);\"f(x;y);#+ei5\u0019\n6fy\n(x+1;y\u00001);#f(x;y);\"\u0000ify\n(x+1;y+2);\"f(x;y);#\n\u0000ify\n(x+1;y+2);#f(x;y);\"+ei5\u0019\n6fy\n(x\u00002;y\u00001);\"f(x;y);#+ei\u0019\n6fy\n(x\u00002;y\u00001);#f(x;y);\"+h:c:i\n; (C7)\nwhere in both Hamiltonians t1,t0\n1denote the \frst neigh-\nbor hoppings and t2denotes the second neighbor hop-\nping.\nThe band structures for speci\fc choices of the hopping\nparameters are plotted in the main text. Clearly, we\nobserve the band touchings at the \u0000, M and K points for\nthe U1A01 state, and band touchings at the \u0000 and M\npoints for the U1A11 state.\nAppendix D: The U1A00 state and the\nspectroscopic results\n1. Free spinon mean-\feld theory\nThe spinon mean-\feld Hamiltonian of the U1A00 state\nis\nHU1A00\nMF =\u0000t1X\nhrr0i;\u000bfy\nr\u000bfr\u000b\u0000t2X\nhhrr0ii;\u000bfy\nr\u000bfr\u000b;(D1)\nfrom which we compute the dynamic spin structure factor\nfor di\u000berent choices t2=t1. The dynamic spin structurefactor is given by\nS(q;!) =1\nNX\nr;r0eiq\u0001(r\u0000r0)Z\ndte\u0000i!t\nh\tU1A00\nMFjS\u0000\nr(t)S+\nr0(0)j\tU1A00\nMFi\n=X\nn\u000e(!\u0000\u0018nq)jhnjS+\nqj\tU1A00\nMFij2;(D2)\nwhereNis the total number of spins, the summation\nis over all mean-\feld states with the spinon particle-hole\nexcitation,\u0018nqis the energy of the n-th excited state with\nthe momentum q. The results are depicted in Fig. 4(a-e)\nand are consistent with the inelastic neutron scattering\nresults36,37. All the results so far are independent from\nany microscopic spin interaction.11\nFIG. 4. (a-e) Dynamic spin structure factor for the free spinon theory of the U1A00 state with di\u000berent values of t2=t1. (f-h)\nThe evolution of SRPA(q;!) as a function of J\u0006\u0006. In all sub\fgures, the energy transfer is normalized against the corresponding\nbandwidth B. The parameter \u000bis de\fned as Jzz=t1.\n2. Variational calculation and random phase\napproximation\nHere we consider the microscopic spin Hamiltonian\nthat was introduced in Refs. 34 and 35,\nHspin=X\nhrr0iJzzSz\nrSz\nr0+J\u0006(S+\nrS\u0000\nr0+S\u0000\nrS+\nr0)\n+J\u0006\u0006(\rrr0S+\nrS+\nr0+\r\u0003\nrr0S\u0000\nrS\u0000\nr0)\n\u0000i\n2Jz\u0006[(\r\u0003\nrr0S+\nr\u0000\rrr0S\u0000\nr)Sz\nr0\n+Sz\nr(\r\u0003\nrr0S+\nr0\u0000\rrr0S\u0000\nr0)]; (D3)\nwhere\rrr0= 1;ei2\u0019=3;e\u0000i2\u0019=3forrr0along thea1;a2\nanda3bonds, respectively. Here, a3=\u0000a1\u0000a2. It\nwas suggested and demonstrated that the anisotropic\nJ\u0006\u0006andJz\u0006interactions compete with the XXZ part of\nthe Hamiltonian and may lead to disordered state34,35,39.\nOur calculation does show the enhancement of quantum\n\ructuation in certain regions of the phase diagram35.\nHere we comment about the choices of the exchange cou-\nplings in the main text and in the following calculation.\nTheJzzandJ\u0006couplings can be determined by theCurie-Weiss temperature measurement on a single crys-\ntal sample. The complication comes from the subtraction\nof the Van Vleck susceptibility. Due to the Ga3+/Mg2+\nexchange disorder in the non-magnetic layers, although\nthese ions do not directly enter the Yb exchange path,\nit may modify the local crystal \feld environment of the\nYb3+ion and thus lead to some complication and varia-\ntion of the Van Vleck susceptibility. As a result, the very\nprecise determination of the JzzandJ\u0006couplings can be\nan issue. That may explain some di\u000berences of the Jzz\nandJ\u0006couplings that were obtained34{37,39. Partly for\nthe same reason, the results on J\u0006\u0006andJz\u0006may also\nbe a\u000bected. However, quantum spin liquid, if it exists as\nthe ground state of our generic model, is expected to be\na phase that covers a \fnite region of the phase diagram.\nTherefore, the very precise value of the couplings may\nnot be quite necessary from this point of view. There-\nfore, we here rely on our previous results of the quantum\n\ructuation for the mean-\feld phase diagram that indi-\ncates strong \ructations in certain parameter regimes. We\nchoose the exchange parameters from these disordered re-\ngions.\nFor this spin Hamiltonian, the mean-\feld variational\nenergy is given as\nEvar=h\tU1A00\nMFjHspinj\tU1A00\nMFi=1\nL2X\nqh\tU1A00\nMFjJzz(q)Sz\nqSz\n\u0000q+ 2J\u0006(q)S+\nqS\u0000\n\u0000qj\tU1A00\nMFi\n=1\nL2X\nq\"\nJzz(q)X\nn\f\fhnjSz\nqj\tU1A00\nMFi\f\f2+ 2J\u0006(q)X\nn\f\fhnjS+\nqj\tU1A00\nMFi\f\f2#\n=1\nL4X\nq2\n4Jzz(q)\n4X\nn;k\f\f\fhnjfy\nk+q;\"fk;\"\u0000fy\nk+q;#fk;#j\tU1A00\nMFi\f\f\f2\n+ 2J\u0006(q)X\nn;k\f\f\fhnjfy\nk+q;\"fk;#j\tU1A00\nMFi\f\f\f23\n5;(D4)12\nwhere we have omitted J\u0006\u0006andJz\u0006because they do not\nconserve spin, therefore their contribution to Evaris zero.\nThis is an artifact of the free spinon theory of HU1A00\nMF\nthat only includes isotropic spinon hoppings for the \frst\ntwo neighbors.\nDue to the isotropic spinon hoppings, HU1A00\nMF does not\nexplicitly re\rect the absence of spin-rotational symmetry\nthat is brought by the J\u0006\u0006andJz\u0006interactions. To in-\ncorporate the J\u0006\u0006andJz\u0006interactions, as we describe\nin the main text, we followed the phenomenological treat-\nment for the \\ t-J\" model in the context of cuprate super-\nconductors60and consider H=HU1A00\nMF +H0\nspin, where\nH0\nspinare theJ\u0006\u0006andJz\u0006interactions. In the parton\nconstruction, H0\nspinis treated as the spinon interactions\nand thus introduces the spin rotational symmetry break-\ning. With a random phase approximation for the inter-\nactionH0\nspin, we obtain the dynamic spin susceptibility60\n\u001fRPA(q;!) =\u0002\n1\u0000\u001f0(q;!)J(q)\u0003\u00001\u001f0(q;!);(D5)\nwhere\u001f0is the free-spinon susceptibility, and J(q) is the\nspin exchange matrix from H0\nspin,\nJ(q) =\n0\nBB@2(uq\u0000vq)J\u0006\u0006\u00002p\n3wqJ\u0006\u0006\u0000p\n3wqJz\u0006\n\u00002p\n3wqJ\u0006\u0006 2(\u0000uq+vq)J\u0006\u0006(uq\u0000vq)Jz\u0006\n\u0000p\n3wqJz\u0006 (uq\u0000vq)Jz\u0006 01\nCCA(D6)\nwithuq= cos(q\u0001a1),vq=1\n2(cos(q\u0001a2) + cos(q\u0001a3)),\nandwq=1\n2(cos(q\u0001a2)\u0000cos(q\u0001a3)). The renormalized\nSRPA(q;!) can be read o\u000b from \u001fRPAviaSRPA(q;!) =\n\u00001\n\u0019Im\u0002\n\u001fRPA(q;!)\u0003+\u0000and is plotted in Fig. 3(b) in the\nmain text.\nThe very precise values of J\u0006\u0006andJz\u0006cannot be de-\ntermined from the existing data-rich neutron scattering\nexperiment in a strong \feld normal to the triangular\nplane. This is partly due to the experimental resolu-\ntion, and is also due to the fact that the linear spin wave\nspectrum for the \feld normal to the plane is indepen-\ndent ofJz\u0006and is not quite sensitive to J\u0006\u000635,39. In\nFig. 3(b) of the main text, instead, we choose ( J\u0006\u0006;Jz\u0006)\nto fall into the disordered region of the phase diagram in\nRef. 35 where the quantum \ructuations are expected to\nbe strong35.\nAppendix E: The U1B states\nIn this section we use PSG to determine the free spinon\nmean-\feld Hamiltonian for the U1B states to the \frst and\nsecond spinon hoppings. In Fig. 5, we further present\ntheir spectroscopic features for comparison. Like the\nnotation for U1As, the U1B states are also labeled by\nU1BnC2nS6.1. The U1B00 state\nFor the\u0019-\rux states, the dynamic spin structure fac-\ntor has an enhanced periodicity due to anticommutative\nlattice translations. One direct consequence of the pe-\nriodicity is that \u0000 and M become equivalent, and the\nV-shaped upper excitation edge in Ref. 36 cannot be re-\nproduced for the U1B states.\nWe choose the spinon basis in the momentum space\nfk;I= (fA;k;\";fB;k;\";fA;k;#;fB;k;#)T, whereAandBde-\nnote the two inequivalent sites in each unit cell due to the\n\u0019\rux.\nThe Hamiltonian is written in terms of the Dirac ma-\ntrices \u0000aand their anticommutators\n\u0000ab= [\u0000a;\u0000b]=(2i): (E1)\nThe representation is chosen to be \u0000(1;2;3;4;5)= (\u001bx\n1;\u001bz\n1;\u001by\n\u001cx;\u001by\n\u001cy;\u001by\n\u001cz). \u0000aand \u0000abis odd\nunder time reversal except when a= 4 orb= 4. The\nHamiltonian is thus\nh(k) =5X\na=1da(k)\u0000a+5X\na<b=1dab(k)\u0000ab: (E2)\nFor the U1B00 state, we have\nd3(k) =t0\n1sin(kx=2\u0000p\n3ky=2);\nd4(k) =t0\n1cos(kx=2 +p\n3ky=2);\nd5(k) =\u00002t0\n1sin(kx);\nd13(k) =\u00002t1sin(kx=2\u0000p\n3ky=2);\nd14(k) =\u00002t1cos(kx=2 +p\n3ky=2);\nd15(k) =\u00002t1sin(kx);\nd23(k) =\u0000p\n3t0\n1sin(kx=2\u0000p\n3ky=2);\nd24(k) =p\n3t0\n1cos(kx=2 +p\n3ky=2);\nd34(k) = 2t2cos(p\n3ky);\nd35(k) = 2t2sin(3kx=2\u0000p\n3ky=2);\nd45(k) = 2t2cos(3kx=2 +p\n3ky=2): (E3)\n2. The U1B01 state\nd3(k) =t2sin(3kx=2 +p\n3ky=2);\nd4(k) =\u0000t2cos(3kx=2\u0000p\n3ky=2);\nd5(k) = 2t2sin(p\n3ky);\nd23(k) =\u0000p\n3t2sin(3kx=2 +p\n3ky=2);\nd24(k) =\u0000p\n3t2cos(3kx=2\u0000p\n3ky=2): (E4)13\nFIG. 5. Dynamic spin structure factor for six free spinon mean-\feld states other than U1A00. Note the U1A10 Hamiltonian is\nidentically zero for the \frst and second neighbor hoppings. None of them is consistent with the spinon Fermi surface picture.\nIn all sub\fgures, the energy transfer is normalized against the corresponding bandwidth B.\n3. The U1B10 state\nd3(k) =\u0000p\n3t1sin\u0002\n(kx\u0000p\n3ky)=2\u0003\n;\nd4(k) =p\n3t1cos\u0002\n(kx+p\n3ky)=2\u0003\n;\nd23(k) =\u0000t1sin\u0002\n(kx\u0000p\n3ky)=2\u0003\n;\nd24(k) =\u0000t1cos\u0002\n(kx+p\n3ky)=2\u0003\n;\nd25(k) = 2t1sinkx: (E5)\n4. The U1B11 state\nd3(k) =\u0000p\n3t2sin\u0002\n(3kx+p\n3ky)=2\u0003\n;\nd4(k) =\u0000p\n3t2cos\u0002\n(3kx\u0000p\n3ky)=2\u0003\n;\nd23(k) =\u0000t2sin\u0002\n(3kx+p\n3ky)=2\u0003\n;\nd24(k) =t2cos\u0002\n(3kx\u0000p\n3ky)=2\u0003\n;\nd25(k) =\u00002t2sin(p\n3ky);\nd34(k) = 2t1cos(kx);\nd35(k) =\u00002t1sin\u0002\n(kx+p\n3ky)=2\u0003\n;\nd45(k) =\u00002t1cos\u0002\n(kx\u0000p\n3ky)=2\u0003\n: (E6)14\n\u0003gangchen.physics@gmail.com\n1William Witczak-Krempa, Gang Chen, Yong Baek Kim,\nand Leon Balents, \\Correlated Quantum Phenomena in\nthe Strong Spin-Orbit Regime,\" Annual Review of Con-\ndensed Matter Physics 5, 57{82 (2014).\n2Je\u000brey G. Rau, Eric Kin-Ho Lee, and Hae-Young Kee,\n\\Spin-Orbit Physics Giving Rise to Novel Phases in Cor-\nrelated Systems: Iridates and Related Materials,\" Annual\nReview of Condensed Matter Physics 7, 195{221 (2016).\n3G. Jackeli and G. Khaliullin, \\Mott Insulators in the\nStrong Spin-Orbit Coupling Limit: From Heisenberg to a\nQuantum Compass and Kitaev Models,\" Phys. Rev. Lett.\n102, 017205 (2009).\n4Gang Chen and Leon Balents, \\Spin-orbit e\u000bects in\nNa4Ir3O8: A hyper-kagome lattice antiferromagnet,\"\nPhys. Rev. B 78, 094403 (2008).\n5Ji \u0014 r\u0013 \u0010 Chaloupka, George Jackeli, and Giniyat Khaliullin,\n\\Kitaev-Heisenberg Model on a Honeycomb Lattice: Pos-\nsible Exotic Phases in Iridium Oxides A2IrO3,\" Phys. Rev.\nLett. 105, 027204 (2010).\n6Dmytro Pesin and Leon Balents, \\Mott physics and band\ntopology in materials with strong spin{orbit interaction,\"\nNature Physics 6, 376{381 (2010).\n7Shigeki Onoda and Yoichi Tanaka, \\Quantum melting of\nspin ice: Emergent cooperative quadrupole and chirality,\"\nPhys. Rev. Lett. 105, 047201 (2010).\n8Lucile Savary and Leon Balents, \\Coulombic quantum liq-\nuids in spin-1 =2 pyrochlores,\" Phys. Rev. Lett. 108, 037202\n(2012).\n9Kate A. Ross, Lucile Savary, Bruce D. Gaulin, and Leon\nBalents, \\Quantum Excitations in Quantum Spin Ice,\"\nPhys. Rev. X 1, 021002 (2011).\n10Yi-Ping Huang, Gang Chen, and Michael Hermele,\n\\Quantum Spin Ices and Topological Phases from Dipolar-\nOctupolar Doublets on the Pyrochlore Lattice,\" Phys. Rev.\nLett. 112, 167203 (2014).\n11Gang Chen and Leon Balents, \\Spin-orbit coupling in d2\nordered double perovskites,\" Phys. Rev. B 84, 094420\n(2011).\n12Hamid R. Molavian, Michel J. P. Gingras, and Benjamin\nCanals, \\Dynamically Induced Frustration as a Route to\na Quantum Spin Ice State in Tb 2Ti2O7via Virtual Crys-\ntal Field Excitations and Quantum Many-Body E\u000bects,\"\nPhys. Rev. Lett. 98, 157204 (2007).\n13Gang Chen and Yong Baek Kim, \\Anomalous enhance-\nment of the Wilson ratio in a quantum spin liquid: The\ncase of Na 4Ir3O8,\" Phys. Rev. B 87, 165120 (2013).\n14R. Applegate, N. R. Hayre, R. R. P. Singh, T. Lin, A. G. R.\nDay, and M. J. P. Gingras, \\Vindication of Yb 2Ti2O7as\na Model Exchange Quantum Spin Ice,\" Phys. Rev. Lett.\n109, 097205 (2012).\n15K. A. Ross, J. P. C. Ru\u000b, C. P. Adams, J. S. Gard-\nner, H. A. Dabkowska, Y. Qiu, J. R. D. Copley, and\nB. D. Gaulin, \\Two-Dimensional Kagome Correlations and\nField Induced Order in the Ferromagnetic XY Pyrochlore\nYb2Ti2O7,\" Phys. Rev. Lett. 103, 227202 (2009).\n16Gang Chen and Michael Hermele, \\Magnetic orders and\ntopological phases from f-dexchange in pyrochlore iri-\ndates,\" Phys. Rev. B 86, 235129 (2012).\n17Lucile Savary, Xiaoqun Wang, Hae-Young Kee, Yong Baek\nKim, Yue Yu, and Gang Chen, \\Quantum spin ice onthe breathing pyrochlore lattice,\" Phys. Rev. B 94, 075146\n(2016).\n18SungBin Lee, Eric Kin-Ho Lee, Arun Paramekanti, and\nYong Baek Kim, \\Order-by-disorder and magnetic \feld re-\nsponse in the Heisenberg-Kitaev model on a hyperhoney-\ncomb lattice,\" Phys. Rev. B 89, 014424 (2014).\n19SungBin Lee, Shigeki Onoda, and Leon Balents, \\Generic\nquantum spin ice,\" Phys. Rev. B 86, 104412 (2012).\n20Eric Kin-Ho Lee, Robert Scha\u000ber, Subhro Bhattacharjee,\nand Yong Baek Kim, \\Heisenberg-kitaev model on the hy-\nperhoneycomb lattice,\" Phys. Rev. B 89, 045117 (2014).\n21Shigeki Onoda and Yoichi Tanaka, \\Quantum \ructua-\ntions in the e\u000bective pseudospin-1\n2model for magnetic py-\nrochlore oxides,\" Phys. Rev. B 83, 094411 (2011).\n22Lieh-Jeng Chang, Shigeki Onoda, Yixi Su, Ying-Jer Kao,\nKu-Ding Tsuei, Yukio Yasui, Kazuhisa Kakurai, and\nMartin Richard Lees, \\Higgs transition from a magnetic\nCoulomb liquid to a ferromagnet in Yb 2Ti2O7,\" Nature\nCommunications 3, 992 (2012).\n23J. S. Gardner, S. R. Dunsiger, B. D. Gaulin, M. J. P. Gin-\ngras, J. E. Greedan, R. F. Kie\r, M. D. Lumsden, W. A.\nMacFarlane, N. P. Raju, J. E. Sonier, I. Swainson, and\nZ. Tun, \\Cooperative Paramagnetism in the Geometrically\nFrustrated Pyrochlore Antiferromagnet Tb 2Ti2O7,\" Phys.\nRev. Lett. 82, 1012{1015 (1999).\n24Fei-Ye Li, Yao-Dong Li, Yue Yu, Arun Paramekanti, and\nGang Chen, \\Kitaev materials beyond iridates: order by\nquantum disorder and weyl magnons in rare-earth double\nperovskites,\" Phys. Rev. B 95, 085132 (2017).\n25Gang Chen, \\\\Magnetic monopole\" condensation of the\npyrochlore ice U(1) quantum spin liquid: Application to\nPr2Ir2O7and Yb 2Ti2O7,\" Phys. Rev. B 94, 205107 (2016).\n26M J P Gingras and P A McClarty, \\Quantum spin ice: a\nsearch for gapless quantum spin liquids in pyrochlore mag-\nnets,\" Reports on Progress in Physics 77, 056501 (2014).\n27Yao-Dong Li and Gang Chen, \\Symmetry enriched U(1)\ntopological orders for dipole-octupole doublets on a py-\nrochlore lattice,\" Phys. Rev. B 95, 041106 (2017).\n28E. Lhotel, S. Petit, S. Guitteny, O. Florea,\nM. Ciomaga Hatnean, C. Colin, E. Ressouche, M. R. Lees,\nand G. Balakrishnan, \\Fluctuations and All-In All-Out\nOrdering in Dipole-Octupole Nd 2Zr2O7,\" Phys. Rev. Lett.\n115, 197202 (2015).\n29Owen Benton, Olga Sikora, and Nic Shannon, \\Seeing\nthe light: Experimental signatures of emergent electromag-\nnetism in a quantum spin ice,\" Phys. Rev. B 86, 075154\n(2012).\n30Anson W. C. Wong, Zhihao Hao, and Michel J. P. Gingras,\n\\Ground state phase diagram of generic xypyrochlore\nmagnets with quantum \ructuations,\" Phys. Rev. B 88,\n144402 (2013).\n31Gang Chen, Rodrigo Pereira, and Leon Balents, \\Exotic\nphases induced by strong spin-orbit coupling in ordered\ndouble perovskites,\" Phys. Rev. B 82, 174440 (2010).\n32S. H. Curnoe, \\Structural distortion and the spin liquid\nstate in Tb 2Ti2O7,\" Phys. Rev. B 78, 094418 (2008).\n33Yuesheng Li, Haijun Liao, Zhen Zhang, Shiyan Li, Feng\nJin, Langsheng Ling, Lei Zhang, Youming Zou, Li Pi,\nZhaorong Yang, Junfeng Wang, Zhonghua Wu, and Qing-\nming Zhang, \\Gapless quantum spin liquid ground state\nin the two-dimensional spin-1/2 triangular antiferromag-15\nnet YbMgGaO 4,\" Scienti\fc Reports 5, 16419 (2015).\n34Yuesheng Li, Gang Chen, Wei Tong, Li Pi, Juanjuan Liu,\nZhaorong Yang, Xiaoqun Wang, and Qingming Zhang,\n\\Rare-Earth Triangular Lattice Spin Liquid: A Single-\nCrystal Study of YbMgGaO4,\" Phys. Rev. Lett. 115,\n167203 (2015).\n35Yao-Dong Li, Xiaoqun Wang, and Gang Chen,\n\\Anisotropic spin model of strong spin-orbit-coupled trian-\ngular antiferromagnets,\" Phys. Rev. B 94, 035107 (2016).\n36Yao Shen, Yao-Dong Li, Hongliang Wo, Yuesheng Li,\nShoudong Shen, Bingying Pan, Qisi Wang, H. C. Walker,\nP. Ste\u000bens, M Boehm, Yiqing Hao, D. L. Quintero-Castro,\nL. W. Harriger, Lijie Hao, Siqin Meng, Qingming Zhang,\nGang Chen, and Jun Zhao, \\Spinon Fermi surface in a tri-\nangular lattice quantum spin liquid YbMgGaO 4,\" Nature\n540, 559{562 (2016).\n37Joseph A. M. Paddison, Zhiling Dun, Georg Ehlers, Yao-\nhua Liu, Matthew B. Stone, Haidong Zhou, and Martin\nMourigal, \\Continuous excitations of the triangular-lattice\nquantum spin liquid YbMgGaO 4,\" Nature Physics, arXiv\npreprint 1607.03231 (2016).\n38Yuesheng Li, Devashibhai Adroja, Pabitra K. Biswas, Pe-\nter J. Baker, Qian Zhang, Juanjuan Liu, Alexander A.\nTsirlin, Philipp Gegenwart, and Qingming Zhang, \\Muon\nSpin Relaxation Evidence for the U(1) Quantum Spin-\nLiquid Ground State in the Triangular Antiferromagnet\nYbMgGaO4,\" Phys. Rev. Lett. 117, 097201 (2016).\n39Yao-Dong Li, Yao Shen, Yuesheng Li, Jun Zhao, and\nGang Chen, \\The e\u000bect of spin-orbit coupling on the\ne\u000bective-spin correlation in YbMgGaO 4,\" arXiv preprint\n1608.06445 (2016).\n40Yao-Dong Li, Xiaoqun Wang, and Gang Chen, \\Hidden\nmultipolar orders of dipole-octupole doublets on a trian-\ngular lattice,\" Phys. Rev. B 94, 201114 (2016).\n41Yao-Dong Li and Gang Chen, \\Detecting spin fractional-\nization in a spinon fermi surface spin liquid,\" Phys. Rev.\nB96, 075105 (2017).\n42Haruki Watanabea, Hoi Chun Po, Ashvin Vishwanath,\nand Michael Zaletel, \\Filling constraints for spin-orbit cou-\npled insulators in symmorphic and nonsymmorphic crys-\ntals,\" PNAS 112, 14551{14556 (2015).\n43Y. Xu, J. Zhang, Y. S. Li, Y. J. Yu, X. C. Hong,\nQ. M. Zhang, and S. Y. Li, \\Absence of Magnetic Ther-\nmal Conductivity in the Quantum Spin-Liquid Candidate\nYbMgGaO4,\" Phys. Rev. Lett. 117, 267202 (2016).\n44Olexei I. Motrunich, \\Variational study of triangular lat-\ntice spin-1=2 model with ring exchanges and spin liquid\nstate in\u0014-(ET)2Cu2(CN)3,\" Phys. Rev. B 72, 045105\n(2005).\n45Sung-Sik Lee and Patrick A. Lee, \\U(1) Gauge Theory of\nthe Hubbard Model: Spin Liquid States and Possible Ap-\nplication to \u0014-(BEDT-TTF)2Cu2(CN)3,\" Phys. Rev. Lett.\n95, 036403 (2005).\n46Patrick A. Lee and Naoto Nagaosa, \\Gauge theory of the\nnormal state of high- Tcsuperconductors,\" Phys. Rev. B\n46, 5621{5639 (1992).\n47Xiao-Gang Wen, \\Quantum orders and symmetric spin liq-\nuids,\" Phys. Rev. B 65, 165113 (2002).\n48See the Supplementary information.\n49Johannes Reuther, Shu-Ping Lee, and Jason Alicea, \\Clas-\nsi\fcation of spin liquids on the square lattice with strong\nspin-orbit coupling,\" Phys. Rev. B 90, 174417 (2014).\n50Gang Chen, Andrew Essin, and Michael Hermele, \\Majo-\nrana spin liquids and projective realization of SU(2) spinsymmetry,\" Phys. Rev. B 85, 094418 (2012).\n51Michael Hermele, \\SU(2) gauge theory of the Hubbard\nmodel and application to the honeycomb lattice,\" Phys.\nRev. B 76, 035125 (2007).\n52Samuel Bieri, Claire Lhuillier, and Laura Messio, \\Projec-\ntive symmetry group classi\fcation of chiral spin liquids,\"\nPhys. Rev. B 93, 094437 (2016).\n53Yi-Zhuang You, Itamar Kimchi, and Ashvin Vishwanath,\n\\Doping a spin-orbit Mott insulator: Topological super-\nconductivity from the Kitaev-Heisenberg model and pos-\nsible application to (Na 2/Li2)IrO 3,\" Phys. Rev. B 86,\n085145 (2012).\n54Robert Scha\u000ber, Subhro Bhattacharjee, and Yong Baek\nKim, \\Spin-orbital liquids in non-Kramers magnets on the\nkagome lattice,\" Phys. Rev. B 88, 174405 (2013).\n55Yuan-Ming Lu, \\Symmetry protected gapless Z2spin liq-\nuids,\" arXiv preprint 1606.05652 (2016).\n56Xiao-Gang Wen, \\Quantum order: a quantum entangle-\nment of many particles,\" Physics Letters A 300, 175 { 181\n(2002).\n57Andrew M. Essin and Michael Hermele, \\Spectroscopic\nsignatures of crystal momentum fractionalization,\" Phys.\nRev. B 90, 121102 (2014).\n58In the previous work36, only the nearest-neighbor spinon\nhopping is included.\n59Sandor Toth, Katharina Rolfs, Rolfs, Andrew R. Wildes,\nand Christian Ruegg, \\Strong exchange anisotropy in\nYbMgGaO 4from polarized neutron di\u000braction,\" arXiv\npreprint arXiv:1705.05699 (2017).\n60Jan Brinckmann and Patrick A. Lee, \\Slave Boson Ap-\nproach to Neutron Scattering in YBa 2Cu3O6+ySupercon-\nductors,\" Phys. Rev. Lett. 82, 2915{2918 (1999).\n61Yuesheng Li, Devashibhai Adroja, Robert I. Bewley, David\nVoneshen, Alexander A. Tsirlin, Philipp Gegenwart, and\nQingming Zhang, \\Crystalline Electric-Field Randomness\nin the Triangular Lattice Spin-Liquid YbMgGaO4,\" Phys.\nRev. Lett. 118, 107202 (2017).\n62Yao Shen, Yao-Dong Li, H. C. Walker, P. Ste\u000bens,\nM. Boehm, Xiaowen Zhang, Shoudong Shen, Hongliang\nWo, Gang Chen, and Jun Zhao, \\Fractionalized exci-\ntations in the partially magnetized spin liquid candidate\nYbMgGaO 4,\" arXiv preprint arXiv:1708.06655 (2017).\n63Changle Liu, Rong Yu, and Xiaoqun Wang, \\Semiclassical\nground-state phase diagram and multi- qphase of a spin-\norbit-coupled model on triangular lattice,\" Phys. Rev. B\n94, 174424 (2016).\n64Qiang Luo, Shijie Hu, Bin Xi, Jize Zhao, and Xiao-\nqun Wang, \\Ground-state phase diagram of an anisotropic\nspin-1\n2model on the triangular lattice,\" Phys. Rev. B 95,\n165110 (2017).\n65Z. Zhu, P.A. Maksimov, S.R. White, and A.L. Cherny-\nshev, \\Disorder-induced Mimicry of a Spin Liquid in\nYbMgGaO 4,\" arXiv preprint arXiv:1703.02971 (2017).\n66M. Hirschberger, J. W. Krizan, R. J. Cava, and N. P.\nOng, \\Large thermal hall conductivity of neutral spin ex-\ncitations in a frustrated quantum magnet,\" Science 348,\n106{109 (2015).\n67S. J. Li, Z. Y. Zhao, C. Fan, B. Tong, F. B. Zhang, J. Shi,\nJ. C. Wu, X. G. Liu, H. D. Zhou, X. Zhao, and X. F.\nSun, \\Low-temperature thermal conductivity of Dy2Ti2O7\nand Yb 2Ti2O7single crystals,\" Phys. Rev. B 92, 094408\n(2015).\n68Q. J. Li, Z. Y. Zhao, C. Fan, F. B. Zhang, H. D. Zhou,\nX. Zhao, and X. F. Sun, \\Phonon-glass-like behavior of16\nmagnetic origin in single-crystal Tb 2Ti2O7,\" Phys. Rev. B\n87, 214408 (2013).\n69Yuji Matsuda, (2017), Talk at workshop on topological\nmaterials, Kyoto University.\n70Tian-Heng Han, Joel S Helton, Shaoyan Chu, Daniel G\nNocera, Jose A Rodriguez-Rivera, Collin Broholm, and\nYoung S Lee, \\Fractionalized excitations in the spin-liquid\nstate of a kagome-lattice antiferromagnet,\" Nature 492,\n406{410 (2012).\n71Y. Shimizu, K. Miyagawa, K. Kanoda, M. Maesato, and\nG. Saito, \\Spin liquid state in an organic mott insulator\nwith a triangular lattice,\" Phys. Rev. Lett. 91, 107001\n(2003).\n72T. Itou, A. Oyamada, S. Maegawa, M. Tamura, and\nR. Kato, \\Quantum spin liquid in the spin-12 triangular\nantiferromagnet EtMe 3Sb[Pd(dmit)2]2,\" Phys. Rev. B 77,\n104413 (2008).\n73Satoshi Yamashita, Yasuhiro Nakazawa, Masaharu Oguni,\nYugo Oshima, Hiroyuki Nojiri, Yasuhiro Shimizu, Kazuya\nMiyagawa, and Kazushi Kanoda, \\Thermodynamic prop-\nerties of a spin-1/2 spin-liquid state in a kappa-type organic\nsalt,\" Nature Physics 4, 459{462 (2008).\n74Y. Q. Liu, S. J. Zhang, J. L. Lv, S. K. Su, T. Dong, Gang\nChen, and N. L. Wang, \\Revealing a triangular latticeising antiferromagnet in a single-crystal CeCd 3As3,\" arXiv\npreprint 1612.03720 (2016).\n75Shohei Higuchi, Yuki Noshima, Naoki Shirakawa, Masami\nTsubota, and Jiro Kitagawa, \\Optical, transport and mag-\nnetic properties of new compound CeCd 3P3,\" Materials\nResearch Express 3, 056101 (2016).\n76Andr T. Nientiedt and Wolfgang Jeitschko, \\The Series\nof Rare Earth Zinc Phosphides RZn 3P3(R=Y, La\u0000Nd,\nSm, Gd\u0000Er) and the Corresponding Cadmium Compound\nPrCd 3P3,\" Journal of Solid State Chemistry 146, 478 {\n483 (1999).\n77A. Yamada, N. Hara, K. Matsubayashi, K. Munakata,\nC. Ganguli, A. Ochiai, T. Matsumoto, and Y. Uwatoko,\n\\E\u000bect of pressure on the electrical resistivity of CeZn 3P3,\"\nJ. Phys.: Conf. Ser. 215, 012031 (2010).\n78Stanislav S. Stoyko and Arthur Mar, \\Ternary Rare-Earth\nArsenides REZn 3As3(RE = La\u0000Nd, Sm) and RECd 3As3\n(RE = La\u0000Pr),\" Inorganic Chemistry 50, 11152{11161\n(2011).\n79M. B. Sanders, F. A. Cevallos, and R. J. Cava, \\Mag-\nnetism in the KBaRE(BO 3)2(RE= Sm, Eu, Gd, Tb, Dy,\nHo, Er, Tm, Yb, Lu) series: materials with a triangular\nrare earth lattice,\" arXiv preprint 1611.08548 (2016)."    },    {        "title": "0905.0611v1.Transport_through_a_band_insulator_with_Rashba_spin_orbit_coupling__metal_insulator_transition_and_spin_filtering_effects.pdf",        "content": "arXiv:0905.0611v1  [cond-mat.mes-hall]  5 May 2009Transport through a band insulator with Rashba spin-orbit\ncoupling:\nmetal-insulator transition and spin-filtering effects\nT. Jonckheere,1G.I. Japaridze,2T. Martin,1,3and R. Hayn4\n1Centre de Physique Th´ eorique, UMR 6207, case 907,\nCampus de Luminy, 13288 Marseille Cedex 9, France\n2Andronikashvili Institute of Physics,\nTamarashvili str. 6, 0177 Tbilisi, Georgia\n3Universit´ e de la Mediterran´ ee, Campus de Luminy, 13288 Ma rseille Cedex 9, France\n4Institut Mat´ eriaux Micro´ electronique Nanosciences de P rovence,\nFacult´ e St. J´ erˆ ome, Case 142, F-13397 Marseille Cedex 20 , France\n(Dated: November 20, 2018)\nAbstract\nWecalculatethecurrent-voltage characteristic ofaone-d imensionalbandinsulatorwithmagnetic\nfield and Rashba spin-orbit coupling which is connected to no nmagnetic leads. Without spin-orbit\ncoupling we find a complete spin-filtering effect, meaning that the electric transport occurs in one\nspin channel only. For a large magnetic field which closes the band gap, we show that spin-orbit\ncoupling leads to a transition from metallic to insulating b ehavior. The oscillations of the different\nspin-components of the current with the length of the transp ort channel are studied as well.\nPACS numbers: 72.25.-b, 71.70.Ej\n1I. INTRODUCTION\nThere is a great interest today to study the phenomena of quantu m transport in low\ndimensional systems, both from a technological and a fundamenta l point of view. Especially\nimportant are questions of spin polarized transport, also known as spintronics.1A famous\nexample is the proposition of the Datta-Das transistor2which uses the rotation of the elec-\ntron spin due to spin-orbit (SO) coupling. There are two sources of spin-orbit coupling in\nquasi one-dimensional systems (1D), an intrinsic one due to the lac k of inversion symmetry\nin certain crystal structures (Dresselhaus term)3and an external one triggered by an applied\nvoltage to surface gates (the Rashba SO coupling).4\nSeveral works studied the SO coupling and electronic transport in q uasi 1D metallic\nsystems.5,6,7,8,9,10,11,12,13,14,15,16In contrast, the influence of SO coupling and magnetic field\non the transport in 1D band insulators is unexplored, and it can be ex pected to be funda-\nmentally different. In the letter band insulators, we will report on tw o interesting effects:\nthe complete spin filtering effect and the SO induced metal-insulator t ransition. An in-\ncomplete spin filtering effect is possible in 1D metallic systems with a pote ntial step or\nadditional impurities,7,14,16but the complete spin filtering as well as the spin-orbit induced\nmetal-insulator transition which will be reported below are specific to 1D band insulators\nand cannot be observed (in principle) in 1D metals.\nA prototype model for a one-dimensional (1D) band insulator is a ha lf-filled ionic chain\nwith alternating on-site energies (energy difference ∆). Such an ion ic chain will be used in\nour study, however the obtained results are expected to be gene ric to any kind of 1D band\ninsulators, including charge transfer insulators and realized in diato mic polymers,17as well\nas the 1D Peierls insulators, such as polyacetylene.18In a wider sense, one-dimensional band\ninsulators may also be realized in carbon-nanotubes. These nanotu bes have the advantage\nthat the value of the gap may be tuned in a very wide range from 600 m eV (for (12,0)\nnanotubes) up to 8 meV (for (13,0) nanotubes) or even smaller valu es.19\nBefore presenting detailed calculations, let us start with some qualit ative arguments.\nWe first discuss transport in 1D band insulators in a magnetic field Band in absence of\nSO interaction. Although the magnetic field induced metal-insulator a nd insulator-metal\ntransitions have been the subject of studies for decades,20in the context of transport in\nmesoscopic systems these effects have not been investigated in de tail. As we show in this\n2paper, in the limit of ultra-low temperatures ( T≪∆) and strong magnetic field ( B≥∆)\nthe field induced insulator-metal transitions lead to the almost comp lete spin filtering effect,\nsince in this case only one spin channel is open for transport at the F ermi level.\nHowever, the metallic phase reached at B≥∆ shows unconventional and substantially\ndifferent properties compared to a normal metal. As we will show, co ntrary to the usual\n1D metallic phase, the Rashba spin-orbit coupling opens up a gap again , leading to a spin-\norbit induced metal-insulator transition. It is important to note tha t both effects, i.e. the\ncomplete spin filtering effect and the metal-insulator transition induc ed by the Rashba spin-\norbit coupling are very specific to 1D band insulators, and may not be observed in 1D\nmetals.\nRather than analyzing the effect of these transitions by computing the bulk transport\nproperties of the chain, such as the conductivity, we choose to co mpute the current of a finite\nchain of such a material, whose extremities are connected to metallic electrodes. A bias is\nimposed between the electrodes in order to induce current flow. On the one hand it allows\nto probe the spin filtering effects in a setup which is close to experimen tal situations, on the\nother hand it also allows to investigate potential fluctuations of the current as a function\nof the chain length in the presence of SO coupling. In particular, we w ill show that a\ncomplex behavior, with several periods and a complicated energy de pendence is obtained in\nthe presence of a band gap ∆ and a magnetic field; this is totally differe nt from the simple\nharmonic oscillations, with a period inversely proportional to the SO c oupling strength,\nobtained in the metallic case.\nThe paper is organized as follows. In Sec. II, we introduce the mode l and in Sec. III we\ndiscuss the spectrum of the infinite chain. In Sec. IV, we discuss th e method which is used\nto obtain the transport properties as well as physical results. We conclude in Sec. V.\nII. THE MODEL\nWe note first that the spin-orbit coupling can be generated by a volt ageVGapplied to\nexternal gates perpendicular to the current. This is known as Ras hba spin-orbit coupling,4\nand defines the device studied in the present paper (Fig. 1). We con sider a finite chain\n(oriented in the ˆ xdirection) connected to metallic leads. Lateral metallic gates are pla ced\nso that to create an electric field which is perpendicular to both the c hain and the magnetic\n3GATEQuantum wire\nVDVGH\nFIG. 1: (Color online) Schematic figure of the transport proc ess studied in the paper. The SO\ncoupling parameter αRis proportional to VG.\nfield(ˆz)direction. Withtheseconventions thefollowingHamiltoniandescribe s themolecular\nchain:\nH=−t/summationdisplay\nn,σ/parenleftbig\nc†\nn,σcn+1,σ+h.c./parenrightbig\n+∆\n2/summationdisplay\nn,σ(−1)nc†\nn,σcn,σ\n−gµBH\n2/summationdisplay\nn,σσc†\nn,σcn,σ\n+αR/summationdisplay\nn/parenleftig\nc†\nn,↑cn+1,↓−c†\nn,↓cn+1,↑+h.c./parenrightig\n. (1)\nHere the first contribution describes the kinetic energy in the tight binding model, the\nsecond one accounts for alternating on-site energies, the third t erm is the Zeeman coupling\n(magnetic field B=gµBH) and the last term is the Rashba SO coupling (strength αR). We\nconsider a finite chain of length Lwhich is connected to left and right leads by tunneling\namplitudes TlandTr, respectively. Note that we investigate here the case of nonmagnetic\nleads. We assume that the SO coupling vanishes in the leads and that the ma gnetic field\nonly affects the central region significantly.\nIII. THE SPECTRUM\nTo understand the magneto transport results it is useful to first consider the spectrum\nof (1). For clarity all spectra are plotted in the reduced Brillouin zon ek∈[−π/2a,π/2a]\nassociated with the presence (possibly small) of alternating on site e nergies. Typically this\n4/MinusΠ\n2/MinusΠ\n4Π\n4Π\n2\n/Minus2/Minus112\n/MinusΠ\n2/MinusΠ\n4Π\n4Π\n2\n/Minus2/Minus112\n/MinusΠ\n2/MinusΠ\n4Π\n4Π\n2\n/Minus2/Minus112\n/MinusΠ\n2/MinusΠ\n4Π\n4Π\n2\n/Minus2/Minus112ΑR/EquΑl0 ΑR/EquΑl0.4\n/CΑΠDeltΑ/EquΑl0\n/CΑΠDeltΑ/EquΑl0.6\nFIG. 2: Spectrum of the tight-binding chain (see Eqs. (1)-(2 )) with magnetic field ( B=gµBH=\n1.3), with and without Rashba coupling αRand ionicity ∆ ( t= 1 has been taken as unit of energy).\nspectrum consists of 4 branches and it can be obtained exactly:\nE±\n1/2(k) =±/radicalig\n4α2\nRsin2k+B2\n4+∆2\n4+4t2cos2k±W\nW=/radicalig\n16α2\nRt2sin2(2k)+4B2t2cos2k+B2∆2\n4(2)\nin the general case with spin-orbit coupling αRand in the presence of a magnetic field. It is\nshown in Fig. 2 for different cases of ∆ and αR, with a non-zero magnetic field B.\nThe upper left corner of Fig. 2 depicts the trivial case of a non dimer ized tight binding\nchain (∆ = 0) in the presence of a magnetic field. The latter gives rise t o a splitting between\nthe spin up and spin down bands. The spectrum has been folded in this reduced Brillouin\nzone to serve as a point of comparison for the other cases, with dim erisation.\nWe now consider the case of a non-zero value for ∆ (bottom left plot of Fig. 2). For\nαR= 0, the spin up and down bands are still separated, but the dimerisa tion opens a gap\nfor each spin band at the boundaries of the Brillouin zone. This implies t hat for energies\nclose to the Fermi level only one spin channel will be open for the tra nsport (complete spin\nfiltering effect, see next Section). As shown on the Figure, the mag netic field can be so\nstrong that the gap closes and the system can become metallic. We n ow switch on the\nRashba coupling in the presence of dimerisation (bottom right corne r of Fig. 2). In this\ncase, the coupling between spin up and spin down gives rise to an antic rossing, so that the\nspin-orbit coupling opens up a gap again.\nOn the other hand, there is no spin filtering effect for a homogeneou s, metallic chain\n5(∆ = 0, top row of Fig. 2). Without magnetic field (not shown), the sp in-orbit coupling can\nbe taken exactly into account by a shift of k→k+arctan(αR/t). As can be easily inferred\nfrom the spin split band structure in a magnetic field (left plot) the de nsity of states for\nspin up and spin down electrons is the same in that case. And the intro duction of spin-orbit\ncoupling (right plot) does not open a gap. This proves that both effe cts, i.e. the complete\nspin filtering effect and the spin-orbit driven metal-insulator transit ion cannot be observed\nin a metallic system (∆ = 0).\nIV. TRANSPORT THROUGH A FINITE CHAIN\nIntheabsence of electronic interactions, the current througha finite chain oflength Lcan\nbe cast exactly in a Landauer type formula, written here for zero t emperature. This current\ndepends on the orientation of electrons spin at the input lead and th e output lead: the\ncurrentIss′for instance, corresponds to electrons which enter with spin s(withs=↑or↓)\nfrom the left lead and leave the current channel with spin s′to the right lead. With this\nconvention,\nIss′(VD) = ΓLΓR/integraldisplayµR\nµLdE/vextendsingle/vextendsingle/vextendsingleGss′\nab(E)/vextendsingle/vextendsingle/vextendsingle2\n. (3)\nThe integration is peformed between the chemical potentials of the left and right leads\n(µL=−VDandµR= 0). The energy dependent transmission is simply proportional to t he\nsquare modulus ofthe total retarded Greenfunction of the chain (which include the coupling\nwith the leads) between both endpoints, noted here aandb. The tunneling rates on the left\nand right side are defined as Γ j≡2πρjT2\nj(j=L,R), whereρjis the (constant) density of\nstates of lead j, andTjthe tunneling amplitude to lead j. The total Green function of the\nchain between the end sites aandb,Gss′\nabcan be obtained from the Green function of the\nbare chain (uncoupled to leads) gss′\nabby solving the Dyson equations:\n\nG↑↑\nab\nG↓↑\nab\nG↑↑\nbb\nG↓↑\nbb\n=\ng↑↑\nab\ng↓↑\nab\ng↑↑\nbb\ng↓↑\nbb\n+\ng↑↑\naag↑↓\naag↑↑\nabg↑↓\nab\ng↓↑\naag↓↓\naag↓↑\nabg↓↓\nab\ng↑↑\nbag↑↓\nbag↑↑\nbbg↑↓\nbb\ng↓↑\nbag↓↓\nbag↓↑\nbbg↓↓\nbb\n\nΣaG↑↑\nab\nΣaG↓↑\nab\nΣbG↑↑\nbb\nΣbG↓↑\nbb\n\nandsimilar equations for theopposite spins, andwhere Σ j=−iΓjis theretarded self-energy\ncoming from the coupling to lead j=L,R. The Green functions of the bare chain gss′\nabare\n6obtained simply by computing the eigenvalues and eigenstates of the finite chain, and using\na spectral representation:\ngss′\nab(E) =/summationdisplay\nnψs\nn(a)/parenleftbig\nψs′\nn(b)/parenrightbig∗\nE−En+i0+(4)\nHere all the Green functions, and consequently the current in Eq. (3), are 2x2 matrices in\nspin space. This is a consequence of the Rashba SO coupling, which co uples the spin-up\nand spin-down channels. Without SO coupling all quantities become dia gonal in spin space,\nand the formula for the total Green function reduces to:\nGss\nab=gss\nab\n(1−ΣLgssaa)(1−ΣRgss\nbb)−ΣLΣRgss\nabgss\nba(5)\nLetusstartthediscussionofournumerical resultswiththecurre nt-voltagecharacteristics\nin a magnetic field with ∆ /negationslash= 0, but without SO coupling (see Fig. 3). The magnetic field\nB=Bcis chosen such that it just closes the gap, but the exact value of th is parameter\nis nevertheless not important for the spin-filtering effect . The transport for drain voltages\nbetweenVD= 0 andVD≃0.6tis only possible for one spin channel. It means that we find\ncomplete spin polarization in the transport channel (connected to nonmagnetic leads) and\na complete spin-filtering. The spin polarization of the current is defin ed in the general case\nas7,14\nP=I↑↑+I↓↑−I↑↓−I↓↓\nI↑↑+I↓↑+I↑↓+I↓↓. (6)\nAs shown on Fig. 3, the spin polarization remains finite (but smaller tha n unity) for larger\nvoltages (between approximatively 0.6 tand2.25t) anddisappears at approximatively 2.25 t\nwhere the current reaches saturation (all the electrons of the t ight-binding band contribute).\nA finite spin polarization means also that the current creates a tota l magnetization M\nin the transport channel of length L. The value of the total magnetization is given by\nM/µB=L(I↑↑+I↓↑−I↑↓−I↓↓)//angbracketleftv/angbracketright, where/angbracketleftv/angbracketrightmeans the average velocity of the electrons\nwhich are active in the transport process (ballistic transport).\nThis spin-filtering effect is expected to work for a wide range of gap v alues. The voltage\nregion where only one spin channel is open is determined by the applied magnetic field.\nThis works also if the magnetic field is not sufficiently strong to close th e gap. Therefore,\neven materials with gap values of about 0.5 eV are possible candidates to show the complete\nspin-filtering effect. The onset of the minority spin channel (at zer o energy in Fig. 3) is\ngiven by the relative position of the chemical potential with respect to the upper band edge\n70.51.01.52.02.5VD 0.000.020.040.060.080.10I\n01200.0250.05\n0.51.01.52.02.5VD0.250.500.751.P\nFIG. 3: Upper plot: total current as a function of the bias vol tageVD, in the spin filtering\nconfiguration. ∆ = 0 .6,B= 0.6, ΓL= ΓR= 0.1 andt= 1, for a chain of 500 sites.The inset shows\nthe separate contributions from the spin-up and spin-down c urrent. Lower plot: spin polarization\n(Eq. (6)) for the same parameters.\nof the valence band which may vary from one experimental situation to another.\nWe now consider the case of non-zero SO coupling. The transition fr om metallic to\ninsulating behavior driven by SO coupling is shown in Fig. 4. The magnetic field is the\nsame as in Fig. 3, i.e. it just closes the gap B=Bc= ∆, and the Rashba SO coupling\nisαR= 0.2t. It is created by an external gate voltage (see Fig. 1). The SO cou pling\nleads to an insulating behavior, as seen in the spectrum (Fig. 2) and in the current-voltage\ncharacteristics (Fig. 4). In contrast to Fig. 3, the presence of t he SO coupling αRleads to\na current on-set at VD≃0.25tcorresponding to half of the gap value for our choice of the\nchemical potential. The different current components Iss′are now all different, and the spin\npolarization (Eq. 6) is different from zero but not complete (0 <P <1).\nNote that the relative values of the different spin-components of t he current in Fig. 4 are\n80.51.01.52.02.5VD 0.000.020.040.060.080.10I\n0120.0.020.04\n0.51.01.52.02.5VD\n/Minus0.4/Minus0.20.2P\nFIG. 4: (Color online) Upper plot: total current as a functio n of the bias voltage VD, in the the\npresence of Rashba spin-orbit coupling, with αR= 0.2, ∆ = 0.6,B= 0.6, ΓL= ΓR= 0.1 andt= 1\nfor a chain of 500 sites. The inset shows the four spin compone nts of the current (in this order\nfrom top to bottom near VD= 2.5):I↓↓(red),I↑↑(black), I↓↑(orange), and I↑↓(blue). Lower\nplot: spin polarization (Eq. (6)) for the same parameters.\ndependent on the chain length. This is due to the Rashba SO coupling, which is known to\ninduce spin precession. Here, this spin precession is made more comp lex due to the presence\nof the magnetic field Band the ionicity ∆. The oscillations of the current components,\nas a function of the chain length L, are shown in Fig. 5, for Lvarying between 500 and\n600. These oscillations have a rather small contrast, show severa l periods and a complicated\ndependence on bias voltage VDin the general case (a dominating period seems to be present\nfor the off diagonal components of the current though). This has to be contrasted with the\npure metallic case ( B= 0 and ∆ = 0, shown in the inset of Fig. 5), where only one period\nLp=π/αis present independently on VD, and where the contrast is maximum.\n95005205405605806000.0050.0100.0150.0200.025\n5205400.0.020.04I/DownArrow/DownArrow\nI/UΠArrow/UΠArrow\nI/DownArrow/UΠArrow\nI/UΠArrow/DownArrow\nL\nFIG. 5: (Color online) Oscillations of the spin components o f the current as a function of the\nchain length (lengths between 500 and 600), for VD= 2.0, when SO Rashba coupling is present\n(αR= 0.2, ∆ = 0 .6,B= 0.6, ΓL= ΓR= 0.05 andt= 1). Inset: the same plot with B= 0 and\n∆ = 0, where I↑↑=I↓↓andI↑↓=I↓↑\nV. CONCLUSIONS\nIn studying the combined effect of magnetic field and SO interaction o n the transport in\n1D band insulators we found two interesting effects. First, already without SO coupling, the\npresence of a magnetic field leads to complete spin filtering. We studie d this effect here by\nconnecting the conduction channels to nonmagnetic leads but the e ffect of magnetic leads\nis easy to imagine, at least qualitatively. Then, spin filtering means high conductance for\nparallel magnetization in the leads and low conductance for antipara llel arrangement.\nWe speculate that the voltage region of the spin filtering effect may b e dramatically\nenhanced by the presence of magnetic impurities in the band insulato r, due to the giant\nZeemann effect. This might be important for the experimental verifi cation of our proposal.\nThe second striking effect of this study appears in band insulators w ith small band gap\nthat may be closed by a magnetic field. In that situation, the SO coup ling leads again to\nan insulating behavior. That is especially interesting for the Rashba s pin orbit coupling\nwhich is tuned by a gate voltage. Therefore, we may propose a devic e in which the metal-\n10insulator transition is controlled by the gate voltage via the Rashba S O term. This is\nin sharp contrast with 1D metallic systems, where the SO coupling doe s not lead to any\nmetal-insulator transition.\nWe also showed the oscillations of the different current components with the chain length.\nWhereas the simple oscillations in metallic systems are easy to underst and, the oscillations\nare much more complex for band insulators. We have let a detailed ana lysis of these oscil-\nlations for further studies. In our calculations the band insulator w as simulated by an ionic\nterm of alternating on-site energies in the Hamiltonian. But we think t hat our results are\ngeneric to any kind of band insulator. On the other hand, the way in w hich Coulomb corre-\nlations influence our results may be different from one microscopic Ha miltonian to another.\nWe expect that the Coulomb correlation just scales the band gap (e ither to larger or to\nsmaller values) and that the presented results should remain valid wit h effective parameters,\nhowever.\nThe authors thank Marc Bescond and Alvaro Ferraz for useful dis cussion.\n1A. Fert, Rev. Mod. Phys. 80, 1517 (2009).\n2S. Datta and B. Das, Appl. Phys. Lett. 56, 665 (1990).\n3G. Dresselhaus, Phys. Rev. 100, 580 (1955).\n4E.I.Rashba, Fiz. Tverd.Tela(Leningrad) 2, 1224 (1960) [Sov. Phys.SolidState 2, 1109(1960)].\n5A.V. Moroz, K.V. Samokhin, and C.H.W. Barnes, Phys. Rev. B 62, 16900 (2000); ibidPhys.\nRev. B62, 16900 (2000).\n6W. H¨ ausler, Phys. Rev. B 63, 121310(R) (2001).\n7P. St˘ reda and P. S˘ eba, Phys. Rev. Lett. 90, 256601 (2003).\n8A. Iucci, Phys. Rev. B 68, 075107 (2003).\n9V. Gritsev, G.I. Japaridze, M. Pletyukhov, and D. Baeriswyl , Phys. Rev. Lett. 94, 137207\n(2005).\n10P. Foldi, B. Molnar, M.G. Benedict, and F.M. Peeters, Phys. R ev. B71, 033309 (2005).\n11F. Cheng and G. Zhou, Journal of Physics: Condensed Matter 19, 136215 (2007).\n12M. Scheid, M. Kohda, Y. Kunihashi, K. Richter, and J. Nitta, P hys. Rev. Lett. 101, 266401\n(2008).\n1113S. Bellucci and P. Onorato Phys. Rev. B 78, 235312 (2008).\n14J.E. Birkholz and V. Meden, Jour. Phys.: Condensed Matter 20, 085226 (2008); ibid, Phys.\nRev. B79, 085420 (2009).\n15G.I. Japaridze, H. Johannesson and A. Ferraz, unpublished, arXiv:0904.1846 (2009).\n16Z. Ristivoyevic, G.I.Japaridze and T. Nattermann, unpubli shed (2009).\n17M.J. Rice and E.J. Mele, Phys. Rev. Lett. 49, 1455 (1982).\n18W.P. Su, J.R. Schrieffer, and A.J. Heeger, Phys. Rev. Lett. 42, 1698 (1979).\n19N. Hamada, S.I. Sawada, and A. Oshiyama, Phys. Rev. Lett. 68, 1579 (1992).\n20N.B. Brandt and E.A. Svistunova, Sov. Phys. Uspekhi 101249 (1970).\n12"    },    {        "title": "1107.2627v3.BCS_BEC_crossover_in_spin_orbit_coupled_two_dimensional_Fermi_gases.pdf",        "content": "arXiv:1107.2627v3  [cond-mat.quant-gas]  4 Jan 2012BCS-BEC crossover in spin-orbit coupled two-dimensional F ermi gases\nGang Chen,1,2,3Ming Gong,1and Chuanwei Zhang1,∗\n1Department of Physics and Astronomy, Washington State Univ ersity, Pullman, Washington, 99164 USA\n2Department of Physics, Shaoxing University, Shaoxing 3120 00, P. R. China\n3State Key Laboratory of Quantum Optics and Quantum Optics De vices,\nCollege of Physics and Electronic Engineering, Shanxi Univ ersity, Taiyuan 030006, P. R. China\nThe recent experimental realization of spin-orbit couplin g for ultra-cold atoms has generated\nmuch interest in the physics of spin-orbit coupled degenera te Fermi gases. Although recently the\nBCS-BEC crossover in three-dimensional (3D) spin-orbit co upled Fermi gases has been intensively\nstudied, the corresponding two-dimensional (2D) crossove r physics has remained unexplored. In\nthis paper, we investigate, both numerically and analytica lly, the BCS-BEC crossover physics in 2D\ndegenerate Fermi gases in the presence of a Rashba type of spi n-orbit coupling. We derive the mean\nfield gap and atom number equations suitable for the 2D spin-o rbit coupled Fermi gases and solve\nthem numerically and self-consistently, from which the dep endence of the ground state properties\n(chemical potential, superfluid pairing gap, ground state e nergy per atom) on the system parameters\n(e.g., binding energy, spin-orbit coupling strength) is ob tained. Furthermore, we derive analytic\nexpressions for these ground state quantities, which agree well with our numerical results within a\nbroad parameter region. Such analytic expressions also agr ee qualitatively with previous numerical\nresults for the 3D spin-orbit coupled Fermi gases, where ana lytic results are lacked. We show that\nwith an increasing SOC strength, the chemical potential is s hifted by a constant determined by the\nSOC strength. The superfluid pairing gap is enhanced signific antly in the BCS limit for strong SOC,\nbut only increases slightly in the BEC limit.\nPACS numbers: 03.75.Ss, 05.30. Fk, 74.20.Fg\nI. INTRODUCTION\nSpin-orbitcoupling(SOC),theinteractionbetweenthe\nspin and orbital degrees of freedom of a particle, has\nplayed important roles in condensed matter as well as\natomic and nuclear physics. For instance, it is known\nthat the coupling between electron spin and its linear\nmomentum in solids leads to many important condensed\nmatter phenomena such as spin and anomalous Hall ef-\nfects [1, 2], topological insulators and and superconduc-\ntors [3], spintronics [4], etc. In atomic physics, the cou-\npling between the electron spin and its motion around an\natomic nucleus is responsible for many of the details of\natomic structure [5].\nSuperfluidity and superconductivity are another im-\nportant phenomena in physics and have been widely\nstudied in many physical systems, including solids, He-\nlium liquids, as well as ultra-cold atomic gases [6]. Al-\nthough particles in superfluids and superconductors usu-\nally possess spins or pseudospins, the effects of SOC\non superfluidity and superconductivity have remained\nlargely unexplored. In this context, the recent experi-\nmental realization of SOC for ultra-cold atoms [7] pro-\nvidesacompletelynewplatformforexploringmany-body\nphenomena in spin-orbit coupled superfluids, including\nboth Bose-Einsteincondensates (BECs) [8, 9] and degen-\nerate Fermi gases [10, 11]. In the presence of SOC, vari-\nousnew andexoticsuperfluid phenomenamayexist since\n∗Electronic address: cwzhang@wsu.eduspins are not conserved during their motion. In particu-\nlar, in spin-orbit coupled BECs, the ground state phase\ndiagrams as well as the collective excitations have been\nstudied [8]. In spin-orbitcoupled degenerateFermi gases,\nthe crossoverphysicsfromthe Bardeen-Cooper-Schrieffer\n(BCS) superfluids of loosely bounded Cooper pairs to\nthe BEC of tightly bounded molecules has been inves-\ntigated extensively, in both uniform and trapped three-\ndimensional (3D) gases [10, 11]. However, the study of\nspin-orbit coupled two-dimensional (2D) Fermi gases is\nstill lacked.\nOnthe otherhand, the 2DdegenerateFermigas(with-\nout SOC) in itself is one of the most active topics in\nultra-cold atomic physics [12]. Experimentally, the 2D\ndegenerate Fermi cold atomic gases have been realized\nrecently using a highly anisotropic pancake-shaped po-\ntential [13], and the interaction energy in this system has\nbeen measured using the radio-frequency spectroscopy\n[14]. In this system, tunable interactions between atoms\nthrough Feshbach resonance [15] allow the exploration of\nthe crossover physics from the BCS superfluids to the\nBEC of molecules in 2D [16]. In addition to the study of\nmany-body physics such as quantum fluctuations, non-\nFermi liquid behavior, etc. [17], the 2D Fermionic cold\natomic gases are particularly interesting because of the\nexistence of exotic topological excitations such as Majo-\nrana fermions with non-Abelian exchange statistics [18].\nMotivated by these recent experimental breakthroughs\nin the realization of SOC and 2D Fermi gases, in this\npaper, we investigate, both numerically and analytically,\nthe BCS-BEC crossover physics in 2D degenerate Fermi\ngases in the presence of a Rashba type of SOC. Our main2\nresults are the following:\n1) We derive the mean field gap and atom number\nequations suitable for the 2D spin-orbit coupled Fermi\ngases.\n2) Thesetwoequationsaresolvednumericallyand self-\nconsistently, from which the dependence of the ground\nstate properties (chemical potential, superfluid pairing\ngap, groundstateenergyperatom)onthesystemparam-\neters (e.g., binding energy, SOC strength) is obtained.\n3) We derive analytic expressions for these ground\nstate physical quantities, which agree well with our nu-\nmerical results within a broad parameter region. Such\nanalytic expressions also agree qualitatively with previ-\nous numerical results for the 3Dspin-orbit coupled Fermi\ngases [10], where analytic results are lacked.\n4)We find that with anincreasingSOCstrength α, the\nchemical potential is shifted by a constant −mα2deter-\nmined by α. The superfluid pairing gap and the ground\nstate energy per atom are affected at the order of α4\nrather than α2, which means that weak SOC does not\naffect the pairing gap and the ground state energy per\natom. In the strong SOC regime, the pairing gap and\nthe ground state energy are enhanced significantly in the\nBCS limit, but only increase slightly in the BEC limit.\nAlthough these analytic results are obtained for the 2D\nFermi gases, they also provide qualitative understanding\nfor the numerical results in 3D Fermi gases [10], where\nsimilar changes of the chemical potential and superfluid\npairing order with respect to the SOC strength are ob-\nserved but analytic results are lacked.\nThe paper is organized as follows. Section II describes\nthe physical system: the spin-orbit coupled 2D degen-\nerate Fermi gases, and the corresponding Hamiltonian.\nIn section III, we derive the mean field gap and atom\nnumber equations. These equations are self-consistently\nsolved in section IV, both numerically and analytically\n(through perturbative methods), to obtain the ground\nstate properties (chemical potential, superfluid order pa-\nrameter, and ground state energy per atom) of the spin-\norbit coupled Fermi gases in the BCS-BEC crossover.\nSection V consists of discussion and conclusion.\nII. THE PHYSICAL SYSTEM AND THE\nHAMILTONIAN\nThe physical system in consideration is a 2D de-\ngenerate Fermi gas. Experimentally, the 2D degener-\nate Fermi gas has been realized using a 1D deep op-\ntical lattice along the third dimension, where the tun-\nneling between different layers is suppressed completely\n[13, 14]. The 1Doptical lattice potential V0sin2(2πz/λw)\ncan be generated using two counter-propagating laser\nbeams (parallel to the zaxis with a wavelength λw).\nIn this case, the two-body binding energy is given\nbyEb≃0.915/planckover2pi1ωLexp(√\n2πlL/as)/π, where ωL=/radicalbig\n8π2V0/(mλ2w) is the effective trapping frequency along\nthezaxis,lL=/radicalbig\n/planckover2pi1/(mωL), andasis the 3D s-wavescattering length [19]. Therefore the two-body binding\nenergyEbcan be tuned by varying the s-wave scattering\nlengthasvia the Feshbash resonance for the study of the\nBCS-BEC crossover physics [16]. For a small attractive\nas→0−,Eb→0, correspondingtotheBCSlimit. While\nfor a small repulsive as→0+,Eb→ ∞, corresponding\nto the BEC limit. When Ebincreases from 0 to ∞, the\nsystem evolves continuously from a BCS superfluid to a\nBEC of molecules.\nThe SOC for cold atoms can be generated by the in-\nteraction between atoms and laser beams, as shown in\nmany previous literatures [20], and demonstrated in a\nrecent benchmark experiment [7]. In this paper, we con-\nsider only a Rashba type of SOC, and the effects of other\ntypes of SOC (e.g. Dresselhaus or the combination of\nboth) can be investigated similarly. For simplicity we\nconsider a uniform Fermi gas and neglect the weak har-\nmonic trap in the 2D plane, whose effects can be incor-\nporated using the local density approximation.\nThe Hamiltonian for this uniform 2D spin-orbit cou-\npled degenerate Fermi gases can be written as\nH=HF+HI+Hsoc, (1)\nwhere\nHF=/summationdisplay\nk,σζkC†\nkσCkσ (2)\nis the single atom Hamiltonian, C†\nkσis the creation oper-\nator for a Fermi atom with the momentum k= (kx,ky),\nσ=↑,↓are the pseudospins of atoms. ζk=ǫk−µwith\nthe kinetic energy ǫk=k2/2m, the chemical potential µ,\nand the atom mass m. Henceforth, we take the Planck\nconstant /planckover2pi1= 1. The s-wave scattering interaction be-\ntween atom\nHI=g/summationdisplay\nkC†\n−k↑C†\nk↓Ck↓C−k↑, (3)\nwheregis the effective scattering interaction parameter.\nIn a 2D Fermi gas,\n1\ng=−/summationdisplay\nk1\n(2ǫk+Eb). (4)\nTheHamiltonianfortheRashbatypeofSOCforatoms\ncan be written as\nHsoc=α/summationdisplay\nk[(ky+ikx)C†\nk↑Ck↓+(ky−ikx)C†\nk↓Ck↑],(5)\nwhereαis the SOC strength. In solid state materials, α\nis generally much smaller than KF/(2m) withKFas the\nFermi vector. However, in ultra-cold neutral atoms, α\ncan reach the order of KF/(2m) [20]. Such strong SOC,\ntogether with tunable interactions through the Feshbach\nresonance, may yield some exotic many-body phenom-\nena that have not been explored in solid state systems.\nRecently, a generalized SOC with the Rashba and the\nDresselhaus terms in the Hamiltonian (1) has been con-\nsidered [11].3\nIII. MEAN FIELD GAP AND ATOM NUMBER\nEQUATIONS\nAs the first step for the eventual understanding of the\n2D spin-orbit coupled Fermi gas, we consider the zero\ntemperature superfluid physics under the mean field ap-\nproximation [16]. Generally, the mean field approxima-\ntion can give qualitatively but not quantitatively correct\nresults [21]. In the mean field approximation, the super-\nfluid order parameter is taken as\n∆ =g/summationdisplay\nk/angbracketleftCk↓C−k↑/angbracketright. (6)\nWith this pairing order parameter, we can rewrite the\ntwo-body interaction Hamiltonian (3) as\nHI=−∆2/g+∆/summationdisplay\nk(Ck↓C−k↑+C†\n−k↑C†\nk↓).(7)\nTherefore the total Hamiltonian can be rewritten as\nHB=1\n2/summationdisplay\nkΨ†(k)MkΨ(k)−∆2\ng+/summationdisplay\nkζk(8)\nunder the Nambu spinor basis Ψ( k) =\n(Ck↑,Ck↓,C†\n−k↓,−C†\n−k↑)T, where Bogoliubov-de-Gennes\noperator\nMk=\nζkαk+∆ 0\nαk−ζk0 ∆\n∆ 0 −ζk−αk+\n0 ∆ −αk−−ζk\n (9)\npreserves the particle-hole symmetry, k±=ky±ikx.\nThe quasiparticle excitation spectrum\nEλ\nk,±=λ/radicalbig\n(ǫk−µ±αk)2+∆2 (10)\nis the eigenvalue of the matrix Mk,λ=±correspond\nto the particle and hole branches of the spectrum. For\neach branch, there are two different excitations due to\nthe existence of the SOC. From Eq. (8), we see the total\nground-state energy is\nEG=−∆2\ng+/summationdisplay\nk[ζk−1\n2(E+\nk,++E+\nk,−)].(11)\nWithout the SOC, the term ( E+\nk,++E+\nk,−)/2 reduces to\nthe well-known form\nEk=/radicalbig\n(ǫk−µ)2+∆2. (12)\nin the BCS theory.\nThe ground-state properties of the 2D spin-orbit cou-\npled Fermi gases can be obtained from the atom number\nequation\nn=−∂EG\n∂µ=/summationdisplay\nk[1+1\n2(∂E+\nk+\n∂µ+∂E+\nk−\n∂µ)],(13)\nand the superfluid gap equation\n∂EG\n∂∆=/summationdisplay\nk[∆\n2ǫk+Eb−1\n4(∂E+\nk+\n∂∆+∂E+\nk−\n∂∆)] = 0.(14)IV. GROUND STATE PROPERTIES\nA. Numerical results\nWe numerically solve the above atom number equa-\ntion (13) and the gap equation (14) self-consistently to\nobtain various ground state quantities. In Fig. 1, we\nplot the dependence of the ground state quantities: the\nchemical potential µ, the superfluid order parameter ∆,\nand the ground state energy per atom E=EG/n, on\nthe physical parameters: the SOC strength αand the\nbinding energy Eb. We see that the chemical potential µ\ndecreaseswiththeincreasingspin-orbitcouplingstrength\nα. With increasing binding energy, the chemical poten-\ntial also decreases, signaling the crossover physics from\nthe BCS superfluids to the BEC molecules. One inter-\nesting feature shown in Fig. 1(a2) is that the shift of the\nchemical potential induced by the SOC depends only on\nthe SOC strength.\nIn Fig. 1(b1), we see the superfluid order parameter\n∆ increases with increasing α. Here ∆ 0is the superfluid\norder parameter without SOC. For a small α, the change\nof ∆/∆0is very small. The growth of ∆ becomes signifi-\ncant only when αKFis large than the Fermi energy EF.\n-1.2-0.8-0.400.40.81.2µ (EF)\n-4.0-3.0-2.0-1.00.01.0\n0.951.001.051.101.151.20∆/∆0\n1.001.011.02\n0.0 0.5 1.0 1.5 2.0\nαKF (EF)-0.6-0.4-0.20.00.2E (EF)\n0 2 4 6 8 10\nEb (EF)-0.50-0.49-0.48-0.47-0.46-0.45(a1) (a2)\n(b1) (b2)\n(c1) (c2)E\nb = 0.1\nαK\nF = 0.5E\nb = 1.0αK\nF = 1.0\nFigure 1: (Color online) (a1-c1) Plot of the chemical poten-\ntialµ, the dimensionless superfluid pairing gap ∆ /∆0, and\nthe ground-state energy per atom Ewith respect to the SOC\nstrength αfor the two-body binding energy Eb= 0.1EF\n(Black line) and 1 .0EF(Red line). (a2-c2) Plot of µ, ∆/∆0,\nandEwith respect to EbforαKF= 0.5EF(Black line) and\n1.0EF(Red line). In all figures, the solid lines represent the\napproximate analytical results obtained in the paper, whil e\nthe open symbols correspond to the exact numerical results.4\nFurthermore, the effects of the SOC are clearly seen for\na smallEb(the BCS side), but not important for a large\nEb(the BEC side).\nThe ground state energy per atom Ehas a similar be-\nhavior as ∆ /∆0. It increases with α, but the growth is\nonly important for a large α. Without SOC E=−1\n2EF,\nand the SOC induces a small correction. The change of\nEis significant in the BCS side (small Eb), but negligible\nin the BEC side (large Eb). We note that similar changes\nofthe chemical potential and the superfluid orderparam-\neter have been observed numerically in a 3D spin-orbit\ncoupled Fermi gases. All these numerical observed phe-\nnomena will be explained in the next subsection where\nthe analytic expressions for these ground state quantities\nare derived.\nB. Analytic results\nAlthough the above numerical results give the depen-\ndence of the ground state quantities on the SOC strength\nfor certain parameters, analytic results are desired for a\nbetter understanding of the underlying physics. In the\nfollowing, we present a perturbative method (with αas\nthe small parameter) to analytically extract the funda-\nmental ground-state properties of the spin-orbit coupled\nFermi gases. For this purpose, we rearrange Eq. (11)\ninto two parts\nEG=E0+Esoc (15)\nwith\nE0=−∆2\ng+/summationdisplay\nk(ζk−Ek) (16)\nas the ground-state energy without SOC. For a 2D Fermi\ngas,E0can be obtained exactly, yielding [16]\nE0=m\n4π[∆2ln(/radicalbig\nµ2+∆2−µ\nEb)−µ(/radicalbig\nµ2+∆2+µ)−∆2\n2].\n(17)\nThe second term in Eq. (15) is given by\nEsoc=/summationdisplay\nk[Ek−1\n2(E+\nk,++E+\nk,−)],(18)\nwhich describes the contribution from the SOC.\nBecause it is difficult to derive a simple analytic ex-\npression for Esocdirectly, we first perform a Taylor ex-\npansion with respect to the SOC strength, and then do\nthe summation over k. Formally, Esoccan be written as\nEsoc=m\n4πCi(µ,∆)ηi(19)\nwithη=mα2/2. The coefficients Ci(µ,∆) can be ob-\ntained, in principle, for any order. Here, we only give thefirst six orders\nC1=−4(/radicalbig\nµ2+∆2+µ), (20)\nC2=−8\n3(1+µ/radicalbig\nµ2+∆2), (21)\nC3=−16∆2\n15(µ2+∆2)3/2, (22)\nC4=32µ∆2\n35(µ2+∆2)5/2, (23)\nC5=64∆2(∆2−4µ2)\n315(µ2+∆2)7/2, (24)\nC6=−128µ∆2(3∆2−4µ2)\n693(µ2+∆2)9/2. (25)\nAlthough the expression for Esocis very complicate,\nthe expressions for the chemical potential µand the su-\nperfluid pairing order ∆ are very simple, as we will show\nlater in the paper. With the ground-state energy EG,\nthe superfluid pair order and the chemical potential can\nbe derived by self-consistently solving the corresponding\ngap and number equations\nln(/radicalbig\nµ2+∆2−µ\nEb) =G∆(η,∆,µ), (26)\n/radicalbig\nµ2+∆2+µ= 2EF−Gµ(η,∆,µ) (27)\nanalytically, where G∆(η,∆,µ) =/summationtext\ni∂Ci(µ,∆)\n∂∆ηi, and\nGµ(η,∆,µ) =/summationtext\ni∂Ci(µ,∆)\n∂µηi.EFis the Fermi energy\nfor a 2D non-interacting Fermi gas without SOC and\nwith the density n=mEF/π. Without SOC ( η= 0),\nG∆(η,∆,µ) =Gµ(η,∆,µ) = 0, and Eqs. (26) and (27)\nbecome\n/radicalbig\nµ2+∆2−µ=Eb,/radicalbig\nµ2+∆2+µ= 2EF.\nInthis case,the superfluidpairingorderandthe chemical\npotential are given exactly by ∆ 0=√2EbEFandµ0=\nEF−Eb/2 [16], as expected.\nIn the presence of SOC ( η/negationslash= 0), the nonlinear equa-\ntions (26) and (27) cannot be solved exactly. However,\napproximate solutions can be derived for the physical\nparameters within current experimentally achievable re-\ngion. Since the ground-state energy depends on ∆2, the\nsolutions of Eqs. (26) and (27) can be assumed to be\nµ=/summationtext\ni=0µiηiand ∆2=/summationtext\ni=0Λiηi. Substituting these\nexpressions into the nonlinear equations (26) and (27)\nand then comparing the coefficients for the same order of\nηi, we obtain Λ iandµirespectively.\n1. Chemical potential\nAfter a straightforwardbut tedious calculation, the co-\nefficients for the chemical potential are given by µ1=5\n−2,µ2= 4Eb/[3(Eb+ 2EF)2] andµ3=−64Eb(Eb−\n4EF)/[15(Eb+ 2EF)4]. Note that the second order µ2\nis already very small for all different values of Ebwhen\nη < EF, therefore the high-order terms do not affect the\nchemical potential significantly. The chemical potential\ncan then be written as\nµ≃EF−Eb\n2−2η. (28)\nFrom Fig. 1a, we see Eq. (28) agrees well with the exact\nvalues of the chemical potential obtained from numeri-\ncally solving Eqs. (13) and (14) self-consistently.\nEquation (28) shows clearly that with the increasing\nSOC strength α, the chemical potential µis decreased\nby 2η=mα2, as shown in Fig. 2. If we define an effec-\ntive chemical potential µ=µ+2η=µ+mα2, Eq. (28)\ncan be rewritten as µ=EF−Eb/2. Similar to the previ-\nous discussion without SOC, we find that in the asymp-\ntotic BCS limit with a weak bound state ( Eb≪EF),\nthe chemical potential µ≃EF. However, in the deep\nBEC regime with a strong bound state ( Eb≫EF), the\nchemical potential µ=−Eb/2. The BCS-BEC crossover\nmay occur around the crossover point µ= 0 [16]. Fur-\nthermore, in the presence of SOC, Eq. (28) can also\nbe written as µ=µf+µ∆, where µf=EF−2η\nis the chemical potential for the free Fermi gas and\nµ∆=−Eb/2 +/summationtext\ni=1µ∆i(Eb)ηireflects the revision of\nthe chemical potential induced by the two-body binding\nenergyEb. Hereµ∆1= 0 implies that the binding energy\nEbhas no effect on the chemical potential µat the order\nofη=mα2/2.\nIt is important to point out that although the chemi-\ncal potential µonly has a simple shift from that without\nSOC, the underlying physics is quite different. Without\nSOC, the pairing wave function is simply singlet. How-\never, in the presence of SOC, each energy band contains\nboth spin up and down components. As a result, the\npairing wave function has a more complicated structure\nwithbothsingletandtripletcomponents[22]. Thetriplet\npairing correlations in s-wave superfluids may be used\ntodetecttheanisotropicFulde-Ferrell-Larkin-Ovchinikov\nstates[23]. Inaddition, the2DFermigaseswithSOCcan\n/s48/s107/s40/s98/s41\n/s107/s69/s40/s107/s41\n/s48/s69/s40/s107/s41\n/s40/s97/s41\n/s61/s109\nFigure 2: (Color online) (a) The chemical potential without\nSOC in the BCS limit. (b) The chemical potential with SOC\nin the BCS limit. For weak SOC, their difference ∆ µ=mα2.exhibitp-wave character in the helicity bases. If a Zee-\nman field in Hamiltonian (8) is added, a novel topolog-\nical phase transition from non-topological superfluids to\ntopologicalsuperfluidscanbe induced. Inthe topological\nsuperfluid phase, there exist Majorana fermions and the\nassociated non-Abelian statistics, which are the critical\ningredients for implementing topological quantum com-\nputing [18].\n2. Superfluid order gaps\nThe effects of SOC are more interesting for the super-\nfluid pairing gap ∆. Applying the same procedure as\nthat for the chemical potential, we find\n∆2≃2EbEF+16EbEF\n3(Eb+2EF)2η2.(29)\nThereisno first-ordercorrectionwith respect to η(∼α2)\nfor ∆2, and the second-order η2(∼α4) is the leading cor-\nrection. Moreover,thesecond-ordercoefficient ∂2∆2/∂η2\nis alwayspositive and hasa maximum 4 η2/3(=m2α4/3)\nwhenEb= 2EF. The high-order coefficients for η3and\nη4are Λ3=−512EbEF(Eb−EF)/[15(Eb+2EF)4] and\nΛ4= 64EbEF(1017E2\nb−3508EbEF+1076E2\nF)/[315(Eb+\n2EF)6] respectively.\nIn order to see the effect of SOC on the superfluid\npairing gap more clearly, we introduce a dimensionless\nquantity\n∆d=∆\n∆0≃/radicaligg\n1+8\n3(Eb+2EF)2η2,(30)\nwhere ∆ 0is the superfluid pairing gap without SOC.\nIn the asymptotic BCS limit with a weak bound state\n(Eb≪EF), ∆d≃/radicalbig\n1+2η2/E2\nF. For a weak SOC\n(η≪EF) in typical solid-state materials, ∆ d≃1,\nwhich means that the SOC does not affect the super-\nfluid pairing gap significantly. However, for a strong\nSOCη∼EFthat has been achieved for ultracold atoms\n[7], this gap can be enhanced greatly. In the deep BEC\nregime with a strong binding energy ( Eb≫EFand\nEb≫η), ∆d≃/radicalbig\n1+8η2/(3E2\nb), therefore the super-\nfluid pairing gap increases only slightly with the increas-\ning SOC strength. These analytic results agree well with\nthe numericalresults shownin Fig. 1b. Note that similar\nbehavior for ∆ dis also observed in the numerical results\nin 3D [10].\n3. Ground-state energy per atom\nIntermsofEqs. (28)and(29), theground-stateenergy\nper Fermi atom E=EG/ncan be obtained\nE≃ −1\n2EF+8EF\n3(Eb+2EF)2η2. (31)6\nThe comparison of Eq. (31) with the direct numerical\nsimulation results is shown in Fig. 1c. The ground-\nstate energy, like the superfluid pairing gap, depends on\nη2(∼α4). In the asymptotic BCS limit ( Eb≪EF),\nE≃ −EF/2+2η2/(3EF), which means that the ground-\nstate energy can be enhanced significantly only for a\nlargeη. In the deep BEC regime ( Eb≫EFand\nEb≫η),E≃ −EF/2 + 8EFη2/(3E2\nb), therefore the\nground-state energy only increases slightly even when\nη∼EF. In addition, the high-order coefficients for η3\nandη4are given by E3=−128EbEF/[5(Eb+ 2EF)4]\nandE4= 128EbEF(491Eb−538EF)/[315(Eb+ 2EF)6]\nrespectively. Note that although such mean field the-\nory gives qualitatively correct results, it may not agree\nquantitatively with the experimental results in the deep\nBEC regime as shown in recent Monte Carlo numerical\nsimulation for the 2D Fermi gases without SOC [21].\nFinally, we want to remarkthat if the high-orderterms\nin the chemical potential µ, the superfluid pairing gap ∆,\nand the ground-state energy per Fermi atom E=EG/n\nare included, the analytical results in Eqs. (28), (30)\nand (31) fit better with the numerical results even for\nEF< αKF<3.0EF.\nV. DISCUSSION AND CONCLUSION\nThe mean field zero temperature BCS-BEC crossover\nphysics discussed above provides the first critical step\ntowards the understanding of the 2D spin-orbit coupled\ndegenerate Fermi gases. Clearly, many issues need be\nfurther explored in the future, as demonstrated by the\ndevelopment along this direction after the initial submis-\nsionofourpaper. Inparticular,thefinitetemperatureef-\nfect need be taken into account in a realistic experiment.\nWithout SOC, it is known that there are no superflu-\nids for 2D degenerate Fermi gas at a finite temperature[24], where relevant physics is the Berezinskii-Kosterlitz-\nThouless(BKT)transition[25], leadingto the generation\nof the vortex-antivortex pairs. Recent some interesting\neffects of the SOC on the BKT transition [26] has been\ninvestigated. In experiments, the Fermi gases are con-\nfined in a 2D harmonic trap, whose effects may be taken\ninto account using the local density approximation, as\nshown in the recent preprint [26]. The trap geometry\nfor the Fermi gases may also be quais-2D instead of the\nstrict 2D. In this case, the confinement along the third\ndirection may affect the critical transition temperature,\nwhich need be further explored.\nIn summary, motivated by the recent experimental\nbreakthrough for the realization of the SOC for cold\natoms and the 2D degenerate Fermi gas, we investigate\nthe zero temperature BCS-BEC crossover physics in 2D\nspin-orbitcoupleddegenerateFermigasesusingthemean\nfield approximation. By solving the corresponding gap\nand atom number equations both numerically and an-\nalytically, we reveal the ground state properties of the\nspin-orbit coupled 2D Fermi gases. Our analytic results\nagreequantitativelywith ournumericalresults in 2Dand\nqualitatively with previous numerical results for the 3D\nspin-orbit coupled Fermi gases, where analytic results\nare lacked. The analytic expressions for various physi-\ncal quantities may provide a powerful tool for engineer-\ning and probing many new topological phenomena in 2D\nFermi gases, including the intriguing Majorana physics\nand the associated non-Abelian statistics.\nAcknowledgements\nWe thank Yongping Zhang, Li Mao, and Wei Yi\nfor helpful discussion. This work is supported by\nDARPA-YFA (N66001-10-1-4025),ARO (W911NF-09-1-\n0248), NSF (PHY-1104546), and AFOSR (FA9550-11-1-\n0313). GC is also supported by the 973 program under\nGrant No. 2012CB921603, the NNSFC under Grant No.\n11074154, and the ZJNSF under Grant No. Y6090001.\n[1] N. Nagaosa, J. Sinova, S. Onoda, A. H. MacDonald, and\nN. P. Ong, Rev. Mod. Phys. 82, 1539 (2010).\n[2] D. Xiao, M. -C. Chang, and Q. Niu, Rev. Mod. Phys. 82,\n1959 (2010).\n[3] X. -L. Qi, and S. -C. Zhang, Rev. Mod. Phys. 83, 1057\n(2011)\n[4] I.˘Zuti´ c, J. Fabian, and S. Das Sarma, Rev. Mod. Phys.\n76, 323 (2004).\n[5] V. B. Berestetskii, E. M. Lifshitz, and L. P.\nPitaevskii, Quantum electrodynamics , (Butterworth-\nHeinemann, 1982).\n[6] A. J. Leggett, Quantum Liquids: Bose condensation\nand Cooper pairing in condensed-matter systems (Oxford\nUniversity Press Inc., New York, 2006).\n[7] Y.-J.Lin, K.Jim´ enez-Garc´ ıa, andI.B.Spielman, Natu re\n(London) 471, 83 (2011); Y. -J. Lin, R. L. Compton1, K.\nJim´ enz-Garc´ ıa, J. V. Porto, and I. B. Spielman, Nature\n(London) 462, 628 (2009).\n[8] C. Wang, C. Gao, C. -M. Jian, and H. Zhai, Phys. Rev.Lett.105, 160403 (2010); T.-L. Ho and S. Zhang, Phys.\nRev. Lett. 107, 150403 (2011); Y. Zhang, L. Mao, and\nC. Zhang, arXiv:1102.4045; H. Hu, H. Pu, and X.-J.Liu,\narXiv: 1108.4233.\n[9] S. K. Yip, Phys. Rev. A 83, 043616, (2011); Z. F. Xu,\nR. L¨ u, and L. You, Phys. Rev. A 83, 053602 (2011); T.\nKawakami, T. Mizushima, and K. Machida, Phys. Rev.\nA84, 011607(R) (2011); C. Jian and H. Zhai, Phys. Rev.\nB84, 060508 (2011); C. Wu, I. Mondragon-Shem, and\nX.-F. Zhou, Chin. Phys. Lett. 28, 097102 (2011).\n[10] J. P. Vyasanakere, S. Zhang, and V. B. Shenoy, Phys.\nRev. B84, 014512 (2011); M. Gong, S. Tewari, and C.\nZhang, Phys. Rev. Lett. 107, 195303 (2011); Z. -Q. Yu,\nand H. Zhai, Phys. Rev.Lett. 107, 195305 (2011); H. Hu,\nL. Jiang, X. -J. Liu, and H. Pu, Phys. Rev. Lett. 107,\n195304 (2011); M. Iskin, and A. L. Subasi, Phys. Rev.\nLett.107, 050402 (2011); Phys. Rev. A 84, 041610 (R)\n(2011); Phys. Rev. A 84, 043621 (2011); W. Yi, and G.\n-C. Guo, Phys. Rev. A 84, 031608 (2011); L. Han, and7\nC. A. R. S´ a de Melo, arXiv:1106.3613.\n[11] L. Dell’Anna, G. Mazzarella, and L. Salasnich, Phys.\nRev. A84, 033633 (2011).\n[12] D. S. Petrov, M. Holzmann, and G. V. Shlyapnikov,\nPhys. Rev. Lett. 84, 2551 (2000); S. S. Botelho, and C.\nA. R. S´ a de Melo, Phys. Rev. Lett. 96, 040404 (2006);\nW. Zhang, G. -D. Lin, and L. -M. Duan, Phys. Rev. A\n77, 063613 (2008); Phys. Rev. A 78, 043617 (2008); I.\nBloch, J. Dalibard, and W. Zwerger, Rev. Mod. Phys.\n80, 885 (2008).\n[13] K. Martiyanov, V. Makhalov, and A. Turlapov, Phys.\nRev. Lett. 105, 030404 (2010).\n[14] B. Fr¨ ohlich, M. Feld, E. Vogt, M. Koschorreck, W. Zw-\nerger, and M. K¨ ohl, Phys. Rev. Lett. 106, 105301 (2011).\n[15] C. Chin, R. Grimm, P. Julienne, and E. Tiesinga, Rev.\nMod. Phys. 82, 1225 (2010).\n[16] M. Randeria, J. -M. Duan, and L. -Y. Shieh, Phys. Rev.\nLett.62, 981 (1989); Phys. Rev. B 41, 327 (1990).\n[17] S. Sachdev, Science 288, 475 (2000); S. E. Korshunov,\nPhys. Usp. 49, 225 (2006).\n[18] S. Tewari, S. Das Sarma, C. Nayak, C. Zhang, and P.\nZoller, Phys. Rev. Lett. 98, 010506 (2007); C. Zhang, S.\nTewari, R. M. Lutchyn, and S. Das Sarma, Phys. Rev.\nLett.101, 160401 (2008); C. Nayak, S. H. Simon, A.\nStern, M, Freedman, and S. D. Sarma, Rev. Mod. Phys.80, 1083 (2008).\n[19] J. Tempere, M. Wouters, and J. T. Devreese, Phys. Rev.\nB75, 184526 (2007).\n[20] J. Ruseckas, G. Juzeli¯ unas, P. ¨Ohberg, and M. Fleis-\nchhauer, Phys. Rev. Lett. 95, 010404 (2005); T. D.\nStanescu, C. Zhang, andV. Galitski, Phys.Rev.Lett. 99,\n110403 (2007); C. Zhang, Phys. Rev. A 82, 021607(R)\n(2010); G. Juzeli¯ unas, J. Ruseckas, and J. Dalibard,\nPhys. Rev. A 81, 053403 (2010); J. D. Sau, R. Sensarma,\nS. Powell, I. B. Spielman, and S. Das Sarma, Phys. Rev.\nB83, 140510 (2011); D. L. Campbell, G. Juzeli¯ unas, and\nI. B. Spielman, Phys. Rev. A 84, 025602 (2011). J. Dal-\nibard, F. Gerbier, G. Juzeliunas, and P. Ohberg, Rev.\nMod. Phys. (accepted for publications)\n[21] G. Bertaina, and S. Giorgini, Phys. Rev. Lett. 106,\n110403 (2011).\n[22] L. P. Gor’kov, and E. I. Rashba, Phys. Rev. Lett. 87,\n037004 (2001).\n[23] I. Zapata, F. Sols, and E. Demler, arXiv:1106.1605.\n[24] P. C. Hohenberg, Phys. Rev. 158, 383 (1967).\n[25] V. L. Berezinskii, Sov. Phys. JETP 32, 493 (1971); J. M.\nKosterlitz and D. Thouless, J. Phys. C 5, L124 (1972).\n[26] L. He, and X. -G. Huang, arXiv:1109.5577."    },    {        "title": "0709.3429v2.Weak_and_strong_coupling_limits_of_the_two_dimensional_Fröhlich_polaron_with_spin_orbit_Rashba_interaction.pdf",        "content": "arXiv:0709.3429v2  [cond-mat.str-el]  25 Jan 2008Weak and strong coupling limits of the two-dimensional Fr¨ o hlich polaron with\nspin-orbit Rashba interaction\nC. Grimaldi\nMax-Planck-Institut f¨ ur Physik komplexer Systeme, N¨ oth nitzer Srt.38, D-01187 Dresden Germany\nLPM, Ecole Polytechnique F´ ed´ erale de Lausanne, Station 1 7, CH-1015 Lausanne, Switzerland\nThe continuous progress in fabricating low-dimensional sy stems with large spin-orbit couplings\nhas reached a point in which nowadays materials may display s pin-orbit splitting energies ranging\nfrom a few to hundreds of meV. This situation calls for a bette r understanding of the interplay\nbetween the spin-orbit coupling and other interactions ubi quitously present in solids, in particular\nwhen the spin-orbit splitting is comparable in magnitude wi th characteristic energy scales such as\nthe Fermi energy and the phonon frequency.\nIn this article, the two-dimensional Fr¨ ohlich electron-p honon problem is reformulated by introduc-\ning the coupling to a spin-orbit Rashba potential, allowing for a description of the spin-orbit effects\non the electron-phonon interaction. The ground state of the resulting Fr¨ ohlich-Rashba polaron is\nstudied in the weak and strong coupling limits of the electro n-phonon interaction for arbitrary val-\nues of the spin-orbit splitting. The weak coupling case is st udied within the Rayleigh-Schr¨ odinger\nperturbation theory, while the strong-coupling electron- phonon regime is investigated by means of\nvariational polaron wave functions in the adiabatic limit. It is found that, for both weak and strong\ncoupling polarons, the ground state energy is systematical ly lowered by the spin-orbit interaction,\nindicating that the polaronic character is strengthened by the Rashba coupling. It is also shown\nthat, consistently with the lowering of the ground state, th e polaron effective mass is enhanced\ncompared to the zero spin-orbit limit. Finally, it is argued that the crossover between weakly and\nstrongly coupled polarons can be shifted by the spin-orbit i nteraction.\nPACS numbers: 71.38.-k, 71.38.Fp, 71.70.Ej\nI. INTRODUCTION\nThe Fr¨ ohlich Hamiltonian describing a single electron\ncoupled to longitudinal optical phonons is a paradig-\nmatic model of the electron-phonon (el-ph) interaction,1\nand has represented in the past, in addition to its in-\nterest for the solid-state physics, an ideal problem for\ntesting many mathematical methods in quantum field\ntheory.2Because of the coupling with the phonon field,\nthe resulting quasi-particle, the polaron, has an effec-\ntive mass larger, and a ground state energy lower, than\nthe free electron. These quantities have been investi-\ngated for the three-dimensional (3D) case by means of\nperturbation theory for the weak-coupling limit,3and of\nvariational treatments for the intermediate,4and strong-\ncoupling cases.5,6The path-integral variational calcula-\ntions of Feynman,7and subsequent refinements of this\nmethod,8have provided a solid description for all val-\nues of the coupling, verified also by improved variational\nmethods,9and by quantum Monte-Carlo studies.10,11\nThe interest aroused some time ago on semiconduc-\ntor heterojunctions, or other low-dimensional systems,\nprompted to modify the Fr¨ ohlich model to accounting for\ntwo-dimensional(2D) and quasi-2Dsystems.12By apply-\ningthe samemethodsderivedforthe3Dcase, theground\nstatepropertiesforthestrictly2Dcasewereevaluatedfor\nweak, strong and intermediate couplings,13,14,15,16and\nthe obtained systematic lowering of the ground state en-\nergy and the enhancing of the effective mass compared\nto the 3D case has pointed out the role of dimensionality\nin enhancing the polaronic character.12,17Concerning the el-ph problem in low dimensions,\nrecent progresses in developing high-quality low-\ndimensional systems and in material engineering provide\nhints that, for a vast class of low-dimensional materi-\nals, the usual 2D Fr¨ ohlich model, as considered in lit-\nerature, may be incomplete. This concern comes about\nby considering 2D systems exhibiting strong spin-orbit\n(SO) splitting of the electronic states due to the inver-\nsion asymmetry in the direction orthogonal to the con-\nducting plane (Rashba SO mechanism). This situation is\nencountered in semiconductor quantum wells with asym-\nmetricconfiningpotentials,18inthesurfacestatesofmet-\nals and semi-metals,19,20,21and in surface alloys such as\nLi/W(110),22Pb/Ag(111),23,24and Bi/Ag(111),25with\nSO splitting energies ranging from a few meV in GaAs\nquantum wells to about 0.2 eV in Bi/Ag(111).25In these\nsystems, therefore, the SO energy may be of the same\norder or even much larger than the typical phonon fre-\nquency, rising the question of how such state of affair\naffects the el-ph interaction, in general, and the Fr¨ ohlich\ncoupling, in particular.\nAs pointed out in several works,26,27,28,29,30,31the\nRashba interaction describing the SO coupling can have\nprofound effects on the low energy properties of the itin-\nerant electrons. Namely, in the low density regime, the\nRashba SO coupling induces a topological change of the\nFermi surface of the free electrons, leading to an effective\nreduction of the dimensionality in the electronic density\nof states (DOS). In this situation, a 2D low density elec-\ntron gas would develop, in the presence of SO Rashba\ncoupling, a phenomenology similar to one-dimensional2\n(1D) systems, triggered by the square-root divergence of\nthe (effectively 1D) DOS at low energies.\nSome interesting consequences of this scenario on\nthe el-ph problem have already been discussed in\nRef.[30], concerning the superconducting transition, and\nin Ref.[31] for the effective mass and the spectral proper-\nties. The picture arising from these works, although be-\ning limited to the momentum-independent Holstein el-ph\ninteraction and to weak-to-moderate couplings, confirms\nthat, for sufficiently low electron densities, the coupling\ntothephononsisamplifiedbytheSOinteractionthrough\nthe 1D-like divergence of the DOS.\nNotwithstanding the relevance of these results for the\nHolstein model, the use of a local el-ph interaction may\nhowever result inadequate in the extremely low elec-\ntron density regime, where the SO effects are more\nevident.30,31Indeed, the lack of effective screening in this\ncase would rather suggest a long-range interaction as be-\ning a more appropriate description of the el-ph coupling.\nIt is therefore natural to consider the 2D Fr¨ ohlich po-\nlaron, and its coupling to the SO interaction, as a model\nbetter describing the unscreened el-ph interaction in 2D\nRashba systems in the low density limit.\nInthisarticle,asingleelectronmovingwithaparabolic\ndispersion in the two-dimensional x-yplane is coupled\nsimultaneously to the Rashba SO potential and to the\nphonon degrees of freedom through a Fr¨ ohlich interac-\ntion term. The total system is then described by the 2D\nFr¨ ohlich-Rashba Hamiltonian H=Hel+Hph+Hel−ph,\nwhere (/planckover2pi1= 1)\nHel=p2\n2m+Ω(p)·σ (1)\nis the Hamiltonian for an electron with mass mand mo-\nmentum operator p=−i∇with components ( px,py,0),\nσis the spin-vector operator with components given by\nthe Pauli matrices, and Ω(p) is the SO vector field which\nin the case of Rashba coupling reduces to:\nΩ(p) =γ\n−py\npx\n0\n, (2)\nwhereγis the SO coupling parameter. The phonon part\nof the Hamiltonian is given by\nHph=ω0/summationdisplay\nqa†\nqaq, (3)\nwherea†\nq(aq) is the creation (annihilation) operator for\na phonon with momentum q= (qx,qy) and optical fre-\nquencyω0. The el-ph interaction Hamiltonian for the\n2Delectroncoupledtolongitudinaloptical(LO)phonons\nis:12,14\nHel−ph=1√\nA/summationdisplay\nq1√q(M0eiq·raq+M∗\n0e−iq·ra†\nq) (4)with\nM0=iω0/parenleftbigg2π2α2\nmω0/parenrightbigg1/4\n, (5)\nwhereα=e2(ǫ−1\n∞−ǫ−1\n0)/radicalbig\nm/2ω0is the dimensionless el-\nph coupling constant, with ebeing the electron charge,\nandǫ∞andǫ0the high frequency and static dielectric\nconstants, respectively.\nIt is worth clarifying here the significance of the 2D\nFr¨ ohlich interaction of Eq.(4) with respect to the char-\nacteristics of specific materials. For quantum wells and\n2D heterostructures, where the electron wave function is\nassumed here to be confined in a sheet of zero thickness,\nEq.(4) describes the coupling of the electron to bulk LO\nphonons, while the coupling to interface phonon modes is\nneglected. The inclusion of such interface phonon contri-\nbutions may be important in describing specific materi-\nals, but it is unnecessary for the present study, where the\nfocus is on the SO effects on the unscreened (long-range)\nel-ph interaction, for which Eq.(4) is a paradigm for the\n2D case. Concerning the el-ph coupling of electronic sur-\nface states, Eq.(4) coincides (apart from a redefinition\nofM0) with the coupling to 2D surface phonons when\nthe coupling to bulk phonons extending below the sur-\nface is negligible.32Such approximation is coherent with\nthe ideal 2D assumption for the electron wave function,\nwhich is physically realized when the electronic surface\nstateshavenegligiblecouplingtothe bulk. Afurther mo-\ntivation of using the 2D Fr¨ ohlich model (4) is that, in the\nabsence of SO interaction, the ground state polaron en-\nergyEPand effective mass m∗havealready been studied\nby several authors,12,13,14,15,16and the exact results ob-\ntained for the weak ( α≪1) and strong ( α≫1) coupling\nlimits provide a useful reference for the effect of nonzero\nSO coupling.\nIn the present work, the 2D Fr¨ ohlich-Rashba Hamil-\ntonian is studied by considering the weak and strong\ncoupling limits of the el-ph interaction, with arbitrary\nstrength of the SO coupling γ. Forα≪1 the polaron\nenergyEPand the effective mass m∗are obtained from\nsecond order perturbation theory in Sec.II, where nu-\nmerical and exact analytical results are presented. It is\nshown that the effect of γ∝negationslash= 0 is qualitatively similar to\nthat observed in the Holstein model,30,31namely, the SO\ncoupling enhances the effective coupling to the phonons.\nInparticular, EPisloweredby γand, simultaneously, the\neffective mass m∗is enhanced. In Sec.III the strong cou-\npling limit α≫1 is treated by the variational method,\nproviding a rigorous upper bound of the ground state\nenergy for arbitrary values of the SO interaction. As for\nthe weak el-ph coupling case, it is found that EP(m∗) is\nlowered (enhanced) by the SO interaction, implying that\nthe Rashba coupling always amplifies the polaronic char-\nacter, regardless of whether the el-ph interaction is weak\nor strong.3\nII. WEAK COUPLING\nIn the presence of SO interaction, the electron wave\nfunction is a spinor and its Green’s function is conve-\nniently represented by a 2 ×2 matrix in the spin sub-\nspace. For α= 0 the free electron Green’s function G0\nis readily obtained from Hel:\nG0(k,ω) =/parenleftbigg\nω−k2\n2m−Ω(k)·σ/parenrightbigg−1\n=1\n2/summationdisplay\ns=±[1+sˆΩ(k)·σ]Gs\n0(k,ω),(6)\nwherekis a 2D electron wavenumber, ˆΩ(k) =\nΩ(k)/|Ω(k)|and\nGs\n0(k,ω) =1\nω−k2/2m−sγk(7)\nis the free electronpropagatorfor the two( s=±1)chiral\nstates characterized by two distinct bands with shifted\nparabolic dispersions k2/2m±γk. The lowest band has\nits minimum value −E0atk=k0, wherek0andE0are\nthe Rashba momentum and energy defined respectively\nby:\nk0=mγ, E 0=m\n2γ2. (8)\nFor later convenience, it is useful to express the electron\nenergy relative to E0, so that the poles of Eq.(7) appear\nat energies:\nE±(k) =1\n2m(k±k0)2. (9)\nThe free electron ground state is then given by the elec-\ntron occupying the lower band at wavenumber k=k0\nwith energy ω= 0.\nIn the weak el-ph coupling limit ( α≪1) the ground\nstate properties are obtained by the electron self-energy\nevaluated in the second order perturbation theory. At\nzerotemperature, the resultingsingleelectronself-energy\nis therefore:\nΣ(k,ω) =|M0|2/integraldisplaydk′\n(2π)21\n|k−k′|G0(k′,ω−ω0).(10)\nBecause of the momentum dependence of the Fr¨ ohlich\ninteraction, and contrary to the Holstein el-ph case con-\nsidered in Ref.[31], the self-energy is not diagonal in the\nspin subspace. However, since the momentum depen-\ndence enter only through the modulus of the momentum\ntransfer, equation (10) can be rewritten in a quite simple\nform. By using ( ˆΩ(k)·σ)2= 1 and\n(ˆΩ(k)·σ)(ˆΩ(k′)·σ) =ˆk·ˆk′+(ˆk׈k′)σxσy,(11)\nthen the quantity ˆΩ(k′)·σappearing in Eq.(10) through\nG0(k′,ω−ω0) can be replaced simply by ( ˆΩ(k)·σ)ˆk·ˆk′because the second term of Eq.(11) vanishes after the\nintegration over k′. In this way, the resulting self-energy\nreduces to:\nΣ(k,ω) = Σd(k,ω)1+Σo(k,ω)ˆΩ(k)·σ,(12)\nwhere1is the unit matrix and Σ dand Σ oare, respec-\ntively, the diagonal and off-diagonal contributions to the\nself-energy, both depending solely on the modulus of k.33\nTheir explicit expressions are:\nΣd(k,ω)=|M0|2\n2/integraldisplaydk′\n(2π)2/summationdisplay\ns1\n|k−k′|1\nω−ω0−Es(k′),\n(13)\nΣo(k,ω)=|M0|2\n2/integraldisplaydk′\n(2π)2/summationdisplay\ns1\n|k−k′|sk·k′\nω−ω0−Es(k′).\n(14)\nIn the limit of zero SO coupling, since Es(k)→k2/2m,\nΣo(k,ω) vanishes because of the summation over s=\n±1 in Eq.(14). Notice also that, independently of γ,\nΣo(k,ω) = 0 when the factor 1 /|k−k′|in Eq.(14) is re-\nplaced by a constant, as in the momentum-independent\nHolstein el-ph coupling model.\nBy using Eq.(12) the Dyson equation for the interact-\ning propagator Greduces to\nG−1(k,ω) =G−1\n0(k,ω)−Σ(k,ω)\n=ω−k2\n2m−Σd(k,ω)−E0\n−[γk+Σo(k,ω)]ˆΩ(k)·σ,(15)\nand the poles ω±ofGare then given by:\nω±=E±(k)+Σd(k,ω±)±Σo(k,ω±).(16)\nNow, the Rayleigh-Schr¨ odinger perturbation theory per-\nmits to evaluate the lower energy pole ω−at the lowest\norder in the el-ph coupling α. This is accomplished by\nreplacingω−by the unperturbed energy E−(k) in the\nenergy variables of Σ dand Σ o. In this way, the lower\npole reduces to ω−=E−(k)+Σ−(k)+O(α2), where\nΣ−(k) = Σd(k,E−(k))−Σo(k,E−(k)).(17)\nFinally, by expanding Σ −(k) up to the second order in\nk−k0, the polaron dispersion in the vicinity of k0can be\nwritten as:\nω−=EP+1\n2m∗(k−k∗\n0)2, (18)\nwhere the polaron ground-state energy EP, the effective\nmassm∗, and the effective Rashba momentum k∗\n0are4\ngiven respectively by:\nEP= Σ−(k0)−m∗\n2Σ′\n−(k0)2\n= Σ−(k0)+O(α2), (19)\nm∗\nm= [1+mΣ′′\n−(k0)]−1\n= 1−mΣ′′\n−(k0)+O(α2), (20)\nk∗\n0\nk0= 1−m∗\nk0Σ′\n−(k0)\n= 1−m\nk0Σ′\n−(k0)+O(α2). (21)\nLet us first consider EPandm∗. In the zero SO limit,\nEqs. (19) and (20) at k0= 0 lead respectively to EP=\nπαω0/2 andm∗/m= 1 +πα/8, which correspond to\nthe results already reported in Refs.[13,14,15]. For finite\nvalues of the SO coupling the ground state energy and\n/s49/s46/s48/s48/s49/s46/s48/s53/s49/s46/s49/s48/s49/s46/s49/s53\n/s32/s32\n/s32/s102\n/s69\n/s80 /s40\n/s48/s41/s40/s97/s41\n/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s49/s50/s51/s52/s53/s32/s102\n/s109/s42/s40\n/s48/s41\n/s48/s40/s98/s41/s48 /s50 /s52 /s54 /s56 /s49/s48/s49/s46/s48/s49/s46/s50/s49/s46/s52/s49/s46/s54\n/s32 /s32/s102\n/s69\n/s80/s40\n/s48/s41\n/s48\n/s48 /s50 /s52 /s54 /s56 /s49/s48/s53/s48/s49/s48/s48/s49/s53/s48/s50/s48/s48\n/s32 /s32/s102\n/s109/s42/s40\n/s48/s41\n/s48\nFIG. 1: (a): ground state energy factor fEP(ε) as a function\nof the SO parameter ε0=E0/ω0. The solid line is the nu-\nmerical calculation, while the dashed line is the weak SO lim it\nEq.(25). Inset: fEP(ε) plotted for a wider range of ε0. (b):\nthe effective mass factor fm∗(ε0) from numerical integration\n(solid line) and from Eq.(26) (dashed line). Inset: fm∗(ε0)\nplotted for a wider range of ε0.the effective mass can be expressed as\nEP=−π\n2αω0fEP(ε0), (22)\nm∗\nm= 1+π\n8αfm∗(ε0), (23)\nwhere the factors fEP(ε0) andfm∗(ε0) contain all the\neffects of the SO interaction and depend solely on the\ndimensionless SO parameter\nε0≡E0\nω0=mγ2\n2ω0. (24)\nIn the weak SO limit, the self-energy terms (13) and (14)\ncanbeexpandedinpowersoftheSOinteraction,allowing\nfor an analytical evaluation of the integrals. In this way,\nup to the linear order in ε0,fEP(ε0) andfm∗(ε0) are\nfound to be:\nfEP(ε0) = 1+ε0\n4+O(ε2\n0), (25)\nfm∗(ε0) = 1+9\n8ε0+O(ε2\n0), (26)\nindicating that the polaronic character is strengthened\nby the SO interaction since, through Eqs. (22) and (23),\nthe polaron energy EPis lowered and, simultaneously,\nthe effective mass m∗is enhanced when ε0>0. This fea-\nture is not limited to the small ε0limit, but holds true\nfor arbitrary strengths of the SO coupling. This is shown\nin Fig. 1 where fEP(ε0) andfm∗(ε0), obtained from a\nnumerical integration of Eqs.(13) and (14), are plotted\nas a function of ε0by solid lines and compared with Eqs.\n(25) and (26) (dashed lines). The same quantities calcu-\nlated for a wider range of ε0are plotted in the insets of\nFig.1 and confirm that the ground state energy EPand\nthe effective mass m∗are continuous functions of ε0and\nare, respectively, further lowered and enhanced by the\nSO coupling. In the strong SO limit ε0≫1, it is found\nthatfEP(ε0)growsasln( ε0)whilefm∗(ε0)growslinearly.\nIt is interesting to note that the Holstein-Rashba model\nstudied in Ref.[31] predicts results qualitatively similar\nto the Fr¨ ohlich model, indicating that the SO interaction\nstrengthen the polaronic character independently of the\nspecific form of the el-ph interaction.34\nIn addition to EPandm∗, the interplay between the\nel-ph coupling and the SO interaction modifies also the\nRashba momentum k0through Eq.(21). In the weak SO\nlimit, the effective quantity k∗\n0is found to be\nk∗\n0\nk0≃1−π\n32αε0, (27)\nindicating a reduction of the bare Rashba momentum k0,\nconfirmed also by the numerical calculation of Eq.(21)\nreported in Fig. 2 by the solid line. As shown in the\ninset, for fixed el-ph coupling α,k∗\n0however does not\ndeviate much from its bare limit k0, even for large values\nof the SO parameter ε0.\nLet us compare now the present results with those\nappeared recently in literature. In Ref.[35] the ground5\n/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s45/s48/s46/s48/s51/s45/s48/s46/s48/s50/s45/s48/s46/s48/s49/s48/s46/s48/s48\n/s32/s32/s40/s107\n/s48/s42/s47/s107\n/s48/s45/s49/s41/s47\n/s48/s48 /s50 /s52 /s54 /s56 /s49/s48/s45/s48/s46/s48/s51/s45/s48/s46/s48/s50/s45/s48/s46/s48/s49/s48/s46/s48/s48\n/s32 /s32/s40/s107\n/s48/s42/s47/s107\n/s48/s45/s49/s41/s47\n/s48\nFIG. 2: Effective Rashba momentum k∗\n0as a function of the\nSO parameter ε0=E0/ω0. The numerical integration of\nEq.(21) (solid line) is compared with the weak SO result (27)\n(dashed line). Inset: the same quantity plotted for a wider\nrange of ε0.\nstate energy of a polaron near a polar-polar semiconduc-\ntor interface with Rashba SO coupling has been evalu-\nated with the Lee-Low-Pines method.4As a function of\nthe SO splitting, the polaron ground state is found to\nbe lowered, in qualitative agreement therefore with the\npresent results. A more quantitative comparison is how-\never precluded by the different model of Ref.[35], where\ncontributions from interface phonon modes and confin-\ning potentials are considered as well. In another work,36\nthe Rayleigh-Schr¨ odinger perturbation theory has been\napplied to the polaron ground state of the 2D Fr¨ ohlich-\nRashba model, permitting therefore a direct comparison\nwith the analysis presented here. Despite that the au-\nthors of Ref.[36] find that the polaron ground state is\nlowered by ε0, their values of EPdiffer from those plot-\nted in Fig. 1(a). In Ref.[36], in fact, the ground state\nenergy factor fEPis found to be fEP(ε0) = 1/√1−ε0,\nwhich implies a small ε0expansiondifferent from Eq.(25)\nand, more importantly, a divergence of EPatε0= 1. In\nFig. 1(a), instead, nothing of special happens at ε0= 1.\nThis discrepancy is easily traced back in the fact that in\nRef.[36] the expansion of Σ −(k), Eq.(17), is made around\nk= 0, instead of k=k0asdone here, which doesnot cor-\nrespond to a perturbative calculation of the ground-state\nenergy.\nThe results presented in this section have been derived\nby assuming a weak coupling to the phonons. However,\nas it is clear from the plots in Fig. 1, the enhancement of\nthe polaronic character driven by ε0for fixedαunavoid-\nably renders the perturbative approach invalid for suffi-\nciently large ε0values. For example, from Eq.(23), the\nvalidity of the weak coupling results for m∗/mare sub-\njected to the condition αfm∗(ε0)≪1, otherwise higher\norder el-ph contributions should be considered for a con-\nsistent description of the SO effects. The question re-\nmains therefore whether the SO enhancement of the po-laronic character survives also for large αvalues, or it is\ninstead limited to the weak coupling limit. In the next\nsection, this problem is studied for the limiting case of\nstrong el-ph interaction α≫1, providing therefore, to-\ngether with the weak coupling results, a global under-\nstanding of the SO effects on the Fr¨ ohlich polaron.\nIII. STRONG COUPLING\nIt is well known that a perturbative scheme such that\nemployed in the previous section fails to describe the\nFr¨ ohlich polaron ground state when the el-ph coupling\nis very large. This is due to the fact that for α≫1\nthe lattice polarization, and resulting “self-trapping” ef-\nfect experienced by the the electron,37renders the plane\nwaverepresentationofthe unperturbed electroninappro-\npriate for obtaining the polaron ground state. Instead,\nas originally proposed in Ref.[5] and rigorously proved in\nRefs.[38,39], the asymptotic description of the polaron\nwave function in the strong coupling limit α≫1 is\nthat of a product between purely electronic, ψ(r), and\npurely phononic, |ξ∝angbracketright, wave functions. Within such adi-\nabatic limit, the ground state energy and the effective\nmass of a 2D Fr¨ ohlich polaron have been calculated in\nRefs.[14,15] by using the variational method with differ-\nent ansatz wave functions. From Ref.[14], one realizes\nthatexponential,gaussianandPekar-typewavefunctions\nprovide increasingly better estimates of EPwith accura-\ncies respectively of 14%, 0 .3%, and 0.03% with respect to\nthe exact ground state energy EP/ω0=−0.40474α2, ob-\ntained by a numerical solution of the integro-differential\nequation for the electron wave function.16In the follow-\ning, the variational method is used to evaluate the SO\neffects on the polaron ground state.\nA. trial wave functions\nFor the nonzero SO case, due to the presence of the\nPauli matrices in Eq.(1), suitable ansatz wave functions\nmust take into account the electron spin degrees of free-\ndom. Hence, in full generality, the strong-coupling po-\nlaron wave function may be represented as: |Ψ,ξ∝angbracketright=\nΨ(r)|ξ∝angbracketright, whereΨ(r) is a two-components spinor for the\nelectron. The corresponding expectation value of the to-\ntal Hamiltonian His:\n∝angbracketleftΨ,ξ|H|Ψ,ξ∝angbracketright=∝angbracketleftΨ|Hel|Ψ∝angbracketright+∝angbracketleftξ|Hph|ξ∝angbracketright\n+1√\nA/summationdisplay\nq1√q(M0ρ(q)∝angbracketleftξ|aq|ξ∝angbracketright+h.c.),\n(28)\nwhere\nρ(q) =∝angbracketleftΨ|eiq·r|Ψ∝angbracketright=/integraldisplay\ndreiq·r|Ψ(r)|2.(29)6\nThe form of Eq.(28) permits to integrate out the phonon\nwave function in the usual way. Hence, by introducing\nthe phonon coherent state |ξ∝angbracketright=NeP\nqξqa†\nq|0∝angbracketright, whereN\nis a normalization factor and ξqa variational parame-\nter, minimization of (28) with respect to ξqleads to the\nfunctional\nE[Ψ] =∝angbracketleftΨ|Hel|Ψ∝angbracketright−|M0|2\nω0/integraldisplaydq\n(2π)21\nq|ρ(q)|2,(30)\nwhere the continuum limit A−1/summationtext\nq→/integraltext\ndq/(2π)2has\nbeen performed. By choosing an appropriate functional\nform for Ψ(r), and by minimizing E[Ψ] with respect\nto the variational parameters defining Ψ(r), an upper\nbound for the ground state energy is then E[Ψ0], where\nΨ0(r) is such that E[Ψ0] = min(E[Ψ]). As done in\nthe previous section, the polaron energy is then obtained\nfrom\nEP=E[Ψ0]+E0, (31)\nwhereE0is the free-electronSOenergydefined in Eq.(8).\nOf course, the functional form of Ψ(r) is decisive for\nobtaining accurate estimates of the ground state energy,\nand a suitable choice must be guided by looking at the\nproperties of the true ground state spinor ΨG(r). These\ncan be deduced by a formal minimization of the func-\ntionalE[Ψ] with respect to Ψ. By introducing the La-\ngrange multiplier ǫto ensure that the wave function is\nnormalized to unity, minimization of (30) leads to:\nHelΨ(r)+V(r)Ψ(r) =ǫΨ(r), (32)\nwhere, by using the definition of ρ(q) given in in Eq.(29):\nV(r) =−2|M0|2\nω0/integraldisplaydq\n(2π)2ρ(q)∗\nqeiq·r\n=−|M0|2\nπω0/integraldisplay\ndr′|Ψ(r′)|2\n|r−r′|. (33)From the above expression of V(r), the functional (30)\ncan be rewritten as E[Ψ] =∝angbracketleftΨ|Hel|Ψ∝angbracketright+¯V/2, where\n¯V=∝angbracketleftΨ|V(r)|Ψ∝angbracketright. Now, if ΨGis the exact ground state\nwave function, with ground state energy EG=E[ΨG],\nthen, from (32) and EG=∝angbracketleftΨ|Hel|Ψ∝angbracketright+¯V/2, it is found\nthatǫ=EG+¯V/2, so that Eq.(32) reduces to:\nHelΨG(r)+[V(r)−¯V/2]ΨG(r) =EGΨG(r).(34)\nAs noted in Ref.[29] (see also Refs.[40,41]), the ground-\nstate wave function of a 2D electron subjected to a SO\nRashba interaction and to a 2D central potential ( i.e.a\npotential depending only upon r=|r|) is of the form\nΨG(r) =/parenleftbigg\nψ1(r)\nψ2(r)eiϕ/parenrightbigg\n, (35)\nwhereϕis the azimuthal angle of r. Now, if Eq.(35)\nis used in Eq.(33), the resulting self-consistent potential\ndepends only upon r,V(r)→V(r), so that Eq.(35) is\nconsistently alsothe correct form for the polaron ground-\nstate wave function. Hence, passing to polar coordi-\nnates, Eq.(34) can be rewritten as a system of integro-\ndifferential equations for the spinor components ψ1and\nψ2:\n/bracketleftbigg\n−1\n2m/parenleftbiggd2\ndr2+1\nrd\ndr/parenrightbigg\n+U(r)/bracketrightbigg\nψ1(r)−γ/parenleftbiggd\ndr+1\nr/parenrightbigg\nψ2(r) =EGψ1(r), (36)\n/bracketleftbigg\n−1\n2m/parenleftbiggd2\ndr2+1\nrd\ndr−1\nr2/parenrightbigg\n+U(r)/bracketrightbigg\nψ2(r)+γd\ndrψ1(r) =EGψ2(r), (37)\nwhereU(r) =V(r)−¯V/2 and the polaron energy is ob-\ntained from EP=EG+E0. By introducing the dimen-\nsionless variable ρ=r/ℓP, whereℓP= 1/α(mω0)1/2is a\nmeasure of the polaron spatial extension in the zero SO\nlimit, and by noticing that EGdoes not depend on the\nsign ofγ, it is straightforward to realize from Eqs.(36)and (37) that the polaron ground state energy scales as\nEP=F/parenleftBigε0\nα2/parenrightBig\nα2ω0, (38)\nwhereε0=E0/ω0is the dimensionless SO energy intro-\nduced in Eq.(24) and Fis a generic function. It is found\ntherefore from Eq.(38) that the dependence of EPon the\nSO interaction is through the effective parameter ε0/α2,7\nwhich is treated in the following as an independent vari-\nable. Although ε0/α2is then formally allowed to vary\nfrom 0 to ∞, it is nevertheless important to estimate the\nrange over which ε0/α2is expected to vary for reason-\nable values of the microscopic parameters E0,ω0, andα.\nTo this end, it must be reminded that the strong cou-\npling limit of a 2D Fr¨ ohlich polaron (in the absence of\nSO interaction) is appropriate only for α/greaterorapproxeql5,12and that\nthe typical phonon energy scale is of the order of few to\ntens meV, say ω0≈5−10 meV. The largest value of the\nRashba energy E0reported so far is of about 0 .2 eV,25\nso thatε0/α2/lessorsimilar1−2 is a rather conservative estimate\ncompatiblewith materialparametersand with the strong\ncoupling polaron hypothesis.\nLetusnowevaluatethebehaviorof ψ1(r) andψ2(r) for\nr≪ℓPandr≫ℓP. By requiring a regular solution at\nthe origin, it turns out by inspection of Eqs.(36) and (37)\nthat the spinor components of (35) behave as ψ1(r) =\nconst.andψ1(r)∝rasr→0, while the behavior for\nr≫ℓPis obtained from the large rlimit of Eqs.(36) and\n(37):\n−1\n2md2ψ1(r)\ndr2−γdψ2(r)\ndr=Wψ1(r),(39)\n−1\n2md2ψ2(r)\ndr2+γdψ1(r)\ndr=Wψ2(r),(40)\nwhere the quantity W=EG+¯V/2 is negative for bound\nstates. Solutions of Eqs.(39) and (40) which are finite\nforr→ ∞are linear combination of exp( −λ+r) and\nexp(−λ−r) with\nλ±=/radicalBig\n−2m(EP+¯V/2)±ik0, (41)\nimplying an exponential decay of the polaron wave func-\ntion, accompanied by periodic oscillations of wavelength\n2π/k0.\nThe informations gathered on the limiting behaviors\nof the ground state wave function are sufficient for guess-\ning some appropriate trial wave functions to be used in\nEq.(30). By assuming that for zero SO coupling the elec-\ntronisinaspin-upstate,thenasimpleansatzcompatible\nwith the limits discussed above is\nΨ(r) =f(r)/parenleftbigg\ncos(br)\nsin(br)eiϕ/parenrightbigg\n, (42)\nwherebis a variationalSO parametervanishing for γ= 0\nandf(r) isanexponentiallydecayingfunction for r→ ∞\nand such that f(0)∝negationslash= 0. The advantage of Eq.(42) is that\none can use exponential or Pekar-type functions for f(r),\nautomatically recovering therefore the known results for\nthe zero SO case.14It should be noted, however, that\nin theU(r)→0 limit Eq.(42) does not reproduce cor-\nrectly the behavior of the exact ground state wave func-\ntion, which is instead given by Eq.(35) with ψ1(r) and\nψ2(r) proportional to the Bessel functions J0(k0r) and\nJ1(k0r), respectively.40,41Hence, Eq.(42) is not expected\ntoprovideareliablegroundstateenergyin the strongSOregime, for which U(r) can be treated as a perturbation.\nTo remedy to this deficiency, the following alternative\nform of the polaron ansatz is proposed:\nΨ(r) =f(r)/parenleftbigg\nJ0(br)\nJ1(br)eiϕ/parenrightbigg\n, (43)\nwhere, as before, bis a variational SO parameter. As it\nwillbeshownbelow, thelowestvalueof EPisgiveneither\nby Eq.(42) or by Eq.(43), depending on the specific form\nconsidered for f(r) and on the value of the SO coupling.\nB. ground state energy\nTo evaluate the polaron ground state energy, three dif-\nferent trial wave functions for f(r) are considered: ex-\nponential, Gaussian and Pekar-type. As shown below,\nthe Gaussian ansatz will provide results comparable to\nthose coming from the exponential and Pekar functions,\ndespite its faster decay for r→ ∞compared to Eq.(41).\nThese three trial wave functions will be used in combi-\nnation with the sinuisodal and the Bessel-type spinors\nof Eqs.(42) and (43), respectively, giving a total of six\ndifferent ansatzes for the Fr¨ ohlich-Rashba polaron wave\nfunction.\nExponential ansatz . Let us start by evaluating the\nfunctionalE[Ψ], Eq.(30), byusingtheexponentialansatz\nf(r) =Aexp(−ar), whereais a variational parameter\nandAis a normalization factor, in combination with the\nsinuisodal trial wave function (42). By introducing the\ndimensionlessquantities ˜ a=aℓP,˜b=bℓP, and ˜γ=k0ℓP,\nfor nonzero SO interaction the functional (30) evaluated\nwith the exponential-sinuisodal ansatz reduces to\nE[Ψ]\nα2ω0=1\n2/bracketleftBigg\n˜a2+˜b2+˜a2ln/parenleftBigg\n1+˜b2\n˜a2/parenrightBigg/bracketrightBigg\n−˜γ˜b/parenleftbigg\n1+˜a2\n˜a2+˜b2/parenrightbigg\n−3π˜a\n8√\n2.(44)\nFor weak SO couplings, Eq.(44) has its minimum at ˜b=\n˜γ=√2ε0/αand ˜a= 3√\n2π/16, so that the resulting\npolaron energy EP=E[Ψ0]+E0becomes\nEP\nα2ω0=−/parenleftbigg3π\n16/parenrightbigg2\n−ε0\nα2+O/parenleftbiggε2\n0\nα4/parenrightbigg\n.(45)\nIn theε0= 0 limit, Eq.(45) reduces to EP/α2ω0=\n−(3π/16)2≃ −0.3469, recovering therefore the result of\nRef.[14], while for ε0>0 the polaron energy is lowered\nby the SO interaction, in qualitative analogy with the\nweak electron-phonon behavior discussed in Sec.II. The\nlowering of EPis confirmed by a numerical minimization\nof Eq.(44) whose results are plotted in Fig.3(a) (open cir-\ncles). Forε0/α2= 1, the polaron energy has dropped to\nEP/α2ω0≃ −0.65, that is about two times lower than\nthe zero SO case. However, upon increasing ε0/α2,EP\ndisplays a minimum at ε0/α2≃3.98 [inset of Fig.3(a)]8\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48/s45/s49/s46/s48/s45/s48/s46/s57/s45/s48/s46/s56/s45/s48/s46/s55/s45/s48/s46/s54/s45/s48/s46/s53/s45/s48/s46/s52/s45/s48/s46/s51\n/s40/s97/s41\n/s32/s32/s69\n/s80/s47/s40\n/s48/s50\n/s41\n/s48/s47/s50/s69/s120/s112/s111/s110/s101/s110/s116/s105/s97/s108/s32/s97/s110/s115/s97/s116/s122\n/s32/s115/s105/s110/s117/s105/s115/s111/s100/s97/s108\n/s32/s66/s101/s115/s115/s101/s108\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48/s45/s49/s46/s48/s45/s48/s46/s57/s45/s48/s46/s56/s45/s48/s46/s55/s45/s48/s46/s54/s45/s48/s46/s53/s45/s48/s46/s52/s45/s48/s46/s51\n/s40/s98/s41\n/s32/s32\n/s48/s47/s50/s71/s97/s117/s115/s115/s105/s97/s110/s32/s97/s110/s115/s97/s116/s122\n/s32/s115/s105/s110/s117/s105/s115/s111/s100/s97/s108\n/s32/s66/s101/s115/s115/s101/s108\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48/s45/s49/s46/s48/s45/s48/s46/s57/s45/s48/s46/s56/s45/s48/s46/s55/s45/s48/s46/s54/s45/s48/s46/s53/s45/s48/s46/s52/s45/s48/s46/s51\n/s40/s99/s41\n/s32/s32\n/s48/s47/s50/s80/s101/s107/s97/s114/s45/s116/s121 /s112/s101/s32/s97/s110/s115/s97/s116/s122\n/s32/s115/s105/s110/s117/s105/s115/s111/s100/s97/s108\n/s32/s66/s101/s115/s115/s101/s108/s48/s46/s49 /s49 /s49/s48 /s49/s48/s48/s45/s50/s46/s48/s45/s49/s46/s53/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48\n/s32 /s32 /s69\n/s80 /s47/s40\n/s48/s50\n/s41\n/s48/s47/s50/s48/s46/s49 /s49 /s49/s48 /s49/s48/s48/s45/s50/s46/s48/s45/s49/s46/s53/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48\n/s32 /s32 /s69\n/s80 /s47/s40\n/s48/s50\n/s41\n/s48/s47/s50/s48/s46/s49 /s49 /s49/s48 /s49/s48/s48/s45/s50/s46/s48/s45/s49/s46/s53/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48\n/s32 /s32 /s69\n/s80 /s47/s40\n/s48/s50\n/s41\n/s48/s47/s50\nFIG. 3: Polaron ground state energy as a function of ε0/α2for different trial wave functions for f(r). (a): exponential; (b):\nGaussian; (c): Pekar. The sinuisodal and the Bessel type of a nstazes are given respectively by Eq.(42) and Eq.(43). Inse t: the\npolaron energy for a wider range of ε0/α2values.\nand for larger values of the SO interaction the polaron\nenergy increases. Eventually, for ε0/α2/greaterorsimilar14 the calcu-\nlated ground state energy becomes larger than the zero\nSO valueEP/α2ω0=−(3π/16)2. Such upturn of EP\nfor largeε0stems from the inadequacy of the sinuiso-\ndal components of (42) in treating the oscillatory be-\nhavior in the strong SO regime which, as pointed out\nabove, should instead be given by Bessel-type functions.\nIndeed when the exponential ansatz for f(r) is used in\nEq.(43), rather than in Eq.(42), not only the resulting\nEPis lower than the previous case, but also the upturn\nofEPdisappears, leading to a monotonous lowering of\nthe polaron energy as ε0/α2increases [filled circles in\nFig.3(a)]. As ε0/α2→ ∞, however, the polaron energy\ndoes not decrease indefinitely but rather approaches a\nlimiting value. Although an accurate numerical evalua-\ntion ofEPforε0/α2>100 has turned out to be difficult,\nthe asymptotic value of EPcan nevertheless be obtained\nanalytically from the strong SO limit of the exponential-\nBessel expression for E[Ψ]:\nE[Ψ]\nα2ω0=˜a2+˜b2\n2−˜b˜γ−π√\n2˜a, (46)\nwhose minimum is at ˜b= ˜γand ˜a=π/√\n2, leading to\nlim\nε0/α2→∞EP\nα2ω0=−π2\n4≃ −2.467.(47)\nGaussian ansatz . The results obtained by using a\nGaussian wavefunction of the form f(r) =Aexp(−a2r2)\nare plotted in Fig. 3(b). Compared to the exponentialwave function, the Gaussian ansatz gives an overall low-\neringofthe polaronenergyfor both sinuisodaland Bessel\nforms of the spinors. In the ε0/α2≪1 limit, and inde-\npendently of which particular spinor is used, the ground\nstate polaron energy is found to be:\nEP\nα2ω0=−π\n8−ε0\nα2+O/parenleftbiggε2\n0\nα4/parenrightbigg\n, (48)\nconfirminginthisregimethelineardependenceontheSO\ncoupling ofEq.(45). For largervalues ofthe SO coupling,\nand contrary to the case shown in Fig. 3(a), the sinuiso-\ndaland Bessel-typespinorsgivebasicallythe samevalues\nofEPfor all SO couplings up to ε0/α2≃1. Beyond this\nvalue, as for the case with the exponential wave function,\nthe polaron energy obtained from the sinuisodal ansatz\nbecomes largerthan that obtained from the Bessel spinor\nand, as shown in the inset of Fig. 3(b), rapidly increases\nwhile the Gaussian-Bessel anstaz gives a monotonous\nlowering of EP. Forε0/α2≫1, the Gaussian-Bessel\nenergy functional has the same form of Eq.(46) with the\nlatter term substituted by −2.279˜a, which implies\nlim\nε0/α2→∞EP\nα2ω0≃ −2.579. (49)\nPekar-type ansatz . Let us now evaluate EPby using\nin Eqs.(42) and (43) the Pekar-type ansatz f(r) =A(1+\na1r+a2r2)exp(−ar). For zero SO coupling, this ansatz\ngivesEP/α2ω0≃ −0.4046,14which is a lower energy\nthan those obtained from the exponential and Gaussian\ntrialwavefunctionsandonly0 .03%higherthanthe exact\nresult−0.40474 of Ref.[16]. As shown in Fig. 3(c), the9\nPekar-type ansatz gives slightly better estimates of EP\nalso for nonzero SO couplings, with an overall behavior\nsimilar to the previous cases. Namely, in the weak SO\nregime one finds\nEP\nα2ω0=−0.4046−ε0\nα2+O/parenleftbiggε2\n0\nα4/parenrightbigg\n,(50)\nand, as before, for stronger SO couplings the energy ob-\ntained from the sinuisodal spinor increases indefinitely\nwithε0/α2. However, contrary to the exponential and\nGaussian ansatzes, the Pekar-type wave function may\ngive a lower polaron energy when used in combination\nwith the sinuisodal spinor. This holds true as long as\nε0/α2/lessorsimilar2.72, while for stronger SO couplings it is the\nBessel-type spinor which gives the lower EP[inset of Fig.\n3(c)]. A numerical minimization of the asymptotic limit\nof the Pekar-Bessel functional for ε0/α2≫1 gives\nlim\nε0/α2→∞EP\nα2ω0≃ −2.91, (51)\nwhichislowerthanthe asymptoticvaluesofEqs.(47) and\n(49).\nThe results plotted in Fig. 3 clearly demonstrate that,\nsincethe variationalmethod providesanupper bound for\ntrue ground state polaron energy, the lowering of EPin-\nducedbythe SOcouplingisarobustfeatureofthe strong\ncoupling Fr¨ ohlich-Rashba polaron. Among the different\nansatzes studied, the lower polaron energy is obtained by\nusing a Pekar-type wavefunction for f(r) in combination\nwith the sinuisodal spinor for weak to moderate values of\nε0/α2orwith the Bessel-typespinor for strongerSO cou-\nplings. Given that, as discussed above, reasonable values\nofε0/α2for strongly-coupled polarons fall in the range\n0≤ε0/α2/lessorsimilar1−2, the Pekar-sinuisodal wave function\nprovides therefore the best description of the Fr¨ ohlich-\nRashba polaron in this regime.\nC. effective mass\nAs demonstrated in Sec.II, the effective mass m∗of\na weakly-coupled polaron is enhanced by the SO in-\nteraction and, given the results above, the same phe-\nnomenon is reasonably expected to occur also for the\nstrong-coupling case. To quantify the polaron mass en-\nhancement within the localized wave function formal-\nism, it is useful to follow the approach of Refs.[42,43,44],\nbriefly described below, where a moving wave packet is\nconstructed from the localized wave function. The quan-\ntity to minimize is\nJυ[Ψ′,ξ′] =∝angbracketleftΨ′,ξ′|H−υ·P|Ψ′,ξ′∝angbracketright,(52)\nwhereυis a Lagrange multiplier, which will turn out to\nbe the mean polaron velocity, and P=p+/summationtext\nqqa†\nqaqis\nthetotalmomentumoperator. Thewavefunction |Ψ′,ξ′∝angbracketright\nis given by the product Ψ′(r)|ξ′∝angbracketrightwhere\nΨ′(r) =eip0·rΨ(r) (53)is the electron wave packet with p0being a variational\nmomentum, Ψ(r) is the ansatz localized wave function,\nand|ξ′∝angbracketright=NeP\nqξ′\nqa†\nq|0∝angbracketright. Minimization of (52) with re-\nspect toξ′\nqgives now the functional\nJυ[Ψ′] =∝angbracketleftΨ′|Hel−υ·p|Ψ′∝angbracketright\n−|M0|2/integraldisplaydq\n(2π)2|ρ(q)′|2\nq1\nω0−q·∝angbracketleftΨ′|υ|Ψ′∝angbracketright,\n(54)\nwherepis the electron momentum operator and ρ(q)′=\n∝angbracketleftΨ′|eiq·r|Ψ′∝angbracketright. By using Eq.(53), it is easily shown that\nJυ[Ψ′] reduces to\nJυ[Ψ′] =∝angbracketleftΨ|Hel|Ψ∝angbracketright+p2\n0\n2m−p0·υ\n−|M0|2/integraldisplaydq\n(2π)2|ρ(q)|2\nq1\nω0−q·υ,(55)\nwhereρ(q) =∝angbracketleftΨ|eiq·r|Ψ∝angbracketright. Equation (55) is minimized\nwithrespectto p0bysetting p0=mυand, byexpanding\nthe last term of Eq.(55) up to the second order in υ, the\ncorresponding minimum Jυ[Ψ] becomes:42\nJυ[Ψ] =E[Ψ]−m\n2υ2/bracketleftbigg\n1+2|M0|2\nmω3\n0/integraldisplaydq\n(2π)2(q·ˆu)2\nq|ρ(q)|2/bracketrightbigg\n,\n(56)\nwhereE[Ψ] is given in Eq.(30). From the above expres-\nsion, it is clear that Jυ[Ψ] differs from J0[Ψ] at least to\norderυ2. Hence, if ΨυandΨ0are the wave functions\nwhich minimize Jυ[Ψ] andJ0[Ψ], respectively, then the\ndifference Ψυ−Ψ0is also of order υ2. As a consequence,\nthe minimum of (56), Jυ[Ψυ], differs from Jυ[Ψ0] only to\norder(Ψυ−Ψ0)2=O(υ4) sothat, by neglectingtermsof\nhigher order than υ2, minimization of (56) is achieved by\nthebestwavefunctionwhichminimizes E[Ψ]. Therefore,\nby usingE[Ψ0] =EP−E0and evaluating ∝angbracketleftΨ0|P|Ψ0∝angbracketright,\nfrom Eqs.(52) and (56) it turns out that\nEP(υ) =EP+m\n2υ2/bracketleftbigg\n1+2|M0|2\nmω3\n0/integraldisplaydq\n(2π)2(q·ˆu)2\nq|ρ0(q)|2/bracketrightbigg\n,\n(57)\npermitting us to identify the quantity within square\nbrackets as the mass enhancement factor m∗/m. By in-\ntegrating over the direction of qand by using (5), m∗/m\nbecomes in the strong-coupling limit\nm∗\nm=√\n2πα\n(mω0)3/2/integraldisplay∞\n0dq\n2πq2|∝angbracketleftΨ0|eiq·r|Ψ0∝angbracketright|2,(58)\nwhich, by replacing the momentum variable by the di-\nmensionless quantity ˜ q=qℓP, gives a mass enhancement\nproportional to α4in the zero SO case. By using the ex-\nponential, Gaussian, and Pekar-type ansatzes in Eq.(58),\nthe resulting mass enhancement factor becomes m∗/m=\n(3/16)3π4α4≃0.6421α4,m∗/m= (π/4)2α4≃0.617α4,\nandm∗/m≃0.73α4, respectively.45\nThe results for nonzero SO coupling are plotted in\nFig.4 for the sinuisodal (open circles) and Bessel (filled10\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48/s50/s46/s53/s51/s46/s48/s51/s46/s53\n/s40/s97/s41\n/s32/s32/s109 /s42/s47/s40/s109 /s52\n/s41\n/s48/s47/s50/s69/s120/s112/s111/s110/s101/s110/s116/s105/s97/s108/s32/s97/s110/s115/s97/s116/s122\n/s32/s115/s105/s110/s117/s105/s115/s111/s100/s97/s108\n/s32/s66/s101/s115/s115/s101/s108\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48/s50/s46/s53/s51/s46/s48/s51/s46/s53\n/s40/s98/s41\n/s32/s32\n/s48/s47/s50/s71/s97/s117/s115/s115/s105/s97/s110/s32/s97/s110/s115/s97/s116/s122\n/s32/s115/s105/s110/s117/s105/s115/s111/s100/s97/s108\n/s32/s66/s101/s115/s115/s101/s108\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48/s50/s46/s53/s51/s46/s48/s51/s46/s53\n/s40/s99/s41\n/s32/s32\n/s48/s47/s50/s80/s101/s107/s97/s114/s45/s116/s121 /s112/s101/s32/s97/s110/s115/s97/s116/s122\n/s32/s115/s105/s110/s117/s105/s115/s111/s100/s97/s108\n/s32/s66/s101/s115/s115/s101/s108\n/s48/s46/s48/s49 /s48/s46/s49 /s49 /s49/s48 /s49/s48/s48/s48/s49/s48/s50/s48/s51/s48/s52/s48/s53/s48/s54/s48\n/s32 /s32 /s109/s42/s47/s40/s109/s52\n/s41\n/s48/s47/s50/s48/s46/s48/s49 /s48/s46/s49 /s49 /s49/s48 /s49/s48/s48/s48/s49/s48/s50/s48/s51/s48/s52/s48/s53/s48/s54/s48\n/s32 /s32 /s109/s42/s47/s40/s109/s52\n/s41\n/s48/s47/s50/s48/s46/s48/s49 /s48/s46/s49 /s49 /s49/s48 /s49/s48/s48/s48/s49/s48/s50/s48/s51/s48/s52/s48/s53/s48/s54/s48\n/s32 /s32 /s109/s42/s47/s40/s109/s52\n/s41\n/s48/s47/s50\nFIG. 4: Polaron mass enhancement m∗/min units of α4as a function of ε0/α2for different ansatz wave functions. (a):\nexponential; (b): Gaussian; (c): Pekar. Inset: m∗/mα4is plotted for a wider range of SO values.\ncircles) spinors evaluated with exponential (a), Gaus-\nsian (b), and Pekar-type (c) wave functions. For all\ncases,m∗/mincreases with ε0/α2without much quan-\ntitative differences between the various ansatzes as long\nasε0/α2/lessorsimilar2. As shown in the insets of Fig. 4, for\nlarger values of the SO coupling the use of the sinuisodal\nspinor largely overestimates the increase of the effective\nmass compared to the Bessel-type spinor results. How-\never, despite of the weaker enhancement of m∗/m, the\nBessel-type spinors give nevertheless an infinite effective\nmass atε0/α2=∞. Indeed, independently of the par-\nticular form of f(r), forε0/α2→ ∞the expectation\nvalue∝angbracketleftΨ0|eiq·r|Ψ0∝angbracketrightappearing in Eq.(58) goes like a/q\nforq→ ∞, rendering the integral over qof Eq.(58) di-\nvergent.\nIV. DISCUSSION AND CONCLUSIONS\nThe results presented in the previous sections consis-\ntently show that, for both the weak and strong coupling\nlimits of the el-ph interaction, the ground state energy\nEPof the Fr¨ ohlich-Rashba polaron is lowered by the\nSO interaction and the mass is enhanced, leading to\nthe conclusion that the Rashba coupling amplifies the\npolaronic character. This scenario suggests also that a\nweak-coupling polaron at ε0= 0 may be turned into a\nstrong-coupling one for ε0>0 or, more generally, that\nthe crossover between weakly and strongly coupled po-\nlarons may be shifted by the SO interaction. This pos-\nsibility can be tested by looking at the curves plotted in\nthe main panel of Fig. 5, where the weak and strong\ncoupling results for EP/ω0are reported as a function ofthe el-ph coupling αfor different ε0values. For ε0= 0,\nthe polaron energy follows EP/ω0≃ −πα/2 for small α\nandEP/ω0≃ −0.4046α2for largeα. These two limiting\nbehaviors are plotted in Fig. 5 by the uppermost curves\nand compared with a numerical solutions of the Feyn-\nman variational path integral for the 2D polaron (filled\ncircles). The largest deviation of the path integral solu-\ntions from the weak and strong coupling approximations\nfalls in the range of intermediate values of αand signals\na region of crossover between the weakly and strongly\ncoupled polaron. A rough estimate of the crossover po-\nsition is given by a “critical” coupling, say α∗, obtained\nby equating the weak and strong coupling results. For\nε0= 0 therefore one has πα/2 = 0.4046α2, which gives\nα∗≃3.9. Now,asshowninFig. 5for ε0= 5andε0= 20,\nthe increase ofthe SO interaction systematically reduces,\nfor fixedα, the polaron ground state energy and, at the\nsame time, shifts the intersection point between the weak\nand strong coupling curves towards smaller values of the\nel-ph interaction. The “critical”value α∗of the crossover\nis therefore reduced by the SO interaction. For ε0= 5\nandε0= 20 it is found that α∗≃3.6 andα∗≃2.7,\nrespectively. The systematic reduction of the crossover\ncoupling by the SO interaction is made evident in the\ninset of Fig. 5, where α∗is plotted as a function of ε0.\nFrom Fig. 5 it is also expected that, beside the reduc-\ntionofα∗, the crossoverregionislikelytobe narrowedby\nε0. Indeed, the intersection between the weak and strong\ncoupling solutions for ε0= 20 is apparently smoother\nthan the case for ε0= 0, suggesting that the true ground\nstate energy would deviate less, and in a narrower region\naroundα∗, from the weak and strong coupling solutions.\nThe scenario illustrated above, and in particular the11\n/s48 /s50 /s52 /s54 /s56 /s49/s48/s45/s51/s53/s45/s51/s48/s45/s50/s53/s45/s50/s48/s45/s49/s53/s45/s49/s48/s45/s53/s48\n/s32/s32/s69\n/s80/s47\n/s48/s32/s32\n/s48/s32/s61/s32/s48\n/s32/s32\n/s48/s32/s61/s32/s53\n/s32/s32\n/s48/s32/s61/s32/s50/s48\n/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48/s50/s46/s48/s50/s46/s53/s51/s46/s48/s51/s46/s53/s52/s46/s48\n/s119/s101/s97/s107/s32\n/s99/s111/s117 /s112/s108/s105/s110 /s103\n/s32/s32/s42\n/s48/s115/s116/s114/s111/s110 /s103/s32\n/s99/s111/s117 /s112/s108/s105/s110 /s103\nFIG. 5: (Color online). Ground state polaron energy EPas\na function of the el-ph coupling αfor different values of the\ndimensionless SO parameter ε0=E0/ω0. The straight lines\nat small αrefer to the weak coupling results, while the curves\nat largeαare the solution of the strong-coupling theory. The\nfilled circles are the solution of the Feynman path integral\nansatz (see text). The point of intersection between the wea k\nand strong coupling curves is a measure of the crossover el-p h\ncoupling α∗. Inset:α∗is plotted as a function of ε0.\nSO effect on the crossover coupling, may be verified\nby quantum Monte-Carlo calculations of the Fr¨ ohlich-\nRashba action or, more simply, by generalizing the Feyn-\nman ansatz for the retarded interaction to ε0>0.7The\nresults presented here on the limiting cases α≪1 and\nα≫1 may then serve as a reference for such more gen-eral calculations schemes for arbitrary values of the el-ph\ncoupling and of the SO interaction.\nLet us discuss, before concluding, possible generaliza-\ntions of the Fr¨ ohlich-Rashba model employed here and\nthe consequences on the polaronic character. Let us re-\nmind that in Ref.[31] it has been demonstrated that also\nfor a momentum independent el-ph interaction model,\nthe Rashba SO term leads to an effective enhancement\nofthe el-phcoupling. TheSOinduced loweringofthe po-\nlaron ground state is therefore robust against the specific\nform of the el-ph interaction, so that a similar behavior\nis expected to occur also when considering the contribu-\ntions from interface or surface phonon modes. However,\na different form of the SO interaction term may lead to\na much weaker effect. Consider for example the situa-\ntion in which, in addition to the Rashba SO coupling,\nthe system lacks also of bulk inversion symmetry, as in\nIII-V semiconductor heterostructures, leading to an ex-\ntra SO term of the Dresselhaus type.18,46When both SO\ncontributions are present, the square root divergence of\nthe DOS at the bottom of the band of the free electron\ndisappears, and it is replaced by a weaker logarithmic\ndivergence at higher energies. In this situation therefore,\nat least for weak el-ph couplings, the SO interaction is\nexpected to have a weaker effect on the polaron ground\nstate, which tends to vanish as the Dresselhaus term be-\ncomes comparable to the Rashba one.\nLet us conclude by noticing that, recently, the possi-\nbility of varying the coupling of 2D Fr¨ ohlich polarons in\na controlled way has been experimentally demonstrated\nby acting on the dielectric polarizability of organic field-\neffect transistors.47The results presented here suggest\nthat tunable 2D Fr¨ ohlich polarons may be achieved also\nby acting on the SO coupling, which can be tuned by\napplied gate voltages in quasi-2D structured materials.\nAcknowledgments\nThe author thanks Emmanuele Cappelluti and Frank\nMarsiglio for valuable comments.\n1H. Fr¨ ohlich, Adv. Phys. 3, 325 (1954).\n2T. K. Mitra, A. Chatterjee, and S. Mukhopadhyay, Phys.\nReports. 153, 91 (1987).\n3H. Fr¨ ohlich, H. Pelzer, and S. Zienau, Phil. Mag. 41, 221\n(1950).\n4T.D.Lee, F.Low, andD.Pines, Phys.Rev. 90, 297(1953).\n5S. I. Pekar, Zh. Eksp. Teor. Fiz. 16, 355 (1946); 16, 341\n(1946).\n6S. J. Miyake, J. Phys. Soc. Japan 38, 181 (1975); 41, 747\n(1976).\n7R. P. Feynman, Phys. Rev. 97, 660 (1955).\n8G. Ganbold and G. V. Efimov, Phys. Rev. B 50, 3733\n(1994); J. Phys.: Condens. Matter 10, 4845 (1998).\n9G. De Filippis, V. Cataudella, V. Marigliano Ramaglia, C.A. Perroni, andD. Bercioux, Eur.Phys. J. B 36, 65(2003).\n10C. Alexandrou, W. Fleischer, and R. Rosenfelder, Phys.\nRev. Lett. 65, 2615 (1990).\n11J. T. Titantah, C. Pierleoni, and S. Ciuchi, Phys. Rev.\nLett.87, 206406 (2001).\n12S. Das Sarma and B. A. Mason, Ann. Phys. (N.Y.), 163,\n78 (1985).\n13W. J. Huybrechts, Solid Sate Commun. 28, 95 (1978).\n14X. Wu, F. M. Peeters, and J. T. Devreese, Phys. Rev. B\n31, 3420 (1985).\n15F. M. Peeters, X. Wu, and J. T. Devreese, Phys. Rev. B\n37, 933 (1988).\n16C. Qinghu, F. Minghu, Z. Qirui, W. Kelin, and W. Shao-\nlong, J. Phys.: Condens. Matter 8, 7139 (1996).12\n17J. T. Devreese, J. Phys.: Condens. Matter. 19, 255201\n(2007).\n18I.ˇZuti´ c, J. Fabian, and S. Das Sarma, Rev. Mod. Phys.\n76, 323 (2004).\n19S. LaShell, B. A. McDougall, and E. Jensen, Phys. Rev.\nLett.77, 3419 (1996).\n20Yu. M. Koroteev, G. Bihlmayer, J. E. Gayone, E. V.\nChulkov, S. Blugel, P. M. Echenique, and P. Hofmann,\nPhys. Rev. Lett. 93, 046403 (2004).\n21K. Sugawara, T. Sato, S. Souma, T. Takahashi, M. Arai,\nand T. Sasaki, Phys. Rev. Lett. 96, 046411 (2006).\n22E. Rotenberg, J. W. Chung, and S. D. Kevan, Phys. Rev.\nLett.82, 4066 (1999).\n23D. Pacil´ e,C. R. Ast, M. Papagno, C. Da Silva, L. Mores-\nchini, M. Falub, Ari P. Seitsonen, and M. Grioni, Phys.\nRev. B73245429 (2006).\n24C. R. Ast, G. Wittich, P. Wahl, R. Vogelgesang, D. Pacil´ e,\nM. C. Falub, L. Moreschini, M. Papagno, M. Grioni, and\nK. Kern, Phys. Rev. B 75, 201401(R) (2007).\n25C. R. Ast, J. Henk, A. Ernst, L. Moreschini, M. C. Falub,\nD. Pacil´ e, P. Bruno, K. Kern, and M. Grioni, Phys. Rev.\nLett.98, 186807 (2007).\n26I. I. Boiko and E. I. Rashba, Sov. Phys. Solid State 2, 1692\n(1960).\n27A. G. Galstyan and M. E. Raikh, Phys. Rev. B 58, 6736\n(1998).\n28C. Grimaldi, Phys. Rev. B 72, 075307 (2005).\n29A. V. Chaplik and L. I. Magarill, Phys. Rev. Lett. 96,\n126402 (2006); Physica E 34, 344 (2006).\n30E. Cappelluti, C. Grimaldi, and F. Marsiglio, Phys. Rev.\nLett.98, 167002 (2007).\n31E. Cappelluti, C. Grimaldi, and F. Marsiglio, Phys. Rev.\nB76, 085334 (2007).\n32E. Evans and D. L. Mills, Phys. ev. B 8, 4004 (1973); H.\nSun and S. -W. Gu, Phys. Rev. B 40, 11576 (1989).\n33The form of the self-energy reported in Eq.(12) is not lim-\nited to the weak-coupling limit. For a more general deriva-tion see Ref.[31].\n34There are however quantitative differences. Indeed, in the\nstrong SO limit, the Holstein-Rashba model gives an effec-\ntive mass enhancement proportional to√ε0, i.e., a weaker\nincrease with respect to the linear dependence found here\nfor the Fr¨ ohlich-Rashba model (see Ref.[31]).\n35J. Liu and J. -L. Xiao, Commun. Theor. Phys. 46, 761\n(2006).\n36Z. Li, Z. Ma, A. R. Wright, and C. Zhang, Appl. Phys.\nLett.90, 112103 (2007).\n37The term “trapping” is used here in a loose way, since\nno real trapping effect exist for the continuum Fr¨ ohlich\npolaron.\n38M. Donsker and S. R. S. Varadhan, Commun. Pure Appl.\nMath.36, 505 (1983).\n39E. H. Lieb and L. E. Thomas, Commun. Math. Phys. 183,\n511 (1997).\n40E. N. Bulgakov and A. F. Sadreev, JETP Lett. 73, 505\n(2001).\n41E. Tsitsishvili, G. S. Lozano, and A. O. Gogolin, Phys.\nRev. B70, 115316 (2004).\n42G. R. Allcock, Adv. Phys. 5, 412 (1956).\n43R. Parker, G. Whitfield, and M. Rona, Phys. Rev. B 10,\n698 (1974).\n44J. da Providencia, M. da Concei¸ c˜ ao Ruivo, and C. A. de\nSousa, Ann. Phys. 91, 366 (1975).\n45Note that the result for the exponential ansatz differs from\nthat of Ref.[15]. The difference stems from an incorrect\nexpression of the exponential decay factor a: in Ref.[15]\na/ℓP= 3π/8 while instead it should be given by a/ℓP=\n3π/16, as in Ref.[14] and in the present paper.\n46G. Dresselhaus, Phys. Rev. 100, 580 (1955).\n47I. N. Hulea, S. Fratini, H. Xie, C. L. Mulder, N. N. Iossad,\nG. Rastelli, S. Ciuchi, and A. F. Morpurgo, Nat. Mater. 5,\n982 (2006)."    },    {        "title": "2401.08966v1.Spin_Orbit_Torque_on_a_Curved_Surface.pdf",        "content": " \n1 \n 1 \nSpin Orbit Torque  on a Curve d Surface  \nSeng Ghee Tan1†,  Che Chun Huang2,  Mansoor  B.A.Jalil3, Zhuobin Siu3 \n(1) Department of Optoelectric Physics, Chinese Culture University, 55 Hwa -Kang Road, Yang -\nMing -Shan, Taipei 11114, Taiwan  \n \n(2) Department of Physics, National Taiwan University, Taipei 10617, Taiwan  \n \n(3) Department of Electrical and Computer Engineering, National University of Singapore, 4 \nEngineering Drive 3, Singapore 117576  \n \n \nABSTRACTS  \nWe provide a general formulation of the spin -orbit coupling on a 2D curved surface.  Considering the \nwide applicability of spin -orbit effect in spinor -based condensed matter physics, a general spin -orbit \nformulation  could aid the  study of  spintronics , Dirac graphene, topological  systems, and quantum \ninformation on curved surfaces. Particular attention is then devoted to the development of an \nimpo rtant spin -orbit quantity known as the spin-orbit  torque.  As devices trend smaller in dimension, \nthe physics of local geometries on spin -orbit torque, hence spin and magnetic dynamics shall not be \nneglected. We derived the general ex pression of a spin-orbit  anisotropy field  for the curved surfaces \nand provided explicit solutions in the special context s of the spherical, cylindrical and flat coordinates. \nOur expression s allow  spin-orbit anisotropy field s and hence spin -orbit torque  to be comp uted over \nthe entire  surface s of devices of any geometry .  \n \n \n \n \n \n \n \nCorresponding author:  \n† Seng Ghee Tan (Prof)  \nDepartment of Optoelectric Physics,  \nChinese Culture University,  \n55, Hwa -Kang Road, Yang -Ming -Shan,  \nTaipei, Taiwan 11114 ROC  \n(Tel: 886 -02-2861 -0511, DID:25221)  \n \n† Email: csy16@ulive.pccu.edu.tw  ; tansengghee@gmail.com   \nPACS:  \n \n  \n2 \n 2 \nINTRODUCTION  \nThe rise of spin physics in condensed matter and nanoscience is so prolific that spinor -based \nexpressions is now a hallmark  formulation  in nearly all the modern research fields like the spintronics  \n[1-5], graphene  [6,7], topological insulators  [8,9], topological Weyl  and Dirac systems  [10, 11], atomic \nand optical physics [12], quantum  computation and  information  science [13, 14]. Central  to the \nsuccess of spin in science  is the advancements in our understanding of the spin-orbit coupling. On the \nlevel of pure science, it  is a Dirac relativistic phenomenon that exists in the non -relativistic limit . From \nthe condensed matter point of view , spin-orbit coupling is a highly accessible  physics that exists  in \nbulk and structural forms in materials  like the semiconductor , semimetal , as well as heterostructures \ncomprising these materials . In applied science, it is simply an effective magnetic field that can be \ncontrolled via electrical mean s to perform electronic functions like in transistors and memory.   In this \narticle, we would focus on the spin orbit coupling in a specific area known as the spin -orbit torque. In \nmagnetic memory, spin current is normally injected into magnetic materia ls to flip the magnetic \nmoments via the force physics  of spin transfer torque  [15]. In nanoscience systems, spin torque is \nusually studied in the ferromagnetic materials, in which the net magnetization, reacts to the injection \nor the “creation” of an accum ulation of spin moments in the material. One example of such “creation” \nis via the spin -orbit effect. With spin-orbit effect  identified for the new role,  the force involved in this \nprocess would  be known as the spin-orbit torque  [16-22].   \n \nThis paper is dedicated to providi ng a general formulation of the spin -orbit coupling and hence the \nspin-orbit torque in a 2D environment that comprises curved surfaces, with the 2D planar surfaces \nbeing a special case of the curve s. Such formulation is im portant because as device becomes smaller, \nthe physics of local geometries on spin -orbit coupling and its torque shall not be neglected. In fact, as \nspin-orbit coupling is versatile in many modern fields, a general formulation is simply right for the \noccasion. For example, curve physics has recently been studied in  Dirac -electronic transport due to \nthe topological surface states [23-26]. Curves were studied for their effects on inducing topological \ntransitions  [27] in a Rashba system , as well as shift ing the  band -inversion point  [28] in the BHZ \ntopological -insulator . In spintronics, the effect of curve on spin current [29-31] spin Hall [32], and spin \nChern number [33] have also been studied.  \n \nIn this paper, we present a general formulation of the spin -orbit torque  in a 2D condensed matter \nsystem. We will begin with a 3D infinitesimal bulk i n which a 2D surfa ce can be  retrieved later by \nsetting 𝑞3→0.  Let the electric field penetrating the bulk be normal to the surface (to be denoted by \n𝒆𝟑) as shown in Fig.1 below.  The spin -orbit energy of  a charged particle contained therein would be \ngiven by  \n𝐻𝑠𝑜𝑐=𝛼3\nℏ𝝈.(𝑷×𝒆𝟑) \n(1) \nwhere 𝝈 is the Pauli matrices , and 𝛼3 is a constant  to characterize the spin -orbit strength .  While \nEquation (1) is  in the general  form, constant 𝛼3 would reflect the material property and the type of \nspin-orbit coupling, e.g. Rashba, Dresselhaus,  in bulk , heterostructures  or topological surface states. \nThe momenta and Pauli matrices on the 2D surface take on coordinates 𝒆𝟏,𝒆𝟐. And the y are related \nto the electric field direction  by 𝒆𝟏×𝒆𝟐=|𝑛3| 𝒆𝟑. It can then be shown that |𝑛3|=√𝐺\n𝐺33 , which leads \nto   \n𝐻𝑠𝑜𝑐=(𝛼3\nℏ)𝜎𝑣𝒆𝒗.[𝑃𝑘𝒆𝒌×(𝒆𝟏×𝒆𝟐)𝐺33\n√𝐺] \n (2)  \nwhere  (𝑣,𝑘) runs over coordinates (𝑞1,𝑞2). 𝐺33=𝜕𝑹\n𝜕𝑞3.𝜕𝑹\n𝜕𝑞3 is the 3D metric of the system  and 𝐺 is the \ndeterminant  given by  𝐺=(𝐺11𝐺22−𝐺12𝐺21)𝐺33.  Note that w ith 𝛼3 being contravariant , the upper \nand lower indices would balance out.   The contravariant property of 𝛼3 will be discussed later.   \n3 \n 3 \n \n \n \n \n \n \n \n \n \n \n \nFIG.1. An infinitesimal bulk can be shrunk along  𝒆𝟑 to a 2D surface embeded in a 3D space . This is tantamont to \ntaking the limit of 𝑞3→0.   \n \nEquation (2) is then wri tten as follows   \n𝐻𝑆𝑂𝐶=(𝛼3\nℏ) 𝑃𝑘𝜎𝑣 𝑅𝑘𝑣3 \n(3) \nwhere 𝑅𝑘𝑣3=𝐺33\n√𝐺(𝐺𝑣1𝐺𝑘2−𝐺𝑣2𝐺𝑘1). Expanding the above with the electric field pre -determined \nalong 𝒆𝟑, discussion is kept to the 2D spin -orbit effect, and one has the  general expression  of  \n \n𝐻𝑠𝑜𝑐=(𝛼3\nℏ) [𝜎1𝑃2−𝜎2𝑃1]√𝐺 \n(4) \nNote that (1,2,3) is the a bbrev iated form of general coordinates  (𝑞1,𝑞2,𝑞3).  A compact expression  \nof the above  is given by  \n𝐻𝑠𝑜𝑐=𝑃1\n𝑚𝑒𝑆1+𝑃2\n𝑚𝑒𝑆2 \n(5) \nwhere  𝑆1=−(𝛼3\nℏ)𝑚𝑒√𝐺 𝜎2 ,    𝑆2=(𝛼3\nℏ)𝑚𝑒√𝐺 𝜎1. Note that  (𝑃1,𝑃2) and (𝜎1,𝜎2) can b e found \nby transforming the ir Cartesian counterparts  to the desired coordinates in a contravariant manner. \nWhen  the general coordinates (𝑞1,𝑞2,𝑞3) take on  the Cartesian  (𝑥,𝑦,𝑧), the term √𝐺 goes to 1.  The \ngeneral Hamiltonian 𝐻𝑆𝑂𝐶 returns to the fami liar spin -orbit system  in the Cartesia n frame,  \n𝐻𝑠𝑜𝑐=(𝛼3\nℏ)(−𝑃𝑥 𝜎𝑦+𝑃𝑦𝜎𝑥) \n(6) \nWe will now examine the physical significance of 𝛼3, the  spin-orbit constant that characterizes its \nstrength. As 𝛼3 captures the material property pertaining directly to the strength of the electric field \nor the effective electric field in the case of the 2D spin -orbit effect e.g. the Rashba  or the Dresellhaus , \nwe will now explicitly express 𝛼3 in terms of i ts electric field as 𝛼3=𝛼′𝐸3.  Eq.(1) can now be \nrewritten to better reflect the actual scenario  whe re the presence of the electric field is explicit  \n𝐻𝑠𝑜𝑐=𝛼3\nℏ𝝈.(𝒑×𝒆𝟑)  →  𝐻𝑠𝑜𝑐=𝛼′\nℏ𝝈.(𝒑×𝐸3𝒆𝟑) \n(7) \nNow 𝐸3 would be the strength of the electric  field  normal to surface  i.e. along  𝒆𝟑. For better clarity, \nwe use the simple spherical surface s for illustration in Fig.2 below . \n \n𝒆𝟏 \n𝒆𝟐 \n 𝒏=𝒆𝟏×𝒆𝟐=|𝑛3| 𝒆𝟑 \n𝑹 \n𝑶 \n𝒆𝟑 \na 3D infinitesimal bulk  \n𝑞3 \n𝒓  \n4 \n 4 \n \n \n \n \n \n \n(a) (b) \nFIG.2 . Illustration of the electric field orientations for the spherical surfaces.  Solid lines are the actual electric \nfields , i.e. in (a)  𝑬=𝐸𝑧𝒆𝒛 , in (b)  𝑬=𝐸𝑅𝒆𝑹. Dotted lines denote the orientation normal to the surface, i.e. 𝒆𝟑=\n𝒆𝑹. \nConsider  the case of a spherical surface now where 𝒆𝟑=𝒆𝑹.  In the event that the actual electric field \nis 𝑬=𝐸𝑧𝒆𝒛, the expression  for the electric field normal to the surface is  𝐸S=(𝐸𝑧 𝒆𝒛) .𝒆𝑹 , and that \nwould result in 𝐸𝑆=𝐸𝑧cos𝜃.  In the case where  actual electric field  is given by  𝑬=𝐸𝑅𝒆𝑹 , one \nwould obtain 𝐸𝑆=(𝐸𝑅 𝒆𝑹) .𝒆𝑹=𝐸𝑅.  Therefore, in the event of a general surface  marked by  𝒆𝟑 or \n𝒆𝟑 and an electric field oriented along 𝒆𝟑, the electric field normal to the surface is  \n \n𝐸𝑆=(𝐸3 𝒆𝟑) .𝒆𝟑=𝐸3 \n(8) \nIt thus  becomes clear that  𝐻𝑠𝑜𝑐=𝛼3\nℏ𝝈.(𝒑×𝒆𝟑), with upper index 3 , generalizes the representation s \nfor electric field s orient ed vertical to a surface 𝒆𝟑. The illustration above lends  a clear physical meaning \nto the expressions of 𝑆1=−(𝛼3\nℏ)𝑚𝑒√𝐺 𝜎2 and  𝑆2=(𝛼3\nℏ)𝑚𝑒√𝐺 𝜎1.  With 𝛼3 and hence (𝑆1,𝑆2) \nproperly understood, a general formulation of the spin -orbit coupling on a 2D curved surface  with \nelectric field normal to the surface  have thus been derived in Eq.(5).   In the event that the electric field \nis not restricted to the normal surface, one could derive using Eq.(2) and letting (𝑣,𝑘) run over \ncoordinates (𝑞1,𝑞2,𝑞3) and obtain   \n𝑆1=(𝛼2\nℏ)𝑚𝑒√𝐺 𝜎3−(𝛼3\nℏ)𝑚𝑒√𝐺 𝜎2 \n(9) \n𝑆2=(𝛼3\nℏ)𝑚𝑒√𝐺 𝜎1−(𝛼1\nℏ)𝑚𝑒√𝐺 𝜎3 \n(10) \n𝑆3=(𝛼1\nℏ)𝑚𝑒√𝐺 𝜎2−(𝛼2\nℏ)𝑚𝑒√𝐺 𝜎1 \n(11) \nOne should , however,  not take for granted  that in the event of 𝛼2, 𝒆𝟐 would be normal to 𝒆𝟏 and 𝒆𝟑, \nand likewise for 𝛼1. The orthogonal property of the space metric  continues to be  enforced by 𝒆𝟏×\n𝒆𝟐=√𝐺\n𝐺33 𝒆𝟑 throughout and the metric of the system is  \n[𝐺11𝐺12𝐺13\n𝐺21𝐺22𝐺23\n𝐺31𝐺32𝐺33]=[𝐺11𝐺120\n𝐺21𝐺220\n001] \n (12) \n𝜃 \n𝑬=𝐸𝑧𝒆𝒛 \n 𝑬=𝐸𝑅𝒆𝑹  \n5 \n 5 \nThe spin -orbit Hamiltonian would simply be  \n𝐻𝑠𝑜𝑐=𝑃1\n𝑚𝑒𝑆1+𝑃2\n𝑚𝑒𝑆2+𝑃3\n𝑚𝑒𝑆3 \n(13) \nSPIN -ORBIT TORQUE  \nIn the context of  the spin-orbit torque, the system under consideration would be a hetero structure  \nthat comprises the ferromagnet and the oxide  layers  (we cite the example of Ta \\CoFeB \\MgO [21]), \nwith the f erromagnetic  interface  hosting a collective density of  spin-orbit -induced spin moment \ndenoted by 𝒔. At the same time, the fe rromagnetic material possesses a collective  density of intrinsic  \nmoment known as 𝒎.  Therefore, t he physics of spin -orbit torque arises due to the simultaneous \npresence of the kinetic energy, the spin -orbit energy and the magnetic energy in the system. The ful l \nHamitonian, must now be presented to incorporate th ese energies as follows    \n𝐻=1\n2𝑚𝑒𝑃𝑎𝑃𝑎+1\n2𝑚𝑒(𝑃𝑎𝑆𝑎+𝑆𝑎𝑃𝑎)+𝜎𝑎𝑚𝑎 \n(14) \nFor generality, we let 𝑎 runs over (𝑞1,𝑞2,𝑞3).  In the above, the kinetic energy would be  1\n2𝑚𝑒𝑃𝑎𝑃𝑎=\n−𝑖ℏ\n2𝑚𝑒(𝜕𝑎+1\n2𝜕𝑎ln𝐺)𝑃𝑎. This is because when 𝑃𝑎 is quantized, the covariant property of the \noperator has to be accounted for .  Now, a  minimal coupling form is presented, which  reflect s \nintuitively  the “forceful” physics  of the Hamiltonian. Written in this way, 𝑆𝑎 relates directly to the non-\nAbelian ga uge that has been studied as an origin of  spin forces and phases in many systems [5]. \n𝐻=1\n2𝑚𝑒(𝑃𝑎+𝑆𝑎)(𝑃𝑎+𝑆𝑎)+𝐺𝑎𝑏𝜎𝑎𝑚𝑏 \n(15) \nExpanding the above,  \n𝐻=−𝑖ℏ\n2𝑚𝑒(𝜕𝑎+1\n2𝜕𝑎ln𝐺)𝑃𝑎𝜓−𝑖ℏ\n2𝑚𝑒(𝜕𝑎+1\n2(𝜕𝑎ln𝐺))𝑆𝑎𝜓+1\n2𝑚𝑒𝑆𝑎𝑃𝑎+𝐺𝑎𝑏𝜎𝑎𝑚𝑏 \n(16) \n \nOne can now perform a local gauge transformation in the spin space suc h that the local frame would \nrotate to “some” axis – the choice of which would pretty much determine the physics to be revealed \nfrom those energies.  The locally transformed system is  given by  \n \n𝐻′=−𝑖ℏ\n2𝑚𝑒(∇𝑎+𝑖𝑒\nℏ𝐴𝑎)(𝑃𝑎+𝑒𝐴𝑎)−𝑖ℏ\n2𝑚𝑒(∇𝑎+𝑖𝑒\nℏ𝐴𝑎) 𝑈𝑆𝑎𝑈†−𝑈𝑆𝑎𝑈†𝑖ℏ\n2𝑚𝑒(𝜕𝑎+𝑖𝑒\nℏ𝐴𝑎)\n+𝐺𝑎𝑏𝑈𝜎𝑎𝑚𝑏𝑈† \n(17) \nwhere ∇𝑎=𝜕𝑎+1\n2𝜕𝑎ln𝐺≡𝜕𝑎+𝛤𝑎 and 𝐴𝑎=−𝑖ℏ\n𝑒𝑈(𝜕𝑎𝑈†) is a gauge potential related to the \nmagnetization and the curved geometry of the device.   Note that the transformed Hamiltonian is still \ngeneral in the sense that as of now, no decision has been taken as to whi ch axis the local frame should \nrotate to. Therefore, the physics that one would like to “see” lies in this important decision, i.e. the \nchoice of the transformation operator 𝑈. And in this paper, s pin-orbit torque  physics would become \napparent in a  frame rotation that rotates the 𝑍 axis to the magnetizatio n axis 𝒆𝒎 as follows  \n \n𝑈𝜂𝑚=𝜂𝑧    ,𝑈𝜎𝑚𝑈†=𝜎𝑧 \n(18) \nwhere 𝜂𝑚 and 𝜂𝑧 are respectively, the eigenstate s along 𝒆𝒎 and 𝒆𝒛.  And 𝑈(𝜃𝑚,𝜙𝑚) operates in the \nspace of the magnetic moment 𝒎. The frame rotation takes place  in the presence of spin -orbit \ncoupling .  Upon transformation, local gauge potentials would appear in the Hamiltonian. Rewriting \nthe Hamiltonian , one has   \n6 \n 6 \n𝐻′=(1\n2𝑚𝑒)(−𝑖ℏ∇𝑎+𝑒𝐴𝑎+𝑈𝑆𝑎𝑈†)(−𝑖ℏ𝜕𝑎+𝑒𝐴𝑎+𝑈𝑆𝑎𝑈†)+𝐺𝑎𝑏𝑈𝜎𝑎𝑚𝑏𝑈† \n(19) \nThe physics of curve, spin -orbit coupling, and magnetism is now captured in the expression of a total \ngauge: ℚ𝑎=𝑒𝐴𝑎+𝑈𝑆𝑎𝑈†. In fact, the rotation gauge 𝑒𝐴𝑎 has previously been associate d with a \nform of adiabatic spin -tansfer torque  [34]  that  w ould not be further elaborated in this article. As  our \ninterest lies in the spin -orbit torque , we will only focus on the transformed spin -orbit gauge also \nknown henceforth as ℱ𝑎=𝑈𝑆𝑎𝑈†.  Relevant to our study are the energy terms as follows : \n \n𝐸=𝜓†(1\n2𝑚𝑒)(−𝑖ℏ∇𝑎+ℱ𝑎)(−𝑖ℏ𝜕𝑎+ℱ𝑎)𝜓 \n(20) \nDropping the  second -order derivative kinetic energy terms,    \n𝐸𝑖𝑛𝑡=(−𝑖ℏ\n2𝑚𝑒)𝜓†[∇𝑎 ℱ𝑎 +ℱ𝑎 𝜕𝑎] 𝜓 \n(21) \nNow, we will examine each energy density (∇𝑎 ℱ𝑎,ℱ𝑎 𝜕𝑎 ) term in details:  \n∇𝑎 ℱ𝑎  →   (−𝑖ℏ\n2𝑚𝑒)[∂𝑎(𝜓†ℱ𝑎𝜓)−(∂𝑎𝜓†)ℱ𝑎𝜓+𝛤𝑎 (𝜓† ℱ𝑎 𝜓)] \n(22) \nℱ𝑎 𝜕𝑎  →   (−𝑖ℏ\n2𝑚𝑒)𝜓†ℱ𝑎 𝜕𝑎 𝜓 \n    (23) \nThe expression ∇𝑎 ℱ𝑎 compri ses three  terms on the RHS. It is reduced to (−𝑖ℏ\n2𝑚𝑒)[−(∂𝑎𝜓†)ℱ𝑎𝜓] when  \nthe first and the third term vanish as surface terms as the energy density is integrated over the entire \ndevice  \n√𝐺∫[∂𝑎+𝛤𝑎](𝜓†ℱ𝑎𝜓)  𝑑𝑉=√𝐺∫(𝜓†ℱ𝑎𝜓).𝑑𝑆=0 \n(24) \nIt suffices  now  to proceed with the remaining terms which combine to make physical sense  in terms \nof 𝑗𝑎 taking on the physical meaning of current density  as follows,  \n(−𝑖ℏ\n2𝑚)[ ℱ𝑎 𝜓†𝜕𝑎𝜓−(𝜕𝑎𝜓†) ℱ𝑎 𝜓 ]  ↔  𝑗𝑎ℱ𝑎 \n(25) \nThe infinitesimal bulk is then compressed along 𝒆𝟑, i.e. taking the operation of 𝑞3→0 where \n(√𝐺)𝑞3→0=√𝑔.   As the Z axis is rotated to the magnetic moment  (𝒎), and s pin is assumed to align \nwith 𝒎 in an adiabat ic manner  and at all times (even as 𝒎 changes spatially) as it propagates, the \neigenspinor of the electron would undergo 𝜂𝑚→𝜂𝑧.  Refering to the formulation [35] for the volume \nand surface integrals,   \n∫𝜓† 𝜓 𝑑𝑉=∫𝜓† 𝜓   𝑑𝑞1 𝑑𝑞2 𝑑𝑞3 √𝐺 \n(26) \nWith 𝑑𝑆=√𝑔 𝑑𝑞1𝑑𝑞2 \n∫𝜓† 𝜓 𝑑𝑉=∫𝜓† 𝜓 𝑓 𝑑𝑆 𝑑𝑞3 \n(27)  \n7 \n 7 \nFrom the above, 𝑑𝑉=𝑓 𝑑𝑆 𝑑𝑞3. Now, writing ∫𝜓† 𝜓 𝑑𝑉=∫ 𝜒†𝜒   𝑑𝑆 𝑑𝑞3 means the wave -\nfunctions are  taken to be  𝜓=𝜒 𝜂𝑧\n√𝑓 ,𝜓†=𝜂𝑧† 𝜒†\n√𝑓  and √𝑓=√𝐺\n√𝑔. Note that 𝜒=𝜒𝑠𝜒𝑛 are separable \nscalar function s where  𝜒𝑠 is the surface wave -function, and 𝜒𝑛 is normal to the surface. The energy \nterms can now be written as follows : \n 𝜓†ℱ𝑎𝜕𝑎𝜓=(𝜂𝑧†  𝜒†\n𝑓 ℱ𝑎 (𝜕𝑎𝜒) 𝜂𝑧+𝜂𝑧†  𝜒†\n√𝑓 ℱ𝑎 𝜒 𝜂𝑧 ( 𝜕𝑎 𝑓−1\n2 )) \n(28) \n−(𝜕𝑎𝜓†) ℱ𝑎 𝜓=( −(𝜕𝑎 𝜂𝑧†  𝜒†)\n𝑓 ℱ𝑎 𝜒 𝜂𝑧 −𝜂𝑧†  𝜒†\n√𝑓(𝜕𝑎 𝑓−1\n2)ℱ𝑎 𝜒 𝜂𝑧) \n(29) \nCombining the above,  \n𝜓†ℱ𝑎𝜕𝑎𝜓−(𝜕𝑎𝜓†) ℱ𝑎 𝜓= 1\n𝑓 ⟨𝜂𝑧|ℱ𝑎  𝐽𝑎 |𝜂𝑧⟩ \n(30) \nwhere 𝐽𝑎=𝜒†𝜕𝑎𝜒−(𝜕𝑎𝜒†) 𝜒 is now t he current density. Since  ℱ𝑎 orginates from the spin -orbit \ncoupling in a curved space, the entire term 𝑗𝑎ℱ𝑎 would be an interaction energy density (𝐸𝑖𝑛𝑡) related \nto the coupling of the current flow  (𝑗𝑎) with the curved spin -orbit physics . As 𝜒=𝜒𝑠(𝑞1,𝑞2) 𝜒𝑛(𝑞3) , \nthe term 𝑗𝑎ℱ𝑎 is decou pled into \n𝑗𝑎ℱ𝑎=(𝑗1ℱ1+𝑗2ℱ2)+𝑗3ℱ3 \n(31) \nwhere 𝑗1ℱ1+𝑗2ℱ2 is the surface current den sity and 𝑗3ℱ3 the normal current density.  Recall  that \nℱ𝑎=𝑈𝑆𝑎𝑈†, and 𝑆𝑎 is summarized below  \n(𝑆1\n𝑆2\n𝑆3)=𝑚𝑒√𝐺\nℏ(0−𝛼3𝛼2\n𝛼30−𝛼1\n−𝛼2𝛼10)(𝜎1\n𝜎2\n𝜎3) \n(32) \n \nTo confine the spin -orbit effect to the 2D which is prevalent in physical systems like the Rashba , 2D \nDres sellhaus , topological surface states, 2D graphene and silicene,  one takes t he limit of 𝑞3→0 and \nnotes that 𝑓→1. It’s worth noting that the term s 𝜂𝑧† 𝜒†\n√𝑓 ( 𝜕𝑎 𝑓−1\n2 )ℱ𝑎 𝜒 𝜂𝑧  would vanish for 𝑎=1,2 \nas limit 𝑞3→0 is taken. The surviving term 𝜂𝑧† 𝜒†\n√𝑓 ( 𝜕3 𝑓−1\n2 )ℱ3 𝜒 𝜂𝑧 would be expected to capture the \nphysics of the confinement effect . But as steps have been taken to ensure Hermiticty in spin-orbit \ncoupling Eq.(14), the confinement terms of Eq.(28) and Eq.(29) cancel one another.  This is an \nunexpected effect for the spin -orbit torque tha t eliminates a curved -surface confinement effect \nbecause of the symmetrization in the current  density .  With the limit taken, the  normal current density \nis discarded and one should from now on ignore the effect of 𝑆3. We will proceed to the spin-orbit \nconstant which is given by 𝛼𝑛=𝛼′𝐸𝑛. We will now let  𝐸1 and 𝐸2 approach zero, so that only  𝛼3 is \nretained . The matrix is reduced back the 2D formalism like in the beginning of the article,  \n \n(𝑆1\n𝑆2)=𝑚𝑒√𝑔\nℏ(0−𝛼3\n𝛼30)(𝜎1\n𝜎2) \n(33) \n \nIt’s worth noting that there’s a caveat in the limit taking, i.e. we have let 𝑞3→0 first so that the \nsurface confinement term 𝜂𝑧† 𝜒†\n√𝑓 ( 𝜕3 𝑓−1\n2 )ℱ3 𝜒 𝜂𝑧 survived although it would vanish  later because of  \n8 \n 8 \nsymmetrization. Had the limit of 𝛼1→0,𝛼2→0 been taken first, the confinement term would not \nhave show n up. Therefore, t he general spin-orbit gauge  potential s can now be  expressed as follows:  \n \nℱ1=−𝑈(𝛼3 𝑚𝑒\nℏ√𝑔𝜎2)𝑈† ,    ℱ2=𝑈(𝛼3 𝑚𝑒\nℏ√𝑔𝜎1)𝑈†  \n (34) \nFinally, the energy density  is \n𝐸𝑖𝑛𝑡=𝑗𝑎ℱ𝑎=(𝛼3𝑚 \n𝑒ℏ√𝑔)[−𝑗1 𝑈𝜎2𝑈†+𝑗2 𝑈𝜎1𝑈†] \n(35) \nThe energy density in magnetic space is an important development for the study of magnetic torque \nand flipping via the concept of the anisotropy field.   We consider a potential in the magnetic space to \narise fr om the energy density 𝐸𝑖𝑛𝑡. The potential is essentially the effective anisotropy field as it is \ncommonly underst ood in magnetic physics:  \n𝜇 𝑯=𝜕𝐸𝑖𝑛𝑡\n𝜕𝒎=𝜕\n𝜕𝒎 (𝑗𝑎ℱ𝑎 \n𝑒) \n(36) \nThe explicit expression of  𝜇 𝑯 is therefore , given as  \n \n𝜇 𝑯=(𝛼3 𝑚𝑒\nℏ√𝑔)[ −𝑗1\n𝑒(𝜕\n𝜕𝒎𝑈𝜎2𝑈†)+𝑗2\n𝑒(𝜕\n𝜕𝒎 𝑈𝜎1𝑈†)] \n(37) \nThe magnetic moment  (𝒎) of the ferromagnetic material  is superimposed on the  curved surface as \nshown in Fig.3. below.  The curved surface can be accessed at every point  in space  by 𝑹=𝒓(𝑞1,𝑞2)+\n𝑞3𝒆𝟑 that charcaterizes the surface on which  𝒎 is located.  Vector  𝒎 will also be  indexed by (𝑚1,𝑚2) \nto reflect its components in the general coordinates of the real space . In that way, the magnetic \ncomponents  will also track the structure of the space  in which it is superimposed . The actual \norientation of 𝒎 is described by (𝜃𝑚,𝜙𝑚,𝑚) with respect to the Cartesian coordinates , where \nsuperscripts of (𝜃𝑚,𝜙𝑚) merely show that these are coordinates for the magnetization.   \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFIG.3 A 2D curve is embeded in a 3D space  and characterized by parameters (𝑞1,𝑞2,𝑞3). The magnetization 𝒎 \nis superimposed on the curved surface.  \n \n𝒆𝟑 \n𝒆𝟐 \n𝒆𝟏 \n𝒎=𝑚𝒆𝒎 \n𝒆𝒙 \n𝒆𝒚 \n𝒆𝒛 \n𝜃 \n𝒓(𝑞1,𝑞2) \n𝒆𝒙 \n𝒆𝒚 \n𝒆𝒛 \n𝜃𝑚  \n9 \n 9 \nWe will now look into the spin physics and focus  our attention  on the Pauli matrices. Originally \nexpressed in the genera l corodinates, the Pauli matrices are now re-expressed in the Cartesian \ncoordinates in which the adiabatic physics will be introduced  at a later stage . In the physics of forces, \nand magnetism, the important physical quantities are: magnetic moment (𝒎), Pauli matrix (𝜎), \ncurrent density (𝑗). There are frame of references under which the physical quantities most relevant \nto our studies are described, e.g. the Cartesian (𝑥,𝑦,𝑧), and the geneal coordinates (𝑞1,𝑞2,𝑞3). \nPhysical quantities and their dimensions  are expressed in  both frames. For example, 𝒎=𝑚1𝒆𝟏+\n𝑚2𝒆𝟐=𝑚𝑣∂𝑣𝑎𝒆𝒂 , implies that a physical quantity remains unchanged when expressed under any \nframe of reference . Explicitly, it will look as follows  \n \n𝒎=𝑚𝑥𝒆𝒙+𝑚𝑦𝒆𝒚+𝑚𝑧𝒆𝒛=𝑚1𝒆𝟏+𝑚2𝒆𝟐+𝑚3𝒆𝟑 \n(38) \nThe two are connected by  the transformaton as follows   \n𝒎=(𝑚1∂1𝑥+𝑚2∂2𝑥+𝑚3∂3𝑥)𝒆𝒙+(𝑚1∂1𝑦+𝑚2∂2𝑦+𝑚3∂3𝑦)𝒆𝒚+(𝑚1∂1𝑧+𝑚2∂2𝑧+𝑚3∂3𝑧)𝒆𝒛 \n(39) \nThe above provides an illustration of coordinate transformatio n. In the actual context, the same \nprinciple is applied to the Pauli matrices to reappear in the Cartesian frame. As a result, t he effective \nanisotropy field would now be,  \n \n𝜇 𝜹𝑯=(𝛼3𝑚𝑒 \nℏ𝑚√𝑔)[−𝑗1\n𝑒𝜕\n𝜕𝒏(∂𝑣2𝑈𝜎𝑣𝑈†)+𝑗2\n𝑒𝜕\n𝜕𝒏(∂𝑣1𝑈𝜎𝑣𝑈†)] \n(40) \nwhere 𝑣 runs over  the Cartesian  (𝑥,𝑦,𝑧).  In the real space, we note once again that the Pauli matrices  \nin the general coordinates have been re -expressed i n the Cartesian  frame  whereby the  curve  effects \nwould be reflected in the quantities of ∂𝑣2 and ∂𝑣1. On the other hand, t he unitary operator 𝑈  is now \napplied to rotate the magnetic axis (𝒆𝒎) to the (𝑥,𝑦,𝑧) frame , thereby providing the 𝒎 an indirect  \nlink in terms of orientation to the general coordinates. It is now possible to  associate the Pauli matices \nwith  𝒎 and perform the operation of 𝜕\n𝜕𝒏 for the anisotropy field  as required in Eq.(38 ).  In the following, \nwe provide an explicit demonstration of the 𝑈 operation.   Recall that 𝑈 is parameteri zed by (𝜃𝑚,𝜙𝑚). \nRefer to Fig.4  below and observe  that 𝑈 rotates the eigenstate along 𝒆𝒎 to the Z  axis about axis 𝒆𝒎𝟐=\n−𝒆𝒚cos𝜙𝑚+𝒆𝒙sin𝜙𝑚.   \n𝑈=(cos𝜃𝑚\n2sin𝜃𝑚\n2𝑒−𝑖𝜙𝑚\n−sin𝜃𝑚\n2𝑒𝑖𝜙𝑚cos𝜃𝑚\n2) \n(41) \nRotation of the magnetic axis (𝒆𝒎) to the (𝑥,𝑦,𝑧) frame  is given by  \n \n𝑈𝝈𝑈†=𝜎𝑧𝒆𝒎+𝜎𝑎𝒆𝒎𝟏+𝜎𝑚2𝒆𝒎𝟐 \n(42) \nRefer to Fig.4 for the d irections implied by the superscripts and subscripts in the equation above. Note \nthat the 𝒎 physical quantities of (𝒆𝒎,𝒆𝒎𝟏,𝒆𝒎𝟐) and (𝜎𝑎,𝜎𝑚2) can all be related to the (𝑥,𝑦,𝑧) \nframe  as follows  \n \n𝒆𝒎=𝒆𝒙sin𝜃𝑚cos𝜙𝑚+𝒆𝒚sin𝜃𝑚sin𝜙𝑚+𝒆𝒛cos𝜃𝑚 \n=𝑛𝑥𝒆𝒙+𝑛𝑦𝒆𝒚+𝑛𝑧𝒆𝒛  \n10 \n 10 \n(43) \n𝒆𝒎𝟏=−𝒆𝒚cos𝜃𝑚sin𝜙𝑚−𝒆𝒙cos𝜃𝑚cos𝜙𝑚+𝒆𝒛sin𝜃𝑚 \n(44) \n𝒆𝒎𝟐=−𝒆𝒚cos𝜙𝑚+𝒆𝒙sin𝜙𝑚 \n(45) \n𝜎𝑎=−𝜎𝑥cos𝜙𝑚−𝜎𝑦sin𝜙𝑚 \n(46) \n𝜎𝑚2=𝜎𝑥sin𝜙𝑚−𝜎𝑦cos𝜙𝑚 \n(47) \n \nThe following is a schematic showing the magnetic moment  vectors in  the Cartesian frame.   \n \n \n \n \n \n \n \nFIG.4. Spin rotation about 𝒆𝒎𝟐 moves the Z axis  to the magnetic moment  axis 𝒆𝒎, and the  𝒆𝒂 axis to  𝒆𝒎𝟏 or vice \nversa.  \nFollowing the mark  of 𝑈𝜎𝑣𝑈† in Eq.(40 ) where 𝑣 runs over (𝑥,𝑦,𝑧), it is handy to write down the \nfollowing for  the sake of easy inspection.  \n \n𝑈𝝈𝑈†＝𝑈𝜎𝑥𝑈†𝒆𝒙+𝑈𝜎𝑦𝑈†𝒆𝒚+𝑈𝜎𝑧𝑈†𝒆𝒛 \n  (48) \n \nUsing E qs. (42) to (47), the process of linking the original magnetic axis through unitary rotation to the \n(𝑥,𝑦,𝑧) frame is complete, where  \n \n𝑈𝜎𝑥𝑈†=sin𝜃𝑚cos𝜙𝑚 𝜎𝑧+𝑓(𝜎𝑎,𝜎m2) \n(49) \n \n𝑈𝜎𝑦𝑈†=sin𝜃𝑚sin𝜙𝑚 𝜎𝑧+𝑔(𝜎𝑎,𝜎m2) \n(50) \n \n𝑈𝜎𝑧𝑈†=cos𝜃𝑚 𝜎𝑧+ℎ(𝜎𝑎) \n(51) \n \nNote that 𝑓,𝑔,ℎ are linear combination s of 𝜎𝑎,𝜎𝑚2 and these terms are non -issue s as they would \nvanish later . At this point, an important physical step is taken, i.e. the adiabatic approximation in which \nit is assumed that electron spin aligns along the new Z axis that has been rotated  to 𝒆𝒎. Under the \nadiabatic spin alignment whereby  the following is resulted:  ⟨𝜂𝑧|𝜎𝑥|𝜂𝑧⟩=0 ,⟨𝜂𝑧|𝜎𝑦|𝜂𝑧⟩=0,\n⟨𝜂𝑧|𝜎𝑧|𝜂𝑧⟩=1, functions 𝑓,𝑔,ℎ vanish while the following remains  \n \n⟨𝜂𝑧|𝑈𝜎𝑥𝑈†|𝜂𝑧⟩=𝑛𝑥 ,⟨𝜂𝑧|𝑈𝜎𝑦𝑈†|𝜂𝑧⟩=𝑛𝑦,⟨𝜂𝑧|𝑈𝜎𝑧𝑈†|𝜂𝑧⟩=𝑛𝑧 \n𝑥 \n𝑦 \n𝑧 \n𝒆𝒎𝟐=−𝒆𝒚cos𝜙𝑚+𝒆𝒙sin𝜙𝑚 \n𝜃𝑚 \n𝜙𝑚 \n𝒎 \n𝜙𝑚 \n𝒆𝒎 \n 𝒆𝒎𝟏 \n𝒆𝒂  \n11 \n 11 \n(52) \nIt thus follows that  \n⟨𝜂𝑧|𝜇 𝜹𝑯|𝜂𝑧⟩=(𝛼3𝑚 \nℏ𝑀𝑠√𝑔)[−𝑗1\n𝑒𝜕\n𝜕𝒏(∂𝑣2𝑛𝑣)+𝑗2\n𝑒𝜕\n𝜕𝒏(∂𝑣1𝑛𝑣)] \n=(𝛼3𝑚 \n𝑒ℏ𝑀𝑠√𝑔)[−𝑗1(∂𝑣2𝒆𝒗)+𝑗2(∂𝑣1𝒆𝒗)] \n(53) \nNote in the above that 𝑛𝑣 carries a contravariant index.  But, 𝑛𝑎 of 𝜕\n𝜕𝒏=𝜕\n𝜕𝑛𝑎𝒆𝒂 is covariant .  As a result,  \n⟨𝜂𝑧|𝜇 𝜹𝑯|𝜂𝑧⟩=(𝛼3𝑚 \n𝑒ℏ𝑀𝑠√𝑔)[−𝑗1𝒆𝟐+𝑗2𝒆𝟏] \n(54) \n \nEquation (54 ) is the  general form of what we called the spin-orbit anisotropy  field . This is another \nimportant result of this paper . Note that (𝑗1,𝑗2) and (𝒆𝟏,𝒆𝟐) can be found by transforming their \nCartesian counterparts to the desired coordinates in a contravariant manner. Such formulation is \nimportant because as device becomes smaller, the physics of local geometries on spin -orbit effect and \nits torque shall not be neglected.  The general formulation allow s spin and magnetic dynamic to be \nstudied o n local curved surfaces. With the spin -orbit anisotropy field derived, the spin -orbit torque is \na straightforward  cross product with the 𝒎  \n𝝉=−𝒎×⟨𝑧|𝜇 𝜹𝑯|𝑧⟩ \n(55) \nAs the formulation is provided in general coordinates, the spin -orbit effective field can be derived for \nany surfaces to be characterized by 𝑹=𝒓′(𝑞1,𝑞2)+𝑞3𝒆𝟑. For illustration, w e will now take the \nexample s of the spherical, cylindrical and flat Cartesian surfaces. In the spherical system , the \ncoordinates are  𝑹=𝒓′(𝜃,𝜙)+𝛿𝑟𝒆𝒓 and  \n⟨𝑧|𝜇 𝜹𝑯|𝑧⟩=(𝛼3𝑚 \n𝑒ℏ𝑀𝑠√𝑔)(−𝜕𝜃\n𝜕𝑣𝜕𝜙\n𝜕𝑣′+𝜕𝜙\n𝜕𝑣𝜕𝜃\n𝜕𝑣′) 𝑗𝑣𝒆𝒗′ \n(56) \nThe current carrying carriers are c onstrained  to the surface , i.e. 𝛿𝑟=0.  The spherical surface is  then  \naccessed by  𝒓′=(𝑟sin𝜃cos𝜙,𝑟sin𝜃sin𝜙 ,𝑟cos𝜃) \n \n( 𝒆𝒙\n𝒆𝒚\n𝒆𝒛)=\n(    𝜕𝜃\n𝜕𝑥𝜕𝜙\n𝜕𝑥𝜕𝑟\n𝜕𝑥\n𝜕𝜃\n𝜕𝑦𝜕𝜙\n𝜕𝑦𝜕𝑟\n𝜕𝑦\n𝜕𝜃\n𝜕𝑧𝜕𝜙\n𝜕𝑧𝜕𝑟\n𝜕𝑧)    \n(𝒆𝜽\n𝒆𝝓\n𝒆𝑹)=\n(   cos𝜙cos𝜃\n𝑟sin𝜙\n−𝑟sin𝜃sin𝜃cos𝜙\nsin𝜙cos𝜃\n𝑟cos𝜙\n𝑟sin𝜃sin𝜃sin𝜙\n−sin𝜃\n𝑟0 cos𝜃)   \n(𝒆𝜽\n𝒆𝝓\n𝒆𝒓) \n(57) \n \nNote that 𝑣,𝑣′ run over (𝑥,𝑦,𝑧) and summation is only non -vanishing for 𝑣≠𝑣′. Meanwhile,  𝑔=\n√(𝑔𝜃𝜃𝑔𝜙𝜙−𝑔𝜃𝜙𝑔𝜙𝜃)𝑔𝑟𝑟=𝑟2sin𝜃.   \n⟨𝑧|𝜇 𝜹𝑯|𝑧⟩=(𝛼3𝑚 \n𝑒ℏ𝑀𝑠√𝑔)(1\n𝑟2tan𝜃(−𝑗𝑥𝒆𝒚+𝑗𝑦𝒆𝒙)+sin𝜙\n𝑟2(𝑗𝑥𝒆𝒛−𝑗𝑧𝒆𝒙)+cos𝜙\n𝑟2(𝑗𝑧𝒆𝒚−𝑗𝑦𝒆𝒛))  \n12 \n 12 \n=(𝛼3𝑚 \n𝑒ℏ𝑀𝑠)(cos𝜃(−𝑗𝑥𝒆𝒚+𝑗𝑦𝒆𝒙)+sin𝜃sin𝜙(𝑗𝑥𝒆𝒛−𝑗𝑧𝒆𝒙)+sin𝜃cos𝜙(𝑗𝑧𝒆𝒚−𝑗𝑦𝒆𝒛)) \n (58) \nWith the above, the spin -orbit anisotropy field can be estimated over the entire spherical surface \nprovided (𝑗𝑥,𝑗𝑦,𝑗𝑧) is computed or measured over the surface. Likewise, with (𝑚𝑥,𝑚𝑦,𝑚𝑧) \ncomputed or measured  over the surface , the spin -orbit torque can be determined at every point of \nthe surface.  Figure 5 illustrates how current flows into and out of the nanoscale structures. D evice s’ \ncentral region can be fabri cated to these structures and one can regard the current to flow from source \nto drain.  \n \n \n \n \n \n \n \n(a) (b) \n \nFIG.5 (a) A spherical surface across which current flows from left to right as indicated by arrows. (b) A cylindrical \nsurface across which current flows along the z direction as shown by the arrow.  \nIn the  cylindrical s ystem , the coordinates are 𝑹=𝒓′(𝜙,𝑧)+𝛿𝑟𝒆𝒓.  The current carrying carriers are \nconstrained to the surface, i.e. 𝛿𝑟=0. The cylindrical  surface is then accessed by  𝒓′=\n(𝑟cos𝜙,𝑟sin𝜙 ,𝑧).  \n( 𝒆𝒙\n𝒆𝒚\n𝒆𝒛)=\n(    𝜕𝜙\n𝜕𝑥𝜕𝑧\n𝜕𝑥𝜕𝑟\n𝜕𝑥\n𝜕𝜙\n𝜕𝑦𝜕𝑧\n𝜕𝑦𝜕𝑟\n𝜕𝑦\n𝜕𝜙\n𝜕𝑧𝜕𝑧\n𝜕𝑧𝜕𝑟\n𝜕𝑧)    \n(𝒆𝝓\n𝒆𝒛\n𝒆𝒓)=\n(  −sin𝜙\n𝑟0cos𝜙\n+cos𝜙\n𝑟0sin𝜙\n010)  (𝒆𝝓\n𝒆𝒛\n𝒆𝒓) \n(59) \nIt follows that  \n⟨𝑧|𝜇 𝜹𝑯|𝑧⟩=(𝛼3𝑚 \n𝑒ℏ𝑀𝑠√𝑔)(sin𝜙\n𝑟(𝑗𝑥𝒆𝒛−𝑗𝑧𝒆𝒙)+cos𝜙\n𝑟(𝑗𝑧𝒆𝒚−𝑗𝑦𝒆𝒛)) \n(60) \nNote that 𝑣,𝑣′ run over (𝑥,𝑦,𝑧) and summation is only non -vanishing for 𝑣≠𝑣′. In the meantime ,  \n𝑔=√(𝑔𝜙𝜙𝑔𝑧𝑧−𝑔𝜙𝑧𝑔𝑧𝜙)𝑔𝑟𝑟=𝑟.  Thus,  \n⟨𝑧|𝜇 𝜹𝑯|𝑧⟩=(𝛼3𝑚 \n𝑒ℏ𝑀𝑠)(sin𝜙(𝑗𝑥𝒆𝒛−𝑗𝑧𝒆𝒙)+cos𝜙(𝑗𝑧𝒆𝒚−𝑗𝑦𝒆𝒛)) \n(61) \n𝑥 \n𝑧 \n𝑦 \n𝜃 \n𝜙 \n𝑧 \n𝑥 \n𝑦 \n𝜙 \n𝑟 \n𝑧  \n13 \n 13 \n \n \n \nLast, i n the Cartesian  system , physical  quantities are re-expressed  in the x-y basis. The results are  \n⟨𝑧|𝜇 𝜹𝑯|𝑧⟩=(𝛼𝑧𝑚 \n𝑒ℏ𝑀𝑠)[−𝑗𝑥𝒆𝒚+𝑗𝑦𝒆𝒙] \n(62) \n \nThe spin -orbit anisotropy field above when substituted in the standard expression for spin torque  in a \nflat surface device , would lead to the to the well -known, experimentally proven [20, 21 ] spin-orbit \ntorque , also known to the experimental community as the field -like spin -orbit torque. This is, however, \nnot the same as another form of spin-orbit torque aka the anti -damping spin -orbit torque [22]. In \nother words, what we have derived is the general formulation for the field -like spin -orbit spin torque.  \nCONCLUSION  \nWe have provided a general formulation of the spin -orbit coupling on a curved surface  in Eq.(5)  with \n𝛼3 and hence (𝑆1,𝑆2) properly defined and understood. We had then consider ed a curved \nferromagnetic and oxide heterostructure to give rise to a spin -orbit torque. A proper choice of the \ntransformation operator 𝑈 that rotates the Z axis to the magnetization axis 𝒆𝒎 leads to the derivation \nof the spin -orbit anisotropy field  and hence the spi n-orbit torque . We note that an unexpected effect  \nthat arises in the symmetriation of the current density  actually eliminates the  curved -surface \nconfinement.  Finally, with the adia batic approximation, we completed the general formluation of the \nspin-orbit anisotropy field  in Eq.(54 ) and hence the spin-orbit torque that can be computed over the \nentire surface of devices of any shape . We provided examples in  spherical, cylindrical and  the \nCartesian surfaces  \n \nAcknowledgments  \nWe would like to thank the National Science and Technology Council of Taiwan for supporting this \nwork under Grant No. 110 -2112 -M-034-001-MY3.  \n \nORCID iD  \nSeng Ghee Tan \n  https://orcid.org/0000 -0002 -3233 -9207  \n \nReferences  \n[1] Igor Žutić, Jaroslav Fabian, and S. Das Sarma, Rev. Mod. Phys. 76, 323 (2004).  \n[2] T. Fujita, M. B. A. Jalil, S. G. Tan, Shuichi Murakami, J. Appl. Phys. [Appl. Phys. Rev.] 110, 1213 01 \n(2011); S. G. Tan, M. B. A. Jalil, Xiong -Jun Liu, and T. Fujita, Phys. Rev. B 78, 245321 (2008).  \n[3] Seng Ghee Tan, Mansoor B.A. Jalil, Introduction to the Physics of Nanoelectronics, Woodhead \nPublishing, Cambridge (Elsevier), U.K.  (2012).  \n[4] Gen Tatara , Physica E: Low -dimensi onal Systems and Nanostructures 106, 208 ( 2019 ).  \n[5] Seng Ghee  Tan, Son -Hsien  Chen, Cong Son Ho, Che -Chun Huang, Mansoor B.A. Jalil, Ching Ray \nChang, Shuichi Murakami, Physics Reports 882, Pages  1-36 (2020).  \n[6] A. H. Castro Neto , F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, Rev. Mod. Phys. 81, \n109 (2009).  \n[7] M.A.H. Vozmediano, M.I.  Katsnelson, F.  Guinea, Physics Reports 496, Pages 109 -148 (2010).  \n[8] M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045 (2010).   \n[9] X.-L. Qi and S. -C. Zhang, Rev. Mod. Phys. 83 1057 (2011).  \n[10] Shengyuan A. Yang,  SPIN 6, 1640003 (2016) . \n[11] N.P. Armitage, E.J. Mele, and Ashvin Vishwanath , Rev. Mod. Phys. 90, 015001  (2018).   \n14 \n 14 \n[12] Jean Dalibard, Fabrice Gerbier, Gediminas Juzeliūnas, and Patrik Öhberg, Rev. Mod. Phys. 83, 1523 \n(2011).  \n[13] Ryszard Horodecki, Paweł Horodecki, Michał Horodecki, and Karol Horodecki , Rev. Mod. Phys. 81, \n865 (2009) . \n[14] Che -Chun Huang , Seng Ghee Tan , Ching -Ray Chang,  New J. Phys. 25 063026  (2023).  \n[15] S. Zhang and Z. Li, Phys. Rev. Lett. 93, 127204 (2004).  \n[16] S. G. Tan, M. B. A. Jalil, and Xiong -Jun Liu, arXiv:0705.3502, (2007).  \n[17] K. Obata and G. Tatara, Phys. Rev. B 77, 214429 (2008).  \n[18] A. Manchon, S. Zhang, Phys. Rev. B 78, 2124 05 (2008).   \n[19] S. G. Tan, M. B. A. Jalil, T. Fujita, and X. J. Liu, Ann. Phys. (NY) 326, 207 (2011).  \n[20] Ioan Mihai Miron, Gilles Gaudin, Stéphane Auffret, Bernard Rodmacq, Alain Schuh l, Stefania \nPizzini, Jan Vogel, Pietro Gambardella Nature Materials 9, 230 (2010) . \n[21] Junyeon Kim, Jaivardhan Sinha, Masamitsu Hayashi, Michihiko Yamanouchi, Shunsuke Fukami, \nTetsuhiro Suzuki, Seiji Mitani , Hideo Ohno , Nature Materials 12, 240 (2013) . \n[22] H. Kurebayashi, Jairo Sinova, D. Fang, A. C. Irvine, T. D. Skinner , J. Wunderlich, V. Novák, R. P. \nCampion, B. L. Gallagher, E. K. Vehstedt, L. P. Zârbo, K. Výborný, A. J. Ferguson & T. Jungwirth, Nature \nNanotechnology 9, 211 (2014).  \n[23] Yositake Takane, Ken -Ichiro Imura, J. Phys. Soc. Japan 82, 074712 (2013).  \n[24] Ken -Ichiro Imura, Yositake Takane, and Akihiro Tanaka, Phys. Rev. B 84, 195406 (2011).  \n[25] V. Parente, P. Lucignano, P. Vitale, A. Tagliacozzo, and F. Guinea, Phys. Rev. B 83, 075424 (2011).  \n[26] Dung -Hai Lee, Phys. Rev. Letts. 103, 196804 (2009) . \n[27] Paola Gentile, Mario Cuoco, Carmine, Phys. Rev. Letts. 115, 256801 (2015).  \n[28] Zhuo Bin Siu, Jian -Yuan Chang, Seng Ghee Tan, Mansoor B. A. Jalil, Ching -Ray Chang, Sci. Reps. 8, \n16497 (2018).  \n[29] Tai -Chung Cheng, Jiann -Yeu Chen, Ching -Ray Chang, Phys. R ev. B 84, 214423 (2011).  \n[30] Ming -Hao Liu, Jhih -Sheng Wu, Son -Hsien Chen, and Ching -Ray Chang, Phys. Rev. B 84, 085307 \n(2011).  \n[31] Z.B. Siu, M.B.A. Jalil, S G Tan, J. Appl. Phys. 115 17C513 (2014).  \n[32] Zhuo Bin Siu, Mansoor B. A. Jalil, and Seng Ghee T an, J. Appl. Phys. 121, 233902 (201 4). \n[33] Zhuo B in Siu, Mansoor B . A. Jalil, and Seng Ghee Tan, Nanotechnology 33, 335703  (2022).  \n[34] Ya. B. Bazaliy, B. A. Jones, Shou -Cheng Zhang, Phys. Rev. B 57, R3213 (1998).  \n[35] R. C. T. da Costa, Phys. Rev. A 23, (1982).  \n "    },    {        "title": "0707.4493v1.Temperature_Dependence_of_Rashba_Spin_orbit_Coupling_in_Quantum_Wells.pdf",        "content": " 1 Temperature Dependence of Rashba Spin-orbit Coupling in Quantum Wells \n \n \nP.S. Eldridge1, W.J.H. Leyland2, P.G. Lagoudakis1, O.Z.Karimov1, M. Henini3, D.Taylor3, \nR.T. Phillips2 and R.T. Harley1 \n \n1School of Physics and Astronomy, University of Southampton, Southampton, SO17 1BJ, UK. \n2Cavendish Laboratory, Madingley Road, Cambridge, CB3 0HE, UK. \n3School of Physics and Astronomy, University of Nottingham, Nottingham, NG7 4RD, UK. \n \n \n \n \nAbstract \nWe perform an all-optical spin-dynamic measurement of the Rashba spin-\norbit interaction in (110)-oriented GaAs/AlGaAs quantum wells. The crystallographic \ndirection of quantum confinement allows us to disentangle the contributions to spin-\norbit coupling from the structural inversion asymmetry (Rashba term) and the bulk \ninversion asymmetry. We observe an unexpected temperature dependence of the \nRashba spin-orbit interaction strength that signifies the importance of the usually \nneglected higher-order terms of the Rashba coupling.        \n pdfMachine by Broadgun Software  - a great PDF writer!  - a great PDF creator! - http://www.pdfmachine.com  http://www.broadgun.com 2 Intense worldwide interest is being focused on new semiconductor spintronic and spin-\noptronic quantum devices in which electronic spin replaces charge for data processing or is used \nto control optical polarisation. Indeed manipulation of electron spins has been signposted as the \npreferred route to quantum computing [1]. Progress towards realistic devices depends on \nengineering the spin-orbit interactions that result in effective magnetic fields seen by the \nelectron spins under application of external electric fields. A classic example is the Datta and \nDas spin transistor, wherein the precession of spin polarised carriers confined in a plane is \ncontrolled by a gate voltage which tunes the Rashba or structural inversion asymmetry (SIA) \ncomponent of spin-orbit coupling [2]. Whereas the ability to tune the Rashba (SIA) coupling \nstrength originates in the field-induced spin-splitting of the electronic bands, other contributions \nto the spin-splitting, from bulk inversion asymmetry (BIA or Dresselhaus coupling) and natural \ninterface asymmetry in heterostructures (NIA), complicate direct characterisation of the Rashba \nterm. Disentangling and evaluating the contribution of the different spin-orbit coupling \nmechanisms to the spin-splitting of the electronic bands is of utmost importance for the \nengineering of spintronic devices. Most of the studies that focus on the characterisation of the \nRashba coupling have either neglected the contribution of the Dresselhaus coupling or have \nmeasured the ratio of the strength of the two mechanisms [3,4]. Here we have designed and \ngrown quantum well heterostructures that utilise crystal asymmetries in a way that allows us, \nthrough combined optical measurements of spin relaxation and of electron mobility, to separate \nthe terms and measure directly the strength of the Rashba coupling. The values are in good \nquantitative agreement with k.p-theoretical estimations but unexpectedly, we observe a \ntemperature dependence of the Rashba coefficient that signifies the importance of usually \nneglected higher order terms in the Rashba spin-orbit coupling [5].  \n    \nThe spin relaxation of a non-equilibrium population of electron spins in non-\ncentrosymmetric semiconductors may involve several mechanisms [1], the dominant one in all \nexcept p-type material is that identified by Dyakonov, Perel and Kachorovskii (DPK)[6,7]. The \ndriving force for spin reorientation and therefore loss of spin memory is the combination of \ninversion asymmetry and the spin-orbit interaction; as an electron propagates its spin tends to \nprecess. The corresponding precession vector (k), which describes conduction band spin-\nsplitting, varies in magnitude and direction according to the electrons wavevector k. Strong \nscattering of the electron wavevector randomises the precession and causes spin relaxation. The \nvector (k) is the sum of the three components described above and denoted BIA(k), SIA(k) \nand NIA(k) [1,8] .  For a quantum well grown on a (110)-oriented substrate the interface \ncomponent NIA(k) is zero [1,9]. Furthermore, since electron motion is confined to the (110) \nplane, BIA(k) is, by symmetry, normal to the plane, parallel or antiparallel to the growth axis \n[110], for all electron wavevectors. Therefore, the BIA term makes no contribution to spin \nrelaxation along the growth axis [7]. Thus using this configuration we can measure directly the \nspin relaxation due to the structural inversion asymmetry alone and derive the strength of the \nRashba coupling, SIA(k).     \nThe DPK mechanism gives the relaxation rate for the component of a spin population \nalong a particular axis, i, as [1] \n),1*||( *,21\n, p pis                             (1) \nwhere <2> is the averaged square component of (k) perpendicular to the axis i taken over \nthe spin-oriented population and p* is the momentum scattering time of an electron. \nFurthermore for a symmetrical quantum well, to first order, the SIA term has the form [8,10] \nSIA(k) = (e/ħ) Fk   (2) \n  3  where  is the Rashba coefficient and F the applied electric field. By applying F along the \ngrowth axis, SIA(k) lies in the quantum well plane and thus causes DPK spin relaxation along \nthe growth axis. Squaring eq. 2 and taking the thermal average assuming kBT>> ħ SIA gives \n \n2TBk*m22F22e22k2F22e22)SIA(\n       (3) \n \nwhere m* is the electron effective mass and kB is Boltzmanns constant.  Substitution eq.3 \nin eq. 1 shows that the spin relaxation rate should be linear in F2 and the Rashba coefficient \nbecomes \n21\n*\npsB2\n)Tk*m2(\neF\n   (4) \n Thus we can obtain the Rashba coefficient  from combined measurements of momentum \nrelaxation time and of spin relaxation in applied electric field.  \n Our sample consists of a GaAs/Al0.4Ga0.6As p+-i-n+ structure grown on a semi-insulating \n(110)-oriented GaAs substrate. The insulating portion of the structure comprises 100nm layers \nof undoped AlGaAs on each side of a stack of 20 7.5 nm undoped GaAs quantum wells with 12 \nnm undoped AlGaAs barriers.  For the pump-probe measurements of the spin-relaxation time, s, \na portion of the wafer is processed into a mesa device 400 microns in diameter with an annular \nmetal contact to the top n+ layer to allow optical access to the quantum wells and a second \ncontact to the lower p+ layer. The electric field is varied by means of applied bias voltage. The \ntotal applied electric field F is obtained as the sum of that due to the bias and that to the built-in \nelectric field of the pin structure which we calculate to be 2.80 106 Vm-1. All measurements are \nmade at temperatures above 80K in order to reveal effects of free electrons rather than excitons \nin the quantum wells [11]. Photocurrent measurements in the pin device as a function of reverse \nbias and of photon energy reveal evidence of resonant tunnelling between n=1 and n=2 confined \nelectron states in adjacent wells above about 3 volts. In the measurements reported here we \ntherefore concentrate on applied bias less than 3 volts equivalent to F 8 106 Vm-1. Under these \nconditions the electrons are resonantly excited into the n=1 confined state of a quantum well and \ncan be considered to remain there until recombination, thermal excitation over the barriers being \nnegligible. The momentum relaxation time, p*, is obtained from measurement of the electron \ndiffusion coefficient using a transient spin-grating method on an unprocessed portion of the \nsame wafer. This gives p* at only one value of transverse field namely the built-in field, 2.80 \n106 Vm-1, and in our analysis we assume that it has no significant dependence on the field.  \nThe spin relaxation of the electrons is investigated using a picosecond-resolution polarised \npump-probe reflection technique (see figure.1a) [12]. Wavelength-degenerate circularly \npolarised pump and delayed linearly polarised probe pulses from a mode-locked Ti-sapphire \nlaser are focused at close to normal incidence on the sample and tuned to the n=1 heavy-hole to \nconduction band transition. The pulse duration is ~1.5 ps and repetition frequency is 75 MHz. \nAbsorption of each pump pulse generates a photoexcited population of electrons spin-polarised \nalong the growth axis. The time evolution of the photoexcited population and of the spin \npolarisation <Sz>(t) are monitored by measuring pump-induced changes of, respectively, probe \nreflection R and of probe polarisation rotation  as functions of probe pulse delay. These \nmeasurements are combined to give the spin relaxation rate s-1. They also show that the \nrecombination time of photoexcited carriers, r, is typically five times longer than s. The pump \nbeam intensity is typically 0.5 mW focused to a 60-micron-diameter spot giving an estimated \nphotoexcited spin-polarised electron density Nex~109 cm−2; the probe intensity is 25% of the \npump. Figure 2 shows s-1 vs F2 for three different temperatures. For fields above ~3 106 V m-1  4 the relationship is linear within experimental uncertainty; the lines represent best fits to the \nexperimental points from which the Rashba coefficient is obtained. For lower values of field, s-1\n \ntends to a constant value which may be associated with imperfections of the interfaces [13] or \nBir-Aronov-Pikus spin relaxation [1] due to accumulation of photoexcited carriers under \nforward bias of the pin structure.  \nThe spin-grating measurements (see figure 1b) [14, 15] are made using twin 0.5 mW pump \nbeams from a 200 fs pulse-length mode-locked Ti-sapphire laser tuned to the n=1 valence-\nconduction band transition. The beams are linearly polarised at 90 degrees to one another and \nincident on the sample at 4.1 degrees to the normal. This produces interference fringes of \npolarisation but not intensity, resulting in a transient grating of spin population with a pitch  ~ \n5.7 microns. The focal spot size on the sample is again of order 60 microns giving excitation \ndensity Nex ~109 cm-2. The decay rate of the amplitude of such a grating is given by [14] \n1\nr1\ns22\ns4D\n     (5) \nwhere Ds is the electron spin-diffusion coefficient. The decay is monitored by measuring first-\norder diffraction, in reflection geometry, using a delayed 0.25 mW linear polarised probe beam \nfrom the same laser, incident on the sample at normal incidence. Signal to noise is enhanced by \nuse of an optical heterodyne detection scheme [15]; the decay rate of the diffracted intensity is \n2. Since the sample is undoped we can equate the electron spin-diffusion coefficient to the \ndiffusion coefficient De and obtain the electron mobility from the Einstein equation  = \n(e/kBT)De. Figure 3 shows an example of a measured decay together with the extracted values of \nelectron mobility as a function of temperature. The grating decay rate (and therefore De) is found \nto be insensitive to temperature so that  ~ T-1 (see Fig.3). This temperature dependence is as \nexpected for a non-degenerate two-dimensional electron system, that is with constant density of \nstates, and dominant phonon scattering with probability ~kBT. From the mobility we obtain the \nensemble momentum relaxation time p= m*/e and since we are dealing with intrinsic material \nwith negligible electron-electron scattering we can equate this to the momentum scattering time \np* [1,16]  \nFigure 4 shows the values of the Rashba coefficient obtained by combining the two sets of \nmeasurements. The value is approximately 0.1 nm2 however there is a clear upward trend which \nis consistent with a linear increase of the Rashba coefficient with electron kinetic energy. The \nsolid curve is based on the 8-band k.p treatment given by Winkler [10]. The spin splitting is \ngiven as \nv2\n0g2\ng222SIAF|k|\n)E(1\nE1\n3PeF|k|e||\n\n\n\n\n \n       (6) \nwhere Eg is the band gap of GaAs, 0 the spin-orbit splitting in the valence band and P is Kanes \nmomentum matrix element [17]. Fv is the effective electric field in the valence band and is given \nby [10] \nF67.1 1FF\ncv\nv \n\n\n\n    (7) \nv and c being the valence and conduction band offsets between GaAs and AlGaAs which we \ntake to be in the ratio 3: 2. The weak temperature dependence of the curve results from the \ntemperature dependence of Eg and recently identified temperature dependence of P [18]. This \ntheoretical estimate is in satisfactory agreement with the magnitude of the experimental values \nof Rashba coefficient but it does not reproduce the observed temperature dependence. We note \nthat extension of the k.p treatment to 14 bands, following  [10,19], increases the calculated \nvalues by less than 2%. The dotted curve in Fig. 4 is the k.p theory plus an empirical term \nlinear in temperature. Within the experimental uncertainties this reproduces the data reasonably  5 well. As the photoexcited electron gas is nondegenerate, the additional linear temperature \ndependence suggests that the Rashba coefficient has an additional approximately linear \ndependence on the electrons kinetic energy. This is reminiscent of the dependence of the \nZeeman splitting on kinetic energy [20] which in turn gives rise to the observed increase of the \neffective electron g-factor with quantum confinement in GaAs/AlGaAs quantum wells [21]. The \nobserved dependence signifies the importance of the usually neglected higher order terms in the \nRashba Hamiltonian. \n In conclusion, by combined measurements of spin relaxation and of electron mobility in an \nundoped and nominally inversion symmetric (110)-oriented quantum wells in a pin structure, we \nhave been able to investigate directly the electric field spin-splitting of the conduction band \nwithout interfering effects from bulk inversion asymmetry. The observed splittings are in \nqualitative agreement with a theoretical k.p calculation [10] but also reveal an unexpected \nsignificant temperature dependence. The observation of the temperature dependence of the \nRashba coefficient is important for developing an understanding of the fundamental interactions \nin semiconductor nanostructures and for engineering spintronic devices.    \n We acknowledge useful discussions with Roland Winkler and Xavier Cartoixa and \nfinancial support of the Engineering and Physical Sciences Research Council (EPSRC). \n \nReferences  \n[1] For a reviews see: F.Meier and B.P.Zakharchenya (ed) 1984 Optical Orientation Modern \nProblems in Condensed Matter Science (Amsterdam: North-Holland); Semiconductor \nSpintronics and Quantum Computation ed D.D.Awschalom et al (Berlin: Springer);  ibid \nchapter 4, Spin Dynamics in Semiconductors  by M.E.Flatté, J.M.Byers and W.H.Lau 2002. \n[2] S.Datta and B.Das, App. Phys. Lett. 56, 665 (1990) \n[3] S.D.Ganichev, V.V.Bel'kov, L.E.Golub, E.L.Ivchenko, Petra Schneider, S. Giglberger, \nJ.Eroms, J.De Boeck, G.Borghs, W.Wegscheider, D.Weiss, and W.Prettl, Phys. Rev. Lett. \n92, 256601 (2004); S.Giglberger, L.E.Golub, V.V.Belkov, S.N.Danilov, D.Schuh, \nC.Gerl, F.Rohlfing, J.Stahl, W.Wegscheider, D.Weiss, W.Prettl, and S.D.Ganichev, \nPhysical Review B75 035327 (2007) \n[4] Takaaki Koga, Junsaku Nitta, Tatsushi Akazaki, and Hideaki Takayanagi, Phys. Rev. Lett. \n89, 046801 (2002) \n[5] X.Cartoixà, L.-W.Wang, D.Z.-Y.Ting, and Y.-C.Chang, Physical Review B73 205341 \n(2006) \n[6] M.I.Dyakonov and V.I.Perel 1971 Sov. Phys. JETP 33 1053 \n[7] M.I.Dyakonov and V.Yu. Kachorovskii 1986 Sov. Phys. Semicond. 20 110 \n[8] Optical Spectroscopy of Semiconductor Nanostructures by E.L.Ivchenko (Alpha Science \n2005) \n[9] K.C.Hall, K.Gründoðdu, E.Altunkaya, W.H.Lau, M.E.Flatté, T.F.Boggess, J.J.Zinck, \nW.B.Barvosa-Carter and S.L.Skeith, Physical Review B68 115311 (2003) \n[10] R.Winkler Physica E22 450 (2004); Spin-orbit Coupling Effects in Two-Dimensional \nElectron and Hole Systems by R. Winkler (Springer 2003) \n[11] A.Malinowski, R.S.Britton, T.Grevatt, R.T.Harley,  D.A.Ritchie, and M.Y.Symmonds, \nPhys.Rev. B62, 13034-39 (2000). \n[12] R.T.Harley, O.Z.Karimov, and M.Henini,  J. Phys. D 36, 2198 (2003). \n[13] O.Z. Karimov, G.H.John, R.T.Harley W.H.Lau M.E.Flatté, M.Henini and R. Airey, \nPhys. Rev. Lett 91 (2003) 246601. \n[14] A.R.Cameron, P.Riblet and A.Miller, Phys. Rev. Lett. 76 4793 (1996) \n[15] W.J.H. Leyland, PhD Thesis (Cambridge, 2007) in preparation.  6 [16]  W. J. H. Leyland, G. H. John, and R. T. Harley, M. M. Glazov, E. L. Ivchenko, D. \nA. Ritchie, I. Farrer,A. J. Shields, M. Henini, Physical Review B75 165309 (2007) and \nreferences therein. \n[17] E.O. Kane, J.Phys.Chem.Solids 1 249 (1957) \n[18] J. Hubner, S. Dohrmann, D. Hagele and M. Oesteich, arXiv:cond-mat/0608534 (2006) \n[19] W. Knap, C. Skierbiszewski, A. Zduniak, E. Litwin-Staszewska, D. Bertho, F. Kobbi, J. \nL. Robert, G. E. Pikus, F. G. Pikus, S. V. Iordanskii, V. Mosser, K. Zekentes, Yu. B. Lyanda-\nGeller, Phys. Rev. B53, 3912 (1996) \n[20] M. A. Hopkins, R. J. Nicholas, P. Pfeffer, W. Zawadzki, D. Gauthier, J. C. Portal and M. \nA. DiForte-Poisson, Semicond. Sci. Technol. 2, 568 (1987) \n[21] M. J. Snelling, G. P. Flinn, A. S. Plaut, R. T. Harley, A. C. Tropper, R. Eccleston and C. \nC. Phillips, Phys. Rev. B 44, 11345 (1991) \n \n \n \n \n  7 pump probe\nsignal \nNex<Sz> pump 1 pump 2 probe\nsignal \nNex<Sz> . (a) (b) \n \nFigure 1. Experimental configurations for measurements of (a) spin \nrelaxation and (b) spin diffusion. Upper diagrams, incident pump and \nprobe beams and their polarisations; lower diagrams, profile of focussed \npump spot with (a) a spin polarised population and (b) spin grating with \nspacing .  8 0 2 4 605101520\n(110)-oriented MQW\n 200K\n 170K\n 80KSpin relaxation rate (ns-1)\n(Electric field)2  (V2m-2x10-13)\nFigure 2. Electric field dependence of spin relaxation rate at three temperatures. \nThe high field linear regions of the graphs extrapolate to the origin and the slopes \nare used to determine the Rashba coefficient  9 60 100 3005000.312Mobility (m2 V-1 s-1)\nTemperature (K)0 100 200 3000.010.11 Spin grating decay\nT=221K\n2=20.4±0.4 ps-1Diffracted power (arb. units)\nProbe delay (ps)T-1 \nFigure 3. Logarithmic plot of the electron mobility, in a sample from the \nsame wafer as data of Fig.2, determined by the spin-grating method. The \nelectric field is the built-in field of the pin structure, 2.80 106 Vm-1. The T-1 \ntemperature dependence is as expected for a non-degenerate two-\ndimensional electron system with dominant phonon scattering. Inset is a \ntypical grating decay signal at 221 K.  10 \n0501001502002503000.000.050.100.15\n  k.p theory\n (k.p)+1.75x10-4TRashba coefft. (nm2)\nTemperature (K)\nFigure 4. Rashba coefficient obtained from combination of spin \nrelaxation and mobility measurements. Solid curve is 8-band k.p \ncalculation including temperature dependence of band edges and of \ninterband momentum matrix element. Dotted curve is k.p theory plus \nan empirical linear-in-T term. "    },    {        "title": "1204.2189v2.Spin_current_absorption_by_inhomogeneous_spin_orbit_coupling.pdf",        "content": "arXiv:1204.2189v2  [cond-mat.mes-hall]  3 Oct 2012Spin-current absorption by inhomogeneous spin-orbit coup ling\nKazuhiro Tsutsui,1,∗Kazuhiro Hosono,2and Takehito Yokoyama1\n1Department of Physics, Tokyo Institute of Technology,\n2-12-1 Ookayama, Meguro-ku, Tokyo 152-8551, Japan\n2International Center of Materials Nanoarchitectonics (WP I-MANA), Namiki 1-1, Tsukuba 305-0044, Japan\n(Dated: August 27, 2018)\nWe investigate the spin-current absorption induced by an in homogeneous spin-orbit coupling due\nto impurities in metals. We consider the system with spin cur rents driven by the electric field or\nthe spin accumulation. The resulting diffusive spin current s, including the gradient of the spin-orbit\ncoupling strength, indicate the spin-current absorption a t the interface, which is exemplified with\nexperimentally relevant setups.\nPACS numbers: 72.25.Ba, 72.25.Mk\nI. INTRODUCTION\nSpintronics aims to utilize not only the charge degree\nof freedom but also the spin degree of freedom1. Spin\ncurrents are an important notion in spintronics from the\naspect of both basic and applied science. Spin currents\nare generated by a current injection into a ferromagnet\nor spin pumping using the magnetization precession2,3.\nOther methods to generate spin currents are the spin\nHall effect4–6or the excitation of the spin wave7. In\nthe development of spintronic devices, techniques to de-\ntect spin currents efficiently are indispensable as well as\nthe spin-current generation. For instance, spin currents\nhave been successfully detected via the inverse spin Hall\neffect8–10. Due to this effect, spin currents are converted\ninto electriccurrentsandthereforeareobservedasavolt-\nage drop. The electric voltage induced by the spin cur-\nrent has been observed in a lateral junction of a ferro-\nmagnetic metal and a nonmagnetic metal with spin-orbit\ncoupling (SOC) such as aluminum8or platinum9,10. In\nparticular, platinum is a typical spin-currentdetector be-\ncause of its strong SOC due to impurities, which is used\nto absorb a spin current7.\nKimura et al.demonstrated the spin-current detec-\ntion via the inverse spin Hall effect and suggested that\nthe absorption of the spin current occurs from the Cu\ncross into the Pt wire10. They explained the spin-current\nabsorption using the resistance mismatch for the spin\ncurrent. Namely, they defined the spin resistance as\nRs=λ/[σS(1−P2)] with the spin diffusion length λ,\nthe spin polarization P, the conductivity σand the ef-\nfective cross-sectional area Sfor the spin current, and\nthen considered that the spin current flowing in a mate-\nrial is absorbed into the adjacent material with smaller\nspin resistance10,11. Since platinum is a strong spin-orbit\ncoupled material while copper is a weak one, the spin\nresistance of Pt is smaller than that of Cu and hence\nthe injected spin current in the Cu cross is partly ab-\nsorbed into the Pt wire. Here, the difference of the SOC\nstrength, i.e., the inhomogeneous SOC is the key to the\nabsorption of spin current. As shown above, the physics\nof the spin-current absorption induced by an inhomo-geneous SOC has been understood only phenomenolog-\nically, in that previous theories involve the phenomeno-\nlogicalparameterssuchasthe spin-diffusionlength orthe\nspin resistance12,13and microscopic theories of the spin-\ncurrent absorption are missing. The spin Hall effect14–16\nand the spin-current generation17due to an inhomoge-\nneous Rashba SOC have been theoretically examined.\nAlso, a spin current generation near an interface due to\ninterfacial spin-orbit coupling has been investigated18.\nIn the present work, we consider the spin-current ab-\nsorption as follows. In general, the spin-continuity equa-\ntion for spin-orbit coupled systems has a source term:\n∂sα\n∂t+∇·jα\ns=Tα. Here,sα,jα\nsandTαrepresent the\nspin density, the spin current and the spin torque (or\nsource term) with αbeing a component in spin space,\nrespectively. Thus, the divergence of spin currents in\nthe steady state is non-zero and therefore leads to the\ngeneration of spin currents. We refer to a spin-current\ngeneration induced near an interface between different\nspin-orbit coupled materials or due to an inhomogeneous\nSOC as the spin-current absorption. On the other hand,\nthe spin-current absorption in previous phenomenologi-\ncal studies is based on the continuity of a spin current\nat an interface, and therefore absorbed spin currents are\nnot generated ones at the interface.\nIn this paper, we theoretically examine the spin-\ncurrent generation induced by an inhomogeneous SOC\ndue to impurities. We consider the spin-current absorp-\ntion in systems where spin currents are generated by an\nexternal electric field in Sec. II. In Sec. III, we also\ninvestigate systems where spin currents are induced by\nspin accumulation. We present analytical expressions of\nthe spin currents as a response to the gradient of the\nSOC strength. Then, we apply these results to the vicin-\nity of the interface between metals with different SOC\nstrengths, verifying the absorption of spin currents. Sec-\ntion IV is devoted to the discussions. We summarize the\npaper in Sec. V.2\nII. ABSORPTION OF SPIN CURRENT\nDRIVEN BY EXTERNAL FIELD\nA. Model and Formalism\nWe consider the ferromagnetic conductor in the pres-\nence of the spin-orbit scattering due to impurities19–22\nand assume that the coupling constant of the spin-orbit\nscattering slowly varies in space. This spatial variation\nof the SOC has not been considered so far. In order\nto address the spin-current absorption, we describe the\ninput spin current by the spin-polarized current flowing\nin a ferromagnet under an external electric field. The\ntotal Hamiltonian is thus composed of the free-electron\npart with the exchange coupling ( HFM), the impurity-\nscattering part (Himp), the SOC due to impurities with\nits minimal substitution ( H0\nSO+HA\nSO), and the interac-\ntion between the vector potential and the electric current\n(Hem):\nHFM=/summationdisplay\nσ=±1/integraldisplay\ndrc†\nσ(r,t)/parenleftbigg\n−/planckover2pi12\n2m∇2−ǫFσ/parenrightbigg\ncσ(r,t),\n(1)\nHimp=/summationdisplay\nσ=±1/integraldisplay\ndrU(r)c†\nσ(r,t)cσ(r,t), (2)\nH0\nSO=/planckover2pi1\n2i/summationdisplay\nσ,σ′=±1/integraldisplay\ndrλSO(r)\n×∇U(r)·c†\nσ(r,t)/parenleftBig← →∇×σσσ′/parenrightBig\ncσ′(r,t),(3)\nHA\nSO=e/summationdisplay\nσ,σ′=±1/integraldisplay\ndrλSO(r)\n×∇U(r)·c†\nσ(r,t)(A(r,t)×σσσ′)cσ′(r,t),(4)\nHem=e/planckover2pi1\n2mi/summationdisplay\nσ=±1/integraldisplay\ndr A(r,t)·c†\nσ(r,t)← →∇cσ(r,t).(5)\nHere,cσ(r,t) (c†\nσ(r,t)) represents the annihilation (cre-\nation) operator of a conduction electron, σαβrepre-\nsents Pauli matrices, e(>0) is the electric charge and\nc†\nσ← →∇cσ≡c†\nσ∇cσ−(∇c†\nσ)cσ.σ= 1 and σ=−1 corre-\nspond to the up ( ↑) and down spins ( ↓), respectively. We\nsetǫFσ≡ǫF+σMwithǫFthe Fermi energy and Mthe\nexchange energy. λSO(r) represents the SOC strength\nwith spatial variation, averaged over the impurity posi-\ntions.A(r,t) is the vectorpotential for the external elec-\ntric field and we adopt the fixed gauge as Eem=−˙A.\nU(r) is the short ranged random impurity potential and\naveraging over the impurity positions is carried out as\n/angb∇acketleftU(r)U(r′)/angb∇acket∇ightimp=u2\n0nimpδ(r−r′), where u0andnimp\nare the strength of the impurity potential and the im-\npurity concentration, respectively. Here, we assume that\nthe exchange field is much stronger than the stray field\nand hence we neglect the effect of the stray (magnetic)\nfield in our model. Note that, in the Hamiltonian H0\nSO,\nthe operator∇does not act on λSO(r).The quantum description of the spin current is ob-\ntained by identifying the Heisenberg equation of mo-\ntion for the spin density with the spin-continuity\nequation22,23. The spin current operator in the i-\ndirection with α-spin polarization reads\n(ˆjα\ns)i≡1\n2m/planckover2pi1\nic†(r,t)σα← →∂ic(r,t)−e\nmAiσαc†(r,t)c(r,t)\n+λSO/summationdisplay\njǫαji∂jU(r)c†(r,t)c(r,t).\n(6)\nThe spin current is thus given by (see also Fig. 1 )\n(jα\ns)i=1\n2m/planckover2pi12\ni/summationdisplay\nk,k′ei(k−k′)·r(k+k′)itr[σαGk,k′(t)]<\n−e/planckover2pi1\nm/integraldisplaydΩ\n2πeiΩt/summationdisplay\nk,k′,qei(k−k′+q)·rAi(q,Ω)tr[σαGk,k′(t)]<\n+i/planckover2pi1/summationdisplay\nj/summationdisplay\nk,k′,u,pei(k−k′+u+p)·rǫαjipjλSO(u)U(p)tr[Gk,k′(t)]<.\n(7)\nHere,Gk,k′(t,t′)≡1\ni/planckover2pi1/angb∇acketleftTc[ck(t)c†\nk′(t′)]/angb∇acket∇ightdenotes the time-\nordered Green’s function (Keldysh Green’s function) of\nthe total Hamiltonian and Gk,k′(t)≡limt′→tGk,k′(t,t′).\n[···]<means taking the lesser component of the Green’s\nfunctions and tr[···] means taking trace over spin.\nWe perturbatively calculate the spin currents induced\nby the inhomogeneity of the SOC strength using the\nKeldyshGreen’sfunction formalism. We consider HFM+\nHimpas the non-perturbative part and H0\nSO,HA\nSOand\nHemas perturbative parts. We remark that the non-\nperturbative Green’s function contains the self energy by\nthe impurity scattering within the first Born approxima-\ntion. Within the linear response with respect to the elec-\ntric field, the second part of the spin currents in Eq.(7)\ncorrespondsto an equilibrium spin current, and therefore\nwe ignore this contribution. The leading contributions of\nthe spin current contain the first order term with respect\nto the SOC strength λSO. Since we are interested in spin\ncurrents generated by the inhomogeneous SOC strength,\nwe focus on the contribution with the spatial derivative\nof the SOC strength. The third part of the spin currents,\non the other hand, does not involve the spatial deriva-\ntive of the SOC strength. Consequently, the first part of\nEq.(7) is relevant to our study.\nB. Local spin current\nWe first treat the local spin current, which is driven\nby the external electric field locally. The diagrams of the\nlocal spin current are shown in Fig. 2. The inhomogene-\nity of the external electric field is required when we dis-\ncuss the leading effect driven by an inhomogeneous SOC3\nFIG. 1. Diagrams of the spin current for the total Hamilto-\nnian. Bold lines denote the total Green’s function. An wavy\nline denotes the vector potential ( A(q,Ω)). A broken line and\ndouble broken line represent the SOC strength ( λSO(u)) and\nthe impurity potential ( U(p)), respectively.\nFIG. 2. Diagrams of the leading contribution of the local spi n\ncurrent induced by the inhomogeneity of the SOC strength.\nSolid lines denote the free Green’s function. Broken lines\ndenote the SOC strength ( λSO(u)) and double broken lines\ndenote the impurity potential ( U(p)). Wavy lines represent\nthe vector potential ( Al(q,Ω)).\nstrength. Namely, the leading contribution of spin cur-\nrentsinvolvingthederivativeoftheSOCstrengthhasthe\nform∂λSO∂Eem. The reason is as follows. Spin current\nand electric field are odd under spatial inversion, while\nλSOand other quantities are even with respect to spatial\ninversion. Therefore, the leading terms including spatial\nderivative of the SOC have the forms of ∂2λSOEemor\n∂λSO∂Eem. We focus on the contributions ∂λSO∂Eemby\nassuming ∂2λSOis negligibly small in this section. Now,\nwe present the resulting local spin currents. (For the de-\ntail of the calculation, refer to Appendix A.) Under the\ncondition ǫFσ≫/planckover2pi1/2τσ(≡ησ) withτσ≡/planckover2pi1/2πu2\n0nimpνσ\nthe spin-dependent relaxation time and νσthe density\nof state with spin index σ=±1, we obtain three types\nof the spin currents induced by an inhomogeneous SOC\n(jα\ns)local\ni= (jα\ns)(1)\ni+(jα\ns)(2)\ni+(jα\ns)(3)\ni:\n(jα\ns)(1)\ni=C(α1eα+α2(eα×ez))·(∇λSO(r)×∂iEem(r)),\n(8)\n(jα\ns)(2)\ni=C(β1(ei×eα)\n+β2(δizeα−δiαez))·∇λSO(r)(∇·Eem(r)),\n(9)\n(jα\ns)(3)\ni=C(γ1eα+γ2(eα×ez))·(∇λSO(r)×∇Ei\nem(r)),\n(10)whereC≡πe/planckover2pi12\n2m(ν↑+ν↓). The detailed expressions of\nαi,βi,andγi(i= 1,2) are given in Appendix A. For\nα=z, the second terms of Eqs. (8), (9) and (10) vanish.\nIt should be noted that even when Mgoes to zero, the\nspin currents still have finite contributions and become\nisotropic in spin space. Therefore, the magnetization is\nnot indispensable for the spin-current generation itself.\nC. Diffusive spin current\n+...\n +...\nFIG. 3. Diagrams of the leading contribution of the diffu-\nsive spin current induced by the inhomogeneity of the SOC\nstrength. (a) Contributions with the vertex correction to t he\nspin-current operator. (b) Contributions with the vertex c or-\nrection to the vector potential. (c) The vertex correction i s\ndefined as the summation of the ladder diagram with respect\nto the normal impurity potential.\nWe calculate the diffusive (or non-local) spin current,\nwhich can be obtained by including the vertex correc-\ntion to the local spin current. Here, the vertex correc-\ntion means the summation of the ladder diagram with\nrespect to the normal impurity potential as shown in\nFig. 3 (c). Since we focus on the diffusive behavior of\nthe spin current induced by the exchange coupling, the\nladder diagram does not involve the spin-orbit coupled\nimpurities. We consider two types of the vertex correc-\ntion, i.e., that for the spin-current operator (Fig. 3 (a))\nand that for the vector potential (Fig. 3 (b)). The vertex4\ncorrection that cuts across the impurity-averaged line is\nnegligible because it is of higher order in ησ22. For the\ndiffusive contribution under the condition ǫFσ≫/planckover2pi1/2τσ,we obtain two types of the spin currents induced by the\ninhomogeneous SOC ( jα\ns)diffusive\ni= (jα\ns)(1′)\ni+(jα\ns)(2′)\ni:\n(jα\ns)(1′)\ni=/braceleftbiggC/summationtext\nσ(ℜ[ασeα·Ki\n(1),σ(r)]+ℑ[ασ(eα×ez)·Ki\n(2),σ(r)]),(α=x,y)\nC/summationtext\nσασez·Ki\n(2),σ(r),(α=z)(11)\nKi\n(n),σ(r)≡/integraldisplay\ndr′/integraldisplay\ndr′′/integraldisplay\ndt′χ(n)\nσ(r−r′,r−r′′,t−t′)(∇λSO(r′)×∂iEem(r′′,t′)),(n= 1,2) (12)\n(jα\ns)(2′)\ni=C/summationdisplay\nσ((ei×eα)β1,σ+(δizeα−δiαez)β2,σ)·∇λSO(r)/integraldisplay\ndr′/integraldisplay\ndt′χσ(r−r′,t−t′)∇·Eem(r′,t′).(13)\nHere,\nχ(1)\nσ(r1,r2,t) =/integraldisplaydΩ\n2π/summationdisplay\nu,qeiΩt+iu·r1+iq·r2\nFσ−iGσ,(14)\nχ(2)\nσ(r1,r2,t) =/integraldisplaydΩ\n2π/summationdisplay\nu,qeiΩt+iu·r1+iq·r2\n(Dσ(u+q)2−iΩ)τσ,(15)\nχσ(r,t) =/integraldisplaydΩ\n2π/summationdisplay\nqeiΩt+iq·r\n(Dσq2−iΩ)τσ,(16)\nrepresent the spin-diffusion propagators. Dσ≡2ǫFτσ\n3mde-\nnotes the diffusion coefficient. The detailed expressions\nofασ,βi,σ,Fσ,andGσ(i= 1,2) are given in Appendix\nB. The vertex corrected spin current ( jα\ns)(1′)\ni((jα\ns)(2′)\ni)\ncorresponds to the local spin current ( jα\ns)(1)\ni((jα\ns)(2)\ni).\nThere does not exist the diffusive-spin current contribu-\ntion which corresponds to ( jα\ns)(3)\ni. (jα\ns)(2′)\nidiffers from\n(jα\ns)(1′)\niin that only the divergence of the external field\npropagates.\nD. Spin-current absorption\nLet us consider a concrete configuration to show the\nabsorption of the spin current by the inhomogeneous\nSOC. We suppose that an external electric field is ap-\nplied to a ferromagnetic metal and therefore a spin-\npolarized current flows in the y-direction as shown in\nFig. 4. Here, we assume that the applied electric\nfield varies smoothly in space from the ferromagnet to\nthe attached non-magnetic metal with SOC, which is\nmodeled as Eem(r) =Ey1+tanh( −z/ξE)\n2eywhereξEis\nof the order of the lattice constant. We also assume\nλSO(r) =λSOθ(z). In the perturbative calculation, we\nhave considered a smoothly varying SOC strength. How-\never, this sharp spatial variation of the SOC would give\na qualitatively correct results.22In the present configu-\nration, a spin-polarized current induced by the external\nelectric field flows in the direction parallel to the inter-\nface. Therefore, our setup is different from that of the\nspin-current injection.The local spin current in the present configuration is\ngenerated when λSOandEemvary in space, i.e., only\nnear the interface between the ferromagnetic metal and\nthe spin-orbit coupled metal. Hence, the local spin\ncurrent is not generated in the bulk spin-orbit coupled\nmetal. We thus focus on the non-local spin current as\nthe spin-current absorption. Since only ( ∇λSO(r′)×\n∂iEem(r′′))j=EyλSOδ(z′)(2ξEcosh2(z′′/ξE))−1δizδjx\nremains non-zero and ∇·Eem= 0, the contribution of\n(jα\ns)(1′)\nm(Eq.(11)) appears in the present configuration.\nFrom Eq. (14), the spin-diffusive propagatorin the static\nlimit (Ω→0) reduces to\nχ(1)\nσ(r1,r2) =π\n2DστσAσe−√\nBσ/Dστσ|r1|\n|r1|δ(r2−r1),\n(17)\nwhere\nAσ≡η2\nση2\n+(η2\n+−3M2)\n(M2+η2\n+)3−iσMη2\nση+(3η2\n+−M2)\n(M2+η2\n+)3,(18)\nBσ≡M(M2+η2\n+)2\nη2ση+(η2\n+(η2\n+−3M2)2+M2(3η2\n+−M2)2)\n×/parenleftbig\n4Mη+(η2\n+−M2)+iσ(6η2\n+M2−M4−η4\n+)/parenrightbig\n,\n(19)\nwithη±≡η↑±η↓\n2. Therefore, we obtain the spin currents\nfor each spin component from Eq. (11):\njx\ns=π2C\n2ξEEyλSO/summationdisplay\nσ1\nDστσℜ/bracketleftbiggασe−i|z|kσ\ns\nAσ√Bσ/bracketrightbigg\ne−|z|/lσ\nsez,\n(20)\njy\ns=π2C\n2ξEEyλSO/summationdisplay\nσ1\nDστσℑ/bracketleftbiggασe−i|z|kσ\ns\nAσ√Bσ/bracketrightbigg\ne−|z|/lσ\nsez,\n(21)\njz\ns=0. (22)\nHere\nlσ\ns≡√Dστσ/vextendsingle/vextendsingle√Bσ/vextendsingle/vextendsinglecos(arg(√Bσ)), (23)\nkσ\ns≡sin(arg(/radicalbig\nBσ)), (24)5\nwhich represent the spin diffusion length and the wave\nnumber for each spin, respectively. Here, we used the\nassumption b≫lσ\nswithbthe diameter of the interface22.\nWe find that spin currents are induced perpendicularly\nto the interface between the ferromagnetic metal and the\nspin-orbit coupled metal, and then decay exponentially\nin an oscillatory fashion as shown in Fig. 4. This ex-\nponential decay is due to the spin-flip scattering by the\nmagnetizationin the ferromagnetand also the oscillatory\nbehaviororiginatesfromthedifferencebetweentheFermi\nwave numbers for each exchange spin-split bands. The\nabove spin currents are spin-polarized perpendicularly to\nthe spin polarization of the input spin-polarized current\n(namely, zcomponent). This reflects the violation of the\nconservation law of spin near the interface, and in other\nwords, spin torque generated at the interface produces\nthe present spin-current absorption.\nIt is also found that the magnitudes of the resulting\nspin currents in Eqs. (20) and (21) at z= 0 are, respec-\ntively, proportionalto η1/2\n+andη−η−1/2\n+in the dirty limit\n(τσ→0(σ=↑,↓)), andη−η1/2\n+andη3/2\n+inthecleanlimit\n(τσ→∞(σ=↑,↓)). Namely, the magnitude of the ab-\nsorbed spin currents gets reduced in clean ferromagnets.\nThis is due to the competition between the two effects of\nimpurity scattering. In general, a SOC due to impurities\nin clean ferromagnets is weak due to low impurity con-\ncentration, leading to a small spin current. On the other\nhand, low impurity concentration leads to a large spin\ncurrent since impurity scattering is suppressed. When\nthe former effect overcomes the latter, spin currents are\nexpected to be suppressedin clean ferromagnets. Finally,\nwe note that for junctions composed of a weak spin-orbit\ncoupledferromagnetandastrongspin-orbitcouplednon-\nmagnet (namely, λSO(r) =λ1\nSOθ(z) +λ2\nSOθ(−z) with\nλ1\nSO> λ2\nSO), spin current absorption predicted in this\npaper also occurs.\nIII. ABSORPTION OF SPIN CURRENT\nDRIVEN BY SPIN ACCUMULATION\nA. Model\nIn this section, we examine the spin-current absorp-\ntion originating from the variation of the SOC strength\nin systems with spin accumulation. These systems are\nmodeled by the conducting electrons in the presence of\nthegradientofthespin-dependentchemicalpotentialand\nthat of the SOC strength. Our Hamiltonian consists of\nFIG. 4. (Color online) Schematic illustration of the spin-\ncurrent absorption by the input spin-polarized current. Th e\nexternal electric field Eemis applied to a ferromagnetic metal\nand therefore a spin-polarized current jz\nsflows in the direc-\ntion ofyaxis. The non-magnetic metal attached to the fer-\nromagnetic metal has a strong SOC λSOdue to impurities.\nThe diffusive spin currents jx,y\nsare induced perpendicularly\nto the interface and then decay exponentially in an oscillat ory\nfashion.\nH0,Hacc,HimpandH0\nSO. Here,\nH0=/summationdisplay\nσ=±1/integraldisplay\ndrc†\nσ(r,t)/parenleftbigg\n−/planckover2pi12\n2m∇2/parenrightbigg\ncσ(r,t), (25)\nHacc=/summationdisplay\nσ=±1/integraldisplay\ndrµσ(r,t)c†\nσ(r,t)cσ(r,t)\n=/summationdisplay\nσ,σ′=±1/integraldisplay\ndrc†\nσ(r,t)(¯µ(r,t)δσσ′+µs(r,t)σz\nσσ′)cσ′(r,t).\n(26)\nHere, wehavedefined thespin accumulationasthediffer-\nence between spin-up and spin-down chemical potentials,\nnamely,µs(r) :=µ↑−µ↓\n2.12,13This spatial distribution of\nspin accumulation leads to a diffusive spin current. In\nthe following, we will also assume the chemical potential\n¯µ(r,t) to be constant.\nB. Spin-current\nWe will calculate the spin current corresponding to\nthe diagrams in Fig. 3 in a similar manner to the pre-\nvious section. Here, we have replaced the vector po-\ntentialA(q,Ω) with the spin accumulation µs(q,Ω) in\nthe diagrams of Fig. 3. We consider H0as the non-\nperturbative part and Hacc,HimpandH0\nSOas pertur-\nbative parts. In this section, we adopt the self energy\nincluding both the normal and the spin-orbit coupled\nimpurity potentials22,24. We also treat only the vertex6\ncorrection for the spin-current operator22. This correc-\ntion provides a finite decay length of the spin current\ndetermined by the SOC strength. After taking the lesser\ncomponent of the Green’s functions and averaging over\nimpurity positions, the spin current shown in Fig. 2 (a)\nand (b) reads\n(jα\ns)m=i/planckover2pi12\nm/planckover2pi1/integraldisplaydΩ\n2πΩeiΩt/summationdisplay\ni,j,k/summationdisplay\nk,k′,u,q\n×ei(u+q)·rλSO(u)µs(q,Ω)/angb∇acketleftU(p)U(−p)/angb∇acket∇ightimp\n×ǫijkℜ(χαijkm)tr/bracketleftbig\nσασkσz/bracketrightbig\n. (27)\nFor the expression of χαijkm, refer to Appendix C. In\nthe same manner as in the previous section, we expand\nthe coefficient χαijkmwith respect to uandq. Ac-\ncording to the inversion operation r→−r, it is found\nthat only the contributions with an odd order of u\nandqremain. Now, we perform the vertex correction\nχαijkmtr/bracketleftbig\nσασkσz/bracketrightbig\n→˜χαijk/summationtext\na,b,c,d,eσα\ndaΓad,cbσk\nbeσz\nec\nwith ˜χαijkbeing the vertex-corrected coefficient and\nΓad,cbbeing the vertex correction corresponding to the\nladder diagram including the normal and the spin or-\nbit coupled impurities22. The contributions from the\nvertex-corrected coefficient with the first order of uor\nqvanish, and hence the leading contribution in the dif-\nfusion regime is of the third order of uandq: ˜χαijk=\n˜χ(2,1)\nαijk+ ˜χ(1,2)\nαijk+···, where the superscript ( i,j) denotes\nthe order of uandq. The contributions from ˜ χ(3,0)\nαijkand\n˜χ(0,3)\nαijkvanish exactly.\nFrom the contribution of ˜ χ(1,2)\nαijkm, we obtain the expres-\nsion of the spin current:\n(˜jαs)(1,2)\nm=ζ(1,2)/integraldisplay\ndr′/integraldisplay\ndr′′/integraldisplay\ndt′D(r−r′,r−r′′,t−t′)\n×[∂αλSO(r′)∂z(∂mµs(r′′,t′))\n−∂zλSO(r′)∂α(∂mµs(r′′,t′))], (28)whereζ(1,2)≡4π/planckover2pi12τν\n3mwithτ≡/planckover2pi1/2πu2\n0nimpνbeing the\nrelaxation time and νbeing the density of state.\nD(r1,r2,t) :=/integraldisplaydΩ\n2π/summationdisplay\nu,qeiΩt+iu·r1+iq·r2\n×Ω(1−κ)(1+3κ)\nDτ((u+q)2+ξ2)+iΩτ(29)\nis the spin-diffusive propagator with κ≡λ2\nSOk4\nF/3 (kF\nis the Fermi wave number), D≡2ǫFτ/3mandξ≡/radicalbig\n4κ/Dτ. As seen from Eq.(29), ξdetermines the de-\ncay length of the spin current.\nAs for the other contribution ˜ χ(2,1)\nαijkm, we obtain the\nspin current of the form:\n(˜jαs)(2,1)\nm=ζ(2,1)/integraldisplay\ndr′/integraldisplay\ndr′′/integraldisplay\ndt′D(r−r′,r−r′′,t−t′)\n×[∂αµs(r′′,t′)∂z(∂mλSO(r′))\n−∂zµs(r′′,t′)∂α(∂mλSO(r′))], (30)\nwhereζ(2,1)≡608π/planckover2pi12τν\n45m/parenleftbigǫFτ\n/planckover2pi1/parenrightbig2. Here, we have focused\non the diffusive spin currents, corresponding to contri-\nbutions with the vertex correction. It is found that if\nthe scales of spatial variations of λSO(r) andµs(r,t)\nare nearly the same, the magnitude of the spin current\n(˜jαs)(2,1)\nmis larger than that of ( ˜jαs)(1,2)\nmby the factor of\nǫFτ//planckover2pi1.\nC. Spin-current absorption\nLet us consider a typical configuration in order to in-\nvestigate the spin-current absorption. We assume that\ntwo non-magnetic metals with different SOC strengths\nare connected at the z= 0 plane as shown in Fig. 5. In\naddition, there exists a gradient of the spin accumulation\nalong the ydirection in one of the metals. In the vicinity\nof thez= 0 plane, we assume that λSO(r) =λSO(z) and\nµs(r,t) =µs(y,z,t), leading to\n(˜jxs)z= 0, (31)\n(˜jy\ns)z=/integraldisplay\ndr′/integraldisplay\ndr′′/integraldisplay\ndt′D(r−r′,r−r′′,t−t′)\n×/bracketleftbig\n−ζ(1,2)∂z′′λSO(z′′)∂z′(∂y′µs(y′,z′,t′))+ζ(2,1)∂y′µs(y′,z′,t′)∂2\nz′′λSO(z′′)/bracketrightbig\n, (32)\n(˜jzs)z= 0. (33)\nThe first term of the right hand side in Eq. (32) rep-\nresents the interplay between the variation of the SOC\nstrength and that of the input spin current near the in-\nterface. Note that spin currentis induced by the gradient\nof the spin accumulation without SOC.13On the other\nhand, the second term comes from the variation of the\nSOC strength and uniform input spin current. Remark\nthataspincurrentpolarizedalongthe y-axisisgeneratedas a response to that polarized along the z-axis. Similar\nto the absorption of spin currents driven by an external\nelectric field, a spin current absorbed into the top metal\nis non-local, and diffusive spin current decays due to the\nspin-orbit coupled scattering as seen from Eq.(29).7\nFIG. 5. (Color online) Schematic illustration of the absorp -\ntion of spin currents driven by the spin accumulation. The\narrows in the bottom metal denote input spin currents ( jz\ns,in)y\nby the spin accumulation. In the top metal, the left arrows\nare normal diffusion of the input spin currents ( jz\ns,in)z, which\nis irrelevant to the variation of a SOC, whereas the middle\nand the right arrows are diffusion of spin currents ( jy\ns,abs.)z\ninduced in the vicinity of the interface, relevant to the diff er-\nence of SOCs.\nIV. DISCUSSION\nIn the present study, we have focused on diffusion of\nspin currents including the gradient of the SOC strength.\nHowever, there also exists a diffusion of spin currents ir-\nrelevant to the difference of SOC strength as shown in\nFig. 5. We can extract the former contribution by com-\nparing spin currents in the triple lateral spin valves us-\ning middle junctions with different SOC11. In fact, the\nabsorption of non-local spin currents has been success-\nfully demonstrated in the spin-valve measurement con-\nsisting of double Py/Cu junctions and middle Au/Cu\njunction10,11.\nSince the argument based on the mismatch of the spin\nresistance by Kimura et al.12assumes the continuity of\na spin current, the absorbed spin currents are not those\ngenerated at an interface. Consequently, the absorbedspin currents in our theory are of different origin from\nthose in the theory based on the spin-resistance mis-\nmatch. Hence, we cannot compare our results with pre-\nvious results directly. Both studies show the spin-current\nabsorption with different mechanisms.\nIn asymmetric structures, conduction electrons at its\ninterface in general feel a Rashba SOC25, leading to the\nspin polarization under an external electric field or cur-\nrent injection26. In the configurations of Figs. 4 and\n5, there would exist a Rashba SOC at their interfaces.\nThe current-induced spin polarization due to the Rashba\nSOC may affect the absorption of spin currents predicted\nin our models.\nV. SUMMARY\nWehaveinvestigatedthegenerationofspincurrentsby\nan inhomogeneous SOC due to impurities, which have\nbeen applied to the interface system to show the spin-\ncurrent absorption. Using the Keldysh Green’s function\nformalism, we have presented analytical expressions of\nthe spin currents with the gradient of the SOC strength\nfor two systems: the system with field-driven spin cur-\nrents and one with spin-accumulation-driven spin cur-\nrents. The resulting spin current indicates the absorp-\ntionofthespin currentat theinterfacebetweenmaterials\nwith different SOC strengths.\nIn the present study, we assumed a homogeneous mag-\nnetization. The extension of our model to an inhomoge-\nneous magnetization is also an interesting future work.\nACKNOWLEDGMENTS\nWe thank Y. Tserkovnyak, G. Tatara and A. Takeuchi\nforhelpful discussions. K. T.thanksS. Murakamifordis-\ncussions. K. H. thanks Y. Nozaki for discussions. This\nwork is supported by Grant-in-Aid for Young Scientists\n(B) (No. 23740236andNo. 24710153)andthe ”Topolog-\nical Quantum Phenomena”(No. 23103505)Grant-in Aid\nforScientific Researchon InnovativeAreasfrom the Min-\nistry of Education, Culture, Sports, Science and Tech-\nnology (MEXT) of Japan. K. T. also acknowledges the\nfinancial support from the Global Center of Excellence\nProgram by MEXT, Japan, through the gNanoscience\nand Quantum Physicsh Project of the Tokyo Institute of\nTechnology.\n∗tsutsui@stat.phys.titech.ac.jp\n1I. Zutic, J. Fabian, and S. Das Sarma, Rev. Mod. Phys.\n76, 323 (2004).\n2R. H. Silsbee, A. Janossy, and P. Monod, Phys. Rev. B 19,\n4382 (1979).\n3Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys.\nRev. Lett. 88, 117601 (2002).4M. I. Dyakonov and V. I. Perel, Phys. Lett. A 35, 459\n(1971).\n5J. E. Hirsch, Phys. Rev. Lett. 83, 1834 (1999).\n6S. Murakami, N. Nagaosa, and S. -C. Zhang, Science 301,\n1348 (2003).\n7Y. Kajiwara, K. Harii, S. Takahashi, J. Ohe, K. Uchida,\nM. Mizuguchi, H. Umezawa, H. Kawai, K. Ando, K.8\nTakanashi, S. Maekawa, and E. Saitoh, Nature 464, 262\n(2010).\n8S.O.ValenzuelaandM. Tinkham, Nature 442, 176(2006).\n9E. Saitoh, M. Ueda, H. Miyajima, and G. Tatara, Appl.\nPhys. Lett. 88, 182509 (2006).\n10T. Kimura, Y. Otani, T. Sato, S. Takahashi, and S.\nMaekawa, Phys. Rev. Lett. 98, 156601 (2007).\n11T. Kimura, Y. Otani, and L. Vila, J. Appl. Phys. 103,\n07F310 (2008).\n12T. Kimura, J. Hamrle, and Y. Otani, Phys. Rev. B 72,\n014461 (2005).\n13S. Takahashi and S. Maekawa, Phys. Rev. B 67, 052409\n(2003).\n14Y. Tserkovnyak, B. I. Halperin, A. A. Kovalev, and A.\nBrataas, Phys. Rev. B 76, 085319 (2007).\n15V. K. Dugaev, M. Inglot, E. Ya. Sherman, and J. Barna´ s,\nPhys. Rev. B 82, 121310(R) (2010).\n16M. M. Glazov, E. Ya. Sherman, and V. K. Dugaev, Physica\nE42, 2157 (2010).\n17K. Tsutsui, A. Takeuchi, G. Tatara, and S. Murakami, J.\nPhys. Soc. Jpn. 80, 084701 (2011).\n18J. Linder and T. Yokoyama, Phys. Rev. Lett. 106, 237201(2011).\n19W. -K. Tse and S. Das Sarma, Phys. Rev. Lett. 96, 056601\n(2006).\n20V. K. Dugaev, P. Bruno, and J. Barna´ s, Phys. Rev. B 64,\n144423 (2001).\n21V. K. Dugaev, P. Bruno, M. Taillefumier, B. Canals, and\nC. Lacroix, Phys. Rev. B 71, 224423 (2005).\n22K. Hosono, A. Yamaguchi, Y. Nozaki, and G. Tatara,\nPhys. Rev. B 83, 144428 (2011).\n23G. Tatara and P. Entel, Phys. Rev. B 78, 064429 (2008).\n24S. Hikami, A. I. Larkin, and Y. Nagaoka, Prog. Theor.\nPhys.63, 707 (1980).\n25E. I. Rashba, Sov. Phys. Solid State 2, 1109 (1960); Yu.\nA. Bychkov and E. I. Rashba, J. Phys. C 17, 6039 (1984).\n26V. M. Edelstein, Solid State Commun. 73, 233 (1990).\n27N. Nagaosa, J. Sinova, S. Onoda, A. H. MacDonald, and\nP. Ong, Rev. Mod. Phys. 82, 1539 (2010).\n28H. Hung and A. P. Jauho, Quantum Kinetics in Trans-\nport and Optics of Semiconductors (Springer-Verlag, Hei-\ndelberg, 1998).\nAppendix A: Calculation of local spin current\nWe perturbatively calculate the spin currents induced by the inhomo geneity of the SOC strength using the Keldysh\nGreen’s function formalism. We first treat the local spin current, w hich is locally driven by the external electric\nfield. The leading contribution of the spin current involves the first o rder with respect to the SOC strength, which\nis diagrammatically shown in Fig. 2. The local spin currents with α-component spin polarization flowing in the m\ndirection, i.e., ( jα\ns)local\nm= (jα\ns)sj\nm+(jα\ns)sk\nmthus read\n(jα\ns)sj\nm=i/planckover2pi12\n2m/planckover2pi1e/planckover2pi1\n2m/integraldisplaydΩ\n2π/integraldisplaydω\n2πΩeiΩt/summationdisplay\ni,j,k,l/summationdisplay\nk,k′,u,qei(u+q)·rλSO(u)/angb∇acketleftU(p)U(−p)/angb∇acket∇ightimpAl(q,Ω)ǫijk2ℜ(χαijklm),(A1)\n(jα\ns)sk\nm=i/planckover2pi12\n2me/integraldisplaydΩ\n2π/integraldisplaydω\n2πΩeiΩt/summationdisplay\ni,j,k/summationdisplay\nk,k′,u,qei(u+q)·rλSO(u)/angb∇acketleftU(p)U(−p)/angb∇acket∇ightimpAj(q,Ω)ǫijk2ℜ(χαikm),(A2)\nwhere the superscripts ”sj” and ”sk” denote the side-jump and t he skew-scattering contributions, respectively,27and\neach coefficients are given by\nχαijklm= +(k−u−k′)i(k+k′)j(2k−2u−q)l(2k−u−q)mtr/bracketleftbig\nσαgR\nkσkgR\nk′gR\nk−ugA\nk−u−q/bracketrightbig\n+(k′−k)i(k−u+k′)j(2k−2u−q)l(2k−u−q)mtr/bracketleftbig\nσαgR\nkgR\nk′σkgR\nk−ugA\nk−u−q/bracketrightbig\n+(k−u−k′)i(k+k′)j(2k′−q)l(2k−u−q)mtr/bracketleftbig\nσαgR\nkσkgR\nk′gA\nk′−qgA\nk−u−q/bracketrightbig\n, (A3)\nχαikm= (k−k′−u−q)i(2k−u−q)mtr/bracketleftbig\nσαgR\nkσkgA\nk′gA\nk−u−q/bracketrightbig\n. (A4)\nHere,gR(A)\nk≡gR(A)\nk,ω=0, andgR\nk,ωandgA\nk,ωdenote the non-perturbative retarded and advanced Green’s fu nc-\ntion, respectively, which are 2 ×2 matrices in spin space. We also use the Langreth’s method to obtain\n[gk1,ω···gki,ωgki+1,ω−Ω···gkn,ω−Ω]<≃ −Ωf′(ω)gR\nk1,ω···gR\nki,ωgA\nki+1,ω···gA\nkn,ωwithf(ω) the Fermi distribution\nfunction28. Here, we set temperature to zero, i.e., f′(ω)≃−δ(ω).\nSince we are interested in spin currents produced by the inhomogen eous SOC strength, we focus on the contribution\nwhich involves the spatial derivative of the SOC strength. After so me algebraic calculations, we obtain the above9\ncoefficients represented as\nχαijklm≃2\n3k2(δjluiqm+δjmuiql)/parenleftbig\ntr/bracketleftbig\nσαgR\nkσkgR\nk′gR\nkgA\nk/bracketrightbig\n+tr/bracketleftbig\nσαgR\nkgR\nk′σkgR\nkgA\nk/bracketrightbig/parenrightbig\n+4\n15/planckover2pi12\nmk4(δjluiqm+δjmuiql+δlmuiqj)/parenleftbig\ntr/bracketleftbig\nσαgR\nkσkgR\nk′gR\nk(gA\nk)2/bracketrightbig\n+tr/bracketleftbig\nσαgR\nkgR\nk′σkgR\nk(gA\nk)2/bracketrightbig/parenrightbig\n+2\n3/parenleftbig\nk′2δjluiqm+k2δjmuiql/parenrightbig\ntr/bracketleftbig\nσαgR\nkσkgR\nk′gA\nk′gA\nk/bracketrightbig\n+8\n9/planckover2pi12\nmk2k′2δjluiqmtr/bracketleftbig\nσαgR\nkσkgR\nk′gA\nk′(gA\nk)2/bracketrightbig\n+4\n9/planckover2pi12\nmk2k′2δjmuiqltr/bracketleftbig\nσαgR\nkσkgR\nk′(gA\nk′)2gA\nk/bracketrightbig\n,(A5)\nχαikm≃(uiqm+qium)/parenleftbigg\ntr/bracketleftbig\nσαgR\nkσkgA\nk′gA\nk/bracketrightbig\n+/planckover2pi12\nmk2tr/bracketleftbig\nσαgR\nkσkgA\nk′(gA\nk)2/bracketrightbig/parenrightbigg\n. (A6)\nHere, weassumedtherotationalsymmetryofthesystem, i.e.,/summationtext\nkkikj=/summationtext\nkk2\n3δijand/summationtext\nkkikjklkm=/summationtext\nkk4\n15(δijδlm+\nδilδjm+δimδjl). We remark that if we do not consider the inhomogeneity of the ext ernal electric field, the present\ncontributions vanish. The inhomogeneity of the external electric fi eld is required to obtain the contributions from the\ninhomogeneous SOC strength.\nNext, we take traces over spin space in Eqs. (A5) and (A6) by using the formula tr[ σαAσkB] =/summationtext\nσ=±1(δαk−\nδαzδkz−iσǫαzk)AσB¯σ+/summationtext\nσ=±1δαzδkzAσBσwithA,BGreen’s functions, and then Eqs. (A1) and (A2) reduce to\n(jα\ns)sj\nm≃e/planckover2pi12\n2mu2\n0nimp/integraldisplaydΩ\n2πΩeiΩt/summationdisplay\nu,q/summationdisplay\ni,j,k,lei(u+q)·rλSO(u)Al(q,Ω)ǫijk/summationdisplay\nσ((δαk−δαzδkz)ℜ+σǫαkℑ)\n×/bracketleftbigg2\n3δjluiqm/parenleftbigg\n(Iσ\n01;00+Iσ\n10;00)/parenleftbigg\nQσ\n11;01+4\n5Sσ\n11;02/parenrightbigg\n+Qσ\n01;01Iσ\n10;01+8\n3Qσ\n01;01Qσ\n10;02/parenrightbigg\n+2\n3δjmuiql/parenleftbigg/parenleftbig\nIσ\n01;00+Iσ\n10;00/parenrightbig/parenleftbigg\nQσ\n11;01+4\n5Sσ\n11;02/parenrightbigg\n+Iσ\n01;01Qσ\n10;01+4\n3Qσ\n01;02Qσ\n10;01/parenrightbigg\n+8\n15δlmuiqj(Iσ\n01;00+Iσ\n10;00)Sσ\n11;02/bracketrightbigg\n+e/planckover2pi12\n2mu2\n0nimp/integraldisplaydΩ\n2πΩeiΩt/summationdisplay\nu,q/summationdisplay\ni,j,k,lei(u+q)·rλSO(u)Al(q,Ω)ǫijk/summationdisplay\nσδαzδkzℜ\n×/bracketleftbigg2\n3δjluiqm/parenleftbigg\n2Iσ\n10;00/parenleftbigg\nQσ\n20;10+4\n5Sσ\n20;20/parenrightbigg\n+Qσ\n10;10Iσ\n10;10+8\n3Qσ\n10;10Qσ\n10;20/parenrightbigg\n+2\n3δjmuiql/parenleftbigg\n2Iσ\n10;00/parenleftbigg\nQσ\n20;10+4\n5Sσ\n20;20/parenrightbigg\n+Iσ\n10;10Qσ\n10;10+4\n3Qσ\n10;20Qσ\n10;10/parenrightbigg\n+16\n15δlmuiqjIσ\n10;00Sσ\n20;20/bracketrightbigg\n, (A7)\n(jα\ns)sk\nm≃e/planckover2pi12\n2mu2\n0nimp/integraldisplaydΩ\n2πΩeiΩt/summationdisplay\ni,j,k/summationdisplay\nu,qei(u+q)·rλSO(u)Aj(q,Ω)ǫijk/bracketleftBigg/summationdisplay\nσ(δαk−δαzδkz)ℜ+σǫαkℑ)\n×uiqmIσ\n00;01(Iσ\n10;01+2Qσ\n10;02)+/summationdisplay\nσδαzδkzuiqmℜIσ\n00;10(Iσ\n10;10+2Qσ\n10;20)/bracketrightBigg\n. (A8)\nHere, we define integrals of the Green’s functions appearing in the a bove expressions as\nIσ\nab;cd≡/summationdisplay\nk(gR\nk,σ)a(gR\nk,¯σ)b(gA\nk,σ)c(gA\nk,¯σ)d, (A9)\nQσ\nab;cd≡/summationdisplay\nkǫk(gR\nk,σ)a(gR\nk,¯σ)b(gA\nk,σ)c(gA\nk,¯σ)d, (A10)\nSσ\nab;cd≡/summationdisplay\nkǫ2\nk(gR\nk,σ)a(gR\nk,¯σ)b(gA\nk,σ)c(gA\nk,¯σ)d, (A11)\nwheregR\nk,σ= [ǫFσ−ǫk+iησ]−1withησ≡/planckover2pi1/2τσandgA\nk,σ= (gR\nk,σ)∗withǫk≡/planckover2pi12k2\n2m. WesumupEqs. (A7)and(A8)and\nconsider the dominant contribution from the integrals of the Green ’s functions under the condition ǫFσ≫/planckover2pi1/2τσ. By10\ncarrying out the integrals of the Green’s functions in Eqs.(A9-A11) and transforming to the real-space representation,\nwe obtain the local spin current of the form\n(jα\ns)m≃πe/planckover2pi12\n2m(ν↑+ν↓)[(α1eα+α2(eα×ez))·(∇λSO(r)×∂mEem(r))\n+(β1(em×eα)+β2(δmzeα−δmαez))·∇λSO(r)(∇·Eem(r))\n+(γ1eα+γ2(eα×ez))·(∇λSO(r)×∇Em\nem(r))]. (A12)\nHere, each coefficients are dimensionless and given by\nαi≡γi+δi, (A13)\nβi≡γi+ǫi, (A14)\nδ1≡/braceleftBigg\n16\n9/parenleftBig\nǫFM2η+\n(M2+η2\n+)2−1\n41\nν↑+ν↓η+\nM2+η2\n+/summationtext\nσǫFσν¯σ/parenrightBig\n,(α=x,y)\n−4\n9ǫF\nη+,(α=z)(A15)\nδ2≡16\n9/parenleftBigg\nǫFM(M2−η2\n+)\n(M2+η2\n+)2−1\n41\nν↑+ν↓M\nM2+η2\n+/summationdisplay\nσǫFσν¯σ/parenrightBigg\n, (A16)\nǫ1≡/braceleftBigg\n8\n9/parenleftBig\nM\nM2+η2\n+/summationtext\nσσǫFσǫFσ\nησ+2\nν↑+ν↓/summationtext\nσǫFσ\nησν¯σ/parenrightBig\n,(α=x,y)\n16\n9ǫFσ\nησ,(α=z)(A17)\nǫ2≡8\n9η+\nM2+η2\n+/summationdisplay\nσσǫFσǫFσ\nησ, (A18)\nγ1≡\n\n2\n15/bracketleftbigg\nMη+\nM2+η2\n+/summationtext\nσσ/parenleftBig\nǫFσ\nησ/parenrightBig2\n+Mη−\nM2+η2\n−/summationtext\nσ/parenleftBig\nǫFσ\nησ/parenrightBig2\n−Mη2\n+\n(M2+η2\n+)2/summationtext\nσσǫFσǫFσ\nησ−ǫFη+\nM2+η2\n+/bracketrightbigg\n,(α=x,y)\n−2\n15ǫF\nη+,(α=z)(A19)\nγ2≡−2\n15/bracketleftBigg\nM2(η2\n+−η2\n−)\n(M2+η2\n+)(M2+η2\n−)/summationdisplay\nσσ/parenleftbiggǫFσ\nησ/parenrightbigg2\n+η+(M2−η2\n+)\n(M2+η2\n+)2/summationdisplay\nσσǫFσǫFσ\nησ+ǫFM\nM2+η2\n+/bracketrightBigg\n, (A20)\nandη±≡η↑±η↓\n2.\nAppendix B: Calculation of diffusive spin current\nWe calculate the diffusive (or non-local) spin current, which is repres ented by the vertex correction to the local spin\ncurrent. Thiscanbeperformedbyreplacing χαijklmandχαikminEqs. (A1)and(A2)with ˜ χαijklm= Γm\nu,qΠu,qχ(1)\nαijkl+\nΓl\nqΠqχ(2)\nαijkmand ˜χαikm= Γm\nu,qΠu,qχαik, respectively, where\nΓm\nσ,¯σ(u) =u2\n0nimp\nV/summationdisplay\nk(2k−u)mgR\nk,ω,σgA\nk−u,ω−Ω,¯σ, (B1)\nΠσ,¯σ(u) =∞/summationdisplay\nn=0/parenleftBigg\nu2\n0nimp\nV/summationdisplay\nkgR\nk,ω,σgA\nk−u,ω−Ω,¯σ/parenrightBiggn\n, (B2)\n(B3)\nχ(1)\nαijkl= +(k−u−k′)i(k+k′)j(2k−2u−q)ltr/bracketleftbig\nσαgR\nkσkgR\nk′gR\nk−ugA\nk−u−q/bracketrightbig\n+(k′−k)i(k−u+k′)j(2k−2u−q)ltr/bracketleftbig\nσαgR\nkgR\nk′σkgR\nk−ugA\nk−u−q/bracketrightbig\n+(k−u−k′)i(k+k′)j(2k′−q)ltr/bracketleftbig\nσαgR\nkσkgR\nk′gA\nk′−qgA\nk−u−q/bracketrightbig\n, (B4)\nχ(2)\nαijkm= +(k−u−k′)i(k+k′)j(2k−u−q)mtr/bracketleftbig\nσαgR\nkσkgR\nk′gR\nk−ugA\nk−u−q/bracketrightbig\n+(k′−k)i(k−u+k′)j(2k−u−q)mtr/bracketleftbig\nσαgR\nkgR\nk′σkgR\nk−ugA\nk−u−q/bracketrightbig\n+(k−u−k′)i(k+k′)j(2k−u−q)mtr/bracketleftbig\nσαgR\nkσkgR\nk′gA\nk′−qgA\nk−u−q/bracketrightbig\n, (B5)\nχαik= (k−k′−u−q)itr/bracketleftbig\nσαgR\nkσkgA\nk′gA\nk−u−q/bracketrightbig\n. (B6)11\nExpanding with respect to uorq, we obtain the leading contributions as follows\nχ(1)\nαijkl≃−2\n3k2δjlui/parenleftbig\ntr/bracketleftbig\nσαgR\nkσkgR\nk′gR\nkgA\nk/bracketrightbig\n+tr/bracketleftbig\nσαgR\nkgR\nk′σkgR\nkgA\nk/bracketrightbig/parenrightbig\n−2\n3k′2δjluitr/bracketleftbig\nσαgR\nkσkgR\nk′gA\nk′gA\nk/bracketrightbig\n−4\n9/planckover2pi12\nmk2k′2δjl(u+q)itr/bracketleftbig\nσαgR\nkσkgR\nk′gA\nk′(gA\nk)2/bracketrightbig\n, (B7)\nχ(2)\nαijkm≃−2\n3k2δjmui/parenleftbig\ntr/bracketleftbig\nσαgR\nkσkgR\nk′gR\nkgA\nk/bracketrightbig\n+tr/bracketleftbig\nσαgR\nkgR\nk′σkgR\nkgA\nk/bracketrightbig/parenrightbig\n−2\n3k2δjmuitr/bracketleftbig\nσαgR\nkσkgR\nk′gA\nk′gA\nk/bracketrightbig\n−4\n9/planckover2pi12\nmk2k′2δimqjtr/bracketleftbig\nσαgR\nkσkgR\nk′(gA\nk′)2gA\nk/bracketrightbig\n, (B8)\nχαik≃−(u+q)i/parenleftbigg\ntr/bracketleftbig\nσαgR\nkσkgA\nk′gA\nk/bracketrightbig\n+1\n3/planckover2pi12\nmk2tr/bracketleftbig\nσαgR\nkσkgA\nk′(gA\nk)2/bracketrightbig/parenrightbigg\n. (B9)\nNext, we take traces over spin space in Eqs. (B7), (B8) and (B9) in a similar manner to the local spin current, and\nthen the diffusive spin currents are reduced to\n(jα\ns)sj\nm≃e/planckover2pi12\n2mu2\n0nimp/integraldisplaydΩ\n2πΩeiΩt/summationdisplay\nu,q/summationdisplay\ni,j,k,lei(u+q)·rλSO(u)Al(q,Ω)ǫijk/summationdisplay\nσ((δαk−δαzδkz)ℜ+σǫαkℑ)\n×/bracketleftbigg2\n3δjluiqmΠσ,¯σ(u+q)u2\n0nimpIσ\n10;01/parenleftbigg\n(Iσ\n01;00+Iσ\n10;00)Qσ\n11;01+Qσ\n01;01Iσ\n10;01+4\n3Qσ\n01;01Qσ\n10;02/parenrightbigg\n+2\n3δjmuiqlΠσ,σ(q)u2\n0nimpIσ\n10;10/parenleftbig\n(Iσ\n01;00+Iσ\n10;00)Qσ\n11;01/parenrightbig\n+Iσ\n01;01Qσ\n10;01/bracketrightbigg\n+e/planckover2pi12\n2mu2\n0nimp/integraldisplaydΩ\n2πΩeiΩt/summationdisplay\nu,q/summationdisplay\ni,j,k,lei(u+q)·rλSO(u)Al(q,Ω)ǫijk/summationdisplay\nσδαzδkzℜ\n×/bracketleftbigg2\n3δjluiqmΠσ,σ(u+q)u2\n0nimpIσ\n10;10/parenleftbigg\n2Iσ\n10;00Qσ\n20;10+Qσ\n10;10Iσ\n10;10+4\n3Qσ\n10;10Qσ\n10;20/parenrightbigg\n+2\n3δjmuiqlΠσ,σ(q)u2\n0nimpIσ\n10;10/parenleftbig\n2Iσ\n10;00Qσ\n20;10+Iσ\n10;10Qσ\n10;10/parenrightbig/bracketrightbigg\n, (B10)\n(jα\ns)sk\nm≃e/planckover2pi12\n2mu2\n0nimp/integraldisplaydΩ\n2πΩeiΩt/summationdisplay\ni,j,k/summationdisplay\nu,qei(u+q)·rλSO(u)Aj(q,Ω)ǫijk\n/bracketleftBigg/summationdisplay\nσ(δαkℜ+σǫαkℑ)(uiqm+qium) Πσ,¯σ(u+q)u2\n0nimpIσ\n10;01/parenleftbigg\nIσ\n00;01Iσ\n10;01+2\n3Iσ\n00;01Qσ\n10;02/parenrightbigg\n+/summationdisplay\nσδαzδkzℜ(uiqm+qium) Πσ,σ(u+q)u2\n0nimpIσ\n10;10/parenleftbigg\nIσ\n00;10Iσ\n10;10+2\n3Iσ\n00;10Qσ\n10;20/parenrightbigg/bracketrightBigg\n. (B11)\nHere\nΠσ,¯σ(u)≃/bracketleftbigg\n1−/parenleftbigg\nu2\n0nimpIσ\n10;01+/planckover2pi1Ωu2\n0nimpIσ\n10;02+4/planckover2pi12\n2m1\n3u2u2\n0nimpQσ\n10;03/parenrightbigg/bracketrightbigg−1\n, (B12)\nΠσ,σ(u)≃/bracketleftbigg\n1−/parenleftbigg\nu2\n0nimpIσ\n10;10+/planckover2pi1Ωu2\n0nimpIσ\n10;20+4/planckover2pi12\n2m1\n3u2u2\n0nimpQσ\n10;30/parenrightbigg/bracketrightbigg−1\n. (B13)\nBy carrying out the above integrals of the Green’s functions and tr ansforming to the real-space representation using\nuλSO(u) =−i/integraltext\ndr′∇r′λSO(r′)e−ir′·uand ΩqmA(q,Ω) =−/integraltext\ndt′/integraltext\ndr′′∂′′\nm˙A(r′′,t′)e−ir′′·qe−iΩt′, we obtain the12\ndiffusive spin current of the form\n(jα\ns)m≃πe/planckover2pi12\n2m(ν↑+ν↓)\n×/bracketleftBigg\n(δαx+δαy)/summationdisplay\nσ(eαℜασ+(eα×ez)ℑασ)·/integraldisplay\ndr′/integraldisplay\ndr′′/integraldisplay\ndt′\n/integraldisplaydΩ\n2π/summationdisplay\nu,qeiΩ(t−t′)+iu·(r−r′)+iq·(r−r′′)\nFσ−iGσ(∇λSO(r′)×∂mEem(r′′,t′))\n+δαz/summationdisplay\nσασez·/integraldisplay\ndr′/integraldisplay\ndr′′/integraldisplay\ndt′\n/integraldisplaydΩ\n2π/summationdisplay\nu,qeiΩ(t−t′)+iu·(r−r′)+iq·(r−r′′)\n(Dσ(u+q)2−iΩ)τσ(∇λSO(r′)×∂mEem(r′′,t′))\n+/summationdisplay\nσ(β1,σ(ei×eα)+β2,σ(δizeα−δiαez))·∇λSO(r)/integraldisplay\ndr′/integraldisplay\ndt′/integraldisplaydΩ\n2π/summationdisplay\nqeiΩ(t−t′)+iq·(r−r′)\n(Dσq2−iΩ)τσ∇·Eem(r′,t′)/bracketrightBigg\n,\n(B14)\nwhere\nασ≡/braceleftBigg\n2\n9Mη+((1+σ)η2\n+−M2)\n(M2+η2\n+)3ǫ2\nFσ+i2\n9η2\n+((1+σ)M2−η2\n+)\n(M2+η2\n+)3ǫ2\nFσ,(α=x,y)\n−1\n9ǫFσ\nη+,(α=z)(B15)\nβ1,σ≡/braceleftBigg\n−2\n3ǫF¯στ¯σ\n/planckover2pi1M2(η2\n+−η2\n−)\n(M2+η2\n+)(M2+η2\n−)+π2τσνσ\n/planckover2pi1ν↑ν↓ǫFσ\nησ,(α=x,y)\nπ2τσνσ\n/planckover2pi1ν↑ν↓ǫFσ\nησ,(α=z)(B16)\nβ2,σ≡2\n3ǫF¯στ¯σ\n/planckover2pi1M(η++ ¯ση−)(η+η−+ ¯σM2)\n(M2+η2\n+)(M2+η2\n−), (B17)\nFσ≡M2\nM2+η2\n++2σMηση2\n+\n(M2+η2\n+)2Ωτσ+η2\nση2\n+(η2\n+−3M2)\n(M2+η2\n+)32ǫF¯στσ\n3m(u+q)2τσ, (B18)\nGσ≡σMη+\nM2+η2\n++ηση+(η2\n+−M2)\n(M2+η2\n+)2Ωτσ+σMη2\nση+(3η2\n+−M2)\n(M2+η2\n+)32ǫF¯στσ\n3m(u+q)2τσ, (B19)\nDσ≡2ǫFτσ\n3m. (B20)\nAppendix C: Calculation of spin current driven by spin accum ulation\nThe spin current shown in Fig. 2 (a) and (b) reads\n(jα\ns)m=−/planckover2pi12\n2m/planckover2pi1/integraldisplaydΩ\n2πΩeiΩt/summationdisplay\ni,j,k/summationdisplay\nk,k′,u,qei(u+q)·rλSO(u)/angb∇acketleftU(p)U(−p)/angb∇acket∇ightimpµs(q,Ω)ǫijktr/bracketleftbig\nσασkσz/bracketrightbig\n2ℜ(χαijkm)\n=−4i/planckover2pi12\n2m/planckover2pi1u2\n0nimp/integraldisplaydω\n2π/summationdisplay\ni,j/summationdisplay\nk,k′,u,qei(u+q)·rλSO(u)µs(q,Ω)(δizδjα−δiαδjz)ℜ(χαijkm), (C1)\nχαijkm= (k−u−k′)i(k+k′)j(2k−u−q)mgR\nkgR\nk′gR\nk−ugA\nk−u−q+(k′−k)i(k−u+k′)j(2k−u−q)mgR\nkgR\nk′gR\nk−ugA\nk−u−q\n+(k−u−k′)i(k+k′)j(2k−u−q)mgR\nkgR\nk′gA\nk′−qgA\nk−u−q. (C2)\nAs for the above coefficient χαijkm, the contribution involving two q’s and one uis calculated as follows;\nχ(1,2)\nαijkm=−2\n3/planckover2pi12\n2m/bracketleftbig\n2k2uiqmqjgR\nk′(gR\nk)2(gA\nk)2+k2uiqmqjgR\nkgR\nk′gA\nk′(gA\nk)2+k′2uiqjqmgR\nkgR\nk′(gA\nk′)2gA\nk/bracketrightbig\n−8\n9/parenleftbigg/planckover2pi12\n2m/parenrightbigg2\nk2k′2qjumqigR\nk′(gA\nk′)2gR\nk(gA\nk)2. (C3)13\nThe resultant spin current reduces to\n(jα\ns)(1,2)\nm=−4/planckover2pi12\n2m/planckover2pi1u2\n0nimp/integraldisplaydΩ\n2πΩeiΩt/summationdisplay\nu,qei(u+q)·rλSO(u)µs(q,Ω)(uαqmqz−uzqmqα)4\n3ℜ/bracketleftbig\nIσ\n10;00Qσ\n20;20+Iσ\n10;10Qσ\n10;20/bracketrightbig\n.\n(C4)\nNow we perform the vertex correction χαijkmtr/bracketleftbig\nσασkσz/bracketrightbig\n→˜χαijk/summationtext\na,b,c,d,eσα\ndaΓad,cbσk\nbeσz\necwith Γ ad,cbbeing the\nvertexcorrectioncorrespondingto the ladderdiagramincluding th e normalimpurity and the SOC due to impurities22,\nand\n˜χαijk= (k−u−k′)i(k+k′)jgR\nkgR\nk′gR\nk−ugA\nk−u−q+(k′−k)i(k−u+k′)jgR\nkgR\nk′gR\nk−ugA\nk−u−q\n+(k−u−k′)i(k+k′)jgR\nkgR\nk′gA\nk′−qgA\nk−u−q. (C5)\nAccording to Ref. 22, the vertex correction is calculated as Γ ad,cb(q,Ω) = Γ C(q,Ω)δadδcb+ΓS(q,Ω)/summationtext\nlσl\nadσl\ncbwith\nΓC(q,Ω) =1+3κ\n2(Dq2+iΩ)τ, (C6)\nΓS(q,Ω) =(1−κ)(1+3κ)\n2(4κ+(Dq2+iΩ)τ), (C7)\nleading to\n/summationdisplay\na,b,c,d,eσα\ndaΓad,cbσk\nbeσz\nec= ΓCtr[σαˆ1]tr[ˆ1σkσz]+/summationdisplay\nlΓStr[σασl]tr[σlσkσz]\n= 4iǫαkzΓS. (C8)\nHere,τ≡/planckover2pi1/2πu2\n0nimpν,κ≡λ2\nSOk4\nF/3 (kFis the Fermi wave number), D≡2ǫFτ/3m, andξ≡/radicalbig\n4κ/Dτ.ˆ1 is the\nidentity matrix. Therefore, we obtain the spin current including the vertex correction of the form\n(˜jαs)(1,2)\nm≃4π/planckover2pi12τν\n3m/integraldisplay\ndr′/integraldisplay\ndr′′/integraldisplay\ndt′/integraldisplaydΩ\n2π/summationdisplay\nu,qeiΩ(t−t′)+iu·(r−r′)+iq·(r−r′′)Ω/τ(1−κ)(1+3κ)\nD((u+q)2+ξ2)+iΩ\n×(∂αλSO(r′)∂z(∂mµs(r′′,t′))−∂zλSO(r′)∂α(∂mµs(r′′,t′))), (C9)\nwithνbeing the density of state. We remark that the spin-diffusion propa gator is given by\nD(r1,r2,t) :=/integraldisplaydΩ\n2π/summationdisplay\nu,qeiΩt+iu·r1+iq·r2Ω/τ(1−κ)(1+3κ)\nD((u+q)2+ξ2)+iΩ\n=−δ(r2−r1)/radicalbiggπ\nDt/parenleftbigg\nDξ2−|r1|2\n4Dt2+1\n2t/parenrightbigg\nexp/bracketleftbigg\n−/parenleftbigg\nDξ2t+|r1|2\n4Dt/parenrightbigg/bracketrightbigg\n. (C10)\nOn the other hand, the contribution involving two u’s and one qis given as follows;\nχ(2,1)\nαijkm=−4\n3/planckover2pi12\n2mk2uiujqm[gR\nkgR\nk′gR\nk)2gA\nk−8\n15/parenleftbigg/planckover2pi12\n2m/parenrightbigg2\nk4(uiujqm+uiuoqoδjm+uiumqj)gR\nkgR\nk′(gR\nk)2(gA\nk)2\n−4\n3/planckover2pi12\n2mk2ui(umqj+qmuj)gR\nkgR\nk′gR\nk(gA\nk)2−1\n3k2(uiumqj+uiujqm)gR\nkgR\nk′gA\nk′(gA\nk)2\n−1\n3k′2uiumqjgR\nkgR\nk′(gA\nk′)2gA\nk−4\n9k2k′2uiumqjgR\nkgR\nk′(gA\nk′)2(gA\nk)2. (C11)\nThe corresponding spin current reduces to\n(jα\ns)(2,1)\nm=−4/planckover2pi12\n2m/planckover2pi1u2\n0nimp/integraldisplaydΩ\n2πΩeiΩt/summationdisplay\nu,qei(u+q)·rλSO(u)µs(q,Ω)/summationdisplay\ni,j,kǫijkǫαkz\n×/parenleftbigg\nuiumqjℜ/bracketleftbigg\n−4\n3Iσ\n10;00Qσ\n20;20−4\n3Iσ\n10;10Qσ\n10;20−16\n9(Qσ\n10;20)2−16\n15Iσ\n10;00Sσ\n30;20/bracketrightbigg\n+uiuoqoδjmℜ/bracketleftbigg\n−16\n15Iσ\n10;00Sσ\n30;20/bracketrightbigg/parenrightbigg\n.\n(C12)\nBy performing the vertex correction, we obtain the final express ion of the spin current\n(˜jαs)(2,1)\nm≃152π/planckover2pi12τν\n45m/parenleftbigg2ǫFτ\n/planckover2pi1/parenrightbigg2/integraldisplay\ndr′/integraldisplay\ndr′′/integraldisplay\ndt′/integraldisplaydΩ\n2π/summationdisplay\nu,qeiΩ(t−t′)+iu·(r−r′)+iq·(r−r′′)Ω/τ(1−κ)(1+3κ)\nD((u+q)2+ξ2)+iΩ\n×(∂αµs(r′,t′)∂z(∂mλSO(r′′))−∂zµs(r′,t′)∂α(∂mλSO(r′′))). (C13)"    },    {        "title": "1202.1604v1.Role_of_spin_orbit_coupling_on_the_electronic_structure_and_properties_of_SrPtAs.pdf",        "content": "arXiv:1202.1604v1  [cond-mat.supr-con]  8 Feb 2012Role of spin-orbit coupling on the electronic structure and properties of SrPtAs\nS. J. Youn1,2, S. H. Rhim2, D. F. Agterberg3, M. Weinert3, A. J. Freeman2\n1Department of Physics Education and Research Institute of N atural Science,\nGyeongsang National University, Jinju 660-701, Korea\n2Department of Physics and Astronomy, Northwestern Univers ity, Evanston, Illinois, 60208-3112, USA and\n3Department of Physics, University of Wisconsin-Milwaukee , Milwaukee, WI 53201-0413, USA\n(Dated: November 9, 2018)\nThe effect of spin-orbit coupling on the electronic structur e of the layered iron-free pnictide su-\nperconductor, SrPtAs, has been studied using the full poten tial linearized augmented plane wave\nmethod. The anisotropy in Fermi velocity, conductivity and plasma frequency stemming from the\nlayered structure are found to be enhanced by spin-orbit cou pling. The relationship between spin-\norbit interaction and the lack of two-dimensional inversio n in the PtAs layers is analyzed within\na tight-binding Hamiltonian based on the first-principles c alculations. Finally, the band structure\nsuggests that electron doping could increase Tc.\nPACS numbers: 74.20.Pq,74.70.Xa,71.20.-b,71.18.+y\nI. INTRODUCTION\nThe recent discovery of superconductivity in pnictides\nhas attracted extensive attention owing to their surpris-\ningly high Tc.1While the highest Tcso far is 56 K for\nGdFeAsO,2a consensus on the pairing mechanism has\nnot yet been reached.3This class of materials share a\ncommon crystal structure, that is, the Fe square lat-\ntice. While most of the superconducting pnictides are\nFe-based, pnictides without iron also exhibit supercon-\nductivity, although Tcis drastically lower than with iron.\nRecently, another superconducting pnictide, SrPtAs, has\nbeen discovered, which is the first non-Fe based super-\nconductor with a hexagonal lattice rather than square\nlattice.4Although Tc=2.4K is lower than those of Fe-\nbased pnictides, it possesses interesting physics associ-\nated with its hexagonal crystal structure.\nSrPtAs crystallizes in a hexagonal lattice of ZrBeSi\ntype with space group P63/mmc(No.194, D4\n6h) — the\nsame(non-symmorphic)spacegroupasthe hcp structure\n— with two formula units per primitive cell. As depicted\nin Fig. 1(a), its structure resembles5that of MgB 2(with\nthe symmorphic space group P6/mmm, No.191, D1\n6h),\nwith a double unit cell along the caxis: the boron layers\nof MgB 2are replaced by PtAs layers, rotated by 60◦in\nsuccessivelayers(responsibleforthenon-symmorphicna-\nture) and Mg is replaced by Sr. Although the crystal as a\nwhole has inversion symmetry (as do the Sr atoms), the\nindividual PtAs layers lack two-dimensional inversion.\nThus, SrPtAs differs from MgB 2in two significant\nways: (i) it exhibits strong spin-orbit coupling (SOC)\nat the Pt ions and ( ii) the PtAs layers individually\nbreak inversion symmetry, exhibiting only C3v(orD3h\nifz-reflection is included) symmetry. These two prop-\nerties play an important role in determining the band\nstructure and also affect the superconducting state. As-\nsuming that the superconductivity is largely determined\nby the two-dimensional PtAs layers, the lack of in-\nversion symmetry in the individual layers (which we\ncall “broken local inversion symmetry”) opens up thepossibility to see the unusual physics associated with\nnon-centrosymmetric superconductors.6With large spin-\norbit coupling, nominally s-wave non-centrosymmetric\nsuperconductors exhibit spin-singlet and spin-triplet\nmixing,7,8enhanced spin susceptibilities,7,9enhanced\nPauli limiting fields,6non-trivial magnetoelectric effects\nand Fulde-Ferrell-Larkin-Ovchinnikov(FFLO)-like states\nin magnetic fields,10–16and Majoranamodes.17The local\ninversion symmetry breaking, together with a SOC that\nis comparable to the c-axis coupling, suggests that SrP-\ntAs will provide an ideal model system to explore related\neffects in centrosymmetric superconductors.18,19\nInthispaperwediscusstheelectronicstructureofSrP-\ntAs, including spin-orbit coupling, which was neglected\nin a previous theoretical study.20In Sec. II, we describe\ndetails of the calculations. The effects of SOC on the\nbands, the Fermi surface, density ofstates, and transport\nproperties at the Fermi surface of SrPtAs are presented\nin Sec. III, along with a tight-binding analysis. Finally,\nwe suggest an enhanced Tcmight be possible via doping.\nFIG. 1. (a) (Color online) Crystal structure of SrPtAs, wher e\nred, blue, and grey spheres denote Pt, As, and Sr atoms,\nrespectively. (b) Brillouin zone of SrPtAs and high symmetr y\nkpoints.2\nII. METHOD\nFirst-principles calculations are performed using\nthe full-potential linearized augmented plane wave\n(FLAPW) method21,22and the local density approxi-\nmation (LDA) for the exchange-correlation functional of\nHedin and Lundqvist.23Then, SOC is included by a\nsecond variational method.24Experimental lattice con-\nstants,a= 4.244˚A andc= 8.989˚A, are used.25Cutoffs\nused for wave function and potential representations are\n196 eV and 1360 eV, respectively. Muffin-tin radii are\n2.6, 2.4, and 2.1 a.u for Sr, Pt, and As, respectively.\nSemicore electrons such Sr 4 pand As 3 dare treated as\nvalence electrons, which are explicitly orthogonalized to\nthe core states.26Brillouin zone summations were done\nwith 90kpoints in the Monkhorst-Pack scheme,27while\nthe density of states are obtained by the tetrahedron\nmethod.28Although most of the calculations were per-\nformed using LDA, some results were also done with the\nGGA as well.29In order to calculate the Fermi velocity\nand plasma frequency for in-plane and out-of-plane con-\ntributions, eigenvalues from self-consistent calculations\nare fitted by a spline method over the whole Brillouin\nzone.30–32\nIII. RESULTS\nThe band structure of SrPtAs is presented in Fig. 2\nalong the symmetry lines shown in Fig. 1(b). Plots in\nthe left (right) column are without (with) spin-orbit cou-\npling, and plots in the upper and lowerrowsare the same\nbut highlighted for As pand Ptdorbitals, respectively.\nOur energy bands and Fermi surfaces without SOC agree\nwith those of Ref. [20]. The main band of Sr 5 sorigin is\nlocated far above the Fermi level ( EF), consistent with\nZintl’s scheme33that Sr donates electrons to the PtAs\nlayer and behaves almost like an inert Sr2+ion.Without\nSOC, bands on the zone boundary face, the kz=π/c\nplane(H−A−L−H), exhibit four fold degeneracy —\ntwo from spin and the other two from two different PtAs\nlayers — as a consequence of the non-symmorphic trans-\nlations along the c-axis, just as for the hcp structure.\nFor thekz= 0 plane ( K−Γ−M−K), there is no such\nsymmetry-dictated degeneracy due to the two equivalent\nlayers, but instead the magnitude of splitting is propor-\ntional to the inter-layer coupling. With SOC, the bands\nchange markedly: The four-fold degeneracy on the zone\nboundary face is reduced to a two fold pseudospin degen-\neracy due to inversion symmetry, whereas bands along\ntheA−Lline keeps the fourfold degeneracy as a conse-\nquence of time-reversal symmetry.34\nIn a simple atomic picture, the SOC Hamiltonian is\nHsoc=δL·σ,whereδrepresentsthestrengthoftheSOC.\nValues of the SOC strength derived from the calculations\nareδPt=0.32 eV for the Pt dorbitals and δAs=0.23 eV\nfor Asporbitals, which are used in laterdiscussions. The\nSOC splitting at the Apoint for Pt dxz,dyz(dxy,dx2−y2)orbitalsis0.59(0.54)eV,whereasforAs px,ythesplitting\n0.28 eV.\nTogaininsightintothe effectofspin-orbitcoupling, we\nconsider a simple tight-binding theory for a single PtAs\nlayer. A single PtAs layer lacks a center of inversion\nsymmetry and therefore allows an anti-symmetric spin-\norbit coupling of the form\nHso=/summationdisplay\nk,s,s′gk·σss′c†\nkscks′(1)\nexists, where c†\nks(cks) creates (annihilates) an elec-\ntron with momentum kand pseudo-spin s, andσde-\nnote the Pauli matrices. Time-reversal symmetry im-\nposesgk=−g−k. Here we find the form of gk\nthrough a consideration of the coupling between the Pt\ndx2−y2,dxyorbitals and the As px,pyorbitals. These\norbitals give rise to the Fermi surfaces with cylindrical\ntopology around the Γ– Aline. On the Pt sites, we define\nstates|d±,s >=|dx2−y2,s >±i|dxy,s >and on the As\nsites we define the states |p±,s >=|px,s >±i|py,s >(s\ndenotes spin). On a single Pt or As site, the spin-orbit\ncoupling L·Shas only LzSzwith non-zero matrix ele-\nments in these subspaces. This splits the local four-fold\ndegeneracyinto twopairs so that forPt (As) |d+,↑>and\n|d−,↓>(|p+,↑>and|p−,↓>) form one degenerate pair\nwhile|d+,↓>and|d−,↑>(|p−,↑>and|p+,↓>) form\nthe other degenerate pair. For both pairs, time-reversal\nsymmetry is responsible for the degeneracy and we can\n(a)\np ,p  x   ypzsEnergy (eV)\n-6-4-2024\nH K ALHK ΓM\n(c)\nd    , d xy      x -y          2   2d   , d   xz      yzdz2\nsEnergy (eV)\n-6-4-2024\nH K ALHK ΓM(b)\nEnergy (eV)\n-6-4-2024\nH K ALHK ΓM\n(d)\nEnergy (eV)\n-6-4-2024\nH K ALHK ΓM\nFIG. 2. (Color online) Band structure of SrPtAs (a),(c) with -\nout and (b),(d) with spin-orbit coupling. In (a) and (b), As\npx,yandpzorbitals are shown in red and green, respectively,\nand Assin blue. In (c) and (d), Pt ( dxy,dx2−y2), (dxz,dyz),\nanddz2orbitals are presented in red, green, and blue, respec-\ntively. Contribution from Pt sis shown in purple.3\nlabel the two degenerate partners of each pair through a\npseudo-spin index. We include a spin-independent near-\nest neighbor hopping between the As and Pt sites. Thisyields the following tight binding Hamiltonian in k-space\nH0=/summationdisplay\nk,sΨ†\ns(k)Hs(k)Ψs(k), (2)\nwhere Ψ s(k) = (ck,d+,s,ck,d−,s,ck,p+,s,ck,p−,s)T,sis\nthe spin label ( s={↑,↓}). For spin up, H↑(k) is (to get\nH↓(k) change the sign of δPtandδAs)\n\nǫd+δPt 0 t(1+ν∗e−ik·T3+νeik·T2)˜t(1+e−ik·T3+eik·T2)\n0 ǫd−δPt˜t(1+e−ik·T3+eik·T2)t(1+νe−ik·T3+ν∗eik·T2)\nt(1+νeik·T3+ν∗e−ik·T2)˜t(1+eik·T3+e−ik·T2) ǫp+δAs 0\n˜t(1+eik·T3+e−ik·T2)t(1+ν∗eik·T3+νe−ik·T2) 0 ǫp−δAs\n\n(3)\nwhereν=ei2π/3,T1=a(1,0),T2=a(−1/2,√\n3/2),\nT3=a(−1/2,−√\n3/2),tand˜tare the hopping param-\neters between the As pand Ptdorbitals, and δPt(δAs)\nis the atomic SOC parameter for Pt d(Asp) orbitals. To\nfind an effective Hamiltonian for the Pt d-orbitals, we as-\nsume that |ǫd−ǫp|is the largest energy scale and treat\nallotherparameters( t,˜t,δPt,δAs) asperturbations. This\nyields the following effective Hamiltonian\nHPt=/summationdisplay\nk,sΨ†\nPt,s(k)HPt,s(k)ΨPt,s(k),(4)where Ψ Pt,s(k) = (ck,d+,s,ck,d−,s)T,sis the spin label,\nand\nHPt,↑=/parenleftbigg\nǫd+δPt+t1/summationtext\nicos(k·Ti)+t2/summationtext\nisin(k·Ti)t3[cos(k·T1)+νcos(k·T2)+ν∗cos(k·T3)]\nt3[cos(k·T1)+ν∗cos(k·T2)+νcos(k·T3)]ǫd−δPt+t1/summationtext\nicos(k·Ti)−t2/summationtext\nisin(k·Ti)/parenrightbigg\n(5)\nwheret1= 2[˜t2+ cos(2π/3)t2]/(ǫd−ǫp),t2=\n−2sin(2π/3)t2/(ǫd−ǫp), andt3=−2t˜t/(ǫd−ǫp),\nand the sum over iis over the three vectors T1,T2,\nandT3. The Hamiltonian HPt,↑yields the disper-\nsionsǫ(k) =t1(k)±/radicalbig\n(δPt+t2(k))2+|t3(k)|2where\nt1(k) =t1/summationtext\nicos(k·Ti),t2(k) =t2/summationtext\nisin(k·Ti), and\nt3(k) =t3[cos(k·T1)+νcos(k·T2)+ν∗cos(k·T3)]. The\nSOC contribution that lifts the four-fold degeneracy into\ntwo bands Ek,±=Ek±|g(k)|atkz=π/ccan be found\nto order δPt//radicalbig\nδ2\nPt+|t2(k)|2+|t3(k)|2and is given by\ng(k) =ˆzt2δPt/radicalbig\nδ2\nPt+|t2(k)|2+|t3(k)|2\n×[sin(k·T1)+sin(k·T2)+sin(k·T3)] (6)\nwhere we have also included the contribution from\nHPt,↓(k). Note that this gleads to pseudo-spin in-\nteraction in Eq. 1 denoted by σz. We emphasize that\nthisσzoperates on pseudo-spin, not actual spin, the\nup and down pseudo-spin states are related by time-\nreversal symmetry (for example, the local states |d+,↑>and|d−,↓>form a pseudo-spin pair). The expres-\nsion forgclearly reveals how the interplay between the\natomic SOC ( δPt) and the broken inversion symmetry\n(g(k) =−g(−k)) of a single PtAs layer leads to the rel-\nevant band SOC. Note that this single-layer band SOC\nwill be of opposite signs for the two inequivalent PtAs\nlayers in the unit cell. Further, the band spin-orbit split-\nting will have additional contributions from terms of or-\nderδPt/(ǫd−ǫp) andδAs/(ǫd−ǫp) that were neglected in\nthe above derivation. However, these additional contri-\nbutions do not qualitatively change the results. Similar\nconsiderations apply for the other Pt d-orbitals.\nThepreviousparagraphconsideredasinglePtAslayer.\nTo complete the description, the coupling between the\ntwo inequivalent layers must be included. For the Pt\ndx2−y2,dxyorbitals considered above, the nearest neigh-\nbor inter-layer hopping matrix between |d+,s >states is\n(the same expression appears for |d−,s >states)\nǫc(k) =tccos(kzc/2)(1+e−ik·T3+eik·T2).(7)\nIncluding this inter-layer hopping leads to the following4\nHamiltonian\nH±,s=/summationdisplay\nkΨ†\n±,s(k)/braceleftig\n[ǫ±(k)−µ]σ0τ0+g(k)·στz\n+Re[ǫc(k)]σ0τx+Im[ǫc(k)]σ0τy/bracerightig\nΨ±(k,s′),(8)\nwhere Ψ ±,s(k) = (c±k↑1,s,c±k↓1,s,c±k↑2,s,c±k↓2,s)T,\n1,2 denote the two inequivalent PtAs layers, σi(τi) are\nPauli matrices that operate on the pseudo-spin (layer)\nspace,ǫ±=t1(k)±/radicalbig\n(t2(k))2+|t3(k)|2, andg(k) is\ngiven in Eq. (6) (the τzmatrix describes the sign change\nofgon the two layers). This Hamiltonian can be di-\nagonalized with resulting dispersion relations ǫ(k) =\nǫ±(k)±/radicalbig\n|ǫc(k)|2+g2(k) and each state is 2-fold degen-\nerate due to time-reversal symmetry (Kramers degener-\nacy). Note that the tight binding theory described above\nsuggests that eigenstates of Szare also eigenstates of the\nsingle electron Hamiltonian. However, inter-layer cou-\npling termscan lead toadditional termsthat donot com-\nmute with Sz. The band structure suggests that these\nterms are not large for the states near the Fermi surface.\nThe SOC found in in Eq. (6) has opposite sign for the\ndifferent layers as well as for the pseudo-spin direction.\nThis is demonstrated in Fig. 3, where the band structure\nis resolved by layer and spin for H–L–H′, whereLis one\nofthe time-reversalinvariantmomentum (TRIM) points,\nand the lines H–LandL–H′are related both by a mirror\nplane and, and morerelevant to the discussion, bya com-\nd    , d      xy      x -y          2   2d    , d   xz      yz\nd   z   2(a)\nEnergy (eV)\n-2.5-2-1.5-1-0.500.5\nH H’ L\n(c)\nEnergy (eV)\nH H’ L-2.5-2-1.5-1-0.500.5(b)\nEnergy (eV)\n-2.5-2-1.5-1-0.500.5\nH H’ L\n(d)\nEnergy (eV)\nH H’ L-2.5-2-1.5-1-0.500.5\nFIG. 3. (Color online) SOC splitting of bands. (a) and (b)\n[(c) and (d)] show components from Pt 5 dorbitals for up-\nper [lower] Pt atoms. Left column[(a) and (c)] is for spin-up\ncomponents and right[(b) and (d)] is for spin-down compo-\nnents. Radius of circles is proportional to the magnitude of\nthe components.bination of inversion and reciprocal lattice vectors. In\nFig. 3, spin up[(a)] and down[(b)] components of the Pt\ndorbital from the upper PtAs layer are marked in differ-\nent colors, while components from the other PtAs layers\nare shown in (c) and (d). As expected from the tight-\nbinding analysis, Figs. 3(a),(d) [and similarly for (b) and\n(c)] appear the same since they correspond to spatial in-\nversion and time-reversal (opposite spin). Despite the\npresence of a global inversion center, the locally broken\ninversion symmetry in PtAs is evident in Fig. 3 where\nthe spin degeneracy is broken in a single layer, for which\nthe consequences are nontrivial.18(This result is not sur-\nprising or unexpected since in the limit that the coupling\nbetween PtAs layers vanishes, i.e., the separation goes to\ninfinity, the single layer result must be recovered.) This\nspin separation, however, does not result in magnetism\nbecause of spin compensation in each layer. These anti-\nsymmetric splittings ( g(k) =−g(−k)) occur not only at\ntheLpoint but also at the other TRIM points35— Γ,A,\nM, and the A-Lline — in the hcp structure.\nAs expected from the layer structure, anisotropy oc-\ncurs in the Fermi surfaces, conductivity, and plasma fre-\nquency. Two-dimensionalcrosssections of the Fermi sur-\nfaces are shown in Fig. 4, both without [(a)-(d)] and\nwith [(e)-(h)] SOC. Contours in the kz=π/c[(a),(e)]\nandkz= 0 [(d),(h)] planes clearly exhibit the conse-\nquence of the crystal symmetry: hexagonal symmetry\naround Γ and Aand trigonal symmetry around Kand\nH. All sheets exhibit almost two-dimensional cylindri-\ncal features except for a small pocket around H. This\nΑ\nL LL L\nHH(a)\n31 32333434 33                     \nΓ MA L(b)\n31 32\n33 34                                    \nΓA H\nKL\nM(c)\n31 32\n33 3434 333334\n34 34                                    \nΓ\nM MM M\nKK(d)\n3132333434\n34                                    Α\nL LL L\nHH(e)\n31323334\n3433                                    \nΓ MA L(f)\n31 32\n33 34                                    \nΓA H\nKL\nM(g)\n31 32\n33 34343334                                    \nΓ\nM MM M\nKK(h)\n31323334\n34                                    \nFIG. 4. Cross section of the Fermi surface (a)-(d): without\nand (e)-(h): with spin-orbit coupling. Cross section along the\nzone boundary face, kz=π/c[(a),(e)] and at the zone center,\nkz= 0 [(d),(h)]. (b),(c) are contours along the vertical plane\nshown with their corner points in the Brillouin zone, where\n(f),(g) are those with SOC included. Numbers next to Fermi\nsurface sheets indicate band indices.5\nanisotropy is further exemplified by transport proper-\nties which will be discussed later. Moreover, because\nof the symmetry-dictated degeneracy due to the non-\nsymmorphic symmetry, there is no spin-orbit splitting\nalong the time-reversal invariant direction, A-L, as seen\nin Fig. 4(e). Comparing Figs. 4(c) and (g), the SOC ap-\npears to make the Fermi surfaces more cylindrical, conse-\nquently enhancing the two dimensional character of the\nFermi surfaces.\nAll the Fermi surfaces are hole-like after turning on\nSOC,in sharpcontrasttootherpnictide superconductors\nwith two electron-like Fermi surfaces and two hole-like\nFermi surfaces.36Instead of electron-like and hole-like\nFermi surfaces, they are distinguished by orbital char-\nacter. Sheets around the Γ- Aline consist of σorbitals\nof the PtAs layer, As px,yand Ptdxy,x2−y2, while sheets\naround the K-Hline are from πorbitls, As pzand Pt\ndxz,yz. Two kinds of Fermi surfaces with different orbital\ncharacter might give rise to a two energy gap supercon-\nductor in SrPtAs.\nThe anisotropy due to the layered structure is further\nmanifested in the average Fermi velocities and plasma\nfrequencies. Neglecting SOC, the in-plane and out-of-\nplane Fermi velocities are /an}bracketle{tv2\nx,y/an}bracketri}ht1/2=3.72×107cm/s and\n/an}bracketle{tv2\nz/an}bracketri}ht1/2=1.02×107cm/s, respectively, and the plasma\nfrequencies are Ω x,y=5.70 eV and Ω z=1.57 eV. The\nanisotropy ratio, defined as a ratio of conductivities be-\ntween in-plane and out-of-plane components, is 13.3, as-\nsuming an isotropic scattering rate. With SOC, the\nanisotropy is enhanced: /an}bracketle{tv2\nx,y/an}bracketri}ht1/2=3.76×107cm/s and\n/an}bracketle{tv2\nz/an}bracketri}ht1/2=6.78×106cm/s; Ω x,y=5.57eV, Ω z=1.00eV, and\nthe anisotropy ratio increases to a much higher value\nof 30.8. The decrease of vzby 33% by SOC is consis-\ntent with the enhanced two dimensional character of the\nFermi surfaces. Table I summarizes contributions from\neach Fermi surface. The largest contribution to the den-\nsity of states at the Fermi level, N(0), comes from the\n34th band at around K, which is due to the low veloc-\nity at the Fermi surface. The anisotropy ratio is usu-\nally much larger than 1 except for the small hole pocket\naroundH.\nTABLE I. Fermi surface properties with SOC included for\neach surface: Density of states, N(0) (states/eV/spin); av-\nerage velocities, /angbracketleftv2\nx/angbracketright1/2,/angbracketleftv2\nz/angbracketright1/2(107cm/s); anisotropy ratio,\n/angbracketleftv2\nx/angbracketright//angbracketleftv2\nz/angbracketright; and plasma frequencies, Ω x, Ωz(eV). The numbers\n31,32,33, and 34 represent band indices and Γ and K in the\nparenthesis represent the locations of the Fermi surface.\n31 32 33(Γ) 34(Γ) 33(K) 34(K) Total\nN(0) 0.085 0.107 0.209 0.346 0.267 0.943 1.898\n/angbracketleftv2\nx/angbracketright1/27.03 6.79 5.92 4.35 1.25 1.68 3.76\n/angbracketleftv2\nz/angbracketright1/21.03 0.67 0.47 1.04 1.37 0.41 0.678\n/angbracketleftv2\nx/angbracketright//angbracketleftv2\nz/angbracketright46.9 103 159 17.3 0.83 16.9 30.8\nΩx2.20 2.39 2.91 2.74 0.69 1.75 5.57\nΩz0.32 0.24 0.23 0.66 0.76 0.43 1.00 0 10 20 30\n      (a)total\n 0 5\n-0.200.2\n 0 0.1 0.2\n      DOS (States/eV)(b)Sr s\nPt s\nAs s\n 0 0.2 0.4 0.6 0.8\n      (c)As pxAs pz\n 0 1 2 3\n-6-4-2024\nEnergy(eV)(d)Pt dxyPt dyzPt dz2\nFIG. 5. (Color online) Density of states (DOS) of SrPtAs:\n(a) total DOS, (b) sorbitals, (c) As porbitals, and (d) Pt d\norbitals.\nThe total density of states (DOS) and orbital decom-\nposed partial DOS are presented in Fig. 5. The states\naroundEFarise mainly from As pand Ptd. The non-\nbonding Pt dz2bands are located at around -2.2 eV, and\nbecausetheyhybridizelittlewithotherorbitals,theygive\nriseto a peak in the DOS. In contrast, π-bonding orbitals\nsuch as As pzand Ptdxz,yzare located around the Fermi\nlevelwith awide band width. In particular, the van Hove\nsingularity(vHS) comingfrom asaddlepoint near Kjust\naboveEF(inset of Fig. 5(a))exhibits two dimensional\ncharactersince a 2D vHS gives rise to a singularity in the\nDOS while a 3D vHS gives only a discontinuity in slope.\nRaisingEFto vHS, which might be realized by electron\ndoping via Sr layers, could potentially increase Tc: As-\nsuming rigid bands, we estimate that electron doping by\nroughly 25% will lift EFto the vHS, enhancing the DOS\nat the Fermi level by 6%. Further, using a weak-coupling\nform for Tc(Tc= 1.14TDe−1/N(0)V), assuming that the\npairing potential Vdoes not change by electron doping,\nand assuming Debye temperature of TD= 229K,37we\nestimate that Tccould be enhanced to as high as 3.8K\nby electron doping.\nApplyinghydrostaticpressurecouldalsoraisetheDOS\nand could, as often happens, enhance Tc. Rh is a can-\ndidate to supply a chemical pressure to SrPtAs since it\nhas a similar electronegativity as Pt but a smaller ionic\nradius. However, the crystal structure is sensitive to the\nconstituent atoms; for example, SrPtSb where As is re-\nplaced by Sb has an AlB 2-type structure, while YPtAs,\nwhere Sr is replaced by Y with more electrons, has a6\nhexagonalstructurewith fourslightlypuckeredPtAs lay-\ners in a unit cell.5\nFinally we consider the possibility of simple collinear\nmagneticsolutions. Theantiferromagnetic(AFM)phase,\nwhere the moments in a PtAs layer are aligned, by an-\ntiparallel to those in adjacent layers, is favored by 0.23\nmeV (0.49 meV in GGA) per formula unit, than the non-\nmagnetic phase; the ferromagnetic orientation converged\nto the non-magnetic solution. Magnetic moments are\ngiven in Table II. While the energy differences and cal-\nculated moments are too small to make definitive state-\nments regardingmagnetic phasesin SrPtAs, the material\nappears to be near a magnetic instability.\nIV. SUMMARY\nFirst-principles calculations of the electronic structure\nof SrPtAs have been presented with SOC fully taken into\naccount. The role of SOC on the electronic structure is\nmanifested in the energy bands and Fermi surfaces. The\nimportant physics originates from two factors: strong\nSOC in Pt atoms and locally broken inversion symme-\ntry in PtAs layers. We have constructed a tight-binding\nHamiltonian based on the self-consistent electronicstruc-\nture that provides insight into the SOC. Sheets of the\nFermi surface are spatially well separated in the Bril-\nlouin zone: cylindrical Fermi surfaces with σ-character\nat the zone center (around Γ- A) and two Fermi sur-faces,i.e.,a pocket and a cylinder, with π-character at\nthe zone corner (around K-H). All the Fermi surfaces\nare hole-like which distinguishes this material from other\npnictide superconductors. The transport properties are\nhighly anisotropic between x,y−andz−directions. Rh\nis suggested for a positive pressure effect to increase Tc.\nFurthermore, the van Hove singularity is shown in the\nDOS above EF. Assuming rigidity of bands, we predict\nthatTCincreases up 3.4 K with 25% doping, which may\nbe achieved by chemical doping in place of the Sr atom.\nACKNOWLEDGMENTS\nSJY and SHR are indebted to Hosub Jin for fruitful\ndiscussions. SJY acknowledges the sabbatical research\nGrant by Gyeongsang National University. SHR and\nAJF are supported by the Department of Energy (DE-\nFG02-88ER45382). DFA is supported by NSF grant\nDMR-0906655. MWissupportedbyNSFDMR-1105839.\nTABLE II. Spin µsand orbital µlmagnetic moments and\ntotal magnetic moment µtin unit of µB.\nµs µl µt\nPt 0.014 0.064 0.078\nAs 0.006 -0.027 -0.021\n1Y. Kamihara, T. Watanabe, M. Hirano, and H. Hosono, J.\nAm. Chem. Soc. 130, 3296 (2008).\n2C. Wang, L. Li, S. Chi, Z. Zhu, Z. Ren, Y. Li, Y. Wang,\nX. Lin, Y. Luo, S. Jiang, et al., Europhys. Lett. 83, 67006\n(2008).\n3I. Mazin and J. Schmalian, Physica C 469, 614 (2009).\n4Y. Nishikubo, K. Kudo, and M. Nohara, J. Phys. Soc. Jpn.\n80, 055002 (2011).\n5R.-D. Hoffmann and R. P¨ ottgen, Z. Kristallogra. 216, 127\n(2001).\n6P. A. Frigeri, D. F. Agterberg, A. Koga, and M. Sigrist,\nPhys. Rev. Lett. 92, 097001 (2004).\n7L. Gor’kov and E. Rashba, Phys. Rev. Lett. 87, 037004\n(2001).\n8H. Yuan, D. Agterberg, N. Hayashi, P. Badica, D. Van-\ndervelde, K. Togano, M. Sigrist, and M. Salamon, Phys.\nRev. Lett. 97, 017006 (2006).\n9P. A. Frigeri, D. Agterberg, and M. Sigrist, New J. Phys.\n6, 115 (2004).\n10V. Barzykin and L. Gor’kov, Phys. Rev. Lett. 89, 227002\n(2002).\n11D. F. Agterberg, Physica C 387, 13 (2003).\n12R. P. Kaur, D. F. Agterberg, and M. Sigrist, Phys. Rev.\nLett.94, 137002 (2005).\n13K. V. Samokhin, Phys. Rev. B 78, 224520 (2008).\n14V. M. Edelstein, J. Phys.: Condens. Matter 8, 339 (1996).\n15S. K. Yip, Phys. Rev. B 65, 144508 (2002).\n16S. Fujimoto, Phys. Rev. B 72, 024515 (2005).17M. Sato and S. Fujimoto, Phys. Rev. B 79, 094504 (2009).\n18S.J. Youn,M. H.Fischer, S.H.Rhim, M.Sigrist, andD.F.\nAgterberg, e-print arXiv:cond-mat/1111.5058 (2011).\n19M. H. Fischer, F. Loder, and M. Sigrist, Phys. Rev. B 84,\n184533 (2011).\n20I. Shein and A. Ivanovskii, Physica C 471, 594 (2011).\n21E. Wimmer, H. Krakauer, M. Weinert, and A. J. Freeman,\nPhys. Rev. B 24, 864 (1981).\n22M. Weinert, E. Wimmer, and A. J. Freeman, Phys. Rev.\nB26, 4571 (1982), and references there in.\n23L. Hedin and B. I. Lundqvist, J. Phys. C. 4, 2064 (1971).\n24A. H. MacDonald, W. E. Pickett, and D. D. Koelling, J.\nPhys. C 13, 2675 (1980).\n25G. Wenski and A. Mewis, Z. Anorg. Allg. Chem. 535, 110\n(1986).\n26M. Weinert, G. Schneider, R. Podloucky, and J. Redinger,\nJ. Phys.: Cond. Matt. 21, 084201 (2009).\n27H. J. Monkhorst and J. D. Pack, Phys. Rev. B 13, 5188\n(1976).\n28P. E. Bl¨ ochl, O. Jepsen, and O. K. Andersen, Phys. Rev.\nB49, 16223 (1994).\n29J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev.\nLett.77, 3865 (1996).\n30D. D. Koelling and J. H. Wood, J. Comp. Phys. 67, 253\n(1986).\n31W. E. Pickett, H. Krakauer, and P. B. Allen, Phys. Rev.\nB38, 2721 (1988).7\n32G. K. H. Madsen and D. J. Singh, J. Comput. Phys.\nComm.175, 67 (2006).\n33H. Sch¨ afer, Ann. Rev. Mater. Sci. 15, 1 (1985).\n34L. F. Mattheiss, Phys. Rev. 151, 450 (1966).35L. Fu, C. L. Kane, and E. J. Mele, Phys. Rev. Lett. 98,\n106803 (2007).\n36D. J. Singh and M.-H. Du, Phys. Rev. Lett. 100, 237003\n(2008).\n37K. Kudo (2011), private communication."    },    {        "title": "0808.2372v1.Dilution_effect_in_correlated_electron_system_with_orbital_degeneracy.pdf",        "content": "arXiv:0808.2372v1  [cond-mat.str-el]  18 Aug 2008Dilutioneffect incorrelated electron systemwithorbital degeneracy\nTakayoshi Tanaka,∗and Sumio Ishihara\nDepartment of Physics, Tohoku University, Sendai 980-8578 , Japan.\n(Dated: November 29, 2018)\nTheory of dilution effect in orbital ordered system is prese nted. The egorbital model without spin degree\nof freedom and the spin-orbital coupled model in a three-dim ensional simple-cubic lattice are analyzed by the\nMonte-Carlo simulation and the cluster expansion method. I n theegorbital model without spin degree of free-\ndom, reductionof the orbitalordering temperature due todi lutionissteeper thanthat inthedilute magnet. This\nisattributedtoamodificationoftheorbitalwave-function aroundvacantsites. Inthespin-orbitalcoupledmodel,\nit is found that magnetic structure is changed from the A-typ e antiferromagnetic order into the ferromagnetic\none. Orbital dependent exchange interaction and a sign chan ge of this interaction around vacant sites bring\nabout thisnovel phenomena. Presentresultsexplaintherec ent experiments intransition-metalcompounds with\norbitaldilution.\nPACS numbers: 71.10.-w,71.23.-k, 75.30.-m\nI. INTRODUCTION\nImpurity effect in correlated electron system is one of the\nattractive themes in recent solid state physics.1,2The well\nknown example is doping of non-magnetic impurity in high\nTc superconducting cuprates; a small amount of substitu-\ntion ofCu by Zn dramaticallydestroysthe superconductivit y.\nNon-magneticimpurityeffect in the low-dimensionalgappe d\nspin system is another example. A few percent doping of Zn\nor Mg, which does not have a magnetic moment, into two-\nleg ladder systems, e.g. SrCu 2O3, and spin-Peierls systems,\ne.g. CuGeO 3, induces long-range orders of antiferromag-\nnetism (AFM).3,4,5,6,7Impurity effect in charge and orbital\nordered state is also studied in the colossal magnetoresist ive\nmanganites.8,9Itisreportedinaso-calledhalf-dopedmangan-\nite La0.5Ca0.5MnO3that a few percent substitution of Mn by\nCrcollapsesthecharge/orbitalorderassociatedwiththeA FM\noneandinducesaferromagneticmetallicstate. Becauseofn o\negelectrons in Cr3+, unlike Mn3+with one egelectron, Cr is\nregardedasan impuritywithoutorbitaldegreeoffreedom.\nRecently, impurity dopingeffect in an orbital orderedstat e\nis examined experimentally in a more ideal material. Mu-\nrakamiet al.have studied substitution effect in an orbital\nordered Mott insulator KCuF 3with the three dimensional\n(3D) Perovskite crystal structure.10A Cu2+ion in the cubic-\ncrystalline field shows the (t2g)6(eg)3electron configuration\nwhere one hole has the orbital degree of freedom. The long-\nrange orbital order (OO), where the dy2−z2- anddz2−x2-like\norbitalsarealignedwithamomentum (π,π,π),wasobserved\nat roomtemperaturesbyseveralexperiments. SincetheAFM\nspin ordering temperature is 39K, which is much lower than\nthe OO temperature ( >1200K), a substitution of Cu by Zn,\nwhichhasanelectronconfiguration (t2g)6(eg)4,isregardedas\nanorbitaldilution. Itwas revealedbytheresonantx-raysc at-\ntering experimentsin KCu 1−xZnxF3that the OO temperature\ndecreaseswithdopingofZnmonotonicallyandthediffracti on\nintensityat (3/23/23/2)disappearsaround x=0.45. Atthe\nsameZnconcentration,thecrystalsymmetryischangedfrom\nthe tetragonal to the cubic one. That is to say, the OO disap-\npearsaround x=0.45. Indilutemagnets,e.g. KMn 1−xMgxF3,\nthexdependenceofthemagneticorderingtemperatureaswellas the critical concentration where the magnetic order van-\nishesarewellexplainedbythepercolationtheory.11,12Onthe\ncontrary, the critical concentration in KCu 1−xZnxF3, where\nthe OO disappears, is much smaller than the site-percolatio n\nthreshold in a 3D simple cubic lattice, xp=0.69. These ex-\nperimentalobservationsimply that the diluteOO maybelong\ntoanewclassofdilutedsystemsbeyondtheconventionalper -\ncolationtheory.\nDilution effect in orbital ordered state was also examined\nexperimentallyin a mothercompoundofthe colossal magne-\ntoresisitivemanganites,LaMnO 3. Thelong-rangeOO,where\nthed3x2−r2- andd3y2−r2-like orbitals align with a momen-\ntum(π,π,0), appears below 780K. The A-type AFM order,\nwhere spins are aligned ferromagnetically in the xyplane\nand are antiferromagnetically along the zaxis, is realized\nat 140K. Substitution of Mn3+by Ga3+, which has a 3 d10\nelectron configuration, corresponds to both the orbital and\nspin dilution.13,14,15,16,17,18From the x-ray diffractionand X-\nrayabsorptionnear-edgestructure(XANES)experiments,t he\ntetragonally distorted MnO 6octahedra become regular cubic\nones around the Ga concentration x=0.6. That is, the OO\ndisappear around x=0.6 which is smaller than the percola-\ntion threshold xp=0.69 for the simple cubic lattice. Differ-\nence between LaMn 1−xGaxO3and KCu 1−xZnxF3is seen in\nthe magnetic structure. Blasco et al.observed by the neu-\ntron diffraction experiments in LaMn 1−xGaxO3that the fer-\nromagnetic (FM) component appears by substitution by Ga\nand increases up to x=0.5. This change of the magnetic\nstructurefromthe A-typeAFM to FM wasalso confirmedby\nthe magnetization measurements. This FM component can-\nnot be attributed to the itinerant electrons through the dou ble\nexchange interaction, since the electrical resistivity in creases\nwith increasing x. These phenomena are in contrast to the\nconventional dilute magnets where the ordering temperatur e\nis reduced, but the magnetic structure is not changed. Far-\nrellandGehringpresentedaphenomenologicaltheoryforth e\nmagnetism in LaMn 1−xGaxO3.13They noticed that a volume\nin a GaO 6octahedron is smaller than that in a MnO 6. Un-\nderan assumptionthat the Mn 3 dorbitalsarounda dopedGa\ntend to be toward the Ga, the magnetic structure change was\nexamined.2\nInthispaper,amicroscopictheoryofdilutioneffectsinth e\negorbital degenerate system is presented. We study the dilu-\ntion effectsin the eg-orbitalHamiltonianwithout the spin de-\ngree of freedom, termed HT[see Eq. (10)], and the spin and\negorbital coupledone, termed HST[see Eq. (9)]. The classi-\ncal Monte-Carlo (MC) method in a finite size cluster, as well\nas the cluster expansion (CE) method is utilized. It is known\nthat, in the classical ground state of HTwithout impurity, a\nmacroscopic number of orbital states are degenerated due to\nfrustrated nature of the orbital interaction. We demonstra te\nnumericallythatthisdegeneracyisliftedatfinitetempera ture.\nIt is shown that the OO temperature decreases rapidly with\nincreasing dilution. From the system size dependence of the\norbital correlation function in the MC method, the OO is not\nrealizedattheimpurityconcentration x=0.2. Theresultsob-\ntainedbytheCEmethodalsoshowrapidquenchingofOOby\ndilutionin comparisonwith dilute spin models. These resul ts\nareinterpretedthat orbitalsaroundimpuritysites arecha nged\nso as to gain the remaining bond energy. This is a conse-\nquence of the bond-direction dependent interaction betwee n\nthe inter-site orbitals. In the analyses of the spin-orbita l cou-\npledmodel,itisshownthattheA-typeAFMstructurerealize d\ninx=0ischangedintoFMonebydilution. Thisisexplained\nby changing a sign of the magnetic exchange interaction due\ntotheorbitalmodificationaroundimpuritysites. Implicat ions\nofthepresentmicroscopictheoryandtheexperimentalresu lts\ninKCu 1−xZnxF3andLaMn 1−xGaxO3are discussed.\nIn Sect. II, the model Hamiltonianfor the egorbitaldegree\nof freedomin a cubic lattice and the spin-orbitalcoupledon e\nare introduced. In Sect. III, the classical MC simulation an d\ntheCE methodarepresented. Resultsofthenumericalanaly-\nses in HTandHSTare presented in Sects. IV and V, respec-\ntively. Section VI is devoted to summary and discussion. A\npartofthenumericalresultsforthe egorbitalmodelhavebeen\nbrieflypresentedin Ref.19.\nII. MODEL\nDoubly degenerate egorbital degree of freedom is treated\nbythepseudo-spin(PS)operatorwithmagnitudeof1/2. This\noperatoris definedby\nTi=1\n2∑\nsγγ′d†\niγsσγγ′diγ′s, (1)\nwherediγsistheannihilationoperatorofanelectronwithspin\ns(=↑,↓)and orbital γ(=3z2−r2,x2−y2)at sitei, andσare\nthePaulimatrices. Occupiedorbitalisrepresentedbyanan gle\nθof PS. The eigen state of the z-component of PS with an\nangleθis\n|θ/an}bracketri}ht=cos/parenleftbiggθ\n2/parenrightbigg/vextendsingle/vextendsingled3z2−r2/angbracketrightbig\n+sin/parenleftbiggθ\n2/parenrightbigg/vextendsingle/vextendsingle/vextendsingledx2−y2/angbracketrightBig\n.(2)\nFor example, θ=0, 2π/3,and4π/3correspondto the states\nwhere the d3z2−r2,d3y2−r2, andd3x2−r2orbitals are occupied\nby an electron, respectively. It is convenient to introduce thelinearcombinationsofthePS operatorsdefinedby\nτl\ni=cos/parenleftbigg2πnl\n3/parenrightbigg\nTz\ni−sin/parenleftbigg2πnl\n3/parenrightbigg\nTx\ni, (3)\nwithl=(x,y,z)andanumericalfactor (nx,ny,nz)=(1,2,3).\nThesearetheeigenoperatorsforthe d3l2−r2orbitals.\nIt is known that dominantorbital interactionsin transitio n-\nmetal compoundsare the electronic exchangeinteractionan d\nphononic one. The former is derived from the generalized\nHubbard-typemodelwiththe doublydegenerate egorbitals;\nHele=∑\n/an}bracketle{tij/an}bracketri}htγγ′s/parenleftBig\ntγγ′\nijd†\niγsdjγ′s+H.c./parenrightBig\n+U∑\niγniγ↑niγ↓\n+1\n2U′∑\niγ/ne}ationslash=γ′niγniγ′+1\n2K∑\niγ/ne}ationslash=γ′ss′d†\niγsd†\niγ′s′diγs′diγ′s,(4)\nwhereniγ=∑sniγs=∑sd†\niγsdiγs. Wedefinetheelectrontrans-\nfer integral tγγ′\nijbetween the a pair of the nearest neighbor-\ning (NN) sites. The intra-orbital Coulomb interaction U, the\ninter-orbital one U′, and the Hund coupling K. Through the\nperturbational expansion with respect to the NN transfer in -\ntegral under the strong Coulomb interaction, the spin-orbi tal\nsuperexchangemodelisobtained.20,21Byassumingarelation\nU=U′+K,forsimplicity,it isgivenas\nHexc=−2J1∑\n/an}bracketle{tij/an}bracketri}ht/parenleftbigg3\n4+Si·Sj/parenrightbigg/parenleftbigg1\n4−τl\niτl\nj/parenrightbigg\n−2J2∑\n/an}bracketle{tij/an}bracketri}ht/parenleftbigg1\n4−Si·Sj/parenrightbigg/parenleftbigg3\n4+τl\niτl\nj+τl\ni+τl\nj/parenrightbigg\n,(5)\nwhereSiis the spin operator at site iwith a mgnitude of\n1/2, andlrepresents a bond direction connecting sites iand\nj. Amplitudes of the superexchangeinteractionsare given as\nJ1[=t2/(U−3K)]andJ2(=t2/U)wheretis the transfer in-\ntegralbetweenthe d3z2−r2orbitalsalongthe zdirection.\nThe phononic interaction between the orbitals is derived\nfromtheorbital-latticecoupledmodelgivenby\nHJT=−gJT∑\nimQm\niTm\ni\n+∑\nkξωkξ\n2/parenleftBig\np∗\nkξpkξ+q∗\nkξqkξ/parenrightBig\n, (6)\nwhereasubscript mtakesxandz. Thefirsttermrepresentsthe\nJahn-Teller (JT) coupling with a coupling constant gJT. Two\ndistortionmodesin a O 6octahedronwith the Egsymmetryis\ndenotedby Qz\niandQx\ni. The second term is for the JT phonon\nwhereqkξandpkξarethephononcoordinateandmomentum,\nrespectively, and ωkξis the phonon frequency. Subscripts k\nandξare the momentumand the phononmode, respectively.\nHere, the spring constant between the NN metal and oxygen\nions are taken into account. The interaction between orbita ls\nandtheuniformstrainandthestrain-energy,whicharenece s-\nsary in study of the cooperative JT effect, are not shown, for\nsimplicity,inthisequation. Forconvenience,thefirstand sec-\nondtermsinEq.6 aredenotedby Horb−lattandHlatt,respec-\ntively. By introducing the canonical transformation define d3\nby\n/tildewideqkξ=qkξ−2\n√ωkξ∑\nmg∗\nkξmTm\n−k, (7)\nand neglectingthe non-commutabilitybetween Hlatand/tildewideqkξ,\nthe orbital and lattice degrees of freedom are decoupled\nas22,23,24,25\nHJT=2g∑\n/an}bracketle{tij/an}bracketri}htτl\niτl\nj+/tildewiderHlatt. (8)\nThe first term in this equation gives the inter-site orbital i n-\nteraction with a coupling constant g=g2\nJT/(3KS)whereKS\nis a spring constant, and /tildewiderHlattis given by the second term in\nEq. (6), i.e. Hlatt, where the phonon coordinate and momen-\ntum are replaced by /tildewideqkξand its canonical conjugate momen-\ntum/tildewidepkξ, respectively.\nThe model Hamiltonian studied in the present paper is\ngiven by a sum of the above two contributions. Quenched\nimpurity without spin and orbital degrees of freedom is de-\nnoted by a parameter εiwhich takes zero (one), when site i\nis occupied(unoccupied)by an impurity. The Hamiltonianis\ngivenas\nHST=−2J1∑\n/an}bracketle{tij/an}bracketri}htεiεj/parenleftbigg3\n4+Si·Sj/parenrightbigg/parenleftbigg1\n4−τl\niτl\nj/parenrightbigg\n−2J2∑\n/an}bracketle{tij/an}bracketri}htεiεj/parenleftbigg1\n4−Si·Sj/parenrightbigg/parenleftbigg3\n4+τl\niτl\nj+τl\ni+τl\nj/parenrightbigg\n+2g∑\n/an}bracketle{tij/an}bracketri}htεiεjτl\niτl\nj. (9)\nNumericalresultsin this Hamiltonianis presentedin Sect. V.\nWealsostudydilutioneffectintheorbitalmodelwithoutsp in\ndegree of freedom. This model is given by taking Si·Sjin\nEq. (9) to be zero. This procedure may be justified in the\ndiluted orbital system of KCu 1−xZnxF3where the N´ eel tem-\nperature (T N) is much below the OO temperature TOO. The\nexplicit form of the egorbital model without spin degree of\nfreedomis givenby\nHT=2J∑\n/an}bracketle{tij/an}bracketri}htεiεjτl\niτl\nj, (10)\nwhereJ(=2g+3J1/4−J2/4)is the effective coupling con-\nstant. Numerical results of this model Hamiltonian are pre-\nsentedinSect. IV.\nIII. METHOD\nIn order to analyze the model Hamiltonian introduced\nabove by using the unbiased method, we adopt mainly the\nclassical MC simulation in finite size clusters. The orbital PS\noperator is treated as a classical vector defined in the Tz−Tx\nplane, i.e. Tz\ni=(1/2)cosθiandTx\ni=(1/2)sinθiwhereθiis\nacontinuousvariable. AswellastheconventionalMetropol is\nalgorithm,the Wang-Landau(WL) methodis utilized.26Thisis suitable for the present spin-orbital coupled model wher e\nthe energy scales of the two degrees are much different with\neachother. Inordertocalculatethedensityofstate, g(E),with\nhigh accuracy in the WL method, we take that the minimum\nenergy edge Emining(E)is higher a little than the ground\nstate energy EGS, and assume g(EGS<E<Emin) =0. As a\nresult, the present MC simulation is valid above a character -\nistic temperature Tminwhich is determined by |Emin−EGS|.\nThis situation will be discussed in Sect. IV in more detail.\nThe simulations have been performed in L×L×Lcubic lat-\ntices (L=12∼18) with the periodic-boundarycondition. In\ntheMetropolismethod,foreachsample,3 ×104−1×105MC\nsteps are spent for measurement after 8 ×103−2×104MC\nstepsforthermalization. Physicalquantitiesareaverage dover\n20−80samplesateachparameterset. IntheWLmethod,the\nfinal modificationfactor26is set to be ffinal=exp(2−27). Af-\ntercalculatingthedensityofstates,2 ×107MCstepsarespent\nformeasurement.\nTo supplement the classical MC simulation, the ordering\ntemperatures are also calculated by utilizing the CE method .\nWe apply the CE method proposed in Ref. 27 to the present\norbital model. For a given impurities configuration {ε}in a\nlatticewith Nsites, theOO parameterisgivenas\nM{ε}=TrN∑\niεiTz\niρN{ε}, (11)\nwiththedensitymatrix\nρN{ε}=e−βH{ε}\nTrNe−βH{ε}, (12)\nwhere Tr Nrepresents the trace over the PS operator at sites\nwithεi=1 in a crystal lattice, and H{ε}is the Hamiltonian\nwith impurity configuration {ε}. The OO parameter per site\nis obtained by averaging about all possible impurity configu -\nration{ε}as\nM=1\n(1−x)N/angbracketleftbig\nM{ε}/angbracketrightbig\n{ε}, (13)\nwherexis the impurity concentration. In the CE method, a\ncluster consisting of msites, termed {m}, is considered, and\nthe PS operatorswhich do not belong to {m}are replaced by\nstochastic variables σi. Here we take (Tx\ni,Tz\ni)=(0,σi). The\neffectiveHamiltonianthusobtainedisdenotedas H{ε}{m}{σ}\nwhere{σ}isasetof σi,andthecorrespondingdensitymatrix\nis\nρ{ε}{m}{σ}=exp/parenleftbig\n−βH{ε}{m}{σ}/parenrightbig\nTr{m}exp/parenleftbig\n−βH{ε}{m}{σ}/parenrightbig,(14)\nwhere Tr {m}represents the trace over the PS operators in a\ncluster{m}. We expand M{ε}intoa seriesofclusteraverages\nasfollows,\nM{ε}=N\n∑\nm=1∑\n{m}m\n∑\nk=1∑\n{k}(−1)k−m\n×Tr{k}/bracketleftBigg/parenleftBigg\n∑\ni∈{k}εiTz\ni/parenrightBigg\n∑\n{σ}ρ{ε}{k}{σ}/bracketrightBigg\n,(15)4\u0001\n\u0002\u0003\n\u0000\n\u0004\n\u0005\n\u0006 \u0007\b \t \n \u000b\nFIG. 1: (a) Scematic picture of the degenerate PS configurati ons\ntermedthe type-(I) degeneracy, and (b) that of the type-(II )one.\nwhere∑{m}is taken overall possible clustersconsisting of m\nsites, and ∑{k}is taken over all subclusters of ksites belong-\ning to a given {m}. The variable σitakes 1/2 or−1/2 by a\nprobabilityof\nP(σi)=δσi,1\n2/parenleftbigg1+2M\n2/parenrightbigg\n+δσi,−1\n2/parenleftbigg1−2M\n2/parenrightbigg\n.(16)\nBy solving Eqs. (13)-(16) self-consistently, the order par am-\neter and the ordering temperature are obtained as a function\nof impurity concentration. In the present study, we adopt th e\nCE method in the two-site cluster approximation,i.e. m=2.\nIt wasshownthat, evenin the two-site clusterapproximatio n,\ntheobtainedresultsshowgoodaccuracyinthecaseofthefer -\nromagneticHeisenbergmodelinasimplecubiclattice;devi a-\ntions fromthe resultsby other reliable methodsare about 2%\nfor the critical impurity concentration.27To compare the re-\nsultsintheclassical MCsimulation,theorderingtemperat ure\nis also calculated in the classical version of the CE method\nwhere the traces in Eqs. (14) and (15) are replaced by inte-\ngrals with respect to the continuous variable Tz\nibetween 1 /2\nand−1/2.\nIV. DILUTIONIN THE egORBITALMODEL\nIn this section, numerical results for dilution effects in t he\negorbital Hamiltonian (10) are presented. First, we show\nthe results in the MC simulation without impurity. It is\nknown that there is a macroscopic degeneracy in the mean-\nfield(MF)groundstatein HTwithoutimpurity.28Thisdegen-\neracy is classified into the following two types: (I) Conside r\na staggered-type OO with two sublattices, termed A and B,\nandmomentum Q=(π,π,π). IntheMFgroundstate,thePS\nanglesinthesublatticesaregivenby( θA,θB)=(θ,θ+π)with\nanyvalueof θ. Suchcontinuousrotationalsymmetryisunex-\npectedfromthe Hamiltonian HTwhereanycontinuoussym-\nmetries do not exist. (II) Consider an OO with Q=(π,π,π)\nand(θA,θB)=(θ0,θ0+π),andfocusononedirectioninthree-\ndimensional simple-cubic lattice, e.g., the zdirection. The\nMF energy is preserved by changing all PS in each layer\nperpendicular to the zaxis independently as ( θ0,θ0+π)→\n\f\n\r \u000e \u000f\n\u0010 \u0011 \u0012\n\u0013 \u0014 \u0015\n\u0016 \u0017 \u0018\n\u0019 \u001a \u001b\u001c\n\u001d\u001e\n\u001f !\"#$\n% &' () *+ ,- . / 0 1 2 3 4 5 6 7 8 9 : ; <\n=\n> ? @\nA B C\nD E F\nG H IJ\nKLM\nNOPQRS T U\nV\nW XY Z[ \\\n] ^ _` a b\nFIG.2: (a)Systemsizedependenceoftheorbitalcorrelatio nfunction\nMOO(x=0), and (b) that of the orbital angle function Mang(x=0).\nTheminimumenergy EminintheWLmethodistakentobe0 .95EGS\nin(a)and 0 .98EGSin(b).\n(−θ0,−θ0−π). These are schematically shown in Fig. 1.\nBothtypesofdegeneracyareunderstoodfromthemomentum\nrepresentationoftheorbitalinteraction,\nHT=2J∑\nkψ†\nkˆE(k)ψk, (17)\nwithψk=[Tz\nk,Tx\nk]and the 2 ×2 matrix ˆE(k). By diagonaliz-\ningˆE(k), weobtainthe eigenvalues\nE±(k) =cx+cy+cz\n±/radicalBig\nc2x+c2y+c2z−cxcy−cycz−czcx,(18)\nwherecl=cosaklwith a lattice constant a. The lower eigen\nvalueJ−(k)has its minima along (π,π,π)−(0,π,π)and\nother two-equivalent directions.29At the point Γ, the two\neigenvalues E+(k)andE−(k)aredegenerate. Thatis,theor-\nbital states correspondingto these momenta are energetica lly\ndegenerateintheMFlevel. Aliftingofthisdegeneracyinth e\nMF ground state has been examined from the view points of\nthe order-by-disorder mechanism by utilizing the spin wave\nanalyses.28,30,31\nHerewedemonstratethedegeneracyliftingandappearance\nofthelong-rageOObytheMCmethod. Weintroduce,forim-\npurity concentration x, the staggered orbital correlation func-\ntion\nMOO(x)=1\nN(1−x)/angbracketleftBigg/braceleftbigg\n∑\ni(−1)iεiTi/bracerightbigg2/angbracketrightBigg1/2\n,(19)5\nandtheanglecorrelationfunction\nMang(x)=1\nN(1−x)/angbracketleftBigg/braceleftbigg\n∑\ni(−1)iεicos3θi/bracerightbigg2/angbracketrightBigg1/2\n,(20)\nwhere/an}bracketle{t.../an}bracketri}htrepresents the MC average and N=L3. The or-\nbitalcorrelationatthemomentum Q=(π,π,π)isrepresented\nbyMOO(x),andtheanglecorrelation Mang(x)takesone,when\ntheorbitalPSangleis2 πn/3withanintegernumber n. There-\nfore,MOO(x)andMang(x)are utilized as monitors for lifting\nof the type-(II) and (I) degeneracies, respectively. Tempe r-\nature dependences of MOO(x=0)for various Lare shown\nin Fig. 2(a). With decreasing temperature, calculated resu lts\nfor allLshow a sharp increasing around T/J=0.35. This\nincreasing becomes sharper with the system size L. Below\nT/J=0.08,MOO(x=0)takes a temperature-independent\nvalue of about 0 .47. This flat behavior is attributed to the\nlowest energy edge Eminfor the density of state calculated in\nthe WL method, as explained in Sect. III. An extrapolated\nvalue ofMOO(x=0)towardT=0 is close to 0.5 which in-\ndicatesthatthetype-(II)degeneracyislifted andthe OO wi th\nthemomentum Q=(π,π,π)isrealized. Temperaturedepen-\ndences of Mang(x=0)presented in Fig. 2(b) increase mono-\ntonicallytowardoneinthelowtemperaturelimit. Almostno -\nsize dependence is seen in Mang(x=0). Therefore, the type-\n(I) degeneracy is also lifted and the PS angle is fixed. Both\nresults indicate the long-range OO where the momentum is\nQ= (π,π,π), and the PS angles are (θA,θB) = (θ0,θ0+π)\nwithθ0=2πn/3.\nThe temperature at which MOO(x=0)andMang(x=0)\nchange abruptly is around T/J=0.33 corresponding to the\nOO temperature TOO(x=0). In more detail, this tempera-\ntureisdeterminedbythefinite-sizescalingforthecorrela tion\nlength. Thisis calculatedbythesecond-momentmethod;\nξ(x)=1\n2sin(akmin/2)/radicalBigg\nMOO(x)2−Mkmin(x)2\nMkmin(x)2,(21)\nwith\nMkmin(x)=1\nN(1−x)/angbracketleftBigg/braceleftBigg\n∑\niei(Q−k)·riεiTi/bracerightBigg2/angbracketrightBigg1/2\n,(22)\nwherekmin=(2π/L,0,0). Thescalingrelationfor ξ(x)is\nξ(x)=LF/bracketleftBig\nL1/ν{T−TOO(x)}/bracketrightBig\n, (23)\nwhereνis the critical exponent for correlation length, and F\nisthescalingfunction. Thecorrelationlengths ξ(x=0)/Lfor\nvarious sizes cross with each other at TOO(x=0). In Fig. 3,\nwe plotξ(x=0)/Las a function of L1/ν[T−TOO(x=0)].\nThe scaling analyses work quite well for L=10, 12, and 14.\nThe OO temperature TOO(x=0)and the critical exponent ν\nare determined by the least-square fitting for the polynomia l\nexpansion. We obtain as TOO(x=0)/J=0.344±0.002 and\nν=0.69−0.81,although statistical errors are not enough to\nobtainthe precisevalueof ν.\nc d e f g h i\nj k l m\nn o p\nq r s t\nu v w\nx y z {|\n} ~     \n   \n\n   \nFIG. 3: Scaling plot of the correlation length ξ(x=0)for the\nstaggered orbital correlation. Numerical data are obtaine d by the\nMetropolis algorithm.       ¡ ¢ £\n¤\n¥ ¦ §\n¨ © ª\n« ¬ \n® ¯ °\n± ² ³´ µ ¶\n·¸¹º»¼\n½ ¾ ¿À Á  ÃÄ Å ÆÇ È É ÊË Ì ÍÎ Ï ÐÑ Ò ÓFIG. 4: Impurity concentration dependence of the staggered orbital\ncorrelationfunction MOO(x). Systemsize istaken tobe L=18. Nu-\nmerical data are obtained bythe Metropolis algorithm.\nNow, we examine impurity effect in the OO. In Fig. 4,\nwe present the staggered orbital correlation function MOO(x)\nfor several impurity concentration x. Numerical data are ob-\ntainedbytheMetropolisalgorithmintheclassicalMCmetho d\nand the system size is chosen to be L=18. First, we fo-\ncus on the region of x≤0.15. As shown above, MOO(x=0)\nabruptly increases at TOO(x=0)∼0.34Jand is saturated to\n0.5 in the low temperature limit. By introducing impurity,\nMOO(x>0)does not reach 0 .5 even at T/J=0.01, and its\nsaturated value in low temperatures gradually decreases wi th\nincreasing x. Although the system sizes are not sufficient to\nestimateMOO(x)in the thermodynamic limit, MOO(x>0)at\nzero temperature does not show the smooth convergence to\n0.5 in contrast to the diluted spin models. Beyond x=0.15,\nresults are different qualitatively; although MOO(x)starts to\nincrease around a certain temperature (e.g. T/J∼0.24 at\nx=0.2), saturated values of MOO(x)in the low temperature\nlimit are rather small. In order to compare the size depen-\ndencesof MOO(x),temperaturedependencesof MOO(x=0.1)\nandMOO(x=0.2)for several system sizes are presented in6Ô Õ Ö × Ø Ù Ú Û Ü Ý\nÞ\nß à á\nâ ã ä\nå æ ç\nè é êë\nìí\nîïðñ\nòó\nôõö\n÷ ø ùú û ü\ný þ ÿ\n \u0000 \u0001\n\u000e \u0002 \u0003 \u0004 \u0005\u0006 \u0007 \b \t \u000b\n\u0010\n\f \r\u000f \u0011\u0012 \u0013\u0014 \u0015 \u0016 \u0017 \u0018 \u0019 \u001a\n\u001b \u001c \u001d \u001e\n\u001f  !\n\" # $ %&' () *+ ,\nFIG.5: (a)Systemsizedependenceoftheorbitalcorrelatio nfunction\nMOO(x)atx=0.1, and(b) that at x=0.2.\nFig.5. InFig.5(a)for x=0.1,MOO(x)forseveralsizescross\naroundT/J=0.25 below which MOO(x)increases with L.\nOn the other hand, In Fig. 5(b) for x=0.2,MOO(x)mono-\ntonically decreases with Lin all temperature range. This dif-\nference above and below x=0.15 is also seen in the results\nof the correlation length. In Fig. 6, a correlation length at\nx=0.15 andx=0.2 are compared. In x=0.15,ξ(x)for\ndifferent sizes cross around T/J=0.25. As shown in the in-\nset of Fig. 6(a), the scaling analyses works well. From this\nanalysesfor ξ(x), theOOtemperaturein x=0.15isobtained\nasTOO(x=0.15)/J=0.248±0.003. On the other hand, in\nx=0.2 [see Fig. 6(b)], ξ(x)for different sizes do not seem\ntocrosswitheachotheratacertaintemperature,andthesca l-\ning analyses does not work. From the above numerical re-\nsults, it is thought that the long-range OO disappears aroun d\n0.15<x<0.2.\nThe impurity concentration xdependence of TOO(x)ob-\ntained by the MC and CE methods are presented in Fig. 7.\nTwo kinds of the CE methods, where the PS operators are\ntreatedasclassicalvectorsandquantumoperators,arecar ried\nout. These are termed the classical and quantum CE meth-\nods,respectively. Inbothcases,weadoptthetwo-sizeclus ter.\nAs a comparison,the N´ eel temperaturesin the 3D XY model\nobtained by the classical MC method, and those in the 3D\nHeisenberg model by the classical CE method are also plot-\nted in the same figure. It is shownthat decrease of TOO(x)by\nthe MC method is much steeper than that of TN(x)in the XY\nand Heisenberg models. As shown in the size dependences\nofMOO(x)andξ(x)atx=0.2, it is thought that the long\nrange OO is not realized at this impurity concentration. A\n- . / 0 1 2 3\n4 5 6\n7 8 9\n: ; <\n= > ?\n@ A B\nC D EF G H\nI\nJKLMN\nOPQRSTU V WX\nY Z[ \\] ^\n_ ` a b c d e f gh i j k l m n o\np q r s t u v w x y z { | } ~ \n  \n     \n   \n   \n  \n   \n   ¡ ¢ £¤ ¥\n¦ § ¨© ª\n«¬®¯°\nFIG.6: (a) System size dependence of the correlation length ξ(x)/L\natx=0.15, and (b) that at x=0.2. The inset of (a) is the scaling plot\nforξ(x)atx=0.1. The OOtemperature andthe critical exponent at\nx=0.15areobtainedtobe TOO(x)=0.248±0.003 andν=0.755±\n0.085, respectively.±²³ ´ µ ¶ · ¸ ¹ º » ¼ ½ ¾ ¿\nÀ\nÁ  Ã\nÄ Å Æ\nÇ È É\nÊ Ë Ì\nÍÎ\nÏ\nÐÑ\nÒÓÔÕÖ\n×Ø\nÙÚÛÜÝÞ ß à á â ã ä\nå æç è é ê ë ì í î ï ð ñò ó ô õ ö ÷ ø ù ú û ü ý þ\nÿ \u0001 \u0000 \u0002 \u0003 \u0004\u0005 \u0006 \u0007 \b \t \n \u000b \f \r \u000e \u000f \u0010 \u0011 \u0012\nFIG.7: Impurityconcentration xdependence of the OOtemperature\nTOO(x). Filled circles are obtained by the MC method. Results by\nthe quantum and classical CE method are shown by broken lines .\nFor comparison, xdependence of the N´ eel temperature TN(x)in the\n3D XY model obtained by the MC method and that in 3D Heisen-\nberg model by the classical CE one are presented by filled tria ngles\nand dotted line, respectively. Thick arrow indicates the pe rcolation\nthreshold ina 3D simple cubic lattice.7\u0013 \u0014 \u0015 \u0016 \u0017 \u0018 \u0019 \u001a\u001b \u001c \u001d \u001e \u001f  ! \"#\n$\n%\n&\nFIG.8: (a)AsnapshotintheMCsimulationforthePSconfigura tion\natx=0.1, and (b)that at x=0.3. Filledcirclesindicate impurities.\nrapid decrease of TOO(x)in comparison with the spin order-\ning temperaturesis also obtainedby the CE method. TheOO\ntemperature monotonically decreases with x, and disappears\naroundx=0.4inthequantumCEcalculation,andaround0 .5\nin the classical CE one. The critical impurity concentratio ns\nobtained by the MC and CE methods are much smaller than\nthe percolation threshold xp=0.69 in the 3D simple-cubic\nlattice.\nLet us explain the physical picture of the orbital dilution.\nSnapshots of the PS configuration in the MC simulation are\nshown in Figs. 8(a) and (b) for x=0.1 and 0.3, respectively.\nThe staggered-type OO with the orbital angle ( θA,θB)=(0,π)\nisseeninthebackgroundofFig.8(a). Attheneighboringsit es\nof the impuritiesindicated by the open circles, PS vectorst ilt\nfrom the angle of (0,π). This deviation of the PS angles is\nnotonlyduetothethermalfluctuation. FocusontheNNsites\nalongthe xdirectionofan impuritywhichoccupiesthe down\nPS sublattice. Inalmost all thesesites, PS anglesare chang ed\nfrom0to a positiveangle δθ. Thiskindoftiltingfrom (0,π)\nbecomes remarkableat x=0.3. Then, we explain the micro-\nscopicmechanismofthisPStiltingduetodilution[seeFig. 9].\nFocus on a PS at a certain site termed i. The interaction act-\ningonthis site isconsideredbythe MF approximationwhere\nweassumethestaggered-typeOOwiththePSangle (0,π)ex-\nceptforthesite iandanimpuritysite. TheHamiltonianwhich\n'\n( ) *+,\n-.\n/\n0\n1\n2 3 4 5 6 78\nFIG. 9: (a) A schematic PS configuration without impurity, an d (b)\nthat withanimpurity. Afilledcirclerepresents animpurity .\nconcernstheinteractionactingonthissite isgivenas\nH(i)\nT=2J∑\nl=(x,y,z)/angbracketleftBig\nεi+ˆelτl\ni+ˆel+εi−ˆelτl\ni−ˆel/angbracketrightBig\nτl\ni\n=−∑\nl=(x,y,z)hl·Ti, (24)\nwhere ˆelis a unit vector along lin the simple cubic lat-\ntice, and hl= (hx\nl,hz\nl)are the MF. In the case of no\ndilution [Fig. 9(a)], the mean-fields are given by hx=\nJ(−√\n3/2,−1/2),hy=J(√\n3/2,−1/2)hz=J(0,−2), and\ntheHamiltonianinEq.(24)isreducedto\nH(i)\nT=3JTz\ni. (25)\nThis implies that the stable PS configuration at the site iis\nθi=π. Then,introduceanimpurityatsite i−ˆexandconsider\nthe PS at site i[Fig. 9(b)]. The x-componentofthe MF in the\ncasewithoutimpurityischangedinto hx=J(−√\n3/4,−1/4),\nand others are not. The effective interaction in Eq. (24) is\ngivenas\nH(i)\nT=J\n4/parenleftBig\n11Tz\ni−√\n3Tx\ni/parenrightBig\n, (26)\nimplyingthatthestableorbitalangleatsite iisθi∼π−0.15.\nThis PS tilting due to dilution is attributed to the fact that\nthe orbital interaction explicitly depends on the bond dire c-\ntion and is the essence of the diluted orbital systems. This\nis highly in contrast to the dilute spin system where dilutio n\ndoesnotcausespecificspintiltingaroundtheimpuritysite but\nsimplyincreasesthermalspin fluctuationsincenumberof th e\ninteractingbondis reduced.\nV. DILUTION INTHE SPIN-ORBITALMODEL\nIn this section, we examine the dilution effect in the spin-\norbital coupled model described by HSTin Eq. (9). First,\nwe briefly introduce the MF calculation for the spin and or-\nbital structures at x=0. The two sublattice structures for\nboth the spin and orbital ordered states are considered, and\nthe PS angles in sublattices A and B are assumed to be\n(θA,θB)=(θ,−θ). Weobtaintheferromagneticspinorderin89 : ; <\n= > ? @\nA B C D\nE F G HI J KL\nMNOP\nQRST\nUVWXYZ[ \\ ]\n^ _\n` a b c de f g h ij k l m n\no p qr s tu v w x y z { | } ~   \n\n  \n  \n  \nFIG. 10: (a) Total energy E, and (b) the A-type AFM correlation\nfunctionMA−AF(x=0)calculatedforseveralvaluesoftheminimum\nenergyEmininthe WLmethod. System size ischosen tobe L=10.\nthe case of J1/J2≥3, and the A-type AFM one in J1/J2<3.\nIn the A-type AFM state, the orbital PS angle is uniquelyde-\ntermined as θ=cos−1{2J2/(5J1−J2+6g)}. By taking the\nMF results into account, for the following MC calculations,\nwe choose the parameter set as (J1/J2,g/J2) = (2.9,5). In\nthesevalues,theOOappearsatmuchhighertemperaturethan\nthe N´ eel one, and the A-type AFM is realized near the phase\nboundary between FM and A-type AFM. These are suitable\nto demonstratethe magnetic structurechange due to dilutio n.\nThe MC simulation results in the realistic parameter set for\nLaMnO 3will be introducedin theSect.VI.\nIntheMCsimulation,weutilizetheWLmethodin L×L×\nLsite cluster ( L=6−10)with the periodic-boundarycondi-\ntion. Thespinoperator SiintheHamiltonianistreatedasa3D\nclassical vector with an amplitude of 1/2. In the simulation ,\n2×107MC steps are spent for measurement after calculating\nthehistogramforthe densityofstates. Physicalquantitie sare\naveraged over 10MC samples at each parameter set. We no-\ntice againthe lowest energyedge Eminin the densityof states\nwhich is introduced in Sec. III. In Fig. 10, we show the Emin\ndependence of the total energy, E, and A-type AFM correla-\ntionfunction, MA−AF(x)definedby\nMA−AF(x)=1\nN(1−x)/angbracketleftBigg/braceleftbigg\n∑\ni,l(−1)ilεiSi/bracerightbigg2/angbracketrightBigg1/2\n(27)\nwhereilforl=(x,y,z)representsthe lcomponentof the co-\nordinateat site i. Theresultsin Fig. 10(a)imply thatthe tem-\nperature below which the total energy is flat is determined\n    \n\n  \n  \n  \n  \n  ¡ ¢ £¤\n¥¦ §\n¨© ª «¬\n®¯ ° ± ²\n³ ´ µ\nFIG.11: Temperature dependence of the orbital correlation function\nMOO(x=0),thePSanglefunction Mang(x=0),theA-typeAFMone\nMA−AF(x=0),andtheFMone MF(x=0). Systemsizeischosento\nbeL=8.\nby an adopted value of Emin. This temperature is denoted\nasTminfrom now on. As shown in Fig. 10(b), in the case\nofEmin=−9.75(−9.84), an obtained MA−AF(x=0)below\nTminis about 55% (75%)of its maximum value of 1/2. That\nis,TminatEmin=−9.84islowerthantheN´ eeltemperatureof\ntheA-typeAFM.Althoughasaturatedvalueof MA−AF(x=0)\nis less than 0.5, this result is enough to examine the orderin g\ntemperature. We chose Emin=−9.84 in the following MC\nsimulationandfocusonchangeofthemagneticorderingtem-\nperatureduetodilution.\nFirst, we show the results without impurities. In Fig. 11,\ncalculated MOO(x=0),MA−AF(x=0),andtheferromagnetic\ncorrelationfunctiondefinedby\nMF(x)=1\nN(1−x)/angbracketleftBigg/parenleftbigg\n∑\niεiSi/parenrightbigg2/angbracketrightBigg1/2\n(28)\nare presented. The staggered-type orbital correlation fun c-\ntionMOO(x=0)abruptlyincreasesaround T/J2=2.5which\ncorresponds to the OO temperature TOO(x=0). This value\nis consistent with the previous results obtained in the mode l\nHamiltonian HT; the effective orbital interaction in the\npresent Hamiltonian HSTwith paramagnetic state is Jorb=\ng+3J1/4−J2/4 whereSi·SjinHSTis replaced by zero.\nThe obtained TOO(x=0) =2.5J2corresponds to 0 .3Jorbin\nthe present parameter set. This value is consistent with\nTOO(x=0) =0.344Jobtained in Sect. IV [see Fig. 2(a)].\nIn Fig. 11, the angle correlation function Mang(x=0)starts\nto increase at TOO(x=0). With decreasing temperature, at\naroundT/J2=0.5[≡TN(x=0)], the second transition oc-\ncurs. The orbital correlation function MOO(x=0)decreases\nabruptly, and MA−AF(x=0)grows up. The ferromagnetic\ncorrelationfunction MF(x=0)showsa small humpstructure\naroundTN(x=0). That is, TN(x=0)is the N´ eel tempera-\nture of A-type AFM. The PS angle correlation Mang(x=0)\ndecreases and almost becomes zero below TN(x=0). This\nresult indicates that, due to the magnetic transition, the P S9¶\n· ¸ ¹\nº » ¼\n½ ¾ ¿\nÀ Á Â\nÃ Ä ÅÆ Ç ÈÉ\nÊ\nËÌ\nÍÎÏÐÑÒ\nÓÔÕÖ\n×ØÙÚÛ\nÜ Ý Þ ß à á â ã ä å æ ç è\né\nê ë ì\ní î ï\nð ñ ò\nó ô õ\nö÷øù ú\nû ü ýþ ÿ \u000f\nFIG. 12: (a) System size dependence of the orbital correlati on func-\ntionMOO(x=0),and(b)thatoftheA-typeAFMcorrelationfunction\nMA−AF(x=0).\nangle is changed into (θA,θB)∼(π/2,−π/2)which is con-\nsistent with the MF results. In Figs. 12, size dependences of\nMA−AF(x=0),MOO(x=0)are presented. With increasing\nL, changes of MOO(x=0)andMA−AF(x=0)atTN(x=0)\nbecome steep, although a saturated values of MA−AF(x=0)\nis still less than 0.5 due to a finite value of |Emin−EGS|as\nmentionedabove.\nImpurity concentration xdependences of MA−AF(x)and\nMF(x)are presented in Fig. 13. With increasing xfrom the\nx=0 case,MA−AF(x)decreases gradually and almost dis-\nappears around x=0.09. On the other hand, MF(x), which\nshowsasmallhumpstructurearound T/J2=0.4atx=0,in-\ncreases with x, and takes about 0.35 in the case of x>0.09.\nThatistosay,themagneticstructureischangedfromA-AFM\ninto FM by dilution. At x=0.06, bothMA−AF(x)andMF(x)\ncoexistdownto the lowest temperaturein the presentsimula -\ntion. This is supposed to be a cant-type magnetic order or a\nmagneticphaseseparationoftheFMandA-typeAFMphases.\nTo clarify the mechanismof the magnetic structurechange\ndue to dilution, the effective magnetic interaction and the PS\nconfiguration are examined. Here, the AFM stacking in the\nA-type AFM structure is chosen to be parallel to the zaxis.\nThe effective magnetic interaction Jl\niis defined such that the\nHamiltonian HSTin Eq. (9) is rewritten as HST=∑/an}bracketle{tij/an}bracketri}htJl\niSi·\nSj. Theexplicitformoftheeffectiveinteractionisgivenas\nJz\ni=2(J1+J2)Tz\niTz\nj+2J2/parenleftBig\nTz\ni+Tz\nj/parenrightBig\n+3\n2J2−1\n2J1,(29)\nwhere we consider a NN pair of sites iandj(=i+ˆez)along\nthezdirection, since we are interested in the magnetic struc-\nture along z. A contour map of the effective interaction Jz\ni,\n\u0001\n\u0000 \u0002 \u0003\n\u0004 \u0005 \u0006\n\u0007 \b \t\n\n \u000b \f\n\r \u000e \u0010\u0011 \u0012 \u0013\u0014\n\u0015\n\u0016\u0017\u0018\u0019\n\u001a\u001b\u001c\u001d\n\u001e\n\u001f !\n\" # $% & '\n( ) * + , - . / 0 1 2 3 4 5 6\n7\n8 9 :\n; < =\n> ? @\nA B C\nD E F\nG H IJ K L\nMN OPQR S T\nUV WXYZ [ \\\n] ^ _`a b cd\nef ghi j kl\nmn o p\nqr s t\nFIG. 13: (a) Impurity concentration xdependence of the A-type\nAFM correlation function MA−AF(x), and (b) that of the FM cor-\nrelationfunction MF(x). Systemsize is chosen tobe L=8.\nand a snapshot of the PS configurations in the same xyplane\nare presented in Fig. 14(a) and (b), respectively. Signs of Jz\ni\ninalmostallregionarepositive(antiferromagnetic),refl ecting\ntheA-AFMstructure. Attheneighboringsitesoftheimpurit y\nalong the ydirection, Jz\nis are negative(ferromagnetic). Away\nfrom the impurity, PS are ordered as ±Txin the staggered-\ntypeOO. Near theimpurity,PS tilt from ±Txand finite com-\nponentsof Tzappears. Thistilting ofPS is seen in theresults\nofHTas explained in Sec. IV. Based on these numerical\nsimulation, we explain mechanism of the magnetic structure\nchange due to dilution. Start from the staggered-type orbit al\norderedstate of (Tx,Tz)=(±1/2,0). Introduceoneimpurity\natasitei0whichbelongstothe Tx=1/2sublattice,andfocus\nonthePSconfigurationandtheeffectiveexchangeinteracti on\nat sitesi0+ˆemandi0+ˆem+ˆezform= (x,y)(see Fig. 15).\nAs explained in Sect. IV, orbital dilution induces the PS til t-\ning so as to gain the energiesof the bondswhere an impurity\ndoesnotoccupy. Thus,PS at site i0+mˆetiltsfrom θ=3π/2\nto 3π/2+δθ(−δθ)form=x(y)with a positive angle δθ.\nSince the orbital interaction is the staggered-type, the bi lin-\near termTz\ni0+ˆemTz\ni0+ˆem+ˆezin Eq. (29) are negative for both the\nm=xandycases. As for the linear term in Eq. (29), there is\na relation (Tz\ni0+ˆex+Tz\ni+ˆex+ˆez) =−(Tz\ni0+ˆey+Tz\ni+ˆey+ˆez). That is,\ncontributionof this linear term to the spin alignment along z,\nwhichisdeterminedbyasumof Jz\ni0+ˆexandJz\ni0+ˆey,iscanceled\nout. Therefore,when the first term in Eq. (29) overcomesthe\npositive constant 3 J2/2−J1/2,Jz\nibecomes negative and the\nferromagneticalignmentalongthe zdirectionisstablearound\ntheimpuritysites.10uv w x y z { |} ~               \n    \n\n\n\n\n\n\n¡\n¢\n£¤\n¥\nFIG. 14: (a) Contour map of Jz\nidefined in Eq. (29), and (b) a snap-\nshot of the PS configuration around an impurity in the xyplane ob-\ntainedintheMCmethod. Afilledcirclerepresentsanimpurit y. Tem-\nperature is chosen tobe T/J2=0.3.\nVI. SUMMARYAND DISCUSSION\nIn this section, we discuss implications of the present nu-\nmerical calculations on the recent experimental results in the\ntransition-metalcompounds. Firstwehaveremarksonthere -\nlationbetweenthecalculatedresultsof HTshowninSect.IV\nandtheexperimentsinKCu 1−xZnxF3.10AsshowninSectIV,\nTOO(x)rapidlydecreaseswithincreasing xincomparisonwith\ndilute magnets (see Fig. 7). Although the critical concentr a-\ntion (x∼0.2−0.5), where the OO disappears, depends on\nthe calculationmethods,that is, MCand CE, these valuesare\nfar belowthe percolationthreshold( xp=0.69). Thisresult is\nconsistentqualitativelywiththeZnconcentrationdepend ence\nof the OO temperature in KCu 1−xZnxF3where OO vanishes\naroundx=0.45. One of the discrepancies between the the-\nory and the experiments are seen in their quantitative value s\nof the critical impurity concentration where OO disappears .\nSome of the reasons of this discrepancy may be attributed to\nthe anharmonic JT coupling and the long-range PS interac-\ntions due to the spring constants beyond the NN ions and so\non,bothofwhicharenottakenintoaccountinthepresentcal -\nculation. Theformersubject,i.e. theanharmonicJTcoupli ng,\n¦§\n¨©ª\n«¬\n®\n¯°±\n²³´µ\n¶·\n¸\n¹\nº\n»¼\n½¾¿\nÀÁ\nÂÃ\nÄÅÆ\nÇÈÉÊ\nËÌ\nÍ\nÎ\nÏ\nFIG.15: AschematicPSconfigurationaroundanimpurityatsi tei0.\nAfilledcirclerepresents animpurity.\ninduces the anisotropy in a bottom of the adiabatic potentia l\noftheQx−Qzplane,andpreventsthePStiltingaroundimpu-\nrity sites. This effect on the reduction of TOO(x)was studied\nbriefly in Ref. 19. It was shown that, in the realistic parame-\nter values, the reduction of TOO(x)becomes moderate by the\nanharmonic coupling, but it is still steeper than that in dil ute\nmagnets. Anotherfactor which may explainsthe discrepancy\nbetweenthetheoryandtheexperimentsisthequantumaspect\nfor the orbital degree of freedom. In the results obtained by\nthe quantum CE method as shown in Fig. 7, the critical xfor\nTOO(x) =0 is larger than the results by other two classical\ncalculations for the orbital model and is close to the experi -\nmental value of x=0.45. This may be due to the fact that\nquantumfluctuationin low temperaturesweakensthe low di-\nmensionalcharacterintheOOstateandpreventsacollapseo f\nOOagainstdilution. Thiskindofquantumeffectsinthedilu te\norbitalsystemwasexaminedbythepresentauthorsinthetwo\ndimensional quantum orbital model.32It was shown that the\nreduction of TOO(x)due to dilution is weaker than that in the\nclassical orbitalmodel.\nWe briefly mention the orbital PS tilting due to dilution.\nSimilar phenomenaare knownas a quadrupolarglass state in\nmolecular crystals where different kind interactions betw een\nmoleculeswithquadruplemomentcoexists.33Akindofglass\nstateintermsofthequadrupolemomentappearswithincreas -\ning randomnessfor the interactions. We suggest a possibili ty\nthatthe presentobservedPS tiltingaccompaniedwith thela t-\nticedistortionofligandionsisabletobedetectedexperim en-\ntally. One of the most adequate experimental techniques are\nthe pair-distribution function method by the neutron diffr ac-\ntionexperiments,andX-rayabsorptionfinestructure(XAFS )\nwheretheincidentx-rayenergyistunedattheabsorptioned ge\nof the impurity ions. This observation may work as a check\nforthe presentscenarioin thediluteorbitalsystem.\nNext we discuss implications of the calculated\nresults in Sect. V to the experimental results in\nLaMn1−xGaxO3.13,14,15,16,17,18By analyzing the spin-\norbital coupled Hamiltonian HST, we find that the magnetic11Ð Ñ Ò Ó Ô Õ Ö × Ø Ù Ú Û Ü\nÝ\nÞ ß à\ná â ã\nä å æ\nç è é\nê ë ìí\nîï\nð ñ ò ó ôõö÷\nø ù ú û üý\nFIG. 16: Impurity concentration dependence of the A-type AF M\ntransition temperature, TN(x), and the FM one, Tc(x), calculated in\nthe realistic parameter values for LaMn 1−xGaxO3. The parameters\nand the system size are chosen to be (J1/J2,g/J2) = (2.5,5)and\nL=8, respectively.\nstructure is changed from the A-AFM order into the FM\none. This calculation qualitatively explains the experime ntal\nresults in LaMn 1−xGaxO3from the macroscopic point of\nview. In Sect. V, the parameter set is chosen to be close to\nthe values for the A-type AFM/FM phase boundary, in order\nto demonstrate clearly the magnetic structure change due to\nthe orbital dilution. Here we briefly introduce the numerica l\nresults obtained in the realistic parametervalues. To eval uate\nthe realistic values, we calculate the OO temperature, the\nN´ eel temperature by the MF approximation, and the spin\nwave stiffness by the spin wave approximation from HST,\nand compare the experimental results in LaMnO 3. Then,\nwe set up the parameters as (J1/J2,g/J2) = (2.5,5). The\nxdependences of the magnetic transition temperatures are\npresented in Fig. 16. With increasing xfromx=0,TN(x)\nof the A-type AFM order gradually decreases, and around\nx=0.2, the A-type AFM is changed into the FM order\nwhich remains at least to x=0.4. In semi-quantitative\nsense,thisresultisconsistentwiththeexperimentalmagn etic\nphase diagram in LaMn 1−xGaxO3. However, one of the\ndiscrepancies is that the canted phase survives up to x=0.4in LaMn 1−xGaxO3. This difference between the theory and\nthe experiments is supposed to be due to the t2gspins in Mn\nsites and the antiferromagnetic superexchange interactio n\nbetweenthemwhicharenotincludedexplicitlyinthepresen t\ncalculation. ThisinteractionstabilizestheA-typeAFMph ase\nin comparison with the FM one, and maintains the canted\nphaseupto ahigher xregion.\nInsummary,wepresentamicroscopictheoryofdilutionef-\nfectsinthe egorbitaldegeneratesystem. Weanalyzethedilu-\ntion effects in the eg-orbital Hamiltonian without spin degree\noffreedom, HT,andthespinand egorbitalcoupledHamilto-\nnian,HST. The classical MC simulation and the CE method\nare utilized. It is shown that the OO temperature decreases\nrapidlywithincreasingdilution. Fromthe systemsize depe n-\ndence of the orbital correlation function in the MC method,\nthe OO is not realized at the impurity concentration x=0.2.\nTilting of orbital PS around impurity is responsible for thi s\ncharacteristic reduction of TOO(x). This is consequence of\nthebonddependentinteractionbetweentheinter-siteorbi tals.\nIn the analyses of the spin-orbital coupled model, the mag-\nneticstructureischangedfromtheA-typeAFMstructureint o\nthe FM one by dilution. This is explained by changing of\nthe magnetic interaction due to the orbital PS tilting aroun d\nthe impurity. The present results explain microscopically the\nnovel dilution effects in KCu 1−xZnxF3and LaMn 1−xGaxO3,\nand provide a unified picture for the dilution effect in the or -\nbitalorderedsystem.\nAcknowledgments\nThe authors would like to thank Y. Murakami, M. Mat-\nsumoto, and H. Matsueda for their valuable discussions. The\nauthors also thank T. Watanabe and J. Nasu for their crit-\nical reading of the manuscript. This work was supported\nby JSPS KAKENHI (16104005),and TOKUTEI (18044001,\n19052001, 19014003) from MEXT, NAREGI, and CREST.\nOne of the authors (T.T,) thanks the financial support from\nJSPS.\n1M. Imada, A. Fujimori,andY. Tokura, Rev. Mod. Phys. 70, 1039\n(1998).\n2S.Maekawa, T.Tohyama,S.E.Barnes,S.Ishihara,W.Koshiba e,\nand G.Khaliullin, Physics ofTransitionMetalOxides , (Springer-\nVerlag, Berlin,2004).\n3M. Azuma, Y. Fujishiro, M. Takano, M. Nohara, and H. Takagi\nPhys. Rev. B 55, R8658 (1997).\n4M. Sigristand A.Furusaki, J.Phys. Soc.Jpn. 65, 2385 (1996).\n5S. B. Oseroff, S-W. Cheong, B. Aktas, M. F. Hundley, Z. Fisk,\nand L.W.Rupp, Jr.,Phys. Rev. Lett. 74, 1450 (1995).\n6M. Hase, K. Uchinokura, R. J. Birgeneau, K. Hirota, and G. Shi -\nrane, J. Phys.Soc. Jpn. 65, 1392 (1996).\n7H. Fukuyama, N. Nagaosa, M. Saitoh, and T. Tanimoto, J. Phys.\nSoc. Jpn. 65, 2377 (1996).\n8A. Barnabe, A. Maignan, M. Hervieu, F. Damay, C.MartinandB.Raveau, Appl. Phys.Lett. 713907 (1997).\n9T. Kimura, Y. Tomioka, R. Kumai, Y. Okimoto, and Y. Tokura,\nPhys. Rev. Lett. 83, 3940 (1999).\n10N. Tatami, Y. Ando, S. Niioka, H. Kira, M. Onodera, H. Nakao,\nK. Iwasa, Y. Murakami, T. Kakiuchi, Y. Wakabayashi, H. Sawa,\nS.Itoh,J.Mag.Mag.Mat. 310,787(2007),andN.Tatami,Master\nthesis inTohoku University(2004).\n11R. J. Elliott, P. Leath, and J. A. Krumhansl, Rev. Mod. Phys. 46,\n465 (1974).\n12R. B. Stinchcombe, Phase Transitions and Critical Phenomen a,\nVol. 7, edited by C. Domb and J. L. Lebowitz, (Academic Press,\nLondon, 1983).\n13J. Farrell,and G.A.Gehring, NewJ. Phys. 6, 168 (2004).\n14J. Blasco, J. Garc ´ia, J. Campo, M. C. S´ anchez, and G. Sub ´ias,\nPhys. Rev. B 66, 174431 (2002).12\n15M. C. Sa´ nchez, G. Subi´ as, J. Garci´ a, and J. Blasco, Phys. R ev. B\n69, 184415 (2004).\n16J. B. Goodenough, A. Wold, R. J. Arnott, and N. Menyuk, Phys.\nRev.124, 373 (1961).\n17J.-S. Zhou, H. Q. Yin, and J. B. Goodenough, Phys. Rev. B 63,\n184423 (2001).\n18J.-S. Zhou and J. B. Goodenough, Phys. Rev. B 68, 144406\n(2003),ibid.Phys.Rev. B 77, 172409 (2008).\n19T. Tanaka, M. Matsumoto and S. Ishihara, Phys. Rev. Lett. 95,\n267041 (2005).\n20K. I.Kugel, andD. I.Khomskii, Sov. Phys.Usp. 25, 231 (1982).\n21S. Ishihara, J. Inoue and S. Maekawa, Phys. Rev. B 55, 8280\n(1997).\n22S. Okamoto, S. Ishihara, and S. Maekawa, Phys. Rev. B 65,\n144403 (2000).\n23J. Kanamori, J.Appl. Phys. 31, 14S (1960).\n24M. Kataoka andJ. Kanamori, J.Phys. Soc.Jpn. 32, 113 (1972).25A. J.Millis,Phys.Rev. B 53, 8434 (1996).\n26F.Wang,andD.P.Landau, Phys.Rev.Lett. 86,2050(2001), ibid.\nPhys. Rev. E 64, 056101 (2001).\n27H. Mano, Prog. Theor.Phys. 57, 1848 (1977).\n28Z. Nussinov, M. Biskup, L. Chayes, and J. van den Brink, Euro.\nPhys. Lett. 67, 990 (2004).\n29S.Ishihara, M. Yamanaka, andN.Nagaosa, Phys.Rev. B 56, 686\n(1997).\n30J. van den Brink, P. Horsch, and F. Mack, Phys. Rev. B 59, 6795\n(1999).\n31K. Kubo, J.Phys. Soc.Jpn. 71, 1308 (2002).\n32T. Tanaka and S.Ishihara, Phys.Rev. Lett. 98, 256402 (2007).\n33K. Binder andJ. D.Reger, Adv. Phys. 41, 547 (1992).\n∗Present address: The Institute for Solid State Physics,\nUniversityofTokyo,Kashiwa,Chiba277-8581,Japan."    },    {        "title": "1201.1607v3.Spin_orbit_coupled_Fermi_liquid_theory_of_ultra_cold_magnetic_dipolar_fermions.pdf",        "content": "arXiv:1201.1607v3  [cond-mat.quant-gas]  10 May 2012Spin-orbit coupled Fermi liquid theory of ultra-cold magne tic dipolar fermions\nYi Li and Congjun Wu\nDepartment of Physics, University of California, San Diego , California 92093, USA\nWe investigate Fermi liquid states of the ultra-cold magnet ic dipolar Fermi gases in the simplest\ntwo-component case including both thermodynamic instabil ities and collective excitations. The\nmagnetic dipolar interaction is invariant under the simult aneous spin-orbit rotation, but not under\neither the spin or the orbit one. Therefore, the correspondi ng Fermi liquid theory is intrinsically\nspin-orbit coupled. This is a fundamental feature of magnet ic dipolar Fermi gases different from\nelectric dipolar ones. The Landau interaction matrix is cal culated and is diagonalized in terms\nof the spin-orbit coupled partial-wave channels of the tota l angular momentum J. The leading\nthermodynamic instabilities lie in the channels of ferroma gnetism hybridized with the ferronematic\norder with J= 1+and the spin-current mode with J= 1−, where + and −represent even and\nodd parities, respectively. An exotic propagating collect ive mode is identified as spin-orbit coupled\nFermi surface oscillations in which spin distribution on th e Fermi surface exhibits a topologically\nnontrivial hedgehog configuration.\nPACS numbers: 03.75.Ss,05.30.Fk,75.80.+q,71.10.Ay\nI. INTRODUCTION\nRecentexperimentalprogressofultracoldelectricdipo-\nlar heteronuclear molecules has become a major focus of\nultracold atom physics1–3. Electric dipole moments are\nessentially classic polarization vectors induced by the ex-\nternal electric field. When they are aligned along the z\naxis, the electric dipolar interaction becomes anisotropic\nexhibiting the dr2−3z2-type anisotropy. In Fermi sys-\ntems, this anisotropy has important effects on many-\nbody physics including both single-particle and collective\nproperties4–14. Fermi surfaces of polarized electric dipo-\nlar fermions exhibit quadrupolar distortion elongated\nalong thezaxis4,5,7,13. Various Fermi surface instabil-\nities have been investigated including the Pomeranchuk\ntype nematic distortions6,7and stripelike orderings10,14.\nThe collective excitations of the zero sound mode exhibit\nanisotropic dispersions: The sound velocity is largest if\nthe propagation wavevector /vector qis along the zaxis, and\nthe sound is damped if /vector qlies in the xyplane7,8. Un-\nder the dipolar anisotropy, the phenomenological Lan-\ndau interaction parameters become tridiagonal matrices,\nwhich are calculated at the Hartree-Focklevel6,7, and the\nanisotropicFermi liquid theoryfor such systemshas been\nsystematically studied7.\nThe magnetic dipolar gases are another type of dipolar\nsystem. Compared to the extensive research on electric\ndipolar Fermi systems, the study on magnetic dipolar\nones is a new direction of research. On the experimen-\ntal side, laser cooling and trapping Fermi atoms with\nlarge magnetic dipole moments (e.g.,161Dy and163Dy\nwithµ= 10µB)15–17have been achieved, which provides\na new opportunity to study exotic many-body physics\nwith magnetic dipolar interactions. There has also been\na great amount of progress for realizing Bose-Einstein\ncondensations of magnetic dipolar atoms17–21.\nAlthough the energy scale of the magnetic dipolar in-\nteraction is much weaker than that of the electric one, it\nis conceptually more interesting if magnetic dipoles arenot aligned by external fields. Magnetic dipole moments\nare proportional to the hyperfine spin up to a Lande fac-\ntor, thus, they are quantum-mechanical operators rather\nthan the nonquantized classic vectors as electric dipole\nmoments are. Furthermore, there is no need to use ex-\nternal fields to induce magnetic dipole moments. In fact,\nthe unpolarized magnetic dipolar systems are isotropic.\nThe dipolar interaction does not conserve spin nor orbit\nangular momentum, but is invariant under simultaneous\nspin-orbit (SO) rotation. This is essentially a spin-orbit\ncoupled interaction. Different from the usual spin-orbit\ncoupling of electrons in solids, this coupling appears at\nthe interaction level but not at the kinetic-energy level.\nThe study of many-body physics of magnetic dipolar\nFermi gases is just at the beginning. For the Fermi liquid\nproperties, although magnetic dipolar Fermi gases were\nstudied early in Refs. [22] and [6], the magnetic dipoles\narefrozen, thus, their behaviorisnot much different from\nthe electric ones. It is the spin-orbit coupled nature that\ndistinguishes non-polarized magnetic dipolar Fermi gases\nfrom polarized electric ones. The study along this line\nwas was pioneered by Fregoso and Fradkin23,24. They\nstudied the coupling between ferromagnetic and ferrone-\nmatic orders, thus, spin polarization distorts the spher-\nical Fermi surfaces and leads to a spin-orbit coupling in\nthe single-particle spectrum.\nSince Cooper pairing superfluidity is another impor-\ntant aspect of the many-body phase, we also briefly sum-\nmarizethecurrentprogressinelectricandmagneticdipo-\nlar systems. For the single-component electric dipolar\ngases, the simplest possible pairing lies in the p-wave\nchannel because s-wave pairing is not allowed by the\nPauli exclusion principle. The dipolar anisotropy se-\nlects thepz-channel pairing25–32. Interestingly, for the\ntwo-component case, the dipolar interaction still favors\nthe triplet pairing in the pzchannel even though the s\nwave is also allowed. It provides a robust mechanism for\nthe triplet pairing to the first order in the interaction\nstrength33–36. The mixing between the singlet and the2\ntripletpairingsiswitharelativephase ±π\n2,whichleadsto\na novel time-reversal symmetry-breaking pairing state33.\nThe investigation of the unconventional Cooper pairing\nsymmetry in magnetic dipolar systems was studied by\nthe authors37. We have found that it provides a robust\nmechanismforanovel p-wave(L= 1)spintriplet( S= 1)\nCooper pairing to the first order in interaction strength.\nIt comesdirectly from the attractivepart ofthe magnetic\ndipolar interaction. In comparison, the triplet Cooper\npairings in3He and solid-state systems come from spin\nfluctuations, which is a second-order effect in interaction\nstrength38,39. Furthermore, that pairing symmetry was\nnot studied in3He systems before in which orbital and\nspin angular momenta of the Cooper pair are entangled\ninto the total angular momentum J= 1. In contrast, in\nthe3He-Bphase40,LandSare combined as J= 0, and\nin the3He-Aphase,LandSare decoupled and Jis not\nwell-defined41,42.\nFermi liquid theory is one of the most important\nparadigms in condensed matter physics on interacting\nfermions38,43. Despitethepioneeringpapers6,22–24, asys-\ntematic study of the Fermi liquid properties of magnetic\ndipolar fermions is still lacking in the literature. In par-\nticular, Landau interactionmatrices havenot been calcu-\nlated, and a systematic analysis of the renormalizations\nfrom magnetic dipolar interactions to thermodynamic\nquantities has not been performed. Moreover, collective\nexcitations in magnetic dipolar ultracold fermions have\nnot been studied before. All these are essential parts of\nFermi liquid theory. The experimental systems of161Dy\nand163Dy are with a very largehyperfine spin of F=21\n2,\nthus the Fermi liquid theory taking into account of all\nthe complicated spin structure should be very challeng-\ning. We take the first step by considering the simplest\ncase of spin-1\n2magnetic dipolar fermions which preserve\nthe essential features of spin-orbit physics and address\nthe above questions.\nIn this paper, we systematically investigate the Fermi\nliquid theory of the magnetic dipolar systems includ-\ning both the thermodynamic properties and the collec-\ntive excitations, focusing on the spin-orbit coupled ef-\nfect. The Landau interaction functions are calculated\nand are diagonalized in the spin-orbit coupled basis.\nRenormalizations for thermodynamic quantities and the\nPomeranchuk-type Fermi surface instabilities are stud-\nied. Furthermore, thecollectivemodesarealsospin-orbit\ncoupled with a topologically non-trivial configuration of\nthe spin distribution in momentum space. Their disper-\nsion relation and configurations are analyzed.\nUponthecompletionofthispaper, webecameawareof\nthe nice work by Sogo et al.44. Reference 44 constructed\nthe Landau interaction matrix for dipolar fermions with\na general value of spin. The Pomeranchuk instabilities\nwereanalyzedforthespecialcaseofspin1\n2, andcollective\nexcitations were discussed. Our paper has some overlaps\non the above topics with Ref. [44] but with a signifi-\ncant difference, including the physical interpretation of\nthe Pomeranchuk instability in the J= 1−channel andour discovery of an exotic propagating spin-orbit sound\nmode.\nThe remaining part of this paper is organized as fol-\nlows. The magnetic dipolar interaction is introduced in\nSec. II. The Landau interaction matrix is constructed at\nthe Hartree-Fock level and is diagonalized in Sec. III. In\nSec. IV, we present the study of the Fermi liquid renor-\nmalization to thermodynamic properties from the mag-\nnetic dipolar interaction. The leading Pomeranchuk in-\nstabilities areanalyzed. In Sec. V, the spin-orbitcoupled\nBoltzmann equation is constructed. We further perform\nthe calculation of propagating spin-orbit coupled collec-\ntive modes. We summarize the paper in Sec. VI.\nII. MAGNETIC DIPOLAR HAMILTONIAN\nWe introduce the magnetic dipolar interaction and the\nsubtlety of its Fourier transform in this section.\nThe magnetic dipolar interaction between two spin-1\n2particles located at /vector r1,2reads\nVαβ;β′α′(/vector r) =µ2\nr3/bracketleftBig\n/vectorSαα′·/vectorSββ′−3(/vectorSαα′·ˆr)(/vectorSββ′·ˆr)/bracketrightBig\n,(1)\nwhere/vectorS=1\n2/vector σ;α,α′,β,β′take values of ↑and↓;/vector r=\n/vector r1−/vector r2and ˆr=/vector r/ris the unit vector along /vector r.\nThe Fourier transform of Eq. (1) is\nVαβ;β′α′(/vector q) =4πµ2\n3/bracketleftBig\n3(/vectorSαα′·ˆq)(/vectorSββ′·ˆq)−/vectorSαα′·/vectorSββ′/bracketrightBig\n,(2)\nwhich depends on the direction along the momentum\ntransfer but not its magnitude. It is singular as /vector q→0.\nMore rigorously, Vαβ,β′α′(/vector q) should be further multiplied\nby a numeric factor7as\ng(q) = 3/parenleftBigj1(qǫ)\nqǫ−j1(qL)\nqL/parenrightBig\n, (3)\nwhereǫis a short range scale cut off, and Lis the long\ndistance cut off at the scale of sample size. The spher-\nical Bessel function j1(x) shows the asymptotic behav-\niorj1(x)→x\n3atx→0, andj1(x)→1\nxsin(x−π\n2) as\nx→ ∞. In the long wavelength limit satisfying qǫ→0\nandqL→ ∞,g(q)→1 and we recover Eq. (2). If /vector q\nis exactly zero, Vαβ;β′α′= 0, because the dipolar inter-\naction is neither purely repulsive nor attractive, and its\nspatial average is zero.\nThe second quantization form for the magnetic dipolar\ninteraction is expressed as\nHint=1\n2V/summationdisplay\n/vectork,/vectork′,/vector qψ†\nα(/vectork+/vector q)ψ†\nβ(/vectork′)Vαβ;β′α′(/vector q)\n×ψβ′(/vectork′+/vector q)ψα′(/vectork), (4)\nwhereVis the volume of the system. The density of\nstates of two-component Fermi gases at the Fermi energy\nisN0=mkf\nπ2/planckover2pi12, and we define a dimensionless parameter3\nλ=N0µ2.λdescribes the interaction strength, which\nequals the ratio between the average interaction energy\nand the Fermi energy up to a factor on the order of 1.\nIII. SPIN-ORBIT COUPLED LANDAU\nINTERACTION\nInthissection, wepresenttheLandauinteractionfunc-\ntions of the magnetic dipolar Fermi liquid, and perform\nthe spin-orbit coupled partial wave decomposition.\nA. The Landau interaction function\nInteraction effects in the Fermi liquid theory are cap-\ntured by the Landau interaction function. It describes\nthe particle-hole channel forward-scattering amplitudes\namong quasiparticles on the Fermi surface. At the\nHartree-Fock level, the Landau function is expressed as\nfαα′,ββ′(ˆk,ˆk′) =fH\nαα′,ββ′(ˆq)+fF\nαα′,ββ′(ˆk,ˆk′),(5)\nwhere/vectorkand/vectork′are at the Fermi surface with the mag-\nnitude ofkfand/vector qis the small momentum transfer in\nthe forward scattering process in the particle-hole chan-\nnel.fH\nαα′,ββ′(/vector q) =Vαβ,β′α′(ˆq) is the direct Hartree in-\nteraction, and fF\nαα′,ββ′(/vectork;/vectork′) =−Vαβ,α′β′(/vectork−/vectork′) is the\nexchange Fock interaction. As /vector q→0,fHis singular,\nthus we need to keep its dependence on the direction of\nˆq. More explicitly,\nfH\nαα′,ββ′(ˆq) =πµ2\n3Mαα′,ββ′(ˆq), (6)\nfF\nαα′,ββ′(ˆk;ˆk′) =−πµ2\n3Mαα′,ββ′(ˆm),(7)\nwhere the tensor is defined as Mαα′,ββ′(ˆq) = 3(/vector σαα′·\nˆq)(/vector σββ′·ˆq)−/vector σαα′·/vector σββ′and ˆmis the unit vector along\nthe direction of the momentum transfer ˆ m=/vectork−/vectork′\n|/vectork−/vectork′|. We\nhave used the following identity:\n3(/vector σαβ′·ˆm)(/vector σβα′·ˆm)−/vector σαβ′·/vector σβα′\n= 3(/vector σαα′·ˆm)(/vector σββ′·ˆm)−/vector σαα′·/vector σββ′(8)\nto obtain Eq. (7).\nB. The spin-orbit coupled basis\nDue to the spin-orbit nature of the magnetic dipolar\ninteraction, we introduce the spin-orbit coupled partial-\nwave basis for the quasiparticle distribution over the\nFermi surface following the steps below.\nTheδnαα′(/vectork) is defined as\nδnαα′(/vectork) =nαα′(/vectork)−δαα′n0(/vectork), (9)wherenαα′(/vectork) =∝an}bracketle{tψ†\nα(/vectork)ψα′(/vectork)∝an}bracketri}htis the Hermitian single-\nparticle density matrix with momentum /vectorkand satisfies\nnαα′=n∗\nα′αandn0(/vectork) is the zero-temperature equilib-\nrium Fermi distribution function n0(/vectork) = 1−θ(k−kf).\nδnαα′(/vectork)isexpandedintermsoftheparticle-holeangular\nmomentum basis as\nδnαα′(/vectork) =/summationdisplay\nSszδnSsz(/vectork)χSsz,αα′\n=/summationdisplay\nSszδn∗\nSsz(/vectork)χ†\nSsz,αα′,(10)\nwhereχSsz,αα′are the bases for the particle-hole singlet\n(density) channel with S= 0 and triplet (spin) channel\nwithS= 1, respectively. They are defined as\nχ00,αα′=δαα′,\nχ10,αα′=σz,αα′, χ1±1,αα′=∓1√\n2(σx,αα′±iσy,αα′),\n(11)\nwhich satisfy the orthonormal condition tr( χ†\nSszχS′s′\nz) =\n2δSS′δszs′z.\nSince quasiparticles are only well defined around the\nFermi surface, we integrate out the radial direction and\narrive at the angular distribution,\nδnαα′(ˆk) =/integraldisplayk2dk\n(2π)3δnαα′(/vectork). (12)\nPlease note that angular integration is not performed in\nEq. (12). We expand δnαα′(ˆk) in the spin-orbit decou-\npled bases as\nδnαα′(ˆk) =/summationdisplay\nLmSs zδnLmSs zYLm(ˆk)χSsz,αα′,\n=/summationdisplay\nLmSs zδn∗\nLmSs zY∗\nLm(ˆk)χ†\nSsz,αα′,(13)\nwhereYLm(ˆk) is the spherical harmonics satisfying the\nnormalization condition/integraltext\ndˆkY∗\nLm(ˆk)YLm(ˆk) = 1.\nWe can also define the spin-orbit coupled basis as\nYJJz;LS(ˆk,αα′) =/summationdisplay\nmsz∝an}bracketle{tLmSs z|JJz∝an}bracketri}htYLm(ˆk)χSsz,αα′,\nY†\nJJz;LS(ˆk,αα′) =/summationdisplay\nmsz∝an}bracketle{tLmSs z|JJz∝an}bracketri}htY∗\nLm(ˆk)χ†\nSsz,αα′,\n(14)\nwhere∝an}bracketle{tLmSs z|JJz∝an}bracketri}htis the Clebsch-Gordon coefficient\nandYJJz;LSsatisfies the orthonormal condition of\n/integraldisplay\ndˆktr[Y†\nJJz;LS(ˆk)YJ′J′z;L′S′(ˆk)] = 2δJJ′δJzJ′zδLL′δSS′.\n(15)4\nUsing the spin-orbit coupled basis, δnαα′(ˆk) is expanded\nas\nδnαα′(ˆk) =/summationdisplay\nJJz;LSδnJJz;LSYJJz;LS(ˆk,αα′)\n=/summationdisplay\nJJz;LSδn∗\nJJz;LSY†\nJJz;LS(ˆk,αα′),(16)\nwhereδnJJz;LS=/summationtext\nmsz∝an}bracketle{tLmSs z|JJz∝an}bracketri}htδnLmSs z.\nC. Partial-wave decomposition of the Landau\nfunction\nWe are ready to perform the partial-wave decomposi-\ntion for Landau interaction functions. The tensor struc-\ntures in Eqs. (6) and (7) only depend on /vector σαα′and/vector σββ′,thus the magnetic dipolar interaction only contributes to\nthe spin-channel Landau parameters, i.e., S= 1. In the\nspin-orbit decoupled basis, the Landau functions of the\nHartree and Fock channels are expanded, respectively, as\nN0\n4πfH,F\nαα′;ββ′(ˆk,ˆk′) =/summationdisplay\nLmsz;L′m′s′zYLm(ˆk)χ1sz(αα′)\n×TH,F\nLm1sz;L′m′1s′zY∗\nL′m′(ˆk′)χ†\n1s′z(ββ′).\n(17)\nFor later convenience, we have multiplied the density of\nstatesN0and the factor of 1 /4πsuch thatTH,Fare di-\nmensionless matrices. Without loss of generality, in the\nHartree channel, we choose ˆ q= ˆz.\nThe matrix elements in Eq. (17) are presented below.\nIn the Hartree channel,\nTH\nLm1sz;L′m′1s′\nz=πλ\n3(2δsz,0−δsz,±1)δL,0δL′,0δm,0δm′,0δszs′\nz; (18)\nand in the Fock channel,\nTF\nLm1sz;L′m′1s′z=−πλ\n2/parenleftBigδLL′\nL(L+1)−δL+2,L′\n3(L+1)(L+2)−δL−2,L′\n3(L−1)L/parenrightBig\n×/integraldisplay\ndΩr[δszs′\nz−4πY1sz(Ωr)Y∗\n1s′\nz(Ωr)]YLm(Ωr)Y∗\nL′m′(Ωr). (19)\nThe magnetic dipolar interaction is isotropic, thus the spin-orbit cou pled basis are the most convenient. In these\nbasis, the Landau matrix is diagonal with respect to the total angu lar momentum Jand itsz-component Jzas\nN0\n4πfαα′;ββ′(ˆk,ˆk′) =/summationdisplay\nJJzLL′YJJz;L1(ˆk,αα′)FJJzL1;JJzL′1Y†\nJJz;L′1(ˆk,ββ′). (20)\nThe matrix kernel FJJzL1;JJzL′1reads as\nFJJzL1;JJzL′1=πλ\n3δJ,1δL,0δL′,0(2δJz,0−δJz,±1)+/summationdisplay\nmsz;m′s′z∝an}bracketle{tLm1sz|JJz∝an}bracketri}ht∝an}bracketle{tL′m′1s′\nz|JJz∝an}bracketri}htTF\nLm1sz;L′m′1s′z.(21)\nWe foundthat up toapositivenumericfactor, the second\nterm in Eq. (21) is the same as the partial-wave matri-\nces in the particle-particle pairing channel, which was\nderived for the analysis of the Cooper pairing instability\nin magnetic dipolar systems37.\nHowever, the above matrix kernel FJJzL1;JJzL′1is\nnot diagonal for channels with the same values of JJz\nbut different orbital angular momentum indices Land\nL′. Moreover, the conservation of parity requires that\neven and odd values of Ldo not mix. Consequently,\nFJJzL1;JJzL′1is either diagonalized or reduced into a\nsmall size of just 2 ×2. For later convenience of study-\ning collective modes and thermodynamic instabilities, we\npresent below the prominent Landau parameters in somelow partial-wave channels. Below, we use ( J±JzLS) to\nrepresent these channels in which ±represents even and\nodd parities, respectively.\nThe parity odd channel of J= 0−only has one possi-\nbility of (0−011) in which\nF0−011;0−011=π\n2λ. (22)\nThere is another even parity density channel with J=\n0+, i.e., (0+000), which receives contribution from short\nranges-wave interaction but no contribution from the\nmagnetic dipolar interaction at the Hartree-Fock level.\nThe parity odd channel of J= 1−only comes from5\n(1−Jz11) in which\nF1−Jz11;1−Jz11=−π\n4λ. (23)\nAnother channel of J= 1−, i.e., (1−Jz10), channel from\nthep-wave channel density interactions, which again re-\nceives no contribution from magnetic dipolar interaction\nat the Hartree-Fock level. These two J= 1−modes are\nspin- and charge-current modes, respectively, and thus,\ndo not mix due to their opposite symmetry properties\nunder time-reversal transformation.\nWe next consider the even parity channels. The\nJ= 1+channels include two possibilities of ( JJzLS) =\n(1+Jz01),(1+Jz21). The former is the ferromagnetism\nchannel, and the latter is denoted as the ferronematic\nchannel in Refs. [6] and [24]. Due to the spin-orbit na-\nture of the magnetic dipolar interaction, these two chan-\nnels are no longer independent but are coupled to each\nother. Because the Hartree term breaks the rotational\nsymmetry, the hybridization matrices for Jz= 0,±1 are\ndifferent. For the case of Jz= 0, it is\nF1+0=/parenleftbigg\nF1001;1001F1001;1021\nF1021;1001F1021;1021/parenrightbigg\n=πλ\n12/parenleftbigg\n8√\n2√\n2 1/parenrightbigg\n,\n(24)\nwhose two eigenvalues and their associated eigenvectors\nare\nw1+0\n1= 0.69πλ, ψ1+0\n1= (0.98,0.19)T,\nw1+0\n2= 0.06πλ, ψ1+0\n2= (−0.19,0.98)T.(25)\nThe hybridization is small. For the case of Jz=±1, the\nLandau matrices are the same as\nF1+1=/parenleftbigg\nF1101;1101F1101;1121\nF1121;1101F1121;1121/parenrightbigg\n=πλ\n12/parenleftbigg\n−4√\n2√\n2 1/parenrightbigg\n.\n(26)\nAgain the hybridization is small as shown in the eigen-\nvalues and their associated eigenvectors\nw1+1\n1=−0.37πλ, ψ1+1\n1= (0.97,−0.25)T,\nw1+1\n2= 0.12πλ, ψ1+1\n2= (0.25,0.97)T.(27)\nLandau parameters, or, matrices, in other high partial-\nwave channels are neglected, because their magnitudes\nare significantly smaller than those above.\nWe need to be cautious on using Eqs. (24) and (26) in\nwhich the Hartree contribution of Eq. 6 is taken. How-\never, Eq. (6) is valid in the limit q≪kfbut should\nbe much larger than the inverse of sample size 1 /L. It\nis valid to use Eqs. (24) and (26) when studying the\ncollective spin excitations in Sec. V below. However,\nwhen studying thermodynamic properties, say, magnetic\nsusceptibility, under the external magnetic-field uniform\nat the scale of L, the induced magnetization is also uni-\nform. In this case, the Hartree contribution is suppressedto zero, thus the Landau matrices in the J= 1+channel\nare the same for all the values of Jzas\nF1+,thm(λ) =/parenleftbigg\nF1Jz01;1Jz01F1Jz01;1Jz21\nF1Jz21;1Jz01F1Jz21;1Jz21/parenrightbigg\nthm\n=πλ\n12/parenleftbigg\n0√\n2√\n2 1/parenrightbigg\n. (28)\nIn this case, the hybridization between these two chan-\nnels is quite significant. The two eigenvalues and their\nassociated eigenvectors are\nw1+\n1=−π\n12λ, ψ1+\n1= (/radicalbigg\n2\n3,−/radicalbigg\n1\n3)T,\nw1+\n2=π\n6λ, ψ1+\n2= (/radicalbigg\n1\n3,/radicalbigg\n2\n3)T. (29)\nIV. THERMODYNAMIC QUANTITIES\nIn this section, we study the renormalizations for ther-\nmodynamic properties by the magnetic dipolar interac-\ntion and investigatethe Pomeranchuk-typeFermi surface\ninstabilities.\nA. Thermodynamics susceptibilities\nThe change in the ground-state energy with respect\nto the variation in the Fermi distribution density matrix\ninclude the kinetic and interaction parts as\nδE\nV=δEkin\nV+δEint\nV. (30)\nThe kinetic-energy variation is expressed in terms of the\nangular distribution of δnαα′(ˆk) as\nδEkin\nV=4π\nN0/summationdisplay\nαα′/integraldisplay\ndˆkδnαα′(ˆk)δnα′α(ˆk)\n=8π\nN0/summationdisplay\nLmSS zδn∗\nLmSs zδnLmSs z,(31)\nwhere the units of δnSsz(ˆk) andδnLmSs zare the same\nas the inverse of the volume. The variation in the inter-\naction energy is\nδEint\nV=1\n2/summationdisplay\nαα′ββ′/integraldisplay/integraldisplay\ndˆkdˆk′fαα′,ββ′(ˆk,ˆk′)δnα′α(ˆk)δnβ′β(ˆk′)\n= 2/summationdisplay\nLmszL′m′s′z;Sδn∗\nLmSs zfLmSs z,L′m′Ss′zδn∗\nL′m′Ss′z.\n(32)\nAdding them together and changing to the spin-orbit\ncoupled basis, we arrive at\nδE\nV=8π\nN0/summationdisplay\nJJz;LL′;Sδn∗\nJJz;LSMJJzLS;JJzL′SδnJJz;L′S,(33)6\nwhere the matrix elements are\nMJJzLS;JJzL′S=δLL′+FJJzLS;JJzL′S.(34)\nIn the presenceofthe externalfield hJJzLS, the ground\nstate energy becomes\nδE\nV= 16π/braceleftBig1\n2χ0/summationdisplay\nJJzLL′Sδn∗\nJJz;LSMJJzLS;JJzL′SδnJJz;L′S\n−/summationdisplay\nJJzLShJJzLSδnJJz;LS/bracerightBig\n, (35)\nwhereχ0=N0is the Fermi liquid density of states. At\nthe Hartree-Fock level, N0receives no renormalization\nfrom the magnetic dipolar interaction. The expectation\nvalue ofδnJJzLSis calculated as\nδnJJzLS=χ0/summationdisplay\nL′(M)−1\nJJzLS;JJzL′ShJJzL′S.(36)\nFor theJ= 1+channel,M−1≈I−F1+,thm(λ) up to\nfirstorderof λinthecaseof λ≪1. Asaresult,theexter-\nnal magnetic field /vectorhalong thezaxis not only induces the\nz-component spin polarization, but also induces a spin-\nnematic order in the channel of ( J+JzLS) = (1+021),\nwhich is an effective spin-orbit coupling term as\nδH=√\n2\n12πλh/summationdisplay\nkψ†\nα(/vectork)/braceleftBig/bracketleftbig\n(k2−3k2\nz)σz\n−3kz(kxσx+kyσy)/bracketrightbig/bracerightBig\nψβ(/vectork). (37)\nApparently, this term breaks time-reversal symmetry,\nand thus cannot be induced by the relativistic spin-orbit\ncouplinginsolidstates. Thismagneticfieldinducedspin-\norbit coupling in magnetic dipolar systems was studied\nby Fregoso et al.6,24\nB. Pomeranchuk instabilities\nEven in the absence of external fields, Fermi surfaces\ncan be distorted spontaneously known as Pomeranchuk\ninstabilities45. Intuitively, we can imagine the Fermi sur-\nface as the elastic membrane in momentum space. The\ninstabilities occur if the surface tension in any of its\npartial-wavechannels becomes negative. In the magnetic\ndipolar Fermi liquid, the thermodynamic stability condi-\ntion is equivalent to the fact that all the eigenvalues of\nthe matrix MJJzLS;JJzL′Sare positive.\nWe next check the negative eigenvalues of the Landau\nmatrix in each partial-wave channel. Due to the absence\nof external fields, the Pomeranchuk instabilities are al-\nlowed to occur as a density wave state with a long wave\nlengthq→0. For the case of J= 1+, it is clear that in\nthe channel of Jz=±1, the eigenvalue w1+1\n1in Eq. (27)\nis negative and the largest among all the channels. Thus\nthe leading channel instability is in the ( JJz) = (1+±1)channel, which occurs at w1+1\n1<−1, or, equivalently,\nλ > λc\n1+1= 0.86. The corresponding eigenvector shows\nthat it is mostly a ferromagnetism order parameter with\nsmall hybridization with the ferronematic channel. A re-\npulsiveshort-range swavescattering, which weneglected\nabove will enhance ferromagnetism and, thus, will drive\nλc\n1+1to a smaller value. The wavevector /vector qof the spin\npolarization should be on the order of 1 /Lto minimize\nthe energy cost of twisting spin, thus, essentially exhibit-\ning a domain structure. The spatial configuration of the\nspin distribution should be complicated by actual bound-\nary conditions. In particular, the three-vector nature of\nspins implies the rich configurations of spin textures. An\ninteresting result is that the external magnetic field actu-\nally weakens the ferromagnetism instability. If the spin\npolarization is aligned by the external field, the Landau\ninteraction matrix changes to Eq. (28). The magnitude\nof the negative eigenvalue is significantly smaller than\nthat of Eq. (26). As a result, an infinitesimal external\nfield cannot align the spin polarization to be uniform but\na finite amplitude is needed.\nFor simplicity, we only consider ferromagnetism with\na single plane wave vector /vector qalong thezaxis, then the\nspin polarization spirals in the xy-plane. Since q∼1/L,\nwe can still treat a uniform spin polarization over a dis-\ntance large comparable to the microscopic length scale.\nWithout loss of generality, we set the spin polariza-\ntion along the xaxis. As shown in Ref. 24, ferro-\nmagnetism induces ferronematic ordering. The induced\nferronematic ordering is also along the xaxis, whose\nspin-orbit coupling can be obtained based on Eq. (37)\nby a permutation among components of /vectorkasH′\nso(/vectork)∝\n(k2−3k2\nx)σx−3kx(kyσy+kzσz). According to Eq. (27),\nferromagnetism and ferronematic orders are not strongly\nhybridized, the energy scale of the ferronematic SO cou-\npling is about 1 order smaller than that of ferromag-\nnetism. An interesting point of this ferromagnetism is\nthatit distortsthe sphericalshapeofthe Fermisurfaceas\npointed by Fregoso and Fradkin24. This anisotropy will\nalso affect the propagationof Goldstone modes. Further-\nmore, spin waves couple to the oscillation of the shape of\nFermi surfaces bringing Landau damping to spin waves.\nThis may result in non-Fermi liquid behavior for fermion\nexcitations, and will be studied in a later paper. This ef-\nfect in the nematic symmetry-breakingFermi liquid state\nhas been extensively studied before in the literature46–51.\nThe next subleading instability is in the J= 1−chan-\nnel withL= 1 andS= 1 as shown in Eq. (23), which\nis a spin-current channel. The generated order parame-\nters are spin-orbit coupled. For the channel of Jz= 0,\nthe generated SO coupling at the single-particle level ex-\nhibits the three-dimensional (3D) Rashba type as\nHso,1−=|nz|/summationdisplay\nkψ†\nα(/vectork)(kxσy−kyσx)αβψβ(/vectork),(38)\nwhere|nz|is the magnitude of the order parameter. The\nsame result was also obtained recently in Ref. 44. In\nthe absence of spin-orbit coupling, the L=S= 1 chan-7\nnel Pomeranchuk instability was studied in Refs. [52]\nand [53], which exhibits the unconventional magnetism\nwith both isotropic and anisotropic versions. They\nare particle-hole channel analogies of the p-wave triplet\nCooper pairings of3He isotropic Band anisotropic A\nphases, respectively. In the isotropic unconventional\nmagnetic state, the total angular momentum of the or-\nder parameter is J= 0, which exhibits the /vectork·/vector σ-type\nspin-orbit coupling. This spin-orbit coupling is gener-\nated from interactions through a phase transition and,\nthus, was denoted as the spontaneous generation of spin-\norbit coupling. In Eq. (38), the spin-orbit coupling that\nappears at the mean-field single-particle level cannot be\ndenoted as spontaneous because the magnetic dipolar in-\nteraction possesses the spin-orbit nature. Interestingly,\nin the particle-particle channel, the dominant Cooper\npairing channel has the same partial-wave property of\nL=S=J= 137.\nThe instability in the J= 1−(spin current) channel\nis weaker than that in the 1+(ferromagnetism) channel\nbecause the magnitude of Landau parameters is larger\nin the former case. The 1−channel instability should\noccur after the appearance of ferromagnetism. Since\nspin-current instability breaks parity, whereas, ferromag-\nnetism does not, this transition is a genuine phase tran-\nsition. For simplicity, we consider applying an external\nmagnetic field along the zaxis in the ferromagnetic state\nto remove the spin texture structure. Even though the\nJ= 1+and 1−channels share the same property un-der rotation transformation, they do not couple at the\nquadratic level because of their different parity proper-\nties. The leading-order coupling occurs at the quartic\norder as\nδF=β1(/vector n·/vector n)(/vectorS·/vectorS)+β2|/vector n×/vectorS|2, (39)\nwhere/vector nand/vectorSrepresent the order parameters in the\nJ= 1−and 1+channels, respectively. β1needs to be\npositive to keep the system stable. The sign of β2de-\ntermines the relative orientation between /vector nand/vectorS. It\ncannot be determined purely from the symmetry analy-\nsis but depends on microscopic energetics. If β2>0, it\nfavors/vector n∝bardbl/vectorS, and/vector n⊥/vectorSis favored at β2<0.\nV. THE SPIN-ORBIT COUPLED COLLECTIVE\nMODES\nIn this section, we investigate another important fea-\ntureoftheFermiliquid, thecollectivemodes, whichagain\nexhibit the spin-orbit coupled nature.\nA. Spin-orbit coupled Boltzmann equation\nWe employ the Boltzmann equation to investigate the\ncollective modes in the Fermi liquid state43\n∂\n∂tn(/vector r,/vectork,t)−i\n/planckover2pi1[ǫ(/vector r,/vectork,t),n(/vector r,/vectork,t)]+1\n2/summationdisplay\ni/braceleftBig∂ǫ(/vector r,/vectork,t)\n∂ki,∂n(/vector r,/vectork,t)\n∂ri/bracerightBig\n−1\n2/summationdisplay\ni/braceleftBig∂ǫ(/vector r,/vectork,t)\n∂ri,∂n(/vector r,/vectork,t)\n∂ki/bracerightBig\n= 0,(40)\nwherenαα′(/vector r,/vectork,t)andǫαα′(/vector r,/vectork,t)arethedensityanden-\nergy matrices for the coordinate ( /vector r,/vectork) in the phase space\nand [,] and{,}mean the commutator and anticommuta-\ntor, respectively. Under small variations in nαα′(/vector r,/vectork,t)\nandǫαα′(/vector r,/vectork,t),\nnαα′(/vector r,/vectork,t) =n0(k)δαα′+δnαα′(/vector r,/vectork,t),\nǫαα′(/vector r,/vectork,t) =ǫ(k)δαα′+/integraldisplayd3k′\n(2π)3fαα′,ββ′(ˆk,ˆk′)\n×δnββ′(ˆk′). (41)\nthe above Boltzmann equation can be linearized. Plug-\nging the plane-wave solution of\nδnαα′(/vector r,/vectork,t) =/summationdisplay\nqδnαα′(/vectork)ei(/vector q·/vector r−ωt),(42)we arrive at\nδnαα′(ˆk)−1\n2cosθk\ns−cosθk/summationdisplay\nββ′/integraldisplay\ndΩk′N0\n4πfαα′,ββ′(ˆk,ˆk′)\n×δnββ′(ˆk′) = 0, (43)\nwheresis the dimensionless parameter ω/(vfq). The\npropagationdirectionofthe wavevector /vector qisdefinedalong\nthez-direction.\nIn the spin-orbit decoupled basis defined as δnLmSs z\nin Sec. IIIB, the linearized Boltzmann equation becomes\nδnLmSs z+ ΩLL′;m(s)FL′m′Ssz;L′′m′′Ss′′zδnL′′m′′Ss′′z= 0,\n(44)\nwhere Ω LL′(s) is equivalent to the particle-hole channel\nFermi bubble in the diagrammatic method as\nΩLL′;m(s) =−/integraldisplay\ndΩˆkY∗\nLm(ˆk)YL′m(ˆk)cosθk\ns−cosθk.(45)8\nFor later convenience, we present Ω LL′;min several chan-\nnels ofLL′andmas follows\nΩ00;0(s) = 1−s\n2ln|1+s\n1−s|+iπ\n2sΘ(s<1),\nΩ10;0(s) = Ω 01;0=√\n3sΩ00;0(s),\nΩ11;0(s) = 1+3s2Ω00;0(s),\nΩ11;1(s) = Ω 11;−1(s) =−1\n2/bracketleftBig\n1−3(1−s2)Ω00;0(s)/bracketrightBig\n.\n(46)\nEquation (44) can be further simplified by using the\nspin-orbit coupled basis δnJJz;LSdefined in Sec. IIIB,\nδnJJz;LS+/summationdisplay\nJ′;LL′KJJzLS;J′JzL′S(s)FJ′JzL′S;J′JzL′′S\n×δnJ′JzL′′S= 0, (47)\nwhere the matrix kernel KJJzLS;J′JzL′Sreads\nKJJzLS;J′JzL′S(s) =/summationdisplay\nmsz∝an}bracketle{tLmSs z|JJz∝an}bracketri}ht∝an}bracketle{tL′mSsz|J′Jz∝an}bracketri}ht\n×ΩLL′;m(s). (48)\nB. The spin-orbit coupled sound modes\nPropagating collective modes exist if Landau parame-\nters are positive. In these collective modes, interactions\namong quasiparticles rather than the hydrodynamic col-\nlisions provide the restoring force. Because only the spin\nchannelreceivesrenormalizationfromthe magnetic dipo-\nlar interaction, we only consider spin channel collectivemodes. The largest Landau parameter is in the (1+001)\nchannel in which the spin oscillates along the direction\nof/vector q. The mode in this channel is the longitudinal spin\nzero sound. On the other hand, due to the spin-orbit\ncoupled nature, the Landau parameters are negative in\nthe transverse spin channels of (1+±1 0±1), and thus\nno propagating collective modes exist in these channels.\nThe hybridization between (1+001) and (1+021) is small\nas shown in Eq. (25), and the Landau parameter in the\n(1+021) channel is small, thus, this channel also is ne-\nglected below for simplicity.\nBecause the propagation wave vector /vector qbreaks the par-\nityand3Drotationsymmetries, the(1+001)channelcou-\nples to other channels with the same Jz. As shown in\nEq. (47), the coupling strengths depend on the magni-\ntudes of Landau parameters. We truncate Eq. (47) by\nkeeping the orbital partial-wave channels of L= 0 and\nL= 1 because Landau parameters with orbital-partial\nwavesL≥2arenegligible. Therearethreechannelswith\nL=S= 1 as(0−011), (1−011), and (2−011). We further\ncheck the symmetry properties of these four modes un-\nder the reflection with respect to any plane containing /vector q.\nThe mode of (1−011)is even and the other three are odd,\nthus it does not mix with them. The Landau parameter\nin the (2−011) channel is calculated asπ\n20λ, which is 1\norder smaller than those in (1+001) and (1−001), thus\nthis channel is also neglected. We only keep these two\ncoupled channels (1+001) and (1−001) in the study of\ncollective spin excitations.\nThe solution of the two coupled modes reduces to a\n2×2 matrix linear equation as\n/parenleftbigg\n1+Ω00;0(s)F1001;1001sΩ00;0(s)F0011;0011\nsΩ00;0(s)F1001;1001 1+Ω00;0(s)F0011;0011/parenrightbigg/parenleftbigg\nδn1001\nδn0011/parenrightbigg\n= 0, (49)\nwhere the following relations are used\nK1001;1001(s) = Ω 00;0(s)\nK1001;0011(s) =K0011;1001(s) =∝an}bracketle{t0010|10∝an}bracketri}ht∝an}bracketle{t1010|00∝an}bracketri}htΩ01;0(s) =sΩ00;0(s)\nK0011,0011(s) =/summationdisplay\nm|∝an}bracketle{t1m1−m|00∝an}bracketri}ht|2Ω11;m(s) =1\n3Ω11;0(s)+2\n3Ω11;1(s) = Ω00;0(s). (50)\nThe condition of the existence of nonzerosolutions of Eq.\n(49) becomes\n(1−s2)Ω2\n00;0(s)+2Ω 00;0(s)F+\nF2\n×+1\nF2\n×= 0,(51)\nwhereF+= (F1001:1001 +F0011;0011)/2 andF×=/radicalbig\nF1001:1001F0011;0011.Let us discuss several important analytical properties\nof its solutions. In order for collective modes to prop-\nagate in Fermi liquids, its sound velocity must satisfy\ns >1, otherwise it enters the particle-hole continuum\nand is damped, a mechanism called Landau damping.9\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48 /s50/s46/s53 /s51/s46/s48 /s51/s46/s53 /s52/s46/s48 /s52/s46/s53 /s53/s46/s48/s49/s46/s48/s49/s46/s53/s50/s46/s48/s50/s46/s53/s51/s46/s48/s51/s46/s53/s52/s46/s48\n/s32/s115\nλ\nFIG. 1: (Color online) The sound velocity sin the unit of\nvfv.s. the dipolar coupling strength λ. At 0 < λ≪1,\ns(λ)≈1 + 0+. On the order of λ≫1,s(λ) becomes linear\nwith the slope indicated in Eq. (56).\nWe can solve Eq. (51) as\nΩ±\n00;0(s) =F+±/radicalBig\nF2\n++(s2−1)F2\n×\n(s2−1)F2\n×.(52)\nOnly the expression of the Ω−\n00;0(s) is consistent with s>\n1 and is kept. The other branch has no solution of the\npropagating collective modes.\nLet us analytically check two limits with large and\nsmall values of λ, respectively. In the case of 0 <λ≪1\nsuch thats→1+0+, Eq. (51) reduces to\nΩ00;0(sλ≪1)≈1−1\n2ln2+1\n2ln(s−1) =−1\n2F+.(53)\nIts sound velocity solution is\nsλ≪1≈1+2e−2/parenleftbig\n1+1\n2F+/parenrightbig\n= 1+2e−2−12\n7πλ.(54)\nThe eigenvector can be easily obtained as1√\n2(1,1)T,\nwhich is an equal mixing between these two modes. On\nthe other hand, in the case of λ≫1, we also expect\ns≫1, and thus Eq. (51) reduces to\nΩ00;0(sλ≫1)≈ −1\nsF×=−1\n3s2, (55)\nwhose solution becomes\nsλ≫1≈F×\n3=π\n3√\n3λ. (56)\nIn our case, F1001is larger than F0011but is on\nthe same order. The eigenvector can be solved as\n1√2F+(√F0011,√F1001)Tin which the weight of the\n(0011) channel is larger.\nThe dispersion of the sound velocity swith respect to\nthe dipolar interaction strength λis solved numerically\nas presented in Fig. 1. Collective sound excitations exist\nfor all the interaction strengths with s >1. In both\nlimits of 0 ≪λ≪1 andλ≫1, the numerical solutions\n0 1 -1-1010\n1-1\nx F/k kk /ky F\nk /kz F\nFIG. 2: (Color online) The spin configuration [Eq. (57)] of\nthe zero-sound mode over the Fermi surface shows hedgehog-\ntype topology at λ= 10. The common sign of u1andu2is\nchosen to be positive, which gives rise to the Pontryagin ind ex\n+1. Although the hedgehog configuration is distorted in the\nzcomponent, its topology does not change for any values of\nλdescribing the interaction strength.\nagree with the above asymptotic analysis of Eqs. (54)\nand (56). In fact, the linear behavior of s(λ) already\nappears atλ∼1, and the slope is around 0 .6. For all the\ninteraction strengths, the (1+001) and (0−011) modes\nare strongly hybridized.\nThis mode is an oscillation of spin-orbit coupled Fermi\nsurface distortions. The configuration of the (0−011)\nmode exhibits an oscillating spin-orbit coupling of the\n/vectork·/vector σtype. This is the counterpart of the isotropic un-\nconventional magnetism, which spontaneously generates\nthe/vectork·/vector σ-type coupling52,53. The difference is that, here,\nit is a collective excitation rather than an instability. It\nstronglyhybridizeswith the longitudinal spin mode. The\nspin configuration over the Fermi surface can be repre-\nsented as\n/vector s(/vector r,/vectork,t) =\nu2sinθ/vectorkcosφ/vectork\nu2sinθ/vectorksinφ/vectork\nu2cosφ/vectork+u1\nei(/vector q·/vector r−sqvft),(57)\nwhere(u1,u2)Tisthe eigenvectorforthe collectivemode.\nWe have checked that for all the values of λ,|u2|>|u1|\nis satisfied with no change in their relative sign, thus the\nspin configuration as shown in Fig. 2 is topologically\nnon-trivial with the Pontryagin index ±1 which periodi-\ncally flips the sign with time and the spatial coordinate\nalong the propagating direction. It can be considered as\na topological zero sound.\nVI. CONCLUSIONS\nTo summairze, we have presented a systematic study\non the Fermi liquid theory with the magnetic dipolar\ninteraction, emphasizing its intrinsic spin-orbit coupled\nnature. Although this spin-orbit coupling does not ex-\nhibit at the single-particle level, it manifests in various10\ninteraction properties. The Landau interaction function\nis calculated at the Hartree-Fock level and is diagonal-\nized by the total angular momentum and parity quan-\ntum numbers. The Pomeranchuk instabilities occur at\nthe strong magnetic dipolar interaction strength gener-\nating effective spin-orbit coupling in the single-particle\nspectrum.\nWe have also investigated novel collective excitations\nin the magnetic dipolar Fermi liquid theory. The Boltz-\nmann transportequations aredecoupled in the spin-orbit\ncoupled channels. We have found an exotic collective ex-citation, which exhibits spin-orbit coupled Fermi surface\noscillations with a topologically nontrivial spin configu-\nration, which can be considered as a topological zero-\nsound-like mode.\nAcknowledgments\nY. L. and C. W. were supported by the AFOSR YIP\nprogram and NSF-DMR-1105945.\n1S. Ospelkaus, K. K. Ni, M. H. G. de Miranda, B. Neyen-\nhuis, D. Wang, S. Kotochigova, P. S. Julienne, D. S. Jin,\nand J. Ye, Faraday Discuss. 142, 351 (2009)..\n2K. K. Ni, S. Ospelkaus, M. H. G. de Miranda, A. Pe’er,\nB. Neyenhuis, J. J. Zirbel, S. Kotochigova, P. S. Julienne,\nD. S. Jin, and J. Ye, Science 322, 231 (2008).\n3A. Chotia, B. Neyenhuis, S. A. Moses, B. Yan, J. P. Covey,\nM. Foss-Feig, A. M. Rey, D. S. Jin, and J. Ye, Phys. Rev.\nLett.108, 080405 (2012).\n4T. Sogo, L. He, T. Miyakawa, S. Yi, H. Lu, and H. Pu,\nNew J. Phys. 11, 055017 (2009).\n5T. Miyakawa, T. Sogo, and H. Pu, Phys. Rev. A 77, 061603\n(2008).\n6B. M. Fregoso, K. Sun, E. Fradkin, and B. L. Lev, New J.\nPhys.11, 103003 (2009).\n7C. K. Chan, C. Wu, W. C. Lee, S. Das Sarma, Phys. Rev.\nA81, 023602 (2010),\n8S. Ronen and J. L. Bohn, Phys. Rev. A 81, 033601 (2010).\n9C. Lin, E. Zhao, and W. V. Liu, Phys. Rev. B 81, 045115\n(2010).\n10Y. Yamaguchi, T. Sogo, T. Ito, T. Miyakawa, Phys. Rev.\nA82, 013643 (2010).\n11Q. Li, E. H. Hwang, S. Das Sarma, Phys. Rev. B 82,\n235126 (2010).\n12C. M. Chang, W. C. Shen, C. Y. Lai, P. Chen, D. W.\nWang, Phys. Rev. A 79, 053630 (2009).\n13J. N. Zhang, S. Yi, Phys. Rev. A 81, 033617 (2010); J. N.\nZhang, S. Yi, Phys. Rev. A 80, 053614 (2009).\n14K. Sun, C. Wu, S. Das Sarma, Phys. Rev. B 82, 075105\n(2010).\n15S. H. Youn, M. Lu, U. Ray, and B. L. Lev, Phys. Rev. A\n82, 043425 (2010).\n16M. Lu, S. H. Youn, and B. L. Lev, Phys. Rev. Lett. 104,\n063001 (2010).\n17M. Lu, N. Q. Burdick, S. H. Youn, B. L. Lev, Phys. Rev.\nLett.107, 190401 (2011).\n18T. Koch, T. Lahaye, J. Metz, B. Frohlich, A. Griesmaier,\nand T. Pfau, Nature Physics 4, 218 (2008).\n19T. Lahaye, C. Menotti, L. Santos, M. Lewenstein, and T.\nPfau, Rep. Prog. Phys. 72, 126401 (2009).\n20T. Lahaye, J. Metz, T. Koch, B. Frohlich, A. Griesmaier,\nand T. Pfau, Pushing the Frontiers of Atomic Physics, 1,\n160 (2009).\n21C. Menotti, M. Lewenstein, T. Lahaye, and T. Pfau, in\nDipolar Interaction in Ultra-Cold Atomic Gases , edited by\nA. Campa, A. Giansanti, G. Morigi and F. S. Labini, AIP\nConf. Proc. No. 970 (AIP, New York, 2008), p. 332.22J. Quintanilla, S. T. Carr, and J. J. Betouras, Phys. Rev.\nA79, 031601 (R) (2009).\n23B. M. Fregoso, E. Fradkin, Phys. Rev. Lett. 103, 205301\n(2009)\n24B. M. Fregoso, E. Fradkin, Phys. Rev. B 81, 214443 (2010).\n25L. You and M. Marinescu, Phys. Rev. A 60, 2324 (1999).\n26M. A. Baranov, M. S. Mar’enko, V. S. Rychkov, and G. V.\nShlyapnikov, Phys. Rev. A 66, 013606 (2002).\n27M. A. Baranov, /suppress L. Dobrek, and M. Lewenstein, Phys. Rev.\nLett.92, 250403 (2004).\n28M. A. Baranov, Physics Reports 464, 71 (2008).\n29G. M. Bruun, E. Taylor, Phys. Rev. Lett. 101, 245301\n(2008).\n30J. Levinsen, N. R. Cooper, and G. V. Shlyapnikov, Phys.\nRev. A84, 013603 (2011).\n31A. C. Potter, E. Berg, D. W. Wang, B. I. Halperin, and E.\nDemler, Phys. Rev. Lett. 105, 220406 (2010).\n32R. M. Lutchyn, E. Rossi, and S. Das Sarma, Phys. Rev. A\n82, 061604 (2010).\n33C. Wu, and J. E. Hirsch, Phys. Rev. B 81, 020508 (R)\n(2010).\n34K. V. Samokhin, and M. S. Mar’enko, Phys. Rev. Lett. 97,\n197003 (2006).\n35T. Shi, J. N. Zhang. C. P. Sun and S. Yi, Phys. Rev. A 82,\n033623, (2010).\n36B. Kain, Hong Y. Ling, Phys. Rev. A 83, 061603(R),\n(2011).\n37Y. Li, C. Wu, Scientific Reports 2, 392 (2012).\n38A. J. Leggett, Rev. Mod. Phys. 47, 331 (1975).\n39G. E. Volovik, “The Universe in a Helium Droplet” , (Ox-\nford University Press, Oxford 2009).\n40R. Balian, N. R. Werthamer, Phys. Rev. 131, 1553 (1963).\n41P. W. Anderson, P. Morel, Phys. Rev. 123, 1911 (1961).\n42W. F. Brinkman, J. W. Serene, and P. W. Anderson, Phys.\nRev. A10, 2386 (1974).\n43W. Negele and H. Orland, Quantum Many-Particle Sys-\ntems, (Perseus Books, New York, 1988).\n44T. Sogo, M. Urban, P. Schuck, T. Miyakawa, Phys. Rev.\nA 85, 031601(R) (2012).\n45I. J. Pomeranchuk, Sov. Phys. JETP 8, 361 (1959).\n46V. Oganesyan, S. A. Kivelson, and E. Fradkin, Phys. Rev.\nB64, 195109 (2001).\n47M. Garst and A. V. Chubukov, Phys. Rev. B 81, 235105\n(2010).\n48M. A. Metlitski and S. Sachdev, Phys. Rev. B 82, 075127\n(2010)\n49P. W¨ olfle, and A. Rosch, Journal of Low Temperature11\nPhysics, 147, 165 (2007).\n50C. A. Lamas, D. C. Cabra, and N. Grandi, Phys. Rev. B\n80, 075108 (2009).\n51C. A. Lamas, D. C. Cabra, and N. E. Grandi, Int. J. of\nMod. Phys. B 25, 3539 (2011).52C. Wu, and S. C. Zhang, Phys. Rev. Lett. 93,\n036403(2004).\n53C. Wu, K. Sun, E. Fradkin, and S. C. Zhang, Phys. Rev.\nB75, 115103 (2007)."    },    {        "title": "1203.6684v1.Soliton_Magnetization_Dynamics_in_Spin_Orbit_Coupled_Bose_Einstein_Condensates.pdf",        "content": "Soliton Magnetization Dynamics in Spin-Orbit Coupled Bose-Einstein Condensates\nO. Fialko,1J. Brand,1and U. Z ulicke2\n1Centre for Theoretical Chemistry and Physics and New Zealand Institute for Advanced Study,\nMassey University, Private Bag 102904 NSMC, Auckland 0745, New Zealand\n2School of Chemical and Physical Sciences and MacDiarmid Institute for Advanced Materials and Nanotechnology,\nVictoria University of Wellington, PO Box 600, Wellington 6140, New Zealand\nRing-trapped Bose-Einstein condensates subject to spin-orbit coupling support localized dark\nsoliton excitations that show periodic density dynamics in real space. In addition to the density\nfeature, solitons also carry a localized pseudo-spin magnetization that exhibits a rich and tunable\ndynamics. Analytic results for Rashba-type spin-orbit coupling and spin-invariant interactions pre-\ndict a conserved magnitude and precessional motion for the soliton magnetization that allows for\nthe simulation of spin-related geometric phases recently seen in electronic transport measurements.\nPACS numbers: 03.75.Lm, 67.85.Fg, 03.65.Vf, 71.70.Ej\nThe recent realization of arti\fcial light-induced gauge\npotentials for neutral atoms [1] has added a power-\nful new instrument to the atomic-physics simulation\ntoolkit [2]. In particular, possibilities to induce Zeeman-\nlike and spin-orbit-type couplings in (pseudo-)spinor\natom gases [3] render them ideal laboratories to investi-\ngate the intriguing interplay of spin dynamics and quan-\ntum con\fnement that has been the hallmark of semicon-\nductor spintronics [4, 5]. At the same time, the unique\naspects of Bose-Einstein-condensed atom gases [6] asso-\nciated, e.g., with their intrinsically nonlinear dynamics,\npromise to give rise to novel behavior under the in\ruence\nof synthetic spin-orbit couplings [7{15].\nOne of the special properties resulting from nonlinear-\nity in Bose-Einstein condensates (BECs) is the existence\nof solitary-wave excitations [16]. Basic types of these\nare distinguished by the shape of their localized density\nfeature: dark (gray) solitons are associated with a full\n(partial) depletion of a uniform condensate density in a\n\fnite region of space, whereas bright solitons are local-\nized density waves on an empty background. A further\ncharacteristic associated with solitons is the phase gra-\ndient of the condensate order parameter centered at the\nposition of the density feature. In multi-component sys-\ntems, the dynamics of soliton excitations is found to be\nenriched by the additional degrees of freedom [17{20].\nWe have studied solitons in ring-trapped pseudo-spin-\n1=2 condensates with spin-invariant repulsive atom-atom\ninteractions subject to a Rashba-type [21, 22] spin-orbit\ncoupling and \fnd that they exhibit a third feature: a\npseudo-magnetization vector with conserved magnitude\nand rich dynamics that unfolds in tandem with the soli-\nton's periodic propagation in real space. Figure 1 shows\nan example and also illustrates the interesting fact that\nthe magnetization directions at the beginning and the\nend of a full cycle of the soliton's motion are generally not\nparallel. The appearance of such a geometric phase [23]\nand the precessional time evolution of the solitonic mag-\nnetization is reminiscent of the spin dynamics of electrons\ntraversing a mesoscopic semiconductor ring [24{28].In the following, we consider several soliton con\fgura-\ntions and obtain analytical results for their density and\nmagnetization pro\fles as well as the magnetization dy-\nnamics associated with their motion. We start by intro-\nducing the basic theoretical description of our system of\ninterest. Using the basis of a spatially varying local spin\nframe [26] for the condensate spinor, the nonlinear Gross-\nPitaevskii equation [6] for the spin-orbit-coupled ring\nBEC turns out to be of Manakov-type [17], making it pos-\nsible to apply standard methods [18, 20] to \fnd solitary-\nwave solutions. Accounting for the presence of spin-orbit\ncoupling adds an important twist: Spinors have to sat-\nisfy non-standard boundary conditions, which introduce\nbackground-density \rows in the local spin frame that\ncontribute to the nontrivial magnetization dynamics ex-\nhibited by the moving solitons in the lab frame.\nWe consider a two-component BEC trapped in the xy\nplane and con\fned to a ring of radius R. The atoms\nare assumed to be in the lowest quasi-onedimensional\nFIG. 1. Time evolution of a gray-bright soliton's magnetiza-\ntion in a ring-trapped BEC with Rashba spin-orbit coupling.\nDuring a full cycle of the soliton's motion on the ring, the\nmagnetization vector follows a trajectory on the surface of a\nsphere. The magnetization vectors at the beginning and the\nend of a cycle (indicated by arrows) di\u000ber by an angle #that is\nrelated to a spin-related geometric phase. Soliton parameters\n(see text):vs=c= 0:5, tan\u0011= 2,g= 100,\u0014=\u00000:01.arXiv:1203.6684v1  [cond-mat.quant-gas]  29 Mar 20122\nsubband [29] and subject to a spin-orbit coupling of the\nfamiliar Rashba form [21, 22] \u000bR[\u001bx(\u0000i@y)\u0000\u001by(\u0000i@x)]\nas well as a spin-rotationally invariant contact inter-\naction. (Here \u001bx;y;z are the spin-1/2 Pauli matrices.)\nThe energy functional of such a system [30] is given by\nE[\t] =R\nd'\ty(H\u0000\u0016) \t, where 'is the azimuthal\nangle,\u0016the chemical potential, \t = (  \"; #)Tthe two-\ncomponent (pseudo-spin-1 =2) spinor order parameter in\nthe representation where the ( z) direction perpendicular\nto the ring's plane is the spin-quantization axis, and\nH=E0\u0014\n\u0000@2\n'+g\n2\ty\t\n+ tan\u0011\u0000\n\u001b+e\u0000i'+\u001b\u0000ei'\u0001\u0010\n\u0000i@'+\u001bz\n2\u0011\u0015\n:(1)\nWe use\u001b\u0006\u0011(\u001bx\u0006i\u001by)=2 to denote raising and lower-\ning operators for spin-1/2 components, E0=~2=(2MR2)\nis the energy scale for quantum con\fnement of atoms\nwith massMin a ring of radius R,E0gis the two-body\ncontact-interaction strength, and tan \u0011= 2MR\u000b R=~2is\na dimensionless measure of the spin-orbit coupling.\nThe e\u000bect of Rashba spin-orbit coupling in a ring ge-\nometry can be elucidated by performing a suitable SU(2)\ntransformation. De\fning \t = U\u001fandHloc=U\u00001HU,\nwithU= e\u0000i'\u001bz=2ei\u0011\u001by=2ei'\u001bz=(2 cos\u0011), we \fnd\nHloc=E0\u0014\n\u0000@2\n'\u0000(tan\u0011)2\n4+g\n2\u001fy\u001f\u0015\n: (2)\nThe transformation U\u00001amounts to a '-dependent rota-\ntion of the pseudo-spin quantization axis [26], followed by\na spin-dependent gauge transformation. We will refer to\nthe original representation where the spin-quantization\naxis coincides with the axis of the ring as the lab frame ,\nwhereas the representation in which the Hamiltonian of\nthe system is diagonal in pseudo-spin space [i.e., given\nbyHlocof Eq. (2)] will be the local spin frame [26]. Note\nthat the spinors \t in the lab frame are periodic func-\ntions of', whereas the spinors \u001f= (\u001f+;\u001f\u0000)Tfrom the\nlocal spin frame have to satisfy the boundary conditions\n\u001f\u0006(') =\u001f\u0006('+ 2\u0019)e\u0006iAwith a spin dependent phase\ntwist originating from the spin-orbit coupling, where\nA=\u0019\u00121\ncos\u0011\u00001\u0013\n: (3)\nKnowledge of the local-spin-frame spinors enables the\ncalculation of expectation values for any observables ac-\ncessible to measurement in the lab frame. The total\ndensityn=j \"j2+j #j2\u0011j\u001f+j2+j\u001f\u0000j2is obviously\nthe same irrespective of which representation is chosen in\nspin space. The pseudo-spin-1/2 projections in the lab\nframe correspond to de\fnite atomic states, hence their\ndensity pro\fles n\"(#)= \ty([1 + (\u0000)\u001bz]=2)\t are of in-\nterest. In addition, we will consider the magnetization-\ndensity vector s= \ty\u001b\t in the lab frame, with \u001b=\n(\u001bx;\u001by;\u001bz) being the vector of Pauli matrices.We analyze the properties of localized excitations in\nspin-orbit-coupled ring-trapped BEC based on the time-\ndependent Gross-Pitaevskii equation [6] \u000eE[\u001f]=\u000e\u001f\u0003\n\u001b=\ni~@\u001f\u001b=@t. After rescaling to use the dimensionless time\nvariable\u001c=tE0=~, it has the form\ni@\u001f\u001b\n@\u001c=h\n\u0000@2\n'+g\u0010\nj\u001f+j2+j\u001f\u0000j2\u0000n0\u0011i\n\u001f\u001b(4)\nfor the two components of the spinor \u001f= (\u001f+;\u001f\u0000)T,\nwheren0= [\u0016+ (tan\u0011)2=4]=(gE0) is the uniform (back-\nground) density consistent with the chemical potential\n\u0016. While the spin-orbit coupling has formally disap-\npeared from the nonlinear equation (4), it is still im-\nplicitly present via the boundary conditions that the in-\ndividual components \u001f\u0006(';\u001c) must satisfy.\nWe have obtained several soliton solutions of Eqs. (4)\nusing established techniques [17, 18, 20] and implemented\nthe appropriate boundary conditions. Before giving fur-\nther details, we like to summarize a few general features.\nThe soliton spinors in the local-spin-frame representation\nturn out to be of the form\n\u001f(s)\n\u001b= \u0007(s)\n\u001b('\u0000vs\u001c) eivb\u001b'=2\u0000iv2\nb\u001b\u001c=4; (5)\nwhere \u0007(s)\n\u001b(\u0018) are complex amplitude functions encoding\nthe speci\fc soliton-like density features, vsis the propa-\ngation speed of the soliton, and vb\u001bare background \row\nvelocities of the individual spinor components that are\nnecessary to implement the boundary conditions arising\ndue to the presence of spin-orbit coupling. The density\nn(s)\n\"(#)and magnetization density s(s)exhibit spatially lo-\ncalized features. Subtracting s(s)from the magnetiza-\ntion density s(s)\nbof the condensate background yields the\nmagnetization density that is associated with the soli-\nton excitation only. Its integral S(s)=R\nd'[s(s)\nb\u0000s(s)]\nis the vector of total soliton magnetization, which is\nan additional property of localized excitations in multi-\ncomponent BECs. For soliton solutions of the form (5),\nS(s)has constant magnitude. Its temporal evolution is\nmost conveniently described by a set of four angles as\nde\fned in Fig. 2(c). While the tilt angles \fand\f0are\ntime-independent, the angles \u000band\u000b0vary linearly in\ntime, signifying the precession of S(s)around tilted z0\naxis with the universal result\n\f=\u0011; \u000b =vs\u001c+\u0019: (6)\nThez0axis is tilted by the angle \u0011characterizing the\nspin-orbit coupling and it rotates around the zaxis with\nthe same angular velocity vsthat characterizes the soli-\nton propagation. The second tilt angle \f0is found to de-\npend only on the soliton pro\fle \u0007 \u0006('), while the preces-\nsion frequency d\u000b0=d\u001chas complicated dependences on\nthe parameters of the soliton solutions. Figure 2 shows\nexemplary magnetization dynamics for gray-bright and3\nFIG. 2. Magnetization dynamics of a gray-bright soliton [panel (a)] and a gray-gray soliton with zero background magnetization\nin the local spin frame [panel (b)]. Parameters used are g= 100 (100), tan \u0011= 0:2 (0:5),vs=c= 0:5 (0.2), and \u0014=\u00000:5 for\nthe gray-bright (gray-gray) case. (c) Angles used to describe the two-step precessional motion of S. The angle \u000b0is measured\nwith respect to an x0axis that is perpendicular to both the zandz0axes.\ngray-gray solitons. Interestingly, we \fnd that the mag-\nnetization vector is usually not parallel to its initial di-\nrection after the soliton has completed a full cycle of its\nmotion around the ring as, e.g. seen in \fgure 1. The an-\ngle#between the magnetization directions at the start\nand the end of a cycle turns out to be \fnite only as a\nconsequence of spin-orbit coupling, as it depends promi-\nnently on the phase Agiven in Eq. (3) that also governs\nspin-dependent interference in mesoscopic ring conduc-\ntors [27].\nIn order to \fnd explicit soliton solutions, we introduce\n\u0018='\u0000u\u001c, whereuis a velocity parameter, and initially\nlook for solutions of the form \u001f\u001b(';\u001c) =p\nn\u001b(\u0018) ei\u0012\u001b(\u0018).\nThis allows us to rewrite Eq. (4) in the form\n\u0000u@n\u001b\n@\u0018+ 2@\n@\u0018\u0012\nn\u001b@\u0012\u001b\n@\u0018\u0013\n= 0;(7a)\nu@\u0012\u001b\n@\u0018+1pn\u001b@2pn\u001b\n@2\u0018\u0000\u0012@\u0012\u001b\n@\u0018\u00132\n\u0000g(n\u0000n0) = 0;(7b)\nwheren=n++n\u0000. Single component solutions for\n\u001b= ~\u001bare easily found by integration of Eqs. (7) to yield\n\u001f~\u001b(\u0018)/\u0007(\u0018) and\u001f\u0000~\u001b= 0, with the well-known dark\nsoliton solution on the in\fnite line [6]\n\u0007(\u0018) =pn0\u0014\niu\nc+\rutanh\u0012\n\ru\u0018\n\u0018D\u0013\u0015\n: (8)\nHere\r2\nu= 1\u0000u2=c2,c2= 2gn0, 1=\u00182\nD=gn0=2. The soli-\nton pro\fle (8) is appropriate for su\u000eciently strong non-\nlinearity, where \u0018D=\ru\u001c2\u0019[31]. However, \u0007( \u0018) does\nnot satisfy the proper boundary condition since it has a\nphase step \u0001 \u0012=\u00002 arccos(u=c). To compensate for the\nphase step and ensure the correct phase shift associated\nwith the gauge transformation U, we perform a Galilean\ntransformation on (8), which yields\n\u001fSC\n~\u001b(';\u001c) = \u0007(\u0018\u0000vb\u001c) eivb'=2\u0000iv2\nb\u001c=4: (9)\nHerevb=\u0000(\u0001\u0012+ ~\u001bA)=\u0019is the background velocity\nimposed by the boundary condition. Thus the single-\ncomponent soliton solution is of the form (5), with \u0007SC\n~\u001b=\n\u0007 and \u0007SC\n\u0000~\u001b= 0, and propagation speed vs=u+vb.A straightforward calculation yields sSC= ~\u001bj\u0007('\u0000\nvs\u001c)j2(\u0000sin\u0011cos';\u0000sin\u0011sin';cos\u0011)Tfor the magne-\ntization density of the single-component soliton solution.\nIn essence, the density depletion at the soliton's posi-\ntion gives rise to a reduction of the magnetization den-\nsitysSC\nb= ~\u001bn0(\u0000sin\u0011cos';\u0000sin\u0011sin';cos\u0011)Tasso-\nciated with the background. Thus sSC\nb\u0000sSCconsti-\ntutes the magnetization density associated with the soli-\nton itself, as it is the change in the background mag-\nnetization density due to the presence of the localized\nexcitation. For the single-component soliton, this corre-\nsponds to a peak in magnetization density at the soli-\nton's position. The total magnetization vector is ob-\ntained by integrating that peak in real space, which yields\nSSC= ~\u001b(\u0000sin\u0011cos(vs\u001c);\u0000sin\u0011sin(vs\u001c);cos\u0011)T. This\nmagnetization vector is precessing in a perfectly synchro-\nnized fashion with the soliton's motion around the ring\n[cf. Fig. 2(c) with Eq. (6) and \f0= 0], i.e.,#SC= 0.\nWe now consider a solution of Eqs. (7) that is a gray-\nbright (GB) soliton in the local spin frame. We assume\nthat the densities approach constant values n+!n0+\n(gray part) and n\u0000!0 (bright part) far away from\nthe soliton's position. To decouple Eq. (7b), we use the\nansatz [20] n\u0000=\u0014(n+\u0000n0+) with\u00001\u0014\u0014\u00140. We\napply a Galilean boost to both components to match the\nphase of the gray part only, hence they are of the form\n(5) with \u0007GB\n+(\u0018) given by \u0007( \u0018) from Eq. (8) but with\nrescaledc2= 2gn0+(1 +\u0014), 1=\u00182\nD=gn0+(1 +\u0014)=2, and\n\u0007GB\n\u0000(\u0018) =p\n\u0000\u0014n0+\rueiu\u0018=2\ncosh(\ru\u0018=\u0018D): (10)\nFurthermore, vb\u0000=vb+andvs=u+vb+. Figure 3\nshows the density pro\fles [panel (a)] and magnetization-\ndensity pro\fle [panel (c)] associated with a GB soliton.\nThe vector SGBof total magnetization for a GB soli-\nton precesses concomitantly with the soliton's motion; cf.\nFig. 2(c) with Eq. (6) and tan \f0= (p\u0000\u0014u\u0019)=[(1\u0000\u0014)c\ru],\n\u000b0=\u0000vs\u001c(1 +A=\u0019). Figures 1(a) and 2(a) show exam-\nples of possible time evolutions of the GB-soliton magne-\ntization. The magnitude of the magnetization vector is4\n/LParen1a/RParen1\n0.00.20.40.60.81.00.00.20.40.60.81.0\n/CurlyPhi/Slash12Π\n/LParen1b/RParen1\n0.00.20.40.60.81.00.00.20.40.60.81.0\n/CurlyPhi/Slash12Π\n/LParen1c/RParen1\n0.00.20.40.60.81.0/Minus0.50.00.51.0\n/CurlyPhi/Slash12Π\n/LParen1d/RParen1\n0.00.20.40.60.81.0/Minus0.50.00.51.0\n/CurlyPhi/Slash12Π\nFIG. 3. Lab-frame spin densities [(a),(b)] and zcompo-\nnent of the magnetization density [(c),(d)] for a stationary\ngray-bright soliton [(a),(c)] and a stationary gray-gray soliton\n[(b),(d)]. Panels (a) and (b) show the total density (solid yel-\nlow curve) and individual-spin (dashed blue = \", dotted red\n=#) densities normalized to the background-density value. In\n(c) and (d), the total density is plotted again for reference as\nthe solid yellow curve, together with the magnetization pro-\n\fless(s)\nz(dashed blue curve), and the magnetization density\ns(s)\nbz\u0000s(s)\nzassociated with the soliton excitations only (dotted\nred curve). Parameters are tan \u0011= 1:0,g= 100, and (for the\ngray-bright soliton) \u0014=\u00000:5.\nfound to be S= 2(1\u0000\u0014)\u0018D\ru=cos\f0. For a GB soliton,\nthe magnetization vector turns out to be notaligned with\nits initial direction after completion of a full cycle of its\nmotion around the ring. A straightforward calculation\nyields sin(#GB=2) = sin\f0sinA. As\f0is a known func-\ntion of the soliton parameters, a measurement of #GBwill\nyield the spin-related geometric phase A.\nThe solutions representing gray-gray (GG) solitons in\nthe local spin frame are obtained by Hirota's method [18].\nThe spinor components are of the form (5) with\n\u0007GG\n\u001b(\u0018) =pn0\u001b\u0014\niu\u001b\nc\u001b+\ru\u001btanh(a\u0018)\u0015\n: (11)\nHere,a2=\r2\nu+=\u00182\nD++\r2\nu\u0000=\u00182\nD\u0000,vs= 2au\u001b=c\u001b\ru\u001b+vb\u001b,\nc2\n\u001b= 2gn0\u001b, 1=\u00182\nD\u001b=gn0\u001b=2. The back-ground \rows\nare given by vb\u001b=\u0000(\u0001\u0012\u001b+\u001bA)=\u0019, where \u0001\u0012\u001b=\n\u00002 arccos(u\u001b=c\u001b). The independent parameters charac-\nterizing a GG soliton are the ratio n0+=n0\u0000(or, equiv-\nalently, the background magnetization in the local spin\nframe) and the speed vsof the soliton. All other param-\neters can be found by solving transcendental equations\ngiven just after Eq. (11). For simplicity, we consider the\ncase of a GG soliton with zero background magnetiza-\ntion in the local spin frame (i.e., n0+=n0\u0000\u0011n0=2).\nFigure 3 shows results for spinor-density [panel (b)] and\nmagnetization-density [panel (d)] pro\fles.\nThe time evolution of the magnetization vector asso-\nciated with a moving GG soliton is characterized by theangles de\fned in Fig. 2(c) with Eq. (6), \f0=\u0019=2, and\n\u000b0=!\u001c+\u0019=2, where!=\u0000(vb+\u0000vb\u0000)vs=2 + (v2\nb+\u0000\nv2\nb\u0000)=4\u0000(1+A=\u0019)vs. Figure 2(b) illustrates this dynam-\nics which, for small \u0011, corresponds to a slow rotation of\nthe magnetization vector in the ring's plane with super-\nimposed fast small-amplitude oscillations in the normal\ndirection. As in the case of the GB soliton, the magneti-\nzation vector does not evolve back to its initial direction\nafter a period of the soliton's ring revolution. The angle\nbetween magnetizations at the start and the end of the\ncycle is found to be #GG= 2\u0019j!j=vs!2AforA\u001c 1.\nAgain, the dependence of #GGonAenables determina-\ntion of the latter by measuring the former.\nIn conclusion, we have investigated the properties\nof soliton excitations in ring-trapped spin-orbit-coupled\nBECs. We \fnd that a magnetization degree of freedom is\ngenerally associated with a soliton, and that the magne-\ntization vector precesses around an axis that is rotating\nsynchronously with the soliton's orbital motion around\nthe ring. The magnetization direction at the end of a\ncycle of revolution does not coincide with the initial di-\nrection for the gray-bright and gray-gray cases, making it\npossible to measure a spin-orbit-related geometric phase.\nOur work opens up new avenues for the realization and\nmanipulation of magnetic soliton excitations in BECs.\nIt also creates the opportunity to study spin-dependent\ninterference and scattering e\u000bects that, until now, were\nonly accessible in semiconductor nanostructures.\nThis work was supported by the Marsden fund (con-\ntract no. MAU0910) administered by the Royal Society\nof New Zealand.\n[1] Y.-J. Lin, R. L. Compton, K. Jimenez-Garcia, J. V.\nPorto, and I. B. Spielman, Nature 462, 628 (2009).\n[2] J. Dalibard, F. Gerbier, Juzeli\u0016 unas, and P. Ohberg, Rev.\nMod. Phys. 83, 1523 (2011).\n[3] Y.-J. Lin, K. Jimenez-Garcia, and I. B. Spielman, Nature\n(London) 471, 83 (2011).\n[4] D. D. Awschalom, D. Loss, and N. Samarth, eds.,\nSemiconductor Spintronics and Quantum Computation\n(Springer, Berlin, 2002).\n[5] I. Zuti\u0013 c, J. Fabian, and S. D. Sarma, Rev. Mod. Phys.\n76, 323 (2004).\n[6] L. Pitaevskii and S. Stringari, Bose-Einstein Condensa-\ntion (Clarendon Press, Oxford, 2003).\n[7] T. D. Stanescu, B. Anderson, and V. Galitski, Phys.\nRev. A 78, 023616 (2008).\n[8] M. Merkl, A. Jacob, F. E. Zimmer, P. Ohberg, and\nL. Santos, Phys. Rev. Lett. 104, 073603 (2010).\n[9] C. Wang, C. Gao, C.-M. Jian, and H. Zhai, Phys. Rev.\nLett.105, 160403 (2010).\n[10] T.-L. Ho and S. Zhang, Phys. Rev. Lett. 107, 150403\n(2011).\n[11] C.-J. Wu, I. Mondragon-Shem, and X.-F. Zhou, Chin.\nPhys. Lett. 28, 097102 (2011).\n[12] S.-K. Yip, Phys. Rev. A 83, 043616 (2011).5\n[13] S. Sinha, R. Nath, and L. Santos, Phys. Rev. Lett. 107,\n270401 (2011).\n[14] H. Hu, B. Ramachandhran, H. Pu, and X.-J. Liu, Phys.\nRev. Lett. 108, 010402 (2012).\n[15] Y. Zhang, L. Mao, and C. Zhang, Phys. Rev. Lett. 108,\n035302 (2012).\n[16] R. Carretero-Gonz\u0013 aez, D. J. Frantzeskakis, and P. G.\nKevrekidis, Nonlinearity 21, R139 (2008).\n[17] S. V. Manakov, Zh. Eksp. Teor. Fiz. 65, 505 (1973), [Sov.\nPhys. JETP 38, 248 (1974)].\n[18] A. P. Sheppard and Y. S. Kivshar, Phys. Rev. E 55, 4773\n(1997).\n[19] P. Ohberg and L. Santos, Phys. Rev. Lett. 86, 2918\n(2001).\n[20] J. Smyrnakis, M. Magiropoulos, G. M. Kavoulakis, and\nA. D. Jackson, Phys. Rev. A 81, 063601 (2010).\n[21] E. I. Rashba, Fiz. Tverd. Tela (Leningrad) 2, 1224 (1960),\n[Sov. Phys. Solid State 2, 1109 (1960)].\n[22] Y. A. Bychkov and E. I. Rashba, J. Phys. C 17, 6039(1984).\n[23] A. Shapere and F. Wilczek, eds., Geometric Phases in\nPhysics (World Scienti\fc, Singapore, 1989).\n[24] D. Loss, P. Goldbart, and A. V. Balatsky, Phys. Rev.\nLett.65, 1655 (1990).\n[25] A. G. Aronov and Y. B. Lyanda-Geller, Phys. Rev. Lett.\n70, 343 (1993).\n[26] J. Splettstoesser, M. Governale, and U. Z ulicke, Phys.\nRev. B 68, 165341 (2003).\n[27] D. Frustaglia and K. Richter, Phys. Rev. B 69, 235310\n(2004).\n[28] F. Nagasawa, J. Takagi, Y. Kunihashi, M. Kohda, and\nJ. Nitta, Phys. Rev. Lett. 108, 086801 (2012).\n[29] This limitation is not crucial, as \fnite-width e\u000bects and\nhigher subbands could be treated straightforwardly.\n[30] M. Merkl, G. Juzeli\u0016 unas, and P. Ohberg, Eur. Phys. J.\nD59, 257 (2010).\n[31] Otherwise periodic solutions involving elliptic functions\nhave to be used. See, e.g., L. D. Carr, C. W. Clark, and\nW. P. Reinhardt, Phys. Rev. A 62, 063610 (2000)."    },    {        "title": "2011.10483v1.Anisotropy_of_the_spin_orbit_coupling_driven_by_a_magnetic_field_in_InAs_nanowires.pdf",        "content": "Anisotropy of the spin-orbit coupling driven by a magnetic \feld in InAs nanowires\nPawe l W\u0013 ojcik,1,\u0003Andrea Bertoni,2,yand Guido Goldoni3, 2,z\n1AGH University of Science and Technology, Faculty of Physics and\nApplied Computer Science, Al. Mickiewicza 30, 30-059 Krakow, Poland\n2CNR-NANO S3, Istituto Nanoscienze, Via Campi 213/a, 41125 Modena, Italy\n3Department of Physics, Informatics and Mathematics, University od Modena and Reggio Emilia, Italy\n(Dated: November 23, 2020)\nWe use the k\u0001ptheory and the envelope function approach to evaluate the Rashba spin-orbit\ncoupling induced in a semiconductor nanowire by a magnetic \feld at di\u000berent orientations, taking\nexplicitely into account the prismatic symmetry of typical nano-crystals. We make the case for the\nstrongly spin-orbit-coupled InAs semiconductor nanowires and investigate the anisotropy of the spin-\norbit constant with respect to the \feld direction. At su\u000eciently high magnetic \felds perpendicular\nto the nanowire, a 6-fold anisotropy results from the interplay between the orbital e\u000bect of \feld\nand the prismatic symmetry of the nanowire. A back-gate potential, breaking the native symmetry\nof the nano-crystal, couples to the magnetic \feld inducing a 2-fold anisotropy, with the spin-orbit\ncoupling being maximized or minimized depending on the relative orientation of the two \felds. We\nalso investigate in-wire \feld con\fgurations, which shows a trivial 2-fold symmetry when the \feld\nis rotated o\u000b the axis. However, isotropic spin-orbit coupling is restored if a su\u000eciently high gate\npotential is applied. Our calculations are shown to agree with recent experimental analysis of the\nvectorial character of the spin-orbit coupling for the same nanomaterial, providing a microscopic\ninterpretation of the latter.\nI. INTRODUCTION\nThe spin-orbit (SO) interaction, which couples the spin\nof electrons with their momentum, is the functioning\nprinciple of many spintronic applications, including spin\ntransistor,1,2spin \flters3{5or spin-orbit qubits.6,7Recent\ninvestigations focus towards semiconductor nanowires\n(NWs) with strong SO interaction8{16as host materials\nfor topological quantum computing based on Majorana\nzero energy modes.17{21These exotic quasi-particles form\nat the ends of a NW as a result of the interplay between\nthe SO coupling, Zeeman spin splitting and s-wave super-\nconductivity induced in the NW by the proximity e\u000bect\nfrom a superconducting shell.22{24\nIn general, a \fnite SO constant originates from the\nlack of the inversion symmetry. In semiconductors, this\ncould either be an intrinsic feature of the crystallographic\nstructure (Dresselhaus SO coupling25) or induced by the\ncon\fnement potential (Rashba SO coupling26,27). In\nzincblende NWs grown along the [111] direction, the crys-\ntal inversion symmetry is preserved and the Dresselhaus\nterm vanishes.10On the other hand, for spintronic ap-\nplications the Rashba term has the essential advantage\nof being tunable by external \felds, e.g., using external\ngates attached to the NW.28In general, external \felds\ninterplay with the overall NW geometry, which is typi-\ncally prismatic, and the value of the SO constant depends\non the position with respect to the underlying substrate,\nthe details of the dielectric con\fguration, as well as on\nthe compositional details of the NW which determine\nthe electronic states.13For example, we have recently\ndiscussed the additional possibilities to engineer the SO\nconstant in core-shell NWs with respect to homogeneous\nsamples.12Since the SO constant depends, in general, on\nthe symmetry and localization of the electronic states, amagnetic \feld may also induce a \fnite SO constant due\nto orbital e\u000bects.\nDespite the number of experiments with measurements\nof the Rashba SO constant in semiconductor NWs,8{10\nthe study of its anisotropy with respect to the magnetic\n\feld orientation is limited. Recently, such a vectorial con-\ntrol was reported for InAs NWs which were suspended in\norder to eliminate the SO contribution originating from\nthe substrate.29In Ref. 29 the authors tracked the non-\ntrivial evolution of the weak anti-localization (WAL) sig-\nnal and determined the SO length as a function of the\nmagnetic \feld intensity and direction. Interestingly, they\nobserved that the average SO coupling is isotropic with\nrespect to the magnetic \feld orientation and does not\nreveal any hallmark of the prismatic symmetry. When\napplying a transverse electric \feld by a gate, however, a\n2-fold anisotropy appears, with the maximal SO length\nwhen Bis perpendicular the electric \feld.\nMotivated by the availability of such experiments, we\nuse the 8\u00028k\u0001pmethod to analyze the dependence of\nthe Rashba SO constant on the magnetic \feld intensity\nand orientation. The full vectorial character of the SO\nconstant is taken into account by evaluating the SO cou-\npling constants separately in di\u000berent directions. While\nthe magnetic \feld perpendicular to the NW axis is able\nto generate a \fnite SO constant which turns out to be\nisotropic at low intensity (below \u00181 T), for larger \felds\nthe SO constant shows a slight 6-fold symmetry with re-\nspect to the \feld orientation, due to the interplay be-\ntween the orbital e\u000bects of the \feld and the prismatic\nsymmetry of the NW. A back-gate potential couples to\nthe magnetic \feld, which maximizes or minimizes the\nSO coupling depending on the relative orientation, lead-\ning to a 2-fold symmetry. We also investigate in-wire\n\feld con\fgurations. The trivial 2-fold symmetry whenarXiv:2011.10483v1  [cond-mat.mes-hall]  20 Nov 20202\nthe \feld is rotated in a plane which contains the axis, is\nalmost completely removed by a gate potential. Our re-\nsults are discussed in light of recent experiments reported\nin Ref. 29.\nThe paper is organized as follows. In Sec. II the Rashba\nSO coe\u000ecients are derived from the 8 \u00028k\u0001pmodel\nwithin the envelope function approximation, including\nthe orbital e\u000bects which originate from the magnetic \feld.\nThe e\u000bective Hamiltonian for the conduction electrons is\nderived in Sec. II A with details on the numerical method\ngiven in Sec. II B. Results of our calculations for homo-\ngeneous InAs NWs are reported in Sec. III, with a dis-\ncussion of recent experiments. Sec. IV summaries our\nresults.\nII. THEORETICAL MODEL\nWe consider a homogeneous InAs NW with hexagonal\ncross-section, grown along the [111] direction for which\nthe Dresselhaus contribution to the SO interaction can be\nneglected.30The NW is subjected to the external mag-\nnetic \feld B=B(cos\u0012sin\u001e;sin\u0012sin\u001e;cos\u001e), with in-\ntensityBand the direction being de\fned by the angle \u001e\nformed with the NW axis along zand the angle \u0012formed\nwith thexaxis, which connects two corners of the NW\nin thex\u0000yplane, see Fig. 1(a). We employ the gauge\nA(r) =B(\u0000ycos\u001e;0;ycos\u0012sin\u001e\u0000xsin\u0012sin\u001e). A back-\ngate is directly attached to the bottom of the NW, along\na facet, generating an electric \feld parallel to the NW\nsection, in the x\u0000yplane.10,13\nBelow we use the 8 \u00028 Kane model to derive the Rashba\nSO constants in terms of a realistic description of the\nquantum states in a magnetic \feld. This allows for quan-\ntitative predictions of SO coe\u000ecients as a function of the\nmagnetic \feld and the gate voltage for di\u000berent electron\nconcentrations.12,13\nA. E\u000bective SO Hamiltonian for conduction\nelectrons\nOur theoretical model is based on the 8 \u00028k\u0001p\nKane Hamiltonian within the envelope function approx-\nimation. We neglect here the spin Zeeman splitting, to\nfocus on the dominating orbital e\u000bects, that is the dis-\ntortion of the envelope function due to the \feld. It is\nstraightforward to add the Zeeman splitting to the elec-\ntron spin levels. The 8 \u00028 Kane Hamiltonian reads31\nH8\u00028=\u0012HcHcv\nHy\ncvHv\u0013\n; (1)\nwhereHcis the Hamiltonian of conduction electrons cor-\nresponding to the \u0000 6cband, while Hvis the Hamiltonian\nof the valence bands, \u0000 8v, \u00007v\nHc=H\u0000612\u00022; (2)\nHv=H\u0000814\u00024\bH\u0000712\u00022: (3)\nFIG. 1. (a) Schematics of a NW with a bottom gate. In our\nsimulations, anisotropy is evaluated with a magnetic \feld B\neither perpendicular to the NW axis ( \u001e=\u0019=2) and rotated\nwith an azimuthal angle \u0012, or with\u0012=\u0019=2 and rotated in\nthey\u0000zplane. (b) Occupation of the lowest subband as a\nfunction of the wave vector kzat each chemical potential \u0016\natB= 4 T. As\u0016increases, the occupation saturates to one\nat anykzbelow the Fermi energy. Of course, in general sev-\neral subbands are occupied. The non-parabolic dispersion is\nclearly appreciated, with the \feld inducing a seemingly Lan-\ndau level dispersion. Two vertical dashed lines mark values\nof\u0016selected for the further analysis.\nIn the above expressions\nH\u00006=\u0000P2\n2m0+Ec+V(r); (4)\nH\u00008=Ec+V(r)\u0000E0; (5)\nH\u00007=Ec+V(r)\u0000E0\u0000\u00010; (6)\nwhere P=p\u0000eA(r),m0is the free electron mass, Ec\nis the conduction band edge, E0is the energy gap, \u0001 0\nis the split-o\u000b gap and V(r) is the potential energy. In\nour target systems, the potential V(r) is the sum of the\nHartee potential energy generated by the electron gas\nand the electrical potential induced by the bottom gate\nattached to NW, V(r) =VH(r) +Vg(r).3\nThe o\u000b-diagonal matrix Hcvin (1) reads\nHcv=P0\n\u0016h0\n@\u0000P+p\n2q\n2\n3PzP\u0000p\n60\u0000Pzp\n3\u0000P\u0000p\n3\n0\u0000P+p\n6q\n2\n3PzP\u0000p\n2\u0000P+p\n3Pzp\n31\nA;\n(7)\nwhereP\u0006=Px\u0006iPyandP0=\u0000i\u0016hhSj^pxjXi=m0is the\nconduction-to-valence band coupling with jSi,jXibeing\nthe Bloch functions at the \u0000 point of Brillouin zone.\nFinally, the folding-down transformation31\nH(E) =Hc+Hcv(Hv\u0000E)\u00001Hy\ncv: (8)\nreduces the 8\u00028 Hamiltonian (1) into the 2 \u00022 e\u000bective\nHamiltonian for the conduction band electrons.\nThe in-plane vector potential is introduced into the nu-\nmerical model through the Peierls substitution.32Note\nthat the \feld does not break translational invariance\nalong the wire axis (the zdirection). Therefore, assuming\n\tn;kz(x;y;z ) = [ \"\nn;kz(x;y); #\nn;kz(x;y)]Teikzzand ex-\npanding the on- and o\u000b-diagonal elements of the Hamil-\ntonian (8) to second order, we obtain\nH=\u0014P2\n2D\n2m\u0003+1\n2m\u0003!2\nc\u0002\n(ycos\u0012\u0000xsin\u0012) sin\u001e\u0000kzl2\nB\u00032\n+Ec+V(x;y)\u0015\n12\u00022+ (\u000bx\u001bx+\u000by\u001by)Pz\n\u0016h; (9)\nwhere P2\n2D=P2\nx+P2\ny= (px+Bycos\u001e)2+p2\ny,!c=\neB=m\u0003,lB=p\n\u0016h=eB is the magnetic length, \u001biare the\nPauli matrices, m\u0003is the e\u000bective mass\n1\nm\u0003=1\nm0+2P2\n0\n3\u0016h2\u00122\nEg+1\nEg+ \u0001 0\u0013\n; (10)\nand\u000bx,\u000byare the SO coe\u000ecients given by\n\u000bx(x;y)\u0019P2\n0\n3\u00121\n(E0+ \u0001 0)2\u00001\nE2\n0\u0013@V(x;y)\n@y;(11)\n\u000by(x;y)\u0019P2\n0\n3\u00121\n(E0+ \u0001 0)2\u00001\nE2\n0\u0013@V(x;y)\n@x:(12)\nB. SO coupling constants calculations\nRepresenting the Hamiltonian (9) in the basis of the in-\nplane envelope functions  n;kz(x;y), calculated without\nSO coupling, i.e., the diagonal part of (9), the matrix\nelements of the SO term are given by\n\u000bnm\ni(kz) =Z Z\n n;kz(x;y)\u000bi(x;y) m;kz(x;y)dxdy:\n(13)\nThese coe\u000ecients de\fne intra- ( n=m) and inter-\nsubband (n6=m) SO constants whose magnetic \feld-\ndependence is studied in Sec. III. Note that the \u000b's co-\ne\u000ecients depend both on the envelope functions and the\ngradient of the potential.Calculations of the  n;kz(x;y)'s is performed by\nthe standard self-consistent Sch odinger-Poisson approach\nwhich includes electron-electron interaction at the mean-\n\feld level. First, the in-plane envelope functions\n n;kz(x;y) are determined from the diagonal term of (9)\n\u0014P2\n2D\n2m\u0003+1\n2m\u0003!2\nc\u0002\n(ycos\u0012\u0000xsin\u0012) sin\u001e\u0000kzl2\nB\u00032\n+Ec+V(x;y)\u0015\n n;kz(x;y) =En;kz n;kz(x;y):(14)\nIn the presence of a magnetic \feld, the subbands are\nnot parabolic and  n;kz(x;y) is explicitly kz-dependent.\nAn example of the non-parabolic dispersion is shown in\nFig. 1(b). Therefore, Eq. (14) is solved at selected kzon\na uniform grid in [ \u0000kmax\nz;kmax\nz], withkmax\nzfairly above\nthe Fermi wave vector. Then, the electron density is\nobtained by\nne(x;y) = 2X\nnZkmax\nz\n\u0000kmaxz1\n2\u0019j n;kz(x;y)j2f(En;k\u0000\u0016;T)dkz;\n(15)\nwhere the factor 2 accounts for spin degeneracy, Tis the\ntemperature, \u0016is the chemical potential and f(En;k\u0000\n\u0016;T) is the Fermi-Dirac distribution given by\nf(En;k\u0000\u0016;T) =1\n1 + exp\u0010\nEn;kz\u0000\u0016\nkBT\u0011: (16)\nFinally, for a given ne(x;y) we solve the Poisson equation\nr2\n2DV(x;y) =\u0000ne(x;y)\n\u000f0\u000f; (17)\nwhere\u000fis the dielectric constant.\nEquations (14) and (17) are solved numerically on a\ntriangular grid assuming Dirichlet boundary conditions.\nThe symmetry of the discretization grid matching the\nsymmetry of the hexagonal integration domain avoids nu-\nmerical artifacts at the boundaries using smaller grid den-\nsities. The procedure of alternately solving Eqs. (14) and\n(17) is repeated until self-consistency is reached, which\nwe consider to occur when the relative variation of the\ncharge density between two consecutive iterations is lower\nthan 0:001 at every point of the discretization domain.\nThen, the self-consistent potential energy pro\fle V(x;y)\nand the corresponding envelope functions  n;kz(x;y) are\nused to determine the SO constants \u000bnm\nifrom Eq. (13).\nFurther details concerning the self-consistent method\nfor hexagonal NWs can be found in our previous\npapers.33,34\nCalculations have been carried out for the material\nparameters corresponding to InAs:35E0= 0:42 eV,\n\u00010= 0:38 eV,m\u0003= 0:0265,EP= 2m0P2=\u0016h2= 21:5 eV,\n\u000f= 15:15,T= 4:2 K, and for the NW width W= 100 nm\n(facet-to-facet). In our calculations we \fx the chemi-\ncal potential. Results will be reported in the follow-\ning section for \u0016= 0:3 eV and\u0016= 0:35 eV, which4\nFIG. 2. (a-c) Intra- ( \u000bnn\ni,n= 1;2;3) and (d,e) inter-subband ( \u000b1m\ni) selected Rashba SO coupling constants as a function of\nthe magnetic \feld Band the wave vector kz. (f)\u000bnn\ni(B;kz) atB= 4 T for the three lowest states. Results are shown for\n\u0016= 0:3 eV and a magnetic \feld perpendicular to the NW axis and along the corner-corner direction ( \u001e=\u0019=2;\u0012= 0).\nare marked by vertical dashed lines in Fig. 1(b). For\nB= 0, these values correspond to the electron concen-\ntrationne= 4:8\u00021016cm\u00003andne= 1:36\u00021017cm\u00003,\nrespectively. Note, however, that an increasing perpen-\ndicular magnetic \feld progressively depletes the NW.36\nTherefore, in a transport experiment the chemical po-\ntential must be set to a su\u000eciently large value. In our\ncalculations, the above two values of \u0016have been cho-\nsen su\u000eciently large as to provide an occupied ground\nstate at the largest magnetic \feld intensity used here,\nB= 4 T [Fig.1(b)]. For a given magnetic \feld, di\u000berent\nvalues of\u0016correspond to di\u000berent occupations, hence a\ndi\u000berent self-consistent potential and charge distribution\nwithin the section of the NW, which in turn a\u000bects the\nSO coupling.\nIII. RESULTS\nWe shall now discuss predictions of the SO constant\nas a function of the magnetic \feld intensity and direc-\ntion. We shall put particular emphasis on the role of the\n\feld-induced orbital e\u000bects and the interplay with the\ngate potential, which also in\ruences electronic states lo-\ncalization and symmetry. We conclude this section by a\ndiscussion of the recent experiment.29A. Perpendicular magnetic \feld with no backgate\npotential\nWe \frst show that a magnetic \feld perpendicular to\nthe NW axis induces a \fnite Rashba SO coe\u000ecients even\nin the absence of any transverse electric \feld ( Vg= 0).\nIn this case only the Hartree term VHcontributes to the\nself-consistent potential.\nForB= 0 the self-consistent potential, having the\nsame hexagonal symmetry of the con\fning potential of\nthe NW, is symmetric with respect to the xandydirec-\ntions. Hence, envelope functions have even or odd parity,\nleading to\u000bnn\nx=\u000bnn\ny= 0 for all electronic states, as im-\nplied by Eq. (9).\nLet us now consider a \fnite magnetic \feld, directed\nalong, e.g, the xaxis (\u001e=\u0019=2;\u0012= 0). The \feld gener-\nates an e\u000bective parabolic potential along y, see Eq. (9),\nremoving the symmetry of the Hamiltonian in this direc-\ntion. This, in turn, induces a \fnite potential gradient and\nakz-dependent displacement of the envelope function,\nhence, \fnite diagonal SO couplings \u000bnn\nx[see Eq. (11)],\nas shown in Fig. 2 (a-c) for selected subbands. For a\nconstant Fermi energy, as assumed in our calculations,\nthe number of occupied subbands changes with magnetic\n\feld. AtB= 1 T,N= 8 subbands are occupied, while\nonlyN= 3 of them are populated at B= 4 T. The\nbehavior of \u000bnn\ni(kz) (i=x;y) for all three subbands is\nboth qualitatively and quantitatively similar, especially5\nfor the high magnetic \feld, as presented in Fig. 2 (f).\nThe maps of \u000bnm\ni(B;kz) in Fig. 2 (d,e) report selected\nSO o\u000b-diagonal couplings between the ground state and\nthe two lowest excited states. Other coe\u000ecients \u000b1m\niare\nfour orders of magnitude lower than \u000b11\nxand are not\nreported here. Note that the suppression of these o\u000b-\ndiagonal matrix elements occurs only for a magnetic \feld\nalong the corner-corner direction, \u0012= 0. For an arbitrary\ndirection of the magnetic \feld, no symmetry applies with\nrespect to the speci\fc x\u0000yreference frame, and all o\u000b-\ndiagonal SO constants have comparable values at kz= 0.\nFIG. 3. Squared envelope functions of the three lowest mag-\nnetic subbands, at kz= 0 and 0:2 nm\u00001, with a transverse\nmagnetic \feld (red arrow) at intensities (a) B= 1 T and (b)\nB= 4 T. Right panels show the electron density neand the\nself-consistent potential pro\fle Vat the corresponding \feld\nintensities.\nThe magnetic \feld dependence of \u000bnm\nican be traced\nto the envelope functions localization and ensuing self-\nconsistent potential, as shown in Fig. 3. For B= 0 (not\nshown) the symmetry of the envelope functions naturally\nleads to\u000bnn\ni= 0.13However, the \feld strongly changes\nthe envelope function symmetry. The magnetic states\nof a NW have been thoroughly investigated in Ref. 36.\nIn short, at kz= 0 these are localized by the \feld in\nthe two corners along the \feld direction, where the verti-\ncal component of the \feld is the strongest, in seemingly\ndispersionless Landau levels (see also Fig. 1(b)). There-\nfore, such states have the inversion symmetry and do\nnot contribute to the SO coupling. At \fnite kzthe elec-\ntron states are localized at one of the facets in dispersive\nstates, which are the analog of the traveling edge states\nin a Hall bar. Accordingly, the SO constant \u000bnn\nxis \fnite,\nit depends on kz, and changes sign at kz= 0, as shown in\nFig. 2 (a-c). Note that \u0006kzstates have opposite localiza-\ntion alongy. Therefore, regardless of the magnetic \feld\nintensity, the self-consistent potential, which is obtained\nby summing states up to the Fermi wavevector, has theinversion symmetry induced by the NW con\fnement, as\nshown in the right panels of Fig. 3.\nFor similar reasons, but with the opposite behavior\ndue to symmetry, the inter-subband SO couplings \u000b1n\niare\nlargest atkz= 0. Its exact value strongly depends on the\n\feld intensity. Note that for the analyzed magnetic \feld\ndirection the symmetry around the y-axis is preserved,\nhence\u000bnn\ny= 0.\nFIG. 4. (a) The intra-subband SO constant \u000b11\nxas a function\nofkzatB= 1 T and B= 4 T and at chemical potentials\n\u0016= 0:30 eV and \u0016= 0:35 eV. (b) The intra-subband SO\nconstant\u000bnn\nx(kF\nn;z); n = 1;2;3 (left axis) calculated at kF\nn;z\nand number of occupied subbands (black line, right axis) as\na function of the magnetic \feld intensity, B.\nWhile a \fnite SO can be induced by a constant mag-\nnetic \feld due to the removal of the inversion symmetry,\nits magnitude also depends on the electric \feld in the\nNW, see Eqs. (11),(12), which in turn depends on the\nelectron concentration via the chemical potential \u0016. At\nsu\u000eciently high electron density, the free charge moves to\nthe corners of the NW to reduce the repulsive Coulomb\nenergy.33The large gradient of the self-consistent poten-\ntial where the envelope function is large generates SO\nconstants\u000bnn\niwhich increase with \u0016. As an example, in\nFig. 4(a) we show the calculated \u000b11\nias a function of the\nwavevector for \u0016= 0:30 eV and\u0016= 0:35 eV. Note that\n\u000b11\niincreases rapidly with kz, but then saturates as the\ncorresponding envelope functions are squeezed more and\nmore to the NW edges.\nIn a transport experiment, electrons are injected in\none of the subbands of the NW with a well de\fned Fermi6\nFIG. 5. Maps of \u000bnn\nxand\u000bnn\nyas a function of \u0012and wave vector kz. Results are shown for \u0016= 0:30 eV and magnetic \felds\n(a)B= 1 T and (b) B= 4 T.\nFIG. 6. Maps of \u000b11as a function of \u0012and wave vector kz.\nResults are shown for \u0016= 0:30 eV and magnetic \felds (a)\nB= 1 T and (b) B= 4 T. Insets under the main panels\nzoom in the kzrange marked by dashed black rectangle of\nthe corresponding panel.\nwave vector, kF\nn;z, which is a function of the magnetic\n\feld intensity due to the \feld induced charge depletion.\nIn Fig. 4(b) we show \u000bnn\ni(Vg) at the Fermi wave vectorkF\nn;z. The strong localization of the electron charge at\nopposite NW edges gives rise to a strong susceptibility\nof\u000bnn\ni(Vg) aroundB= 0, analogously to what happens\nwhen a gate potential is switched on, as we discussed\nin Ref. 13. On the other hand, \u000bnn\nisaturates for high\nmagnetic \felds due to the orbital e\u000bect which squeezes\nthe envelope functions to NW edges. Slight oscillations\nof\u000bnn\nx(B) correspond to changes in the self-consistent\npotential due to depopulation of subsequent subbands\nwhen increasing \feld [see the black line in Fig. 4(b)].\nWe next analyze the anisotropy of the SO constant\nwith respect to the transverse \feld direction. Indeed, as\na \fnite\u000bnn\nioriginates from the con\fnement induced by\nthe \feld, it is expected that the latter intertwines with\nthe natural con\fnement of the electron charge at the NW\nedges, as discussed above. Therefore, we expect a 6-fold\nanisotropy with respect to \u0012.\nThe angular dependence of the intra-subband SO cou-\nplings is shown in Fig. 5 for the three lowest subbands\nand di\u000berent magnetic \feld intensities. Note these sub-\nbands exhaust the occupied states at B= 4 T, but they\nare only a subset of the N= 8 occupied subbands at\nB= 1 T [see also Fig. 4(b)]. Subbands with N > 3 are\nnot shown here, however, as they do not add information.\nIn Fig. 6 we show \u000bnn=q\n(\u000bnnx)2+ (\u000bnny)2calculated7\nfor the ground state n= 1. The SO coupling \u000b11ap-\npears isotropic and una\u000bected by the magnetic \feld ori-\nentation. However, a very weak dependence on \u0012can\nbe observed in the bottom subpanels which zoom in the\nkzrange marked by the dashed rectangular at the main\ngraph. A similar weak 6-fold anisotropy is shown by all\nthe occupied states and corresponds to the hexagonal ge-\nometry of NW. It is due to the slight reshaping of the en-\nvelope functions which localize alternately on facets and\ncorners as the magnetic \feld is rotated around the NW\n(see Fig. 7).\nInterestingly, at B= 4 T the SO coupling shows a\n\rower-like pattern around kz= 0 forn= 1;2, see\nFig. 5(b). This behavior emerges in the low kzrange,\nwhere the \feld drives the electron charge around the NW\ndue to the parabolic well generated by the \feld. However,\na smallkz-dependent term slightly removes the symme-\ntry, displacing the envelope function on one side and in-\nterplaying with the hexagonal potential. In Fig. 7(a) the\n\u0012= 28\u000ecase is much more symmetric than the other two\ndirections, due to the larger tunneling energy between the\nlobes, which makes the symmetric con\fguration more ro-\nbust. In Fig. 7(b), instead, the envelope function of the\nground state for kz= 0:4 nm\u00001is strongly localized by\nthe \feld near the edges. In this case the symmetry of\nthe envelope function is strongly removed, regardless of\nthe \feld direction, and only a weak anisotropy is present\nthereof.\nFIG. 7. Squared envelope function of the ground state at\nselected angles \u0012atB= 4 T. (a) kz= 0:04 nm\u00001(b)kz=\n0:4 nm\u00001.\nIn Fig. 8 (a-c) we report polar diagrams of the intra-\nsubband SO constant calculated at the Fermi wave vector\nkF\nn;zfor all occupied states ( N= 3) atB= 4 T. The x-\nandy\u0000components and the modulus \u000bnnare shown sep-\narately. The value of SOC is the largest for the ground\nstate, panel (a), which is almost isotropic. On the con-\ntrary, other electronic bands have a smaller values but a\nstronger anisotropy. The total SOC, \u000btot, averaged over\nall occupied subbands, panel (d), to be compared with\nthe observed value in the magnetotransport experiment,\nFIG. 8. (a-c) Angular dependence of the x\u0000(blue) and\ny\u0000(red) components of intra-subband SO coupling constant\n(in units of meVnm) at kF\nn;ztogether with the modulus\n\u000bnn=p\n(\u000bnnx)2+ (\u000bnny)2(black) for the three occupied\nstates. Panel (d) presents the total SO coupling constant,\n\u000btot, averaged over all occupied states. Results for B= 4 T,\n\u0016= 0:3 eV.\nshows a slight 6-fold anisotropy, with the smaller value\nalong the corner-corner direction and the larger value\nalong the facet-facet direction.\nThe total SOC for di\u000berent Band\u0016is shown in Fig. 9.\nAt the lowest magnetic \feld B= 0:1 T, panel (a), we do\nnot observe any anisotropy. A slight 6-fold anisotropy\ncan be appreciated at B= 1 T, in panel (b). In this case\na di\u000berent behaviour of the SOC as compared to that ob-\ntained atB= 4 T is due to the averanging over a larger\nnumber of subbands ( N= 8), including higher excited\nstates whose angular dependence is a combined e\u000bect of\nthe orbital e\u000bects and the envelope function symmetry.\nAlthough the orbital e\u000bects for these higher excited states\nare suppressed due to low kn;F, and therefore the contri-\nbution of them to the SOC is reduced, they cause a visible\nripples of SOC, but still with the lowest SOC along the\ncorner-corner line.\nThe observed 6-fold anisotropy of SOC is actually\nexpected. Due to external con\fnement and the self-\nconsistent \feld arising from Coulomb interaction, the\nelectron gas is strongly localized near the edges of NW\nfor lowB. A weak magnetic \feld cannot perturbate\nthe symmetry of such strongly localized states. For\nhigher magnetic \feld the Coulomb interaction weakens\ndue to the magnetically induced charge depletion (see\nFig.4(b)). Therefore a su\u000eciently strong magnetic \feld\nmay squeezee the envelope functions to the surface in8\nFIG. 9. The angular dependence of the total SO coupling\nconstant (in units of meVnm), \u000btot, averaged over all Noc-\ncupied subbands at kF\nn;z. (a)B= 0:1 T,\u0016= 0:3 eV (N= 8)\nand (b)B= 1 T,\u0016= 0:3 eV (N= 8). (c)\u000b11at\u0016= 0:3 eV\n(dashed line) and \u0016= 0:35 eV (solid line).\na way which depends on the relative orientation of the\nsurface and the \feld. Note that the localization of the\nwave function at the surface is enhanced by the Coulomb\nrepulsion at the high concentration regime. Indeed, as\npresented in Fig. 9(c), the 6-fold anisotropy of \u000b11(for\nthe ground state) is somewhat larger for higher \u0016.\nOur results qualitatively agree with experimental ev-\nidence in Ref. 29 where the SO coupling was measured\nto be isotropic in a suspended hexagonal InAs NW. This\nnegative result is expected in the low magnetic \feld used\nin the experiments ( B < 0:1 T). Evaluating the \feld\nintensity at which anisotropy is exposed is a non trivial\nissue. The reason is that increasing the \feld enhances the\norbital e\u000bects on the charge density, which at zero \feld\ntends to be localized near to the surface, but it also de-\npletes the NW from free charge, which makes the charge\nto delocalize, due to the small Coulomb repulsion, and\nless sensitive to the anisotropy of the NW.\nB. Perpendicular magnetic \feld with a \fnite\nbackgate potential\nNext we consider the e\u000bect of a bottom gate attached\nto the NW (see Fig. 1). As in the previous section, the\nmagnetic \feld is perpendicular to the NW axis. We \frst\nconsider the \u0012= 0 (corner-to-corner) direction, hence the\ntwo \felds are orthogonal to each other.\nThe total intra-subband SO coupling \u000btotaveraged\nFIG. 10. The total intra-subband SO, \u000btot, as a function of\nVgat selected magnetic \felds B= 0;1;4 T directed in the\n\u0012= 0 (corner-to-corner) direction. Results are shown for\n\u0016= 0:30 eV.\nover all occupied states at the Fermi wave vector kF\nn;z\nis shown in Fig. 10 as a function of the back-gate poten-\ntialVgat selected \feld intensities. For the present \felds\ncon\fguration the symmetry around the y-axis is not bro-\nken, hence \u000bnn\ny(Vg) = 0. Figure 10 shows that \u000btot(Vg),\nwhich is \fnite due to the broken symmetry along x, in-\ncreases with BforVg>0.\u000btottakes o\u000b at a threshold\nVgwhich moves toward negative gate voltages with in-\ncreasing magnetic \feld.\nThe strong asymmetry shown in Fig. 10 between posi-\ntive and negative voltages is easily understood. For pos-\nitive voltages the electron charge is pulled toward the\ngates, where the self-consistent \feld has the largest gra-\ndient. For negative voltages, instead, electrons are pulled\nfar from the gate, where the potential is almost \rat.13\nNote, however, the opposite e\u000bect of the magnetic \feld.\nHere, the electric and magnetic \felds are orthogonal,\n\u0012= 0. Therefore, for positive voltages both the gate po-\ntential and the magnetic \feld push electrons toward the\nbottom edge, hence the magnetic \feld reinforces the back\ngate e\u000bect, increasing the SO coupling. The opposite is\ntrue forVg<0; in this case, electric and magnetic \feld\npush the electrons on opposite sides, and the magnetic\n\feld weakens the SO coupling. Of course, the opposite\nsituation takes place when the magnetic \feld is directed\nat\u0012= 180\u000e. Therefore, for a \fxed Vg, we expect a strong\nanisotropy with respect to the magnetic \feld orientation,\nas shown below.\nFigure 11 shows the polar plot of \u000btotaveraged over\nkF\nn;zforVg= 0:1 V together with \u000b11\ni. In the absence of\na magnetic \feld, the electronic charge is strongly local-\nized by the electric \feld at the edge of the NW, near to\nthe backgate. At a small magnetic \feld [ B= 0:1 T in\npanel (a)], the orbital e\u000bects are negligible, and the SO\ncoupling is isotropic. If we increase the magnetic \feld\n(panel (b)), however, \u000btot(as well as\u000b11\nx) shows a 2-fold\nanisotropy, as expected from the interplay between the9\nFIG. 11. The angular dependence of the x\u0000(blue) and y\u0000\ncomponent (red) of the intra-subband SO constant (in units\nof meVnm) \u000b11\nicalculated at the Fermi wave vector kF\n1;zfor\nthe lowest subband and the total SO constant averaged over\nall occupied states at kF\nn;z(magenta line). Insets in panel (b)\nshow the squared envelope functions of the lowest subband at\nkF\n1;zfor the magnetic \feld with \u0012= 0 and\u0012= 180\u000e. Calcula-\ntions are performed with \u0016= 0:30 eV andVg= 0:1 eV.\ntwo \felds. Note that at \u0012= 180\u000e, the SO coupling of the\nground state is nearly zero as the orbital e\u000bects local-\nizes the electron wave function near the upper facet (see\nthe inset), overcoming the gate e\u000bect. There, the electric\n\feld is weak due to the distance from the gate, and the\ngradient is almost vanishing.13The nonzero value of \u000btot\nin this case results from the other states which contribute\nto the total SOC. Further increasing the \feld intensity B\nenhances the orbital e\u000bect enhancing the anisotropy due\nto suppressing \u000bnn\nxin a wide angular range, as shown in\npanel (c) for the ground state.\nA similar 2-fold anisotropy has been reported in Ref. 29\nwith a di\u000berent gate con\fguration, but with the same\nsymmetry. We postpone the detailed analysis of this ex-\nperiment to Sec. III D.\nC. Axial magnetic \feld\nWe now consider the SO coupling constants under a\nmagnetic \feld with a component along the NW axis.\nThis is the relevant con\fguration in the context of Majo-\nrana states engineering, which requires the axially mag-\nnetic \feld and the SO interaction to create Majorana\nzero energy modes at the ends of a NW. The question\nconcerning the relative relationship between the SO cou-pling and the magnetic \feld is still an open issue.37\nFIG. 12. The total intra-subband SO constant \u000btotas a func-\ntion of the gate voltage Vgfor di\u000berent axial magnetic \felds.\nInset: squared envelope functions of the lowest subband for\ndi\u000berent magnetic \felds at Vg= 0.\nFIG. 13. Angular dependence of the x\u0000(blue) and y\u0000(red)\ncomponent of the intra-subband SO (in units of meVnm) of\nthe ground state \u000b11\nicalculated at kF\n1;ztogether with the total\nSOC\u000btot(magenta). The magnetic \feld is rotated in the y\u0000z\nplane. Results for \u0016= 0:30 V,B= 1 T and (a) Vg= 0 and\n(b)Vg= 0:1 V.\nFigure 12 shows the calculated \u000btot(Vg)vs\feld inten-\nsityBwith an axial \feld ( \u001e= 0). Clearly, the axial mag-\nnetic \feld a\u000bects the SO coupling to a slight extent up to\nB= 16 T. This is in agreement with previous calculations\nwithin the Spin Density Functional formalism.38Indeed,\nin the axial \feld con\fguration, the inversion symmetry\nis not removed (see Eq. 9), although the orbital e\u000bect is\nstill visible in the inset of Fig. 12, where the envelope\nfunction is shown to localize further at the edges with\nthe \feld. There is almost no \feld-induced depletion ef-\nfect here, which is only due to the part of the orbital\ne\u000bect related with the \feld-induced quadratic terms in\nEq. 9. Note the strong asymmetry with respect to the\ngate potential, which has the same explanation as the\none in Fig. 10.\nNext, we consider a magnetic \feld rotating in the y\u000010\nzplane, see Fig. 13, which shows a 2-fold anisotropy.\nHowever, the anisotropy is almost removed by the gate\npotential, with the SO constant being only slightly larger\nfor the axially magnetic \feld.\nThe behaviour shown in Fig. 13 is easily traced to the\nwave function localization. At Vg= 0, SOC is trivially\nzero if the magnetic \feld is in the axial direction (inver-\nsion symmetry holds), while it is at maximum with the\n\feld in the orthogonal direction, \u001e=\u0019=2, as discussed in\nthe previous paragraphs. If Vg= 0:1 V, instead, the wave\nfunction is localized near to the bottom edge, where the\nelectric \feld is the largest, and the SO coupling is large as\nwell. AtB= 1 T the magnetic \feld does not change the\nlocalization, although if the magnetic \feld is perpendic-\nular to NW the orbital e\u000bects squeezes the wave function\nto the side edges (either to the right or to the left) where\nthe electric \feld is lower, slightly lowering the SO cou-\npling. Hence, a small gate potential restores the y\u0000z\nisotropy.\nD. Comparison with experiment [Ref.29]\nIn Ref. 29 the authors used magnetotransport experi-\nments to determine the SO coupling in suspended InAs\nNWs. Using a vectorial magnet, the non-trivial evolution\nof weak anti-localization (WAL) is tracked and the SO\nlength is determined as a function of the magnetic \feld\nintensity and direction. This study shows no anisotropy\nrelated to the geometrical con\fnement in a low \feld\nregime. The isotropy of SO coupling is however removed\nin the presence of an external electric \feld induced by\nside gates. In this case, the SO coupling demonstrates\na 2-fold periodic angular modulation when the magnetic\n\feld is rotated in both the y\u0000zandx\u0000yplane.\nTo simulate the experimental conditions, we consider a\nInAs NW attached to two side electrodes located 200 nm\nfrom the NW, see Fig. 14(a). Potentials applied to\nthe gates generate an electric \feld which is assumed to\nchange linearly in the region between the electrodes. All\nparameters are taken from the experiment. We assume\nW= 100 nm (facet-facet) and ne= 2\u00021018cm\u00003, which\nfor the considered NW geometry, gives EF= 0:935 eV.\nIn order to keep the electron density constant, the \feld\nis induced by applying an asymmetric potential VSG1=\n\u000bgVSG2, where\u000bgis determined separately for each Vg,\nas to keep the density constant. We consider only the\ncase with the magnetic \feld directed perpendicular to\nthe NW and rotating in the x\u0000yplane, with B= 0:1 T\nas used in the experiment.\nThexandycomponents of the intra-subband SO cou-\npling for the ground state \u000b11\nicalculated at Vg= 0 is pre-\nsented in Fig. 14(b,c). The rapid switch between the two\ncomponents results from the Coulomb interaction. At\nthe considered high electron concentrations the electron-\nelectron repulsion localizes the charge in quasi-1D chan-\nnels at the corners.33When the magnetic \feld rotates the\nlocalization of the ground state suddenly moves betweenthe corners resulting in a step-like change between the\nx\u0000andy\u0000components which swap their intensities.\nThe total SO coupling constant averaged over all oc-\ncupied states at kz\nn;Fis presented in panels (d) and\n(e) for two di\u000berent gate voltages. The total SO cou-\npling atVg= 0, panel (d), is nearly isotropic exhibiting\nslight oscillations with the 6-fold symmetry due the pris-\nmatic symmetry of the NW which, in the considered high\nelectron density regime, is more pronounced due to the\nstrong localization of electrons at the six corners. Note\nthat in Ref. 29 the authors reported full isotropic be-\nhaviour of SOC at Vg= 0 without the oscillations. This\ninconsistency remains to be clari\fed. It may be the re-\nsult of the speci\fc extraction of the SO length used in\nRef. 29 which includes the correction from the e\u000bective\nNW width. Alternatively, a low resolution of the magne-\ntotransport measurement might not be able to capture\nsmall changes of SOC.\nFinally, we apply a potential Vg= 2 V, as in the ex-\nperiments, to the side electrodes ( \u000bg= 0:96). In this\ncon\fguration, the y\u0000component of SO coupling becomes\ndominant and is barely a\u000bected by the magnetic \feld ori-\nentation. For such a high gate potential the wave func-\ntion of the ground state is strongly localized in the right\ncorner [see the inset Fig. 14(e)] and it is only slightly dis-\nturbed by the orbital e\u000bects originating from the weak\nmagnetic \feld used in the experiment ( B= 0:1 T). This\nresults in the slight 2-fold anisotropy of SOC, shown in\npanel (e), similarly as reported in the experiment.29Note\nhowever that the experimental evidence shows a 2-fold\nanisotropy with respect to the magnetic \feld orientation\nin they\u0000zplane (although authors suggested its exis-\ntence also in the x\u0000ymagnetic \feld rotation) and its\nintensity is much stronger.\nAlthough we did not perform explicit calculations in\nthis con\fguration for such a high electron density, which\nimplies a very large number of subbands ( \u0018100) and a\ncorrespondingly large numerical e\u000bort, results presented\nin Fig. 13 for a lower electron density and higher mag-\nnetic \feld agree with the experimental result and support\nthe interpretation. Note however that at Vg= 0 and the\naxially directed magnetic \feld, the inversion symmetry\naround either the xandyaxis is not broken, which re-\nsults in\u000btot= 0 as presented in Fig. 13(a). This sce-\nnario is however not supported by the experimental data\nwhich exhibit nonzero SOC even for the axially magnetic\n\feld. This strongly suggests the presence in the samples\nof an intrinsic electric \feld of an unknown origin, which\nis a source of SO coupling whose distortion by the weak\nmagnetic \feld used in the experiment ( B= 0:1 T) is\nnot possible, resulting in the isotropic SOC. An intrin-\nsic electric \feld would explain also the absence of the\nSO coupling angular oscillations [as in Fig. 14(a)] and\nthe slightly lower value of SOC from the calculations,\n\u000btot\u001910 meVnm, as compared with the corresponding\nexperimental value \u000bexp\ntot\u001915 meVnm. Interestingly, it\nmight also explain the observed unexplained phase shift\nin the magnetoconductance measurement [see Fig. 3(c,d)11\nFIG. 14. (a) Schematic illustration of the experimental setup. (b,c) The x\u0000andy\u0000component of the intra-subband SO\ncoupling\u000b11\nias a function of the angle \u0012and the wave vector kz. The magnetic \feld is rotated in the x\u0000yplane. (d,e) The\nangular dependence of the total SO constant, \u000btot. Results for B= 0:1 T and\u001e=\u0019=2.\nin Ref. 29] in terms of the relative alignment between the\nmagnetic \feld and the resultant electric \feld (sum of the\nnon-collinear intrinsic and extrinsic electric \feld) which\nchanges depending on the applied voltage.\nIV. SUMMARY\nBased on the k\u0001ptheory within the envelope function\napproximation, we have analyzed the orbital e\u000bects of a\nmagnetic \feld on the Rashba SO coupling in InAs homo-\ngeneous semiconductor NWs. The full vectorial character\nof the SO constant has been studied under the magnetic\n\feld magnitude and orientation.\nThe Rashba SO interaction of conduction electrons in a\nNW is determined by the position and symmetry of the\nelectron's wave function, which can be tuned by gate-\ninduced electric \felds as well as by the the orbital e\u000bects\ninduced by a magnetic \feld. Speci\fcally, when we ap-\nply the magnetic \feld perpendicular to NW the inver-\nsion symmetry of the envelope functions is broken and\nthe wave functions is squeezed to the NW surface by a\nkz-dependent e\u000bective potential. This e\u000bect results in a\n\fnite SO coupling, which is also sensitive to the geomet-\nrical con\fnement. As we have shown, at low magnetic\n\feld (<1 T for the considered NW), when orbital e\u000bects\nare weak, the SO coupling is isotropic with respect tothe magnetic \feld in the NW section. Interestingly, the\nslight 6-fold anisotropy appears at higher magnetic \felds\n(or high electron concentration), when the wave function\nis squeezed to the NW edges to a larger extent.\nWhen a gate potential is applied in the direction or-\nthogonal to the magnetic \feld, the two \felds intertwin in\na way which may enhance or suppress the SO coupling,\ndepending on the relative direction, leading to a 2-fold\nanisotopy with respect to the magnetic \feld rotation in\nboth thex\u0000yplane.\nFinally, in light of our simulations, we have analyzed\nqualitatively recent experiments with suspended InAs\nNWs29and good agreement with the experimental data\nhas been found. However, we suggest that an unintended\nelectric \feld is present in the sample, which would rec-\noncile observations with our predictions.\nAs a \fnal remark, we note that in real devices a dielec-\ntric spacer often separates the gate from the NW, which\nreduces the SO constant. However, a spacer layer could\nchange the cancellation e\u000bect, as it only lowers the inter-\nnal electric \feld. Importantly, our study has shown no\nsigni\fcant changes of the SO coupling with the axially\nmagnetic \feld.12\nV. ACKNOWLEDGEMENT\nThis work was supported by the AGH UST statutory\ntasks No.11.11.220.01/2 within subsidy of the Ministryof Science and Higher Education in part by PL-Grid In-\nfrastructure.\n\u0003pawel.wojcik@\fs.agh.edu.pl\nyandrea.bertoni@nano.cnr.it\nzguido.goldoni@unimore.it\n1H. C. Koo, J. H. Kwon, J. Eom, J. Chang, S. H. Han, and\nM. Johnson, Science 325, 1515 (2009).\n2P. W\u0013 ojcik, J. Adamowski, B. J. Spisak, and M. Wo loszyn,\nJ. Appl. Phys. 115, 104310 (2014).\n3P. W\u0013 ojcik and J. Adamowski, Sci Rep 7, 45346 (2017).\n4A. T. Ngo, P. Debray, and S. E. Ulloa, Phys. Rev. B 81,\n115328 (2010).\n5M. Kohda, S. Nakamura, Y. Nishihara, K. Kobayashi,\nT. Ono, J. O. Ohe, Y. Tokura, T. Mineno, and J. Nitta,\nNature Communications 3, 1082 (2012).\n6S. Nadj-Perge, S. M. Frolov, E. P. A. M. Bakkers, and\nL. P. Kouwenhoven, Nature 468, 1084 (2010).\n7J. W. G. van den Berg, S. Nadj-Perge, V. S. Pribiag, S. R.\nPlissard, E. P. A. M. Bakkers, S. M. Frolov, and L. P.\nKouwenhoven, Phys. Rev. Lett. 110, 066806 (2013).\n8J. Kammhuber, M. C. Cassidy, F. Pei, M. P. Nowak,\nA. Vuik, D. Car, S. R. Plissard, E. P. A. M. Bakkers,\nM. Wimmer, and L. P. Kouwenhoven, Nat Commun. 8,\n478 (2017).\n9S. Heedt, N. Traverso Ziani, F. Cr\u0013 epin, W. Prost, S. Trel-\nlenkamp, J. Schubert, D. Gr utzmacher, B. Trauzettel, and\nT. Sch apers, Nature Phys. 13, 563 (2017).\n10I. van Weperen, B. Tarasinski, D. Eeltink, V. S. Pribiag,\nS. R. Plissard, E. P. A. M. Bakkers, L. P. Kouwenhoven,\nand M. Wimmer, Phys. Rev. B 91, 201413(R) (2015).\n11T. Campos, P. E. Faria Junior, M. Gmitra, G. M. Sipahi,\nand J. Fabian, Phys. Rev. B 97, 245402 (2018).\n12P. W\u0013 ojcik, A. Bertoni, and G. Goldoni, Appl. Phys. Lett.\n114, 073102 (2019).\n13P. W\u0013 ojcik, A. Bertoni, and G. Goldoni, Phys. Rev. B 97,\n165401 (2018).\n14A. Bringer, S. Heedt, and T. Sch apers, Phys. Rev. B 99,\n085437 (2019).\n15X. W. Zhang and J. B. Xia, Phys. Rev. B 74, 075304\n(2006).\n16K. Takase, K. Tateno, and S. Sasaki, Applied Physics\nExpress 12, 117002 (2019).\n17J. Alicea, Rep. Prog. Phys. 75, 076501 (2012).\n18A. Kitaev, Ann. Phys. 303, 2 (2003).\n19S. Das Sarma, M. Freedman, and C. Nayak, npj Quantum\nInformation 1, 15001 (2015).20Y. Oreg, G. Refael, and F. von Oppen, Phys. Rev. Lett.\n105, 177002 (2010).\n21K. O. Klausen, A. Sitek, S. I. Erlingsson, and\nA. Manolescu, Nanotechnology 31, 354001 (2020).\n22V. Mourik, K. Zuo, S. M. Frolov, S. R. Plissard, E. P.\nA. M. Bakkers, and L. P. Kouwenhoven, Science 336,\n1003 (2012).\n23S. M. Albrecht, A. P. Higginbotham, M. Madsen, F. Kuem-\nmeth, T. S. Jespersen, J. Nyg\u0017 aard, P. Krogstrup, and\nC. M. Marcus, Nature 531, 206 (2016).\n24J. D. Sau, S. Tewari, and S. Das Sarma, Phys. Rev. B 85,\n1 (2012).\n25G. Dresselhaus, Phys. Rev. 100, 580 (1955).\n26E. I. Rashba, Sov. Phys. Solid State 2, 1109 (1960).\n27A. Manchon, C. Koo, H, J. Nitta, M. Frolov, S, and\nA. Duine, R, Nature Materials 14, 871 (2015).\n28S. Nadj-Perge, V. S. Pribiag, J. W. G. van den Berg,\nK. Zuo, S. R. Plissard, E. P. A. M. Bakkers, S. M. Frolov,\nand L. P. Kouwenhoven, Phys. Rev. Lett. 108, 166801\n(2012).\n29A. Iorio, M. Rocci, L. Bours, M. Carrega, V. Zannier,\nL. Sorba, S. Roddaro, F. Giazotto, and E. Strambini,\nNano Lett. 19, 652 (2019).\n30J.-W. Luo, L. Zhang, and A. Zunger, Phys. Rev. B 84,\n121303 (2011).\n31J. Fabian, A. Matos-Abiague, C. Ertler, P. Stano, and\nI.\u0014Zuti\u0013 c, Acta Physica Slovaca 57, 565 (2007).\n32M. P. Nowak and P. W\u0013 ojcik, Phys. Rev. B 97, 045419\n(2018).\n33A. Bertoni, M. Royo, F. Mahawish, and G. Goldoni, Phys.\nRev. B 84, 205323 (2011).\n34M. Royo, A. Bertoni, and G. Goldoni, Phys. Rev. B 89,\n155416 (2014).\n35I. Vurgaftman, R. Meyer, J, and R. Ram-Mohan, L, Appl.\nPhys. Lett. 89, 5815 (2001).\n36M. Royo, A. Bertoni, and G. Goldoni, Phys. Rev. B 87,\n115316 (2013).\n37G. W. Winkler, D. Varjas, R. Skolasinski, A. A. Soluyanov,\nM. Troyer, and M. Wimmer, Phys. Rev. Lett. 119, 037701\n(2017).\n38M. Royo, C. Segarra, A. Bertoni, G. Goldoni, and\nJ. Planelles, Phys. Rev. B 91, 115440 (2015)."    },    {        "title": "1710.10579v2.Contact_theory_for_spin_orbit_coupled_Fermi_gases.pdf",        "content": "arXiv:1710.10579v2  [cond-mat.quant-gas]  24 Dec 2017Contact theory for spin-orbit-coupled Fermi gases\nShi-Guo Peng1, Cai-Xia Zhang1,4, Shina Tan2,3,∗and Kaijun Jiang1,3†\n1State Key Laboratory of Magnetic Resonance and Atomic and Mo lecular Physics,\nWuhan Institute of Physics and Mathematics, Chinese Academ y of Sciences, Wuhan 430071, China\n2School of Physics, Georgia Institute of Technology, Atlant a, Georgia 30332, USA\n3Center for Cold Atom Physics, Chinese Academy of Sciences, W uhan 430071, China and\n4School of Physics, University of Chinese Academy of Science s, Beijing 100049, China\n(Dated: December 27, 2017)\nWe develop the contact theory for spin-orbit-coupled Fermi gases. By using a perturbation\nmethod, we derive analytically the universal two-body beha vior at short distance, which does not\ndepend on the short-range details of interatomic potential s. We find that two new scattering pa-\nrameters need to be introduced because of spin-orbit coupli ng, besides the traditional s- andp-wave\nscattering length (volume) and effective ranges. This is a ge neral and unique feature for spin-\norbit-coupled systems. Consequently, two new adiabatic en ergy relations with respect to the new\nscattering parameters are obtained, in which a new contact i s involved because of spin-orbit cou-\npling. In addition, we derive the asymptotic behavior of the large-momentum distribution, and find\nthat the subleading tail is corrected by the new contact. Thi s work paves the way for exploring the\nprofound properties of spin-orbit-coupled many-body syst ems, according to two-body solutions.\nIntroduction .—Universality, referring to observations\nindependent of short-range details, is one of the most\nfascinating and intriguing phenomena in modern physics.\nIn ultracold atoms, a set of universal relations, following\nfrom the short-range behavior of the two-body physics\nare discovered [1]. These relations are connected sim-\nply by a universal contact parameter, which overarches\nbetween microscopic and macroscopic properties of a\nstrongly interacting many-body system. Nowadays, the\ncontact theory becomes significantly important in ultra-\ncold atomic physics, and has systematically been verified\nand investigated both experimentally and theoretically\n[2–7]. Nevertheless, the contact theory for spin-orbit-\ncoupled systems is still unexplored till now, even though\nthe spin-orbit (SO) coupling was realized in cold atoms\nseveral years ago [8–10], and resulted unique phenomena\nhave attracted a great deal of interest, such as topological\ninsulators and superconductors [11–14].\nIn this letter, for the first time, we generalize the\ncontact theory to strongly interacting spin-orbit-couple d\nFermi gases, and the single-particle Hamiltonian takes\nthe form,\nˆH1=/planckover2pi12ˆk2\n1\n2M+/planckover2pi12λ\nMˆk1·ˆσ+/planckover2pi12λ2\n2M, (1)\nwhereˆk1=−i∇andˆσare respectively the single-\nparticle momentum and spin operators, λ >0is the\nstrength of SO coupling, Mis the atomic mass, and\n/planckover2pi1is the Planck’s constant divided by 2π. Here, the\nSO coupling is assumed to be isotropic for simplicity,\nand the possible scheme for the realization of the three-\ndimensional (3D) isotropic SO coupling is proposed in\n[15]. Because of SO coupling, the orbital angular momen-\ntum of the relative motion of two fermions is no longer\nconserved, and then all the partial-wave scatterings are\ncoupled [16]. Fortunately, the total momentum Kof twofermions is still conserved as well as the total angular\nmomentum J. Therefore, we may reasonably focus on\nthe two-body problem in the subspace of K= 0and\nJ= 0for simplicity, and then only s- andp-wave scatter-\nings are coupled [16, 17]. Consequently, the two spin-half\nfermions in the subspace of K= 0andJ= 0is described\nby the following two-body Hamiltonian\nˆH2=/planckover2pi12ˆk2\nM+/planckover2pi12λ\nMˆk·(ˆσ2−ˆσ1)+/planckover2pi12λ2\nM+V(r),(2)\nwhereˆkis the momentum operator for the relative mo-\ntionr=r2−r1,ˆσiis the spin operator of the ith atom,\nandV(r)is the interatomic potential with a finite range\nǫ. Our theory may also be generalized to the case of\nK/ne}ationslash= 0andJ/ne}ationslash= 0, and then more partial waves should\nbe involved.\nOne of the most daunting challenges for establishing\nthe contact theory is how to obtain the universal two-\nbody behavior at short distance for a SO-coupled Fermi\ngas. Although the SO-coupled two-body problem was\nconsidered recently by using a spherical square-well po-\ntential [16–18], the general form of such universal behav-\nior for any interatomic potential still remains elusive til l\nnow. In this work, we develop a perturbation method\nto construct the short-range asymptotic form of the two-\nbody wave function for a SO-coupled system. We find\nthat two new scattering parameters u,vneed to be in-\ntroduced in the short-range behavior of two-body wave\nfunctions, besides the traditional scattering length (vol -\nume) and effective ranges. The obtained universal be-\nhavior does not depend on the short-range details of the\ninteratomic potentials, and thus is feasible for any inter-\natomic potential with short range. Two new adiabatic\nenergy relations are accordingly found with respect to2\nthe new scattering parameters, i.e.,\n∂E\n∂u=/planckover2pi12λ\n32π2M/parenleftbigg\nC(0)\na−λPλ\n2/parenrightbigg\n, (3)\n∂E\n∂v=λ3/planckover2pi12C(1)\na\n32π2M, (4)\nin which we hold all the other two-body parameters un-\nchanged in the partial derivatives. Here, C(0)\nais the well\nknowns-wave contact, C(1)\nais thep-wave contact corre-\nsponding to the p-wave scattering volume [3, 6]. In addi-\ntion,Pλis the new contact introduced by SO coupling.\nFurther, we derive the asymptotic behavior of the large-\nmomentum distribution from the universal two-body be-\nhavior at short distance,\nn(q) =C(1)\na\nq2+/parenleftBig\nC(0)\na+C(1)\nb+λPλ/parenrightBig1\nq4+O/parenleftbig\nq−6/parenrightbig\n,(5)\nin which C(1)\nbis thep-wave contact corresponding to the\np-wave effective range. We find that the subleading tail\n(q−4) of the large-momentum distribution is amended by\nthe new contact Pλbecause of SO coupling.\nUniversal short-range behavior of two-body wave func-\ntions.—Let us consider the two-body problem of a SO-\ncoupled system in the subspace of K= 0andJ=\n0, and the corresponding Hamiltonian takes the form\nof Eq.(2). The subspace is spanned by two orthog-\nonal basis, i.e., Ω0(ˆr) =Y00(ˆr)|S/an}b∇acket∇i}htandΩ1(ˆr) =\n−i[Y1−1(ˆr)|↑↑/an}b∇acket∇i}ht+Y11(ˆr)|↓↓/an}b∇acket∇i}ht−Y10(ˆr)|T/an}b∇acket∇i}ht]/√\n3, where\nYlm(ˆr)is the spherical harmonics, ˆrdenotes the an-\ngular part of the coordinate r, and|S/an}b∇acket∇i}ht=(|↑↓/angbracketright−|↓↑/angbracketright )√\n2\nand/braceleftBig\n|↑↑/an}b∇acket∇i}ht,|↓↓/an}b∇acket∇i}ht,|T/an}b∇acket∇i}ht=|↑↓/angbracketright+|↓↑/angbracketright√\n2/bracerightBig\nare the spin-singlet and\nspin-triplet states with total spin 0and1, respec-\ntively. The two-body solution can formally be written\nin the basis of {Ω0(ˆr),Ω1(ˆr)}asΨ(r) =ψ0(r)Ω0(ˆr)+\nψ1(r)Ω1(ˆr)[16, 17].\nSince the SO effect exists even inside the interatomic\npotential, it should modify the short-range behavior of\nthe two-body wave function dramatically [19]. However,\nin current experiments of ultracold atoms [20], the SO-\ncoupling strength λis of the order µm−1, pretty small\ncompared to the inverse of the range of interatomic po-\ntentialǫ−1(of the order nm−1). Moreover, the momen-\ntumk=/radicalbig\nME//planckover2pi12is also much smaller than ǫ−1in\nthe low-energy scattering limit. Therefore, when two\nfermions get as close as the range ǫ, we may deal with\nthe SO coupling perturbatively as well as the energy, and\nassume that the form of the two-body solution has the\nfollowing structure,\nΨ(r)≈φ(r)+k2f(r)−λg(r) (6)\nasr∼ǫ. Here, we keep up to the first-order terms of\nk2andλ. The advantage of this ansatz is that the func-\ntionsφ(r),f(r)andg(r)are all independent on theenergy and SO-coupling strength. Therefore, they are\ndetermined only by the short-range details of the inter-\naction, and characterize the intrinsic properties of the\ninteratomic potential. We expect that the traditional\nscattering length or volume in the absence of SO cou-\npling are included in the zero-order term φ(r), while the\neffective ranges are involved in f(r), the coefficient of\nthe first-order term of k2. Interestingly, new scattering\nparameters should appear in the first-order term of λ\n(ing(r)), which are introduced by SO coupling. Conve-\nniently, more scattering parameters may be introduced\nif higher-order terms of k2andλare perturbatively con-\nsidered. Inserting the ansatz (6) into the Schrödinger\nequation, and comparing the corresponding coefficients\nofk2andλ, we obtain\n/bracketleftbigg\n−∇2+M\n/planckover2pi12V(r)/bracketrightbigg\nφ(r) = 0, (7)\n/bracketleftbigg\n−∇2+M\n/planckover2pi12V(r)/bracketrightbigg\nf(r) =φ(r), (8)\n/bracketleftbigg\n−∇2+M\n/planckover2pi12V(r)/bracketrightbigg\ng(r) =Q(r)φ(r), (9)\nwhereQ(r) =−i∇ ·(ˆσ2−ˆσ1). These equations can\nanalytically be solved for r>ǫ, and simply yield\nΨ(r) =α0/bracketleftbigg1\nr+/parenleftbigg\n−1\na0+b0\n2k2+uλ/parenrightbigg\n−k2\n2r/bracketrightbigg\nΩ0(ˆr)\n+α1/bracketleftbigg1\nr2+/parenleftbiggk2\n2+α0\nα1λ/parenrightbigg\n+/parenleftbigg\n−1\n3a1+b1\n6k2+vλ/parenrightbigg\nr/bracketrightbigg\nΩ1(ˆr)\n+O/parenleftbig\nr2/parenrightbig\n,(10)\nwhereα0andα1are two complex superposition coeffi-\ncients. Apparently, a0,b0are thes-wave scattering length\nand effective range, and a1,b1are thep-wave scattering\nvolume and effective range without SO coupling, respec-\ntively. For simplicity, we may only consider the case with\nb0≈0for broads-wave resonances throughout the paper.\nWe can see that the s-wave component is hybridized in\nthep-wave channel by SO coupling as manifested as the\ntermα0λ/α1. Interestingly, two new scattering parame-\ntersuandvas we anticipate are involved. They are the\ncorrections from SO coupling to the short-range behav-\nior of the two-body wave function in s- andp-wave chan-\nnels, respectively. If λ= 0, thes- andp-wave scatterings\ndecouple, and the asymptotic form of Ψ(r)at small r,\ni.e., Eq.(10), simply reduces to the ordinary s- andp-\nwave short-range boundary conditions, respectively. The\nderivation above doesn’t depend on the short-range de-\ntails of the interaction, and thus is universal and appli-\ncable for all kinds of neutral fermionic atoms.\nIn general, the s- andp-wave scatterings in different\nspin channels should both be taken into account because\nof SO coupling. We may roughly estimate which partial\nwave is more important as follows. Without SO coupling,\nand away from any resonances—in the weak interacting3\nlimit, the two-body wave function should well behave as\nr→0asΨ(r)∼(α0/a0)Ω0(ˆr) + (α1r/3a1)Ω1(ˆr). If\nwe assume that the atoms are initially prepared equally\nin the spin channels Ω0(ˆr)andΩ1(ˆr), we haveα0/α1∼\na0r/3a1. When interatomic interactions are turned on,\nthe two-body wave function becomes divergent as r→\n0(>ǫ),α0r−1andα1r−2fors- andp-wave scatterings, re-\nspectively. This divergent behavior is unchanged even in\nthe presence of SO coupling. Then the ratio between the\nstrengths of s- andp-wave scatterings at small rbecomes/parenleftbig\nα0r−1/parenrightbig\n//parenleftbig\nα1r−2/parenrightbig\n≈a0r2/3a1. Nears-wave resonances,\nwe havea0∼k−1\nf,a1∼ǫ3,r∼ǫ, wherekfis the Fermi\nwavenumber, and then this ratio is approximately of the\norder(kfǫ)−1≫1. Therefore, the s-wave interaction\ndominates the two-body scattering. By noticing Ω0(ˆr) =\n|S/an}b∇acket∇i}ht/√\n4π, andΩ1(ˆr) =−i(ˆσ2−ˆσ1)·(r/r)|S/an}b∇acket∇i}ht/√\n16π,\nand if thep-wave interaction could be ignored near broad\ns-wave resonances, Eq.(10) becomes (up to a prefactor\nα0/√\n4π),\nΨ(r) =/parenleftbigg1\nr−1\na0+uλ/parenrightbigg\n|S/an}b∇acket∇i}ht−iλ\n2(ˆσ2−ˆσ1)·r\nr|S/an}b∇acket∇i}ht+O(r),\n(11)\nwhich exactly recovers the result of [19] (see Eq.(31) of\n[19]) witha−1\nR=a−1\n0−uλ.\nNearp-wave resonances, for example, the p-wave Fes-\nhbach resonance at B0= 185.09G in6Li [21], we have\na0∼ǫ,a1∼k−3\nf,r∼ǫ, then the ratio between the\nstrengths of s- andp-wave scatterings is roughly of the\norder(kfǫ)3≪1. In this case, the p-wave scattering\nbecomes significantly important.\nLarge-momentum distribution. —For a many-body sys-\ntem withNspin-half fermions, if only two-body correla-\ntions are taken into account, the many-body wave func-\ntionΨNcan approximately be written as the form of\nEq.(10), when fermions (i,j)get close while all the others\nare far away. In this case, r=ri−rj, and the arbitrary\ncomplex numbers α0andα1become the functions of the\nvariableX, which involves both the center-of-mass (c.m.)\ncoordinate of the pair being considered and the coordi-\nnates of all the other fermions. Further, α0andα1should\nbe constrained by the normalization of the many-body\nwave function. Using the asymptotic form of the many-\nbody wave function ΨNat small r, we can easily obtain\nthe behavior of the tail of the single-particle momentum\ndistribution at large q(but smaller than ǫ−1), which is\ndefined asn(q)≡/summationtextN\ni=1´/producttext\nj/negationslash=idrj/vextendsingle/vextendsingle´\ndriΨNe−iq·ri/vextendsingle/vextendsingle2.\nAfter straightforward algebra, we easily obtain the mo-\nmentum distribution n(q)taking the form of Eq.(5) at\nlargeq/parenleftbig\n<ǫ−1/parenrightbig\n. Here, we are only interested in the de-\npendence of the momentum distribution on the ampli-\ntude ofq, and have already integrated over the angular\npart ofq. We find that\nC(ν)\na= 32π2Nˆ\ndX|αν(X)|2,(ν= 0,1), (12)C(1)\nb=64π2MN\n/planckover2pi12ˆ\ndXα∗\n1(X)/bracketleftBig\nE−ˆT(X)/bracketrightBig\nα1(X)(13)\nare the conventional s- andp-wave contacts [6], where\nˆT(X)denotes the operators of the c.m. motion of\nthe pair (i,j)and all the other fermions, and N=\nN(N−1)/2is the number of all the possible ways to\npair atoms. Besides, a new contacts Pλresulted from\nSO coupling appears, which is defined as\nPλ≡64π2Nˆ\ndXα∗\n0(X)α1(X)+c.c..(14)\nObviously, this new contact describes the interplay of the\ns- andp-wave scatterings because of SO coupling.\nSince the momentum distribution at large qis only\ncharacterized by the short-range behavior of the two-\nbody physics, we may roughly estimate the order of all\nthe quantities in the large- qbehavior of the momentum\ndistribution simply according to the two-body picture\nas before. Near s-wave resonances, if initially without\nSO coupling and away from any resonances, the atoms\nare prepared equally in the spin states Ω0(ˆr)andΩ1(ˆr),\nwe haveα0/α1∼a0r/3a1, and then C(1)\naq−2/C(0)\naq−4≈\n9a2\n1q2/a2\n0r2, which is roughly of the order (kfǫ)4≪1.\nBesides, we may also find C(1)\nb/C(0)\na∼(kfǫ)4≪1. This\nmeans that the p-wave contribution to the tail of momen-\ntum distribution at large qmay reasonably be ignored,\nwhich is consistent with the discussion before. However,\nthe SO-coupling correction is notable compared to the\np-wave contact in the subleading tail of the momentum\ndistribution, i.e., λPλ/C(1)\nb∼(kfǫ)−2≫1.\nNearp-wave resonances, the leading tail q−2of the\nlarge-momentum distribution becomes important, be-\ncauseC(1)\naq−2/C(0)\naq−4∼(kfǫ)−4≫1. In the sub-\nleading tail of q−4, we find C(1)\nb/C(0)\na∼(kfǫ)−4≫1,\nthus thes-wave contribution may be ignored. Conse-\nquently, the momentum distribution at large qbehaves\nasC(1)\naq−2+/parenleftBig\nC(1)\nb+λPλ/parenrightBig\nq−4with a considerable cor-\nrection ofλPλin the subleading tail due to SO coupling,\ncompared to the s-wave contribution, i.e. λPλ/C(0)\na∼\n(kfǫ)−2≫1.\nAdiabatic energy relations. —The thermodynamics of\nmany-body systems, which is seemingly uncorrelated to\nthe momentum distribution, is also characterized by the\ncontacts defined above. A set of adiabatic energy rela-\ntions describe how the energy of a many-body system\nchanges as the two-body interaction is adiabatically ad-\njusted. Let us consider two many-body wave functions\nΨNandΨ′\nNcorresponding to different interatomic inter-\naction strengths. From the Schrödinger equations satis-4\nfied byΨNandΨ′\nN, we easily obtain\n(E−E′)˚\nDǫdr1dr2···drNΨ′∗\nNΨN=\n−/planckover2pi12N\nM\"\nr=ǫI·ˆndΣ+/planckover2pi12λN\n2πM\"\nr=ǫF·ˆndΣ,(15)\nwhereI≡Ψ′∗∇Ψ−(∇Ψ′∗)Ψ,F≡(ψ′∗\n1ψ0−ψ′∗\n0ψ1)ˆer\nwith the unit radial vector ˆerofr, andΨN(X,r) =\nψ0(X,r)Ω0(ˆr)+ψ1(X,r)Ω1(ˆr). Here, the domain Dǫis\nthe set of all configurations (ri,rj)withr=|ri−rj|>ǫ,\nΣis the surface in which the distance between the two\natoms in the pair (i,j)isǫ, andˆnis the direction normal\ntoΣbut is opposite to the radial direction. Using the\nasymptotic form of the many-body wave function ΨNat\nsmallr, we find\nδE=−/planckover2pi12\n32π2M/bracketleftbigg/parenleftbigg\nC(0)\na−λPλ\n2/parenrightbigg\nδa−1\n0+C(1)\naδa−1\n1\n−C(1)\nb\n4δb1−λ/parenleftbigg\nC(0)\na−λPλ\n2/parenrightbigg\nδu−3λC(1)\naδv/bracketrightBigg\n,(16)\nwhich characterizes how the energy of the system varies\nas the scattering parameters adiabatically change. In the\nabsence of SO coupling, Eq.(16) simply reduces to the or-\ndinary form of the adiabatic energy relations for s- and\np-wave interactions [6, 22], with respect to the scattering\nlength (or volume) as well as effective range. Because of\nSO coupling, two new scattering parameters come into\nthe problem, and then additional new adiabatic energy\nrelations appear, i.e., Eqs.(3)-(4). These adiabatic en-\nergy relations demonstrate how the macroscopic thermo-\ndynamics of SO-coupled many-body systems varies with\nmicroscopic two-body scattering parameters.\nContacts in a two-body problem .—On behalf of the\nfuture experiments and calculations, we may explicitly\nevaluate the contacts defined above for a two-body bound\nstate, the wave function of which may be written as a col-\numn vector in the basis of {Ω0(ˆr),Ω1(ˆr)}as [17]\nΨb(r) =Bκ−/bracketleftBigg\nh(1)\n0(κ−r)\n−h(1)\n1(κ−r)/bracketrightBigg\n+Dκ+/bracketleftBigg\nh(1)\n0(κ+r)\nh(1)\n1(κ+r)/bracketrightBigg\n,\n(17)\nwhereκ±=iκ±λ, andκ=/radicalbig\n−ME//planckover2pi12. The bind-\ning energy Ecan be determined by expanding Ψb(r)at\nsmallrand comparing with the short-range boundary\ncondition (10), then the two-body contacts are easily ob-\ntained according to the adiabatic relations. Near s-wave\nresonances, we find\nE=−/planckover2pi12\nMa2\n0+2/planckover2pi12u\nMa0λ+O/parenleftbig\nλ2/parenrightbig\n, (18)\nwhich simply reduces to the result E=−/planckover2pi12/Ma2\n0in the\nabsence of SO coupling. Then we immediately obtainC(0)\na= 64π2/a0andPλ= 128π2uby using adiabatic\nrelations. Near p-wave resonances, we find\nE=2/planckover2pi12\nMa1b1−6/planckover2pi12v\nMb1λ+O/parenleftbig\nλ2/parenrightbig\n, (19)\nwhich is consistent with that without SO coupling\n[3], and then it yields C(1)\na=−64π2/b1andC(1)\nb=\n−256π2/parenleftbig\na−1\n1−3vλ/parenrightbig\n/b2\n1.\nGrand canonical potential and pressure relation .—The\nadiabatic relations as well as the large-momentum dis-\ntribution we obtained above is valid for any pure energy\neigenstate. Therefore, they should still hold for any in-\ncoherent mixed state statistically at finite temperature.\nThen the energy, particle number density and contacts\nbecome their statistical average values. It should be in-\nteresting to discuss how the results presented above affect\nthe finite-temperature thermodynamics. To this end, let\nus look at the grand canonical potential, which is de-\nfined asJ ≡ −PV[23], where Pis the pressure and Vis\nthe volume of the system. According to straightforward\ndimensional analysis [24, 25], we can obtain\nJ=−2\n3E−/planckover2pi12\n96π2Ma0/parenleftbigg\nC(0)\na−λPλ\n2/parenrightbigg\n−/planckover2pi12C(1)\na\n32π2Ma1+/planckover2pi12b1C(1)\nb\n384π2M+λ/planckover2pi12vC(1)\na\n16π2M,(20)\nwhich alternatively yields the pressure relation by divid-\ning both sides of Eq.(20) by −V.\nConclusions. —We systematically study the contact\ntheory for spin-orbit-coupled Fermi gases. The univer-\nsal two-body behavior at short distance is analytically\nderived, by introducing a perturbation method, which\ndoesn’t depend on the short-range details of interatomic\npotentials. For simplicity, we focus on the s- andp-wave\nscatterings in the subspace of vanishing center-of-mass\nmomentum and total angular momentum. Interestingly,\ntwo new microscopic scattering parameters appear in the\nshort-range behavior of two-body wave functions because\nof spin-orbit coupling. We claim that this is a general\nand unique feature for spin-orbit-coupled systems, and\nthus the obtained universal short-range behavior of two-\nbody wave functions is feasible for all kinds of neutral\nfermionic atoms. Consequently, a new contact is intro-\nduced originated from spin-orbit coupling, which, com-\nbining with conventional s- andp-wave contacts, char-\nacterizes the universal properties of spin-orbit-coupled\nmany-body systems. In general, more partial-wave scat-\nterings should be taken into account for nonzero center-\nof-mass momentum and nonzero total angular momen-\ntum. Then more contacts should appear. Our method\ncould conveniently be generalized to other kinds of spin-\norbit couplings as well as to low dimensions. Besides, our\nmethod could also be applied to bosons. In the presence\nof spin-orbit coupling, we expect that additional contacts\nwould be introduced for bosonic systems.5\nS. G. P and K. J are supported by the NKRDP (Na-\ntional Key Research and Development Program) under\nGrant No. 2016YFA0301503 and NSFC under Grant No.\n11474315, 11674358, 11434015, 91336106. S. T is sup-\nported by the US National Science Foundation CAREE\naward Grant No. PHY-1352208.\nS. G. P and C. X. Z contributed equally to this work.\n∗shina.tan@physics.gatech.edu\n†kjjiang@wipm.ac.cn\n[1] S. Tan, Ann. Phys. 323, 2952 (2008); S. Tan, Ann. Phys.\n323, 2971 (2008); S. Tan, Ann. Phys. 323, 2987 (2008).\n[2] W. Zwerger, The BCS-BEC Crossover and the Unitary\nFermi Gas , Col 836 of Lecture Notes in Physics (Springer,\nBerlin, 2011); Please see Chapter 6 for a brief review.\n[3] Z. H. Yu, J. H. Thywissen, and S. Z. Zhang, Phys. Rev.\nLett. 115, 135304 (2015); Z. Yu, J. H. Thywissen, and S.\nZhang, Phys. Rev. Lett. 117, 019901(E) (2016).\n[4] M. Y. He, S. L. Zhang, H. M. Chan, and Q. Zhou, Phys.\nRev. Lett. 116, 045301 (2016).\n[5] C. Luciuk, S. Trotzky, S. Smale, Z. Yu, S. Zhang, and J.\nH. Thywissen, Nat. Phys. 12, 599 (2016).\n[6] S. G. Peng, X. J. Liu, and H. Hu, Phys. Rev. A 94, 063651\n(2016).\n[7] R. J. Fletcher, R. Lopes, J. Man, N. Navon, R. P. Smith,\nM. W. Zwierlein, and Z. Hadzibabic, Science 355, 377\n(2017).\n[8] Y. J. Lin, K. Jimenez-Garcia, and I. B. Spielman, Nature\n471, 83 (2011).\n[9] P. J. Wang, Z. Q. Yu, Z. K. Fu, J. Miao, L. H. Huang,\nS. J. Chai, H. Zhai, and J. Zhang, Phys. Rev. Lett. 109,\n095301 (2012).\n[10] L. W. Cheuk, A. T. Sommer, Z. Hadzibabic, T. Yefsah,\nW. S. Bakr, and M. W. Zwierlein, Phys. Rev. Lett. 109,\n095302 (2012).\n[11] X. L. Qi and S. C. Zhang, Phys. Today 63, 33 (2010).\n[12] M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045(2010).\n[13] J. Dalibard, F. Gerbier, G. Juzeliunas, and P. Ohberg,\nRev. Mod. Phys. 83, 1523 (2011).\n[14] J. Zhang, H. Hu, X.-J. Liu, and H. Pu, Annual Review of\nCold Atoms and Molecules , Vol. 2, 81 (World Scienific,\n2014).\n[15] B. M. Anderson, G. Juzeliunas, V. M. Galitski, and I. B.\nSpielman, Phys. Rev. Lett. 108, 235301 (2012).\n[16] X. L. Cui, Phys. Rev. A 85, 022705 (2012).\n[17] Y. X. Wu and Z. H. Yu, Phys. Rev. A 87, 032703 (2013).\n[18] X. L. Cui, Phys. Rev. A 95, 030701 (2017).\n[19] P. Zhang, L. Zhang, and Y. J. Deng, Phys. Rev. A 86,\n053608 (2012).\n[20] P. J. Wang, Z. Q. Yu, Z. K. Fu, J. Miao, L. H. Huang,\nS. J. Chai, H. Zhai, and J. Zhang, Phys. Rev. Lett. 109,\n095301 (2012); L. W. Cheuk, A. T. Sommer, Z. Hadz-\nibabic, T. Yefsah, W. S. Bakr, and M. W. Zwierlein,\nPhys. Rev. Lett. 109, 095302 (2012).\n[21] J. Zhang, E. G. M. van Kempen, T. Bourdel, L.\nKhaykovich, J. Cubizolles, F. Chevy, M. Teichmann, L.\nTarruell, S. J. J. M. F. Kokkelmans, and C. Salomon,\nPhys. Rev. A 70, 030702 (2004); C. H. Schunck, M. W.\nZwierlein, C. A. Stan, S. M. F. Raupach, W. Ketterle, A.\nSimoni, E. Tiesinga, C. J. Williams, and P. S. Julienne,\nPhys. Rev. A 71, 045601 (2005).\n[22] For the s-wave interaction, there is a difference of the\nfactor8πfrom the well-known form of adiabatic rela-\ntions (see Eq.(36)-(37) of [6]). This is because here we\ninclude the spherical harmonics Y00(ˆr) = 1/√\n4πin the\ns-partial wave function. Besides, an additional factor 1/2\nis introduced in order to keep the definition of the con-\ntacts consistent with those in the tail of the momentum\ndistribution at large q.\n[23] L. D. Landau, and E. M. Lifshitz, Statistical Physics,\nPart I (Elsevier, Singapore, Third Edition, 2007).\n[24] E. Braaten and L. Platter, Physical Review Letters 100,\n205301 (2008); E. Braaten, D. Kang, and L. Platter,\nPhysical Review A 78, 053606 (2008).\n[25] M. Barth and W. Zwerger, Annals of Physics 326, 2544\n(2011)."    },    {        "title": "1807.05106v2.Spin_orbit_coupling_and_correlations_in_three_orbital_systems.pdf",        "content": "Spin-orbit coupling and correlations in three-orbital systems\nRobert Triebl,1,\u0003Gernot J. Kraberger,1Jernej Mravlje,2and Markus Aichhorn1\n1Institute of Theoretical and Computational Physics,\nGraz University of Technology, NAWI Graz, 8010 Graz, Austria\n2Jozef Stefan Institute, Jamova 39, SI-1000 Ljubljana, Slovenia\n(Dated: November 30, 2018)\nWe investigate the in\ruence of spin-orbit coupling \u0015in strongly-correlated multiorbital systems\nthat we describe by a three-orbital Hubbard-Kanamori model on a Bethe lattice. We solve the\nproblem at all integer \fllings Nwith the dynamical mean-\feld theory using the continuous-time\nhybridization expansion Monte Carlo solver. We investigate how the quasiparticle renormalization\nZvaries with the strength of spin-orbit coupling. The behavior can be understood for all \fllings\nexceptN= 2 in terms of the atomic Hamiltonian (the atomic charge gap) and the polarization\nin thej-basis due to spin-orbit induced changes of orbital degeneracies and the associated kinetic\nenergy. At N= 2,\u0015increasesZat smallUbut suppresses it at large U, thus eliminating the\ncharacteristic Hund's metal tail in Z(U). We also compare the e\u000bects of the spin-orbit coupling to\nthe e\u000bects of a tetragonal crystal \feld. Although this crystal \feld also lifts the orbital degeneracy,\nits e\u000bects are di\u000berent, which can be understood in terms of the di\u000berent form of the interaction\nHamiltonian expressed in the respective diagonal single-particle basis.\nI. INTRODUCTION\nStrongly-correlated electronic systems with sizable\nspin-orbit coupling (SOC) are a subject of intense cur-\nrent interest. We stress a few aspects: (i) In the limit\nof strong interactions, the associated \\spin\" models are\ncharacterized by unusual exchange and are argued to lead\nto exotic phases such as spin-liquid ground states [1{12].\n(ii) The electronic structure of layered iridate Sr 2IrO4,\nwhich features both SOC and sizable electronic repul-\nsion, is (at low energies) similar to the one of layered\ncuprates and is argued to lead to high-temperature su-\nperconductivity [13{20]. (iii) In Sr 2RuO 4, a compound\nin which the correlations are driven by the Hund's rule\ncoupling, the SOC a\u000bects the Fermi surface [21, 22] and\nplays an important role in the ongoing discussion regard-\ning the superconducting order parameter [23, 24]. (iv)\nLast, but not least, the development and improvement of\nmultiorbital dynamical mean-\feld theory (DMFT) tech-\nniques (also driven by the interest in multiorbital com-\npounds following the discovery of superconductivity in\niron-based superconductors) has lead to a detailed and\nto a large extent even quantitative understanding of sev-\neral correlated multiorbital materials. Particular empha-\nsis has been put on the importance of the Hund's rule\ncoupling for electronic correlations [25{27]. A question\nthat is imminent in this respect is how this picture is\na\u000bected by the SOC.\nLet us \frst summarize the key results for the three-\norbital models without SOC. The overall behavior was\nin part understood in terms of the atomic criterion, com-\nparing the atomic charge gap \u0001 atto the kinetic energy.\nThis criterion failed for an occupancy of N= 2, where\nthe additional suppression of the coherence scale is im-\nportant [25{27]. This suppression coincides with the\nslowing down of the spin \ructuations [28] and was ex-\nplained from the perspective of the impurity model that\nis in\ruenced by a reduction of the spin-spin Kondo cou-pling due to virtual \ructuations to a high-spin multiplet\nat half \flling [29{32]. The occurrence of strong corre-\nlations atN= 2 for moderate interactions was also in-\nterpreted (in the context of iron-based superconductors)\nas a consequence of the proximity to a half-\flled (in our\ncaseN= 3) Mott insulating state [33{36], for which the\ncritical interaction is very small due to the Hund's rule\ncoupling. The compounds characterized by the behavior\ndiscussed above were dubbed Hund's metals.\nIn each case, the SOC modi\fes all aspects of this pic-\nture. First, the local Hamiltonian changes, and as a re-\nsult the atomic charge gap also changes. Second, the\nSOC reduces the ground-state degeneracy and hence the\nkinetic energy. Therefore, both the qualitative picture\ninferred from the atomic criterion, as well as quantita-\ntive results, can be expected to be strongly a\u000bected by\nthe SOC.\nIn this work, we use multiorbital DMFT to investi-\ngate the role of SOC in a three-orbital model with semi-\ncircular noninteracting density of states and Kanamori\ninteractions. We are particularly interested in the elec-\ntronic correlations and aim to establish the key properties\nthat control their strength, similarly to what has been\nachieved for the materials without SOC in earlier works.\nFor this purpose, we calculate the quasiparticle residue Z\nand investigate its behavior as a function of interaction\nparameters and SOC for di\u000berent electron occupancies.\nWe \fnd rich behavior, where, depending on the occu-\npancy and the interaction strength, the SOC increases\nor suppresses Z. Partly, this is understood in terms of\nthe in\ruence of the SOC on the atomic charge gap \u0001 at\nand the associated changes of the critical interaction for\nthe Mott transition [26]. In the Hund's metal regime,\nwhere the SOC leads to a disappearance of the charac-\nteristic Hund's metal tail, this criterion fails. Instead,\nwe interpret the behavior in terms of the suppression of\nthe half-\flled Mott insulating state in the phase diagram.\nWe discuss also the e\u000bects of the electronic correlationsarXiv:1807.05106v2  [cond-mat.str-el]  28 Nov 20182\non the SOC.\nEarlier DMFT work investigated some aspects of the\nSOC, for instance its in\ruence on the occurence of dif-\nferent magnetic ground states at certain electron \fll-\nings [37{39]. Zhang et al. successfully applied DMFT\nto Sr 2RuO 4and pointed out an increase of the e\u000bective\nSOC by correlations [21], discussed also in LDA+U [40]\nand slave-boson/Gutzwiller approaches [41, 42]. Kim et\nal.also investigated Sr 2RuO 4and reconciled the Hund's\nmetal picture with the presence of SOC in this com-\npound [22, 43]. In an important work Kim et al. looked\nat the semicircular model [44], as in the present work\nbut did not systematically investigate the evolution of\nthe quasiparticle residue. The e\u000bects of the SOC were\nstudied also with the rotationally invariant slave boson\nmethods [45, 46]. Notably, Ref. [45] that studied a \fve or-\nbital problem also found the disappearance of the Hund's\nmetal tail due to the SOC.\nThis paper is structured as follows. In Sec. II, we start\nby describing the model and the methods used. In Sec. III\nwe give a qualitative discussion of the expected behavior\nin terms of the atomic problem. In Sec. IV we discuss\nthe results of the DMFT calculations and put them into\ncontext of real materials. We end with our conclusions in\nSec. V. In Appendix A we discuss the atomic Hamiltonian\nfor small and large SOC, and in Appendix B we discuss\nthe enhancement of the e\u000bects of SOC by electronic cor-\nrelations in the large- and in the small-frequency limits.\nII. MODEL AND METHOD\nWe consider a three-orbital problem with the (non-\ninteracting) semicircular density of states \u001a(\u000f) =\n2\n\u0019D2p\nD2\u0000\u000f2. We use the half bandwidth Das the en-\nergy unit. Such a density of states pertains to the Bethe\nlattice, for which the DMFT provides an exact solution.\nFor real materials, however, this density of states, as well\nas the DMFT itself, is only an approximation. Neverthe-\nless, qualitative aspects of the results reported here can\nbe expected to apply to real materials, see also Sec. IV F\nbelow.\nThe e\u000bects of spin-orbit coupling are, in general, de-\nscribed by the one-particle operator\nH\u0015=\u0015l\u0001s (1)\nwhere landsare the orbital angular momentum and the\nspin of the respective electron. Our three-orbital model is\nmotivated by cases where the eg-t2gcrystal-\feld splitting\nwithin the dmanifold of a material is large. Therefore,\none retains only the three t2gorbitalsdxy,dxz, anddyz.\nThe matrix representations of the l= 2 operators lx,ly,\nandlzin the cubic basis within the t2gsubspace are up\nto a sign equal to the ones for the l= 1 operators in\ncubic basis, which is called TP correspondence [19, 47].\nTo be more precise, the dxyorbital corresponds to the\npzorbital,dxztopy, anddyztopx. Therefore, the SOCoperator reads\nH\u0015=\u0015lt2g\u0001s=\u0000\u0015lp\u0001s=\u0000\u0015=2 (j2\ne\u000b\u0000l2\np\u0000s2);(2)\nwhere lpare the generators of the l= 1 orbital angu-\nlar momentum and je\u000bis the e\u000bective total one-particle\nangular momentum je\u000b=lp+s. In order to keep the no-\ntation light, we will drop the index \\e\u000b\" in the following,\nand denote the total one-electron angular momentum by\nj. With the eigenvalues lp= 1 ands= 1=2 (\u0016h= 1),j\ncan be 1=2 or 3=2 andmj=\u0000j;\u0000j+ 1;:::; +j. The\neigenvalues of H\u0015are thus\u0000\u0015=2 forj= 3=2 and\u0015for\nj= 1=2, leading to a spin-orbit splitting of3\n2\u0015. Note\nthat in contrast to porbitals, the j= 3=2 band is lower\nin energy because of the minus sign in the TP correspon-\ndence. Therefore, the noninteracting electronic structure\nconsists of four degenerate j= 3=2 bands and two de-\ngeneratej= 1=2 bands, the latter higher in energy.\nIn the second-quantization formalism, the SOC Hamil-\ntonian reads\nH\u0015=\u0015X\nmm0\u001b\u001b0hm\u001bjlt2g\u0001sjm0\u001b0icy\nm\u001bcm0\u001b0\n=\u0000\u0015X\nmm0\u001b\u001b0hmjlpjm0i\u0001h\u001bjsj\u001b0icy\nm\u001bcm0\u001b0\n=i\u0015\n2X\nmm0m00\u001b\u001b0\u000fmm0m00\u001cm00\n\u001b\u001b0cy\nm\u001bcm0\u001b0;(3)\nwhere we expressed the orbital state in the cubic t2gbasis,\nthuscy\nm\u001bcreates an electron in orbital m2fxy;xz;yzg\nwith spin\u001b2f\";#g. The matrix elements of the spin op-\nerators sare given by \u001c=2, where \u001cis the vector of Pauli\nmatrices. The matrix elements of of the components of\nthe orbital angular momentum operator are in case of\ntheporbitalshmjlk\npjm0i=\u0000i\u000fkmm0, wherek;m;m02\nfx;y;zg. In case of t2gorbitals, this notation takes use\nof the TP correspondence fx;y;zgb=fyz;xz;xyg.\nThe atomic interaction is described in terms of the\nKanamori Hamiltonian, which reads in the second quan-\ntization formalism\nHI=X\nmUnm\"nm#+U0X\nm6=m0nm\"nm0#\n+ (U0\u0000JH)X\nm<m0;\u001bnm\u001bnm0\u001b\n+JHX\nm6=m0cy\nm\"cy\nm0#cm#cm0\"\n+JHX\nm6=m0cy\nm\"cy\nm#cm0#cm0\":(4)\nWe setU0=U\u00002JHto make the Hamiltonian ro-\ntationally invariant in orbital space. One can express\nHIin terms of the total number of electrons N=P\nm\u001bnm\u001b, the total spin S=P\nmP\n\u001b\u001b0cy\nm\u001bs\u001b\u001b0cm\u001b0,\nand the total orbital isospin Lwith components Lk=3\nP\nmm0\u001bhmjlk\npjm0icy\nm\u001bcm0\u001b,\nHI= (U\u00003JH)N(N\u00001)\n2+5\n2JHN\n\u00002JHS2\u0000JH\n2L2:(5)\nIn thet2gbasis, again the generators of the por-\nbitals and the TP correspondence are used. The \frst\ntwo Hund's rules are manifest in this form.\nThe full problem is solved by the DMFT [48, 49], where\nthe Hamiltonian is mapped self-consistently to an Ander-\nson impurity model. This impurity problem is solved by\nthe continuous-time quantum Monte Carlo hybridization\nexpansion method [50]. We performed the calculations\nusing the TRIQS package [51, 52]. In the j-basis, which\nis de\fned to diagonalize the local Hamiltonian H\u0015, also\nthe hybridization is diagonal, hence one can use real-\nvalued imaginary-time Green's functions for the calcula-\ntions. This is convenient because it reduces the fermionic\nsign problem and makes the calculations feasible [37, 44].\nHowever, the sign problem still remains a limiting factor\nfor large Hund's couplings and small temperatures. All\nresults reported in this paper were calculated at an in-\nverse temperature \fD= 80.\nAll calculations are done in the paramagnetic state,\nas we focus on the e\u000bect of the SOC in the corre-\nlated metallic regime. Note that di\u000berent kinds of in-\nsulating states occur because antiferromagnetic and ex-\ncitonic order parameters do not vanish in some parameter\nregimes [9, 37, 38, 53{55].\nIII. CRYSTAL FIELD ANALOGY AND THE\nATOMIC PROBLEM\nThe ground-state energies and the atomic charge gaps\nfor a Kanamori Hamiltonian with spin-orbit coupling\nhave been already analyzed in the supplementary mate-\nrial of Ref. [44]. Here, we brie\ry recapitulate certain lim-\nits and compare them to the case of a tetragonal crystal-\n\feld splitting. The SOC lowers the energy of the j= 3=2\nbands by\u0015=2 and increases the energy of the j= 1=2\norbitals by \u0015. Therefore, the crystal-\feld splitting pa-\nrameter \u0001 cfis chosen such that it increases the on-site\nenergy of one orbital by \u0001 cfand that it lowers the en-\nergy of the other two by \u0001 cf=2 in accordance with the\ne\u000bect of\u0015. Physically, this crystal \feld corresponds to\na tetragonal tensile distortion in the zdirection. Both\n\u0015and \u0001 cfare supposed to be positive; a negative sign\nwould correspond to a particle-hole transformation. In\nFig. 1 we illustrate the e\u000bects of the SOC and the tetrag-\nonal crystal \feld on the energy levels and also include a\nreal-space representation of the respective orbitals. Al-\nthough the SOC and the considered crystal \feld give an\nidentical splitting of the single-electron energy levels, the\ncorresponding orbitals and hence also the corresponding\nmatrix elements are di\u000berent, which has important con-\nsequences as discussed below.\n3/2 ∆ cf 3/2λdxy\ndxz\ndyzj= 1/2\nj= 3/2\nHI=Unxz↑nxz↓+...HI=Unxy↑nxy↓HI=/parenleftbig\nU−4\n3JH/parenrightbig\nn1\n2,1\n2n1\n2,−1\n2\nHI=(U−JH)n3\n2,3\n2n3\n2,−3\n2+...crystal field splitting spin-orbit coupling\nspin up spin down\nFIG. 1. Energy levels of the considered models. For both\nSOC and tetragonal crystal-\feld splitting, the orbitally three-\nfold degenerate t2glevel splits into a twodfold degenerate and\na onefold degenerate level. Each level has an additional spin\ndegeneracy. In case of the crystal \feld, the dxyorbital is\nhigher in energy, whereas it is the j= 1=2 orbital in case of\nSOC. The respective orbitals are plotted left (crystal \feld)\nand right (SOC) of the energy levels. The color denotes the\nspin. The fact that the interaction matrix elements in the j\nbasis di\u000ber from the ones in the cubic t2gbasis is also indi-\ncated in the \fgure.\nBefore discussing the issue of the interactions, let us\nbrie\ry discuss the noninteracting case. Since both SOC\nand tetragonal crystal \feld lift the orbital degeneracy,\nthey change the kinetic energy in the system. Without in-\nteractions where SOC and crystal \feld are equivalent, the\nkinetic energy can be readily calculated from the semi-\ncircular density of states, EK=R\n\u000f\u001a(\u000f)f(\u000f)d\u000f, withf(\u000f)\nthe Fermi function at T= 0. The SOC suppresses the\nnoninteracting kinetic energy. In the large- \u0015limit we \fnd\nEK(0)=EK(\u0015!1 ) to be 1.13, 1.34, 1.96, and 1.73 for\nthe casesN= 1, 2, 3, and 5, respectively (for N= 4, the\nlarge-\u0015limit corresponds to a band insulator with a van-\nishing kinetic energy). The reduction of kinetic energy\ndue to the SOC was discussed in the case of the N= 3\ncompound NaOsO 3, where even a somewhat larger re-\nduction of 2.3 was found in a realistic density-functional\nsimulation [56].\nWe now turn to the atomic problem with interactions.\nIt is instructive to rewrite the Kanamori Hamiltonian to\nthejbasis,\nHI=X\nabcdUabcdcy\nacy\nbcdcc=X\n\u000b\f\r\u000e~U\u000b\f\r\u000edy\n\u000bdy\n\fd\u000ed\r (6)\nwith\n~U\u000b\f\r\u000e =X\nabcdUabcdA\u0003\n\u000baA\u0003\n\fbA\rcA\u000ed; (7)\nwhereAis the unitary transformation between the cubic\nt2gand thejbasis [57]. The Latin indices are combined\nindices of orbital and spin; the Greek indices are com-\nbined indices of jandmj. As the Kanamori Hamilto-\nnian is invariant under this transformation for JH= 04\n[seen easily from Eq. (5)], the result of the crystal-\feld\nsplitting and the SOC is identical in this case.\nOn the other hand, for a \fnite Hund's coupling, the\ncrystal \feld and SOC lead to di\u000berent results. The trans-\nformed Hamiltonian in the jbasis di\u000bers from its form\nin the cubic basis (4). We can split it into a pure j= 1=2\npart, a pure j= 3=2 part, and a part that mixes the\nj= 1=2 andj= 3=2 parts,\nHI=Hj=1\n2+Hj=3\n2+Hmix: (8)\nThe \frst two terms read\nHj=1\n2=\u0012\nU\u00004\n3JH\u0013\nn1\n2;1\n2n1\n2;\u00001\n2; (9)\nHj=3\n2= (U\u0000JH)\u0010\nn3\n2;3\n2n3\n2;\u00003\n2+n3\n2;1\n2n3\n2;\u00001\n2\u0011\n+\u0012\nU\u00007\n3JH\u0013\u0010\nn3\n2;\u00003\n2n3\n2;\u00001\n2+n3\n2;3\n2n3\n2;1\n2\u0011\n+\u0012\nU\u00007\n3JH\u0013\u0010\nn3\n2;\u00003\n2n3\n2;1\n2+n3\n2;3\n2n3\n2;\u00001\n2\u0011\n+4\n3JHdy\n3\n2;\u00003\n2dy\n3\n2;3\n2d3\n2;\u00001\n2d3\n2;1\n2\n+4\n3JHdy\n3\n2;\u00001\n2dy\n3\n2;1\n2d3\n2;\u00003\n2d3\n2;3\n2;\n(10)\nthe density-density part of Hmixis\nHmix, dd =\u0012\nU\u00005\n3JH\u0013\u0010\nn1\n2;1\n2n3\n2;3\n2+n1\n2;\u00001\n2n3\n2;\u00003\n2\u0011\n+ (U\u00002JH)\u0010\nn1\n2;1\n2n3\n2;1\n2+n1\n2;\u00001\n2n3\n2;\u00001\n2\u0011\n+\u0012\nU\u00007\n3JH\u0013\u0010\nn1\n2;1\n2n3\n2;\u00001\n2+n1\n2;\u00001\n2n3\n2;1\n2\u0011\n+\u0012\nU\u00008\n3JH\u0013\u0010\nn1\n2;1\n2n3\n2;\u00003\n2+n1\n2;\u00001\n2n3\n2;3\n2\u0011\n:\nThe convention is that n1\n2;1\n2, for example, means\nnj=1\n2;mj=1\n2.Hmixcontains 30 more terms that are not\nshown here.\nHj=1\n2is a one-band Hubbard Hamiltonian with an ef-\nfective interaction Ue\u000b=U\u00004=3JH. For the density-\ndensity part of Hj=3\n2, one observes that the terms with\nthe samejmjj's have prefactors U\u0000JH, whereas terms\nwith di\u000berentjmjj's have prefactors U\u00007=3JH. If one\nusesjmjjas the orbital index and the sign of mjas the\nspin, the density-density part of this Hamiltonian is sim-\nilar to the density-density part of a two-band Kanamori\nHamiltonian, but with di\u000berent prefactors. Importantly,\nthere is only one kind of prefactor for interorbital interac-\ntions, namely U\u00007=3JH, instead of U\u00002JHandU\u00003JH\nin Eq. (4). This in\ruences the electronic correlations,\nas we will see below in the case of N= 2. Following\nthis interpretation of the mj's, the last two terms are\npair-hopping-like expressions with an e\u000bective strengthof 4=3JH. A detailed analysis of this Hamiltonian can be\nfound in Appendix A.\nIt is useful to characterize the atomic Hamiltonian\nHloc=HI+H\u0015in terms of the atomic charge gap\n\u0001at=E0(N+ 1) +E0(N\u00001)\u00002E0(N); (11)\nwhereE0(N) is the ground state of a system with Nelec-\ntrons [27]. According to the Mott-Hubbard criterion, the\nmetal-insulator transition takes place when \u0001 atexceeds\nthe kinetic energy. Hence, the proximity of interaction\nparameters to the associated critical value Uccan be used\nto anticipate the strength of electronic correlations.\nWe start with a discussion of the crystal-\feld split-\nting [58{61]. For \fllings N= 1, 2, and 5, the ground\nstate does not change with the crystal-\feld splitting. For\nN= 3 andN= 4, there is a level crossing with a tran-\nsition from a high-spin to a low-spin state (e.g., from\nj\";\";\"itoj\"#;\";0i), which is responsible for di\u000berences\nin the atomic charge gap for small and large \u0001 cf. The\nrespective values for the charge gap in the limits of small\nand large \u0001 cfare listed in Tables I and II. Note that\nin the large \u0001 cflimit, the relevant Hamiltonian is a two-\norbital one for \fllings N= 1;2;and 3, and a one-orbital\none forN= 5. For the Kanamori Hamiltonian with \u0017\norbitals, the charge gap depends on the relative \flling; at\nhalf \flling it is \u0001 at=U+ (\u0017\u00001)JH, otherwise U\u00003JH.\nThe \flling N= 4 is special as an electron can only be\nadded by paying additionally crystal-\feld splitting en-\nergy.\nWe now turn to the discussion of SOC. Note that the\nlimits\u0015\u001cJHand\u0015\u001dJHcorrespond to the LSandjj\ncoupling scheme, respectively. A look at Tables I and II\nreveals that practically all entries are di\u000berent from the\ncorresponding crystal-\feld ones. The values for a large\nSOC can be obtained from the Hamiltonian expressed\nin thejbasis discussed above. For N= 5, where the\ne\u000bective model is a single-orbital model, the interaction\nparameter is U\u00004\n3JH, as seen from Eq. (9), in contrast\nto the crystal \feld result, where one obtains simply U,\ninstead. In the case of N= 2, it is interesting to note\nthat the dependence of the charge gap on JHis di\u000berent\nin sign for the SOC and the crystal \feld. This follows\nfrom Eq. (10), which does not favor the alignment of the\nangular momenta jzof the respective orbitals (see also\nAppendix A). This opposite behavior is also re\rected in\nthe full DMFT solution, as we discuss below. We will see\nthat forN= 2, there are parameter regimes, where the\ncorrelation strength increases with crystal-\feld splitting,\nbut it decreases with SOC.\nIV. DMFT RESULTS\nWe now turn to the DMFT results. We focus on\nthe interplay between the SOC and electronic correla-\ntions, which we follow by calculating the Matsubara self-\nenergies. Due to the symmetry, the Green's functions5\nTABLE I. Comparison of the atomic charge gap \u0001 atobtained\nfrom a spin-orbit coupling \u0015or a tetragonal crystal-\feld split-\nting \u0001 cfin the limit \u0015;\u0001cf\u001cJH.\nN SOC crystal \feld\n1U\u00003JH+ 1=2\u0015U\u00003JH\n2U\u00003JH+ 1=2\u0015U\u00003JH+ 3=2 \u0001cf\n3U+ 2JH\u00003=2\u0015U+ 2JH\u00003=2 \u0001cf\n4U\u00003JH+\u0015 U\u00003JH\n5U\u00003JH+\u0015U\u00003JH+ 3=2 \u0001cf\nTABLE II. Comparison of the atomic charge gap \u0001 atobtained\nfrom a spin-orbit coupling \u0015or a tetragonal crystal-\feld split-\nting \u0001 cfin the limit \u0015;\u0001cf\u001dJH.\nN SOC crystal \feld\n1U\u00007=3JH U\u00003JH\n2U\u0000JH U+JH\n3U\u00007=3JH U\u00003JH\n4U\u00003JH+ 3=2\u0015U\u00005JH+ 3=2 \u0001cf\n5U\u00004=3JH U\nand the self-energies are diagonal in the jbasis with two\nindependent components \u0006 1=2and \u0006 3=2.\nFigure 2 displays the calculated self-energies for the\nN= 1 case. One can see that due to the SOC jIm\u0006 3=2j\nis larger and its slope at low energies that determines the\nquasiparticle residue\nZ\u0017= lim\ni!n!0\u0014\n1\u0000@Im\u0006\u0017(i!n)\n@i!n\u0015\u00001\n(12)\nis larger. The origin of that is discussed below, where\nwe investigate the evolution of Z\u0017with\u0015for all integer\noccupancies, but let us \frst discuss the other part of the\ninterplay, namely the in\ruence of the electronic correla-\ntions on the SOC.\nA. In\ruence of electronic correlations on the SOC;\ne\u000bective SOC\nFor this purpose it is convenient to introduce the av-\nerage self-energy\n\u0006a=2\n3\u00063\n2+1\n3\u00061\n2(13)\nand the di\u000berence\n\u0006d= \u0006 1\n2\u0000\u00063\n2: (14)\nIn terms of \u0006 a,dthe self-energy matrix can be written in\nthe form\n\u0006 = \u0006 a1+2\n3\u0006dlt2g\u0001s; (15)\n1.41.61.82.0ReΣ(a)\nj= 3/2\nj= 1/2\nλ= 0\n0 1 2 3\nωn−0.6−0.5−0.4−0.3−0.2−0.10.0ImΣ\n(b)fitj= 3/2\nfitj= 1/2\nfitλ= 0FIG. 2. Real (a) and imaginary (b) part of the self-energy for\nthe parameters N= 1,\u0015= 0:1,U= 3, andJH= 0:1U. The\ngreen squares display the results without SOC for comparison.\nThe lines show a polynomial \ft of degree four through the \frst\nsix Matsubara frequencies.\nwhich holds in any basis (see Appendix B). This form is\nalso convenient as one can directly see that \u0006 ddetermines\nthe in\ruence of electronic correlations on the physics of\nSOC. In particular, because the Green's function is\nG(k;i!n) = [i!n+\u0016\u0000H0(k)\u0000\u0006(i!n)]\u00001(16)\nwithH0(k) the noninteracting Hamiltonian that includes\nthe SOC, the real part of the self-energy can be used to\nde\fne the e\u000bective spin-orbit-coupling constant\n\u0015e\u000b=\u0015+2\n3Re\u0006 d(i!n!0): (17)\nFor all cases we looked at (some data is shown in Ap-\npendix B), we \fnd that the real part of \u0006 d(i!n) is pos-\nitive for all !n(as long as the system is metallic) and\nits e\u000bect hence adds up to the bare SOC Hamiltonian so\nthat\u0015e\u000b>\u0015, as found also in realistic studies [21, 22, 40].\nNotice that there is also a further renormalization of the\noverall bandstructure due to the frequency dependence of\nthe self-energy [22, 41]. The e\u000bects on the quasiparticle\ndispersions, for instance on the liftings of the quasipar-6\nticle degeneracies, can be phrased in terms of the quasi-\nparticle SOC constant \u0015\u0003=Z\u0015e\u000b[22] with quasiparticle\nrenormalization Z < 1, hence\u0015\u0003can be smaller or larger\nthan the bare \u0015. However, relative to the other features\nof the quasiparticle dispersions that are obviously renor-\nmalized by Z, too, the SOC splittings are enhanced due\nto the e\u000bect of \u0006 d.\nB. In\ruence of SOC on electronic correlations:\nOne and \fve electrons\nIn the remainder of the paper we investigate how the\nSOC in\ruences the electronic correlations, which is fol-\nlowed by calculating the j-orbital occupations and the\nquasiparticle residues Z\u0017. These are calculated by \ftting\nsix lowest frequency points of Matsubara self-energies to\na fourth order polynomial, as shown in Fig. 2(b).\nWithout SOC, one electron and one hole (\fve elec-\ntrons) in the system are equivalent due to the particle-\nhole symmetry, but the SOC breaks this symmetry. For\nlarge\u0015, only thej= 3=2 (j= 1=2) orbitals are partially\noccupied for N= 1 (N= 5). Hence, these are more in-\nteresting regarding electronic correlations. In Figs. 3(a)\nand 3(b), we show how the quasiparticle weights and\nthe \fllings of these orbitals change when the SOC is in-\ncreased. The corresponding atomic charge gap is also\nplotted, Fig. 3(c).\nThe change in orbital polarization in\ruences the corre-\nlation strength. This is best seen for JH= 0, since then\nthe e\u000bective repulsion is simply U, independent of the\nSOC. The quasiparticle weight of the relevant orbitals is\nreduced by the SOC as the polarization increases, which\nis shown in Fig. 3(b) for U= 3 (circles). The reduction\nis weak for N= 1 but strong for N= 5, which is due\nto the lower kinetic energy of one hole in one j= 1=2\norbital compared to the energy of one electron in two\nj= 3=2 orbitals. In the case of U= 3 andJH= 0, even\na metal-insulator transition takes place.\nThe Hund's coupling reduces the correlation strength\n(stars, crosses). This happens for two reasons: JHre-\nduces the polarization, and it decreases the atomic charge\ngap. The latter is expected for N= 1, where the e\u000bec-\ntive number of orbitals reduces with increasing \u0015from\nthree to two. In this case, a \fnite exchange interaction\nJHleads to a reduction of the repulsion between electrons\nin di\u000berent orbitals.\nInterestingly, JHalso decreases the strength of corre-\nlations for N= 5 in the limit of large \u0015, although the\ne\u000bective number of orbitals is one and interorbital e\u000bects\nare thus suppressed. However, the transformation from\nthe cubic Kanamori Hamiltonian to its jbasis equivalent\nmixes inter- and intraorbital interactions, so that the ef-\nfectivej= 1=2 interaction strength is U\u00004=3JH, as\nexplained in Sec. III. In contrast, in the case of a large\ntetragonal crystal-\feld splitting, the atomic charge gap\nis indeed simply given by UforN= 5.\nIt is also interesting to compare the dependence of\n0.00.20.40.60.81.0ZN= 5,j= 1/2 N= 1,j= 3/2(a)JH= 0.0U\nJH= 0.1UJH= 0.2U\nnoninteracting\n0.00.20.40.6n(b)\n1.0 0 .5 0 0 .5 1 .0\nλ1234∆at(c)FIG. 3. In\ruence of the spin-orbit coupling for a \flling of\nN= 1 (right column) and N= 5 (left column) for U= 3. (a)\nQuasiparticle weight Zof thej= 3=2 orbitals (for N= 1)\nand of thej= 1=2 orbitals (for N= 5). (b) Electron density\nnof thej= 3=2 orbitals ( N= 1) and hole density of the\nj= 1=2 orbitals (N= 5) to allow for a better comparability.\nThe green dotted line displays the respective noninteracting\nresults. (c) Atomic charge gap \u0001 at.\nthe respective orbital occupation nwith the noninter-\nacting result [green dotted line in Fig. 3(b)]. One can\nsee that the correlations increase the orbital polarization\nn3=2\u0000n1=2, in line of what one would expect from the\nenhancement of the SOC physics by electronic correla-\ntions discussed above. As shown below, we \fnd similar\nbehavior also for other \fllings, but not for N= 3 when\nthe Hund's coupling is large.7\n1 2 3 4\nU0.00.20.40.60.81.0Z3/2λ= 0.0\nλ= 0.5\nλ= 1.0\nλ=∞\nFIG. 4. Quasiparticle weight Z3=2of thej= 3=2 orbital as\na function of UforJH= 0:1Uand a total \flling of N= 3.\nC. Half \flling\nIn Fig. 4 we display the quasiparticle weight of the\nj= 3=2 orbitals (again, the j= 1=2 are emptied out\nwith SOC and are therefore not discussed here) at N= 3\nfor several \u0015. One can see that \u0015strongly increases Uc\nand changes the behavior drastically. To understand why\nthis occurs, \frst recall that at \u0015= 0, Hund's coupling\nstrongly reduces the kinetic energy since it enforces the\nhigh-spin ground state [25]. Hence, the Hund's coupling\nleads to a drastic reduction of the critical interaction\nstrength [26]. This causes a steep descent of Zas a func-\ntion ofUwhen the critical Uis approached (see Fig. 4\nfor\u0015= 0 andJH= 0:1U).\nAs\u0015is large, this physics does not apply any more.\nThe \flling of the j= 3=2 orbitals increases to three\nelectrons in two orbitals. Since the Hamiltonian of the\nj= 3=2 orbitals alone is particle-hole symmetric, this\nlarge\u0015limit shows identical physics to the large \u0015limit\nin the case of N= 1. As described above in Sec. IV B,\nthis\u0015!1 system is characterized by an increase of\nZwith increasing JH. This is opposite to the half-\flled\nN= 3 case at \u0015= 0, where Zdecreases with JH.\nIn Figs. 5(a)-5(c) we show how the quasiparticle\nweight, the orbital polarization, and the atomic charge\ngap vary with \u0015, respectively. We \fnd that Zin-\ncreases for physically relevant Hund's couplings (e.g.,\nJH= 0:1U,JH= 0:2U). Furthermore, the qualitative\ndi\u000berence between the small and the large \u0015limits dis-\ncussed above results in crossings of the Z(\u0015) curves for\ndi\u000berent Hund's couplings [see Fig. 5(a)]. These crossings\nare already expected from the atomic charge gap, which\nisU+ 2JHfor\u0015= 0 and drops to U\u00007=3JHfor\u0015!1 ,\nas shown in Tables I and II as well as in Fig. 5(c).\nThe results in Fig. 5 show that SOC can strongly mod-\nify the correlation strength. One needs to notice, though,\nthat it takes a quite large \u0015for these changes to occur; for\ninstance, full polarization is reached at \u0015\u00191, whereas\nit occurs at \u0015\u00190:3 in the case of N= 1 andU= 3\n0.00.20.40.60.81.0Z3/2(a)JH= 0.0U\nJH= 0.1UJH= 0.2U\nnoninteracting\n0.40.50.60.70.8n3/2(b)\n0.0 0 .5 1 .0 1 .5 2 .0\nλ1.01.52.02.53.0∆at(c)FIG. 5. Quasiparticle weight Z3=2(a) and \flling n3=2(b)\nof the electrons in the j= 3=2 orbitals as functions of \u0015\nforU= 2. The green dotted line displays the respective\nnoninteracting results. (c) Atomic charge gap \u0001 at.\n(compare Fig. 5 with Fig. 3). In this respect we notice\nalso that in contrast to the N= 1 case, the electronic\ncorrelations increase the orbital polarization at N= 3\nas compared to the noninteracting result only for small\nvalues ofJH.\nD. Two electrons\nWe now discuss the interesting case of two electrons. In\nthe absence of SOC, this is the case of a Hund's metal.\nFigure 6 shows the dependence of ZonUfor several\nvalues of\u0015andJH=U= 0:2. The data at small \u0015ex-\nhibit a tail with small Z, which is characteristic for the8\n0 1 2 3 4 5 6\nU0.00.20.40.60.81.0Zλ= 0.0\nλ= 0.5\nλ=∞\n∆cf=∞\nFIG. 6. Quasiparticle weight Zof thej= 3=2 orbital as a\nfunction of UforJH= 0:2UandN= 2. The dashed line\nshows the corresponding Zof thedxzorbital in the case of an\nin\fnite tetragonal crystal-\feld splitting.\nHund's metal regime. The SOC has a drastic e\u000bect here;\nincreasing\u0015suppresses the Hund's metal behavior and\nleads to a featureless, almost linear, approach of Zto-\nwards 0 with increasing U. Interestingly, the in\ruence\nof\u0015onZis opposite at small Uwhere increasing \u0015in-\ncreasesZ, thus making the system less correlated, and\nat a highU, whereZdiminishes with \u0015and hence cor-\nrelations become stronger.\nThe latter behavior is easy to understand. A strong\nSOC reduces the number of relevant orbitals from three\nto two, and leads to the increase of the atomic charge gap\nfromU\u00003JHtoU\u0000JH[see Fig. 8(c) and Sec. III]. Both\nthe reduction of the kinetic energy due to the reduced\ndegeneracy and the increase of the atomic charge gap\nwith\u0015contribute to a smaller critical U, which is indeed\nseen on the plot. We want to note here that the reduc-\ntion of the critical Uis even stronger for the crystal-\feld\ncase (shown as a dashed line in Fig. 6), since there the\ncorresponding atomic gap is larger ( U+JH, see Sec. III).\nWe turn now to the small- Uregime where the SOC\nreduces the electronic correlations. One can rationalize\nthis from a scenario that pictures Hund's metals as doped\nMott insulators at half \flling [33{36]. Figure 7 presents\nthe values of Uwhere a Mott insulator occurs. Let us\n\frst discuss the case without SOC, i.e., the left panel\nof Fig. 7. In this picture of doped Mott insulators, the\ncorrelations for small interactions at N= 2 are due to\nproximity to a half-\flled insulating state. For interac-\ntion parameters UandJHthat lead to a Mott insulator\nat half \flling, doping with holes leads to a metallic state\nwith low quasiparticle weight. This low- Zregion persists\nto doping concentrations of more than one hole per atom,\nas can be seen from Fig. 2 in Ref. [26]. As a result, for an\ninteraction Uin between the critical values for two and\nthree electrons Uc(N= 3)<U <U c(N= 2), the quasi-\nparticle weight is small, but not zero. As one increases\nnow\u0015, the critical UatN= 3 increases strongly, and\n0 1 2 3 4 5 6\nN0246810U\n(a)λ= 0\n0 1 2 3 4 5 6\nN0246810\n(b)λ=∞FIG. 7. The Mott insulator occurs for values of Uindicated\nby bars for a Hund's coupling of JH= 0:2U. The left picture\n(a) shows the case without SOC, the right (b) with an in\fnite\nSOC. Note that in the latter case no Mott insulator occurs for\nN= 4 since this case is a band insulator. The critical values\nfor\u0015= 0 are taken from Ref. 26. The red crosses indicate\nthe critical Uin the case where a tetragonal crystal \feld is\napplied instead of the SOC.\nthe insulating state appears only for large values of U, see\nthe right panel of Fig. 7. Consequently, the N= 2 state\ncannot be viewed as a doped N= 3 Mott insulator any\nmore. In fact, for a large SOC, the critical interaction\nstrengthUcfor a Hund's coupling of JH=U= 0:2 is low-\nest forN= 2, as displayed in Fig. 7. As a consequence,\nthe Hund's tail disappears (this was earlier noted also in\na rotationally-invariant slave boson study of a \fve orbital\nproblem [45]), as highlighted in Fig. 6, and the quasipar-\nticle weight increases with SOC in the case of a small\nUand large Hund's couplings [see Fig. 8(a)]. In pass-\ning we note that the DMFT self-consistency is essential\nto account for the increase of Zin the small Uregime.\nCalculations for an impurity model found a suppression\nof the Kondo temperature (and hence a suppression of\nZ) with increased \u0015[43], which is di\u000berent from what we\n\fnd in the DMFT results here.\nFigure 8(b) shows the orbital occupancy as a function\nof\u0015. Like inN= 1,N= 5, and, for small enough JH,\nalsoN= 3, from a comparison with the noninteracting\nresult one \fnds that the SOC usually leads to a larger\norbital polarization when the interactions are present.\nLooking at the data more precisely, this ceases to hold\nin the large- \u0015regime. We actually \fnd this at other \fll-\nings, too. At values of \u0015where the noninteracting result\nis already fully polarized, the electronic correlations rein-\ntroduce some charge in the empty/fully polarized orbital.\nIn Fig. 8, we also compare the in\ruence of the SOC9\n0.00.20.40.60.81.0Z(a)JH= 0.0U, spin-orbit coupling\nJH= 0.1U, spin-orbit coupling\nJH= 0.2U, spin-orbit coupling\nJH= 0.0U, crystal-field splitting\nJH= 0.1U, crystal-field splitting\nJH= 0.2U, crystal-field splitting\nnoninteracting\n0.20.30.40.50.6n(b)\n0.00 0 .25 0 .50 0 .75 1 .00 1 .25 1 .50\nλ,∆cf0123∆at(c)\nFIG. 8. Quasiparticle weight of the electrons (a), \flling (b),\nand the atomic charge gap (c) for N= 2 andU= 2. Solid\nlines correspond to the SOC case, and j= 3=2 quantities\nare plotted as functions of \u0015. Dashed lines are the results\nfor a crystal-\feld splitting, where we plot dxz=yz quantities as\nfunctions of \u0001 cf.\nto that of a tetragonal crystal \feld. One sees that the\ncrystal \feld always increases the correlation strength. To\nunderstand this it is convenient to recall that the atomic\ngaps are di\u000berent, and as a result, also the critical U's are\ndi\u000berent. For an in\fnite crystal \feld, they are marked\nwith crosses in Fig. 7(b). In particular, the critical inter-\naction atN= 2 in the case of an in\fnite crystal \feld is\nonly slightly larger than the critical interaction at N= 3\nwithout any splitting. Therefore, Hund's metals with in-\nteractions in the range Uc(N= 3)< U < U c(N= 2),\nbecomes insulating, as the interaction driven Mott tran-\nsition atN= 2 is pushed to such small values of Uc\n0 1 2 3 4 5 6\nU0.00.20.40.60.81.0Z3/2\nJH= 0.0U\nJH= 0.1U\nJH= 0.2U\nJH= 0.3U−2.5 0.0 2 .5\nω0.000.250.50AFIG. 9. Quasiparticle weight Z3=2of thej= 3=2 orbital\nas a function of Ufor\u0015!1 and a total \flling of N= 2.\nThe inset shows the respective impurity spectral functions\nforU= 3 andJH= 0 (blue) and JH= 0:2U(black). As\nthe Hund's coupling JHincreases, the quasiparticle weight (=\narea of the quasiparticle peak) stays the same, whereas the\nposition of the Hubbard bands changes due to di\u000berent charge\ngaps. To obtain the spectral functions, imaginary-time data\nhas been analytically continued using a maximum entropy\nmethod [62] with an alternative evidence approximation [63]\nand the preblur formalism [64].\nby the large \u0001 cf. Another di\u000berence is the ground state\ndegeneracy, which is three for the S= 1 ground state\nof the two-orbital Kanamori and \fve in the case of the\nJ= 2 ground state of Hj=3=2, see Appendix A, which\nalso points to weaker correlations in the SOC case.\nAnother interesting observation from Fig. 8(a) is that\nthe quasiparticle weight is almost independent of Hund's\ncoupling in the limit of large \u0015forU= 2. In Fig. 9, we\nshow that the weak dependence on JHis also apparent\nfor other values of U, and only becomes signi\fcant when\nthe Hund's coupling is exceeding JH>0:2U. However,\nsince the atomic gap does depend on JH, the position of\nthe Hubbard bands are di\u000berent, even though Zis the\nsame, as shown in the inset of Fig. 9.\nE. Four electrons\nThe \flling of four electrons is special because strong\nSOC leads to a band insulator with fully occupied j=\n3=2 orbitals and empty j= 1=2 orbitals, with no renor-\nmalization ( Z= 1) for both orbitals in the large \u0015regime.\nFigure 10(a) shows the quasiparticle renormalization of\nboth orbitals in the metallic phase as a function of \u0015. One\ncan see that Z3=2is hardly a\u000bected, and Z1=2increases\nonly slightly for the given parameters U= 2 andJH=\n0:2U, indicating that the orbital polarization a\u000bects only\nweakly the correlation strength, unless in close vicinity\nto the metal-insulator transition.\nA comparison to the crystal-\feld results shows two ma-10\njor di\u000berences: First, the orbital polarization, displayed\nin Fig. 10(b), is smaller in the case of the crystal \feld, as\ncompared to the SOC case, and a larger value of crystal-\n\feld splitting is needed to reach a band insulator. The\nreason for this is a larger atomic gap in the SOC case\n[see Fig. 10(c) and Tables I and II]. Second, the quasi-\nparticle renormalization of the less occupied (in the case\nof crystal \feld dxy) orbital is lowest when its \flling is\naround 1/2. This enhancement of correlation e\u000bects at\nhalf \flling is absent for the j= 1=2 orbital.\nF. Discussion\nIt is interesting to discuss our results in the context of\nreal materials and to consider which parameter regimes\nare realized (see also Refs. 19 and 44). One can \frst recall\nthe atomic values \u0010for the SOC that roughly increase\nwith the fourth power of the atomic number. It takes\nsmall values in 3 d(Mn: 0:04 eV, Co: 0 :07 eV), intermedi-\nate values in 4 d(Ru: 0:13 eV, Rh: 0 :16 eV), and reaches\nconsiderable strength in 5 d(Os: 0:42 eV, Ir: 0 :48 eV)\natoms [65]. These atomic values are representative also\nfor the values of SOC \u0015found in corresponding oxides.\nRegarding interaction parameters, one can roughly take\nthatJH=U= 0:1 and values of Uthat diminish from\n4 eV(in 3d), 3 eV(4d), 2 eV(5d). Finally, the bandwidth\nwill vary from case to case, since it depends the most on\nstructural details among all the microscopic parameters.\nAs a rule of thumb, however, it increases with the prin-\nciple quantum number, giving values of half bandwidth\nfromD=1 eV(3d), 1:5 eV(4d), 2 eV(5d). These all are of\ncourse only rough estimates, meant to indicate trends.\nThe clear-cut case with strong in\ruence of SOC are 5 d\noxides atN= 5. In iridates, \u0015=D ranges from 0.26 in\nSr2IrO4up to 2.0 in Na 2IrO3due to the small bandwidth\nin this compound [44]. Inspecting now Fig. 3, one sees\nthat the SOC leads to a strong orbital polarization and\nstrongly a\u000bects the correlations at those values of \u0015=D.\nActually, the sensitivity to SOC at N= 5 is so strong\nthat one can expect signi\fcant impact also in 4 d5com-\npounds, like rhodates, too, although \u0015is by a factor of\nthree smaller there. Indeed, the enhancement of corre-\nlations has been observed in a material-realistic DMFT\nstudy of Sr 2RhO 4[18, 19]. Rather small SOC leads also\nto a large polarization in the particle-hole transformed\ncounterpart N= 1 (with potentially important conse-\nquences for the magnetic ordering [66]), but the increase\nof the quasiparticle renormalization is weak, see Fig. 3(a).\nOpposite to the N= 1 andN= 5 cases, the SOC at\nN= 3 makes the electronic correlations weaker. Also in\ncontrast to the former two cases, the e\u000bect of SOC on\npolarization and quasiparticle renormalization becomes\npronounced only at larger values of \u0015. From Fig. 5(b)\nwe can infer that for full polarization \u0015=D > 0:5 is nec-\nessary. Large values of \u0015=D can be obtained in dou-\nble perovskites based on 5 delements. In Sr 2ScOsO 6,\nfor instance, quite a substantial reduction of correlations\n0.00.20.40.60.81.0Z(a)j= 3/2, spin-orbit coupling\nj= 1/2, spin-orbit coupling\ndxz,dyz, crystal-field splitting\ndxy, crystal-field splitting\nnoninteracting\n0.00.20.40.60.81.0n(b)\n0.00 0 .25 0 .50 0 .75 1 .00 1 .25 1 .50\nλ,∆cf0123∆at(c)FIG. 10. Quasiparticle renormalization (a), \flling (b), and\natomic charge gap (c) of the orbitals as functions of spin-orbit\ncoupling (full lines) and crystal-\feld splitting (dashed lines)\nforN= 4,U= 2,JH= 0:2U. Full dots indicate insulating\nphases. In the case of SOC, all calculations with \u0015\u00150:7 are\ninsulating, whereas in the case of a crystal\feld only the last\npoint shown (\u0001 cf= 1:5) is insulating. The green dotted lines\nshows the orbital \fllings in the noninteracting case. Then,\ncrystal \feld and SOC are equivalent.\noccurs with SOC [67]. In case of the single perovskite\nNaOsO 3, the SOC modi\fes the band structure [68] too,\nwhich leads to an important suppression of kinetic en-\nergy [56], as discussed also in Sec. III. In the case of 4 d\nelements, typically \u0015=D < 0:2; therefore we expect only\nsmall e\u000bects of the SOC on the correlation strength in\nthese materials.\nFor the \flling N= 2, we show in Fig. 6(a) a system-\natic suppression of the Janus-faced behavior with SOC,11\nmaking the Hund's tail disappear. This e\u000bect is already\nsizable for \u0015=D\u00190:5 and should, hence, be present in\nmany 5dsystems. Indeed, it has been seen in calculations\nfor the 5d2compound Sr 2MgOsO 6[67]. For a smaller\nSOC of\u0015=D\u00190:1, which is a good estimate for many 4 d\nmaterials, we do not \fnd a substantial change of Z[see,\nfor example, Fig. 8(a)]. Therefore, we think the SOC\nonly weakly a\u000bects the correlation strength in materials\nwith 4d2con\fguration, such as Sr 2MoO 4[69{71].\nForN= 4, our model calculations predict that the\nSOC a\u000bects the correlation strength only a little, pro-\nvided it is small enough such that the system remains\nin the metallic phase. If it exceeds a certain magnitude,\nthough, a metal-insulator transition occurs. The critical\n\u0015decreases with increasing U. Examples for this behav-\nior are on one hand Sr 2RuO 4(\u0015= 0:10 eV), where the\nquasiparticle renormalization hardly changes as the SOC\nis turned on [22], and, on the other hand, NaIrO 3(\u0015=\n0:33 eV), where the interplay of SOC and Uleads to an\ninsulating state [72].\nV. CONCLUSION\nIn this paper we investigated the in\ruence of the SOC\non the quasiparticle renormalization Zin a three-orbital\nmodel on a Bethe lattice within DMFT. Depending on\nthe \flling of the orbitals (and for N= 2 also the inter-\naction strength), the SOC can decrease or increase the\nstrength of correlations. The behavior can be understood\nin terms of the SOC-induced changes of the e\u000bective de-\ngeneracy, the \fllings of the relevant orbitals, and the in-\nteraction matrix elements in the low-energy subspace.\nThe spin-orbital polarization leads to an increase of the\ncorrelation strength for N= 1 and 5, with particularly\nstrong e\u000bect for N= 5, where a half-\flled single-band\nproblem is realized, relevant for iridate compounds. For\nthe nominally half-\flled case N= 3, the opposite trend\nis observed. Here, turning on SOC makes the system\nless correlated, and the critical interaction strength Uc\nfor a Mott transition is increased. For the N= 2 Hund's\nmetallic phase, the in\ruence of SOC is more involved.\nWe \fnd that there are two regimes as a function of U\nwith opposite e\u000bect of SOC. For small U, the inclusion\nof SOC increases Z, whereas for large Uit decreases Z,\nand in turn also the critical interaction Ucdecreases. As\na result, the so-called Hund's tail with small quasiparticle\nrenormalization for a large region of interaction values,\ndisappears.\nWe also considered the e\u000bects of the electronic cor-\nrelations on SOC and found that in the cases where the\nsystem remains metallic, correlations always enhance the\ne\u000bective SOC.ACKNOWLEDGMENTS\nWe thank Michele Fabrizio, Antoine Georges, Alen\nHorvat, Minjae Kim, Andrew Millis, and Hugo Strand\nfor helpful discussion. We acknowledge \fnancial support\nfrom the Austrian Science Fund FWF, START program\nY746. Calculations have been performed on the Vienna\nScienti\fc Cluster. J.M. acknowledges the support of the\nSlovenian Research Agency (ARRS) under Program P1-\n0044.\nAppendix A: Atomic Hamiltonian in the limit of\nsmall and large spin-orbit couplings\nThe full local Hamiltonian reads [see also Eq. (5)]\nHloc=HI+H\u0015+H\u000f\n= (U\u00003JH)N(N\u00001)\n2+\u00125\n2JH+\u000f\u0013\nN\n\u00002JHS2\u0000JH\n2L2+\u0015lt2g\u0001s;(A1)\nwith an SOC \u0015and an on-site energy \u000f. Note that this\nHamiltonian contains both two-particle terms like N2,\nL2, and S2, as well as one-particle terms like Nandlt2g\u0001\ns. For\u0015= 0, the total spin Sand the total orbital\nangular momentum Lare good quantum numbers and\ndetermine together with the total number of electrons\nNthe eigenenergies. As \u0015is \fnite, the energy levels\nsplit according to their total angular momentum J. For\nexample, the nine-fold degenerate S= 1,L= 1 ground\nstate in the N= 2 sector splits into a J= 2, aJ= 1, and\naJ= 0 sector. The respective degeneracies are 2 J+ 1.\nThe total angular momentum Jis for all values of \u0015a\ngood quantum number, in contrast to the total spin S\nand the total orbital angular momentum L.\nFor a small SOC ( \u0015\u001cJH), one can use \frst-order per-\nturbation theory in order to calculate the level splitting\ndue to the SOC. In this approximation, the spin-orbit\nterm is approximated by C\u0015L\u0001S. The constant Cde-\npends on the number of electrons and is C= 1;1=2 for\none and two electrons, and C=\u00001;\u00001=2 for one and\ntwo holes. For three electrons, L= 0, and the \frst-order\nperturbation theory gives no energy correction. Since the\ntotal angular momentum is approximated by J=L+S,\nthis regime is known as LScoupling regime.\nIn the limit of large SOC ( \u0015\u001dJH), the spin-orbit term\nis the dominant term that is solved exactly, whereas S2\nandL2may be treated perturbatively. The many-body\neigenstates of the unperturbed system are then the Slater\ndeterminants of j= 1=2 andj= 3=2 one-electron states.\nFollowing Eq. (2), the matrix elements of \u0015lt2g\u0001sdepend\nin this unperturbed eigenbasis only on the number of elec-\ntrons in the j= 3=2 and thej= 1=2 orbitals. The total\nangular momentum is J=P\niji, therefore, this regime\nis thejjcoupling regime. For \fllings N\u00144, only the\nj= 3=2 orbitals are occupied in the ground state. The12\nTABLE III. Eigenenergies of the Hamiltonian Hj=3\n2of the\nj= 3=2 orbitals, Eq. (10).\nNJEj=3=2\n00 0\n13/2 \u000f\n222\u000f+U\u00007=3JH\n202\u000f+U+ 1=3JH\n33/23\u000f+ 3U\u000017=3JH\n404\u000f+ 6U\u000034=3JH\nTABLE IV. Full list of quantum numbers and eigenenergies in\nthe two-particle sector of a two-orbital system. We compare\nenergiesEegof the ordinary Kanamori Hamiltonian for eg\norbitals with energies Ej=3=2for the e\u000bective j= 3=2 Hamil-\ntonian stemming from a large SOC in t2gorbitals.\nNTTy~S~SzEegEj=3=2\n2001-1U\u00003JHU\u00007=3JH\n20010U\u00003JHU\u00007=3JH\n20011U\u00003JHU\u00007=3JH\n21-100U\u0000JHU\u00007=3JH\n21000U+JHU+ 1=3JH\n21100U\u0000JHU\u00007=3JH\nspin-orbit term is then proportional to the particle num-\nberNand can be absorbed in the one-electron energy\n\u000f.\nCalculating the matrix elements of S2andL2for Slater\ndeterminants with di\u000berent NandJusing Clebsch-\nGordan coe\u000ecients, one can \fnd the eigenenergies of the\nHamiltonian in the jjcoupling regime. This approach\nis equivalent to looking for the eigenvalues of Hj=3\n2pre-\nsented in Eq. (10) in the main text, where all contribu-\ntions of the j= 1=2 orbitals are neglected. The eigenen-\nergies ofHj=3\n2, including an on-site energy \u000f, are shown\nin Table III.\nIt is possible to bring the Hamiltonian Hj=3\n2into a\nmore symmetric form if one assigns the absolute value of\nmjas orbitals and its sign as spin, e.g., d3\n2;1\n27!c1\"and\nd3\n2;\u00003\n27!c2#. It reads then\nHj=3\n2=\u0012\nU\u00005\n3JH\u0013N(N\u00001)\n2\u00001\n3JHN\n+4\n3JH\u0000\nT2\u00002T2\ny\u0001(A2)\nwith a total spin\n~S=1\n2X\nmX\n\u001b\u001b0cy\nm\u001b\u001c\u001b\u001b0cm\u001b0 (A3)\nand the two-orbital isospin\nT=1\n2X\n\u001bX\nmm0cy\nm\u001b\u001cmm0cm0\u001b (A4)Note that ~Sis not a physical spin, since it stems from\nmapping the sign of mjto an arti\fcial spin.\nHamiltonian (A2) has the structure of a generalized\nKanamori Hamiltonian, where the spin-\rip and pair-\nhopping parameters JSFandJPHare not restricted to\nbe equal to the Hund's coupling JHas in the ordinary\nKanamori Hamiltonian (4). In terms of Tand ~S, the\ngeneralized Kanamori Hamiltonian reads [27]\nHGK= (U+U0\u0000JH+JSF)N(N\u00001)\n4\n\u0000(U\u0000U0\u0000JH+ 3JSF)N\n4\n+ (JSF+JPH)T2\nx+ (JSF\u0000JPH)T2\ny\n+ (U\u0000U0)T2\nz+ (JSF\u0000JH)~S2\nz:(A5)\nIn order that Hj=3\n2\fts into the structure of the gener-\nalized Hamiltonian, one has to replace the parameters of\nHGKbyU7!U\u0000JH,JH7!0,JSF7!0,JPH7!4\n3JH,\nandU07!U\u00007\n3JH.\nHamiltonian (A5) with the parameters of the usual\nKanamori Hamiltonian, U0=U\u00002JH,JSF=JPH=\nJH, is the symmetric form of the two-band Hamiltonian\ndescribingegbands [27]\nHeg= (U\u0000JH)N(N\u00001)\n2\u0000JHN\n+ 2JH\u0000\nT2\u0000T2\ny\u0001\n:(A6)\nWhileHj=3\n2is the Hamiltonian relevant for the two\nj= 3=2 orbitals of a three orbital system with in\fnite\nSOC,Hegis its counterpart describing the dxzanddxy\norbitals when the tetragonal crystal-\feld splitting is in\f-\nnite. The di\u000berence between these two operators is thus\nresponsible for the qualitative di\u000berent behavior of crys-\ntal \feld and SOC in the N= 2 case (see Sec. IV D). The\noperators (A2) and (A6) are of similar form, but have\ndi\u000berent prefactors.\nA complete set of commuting operators for both\nHamiltonians is N,T2,Ty,~S2, and ~Sz. The full list of\nquantum numbers and the eigenenergies of the two oper-\nators are shown in Table IV for N= 2. For the j= 3=2\norbitals, one sees that due to the prefactors, the ~S= 1\nground state is degenerate with two ~S= 0 states. This\nis related to the fact that spin-\rip and Hund's coupling\nterms vanish in the related generalized Kanamori Hamil-\ntonian so that the relative orientation of pseudo-spins of\ntwo electrons in di\u000berent orbitals has no in\ruence on the\nenergy. The physical reason for this is that all \fve states\nbelong to the J= 2 ground state manifold that is found\nin the picture of jjcoupling and therefore have to be\ndegenerate. As a consequence, charge \ructuations to dif-\nferent values of pseudospin ~Sare still possible for large\nHund's couplings, in contrast to an ordinary Kanamori\nHamiltonian, where JHsplits energy levels of di\u000berent\nspins.13\nAppendix B: E\u000bective spin-orbit coupling\nThe SOC (2) leads to o\u000b-diagonal elements in the non-\ninteracting Hamiltonian in the cubic basis. If both inter-\nactions and SOC are present, the self-energy will have o\u000b-\ndiagonal elements as well, changing the e\u000bective strength\n\u0015e\u000bof the SOC.\nThe structure of the o\u000b-diagonal elements can be un-\nderstood in the case of our degenerate three-orbital model\nsystem using simple analytical considerations. In the j\nbasis, both the local Hamiltonian and the hybridization\nfunction are diagonal, hence \u0006 is diagonal as well, with\ndi\u000berent values for the j= 3=2 and thej= 1=2 orbitals.\nThis diagonal matrix can be split into a term propor-\ntional to the unit matrix and a term proportional to the\nmatrix representation of the lt2g\u0001soperator, which is\ndiagonal in the jbasis with elements \u00000:5 in the case of\nj= 3=2 and 1 in the case of j= 1=2. Therefore,\n\u0006 = \u0006 a1+2\n3\u0006dlt2g\u0001s; (B1)\nwith an average self-energy\n\u0006a=2\n3\u00063\n2+1\n3\u00061\n2(B2)\nand the di\u000berence\n\u0006d= \u0006 1\n2\u0000\u00063\n2: (B3)\nThe e\u000bective SOC can be de\fned as\n\u0015e\u000b=\u0015+2\n3Re\u0006 d(i!n!0): (B4)\nIn the cubic basis, the diagonal elements of the self-\nenergy are given by \u0006 a, the o\u000b-diagonal elements up to\na phase by 2 =3 \u0006d.\nLet us have a look now at the frequency dependence\nof the self-energy. For large frequencies, the values of \u0006 d\nare given by the Hartree-Fock values. Using Eq. (8), the\nHartree-Fock values in the jbasis are\n\u0006HF\n1\n2=*\n@HI\n@n 1\n2;1\n2+\n=\u0012\nU\u00004\n3JH\u0013\nn1\n2(B5)\n+\u0012\n4U\u000026\n3JH\u0013\nn3\n2(B6)\n\u0006HF\n3\n2=*\n@HI\n@n 3\n2;3\n2+\n=\u0012\n2U\u000013\n3JH\u0013\nn1\n2(B7)\n+\u0012\n3U\u000017\n3JH\u0013\nn3\n2; (B8)\nhence\n\u0006d(!!1 ) = \u0006HF\nd= (U\u00003JH)\u0010\nn3\n2\u0000n1\n2\u0011\n:(B9)\nThe e\u000bective SOC for large frequencies is therefore deter-\nmined by an e\u000bective correlation strength U\u00003JHand\n0 1 2 3 4 5\nωn−0.050.000.050.100.150.20Σd(a)ReΣd,JH=0.2U\nImΣd,JH=0.2U\n0 1 2 3 4 5\nωn−0.050.000.050.100.150.20Σd(b)\nReΣd,JH=0.1U\nImΣd,JH=0.1U\n0.00 0 .05 0 .10 0 .15 0 .20 0 .25 0 .30\nJH/U0.00.10.20.30.4ReΣd(c) ΣHF\nd\nΣd(iω0)FIG. 11. Di\u000berence of the self-energies \u0006 d= \u0006 1\n2\u0000\u00063\n2for\nN= 4,\u0015= 0:1, andU= 2. Subplots (a) and (b) show \u0006 das\na function of Matsubara frequencies !nfor Hund's couplings\nJH= 0:2UandJH= 0:1U, respectively. The dashed lines\nare the corresponding Hartree-Fock values. Subplot (c) shows\nRe\u0006 d(i!0)\u0019Re\u0006 d(i!n!0) (full line) and the Hartree-Fock\nvalues \u0006HF\ndequivalent to \u0006 d(i!n!1 ) (dashed) as a function\nofJH. While the Hartree-Fock value strongly decreases with\nJH, \u0006d(i!0) is hardly in\ruenced.\nthe orbital polarization. Since the j= 3=2 orbital is lower\nin energy, its occupation is higher, and \u0006HF\ndis always pos-\nitive as long as the e\u000bective interaction is repulsive. As\na consequence, the correlations usually enhance the SOC\nat large frequencies.\nAt low frequencies and temperatures, assuming a\nmetal, the values of \u0006 are related to electronic occupan-\ncies, too. Namely, j= 1=2 andj= 3=2 problems are\nindependent and the corresponding Fermi surface must,14\n0.0 0 .2 0 .4 0 .6 0 .8 1 .0\nλ0.00.10.20.30.40.50.60.70.8ReΣd(iω0)N= 1\nN= 2\nN= 3\nN= 4\nN= 5\nFIG. 12. Increase of the \frst Matsubara self-energy\n\u0006d(i!0)\u0019\u0006d(!= 0) with the SOC for U= 2,JH= 0:1U,\nand all integer \fllings. For N= 3 and\u0015<0:3, the system is\na Mott insulator, and for N= 4 and\u0015>0:3 a band insulator.\nThe data points are not shown for these parameters.\nby Luttinger theorem, contain the correct number of elec-\ntrons. At the Fermi surface, \u0016+\u000fk\u0000Re\u0006 = 0, which\ncan be used to relate the di\u000berence of \u000fkto the di\u000berence\nof \u0006. Assuming that the electronic density of states is\na constant\u001aindependent of energy (square shaped func-\ntion), the result is \u0006 d(0) = 1=\u001a\u0000\nn3=2\u0000n1=2\u0001\n\u00003=2\u0015.\nIn general, \u0006 d(0) depends on the density of states, the\nSOC, and the orbital polarization, but not explicitly onthe interaction parameters UandJH. Since the Hartree-\nFock value does depend on the interaction parameters,\nthe large frequency and small frequency values of \u0006 dcan\nbe quite di\u000berent, as shown in Fig. 11. In contrast to the\nHartree-Fock value valid at large frequencies, \u0006 d(!= 0)\ncannot be given in a closed form. However, for all metal-\nlic solutions we veri\fed numerically that \u0006 d(i!0) is pos-\nitive, hence the e\u000bective SOC is also increased for low\nfrequencies [41]. The results for U= 2,JH= 0:1Uare\nshown in Fig. 12.\nIn the case of Sr 2RuO 4, the DMFT work of Ref. [22]\nand Ref. [21] found that the real part of \u0006 dwas to a good\napproximation a constant and the imaginary part nearly\nvanishing, which motivated the introduction of \u0015e\u000b. We\nreproduce this result in a DMFT calculation with param-\netersN= 4,U= 2,JH= 0:2U, and\u0015= 0:1, which cor-\nrespond approximately to the values in Sr 2RuO 4. How-\never, if the parameters are changed, for example to a\nHund's coupling of JH= 0:1U, the o\u000b-diagonal elements\nof \u0006 start to show a more pronounced frequency depen-\ndence, as shown in Fig. 11. The reason for this is the\nstrong direct dependence of \u0015e\u000bon the interaction param-\neters in the Hartree-Fock limit, which is not present at\nlow frequencies. In Fig. 11(c), one sees that the Hartree-\nFock value strongly decreases with the Hund's coupling,\nwhereas the static value at != 0 only changes slightly.\nThe stronger frequency dependence of \u0006 dimplies that\nthe accuracy of describing the e\u000bects of correlations on\nthe SOC physics in terms of \u0015e\u000bis in general restricted\nto low energies only.\n\u0003robert.triebl@tugraz.at\n1M. Imada, A. Fujimori, and Y. Tokura, Rev. Mod. Phys.\n70, 1039 (1998).\n2Y. Zhou, K. Kanoda, and T.-K. Ng, Rev. Mod. Phys. 89,\n025003 (2017).\n3Z. Nussinov and J. van den Brink, Rev. Mod. Phys. 87, 1\n(2015).\n4G. Jackeli and G. Khaliullin, Phys. Rev. Lett. 102, 017205\n(2009).\n5J. c. v. Chaloupka, G. Jackeli, and G. Khaliullin, Phys.\nRev. Lett. 105, 027204 (2010).\n6G. Chen, R. Pereira, and L. Balents, Phys. Rev. B 82,\n174440 (2010).\n7G. Chen and L. Balents, Phys. Rev. B 84, 094420 (2011).\n8O. N. Meetei, W. S. Cole, M. Randeria, and N. Trivedi,\nPhys. Rev. B 91, 054412 (2015).\n9G. Khaliullin, Phys. Rev. Lett. 111, 197201 (2013).\n10J. c. v. Chaloupka and G. Khaliullin, Phys. Rev. B 92,\n024413 (2015).\n11K. W. Plumb, J. P. Clancy, L. J. Sandilands, V. V.\nShankar, Y. F. Hu, K. S. Burch, H.-Y. Kee, and Y.-J.\nKim, Phys. Rev. B 90, 041112 (2014).\n12S. M. Winter, Y. Li, H. O. Jeschke, and R. Valent\u0013 \u0010, Phys.\nRev. B 93, 214431 (2016).\n13B. J. Kim, H. Ohsumi, T. Komesu, S. Sakai, T. Morita,\nH. Takagi, and T. Arima, Science 323, 1329 (2009).\n14F. Wang and T. Senthil, Phys. Rev. Lett. 106, 136402(2011).\n15H. Zhang, K. Haule, and D. Vanderbilt, Phys. Rev. Lett.\n111, 246402 (2013).\n16Z. Y. Meng, Y. B. Kim, and H.-Y. Kee, Phys. Rev. Lett.\n113, 177003 (2014).\n17J. c. v. Chaloupka and G. Khaliullin, Phys. Rev. Lett. 116,\n017203 (2016).\n18C. Martins, M. Aichhorn, L. Vaugier, and S. Biermann,\nPhys. Rev. Lett. 107, 266404 (2011).\n19C. Martins, M. Aichhorn, and S. Biermann, Journal of\nPhysics: Condensed Matter 29, 263001 (2017).\n20C. Martins, B. Lenz, L. Perfetti, V. Brouet, F. m. c.\nBertran, and S. Biermann, Phys. Rev. Materials 2, 032001\n(2018).\n21G. Zhang, E. Gorelov, E. Sarvestani, and E. Pavarini,\nPhys. Rev. Lett. 116, 106402 (2016).\n22M. Kim, J. Mravlje, M. Ferrero, O. Parcollet, and\nA. Georges, Phys. Rev. Lett. 120, 126401 (2018).\n23B. Kim, S. Khmelevskyi, I. Mazin, D. Agterberg, and\nC. Franchini, npj Quantum Materials 2, 37 (2017).\n24A. P. Mackenzie, T. Sca\u000edi, C. W. Hicks, and Y. Maeno,\nnpj Quantum Materials 2, 40 (2017).\n25K. Haule and G. Kotliar, New J. Phys. 11, 025021 (2009).\n26L. de Medici, J. Mravlje, and A. Georges, Physical Review\nLetters 107, 256401 (2011).\n27A. Georges, L. d. Medici, and J. Mravlje, Annual Review\nof Condensed Matter Physics 4, 137 (2013).15\n28P. Werner, E. Gull, M. Troyer, and A. J. Millis, Phys.\nRev. Lett. 101, 166405 (2008).\n29Z. P. Yin, K. Haule, and G. Kotliar, Phys. Rev. B 86,\n195141 (2012).\n30C. Aron and G. Kotliar, Phys. Rev. B 91, 041110 (2015).\n31K. M. Stadler, Z. P. Yin, J. von Delft, G. Kotliar, and\nA. Weichselbaum, Phys. Rev. Lett. 115, 136401 (2015).\n32A. Horvat, R. Zitko, and J. Mravlje, Phys. Rev. B 94,\n165140 (2016).\n33H. Ishida and A. Liebsch, Phys. Rev. B 81, 054513 (2010).\n34T. Misawa, K. Nakamura, and M. Imada, Phys. Rev. Lett.\n108, 177007 (2012).\n35L. de' Medici, G. Giovannetti, and M. Capone, Phys. Rev.\nLett. 112, 177001 (2014).\n36L. Fanfarillo and E. Bascones, Phys. Rev. B 92, 075136\n(2015).\n37T. Sato, T. Shirakawa, and S. Yunoki, Phys. Rev. B 91,\n125122 (2015).\n38T. Sato, T. Shirakawa, and S. Yunoki, arXiv preprint\narXiv:1603.01800 (2016).\n39M. Kim, Phys. Rev. B 97, 155141 (2018).\n40G.-Q. Liu, V. N. Antonov, O. Jepsen, and O. K. Ander-\nsen., Phys. Rev. Lett. 101, 026408 (2008).\n41J. B unemann, T. Linneweber, U. L ow, F. B. Anders, and\nF. Gebhard, Phys. Rev. B 94, 035116 (2016).\n42M. Behrmann, C. Piefke, and F. Lechermann, Phys. Rev.\nB86, 045130 (2012).\n43A. Horvat, R. Zitko, and J. Mravlje, Phys. Rev. B 96,\n085122 (2017).\n44A. J. Kim, H. O. Jeschke, P. Werner, and R. Valent\u0013 \u0010, Phys.\nRev. Lett. 118, 086401 (2017).\n45C. Piefke and F. Lechermann, Phys. Rev. B 97, 125154\n(2018).\n46J. I. Facio, J. Mravlje, L. Pourovskii, P. S. Cornaglia, and\nV. Vildosola, Phys. Rev. B 98, 085121 (2018).\n47S. Sugano, Y. Tanabe, and H. Kamimura, Multiplets\nof transition-metal ions in crystals , Pure and Applied\nPhysics, Vol. 33 (Academic Press, New York, 1970).\n48A. Georges, G. Kotliar, W. Krauth, and M. J. Rozenberg,\nRev. Mod. Phys. 68, 13 (1996).\n49W. Metzner and D. Vollhardt, Phys. Rev. Lett. 62, 324\n(1989).\n50P. Werner and A. J. Millis, Phys. Rev. B 74, 155107 (2006).\n51O. Parcollet, M. Ferrero, T. Ayral, H. Hafermann,\nI. Krivenko, L. Messio, and P. Seth, Computer Physics\nCommunications 196, 398 (2015).\n52P. Seth, I. Krivenko, M. Ferrero, and O. Parcollet, Com-\nput. Phys. Commun. 200, 274 (2016).\n53N. Kaushal, J. Herbrych, A. Nocera, G. Alvarez, A. Moreo,\nF. A. Reboredo, and E. Dagotto, Phys. Rev. B 96, 155111(2017).\n54J. Kunes, Journal of Physics: Condensed Matter 27,\n333201 (2015).\n55A. Akbari and G. Khaliullin, Phys. Rev. B 90, 035137\n(2014).\n56B. Kim, P. Liu, Z. Erg onenc, A. Toschi, S. Khmelevskyi,\nand C. Franchini, Phys. Rev. B 94, 241113 (2016).\n57L. Du, L. Huang, and X. Dai, The European Physical\nJournal B 86, 94 (2013).\n58P. Werner and A. J. Millis, Physical Review Letters 99,\n126405 (2007).\n59P. Werner, E. Gull, and A. J. Millis, Physical Review B\n79, 115119 (2009).\n60T. Kita, T. Ohashi, and N. Kawakami, Physical Review\nB84, 195130 (2011).\n61T. Kita, T. Ohashi, and N. Kawakami, Journal of the\nPhysical Society of Japan 80, SA142 (2011).\n62M. Jarrell and J. Gubernatis, Physics Reports 269, 133\n(1996).\n63W. von der Linden, R. Preuss, and V. Dose, \\The Prior-\nPredictive Value: A Paradigm of Nasty Multi-Dimensional\nIntegrals,\" in Maximum Entropy and Bayesian Methods ,\nedited by W. von der Linden, V. Dose, R. Fischer, and\nR. Preuss (Kluwer Academic Publishers, Dortrecht, 1999)\npp. 319{326.\n64J. Skilling, \\Fundamentals of MaxEnt in data analysis,\" in\nMaximum Entropy in Action , edited by B. Buck and V. A.\nMacaulay (Clarendon Press, Oxford, 1991) pp. 19{40.\n65M. Montalti, A. Credi, L. Prodi, and M. T. Gandol\f,\nHandbook of Photochemistry , 3rd ed. (CRC press, 2006).\n66K.-W. Lee and W. E. Pickett, EPL (Europhysics Letters)\n80, 37008 (2007).\n67G. Giovannetti, arXiv preprint arXiv:1611.06482 (2016).\n68Y. G. Shi, Y. F. Guo, S. Yu, M. Arai, A. A. Belik, A. Sato,\nK. Yamaura, E. Takayama-Muromachi, H. F. Tian, H. X.\nYang, J. Q. Li, T. Varga, J. F. Mitchell, and S. Okamoto,\nPhys. Rev. B 80, 161104 (2009).\n69S.-I. Ikeda, N. Shirakawa, H. Bando, and Y. Ootuka, Jour-\nnal of the Physical Society of Japan 69, 3162 (2000).\n70I. Nagai, N. Shirakawa, S.-i. Ikeda, R. Iwasaki,\nH. Nishimura, and M. Kosaka, Applied Physics Letters\n87, 024105 (2005).\n71H. Wadati, J. Mravlje, K. Yoshimatsu, H. Kumigashira,\nM. Oshima, T. Sugiyama, E. Ikenaga, A. Fujimori,\nA. Georges, A. Radetinac, K. S. Takahashi, M. Kawasaki,\nand Y. Tokura, Phys. Rev. B 90, 205131 (2014).\n72L. Du, X. Sheng, H. Weng, and X. Dai, EPL (Europhysics\nLetters) 101, 27003 (2013)."    },    {        "title": "1003.1618v1.Spin_orbit_coupling_in_a_graphene_bilayer_and_in_graphite.pdf",        "content": "Spin-orbit coupling in a graphene bilayer and in graphite\nF. Guinea1\n1Instituto de Ciencia de Materiales de Madrid, CSIC,\nSor Juana In\u0013 es de la Cruz 3, E28049 Madrid, Spain\nThe intrinsic spin-orbit interactions in bilayer graphene and in graphite are studied, using a tight\nbinding model, and an intraatomic ~L~Scoupling. The spin-orbit interactions in bilayer graphene and\ngraphite are larger, by about one order of magnitude, than the interactions in single layer graphene,\ndue to the mixing of \u0019and\u001bbands by interlayer hopping. Their value is in the range 0 :1\u00001K. The\nspin-orbit coupling opens a gap in bilayer graphene, and it also gives rise to two edge modes. The\nspin-orbit couplings are largest, \u00181\u00004K, in orthorhombic graphite, which does not have a center\nof inversion.\nINTRODUCTION\nThe isolation and control of the number of carriers\nin single and few layer graphene \rakes[1, 2] has lead to\na large research activity exploring all aspects of these\nmaterials[3]. Among others, the application of graphene\nto spintronic devices[4{10] and to spin qubits[11{13] is\nbeing intensively studied. The understanding of these\ndevices requires a knowledge of the electronic spin-orbit\ninteraction. In principle, this interaction turns single\nlayer graphene into a topological insulator[14], which\nshows a bulk gap and edge states at all boundaries.\nThe magnitude of the spin-orbit coupling in single layer\ngraphene has been studied[15{18]. The calculated cou-\nplings are small, typically below 0.1K. The observed spin\nrelaxation[8, 19] suggests the existence of stronger mech-\nanisms which lead to the precession of the electron spins,\nlike impurities or lattice deformations[20{22].\nBilayer graphene is interesting because, among other\nproperties, a gap can be induced by electrostatic means,\nleading to new ways for the con\fnement of electrons[23].\nThe spin-orbit interactions which exist in single layer\ngraphene modulate the gap of a graphene bilayer[24].\nThe unit cell of bilayer graphene contains four carbon\natoms, and there are more possible spin-orbit couplings\nthan in single layer graphene.\nWe analyze in the following the intrinsic and extrinsic\nspin-orbit couplings in bilayer graphene, using a tight\nbinding model, and describing the relativistic e\u000bects\nresponsible for the spin-orbit interaction by a ~L~Sin-\ntraatomic coupling. We use the similarities between the\nelectronic bands of a graphene bilayer and the bands of\nthree dimensional graphite with Bernal stacking to gen-\neralize the results to the latter.\nTHE MODEL\n. We describe the electronic bands of a graphene bi-\nlayer using a tight binding model, with four orbitals, the\n2sand the three 2 porbitals, per carbon atom. We con-\nsider hoppings between nearest neighbors in the sameplane, and nearest neighbors and next nearest neighbors\nbetween adjacent layers, see[25]. The couplings between\neach pair of atoms is parametrized by four hoppings,\nVss;Vsp;Vpp\u0019andVpp\u001b. The model includes also two in-\ntraatomic levels, \u000fsand\u000fp, and the intraatomic spin-orbit\ncoupling\nHso\u0011\u0001soX\ni~Li~Si (1)\nThe parameters used to describe the \u0019bands of\ngraphite[26, 27], \r0;\r1;\r2;\r3;\r4;\r5and \u0001, can be de-\nrived from this set of parameters. We neglect the di\u000ber-\nence between di\u000berent hoppings between atoms which are\nnext nearest neighbors in adjacent layers, which are re-\nsponsible for the di\u000berence between the parameters \r3\nand\r4. We also set the di\u000berence in onsite energies\nbetween the two inequivalent atoms, \u0001 to zero. The\nparameters \r2and\r5are related to hoppings between\nnext nearest neighbor layers, and they do not play a role\nin the description of the bilayer. The total number of\nparameters is 15, although, without loss of generality,\nwe set\u000fp= 0. We do not consider hoppings and spin\norbit interactions which include dlevels, although they\ncan contribute to the total magnitude of the spin-orbit\ncouplings[18, 28]. The e\u000bects mediated by dorbitals do\nnot change the order of magnitude of the couplings in\nsingle layer graphene, and their contribution to interlayer\ne\u000bects should be small.\nThe main contribution to the e\u000bective spin-orbit at\nthe Fermi level due to the interlayer coupling is due to\nthe hoppings between porbitals in next nearest neigh-\nbor atoms in di\u000berent layers. This interaction gives rise\nto the parameters \r3and\r4in the parametrization of\nthe bands in graphite. For simplicity, we will neglect\ncouplings between sandporbitals in neighboring layers.\nThe non zero hoppings used in this work are listed in\nTable I.\nThe hamiltonian can be written as a 32 \u000232 matrix\nfor each lattice wavevector. We de\fne an e\u000bective hamil-\ntonian acting on the \u0019, orpz, orbitals, by projecting out\nthe rest of the orbitals:\nHeff\n\u0019\u0011H\u0019+H\u0019\u001b(!\u0000H\u001b\u001b)\u00001H\u001b\u0019 (2)arXiv:1003.1618v1  [cond-mat.mes-hall]  8 Mar 20102\n\u000fs-7.3\nt0\nss2.66\nt0\nsp4.98\nt0\npp\u001b2.66\nt0\npp\u0019-6.38\nt1\npp\u0019 0.4\nt2\npp\u001b 0.4\nt2\npp\u0019-0.4\n\u0001so0.02\nTABLE I: Non zero tight binding parameters, in eV, used in\nthe model. The hoppings are taken from[29, 30], and the spin-\norbit coupling from[31]. Superindices 0,1, and 2 correspond to\natoms in the same layer, nearest neighbors in di\u000berent layers,\nand next nearest neighbors in di\u000berent layers.\nA1,B2\nB1A2\nFIG. 1: (Color online). Unit cell of a graphene bilayer. La-\nbels A and B de\fne the two sublattices in each layer, while\nsubscripts 1 and 2 de\fne the layers.\nWe isolate the e\u000bect of the spin-orbit coupling by de\fn-\ning:\nHso\n\u0019\u0010\n~k\u0011\n\u0011Heff\n\u0019(\u0001so)\u0000Heff\n\u0019(\u0001so= 0) (3)\nNote thatHso\n\u0019depends on the energy, !.\nWe analyzeHso\n\u0019at theKandK0points. The two\nmatrices have a total of 16 entries, which can be labeled\nby specifying the sublattice, layer, spin, and valley. We\nde\fne operators which modify each of these degrees of\nfreedom using the Pauli matrices ^ \u001b;^\u0016;^s, and ^\u001c. The unit\ncell is described in Fig. 1.\nThe hamiltonian has inversion and time reversal sym-\nmetry, and it is also invariant under rotations by 120\u000e.\n-0.4-0.20.20.4EHeVL0.01400.01440.0148Èl1ÈHmeVL\n-0.4-0.20.20.4EHeVL0.0070.0080.009Èl2ÈHmeVL\n-0.4-0.20.00.20.4EHeVL0.00500.00550.00600.0065Èl3ÈHmeVL\n-0.4-0.20.20.4EHeVL0.480.500.52Èl4ÈHmeVLFIG. 2: (Color online). Dependence on energy of the spin-\norbit couplings, as de\fned in eq. 5.\nThese symmetries are de\fned by the operators:\nI\u0011\u001bx\u0016x\u001cx\nT \u0011isy\u001cxK\nC120\u000e\u0011 \n\u00001\n2+ip\n3\n2sz!\n\u0002 \n\u00001\n2\u0000ip\n3\n2\u001cz\u0016z!\n\u0002\n\u0002 \n\u00001\n2+ip\n3\n2\u001cz\u001bz!\n(4)\nwhereKis complex conjugation.\nThe possible spin dependent terms which respect these\nsymmetries were listed in[32], in connection with the\nequivalent problem of three dimensional Bernal graphite\n(see below). In the notation described above, they can\nbe written as\nHso\n\u0019=\u00151\u001bz\u001czsz+\u00152\u0016z\u001czsz+\u00153\u0016z(\u001bysx\u0000\u001cz\u001bxsy) +\n+\u00154\u001bz(\u0016ysx+\u001cz\u0016xsy) (5)\nThe \frst term describes the intrinsic spin-orbit coupling\nin single layer graphene. The other three, which involve\nthe matrices \u0016i, are speci\fc to bilayer graphene. The\nterm proportional to \u00153can be viewed as a Rashba cou-\npling with opposite signs in the two layers.\nRESULTS\n.\nBilayer graphene\n.\nThe energy dependence of the four couplings in eq. 5\nis shown in Fig. 2. The values of the couplings scale\nlinearly with \u0001 so. This dependence can be understood\nby treating the next nearest neighbor interlayer coupling\nand the intratomic spin-orbit coupling as a perturbation.3\n-0.10.1EgHeVL0.01400.0144Èl1ÈHmeVL\n-0.10.1EgHeVL0.00760.0080Èl2ÈHmeVL\n-0.10.1EgHeVL0.005450.005460.00547Èl3ÈHmeVL\n-0.10.1EgHeVL0.480.490.50Èl4ÈHmeVL\nFIG. 3: (Color online). Dependence on interlayer gap, Eg, of\nthe spin-orbit couplings, as de\fned in eq. 5.\nThe spin-orbit coupling splits the spin up and spin down\nstates of the \u001bbands in the two layers. The interlayer\ncouplings couple the \u0019band in one layer to the \u001bband in\nthe other layer. Their value is of order \r3. The\u0019states\nare shifted by:\n\u000e\u000f\u0019\u0006\u0018\u0000\r2\n3\nj\u000f\u001b\u0006j/\u0007\u0001so\u0012\r3\n\u000f0\u001b\u00132\n(6)\nwhere\u000f0\n\u001bis an average value of a level in the \u001bband.\nThe model gives for the only intrinsic spin-orbit cou-\npling in single layer graphene the value\n\f\f\u0015SLG\n1\f\f= 0:0065meV (7)\nThis coupling depends quadratically on \u0001 so,\u000e\u000f\u0019\u0006\u0018\n\u0006\u00012\nso=\u000f0\n\u001b[15].\nThe band dispersion of bilayer graphene at low ener-\ngies, in the absence of spin-orbit couplings is given by\nfour Dirac cones, because of trigonal warping e\u000bects as-\nsociated with \r3[23]. Hence, we must to consider the\ncouplings for wavevectors ~kslightly away from the K\nandK0points. We have checked that the dependence of\nthe couplings \u0015ion momentum, in the range where trig-\nonal warping is relevant, is comparable to the changes\nwith energy shown in Fig. 2.\nA gap,Eg, between the two layers breaks inversion\nsymmetry, and can lead to new couplings. The calcula-\ntions show no new coupling greater than 10\u00006meV for\ngaps in the range \u00000:1eV\u0014Eg\u00140:1eV. The depen-\ndence of the couplings on the value of the gap is shown\nin Fig. 3. This calculation considers only the e\u000bect in the\nshift of the electrostatic potential between the two layers.\nThe existence also of an electric \feld will mix the pzands\norbitals within each atom, leading to a Rashba term sim-\nilar to the one induced in single layer graphene[15, 16].\nThe e\u000bect of \u00151is to open a gap of opposite sign in\nthe two valleys, for each value of sz. The system will be-\ncome a topological insulator[14, 33]. The number of edge\nstates is two, that is, even. The spin Hall conductivity\n02p\n34p\n32pkzc0.010.020.03Èl1ÈHmeVL\n02p\n34p\n32pkzc0.010.020.03Èl2ÈHmeVL\n02p\n34p\n32pkzc0.010.02Èl3ÈHmeVL\n02p\n34p\n32pkzc0.40.8Èl4ÈHmeVLFIG. 4: (Color online). Dependence on momentum perpen-\ndicular to the layers in Bernal graphite of the spin-orbit cou-\nplings, as de\fned in eq. 5.\nis equal to two quantum units of conductance. A per-\nturbation which preserves time reversal invariance can\nhybridize the edge modes and open a gap. Such pertur-\nbation should be of the form \u001cxsy.\nThe terms with \u00153and\u00154describe spin \rip hoppings\nwhich involve a site coupled to the other layer by the\nparameter \r1. The amplitude of the wavefunctions at\nthese sites is suppressed at low energies[23]. The shifts\ninduced by \u00153and\u00154in the low energy electronic levels\nwill be of order \u00152\n3=\r1;\u00152\n4=\r1.\nBulk graphite\n. The hamiltonian of bulk graphite with Bernal stack-\ning can be reduced to a set of bilayer hamiltonians with\ninterlayer hoppings which depend on the momentum\nalong the direction perpendicular to the layers, kz. We\nneglect in the following the (small) hoppings which de-\nscribe hoppings between next nearest neighbor layers, \r2\nand\r5, and the energy shift \u0001 between atoms in di\u000berent\nsublattices. At the KandK0points of the three dimen-\nsional Brillouin Zone (2 kzc= 0, where cis the interlayer\ndistance) the hamiltonian is that of a single bilayer where\nthe value of all interlayer hoppings is doubled. At the H\nandH0points, where 2 kzc=\u0019, the hamiltonian reduces\nto two decoupled layers, and in the intermediate cases the\ninterlayer couplings are multiplied by j2 cos(kzc)j. Carry-\ning out the calculations described in the previous section,\nkzdependent e\u000bective couplings, \u0015i(kz), can be de\fned.\nThese couplings are shown in Fig. 4. The results for\nbilayer graphene correspond to kzc= 2\u0019=3;4\u0019=3. The\nlayers are decoupled for kzc=\u0019. In this case, the only\ncoupling is\u00151, which gives the coupling for a single layer,\ngiven in eq. 7.\nThe signi\fcant dispersion as function of momentum\nparallel to the layers shown in Fig. 4 implies the exis-\ntence of spin dependent hoppings between layers in di\u000ber-\nent unit cells. This is consistent with the analysis which4\np4p\n35p\n3kya30.060.080.100.12Èl1ÈHmeVL\np4p\n35p\n3kya30.200.25Èl2ÈHmeVL\nFIG. 5: (Color online). Dependence on wavevector, 2 ky, of\nthe spin-orbit couplings for orthorhombic graphite, as de\fned\nin eq. 9. The point kx= 0;kyap\n3 = 4\u0019=3 corresponds to the\nKpoint (ais the distance between carbon atoms in the plane).\nshowed that the spin-orbit coupling in a bilayer has a\ncontribution from interlayer hopping, see eq. 6.\nThe spin-orbit couplings can be larger in bulk graphite\nthan in a graphene bilayer. The bands in Bernal graphite\ndo not have electron-hole symmetry. The shift in the\nFermi energy with respect to the Dirac energy is about\nEF\u001920meV\u001d\u00151;\u00153[34]. Hence, the spin-orbit cou-\npling is not strong enough to open a gap throughout the\nentire Fermi surface, and graphite will not become an\ninsulator.\nA similar analysis applies to orthorhombic graphite,\nwhich is characterized by the stacking sequence\nABCABC\u0001\u0001\u0001[35]. The electronic structure of this al-\nlotrope at low energies di\u000bers markedly from Bernal\ngraphite[36, 37], and it can be a model for stacking\ndefects[36{38]. If hoppings beyond nearest neighbor lay-\ners are neglected, the hamiltonian can be reduced to an\ne\u000bective one layer hamiltonian where all sites are equiv-\nalent. The e\u000bective hamiltonian which describes the K\nandK0valleys contains eight entries, which can be de-\nscribed using the matrices \u001bi;si, and\u001ci. Orthorhombic\ngraphene is not invariant under inversion, and a Rashba\nlike spin-orbit coupling is allowed. The spin-orbit cou-\npling takes the form:\nHso\northo\u0011\u0015ortho\n1\u001bzsz\u001cz+\u0015ortho\n2 (\u001bysx\u0000\u001cz\u001bxsy) (8)\nAs in the case of Bernal stacking, the couplings have a\nsigni\fcant dependence on the momentum perpendicular\nto the layers, kz, and interlayer hopping terms are in-\nduced. For != 0;~k= 0 andkz= 0, we \fnd:\n\u0015ortho\n1 = 0:134meV\n\u0015ortho\n2 = 0:275meV (9)\nIn orthorhombic graphite the Fermi level is away from\ntheKandK0points, in the vicinity of a circle de\fned\nbyj~kj=\r1=vF[36, 37]. The variation of the couplings as\nfunction of wavevector is shown in Fig. 5.\nCONCLUSIONS\nWe have studied the intrinsic spin-orbit interactions in\na graphene bilayer and in graphite. We assume that theorigin of the couplings is the intraatomic ~L~Sinteraction,\nand we use a tight binding model which includes the 2 s\nand 2patomic orbitals.\nThe intrinsic spin-orbit couplings in a graphene bilayer\nand in graphite are about one order of magnitude larger\nthan in single layer graphene, due to mixing between\nthe\u0019and\u001bbands by interlayer hoppings. Still, these\ncouplings are typically of order 0 :01\u00000:1meV, that is,\n0:1\u00001K.\nBilayer graphene becomes an insulator with an even\nnumber of edge states. These states can be mixed by per-\nturbations which do not break time reversal symmetry.\nThese perturbations can only arise from local impurities\nwith strong spin-orbit coupling, as a spin \rip process and\nintervalley scattering are required.\nThe interplay of spin-orbit coupling and interlayer hop-\nping leads to spin dependent hopping terms. The spin-\norbit interactions are largest in orthorhombic graphite,\nwhich does not have inversion symmetry.\nACKNOWLEDGEMENTS\nFunding from MICINN (Spain), through grants\nFIS2008-00124 and CONSOLIDER CSD2007-00010 is\ngratefully acknowledged.\n[1] K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang,\nY. Zhang, S. V. Dubonos, I. V. Grigorieva, , and A. A.\nFirsov, Science 306, 666 (2004).\n[2] K. S. Novoselov, D. Jiang, F. Schedin, T. J. Booth, V. V.\nKhotkevich, S. V. Morozov, and A. K. Geim, Proc. Natl.\nAcad. Sci. U.S.A. 102, 10451 (2005).\n[3] A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S.\nNovoselov, and A. K. Geim, Rev. Mod. Phys. 81, 109\n(2009).\n[4] E. W. Hill, A. K. Geim, K. S. Novoselov, F. Schedin, and\nP. Blake, IEEE Trans. Magn. 42, 2694 (2006).\n[5] S. J. Cho, Y.-F. Chen, and M. S. Fuhrer, Appl. Phys.\nLett. 91, 123105 (2007).\n[6] M. Nishioka and A. M. Goldman, Appl. Phys. Lett. 90,\n252505 (2007).\n[7] N. Tombros, C. J\u0013 ozsa, M. Popinciuc, H. T. Jonkman,\nand B. J. van Wees, Nature 448, 571 (2007).\n[8] N. Tombros, S. Tanabe, A. Veligura, C. J\u0013 ozsa, M. Popin-\nciuc, H. T. Jonkman, and B. J. van Wees, Phys. Rev.\nLett. 101, 046601 (2008).\n[9] W. H. Wang, K. Pi, Y. Li, Y. F. Chiang, P. Wei, J. Shi,\nand R. K. Kawakami, Phys. Rev. B 77, 020402 (2008).\n[10] M. Popinciuc, C. J. P. J. Zomer, N. Tombros, A. Veligura,\nH. T. Jonkman, and B. J. van Wees, Phys. Rev. B 80,\n214427 (2009).\n[11] B. Trauzettel, D. V. Bulaev, D. Loss, and G. Burkard,\nNature Phys. 3, 192 (2007).\n[12] J. Fischer, B. Trauzettel, and D. Loss, Phys. Rev. B 80,\n155401 (2009).5\n[13] W. L. Wang, O. V. Yazyev, S. Meng, and E. Kaxiras,\nPhys. Rev. Lett. 102, 157201 (2009).\n[14] C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 226801\n(2005).\n[15] D. Huertas-Hernando, F. Guinea, and A. Brataas, Phys.\nRev. B 74, 155426 (2006).\n[16] H.Min, J. E. Hill, N. Sinitsyn, B. Sahu, L. Kleinman, and\nA. MacDonald, Phys. Rev. B 74, 165310 (2006).\n[17] Y. Yao, F. Ye, X.-L. Qi, S.-C. Zhang, and Z. Fang, Phys.\nRev. B 75, 041401 (2007).\n[18] M. Gmitra, S. Konschuh, C. Ertler, C. Ambrosch-Draxl,\nand J. Fabian, Phys. Rev. B 80, 235431 (2009).\n[19] C. J\u0013 ozsa, T. Maassen, M. Popinciuc, P. J. Zomer,\nA. Veligura, H. T. Jonkman, and B. J. van Wees, Phys.\nRev. B 80, 241403 (2009).\n[20] A. H. Castro Neto and F. Guinea, Phys. Rev. Lett. 103,\n026804 (2009).\n[21] D. Huertas-Hernando, F. Guinea, and A. Brataas, Phys.\nRev.Lett. 103, 146801 (2009).\n[22] C. Ertler, S. Konschuh, M. Gmitra, and J. Fabian, Phys.\nRev. B 80, 041405 (2009).\n[23] E. McCann and V. I. Fal'ko, Phys. Rev. Lett. 96, 086805\n(2006).\n[24] R. van Gelderen and C. M. Smith (2009),\narXiv:0911.0857.\n[25] L. Chico, M. P. L\u0013 opez-Sancho, and M. C. Mu~ noz, Phys.Rev. B 79, 235423 (2009).\n[26] J. W. McClure, Phys. Rev. 108, 612 (1957).\n[27] J. C. Slonczewski and P. R. Weiss, Phys. Rev. 109, 272\n(1958).\n[28] J. W. McClure and Y. Yafet, in 5th Conference on Car-\nbon(Pergamon, University Park, Maryland, 1962).\n[29] D. Tom\u0013 anek and S. G. Louie, Phys. Rev. B 37, 8327\n(1987).\n[30] D. Tom\u0013 anek and M. A. Schluter, Phys. Rev. Lett. 67,\n2331 (1991).\n[31] J. Serrano, M. Cardona, and J. Ruf, Solid St. Commun.\n113, 411 (2000).\n[32] G. Dresselhaus and M. S. Dresselhaus, Phys. Rev. 140,\nA401 (1965).\n[33] F. D. M. Haldane, Phys. Rev. Lett. 61, 2015 (1988).\n[34] M. S. Dresselhaus and J. Mavroides, IBM Journ. of Res.\nand Development 8, 262 (1964).\n[35] J. W. McClure, Carbon 7, 425 (1969).\n[36] F. Guinea, A. H. C. Neto, and N. M. R. Peres, Phys.\nRev. B 73, 245426 (2006).\n[37] D. P. Arovas and F. Guinea, Phys. Rev. B 78, 245716\n(2008).\n[38] N. B. Brandt, S. M. Chudinov, and Y. G. Ponomarev, in\nSemimetals I: Graphite and Its Compounds (North Hol-\nland, Amsterdam, 1988), vol. 20.1."    },    {        "title": "2210.01700v2.Spin_orbit_enhancement_in_Si_SiGe_heterostructures_with_oscillating_Ge_concentration.pdf",        "content": "Spin-orbit enhancement in Si/SiGe heterostructures with oscillating Ge concentration\nBenjamin D. Woods,1M. A. Eriksson,1Robert Joynt,1and Mark Friesen1\n1Department of Physics, University of Wisconsin-Madison, Madison, Wisconsin 53706, USA\nWe show that Ge concentration oscillations within the quantum well region of a Si/SiGe het-\nerostructure can significantly enhance the spin-orbit coupling of the low-energy conduction-band\nvalleys. Specifically, we find that for Ge oscillation wavelengths near \u0015= 1:57nm with an aver-\nage Ge concentration of \u0016nGe= 5%in the quantum well region, a Dresselhaus spin-orbit coupling\nis induced, at all physically relevant electric field strengths, which is over an order of magnitude\nlarger than what is found in conventional Si/SiGe heterostructures without Ge concentration oscil-\nlations. This enhancement is caused by the Ge concentration oscillations producing wave-function\nsatellite peaks a distance 2\u0019=\u0015away in momentum space from each valley, which then couple to\nthe opposite valley through Dresselhaus spin-orbit coupling. Our results indicate that the enhanced\nspin-orbit coupling can enable fast spin manipulation within Si quantum dots using electric dipole\nspin resonance in the absence of micromagnets. Indeed, our calculations yield a Rabi frequency\n\nRabi=B> 500MHz/T near the optimal Ge oscillation wavelength \u0015= 1:57nm.\nI. INTRODUCTION\nFollowing the seminal work of Loss and DiVincenzo\n[1], quantum dots in semiconductors have emerged as a\nleading candidate platform for quantum computation [2–\n5]. Gate-defined quantum dots in silicon [6, 7] are par-\nticularly attractive due to their compatibility with the\nmicroelectronics fabrication industry. Moreover, in con-\ntrast with GaAs, for which coherence times are limited\nby unavoidable hyperfine interactions with nuclear spins\n[8], isotropic enrichment dramatically suppresses these\ninteractions in Si, enabling long coherence times [9].\nWhile recent progress in Si quantum dots has been\nquite promising, many of the leading qubit architectures\nrely on synthetic spin-orbit coupling arising from micro-\nmagnets [10–13], leading to challenges for scaling up to\nsystems with many dots. An alternative approach is to\nuse intrinsic spin-orbit coupling for qubit manipulation,\nfor example, through the electric dipole spin resonance\n(EDSR) mechanism [14, 15]. While this possibility has\nbeen considered for Ge and Si hole-spin qubits, where the\ndegeneracyofthe p-orbital-dominatedvalencebandleads\nto strong spin-orbit coupling [16, 17], the weak spin-orbit\ncoupling of the Si conduction band appears unfavorable\nfor electron-spin qubits.\nIn this work, we show how the spin-orbit coupling in\nSi/SiGe quantum well heterostructures can be enhanced\nby more than an order of magnitude by incorporating\nGe concentration oscillations insidethe quantum well,\nleading to the possibility of exploiting intrinsic spin-orbit\ncouplinginSiquantumdotsforfastgateoperations. Fig-\nure1(a)showsaschematicofthesystemwhichconsistsof\na Si-dominated quantum well region sandwiched between\nSi0:7Ge0:3barrier regions, where the growth direction is\ntaken along the [001]crystallographic axis. In contrast to\n“conventional” Si/SiGe quantum wells, the quantum well\nregion contains a small amount of Ge with concentration\noscillations of wavelength \u0015, as shown in Fig. 1(b). For\ncomparison, Fig. 1(c) shows the Ge concentration pro-\nfile of a conventional Si/SiGe quantum well. Previous\n(a)\nSi0.7Ge0.3Si0.7Ge0.3\nSi well \n with Ge oscillations\n01530\nnGe (%)growth direction, z(b)\n01530\nnGe (%)growth direction, z(c)FIG. 1. (a) Schematic of the Si/SiGe heterostructure consid-\nered here, consisting of a Si-dominated quantum well sand-\nwiched between Si 0.7Ge0.3barrier regions. Note that the\ngrowth direction is along the [001]crystallographic axis. (b)\nGe concentration profile along the growth ( z) direction of a\nwiggle well. Ge concentration oscillations of wavelength \u0015in-\nside of the quantum well region lead to spin-orbit coupling\nenhancement for a proper choice of \u0015. (c) Ge concentration\nprofile of a “conventional” Si/SiGe quantum well, for compar-\nison.\nworks [18, 19] have studied such a structure, which has\nbeen named the wiggle well, and found that the periodic\nGe concentration leads to an enhancement of the valley\nsplitting. Here, we develop a theory of spin-orbit cou-\npling within such structures and show that the periodic\nnature of the device, along with the underlying diamond\ncrystal structure and degeneracy of the Si zvalleys, also\ngives rise to an enhancement in spin-orbit coupling. Im-\nportantly, we find that the wavelength \u0015must satisfy a\nresonance condition to give rise to this spin-orbit cou-\npling enhancement. As discussed in detail in Sec. IV,\nthis involves a two-step process that can be summarized\nas follows. First, the periodic potential produced by\nthe Ge concentration oscillations produces wave-function\nsatellites a distance 2\u0019=\u0015away in momentum space from\neach valley. Then, a satellite of a given valley couples\nstrongly to the opposite valley through Dresselhaus spin-arXiv:2210.01700v2  [cond-mat.mes-hall]  16 Jan 20232\norbit coupling, provided that the satellite-valley separa-\ntion distance in momentum space is 4\u0019=a, corresponding\nto the condition \u0015= 1:57nm.\nFrom the outset, it is important to remark that the\nspin-orbit coupling introduced by the Ge concentration\noscillations is fundamentally distinct from the spin-orbit\ncoupling of conventional Si/SiGe quantum wells. For a\ngiven subband of a conventional Si/SiGe quantum well\nimmersed in a vertical electric field, the C2vpoint group\nsymmetry of the system allows for both Rashba and\n“Dresselhaus-type” linear- kkterms of the form [20, 21],\nHSO=\u000b(ky\u0016\u001bx\u0000kx\u0016\u001by) +\f(kx\u0016\u001bx\u0000ky\u0016\u001by);(1)\nwhere \u0016\u001bjare the Pauli matrices acting in (pseudo)spin\nspace and\u000band\fare the Rashba and Dresselhaus coef-\nficients, respectively, of the subband. The presence of\nRasbha spin-orbit coupling is unsurprising due to the\nstructural asymmetry provided by the electric field [22],\nwhile the presence of the Dresselhaus-type term is ini-\ntially surprising since the diamond lattice of Si/SiGe\nquantum wells possesses bulk inversion symmetry [23].\nHowever, these systems still support a Dresselhaus-type\nterm of the same form \f(kx\u0016\u001bx\u0000ky\u0016\u001by), due to the broken\ninversion symmetry caused by the quantum well inter-\nfaces [20, 21, 24, 25]. This is in stark contrast to the\ntrue Dresselhaus spin-orbit coupling in III-V semicon-\nductors, where the asymmetry of the anion and cation\nin the unit cell leads to bulk inversion asymmetry [23].\nImportantly, we find in Sec. IIIB that the spin-orbit cou-\npling of the wiggle well does not rely upon the presence\nof an interface. Rather, it is an intrinsic property of a\nbulksystem with Ge concentration oscillations. In this\nsense, the spin-orbit coupling investigated here is more\nakin to the true Dresselhaus spin-orbit coupling of III-\nV semiconductors than the Dresselhaus-type spin-orbit\ncoupling of conventional Si/SiGe quantum wells brought\nabout by interfaces. Indeed, the only requirement for\nlinear- kkDresselhaus spin-orbit coupling in a wiggle well\nwith an appropriate \u0015is confinement in the growth di-\nrection (even symmetric confinement), to allow for the\nformation of subbands. For simplicity in the remainder\nof this work, we simply refer to this form of spin-orbit\ncoupling as Dresselhaus.\nThe rest of this paper is organized as follows. In Sec. II\nwe describe our model used to study the quantum well\nheterostructure. Section III then presents our numerical\nresults for the spin-orbit coefficients. This also includes\nthe calculation of the EDSR Rabi frequency and studies\nthe impact of alloy disorder on the spin-orbit coefficients.\nIn Sec. IV we provide an extensive explanation of the\nmechanism behind the spin-orbit coupling enhancement.\nFinally, we conclude in Sec. V.\nII. MODEL\nIn this section, we outline the model used to study\nour Si/SiGe heterostructure along with the methods usedto calculate the spin-orbit coefficients. In Sec. IIA, we\ndescribe the tight binding model used to model generic\nSiGe alloy systems. Next, in Sec. IIB we employ a vir-\ntual crystal approximation to impart translation invari-\nance in the plane of the quantum well, allowing us to\nreduce the problem to an effective one-dimensional (1D)\nHamiltonian parametrized by in-plane momentum kk. In\nSec.IIC,weexpandthemodelaround kk= 0toseparate\nout the Hamiltonian components that give rise to Rashba\nand Dresselhaus spin-orbit coupling, respectively, and we\nexplain the important differences between the two com-\nponents. Finally, in Sec. IID, we transform the Hamil-\ntonian into the subband basis, which allows us to obtain\nexpressions for the Rashba and Dresselhaus spin-orbit\ncoefficients in each subband.\nA. Model of SiGe alloys\nTo study the spin-orbit physics of our system we use\nthe empirical tight-binding method [26], where the elec-\ntronic wave function is written as a linear combination\nof atomic orbitals:\nj i=X\nn;j;\u0017;\u001bjnj\u0017\u001bi nj\u0017\u001b: (2)\nHere,hrjnj\u0017\u001bi=\u001e\u0017(r\u0000Rn;j)j\u001biis an atomic orbital\ncentered at position Rn;j, corresponding to atom jof\natomic layer nalong the growth direction [001],\u0017is a\nspatial orbital index, and j\u001biis a two component spinor\nwith\u001b=\";#indicating the spin of the orbital. We\nuse an sp3d5s*basis set with 20 orbitals per atom, on-\nsite spin-orbit coupling, nearest-neighbor hopping, and\nstrain. Note that nearest-neighbor sp3d5s*tight-binding\nmodels are well established for accurately describing the\nelectronic structure of semiconductor materials over a\nwide energy range [27]. Explicitly, \u0017is a spatial orbital\nindex from the set including s,s\u0003,pi(i=x;y;z), and\ndi(i=xy;yz;zx;z2;x2\u0000y2) orbitals, which are meant\nto model the outer-shell orbitals of individual Si and\nGe atoms that participate in chemical bonding. Addi-\ntionally, these orbitals possess certain spatial symmetries\nthat, combined with the diamond crystal structure of the\nSiGe alloy, dictate the forms of the nearest neighbor cou-\nplings, as first explained in the work of Slater and Koster\n[26]. The free parameters of the tight-binding model (in-\ncluding onsite orbital energies, nearest-neighbor hopping\nenergies, strain parameters, etc.) are then chosen such\nthat the band structure of the system agrees as well as\npossible with experimental and/or ab initio data. In this\nwork, we use the tight binding model and parameters\nof Ref. [28], which allows for the modeling of strained,\nrandom SiGe alloys with any Ge concentration profile.\nThe Hamiltonian of an arbitrary SiGe alloy takes the3\nform,\nHmi;nj\n\u0016\u001b;\u0017\u001b =\u000enj\nmih\n\u000e\u0017\u001b0\n\u0016\u001b\u0010\n\"(nj)\n\u0017+Vn\u0011\n+\u000e\u001b0\n\u001bC(nj)\n\u0016\u0017+S(nj)\n\u0016\u001b;\u0017\u001b0i\n+\u000e\u001b0\n\u001b\u0010\n\u000en+1\nmT(n)\ni\u0016;j\u0017 +\u000en\u00001\nmT(m)y\ni\u0016;j\u0017\u0011\n;(3)\nwhereHmi;nj\n\u0016\u001b;\u0017\u001b0=hmi\u0016\u001bjHjnj\u0017\u001b0iand\u000eequals 1if its\nsubscripts match its superscripts and 0otherwise. The\nfirst line in Eq. (3) contains intra-atomic terms, where\n\"(nj)\n\u0017is the onsite energy of orbital \u0017, for atom jin\natomic layer n,Vnis the potential energy due to the ver-\ntical electric field, C(nj)\n\u0016\u0017accounts for onsite energy shifts\nand couplings caused by strain, and S(nj)\n\u0016\u001b;\u0017\u001b0accounts for\nspin-orbit coupling. The matrix C(nj)\n\u0016\u0017is determined by\nthe deformation of the lattice due to strain as detailed\nin Ref. [28] and arises from changes in the onsite poten-\ntial of the atom due to the displacement of its neighbors.\nIn addition, spin-orbit coupling S(nj)is an intra-atomic\ncoupling between porbitals [29] and is the only term in\nEq. (3) that does not conserve spin \u001b. (See Appendix B\nfor the explicit form of S(nj).) Note that the superscripts\n(nj), which index the atoms, are needed here because the\nintra-atomic terms depend on whether an atom is Si or\nGe, as well as the local strain environment. The second\nline in Eq. (3) contains inter-atomic terms describing the\nhopping between atoms on adjacent atomic layers, where\nT(n)is the hopping matrix from atomic layer nto atomic\nlayern+1. Nearlyallelementsof T(n)arezero, withnon-\nzero hoppings occurring only between nearest-neighbor\natoms. A non-zero hopping matrix element T(n)\ni\u0016;j\u0017then\ndepends on three things: (1) the orbital indices \u0016and\n\u0017, (2) the types of atoms involved, and (3) the direction\nandmagnitudeofthevector Rn+1;i\u0000Rn;jconnectingthe\natoms. We then use the Slater-Koster table in Ref. [26]\nalong with the parameters of Ref. [28] to calculate T(n)\ni\u0016;j\u0017.\nWe note that strain affects the hopping elements by al-\ntering the direction and length of the nearest-neighbor\nvectors (i.e., the crystalline bonds) [28, 30].\nIn this work, we let the Ge concentration vary be-\ntween layers, as shown in Figs. 1(b) and 1(c), but as-\nsume it to be uniform within a given layer. Note that\nthe large difference in Ge concentration between the bar-\nrier and well regions results in a large conduction band\noffset that traps electrons inside the quantum well. This\noccurs naturally in the tight binding model of Eq. (3)\nbecause\"(nj);C(nj), andT(n)\nijare different for Si and Ge\natoms. Finally, we point the reader to Appendix A for a\ndescription of the lattice constant dependence on strain.\nB. Virtual crystal approximation and pseudospin\ntransformation\nWhile the Hamiltonian in Eq. (3) provides an accurate\ndescription of SiGe alloys, it lacks translation invariance\nwhen alloy disorder is present. This makes the modelcomputationally expensive to solve, and it obscures the\nphysics of the spin-orbit enhancement coming from the\naveraged effects of the inhomogeneous Ge concentration\nprofile. We therefore employ a virtual crystal approxima-\ntion where the Hamiltonian matrix elements are replaced\nby their value averaged over all alloy realizations. Specif-\nically, we define a virtual crystal Hamiltonian HVCwith\nelements (HVC)mi;nj\n\u0016\u001b;\u0017\u001b0=D\nHmi;nj\n\u0016\u001b;\u0017\u001b0E\n, whereh:::iindicates\nanaverageoverallpossiblealloyrealizations. TheHamil-\ntonian is then translation invariant within the plane of\nthe quantum well. In addition, it is useful to move be-\nyond the original orbital basis, where the spin is well-\ndefined, to a pseudospin basis defined by\njnj\u0016\u0017\u0016\u001bi=X\n\u0017\u001bjnj\u0017\u001biU(n)\n\u0017\u001b;\u0016\u0017\u0016\u001b; (4)\nwhere \u0016\u0017are the hybridized orbital states, \u0016\u001b=*;+are the\npseudospins, and U(n)is the transformation matrix for\nlayern. Full details of this basis transformation can be\nfound in Appendix B. Here, we mention three important\nfeatures of Eq. (4). First, the basis transformation diag-\nonalizes the onsite spin-orbit coupling, making the spin-\norbit physics more transparent. Second, the pseudospin\nstates represent linear combinations of orbitals including\nboth spin\"and spin#. Third, the transformation matrix\nU(n)can be shown to satisfy\nU(n)=(\nU(0); n2Zeven;\nU(1); n2Zodd:(5)\nThis alternating structure for the transformation matrix\nis crucial to the results that follow, and results from the\npresence of two sublattices in the diamond crystal struc-\nture of Si, as shown in Fig. 2(a).\nMaking use of the virtual crystal approximation, we\nnow convert our three-dimensional (3D) Hamiltonian\ninto an effective 1D Hamiltonian. To begin, we note that\nthe virtual crystal Hamiltonian takes the simplified form,\n(HVC)mi;nj\n\u0016\u0016\u0016\u001b;\u0016\u0017\u0016\u001b0=\u000enj\u0016\u001b0\nmi\u0016\u001bh\n\u000e\u0016\u0017\n\u0016\u0016\u0010\n\u0016\"(n)\n\u0016\u0017+Vn\u0011\n+\u0016C(n)\n\u0016\u0016\u0016\u0017i\n+\u000en+1\nm\u0016T(n)\ni\u0016\u0016\u0016\u001b;j\u0016\u0017\u0016\u001b0+\u000en\u00001\nm\u0016T(m)y\ni\u0016\u0016\u0016\u001b;j\u0016\u0017\u0016\u001b0;(6)\nwhere (HVC)mi;nj\n\u0016\u0016\u0016\u001b;\u0016\u0017\u0016\u001b0=hmi\u0016\u0016\u0016\u001bjHVCjnj\u0016\u0017\u0016\u001b0i,\u0016\"(n)\n\u0016\u0017is the on-\nsite energy of pseudospin orbital \u0016\u0017, and \u0016C(n)and \u0016T(n)\nare the onsite strain and hopping matrices, respectively,\ntransformed into the pseudospin basis and averaged over\nalloy realizations. Note that \u0016\"(n)includes contributions\nfrom diagonalizing the onsite spin-orbit coupling. Impor-\ntantly, \u0016\"(n)and \u0016C(n)maintain a dependence on the layer\nindexndue to the non-uniform Ge concentration pro-\nfile. In contrast, the dependence of intra-atomic terms\non theintra-layer atom index jhas vanished due to the\nvirtual crystal approximation. Moreover, the translation\ninvariance of the virtual crystal approximation implies\nthat hopping matrix elements between any two layers\nonly depend upon the relative position of atoms, i.e.,4\n\u0016T(n)\ni\u0016\u0016\u0016\u001b;j\u0016\u0017\u0016\u001b0=\u0016T(n)\n\u0016\u0016\u0016\u001b;\u0016\u0017\u0016\u001b0(Rn+1;i\u0000Rn;j). We therefore intro-\nduce in-plane momentum kkas a good quantum number\nand Fourier transform our Hamiltonian. To do so, we\ndefine the basis state\n\f\fkkn\u0016\u0017\u0016\u001b\u000b\n=1pNkX\njeikk\u0001Rnjjnj\u0016\u0017\u0016\u001bi;(7)\nwhere kk= (kx;ky)andNkis the number of atoms\nwithin each layer. The Hamiltonian has matrix elements\neHmn\n\u0016\u0016\u0016\u001b;\u0016\u0017\u0016\u001b0(kk) =\u000en\u0016\u001b0\nm\u0016\u001bh\n\u000e\u0016\u0017\n\u0016\u0016\u0010\n\u0016\"(n)\n\u0016\u0017+Vn\u0011\n+\u0016C(n)\n\u0016\u0016\u0016\u0017i\n+\u000en+1\nmeT(n)\n\u0016\u0016\u0016\u001b;\u0016\u0017\u0016\u001b0\u0000\nkk\u0001\n+\u000en\u00001\nmeT(m)y\n\u0016\u0016\u0016\u001b;\u0016\u0017\u0016\u001b0\u0000\nkk\u0001\n;(8)\nwhereeHmn\n\u0016\u0016\u0016\u001b;\u0016\u0017\u0016\u001b0(kk) =\nkkm\u0016\u0016\u0016\u001b\f\fHVC\f\fkkn\u0016\u0017\u0016\u001b0\u000b\n, and\neT(n)\n\u0016\u0016\u0016\u001b;\u0016\u0017\u0016\u001b0\u0000\nkk\u0001\nis the Fourier-transformed hopping matrix\ngiven by\neT(n)\n\u0016\u0016\u0016\u001b;\u0016\u0017\u0016\u001b0\u0000\nkk\u0001\n=2X\nl=1e\u0000ikk\u0001r(n)\nl\u0016T(n)\n\u0016\u0016\u0016\u001b;\u0016\u0017\u0016\u001b0(r(n)\nl);(9)\nwhere r(n)\nlis a nearest-neighbor vector from a reference\natom in layer nto one of its nearest neighbors in layer\nn+ 1. For a diamond lattice, each atom has only has\ntwo such bonds, as indicated in Fig. 2(a). Note that\nthe Hamiltonian matrix elements vanish between states\nwith different momenta due to translational invariance.\nHence, we obtain an effective 1D Hamiltonian as a func-\ntion of kk.\nAn important feature of the Fourier-transformed hop-\npingmatrixeT(n)(kk)isthatitdependsonthelayerindex\nnfor two reasons. First, the inhomogeneous Ge concen-\ntration along the growth axis causes the hopping param-\neters to change slightly from layer to layer. Second, and\nmore importantly, the diamond crystal structure is com-\nposed of two interleaving face-centered-cubic sublattices\nwhich each contribute an inequivalent atom to the prim-\nitive unit cell. This is illustrated in Fig. 2(a) where the\natoms belonging to the two sublattices are colored red\nand blue, respectively. Indeed, the atoms for n2Zeven\nandn2Zoddbelong to sublattice 1 and 2, respectively,\nand have different nearest neighbor vectors. It is there-\nfore useful to define\neT(n)\u0000\nkk\u0001\n=(eT(n)\n+\u0000\nkk\u0001\n; n2Zeven\neT(n)\n\u0000\u0000\nkk\u0001\n; n2Zodd(10)\nasthehoppingmatricesforthetwosublattices. Westress\nthat the dependence of eT(n)\n+(kk)andeT(n)\n\u0000(kk)on the\nlayer index nis due to the inhomogeneous Ge concentra-\ntion profile, and that eT(n)\n+(kk)andeT(n)\n\u0000(kk)differ due\nto the diamond crystal structure having two inequiva-\nlent atoms in its primitive unit cell. We can therefore\nvisualize the system, for any given kk, as a 1D, multi-\norbital tight binding chain, as shown in Fig. 2(b), where\nthe hopping terms alternate in successive layers.\n(a) z\n (b)\n/tildewideT(0)\n+/parenleftbig\nk∥/parenrightbig/tildewideT(1)\n−/parenleftbig\nk∥/parenrightbig/tildewideT(2)\n+/parenleftbig\nk∥/parenrightbig/tildewideT(3)\n−/parenleftbig\nk∥/parenrightbigzFIG. 2. (a) Diamond crystal structure of silicon. The dashed\nlines outline the conventional unit cell of the face centered\ncubic lattice. Both red and blue atoms are silicon but be-\nlong to different sublattices. Notice that the vectors connect-\ning an atom to its four nearest neighbors are fundamentally\ndifferent for the red and blue atoms, giving rise to the al-\nternating hopping structure shown in (b). (b) Effective 1D\ntight-binding chain, with hopping matrix terms alternating\nbetween eT(n)\n+(kk)and eT(n)\n\u0000(kk). Note that each site has 20\norbitals, and only the forward hopping terms are shown. On-\nsite and backward hopping terms are not shown. For a SiGe\nalloy in the virtual crystal approximation, the two-sublattice\nstructure is retained, but the atoms are replaced by virtual\natoms, with averaged properties consistent with the Ge con-\ncentration of a given layer.\nC. Expansion around kk= 0\nOur goal is to understand the spin-orbit physics of low-\nenergy conduction band states near the Fermi level. In\nstrained Si/SiGe quantum wells, these derive from the\ntwo degenerate valleys near the Z-point of the strained\nBrillouin zone [31]. Therefore, the low-energy states have\nsmalljkkj, and we can understand the spin-orbit physics\nby expanding the Fourier-transformed hopping matrices\neT(n)\n\u0006\u0000\nkk\u0001\nto linear order. We find that\neT(n)\n\u0006(kk) =eT(n)\n0+eT(n)\nR(kk)\u0006eT(n)\nD(kk) +O(k2\nk);(11)\nwhereeT(n)\n0is the hopping matrix for kk= 0, andeT(n)\nR\nandeT(n)\nDcontain the linear kkcorrections. These hop-\nping matrix components are found to be\neT(n)\n0= \n(n)\u0016\u001b0; (12)\neT(n)\nR(kk) = \b(n)(ky\u0016\u001bx\u0000kx\u0016\u001by); (13)\neT(n)\nD(kk) = \b(n)(kx\u0016\u001bx\u0000ky\u0016\u001by); (14)\nwhere \n(n)and\b(n)are real-valued 10\u000210matrices,\nand\u0016\u001bjarethePaulimatricesactingonpseudospinspace,\nwithj= 0;x;y;z.\nThere are several features to remark on in Eqs. (12)-\n(14). First, the momentum-spin structure of eT(n)\nRand\neT(n)\nDhave the familiar forms of Rasbha and Dresselhaus\nspin-orbit coupling; hence, we apply the subscript labels5\nRandD. Second, the hopping matrices for the two sub-\nlattices,eT(n)\n+andeT(n)\n\u0000, differ by the sign in front of eT(n)\nD,\nwhile thesign of eT(n)\nRis sublatticeindependent. Asnoted\nabove, this alternating sign is a consequence of the two\nsublattices of the diamond crystal being inequivalent. As\nweshallshowinSec.IVC,thisalternatinghoppingstruc-\nture ofeT(n)\nDis key to explaining the mechanism behind\nthe enhanced spin-orbit coupling. In Appendix C, we\nalso provide a symmetry argument for why the system\nhas this particular alternating hopping structure for the\nRashba and Dresselhaus terms. Third, for the special\ncase of kk= 0, the two sublattices become equivalent\n(i.e., there is no even/odd structure due to the vanishing\nofeT(n)\nRandeT(n)\nDatkk= 0). Furthermore, the pseu-\ndospin sectors are uncoupled and equivalent at kk= 0,\nwhich implies that all eigenstates of our Hamiltonian are\ndoubly degenerate at kk= 0as is expected since the sys-\ntemhastime-reversalsymmetry. Fourth, thedependence\nof\n(n)and\b(n)on the layer index narises only from the\ninhomogeneous Ge concentration profile. However, note\nthat neither matrix vanishes if the Ge concentration is\nuniform. For interested readers, we provide expressions\nfor\n(n)and\b(n)in Appendix D, for the case of a pure\nSi structure. Finally, we note that the diagonal elements\nof\b(n)all vanish.\nTo linear order in kk, the Hamiltonian then takes the\ncompact form\nH=H(z)\n0\u0016\u001b0+H(z)\nR(ky\u0016\u001bx\u0000kx\u0016\u001by)\n+H(z)\nD(kx\u0016\u001bx\u0000ky\u0016\u001by) +O(k2\nk);(15)\nwhereH(z)\n0(which we call the subband Hamiltonian) de-\nscribes the physics at kk= 0, andH(z)\nRandH(z)\nDdescribe\nthe linear kkperturbations arising from eT(n)\nRandeT(n)\nD,\nrespectively. These take the form\nhm\u0016\u0016jH(z)\n0jn\u0016\u0017i=\u000en\nmh\n\u000e\u0016\u0017\n\u0016\u0016\u0010\n\u0016\"(n)\n\u0016\u0017+Vn\u0011\n+\u0016C(n)\n\u0016\u0016\u0016\u0017i\n+\u000en+1\nm\n(n)\n\u0016\u0016\u0016\u0017+\u000en\u00001\nm\n(m)T\n\u0016\u0016\u0016\u0017;(16)\nhm\u0016\u0016jH(z)\nRjn\u0016\u0017i=\u000en+1\nm\b(n)\n\u0016\u0016\u0016\u0017+\u000en\u00001\nm\b(m)T\n\u0016\u0016\u0016\u0017; (17)\nhm\u0016\u0016jH(z)\nDjn\u0016\u0017i=(\u00001)n\u0010\n\u000en+1\nm\b(n)\n\u0016\u0016\u0016\u0017\u0000\u000en\u00001\nm\b(m)T\n\u0016\u0016\u0016\u0017\u0011\n;(18)\nwhere the superscript zindicates that only the orbital\ndegrees of freedom in the zdirection (i.e., the growth\ndirection) are acted upon. Hence, the momentum kkand\npseudospin \u0016\u001bindices are both dropped in Eqs. (16)-(18).\nAlso note that the alternating \u0006factor in front of eT(n)\nD\nin Eq. (11) is reflected in the (\u00001)nfactor in Eq. (18).\nD. Transformation to the subband basis\nThe largest term in Hamiltonian (15) (by far) is the\nsubband Hamiltonian H(z)\n0. The eigenstates of H(z)\n0arereferred to as the orbital subbands of the quantum well,\nincluding two distinct valley states per subband. The\nsubband and valley states, in turn, serve as a natural\nbasis for representing the Hamiltonian, since the lateral\nconfinement associated with a quantum dot barely per-\nturbs this subband designation, although disorder may\ncause hybridization of the valley states. It is therefore\nthe properties of the individual subbands that largely de-\ntermine the properties of quantum dot states, including\ntheir spin-orbit behavior.\nTo perform a subband basis transformation, we de-\nfinej'`ias the`theigenstate of H(z)\n0with energy E`.\n(Here, for convenience, we include both subband and val-\nley states in the set f`g.) Generically we can write\nj'`i=X\nn\u0016\u0017jn\u0016\u0017iQn\u0016\u0017;`; (19)\nwhereQis an orthogonal matrix, defined such that\nH(z)\n0j'`i=E`j'`i (20)\nfor each`. Using these eigenstates as a basis, the Hamil-\ntonian can then be expressed as\nH=\u0016\u0003\u001b0+ \u0016\u000b(ky\u0016\u001bx\u0000kx\u0016\u001by)\n+\u0016\f(kx\u0016\u001bx\u0000ky\u0016\u001by) +O(k2\nk);(21)\nwhere \u0016\u0003,\u0016\u000b, and \u0016\fare real-symmetric matrices acting in\nsubband space with elements\n\u0016\u0003``0=\u000e``0E`; (22)\n\u0016\u000b``0=h'`jH(z)\nRj'`0i; (23)\n\u0016\f``0=h'`jH(z)\nDj'`0i: (24)\nThe matrix elements \u0016\u000b``0and \u0016\f``0are referred to as the\nRashba and Dresselhaus spin-orbit coupling coefficients,\nrespectively [16]. The diagonal elements are of particular\nimportance since they determine the linear dispersion of\na given subband near kk= 0. Indeed, the diagonal ele-\nments \u0016\u000b``and\u0016\f``themselves are often referred to in the\nliterature as the Rasbha and Dresselhaus spin-orbit cou-\npling coefficients, respectively, and are typically denoted\nsimply as\u000band\f. Furthermore, we focus on the diag-\nonal elements of the ground ( `= 0) and excited ( `= 1)\nvalleystatescorrespondingtothelowestorbitalsubband,\nwhich we henceforth refer to as simply the ground and\nexcited valley states. These represent the lowest-energy\nconduction subbands, which are nearly degenerate due\nto the wide separation of the two degenerate zvalleys\nwithin the Brillouin zone of Si [32], thus playing a domi-\nnating role in the physics of Si spin qubits. In some cases,\nwe may also be interested in the spin-orbit coupling be-\ntween the ground and excited valleys, often referred to\nas spin-valley coupling, since the valley states are much\ncloser in energy than the orbitally excited subbands. We\nnote that confinement in the growth direction is a cru-\ncialingredientforobtainingnonzerovaluesof \u0016\u000b``and\u0016\f``.6\nWhile this latter fact is not obvious from the structure\nof the Hamiltonian, it can be shown to be true, using the\nfact that \b(n)has vanishing diagonal elements and the\nstructure of the \n(n)and\b(n)matrices described in Ap-\npendix D. Finally, we also mention that one can arrive at\nan effective 2D theory, similar to previous SU (2)\u0002SU(2)\napproaches to spin-valley physics in Si [33], by projecting\nthe Hamiltonian in Eq. (21) onto the subspace contain-\ning the ground ( `= 0) and excited ( `= 1) valleys.\nE. Summary of calculation procedure\nTo conclude this section, we present a brief summary\nof the procedure used in a typical calculation, like those\nreported in Sec. III. First, we specify a Ge concentra-\ntion profile as a function of layer index n. Second, we\nconstruct the subband Hamiltonian H(z)\n0using Eq. (16).\nThird, we diagonalize the subband Hamiltonian to ob-\ntain a set of eigenstates fj'`ig. Finally, we construct the\nH(z)\nRandH(z)\nDmatrices in Eqs. (17) and (18) and cal-\nculate the matrix elements in Eqs. (23) and (24), which\nyields the spin-orbit coupling coefficients.\nIII. NUMERICAL RESULTS\nIn this section, we present our numerical results. We\nfirst present in Sec. IIIA spin-orbit coupling results for a\n“conventional” Si/SiGe quantum well system withoutGe\noscillations included in the quantum well region. These\nresults serve as a baseline for comparison. Next, we\nprovide spin-orbit coupling results in Sec. IIIB for the\nSi/SiGe quantum well system withGe oscillations in-\ncluded in the quantum well region, namely, a wiggle well.\nIn this case, we observe significant enhancement of the\nDresselhaus spin-orbit coupling for appropriate Ge con-\ncentration oscillation wavelengths \u0015. In Sec. IIIC, we\nshow that the enhanced spin-orbit coupling allows for\nfast Rabi oscillations using EDSR. Finally, we study in\nSec. IIID the impact of alloy disorder on the spin-orbit\ncoupling.\nA. Spin-orbit coupling in Si/SiGe quantum wells\nwithoutGe concentration oscillations\nWe first calculate spin-orbit coupling for the conven-\ntional Si/SiGe quantum well shown in Fig. 1(c). We as-\nsume barrier regions with a uniform Ge concentration of\nnGe,bar = 30%and a quantum well width of Lz\u001920nm,\nconsisting of an even number of atomic layers; the lat-\nter value was chosen to ensure that the wave functions\nhave negligible weight at the bottom barrier except in\nthe limit of very weak electric fields. In addition, the\ninterfaces are given a nonzero width of Lint\u00190:95nm\n(7atomic layers), in which the Ge concentration linearly\ninterpolates between values appropriate for the barrier\n25\n025 (eV nm)\n(a)\n0 2 4 6 8 10\nFz (mV/nm)4\n2\n0 (eV nm)\n(b)FIG. 3. Diagonal Dresselhaus \u0016\f``(a) and Rashba \u0016\u000b``(b)\nspin-orbit coupling coefficients as a function of vertical elec-\ntric fieldFz, for the ground (solid blue, \u0016\f00and\u0016\u000b00) and ex-\ncited (dashed red, \u0016\f11and\u0016\u000b11) valleys of the “conventional”\nSi/SiGe quantum well shown in Fig. 1(c). Notice that the\nDresselhaus coefficients \u0016\f``are\u00187times larger in magnitude\nthan the Rashba coefficients \u0016\u000b``.\nand well regions. Such finite-width interfaces occur in\nrealistic devices, and are known to significantly impact\nimportant properties of the quantum well such as the\nvalley splitting [34].\nThe diagonal Dresselhaus and Rashba spin-orbit cou-\nplingcoefficientsarecalculatedfortheground(solidblue,\n\u0016\f00and\u0016\u000b00) and excited (dashed red, \u0016\f11and\u0016\u000b11) valley\nstates, andareplottedinFigs.3(a)and3(b)asafunction\nof vertical electric field Fz. At zero field, Fz= 0, the di-\nagonal spin-orbit coupling coefficients in Fig. 3 all vanish,\nas consistent with the system being inversion symmetric\n[20, 21]. By turning on the electric field we break the\nstructural inversion symmetry, and the resulting spin-\norbit coefficients vary linearly over the entire field range\nconsidered here. Note again that Dresselhaus spin-orbit\ncoupling in SiGe requires the presence of a broken struc-\ntural inversion symmetry [20, 21, 24, 25], in contrast with\nGaAs, which requires a broken bulk inversion symme-\ntry [23]. As consistent with previous studies [20, 35–37],\ntheDresselhauscoefficientsofthegroundandexcitedval-\nleys are found to be approximately opposite in sign, with\nthe Dresselhaus coefficient of the excited valley being\nslightly smaller in magnitude. Additionally, the diagonal\nRashba matrix elements are seen to be much smaller in\nmagnitude than the Dresselhaus elements. Here, we find\nthatj\u0016\f``j\u00197j\u0016\u000b``jfor both low-energy valleys ( `= 0;1).\nInaddition, themagnitudesofthespin-orbitelementsare\nfound to quantitatively agree with Ref. [20], in the large\nelectric field regime. (The system studied in Ref. [20]\nassumed a narrower quantum well, resulting in different\nbehavior at low electric fields.) We note, however, that\nthe Rashba coefficients of the ground and excited valley\nstateswerefoundtohaveoppositesignsinRef.[20], while\nwe find them to have the same sign here. This difference\nin our results occurs because we have used a softened in-\nterface where the Ge concentration interpolates between7\n400\n0400 (eV nm)\n(a)\n0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 (nm)\n5\n05 (eV nm)\n(b)\n0.25 0.29\n (nm)\n600\n300\n0300 (eV nm)\n(c)\n0.44 0.48\n (nm)\n10\n010 (eV nm)\n(d)\nFIG. 4. Dresselhaus \u0016\f``0(a) and Rashba \u0016\u000b``0(b) spin-orbit\nmatrix elements as a function of Ge oscillation wavelength\n\u0015for an electric field of strength Fz= 10mV/nm. Coeffi-\ncients for the ground and excited valleys are shown in blue\n(solid, \u0016\f00and \u0016\u000b00) and red (dashed, \u0016\f11and \u0016\u000b11), respec-\ntively. The off-diagonal elements ( \u0016\f01and\u0016\u000b01), often referred\nto as the spin-valley coefficients, are shown as purple dashed-\ndotted lines. A wide bump centered at \u0015\u00191:57nm occurs\nin (a), corresponding to a dramatically enhanced Dresselhaus\nspin-orbit coupling. At the center of the bump, j\u0016\f``jis\u001815\ntimes larger than the “conventional” Si/SiGe system at the\nsame electric field [See Fig. 3(a).] Narrow bumps for the di-\nagonal Dresselhaus and Rashba coefficients at small \u0015values\nareshownin(c)and(d). ThecorrespondingGeconcentration\nprofile is shown in Fig. 1(b), where the average Ge concentra-\ntion in the quantum well region is \u0016nGe= 5%.\nthe barrier and well regions over a finite width, whereas\nRef. [20] used a completely sharp interface. We have\nnumerically confirmed our results for the diagonal spin-\norbit matrix elements using the computational scheme of\nRef. [20], which is unrelated to our scheme, summarized\nin Eqs. (23) and (24).\nB. Spin-orbit coupling in Si/SiGe quantum wells\nwithGe concentration oscillations\nWe now calculate spin-orbit coefficients for the wiggle\nwell geometry shown in Fig. 1(b), with a sinusoidally\nvarying Ge concentration ranging from nGe= 0%to a\nmaximum amplitude of nGe= 10%, and an oscillation\nwavelength of \u0015. These parameters were chosen to match\nthose of an experimental device reported in Ref. [18].\nHere, the system parameters, nGe,bar,Lz, andLintare\nthe same as before, and we apply an electric field of Fz=\n10mV/nm.\nThe resulting diagonal Dresselhaus \u0016\f``and Rashba \u0016\u000b``\nspin-orbit coefficients for the ground (solid blue, \u0016\f00and\n400\n0400 (eV nm)\n(a)\n0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 (nm)\n5\n05 (eV nm)\n(b)\n0.25 0.29\n (nm)\n600\n300\n0300 (eV nm)\n(c)\n0.44 0.48\n (nm)\n10\n010 (eV nm)\n(d)FIG. 5. Same as Fig. 4 except with an electric field strength\nofFz= 2mV/nm. Notice that the magnitude of \u0016\f``at the\npeaks of the bumps are nearly the same as in Fig. 4, but that\nthe features are narrower than in Fig. 4.\n\u0016\u000b00) and excited (dashed red, \u0016\f11and\u0016\u000b11) valley states\nare shown in Figs. 4(a) and 4(b) as a function of the Ge\noscillation wavelength \u0015. We find that the spin-orbit co-\nefficients vary nontrivially with the choice of oscillation\nwavelength \u0015, exhibiting bumps where the coefficients\nare significantly enhanced. In particular, we observe a\nbroad bump in Fig. 4(a), centered at \u0015\u00191:57nm, in\nwhich the Dresselhaus spin-orbit coefficient reaches the\nvaluesj\u0016\f00j\u0019365\u0016eV\u0001nm for the ground valley and\nj\u0016\f11j\u0019357\u0016eV\u0001nm for the excited valley. Note that this\npeak location corresponds to the same wavelength as the\nlong-period wiggle well predicted in Refs. [18, 19] to en-\nhance the valley splitting. (Note that a slightly different\n\u0015value was predicted in Refs. [18, 19], due to the valley\nminima residing at slightly different momenta \u0006k0com-\npared to our model.) Similar to the conventional Si/SiGe\nquantum well shown in Fig. 4, the ground and excited\nvalleys have approximately opposite Dresselhaus coeffi-\ncients for the range of \u0015values considered here, except\nat very small \u0015. At the peak of the bump, however, the\ndiagonal Dresselhaus coefficients \u0016\f``for the wiggle well\nare\u001815times larger than those of the conventional sys-\ntem at the same electric field strength. [See Fig. 3(a) at\nFz= 10mV/nm.] Moreover, the region of enhancement\nappears very broad, with a full-width at half-maximum\nof\u0001\u0015= 0:55nm. In contrast to the diagonal Dres-\nselhaus coefficients \u0016\f``, the diagonal Rashba coefficients\n\u0016\u000b``show no enhancement features except at small wave-\nlengths. Indeed, for \u0015&0:5nm, the Rashba coefficients\nare essentially independent of \u0015with \u0016\u000b00\u0019\u00003:7\u0016eV\u0001nm\nand\u0016\u000b11\u0019\u00003:3\u0016eV\u0001nm for the ground and excited val-\nleys, respectively. Comparing this to the results of the\nconventional system in Fig. 3(b) for the same electric8\nfield, we see that the Rashba elements are nearly identi-\ncal in the two cases. Evidently, the Rashba coefficients\nare unaffected by the Ge concentrations oscillations for\nall but the shortest \u0015values.\nWe also plot the off-diagonal spin-orbit coupling el-\nements (\u0016\f01and \u0016\u000b01)as purple dashed-dotted lines in\nFigs. 4(a) and 4(b). Note that these quantities are of-\nten referred to as spin-valley coupling. Interestingly, the\noff-diagonal Dresselhaus coefficient \u0016\f01vanishes at the\ncenter of the \u0015\u00191:57nm feature. Except at this partic-\nular wavelength for the Dresselhaus spin-orbit coupling,\nthe off-diagonal spin-orbit coefficients are comparable in\nsize to the diagonal coefficients.\nTo illustrate the features at small wavelengths, we plot\nthe diagonal Dresselhaus and Rashba coefficients over se-\nlected, narrow ranges of \u0015in Figs. 4(c) and 4(d). Here we\nobserveanenhancementintheDresselhauscoefficient \u0016\f``\ncentered at \u0015=a=2\u00190:27nm, whereais the size of the\ncubic unit cell in Fig. 2(a) with an amplitude about twice\nthat of the \u0015= 1:57nm peak. This \u0015value corresponds\nto having a Ge concentration profile that alternates on\nevery other atomic layer, essentially transforming the di-\namondlatticeofSiintoazinc-blendelattice. Inthiscase,\nthe system can be thought of as a III-V semiconductor\nwith the cation and anion corresponding to different con-\ncentrations of Ge. Interestingly, we find that the ground\nand excited valleys have the same sign of \u0016\f``here, in\ncontrast to the bump centered at \u0015\u00191:57nm. We also\nobserve a enhancement in the Rashba coefficient \u0016\u000b``near\n\u0015= 0:46nm as shown in Fig. 4(d). However, note that\nthe magnitude of this peak is still more than an order\nof magnitude smaller than the \u0016\f``peaks. Indeed, the\nmagnitude of this peak is even smaller than the Dressel-\nhaus coefficients \u0016\f``shown in Fig. 3(a) for the conven-\ntional Si/SiGe system at the same electric field strength.\nFinally, we note that these small- \u0015bumps have a much\nnarrower width than the \u0016\f``bump at\u0015= 1:57nm, which\nhas important consequences for practical applications.\nTo study the tunability of the spin-orbit physics, we\nalso calculate the spin-orbit coefficients of a wiggle well\nfor a weaker vertical electric field. These results are\nshown in Fig. 5, which is the same system as Fig. 4, ex-\ncept withFz= 2mV/nm. The results are qualitatively\nsimilar to the stronger electric field case. However, there\nexists an important quantitative similarity and difference\nbetween the two cases, which we address in the following\ntwo paragraphs.\nThe remarkable similarity is that the Dresselhaus co-\nefficients \u0016\f``at the peaks of the bumps are nearly iden-\ntical in the two cases. Note that this is true for both\nthe\u0015= 1:57nm and\u0015= 0:27nm bumps. Evidently, at\nthe center of the bumps, the electric field plays a minor\nrole. This is in stark contrast with the Dresselhaus coef-\nficients for the conventional system, where the diagonal\nDresselhaus coefficients are proportional to Fzas shown\ninFig.3(a). Thishighlightstheimportantdifferencefirst\ndiscussedintheIntroductionbetweenspin-orbitcoupling\nin conventional Si/SiGe quantum wells and wiggle wells,namely, that the conventional system requires the pres-\nence of an interface and structural asymmetry, while the\nwiggle well fundamentally does not. In the latter case,\nthe spin-orbit coupling is an intrinsic property of the bulk\nsystem. Indeed, we have checked that the Dresselhaus\nspin-orbit coupling persists in a wiggle well of wavelength\n\u0015= 1:57nm, in the absence of an interface, by cal-\nculating the spin-orbit coefficients for a system without\nbarriers, but instead immersed in a harmonic potential,\nVn=V0z2\nn, obtaining similar results.\nThe main difference between the two cases is that the\nfeatures are narrower in \u0015space for the weak electric\nfield of Fig. 5, compared to the strong electric field of\nFig. 4. This represents an important advantage of strong\nelectric fields for the wiggle well system, since it can be\nchallenging to grow heterostructures with perfect oscil-\nlation periods. We conclude that stronger electric fields\nprovide more reliable access to the enhanced spin-orbit\ncoupling provided by Ge concentration oscillations, since\nthe control of \u0015does not need to be as precise during\nthe growth process. The reason for narrower features in\nweaker electric field will be explained in Sec. IVC. Fi-\nnally, we note that stronger electric fields also provide\nlarger valley splittings [38].\nC. Electric dipole spin resonance\nAside from being of interest from a purely scientific\nstandpoint, the presence of spin-orbit coupling can be\nexploited to perform gate operations within a quantum\ncomputation context. In particular, electric dipole spin\nresonance (EDSR) is a powerful technique to manipu-\nlate individual spins through all-electrical means [14, 15].\nHere, we calculate the EDSR Rabi frequency for a single\nelectron in a quantum dot embedded in a wiggle well.\nAs shown in Ref. [15], applying an AC, in-plane electric\nfield of amplitude Fxwith frequency !dacross a quan-\ntum dot with spin-orbit coupling leads to an effective AC\nin-plane magnetic field. For valley `, the magnitude of\nthis effective AC magnetic field is given by\nBeff(t) =2eFx\n~!\u0010!d\n!\u0011q\n\u0016\f2\n``+ \u0016\u000b2\n``\ng\u0016Bsin (!dt);(25)\nwhere~!is the level spacing characteristic of the dot’s\nharmonicconfinementpotential, gisthe g-factor, and \u0016B\nis the Bohr magneton. Note that for j\u0016\f``j\u001dj \u0016\u000b``j, as is\nthe case for Si/SiGe systems, the effective AC magnetic\nfield is parallel to the in-plane electric field Fx. Apply-\ning a static, out-of-plane magnetic field B, the effective\nHamiltonianforthequantumdotrestrictedtotheorbital\nground state is then Heff(t) =1\n2g\u0016B[B\u001bz+Beff(t)\u001bx].\nFor a system initialized in the spin- \"ground valley state\nand driven at the resonance frequency !d=g\u0016BB=~set\nby the external magnetic field, the probability of find-\ning the electron in the spin- #state is given by P#(t) =\nsin2(\nRabit=2)[39], where the Rabi frequency is found to9\nFIG.6. Electricdipolespinresonance(EDSR)Rabifrequency\n\nRabias a function of magnetic field Band Ge concentra-\ntion oscillation wavelength \u0015for a quantum dot with confine-\nment energy ~!= 1meV and an in-plane AC electric field\namplitudeFx= 10\u00002mV/nm. The vertical electric field is\nFz= 10mV/nm.\nbe\n\nRabi=eFxg\u0016Bq\n\u0016\f2\n``+ \u0016\u000b2\n``\n~(~!)2B: (26)\nThe EDSR Rabi frequency \nRabiof a quantum dot in\na wiggle well is plotted in Fig. 6 as a function of mag-\nnetic field and Ge oscillation wavelength \u0015for the real-\nistic parameters of ~!= 1meV andFx= 10\u00002mV/nm.\nWe see that for oscillation wavelengths near the peak of\nthe the spin-orbit enhancement at \u0015\u00191:57nm, we can\nobtain Rabi frequencies of 100’s of MHz for moderate\nmagnetic field strengths. Indeed, a Rabi frequency of\n\nRabi= 1GHz is achieved at the peak of the spin-orbit\nenhancement for a magnetic field B\u00191:5T. We there-\nforeconcludethatincludingGeconcentrationoscillations\nin Si/SiGe quantum wells enables a dramatic speedup of\nsingle qubit gates using EDSR.\nD. Impact of alloy disorder\nSiGe is a random alloy, and the resulting alloy disor-\nderisknowntohaveaneffectonquantumwellproperties\nsuch as valley splitting, both for conventional quantum\nwells [34] and wiggle wells [18, 19]. It is therefore impor-\ntant to explore the effects of alloy disorder on spin-orbit\ncoupling. Recall from Sec. IIB that we have employed a\nvirtual crystal approximation in our model, which aver-\nages over all possible alloy realizations. While providing\ntractability to our calculations, this approximation ig-\nnores the fluctuations arising from the random nature of\nthe Ge atom arrangements in the SiGe alloy. In this sec-\ntion, we explain how Ge concentration fluctuations can\nbe reintroduced into our model, to explore the effects of\nalloy disorder.\nA full 3D calculation including alloy disorder is com-\nputationally expensive due to the loss of translation in-\nvariance, and is beyond the scope of this work. We canstill, however, include the effects of alloy disorder ap-\nproximately within our 1D effective model by allowing\nfor fluctuations in the Ge concentration in each atomic\nlayer. Thisisaccomplishedusingtheproceduredescribed\nin Ref. [34], which can be summarized as follows. First,\nwe assume a dot of radius adot=p\n~=mk!= 20nm in\nthe plane of the quantum well, where mk= 0:19meis the\nin-plane effective mass and ~!= 1meV is the orbital ex-\ncitation energy characterizing the parabolic confinement\nof the dot. We then calculate the effective Ge concentra-\ntionneff\nGe;ninlayernofourdisorderedsystembycounting\nthe number of Ge atoms withinour dot. Here, the prob-\nability of any given atom in layer nbeing a Ge atom\nisnGe;n, wherenGe;nis the average germanium concen-\ntration throughout the entire layer n, and the number\nof atoms in our dot is Neff= 4\u0019a2\ndot=a2\u001917100, where\na= 0:543nmisthecubiclatticeconstantofSi. Theeffec-\ntive Ge concentration neff\nGe;nin layerncan then be drawn\nfrom the distribution neff\nGe;n=N\u00001\neffBinom (Neff;nGe;n),\nwhere Binom (n;p)is the binomial distribution with n\ntrials and probability of success p. In the limit of\nNeff!1, the resulting, randomized effective Ge con-\ncentrationneff\nGe;napproaches the ideal Ge concentration\nnGe;n, but for smaller dots, fluctuations from this ideal\nlimit become more pronounced. We then calculate the\nspin-orbit coefficients for our effective 1D model as in\nprevious sections but with the Ge concentration profile\ngiven byneff\nGe;ninstead ofnGe;n. Note that this method\nof including alloy disorder in the 1D effective model was\nshown in Ref. [34] to yield valley splitting distributions\nin good agreement with 3D calculations.\nWe now calculate the diagonal Dresselhaus coefficients\n\u0016\f``for the same Si/SiGe system as Fig. 4, with Ge con-\ncentration oscillations of wavelength \u0015= 1:57nm, but\nnow with Ge concentration fluctuations included. Note\nthat this\u0015value corresponds to the peak of the main\nspin-orbit enhancement bump in Fig. 4(a). The distribu-\ntion of the spin-orbit coefficients is plotted in Fig. 7(a)\nfor1000random-alloy realizations, for the ground (blue,\n\u0016\f00) and excited (red, \u0016\f11) valleys. Unsurprisingly, we\nsee that the alloy fluctuations affect the spin-orbit co-\nefficients, with the ground valley coefficient spanning\nthe range\u0000377<\u0016\f00<355\u0016eV\u0001nm. The distribu-\ntions are highly peaked, however, near the barevalues\n\u0016\f00=\u0000365\u0016eV\u0001nm and \u0016\f11= 357\u0016eV\u0001nm of the\ndisorder-free system (see Fig. 4). Indeed, we find that\n86%of the alloy realizations have j\u0016\f00j>200\u0016eV\u0001nm.\nWe therefore conclude that the spin-orbit enhancement\narisingfromtheGeconcentrationoscillationswithinwig-\ngle wells is robust against alloy fluctuations.\nWe gain further insight into the effects of the alloy\nfluctuations by studying the inter-valley spin-orbit coef-\nficient \u0016\f01. Figure 7(b) presents a scatter plot showing\nboththediagonal( \u0016\f00and\u0016\f11)andoff-diagonal( \u0016\f01)co-\nefficients of all 1000alloy realizations. Interestingly, all\nrealizations yield coefficients that land in a narrow semi-\ncircular region of parameter space. This is an indication\nthat the alloy disorder is essentially mixing the original10\n400\n 200\n 0 200 400\n (eV nm)\n020040001 (eV nm)\n(b)0100200counts(a)\n00\n11\nFIG. 7. (a) Distribution of the diagonal Dresselhaus coef-\nficients \u0016\f``of the wiggle well when including alloy disor-\nder as described in the main text. System parameters are\n\u0015= 1:57nm andFz= 10mV/nm. The data includes 1000\nalloy realizations. We see that the spin-orbit coupling en-\nhancement is robust against alloy fluctuations with 86%of\nalloy realizations having j\u0016\f00j>200\u0016eV\u0001nm. (b) Scatter\nplot of the diagonal [ \u0016\f00(blue) and \u0016\f11(red)] and off-diagonal\n(\u0016\f01) spin-orbit coefficients of all alloy realizations. This re-\nsult indicates that the main effect of the alloy disorder is to\nmix the ground and excited valley states.\nground and excited valley states of the system without\ndisorder. Note that perturbations arising from higher\norbital subbands are insignificant, except for inducing a\nsmall width to the distribution. This is not unexpected\nsince the energy difference between the ground and ex-\ncited valley states is <1meV, while the lowest orbital\nexcitation energy is &20meV.\nFor completeness, we also calculate the distribution\nof Dresselhaus spin-orbit coefficients in conventional\nSi/SiGe quantum wells that include alloy fluctuations.\nThese results are shown in Fig. 8(a) for a system with no\nGe in the well region and electric field Fz= 10mV/nm.\nHere, the Ge concentration profile is shown in the in-\nset. Again, we obtain \u0016\f``distributions centered at the\nsame values as the disorder-free system. (See Fig. 3,\nwithFz= 10mV/nm.) Here, however, the spread is\nsignificantly narrower than the wiggle well results shown\nin Fig. 7(a), with the full-width-at-half-maximum being\n\u001810\u0016eV\u0001nm. This is because no alloy fluctuations occur\nin the well region where the majority of the wave func-\ntion resides. We also consider a system with nGe= 5%\ndistributed uniformly throughout the well region, tak-\ning into account the effects of random alloy fluctuations.\n[Here, we do not include intentional Ge concentration os-\ncillations; the resulting Ge concentration profile is shown\nin the inset of Fig. 8(b).] Note that this system con-\ntains the same amount of Ge in the well region as the\nwiggle well system shown in Fig. 7. Unsurprisingly, we\nfind that the distribution of the \u0016\f``coefficients spreads\nconsiderably, compared to the results in Fig. 8(a), due\nto the presence of alloy fluctuations inside of the well.\n0100200counts(a)00\n11\n40\n 20\n 0 20 40\n (eV nm)\n0100200counts(b)\nz030nGe (%)z030nGe (%)FIG. 8. (a) Distribution of the diagonal Dresselhaus coeffi-\ncients \u0016\f``of a conventional Si/SiGe system with a pure Si\nquantum well when including alloy disorder as described in\nthe main text. The electric field is Fz= 10mV/nm. (b)\nSame as (a) except that nGe;n= 5%throughout the well re-\ngion. Insets show the nGe;nprofiles. We see that inclusion of\nGe inside of the well region widens the distribution of \u0016\f``but\ndoes not on average increase its magnitude.\nImportantly, however, a uniform Ge concentration in the\nwell region does not on average increase the spin-orbit\ncoupling, in contrast to the effect on valley splitting. In-\ndeed, we find that\nj\u0016\f00j\u000b\n= 26\u0016eV\u0001nm for the data\nin Fig. 8(a), while\nj\u0016\f00j\u000b\n= 22\u0016eV\u0001nm for the data\nin Fig. 8(b). This highlights that fact that in order to\nenhance spin-orbit coupling, it is not enough to simply\ninclude Ge in the well region, but rather the Ge concen-\ntration must oscillate with the appropriate wavelength\n\u0015. We note that this result is consistent with the experi-\nmentalobservationthatthe g-factormeasuredinuniform\nSi1\u0000xGexalloys is only slightly altered by changing the\nGe concentration [40, 41].\nIV. MECHANISM BEHIND THE SPIN-ORBIT\nENHANCEMENT\nHaving shown through simulations that the inclusion\nof Ge concentration oscillations of an appropriate wave-\nlength can significantly enhance the spin-orbit coupling,\nthe natural question is what mechanism leads to this en-\nhancement? In this section, we provide an explanation\nof this mechanism. To begin, we describe in Sec. IVA a\nsimplified version of our model that allows for easier un-\nderstanding of the spin-orbit enhancement mechanism.\nNext, we study in Sec. IVB the real-space representa-\ntion of the ground valley wave function in the absence\nand presence of Ge concentration oscillations. We find\nthat a real-space picture is inadequate in explaining the\nspin-orbit enhancement. We therefore study in Sec. IVC\nthe structure of the wave function in momentum space.\nWe show how the combination of the oscillating potential\nproduced by the Ge concentration oscillations and the se-\nlection rules of Dresselhaus spin-orbit coupling leads to11\nthe spin-orbit coupling enhancement.\nA. Simplified Model\nTo focus on the essential physics for the spin-orbit\ncoupling enhancement, we use in this section a model\nfor SiGe alloys that is slightly simplified compared to\nthe model presented in Sec. II and used in our numeri-\ncal calculations in Sec. III. In this model, Ge atoms are\nassumed to be identical to Si atoms, except for their\norbitals being shifted up in energy by a constant, i.e.\n\u0016\"(Ge)\n\u0016\u0017= \u0016\"(Si)\n\u0016\u0017+EGewhereEGe= 0:8eV is the extra en-\nergy of every Ge orbital. This is meant to capture at the\nsimplest level that inclusion of Ge increases the energy\nof the conduction band minima. In particular, the cho-\nsen value produces a band offset of 0:24eV between a\npure Si region and a barrier region with a nominal 30%\nGe concentration. Note, however, that the precise value\nis not important since we are only using this simplified\nmodel to understand the spin-orbit enhancement mecha-\nnism, leaving quantitative questions to the more accurate\nmodel of Sec. II. In addition, we also neglect the effects\nof strain, such that \u0016C(n)!0, since they are not crucial\ninunderstandingthespin-orbitenhancementmechanism.\nWith these simplifications, the addition of Ge is equiv-\nalent to adding a term to the potential energy Vof a\npure Si system. For simplicity, we define a new potential\nVn=eFzzn+EGenGe;n, that includes both the electric\nfieldFzand the energy shift from the Ge concentration\nnGe;nof the layer, and we let all orbitals energies take\nvalues appropriate for Si: \u0016\"(n)\n\u0016\u0017!\u0016\"(Si)\n\u0016\u0017. Here,zn=na=4\nis thezcoordinate of atomic layer n, anda= 0:543nm\nis the cubic lattice constant of Si. Importantly, in this\nsimplified model, the onsite orbital energies and hop-\nping matrices all lose their dependence on the layer\nindexn,n\n\u0016\"(n)\n\u0016\u0017;eT(n)\n0;eT(n)\nR(kk);eT(n)\nD(kk);\n(n);\b(n)o\n!n\n\u0016\"(Si)\n\u0016\u0017;eT0;eTR(kk);eTD(kk);\n;\bo\n. The Hamiltonian com-\nponents then take the simplified forms\nhm\u0016\u0016jH(z)\n0jn\u0016\u0017i=\u000en\u0016\u0017\nm\u0016\u0016\u0010\n\u0016\"(Si)\n\u0016\u0017+Vn\u0011\n+\u000en+1\nm\n\u0016\u0016\u0016\u0017+\u000en\u00001\nm\nT\n\u0016\u0016\u0016\u0017; (27)\nhm\u0016\u0016jH(z)\nRjn\u0016\u0017i=\u000en+1\nm\b\u0016\u0016\u0016\u0017+\u000en\u00001\nm\bT\n\u0016\u0016\u0016\u0017; (28)\nhm\u0016\u0016jH(z)\nDjn\u0016\u0017i=(\u00001)n\u0000\n\u000en+1\nm\b\u0016\u0016\u0016\u0017\u0000\u000en\u00001\nm\bT\n\u0016\u0016\u0016\u0017\u0001\n:(29)\nNote that the numerical values of the \nand\bmatrices\nused in the above equations are specified in Appendix D.\nB. Real-space wave functions\nLet us now study the effects of the Ge concentration\noscillations on the wave functions of the subband Hamil-\ntonianH(z)\n0. To do so, we first calculate the ground\nvalley wave function of a conventional Si/SiGe quantum\n||2 \n(a)\n0 5 10 15\nz (nm)||2 \n(b)01530\nnGe (%)\n01530\nnGe (%)FIG. 9. Comparison of the ground valley wave function for a\n“conventional” Si/SiGe system with a pure Si well region (a)\nandawigglewell(b)withoscillationwavelength \u0015= 1:62nm.\nBlue lines show the Ge concentration profiles while red filled\nin curves are the wave functions j j2. Note that we sum over\norbital indices. The electric field is Fz= 5mV/nm in both (a)\nand (b). We see in (b) that the wave function is suppressed\nin regions of high Ge concentration, consistent with the fact\nthattheconductionbandminimaarehigherinenergyinthose\nregions.\nwell that has the Ge concentration profile shown as the\nblue line in Fig. 9(a). The ground valley wave func-\ntionj j2of the subband Hamiltonian H(z)\n0is shown in\nred in Fig. 9(a), where the state is pushed up against\nthe barrier-well interface by an electric field of strength\nFz= 5mV/nm. The wave function also exhibits fast\noscillations characteristic of the superposition of the two\nvalley minima [32]. We next consider the same system,\nexcept with Ge concentration oscillations of wavelength\n\u0015= 1:62nm included in the well region, as shown by the\nblue line in Fig. 9(b). Comparing the two cases, we see\nthat the wave function is suppressed in regions of high\nGe concentration, as consistent with the fact that the\nconduction-band minima (and thus the local potential\nenergies) are higher in energy in those regions. However,\nthese qualitative observations do not directly explain the\nenhancement of Dresselhaus spin-orbit coupling observed\nin Figs. 4 and 5.\nC. Momentum-space wave functions\nWegainaclearerunderstandingoftheeffectsoftheGe\nconcentration oscillations by studying the momentum-\nspace representation of the ground valley. To begin, let\nus first investigate the band structure of Si at kk= 0,\nwhich is shown as the orange dashed lines in Fig. 10.\nA central feature of the band structure is the presence\nof two degenerate valleys near zero energy that give rise\nto the ground and excited valley states in a quantum\nwell. In addition, the valley minima are located only\na short distance of 0:17(2\u0019=a)away from the Brillouin\nzone edge at kz=\u00062\u0019=a. Interestingly, we notice that\natkz=\u00062\u0019=a, there are no band anti-crossings (only12\n4/a\n2/a\n02/a\n4/a\nkz10\n5\n05E (eV)\nFIG. 10. The band structure of Si for kk= 0. The band\nstructure for the conventional Brillouin zone extending to\nkz=\u00062\u0019=ais shown by dashed orange lines. The presence\nof bandcrossings at the zone edge kz=\u00062\u0019=aindicates that\nthe Brillouin zone can be enlarged. Indeed, the equivalence\nof the two sublattices of Si for kk= 0allows us to extend the\nBrillouin zone to kz=\u00064\u0019=a. The band structure for the\nextended zone is shown as solid black lines. See the main text\nfor details on the green, blue, and red points.\ncrossings). This is an indication that the Brillouin zone\ncan be enlarged for kk= 0. Indeed, for the special case\nofkk= 0, the two sublattice hopping matrices become\nequal,eT+(0) =eT\u0000(0), and the primitive unit cell of the\n1D chain shown in Fig. 2(b) reduces from two sites to\none. The Brillouin zone should therefore extend to kz=\n\u00064\u0019=ainstead ofkz=\u00062\u0019=a. Here, we will refer to\nthe Brillouin zone that extends to kz=\u00064\u0019=aas the\nextended zone , while the zone extending only to kz=\n\u00062\u0019=aas theconventional zone .\nTo calculate the band structure in the extended zone,\nwe define a plane wave basis state with momentum kzas\njkz\u0016\u0017i=1pNzX\nneikzznjn\u0016\u0017i; (30)\nwhereNzis the number of sites in the 1D chain and\n\u00004\u0019=a < kz\u00144\u0019=a, wherekz= 8\u0019n=(Nza). The sub-\nband Hamiltonian H(z)\n0then has matrix elements,\nhkz\u0016\u0016jH(z)\n0jk0\nz\u0016\u0017i=\u000ek0\nz\nkz\u0010\n\u000e\u0016\u0017\n\u0016\u0016\u0016\"(Si)\n\u0016\u0017+e\u0000ikza\n4\n\u0016\u0016\u0016\u0017+eikza\n4\nT\n\u0016\u0016\u0016\u0017\u0011\n+\u000e\u0016\u0017\n\u0016\u0016eV(kz\u0000k0\nz);\n(31)\nwhereeV(kz\u0000k0\nz)is the Fourier transform of the poten-\ntialVand is given by eV(qz) =N\u00001\nzP\nnexp(\u0000iqzzn)Vn.\nNotice that in the absence of the potential [ eV(qz)!0],\nkzis a good quantum number for H(z)\n0, and Eq. (31)\nrepresents a Bloch Hamiltonian. The spectrum of H(z)\n0\nforeV(qz) = 0is shown as the solid black lines in Fig. 10,\nwhichwerefertoastheextendedbandstructure. Within\nthe conventional zone, we see that the extended band\n|| \n(a)\n4/a\n2/a\n02/a\n4/a\nkz || \n2\n4/a\n(b)FIG. 11. Wave-function profiles of the ground valley in the\nplane-wave representation. (a), (b) Correspond to Ge profiles\n(a) and (b) in Fig. 9, respectively. Both states show main\npeaks centered at the valley minima, kz\u0019\u00060:83(2\u0019=a), as\nshown in Fig. 10. As shown in (b), however, the Ge concen-\ntrationoscillationsproducewave-functionsatellitesadistance\n2\u0019=\u0015awayfromthemainpeaks. Thewavevectorfromamain\npeak to the opposite outer satellite is 4\u0019=afor this choice of\n\u0015. Since Dresselhaus spin-orbit coupling connects locations in\nreciprocal space separated by 4\u0019=a, spin-orbit enhancement\noccurs when a satellite peak is separated from a main peak\nby4\u0019=a.\nstructure aligns perfectly with half of the conventional\n(orange) bands, while the other half of the bands have\nbeen removed from the conventional zone and instead re-\nside in the regions between kz=j2\u0019=ajandj4\u0019=aj. Fur-\nthermore, we observe that the conventional bands that\ndo not match with the extended band structure can be\nmade to match by shifting them by a reciprocal lattice\nvectorGz=\u00064\u0019=aof the lattice containing two sites.\nFor example, the green point at kz=\u0000\u0019=ain Fig. 10\ngets shifted to the blue point at kz= 3\u0019=a, which coin-\ncides with a band of the extended band structure. This\nis expected since the points kzandkz+Gzare equivalent\nfrom the point of view of the conventional zone [42].\nWe stress that the extended band structure contains\nmore information than the conventional band structure.\nIndeed, the extended scheme clarifies which states can\nbe coupled by a given Fourier component qzof the po-\ntentialeV(qz). For example, let us consider what Fourier\ncomponent of the potential could couple the states of the\nconventional band structure marked by the green and\nred points at kz=\u0000\u0019=aand\u00001:3\u0019=a, respectively, in\nFig. 10. Looking at the conventional band structure, one\nmay initially believe the eV(\u00000:3\u0019=a)Fourier component\ncould couple the states. Looking at these states in the\nextended band structure (red and blue points in Fig. 10),\nhowever, we immediately see that these states are instead\ncoupled by the eV(3:7\u0019=a)Fourier component. This ad-\nditional information will be crucial in understanding the\nenhanced spin-orbit coupling mechanism below.\nReturning to our comparison of the systems with and\nwithout Ge concentration oscillations, we now plot the\nground valley wave functions in the plane-wave repre-13\nsentation. These are shown in Figs. 11(a) and 11(b)\nand correspond to the real-space wave functions shown\nin Figs. 9(a) and 9(b), respectively. That is, we plot\n(P\n\u0016\u0017jhkz\u0016\u0017j'`ij2)1=2as a function of kzwherej'`iis the\nground valley wave function. As shown in Fig. 11(a), the\nground valley wave function of the conventional Si/SiGe\nheterostructure consists of two peaks centered at kz\u0019\n\u00060:83(2\u0019=a). These coincide with the conduction-band\nminima in Fig. 10(a) as expected. Note that the posi-\ntions of these minima in the band structure of Si depends\non the exact tight binding parameters used, and differs\nslightly from other models in the literature. As shown\nin Fig. 11(b), the ground valley wave fuction in the pres-\nence of the Ge oscillations still has its main peaks but\nalso includes surrounding satellite features. The location\nof these satellites with respect to the central peaks is de-\ntermined by the wavelength of the Ge oscillations, with a\npeak-satellite separation of 2\u0019=\u0015, as shown in Fig. 11(b).\nThe second step in explaining the enhanced spin-orbit\ncoupling requires a mechanism for coupling different re-\ngions of the Brillouin zone. We first express the ma-\ntrix elements of the Rashba H(z)\nRand Dresselhaus H(z)\nD\nHamiltonian components in the plane wave basis, giving\nhkz\u0016\u0016jH(z)\nRjk0\nz\u0016\u0017i=\u000ek0\nz\nkz\u0000\ne\u0000ikza\n4\b\u0016\u0016\u0016\u0017+eikza\n4\bT\n\u0016\u0016\u0016\u0017\u0001\n;(32)\nhkz\u0016\u0016jH(z)\nDjk0\nz\u0016\u0017i=\u000e4\u0019=a\njkz\u0000k0zj\u0010\ne\u0000ikza\n4\b\u0016\u0016\u0016\u0017\u0000eikza\n4\bT\n\u0016\u0016\u0016\u0017\u0011\n:\n(33)\nThe key features to notice here are the selection rules be-\ntweenkzandk0\nz: whileH(z)\nRconserveskz,H(z)\nDcouples\nstates with momenta differing by 4\u0019=a. The latter result\nis obtained by Fourier transforming the (\u00001)nfactor in\nEq. (29), which itself is a manifestation of the alternat-\ning sign in front of eT(n)\nDin Eq. (11). To our knowledge,\nthese selection rules and their relation to Rashba and\nDresselhaus spin-orbit couplings have not been noticed\npreviously, although we speculate that they could be de-\nduced from group-theory methods, such as the method\nof invariants [16, 43]. We emphasize, however, that the\nextended zone scheme is key to obtaining such results,\nsince in the conventional zone scheme, kzvalues differing\nby4\u0019=aare equivalent, and cannot yield a selection rule\nlike Eq. (33). We also note that Eqs. (32) and (33) apply\nto any system with a diamond crystal structure.\nThe Dresselhaus momentum selection rule in Eq. (33)\nindicates that the spin-orbit coupling will be enhanced\nwhen wave-function peaks associated with two different\nvalleys are separated by 4\u0019=a. In Fig. 11(b), we see that\nthis can only occur for coupling between a central valley\npeak and the outer satellite associated with the oppo-\nsite valley. Since the central valley peaks are located at\nkz\u0019\u00060:83(2\u0019=a), we see that the resonance condition\nfor the oscillation wavelength to enhance the Dresselhaus\nspin-orbit is given by \u0015res= 2:94a= 1:59nm. Uncoinci-\ndentally, thiswavelengthvalueisveryclosetothepeakof\nthebumpcenteredat \u0015\u00191:57nmofournumericalresult\nin Fig. 4(a) where the Dresselhaus spin-orbit coupling issignificantly enhanced. Note that similar considerations\napply to the excited valley.\nOur understanding of the spin-orbit enhancement\nmechanism also explains the narrower features observed\nin Fig. 5 for a weak vertical electric field Fz= 2mV/nm,\nas compared to the wider features observed in Fig. 4\nfor a strong electric field Fz= 10mV/nm. Essen-\ntially, a weaker electric field produces wave functions\nwith narrower features in the plane-wave representation\nthan those from a strong electric field. These thinner\nfeatures make it more difficult to satisfy the Dresselhaus\nresonance condition since the narrower peaks need to be\nsituated more precisely in kzspace.\nFinally, we comment that the spin-orbit enhancement\nmechanism relies fundamentally on the degeneracy of the\ntwozvalleys in the band structure of strained Si. Such a\nsituation could not occur if, for example, Si was a direct\nbandgap semiconductor with a single non-degenerate val-\nley at the \u0000point, since the key coupling in Fig. 11(b)\noccurs between the central peak of one valley and the\nouter satellite of the opposite valley. Interestingly, in this\ncase the valley degeneracy of Si can be considered as ben-\neficial, while in other scenarios it is often considered to\nbe problematic. In addition, the spin-orbit enhancement\nmechanism has similarities to holes in semiconductors,\nfor which the enhancement arises due to degeneracy at\nthe valence-band edge. In contrast to the valence-band\ncase, however, which has degenerate bands at the same\nmomentum, the degenerate zvalleys in Si require a peri-\nodicpotentialintheformofGeconcentrationoscillations\nto enhance the coupling strength.\nV. CONCLUSIONS\nWe have shown that the inclusion of periodic Ge con-\ncentrations oscillations within the quantum well region of\na Si/SiGe heterostructure leads to enhanced spin-orbit\ncoupling when the oscillation wavelength \u0015is properly\nchosen. Specifically, we find that the Dresselhaus spin-\norbit coupling coefficient is enhanced by over an order of\nmagnitude when \u0015\u00191:57nm, as shown in Figs. 4 and\n5. We have provided a detailed explanation for this be-\nhavior: the Ge concentration oscillations produce wave\nfunction satellites in momentum space which can cou-\nple strongly to the valley minima through Dresselhaus\nspin-orbit coupling provided that the satellite-valley sep-\naration is approximately 4\u0019=ain the extended Brillouin\nzone as shown in Fig. 11. Importantly, the region of en-\nhancement in Fig. 4 is quite wide in \u0015space, which has\nthe important implication that the wiggle well structure\nshould allow for rather large growth errors in the Ge con-\ncentration profile while maintaining the enhanced spin-\norbit effect. Additionally, our results indicate that the\nspin-orbit enhancement is robust against alloy disorder,\nas shown in Fig. 7.\nEnhancement of both the Dresselhaus and Rashba co-\nefficients at smaller \u0015values have also been found in14\nFigs. 4(c) and 4(d), although these bumps are much nar-\nrower in width than the \u0015\u00191:57nm bump, making\nsuch structures more challenging to fabricate. Assuming\nthat the wiggle well with \u0015\u00190:27nm can be practi-\ncally realized, however, this period is quite attractive as\nit could provide the enhanced Dresselhaus spin-orbit cou-\npling studied here along with a huge deterministic valley\nsplitting [18, 19].\nWith regards to possible applications, the enhanced\nspin-orbit coupling of the wiggle well indicates that\nEDSR can be used for fast, electrically-driven manipula-\ntions of single-spin, Loss-DiVincenzo qubits without the\nuse of micromagnets. Indeed, a fast, spin-orbit driven\nEDSR capability is one of the main attractive features\nof hole-spin qubits [17, 44–47], and has recently also\nattracted interest in Si electron-spin qubits [48]. This\npossibility is supported by our calculations in Sec. III,\nwhere it was shown that an EDSR Rabi frequency of\n\nRabi=B > 500MHz/T can be obtained near the opti-\nmal Ge oscillation wavelength \u0015= 1:57nm. It is also\npossible that the enhanced spin-orbit coupling between\nthe valleys may be used to drive fast singlet–triplet ro-\ntations near the valley-Zeeman hot spot [49]. Finally, we\nmention that the enhanced and spatially varying spin-\norbit coupling may have interesting effects on many-body\nphysics in multi-electron dots [33, 50–53].\nACKNOWLEDGEMENTS\nResearch was sponsored in part by the Army Research\nOffice (ARO) under Award No. W911NF-17-1-0274 and\nNo. W911NF-22-1-0090. The views, conclusions, and\nrecommendationscontainedinthisdocumentarethoseof\nthe authors and are not necessarily endorsed nor should\nthey be interpreted as representing the official policies,\neither expressed or implied, of the Army Research Of-\nfice (ARO) or the U.S. Government. The U.S. Govern-\nment is authorized to reproduce and distribute reprints\nfor Government purposes notwithstanding any copyright\nnotation herein.\nAppendix A: Lattice constants in strained Si/SiGe\nquantum wells\nTo determine the lattice constants of the strained\nSi/SiGe heterostructure, we use pseudomorphic bound-\nary conditions [31, 54], where the in-plane lattice con-\nstantakthroughout the system is given by the relaxed\nlattice constant of the Si 0.7Ge0.3barrier regions. This\nthen also sets the lattice spacing along the growth direc-\ntion as described below.\nWe take the relaxed lattice constant of a Si 1-xGexalloy\nas\nao(x) = (1\u0000x)aSi+xaGe; (A1)whereaSi= 0:5431nm andaGe= 0:5657nm are\nthe relaxed lattice constants of Si and Ge, respectively.\nOur structure therefore has an in-plane lattice constant\nak=ao(0:3) = 0:5499nm throughout the entire sys-\ntem. Atomic layers in our system with Ge concentration\nnGe;n6= 0:3are therefore strained. Explicitly, the in-\nplane strain \u000fk;nof layernis\n\u000fk;n=ak\u0000ao(nGe;n)\nao(nGe;n): (A2)\nFor a bulk Si1-x0Gex0alloy under biaxial stress perpen-\ndicular to the [001], we have strains \u000fxx=\u000fyy=\u000fkand\n\u000fzz=\u000f?that are related by [54]\n\u000f?(x0) =\u00002C12(x0)\nC11(x0)\u000fk; (A3)\nwhereC12andC11are elastic constants that depend\non the Ge concentration x0. For pure Si ( x= 0), we\nhaveC11(0) = 165:8GPa andC12(0) = 63:9GPa, while\nfor pure Ge ( x= 1), we have C11(1) = 131:8GPa and\nC12(1) = 48:3GPa [28]. For simplicity, we assume that\nthe elastic constants vary linearly with the Ge concentra-\ntion For a well region composed of Si1-x0Gex0, we would\nthen have the out-of-plane lattice constant,\na?(x0) =\u0010\n1 +\u000f?(x0)\u0011\nao(x0): (A4)\nIn our system with its inhomogeneous Ge concentration\nprofile, we take the lattice spacing between layers nand\nn+ 1with Ge concentrations nGe;nandnGe;n+1, respec-\ntively, asa(n+1;n)\n?=4, where\na(n+1;n)\n?=1\n2\u0010\na?(nGe;n+1) +a?(nGe;n)\u0011\n;(A5)\nis the average of the out-of-plane lattice constants ex-\npected for strained regions with Ge concentration nGe;n\nandnGe;n+1, respectively.\nAppendix B: Pseudospin basis transformation details\nIn this appendix, we provide details of the pseudospin\nbasis introduced in Sec. IIB of the main text. The pseu-\ndospin basis is defined as\n\f\fkkn\u0016\u0017\u0016\u001b\u000b\n=X\n\u0017\u001b\f\fkkn\u0017\u001b\u000b\nU(n)\n\u0017\u001b;\u0016\u0017\u0016\u001b; (B1)\nwhere \u0016\u0017and\u0016\u001blabel the new orbitals with \u0016\u001b=*;+being\napseudospin label,\u0017and\u001bare indices of the original\nbasis with\u001b=\";#simply denoting spin, and U(n)is the\ntransformation matrix of layer n.\nOur first requirement of the new basis is that it diag-\nonalizes the onsite spin-orbit coupling. Following Chadi\n[29], spin-orbit coupling is taken to be an intra-atomic\n(onsite) coupling between porbitals and enters into the15\nHamiltonian as the matrix S. The explicit form of the\nSmatrix of atom jin atomic layer nin thep-orbital\nsubspacefjpz\"i;jpx#i;jpy#i;jpz#i;jpx\"i;jpy\"igis\nS(nj)=\u0001(nj)\nSO\n30\nBBBBB@0\u00001i0 0 0\n\u00001 0i0 0 0\n\u0000i\u0000i0 0 0 0\n0 0 0 0 1 i\n0 0 0 1 0\u0000i\n0 0 0\u0000i i 01\nCCCCCA;(B2)\nwhere \u0001(nj)\nSOis the spin-orbit energy. All matrix elements\ninvolvings,s\u0003, anddorbitals are set to zero in Sand\nare not shown. Note that spin-orbit coupling, in prin-\nciple, does exist between d-orbitals, but is much smaller\nthan thep-orbital couplings and is typically neglected\n[27]. Also note that the spin-orbit energy depends on\nif the atom is Si or Ge, but the form of the spin-orbit\ncoupling matrix is independent of atom type. It turns\nout that the Smatrix is diagonalized by the eigenstates\nof total angular momentum [16]. Within the p-orbital\nsubspace, these states are given by\njp1*i=ip\n2(jpx#i\u0000ijpy#i); (B3)\njp2*i=1p\n6(2jpz\"i\u0000jpx#i\u0000ijpy#i);(B4)\njp3*i=1p\n3(jpz\"i+jpx#i+ijpy#i);(B5)\njp1+i=ip\n2(jpx\"i+ijpy\"i); (B6)\njp2+i=1p\n6(2jpz#i+jpx\"i\u0000ijpy\"i);(B7)\njp3+i=1p\n3(jpz#i\u0000jpx\"i+ijpy\"i);(B8)\nwhere*;+arepseudospin labels. Wefindjp1\u0016\u001biandjp2\u0016\u001bi\nboth have an Smatrix eigenvalue of \u0001(nj)\nSO=3, whilejp3\u0016\u001bi\nhas an eigenvalue of \u00002\u0001(nj)\nSO=3. We also define pseu-\ndospinsands\u0003orbitals as\njs*i=js\"i; (B9)\njs\u0003*i=js\u0003\"i; (B10)\njs+i=js#i; (B11)\njs\u0003+i=js\u0003#i; (B12)\nand pseudospin d-orbitals as\njd1*i=ip\n2(jdzx#i\u0000ijdyz#i); (B13)\njd2*i=1p\n6(2jdz2\"i\u0000jdzx#i\u0000ijdyz#i);(B14)\njd3*i=1p\n3(jdz2\"i+jdzx#i+ijdyz#i);(B15)\njd4*i=jdxy\"i; (B16)\njd5*i=\f\fdx2\u0000y2\"\u000b\n; (B17)jd1+i=ip\n2(jdzx\"i+ijdyz\"i); (B18)\njd2+i=1p\n6(2jdz2#i+jdzx\"i\u0000ijdyz\"i);(B19)\njd3+i=1p\n3(jdz2#i\u0000jdzx\"i+ijdyz\"i)(B20)\njd4+i=jdxy#i; (B21)\njd5+i=\f\fdx2\u0000y2#\u000b\n: (B22)\nNote that all pseudospin s,s\u0003, andd-orbitals are triv-\nially eigenstates of the spin-orbit matrix Swith eigen-\nvalue 0. In addition, note that for j= 1;2;3,jdj*iand\njdj+iare found fromjpj*iandjpj+i, respectively, by\nlettingpx;py;pz!dzx;dyz;dz2. The other pseudospin\ndorbitals and sorbitals are trivially related to the orig-\ninal basis. Note that we adopt these altered d-orbital\npseudospin states even though they possess no spin-orbit\ncoupling such that we obtain the pseudospin structure\nof the hopping matrices in Eqs. (12 - 14) of the main\ntext. Failure to adopt these d-orbital pseudospin states\nwould result in coupling between the pseudospin sectors\nforkk= 0.\nSecondly, we require the pseudospin basis to transform\nthe Hamiltonian in such a way that the minimum unit\ncell (in the absence of an inhomogenous potential Vn)\ndecreases from two sites to one site for kk= 0. In other\nwords, the Fourier-transformed hopping matrix, which is\nintroduced in Eq. (9) of the main text, must become site\nindependent for kk= 0. Naively adopting the orbitals\ndefined in Eqs. (B3 - B22) for every site does not fulfill\nthis requirement. However, this requirement is fulfilled if\nwe adopt the orbitals defined in Eqs. (B3 - B22) if the\njp1*i,jp1+i,jd1*i,jd1+i,jd4*i, andjd4+i, orbitals\nare multiplied by (\u00001)n, wherenis the site index in the\n1D chain. In other words, we flip the sign of these select\norbitals on every other site. As stated in the main text,\nthis alternating structure for the transformation matrix\nis due to the presence of two sublattices in the diamond\ncrystal structure of Si as shown in Fig. 2 (a).\nFor clarity, we now provide the explicit form of U(n).\nWe writeU(n)as\nU(n)=0\nB@U(n)\ns 0 0\n0U(n)\np 0\n0 0U(n)\nd1\nCA; (B23)\nwhereU(n)\ns,U(n)\np, andU(n)\ndare matrix blocks which de-\nscribehowthe s,p, anddorbitalstransform, respectively.\nHere,U(n)\ns=I4\u00024is just identity. The pblock is given16\nby\nU(n)\np=0\nBBBBBBBBB@02p\n61p\n30 0 0\n0 0 0i(\u00001)n\np\n21p\n6\u00001p\n3\n0 0 0(\u00001)n+1\np\n2\u0000ip\n6ip\n3\n0 0 0 02p\n61p\n3\ni(\u00001)n\np\n2\u00001p\n61p\n30 0 0\n(\u00001)n\np\n2\u0000ip\n6ip\n30 0 01\nCCCCCCCCCA;(B24)\nwhere the columns correspond to the pseudospin orbitals\nin the orderfp1*;p2*;p3*;p1+;p2+;p3+g, and the\nrows correspond to the “standard” orbitals in the order\nfpz\";px\";py\";pz#;px#;py#g. Finally, the dblock is\nU(n)\nd=0\nBBBBBBBBBBBBBBBBB@02p\n61p\n30 0 0 0 0 0 0\n0 0 0 0 0i(\u00001)n\np\n21p\n6\u00001p\n30 0\n0 0 0 0 0(\u00001)n+1\np\n2\u0000ip\n6ip\n30 0\n0 0 0 (\u00001)n0 0 0 0 0 0\n0 0 0 0 1 0 0 0 0 0\n0 0 0 0 0 02p\n61p\n30 0\ni(\u00001)n\np\n2\u00001p\n61p\n30 0 0 0 0 0 0\n(\u00001)n\np\n2\u0000ip\n6ip\n30 0 0 0 0 0 0\n0 0 0 0 0 0 0 0 ( \u00001)n0\n0 0 0 0 0 0 0 0 0 11\nCCCCCCCCCCCCCCCCCA; (B25)\nwhere the columns correspond to the pseudospin orbitals\nin the orderfd1*;d2*;d3*;d4*;d5*;d1+;d2+;d3+\n;d4+;d5+g, and the rows correspond to the “standard”\norbitals in the order fdz2\";dzx\";dyz\";dxy\";dx2\u0000y2\"\n;dz2#;dzx#;dyz#;dxy#;dx2\u0000y2#g. Notice the (\u00001)n\nfactors that flips the signs of the jp1*i,jp1+i,jd1*i,\njd1+i,jd4*i, andjd4+iorbitals on every odd site. This\nallows for a unit cell containing only one site for kk= 0\nas described above.\nAppendix C: Symmetry argument for the hopping\nstructure of the Rashba and Dresselhaus\nHamiltonian components\nIn Sec. IIC of the main text, we found the Fourier\ntransformed hopping matrix eT(n)\n\u0006(kk)to have the form\neT(n)\n\u0006(kk) =eT(n)\no+eT(n)\nR(kk)\u0006eT(n)\nD(kk) +O(k2\nk);(C1)\nwhereeT(n)\nR(kk)andeT(n)\nD(kk)are the Rashba and Dres-\nselhaus hopping matrices, respectively, and are given by\neT(n)\nR(kk) = \b(n)(ky\u001bx\u0000kx\u001by);(C2)\neT(n)\nD(kk) = \b(n)(kx\u001bx\u0000ky\u001by);(C3)with \b(n)being a real-valued 10\u000210matrix with van-\nishing diagonal elemental. Importantly, the sign in front\nof the Rashba hopping matrix eT(n)\nRis site independent,\nwhile the sign of the Dresselhaus hopping matrix eT(n)\nD\nchanges sign between every site, as indicated by the \u0006in\nEq. (C1). This can be understood as originating from\nthe diamond crystal structure of the Si by the follow-\ning symmetry argument; Let us consider the case of pure\nSi such thatn\n\b(n);eT(n)\nR;eT(n)\nDo\n!n\n\b;eTR;eTDo\nall lose\ntheir dependence on the layer index. Next, note that\nunder aC4rotation about the z-axis (growth axis), we\nhavefkx;ky;\u001bx;\u001byg!fky;\u0000kx;\u001by;\u0000\u001bxg. This leaves\ninvariant the Rashba term in Eq. (C2) and flips the sign\nof the Dresselhaus term in Eq. (C3). Finally, performing\nthe sameC4rotation on our diamond crystal structure\nin Fig. 2 (a) transforms red atoms into blue atoms and\nvice versa in the sense that the nearest neighbor vectors\nof even and odd atomic layers swap. In other words, the\ntwo sublattices of the Si lattice swap. This in turn, swaps\neT+andeT\u0000within the tight binding chain. Clearly then,\nthe Rashba hopping term eTR, being invariant under a\nC4rotation, should contain the part common to eT+and\neT\u0000. Incontrast, theDresselhaushoppingterm eTDshould\ncontain the part which is different between eT+andeT\u0000,17\nsince it flips sign under a C4rotation. This then explains\nthe\u0006in front ofeT(n)\nDin Eq. (C1).\nAppendix D: \nand \bmatrices\nIn Sec. IIC of the main text, we introduced the 10\u000210\nmatrices \n(n)and\b(n)as components of the hopping\nmatrices in Eqs. (12 - 14). These matrices can be further\ndecomposed into the block forms,\n\n(n)=2\n64\n(n)\n00 \n(n)\n01 0\n\u0000\n(n)T\n01 \n(n)\n11 0\n0 0 \n(n)\n223\n75; (D1)\n\b(n)=2\n64\b(n)\n00 \b(n)\n01\b(n)\n02\n\b(n)T\n01 \b(n)\n11\b(n)\n12\n\u0000\b(n)T\n02 \b(n)T\n12 03\n75;(D2)\nwhere the shapes of the diagonal blocks are 5\u00025,\n4\u00024, and 1\u00021, respectively, and the diagonal block\nmatrices satisfy \n(n)T\nii = \n(n)\niiand \b(n)T\nii =\u0000\b(n)\nii.\nNote that this implies that all diagonal elements of\n\b(n)are zero. Here, the orbital ordering used is\nfs;s\u0003;p1;d2;d3;p2;p3;d1;d4;d5g. Generically, these ma-\ntrices depend on the layer index ndue to Ge concentra-\ntion changing from layer to layer. In the case of a uni-\nform Ge concentration, however, this layer dependence\ngoes away,\b\n\n(n);\b(n)\t\n!f\n;\bg. In the particular case\nof an unstrained Si system, the \nmatrix blocks (in eV)are given by\n\n00=0\nBBB@\u00003:73\u00002:78 0 0 0\n\u00002:78\u00009:03 0 0 0\n0 0\u00000:73\u00001:71 2:42\n0 0\u00001:71 1:63 2:03\n0 0 2 :42 2:03 0:191\nCCCA;(D3)\n\n11=0\nB@0:73 0 1 :71\u00002:42\n0 0:73\u00002:42\u00001:71\n1:71\u00002:42 1:24 0\n\u00002:42\u00001:71 0 1 :241\nCA; (D4)\n\n22= 3:06; (D5)\n\n01=0\nBBB@2:74 1:94 0\u00002:59\n2:89 2:05 0\u00000:90\n2:15\u00003:04 0:19 0\n\u00002:07\u00001:60 0:70\u00002:92\n\u00001:60 0:94\u00000:99\u00002:071\nCCCA; (D6)\nand the \bmatrix blocks (in eV \u0001Å) are given by\n\b00=0\nBBB@0 0 3 :23 1:43\u00002:03\n0 0 3 :40 0:50\u00000:70\n\u00003:23\u00003:40 0\u00001:25\u00000:89\n\u00001:43\u00000:50 1:25 0 1 :72\n2:03 0:70 0:89\u00001:72 01\nCCCA;(D7)\n\b11=0\nB@0 3:57\u00000:15\u00000:10\n\u00003:57 0\u00000:1 0:15\n\u00000:15\u00000:1 0\u00001:16\n\u00000:10\u00000:15 1:16 01\nCA; (D8)\n\b02=\u00000 0 2:66 1:72 2:43\u0001\n; (D9)\n\b12=\u0000\u00001:54 2:17 2:98 0\u0001\n: (D10)\nNote that these \nand\bare precisely the matrices used\nin the simplified model of Sec. IV, where Ge atoms are\ntreated as Si atoms with orbitals shifted up by a constant\nenergyEGe.\n[1] D. Loss and D. P. DiVincenzo, Quantum computation\nwith quantum dots, Phys. Rev. A 57, 120 (1998).\n[2] R. Hanson, L. P. Kouwenhoven, J. R. Petta, S. Tarucha,\nand L. M. K. Vandersypen, Spins in few-electron quan-\ntum dots, Rev. Mod. Phys. 79, 1217 (2007).\n[3] G. Burkard, T. D. Ladd, J. M. Nichol, A. Pan, and\nJ.R.Petta,Semiconductorspinqubits,arXiv:2112.08863\n(2021).\n[4] A.Chatterjee, P.Stevenson, S.D.Franceschi, A.Morello,\nN. P. de Leon, and F. Kuemmeth, Semiconductor qubits\nin practice, Nature Reviews Physics 3, 157 (2021).\n[5] X. Zhang, H.-O. Li, G. Cao, M. Xiao, G.-C. Guo, and G.-\nP. Guo, Semiconductor quantum computation, National\nScience Review 6, 32 (2019).\n[6] F. A. Zwanenburg, A. S. Dzurak, A. Morello, M. Y. Sim-\nmons, L. C. L. Hollenberg, G. Klimeck, S. Rogge, S. N.\nCoppersmith, and M. A. Eriksson, Silicon quantum elec-\ntronics, Rev. Mod. Phys. 85, 961 (2013).[7] M. F. Gyure, A. A. Kiselev, R. S. Ross, R. Rahman, and\nC. G. V. de Walle, Materials and device simulations for\nsilicon qubit design and optimization, MRS Bulletin 46,\n634 (2021).\n[8] J. R. Petta, A. C. Johnson, J. M. Taylor, E. A. Laird,\nA. Yacoby, M. D. Lukin, C. M. Marcus, M. P. Hanson,\nand A. C. Gossard, Coherent manipulation of coupled\nelectron spins in semiconductor quantum dots, Science\n309, 2180 (2005).\n[9] L. V. C. Assali, H. M. Petrilli, R. B. Capaz, B. Koiller,\nX. Hu, and S. Das Sarma, Hyperfine interactions in sili-\ncon quantum dots, Phys. Rev. B 83, 165301 (2011).\n[10] M. Pioro-Ladrière, T. Obata, Y. Tokura, Y.-S. Shin,\nT.Kubo,K.Yoshida,T.Taniyama,andS.Tarucha,Elec-\ntrically driven single-electron spin resonance in a slanting\nZeeman field, Nat. Phys. 4, 776 (2008).\n[11] X. Xue, M. Russ, N. Samkharadze, B. Undseth, A. Sam-\nmak, G.Scappucci,andL.M.K.Vandersypen,Quantum\nlogicwithspinqubitscrossingthesurfacecodethreshold,18\nNature601, 343 (2022).\n[12] A.Noiri,K.Takeda,T.Nakajima,T.Kobayashi,A.Sam-\nmak, G. Scappucci, and S. Tarucha, Fast universal quan-\ntum gate above the fault-tolerance threshold in silicon,\nNature601, 338 (2022).\n[13] A. R. Mills, C. R. Guinn, M. J. Gullans, A. J. Sigillito,\nM. M. Feldman, E. Nielsen, and J. R. Petta, Two-qubit\nsilicon quantum processor with operation fidelity exceed-\ning 99%, Science Advances 8, eabn5130 (2022).\n[14] V.N.Golovach,M.Borhani,andD.Loss,Electric-dipole-\ninduced spin resonance in quantum dots, Phys. Rev. B\n74, 165319 (2006).\n[15] E. I. Rashba, Theory of electric dipole spin resonance in\nquantum dots: Mean field theory with gaussian fluctua-\ntions and beyond, Phys. Rev. B 78, 195302 (2008).\n[16] R. Winkler, Spin-Orbit Coupling Effects in Two-\nDimensional Electron and Hole Systems (Springer,2003).\n[17] L. A. Terrazos, E. Marcellina, Z. Wang, S. N. Copper-\nsmith, M. Friesen, A. R. Hamilton, X. Hu, B. Koiller,\nA. L. Saraiva, D. Culcer, and R. B. Capaz, Theory of\nhole-spin qubits in strained germanium quantum dots,\nPhysical Review B 103, 125201 (2021).\n[18] T. McJunkin, B. Harpt, Y. Feng, M. P. Losert, R. Rah-\nman, J. P. Dodson, M. A. Wolfe, D. E. Savage, M. G.\nLagally, S. N. Coppersmith, M. Friesen, R. Joynt, and\nM. A. Eriksson, SiGe quantum wells with oscillating Ge\nconcentrations for quantum dot qubits, Nature Commu-\nnications13, 7777 (2022).\n[19] Y. Feng and R. Joynt, Enhanced valley splitting in Si\nlayers with oscillatory Ge concentration, Phys. Rev. B\n106, 085304 (2022).\n[20] M. O. Nestoklon, E. L. Ivchenko, J.-M. Jancu, and\nP. Voisin, Electric field effect on electron spin splitting in\nSiGe=Siquantum wells, Phys. Rev. B 77, 155328 (2008).\n[21] M. Prada, G. Klimeck, and R. Joynt, Spin–orbit split-\ntings in Si/SiGe quantum wells: from ideal Si membranes\nto realistic heterostructures, New Journal of Physics 13,\n013009 (2011).\n[22] Y. A. Bychkov and E. I. Rashba, Oscillatory effects and\nthe magnetic susceptibility of carriers in inversion layers,\nJournalofPhysicsC:SolidStatePhysics 17,6039(1984).\n[23] G. Dresselhaus, Spin-orbit coupling effects in zinc blende\nstructures, Phys. Rev. 100, 580 (1955).\n[24] E. L. Ivchenko, A. Y. Kaminski, and U. Rössler, Heavy-\nlight hole mixing at zinc-blende (001) interfaces under\nnormal incidence, Phys. Rev. B 54, 5852 (1996).\n[25] L. Vervoort, R. Ferreira, and P. Voisin, Effects of in-\nterface asymmetry on hole subband degeneracies and\nspin-relaxation rates in quantum wells, Phys. Rev. B 56,\nR12744 (1997).\n[26] J. C. Slater and G. F. Koster, Simplified LCAO method\nfor the periodic potential problem, Phys. Rev. 94, 1498\n(1954).\n[27] J.-M. Jancu, R. Scholz, F. Beltram, and F. Bassani, Em-\npirical spds\u0003tight-binding calculation for cubic semicon-\nductors: Generalmethodandmaterialparameters,Phys.\nRev. B57, 6493 (1998).\n[28] Y. M. Niquet, D. Rideau, C. Tavernier, H. Jaouen, and\nX. Blase, Onsite matrix elements of the tight-binding\nhamiltonian of a strained crystal: Application to silicon,\ngermanium, and their alloys, Phys. Rev. B 79, 245201\n(2009).\n[29] D. J. Chadi, Spin-orbit splitting in crystalline and com-\npositionally disordered semiconductors, Phys. Rev. B 16,790 (1977).\n[30] S. Y. Ren, J. D. Dow, and D. J. Wolford, Pressure de-\npendence of deep levels in GaAs, Phys. Rev. B 25, 7661\n(1982).\n[31] F. Schäffler, High-mobility Si and Ge structures, Semi-\nconductor Science and Technology 12, 1515 (1997).\n[32] M. Friesen, S. Chutia, C. Tahan, and S. N. Coppersmith,\nValley splitting theory of SiGe=Si=SiGequantum wells,\nPhys. Rev. B 75, 115318 (2007).\n[33] C. Yannouleas and U. Landman, Valleytronic full\nconfiguration-interaction approach: Application to the\nexcitation spectra of Si double-dot qubits, Phys. Rev. B\n106, 195306 (2022).\n[34] B. P. Wuetz, M. P. Losert, S. Koelling, L. E. A. Ste-\nhouwer, A.-M. J. Zwerver, S. G. J. Philips, M. T.\nMądzik, X. Xue, G. Zheng, M. Lodari, S. V. Amitonov,\nN. Samkharadze, A. Sammak, L. M. K. Vandersypen,\nR. Rahman, S. N. Coppersmith, O. Moutanabbir,\nM. Friesen, and G. Scappucci, Atomic fluctuations lifting\nthe energy degeneracy in si/sige quantum dots, Nature\nCommunications 13, 7730 (2022).\n[35] M. O. Nestoklon, L. E. Golub, and E. L. Ivchenko, Spin\nand valley-orbit splittings in SiGe=Siheterostructures,\nPhys. Rev. B 73, 235334 (2006).\n[36] M. Veldhorst, R. Ruskov, C. H. Yang, J. C. C. Hwang,\nF. E. Hudson, M. E. Flatté, C. Tahan, K. M. Itoh,\nA. Morello, and A. S. Dzurak, Spin-orbit coupling and\noperation of multivalley spin qubits, Phys. Rev. B 92,\n201401(R) (2015).\n[37] R. Ruskov, M. Veldhorst, A. S. Dzurak, and C. Tahan,\nElectrong-factor of valley states in realistic silicon quan-\ntum dots, Phys. Rev. B 98, 245424 (2018).\n[38] T. B. Boykin, G. Klimeck, M. A. Eriksson, M. Friesen,\nS. N. Coppersmith, P. von Allmen, F. Oyafuso, and\nS. Lee, Valley splitting in strained silicon quantum wells,\nApplied Physics Letters 84, 115 (2004).\n[39] J. S. Townsend, A Modern Approach to Quantum Me-\nchanics(University Science Books, 2012).\n[40] W. Jantsch, Z. Wilamowski, N. Sandersfeld,\nM. Mühlberger, and F. Schäffler, Spin lifetimes and\ng-factor tuning in Si/SiGe quantum wells, Physica E:\nLow-dimensional Systems and Nanostructures 13, 504\n(2002).\n[41] H. Malissa, W. Jantsch, M. Mühlberger, F. Schäffler,\nZ. Wilamowski, M. Draxler, and P. Bauer, Anisotropy of\ng-factor and electron spin resonance linewidth in modu-\nlation doped sige quantum wells, Applied Physics Letters\n85, 1739 (2004).\n[42] C. Kittel and P. McEuen, Introduction to solid state\nphysics(John Wiley & Sons, 2019).\n[43] P. Li and H. Dery, Spin-orbit symmetries of conduction\nelectrons in silicon, Phys. Rev. Lett. 107, 107203 (2011).\n[44] R. Maurand, X. Jehl, D. Kotekar-Patil, A. Corna, H. Bo-\nhuslavskyi, R. Laviéville, L. Hutin, S. Barraud, M. Vinet,\nM. Sanquer, and S. D. Franceschi, A CMOS silicon spin\nqubit, Nature Communications 7, 13575 (2016).\n[45] H. Watzinger, J. Kukučka, L. Vukušić, F. Gao, T. Wang,\nF. Schäffler, J.-J. Zhang, and G. Katsaros, A germanium\nhole spin qubit, Nature Communications 9, 3902 (2018).\n[46] K. Wang, G. Xu, F. Gao, H. Liu, R.-L. Ma, X. Zhang,\nZ. Wang, G. Cao, T. Wang, J.-J. Zhang, D. Culcer,\nX. Hu, H.-W. Jiang, H.-O. Li, G.-C. Guo, and G.-P. Guo,\nUltrafast coherent control of a hole spin qubit in a ger-\nmanium quantum dot, Nature Communications 13, 20619\n(2022).\n[47] C. Kloeffel and D. Loss, Prospects for spin-based quan-\ntum computing in quantum dots, Annual Review of Con-\ndensed Matter Physics 4, 51 (2013).\n[48] W.Gilbert, T.Tanttu, W.H.Lim, M.Feng, J.Y.Huang,\nJ. D. Cifuentes, S. Serrano, P. Y. Mai, R. C. C. Leon,\nC. C. Escott, K. M. Itoh, N. V. Abrosimov, H.-J. Pohl,\nM. L. W. Thewalt, F. E. Hudson, A. Morello, A. Laucht,\nC. H. Yang, A. Saraiva, and A. S. Dzurak, On-demand\nelectricalcontrolofspinqubits,arXiv:2201.06679 (2022).\n[49] R. M. Jock, N. T. Jacobson, M. Rudolph, D. R. Ward,\nM.S.Carroll,andD.R.Luhman,Asiliconsinglet–triplet\nqubit driven by spin-valley coupling, Nature Communi-\ncations13, 641 (2022).\n[50] M.Governale,QuantumdotswithRashbaspin-orbitcou-\npling, Phys. Rev. Lett. 89, 206802 (2002).[51] C. F. Destefani, S. E. Ulloa, and G. E. Marques, Spin-\norbit coupling and intrinsic spin mixing in quantum dots,\nPhys. Rev. B 69, 125302 (2004).\n[52] J. Corrigan, J. P. Dodson, H. E. Ercan, J. C. Abadillo-\nUriel, B. Thorgrimsson, T. J. Knapp, N. Holman,\nT. McJunkin, S. F. Neyens, E. R. MacQuarrie, R. H.\nFoote, L. F. Edge, M. Friesen, S. N. Coppersmith, and\nM. A. Eriksson, Coherent control and spectroscopy of a\nsemiconductorquantumdotWignermolecule,Phys.Rev.\nLett.127, 127701 (2021).\n[53] H. E. Ercan, S. N. Coppersmith, and M. Friesen, Strong\nelectron-electron interactions in Si/SiGe quantum dots,\nPhys. Rev. B 104, 235302 (2021).\n[54] C. G. Van de Walle and R. M. Martin, Theoretical cal-\nculations of heterojunction discontinuities in the Si/Ge\nsystem, Phys. Rev. B 34, 5621 (1986)."    },    {        "title": "1307.4668v1.Non_monotonic_spin_relaxation_and_decoherence_in_graphene_quantum_dots_with_spin_orbit_interactions.pdf",        "content": "Non-monotonic spin relaxation and decoherence in graphene quantum dots with spin-orbit\ninteractions\nMarco O. Hachiya,1Guido Burkard,2and J. Carlos Egues1\n1Instituto de F´ ısica de S˜ ao Carlos, Universidade de S˜ ao Paulo, 13560-970 S˜ ao Carlos, S˜ ao Paulo, Brazil\n2Department of Physics, University of Konstanz, D-78457 Konstanz, Germany\n(Dated: August 17, 2021)\nWe investigate the spin relaxation and decoherence in a single-electron graphene quantum dot with Rashba\nand intrinsic spin-orbit interactions. We derive an e \u000bective spin-phonon Hamiltonian via the Schrie \u000ber-Wol \u000b\ntransformation in order to calculate the spin relaxation time T1and decoherence time T2within the framework\nof the Bloch-Redfield theory. In this model, the emergence of a non-monotonic dependence of T1on the exter-\nnal magnetic field is attributed to the Rashba spin-orbit coupling-induced anticrossing of opposite spin states.\nA rapid decrease of T1occurs when the spin and orbital relaxation rates become comparable in the vicinity of\nthe spin-mixing energy-level anticrossing. By contrast, the intrinsic spin-orbit interaction leads to a monotonic\nmagnetic field dependence of the spin relaxation rate which is caused solely by the direct spin-phonon coupling\nmechanism. Within our model, we demonstrate that the decoherence time T2'2T1is dominated by relaxation\nprocesses for the electron-phonon coupling mechanisms in graphene up to leading order in the spin-orbit inter-\naction. Moreover, we show that the energy anticrossing also leads to a vanishing pure spin dephasing rate for\nthese states for a super-Ohmic bath.\nPACS numbers: Valid PACS appear here\nI. INTRODUCTION\nCarbon-based materials such as graphene and carbon\nnanotubes are of recognized importance for their poten-\ntial spintronic and quantum computation applications. No-\ntably, single-layer graphene, a one-carbon-atom-thick layer\narranged in a honeycomb crystal lattice, has attracted much\ninterest in the last decade due to its unique electronic\nproperties1. The electron spin degree of freedom in graphene\nquantum dots makes them promising candidates for univer-\nsal scalable quantum computing2,3, which would rely on spin\nrelaxation and decoherence times much longer than the gate\noperation times4. Graphene has a relatively weak hyperfine\ninteraction and spin-orbit (SO) couplings. A graphene sheet\nis composed naturally of 99% of12Cwith nuclear spin 0, and\nof 1%13Cwith nuclear spin 1 =2, leading us to long dephas-\ning times in carbon-based quantum dots due to a weak hy-\nperfine interaction5. Thus graphene emerges as a good candi-\ndate to host a spin qubit, in contrast to GaAs quantum dots,\nwhose spin dynamics is strongly modified by the nuclear spin\nbath. Moreover, the weak SO couplings in graphene gener-\nates a spin-splitting on the order of tens of \u0016eVdue to the\nlow atomic weight of carbon atoms6,7. Long spin relaxation\ntimes are expected since the mechanisms that enable relax-\nation channels arise as a combined e \u000bect of non-piezo-electric\nelectron-phonon interaction and weak SO coupling.\nDespite the lack of measurements of the spin relaxation\nand dephasing times in graphene quantum dots, experimen-\ntal results have already been reported in a two-electron13C\nnanotube double quantum dot8that has been isotopically-\nenriched. These results showed a non-monotonic magnetic\nfield dependence of the spin relaxation time near the energy\nanticrossing. In this case, the spin relaxation minimum is re-\nlated to the coupling between electron spin in the quantum dot\nand the nanotube deflection9,10.\nIn this paper, we derive a spin-phonon Hamiltonian us-\nFigure 1: Schematic of a gate-tunable circular graphene quantum\ndot setup. An homogeneous magnetic field is applied perpendicu-\nlarly to the gapped graphene sheet. A metallic gate put on top of\nthe graphene defines the confinement potential for a single-electron.\nFigure not drawn to scale.\ning the Schrie \u000ber-Wol \u000btransformation for all mechanisms\nof electron-phonon and spin-orbit interactions. This e \u000bec-\ntive Hamiltonian captures the combined e \u000bect of the SO\ninteraction and electron-phonon-induced potential fluctua-\ntions. Within the Bloch-Redfield theory, we find that a non-\nmonotonic behavior of the spin relaxation time occurs as a\nfunction of the external magnetic field around the spin mix-\ning energy-level anticrossing by the Rashba SO coupling in\ncombination with the deformation potential and bond-length\nchange electron-phonon mechanisms. We predict that the\nmininum of the spin relaxation time T1could be experimen-\ntally observed in graphene quantum dots. This energy anti-\ncrossing takes place between the first two excited energy lev-\nels at the accidental degeneracy for a certain value B\u0003of the\nexternal magnetic field. We treat the accidental degeneracy\nmixed by the Rashba SO coupling using degenerate-state per-arXiv:1307.4668v1  [cond-mat.mes-hall]  17 Jul 20132\nturbation theory. T1strongly increases at the energy anticross-\ning, reaching the same order as the orbital relaxation time11–13.\nIn contrast with carbon nanotubes, the intrinsic SO does not\ncouple these states due to the selection rules in a circular quan-\ntum dot, exhibiting a monotonic magnetic field dependence\nofT1due to direct spin-phonon coupling (deflection coupling\nmechanism). We also demonstrate that pure spin dephasing\nrates vanish in the leading order of the electron-phonon inter-\naction and SO interactions causing a decoherence dominated\nby relaxation processes, i.e. T2=2T1. Moreover, we find\na vanishing spin dephasing rate for a super-Ohmic bath as a\ngeneral property of the energy anticrossing spectrum.\nThis paper is organized as follows: In Sec. II, we introduce\nthe model to describe a circular graphene quantum dot. In\nSec. III, we derive the e \u000bective spin-phonon Hamiltonian. In\nSec. IV, we present a calculation of the spin relaxation time\nT1within the Bloch-Redfield theory. In Sec. V, we discuss the\nvanishing spin dephasing rate within our model. Finally, we\nsummarize our results and draw our conclusions in Sec. VI.\nII. THE MODEL\nIn this section, we introduce the model for a circular and\ngate-tunable graphene quantum dot. Within our model, we\nconsider a gapped graphene taking into account electron-\nphonon coupling mechanisms and spin-orbit interactions. We\nalso analyze the energy spectrum of the quantum dot and its\nenergy-level degeneracy. The degenerate levels are mixed by\nthe Rashba SO coupling, and the energy crossings are re-\nmoved using the standard degenerate perturbation theory.\nA. Graphene quantum dots\nThe low-energy e \u000bective Hamiltonian for graphene is anal-\nogous to the two-dimensional massless Dirac equation. The\ncharacteristic linear dispersion for massless fermions occurs\nat the two non-equivalent points KandK0(valleys), in the\nhoneycomb lattice Brillouin zone. The graphene energy\nbands in the vicinity of these high-symmetry points consti-\ntute a solid-state realization of relativistic quantum mechan-\nics. However, confining electrons in graphene quantum dots\nis a di \u000ecult task, since the particles tend to escape from\nthe electrostatic confinement potential due to Klein tunnel-\ning. This problem can be overcome by putting graphene on\ntop of a substrate, such as SiC14and BN15,16, that induces a\nnon-equivalent potential for each atom of the two carbon sub-\nlattices and adds a mass term to the Hamiltonian17. The sub-\nlattice A(B) will feel a potential parametrized by +(\u0000)\u0001which\nbreaks inversion symmetry, opening a gap 2 \u0001in the electron-\nhole energy spectrum. Combined with the mass term, an ex-\nternal magnetic field Bis necessary to break the time-reversal\nsymmetry and lift the valley degeneracy. Thus it is reasonable\nto confine a single electron in a quantum dot with the restric-\ntion of its being localized in a single valley.\nConsider then, a circular and gate-tunable graphene quan-\ntum dot in an external magnetic field with SO interactionsand the electron-phonon interaction described by the follow-\ning low-energy Hamiltonian for the Kvalley17,\nH=Hd+HZ+HSO+Hph+He\u0000ph; (1)\nwith the quantum dot Hamiltonian Hdand the Zeeman term\nHZ, respectively, given by\nHd=~vF\u0005\u0001\u001b+U(r)+ \u0001\u001bz;HZ=1\n2g\u0016BB\u0001s;(2)\nwhere\u0005=p\u0000eAis the canonical momentum. The vector\npotential is chosen such that B=r\u0002A=(0;0;B), i.e., per-\npendicular to the graphene sheet. Here, vF=106m=s is the\nFermi velocity, U(r)=U0\u0002(r\u0000R) is the circular-shaped elec-\ntrostatic potential, with \u0002(x)=1 for x\u00150 and \u0002(x)=0 for\nx<0. The operator \u001bacts on the pseudospin subspace (A,B\nsublattices), while sacts on the real spin. Both operators \u001b\nandsare represented by Pauli matrices.\nThe SO Hamiltonian for the Kvalley reads18\nHSO=Hi+HR=\u0015i\u001bzsz+\u0015R(\u001bxsy\u0000\u001bysx); (3)\nwhereHiandHRdenote the intrinsic and Rashba SO e \u000bec-\ntive Hamiltonians6, respectively. The intrinsic SO coupling\noriginates from the local atomic SO interaction. At first, only\nthe contribution from the \u001b\u0000\u0019orbital coupling was con-\nsidered, resulting in a second-order term to the intrinsic SO\ncoupling strength \u0015i7. However, some dorbitals hybridize\nwith pzforming a\u0019-band that gives a first-order contribution\nwhich plays a major role in the spin-orbit-induced gap19. The\nRashba SO coupling, also called the extrinsic contribution,\narises when an electric field is applied perpendicular to the\ngraphene sheet. The major contribution of the SO coupling\n\u0015Rcomes from the \u001b\u0000\u0019hybridization7, in contrast with the\nintrinsic case. The Rashba SO could also be enhanced by cur-\nvature e \u000bects in the graphene sheet20.\nThe free phonon Hamiltonian is given by\nHph=X\nq;\u0016~!q;\u0016by\nq;\u0016bq;\u0016 (4)\nwith the dispersion relation !q;\u0016=s\u0016jqjm, where s\u0016is the\nsound velocity and m=1;2 depending on the type of phonon\nbranch.\nFinally, we have the electron-phonon interaction He\u0000ph.\nWe consider long-wavelength acoustic phonons represented\nby two main mechanisms: the deformation potential and the\nbond-length change mechanism21. The former is an e \u000bective\npotential generated by static distortions of the lattice. It is rep-\nresented in the sublattice space as a diagonal energy shift in\nthe band structure. The latter are o \u000b-diagonal terms due to\nmodifications of the bond-length between neighboring carbon\natoms, which causes changes in the hopping amplitude. The\nelectron-phonon interaction in the sublattice space is given\nby21\nHe\u0000ph=X\nq;\u0016qpA\u001a!q;\u0016 \ng1a1g2a\u0003\n2\ng2a2g1a1!\n(eiqrby\nq;\u0016\u0000e\u0000iqrbq;\u0016);\n(5)3\nwhere g1andg2are the deformation potential and bond-length\nchange coupling constants. Here, Ais the area of the graphene\nlayer and\u001ais the mass area density. The constants a1,a2\nand the sound velocities sLA,sTAfor the longitudinal-acoustic\n(\u0016=LA) and transverse-acoustic ( \u0016=TA) modes are given\nin Table I. Both phonon branches have a linear dispersion re-\nlation given by !q;\u0016=s\u0016jqj. Optical phonons are not taken\ninto account in this work, since their energies do not match\nthe Zeeman splitting for typical laboratory fields. The out-\nof-plane phonons ( \u0016=ZA) will be discussed further below.\nNotice that the electron-phonon interaction is spin indepen-\ndent and can only cause a spin relaxation when assisted by the\nSO interaction.\nIn the following subsection, we analyze the bare quantum\ndot spectrum and perform a perturbation theory calculation\nfor degenerate levels treating the SO Hamiltonian as a pertur-\nbative term.\nB. Degenerate state perturbation theory\nIn order to calculate T1andT2, we use the quantum dot\neigenstates perturbed by the SO interaction. Before doing so,\nwe have to get rid of the degeneracies in the quantum dot spec-\ntrum by applying degenerate state perturbation theory. This\nprocedure makes it clearer to define which states constitute\nour spin qubit and where the spin relaxation occurs.\nDue to the selection rules for the matrix elements of the SO\ninteraction22, only the Rashba SO term couples states from the\ndegenerate subspace. Thus we intend to find a linear combina-\ntion of eigenstates from the degenerate subspace of the quan-\ntum dot such that these states are not coupled by the Rashba\nSO HamiltonianHR.\nConsider then, first the bare quantum dot Hamiltonian in\ntheKvalleyHd, withHdjj;\u0017;si=Ej;\u0017jj;\u0017;siand the quantum\ndot wave functions17\nhr;\u001ejj;\u0017;si= j;\u0017;s(r;\u001e)=ei(j\u00001=2)\u001e \n\u001fj;\u0017;s\nA(r)\n\u001fj;\u0017;s\nB(r)ei\u001e!\n: (6)\nThe spinor components \u001fj;\u0017;s\nA;B(r) are proportional to the con-\nfluent hypergeometric functions and are described by the set\nj;\u0017;s, where we introduce the angular ( j=\u00061=2;\u00063=2;:::),\nradial (\u0017=1;2;3;:::) and spin s=\";#quantum numbers.\nMatching the spinors at r=Rresults in a transcenden-\ntal equation for the eigenvalues Ej;\u0017which can be obtained\nnumerically17. Since we are going to calculate the spin relax-\nation rates due to transitions between the lowest three energy\nTable I: Electron-phonon constants and sound velocities for longitu-\ndinal (LA) and transverse (TA) acoustic phonons. The phonon emis-\nsion angle is denoted by \u001eq.\na1 a2 s\u0016(104m=s)\nLA i ie2i\u001eq 1.95a\nTA 0 e2i\u001eq 1.22a\naFrom Ref. 31.\n5.05.56.0B*=6.4T7.07.58.00.00.020.040.060.080.10\nB@TDEd\nGg0¬g1Gg0¬g2\nEg0-Eg0Eg1-Eg0Eg2-Eg0\nB*0.0390.040Figure 2: Magnetic field dependence of the energy di \u000berence be-\ntween the perturbed three lowest energy levels and the ground state in\na circular graphene quantum dot. Our spin qubit is composed by the\nground state and the first excited state with opposite spin orientation.\nSequentially from bottom to top, E\r0\u0000E\r0(solid), E\r1\u0000E\r0(dashed)\nandE\r2\u0000E\r0(dot-dashed). The Rashba SO interaction-induced anti-\ncrossing of the bare quantum dot states E1=2;1;\"andE\u00001=2;1;#, atB=B\u0003\n(solid lines in the inset). The spin relaxation rate takes place between\nthe statesj\r0iandj\r1i(\u0000#\"= \u0000\r0 \r1) before the anticrossing, and\nbetween the states j\r0iandj\r2i(\u0000#\"= \u0000\r0 \r2) after the anticross-\ning. Inset: Blowup of the energy levels in the vicinity of the crossing\nregion.\nlevels of the quantum dot, we restrict ourselvels to the anal-\nysis of the subspace fj+1=2;1;#i;j1=2;1;\"i;j\u00001=2;1;#ig. In-\ncluding the Zeeman spin-splitting, it leads to a crossing of the\nenergy levels E1=2;1;\"andE\u00001=2;1;#, for a certain magnetic field\nB\u0003depending on the size of the quantum dot. The ground state\nj+1=2;1;#iis not degenerate for any value of B. The Rashba\nSO interactionHRcouples two of these states j+1=2;1;\"iand\nj\u00001=2;1;#idue to its selection rule for the angular quantum\nnumber j22, which is given by jj\u0000j0j=1. By contrast, the\nintrinsic SO interaction Hidoes not couple them since its se-\nlection rule isjj\u0000j0j=0. Now, we have to find an appropriate\nlinear combination of the states from the degenerate subspace\nj+1=2;1;\"i;j\u00001=2;1;#iin whichHRbecomes diagonal in or-\nder to remove the accidental energy level degeneracy from the\ndenominator in the usual non degenerate perturbation theory.\nThen, performing standard degenerate state perturbation the-\nory, we obtain the zero-order eigenstates for the three lowest\nenergy levels are given by\n26666666666666666664j\r0i\nj\r1i\nj\r2i37777777777777777775=266666666666666666641 0 0\n0 cos(#=2)ei\u000e\u0000sin(#=2)\n0 sin(#=2)ei\u000ecos(#=2)3777777777777777777526666666666666666664j1=2;1;#i\nj1=2;1;\"i\nj\u00001=2;1;#i37777777777777777775;(7)4\nwith the associated first-order eigenvalues\nE\r0=E1=2;1\u0000~!Z\n2;E\r1;\r2=\u000f+\u0007q\n\u000f2\n\u0000+j\u0001SOj2; (8)\nplotted in Fig. 2. We define \u000f+=(E1=2;1+E\u00001=2;1)=2 and\n\u000f\u0000=(E1=2;1\u0000E\u00001=2;1+~!Z)=2,~!Z=g\u0016BBis the Zeeman\nenergy splitting. Here, \u0001SO=h1=2;1;\"jH Rj\u00001=2;1;#i=\n4\u0019i\u0015RR\ndrr\u001f1=2;1\nA(r)\u001f\u00001=2;1\nB(r), tan#= \u0001 SO=\u000f\u0000and tan\u000e=\nI[\u0001SO]=R[\u0001SO], where I[x] is the imaginary part and R[x]\nthe real part of x. As a result, the Rashba SO induces an en-\nergy gap 2 \u0001SOat the energy anticrossing ( \u000f\u0000=0), as shown\nin Fig. 2. We have two dominant spin components for j\r1i\nandj\r2idepending on whether the spin relaxation takes place\nbefore or after the energy anticrossing region. Before the en-\nergy anticrossing \u0001SO=\u000f\u0000>0,j\r1i\u0019j 1=2;1;\"i+O(\u0001SO=\u000f\u0000)\nandj\r2i\u0019j\u0000 1=2;1;#i+O(\u0001SO=\u000f\u0000). Increasing the magnetic\nfield we go through the energy anticrossing region such that\n#!\u0019=2 when\u000f\u0000=0. As a result, the states from the degen-\nerate subspace hybridize j\r1i\u0019(j1=2;1;\"i\u0000j\u0000 1=2;1;#i)=p\n2\nandj\r2i \u0019 (j1=2;1;\"i+j\u00001=2;1;#i)=p\n2. After the energy\nanticrossing \u0001SO=\u000f\u0000<0,j\r1i\u0019j\u0000 1=2;1;#i+O(\u0001SO=\u000f\u0000) and\nj\r2i\u0019j 1=2;1;\"i+O(\u0001SO=\u000f\u0000). Thus before the energy anti-\ncrossing, the spin relaxation takes place between j\r1i!j\r0i\nand after the energy anticrossing between j\r2i!j\r0i. At the\nenergy anticrossing, the spin up and down are equivalently\nmixed and the orbital relaxation rate dominates over the spin\nrelaxation rate, since the latter is a higher-order process as-\nsisted by the SO interaction11–13. These results will be used\nto study the energy relaxation with spin-flip between excited\nstates and the ground state.\nIII. EFFECTIVE SPIN-PHONON HAMILTONIAN\nThe electron-phonon coupling allows for energy relaxation\nbetween the Zeeman levels via the admixed states with oppo-\nsite spin due to the presence of the SO interaction. To study\nthis admixture mechanism we derive an e \u000bective Hamilto-\nnian describing the coupling of spin to potential fluctuations\ngenerated by the electron-phonon coupling. We perform a\nSchrie \u000ber-Wol \u000btransformation in order to eliminate the SO\ninteraction in leading order23,24,\neH=eSHe\u0000S=Hd+HZ+Hph+He\u0000ph+h\nS;He\u0000phi\n;(9)\nwhere we have retained terms up to O(HSO)25. The oper-\natorSobeys the commutator\u0002Hd+HZ;S\u0003=HSO, with\nS\u0018O (HSO). The termh\nS;He\u0000phi\nrepresents the coupling\nof the electron spin to the charge fluctuations induced by\nthe electron-phonon interaction via the SO interaction (ad-\nmixture mechanism). The operator Scan be rewritten as\nS=(Ld+LZ)\u00001HSOwhere ˆLiis the Liouvillian superoper-\nator defined as LiA=\u0002Hi;A\u0003, where A denotes an arbitrary\noperator. Here, we make the distinction S=SR+Si, where\nSi/\u0015iandSR/\u0015R.For the Rashba SO coupling, we have to consider the new\nbasisfj\r1i;j\r2igcalculated in Sec. II B using perturbation the-\nory for the degenerate levels. As explained in Sec. II B, we\nare interested in transitions from the excited states j\rkito the\nground statej\r0i. In this case, we calculate the matrix element\nof the e \u000bective spin-phonon Hamiltonian h\r0jHR\ns\u0000phj\rki=\nh\r0jHe\u0000ph+h\nSR;He\u0000phi\nj\rki, where\rk=\r1;\r2. We find that\nh\r0jHR\ns\u0000phj\rki=h\r0jHe\u0000phj\rki (10)\n+X\nn;s,\r0\n1(\r0;n;\rk)\nE\r0\u0000En\n+X\nn;s,D\n2(\r0;n;\rk)\nE\rk\u0000En;\nwhere the degenerate subspace is given by D =\nfj+1=2;1;\"i;j\u00001=2;1;#ig. Here, we have defined the product\nof the matrix elements as\n\n1(\r0;n;s;\rk)=h\r0jHRjn;sihn;sjHe\u0000phj\rki; (11)\n\n2(\r0;n;s;\rk)=h\r0jHe\u0000phjn;sihn;sjHRj\rki: (12)\nThe matrix elements of the Rashba SO coupling give the se-\nlection rulejj\u0000j0j=126. These transitions are compati-\nble with the selection rules of the electron-phonon interaction\nmechanisms depending on the order of the dipole expansion\nconsidered in the term e\u0006iq\u0001r22. In this instance, the selection\nrules matchjj\u0000j0j=1 for the first order and zero order of\nthe dipole expansion of the deformation potential (LA) and\nbond-length change (LA, TA), respectively.\nFor the intrinsic SO, the matrix element of the spin-\nphonon Hamiltonian is given by hn0;#jHi\ns\u0000phjn0;\"i=\nhn0;#jh\nSi;He\u0000phi\njn0;\"i, with the ground state set of angular\nand radial quantum numbers n0=(1=2;1), sinceHidoes not\nconnect the quantum states related with the crossed energy\nlevels. Explicitly, we have\nhn0;#jHi\ns\u0000phjn0;\"i/X\nn0,n0\u000ej;j0\u0010\nNAA\nn0n0\u0000NBB\nn0n0\u0011\n; (13)\nwhere NAA\nnn0=R\ndrr\u001fn\nA(r)\u001fn0\nA(r) and NBB\nnn0=R\ndrr\u001fn\nB(r)\u001fn0\nB(r).\nThe selection rule of the intrinsic SO is jj\u0000j0j=0 which\nis compatible with the the zero order and first order of the\ndipole expansion of the deformation potential (LA) and bond-\nlength change (LA, TA), respectively. The functions \u001fn\nA(r)\nand\u001fn\nB(r) are respectively, purely real and purely imaginary.\nThusHi\ns\u0000phcan be rewritten as proportional to hj;\u001djj;\u001d0iwith\n\u001d,\u001d0which is identically zero. Consequently, the admixture\nmechanism due to the intrinsic SO does not contribute to the\nspin relaxation and dephasing process within our model.\nIn addition to the admixture mechanism, the spin relaxation\ncan also take place due to the direct coupling of spin and local5\nout-of-plane deformations of the graphene sheet (deflection\ncoupling mechanism)10,22. Assuming small amplitudes for the\ndisplacement compared to the phonon wavelength, the normal\nvector to the graphene sheet is ˆ n(z)\u0019ˆz+ru(x;y). The dis-\nplacement operator is given by uz=p1=A\u001a!q(eiqrby\u0000e\u0000iqrb),\nwhere we consider linear and quadratic behaviors to the dis-\npersion relation ~!q=~sq+~\u0016q2, where\u0016=p\n\u0014=\u001a, with\nthe bending rigidity \u0014=1:1 eV . The matrix element of the ef-\nfective Hamiltonian containing only the terms connecting the\nZeeman levels of the ground state reads\nhn0;#jHZA\ns\u0000phjn0;\"i=i\u0015ipA\u001a!q\u0010\nqx+iqy\u0011\n(14)\n\u0002\u0010\nNAA\nn0n0+NBB\nn0n0\u0011\n;\nwhere sZA=0:25\u0002103m=s is the sound velocity. Here, only\nthe lowest order of the dipole approximation gives a nonzero\ncontribution.\nThe spin-phonon terms presented here will be used to cal-\nculate the spin relaxation and dephasing rates in the following\nsections.\nIV . SPIN RELAXATION RATES\nIn this section, we calculate the spin relaxation time using\nthe e\u000bective spin-phonon Hamiltonian derived in the previous\nsection. First, we introduce the Bloch-Redfield theory28,29,\nwhich allows us to derive the general expression for the spin\nrelaxation and decoherence times.\nConsider a general Hamiltonian given by H=HS+HB+\nHS B, whereHSdescribes the system, HBa reservoir in ther-\nmal equilibrium (bath) and HS Bdescribes the interaction be-\ntween them. This general Hamiltonian His analogous to\nthe one derived in Sec. III for all electron-phonon mecha-\nnisms and SO interactions via the mapping, HS!H d+HZ,\nHB!H phandHS B!H s\u0000ph. The system and the bath are\nuncorrelated initially, i.e., their spin matrices \u001acan be sepa-\nrated as\u001a(0)=\u001aS(0)\u001aB(0). Nevertheless, as time goes by,\nthe system and the bath become correlated via the interaction\ntermHs\u0000ph. This system dynamics is described by an equa-\ntion of motion for the density matrix in the interaction picture\n(ˆ\u001a=ei(Hd+HZ+Hph)t=~\u001ae\u0000i(Hd+HZ+Hph)t=~) with the bath variables\ntraced out ˆ\u001aS=TrB\u0002ˆ\u001a\u0003as\nd\ndtˆ\u001aS(t)=\u0000i\n~Zt\n0dt0TrBhˆHs\u0000ph(t);hˆHs\u0000ph(t0);ˆ\u001aS(t0)ˆ\u001aB(0)ii\n(15)\nThis equation of motion for the reduced density matrix is\ncalled the Nakajima-Zwanzig equation29. If we assume that\nthe coupling system-bath is weak, this equation can be fur-\nther simplified by neglecting terms up to O(H2\ns\u0000ph) in Eq. (15),\nwhich is equivalent to approximating the density matrix in the\nintegral as\u001a(t)=\u001aS(t)\u001aB(0)+O(Hs\u0000ph) (Born approximation).\nConsidering a phonon bath, we assume that the time evolution\nof the\u001aS(t) depends only on its present value and not on itspast state (Markov approximation), i.e., ˆ \u001a(t0)!ˆ\u001a(t) in the\nintegral of Eq. (15). Taking the matrix elements of Eq. (15)\nbetween the eigenstates of HS, we have that\nd\ndtˆ\u001aS mn(t)=\u0000i\n~!mn\u001amn\u0000X\nk;lRnmkl\u001akl(t) (16)\nwhere\u001amn=hmj\u001ajniand!nm=!n\u0000!m. The term Rnmklis\nthe Redfield tensor\nRnmkl=\u000enmX\nr\u0000+\nnrrk+\u000enkX\nr\u0000\u0000\nlrrm\u0000\u0000+\nlmnk\u0000\u0000\u0000\nlmnk;(17)\nwhere \u0000+\nlmnk=R1\n0dte\u0000i!nkthljHs\u0000phjmihnjHs\u0000ph(t)jki, with\n\u0000+\nlmnk=\u0010\n\u0000\u0000\nknml\u0011\u0003. Here, the overbar denotes the average\nover a phonon bath in thermal equilibrium at temperature T.\nUsing Eq. (16) in the secular approximation where Rnmklis\napproximatedly given by a diagonal tensor and hdSz=dti=\nTr[(d\u001a=dt)S], we can derive the di \u000berential equation describ-\ning time evolution of the average values of the spin compo-\nnents, also known as Bloch equations. The solution for the\nhSzicomponent with a magnetic field applied along the same\ndirection ishSzi(t)=S0\nz\u0000(S0\nz\u0000Sz(0))e\u0000t=T1, where S0\nzis the\nequilibrium spin polarization (ensemble of spin-down elec-\ntrons) and Sz(0) is the initial non-equilibrium spin alignment\nconsidered in the problem (ensemble of spin-up electrons).\nExplicitly, the spin relaxation rate is given by29\n\u0000#\"=1\nT1=2R\u0010\n\u0000+\n\r0\rk\rk\r0+ \u0000+\n\rk\r0\r0\rk\u0011\n; (18)\nEquation (18) can be simplified to\n1\nT1=2\u0019\n~X\nq\f\f\fh\r0jHs\u0000phj\rki\f\f\f2\u000e(~!\r0\rk\u0000~!q)coth ~!\r0\rk\n2kbT!\n:\n(19)\nThe spin relaxation rate is then calculated combining\nEqs. (19) and (10). The contribution due to the deforma-\ntion potential (LA) combined with the Rashba SO coupling is\ngiven by\n\u0000g1:LA\n\r0 \rk=\u0019\n2g2\n1\n~\u001as2\nLA E\rk\u0000E\r0\n~sLA!4Z2\u0019\n0d\u001eqh\n\u0003k\ni(Ag1)i2:(20)\nAnd those due to the bond-length change mechanism for \u0016=\nLA;TA,\n\u0000g2:LA;T A\n\r0 \rk=2\u0019g2\n2\n~\u001as2\u0016 E\rk\u0000E\r0\n~s\u0016!2Z2\u0019\n0d\u001eqh\n\u0003k\ni(Ag2)i2;(21)\nwhere we imply summation over the repeated index i=1;2;3.\nIn the above we have define\n\u0003k\n1(Ag1;g2)=\u0015n\n1h1=2;1;#jAg1;g2j\u00001=2;1;\"i\u001ak; (22)6\n\u0003k\n2(Ag1;g2)=X\nn,(1=2;1)\u0015n\n2h1=2;1;#jAg1;g2jn;#i (23)\n\u0002hn;#jHRj1=2;1;\"i\u001bk;\n\u0003k\n3(Ag1;g2)=X\nn,(1=2;1)\u0015n\n3h1=2;1;#jHRjn;\"i (24)\n\u0002hn;\"jAg1;g2j1=2;1;\"i\u001bk;\nwhere Ag1=a1112x2,Ag2=g2\u0010\n\u001b+a\u0003\n2+\u001b\u0000a2\u0011\n, with\u001b\u0006=(\u001bx\u0006\ni\u001by)=2. Their respective matrix elements are given by\nhnjAg1jn0i=Mnn0\u0010\n\u000ej;j0+1e\u0000i\u001eq+\u000ej;j0\u00001e+i\u001eq\u0011\n;(25)\nwith Mnn0=R\ndrr2\u0010\n\u001fn\nA\u0003\u001fn0\nA+\u001fn\nB\u0003\u001fn0\nB\u0011\n, and\nhnjAg2jn0i=\u0010\ng2a\u0003\n2\u000ej;j0+1NAB\nnn0+g2a2\u000ej;j0\u00001NAB\nn0n\u0011\n;(26)\nwhere NAB\nnn0=R\ndrr\u001fn\nA(r)\u001fn0\nB(r). Here,\u001a\r1=\u0000sin(#=2),\u001b\r1=\ncos(#=2) and\u001a\r2=cos(#=2),\u001b\r2=sin(#=2). The energy-\ndependent denominators are given by \u0015n\n1=1,\u0015n\n2=1=Ek\u0000\nEn+g\u0016BB=2,\u0015n\n3=1=E1=2;1\u0000En\u0000g\u0016BB=2.\nAs stated in Sec. II B, the energy relaxation accompanied\nby a spin-flip transition occurs between the states j\r0iandj\r1i\nbefore the energy anticrossing \u0000#\"= \u0000\r0 \r1, and between the\nstatesj\r0iandj\r2iafter the energy anticrossing \u0000#\"= \u0000\r0 \r2,\nfor all electron-phonon mechanisms \u0000R\n#\"= \u0000g1:LA\n\r0 \rk+ \u0000g2:LA\n\r0 \rk+\n\u0000g2:T A\n\r0 \rk.\nThe contribution from the out-of-plane flexural phonons via\nthe deflection coupling mechanism, calculated using Eq. (19)\ncombined with Eq. (15), is\n\u0000ZA\n#\"=4\u00192\n\u001a\u00152\ni\ng\u0016BB1\nQ(B) \u0000sZA+Q(B)\n2\u0016!3\n(27)\n\u0002\f\f\f\f\fZ\ndr r\u0012\f\f\f\u001fn\nA\f\f\f2\u0000\f\f\f\u001fn\nB\f\f\f2\u0013\f\f\f\f\f2\n;\nwhere we define Q(B)=q\ns2\nZA+4\u0016(g\u0016BB=~), with sZA=\n0:25\u0002103m=s. In the low magnetic field limit, the term \u0000ZA\n#\"\nsimplifies to\n\u0000ZA\n#\"=4\u00192\u00152\ni\n\u001a1\ns5\nZA(g\u0016BB)2\u0002\f\f\f\f\fZ\ndr r\u0012\f\f\f\u001fn\nA\f\f\f2\u0000\f\f\f\u001fn\nB\f\f\f2\u0013\f\f\f\f\f2\n:(28)\nThe magnetic field dependence of T1=(\u0000R\n#\"+ \u0000ZA\n#\")\u00001with all\nthe mechanisms considered in this work is evaluated numeri-\ncally and is presented in Fig. 3. It can be observed that at the\nenergy anticrossing region, the spin relaxation time rapidly\ndecreases, characterizing its non-monotonic behavior induced\nby an external electric field via the Rashba SO interaction.\nNotice that if no external electric field is applied, the spin re-\nlaxation time is monotonic with contributions from only the\nintrinsic SO interaction via deflection coupling mechanism.The magnetic field dependence of the spin relaxation rate\nfor each electron-phonon coupling mechanism can be under-\nstood using the spectral density of the system-bath interaction\nJ\r0\rk(!)=Z1\n\u00001dte\u0000i!th\r0jHs\u0000ph(0)j\rkih\rkjHs\u0000ph(t)j\r0i:(29)\nFurther simplifications in Eq. 18 allow us to find the\nfollowing relation 1 =T1/J\r0\rk(!\r0\rk), where!\r0\rk/\n!Z/g\u0016BB. In a general form, we have that 1 =T1/P\nqKqh\r0jeiq\u0001rj\rkih\rkjHSOj\r0i\u000e(!q\u0000!\r0\rk), where Kq=\nq=p!qsinceHe\u0000ph/Kqe\u0006iq\u0001r. Also,P\nq/R\ndqqd\u00001,\nwhere d=2 is the dimensionality of graphene. Each SO\ncoupling defines the selection rule for the quantum number\njand consequently, the order of the dipole expansion as ex-\nplained in Sec. III. We find that for the Rashba SO coupling,\nJ\r0\rk(!\r0\rk)/!s\nZwith s=4 for the deformation potential\n(LA) and s=2 for the bond-length change mechanism (LA,\nTA). Also, for the intrinsic SO, s\u00152 for the direct spin-\nphonon coupling (ZA). Therefore the spectral density of the\nsystem-bath interaction is super-Ohmic ( s>1) with a strong\ndependence with the bath frequency for all phonons consid-\nered in graphene.\nV . SPIN DEPHASING RATES\nNext we evaluate the spin dephasing rates for all the\nelectron-phonon mechanisms introduced in Sec. III. Within\nthe Bloch-Redfield theory, we can also solve the Bloch equa-\ntions for the spin components perpendicular to the magnetic\nfield, which are given by hSxi(t)=S0\nxcos(!Zt)e\u0000t=T2andD\nSyE\n(t)=S0\nysin(!Zt)e\u0000t=T2, where S0\nx;yare the initial spin po-\nlarizations along the x;ydirections. The decoherence time\nTable II: Parameters for the numerical evaluation of the spin relax-\nation rates. The electron-phonon coupling constants for the deforma-\ntion potential g1and for the bond-length change mechanism g2and\nthe coupling strengths for Rashba \u0015Rfor an external electric field E\nand the intrinsic \u0015iSO couplings. The graphene layer is character-\nized by its mass area density \u001a. The quantum dot parameters are its\nradius R, potential height U0and the substrate-induced energy gap\n\u0001. The system is assumed to be in thermal equilibrium with the bath\nat temperature T.\ng1 30 eVa\ng2 1:5 eVa\n\u0015R 11\u0016eVb\nE 50 V/300 nmc\n\u0015i 12\u0016eVd\n\u001a 7:5\u000210\u00007kg=m2e\nR 35nm\nU0= \u0001 260 meV\nT 100 mK\naFrom Ref. 21.\ncFrom Ref. 6.\ncFrom Ref. 7.\ndFrom Ref. 19.\neFrom Ref. 31.7\n024B*=6.4T81010-1110-910-710-510-310-1\nB@TDT1@sD\n0.81.31.810-310-2\nFigure 3: Magnetic field dependence of the spin relaxation time.\nParameters used in the numerical evaluation are given in Table II.\nContributions from the deformation potential g 1: LA (dark, dotted),\nbond-length change mechanism g 2: LA (dark, dotted) and g 2: TA\n(light, dashed) and the out-of-plane phonons ZA (light, dot-dashed).\nDark solid: the sum of all processes. The minimum in T1occurs at\nthe energy-level anticrossing at B\u0003. Inset: Blowup of the low mag-\nnetic field regime. Competition between the two electron-phonon\ndominant mechanisms: deformation potential and flexural phonons.\nThe absence of Van Vleck cancellation22,30leads to a finite value for\nT1atB=0.\ncan be separated into two contributions: the spin relaxation\nand the pure spin dephasing 1 =T2=1=2T1+1=T\u001e, where the\npure spin dephasing rate is29\n\u0000\u001e=1\nT\u001e=R\u0010\n\u0000+\n\r0\r0\r0\r0+ \u0000+\n\rk\rk\rk\rk\u00002\u0000+\n\r0\r0\rk\rk\u0011\n: (30)\nIn the low-temperature limit, we find that\n1\nT\u001e=lim\n!!0\f\f\fh\r0jHs\u0000phj\r0i\u0000h\rkjHs\u0000phj\rki\f\f\f2\u000e(~!\u0000~!q)2\u0019kbT\n~!:\n(31)\nThe dephasing time can also be rewritten in terms of the\nspectral density of the system-bath interaction as 1 =T\u001e/\nlim!!0J(!)coth (~!=2kbT)/lim!!0J(!)=!.\nAs we have shown in Sec. III, the spectral function for\nall electron-phonon coupling mechanisms considered in this\nwork are super-Ohmic. Thus the spin dephasing vanishes in\nall cases, since 1 =T\u001e/lim!!0!s=!!0, with s>1. In\nother words, there are no phonons available in leading order to\ncause dephasing in graphene quantum dots. The decoherence\ntime T2is determined only by the relaxation contribution, i.e.,\nT2=2T1. Notice that this relation is no longer necessarily\ntrue considering two-phonon processes since the combination\nof emission and absorption energies can fulfill the energy con-\nservation requirement.24\nAditionally, the spin dephasing rate could also vanish at\nthe energy anticrossing for a super-Ohmic bath. Within the\nsubspace spanned by the states fj+1=2;1;\"i;j\u00001=2;1;#ig, the\nHamiltonianHcan be rewritten as H\u001e= \u0001 +(B)1+ \u0001\u0000(B)\u001cz,\nwhere\u001czdenote a Pauli matrix and \u0001\u0006=(E\r3\u0006E\r2)=2.\nThis magnetic field can be divided into two contributions\nB=B0+\u000eB(t): an external source B0and an internal con-tribution\u000eB(t) due to the bath. For small fluctuation of \u000eB(t),\ntheH\u001eis approximatedly given by\nH\u001e=(\u0001\u0000(B0)+@B\u0001\u0000(B0)\u000eB(t))\u001cz; (32)\nwhere we have not included the term proportional to \u0001+112x2\nsince it does not cause spin dephasing. Calculating the spin\ndephasing rate within the Bloch-Redfield theory using Eq. 30,\nwe find that\n1\nT\u001e= 2\n~@B\u0001\u0000(B0)!2\nlim\n!!0RZ1\n0dt0e\u0000i!th\u000eB(0)\u000eB(t0)i;(33)\nwherehA(t)iis the thermal equilibrium expectation value of\nthe operator A(t) on the bath. Therefore the spin dephasing\nrate goes to zero at the energy anticrossing, since @B\u0001\u0000(B0)!\n0. This condition is valid under the assumption that the ther-\nmal average of the fluctuating magnetic field does not diverge.\nFollowing the result given by Eq. (31), the spin dephasing rate\nstill vanishes as long as the spectral density of the system-bath\ninteraction is super-Ohmic, i.e., J(!)/!s, with s>1.\nVI. CONCLUSION\nIn summary, we find a minimum in the spin relaxation time\nas a function of the magnetic field that is induced by the\nRashba SO coupling and is controllable by an external electric\nfield. In larger quantum dots, the intrinsic SO dominates the\nspin relaxation over the Rashba SO contribution at low mag-\nnetic fields. As the magnetic field increases, the extrinsic con-\ntribution takes over, generating a non-monotonic behaviour of\nT1due to the Rashba SO interaction-induced level anticross-\ning. We have also analyzed the spectral density of the system-\nbath interaction for the first-order electron-phonon interaction\nand we have identified a vanishing contribution to the energy-\nconserving dephasing process. Therefore the phonon-induced\npure spin dephasing rate is of the same order of magnitude as\nthe spin relaxation rate, i.e., T2=2T1, in the leading order\nof the electron-phonon interaction. Other mechanisms such\nas nuclear spins from the13Catoms and charge noise com-\nbined with SO interaction could lead to a non-vanishing spin\ndephasing rate. Nevertheless, these mechanism are expected\nto be weak in graphene3,5. Moreover, we have shown that any\nsuper-Ohmic bath has a vanishing spin dephasing rate at the\nenergy anticrossing.\nAcknowledgments\nWe wish to acknowledge useful discussions with P. R.\nStruck and Peter Stano. Funding for this work was provided\nby CNPq, FAPESP and PRP /USP within the Research Sup-\nport Center initiative (NAP Q-NANO) (MOH and JCE) and\nDFG and ESF under grants FOR912, SPP1285, and Euro-\nGraphene (CONGRAN) (GB).8\n1A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov,\nand A. K. Geim, Rev. Mod. Phys. 81, 109 (2009).\n2D. Loss and D. P. DiVincenzo, Phys. Rev. A 57, 120 (1998).\n3B. Trauzettel, D. V . Bulaev, D. Loss, and G. Burkard, Nature\nPhysics 3, 192 (2007).\n4D. P. DiVincenzo, Fortschr. Phys. 48, 771 (2000).\n5J. Fischer, B. Trauzettel, and D. Loss. Phys. Rev. B 80, 155401\n(2009).\n6C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 226801 (2005).\n7H. Min, J. E. Hill, N. A. Sinitsyn, B. R. Sahu, L. Kleinman, and\nA. H. MacDonald, Phys. Rev. B 74, 165310 (2006).\n8H. O. H. Churchill, F. Kuemmeth, J. W. Harlow, A. J. Bestwick, E.\nI. Rashba, K. Flensberg, C. H. Stwertka, T. Taychatanapat, S. K.\nWatson, and C. M. Marcus, Phys. Rev. Lett. 102, 166802 (2009).\n9D. V . Bulaev, B. Trauzettel, and D. Loss, Phys. Rev. B 77, 235301\n(2008).\n10M.S. Rudner and E.I. Rashba, Phys. Rev. B 81, 125426 (2010).\n11D.V . Bulaev and D. Loss, Phys. Rev. B 71, 205324, (2005).\n12P. Stano and J. Fabian, Phys. Rev. B 72, 155410, (2005).\n13P. Stano and J. Fabian, Phys. Rev. B 74, 045320, (2005).\n14S. Y . Zhou, G.-H. Gweon, A. V . Fedorov, P. N. First, W. A. de\nHeer, D.-H. Lee, F. Guinea, A. H. Castro Neto, and A. Lanzara,\nNature Materials 6, 770 (2007).\n15G. Giovannetti, P. A. Khomyakov, G. Brocks, P. J. Kelly, and J.\nvan den Brink, Phys. Rev. B 76, 073103 (2007).\n16J. Slawinska, I. Zasada, P. Kosiski, and Z. Klusek. Phys. Rev. B\n82, 085431 (2010).\n17P. Recher, J. Nilsson, G. Burkard, B. Trauzettel, Phys. Rev. B 79,085407 (2009).\n18The tensor product notation is implicit, i.e., \u001bisj\u0011\u001bi\nsj, for\ni;j=x;y;z.\n19M. Gmitra, S. Konschuh, C. Ertler, C. Ambrosch-Draxl, J. Fabian,\nPhys. Rev. B 80, 235431 (2009).\n20Huertas-Hernando, F. Guinea, and A. Brataas, Phys. Rev. B 74\n155426 (2006).\n21T. Ando, J. Phys. Soc. Jpn. 74, 777 (2005).\n22P.R. Struck and Guido Burkard, Phys. Rev. B 82, 125401 (2010).\n23R. Winkler, Spin-Orbit Coupling E \u000bects in Two-Dimensional\nElectron and Hole Systems (Springer-Verlag, Berlin, 2003).\n24V . N. Golovach, A. Khaetskii, and D. Loss, Phys. Rev. Lett. 93,\n016601 (2004).\n25We do not carry the termh\nS;Hphi\nsince it does not con-\ntribute to the e \u000bective spin-phonon Hamiltonian, i.e., h\r0jH ph+h\nS;Hphi\nj\rki=0.\n26The matrix elements of the Rashba SO and all the electron-phonon\ncoupling mechanisms are presented in Ref. [22].\n27A. V . Khaetskii and Yu. V . Nazarov, Phys. Rev. B 64, 125316,\n(2001).\n28K. Blum, Density Matrix Theory and Applications (Springer, New\nYork, 2012), 2nd ed.\n29L. Chirolli and G. Burkard, Adv. Phys. 57, 225 (2008)\n30J. H. van Vleck, Phys. Rev. 54, 426, (1940).\n31L. A. Falkovsky, Phys. Lett. A 3725189 (2008); J. Exp. Theor.\nPhys. 105, 397 (2007)."    },    {        "title": "1505.07937v1.Topological_Surface_States_Originated_Spin_Orbit_Torques_in_Bi2Se3.pdf",        "content": "1\nTopological Surface States Orig inated Spin-Orbit Torques in Bi 2Se3 \n \nYi Wang,1 Praveen Deorani,1 Karan Banerjee,1 Nikesh Koirala,2 Matthew Brahlek,2 \nSeongshik Oh,2 and Hyunsoo Yang1,* \n \n1Department of Electrical and Computer Engineering, National University of Singapore, \n117576, Singapore \n2Department of Physics & Astronomy, Rutgers Cent er for Emergent Materials, Institute for \nAdvanced Materials, Devices and Nanotechnology, The State University of New Jersey, New \nJersey 08854, USA \n\nThree dimensional topological insulator bismuth selenide (Bi 2Se3) is expected to possess \nstrong spin-orbit coupling and spin-textured to pological surface states , and thus exhibit a \nhigh charge to spin current conversion efficiency. We evaluate spin-orbit torques in Bi\n2Se3/Co 40Fe40B20 devices at different temperatures by spin torque ferromagnetic resonance \nmeasurements. As temperature decreases, the sp in-orbit torque ratio increases from ~ 0.047 \nat 300 K to ~ 0.42 below 50 K. Moreover, we observe a significant out-of-plane torque at \nlow temperatures. Detailed analysis indicates that the origin of the observed spin-orbit torques is topological surface states in Bi\n2Se3. Our results suggest that topological insulators \nwith strong spin-orbit coupling could be promising candidates as highly efficient spin current sources for exploring next generation of spintronic applications. \n \n*eleyang@nus.edu.sg \n2\n    The realization of functional devices su ch as the non-volatile memories and spin logic \napplications is of key importan ce in spintronic research [1].  The functions of these magnetic \ndevices require highly effici ent magnetization manipulation in a ferromagnet (FM), which \ncan be achieved by an external magnetic field or  a spin polarized current by spin transfer \ntorque (STT). Recent advances have demonstrat ed that pure spin currents resulting from \ncharge currents via spin-orbit coupling in heavy metals, such as Pt [2-7], Ta [8-10], and W \n[11], can produce strong spin-orbit torques on the adjacent magnetic layers. The reported \namplitude of spin Hall angles (i.e. efficiency of spin-orbit torques) in Pt and Ta is in the range of ~ 0.012 to ~ 0.15, and in W is ~ 0.33. The exploration for new materials exhibiting new \nphysics and possessing an even higher conversion efficiency between the charge current \ndensity ( J\nc) and spin current density ( Js) is crucial to exploit next generation spintronic \ndevices. \nThe three dimensional (3D) topological insula tors (TI) are a new class of quantum state \nof materials that have an insulating bulk an d spin-momentum-locked metallic surface states \n[12-14]. They exhibit strong spin-orbit coupling  and are expected to show a high charge to \nspin current conversion efficiency. So far, by extensively employing angle-resolved photoemission spectroscopy (ARPES) and spin-resolved ARPES, the Dirac cones and the helical spin polarized topological surface states (TSS) have been observed and the \ntopological nature has been confirmed in TIs [15,16]. The surface state dominant conduction \nhas also been confirmed by thickness dependent transport measurements in Bi\n2Se3 [17].  \nThe TSS in TI is immune to the nonmagnetic impurities due to the time reversal \nsymmetry protection. Although a gap opening in the TSS dispersion was reported in Bi 2Se3 3\ndoped with Fe in the bulk [18], most recently repor ts have confirmed that the TSS is intact in \nBi2Se3 covered with Fe [19,20] or Co [21] with in-plane magnetic anisotropy. The spin \ndependent transport is known to be significant near the Fermi level in the Bi 2Se3 surface \nstates. However, limited spin dependent transport experiments have been focused on TI/FM heterostructures. Only recently, spin-orbit effects have been reported by spin pumping \nmeasurements [22-24] and magnetor esistance measurements [25,26].  Direct charge current \ninduced spin-orbit torque on the FM layer has been demonstrated by spin torque ferromagnetic resonance (ST-FMR) measurement only at room temperature [27] and magnetization switching at cryogenic temperature [28].  It is known that for Bi\n2Se3 the bulk \nchannel provides an inevitable contribution to transport at room temperature and may \ndiminish the signals of spin-orbit torques ar ising from surface states . At low temperatures, \nhowever, the surface contribution should become significant [17], and spin-orbit torques in \nTI/FM heterostructures should be enhanced [28]. \nIn this work, we adopt extensively studied Bi 2Se3 as the TI layer and investigate the \ntemperature dependence of charge-spin conversion efficiency, spin-orbit torque ratio ( || = \nJs/Jc), by the ST-FMR technique in Bi 2Se3/Co 40Fe40B20 heterostructures. In this structure, the \nspin currents generated from ch arge currents flowing in Bi 2Se3 are injected into \nferromagnetic Co 40Fe40B20 layer and exert torques on it. It must be pointed out that the \nspin-orbit torques could be attributed to either the spin Hall effect (SHE) in the Bi 2Se3 bulk, \nRashba-split states at the interface [29-31], or Bi 2Se3 topological surface states [23,27,28,31]. \nWe find that || drastically increases when the temperature decreases to ~ 50 K. As the \ntemperature decreases furthermore, || reaches up to ~ 0.42, which is ~ 10 times larger than 4\nthat at 300 K. In addition, a significant out-of-plane torque is extracted at low temperatures. \nWe argue that our observations could be correlated with the TSS in our Bi 2Se3/Co 40Fe40B20 \nheterostructures.  \n20 quintuple layer (QL, 1 QL  1 nm) of Bi 2Se3 films are grown on Al 2O3 (0001) \nsubstrates using a custom designed SVTA MOSV-2 molecular beam epitaxy (MBE) system \nwith a base pressure < 3 × 10-10 Torr. The detailed procedures for Bi 2Se3 thin film growth can \nbe found in previous reports [17,32]. The temperature dependent resistivity of Bi 2Se3 film is \nmeasured by four probe method. Figure 1(a)  shows a typical characteristic of Bi 2Se3 that the \nsheet resistivity decreases as temperature decreases and then saturates at temperature < 30 K \n[17,33]. High resistivity Co 40Fe40B20 (CFB) is chosen as the FM layer in order to minimize \nthe current shutting effect thru the FM layer. We have prepared five Bi 2Se3/CFB ( t) samples \n(thickness t = 1.5, 2, 3, 4 and 5 nm) and measured their magnetization response as a function \nof external magnetic field as plotted in Fig. 1(b). From the inset of Fig. 1(b), the CFB dead \nlayer in Bi 2Se3/CFB samples is estimated to be 1.36 nm, similar to a recent report in which \nthe Co dead layer at the interface of Bi 2Se3/Co is ~ 1.2 nm [34]. \nThe ST-FMR devices are fabricated by the following process. First, a 5 nm CFB layer is \nsputtered onto the Bi 2Se3 film at room temperature with a base pressure of 3×10-9 Torr \nfollowed by a MgO (1 nm)/SiO 2 (3 nm) capping layer to prevent CFB from oxidation. Then \nthe film is patterned into rectangular shaped mi crostrips (dotted blue line) with dimensions of \nL (130 µm) × W (10  20 µm) by photolithography and Ar ion milling as shown in Fig. 2(a). \nIn the next step, coplanar waveguides (CPWs) are fabricated. Different gaps (10  55 µm) \nbetween ground (G) and signal (S) electrodes are designed to tune the device impedance ~ 50 5\nΩ. A radio frequency (RF) current ( Irf) with frequencies from 7 to 10 GHz and a nominal \npower of 15 dBm from a signal generator (SG, Agilent E8257D) is applied to the \nBi2Se3/CFB bilayer via a bias-tee, and the ST-FMR signal ( Vmix) is detected simultaneously \nby a lock-in amplifier. An in-plane external magnetic field ( Hext) is applied at a fixed angle \n(H) of 35º with respect to the microstrip length direction [6]. We present the data from three \ndifferent devices, denoted as D1, D2 and D3.  \nFigure 2(b) shows the measured ST-FMR signals from D1 at different temperatures \nranging from 20 to 300 K. Vmix can be fitted by a sum of symmetric and antisymmetric \nLorentzian functions, mix s sym ext a asym ext () () VV F H V F H [3,6,27].  From fitting, the symmetric \ncomponent Vs (corresponding to in-plane torque τ|| on CFB) and antisymmetric component Va \n(corresponding to total out-of-plane torque τ) can be determined, simultaneously. \nThe spin-orbit torque ratio from ST-FMR me asurements can be characterized by two \nmethods.  One is to obtain || from the analysis of Vs/Va via \n12\nsa 0s e f f e x t(/) ( / ) [ 1 + ( 4 / ) ]/VV e M t d M H    [3], where t and d represent the thickness of the \nCFB and Bi 2Se3 layer, respectively. Ms is the saturation magnetization of CFB and Meff is the \neffective magnetization. This method (denoted as ‘by Vs/Va’ hereafter) is to date widely used \nin ST-FMR measurements of heavy metals Pt (or Ta)/FM bilayers [3,6,8].  However, one \nassumption of this method is that the Va is only attributed to the Oersted field induced \nout-of-plane torque. However, in the case of a TI , the TSS in TI and/or Rashba-split states at \nthe interface could also contribute to Va, therefore, we cannot estimate the actual || value by \nVs/Va. On the other hand, the second method can avoid such an issue by analyzing only the \nsymmetric component Vs (denoted as ‘by Vs only’ hereafter) using the following equations:6\nrf H\nsym ext\nHs1()4I cos dRFHdV , ss s //JEM t E  , and s/  [6,27],  where Irf \nis the RF current flowing through the device, H/dR d is the angular dependent \nmagnetoresistance at H = 35,  is the linewidth of ST-FMR signal, Fsym (Hext) is a \nsymmetric Lorentzian, τ|| is the in-plane spin-orbit torque on unit CFB moment at H = 0, s \nis the Bi 2Se3 spin Hall conductivity,  is the Bi 2Se3 conductivity,  and E is the microwave \nfield across the device. The second met hod avoids the possible contamination to || arising \nfrom Va, therefore we can extract the || values in Bi 2Se3 by analyzing only  Vs. At the same \ntime, the total out-of-plane torque τ can be derived by using \n12\nrf H 0 eff ext\nasym ext\nHa[1 ( / )]()4/I cos dR M HFdVH   [27], where Fasym (Hext) is an \nantisymmetric Lorentzian. \nFigure 3(a-b) show the τ|| and τ as functions of temperature, respectively, using the 2nd \nmethod. Here, the τ|| (τ) represents the mean value for different RF frequencies. At 300 K, \nthe τ|| is ~ 0.43 Oe for D1 (~ 0.84 Oe for D2 and ~ 0.48 Oe for D3). As the temperature \ndecreases from 300 to 100 K, τ|| for all three devices gradually increases. At ~ 50 K, τ|| shows \na steep increase and finally reaches ~ 5.25 Oe for D1 (~ 4.11 Oe for D2 and ~ 2.26 Oe for D3), which is ~ 10 times larger than that at 300 K. It is noteworthy that the observed drastic temperature dependent behavior of τ\n|| is different from the recently reported results in heavy \nmetals such as Ta [10,35] as well as Pt [6,36,37], where the damping-like torque (equivalent \nto τ|| here), often argued to arise mainly from the SHE, shows a weak temperature \ndependence. This difference indicates the SH E mechanism may not account for the observed \nτ|| in our Bi 2Se3/CFB. Moreover, the τ shows a similar temperature dependent behavior as τ|| 7\nshown in Fig. 3(b). It is worth noting that the difference in τ|| (and τ) among D1, D2 and D3 \ncan be attributed to the slight variation of the Bi 2Se3/CFB interface during the fabrication \nprocess considering recent challenges in TI film  growth and device fabrication. However, a \nqualitatively similar temperature dependence of torques is observed in all devices. \nThe || values as a function of temperature determined by above two methods have been \nshown in Fig. 3(c). From analysis by Vs only, || is ~ 0.047 for D1 (~ 0.113 for D2 and ~ \n0.072 for D3) at 300 K, and increases to ~ 0.158 for D1 (~ 0.225 for D2 and ~ 0.149 for D3) \nas temperature decreases to 100 K. In  this temperature range (100 - 300 K), || has similar \namplitudes as the spin Hall angle in heavy metals such as Pt, Ta, and W [3,8,11,42-44].  \nHowever, || increases sharply as temperature decreases to ~ 50 K and reaches maximum \nvalues of ~ 0.42 for D1 (~ 0.44 for D2 and ~ 0.30 for D3) at lower temperatures, respectively. \nRemarkably, || increases ~ 10 times compared to that at 300 K for D1. Similarly, from the \nanalysis by Vs/Va, || also shows an abrupt increase as temperature decreases to ~ 50 K in Fig. \n3(c). It is worth noting that we use the effective CFB thickness of t = 3.64 nm due to the dead \nlayer for || estimation by Vs/Va at different temperatures. Interestingly, as shown in Fig. 3(d), \nthe ratio of [ || (by Vs only) ‒ || (by Vs/Va)]/|| (by Vs/Va) obtained by two different methods \nincreases as temperature decreases and becomes more significant below ~ 50 K, as discussed \nlater. \nIn the context of spin Hall mechanism, the spin Hall angle ( sh) is found to be almost \nindependent of temperature from Pt [6,36], Ta [45], Cu 99.5Bi0.5, and Ag 99Bi1 [46], which is \nattributed to the extrinsic mechanisms. In some cases, sh shows a gradual increase as the \ntemperature decreases, which behaves as a typical intrinsic mechanism based on the 8\ndegeneracy of d-orbits by spin-orbit coupling [47,48]. In contrast, in our Bi 2Se3/CFB, the \nspin-orbit torque ratio ( ||) shows an abrupt and nonlinear increase as temperature decreases, \nespecially below ~ 50 K. Therefore, the SHE from the Bi 2Se3 bulk is probably not the \ndominant mechanism for our observation of temperature dependent spin-orbit torque (ratio) in Bi\n2Se3/CFB. From the measured ST-FMR signals as shown in Fig. 2(b), we also find that \nthe Rashba-split state at the Bi 2Se3/CFB interface is not the main mechanism for our \nobservations, since the Rashba-split states lead  to opposite direction (and sign) of charge \ncurrent-induced spin polarization (and ||) on the basis of the spin structure [27,31]. Instead, \nwe ascertain that the direction of in-plane spin polarization to the electron momentum in our \nBi2Se3/CFB is consistent with expectations of the TSS of TIs (spin-momentum locking) \n[12-14,27,31,37].  From further analysis [37], we have found that in our devices a large \nportion of the charge current flows through the TSS in Bi 2Se3. The effective || attributed to \nonly TSS is in the range from ~ 1.62 ± 0.18 to ~ 2.1 ± 0.39. \nAs mentioned before, the temperature dependent || obtained from the above two \nmethods shown in Fig. 3(c) should not show any difference, if Va is attributed to only the \ncharge current induced Oersted field. Therefore, the observed difference implies the \nexistence of other contributions to Va (i.e. to τ). For the Bi 2Se3/CFB system, the difference \ncan be attributed to the TSS in Bi 2Se3 [23,27,28,31] and/or Rashba-split states at the \nBi2Se3/CFB interface [29-31]. We analyze τ = τ  τOe as the other contributions to the \nout-of-plane torque, where τ is the total out-of-plane torque as shown in Fig. 3(b),  and τOe is \na partial out-of-plane torque fro m charge current (flowing in Bi 2Se3) induced Oersted field. \nBy using the measured || by Vs only, we can deduce τOe and thus τ by 9\n12\nsa 0s e f f e x t(/ ) ( / ) [ 1 + ( 4 / ) ]/VV e M t d M H   , and  \n12\nrf H 0 eff ext\nOe asy e t\nHa mx[1 ( / )]()4/Ic o s d R MHFHdV    [3,27], where Va is the \nequivalent antisymmetric component only due to the current induced Oersted field ( τOe). As \nshown in Fig. 4(a), the out-of-plane torque ( τ) in all three devices becomes much larger at \nlow temperatures < 50 K, compared to the τ at high temperatures (100 – 300 K). \nConsequently, we can obtain the out-of-plane spin-orbit torque ratio ( ) as a function of \ntemperature by using the same method by which we deduce || from τ|| above. As shown in \nFig. 4(b), we find that  in all three devices also becomes more significant at low \ntemperatures (< 50 K). More interestingly, the  almost has the same order of magnitude \ncompared to ||. \nWe now discuss the origin of the out-of-plane torque. As has been reported recently, a \nRashba-split surface state in two dimensional electron gas (2DEG) coexists with TSS in the \nBi2Se3 surface due to the band bending and structural inversion asymmetry [29,30,49-52]. \nThe Rashba effective magnetic field can be written as TR /( z )ˆ H k  [49-51], where zˆ \nis a unit vector normal to film plane, k is the average electron Fermi wavevector, and αR is a \ncharacteristic parameter of the strength of Rashba splitting in 2DEG. Since the electron \nFermi wavevector can be assumed to show  a weak temperature dependence and the αR \ndecrease as temperature decreases in a typical 2DEG [53,54], HT is expected to decrease as \ntemperature decreases in these semiconductor sy stems. In addition, the similar temperature \ndependent behavior of HT has been recently reported in Ta/CoFeB heterostructures, where HT \ndecreases and eventually almost reaches to zero at low temperatures [10,35]. However, the \nobserved τ (equivalent to HT) in our Bi 2Se3/CFB presents the opposite temperature 10\ndependent behavior which is not in line with  the reports about Rashba induced torques. \nTherefore, we conclude that the Rashba-split surface state in 2DEG of Bi 2Se3 is not the main \nmechanism for the out-of-plane torque ( τ).  \nOn the other hand, a possible out-of-plane spin polarization in the TSS has been \ntheoretically predicted [55,56] and experimentally observed in Bi 2Se3 [57,58], which is \nattributed to the hexagonal warping effect in the Fermi surface [55,59]. This out-of-plane \nspin polarization in the TSS can account for the observed τ especially in the low \ntemperature range (< 50 K) and the τ adds to the τOe [27,31]. Moreover, as shown in Fig. 3(a) \nand 4(a), the out-of-plane torque ( τ) has the same order of magnitude comparable to \nin-plane torque ( τ||) below 50 K ( τ/τ|| ~ 60%) [37], which is in agreement with the behavior \nof hexagonal TSS in TI [55,56]. With the analysis from different aspects, our findings \nespecially in the low temperature range (< 50 K) indicate a TSS origin of spin-orbit torques in Bi\n2Se3/CFB.  \nIn summary, we have studied the temperature dependence of spin-orbit torques in \nBi2Se3/CoFeB heterostructures. As temperature decreases, the spin-orbit torque ratio \nincreases drastically and eventually reaches a maximum value of ~ 0.42, which is almost 10 \ntimes larger than that at 300 K. A significant out-of-plane torque ( τ), in addition to charge \ncurrent induced Oersted field torque ( τOe), can be observed below 50 K. The observed \nspin-orbit torques are attributed to the topological surface states in Bi 2Se3. Our results \nsuggest that topological insulators with strong spin-orbit coupling and spin-momentum locking are promising spin current sources for next generation of spintronic devices. \n 11\nThis work was partially supported by the National Research Foundation (NRF), Prime \nMinister’s Office, Singapore, under its Competitive Research Programme (CRP award no. \nNRFCRP12-2013-01), the Ministry of Educatio n-Singapore Academic Research Fund Tier 1 \n(R-263-000-A75-750) & Tier 2 (R-263-00 0-B10-112), Office of Naval Research \n(N000141210456), and by the Gordon and Betty Moore Foundation’s EPiQS Initiative \nthrough Grant GBMF4418. 12\nReferences \n[1]I.Zutic,J.Fabian,andD.Sarma,Rev.Mod.Phys.76,323(2004).\n[2]I.M.Mironetal.,Nature476,189(2011).\n[3]L.Q.Liu,T.Moriyama, D.C.Ralph,andR.A.Buhrman, Phys.Rev.Lett.106,036601(2011).\n[4]L.Q.Liu,O.J.Lee,T.J.Gudmundsen, D.C.Ralph,andR.A.Buhrman, Phys.Rev.Lett.109,096602(2012).\n[5]T.D.Skinner,M.Wang,A.T.Hindmarch, A.W.Rushforth, A.C.Irvine,D.Heiss,H.Kurebayashi, andA.J.\nFerguson, Appl.Phys.Lett.104,062401(2014).\n[6]Y.Wang,P.Deorani,X.P.Qiu,J.H.Kwon,andH.Yang,Appl.Phys.Lett.105,152412(2014).\n[7]X.Qiu,K.Narayanapillai, Y.Wu,P.Deorani,D.H.Yang,W.S.Noh,J.H.Park,K.J.Lee,H.W.Lee,andH.\nYang,NatureNanotechnol. 10,333(2015).\n[8]L.Q.Liu,C.F.Pai,Y.Li,H.W.Tseng,D.C.Ralph,andR.A.Buhrman, Science336,555(2012).\n[9]J.Kim,J.Sinha,M.Hayashi,M.Yamanouchi, S.Fukami,T.Suzuki,S.Mitani,andH.Ohno,NatureMater.12,\n240(2013).\n[10]X.Qiu,P.Deorani,K.Narayanapillai, K.S.Lee,K.J.Lee,H.W.Lee,andH.Yang,Sci.Rep.4,4491(2014).\n[11]C.‐F.Pai,L.Liu,Y.Li,H.W.Tseng,D.C.Ralph,andR.A.Buhrman, Appl.Phys.Lett.101,122404(2012).\n[12]J.E.Moore,Nature464,194(2010).\n[13]M.Z.HasanandC.L.Kane,Rev.Mod.Phys.82,3045(2010).\n[14]X.‐L.QiandS.‐C.Zhang,Rev.Mod.Phys.83,1057(2011).\n[15]D.Hsiehetal.,Science323,919(2009).\n[16]A.Nishideetal.,Phys.Rev.B81,041309(2010).\n[17]N.Bansal,Y.S.Kim,M.Brahlek,E.Edrey,andS.Oh,Phys.Rev.Lett.109,116804(2012).\n[18]Y.L.Chenetal.,Science329,659(2010).\n[19]M.R.Scholz,J.Sánchez‐Barriga,D.Marchenko, A.Varykhalov, A.Volykhov, L.V.Yashina,andO.Rader,\nPhys.Rev.Lett.108,256810(2012).\n[20]J.Honolkaetal.,Phys.Rev.Lett.108,256811(2012).\n[21]M.Yeetal.,Phys.Rev.B85,205317(2012).\n[22]P.Deorani,J.Son,K.Banerjee, N.Koirala,M.Brahlek,S.Oh,andH.Yang,Phys.Rev.B90,094403(2014).\n[23]Y.Shiomi,K.Nomura,Y.Kajiwara,K.Eto,M.Novak,K.Segawa,Y.Ando,andE.Saitoh,Phys.Rev.Lett.113,\n196601(2014).\n[24]M.Jamali,J.S.Lee,Y.Lv,Z.Y.Zhao,N.Samarth,andJ.P.Wang,arXiv:1407.7940. \n[25]C.H.Li,O.M.J.van'tErve,J.T.Robinson, Y.Liu,L.Li,andB.T.Jonker,NatureNanotechnol. 9,218(2014).\n[26]Y.Ando,T.Hamasaki, T.Kurokawa, K.Ichiba,F.Yang,M.Novak,S.Sasaki,K.Segawa,andM.Shiraishi,\nNanoLett.14,6226(2014).\n[27]A.R.Mellniketal.,Nature511,449(2014).\n[28]Y.Fanetal.,NatureMater.13,699(2014).\n[29]P.D.C.Kingetal.,Phys.Rev.Lett.107,096802(2011).\n[30]H.M.Benia,A.Yaresko,A.P.Schnyder, J.Henk,C.T.Lin,K.Kern,andC.R.Ast,Phys.Rev.B88,081103(R) \n(2013).\n[31]M.H.Fischer,A.Vaezi,A.Manchon, andE.‐A.Kim,arXiv:1305.1328. \n[32]N.Bansaletal.,ThinSolidFilms520,224(2011).\n[33]J.Son,K.Banerjee, M.Brahlek,N.Koirala,S.‐K.Lee,J.‐H.Ahn,S.Oh,andH.Yang,Appl.Phys.Lett.103,\n213114(2013).\n[34]J.Li,Z.Y.Wang,A.Tan,P.A.Glans,E.Arenholz, C.Hwang,J.Shi,andZ.Q.Qiu,Phys.Rev.B86,054430\n(2012).13\n[35]J.Kim,J.Sinha,S.Mitani,M.Hayashi,S.Takahashi, S.Maekawa, M.Yamanouchi, andH.Ohno,Phys.Rev.\nB89,174424(2014).\n[36]L.Vila,T.Kimura,andY.Otani,Phys.Rev.Lett.99,226604(2007).\n[37]SeeSupplemental MaterialatURLfordetailsonspin‐momentum locking,additional dataaboutspin‐orbit\ntorquesandspin‐orbittorqueratiosandanalysis,whichincludesRefs.[38‐41].\n[38]J.Linder,T.Yokoyama, andA.Sudbø,Phys.Rev.B80,205401(2009).\n[39]L.A.Wray,Su‐YangXu,Y.Xia,D.Hsieh,A.V.Fedorov,Y.S.Hor,R.J.Cava,A.Bansil,H.Lin,andM.Z.Hasan,\nNaturePhys.7,32(2011).\n[40]T.Valla,Z.‐HPan,D.Gardner,Y.S.Lee,andS.Chu,Phys.Rev.Lett.108,117601(2012).\n[41]E.Wang,P.Tang,G.Wan,A.V.Fedorov,I.Miltkowski, Y.P.Chen,W.Duan,andS.Zhou,NanoLett.15,2031\n(2015).\n[42]A.Hoffmann, IEEETrans.Magn.49,5172(2013).\n[43]L.Bai,P.Hyde,Y.S.Gui,C.M.Hu,V.Vlaminck, J.E.Pearson,S.D.Bader,andA.Hoffmann, Phys.Rev.Lett.\n111,217602(2013).\n[44]S.Kasai,K.Kondou,H.Sukegawa, S.Mitani,K.Tsukagoshi, andY.Otani,Appl.Phys.Lett.104,092408\n(2014).\n[45]P.Deorani,Y.Wang,andH.Yang,International Magnetics Conference (INTERMAG), GE‐11,Beijing,May\n2015.\n[46]Y.Niimi,H.Suzuki,Y.Kawanishi, Y.Omori,T.Valet,A.Fert,andY.Otani,Phys.Rev.B89,054401(2014).\n[47]G.Y.Guo,S.Murakami, T.W.Chen,andN.Nagaosa,Phys.Rev.Lett.100,096401(2008).\n[48]T.Tanaka,H.Kontani,M.Naito,T.Naito,D.Hirashima, K.Yamada,andJ.Inoue,Phys.Rev.B77,165117\n(2008).\n[49]Yu.A.BychkovandE.I.Rashba,J.Exp.Theor.Phys.Lett.39,78(1984).\n[50]A.Manchon andS.Zhang,Phys.Rev.B78,212405(2008).\n[51]I.M.Miron,G.Gaudin,S.Auffret,B.Rodmacq, A.Schuhl,S.Pizzini,J.Vogel,andP.Gambardella, Nature\nMater.9,230(2010).\n[52]M.Bianchi,D.Guan,S.Bao,J.Mi,B.B.Iversen,P.D.King,andP.Hofmann, NatureCommun. 1,128\n(2010).\n[53]P.Eldridge,W.Leyland,P.Lagoudakis, O.Karimov,M.Henini,D.Taylor,R.Phillips,andR.Harley,Phys.Rev.\nB77,125344(2008).\n[54]M.A.Leontiadou etal.,J.Phys.:Condens. Matter23,035801(2011).\n[55]L.Fu,Phys.Rev.Lett.103,266801(2009).\n[56]O.V.Yazyev,J.E.Moore,andS.G.Louie,Phys.Rev.Lett.105,266806(2010).\n[57]Y.H.Wang,D.Hsieh,D.Pilon,L.Fu,D.R.Gardner,Y.S.Lee,andN.Gedik,Phys.Rev.Lett.107,207602\n(2011).\n[58]M.Nomura,S.Souma,A.Takayama, T.Sato,T.Takahashi, K.Eto,K.Segawa,andY.Ando,Phys.Rev.B89,\n045134(2014).\n[59]K.Kurodaetal.,Phys.Rev.Lett.105,076802(2010).\n 14\nFigure captions  \n \nFIG. 1. (a) Temperature dependent sheet resistivity of Bi 2Se3 films (20 QL). (b) The \nmagnetization versus field ( H) for Bi 2Se3 (20 QL)/CFB ( t) (nominal  thickness t =1.5, 2, 3, 4 \nand 5 nm) at room temperature. The inset show s the magnetization per unit area versus CFB \nthickness. \n \nFIG. 2. (a) The schematic diagram of the ST-FMR measurement, illustrating a bias-tee, lock-in amplifier, RF signal generator (SG), and ST-FMR device with a Bi\n2Se3/CFB (5 nm). \nMicro-strip is denoted by a dashed blue rectangle. (b) The measured ST-FMR signals from a \nBi2Se3/CFB (5 nm) device (D1) at different temperatures.  \n \nFIG. 3. Temperature dependence of (a) τ||, (b) τ, (c) ||, and (d) [ || (by Vs only) ‒ || (by \nVs/Va)]/[|| (by Vs/Va)] in Bi 2Se3/CFB (5 nm) for D1, D2, and D3. The || is analyzed by two \ndifferent methods, by ‘ Vs only’ and by ‘ Vs/Va’. \n \nFIG. 4. (a) Temperature dependent out-of-plane torque ( τ = τ  τOe) and (b) out-of-plane \ntorque ratio ( ) in Bi 2Se3/CFB (5 nm) devices. \n  15\n \nFIG. 1 \n  -200 0 200 400-400-2000200400 CFB 5 nm\n CFB 4 nm\n CFB 3 nm\n CFB 2 nm\n CFB 1.5 nm\n  M/area (emu/cm2)\n  \nH (Oe)123450100200300400\n  \n  M/area\nCFB (nm)MDL = 1.36 nm\n1 10 100200300400\n  \n T(K)Rxx () Bi2Se3 20 QL(a) (b)16\n \nFIG. 2 \n  \n  (a) (b)\n-1000 0 1000-4-20246  50 K\n 20 K\nf = 8 GHz 300 K\n 200 K\n 100 KV (V)\nH (Oe)\nBias-Tee\n\n100 µmSGH\nxy\nHext\nGS\nRef Signal\nLock-in17\n\nFIG. 3 \n  0 50 100 150 200 250 3000.00.40.81.21.6  (,Vs-,Vs/Va)/,Vs/Va D 1\n D 2\n D 3\nT (K)(a) (b)\n(c) (d)\n0 50 100 150 200 250 3000.00.10.20.30.4 D 1\n D 2\n D 3By Vs Only\n D 1\n D 2\n D 3 \n \nT (K)By Vs/Va0 50 100 150 200 250 300012345|| (Oe)\nT (K)\n  \n D 1\n D 2\n D 3\n0 50 100 150 200 250 300012345 (Oe)\nT (K) D 1\n D 2\n D 318\n \nFIG. 4 \n \n0 50 100 150 200 250 3000123\n   \n T (K) (Oe) D 1\n D 2\n D 3(a) (b)\n0 50 100 150 200 250 3000.00.10.20.30.4\nT (K)  D 1\n D 2\n D 3"    },    {        "title": "1908.02232v1.Spectral_properties_of_spin_orbital_polarons_as_a_fingerprint_of_orbital_order.pdf",        "content": "Spectral properties of spin-orbital polarons as a fingerprint of orbital order\nKrzysztof Bieniasz,1, 2, 3,∗Mona Berciu,2, 3and Andrzej M. Ole´ s4, 1,†\n1Marian Smoluchowski Institute of Physics, Jagiellonian University, Prof. S. /suppress Lojasiewicza 11, PL-30348 Krak´ ow, Poland\n2Department of Physics and Astronomy, University of British Columbia, Vancouver, British Columbia, Canada V6T 1Z1\n3Quantum Matter Institute, University of British Columbia, Vancouver, British Columbia, Canada V6T 1Z4\n4Max Planck Institute for Solid State Research, Heisenbergstraße 1, D-70569 Stuttgart, Germany,\n(Dated: 15 May 2019)\nTransition metal oxides are a rich group of materials with very interesting physical properties that\narise from the interplay of the charge, spin, orbital, and lattice degrees of freedom. One interesting\nconsequence of this, encountered in systems with orbital degeneracy, is the coexistence of long\nrange magnetic and orbital order, and the coupling between them. In this paper we develop and\nstudy an effective spin-orbital superexchange model for e3\ngsystems and use it to investigate the\nspectral properties of a charge (hole) injected into the system, which is relevant for photoemission\nspectroscopy. Using an accurate, semi-analytical, magnon expansion method, we gain insight into\nvarious physical aspects of these systems and demonstrate a number of subtle effects, such as orbital\nto magnetic polaron crossover, the coupling between orbital and magnetic order, as well as the orbital\norder driving the system towards one-dimensional quantum spin liquid behavior. Our calculations\nalso suggest a potentially simple experimental verification of the character of the orbital order in the\nsystem, something that is not easily accessible through most experimental techniques.\nI. INTRODUCTION\nIt is a well established fact that the ground state and ex-\ncitations of a Hubbard-like model in the regime of strong\nCoulomb interactions are faithfully reproduced by an ef-\nfective model, derived using second order perturbation\ntheory, which describes almost localized electrons with\nsuppressed charge fluctuations. The simplest and the most\nextensively studied of such models is the t-Jmodel [ 1],\nwhich describes an antiferromagnetic (AF) Heisenberg ex-\nchange interaction between localized spins. Doping away\nfrom half-filling generates an electron (or hole) hopping\nin the subspace without double occupancies, a formidable\nmany-body problem. Notably, this model predicts that\na charge added to the system will produce a string of\nmisaligned spins, when the N´ eel AF state is considered,\nthat would trap it in a linear string potential [ 2–5], while\non the other hand it allows for coherent charge propa-\ngation by means of spin fluctuations [ 6–8] which remove\nthe spin excitations produced by the charge. As such,\nthis is a simple demonstration of a quasiparticle (QP), in\nwhich the charge can only move freely if it couples to the\nmagnetic background of the system.\nIn systems with active orbital degrees of freedom, such a\nlow-energy effective model includes superexchange interac-\ntions between spins and orbitals [ 9,10]. The development\nof multiorbital Hubbard models [ 11,12], most commonly\nemployed in the description of transition metal oxides\nwithdorbital degeneracy, led to the derivation of spin-\norbital superexchange models [ 9], which are t-J-like model\ngeneralizations which accommodate the orbital degrees of\nfreedom on equal footing with electron spins [ 13–26]. Such\n∗krzysztof.t.bieniasz@gmail.com\n†a.m.oles@fkf.mpi.de, corresponding authormodels are composed of products of a spin term, charac-\nterized by the common SU(2) symmetry, and the orbital\npseudospin part of a lower symmetry [ 20], reflecting the or-\nbitals’ spatial extent and their interdependence on lattice\nsymmetry. These models allow not only spin but also or-\nbital long range order in the system, and predict coherent\norbital excitations (orbitons) akin to magnons, to which\na charge can couple in a similar fashion [ 27–29]. However,\nthe unusual properties of orbitons and their interaction\nwith the spin degree of freedom make this problem even\nmore challenging than the one described above. It is for\nthis reason that these models have remained a challenge\nthat requires novel theoretical approaches.\nHere we are primarily interested in egsystems, which\nrealize a pseudospin T=1/2interactions and are thus\nthe closest analogue of the t-Jmodel with S=1/2spins.\nHowever, due to non-conservation of the orbital quantum\nnumber, free propagation of charge will be permitted by\nthe kinetic Hamiltonian, and the interaction with orbitons\nwill primarily make the resulting QP heavier, especially\nin view of the much smaller role played by orbital fluctu-\nations. It was nonetheless suggested that the importance\nof the fluctuations increases with the dimensionality of\ntheegproblem in the case of ferromagnetic (FM) spin\norder, with one-dimensional (1D) alternating orbital (AO)\nsystems being Ising-like [30].\nOn the other hand, for an AF system hole dynamics\nis dominated by orbital excitations which leads to quasi-\nlocalization when AF and AO order coexist [ 31,32]. Here\nwe shall address the interesting complementary question\nof what happens in an intermediate state where AF and\nAO orders exist simultaneously, but in orthogonal di-\nrections, such that the system can be decomposed into\n1D AF chains and orthogonal two-dimensional (2D) AO\nplanes. Such a situation occurs in numerous real three-\ndimensional (3D) systems, in particular in copper-fluoride\nperovskite KCuF 3[33], and in the perovskite manganitearXiv:1908.02232v1  [cond-mat.str-el]  6 Aug 20192\nLaMnO 3[34,35]. Both of these systems are of high inter-\nest either from the point of view of basic research, or novel\nphenomena triggered by spin-orbital interplay. KCuF 3is\na rare example of a nearly perfect 1D spin liquid [ 36,37],\nwhile LaMnO 3has almost perfect orbital order and ap-\nplications stemming from the colossal magnetoresistance\nare found in doped La 1−xSrxMnO 3[38, 39].\nIt is the type of orbital order in spin-orbital systems\nwhich is very intriguing. The orbitals occupied by elec-\ntrons in LaMnO 3are tuned by the tetrahedral field which\nsplits theegorbitals [ 16,40]. It has been realized long ago\nthat the photoemission spectra in LaMnO 3strongly de-\npend on the type of orbital order in the ground state [ 41],\nbut there is no systematic method to measure this order\nexperimentally. Resonance Raman spectroscopy [ 42] and\noptical properties [ 43,44] were proposed to investigate\nthe orbital order but one has to realize that the orbitals\ncouple rather strongly to spins [ 45] and it is thus chal-\nlenging to investigate the hole coupling to spin-orbital\nexcitations in a systematic way. In the regime of interme-\ndiate coupling, the spectral functions could be obtained\nusing the generalized gradient approximation with dy-\nnamical mean-field theory (GGA+DMFT) [ 46]. Below\nwe use the strong coupling approach and show that the\nspectral functions of spin-orbital polarons, obtained from\nthe respective Green’s function, may be used to identify\nthe orbitals occupied in the ground state.\nThe remainder of this paper is organized as follows.\nWe introduce the spin-orbital model with egdegrees of\nfreedom in Sec. II. The variational momentum average\nmethod used to generate the spectra with increasing num-\nber of excitations is described in Sec. III. In Sec. IV we\npresent and discuss the numerical results obtained for two\nrepresentative types of orbital order in the intermediate\nphase with AF/AO order. The paper is summarized with\nmain conclusions in Sec. V. Finally, we present the de-\ntails of the derivation of the mean field phase diagram\nin Appendix A, and some of the more involved steps of\nthe derivation of the fermion-boson polaronic model in\nAppendix B.\nII. THE SPIN-ORBITAL MODEL\nKCuF 3is a tetragonal system (pseudo-cubic to first\napproximation), with Cu( d9) ions placed in octahedral\ncages of fluorides. The crystal-field splitting splits the 3 d\norbitals into the low-lying t2gfilled states and the active eg\nstates. Thus, the copper configuration can be equivalently\ndescribed as e3\ngin terms of electron occupation, or e1\ngin\nterms of hole occupation.\nThe kinetic part of the Hamiltonian includes the\nelectron hopping tbetween two directional orbitals\n|zγ/angbracketright= (3z2\nγ−r2)/√\n6, located on nearest neighbor (NN)\nCu(3d9) sites, where zγ≡x/y/z is parallel to the main\ncubic directions a/b/c of the system [ 47]. The comple-\nmentary orbitals |¯zγ/angbracketright= (x2\nγ−y2\nγ)/√\n2do not contribute\nbecause they are orthogonal to the intermediary ligandF(2p6) orbitals. The above definition of the hopping is\nnot practical, however, due to the orbital basis changing\nwith the hopping direction. Transforming all terms into\nthe{|z/angbracketright,|¯z/angbracketright}basis we find:\nHt=−t\n4/summationdisplay\n/angbracketleftij/angbracketright⊥c/parenleftBig\nd†\nizσ∓√\n3d†\ni¯zσ/parenrightBig/parenleftBig\ndjzσ∓√\n3dj¯zσ/parenrightBig\n−t/summationdisplay\n/angbracketleftij/angbracketright/bardblcd†\nizσdjzσ+ H.c., (1)\nwhere the upper/lower sign corresponds to the in-plane\ndirectionsa/b, respectively. Here, d†\nizσandd†\ni¯zσcreate\nelectrons with spin σin the|z/angbracketrightor the|¯z/angbracketrightorbital, respec-\ntively, at site i.\nThe electron interactions are described using a multi-\norbital Hubbard-like model, including on-site Coulomb\nrepulsionUand Hund’s exchange interaction JHwhich\ndrives the site towards maximal spin. We are interested\nin the strongly correlated limit U/greatermucht, which, when con-\nsidering virtual excitations, e3\nge3\ng\ne2\nge4\ng, leads to an\neffective superexchange model [ 9]. Due to the proximity\nof degeneracy of the egorbitals, one needs to consider the\nmultiplet structure of the e2\ngion. The spectrum of these\nexcitations has four eigenenergies U−3JH,U−JH(dou-\nble), andU+JH[48]. Taking all this into consideration\nleads to the following superexchange Hamiltonian:\nHγ\n1=−2Jr1/summationdisplay\n/angbracketleftij/angbracketright/bardblγ/parenleftbigg\nSi·Sj+3\n4/parenrightbigg/parenleftbigg1\n4−τγ\niτγ\nj/parenrightbigg\n,(2a)\nHγ\n2= 2Jr2/summationdisplay\n/angbracketleftij/angbracketright/bardblγ/parenleftbigg\nSi·Sj−1\n4/parenrightbigg/parenleftbigg1\n4−τγ\niτγ\nj/parenrightbigg\n, (2b)\nHγ\n3= 2Jr3/summationdisplay\n/angbracketleftij/angbracketright/bardblγ/parenleftbigg\nSi·Sj−1\n4/parenrightbigg/parenleftbigg1\n2−τγ\ni/parenrightbigg/parenleftbigg1\n2−τγ\nj/parenrightbigg\n,\n(2c)\nHγ\n4= 2Jr4/summationdisplay\n/angbracketleftij/angbracketright/bardblγ/parenleftbigg\nSi·Sj−1\n4/parenrightbigg/parenleftbigg1\n2−τγ\ni/parenrightbigg/parenleftbigg1\n2−τγ\nj/parenrightbigg\n,\n(2d)\nwhere the{ri}coefficients serve to impose the multiplet\nstructure at finite Hund’s exchange JH>0,\nr1=1\n1−3η, r 2=r3=1\n1−η, r 4=1\n1 +η,(3)\nwith\nη=JH/U, (4)\nwhileτγ\niare bond-direction-dependent orbital operators\nfor the principal cubic axes, which can be expressed using\nthe pseudospin operators in the following way:\nτa/b\ni=−1\n2/parenleftBig\nTz\ni∓√\n3Tx\ni/parenrightBig\n, τc\ni=Tz\ni, (5)\nunder the standard convention,\n|¯z/angbracketright≡|↑/angbracketright,|z/angbracketright≡|↓/angbracketright. (6)3\nIt can be shown that assuming a FM spin state in the\nabplanes and under a purely octahedral crystal field, the\norbital order preferred by the superexchange Hamiltonian\nis AO, with the\n|±/angbracketright= (|¯z/angbracketright±|z/angbracketright)/√\n2 (7)\nstates occupied. However, this need not be the case for\nother magnetic orders. In the general case, the occupied\norbitals are given by rotation of the basis, which is most\nconveniently parametrized with an angle ±(π/2 +φ),\nwhere the sign depends on the orbital sublattice, with\nφ= 0 corresponding to the {|+/angbracketright,|−/angbracketright} reference basis (7).\nFor further convenience, we also introduce an orbital\ncrystal field into the Hamiltonian, which serves to remove\nthe orbital degeneracy of the system [ 16], and to make\nthe model more realistic [49]:\nHz=−Ez/summationdisplay\niTz\ni. (8)\nThis term simulates an axial pressure along the caxis,\nand for large values of |Ez|it supports ferro-orbital (FO)\norder, with occupied states either |¯z/angbracketright(forEz>0) or|z/angbracketright\n(forEz<0). Tuning the orbital field thus allows one to\ndrive the system from AO all the way to FO order in a\ncontinuous manner, although we will not be interested in\nthis extreme limitof the superexchange terms are similar to\nthe crystal field in that they are linear in the τγoperators,\nand thus when these are active ( i.e., when the magnetic\norder is not assumed to be FM) there is an internal orbital\nfield already present in the superexchange Hamiltonian.\nThus, the external field will work either to counter or to\nenhance these terms, in turn affecting the magnetic order.\nIn this way the system incorporates spin-orbit coupling\nthrough indirect means, allowing for the magnetic and\norbital orders to affect each other and, furthermore, to be\ncontrolled through external parameters, such as an axial\npressure.\nIn order to derive an effective polaronic Hamiltonian for\na single charge doped into the system, we need to perform\na series of rather involved steps: (i) determine the classi-\ncal ground state by calculating the mean field energy and\nminimizing it with respect to the crystal field Ez, for more\ndetails see Appendix A; (ii) transform the kinetic part\nof the Hamiltonian (1)to the orbital basis correspond-\ning to the classical ground state; (iii) introduce magnons\nand orbitons (to represent magnetic and orbital excita-\ntions above the classical ground state) as slave bosons\nby means of a Holstein-Primakoff transformation. As\nthese operations are rather tedious and unlikely to be of\nmuch interest to the general audience, we relegate this\nderivation of the polaronic Hamiltonian to Appendix B.\nIt is only important to notice that from this point onward\nwe will be mostly relying on the outlined formalism, and\nthus we will be referring to magnetic and orbital excita-\ntions as magnons (denoted with the operators b†\ni) and\norbitons (denoted as a†\ni), respectively, and treating them\nas well-defined, spinless bosons, while the charge degree\nof freedom will be represented by the spinless fermion f†\ni.The final Hamiltonian consists of the exchange term\nHJ, and the kinetic term Ht. It is important for the\nunderstanding of the paper what physical processes are\nrealized by each of those terms. The exchange term,\nHJ≡H I+HII, is of course responsible for the spin-\norbital order in the presence of the crystal field; here we\nhave conveniently divided it into the terms quadratic in\n(pseudo)spin operators, included in HI, and the linear\n(crystal field like) terms included in HII, see the Ap-\npendix B. After the Holstein-Primakoff transformation\nthese terms are purely bosonic operators, and include the\nIsing terms which only serve to “count” the bosonic en-\nergy, and the fluctuation terms which create and destroy\nthe various bosons without involving the doped charge,\nsimilar to the spin polaron in the t-Jmodel [7].\nThe kinetic Hamiltonian, Ht≡T +V⊥\nt+V/bardbl\nt, on the\nother hand, contains all of the charge dynamics, as shown\nin the Appendix B. The free hopping term Tis restricted\nto the FMabplanes due to spin conservation—any hop-\nping out of plane necessarily produces magnons. The Vt\nterm includes all the processes responsible for the electron-\nboson coupling and constitute the actual interaction in\nour model. Because of the in-plane FM order, the per-\npendicular term, V⊥\nt, can only produce orbitons, while\nits influence on magnons is limited to a fermion-magnon\nswap term. Finally, the out of plane term V/bardbl\ntdescribes\nhole dynamics by the coupling to both magnons and or-\nbitons at the same time. Altogether, these terms represent\nall the fermion-boson coupling processes possible in this\nsystem and include terms as complicated as five particle\ninteractions. Our variational technique, which we will\nbriefly describe in the next section, allows us to include\nall of those terms, something that would not be possible\nto do in more standard polaronic methods relying on the\nlinear spin wave (LSW) approximation.\nIt needs to be emphasized, however, that the present\nmodel employs a number of idealizations ( e.g., we neglect\nthe intermediary oxygen orbitals and proper Jahn-Teller\ninteractions, and ignore any resulting structural transi-\ntions that might occur in the system) and is not intended\nto produce a realistic low energy excitation spectrum, but\nrather to study the effects of spin and orbital excitations\non the charge dynamics in systems with the A-AF/C-AO\nground state, as encountered in KCuF 3and LaMnO 3.\nThe results presented here are therefore not meant to\ndirectly address the experimental results, although some\nof the observed qualitative effects could be relevant to\ninterpret or guide the experiment.\nIII. THE MOMENTUM AVERAGE METHOD\nWe use the well-established momentum average\n(MA) variational method [ 52–55] to determine the one-\nelectron Green’s function, G(k,ω) =/angbracketleftk|G(ω)|k/angbracketright, where\nG(ω) = [ω+iη−H]−1is the resolvent operator and\n|k/angbracketright=f†\nk|0/angbracketrightis the Bloch state for an electron injected4\nFIG. 1. The mean-field phase diagram of the 3D Kugel-\nKhomskii model. We focus on the A-AF/C-AO spin-orbital\norder for which we determine the spectral function (10) occurs\nbetween two AF phases with FO order (white areas), AF z\n(left) and AF ¯z(right). The color scale indicates the detuning\nangleφin degrees. The values of φ= 0 andφ=π/6, found\natη= 0.16 (red dashed line), used to investigate the spectral\nfunctions in the present study, are indicated by ×and +,\nrespectively. Note that a more complete mean-field phase\ndiagram including possible phases described by variational\nwave functions with short-range order was presented before in\nRef. [58].\ninto the undoped, semiclassical ground state |0/angbracketright. The\nHamiltonianHis divided intoH0=T+Hz\nJ, whereHz\nJis\nthe Ising part of the exchange terms in Eq. (A2) (usually,\nthe quantum fluctuations are of little importance and\ncan be ignored, see also Ref. [ 56]), and the interaction,\nV=V⊥\nt+V/bardbl\nt, which might also be extended to include\nthe spin fluctuation terms of the exchange Hamiltonian.\nThe variational MA method uses Dyson’s identity,\nG(ω) =G0(ω) +G(ω)VG0(ω), (9)\nto generate the equations of motion (EOMs) for the\nGreen’s functions, within a chosen variational space.\nSpecifically, evaluation of V|k/angbracketrightin real space links to gener-\nalized propagators that involve various bosons beside the\nfermion; the variational expansion controls which such\nconfigurations are included in the calculation. The EOMs\nfor these generalized Green’s functions are then obtained\nusing the same procedure and the process is continued\nuntil all the variational configurations are exhausted, at\nwhich point this hierarchy of coupled EOMs automatically\ntruncates. The validity and accuracy of the approxima-\ntion is determined by how appropriate is the choice of the\nvariational space; this is usually based on some physically-\nmotivated criterion restricting the spatial spread of the\nbosonic cloud, as exemplified below. The accuracy of the\nresults can be systematically improved by increasing the\nvariational space until convergence is achieved.\nIn this way we generate analytical EOMs that easilyallow for exact implementation of the local constraints\n(i.e., charge and bosons are forbidden from being at the\nsame site simply by removing from the variational space\nthe configurations which violate this constraint). Once\ngenerated, the EOMs form an inhomogeneous system of\nlinearly coupled equations, which is solved numerically to\nyield all the Green’s functions, and in particular G(k,ω)\nfrom which we determine the spectral function,\nA(k,ω) =−1\nπ/IfracturG(k,ω). (10)\nThis quantity is directly measured through angle resolved\nphotoemission spectroscopy for LaMnO 3, or inverse pho-\ntoemission for KCuF 3.\nWe shall be interested in the spectral function obtained\nfor theA-AF/C-AO spin-orbital order phase where both\nmagnon and orbiton excitations may couple to the moving\ncharge. The mean field analysis of this phase includes\nthe energy minimization to select the optimal value of\nthe detuning angle φ, as described in Appendix A. We\ninvestigate two ground states with φ= 0 andφ=π/6\nfound atη= 0.16, shown by the respective symbols in\nFig. 1, and take t≡1.0 as the energy unit.\nOur method, while highly accurate and versatile, does\nnot come without its limitations. The most important\nstems from the very basis of the expansion, namely the\ncut-off criterion being implemented in real space. As a\nconsequence, only local processes can be treated exactly,\nwhile other interactions have to be approximated in a way\ncompatible with this methodology. As such, this method\nis especially well-suited to polaronic problems, where a\ncharge couples to bosonic excitations either on-site or on\nthe nearest-neighboring site, such as in this paper. The\nmost common obstacle here is the treatment of quantum\nfluctuations, which are not tied to the itinerant charge\nand are therefore completely non-local. These are gen-\nerally treated by being included only in the immediate\nneighborhood of the electron, the logic behind this being\nthat only then will they affect the properties of the arising\nQP. This works as long as the classical ground state is not\ntoo different from the true quantum ground state, i.e., the\nclassical state is a good starting point for the expansion.\nThis would make our method tricky to use in 1D, but\nany higher dimensional problem is easily treatable. An-\nother limitation comes from the use of real space Green’s\nfunctions, which are hard to calculate already for a single\nelectron. Treatment of multi-electron problems is an on-\ngoing, highly challenging effort, although this is certainly\ntrue of all semi-analytical Green’s function methods. Here\nwe only focus on single-electron spectral functions, which\nare relevant for photoemission spectroscopies.\nIV. RESULTS AND DISCUSSION\nWe carry out the MA calculation in the variational\nspace defined by configurations with up to 4 bosons\npresent. Because the calculation is done for a 3D system5\nFIG. 2. The spectral functions A(k,ω) (shown by intensity of brown/yellow color) in partial and full variational space for φ= 0\nandJ= 0.1 (left) and J= 0.5 (right). The dashed blue line indicates the free charge dispersion, /epsilon1kφ. The numbers in the\nupper-left corner indicate the maximal number of magnons and orbitons, respectively. The number in the upper-right corner\ngives the size of the variational space. The high-symmetry points are: Γ = (0 ,0),X= (π,0),Y= (0,π),S= (π/2,π/2), and\nM= (π,π). Parameter: η= 0.16.\nwith full treatment of the charge coupling to bosonic de-\ngrees of freedom, the branching factor for the EOMs is\nfar too great to allow us to include more configurations.\nNevertheless, based on our previous research within simi-\nlar models [ 56,57], we expect this choice to be sufficient\nfor the ground state convergence to be satisfactory.\nIn order to distinguish the physical effects arising due to\nthe coupling to magnons and orbitons, we have performed\nthe calculation not only in the variational space with up\nto four bosons of any kind, but also in subspaces where we\nfurther restrict the number of individual bosonic flavors\n(e.g., up to three orbitons and up to one magnon). This\nallows us, to some extent, to trace the evolution of the\nspectral function depending on the bosonic content of\nthe QP’s cloud in its ground state. By comparing these\nsubspace projections to the full calculation, we can infer\nwhich bosons dominate the QP dynamics.\nThe spectral functions (10) were obtained for two rep-\nresentative mixing angles φwith coexisting A-AF/C-AO\nspin-orbital order, φ= 0 andφ=π/6. They occur at\nfinite Hund’s exchange η>0 near the orbital degeneracy,\nEz≈0. We have selected η= 0.16 which is represen-\ntative for the AO order in KCuF 3considered here and\nclose to what is reported in earlier studies [ 20,58–62].\nThis value ensures that both the φ= 0 andφ=π/6\nA-AF/C-AO phases appear as the actual ground states\nwithin the range of variation of the crystal field [58].\nThe firstA-AF/C-AO spin-orbital phase is obtained\nforEz>0 close to the boundary between A-AF and AF ¯z\nphases, see Fig. 1. It is characterised by symmetric and\nantisymmetric linear combinations of the basis orbitals,\n{|¯z/angbracketright,|z/angbracketright}; a finite value of Ez>0 is needed because of the\nspin order which is AF in the ( a,b) planes and FM along\nthecaxis. The second spin-orbital phase discussed belowhas the orbital angle φ=π/6 (A1), which is obtained for\nEz<0, see Fig. 1. It corresponds to the other extreme\ncharacterized by the external orbital field favoring the\nKugel-Khomskii orbitals.\nWe start by analyzing the spectral functions for the\nφ= 0 phase in the Ising limit, see Fig. 2. The occupied\n|±/angbracketrightorbitals (7) form an AO state shown in Fig. 3.The\nIsing limit used here is defined by neglecting both spin and\norbital fluctuations, i.e., discarding all terms containing\noperators other than SzorTz. Note that the spectral\nfunction density maps are presented in a nonlinear ∝tanh\nscale which allows us to highlight the low amplitude states\nthat would otherwise not be visible. The results are shown\nfor two values of the superexchange constant, J= 0.1\n(canonical value, note that the definition of J≡t2/Udoes\nnot include here the factor of 4, conventionally present in\nthe standard t-Jmodel) and J= 0.5 (weak interaction\nregime, this is not a physically relevant limit but it is\nFIG. 3. The in-plane orbital arrangement of the φ= 0 phase.6\nFIG. 4. The extracted QP ground state energies E(k) (left)\nand spectral weights Z(k) for the full fourth order expansion\n(right), and the respective subspace expansions for the ex-\nchange constants J= 0.1 (upper panels) and J= 0.5 (lower\npanels). Parameter: η= 0.16. Labeling conventions are the\nsame as in Fig. 2.\nuseful for exploring the interdependence between orbitons\nand magnons in the system, and its effect on the polaronic\nphysics).\nEach panel in Fig. 2 is marked in the upper-left cor-\nner with the maximal number of magnons and orbitons,\nrespectively, allowed in a given subspace, and in the\nupper-right corner with the size of the variational Hilbert\nspace. The lower-right panel marked with the word “Full”\npresents the full expansion for up to 4 bosons (without\nfurther specifying individual bosonic flavors). The dashed\nblue line indicates the free charge dispersion /epsilon1kφ, and\nserves as a reference energy for the QP state. As ex-\npected, the dressing with bosons creates a QP which\nis energetically more stable than the free particle, how-\never this comes at the cost of an increased effective mass\nand decreased mobility. Note that this is all consistent\nwith standard polaronic physics. The renormalization is\nmuch smaller for the large Jlimit. This can be easily\nunderstood because the cost of creating any boson is pro-\nportional to J, so the bigger Jis, the more expensive it\nis to create a big bosonic cloud. Thus, for large Jthere\nwill be fewer bosons in the cloud, resulting in smaller\nrenormalization of physical properties.\nRemarkably, by comparing the full results against the\nFIG. 5. The full and partial spectral functions A(k,ω) for the\nφ=π/6 phase. Parameters: J= 0.1 andη= 0.16. Notation\nand conventions are the same as in Fig. 2.\npartial results, we can see that in the strong interaction\ncase (J= 0.1) the QP behaves predominantly like in\nthe orbiton rich cases (1,3) and (2,2). To highlight this\neffect we extract the ground state energy and spectral\nweight for all these solutions and plot them against each\nother, see Fig. 4. As is evident, the full solution tends to\ninclude more orbitons and fewer magnons. Having said\nthat though, a cloud consisting of only orbitons would\nnot be sufficient to achieve the optimal QP energy, either.\nThus, we can already see that this is an intrinsically spin-\norbital system, where the interaction of alldegrees of\nfreedom (charge, spin, and orbital) is crucial to achieve\nthe complete understanding of underlying physics.\nEven more interestingly, if we now make the same\ncomparison for the weak interaction limit ( J= 0.5), we see\nthat this time the QP band behaves most like the magnon-\nrich solutions (3,1) and (2,2). This suggests a crossover,\ncontrolled by the exchange parameter J, between orbiton-\nrich and magnon-rich QP clouds. This happens because\nmagnons have lower energy and are cheaper to create than\norbitons. In the large Jlimit, only very few bosons are\ncreated and they are more likely to be magnons, which\ntherefore dominate the dynamics of the resulting QP. In\ncontrast, for small Jall bosons are cheap(er) and orbitons\ndominate by means of geometric effects, i.e., the fact that\nthe charge can couple to them by moving in any of the\nthree principal cubic directions, in contrast to magnons\nwhich couple only when the particle moves along the\nsingle AFcdirection [57].\nFigure 5 shows the spectral functions for φ=π/6 with\nJ= 0.1. The orbital order itself is depicted in Fig. 6.The\nfirst striking observation is that the bands show hardly\nany dispersion at all, except for the purely orbitonic\nsolution (0,4). This is easily understood if we look at the\nfree charge dispersion /epsilon1kφ, which vanishes for φ=π/6,\nas illustrated by the flat dashed blue reference line in7\nFIG. 6. The in-plane orbital arrangement of the φ=π/6\nKugel-Khomskii phase.\nFig. 5. In other words,the unrenormalized particle is\ncompletely localized, and the coupling to bosons does not\nchange that in any substantial way. The tiny dispersion\nobserved in the orbitonic solution is due to Trugman\nloops [ 2], which require a 2D AO order, just like we\nhave in this system, and the existence of at least three-\nboson clouds, hence its appearance in the purely orbitonic\nsolution. In fact, a very tiny dispersion can also be seen\nin the (1,3) panel, however there the interference between\norbitons and magnons clearly suppresses the Trugman\nprocesses [ 2], again underlining the crucial role of orbiton-\nmagnon interplay in the physics of these systems.\nThe lack of dispersion in this orbital phase is a straight-\nforward consequence of a special symmetry of the orbital\norder in the Kugel-Khomskii state. Namely, as evident\nfrom Fig. 6, the φ=π/6 detuning corresponds to the\noccupation of AO y2−z2/z2−x2, so the hopping pro-\ncess would require the charge to move from a lobe of one\nsuch orbital to the nodal point of the neighboring orbital,\nwhich is forbidden by symmetry of the wave function.\nThe results presented thus far point to an interesting\nexperimental possibility. Namely, the orbital order should\nbe discernible from a spectral experiment: the flatter the\nQP band, the closer the occupied orbitals should be to the\nφ=π/6 phase. Naturally, determining the exact phase\nmight not be simple, however, verifying the validity of the\nφ=π/6 case to which most local density approximation\n(LDA) studies seem to point [ 60–64] should be possible\nowing to the dispersionless character of this phase. Having\nsaid that, the issue of an insulating sample and thus\nstrong charging during an angle resolved photoemission\nspectroscopy (ARPES) experiment might pose a barrier\neven to this verification.\nThere is another possibility, however, owing to the\nQP mass renormalization. Going back to Fig. 2 and\ncomparing the QP vs.the free charge dispersion, we see\nthat not only is there a difference in bandwidth between\nthe two cases, but also the symmetry between the Γ\nandMpoints is significantly suppressed, with the QP\nband at the Mpoint being much flatter and having a\nFIG. 7. Comparison of the density of states for the two major\norbital phases discussed in this paper, φ= 0 andφ=π/6;\nParameters: J= 0.1 andη= 0.16.\ngreatly reduced spectral weight. If we would now integrate\nthe spectrum to produce the density of states (DOS)\nfor this system, we would see that the QP DOS for a\ndispersive phase should be highly asymmetric, whereas\nthe dispersionless phase should be characterized by a\nsharp and completely symmetric QP DOS, as verified in\nFig. 7. Thus, the orbital phase could be inferred, even\nif only approximately, from the shape and asymmetry of\nthe QP DOS. In turn, the DOS can be obtained from\na scanning tunneling microscope experiment for which\nsample charging might be less problematic.\nIn all of the above we have assumed an Ising interac-\ntion, or to put it differently, that the effect of quantum\nfluctuations is negligible. This is a reasonable assumption\nbecause fluctuations generally are less important in higher\ndimensions and here we are dealing with an ostensibly\n3D system. Having said that, however, the AO order\ncan at the same time be thought as 2D and the AF or-\nder as 1D. While it has been established that the role\nof fluctuations for egorbital pseudospin in a 2D planar\nsubsystem is indeed negligible [ 56], the same assumption\nseems less justified for magnetic excitations. Apart from\nthe dimensionality of the corresponding order, another\nargument is that the relative lack of importance of or-\nbitonic fluctuations comes from the fact that the orbiton\nspectrum is gapped, which is not the case for magnons.\nThis is why it is reasonable to neglect the orbital fluctua-\ntions while including magnetic fluctuations, in order to\nexplicitly establish whether they are relevant or not.\nFortunately, magnetic fluctuations may be fairly easily\nincluded within MA by allowing arbitrary fluctuations\nbut only in the vicinity of the propagating charge (the\nvariational space cutoff is controlled with exactly the same\ncloud spatial criteria as before), since these are the only\nones which will affect the QP dynamics. Fluctuations\noccurring far from the charge will instead only affect the8\nFIG. 8. Comparison of the spectral functions in the Ising\napproximation (ising) and the one including full magnetic\nfluctuations (mfluct), and for the two orbital phases, φ= 0 and\nφ=π/6. To highlight the effect we take the weak interaction\nregimeJ= 0.5. Notation and conventions are the same as in\nFig. 2. Parameter: η= 0.16.\nnature of undoped regions far from the particle, affecting\nthe overall energy. However, as long as the classical\nground state is close enough to the true quantum state\nrealized for a given set of parameters, this would only be\nreflected by a constant shift of the entire spectrum, which\nis not a physically significant effect.\nTo illustrate the role of fluctuations, we present a com-\nparison between the Ising solution and the one including\nlocal fluctuations for both angles, φ= 0 andφ=π/6,\nsee Fig. 8. As their effect proves to be rather elusive,\nwe focus on the weak interaction limit J= 0.5, where\nthe changes can be more readily observed. One immedi-\nately sees the huge difference in the size of the variational\nspaces, even though only magnetic fluctuations (albeit\nin all three cubic directions) are considered here. This,\nhowever, has surprisingly little overall effect on the QP\ndispersion. While their effect seems more considerable for\nthe excited states, these are likely not fully converged any-\nway, so that part of the spectra is not sufficiently reliable\nfor comparison. A tiny dispersive effect can also be ob-\nserved, most readily visible in the φ=π/6 phase, however\nit is much too small to be of any practical importance.\nThere is however one interesting feature, namely the\npure gain in energy experienced by the QP ground state,\nindicative of a stronger binding of the QP, which however\ndoes not affect its dynamical properties. In particular,while the gain for the φ= 0 phase is relatively small, the\none observed for φ=π/6 is considerable. This difference\ncould be indicative of a subtle quantum effect arising\nfrom the spin-orbital coupling in the system. Clearly, the\nimportance of the magnetic fluctuations strongly depends\non the orbital phase in the system, and in particular,\nthe fluctuations in the φ=π/6 phase grow particularly\nstrong. This indicates that the magnetic order becomes\nless classical in character, which is likely caused by the\nsystem decoupling into 1D AF chains. There is ample\nevidence of the actual KCuF 3exhibiting a 1D quantum\nAF character [ 36], so this would seem to point to the\nactual orbital phase in that system being close to φ=π/6,\nsomething that was long proposed based on electronic\nstructure calculations using LDA. Here we were able to\narrive at similar conclusions through indirect means and\nby a completely different methodology.\nV. SUMMARY AND CONCLUSIONS\nWe have developed an effective spin-orbital superex-\nchange model for an e3\ngsystem, and computed the single-\npolaron spectrum resulting when a single charge is doped\nin the system by using the semi-analytical, variational\nmomentum average method for calculating Green’s func-\ntions. This allowed us not only to obtain the relevant\nspectral functions, but also to gain insight into the nature\nof the magnetic and orbital order in the system. Thus we\nwere able to demonstrate a number of subtle quantum\neffects arising from the interaction between the charge,\norbital, and magnetic degrees of freedom.\nOne such effects is the change of the character of the\npolaronic quasiparticle cloud from being dominated by\norbitons to being dominated by magnons; this is controlled\nby the strength of the superexchange interaction J. This\nbehavior, although only a theoretical prediction due to the\nimpossibility of experimentally tuning the parameter Jin\nsuch a wide range of values, nonetheless points towards\na strong interplay between orbital and magnetic degrees\nof freedom in this model. It should come as no surprise,\nthen, that their intermingling should also crop up in other\nproperties of the system, some of which might be more\nreadily accessible to experiment.\nOne possible experimental consequence lies in the quasi-\nparticle dispersion being strongly dependent on the orbital\norder in the system which, coupled with the polaronic sup-\npression of the symmetry between the Γ and Mpoints of\nthe Brillouin zone, suggests that the quasiparticle density\nof states should be particularly sensitive to the orbital\norder in these systems. In turn, this would point towards\nScanning Tunneling Microscopy as a promising tool for\nan experimental probe of the orbital order type. We thus\npropose that the orbital order could be inferred by inves-\ntigating the amplitude to width ratio and the asymmetry\nof the density of states peaks.\nFinally, we point out that the orbital order around the\ndetuning angle φ=π/6 seems to drive the magnetic sys-9\ntem closer towards the 1D AF chain. Indeed, the already\navailable results of neutron scattering experiments [ 36]\ndemonstrate a nearly ideal 1D spin liquid behavior. We\nsuggest that this is a strong indication that the orbital\norder in KCuF 3is likely to be close to φ=π/6.\nACKNOWLEDGMENTS\nK. B. and A. M. O. kindly acknowledge support\nby UBC Stewart Blusson Quantum Matter Insti-\ntute (SBQMI), by Natural Sciences and Engineering\nResearch Council of Canada (NSERC), and by Naro-\ndowe Centrum Nauki (NCN, Poland) under Projects\nNos. 2016/23/B/ST3/00839 and 2015/16/T/ST3/00503.\nM. B. acknowledges support from SBQMI and NSERC.\nA. M. Ole´ s is grateful for the Alexander von Humboldt\nFoundation Fellowship (Humboldt-Forschungspreis).\nAppendix A: The mean field ground state\nIn this Section we provide the more technical details\nconcerning the derivation of the effective polaronic spin-\norbital model that is the basis of our calculation. In order\nto find the classical orbital ground state, we parametrizethe orbital basis in terms of a standard rotation of the\n{¯z,z}basis. However, since the reference, field-free order\nis composed of alternating |±/angbracketrightstates (7), the rotation is\nmost conveniently parametrized with an angle ±(π/2+φ),\nwhere the sign depends on the orbital sublattice,\n|φA/angbracketright= cos/parenleftbiggπ\n4+φ\n2/parenrightbigg\n|¯z/angbracketright+ sin/parenleftbiggπ\n4+φ\n2/parenrightbigg\n|z/angbracketright,\n|φB/angbracketright= cos/parenleftbiggπ\n4+φ\n2/parenrightbigg\n|¯z/angbracketright−sin/parenleftbiggπ\n4+φ\n2/parenrightbigg\n|z/angbracketright.(A1)\nThis choice also serves to transform the underlying AO\norder to an FO order, effectively eliminating the bipartite\ndivision of the lattice in the reference state. It should\nbe stressed that this operation does not affect the ac-\ntual ground state, it merely changes its representation\nto one that is more convenient—it spares us the trouble\nof distinguishing between bosons on different sublattices\n(cf.Ref.7).Nowφwill indicate a detuning from the field-\nfree orbital order, and the angle between the occupied\norbitals on the two sublattices will be 2 φ(i.e., in general,\nthe bases on different sublattices will not be mutually\northogonal).\nTo find the relation between the orbital field Ezand\nthe detuning angle φin Eqs. (A1), we start by writing\nout the superexchange Hamiltonian in the new basis:\nH⊥\nI=J/summationdisplay\n/angbracketleftij/angbracketright⊥c/parenleftbig\nAηSi·Sj+1\n4Bη/parenrightbig/braceleftBig\n2Cη\nAη−1−(2 cos 2φ+ 1)Tz\niTz\nj−(2 cos 2φ−1)Tx\niTx\nj\n+2eiQRisin 2φ(Tz\niTx\nj−Tx\niTz\nj)∓√\n3 (Tx\niTz\nj+Tz\niTx\nj)/bracerightBig\n, (A2a)\nH/bardbl\nI= 2J/summationdisplay\n/angbracketleftij/angbracketright/bardblc/parenleftbig\nAηSi·Sj+1\n4Bη/parenrightbig/braceleftBig\nCη\nAη−1\n2+ 2 sin2φTz\niTz\nj+ 2 cos2φTx\niTx\nj−eiQRisin 2φ(Tx\niTz\nj+Tz\niTx\nj)/bracerightBig\n,(A2b)\nH⊥\nII=−JCη/summationdisplay\n/angbracketleftij/angbracketright⊥c/parenleftbig\nSi·Sj−1\n4/parenrightbig/braceleftBig/bracketleftBig\nsinφ(Tz\ni+Tz\nj)∓√\n3eiQRicosφ/bracketrightBig\n(Tz\ni−Tz\nj)\n−eiQRicosφ(Tx\ni−Tx\nj)∓√\n3 sinφ(Tx\ni+Tx\nj)/bracerightBig\n, (A2c)\nH/bardbl\nII= 2JCη/summationdisplay\n/angbracketleftij/angbracketright/bardblc/parenleftBig\nSi·Sj−1\n4+Ez\n4JCη/parenrightBig/bracketleftbig\nsinφ(Tz\ni+Tz\nj)−eiQRicosφ(Tx\ni+Tx\nj)/bracketrightbig\n, (A2d)\nwhere the last term incorporates the orbital field Hz. Here,\nQ= (π,π,0) is the ordering vector for the C-AO state,\nand the resulting phase factor encodes the alternating\nnature of the orbital order. The symbol ⊥//bardblrefers to\nthe cubic directions with respect to the c-axis. Note that\nthe various superexchange terms of Eq. (2)have been\nsplit into terms quadratic in {Tz\ni}operators (HI) and\nlinear in{Tz\ni}operators (HII). The Hund’s exchange (4)\nis now encoded in the three prefactors (if η= 0, one findsA0=B0=C0= 1):\nAη=1−η\n(1 +η)(1−3η), (A3a)\nBη=1 + 3η\n(1 +η)(1−3η), (A3b)\nCη=1\n1−η2, (A3c)\nwhich themselves result from various combinations of\ntherimultiplet parameters listed above. Note that the\nexchange Hamiltonian has also been shifted in energy so10\nthat the Ising energy for the ground state of the system is\nset to zero. This is done merely for reasons of convenience,\nso that the excitation energies are easier to track once we\nstart considering excitations in the system.\nNext we evaluate the mean field energy assuming the\nclassical ground state to be A-AF/ C-AO, as is known to\nbe the case in KCuF 3. We find:\nEMF=1\n4J(Bη−Aη) sin2φ−1\n8J(Aη+Bη)(2 cos 2φ+1)\n−J/parenleftbigg\nCη−Ez\n2J/parenrightbigg\nsinφ.(A4)\nThis expression is then minimized with respect to the\ndetuning angle φ, yielding the relation\nEz=J[2Cη−(Aη+ 3Bη) sinφ]. (A5)\nThis identity can now be used to eliminate Ezfrom the\nHamiltonian by replacing it with the detuning angle φ.\nNote that if we now set φ= 0, we will, seemingly para-\ndoxically, get Ez= 2JCη,i.e., a finite orbital field corre-\nsponding to the field-free case. This is due to the fact that\nthe superexchange Hamiltonian already includes terms\nlinear in pseudospin operators which behave like an or-\nbital field, and the external field works to compensate\nthese terms. In other words, the exchange Hamiltonian\nbreaks cubic symmetry by itself, and the above estimate\nis the orbital field needed to restore it. On the other hand,\nthe caseEz= 0 corresponds to φ=π/6 in the 2D orbital\nmodel [ 50], which is the Kugel-Khomskii state composed\nof alternating y2−z2/z2−x2states. These two limits are\ncommonly cited as the extreme possibilities for the orbital\norder in this system. The actual orbital order realized in\nthe system will be bounded by these two extremes, and\nin fact could be, to some extent, tuned by means of an\naxial pressure∝Ezapplied along the caxis.\nAppendix B: Fermion-boson polaronic model\nTo go beyond the mean field ground state, we derive\nthe effective Hamiltonian transforming the physics of a\nsingle charge doped into a spin-orbital model to a fermion-\nboson many-body problem. To that end, we transform\nthe kinetic Hamiltonian to the same basis as the one\nconsidered in the last Section, so that the entire model isexpressed in compatible representations. This leads to\nH⊥\nt=−t\n4/summationdisplay\n/angbracketleftij/angbracketright⊥c,σ/braceleftBig/bracketleftBig\n(1−2 sinφ)d†\niσ0djσ0\n−2eiQRicosφ(d†\niσ0djσ1−d†\niσ1djσ0)\n∓√\n3 (d†\niσ0djσ1+d†\niσ1djσ0)\n−(1 + 2 sinφ)d†\niσ1djσ1/bracketrightBig\n+ H.c./bracerightBig\n, (B1a)\nH/bardbl\nt=−t\n2/summationdisplay\n/angbracketleftij/angbracketright/bardblc,σ/braceleftBig/bracketleftBig\n(1 + sinφ)d†\niσ0dj¯σ0\n−eiQRicosφ(d†\niσ0dj¯σ1+d†\niσ1dj¯σ0)\n−(1−sinφ)d†\niσ1dj¯σ1/bracketrightBig\n+ H.c./bracerightBig\n, (B1b)\nwhere the 0 (1) indices denote the ground (excited) orbital\nstates, respectively.\nFinally, following Mart´ ınez and Horsch [ 7], we repre-\nsent the spin and orbital degrees of freedom using a slave\nboson representation. This is achieved by expanding the\n(pseudo)spin operators around the assumed mean field\nground state by means of a Holstein-Primakoff transfor-\nmation,\nd†\ni↑0=f†\ni,\nd†\ni↓0=f†\nibi,d†\ni↑1=f†\niai,\nd†\ni↓1=f†\niaibi,(B2)\nwhereb†\nicreates a spin excitation at site i,a†\nicreates an\norbital excitation, and f†\nicreates a spinless fermion which\nrepresents the charge degree of freedom, where 0 indicates\nthe siteiin the ground state, while 1 means that the\nrespective site hosts an excited state. Thus, a charge can\nbe added to the system only if it is locally in its (classical)\nground state, otherwise if a boson occupied the considered\nsite, first it has to be removed before the charge can be\nadded. Also note that the on-site bosonic Hilbert space\nis restricted to (2 S+ 1) states, and since both the spin\nand the pseudospin have length 1/2, each site can host not\nmore than one boson of each kind. This local constraint\napplies to every site and is fully taken into account in our\ncalculations through the variational technique employed.\nThe exchange Hamiltonian also has to be transformed\ninto its bosonic representation, which is done by means\nof the Holstein-Primakoff transformation of the spin op-\nerators,\nSz\ni=1\n2−b†\nibi, S+\ni=/radicalBig\n1−b†\nibibi, S−\ni=b†\ni/radicalBig\n1−b†\nibi,\n(B3)\nand similarly for the pseudospin operators, but in terms\nof orbiton operators/braceleftBig\nai,a†\ni/bracerightBig\n. There are two issues that\nare still worth pointing out concerning this transforma-\ntion. Firstly, the Sz\nioperators are the ones that most\nreadily introduce higher order terms into the Hamilto-\nnian, and thus are principally responsible for inter-bosonic\ninteractions, which can have profound effects for low-\ndimensional physics [ 51] but which nonetheless are all11\ntoo often neglected in techniques reetalying on the LSW\napproximation. It is therefore worth mentioning that in\nour variational method this part of the transformation is\nnot strictly necessary, as the Sz\nioperators merely “count”\nthe Ising energy of the bosons and thus their effect can be\ndiscerned directly from the configuration of the system.\nOr, to put it differently, in our method it would actu-\nally be more cumbersome (although possible) to calculate\nthe energy in the LSW approximation than it is to do it\nexactly.\nSecondly, the square root factors in the fluctuation\noperatorsS±\niare conventionally treated by Taylor ex-\npansion and truncation at second order, to be consistent\nwith the LSW approximation. However, here again, one\nshould realize that these factors merely serve to impose\nthe restriction of a single boson per site, and thus thisconstraint can be taken into account by excluding from\nthe variational space the configurations which violate it.\nTherefore, with our variational technique we can bosonize\nthe exchange interactions without the need to abandon\nany of the inter-bosonic interactions or constraints.\nApplying these transformations decouples the original\nfermions into their constituent charge, spin, and orbital\ndegrees of freedom. The free charge propagation Ht, as\nwell as its coupling to the bosonic degrees of freedom, will\nnow be described by the kinetic Hamiltonian, which after\nthe above transformations reads,\nHt=T+V⊥\nt+V/bardbl\nt, (B4)\nwhere:\nT=−t\n4/summationdisplay\n/angbracketleftij/angbracketright⊥c(1−2 sinφ)/parenleftBig\nf†\nifj+ H.c./parenrightBig\n=/summationdisplay\nk/epsilon1kφf†\nkfk, (B5a)\nV⊥\nt=t\n4/summationdisplay\n/angbracketleftij/angbracketright⊥c/braceleftBig/bracketleftBig\n2eiQRicosφ(a†\nj−ai)±√\n3(a†\nj+ai) + (1 + 2 sin φ)a†\njai/bracketrightBig\n(1 +bib†\nj)f†\nifj+ H.c./bracerightBig\n−t\n4/summationdisplay\n/angbracketleftij/angbracketright⊥c/bracketleftBig\n(1−2 sinφ)bib†\njf†\nifj+ H.c./bracketrightBig\n, (B5b)\nV/bardbl\nt=−t\n2/summationdisplay\n/angbracketleftij/angbracketright/bardblc/braceleftBig/bracketleftBig\n(1 + sinφ)−eiQRicosφ(ai+a†\nj)−(1−sinφ)aia†\nj/bracketrightBig\n(bi+b†\nj)f†\nifj+ H.c./bracerightBig\n, (B5c)\nand/epsilon1kφ=−1\n2t(1−2sinφ)(coskx+cosky) is the free\nelectron dispersion. We emphasize that it depends on the\norbital order (A1) through the angle φ, and vanishes when\nφ=π/6. The free charge hopping term Tis restricted to\ntheabplanes because of the magnetic order alternating in\nthe perpendicular direction c, which thus requires creation\nor annihilation of a magnon when the charge hops in that\ndirection.The remaining terms couple the charge to the bosonic\ndegrees of freedom, and some of them are high-order\nmany-body interactions, sometimes coupling the charge\nto multiple bosons at the same time. Treating these\ninteractions with a technique relying on the LSW approx-\nimation would be impossible, whereas this can be done\nwithin the variational momentum average approach, as\npresented in Sec. IV.\n[1]K. A. Chao, J. Spa/suppress lek, and A. M. Ole´ s, J. Phys. C 10,\nL271 (1977).\n[2] S. A. Trugman, Phys. Rev. B 37, 1597 (1988).\n[3]Z. Liu and E. Manousakis, Phys. Rev. B 45, 2425 (1992).\n[4] E. Manousakis, Phys. Rev. B 75, 035106 (2007).\n[5]F. Grusdt, M. K´ anasz-Nagy, A. Bohrdt, C. S. Chiu, G. Ji,\nM. Greiner, D. Greif, and E. Demler, Phys. Rev. X 8,\n011046 (2018).\n[6]C. L. Kane, P. A. Lee, and N. Read, Phys. Rev. B 39,\n6880 (1989).\n[7]G. Mart´ ınez and P. Horsch, Phys. Rev. B 44, 317 (1991).\n[8]P. A. Lee, N. Nagaosa, and X.-G. Wen, Rev. Mod. Phys.\n78, 17 (2006).\n[9]K. I. Kugel and D. I. Khomskii, Sov. Phys. Usp. 25, 231(1982).\n[10] Y. Tokura and N. Nagaosa, Science 288, 462 (2000).\n[11] A. M. Ole´ s, Phys. Rev. B 28, 327 (1983).\n[12]S. Hoshino and P. Werner, Phys. Rev. B 93, 155161\n(2016).\n[13]L. F. Feiner, A. M. Ole´ s, and J. Zaanen, Phys. Rev. Lett.\n78, 2799 (1997).\n[14]S. Ishihara, J. Inoue, and S. Maekawa, Phys. Rev. B 55,\n8280 (1997).\n[15]L. F. Feiner and A. M. Ole´ s, Phys. Rev. B 59, 3295 (1999).\n[16]M. Snamina and A. M. Ole´ s, Phys. Rev. B 97, 104417\n(2018).\n[17]G. Khaliullin and S. Maekawa, Phys. Rev. Lett. 85, 3950\n(2000).12\n[18]G. Khaliullin, P. Horsch, and A. M. Ole´ s, Phys. Rev.\nLett. 86, 3879 (2001); Phys. Rev. B 70, 195103 (2004).\n[19]G. Khaliullin, Prog. Theor. Phys. Suppl. 160, 155 (2005).\n[20]A. M. Ole´ s, G. Khaliullin, P. Horsch, and L. F. Feiner,\nPhys. Rev. B 72, 214431 (2005).\n[21]J. Chaloupka and G. Khaliullin, Phys. Rev. Lett. 100,\n016404 (2008).\n[22]P. Horsch, A. M. Ole´ s, L. F. Feiner, and G. Khaliullin,\nPhys. Rev. Lett. 100, 167205 (2008).\n[23]B. Normand and A. M. Ole´ s, Phys. Rev. B 78, 094427\n(2008); B. Normand, Phys. Rev. B 83, 064413 (2011);\nJ. Chaloupka and A. M. Ole´ s, Phys. Rev. B 83, 094406\n(2011).\n[24]J. Sirker, A. Herzog, A. M. Ole´ s, and P. Horsch, Phys.\nRev. Lett. 101, 157204 (2008); A. Herzog, P. Horsch,\nA. M. Ole´ s, and J. Sirker, Phys. Rev. B 83, 245130\n(2011).\n[25]W. Brzezicki, A. M. Ole´ s, and M. Cuoco, Phys. Rev. X\n5, 011037 (2015).\n[26] W. Brzezicki, arXiv:1904.11772 (2019).\n[27]J. van den Brink, P. Horsch, and A. M. Ole´ s, Phys. Rev.\nLett. 85, 5174 (2000).\n[28]S. Ishihara, Y. Murakami, T. Inami, K. Ishii, J. Mizuki,\nK. Hirota, S. Maekawa, and Y. Endoh, New J. Phys. 7,\n119 (2005).\n[29] S. Ishihara, Phys. Rev. Lett. 94, 156408 (2005).\n[30]M. Daghofer, A. M. Ole´ s, and W. von der Linden, Phys.\nRev. B 70, 184430 (2004).\n[31]K. Wohlfeld, A. M. Ole´ s, and P. Horsch, Phys. Rev. B\n79, 224433 (2009).\n[32] M. Berciu, Physics 2, 55 (2009).\n[33]A. Okazaki and Y. Suemune, J. Phys. Soc. Japan 16, 176\n(1961).\n[34]J.-S. Zhou and J. B. Goodenough, Phys. Rev. Lett. 96,\n247202 (2006).\n[35]T. Kimura, S. Ishihara, H. Shintani, T. Arima, K. T.\nTakahashi, K. Ishizaka, and Y. Tokura, Phys. Rev. B 68,\n060403 (2003).\n[36]B. Lake, D. A. Tennant, C. D. Frost, and S. E. Nagler,\nNature Mat. 4, 329 (2005).\n[37]B. Lake, D. A. Tennant, and S. E. Nagler, Phys. Rev. B\n71, 134412 (2005).\n[38]G. H. Jonker and J. H. van Santen, Physica 16, 337\n(1950).\n[39]Y. Tokura, Reports on Progress in Physics 69, 797 (2006).\n[40]K. Ro´ sciszewski and A. M. Ole´ s, Phys. Rev. B 99, 155108\n(2019).\n[41]J. Ba/suppress la, G. A. Sawatzky, A. M. Ole´ s, and A. Macridin,Phys. Rev. Lett. 87, 067204 (2001).\n[42]R. Kr¨ uger, B. Schulz, S. Naler, R. Rauer, D. Budelmann,\nJ. B¨ ackstr¨ om, K. H. Kim, S.-W. Cheong, V. Perebeinos,\nand M. R¨ ubhausen, Phys. Rev. Lett. 92, 097203 (2004).\n[43]M. W. Kim, J. H. Jung, K. H. Kim, H. J. Lee, J. Yu,\nT. W. Noh, and Y. Moritomo, Phys. Rev. Lett. 89,\n016403 (2002).\n[44]N. N. Kovaleva, A. M. Ole´ s, A. M. Balbashov, A. Maljuk,\nD. N. Argyriou, G. Khaliullin, and B. Keimer, Phys. Rev.\nB81, 235130 (2010).\n[45]M. Snamina and A. M. Ole´ s, New Journal of Physics 21,\n023018 (2019).\n[46]J. Kuneˇ s, I. Leonov, M. Kollar, K. Byczuk, V. Anisimov,\nand D. Vollhardt, Eur. Phys. J. Special Topics 180, 5\n(2010).\n[47]L. F. Feiner and A. M. Ole´ s, Phys. Rev. B 71, 144422\n(2005).\n[48]A. M. Ole´ s, L. F. Feiner, and J. Zaanen, Phys. Rev. B\n61, 6257 (2000).\n[49]W. Brzezicki, J. Dziarmaga, and A. M. Ole´ s, Phys. Rev.\nLett. 109, 237201 (2012).\n[50]P. Czarnik, J. Dziarmaga, and A. M. Ole´ s, Phys. Rev. B\n96, 014420 (2017).\n[51]K. Bieniasz, P. Wrzosek, A. M. Ole´ s, and K. Wohlfeld,\narXiv:1809.07120 (2018).\n[52] M. Berciu, Phys. Rev. Lett. 97, 036402 (2006).\n[53]D. J. J. Marchand, G. De Filippis, V. Cataudella,\nM. Berciu, N. Nagaosa, N. V. Prokof’ev, A. S. Mishchenko,\nand P. C. E. Stamp, Phys. Rev. Lett. 105, 266605 (2010).\n[54]M. Berciu and H. Fehske, Phys. Rev. B 84, 165104 (2011).\n[55]H. Ebrahimnejad, G. A. Sawatzky, and M. Berciu, J.\nPhys.: Cond. Mat. 28, 105603 (2016).\n[56]K. Bieniasz, M. Berciu, M. Daghofer, and A. M. Ole´ s,\nPhys. Rev. B 94, 085117 (2016).\n[57]K. Bieniasz, M. Berciu, and A. M. Ole´ s, Phys. Rev. B\n95, 235153 (2017).\n[58]W. Brzezicki, J. Dziarmaga, and A. M. Ole´ s, Phys. Rev.\nB87, 064407 (2013).\n[59]A. I. Liechtenstein, V. I. Anisimov, and J. Zaanen, Phys.\nRev. B 52, R5467 (1995).\n[60] M. Kataoka, J. Phys. Soc. Jpn. 73, 1326 (2004).\n[61]E. Pavarini, E. Koch, and A. I. Lichtenstein, Phys. Rev.\nLett. 101, 266405 (2008).\n[62]I. Leonov, D. Korotin, N. Binggeli, V. I. Anisimov, and\nD. Vollhardt, Phys. Rev. B 81, 075109 (2010).\n[63]N. Binggeli and M. Altarelli, Phys. Rev. B 70, 085117\n(2004).\n[64]E. Pavarini and E. Koch, Phys. Rev. Lett. 104, 086402\n(2010)."    },    {        "title": "1912.05181v2.Selective_tuning_of_spin_orbital_Kondo_contributions_in_parallel_coupled_quantum_dots.pdf",        "content": "Selective tuning of spin-orbital Kondo contributions in parallel-coupled quantum dots\nHeidi Potts,1,\u0003Martin Leijnse,1Adam Burke,1Malin Nilsson,1\nSebastian Lehmann,1Kimberly A. Dick,1, 2and Claes Thelander1,y\n1Division of Solid State Physics and NanoLund, Lund University, SE-221 00 Lund, Sweden\n2Centre for Analysis and Synthesis, Lund University, SE-221 00 Lund, Sweden\n(Dated: March 18, 2020)\nWe use co-tunneling spectroscopy to investigate spin-, orbital-, and spin-orbital Kondo transport\nin a strongly con\fned system of InAs double quantum dots (QDs) parallel-coupled to source and\ndrain. In the one-electron transport regime, the higher symmetry spin-orbital Kondo e\u000bect manifests\nat orbital degeneracy and no external magnetic \feld. We then proceed to show that the individual\nKondo contributions can be isolated and studied separately; either by orbital detuning in the case\nof spin-Kondo transport, or by spin splitting in the case of orbital Kondo transport. By varying\nthe inter-dot tunnel coupling, we show that lifting of the spin degeneracy is key to con\frming the\npresence of an orbital degeneracy, and to detecting a small orbital hybridization gap. Finally, in the\ntwo-electron regime, we show that the presence of a spin-triplet ground state results in spin-Kondo\ntransport at zero magnetic \feld.\nI. INTRODUCTION\nThe Kondo e\u000bect is a widely studied many-body\nphenomenon that has increased the understanding of\nstrongly correlated electron systems. Experimentally\nit can be investigated using quantum dots (QDs) with\nhighly transparent tunnel barriers, and manifests as a\nzero-bias conductance resonance. Most studies focus on\nthe spin-1/2 Kondo e\u000bect, where an unpaired spin is\nscreened in the absence of a magnetic \feld1{3. Two-\nelectron spin states represent another common system\nfor Kondo studies, where resonances arise from a vanish-\ning singlet-triplet exchange energy4{6, a magnetic \feld\ninduced singlet-triplet crossing7{12, or a triplet ground\nstate13{15. More recently, the orbital Kondo e\u000bect, which\nrelies on two degenerate orbitals, has received consider-\nable attention. In particular, by combining spin and or-\nbital degrees of freedom, the SU(4) Kondo e\u000bect can be\nstudied16. The presence of such a higher symmetry can\nbe identi\fed by di\u000berent temperature scaling17, and an\nexpected enhancement in the shot-noise properties18{20.\nIn single QDs, orbital and spin-orbital Kondo trans-\nport has been observed in carbon nanotubes18{23, and\nsilicon based devices24,25. However, in such systems,\nthe orbital degeneracy is an inherent material property\nand the tunability of the orbital alignment is therefore\nlimited. An alternative approach is to fabricate two\nparallel-coupled QDs for which the orbitals can be\ntuned independently. Typically, this is realized in two-\ndimensional electron gases17,26{30, but at the expense of\nlimited Zeeman splitting of spin states.\nIn this work, we use parallel-coupled quantum dots\nin indium-arsenide (InAs) nanowires to investigate the\nspin- and orbital Kondo e\u000bect. We show that each\ndegeneracy here can be induced and lifted selectively,\nwhich makes it an ideal system for studies of higher\nKondo symmetries. We start by discussing Kondo res-\nonances in the one-electron regime, and present resultson the spin-, orbital-, and combined spin-orbital Kondo\ne\u000bect. The individual contributions to the transport\nare isolated by electric \feld-induced orbital detuning,\nand magnetic \feld-induced spin splitting. Furthermore,\nby controlling the inter-dot tunnel coupling, t, we\ndemonstrate that the formation of hybridized states\ninhibits the orbital Kondo e\u000bect. If tis small, the\nenergy gap of the avoided crossing can only be resolved\nif the spin-degeneracy is lifted. This underlines the\nimportance of isolating di\u000berent Kondo mechanisms\nwhen studying their contribution to Kondo e\u000bects of\nhigher symmetry. Finally, we investigate the Kondo\ne\u000bect in the two-electron regime when each of the QDs\ncontains one unpaired electron, and a \fnite tunnel\ncoupling between the QDs gives rise to two-electron\nstates. A Kondo resonance at zero magnetic \feld is here\nobserved due to a spin-triplet ground state.\nII. EXPERIMENT\nOur study is based on InAs nanowires where a QD\nis de\fned by a thin zinc-blende (ZB) section between\ntwo wurtzite (WZ) barriers31. Figure 1(a) shows a\nschematic representation of the nanowire and a scanning\nelectron microscopy (SEM) image of a representative de-\nvice. Metal contacts (Ni/Au 25 nm/75 nm) are placed\non the outer ZB sections as source and drain32. Trans-\nport measurements are performed in a dilution refriger-\nator with an electron temperature Te\u001970 mK. If ap-\nplied, the magnetic \feld, B, is aligned perpendicular to\nthe substrate plane in this study. Previous studies have\nshown that two sidegates ( VL,VR) and a global backgate\n(VBG) allow to split the QD into two parallel-coupled\nQDs, and the electron population on the two QDs can\nbe controlled independently. The system is highly tun-\nable, and the small e\u000bective electron mass provides large\nintra-dot orbital separations32{34. In recent works, we\nhave furthermore demonstrated the formation of ring-likearXiv:1912.05181v2  [cond-mat.mes-hall]  17 Mar 20202\nZB WZ\nQDsEc\nVR (V)3 4 5 6 7(0,0)(1,0)\n(0,1)VL (V)-5.0\n-5.5\n-6.0\n-6.5-7.0\n-7.5 00.2\n0.1\n500 nm(a) (b)\nsourcedrainVL\nVRQDs\nG\n(e2/h)\n(1,1)\nVsd = 25 μV\nB = 1 T\nFIG. 1. (a) Top: Schematic of the nanowire and conduction\nband alignment. Bottom: 45\u000etilted SEM image of a represen-\ntative device. (b) Conductance ( G) as a function of sidegate\nvoltages at B= 1 T. The electron populations on the left and\nright QD (Nleft,Nright) after subtracting 2 Nelectrons are in-\ndicated. Important gate vectors are shown in red, green, and\norange.\nstates when coupling the QDs in two points, resulting in\na vanishing one-electron hybridization energy35, and a\nspin-triplet ground state when each QD contains an un-\npaired electron.\nIII. RESULTS AND DISCUSSION\nIn this article, we investigate transport in a regime\nwhere two orbitals (one from each QD) cross in energy\nand interact, as shown in Fig. 1(b). The electron\npopulation of the left and right dot ( Nleft,Nright) is\nindicated. Here, 2 Nelectrons were subtracted on\nboth QDs, as the contribution of \flled orbitals can\nbe neglected due to strong con\fnement (a large-range\noverview measurement can be found in the supporting\ninformation). In such an orbital crossing, Kondo\ntransport due to both spin and orbital degeneracies is\npossible. The spin-1/2 Kondo e\u000bect is expected when\none of the QDs contains an unpaired electron, which\nis the case for transport involving the (1,0) and (0,1)\norbitals. Assuming a vanishing hybridization energy\nfor these two orbitals, we additionally expect an orbital\nKondo e\u000bect where they cross in energy. Additionally,\nin the (1,1) regime, two-electron spin states provide\ndegeneracies that can also result in Kondo transport. In\norder to study these di\u000berent types of Kondo origins, we\nwill focus on transport along the gate vectors indicated\nin green, red, and orange in Fig. 1(b).\nA. Spin-orbital Kondo e\u000bect\nFigure 2(a) shows a measurement of di\u000berential\nconductance, d I/dVsd, versus the source-drain voltage,\nVsd, along the green gate vector indicated in Fig. 1(b).\nThe gate vector is chosen such that the (1,0) and\n(0,1) orbitals are approximately degenerate along thisvector in the one-electron regime. A zero-bias peak\nin the di\u000berential conductance can be observed within\nthe outlined Coulomb diamonds corresponding to the\none-electron (1e), two-electron (2e), and three-electron\n(3e) regimes. Next, we investigate the 1e regime in more\ndetail by detuning the orbitals of the left and right QD\nwith respect to each other (red gate vector). Due to\nthe large interdot Coulomb energy U1;2\u00193 meV, the\nsystem holds one electron along the whole vector, and\nall transport features are related to co-tunneling events.\nThe transport measurement at B= 0 T in Fig. 2(b)\nshows a zero-bias peak36, as well as a step in di\u000berential\nconductance that approaches zero bias at zero detuning.\nComparing with a schematic representation of the states\nand resulting onset of co-tunneling transport (Fig.\n2(c)), we can distinguish two transport processes: 1)\nthe spin-1/2 Kondo e\u000bect on the populated QD, and 2)\nco-tunneling involving both QDs, which we will refer\nto as orbital co-tunneling. For the latter, an electron\ntunnels out of the left (right) QD and is replaced by\nan electron on the right (left) QD (with or without a\nspin-\rip), and the energy cost of this process corresponds\nto the detuning of the two QDs. At zero detuning of\nthe QDs (\u0001 Eorb= 0), the onset of orbital co-tunneling\ncrosses zero bias. However without further investigation\nit is unclear whether both the spin and the orbital\ndegeneracy contribute to the Kondo resonance.\nTransport measurements at di\u000berent temperatures\nwere performed in order to extract the Kondo temper-\natures in di\u000berent locations in the honeycomb diagram.\nThe intensity of a Kondo zero-bias peak shows a charac-\nteristic decay with temperature, which can be described\nby the phenomenological expression1,37\nG(T) =G0\u0014\n1 + (21=s\u00001)\u0012T\nTK\u0013n\u0015\u0000s\n+G1(1)\nwhereTKis the Kondo temperature, G0is the Kondo\nconductance at T= 0 K, and G1corresponds to a\nconstant background conductance. The parameters s\nandndepend on the symmetry of the Kondo state.\nFor the spin-1/2 Kondo e\u000bect s= 0:22 andn= 2 is\ncommonly used (SU(2) parameters), while Keller et al.\nhave introduced s= 0:2 andn= 3 for the spin-orbital\nKondo e\u000bect with SU(4) symmetry17.\nFigure 2(d) shows the temperature dependence of the\nconductance at Vsd= 0 and \u0001 Eorb= 0 in the 1e\nregime. Assuming that both the spin and orbital de-\ngeneracies contribute to the Kondo resonance, we use\nthe SU(4) \ftting parameters, and extract TK= 610 mK\n(G0= 0:25e2/h,G1= 0:15e2/h) using Eq. 1. However,\nalso the standard parameters for the SU(2) Kondo e\u000bect\nprovide a good \ft, and would give a Kondo temperature\nofTK= 950 mK ( G0= 0:33e2/h,G1= 0:08e2/h), thus\nmaking it impossible to conclude the symmetry of the\nKondo e\u000bect by the temperature dependence alone.3\nSU(2) TK = 350 mK0.050.100.15\nSU(2) TK = 950 mK\nSU(4) TK = 610 mK(1,0) (0,1)\n(d) (e)(b)(d) (e)\n(1,0) (0,1)SU(2) TK = 470 mK\n0.20.30.40.5\n0.1\nT (K)0.01 0.1 1(g) (i)-6.8,\n5.4-6.3,4.0\n-6.7,5.2-6.2,4.0Vsd (mV)1.0\n0\n-1.0\n0.51.5\n-0.5\n-1.5(0,0) (1,1) (2,2) 1e 3eB = 0\ndI/dVsd\n(e2/h)\n0 0.4 0.2Vsd (mV)1.0\n0\n-1.00.51.5\n-0.5\n-1.5\n0 0.4 0.2(a)\n(f)VL,R (V)-6.8,\n4.2-5.6,5.6\n-6.8,4.3-5.5,5.40.05 0.4\nB = 0, t = 0\n(1,0)\n(0,1)E E-EGS\n(0,1)    (1,0)(1,0)    (0,1)\nspin-flip spin-flip\ndetuning, ΔEorb(c)\n(1,0)\n(0,1)\nB = 1 T, t = 0\n(1,0)(0,1)\n(1,0)(0,1)\nspin-flip spin-flip(0,1)\n(1,0)EZ\n(1,0)(0,1)(h)\n(1,0)\n(0,1)B = 0\n-0.40.4\n0Vsd (mV)1 K\nVR (V)VL (V)\n(1,0)\n(0,1)70 mK\n 1 K\nVR (V)VL,R (V)\nVL,R (V) VL,R (V)0.20.30.40.5\n0.1dI/dVsd (e2/h)\nT (K)0.01 0.1 1\nT (K)0.01 0.1 1\n70 mK 300 mK(j)\n(k)B = 1 T B = 1 T\n0.05 0.3\nE E-EGSdI/dVsd\n(e2/h)\ndI/dVsd (e2/h)\ndetuning, ΔEorb\nFIG. 2. Spin-orbital and isolated orbital Kondo transport in the one-electron regime. (a) Measurement of d I/dVsdversusVsd\nalong the green gate vector indicated in Fig. 1(b) for B= 0. A zero-bias peak can be observed in the 1e, 2e, and 3e regimes. (b)\nCorresponding measurement in the 1e regime, when detuning the orbitals along the red gate vector. The dashed lines represent\nthe sidegate voltages where the temperature sweeps are performed. (c) Schematic representation of the electron energy levels\nand resulting onset of co-tunneling when detuning the QD orbitals (assuming t= 0). (d) Temperature dependence of the\nzero-bias conductance peak at orbital degeneracy. Blue and dashed black lines show the \ft using Eq. 1 and the parameters for\nSU(2) and SU(4) Kondo scaling, respectively. (e) Temperature dependence and SU(2) Kondo \ft of the zero-bias conductance\npeak in the (0,1) state. (f-i) Corresponding measurements and schematic for B= 1T. The spin degeneracy is lifted, but the\norbital degeneracy at zero detuning in the 1e regime remains. (j) Same as (f) but for di\u000berent temperatures (only 1e regime).\n(k) Conductance as a function of sidegate voltages ( Vsd= 25\u0016V) for 70 mK and 1 K (1e regime).\nB. Isolating the spin-1/2 Kondo e\u000bect\nNext, we isolate spin-Kondo transport in the right\nQD by electrostatically detuning the system to the (0,1)\nregime, and study its temperature dependence (Fig.\n2(e)). For the chosen sidegate voltages, TK= 350 mK\ncan be extracted using Eq. 1 and standard SU(2) pa-\nrameters (G0= 0:06e2/h,G1= 0:06e2/h). The fact\nthatG0\u001c2e2/hcan be explained by an asymmetry\nin the tunnel couplings to source and drain (Appendix\nD). In our QD system, the asymmetry can be due to a\ndi\u000berence in barrier thickness and shape of the QDs.\nC. Isolating the orbital Kondo e\u000bect\nIn order to unambiguously verify the orbital contribu-\ntion to the Kondo e\u000bect at \u0001 Eorb= 0, we isolate the ef-\nfect from spin-Kondo transport using Zeeman spin split-\nting atB= 1 T. The large e\u000bective g-factor (g\u0003) of InAs\n(-14.7 in bulk) facilitates lifting of the spin degeneracy by\nthe Zeeman energy EZ=g\u0003\u0016BB, where\u0016Bis the Bohr\nmagneton. In Fig. 2(f) a resulting gap is observed in the2e and 3e regimes, where spin-\rip transport by inelastic\nco-tunneling is possible when the source-drain voltage is\nlarger than the gap energy. Preliminarily, the absence of\nzero-bias peaks in the 2e and 3e regimes indicates that\nthe observed Kondo peaks in Figs. 2(a) were due to a\nspin-related Kondo e\u000bect. However, in the 1e region the\nzero-bias peak remains, indicating that a di\u000berent degen-\neracy is still present38.\nDetuning the orbitals along the red gate vector allows\nto distinguish the Zeeman gap of the single QD orbitals\nfrom the onset of orbital co-tunneling (Figs. 2(g,h)). In\nthe (1,0) and (0,1) regimes, we observe EZ\u00190:5 meV,\ncorresponding to g\u0003\u00199. The step in di\u000berential conduc-\ntance, originating from the onset of orbital co-tunneling\ntransport, crosses in the Zeeman gap at \u0001 Eorb= 0. A\nzero-bias peak can be observed, which corresponds to\nthe orbital Kondo e\u000bect. Temperature dependent con-\nductance data at this point (Fig. 2(i)) can be \ftted\nwith Eq. 1 and standard SU(2) parameters, resulting in\nTK= 470 mK ( G0= 0:28e2/h,G1= 0:08e2/h). Com-\nparing the Kondo temperature for the spin-orbital Kondo\nresonance with that of the pure spin-1/2 or the pure or-\nbital Kondo e\u000bect, we \fnd a higher TKandG0when both\ndegeneracies are present, which is in agreement with a4\nB = 1 T, t > 0\n(2,1)(1,2)\n(2,1)(1,2)\nspin-flip spin-flip(1,2)\n(2,1)EZ\n(2,1)(1,2)Vsd (mV)0.8\n0\n-0.80.4\n-0.4Vsd (mV)0.8\n0\n-0.80.4\n-0.4\nVR (V)VL (V)\n-2.5-2.0-1.5-1.0-0.5\n-9 -8 -7 -6(1,1)(2,1)\n(2,2)\n(0,0)dI/dVsd\n(e2/h)\n0.05\n0.10.10.2\n(a) (b)\n(c) (d)-1.5,\n-6.6-1.3,-7.2\n-1.5,-6.6-1.3,-7.2\n(2,1)\n(1,2)B = 0\nB = 1 TVL,R (V)\nVL,R (V)00.10.20.3\nG\n(e2/h)\nVsd = 25 μV\nB = 1 T\nE E-EGSdI/dVsd\n(e2/h)\ndetuning, ΔEorb(1,2)\nFIG. 3. The e\u000bect of a \fnite tunnel coupling between the QDs\nin another orbital crossing. (a) Overview measurement with\nthe electron number on the left and right QD ( Nleft,Nright)\nindicated. (b,c) Measurement of d I/dVsdversusVsdrecorded\nalong the pink gate vector at B= 0 T, and B= 1 T. (d)\nSchematic representation of the energy levels and resulting\nonset of co-tunneling when detuning the QD orbitals, assum-\ning \fnite tunnel coupling between the two QD levels.\nhigher symmetry.\nIn Figs. 2(j-k), measurements at di\u000berent tempera-\ntures are presented for B= 1 T. The bias-dependent\nmeasurements of d I/dVsdalong the green gate vector in\nthe 1e regime (Fig. 2(j)) show that the zero-bias peak\ndisappears with increasing temperature, while transport\ncorresponding to sequential tunneling only exhibits a\nweak temperature dependence. Accordingly, the con-\nductance line at orbital degeneracy strongly decays with\ntemperature in Fig. 2(k).\nWe note that the presence of the orbital Kondo e\u000bect\nimplies that the orbital quantum number is also a good\nquantum number in the leads (similar to spin). In this\nwork, the results are discussed in the picture of coupled\ndouble QDs, where the orbital degeneracy is commonly\nattributed to zero tunnel coupling between the two\nQDs. An orbital Kondo resonance therefore requires two\ntransport channels in the leads, each of them coupling\npredominantly only to one of the QDs. However, an\norbital degeneracy would also be present if the two QDs\nare coupled into ring-like states (c.f.35), if the hybridiza-\ntion gap due to spin-orbit interaction and backscattering\nis smaller than the Kondo temperature18,21,39.D. E\u000bect of orbital hybridization\nIn the following, we will discuss the e\u000bect of inter-\ndot tunnel coupling, based on data from the same device\nbut from a di\u000berent crossing of QD orbitals (Fig. 3(a)).\nWhen detuning the orbitals along the pink gate vector\nforB= 0, both a zero-bias peak, and a sloped step in\ndi\u000berential conductance due to orbital co-tunneling can\nbe observed (Fig. 3(b)). From this data it appears as if\nthe onset of orbital co-tunneling crosses at zero bias for\n\u0001Eorb= 0, similar to what has been shown in Fig. 2(b).\nHowever, when the spin degeneracy is lifted ( B= 1 T,\nFig. 3(c)), a \fnite gap at zero detuning is visible instead\nof an orbital Kondo resonance. This can be understood\nconsidering an avoided crossing due to inter-dot tunnel\ncoupling, as schematically presented in Fig. 3(d). In this\ncase the hybridization gap is \u001870\u0016eV, corresponding to\na temperature of \u0018800 mK. Since the gap is comparable\nto the observed Kondo temperatures, it can only be\ndetected by \frst lifting the spin-degeneracy. A similar\nbehavior but with a much larger avoided crossing is\nfound for the 3e regime of Fig. 1(b) (see Appendix Fig.\n5).\nE. Spin-1 Kondo transport\nFinally, we investigate Kondo transport in the 2e\nregime by detuning the orbitals along the orange gate\nvector in Fig. 1(b). A zero-bias peak can be observed\nin the (1,1) regime at B= 0 T (Fig. 4(a)), indicating a\nKondo resonance due to a degenerate ground state when\neach QD contains one electron. At B= 1 T (Fig. 4(b))\nthis degeneracy is lifted, resulting in a gap of \u0018800\u0016eV\nin the (1,1) regime. A second excited state can be\nobserved \u0018350\u0016eV higher in energy than the \frst\nexcited state, which implies the existence of two-electron\nstates due to hybridization, rather than non-interacting\norbitals on each QD. To study the two-electron states\nin more detail, bias-dependent transport as function of\nmagnetic \feld was performed in the center of (1,1), as\nshown in Fig. 4(c). The zero-bias peak Zeeman splits\nwithg\u0003\u00197, which is slightly di\u000berent compared to\ng\u0003\u00199 in the 1e regime, likely due to the interaction\nwith another orbital40. The second onset of co-tunneling\nevolves parallel to the \frst excited state. These observa-\ntions are in agreement with a spin-triplet ground state\nand a spin-singlet excited state, with an exchange energy\nofJ\u0019350\u0016eV, as schematically depicted in Fig. 4(d).\nThe spin-triplet ground state could be explained by the\nformation of a quantum ring41, which can form when two\nquantum dots couple in two points, as previously shown\nfor this material system35. We \fnd that the intensity\nof the zero-bias peak monotonically decreases both with\nB-\feld (Fig. 4(c)) and temperature (Fig. 4(d)), and\ncan therefore be interpreted as an underscreened spin-1\nKondo e\u000bect15,42,43. The temperature scaling can be5\nVsd (mV)1.0\n0\n-1.00.51.5\n-0.5\n-1.5\n Vsd (mV)0.8\n0\n-0.80.4\n-0.4\nB (T)00.4 0.8(2,0) (1,1) (0,2)(a) (b)\n(c)\n0.100.150.20\nSU(2) \nTK = 780 mK(e)\n0.10.20.3\n-7.0,\n7.1-5.7,3.5-7.1,7.4-5.5,3.0B = 0 B = 1 T\nVL,R (V) VL,R (V)\nT (K)0.01 0.1 1dI/dVsd\n(e2/h)dI/dVsd (e2/h)E E-EGS\nBS\nT0\nT\n+T-J\nT+      T0T+      S\n0.060.100.14dI/dVsd\n(e2/h)(d)\nFIG. 4. Kondo transport in the two-electron regime. (a)\nMeasurement of d I/dVsdversusVsdrecorded along the or-\nange gate vector at B= 0 T, showing a Kondo resonance\nin (1,1). (b) Corresponding measurement at B= 1 T. The\nground state degeneracy is lifted, and an additional onset of\nco-tunneling is observed with an energy of \u0018350\u0016eV higher\nthan the \frst excited state. (c) Magnetic \feld sweep in the\ncenter of (1,1). (d) Schematic representation of the 2e state\nenergies and resulting onset of co-tunneling as a function of\nB-\feld. The triplet GS Zeeman splits into T +, T0and T\u0000.\nAt \fniteB-\feld, T +is the GS, and we observe transitions to\nT0and to the singlet state, S. (e) Temperature dependence\nof the zero-bias conductance peak in the center of (1,1) at\nB= 0 T.\ndescribed by Eq. 1, and corresponds to TK= 780 mK\n(G0= 0:07e2/h,G1= 0:08e2/h) if using standard\nSU(2) parameters.\nIV. SUMMARY AND CONCLUSION\nIn summary, we have studied Kondo transport in\nparallel-coupled QDs in InAs nanowires. In the 1e regime\nwe observe the spin-1/2 Kondo e\u000bect, the combined\nspin-orbital Kondo e\u000bect, and the orbital Kondo e\u000bect\nwhen the spin degeneracy is lifted. We demonstrate\nthat \fnite inter-dot tunnel coupling inhibits the orbital\nKondo e\u000bect by hybridizing the orbitals, and emphasize\nthat the presence of a small energy gap can only be\ndetected when the spin degeneracy is selectively lifted.\nThe 2e regime exhibits a triplet ground state likely\ndue to the formation of ring-like states, leading to a\nKondo resonance at B= 0. The possibility to isolate\nthe di\u000berent degeneracies makes this an ideal material\nsystem for studies of higher symmetry Kondo e\u000bects.ACKNOWLEDGMENTS\nThe authors thank I-J. Chen and J. Paaske for fruit-\nful discussions. This work was carried out with \fnan-\ncial support from the Swedish Research Council (VR),\nNanoLund, the Knut and Alice Wallenberg Foundation\n(KAW), and the Crafoord Foundation. H.P. thank-\nfully acknowledges funding from the Swiss National Sci-\nence Foundation (SNSF) via Early PostDoc Mobility\nP2ELP2 178221.\nAppendix A: Overview conductance measurement\nFigure 5(a) shows conductance ( G) for a wide range\nof sidegate voltages ( VL,VR). The overview measurement\nshows that honeycombs with no discernible hybridiza-\ntion can be observed for speci\fc orbital crossings. We\nexplain this by the formation of ring-like states when\nthe orbitals are coupled in two points, as investigated in\ndetail in Ref.35. The orbital crossing showing the spin-\norbital Kondo e\u000bect (c.f. Fig. 2 of the main article) is\nhighlighted with a red square, while the additional cross-\ning where the orbital Kondo e\u000bect is suppressed due to\na small hybridization gap (c.f. Fig. 3 of the main ar-\nticle) is highlighted with a green square. We note that\nthe data presented in Fig. 3 was obtained at a backgate\nvoltage ofVBG=\u00000:8 V, while the overview here was\nobtained at VBG=\u00001 V. In this supplementary mate-\nrial, we present additional measurements for the orbital\ncrossing highlighted in red. The gate vectors which will\nbe discussed in the following are indicated in the conduc-\ntance measurement in Fig. 5(b) (same as Fig. 1(b) of the\nmain article).\nAppendix B: Extraction of TKfrom the voltage\ndependence\nIn the main article, the red gate vector where the or-\nbitals are detuned from the (1,0) to the (0,1) regime is\ndiscussed in great detail, and the Kondo temperature\n(TK) is extracted from the temperature dependence of\nthe zero-bias conductance peak. Alternatively, TKcan\nbe determined from the source-drain voltage dependence\nof the zero-bias peak. Figure 1(c) shows di\u000berential con-\nductance (d I/dVsd) versus the source-drain voltage ( Vsd)\nalong the red gate vector at a magnetic \feld of B= 1 T\n(same as Fig. 2(g) of the main article). Di\u000berential\nconductance as a function of applied voltage at the or-\nbital degeneracy point (highlighted with a dashed line)\nis presented in Fig. 5(d). Note that the x-axis corre-\nsponds to the voltage applied across the device ( VDUT),\nafter subtracting the voltage drop across the series re-\nsistance of the ampli\fer and cryostat wiring and \flters\n(Rs= 16:5 k\n). For the spin-1/2 Kondo e\u000bect, Ple-\ntyukhov et. al. have introduced that the voltage depen-\ndence of the di\u000berential conductance can be described6\nG\n(e2/h)\n01VL (V)\nVR (V)-12-8-44\n-15 -10 -5 00VBG = -1 V\n5(a)\nVR (V)3 4 5 6 7(0,0)(2,2)\n(1,0)\n(0,1)Vsd = 25 μV\nB = 1 TVL (V)-5.0\n-5.5\n-6.0\n-6.5\n-7.0\n-7.5 00.3(b)\nG\n(e2/h)\n(1,1)\nVRVL\n(0,0)\n(0,1)(1,1)(1,0)(2,1)\n(2,2)\n(1,2)seq. tunneling left QD\nseq.tunneling right QDspin-orbital Kondo   spin-1/2 Kondo   spin-1 Kondo(h)\nTK = 800  mK\n0.10.20.3\nVDUT (mV)-0.2 -0.1 00.1 0.20.4(d)\n(1,0) (0,1)(c)\n-6.7,\n5.2-6.2,4.0V\nL,R (V)B = 1 T\nVsd (mV)1.0\n0\n-1.00.51.5\n-0.5\n-1.5B = 0 B = 1 T0.050.3\n(e) (f)\n(2,1) (1,2)\n-6.6\n6.9-5.8,4.6V\nL,R (V)-6.4,\n6.5-5.7,4.8V\nL,R (V)\nVsd (mV)1.0\n0\n-1.00.51.5\n-0.5\n-1.5\n0.050.3dI/dVsd\n(e2/h)\ndI/dVsd\n(e2/h)\ndI/dVsd (e2/h)\nVL,R (V)-6.8,\n4.2-5.6,5.6GVsd=0 (e2/h)\n0.10.20.30.4\n0(g)\n(0,0)\n(1,1)1e 3eB = 0\nFIG. 5. (a) Conductance as a function of sidegate voltages. The red and green square indicate the orbital crossings which are\nstudied. (b) Magni\fed conductance plot of the red orbital crossing, where relevant gate vectors are indicated. (c) Di\u000berential\nconductance versus source-drain voltage recorded along the red gate vector at B= 1 T. (d) Di\u000berential conductance as a\nfunction of voltage across the device measured at the sidegate voltages corresponding to the dashed line in (c). The data\nis \ftted with Eq. B144. (e)-(f) Transport recorded in the 3e regime along the cyan gate vector at B= 0 T, and B= 1 T,\nrespectively. (g) Conductance Gat zero bias along the green gate vector. (h) Schematic representation of the orbital crossing\nand the di\u000berent transport mechanisms at B= 0 T.\nby\ndI=dVsd(Vsd) = dI=dVsd(0)\u0014\n1 +\u00172(21=s1\u00001)\n(1\u0000b+b\u0017s2)\u0015\u0000s1\n(B1)\nwiths1= 0:32,b= 0:05,s2= 1:26 and\u0017=e(Vsd\u0000\nV0)=(kBT\u0003\nK). The di\u000berential conductance at zero bias\n(dI/dVsd(0)), the Kondo temperature ( T\u0003\nK), andV0are\n\ftting parameters44. Using this equation to \ft our exper-\nimental data for the orbital Kondo resonance, and after\nsubstituting T\u0003\nK= 1:8TK, as suggested by Pletyukhov et.\nal.44, we extract a Kondo temperature of TK= 800 mK.\nWe note that the Kondo temperature extracted from the\nvoltage dependence is slightly higher compared to the one\nobtained by the temperature dependent data.\nAppendix C: Transport in the 3e regime\nIn the main article, we showed that the orbital Kondo\ne\u000bect is absent if tunnel coupling between the two QDs\nleads to an avoided level crossing at \u0001 Eorb= 0 (c.f. Fig.\n3 of the main article). In order to illustrate that lift-\ning the spin degeneracy can be key to detecting a small\nhybridization gap, we presented data from a di\u000berent or-\nbital crossing (highlighted with a green square in Fig.\n5(a)). Here, we show transport measurements for the\n3e regime of the orbital crossing highlighted with a redsquare in Fig. 5(a). Figures 5(e-f) show d I/dVsdas a\nfunction of VsdforB= 0 T and B= 1 T, when de-\ntuning the orbitals along the cyan gate vector. We ob-\nserve a relatively large hybridization gap which can be\nseen both with and without magnetic \feld. This \fnding\nshows that adding electrons to the QD orbitals slightly\nalters the tunnel coupling between them. In the 1e regime\nthe gate voltages were tuned in order to have the orbital\nKondo e\u000bect (which requires the absence of a hybridiza-\ntion gap). In the 2e regime the two electrons strongly\ninteract, resulting in two-electron states (c.f. Fig 4 of\nthe main article). Finally, in the 3e regime the electrons\nalso strongly interact, which we interpret as a gradual in-\ncrease in tunnel coupling strength with particle number.\nFigure 5(g) shows the conductance at zero bias along the\ngreen gate vector (linecut at Vsd= 0 in Fig. 2(a) of the\nmain article). The single electron peaks are clearly visi-\nble, indicating that the sample is not in the mixed valence\nregime for sidegate voltages in the center of the valleys23.\nThis will be con\frmed by estimating the tunnel coupling\nin Appendix D.\nBased on our observations, the di\u000berent mechanisms\nleading to transport through this particular orbital cross-\ning are summarized in Fig. 5(h): The red and green lines\ncorrespond to sequential tunneling through the left and\nright QD, respectively. The spin-1/2 Kondo e\u000bect is ob-\nserved in the (0,1), (1,0), (1,2), and (2,1) regimes (yellow\nshading). In the 1e regime, the tunnel coupling between7\nVsd (mV)1.0\n0\n-1.00.51.5\n-0.5\n-1.5\n-5.0,\n4.8-5.6,3.6B = 0\nV\nL,R (V)-5.5,\n3.4-6.3,3.0V\nL,R (V)-4.5,\n3.4-6.2,2.2V\nL,R (V)B = 0 B = 1 T\nVgate (meV)0.10.2\n0 1.0 2.0 0.5 1.5Left QD Right QD Right QD Left QD Left QD Right QD\nVgate (meV)G (e2/h)\n0.10.20.30.5\n0 1 2 3FWHM = 0.8 meV FWHM = 1.0 meV(a)(b) (c)(d) (e)\n00.3\n(0,0) (1,0) (2,0) (2,0) (2,1) (2,2) (0,0) (1,0) (2,0) (2,1) (2,2)dI/dVsd\n(e2/h)0.4\nFIG. 6. (a) Di\u000berential conductance versus source drain voltage as a function of gate vector through the unperturbed left and\nright QD levels (solid and dashed orange gate vector in Fig. 5(b)) for B= 0 T. (c) Corresponding measurement along the\nyellow gate vector for B= 1 T. (d)-(e) Conductance as a function of gate voltage for sequential tunneling through the left and\nright QD (purple and pink gate vector in Fig. 5(b)). The data is \ftted using Eq. D1 to extract \u0000.\nthe two QDs is zero, leading to the spin-orbital Kondo ef-\nfect at orbital degeneracy (blue line). In the (1,1) regime,\na Kondo resonance is found due to a degenerate triplet\nground state (orange shading). No orbital Kondo e\u000bect\nis found between the (2,0)-(1,1)-(0,2) regimes, explained\nby a small spin-orbit-induced mixing (see small avoided\ncrossings in Fig. 4(b)).\nAppendix D: Extraction of the leverarms and tunnel\ncoupling\nIn Fig. 6(a,b) we present d I/dVsdversusVsdfor gate\nvectors through the unperturbed levels of the left and\nright QD (corresponding to the solid/dashed orange vec-\ntors in Fig. 5(b)) at B= 0 T. A zero-bias peak can be\nobserved due to the spin-1/2 Kondo e\u000bect in the (1,0)\nand (2,1) regimes, when the electron population on one\nof the QDs is odd. At B= 1 T (Fig. 6(c)) the spin de-\ngeneracy is lifted, resulting in a Zeeman gap. From the\nheight of the Coulomb diamonds in Figs. 6(a)-(c), the\ncharging energy Ecof the QDs can be extracted. The\nwidth of the Coulomb diamonds is then used to calculate\nthe leverarms of the QDs with respect to each of the two\nsidegate voltages. We note that the Coulomb diamonds\nwere recorded by sweeping both sidegates simultaneously\n(as indicated on the x-axis of Figs. 6(a)-(c)), and we used\nthe slope of conductance lines in Fig. 5(b) in order to\ncalculate an e\u000bective gate vector which depends only on\none sidegate voltage. The relevant parameters for both\nQDs are summarized in Table I. Here, \u000bVL=Ldenotes\nthe leverarm of the left QD (L) with respect to the left\nsidegate voltage VL, and the other leverarms are labeled\ncorrespondingly.\nNext, we extract the tunneling coupling (\u0000) to the two\nQDs. This is relevant, since strictly speaking the em-\npirical equation to calculate TKfrom the temperature\ndependence is only valid as long as \u000f0=\u0000<\u00000:5, where\n\u000f0is the energy di\u000berence of the nearest lower energy\nstate relative to the Fermi level of the contacts1.Ec\u000bVL=L\u000bVR=L\u000bVL=R\u000bVR=R\n(meV) (meV/V) (meV/V) (meV/V) (meV/V)\nLeft QD 6 14 4 - -\nRight QD 8 - - 14 8\nTABLE I. Approximate values of the charging energy and the\nQD leverarms.\n\u0000 can be estimated from the conductance due to se-\nquential tunneling through a QD level. At low source-\ndrain voltage and low temperature ( kBT\u001c\u0000), the con-\nductanceGas a function of gate voltage ( Vgate) can be\ndescribed by a Lorentzian\nG=Gmax\n1 + (2\u0001Vgate=\u0000)2+c (D1)\nwhereGmaxis the peak conductance, \u0001 Vgate=Vgate\u0000\nVgate;0is the detuning from the QD resonance centered\natVgate;0, \u0000 is the full width half maximum (FWHM) of\nthe Lorentzian, and cis a constant o\u000bset. The FWHM\ncorresponds to the sum of the tunnel couplings to source\nand drain \u0000 = \u0000 S+ \u0000 D. Figures 6(d,e) show linecuts\nthrough an orbital of the left and right QD, correspond-\ning to the purple and pink gate vector in Fig. 5(b).\nWe note that the gate vectors were chosen such that\nthey are perpendicular to the conductance lines of the\nQD levels, after having converted the sidegate voltages\ninto energies using the leverarms. Using equation D1,\n\u0000\u00190:8 meV and \u0000 \u00191 meV can be extracted for the\nleft and right QD, respectively. We note that the se-\nquential tunneling lines were measured at Vsd= 25\u0016eV,\nleading to an additional broadening of the peak. The cal-\nculated \u0000 therefore can be considered an upper bound.\nFor the orbital Kondo e\u000bect, the measurement was done\nat orbital degeneracy in the 1e in the middle of the two\ncharge state resonances (see Fig. 2(a)). We therefore\nhave\u000f0=\u00000:5\u0001U1;2=\u00001:5 meV, where U1;2= 3 meV is\nthe inter-dot Coulomb energy. With \u000f0=\u0000 =\u00001:5<\u00000:5\ntransport is in the Kondo regime, and the empirical equa-8\ntion to extract the Kondo temperature from the temper-\nature dependence is therefore valid.\nFinally, we extract the asymmetry \u0000 S=\u0000Dfor the two\nQDs based on the sequential tunneling conductance in\nFigs. 6(d)-(e). Using Gmax= (4\u0000 S\u0000D)=(\u0000S+ \u0000 D)2e2/h,\nwe extract \u0000 S=\u0000D\u001920 and \u0000 S=\u0000D\u00198 for the left andright QD, respectively. We note that an estimation of\nthe asymmetry based on G0in the Kondo regime would\nbe less accurate since the electron temperature is com-\nparable toTK45. The asymmetry of the tunnel couplings\nof each QD also depends on the sidegate voltages due to\na change of con\fnement potential.\n\u0003heidi.potts@ftf.lth.se\nyclaes.thelander@ftf.lth.se\n1D. Goldhaber-Gordon, J. G ores, M. A. Kastner, H. Shtrik-\nman, D. Mahalu, and U. Meirav, Phys. Rev. Lett. 81, 5225\n(1998).\n2S. M. Cronenwett, T. H. Oosterkamp, and L. P. Kouwen-\nhoven, Science 281, 540 (1998).\n3J. Schmid, J. Weis, K. Eberl, and K. v. Klitzing, Physica\nB: Condensed Matter 256-258 , 182 (1998).\n4J. Schmid, J. Weis, K. Eberl, and K. v. Klitzing, Phys.\nRev. Lett. 84, 5824 (2000).\n5A. Kogan, G. Granger, M. A. Kastner, D. Goldhaber-\nGordon, and H. Shtrikman, Phys. Rev. B 67, 113309\n(2003).\n6J. Paaske, A. Rosch, P. Woel\re, N. Mason, C. M. Marcus,\nand J. Nygard, Nature Physics 2, 460 (2006).\n7S. Tarucha, D. G. Austing, Y. Tokura, W. G. van der Wiel,\nand L. P. Kouwenhoven, Phys. Rev. Lett. 84, 2485 (2000).\n8M. Pustilnik, Y. Avishai, and K. Kikoin, Phys. Rev. Lett.\n84, 1756 (2000).\n9S. Sasaki, S. De Franceschi, J. Elzerman, W. van der Wiel,\nM. Eto, S. Tarucha, and L. Kouwenhoven, Nature 405,\n764 (2000).\n10D. Giuliano, B. Jouault, and A. Tagliacozzo, Phys. Rev.\nB63, 125318 (2001).\n11W. G. van der Wiel, S. De Franceschi, J. M. Elzerman,\nS. Tarucha, L. P. Kouwenhoven, J. Motohisa, F. Nakajima,\nand T. Fukui, Phys. Rev. Lett. 88, 126803 (2002).\n12S. J. Chorley, M. R. Galpin, F. W. Jayatilaka, C. G. Smith,\nD. E. Logan, and M. R. Buitelaar, Phys. Rev. Lett. 109,\n156804 (2012).\n13W. Liang, M. Bockrath, and H. Park, Phys. Rev. Lett.\n88, 126801 (2002).\n14C. H. L. Quay, J. Cumings, S. J. Gamble, R. dePicciotto,\nH. Kataura, and D. Goldhaber-Gordon, Phys. Rev. B 76,\n245311 (2007).\n15N. Roch, S. Florens, T. A. Costi, W. Wernsdorfer, and\nF. Balestro, Phys. Rev. Lett. 103, 197202 (2009).\n16D. Krychowski and S. Lipi\u0013 nski, Phys. Rev. B 93, 075416\n(2016).\n17A. J. Keller, S. Amasha, I. Weymann, C. P. Moca,\nI. G. Rau, J. A. Katine, H. Shtrikman, G. Zar\u0013 and, and\nD. Goldhaber-Gordon, Nature Physics 10, 145 (2013),\n1306.6326.\n18T. Delattre, L. G. Herrmann, P. Mor\fn, J. Berroir,\nG. F\u0012 eve, B. Pla\u0018 cais, D. C. Glattli, M. Choi, C. Mora, and\nT. Kontos, Nature Physics 5, 208 (2009).\n19M. Ferrier, T. Arakawa, T. Hata, R. Fujiwara, R. De-\nlagrange, R. Deblock, R. Sakano, A. Oguri, and\nK. Kobayashi, Nature Physics 12, 230 (2015).\n20M. Ferrier, T. Arakawa, T. Hata, R. Fujiwara, R. Dela-\ngrange, R. Deblock, Y. Teratani, R. Sakano, A. Oguri,\nand K. Kobayashi, Phys. Rev. Lett. 118, 196803 (2017).21P. Jarillo-Herrero, J. Kong, H. S. J. v. d. Zant, and\nC. Dekker, Nature 434, 484 (2005).\n22A. Makarovski, A. Zhukov, J. Liu, and G. Finkelstein,\nPhys. Rev. B 75, 241407(R) (2007).\n23A. Makarovski, J. Liu, and G. Finkelstein, Phys. Rev.\nLett. 99, 066801 (2007).\n24G. P. Lansbergen, G. C. Tettamanzi, J. Verduijn, N. Col-\nlaert, S. Biesemans, M. Blaauboer, and S. Rogge, Nano\nLetters 10, 455 (2010).\n25G. C. Tettamanzi, J. Verduijn, G. P. Lansbergen,\nM. Blaauboer, M. J. Calder\u0013 on, R. Aguado, and S. Rogge,\nPhys. Rev. Lett. 108, 046803 (2012).\n26U. Wilhelm, J. Schmid, J. Weis, and K. Klitzing, Physica\nE: Low-dimensional Systems and Nanostructures 14, 385\n(2002).\n27A. W. Holleitner, A. Chudnovskiy, D. Pfannkuche,\nK. Eberl, and R. H. Blick, Phys. Rev. B 70, 075204 (2004).\n28S. Sasaki, S. Amaha, N. Asakawa, M. Eto, and S. Tarucha,\nPhys. Rev. Lett. 93, 017205 (2004).\n29S. Amasha, A. J. Keller, I. G. Rau, A. Carmi, J. A. Katine,\nH. Shtrikman, Y. Oreg, and D. Goldhaber-Gordon, Phys.\nRev. Lett. 110, 046604 (2013).\n30Y. Okazaki, S. Sasaki, and K. Muraki, Phys. Rev. B 84,\n161305(R) (2011).\n31M. Nilsson, L. Namazi, S. Lehmann, M. Leijnse, K. A.\nDick, and C. Thelander, Phys. Rev. B 93, 195422 (2016).\n32M. Nilsson, I.-J. Chen, S. Lehmann, V. Maulerova, K. A.\nDick, and C. Thelander, Nano Letters 17, 7847 (2017).\n33M. Nilsson, F. Vi~ nas Bostr om, S. Lehmann, K. A. Dick,\nM. Leijnse, and C. Thelander, Phys. Rev. Lett. 121,\n156802 (2018).\n34C. Thelander, M. Nilsson, F. Vi~ nas Bostr om, A. Burke,\nS. Lehmann, K. A. Dick, and M. Leijnse, Phys. Rev. B\n98, 245305 (2018).\n35H. Potts, I.-J. Chen, A. Tsintzis, M. Nilsson, S. Lehmann,\nK. A. Dick, M. Leijnse, and C. Thelander, Nature Com-\nmunications 10, 5740 (2019).\n36(2019), we also note a \fnite splitting of the spin-1/2 Kondo\nresonance in the (1,0) regime. This could imply spin-\npolarization of the electrons in the InAs nanowire segments\nthat act as source and drain due to the Ni contacts, similar\nto observations in carbon nanotubes with Ni contacts46.\n37T. A. Costi, A. C. Hewson, and V. Zlatic, Journal of\nPhysics: Condensed Matter 6, 2519 (1994).\n38(2019), we note that the small gap in the zero-bias peak\nin the 1e regime can be explained by the gate vector being\nnot exactly aligned with the degeneracy of the QD levels.\n39J. Paaske, A. Andersen, and K. Flensberg, Phys. Rev. B\n82, 081309(R) (2010).\n40S. Csonka, L. Hofstetter, F. Freitag, S. Oberholzer,\nC. Schnenberger, T. S. Jespersen, M. Aagesen, and J. Ny-\ngrd, Nano Letters 8, 3932 (2008).\n41B. Baxevanis and D. Pfannkuche, \\Coulomb interaction in9\n\fnite-width quantum rings,\" in Physics of Quantum Rings\n(Springer Berlin Heidelberg, 2014) Chap. 15, pp. 381{408.\n42M. Pustilnik and L. I. Glazman, Phys. Rev. Lett. 87,\n216601 (2001).\n43A. Posazhennikova and P. Coleman, Phys. Rev. Lett. 94,\n036802 (2005).44M. Pletyukhov and H. Schoeller, Phys. Rev. Lett. 108,\n260601 (2012).\n45M. Pustilnik and L. Glazman, Journal of Physics: Con-\ndensed Matter 16, R513 (2004).\n46J. R. Hauptmann, J. Paaske, and P. E. Lindelof, Nature\nPhysics 4, 373 (2008)."    },    {        "title": "1311.0965v1.Spin_accumulation_detection_of_FMR_driven_spin_pumping_in_silicon_based_metal_oxide_semiconductor_heterostructures.pdf",        "content": "1 \n Spin accumulation detection of FMR driven  spin pumping  in silicon -based \nmetal -oxide -semiconductor hetero structures  \n \nY. P u1, P. M. Odenthal2, R. Adur1, J. Beardsley1, A. G. Swartz2, D. V. Pelekhov1, R. K. \nKawakami2, J. Pelz1, P. C. Hammel1, E. Johnston -Halperin1  \n \n1Department of Physics, The Ohio State University , Columbus, Ohio 43210  \n2Department of Physics and Astronomy, University of California, Riverside, California 92521  \n \nThe use of the spin  Hall effect and its inverse  to electrically detect and manipulate dynamic \nspin currents generated via ferromagnetic resonance (FMR) driven spin pumping has \nenabled the investigation of  these dynamically injected currents across a wide variety  of \nferromagnetic materials.  However, while this approach has proven to be an invaluable \ndiagnostic for exploring the spin pumping process it requires  strong spin -orbit coupling , \nthus substantially  limit ing the materials basis available for  the detector/ch annel material \n(primarily Pt, W and Ta) . Here , we re port FMR driven  spin pumping into a weak spin -\norbit channel through the measurement of a spin  accumulation voltage in a Si-based  metal -\noxide -semiconductor (MOS)  heterostructure . This alternate experimental approach \nenables the investigation of dynamic  spin pumping in a broad  class of materials with weak \nspin-orbit coupling and long spin  lifetime  while providing additional information regarding \nthe phase evolution of the injected s pin ensemble  via Hanle -based measurements of the \neffective spin lifetime .  \n 2 \n The creation and manipulation of non -equilibrium spin populations in non -magnetic \nmaterials (NM) is one of the cornerstones of modern spintronics. These excitations have to date \nrelied primarily on charge based phenomena, either via direct electrical injection from a \nferromagnet (FM)1-7 or through the exploitation of the spin -orbit interaction8-10. Ferromagnetic \nresonance (FMR) driven spin pumping11-22 is an emerging method to dynam ically inject pure \nspin current  into a NM with no need for an accompanying charge current, implying substantial \npotential impacts on low energy cost , high efficiency spintronics . However, while the creation  of \nthese non -equilibrium spin currents does not require a charge current , previous studies of \ntransport -detected spin pumping do rely on a strong spin -orbit interaction in the NM to convert \nthe spin current into a charge current  in the detector via the  inverse spin -Hall effect (ISHE) 12-20. \nThis approach has proven  to be an  instrumental diagnostic , but it does carry with it several \nlimitations; specifically, the ISHE measures spin current  not spin density , is only sensitive to a \nsingle component of the ful l spin vector  and is only effective in materials with strong spin -orbit \ncoupling . Here we demonstrate an alternate detection geometry relying on the measurement of a \nspin accumulation voltage using a ferromagnetic electrode, similar to the three -terminal  \ngeometry pioneered for electrically -driven spin injection4,23-28. This approach dramatically \nexpands the materials basis for FMR  driven spin pumping, allows for the direct measurement of \nspin accumulation in the channel and enables the phase -sensitive meas urement of the injected \nspin population.  \n Our study is performed in  a silicon -based metal -oxide -semiconductor (MOS) structure \ncompatible with current semiconductor logic technologies.  Using  a Fe/MgO/p -Si tunnel diode \nwe achieve spin pumping into  a semiconductor across an insulat ing dielectric. This approach \nallows voltage -based detection of the spin accumulation under the electrode4-6,23-29. Further, we 3 \n demonstrate sensitivity to the phase of the injected spin via the observation of Hanle dephasing  \nin the presence of an out -of-plane magnetic field . These results establish a bridge between the \npure spin currents generated by FMR driven spin pumping and traditional charge -based spin \ninjection,  laying the foundation for a new class of experimental probes and promising the \ndevelopment of novel spin -based devices compatible with current CMOS technologies.  \n Tunne l diodes are fabricated from  Fe(10nm)/ MgO( 1.3nm)/ Si(100) heterostructures  \ngrown by mo lecular beam epitaxy (MBE). The p -type Si substrates are semiconductor on \ninsulator (SOI) wafers with a 3 μm thick Si device layer containing 5×1018 cm-3 boron dopants , \nproducing a room temperature resistivity of 2× 10-2 Ωcm. The device is  patterned by conv entional \nphotolithography technique s into  a Fe/MgO/Si tunnel contact  of 500  μm × 500 μm lateral size, \nplaced 1 mm away from Au reference contacts for voltage measurements.  Spin pumping and \nFMR measurements are performed in the center of a radio frequency (RF) microwave cavity \nwith f = 9.85GHz  with a DC magnetic field , \nDCH , applied along the x-axis, as sketched in Fig.  \n1a.   On resonance , a pure sp in current is injected into the silicon channel via coupling between  \nthe precessing magnetization  of the ferromagnet,  M, and the conduction electrons in the silicon. \nThis spin current  induces an imbalance in the spin -resolved electrochemical potential and \nconsequent  spin accumulation  given by\n S , where  \nand \n are the chemical \npotentials of up and down spin s, respectively. Using a standard electrical  spin detection \ntechnique4-6,23-29 the spin accumulation can be detected electrically via the relationship : \n)1(2eP VS\nS\n \nwhere \nSV  is the spin -resolved voltage between Fe and Si, \nP is the spin polarization of Fe , \n is \nthe spin detection efficiency , and \nS  is assumed to be proportional to the component of the net 4 \n spin polarization parallel to M30. Figure 1c shows the magnetic field dependence of the FMR \nintensity (upper panel) and the spin accumulation induced voltage VS (lower panel), clearly \ndemonstrating  spin accumulation at the ferromagnetic resonance . \n Figure 2a show s the RF power , PRF, dependence of the FMR intensity  (upper panel)  and \nSV\n (lower panel) on resonance ; the former is proportional to the square root of  \nRFP  and latter is \nlinear with\nRFP , consistent with ISHE detected spin pumping18-20. As shown in Fig.2b, \nSV  is \nconstant  when M reverses, consistent with our local detection geometry wherein the injected spin \nis always parallel to the magnetization of the FM electrode . Note that this is in contrast to the \nmagnetization dependence of ISHE detection, wherein the sign of the ISHE gives a  measure of \nthe spin orientation relative to the  detection electrode . As a result, o ur technique distinguishes the \nspin accumulation signal from  artifacts due to magneto -transport or spin transport, such as the \nanomalous Hall effect, ISHE or spin Seebeck effect , which depend on the direction of M. In \naddition, the current -voltage characteristic ( I-V) of the  tunnel contact is linear at room \ntemperature (see Supplementary Information ), implying at best a weak rectification of any RF -\ninduced pickup currents. This expectation is confirmed by the small offset (below 10 V) \nobserved in spin accumulation voltage measurement s.  Our technique  also rules out potential  \nspurious signal s due to magneto -electric transport such as tunneling anisotropic magneto -\nresistance (TAMR)  that would  require  a rectified  bias voltage of at least mV scale to give the  \nobserved  V scale signal  observed  at resonance.      \n In order to probe the dynamics of the  observed spin accumulation, the applied magnetic \nfield is rotated towards the sample normal (within the xz-plane) by an angl e \n, introducing an \nout-of-plane component,  \nzH. Due to the strong  demagnetization field of our thin -film geometry \n(~2.2 T) the orientation of the magnetization lags the orientation of the applied field,  remaining 5 \n almost entirely in -plane ( the maximum estimated deviation is 2°). As a result , the injected spins \n(parallel to M) precess due to the applied field.  As the magnitude of \nzH  increases this precession \nwill lead to a dephasing of the spin ensemble  and consequent decrease in its net magnetization , \n(the Hanle effect3-6,23-29). Figure 3a shows the FMR spectrum for different angles \n ; the \nresonant field \nFMRH  changes from 250G to 400G as \n  changes from 0 to 40 degrees. This \nincrease is consistent with the fact that the in -plane component of H primarily determines the \nresonance condition, so as \n  increases a larger total applied field is therefore required to drive \nFMR  (see Supplementa ry Information ). Figure 3b shows \nSV  vs. \nDCH  over the same angular \nrange. The peak position of \nSV  shifts in parallel with the FMR spectrum, but the peak value \ndecreases with increasing  \nzH , as expected for Hanle -induced dephasing in an ensemble of \ninjected spins3-6,23-29.  \nFor an isotropic ensemble of spins precessing in a uniform field perpendicular to M the \neffect of this dephasing on \nSV  can be described by a simple Lorentzian function:  \n)2()(1)(20\n SHS Vz x S S\n \nwhere  \n  is the Larmor frequency given by \n/z B effH g , \neffg  is the effective Land é g-factor, \nB\n is the Bohr magneton ,\n is the Plank’s constant , and \n  is the spin lifetime. In the more \ngeneral case that H is not perpendicular to M, as is the case here, then Eq.  (2) should be replaced \nby the more general function:  \n)3() (11\n2 22 2\n22\n0\n\n  \n\ntotal totalz y\ntotalx\nx S S S V\n \nwhere \n),, (/ zyxi gHB effi i    23,27.  If H is in the xz-plane, this reduces to  6 \n \n)4() (11sin cos22 2\n0\n\n\n \ntotalx S S S V \n This general behavior has been observed in previous studies of three terminal  electrical \nspin injection4,23-28 (Fig. 4a). However, it has been widely reported in electrically detected spin \ninjection experiments  that spatially varying local fields due to the magnetic electrodes, coupling \nto interfacial spin states and other non -idealities generate spin dynamics that are not well \ndescribed by this simple model. As a result \n  is generally understo od to represent an  effective \nspin lifetime, \neff , and while there are some initial efforts to more quantitatively account for the \nreal sample environment, such as the so -called “inver ted-Hanle” measurement23,26,27, a detailed \nmodel of these interactions is currently lacking.  \nWe explore the functional dependence of the dephasing of our FMR driven spin current \nby plotting the peak spin accumulation voltage, \npeak\nSV , as a function of \nzH  (Fig. 4b, solid \ncircles). The suppression of the spin accumulation at high magnetic fields seen in Fig. 3 is a clear \nindication of the dephasing of the injected spin ensemble; however, attempts to fit this behavior \nto Eq. (4) reveal that this simple, isotropic model fails to accurately reproduce our data (Fig. 4b, \nblack dashed  lines). In particular, it is a feature of the isotropic model that for a magnetic field \nthat is not parallel to z that the spin polarization along M, and therefore the measured \naccumulation voltage, does not go to zero at high field even for infinite spin lifetime.  This \ndiscrepancy likely arise s from  contributions due to the various non -idealities discussed above ; in \nparticular, as we discuss below, we believe that the coupling to localized states and the impact of \nbulk spin diffusion may play a more central role in this experimental geometry.  \nWe note that our data is well described by  a simple Lorentzian  (Eq. 2), though the \nrelationship between the effective spin lifetime extracted from this fit  (0.6 ns) , which we label 7 \n \nFMR, to the \neff  defined in Eq. (4) is not clear. For comparison, the intensity of the FMR signal \nis found to be constant  to within roughly 10% (solid red triangles) , suggesting  that the spin \ncurrent is roughly  constant and indicating that FMR -driven heating19,20, if present, does not \ncontribute significantly to the field -induced suppression of  \npeak\nSV  seen i n Fig. 4b.  \nA key advantage of our experimental geometry is that it allows direct comparison of this \ndephasing with the more traditional three -terminal electrical injectio n within the same device26. \nFigure 4b (open purple circles) shows the spin accumulation voltage measured in the three -\nterminal geometry as a function of a perpendicular applied field, \nzH . The dephasing in this \ngeometry is clearly much slower than in FMR driven s pin pumping. This observation is \nsupported by the Lorentzian fit to Eq. (2) indicated by the solid purple line, yielding an effective \nlifetime of 0.11 ns, consistent with previous reports by our group and others4,23-27. In considering \nthe origin of this discrepancy in observed lifetime a natural suspicion falls on the different \nexperimental methodologies . Specifically , for the spin pumping case the field is applied at an \nangle \n , resulting in both in -plane and perpendicular components to the field, while for the DC \ncurrent injection only the perpendicular component is present.  \nThe consequence of this vector magnetic field is twofold: first, it will rotate the \nprecession axis of the injected spins away from the perpendicular case implicit in the simple \nHanle model as described above, and second, it will generate an “inver ted” Hanle effect that has \nbeen proposed to derive from the interplay between an in -plane applied magnetic fiel d and some \nfinite inhomogeneity in the local fields due to the magnetization of the electrode.  In Fig. 4c w e \nexplore this behavior in a control sample wherein we perform both traditional Hanle (i.e. wherein \nthe only applied magnetic field is\nzH ) and rotating Hanle measurements (i.e. wherein the \nmagnetic field is rotated by an angle \n , yielding both in -plane and perpendicular components to  8 \n \nH), see Supplementary Information.  The field values for the rotating Hanle experiment are \nchosen to correspond to the values o f \nzH  from Fig. 4b. As expected, the traditional Hanle \nmeasurement again yields an effective lifetime of roughly 0.1 4 ns (open purple circles) . While \nthe rotating Hanle geometry does yield a slightly shorter lifetime of  0.09 ns (solid blue squares) , \nthis variation is too small  (and in the wrong direction)  to account for  the longer effective lifetime  \nobserved for FMR driven spin pumping. We therefore conclude that th e enhanced dephasing rate \nobserved in Fig. 4b indicates the FMR -driven  and electrically -driven spin injection  processes \ndiffer . \nWhile the origin of this discrepancy  is still an active area of investigation, we note that \nthe observed  \nFMR  of 0.6 ns is consistent with previous measurements of the spin lifetime in the \nbulk silicon channel at these doping level4,23-25. Further, the requirement that there be no net \ncharge flow during FMR driven spin injection implies that any forward propagati ng tunneling \nprocess be balanced by an equal and opposite back tunneling process. As shown by the band \ndiagram of spin pumping  in Fig. 4a this opens up a potential pathway for coupling from  the bulk \nSi states into the intermediate states that dominate the three -terminal accumulation voltage26,28. \nThe band diagram of electrical spin injection (upper panel of Fig. 4a) shows that  this process is \nstrongly suppressed in electrical spin injection due to the finite bias ( 5 mV in this case) present \nacross the tunnel junction . If we assume for the sake of argument that  the spin pumping \nmeasurement is in fact sensitive to the bulk spin polarization in silicon, we can calculate the spin \ncurrent  using the relation \nsf S SeJ ; according to Eq . (1) with  \nP=0.4 for Fe and assuming  \n\n=0.5, the spin current density is \nSJ  ~ 2 ×  105 Am-2. This spin current is roughly one order of \nmagnitude smaller than previous reports of spin pumping from conventional ferromagnets into \nmetals.  9 \n  In summary, we demonstrate spin pumping into  a semiconductor through an insulat ing \ndielectric . This approach allows observation of  precession  of the dynamically injected spins an d \ncharacteriz ation of the  effective spin lifetime . Our results directly probe the coherence and phase \nof the dynamically injected spins and the spin manipulation  of that spin ense mble via spin \nprecession, lay ing the foundation for novel spin pumping based spintronic applications . \n \nAcknowledgements  \nThis work is supported by the Center for Emergent Materials at the Ohio State University, \na NSF Materials Research Science and Engineering Center (DMR -0820414) (YP, PO, JB, AS, \nRK, JP and EJH) and by the Department of Energy through grant DE -FG02 -03ER46054 (RA \nand PCH). Technical  support is provided by the NanoSystems Laboratory at the Ohio State \nUniversity.  The aut hors thank Andrew Berger  and Steven Tjung  for discussion s and assistance.  \n \nReference and notes   \n1. Žutić, I. , Fabian , J. & Das Sarma , S. Spintronics: Fundamentals and applications. Rev. Mod. \nPhys.  76, 323 (2004).  \n2. Awschalom , D. D.  & Flatté , M. E. Challenges for semiconductor spintronics. Nature Phys. \n3, 153 (2007).  \n3. Appelbaum , I., Huang , B., & Monsma, D. J. Electronic measurement and control of spin \ntransport in silicon . Nature  447, 295 (2007).  \n4. Dash, S. P. , Sharma,  S., Patel, R. S. , de Jong, M. P. & Jansen, R. Electrical creation of spin \npolarization in silicon at room temperature . Nature  462, 491 (2009).  10 \n 5. Jedema, F. J., Heersche, H. B., Filip, A. T., Baselmans, J. J. A. & van Wees, B. J. Electrical \ndetection of spin precession in a metallic mesoscopic spin valve. Nature  416, 713 (2002).  \n6. Lou, X. et al.  Electrical detection of spin transport in lateral  ferromagnet  semiconductor \ndevices. Nature Phys.  3, 197  (2007).  \n7. Jonker,  B. T. , Kioscog lou, G., Hanbicki, A. T., Li, C. H., & Thompson, P. E.  Electrical spin -\ninjection into silicon from a ferromagnetic metal /tunnel barrier contact . Nature Phys.  3, 542  \n(2007).  \n8. Kato , Y. K., Myers , R. C., Gossard , A. C.  & Awschalom , D. D. Observation of the Spin Hall \nEffect in Semiconductors . Science  306, 1910 (2004).  \n9. Valenzuela , S. O.  & Tinkham , M. Direct electronic measurement of the spin Hall  effect . \nNature  442, 176 (2006).  \n10. Liu, L. et al.  Spin-Torque Switching with the Giant Spin Hall Effect of Tantalum. Science  \n336, 555 (2012).  \n11. Tserkovnyak Y., Brataas, A. & Bauer, G. E.  W. Enhanced Gilbert damping in thin \nferromagnetic Films. Phys. Rev. Lett . 88, 117601 (2002).  \n12. Saitoh, E., Ueda, M., Miyajima, H. & Tatara, G. Conversion of spin current into charge \ncurrent at room temperature: Inverse spin -Hall effect. Appl. Phys. Lett.  88, 182509 (2006).  \n13. Tserkovnyak Y., Brataas, A. & Bauer, G. E.W. & Halperin, B. I. Nonlocal magnetization \ndynamics in ferromagnetic heterostructures. Rev. Mod. Phys.  77, 1375 (2005).  \n14. Kajiwara , Y. et al . Transmission of electrical signals by spin -wave interco nversion in a \nmagnetic insulator . Nature  464, 262 (2010).  \n15. Kurebayashi , H. et al . Controlled enhancement of spin -current emission  by three -magnon \nsplitting . Nature Materials  10, 660 (2011).  11 \n 16. Sandweg, C. W . et al. Spin Pumping by Parametrically Excited Exchange Magnons . Phys. \nRev. Lett.  106, 216601 (2011).  \n17. Czeschka , F. D. et al. Scaling behavior of the spin pumping effect in ferromagnet -platinum \nbilayers . Phys. Rev. Lett.  107, 046601 (2011).  \n18. Ando , K. et al.  Inverse spin -Hall effect induced by spin pumping in metallic system . J. Appl. \nPhys.  109, 103913 (2011).  \n19. Ando , K. et al . Electrically tunable spin injector free from the  impedance mismatch \nproblem . Nature Materials  10, 655 (2011).  \n20. Ando , K. & Saitoh, E.  Observation of the inverse spin Hall effect i n silicon. Nature Comm.  \n3, 629 (2012 ). \n21. Costache, M. V. et al.  Electrical detection of spin pumping due to the precessing \nmagnetization of a single ferromagnet. Phys. Rev. Lett.  97, 216603 (2006).  \n22. Heinrich , B. et al . Spin pumping at the magnetic insulator (YIG)/normal metal (Au) \ninterfaces . Phys. Rev. Lett.  107, 066604 (2011) . \n23. Dash, S. P. et al. Spin precession and inverted Hanle effect in a semiconductor near a finite -\nroughness ferromagnetic interface . Phys. Rev. B  84, 054410  (2011).  \n24. Li, C. H., van‘t Erve , O. & Jonker , B. T. Electrical injection and detection of spin  \naccumulation in silicon at 500K with magnetic metal/silicon dioxide contacts . Nature \nComm.  2, 245 (2011 ). \n25. Gray , N. W. & Tiwaria , A. Room temperature electrical injection and detection of spin \npolarized  carriers in silicon using MgO tunnel barrier . Appl. Phys. Lett.  98, 102112 (2011 ). \n26. Pu, Y . et al. Correlation of electrical spin injection and non -linear charge -transport in \nFe/MgO/Si . Appl. Phys. Lett.  103, 012402 (2013 ). 12 \n 27. Jeon, K. et al. Electrical investigation of the oblique Hanle effect in ferromagnet / oxide / \nsemiconductor c ontacts . arXiv  1211.3486 (2013).  \n28. Tran, M. et al. Enhancement of the spin accumulation at the i nterface  between a spin-\npolarized tunnel junction and a semiconductor . Phys. Rev. Lett. 102, 036601 (2009 ). \n29. Sasaki, T., Oikawa,  T., Suzuki, T., Shiraishi,  M., Suzuki, Y. & Noguchi,  K. Comparison of \nspin signals in silicon between nonlocal four -terminal  and three -terminal methods. Appl. \nPhys. Lett.  98, 012508 (2011).  \n30. As discussed in Ref. 26, Eq. (1) is derived assuming a linear tunneling model, and might \nunderestimate the actual value of \nS  at higher  bias if the current depends super -linearly on \napplied bias.   \n  13 \n Figure legends  \n \nFigure 1 | Experimental setup  \n(a) Schematic of experimental setup. ( b) Diagram of spin accumulation and spin -resolved  \nvoltage  \nSV. (c) FMR intensity (upper panel; arrows indicate state of the Fe magnetiza tion) and \nspin-resolved voltage  \nSV (lower panel) as a function of \nDCH .  \n \nFigure 2  | RF power - and magnetic field - dependence  \n(a) FMR intensity (upper panel) and spin-resolved voltage (lower panel) as a function of RF \npower; ( b) Solid symbols: \nSV  vs. \nFMR DC H H  when \nDCH  is parallel or anti -parallel with the x-\naxis; open symbols indicate the voltage between two Au/Si reference contacts; all data is \nmeasured under the same experimental conditions.  A background offset of ~2 V has been \nsubtracted from all data.  \n \nFigure 3 | Experiments with increasing Hz \n(a) FMR intensity spectra at various magnetic field orientations \n  as described in the text; ( b) \nSV\n vs. \nDCH measured at the same set of magnetic field orientations. The shift in FMR center \nfrequency tracks the expected magnetization anisotropy of the Fe thin film, see text.  \n \nFigure 4 | Hanle effect measurements  14 \n (a) Schematics of experimental setup and band diagram for three -terminal electrical spin \ninjection (upper panel) and spin pumping (lower panel); ( b) Hanle effect as a function of \nzH  for \nthree -terminal (open circles) and spin pumping (solid circles), solid lines are Lorentzian fits \nyielding 0.11 ns and 0.6 ns, respectively; solid triangles are FMR absorption as a function  of H z, \nthe red dashed line is a guide to the eye and the scale bar represents 10% variation; Black  dashed \nlines are simulated using Eq. (4) with \n , \nxH = 248G ( dash dot ) and 310G ( dash), \nrespectively , see Supplementary Information . (c) Hanle effect measured in a control sample by \nthe three -terminal method, open circles are measured with magnetic field applied out of plane, \nsolid squares are obtained using same magnetic field configuration as for FMR driven spin \npumping;  lines are Lorentzian fits yielding 0.14 ns and 0.09 ns, respectively.   15 \n Figures  \n \n \n \nFigure 1  Y. Pu et al.   \n16 \n  \n \n \n \nFigure 2   Y. Pu et al.  \n  \n17 \n  \n \n \n \nFigure 3  Y. Pu et al.  \n  \n18 \n  \n \n \n \nFigure 4   Y. Pu et al.   \n19 \n Supplementary Information  \n \nSpin accumulation detection of FMR driven spin pumping in silicon -based \nmetal -oxide -semiconductor heterostructures  \n \nY. Pu1, P. M. Odenthal2, R. Adur1, J. Beardsley1, A. G. Swartz2, D. V. Pelekhov1, R. K. \nKawakami2, J. Pelz1, P. C. Hammel1, E. Johnston -Halperin1  \n \n1Department of Physics, The Ohio State University, Columbus, Ohio 43210  \n2Department of Physics and Astronomy, University of California, Riverside, California 92521  \n \nA. Linear I -V of Fe/MgO/Si contact at room temperature  \n The I-V characteristic of the Fe/MgO/p -Si contact is linear at room temperature , as \nshown in Fig. S1, indicating that in this regime  the contact resistance is domin ated by the MgO \ninsulating barrier and contribution from the Schottky barrier is negligible. The linear fit gives \n40.9 resistance with about 25m  uncertainty.   \n \nB.  Impact of interface roughness and in -plane external magnetic field  \n  In contrast to traditional three -terminal spin accumulation measurements, for the FMR \ndriven measurements described in Figs. 3 and 4 it is necessary to apply an external magnetic \nfield both parallel and perpendicular to the magnetization. As described in the main text, the \nparallel component of the field is necessary to satisfy the conditions of magnetic resonance and 20 \n the perpendicular component contributes to  the decay of spin accumulation, allowing for a \nHanle -style measurement of the effective spin life time. However, in consulting the literature23,27 \nit becomes evident that this geometry potentially raises an additional concern regarding the \ninterpretation of this data. Specifically, the in plane component of the magnetic field will itself \ninduce an “inv erted -Hanle” effect wherein the measured spin accumulation rises  with the \nmagnitude of the parallel component of the magnetic field (Fig. S2a). The accepted interpretation \nof this effect is an annealing of fluctuations in the magnetization induced by surfa ce roughness of \nthe magnetic layer in large external fields23,27.   \n We measure the spin accumulation on a control sample via 3T electrical spin injection, as \nshown in Fig. S2a, where magnetic field is applied in xz -plane with orientation ranging from in -\nplane ( H//x, 0 degree) to out of plane (H//z, 90 degree). Using the 3T data of \n) (FMR S H HV , \nwhere \nFMRH  for given magnetic field orientation is obtained by spin pumping experiment as \nshown in Figure 3, we can first directly compare electrical spin injection and spin pumping under \nthe same experimental configuration, as discussed in main text.  \nTo get a quantitative understanding of the angular dependence shown in Fig. S2a, one can \nstart with a general formula:  \nSSωSSDt\n      (S1) \nwhere  \n  is the Larmor frequency , \nD is the spin diffusion constant and \n  is the spin lifetime. In \nthe 3T geometry the impact of spin diffusion is usually believed to be negligible23,27, from this \none can obtain an analytical solution under arbitrary applied magnetic field:  21 \n  \n\n\n 2 22 2\n22\n0) (11\n \n\ntotal totalz y\ntotalx\nxS S       (S2) \nwhere \n),, (/ zyxi gHB effi i     , representing each component of the applied magnetic field.  \nFigure S2b shows the simulation according to Equation (S2) with \n = 0.14ns as obtained by \nLorentzian  fit. Clearly the model fails to explain the data shown in Fig. S2a, especially  the \nobservation that at certain orientations the measured spin accumulation rises  with the applied \nmagnetic field increasing.  \n As pointed out by previous studies23,27 stray magnetic fields from the injector due to \ninterface roughness should strongly impact on the Hanle -style measurements. The total magnetic \nfield should be taken as\n),, ( zyxi H H Hms\niext\ni i  , which re presents the contribution from \nexternal and magnetic -stray fields. The stray field strength is taken to have a spatial variation \n)/2cos()0( )( x Hx Hms\nims\ni \n, where \n  is the typical length scale (~20nm) of the surface \nroughness23,27.  Assuming the spin diffusion length is much longer than \n  we average the total \nmagnetic field over a full period of \n , i.e. \n2 2 2) () (ms\niext\ni i H H H  , we therefore have a formula:  \n\n\n\n2 2 2 22 2\n22 2\n0)/ (11 ) () () () (\n total B totalext\nzms\ntotalext\nxms\nx\nxH g HH H\nHH HS S\n      (S3) \nwhere \nms\nxH  and\nmsH  represent the averaged stray field parallel or perpendicular to the injected \nspins, respectively.  Figure S2c shows the simulation according to Equation (S3) with parameter s \n\n= 0.9ns, \nms\nxH 270G and \nmsH 440G.  The simulation qualitatively agrees with the \nexperiment, but shows some systematic deviations especially in the low -field regime.  22 \n  Although the Equation (S1) is generally accepted, and in principle rigorous analysis can \nbe done with spin precessi on, spin diffusion and spin flip involved, a well -established approach \nto determine the intrinsic spin lifetime using the local spin detection geometry is still lacking. A \nprecise determination of the intrinsic spin lifetime in our sample is beyond the sco pe of this \nreport; we treat the lifetime obtained by the simple Lorentzian fit as an effective spin lifetime or \nspin dephasing time, which represents the decay rate of average spin accumulation under applied \nperpendicular magnetic field.  \n \nC. Simulation on the Hanle effect under FMR condition  \nAs indicated by the FMR spectrum, at FMR there are varying x- and z- components of the \napplied magnetic field with different field orientation \n . The table below is a summary:  \n (deg.)  0 10 20 25 30 40 60 \nHx (G) 250 276 287 295 310 302 248 \nHz (G) 0 49 105 138 179 253 430 \n \nAs shown in the table, \nzH   increases monotonically with \n  and \nxH  is in range of 248 – 310 G \n(roughly constant to maintain the conditions for magnetic resonance), both should impact on the \nHanle effect. The black dashed curves shown in Fig. 4(b) are simulated using Eq. (4) with \n\nand \nxH = 248G, 310G respectively. In the situation that Eq. (4) is valid, finite values of the spin \nlifetime should give a weaker H z-dependence than the simulation curve.  23 \n  \n \nFigure S1: Plot of current vs. voltage of the Fe/MgO/p -Si contact at room temperature, symbol s \nare data and the solid line is a linear fit.  \n \n \n \n  \n24 \n  \n \nFigure S 2: (a) Spin accumulation from 3T electrical spin injection, magnetic field is applied in \ndifferent orientations, ranging from in -plane ( H//x, 0 degree, top curve) to out of plane ( H//z, 90 \ndegree, bottom curve); (b) Simu lations according to Equation (S2)  with \n= 0.14ns, assuming no \nstray field; ( c) Simulations  using Equation (S3 ) with parameters  \n= 0.9ns, \nms\nxH 270G and \nmsH\n440G . \n \n  \n \n"    },    {        "title": "1907.11433v1.Universal_relations_for_spin_orbit_coupled_Fermi_gases_in_two_and_three_dimensions.pdf",        "content": "arXiv:1907.11433v1  [cond-mat.quant-gas]  26 Jul 2019Universal relations for spin-orbit-coupled Fermi gases in two and three dimensions\nCai-Xia Zhang\nGuangdong Provincial Key Laboratory of Quantum Engineerin g and Quantum Materials,\nGPETR Center for Quantum Precision Measurement and SPTE,\nSouth China Normal University, Guangzhou 510006, China\nShi-Guo Peng∗and Kaijun Jiang†\nState Key Laboratory of Magnetic Resonance and Atomic and Mo lecular Physics,\nWuhan Institute of Physics and Mathematics, Chinese Academ y of Sciences, Wuhan 430071, China and\nCenter for Cold Atom Physics, Chinese Academy of Sciences, W uhan 430071, China\n(Dated: July 29, 2019)\nWe present a comprehensive derivation of a set of universal r elations for spin-orbit-coupled Fermi\ngases in three or two dimension, which follow from the short- range behavior of the two-body physics.\nBesides the adiabatic energy relations, the large-momentu m distribution, the grand canonical po-\ntential and pressure relation derived in our previous work f or three-dimensional systems [Phys. Rev.\nLett. 120, 060408 (2018)], we further derive high-frequenc y tail of the radio-frequency spectroscopy\nand the short-range behavior of the pair correlation functi on. In addition, we also extend the deriva-\ntion to two-dimensional systems with Rashba type of spin-or bit coupling. To simply demonstrate\nhow the spin-orbit-coupling effect modifies the two-body sho rt-range behavior, we solve the two-\nbody problem in the sub-Hilbert space of zero center-of-mas s momentum and zero total angular\nmomentum, and perturbatively take the spin-orbit-couplin g effect into account at short distance,\nsince the strength of the spin-orbit coupling should be much smaller than the corresponding scale\nof the finite range of interatomic interactions. The univers al asymptotic forms of the two-body\nwave function at short distance are then derived, which do no t depend on the short-range details\nof interatomic potentials. We find that new scattering param eters need to be introduced because of\nspin-orbit coupling, besides the traditional s- andp-wave scattering length (volume) and effective\nranges. This is a general and unique feature for spin-orbit- coupled systems. We show how these two-\nbody parameters characterize the universal relations in th e presence of spin-orbit coupling. This\nwork probably shed light for understanding the profound pro perties of the many-body quantum\nsystems in the presence of the spin-orbit coupling.\nI. INTRODUCTION\nUnderstanding strongly-interacting many-body sys-\ntems is one of the most daunting challenges in modern\nphysics. Owing to the development of the experimental\ntechnique, ultracold atomic gases acquire a high degree\nof control and tunability in interatomic interaction, ge-\nometry, purity, atomic species, and lattice constant (of\noptical lattices) [1–5]. To date, ultracold quantum gases\nhave emerged as a versatile platform for exploring a broad\nvariety of many-body phenomena as well as offering nu-\nmerous examples of interesting many-body states [6–8].\nUnlike conventional electric gases in condensed matters,\natomic quantum gases are extremely dilute, and the mean\ndistance between atoms is usually very large (on the or-\nder ofµm), while the range of interatomic interactions\nis several orders smaller (on the order of several tens of\nnm). Therefore, the two-body correlations characterize\nthe key properties of such many-body systems near scat-\ntering resonances, where the two-body interactions are\nsimply described by the scattering length and become\nirrelevant to the specific form of interatomic potentials.\n∗pengshiguo@gmail.com\n†kjjiang@wipm.ac.cnA set of universal relations, following from the short-\nrange behavior of the two-body physics, govern some cru-\ncial features of ultracold atomic gases, and provide pow-\nerful constraints on the behavior of the system. Many\nof these relations were first derived by Shina Tan, such\nas the adiabatic energy relation, energy theorem, general\nvirial theorem and pressure relation [9–11]. Afterwards,\nmore universal behaviors were obtained by others, such\nas the radio-frequency (rf) spectroscopy, photoassocia-\ntion, static structure factors and so on [12]. All these\nrelations are characterized by the only universal quan-\ntity named contact , and therefore known as the contact\ntheory. During past few years, the concept of contact\ntheory was further generalized to higher-partial-wave in-\nteractions [13–20] as well as to low dimensions [21–29],\nand more contacts appear when additional two-body pa-\nrameters are involved.\nThe reason why the contact theory is significantly im-\nportant in ultracold atoms is attributed to its direct\nconnection to the experimental measurements. Some of\nthe universal relations were experimentally confirmed, in-\nvolving various measurements of the contact itself. For\ntwo-component Fermi gases with s-wave interactions, D.\nS. Jin’s group measured the contact according to three\ndifferent methods, i.e., the momentum distribution, pho-\ntoemission spectroscopy, and rf spectroscopy, and tested\nthe adiabatic energy relation when the interatomic in-2\nteraction was adiabatically swept [30]. The asymptotic\nbehavior of the static structure factor at large momen-\ntum was confirmed by C. J. Vale’s group, by using Bragg\nspectroscopy technique [31, 32]. Recently, the tempera-\nture evolution of the contact was resolved independently\nby M. Zwierlein’s group and C. J. Vale’s group, espe-\ncially the characteristic behavior of the contact across\nthe superfluid transition [33, 34]. For single-component\nFermi gases with p-wave interactions, the feasibility of\ngeneralizing the contact theory for higher-partial-wave\nscatterings was confirmed experimentally by Thywissen’s\ngroup [35], in which the anisotropic p-wave interaction\nwas tuned according to the magnetic vector [36]. Nowa-\ndays, the contact gradually becomes one of fundamental\nconcepts in ultracold atomic physics both theoretically\nand experimentally.\nIn the past decade, the realizations of the spin-orbit\n(SO) coupling in ultracold neutral atoms have sparked a\ngreat deal of interest [37–44]. It provides an ideal plat-\nform on which to study novel quantum phenomena re-\nsulted from SO coupling in a highly controllable and tun-\nable way, such as topological insulators and superconduc-\ntors [6, 7], and (spin) Hall effect [45–47]. Nevertheless,\nit is still challenging to theoretically deal with the many-\nbody correlations for SO-coupled systems. Unlike the\nsituation in condensed matters, the intrinsic short-range\nfeature of interatomic potentials is unchanged for neutral\natoms even in the presence of SO coupling. The natu-\nral question may be raised, from the point of view of the\ncontact theory, as to whether the two-body physics could\nprovide crucial constraints on many-body behaviors of\nSO-coupled atomic systems. In addition, it was pointed\nout that although the short-range feature remains, the\nSO-coupling effect does modify the short-range behav-\nior of the two-body wave function [48]. Therefore, the\nexistence and exact forms of universal relations for SO-\ncoupled atomic systems attract a great deal of attention.\nIn [49], we preliminarily discussed some of the universal\nrelations for three-dimensional (3D) Fermi gases in the\npresence of 3D isotropic SO coupling. We proposed a\nsimple way to construct the short-range wave function,\nin which the SO coupling effect could be taken into ac-\ncount perturbatively. Since SO-coupling in general cou-\nples different partial waves of the two-body scatterings,\nadditional contact parameters appear in universal rela-\ntions. Before long, our theory was verified by different\ngroups near s-wave resonances [50, 51].\nSo far, the generalization of the contact theory in the\npresence of SO-coupling is mostly discussed in 3D, while\nthe derivation of these universal relations is still elusiv e in\ntwo-dimensional (2D) systems. The short-range behavior\nof the two-body physics in 2D is different from that in 3D:\nthe two-body wave function in 3D is power-law divergent,\nwhile one has to deal with the logarithmic divergence in\n2D. From the point of view of the contact theory, differ-\nent short-range correlations in two-body physics result\nin different forms of universal relations. Therefore, it re-\nquires a direct extension to 2D in the similar manner asin 3D in the presence of SO coupling.\nThe purpose of this article is to present a compre-\nhensive derivation of universal relations for SO-coupled\nFermi gases. Besides the adiabatic energy relations, the\nlarge-momentum distribution, the grand canonical po-\ntential and pressure relation derived in our previous work\nfor 3D systems [49], we further derive high-frequency tail\nof the rf spectroscopy and the short-range behavior of the\npair correlation function. Then we generalize the deriva-\ntion of universal relations for 3D systems to 2D case with\nRashba SO coupling in a similar way. For the convenience\nof the presentation, we still construct the short-range be-\nhavior of the two-body wave function in the sub-Hilbert\nspace of zero center-of-mass (c.m.) momentum and zero\ntotal angular momentum as before, and then only s- and\np-wave scatterings are coupled [49, 52, 53]. Our results\nshow that the SO coupling introduces a new contact and\nmodifies the universal relations of many-body systems.\nThe remainder of this paper is organized as follows.\nIn the next section, we present the derivations of the\nshort-range behavior of two-body wave functions for SO-\ncoupled Fermi gases in three and two dimensions, respec-\ntively. Subsequently, with the short-range behavior of\nthe two-body wave functions in hands, we derive a set\nof universal relations for a 3D SO-coupled Fermi gases\nin Sec. III, and then generalize them to 2D SO-coupled\nFermi gases in Sec. IV, including adiabatic energy rela-\ntions, asymptotic behavior of the large-momentum dis-\ntribution, the high-frequency behavior of the rf response,\nshort-range behavior of the pair correlation function,\ngrand canonical potential and pressure relation. Finally,\nthe main results are summarized in Sec. V.\nII. UNIVERSAL SHORT-RANGE BEHAVIOR\nOF TWO-BODY WAVE FUNCTIONS\nThe ultracold atomic gases are dilute, while the range\nof interatomic potentials is extremely small. When two\nfermions get close enough to interact with each other,\nthey usually far away from the others. If only these\ntwo-body correlations are taken into account, some key\nproperties of many-body systems are characterized by the\nshort-range two-body physics, which is the basic idea of\nthe contact theory. In this section, we are going to discuss\nthe short-range behavior of two-body wave functions for\n3D Fermi gases in the presence of 3D SO coupling and 2D\nFermi gases in the presence of 2D SO coupling, respec-\ntively. Let us consider spin-half SO-coupled Fermi gases,\nand the Hamiltonian of a single fermion is modeled as\nˆH1=/planckover2pi12ˆk2\n1\n2M+/planckover2pi12λ\nMˆχ+/planckover2pi12λ2\n2M, (1)\nwhereˆk1=−i∇is the single-particle momentum oper-\nator,Mis the atomic mass, /planckover2pi1is the Planck’s constant\ndivided by 2π. Here, the SO coupling is described by\nthe term /planckover2pi12λˆχ/M with the strength λ>0, andˆχtakes3\nthe isotropic form of ˆk1·ˆσin 3D or the Rashba form of\nˆσ׈k1·ˆ nin 2D [54], where ˆσis the Pauli operator, and\nˆ nis the unit vector perpendicular to the ( x−y) plane.\nBecause of SO coupling, the orbital angular momen-\ntum of the relative motion of two fermions is no longer\nconserved, and then all the partial-wave scatterings are\ncoupled [52]. Fortunately, the c.m. momentum Kof two\nfermions is still conserved as well as the total angular\nmomentum J. For simplicity, we may reasonably focus\non the two-body problem in the subspace of K= 0and\nJ= 0, and then only s- andp-wave scatterings are in-\nvolved [52, 53]. Consequently, the Hamiltonian of two\nspin-half fermions can be written as\nˆH2=/planckover2pi12ˆk2\nM+/planckover2pi12λ\nMˆQ(r)+/planckover2pi12λ2\nM+V(r), (2)\nwhereˆk=/parenleftBig\nˆk2−ˆk1/parenrightBig\n/2is the momentum operator for\nthe relative motion r=r2−r1,V(r)is the short-range\ninteratomic interaction with a finite range ǫ,ˆQ(r) =\n(ˆσ2−ˆσ1)·ˆkin 3D or ˆQ(r) = (ˆσ2−ˆσ1)׈k·ˆ nin 2D,\nandˆσiis the spin operator of the ith atom. In the fol-\nlows, let us consider the two-body problems in the 3D\nsystems with 3D SO coupling and 2D systems with 2D\nSO coupling, respectively.\nA. For 3D systems with 3D SO coupling\nIn the subspace of K= 0andJ= 0, we may choose\nthe common eigenstates of the total Hamiltonian ˆH2and\ntotal angular momentum J(= 0) as the basis of Hilbert\nspace, which take the forms of\nΩ0(ˆ r) =Y00(ˆ r)|S/an}b∇acket∇i}ht, (3)\nΩ1(ˆ r) =−i√\n3[Y1−1(ˆ r)|↑↑/an}b∇acket∇i}ht\n+Y11(ˆ r)|↓↓/an}b∇acket∇i}ht−Y10(ˆ r)|T/an}b∇acket∇i}ht], (4)\nwhereYlm(ˆ r)is the spherical harmonics, ˆ r≡\n(θ,ϕ)denotes the angular degree of freedom of\nthe coordinate r, and|S/an}b∇acket∇i}ht= (|↑↓/an}b∇acket∇i}ht−|↓↑/an}b∇acket∇i}ht )/√\n2and/braceleftbig\n|↑↑/an}b∇acket∇i}ht,|↓↓/an}b∇acket∇i}ht,|T/an}b∇acket∇i}ht= (|↑↓/an}b∇acket∇i}ht+|↓↑/an}b∇acket∇i}ht)/√\n2/bracerightbig\nare the spin-singlet\nand spin-triplet states with total spin S= 0and1, re-\nspectively. Then the two-body wave function can for-\nmally be written in the basis of {Ω0(ˆ r),Ω1(ˆ r)}as\nΨ(r) =ψ0(r)Ω0(ˆ r)+ψ1(r)Ω1(ˆ r), (5)\nwhereψi(r) (i= 0,1)is the radial part of the wave func-\ntion. Note that we here consider an isotropic p-wave in-\nteraction and the radial wave function ψ1(r)is identical\nfor three scattering channels, i.e., m= 0,±1.\nTypically, the SO coupling strength (of the order\nµm−1) is pretty small compared to the inverse of theinteraction range (of the order nm−1) [38, 39], i.e., λ≪\nǫ−1. Moreover, in the low-energy scattering limit, the\nrelative momentum k=/radicalbig\nME//planckover2pi12is also much smaller\nthanǫ−1. Thus, when two fermions get as close as the\nrange of the interaction, i.e., r∼ǫ, the SO coupling can\nbe treated as perturbation as well as the energy. We as-\nsume that the two-body wave function may take the form\nof the following ansatz [49]\nΨ(r)≈φ(r)+k2f(r)−λg(r), (6)\nas the distance of two fermions approaches ǫ. Here, we\nkeep up to the first-order terms of the energy ( k2) and\nSO coupling strength ( λ). The advantage of this ansatz is\nthat the functions φ(r),f(r), andg(r)are all indepen-\ndent onk2andλ. These functions are determined only\nby the short-range details of the interaction, and thus\ncharacterize the intrinsic properties of the interatomic\npotential. We expect that in the absence of SO coupling\nthe conventional scattering length or volume is included\nin the zero-order term φ(r), while the effective range is\nincluded in f(r), the coefficient of the first-order term of\nk2. Interestingly, we may anticipate that new scattering\nparameters resulted from SO coupling appear in the first-\norder term of λ[ing(r)]. Conveniently, more scattering\nparameters may be introduced if higher-order terms of\nk2andλare perturbatively considered.\nInserting the ansatz (6) into the Schrödinger equation\nˆH2Ψ(r) =EΨ(r), and comparing the corresponding co-\nefficients of k2andλ, we find\n/bracketleftbigg\n−∇2+MV(r)\n/planckover2pi12/bracketrightbigg\nφ(r) = 0, (7)\n/bracketleftbigg\n−∇2+MV(r)\n/planckover2pi12/bracketrightbigg\nf(r) =φ(r), (8)\n/bracketleftbigg\n−∇2+MV(r)\n/planckover2pi12/bracketrightbigg\ng(r) =ˆQ(r)φ(r). (9)\nThese coupled equations can easily be solved for r > ǫ,\nand we obtain\nφ(r) =α0/parenleftbigg1\nr−1\na0/parenrightbigg\nΩ0(ˆr)\n+α1/parenleftbigg1\nr2−1\n3a1r/parenrightbigg\nΩ1(ˆr)+O/parenleftbig\nr2/parenrightbig\n, (10)\nf(r) =α0/parenleftbigg1\n2b0−1\n2r/parenrightbigg\nΩ0(ˆr)\n+α1/parenleftbigg1\n2+b1\n6r/parenrightbigg\nΩ1(ˆr)+O/parenleftbig\nr2/parenrightbig\n, (11)\ng(r) =−α1uΩ0(ˆr)−α0(1+vr)Ω1(ˆr)+O/parenleftbig\nr2/parenrightbig\n,(12)\nwhereα0andα1are two complex superposition coeffi-\ncients,aiandbiares-wave scattering length and effec-\ntive range for i= 0, andp-wave scattering volume and\neffective range for i= 1, respectively. Interestingly, two4\nnew scattering parameters uandvare involved as we\nanticipate. They are corrections from SO coupling to the\nshort-range behavior of the two-body wave function in s-\nandp-wave channels, respectively.\nIn the absence of SO coupling, if atoms are initially\nprepared near an s-wave resonance, the contribution from\nthep-wave channel could be ignored, and we have α1≈0.\nNaturally, the two-body wave function Ψ(r)reduces to\nthe known s-wave form of (up to a constant α0)\nΨ(r) =/parenleftbigg1\nr−1\na0+b0k2\n2−k2\n2r/parenrightbigg\nΩ0(ˆr)+O/parenleftbig\nr2/parenrightbig\n(13)\nat short distance r/apprgeǫ. Subsequently, when SO coupling\nis switched on near the s-wave resonance, a considerable\np-wave contribution is involved, and the two-body wave\nfunction becomes\nΨs(r) =/parenleftbigg1\nr−1\na0+b0k2\n2−k2\n2r/parenrightbigg\nΩ0(ˆr)\n+(1+vr)λΩ1(ˆr)+O/parenleftbig\nr2/parenrightbig\n,(14)\nwhich recovers the modified Bethe-Peierls boundary con-\ndition of [48] by noticing Ω0(ˆr) =|S/an}b∇acket∇i}ht/√\n4πandΩ1(ˆr) =−i(ˆσ2−ˆσ1)·(r/r)|S/an}b∇acket∇i}ht/√\n16π. We can see that the pa-\nrametervcharacterizes the hybridization of the p-wave\ncomponent into the s-wave scattering due to SO coupling.\nIf atoms are initially prepared near a p-wave resonance\nwithout SO coupling, the s-wave scattering could be ig-\nnored, then we have α0≈0. The two-body wave function\nΨ(r)takes the known p-wave form at short distance, i.e.,\nΨ(r) =/parenleftbigg1\nr2−1\n3a1r+k2\n2+b1k2\n6r/parenrightbigg\nΩ1(ˆr)+O/parenleftbig\nr2/parenrightbig\n.\n(15)\nIn the presence of SO coupling near the p-wave resonance,\nans-wave component is introduced, and the two-body\nwave function becomes\nΨp(r) =/bracketleftbigg1\nr2+k2\n2+/parenleftbigg\n−1\n3a1+b1k2\n6/parenrightbigg\nr/bracketrightbigg\nΩ1(ˆr)\n+uλΩ0(ˆr)+O/parenleftbig\nr2/parenrightbig\n(16)\nat short distance. We can see that the parameter ude-\nscribes the hybridization of the s-wave component into\nthep-wave scattering due to SO coupling. In general,\nboths- andp-wave scatterings exist between atoms in the\nabsence of SO coupling. Therefore, when SO coupling is\nintroduced, the two-body wave function is generally the\narbitrary superposition of Eqs.(14) and (16), and can be\nwritten as\nΨ3D(r) =α0/parenleftbigg1\nr−1\na0+b0k2\n2+α1\nα0uλ−k2\n2r/parenrightbigg\nΩ0(ˆr)\n+α1/bracketleftbigg1\nr2+k2\n2+α0\nα1λ+/parenleftbigg\n−1\n3a1+b1k2\n6+α0\nα1vλ/parenrightbigg\nr/bracketrightbigg\nΩ1(ˆr)+O/parenleftbig\nr2/parenrightbig\n(17)\nat short distance r/apprgeǫ. Eq.(17) can be treated as the\nshort-range boundary condition for two-body wave func-\ntions in 3D in the presence of 3D SO coupling, when both\ns- andp-wave interactions are considered.\nB. For 2D systems with 2D SO coupling\nLet us consider two spin-half fermions scattering in the\nx−yplane. We easily find that the total angular momen-\ntumJperpendicular to the x−yplane is conserved as\nwell as the c.m. momentum K. Therefore, we may still\nfocus on the two-body problem in the subspace of K=0\nandJ=0, which is spanned by the following three or-\nthogonal basisΩ0(ϕ) =1√\n2π|S/an}b∇acket∇i}ht, (18)\nΩ−1(ϕ) =e−iϕ\n√\n2π|↑↑/an}b∇acket∇i}ht, (19)\nΩ1(ϕ) =eiϕ\n√\n2π|↓↓/an}b∇acket∇i}ht, (20)\nwhereϕis the azimuthal angle of the relative coordinate\nr. Then the two-body wave function can formally be\nexpanded as\nΨ(r) =/summationdisplay\nm=0,±1ψm(r)Ωm(ϕ), (21)\nandψm(r)is the radial wave function. Analogously, the\nstrength of SO coupling as well as the energy can be\ntaken into account perturbatively at short distance. We\nassume that the two-body wave function has the form5\nof the ansatz (6), and the corresponding functions to be\ndetermined can easily be solved out from the Schrödinger\nequation outside the range of the interatomic potential,\ni.e.,r/apprgeǫ. After straightforward algebra, we obtain\nφ(r) =α0/parenleftbigg\nlnr\n2a0+γ/parenrightbigg\nΩ0(ϕ)\n+/parenleftbigg1\nr−π\n4a1r/parenrightbigg/summationdisplay\nm=±1αmΩm(ϕ)+O/parenleftbig\nr2/parenrightbig\n,(22)\nf(r) =−α0/parenleftbiggπ\n4b0+1\n4r2lnr\n2a0/parenrightbigg\nΩ0(ϕ)\n+/parenleftbigg1−2γ\n4r−1\n2rlnr\n2b1/parenrightbigg/summationdisplay\nm=±1αmΩm(ϕ)+O/parenleftbig\nr2/parenrightbig\n,\n(23)g(r) =−/parenleftBigg/summationdisplay\nm=±1αm/parenrightBigg\nuΩ0(ϕ)\n−α0/parenleftbigg\nvr+r√\n2lnr\n2b1/parenrightbigg/summationdisplay\nm=±1Ωm(ϕ)+O/parenleftbig\nr2/parenrightbig\n(24)\nforr/apprgeǫ, whereγis Euler’s constant, αm(m= 0,±1)\nis complex superposition coefficients, amandbmares-\nwave scattering length and effective range for m= 0, and\np-wave scattering area and effective range for |m|= 1,\nrespectively. Here, we have assumed that the p-wave in-\nteraction is isotropic and thus is the same in m=±1\nchannels, and applied the p-wave effective-range expan-\nsion of the scattering phase shift, i.e., k2cotδ1=−1/a1+\n2k2ln(kb1)/π[55]. We find that two new scattering pa-\nrameters are similarly introduced, and they demonstrate\nthe hybridization of s- andp-wave scattering in the pres-\nence of Rashba SOC in 2D. Finally, the asymptotic form\nof the two-body wave function at short distance can be\nwritten as\nΨ2D(r) =α0/bracketleftBigg\nlnr\n2a0+γ−π\n4b0k2+/parenleftBigg/summationdisplay\nm=±1αm\nα0/parenrightBigg\nuλ−k2\n4r2lnr\n2a0/bracketrightBigg\nΩ0(ϕ)\n+/summationdisplay\nm=±1αm/bracketleftbigg1\nr+/parenleftbigg\n−π\n4a1+1−2γ\n4k2+α0\nαmvλ/parenrightbigg\nr+/parenleftbigg\n−k2\n2+α0\nαmλ√\n2/parenrightbigg\nrlnr\n2b1/bracketrightbigg\nΩm(ϕ)+O/parenleftbig\nr2/parenrightbig\n(25)\nforr/apprgeǫ. It is apparent that Ψ2D(r)naturally decouples\nto thes- andp-wave short-range boundary conditions\nin the absence of SO coupling. However, Rashba SO\ncoupling mixes the s- andp-wave scatterings, and two\nnew scattering parameters uandvare introduced. We\nshould note that the short-range behaviors of the two-\nbody wave function, i.e., Eqs.(17) and (25), are universal\nand does not depend on the specific form of interatomic\npotentials.\nIII. UNIVERSAL RELATIONS IN THE\nPRESENCE OF ISOTROPIC 3D SO COUPLING\nIn the previous section, we have discussed the two-\nbody problem in the presence of SO coupling, and ob-\ntained the short-range behaviors of the two-body wave\nfunctions. Then, we are ready to consider Tan’s univer-\nsal relations of SO-coupled many-body systems, if only\ntwo-body correlations are taken into account. Owing to\nthe short-range property of interactions between neutral\natoms, when two fermions ( iandj) get as close as the\nrange of interatomic potentials, all the other atoms are\nusually far away. In this case, the many-body wave func-\ntions approximately take the forms of Eq.(17) in 3D sys-\ntems with 3D SO coupling, when the fermions iandjapproach to each other. We need to pay attention that\nthe arbitrary superposition coefficient αm(X)then be-\ncomes the functions of the c.m. coordinates of the pair\n(i,j)as well as those of the rest of the fermions, which\nwe include into the variable X. In the follows, we de-\nrive a set of universal relations for SO-coupled many-\nbody systems by using Eqs.(17) for 3D SO-coupled Fermi\ngases. These relations include adiabatic energy relations ,\nthe large-momentum behavior of the momentum distri-\nbution, the high-frequency tail of the rf spectroscopy,\nthe short-range behavior of the pair correlation function,\nthe grand canonical potential and pressure relation. Let\nus consider a strongly interacting two-component Fermi\ngases with total atom number N. For simplicity, we con-\nsider the case with b0≈0for broads-wave resonances in\nthe follows.\nA. Adiabatic energy relations\nIn order to investigate how the energy varies with\nthe two-body interaction, let us consider two many-\nbody wave functions ΨandΨ′, corresponding to differ-\nent interatomic interaction strengths. They satisfy the\nSchrödinger equation with different energies6\nN/summationdisplay\ni=1ˆH(i)\n1Ψ =EΨ, (26)\nN/summationdisplay\ni=1ˆH(i)\n1Ψ′=E′Ψ′, (27)\nif there is not any pair of atoms within the range of the\ninteraction, where ˆH(i)\n1denotes the single-atom Hamilto-\nnian (1) for the ith fermion. By subtracting [27]∗×Ψ\nfromΨ′∗×[26], and integrating over the domain Dǫ, the\nset of all configurations (ri,rj)in whichr=|ri−rj|>ǫ,\nwe arrive at\n(E−E′)ˆ\nDǫN/productdisplay\ni=1driΨ′∗Ψ =\n−/planckover2pi12\nMNˆ\nr>ǫdXdr/bracketleftbig\nΨ′∗∇2\nrΨ−/parenleftbig\n∇2\nrΨ′∗/parenrightbig\nΨ/bracketrightbig\n+/planckover2pi12λ\nMNˆ\nr>ǫdXdr/bracketleftBig\nΨ′∗/parenleftBig\nˆQΨ/parenrightBig\n−/parenleftBig\nˆQΨ′/parenrightBig∗\nΨ/bracketrightBig\n,(28)\nwhereN=N(N−1)/2is the number of all the possible\nways to pair atom. Using the Gauss’ theorem, the first\nterm on the right-hand side (RHS) can be written as\n−/planckover2pi12\nMNˆ\nr>ǫdXdr/bracketleftbig\nΨ′∗∇2\nrΨ−/parenleftbig\n∇2\nrΨ′∗/parenrightbig\nΨ/bracketrightbig\n=−/planckover2pi12\nMN\"\nr=ǫ[Ψ′∗∇rΨ−(∇rΨ′∗)Ψ]·ˆndS\n=/planckover2pi12ǫ2\nMN1/summationdisplay\ni=0ˆ\ndX/parenleftbigg\nψ′∗\ni∂\n∂rψi−ψi∂\n∂rψ′∗\ni/parenrightbigg\nr=ǫ,(29)\nwhereSis the surface in which the distance between the\ntwo atoms in the pair ( i,j) isǫwith,ˆnis the direction\nnormal to Sbut opposite to the radial direction, and ψ0\n(ψ1) is thes-wave (p-wave) component of the radial two-\nbody wave function. In addition, for the second term on\nthe RHS of Eq.(28), we have\nˆQ(r)Ψ =−2\nr2∂\n∂r/parenleftbig\nr2ψ1/parenrightbig\nΩ0(ˆ r)+2∂ψ0\n∂rΩ1(ˆ r),(30)\nthen it becomes\n/planckover2pi12λ\nMNˆ\nr>ǫdXdr/bracketleftBig\nΨ′∗/parenleftBig\nˆQ(r)Ψ/parenrightBig\n−/parenleftBig\nˆQ(r)Ψ′/parenrightBig∗\nΨ/bracketrightBig\n=2λ/planckover2pi12ǫ2\nMNˆ\ndX(ψ′∗\n0ψ1−ψ′∗\n1ψ0)r=ǫ.(31)Combining Eqs.(28), (29) and (31), we obtain\n(E−E′)ˆ\nDǫN/productdisplay\ni=1driΨ′∗Ψ\n=/planckover2pi12ǫ2\nMN1/summationdisplay\ni=0ˆ\ndX/parenleftbigg\nψ′∗\ni∂\n∂rψi−ψi∂\n∂rψ′∗\ni/parenrightbigg\nr=ǫ\n+2λ/planckover2pi12ǫ2\nMNˆ\ndX(ψ′∗\n0ψ1−ψ′∗\n1ψ0)r=ǫ.(32)\nInserting the asymptotic form of the many-body wave\nfunction Eq.(17) into Eq.(32), and letting E′→Eand\nΨ′→Ψ, we find\nδE·ˆ\nDǫN/productdisplay\ni=1dri|Ψ|2=−/planckover2pi12\nM/parenleftBig\nI(0)\na−λIλ/parenrightBig\nδa−1\n0\n−/planckover2pi12I(1)\na\nMδa−1\n1+E1\n2δb1+3λ/planckover2pi12\n2MIλδv\n−λ/planckover2pi12\nM/parenleftbigg\n2λI(1)\na−1\n2Iλ/parenrightbigg\nδu+/parenleftbigg1\nǫ+b1\n2/parenrightbigg\nI(1)\naδE, (33)\nwhere\nI(m)\na=Nˆ\ndX|αm(X)|2, (34)\nEm=Nˆ\ndXα∗\nm(X)/bracketleftBig\nE−ˆT(X)/bracketrightBig\nαm(X),(35)\nIλ=Nˆ\ndXα∗\n0(X)α1(X)+c.c., (36)\nEλ=Nˆ\ndXα∗\n0(X)/bracketleftBig\nE−ˆT(X)/bracketrightBig\nα1(X)+c.c.(37)\nform= 0,1, andˆT(X)is the kinetic operator including\nthe c.m. motion of the pair (i,j)and those of all the\nrest fermions. Using the normalization of the many-body\nwave function (see appendix A)\nˆ\nDǫN/productdisplay\ni=1dri|Ψ|2= 1 +/parenleftbigg1\nǫ+b1\n2/parenrightbigg\nI(1)\na,(38)\nwe can further simplify Eq. (33) as\nδE=−/planckover2pi12\nM/parenleftBig\nI(0)\na−λIλ/parenrightBig\nδa−1\n0−/planckover2pi12I(1)\na\nMδa−1\n1\n+E1\n2δb1+3λ/planckover2pi12Iλ\n2Mδv+λ/planckover2pi12\n2M/parenleftBig\nIλ−4λI(1)\na/parenrightBig\nδu,(39)\nwhich yields the following set of adiabatic energy rela-\ntions7\n∂E\n∂a−1\n0=−/planckover2pi12\nM/parenleftBig\nI(0)\na−λIλ/parenrightBig\n, (40)\n∂E\n∂a−1\n1=−/planckover2pi12I(1)\na\nM, (41)\n∂E\n∂b1=E1\n2, (42)\n∂E\n∂u=λ/planckover2pi12\n2M/parenleftBig\nIλ−4λI(1)\na/parenrightBig\n, (43)\n∂E\n∂v=3λ/planckover2pi12Iλ\n2M. (44)\nInterestingly, two additional new adiabatic energy re-\nlations appear, i.e. Eqs. (43) and (44), which origi-\nnate from new scattering parameters introduced by SO\ncoupling. These relations demonstrate how the macro-\nscopic internal energy of an SO-coupled many-body sys-\ntem varies with microscopic two-body scattering param-\neters.\nB. Tail of the momentum distribution at large q\nLet us then study the asymptotic behavior of the large\nmomentum distribution for a many-body system with N\nfermions. The momentum distribution of the ith fermion\nis defined as\nni(q) =ˆ/productdisplay\nt/negationslash=idrt/vextendsingle/vextendsingle/vextendsingle˜Ψi(q)/vextendsingle/vextendsingle/vextendsingle2\n, (45)\nwhere˜Ψi(q)≡´driΨ3De−iq·ri, and then the to-\ntal momentum distribution can be written as n(q) =/summationtextN\ni=1ni(q). When two fermions (i,j)get close while all\nthe other fermions are far away, we may write the many-\nbody function Ψ3Datr=|ri−rj| ≈0as the following\nansatz\nΨ3D(X,r) =/bracketleftbiggα0(X)\nr+B0(X)+C0(X)r/bracketrightbigg\nΩ0(ˆr)\n+/bracketleftbiggα1(X)\nr2+B1(X)+C1(X)r/bracketrightbigg\nΩ1(ˆr)+O/parenleftbig\nr2/parenrightbig\n,(46)\nwhereαm,BmandCm(m= 0,1) are all regular func-\ntions. Comparing Eq. (17) with (46) at small r, we find\nB0(X) =−α0\na0+α1uλ, (47)\nB1(X) =α1k2\n2+α0λ, (48)\nC0(X) =−α0k2\n2, (49)\nC1(X) =−α1\n3a1+α1b1k2\n6+α0vλ. (50)The asymptotic form of the momentum distribution at\nlargeqbut still smaller than ǫ−1is determined by the\nasymptotic behavior at short distance with respect to the\ntwo interacting fermions, then we have\n˜Ψi(q)≈\nq→∞ˆ\ndrΨ3D(X,r∼0)e−iq·r.(51)\nUsing∇2/parenleftbig\nr−1/parenrightbig\n=−4πδ(r), we have the identity\nf(q)≡ˆ\ndre−iq·r\nr=4π\nq2, (52)\nso that\nˆ\ndrα0(X)\nrΩ0(ˆr)e−iq·r=4π\nq2α0(X)Ω0(ˆq),(53)\nˆ\ndrB0(X)Ω0(ˆr)e−iq·r= 0, (54)\nˆ\ndrC0(X)rΩ0(ˆr)e−iq·r=−8π\nq4C0(X)Ω0(ˆq),(55)\nˆ\ndrα1(X)\nr2Ω1(ˆr)e−iq·r=−i4π\nqα1(X)Ω1(ˆq),(56)\nˆ\ndrB1(X)Ω1(ˆr)e−iq·r=−i8π\nq3B1(X)Ω1(ˆq),(57)\nˆ\ndrC1(X)rΩ1(ˆr)e−iq·r= 0. (58)\nInserting Eqs. (53)-(58) into (51), and then into Eq.\n(45), we find that the total momentum distribution n(q)\nat largeqtakes the form\nn3D(q)≈ Nˆ\ndX32π2α1α∗\n1Ω1(ˆq)Ω∗\n1(ˆq)\nq2\n+i32π2[α0α∗\n1Ω0(ˆq)Ω∗\n1(ˆq)−α∗\n0α1Ω∗\n0(ˆq)Ω1(ˆq)]\nq3\n+/bracketleftbig\n32π2α0α∗\n0Ω0(ˆq)Ω∗\n0(ˆq)+64π2k2α1α∗\n1Ω1(ˆq)Ω∗\n1(ˆq)\n+64π2λ(α0α∗\n1+α∗\n0α1)Ω1(ˆq)Ω∗\n1(ˆq)/bracketrightbig1\nq4+O/parenleftbig\nq−5/parenrightbig\n.\n(59)\nIf we are only interested in the dependence of the momen-\ntum distribution on the amplitude of q, we may integrate\nover the direction of q, and then we find all the odd-order\nterms ofq−1vanish. Finally, we obtain\nn3D(q) =C(1)\na\nq2+/parenleftBig\nC(0)\na+C(1)\nb+λPλ/parenrightBig1\nq4+O/parenleftbig\nq−6/parenrightbig\n,\n(60)\nwhere the contacts are defined as\nC(m)\na= 32π2I(m)\na,(m= 0,1), (61)\nC(1)\nb=64π2M\n/planckover2pi12E1, (62)\nPλ= 64π2Iλ. (63)8\nWith these definitions in hands, the adiabatic energy re-\nlations (40)-(44) can alternatively be written as\n∂E\n∂a−1\n0=−/planckover2pi12C(0)\na\n32π2M+λ/planckover2pi12Pλ\n64π2M, (64)\n∂E\n∂a−1\n1=−/planckover2pi12C(1)\na\n32π2M, (65)\n∂E\n∂b1=/planckover2pi12C(1)\nb\n128π2M, (66)\n∂E\n∂u=λ/bracketleftBigg\n/planckover2pi12Pλ\n128π2M−λ/planckover2pi12C(1)\na\n16π2M/bracketrightBigg\n, (67)\n∂E\n∂v=3λ/planckover2pi12Pλ\n128π2M. (68)\nIn the absence of SO coupling, Eqs. (64), (65) and (66)\nsimply reduce to the ordinary form of the adiabatic en-\nergy relations for s- andp-wave interactions [10, 17], with\nrespect to the scattering length (or volume) as well as ef-\nfective range. We should note that for the s-wave inter-\naction, there is a difference of the factor 8πfrom the well-\nknown form of adiabatic energy relations. This is because\nwe include the spherical harmonics Y00(ˆ r) = 1/√\n4πin\nthes-partial wave function. Besides, an additional fac-\ntor1/2is introduced in order to keep the definition of\ncontacts consistent with those in the tail of the momen-\ntum distribution at large q. In the presence of SO cou-\npling, two additional adiabatic energy relations appear,\ni.e., Eqs. (67) and (68), and a new contact Pλis intro-\nduced.\nC. The high-frequency tail of the rf spectroscopy\nNext, we discuss the asymptotic behavior of the rf\nspectroscopy at high frequency. The basic ideal of the\nrf transition is as follows. For an atomic Fermi gas with\ntwo hyperfine states, denoted as |↑/an}b∇acket∇i}htand|↓/an}b∇acket∇i}ht, the rf field\ndrives transitions between one of the hyperfine states (\ni.e.|↓/an}b∇acket∇i}ht) and an empty hyperfine state |3/an}b∇acket∇i}htwith a bare\natomic hyperfine energy difference /planckover2pi1ω3↓due to the mag-\nnetic field splitting [56, 57]. The universal scaling behav-\nior at high frequency of the rf response of the system is\ngoverned by contacts. In this subsection, we are going to\nshow how the contacts defined by the adiabatic energy\nrelations characterize such high-frequency scalings of th e\nrf transition in 3D Fermi gases with 3D SO coupling.\nHere, we will present a two-body derivation first, which\nmay avoid complicated notations as much as possible,\nand the results can easily be generalized to many-body\nsystems later. The rf field driving the spin-down particle\nto the state |3/an}b∇acket∇i}htis described by\nHrf=γrf/summationdisplay\nk/parenleftBig\ne−iωtc†\n3kc↓k+eiωtc†\n↓kc3k/parenrightBig\n, (69)whereγrfis the strength of the rf drive, ωis the rf fre-\nquency, and c†\nσkandcσkare respectively the creation and\nannihilation operators for fermions with the momentum\nkin the spin states |σ/an}b∇acket∇i}ht.\nFor any two-body state |Ψ2b/an}b∇acket∇i}ht, we may write it in the\nmomentum space as\n|Ψ2b/an}b∇acket∇i}ht=/summationdisplay\nσ1σ2/summationdisplay\nk1k2˜φσ1σ2(k1,k2)c†\nσ1k1c†\nσ2k2|0/an}b∇acket∇i}ht,(70)\nwhere˜φσ1σ2(k1,k2)is the Fourier transform of\nφσ1σ2(r1,r2)≡ /an}b∇acketle{tr1,r2;σ1,σ2|Ψ2b/an}b∇acket∇i}ht, i.e.,\n˜φσ1σ2(k1,k2) =ˆ\ndr1dr2φσ1σ2(r1,r2)e−ik1·r1e−ik2·r2,\n(71)\nandσi=↑,↓denotes the spin of the ith particle. The spe-\ncific form of ˜φσ1σ2(k1,k2)can easily be obtained by using\nthat of the two-body wave function /an}b∇acketle{tr1,r2;σ1,σ2|Ψ2b/an}b∇acket∇i}htin\nthe coordinate space, i.e., Eq.(5). Acting Eq.(69) onto\n(70), we obtain the two-body wave function after the rf\ntransition,\nHrf|Ψ2b/an}b∇acket∇i}ht=γrfe−iωt×\n/summationdisplay\nk1k2/bracketleftBig\n˜φ↓↑(k1,k2)c†\n3k1c†\n↑k2−˜φ↑↓(k1,k2)c†\n3k2c†\n↑k1\n+˜φ↓↓(k1,k2)/parenleftBig\nc†\n3k1c†\n↓k2−c†\n3k2c†\n↓k1/parenrightBig/bracketrightBig\n|0/an}b∇acket∇i}ht.(72)\nThe physical meaning of Eq.(72) is apparent: after the rf\ntransition, the atom with initial spin state |↓/an}b∇acket∇i}htis driven to\nthe empty spin state |3/an}b∇acket∇i}ht, while the other one remains in\nthe spin state |↑/an}b∇acket∇i}ht. Therefore, there are totally four possi-\nble final two-body states with, respectively, possibilitie s\nof/vextendsingle/vextendsingle/vextendsingle˜φ↓↑/vextendsingle/vextendsingle/vextendsingle2\n,/vextendsingle/vextendsingle/vextendsingle˜φ↑↓/vextendsingle/vextendsingle/vextendsingle2\n,/vextendsingle/vextendsingle/vextendsingle˜φ↓↓/vextendsingle/vextendsingle/vextendsingle2\n, and/vextendsingle/vextendsingle/vextendsingle˜φ↓↓/vextendsingle/vextendsingle/vextendsingle2\n. Taking all these final\nstates into account, and according to the Fermi’s golden\nrule [29], the two-body rf transition rate is therefore give n\nby the Franck-Condon factor,\nΓ2(ω) =2πγ2\nrf\n/planckover2pi1×\n/summationdisplay\nk1k2/parenleftbigg/vextendsingle/vextendsingle/vextendsingle˜φ↓↑/vextendsingle/vextendsingle/vextendsingle2\n+/vextendsingle/vextendsingle/vextendsingle˜φ↑↓/vextendsingle/vextendsingle/vextendsingle2\n+2/vextendsingle/vextendsingle/vextendsingle˜φ↓↓/vextendsingle/vextendsingle/vextendsingle2/parenrightbigg\nδ(/planckover2pi1ω−△E),(73)\nwhere△Eis the energy difference between the final and\ninitial states, and takes the form of\n∆E=/planckover2pi12k2\nM−/planckover2pi12q2\nM+/planckover2pi1ω3↓, (74)\nwherek= (k1−k2)/2,/planckover2pi12q2/Mis the relative energy of\ntwo fermions in the initial state, and ω3↓≡ω3−ω↓is the\nbare hyperfine splitting between the spin states |3/an}b∇acket∇i}htand\n|↓/an}b∇acket∇i}ht, and can be set to 0without loss of generality. Now,\nwe are interested in the asymptotic form of Γ2(ω)at large9\nωbut still small compared to /planckover2pi1/Mǫ2, which is determined\nby the short-range behavior when two fermions get as\nclose asǫ. Combining Eqs.(71) and (73), as well as the\nasymptotic form of the two-body wave function (17) at\nr=r1−r2∼0, we finally obtain the asymptotic behavior\nof the rf response of 3D SO-coupled Fermi gases at large\nω,\nΓ2(ω) =Mγ2\nrf\n16π2/planckover2pi13/bracketleftBigg\nc(1)\na\n(Mω//planckover2pi1)1/2+\nc(0)\na+3c(1)\nb/4+λpλ\n(Mω//planckover2pi1)3/2/bracketrightBigg\n,(75)\nwherec(0)\na,c(1)\na,c(1)\nbandpλare contacts for a two-body\nsystem with N= 1in the definitions (61)-(63).\nFor many-body systems, all possible N=\nN(N−1)/2pairs may contribute to the high-frequency\ntail of the rf spectroscopy, while high-order contribution s\nfrom more than two fermions are ignored. Then we can\ngeneralize the above two-body picture to many-body\nsystems by simply redefining the constant Ninto the\ncontacts, and then obtain\nΓN(ω) =Mγ2\nrf\n16π2/planckover2pi13/bracketleftBigg\nC(1)\na\n(Mω//planckover2pi1)1/2+\nC(0)\na+3C(1)\nb/4+λPλ\n(Mω//planckover2pi1)3/2/bracketrightBigg\n,(76)\nwhereC(0)\na,C(1)\na,C(1)\nbandPλare corresponding contacts\nfor many-body systems. In the absence of SO coupling,\nEq. (76) simply reduces to the ordinary asymptotic be-\nhaviors of the rf response for s- andp-wave interactions,\nrespectively [13, 58].\nD. Pair correlation function at short distances\nThe pair correlation function g2(s1,s2)gives the prob-\nability of finding two fermions with one at position s1\nand the other one at position s2simultaneously, i.e.,\ng2(s1,s2)≡ /an}b∇acketle{tˆρ(s1) ˆρ(s2)/an}b∇acket∇i}ht, where ˆρ(s) =/summationtext\niδ(s−ri)\nis the density operator at the position s. For a pure\nmany-body state |Ψ/an}b∇acket∇i}htofNfermions, we have [29]\ng2(s1,s2) =ˆ\ndr1dr2···drN/an}b∇acketle{tΨ|ˆρ(s1) ˆρ(s2)|Ψ/an}b∇acket∇i}ht\n=N(N−1)ˆ\ndX′|Ψ(X,r)|2, (77)\nwherer=s1−s2is relative coordinates of the pair\nfermions at positions s1ands2, andX′denotes the de-\ngrees of freedom of all the other fermions. If we further\nintegrate over the c.m. coordinate of the pair, we candefine the spatially integrated pair correlation function\nas\nG2(r)≡N(N−1)ˆ\ndX|Ψ(X,r)|2, (78)\nandXincludes the c.m. coordinate R= (s1+s2)/2of\nthe pair besides X′. Inserting the short-range form of\nmany-body wave functions for SO coupled Fermi gases,\ni.e. Eq.(17) into Eq. (78), we find\nG2(r)≈N(N−1)ˆ\ndX/braceleftbiggα1α∗\n1Ω1Ω∗\n1\nr4\n+α∗\n0α1Ω∗\n0Ω1+α0α∗\n1Ω0Ω∗\n1\nr3+[α0α∗\n0Ω0Ω∗\n0\n+k2α1α∗\n1Ω1Ω∗\n1+λ(α∗\n0α1+α0α∗\n1)Ω∗\n1Ω1\n+λuα∗\n1α1(Ω∗\n0Ω1+Ω0Ω∗\n1)\n−α∗\n0α1Ω∗\n0Ω1+α0α∗\n1Ω0Ω∗\n1\na0/bracketrightbigg1\nr2+O/parenleftbig\nr−1/parenrightbig/bracerightbigg\n.\n(79)\nFurther, if we are only care about the dependence of\nG2(r)on the amplitude of r=|r|, we can integrate over\nthe direction of r, and use the definitions of contacts\n(61)-(63), then it yields\nG2(r)≈1\n16π2/bracketleftBigg\nC(1)\na\nr4+/parenleftBigg\nC(0)\na+C(1)\nb\n2+λPλ\n2/parenrightBigg\n1\nr2\n+/parenleftBigg\n−2C(0)\na\na0−2C(1)\na\n3a1+b1C(1)\nb\n6+λ(u+v)Pλ\n2/parenrightBigg\n1\nr/bracketrightBigg\n.\n(80)\nwhich reduces to the results in the absence of the SO\ncoupling for s- andp-wave interactions, respectively[9,\n13, 31, 59, 60].\nE. Grand canonical potential and pressure relation\nThe adiabatic energy relations as well as the large-\nmomentum distribution we obtained is valid for any pure\nenergy eigenstate. Therefore, they should still hold for\nany incoherent mixed state statistically at finite tempera-\nture. Then the energy Eand contacts then become their\nstatistical average values. Now, let us look at the grand\nthermodynamic potential Jfor a homogeneous system,\nwhich is defined as [61]\nJ ≡ −PV=E−TS−µN, (81)\nwhereP,V,T,S,µ,Nare, respectively, the pressure,\nvolume, temperature, entropy, chemical potential, and\ntotal particle number. The grand canonical potential J10\nis the function of V,T,S, and takes the following differ-\nential form\ndJ=−PdV−SdT−Ndµ. (82)\nFor the two-body microscopic parameters, we may evalu-\nate their dimensions as a0∼Length1,a1∼Length3,b1∼\nLength−1,u∼Length−1, andv∼Length−1. There-\nfore, there are basically following energy scales in the\ngrand thermodynamic potential, i.e., kBT,µ,/planckover2pi12/MV2/3,\n/planckover2pi12/Ma2\n0,/planckover2pi12/Ma2/3\n1,/planckover2pi12b2\n1/M,/planckover2pi12u2/M,/planckover2pi12v2/M. Then\nwe may express the thermodynamic potential Jin the\nterms of a dimensionless function ¯Jas [21, 62]\nJ(V,T,µ,a 0,a1,b1,u,v)\n=kBT¯J/parenleftbigg/planckover2pi12/MV2/3\nkBT,µ\nkBT,/planckover2pi12/Ma2\n0\nkBT,\n/planckover2pi12/Ma2/3\n1\nkBT,/planckover2pi12b2\n1/M\nkBT,/planckover2pi12u2/M\nkBT,/planckover2pi12v2/M\nkBT/parenrightBigg\n.(83)\nConsequently, one can deduce the simple scaling law\nJ/parenleftBig\nγ−3/2V,γT,γµ,γ−1/2a0,γ−3/2a1,γ1/2b1,γ1/2u,γ1/2v/parenrightBig\n=γJ(V,T,µ,a 0,a1,b1,u,v).(84)\nThe derivative of Eq.(84) with respect to γatγ= 1\nsimply yields\n/parenleftbigg\n−3V\n2∂\n∂V+T∂\n∂T+µ∂\n∂µ−a0\n2∂\n∂a0\n−3a1\n2∂\n∂a1+b1\n2∂\n∂b1+u\n2∂\n∂u+v\n2∂\n∂v/parenrightbigg\nJ=J,(85)\nwhere all the partial derivatives are to be understood as\nleaving all other system variables constant. Since\nJ −T∂J\n∂T−µ∂J\n∂µ=J+TS+µN=E, (86)\nand the variation of the grand thermodynamic potential\nδJwith respect to the two-body parameters at fixed vol-\numeV, temperature Tand chemical potential µis equal\nto that of the energy δEat fixed volume V, entropyS\nand particle number N, i.e.,(δJ)V,T,µ= (δE)V,S,Nand\nV∂J\n∂V=J, we easily obtain from Eqs.(85) and (86)\n−3\n2J −a0\n2∂E\n∂a0−3a1\n2∂E\n∂a1+b1\n2∂E\n∂b1+u\n2∂E\n∂u+v\n2∂E\n∂v=E,\n(87)\nFurther by using adiabatic energy relations, Eq.(87) be-\ncomesJ=−2\n3E−/planckover2pi12\n96π2Ma0/parenleftbigg\nC(0)\na−λPλ\n2/parenrightbigg\n−/planckover2pi12C(1)\na\n32π2Ma1+/planckover2pi12b1C(1)\nb\n384π2M\n−λu/planckover2pi12\n48π2M/parenleftbigg\nλC(1)\na−Pλ\n8/parenrightbigg\n+λvPλ/planckover2pi12\n128π2M,(88)\nor the pressure relation by dividing both sides of Eq.\n(88) by−V, which respectively reduces to the well-known\nresults in the absence of the spin-orbit coupling\nP=2E\n3V+/planckover2pi12C(0)\na\n96π2MVa0(89)\nfors-wave interactions, which is consistent with the result\nof Ref.[11, 60, 63], and\nP=2E\n3V+/planckover2pi12C(1)\na\n32π2MVa1−b1/planckover2pi12C(1)\nb\n384π2MV(90)\nforp-wave interactions, which is consistent with the re-\nsult of Ref.[13].\nIV. UNIVERSAL RELATIONS IN 2D SYSTEMS\nWITH RASHBA SO COUPLING\nThe derivation of the universal relations for 3D Fermi\ngases with 3D SO coupling can directly be generalized\nto those for 2D systems with 2D SO coupling. In this\nsection, with the short-range form of the two-body wave\nfunction for 2D systems with 2D SO coupling in hands,\ni.e., Eq.(25), we are going to discuss Tan’s universal rela-\ntions for 2D Fermi gases with 2D SO coupling , by taking\ninto account only two-body correlations.\nA. Adiabatic energy relations\nLet us consider how the energy of the SO-coupled sys-\ntem varies with the two-body interaction in 2D systems\nwith 2D SO coupling. The two wave functions of a many-\nbody system Ψ(r)andΨ′(r), corresponding to different\ninteratomic interaction strengths, satisfy the Schröding er\nequation with different energies, i.e. formally as Eqs.\n(26) and (27). Analogously, by subtracting [27]∗×Ψ\nfromΨ′∗×[26], and integrating over the domain Dǫ, the\nset of all configurations (ri,rj)in whichr=|ri−rj|>ǫ,\nwe obtain11\n(E−E′)ˆ\nDǫN/productdisplay\ni=1driΨ′∗Ψ =\n−/planckover2pi12\nMNˆ\nr>ǫdXdr/bracketleftbig\nΨ′∗∇2\nrΨ−/parenleftbig\n∇2\nrΨ′∗/parenrightbig\nΨ/bracketrightbig\n+/planckover2pi12λ\nMNˆ\nr>ǫdXdr/bracketleftBig\nΨ′∗/parenleftBig\nˆQΨ/parenrightBig\n−/parenleftBig\nˆQΨ′/parenrightBig∗\nΨ/bracketrightBig\n,(91)\nwhereN=N(N−1)/2is again the number of all the\npossible ways to pair atom. Using the Gauss’ theorem,\nthe first term on the right-hand side (RHS) can be writ-\nten as\n−/planckover2pi12\nMNˆ\nr>ǫdXdr/bracketleftbig\nΨ′∗∇2\nrΨ−/parenleftbig\n∇2\nrΨ′∗/parenrightbig\nΨ/bracketrightbig\n=−/planckover2pi12\nMN˛\nr=ǫ[Ψ′∗∇rΨ−(∇rΨ′∗)Ψ]·ˆndS,\n=/planckover2pi12ǫ\nMNˆ\ndX/summationdisplay\nm=0,±1/parenleftbigg\nψ′∗\nm∂\n∂rψm−ψm∂\n∂rψ′∗\nm/parenrightbigg\nr=ǫ,\n(92)\nwhereSis the boundary of Dǫthat the distance between\nthe two fermions in the pair (i,j)isǫ,ˆnis the direction\nnormal to S, but is opposite to the radial direction, and\nψ0(ψ±1) is thes-wave (p-wave) component of the two-\nbody wave function as defined in Eq.(21). Since\nˆQ(r)Ψ =/summationdisplay\nm=±1/bracketleftBigg\n−√\n2\nr∂\n∂r(rψm)Ω0(ˆ r)\n+√\n2∂ψ0\n∂rΩm(ˆ r)/bracketrightbigg\n,(93)\nwe find that the second term on the right-hand side\n(RHS) of Eq. (91) can be written as\n/planckover2pi12λ\nMNˆ\nr>ǫdXdr/bracketleftBig\nΨ′∗/parenleftBig\nˆQ(r)Ψ/parenrightBig\n−/parenleftBig\nˆQ(r)Ψ′/parenrightBig∗\nΨ/bracketrightBig\n=√\n2λ/planckover2pi12ǫ\nMNˆ\ndX/summationdisplay\nm=±1(ψ′∗\n0ψm−ψ′∗\nmψ0)r=ǫ.(94)\nCombining Eqs.(91), (92) and (94), we have\n(E−E′)ˆ\nDǫN/productdisplay\ni=1driΨ′∗Ψ\n=/planckover2pi12ǫ\nMNˆ\ndX/summationdisplay\nm=0,±1/parenleftbigg\nψ′∗\nm∂\n∂rψm−ψm∂\n∂rψ′∗\nm/parenrightbigg\nr=ǫ\n+√\n2λ/planckover2pi12ǫ\nMNˆ\ndX/summationdisplay\nm=±1(ψ′∗\n0ψm−ψ′∗\nmψ0)r=ǫ.(95)Inserting the asymptotic form of the many-body wave\nfunction Eq.(25) into Eq.(95), and letting E′→Eand\nΨ′→Ψ, we arrive at\nδE·ˆ\nDǫN/productdisplay\ni=1dri|Ψ|2\n=/planckover2pi12\nM/parenleftBigg\nI(0)\na+/summationdisplay\nm=±1√\n2\n2λI(m)\nλ/parenrightBigg\nδlna0\n+/summationdisplay\nm=±1/braceleftBigg\n−π/planckover2pi12I(m)\na\n2Mδa−1\n1+/parenleftBigg\nEm−λ/planckover2pi12I(m)\nλ√\n2M/parenrightBigg\nδlnb1\n−/planckover2pi12\nM/bracketleftBigg/parenleftbigg√\n2λ2I(m)\na+λ\n2I(m)\nλ/parenrightbigg\n+√\n2λ2\n2Ip/bracketrightBigg\nδu\n+λ/planckover2pi12\nMI(m)\nλδv−/parenleftbigg\nlnǫ\n2b1+γ/parenrightbigg\nI(m)\naδE/bracerightbigg\n,(96)\nwhere\nI(m)\na=Nˆ\ndX|αm|2, (97)\nEm=Nˆ\ndXα∗\nm/parenleftBig\nE−ˆT/parenrightBig\nαm (98)\nform= 0,±1,\nI(±1)\nλ=Nˆ\ndXα∗\n0α±1+c.c, (99)\nE(±1)\nλ=Nˆ\ndXα∗\n0/parenleftBig\nE−ˆT/parenrightBig\nα±1+c.c, (100)\nIp=Nˆ\ndXα∗\n−1α1+c.c., (101)\nandˆT(X)is the kinetic operator including the c.m. mo-\ntion of the pair as well as those of all the rest fermions.\nUsing the normalization of the wave function (see ap-\npendix B)\nˆ\nDǫN/productdisplay\ni=1dri|Ψ|2= 1−/summationdisplay\nm=±1/parenleftbigg\nlnǫ\n2b1+γ/parenrightbigg\nI(m)\na,(102)\nwe can further simplify Eq. (96) as\nδE=/planckover2pi12\nM/parenleftBigg\nI(0)\na+/summationdisplay\nm=±1λI(m)\nλ√\n2/parenrightBigg\nδlna0\n+/summationdisplay\nm=±1/braceleftBigg\n−π/planckover2pi12I(m)\na\n2Mδa−1\n1+/parenleftBigg\nEm−λ/planckover2pi12I(m)\nλ√\n2M/parenrightBigg\nδlnb1\n−λ/planckover2pi12\nM/bracketleftBigg\n√\n2λI(m)\na+I(m)\nλ\n2+λIp√\n2/bracketrightBigg\nδu+λ/planckover2pi12\nMI(m)\nλδv/bracerightbigg\n,\n(103)12\nwhich characterizes how the energy of a 2D system with\n2D SO coupling varies as the scattering parameters adia-\nbatically change, and yields the following set of adiabatic\nenergy relations\n∂E\n∂lna0=/planckover2pi12\nM/parenleftBigg\nI(0)\na+λ√\n2/summationdisplay\nm=±1I(m)\nλ/parenrightBigg\n, (104)\n∂E\n∂a−1\n1=−π/planckover2pi12\n2M/summationdisplay\nm=±1I(m)\na, (105)\n∂E\n∂lnb1=/summationdisplay\nm=±1/parenleftBigg\nEm−λ/planckover2pi12I(m)\nλ√\n2M/parenrightBigg\n, (106)\n∂E\n∂u=−/planckover2pi12λ√\n2M/summationdisplay\nm=±1/bracketleftBigg\nI(m)\nλ√\n2+λ/parenleftBig\n2I(m)\na+Ip/parenrightBig/bracketrightBigg\n,(107)\n∂E\n∂v=/planckover2pi12λ\nM/summationdisplay\nm=±1I(m)\nλ. (108)\nObviously, there are additional two new adiabatic energy\nrelations appear, i.e. Eqs. (107) and (108), which orig-\ninate from new scattering parameters introduced by SO\ncoupling.\nB. Tail of the momentum distribution at large q\nIn general, the momentum distribution at large qis de-\ntermined by the short-range behavior of the many-body\nwave function when the fermions iandjare close. Simi-\nlarly as in the 3D case, we can formally write the many-\nbody wave function Ψ2Datr≈0as the following ansatz\nΨ2D(X,r) =/bracketleftbig\nα0lnr+B0+C0r2lnr/bracketrightbig\nΩ0(ˆr)\n+/summationdisplay\nm/bracketleftBigαm\nr+Bmrlnr+Cmr/bracketrightBig\nΩm(ˆr)+O/parenleftbig\nr2/parenrightbig\n,(109)\nwhereαj,BjandCj(j= 0,±1) are all regular functions\nofX. Comparing Eqs. (25) and (109) at small r, we find\nthat\nB0(X) =α0(γ−ln2a0)+/summationdisplay\nm=±1αmλu, (110)\nBm(X) =−αmk2\n2+λα0√\n2, (111)\nC0(X) =−α0k2\n4, (112)\nCm(X) =αm/parenleftbigg\n−π\n4a1+1−2γ\n4k2/parenrightbigg\n+α0λv+/parenleftbiggαmk2\n2−λα0√\n2/parenrightbigg\nln2b1.(113)\nIn the follows, we derive the momentum distribution at\nlargeqbut still smaller than ǫ−1. With the help of the\nplane-wave expansioneiq·r=√\n2π∞/summationdisplay\nm=0/summationdisplay\nσ=±ηmimJm(qr)e−iσmϕ qΩ(σ)\nm(ϕ),\n(114)\nwhereηm= 1/2form= 0, andηm= 1form≥1, and\nϕqis the azimuthal angle of q, we have\nˆ\ndrα0lnrΩ0(ˆr)e−iq·r=−2π\nq2α0Ω0(ˆq), (115)\nˆ\ndrB0Ω0(ˆr)e−iq·r= 0, (116)\nˆ\ndrC0r2lnrΩ0(ˆr)e−iq·r=8π\nq4C0Ω0(ˆq), (117)\nˆ\ndrαm\nrΩm(ˆr)e−iq·r=−i2π\nqαmΩm(ˆq),(118)\nˆ\ndrBmrlnrΩm(ˆr)e−iq·r=i4π\nq3BmΩm(ˆq),(119)\nˆ\ndrCmrΩm(ˆr)e−iq·r= 0, (120)\nwhereˆqis the angular part of q. Inserting Eqs. (115)-\n(120) into (45), we find that the total momentum distri-\nbutionn2D(q)at largeqtakes the form of\nn2D(q)≈ Nˆ\ndX/summationdisplay\nm,m′αmα∗\nm′Ωm(ˆq)Ω∗\nm′(ˆq)8π2\nq2\n+i/summationdisplay\nm[α∗\n0αmΩ∗\n0(ˆq)Ωm(ˆq)−α0α∗\nmΩ0(ˆq)Ω∗\nm(ˆq)]8π2\nq3\n+\n\nα0α∗\n0Ω0(ˆq)Ω∗\n0(ˆq)+/summationdisplay\nm,m′/bracketleftBig\n−√\n2λ(α0α∗\nmΩm′(ˆq)Ω∗\nm(ˆq)\n+α∗\n0αmΩm(ˆq)Ω∗\nm′(ˆq))+2k2αmα∗\nm′Ωm(ˆq)Ω∗\nm′(ˆq)/bracketrightbig/bracerightbig\n×8π2\nq4+O/parenleftbig\nq−5/parenrightbig\n(121)\nand the summations are over m,m′=±1. If we are only\ninterested in the dependence of n2D(q)on the amplitude\nofq, the expression can further be simplified by integrat-\ningn2D(q)over the direction of q, and all the odd-order\nterms ofq−1vanish. Finally, we arrive at\nn2D(q) =/summationtext\nm=±1C(m)\na\nq2\n+/bracketleftBigg\nC(0)\na+/summationdisplay\nm=±1/parenleftBig\nC(m)\nb−λP(m)\nλ/parenrightBig/bracketrightBigg\n1\nq4+O/parenleftbig\nq−6/parenrightbig\n,\n(122)\nwhere the contacts are defined as\nC(j)\na= 8π2I(j)\na (123)13\nforj= 0,±1, and\nC(m)\nb=16π2M\n/planckover2pi12Em, (124)\nP(m)\nλ= 8√\n2π2I(m)\nλ(125)\nform=±1. With these definitions in hands, the adi-\nabatic energy relations (104)-(108) can alternatively be\nwritten as\n∂E\n∂lna0=/planckover2pi12\n8π2M/parenleftBigg\nC(0)\na+λ\n2/summationdisplay\nm=±1P(m)\nλ/parenrightBigg\n,(126)\n∂E\n∂a−1\n1=−/planckover2pi12\n16πM/summationdisplay\nm=±1C(m)\na, (127)\n∂E\n∂lnb1=/planckover2pi12\n16π2M/summationdisplay\nm=±1/parenleftBig\nC(m)\nb−λP(m)\nλ/parenrightBig\n,(128)\n∂E\n∂u=−/planckover2pi12λ\n16√\n2π2M/summationdisplay\nm=±1P(m)\nλ, (129)\n∂E\n∂v=/planckover2pi12λ\n8√\n2π2M/summationdisplay\nm=±1P(m)\nλ. (130)\nIn the absence of SO coupling, Eqs. (126), (127) and\n(128) simply reduce to the ordinary form of the adiabatic\nenergy relations for s- andp-wave interactions [24, 29],\nwith respect to the scattering length (or area) as well\nas effective range. And for the s-wave interaction, there\nis a difference of the factor 2πfrom the Ref.[24], which\nis because we include the angular part 1/√\n2πin thes-\npartial wave function. In addition, two additional new\nadiabatic energy relations, i.e., Eqs. (129) and (130),\nand new contacts P(m)\nλappear, due to SO coupling.\nC. The high-frequency tail of the rf spectroscopy\nWe may carry out the analogous procedure as that in\n3D systems with 3D SO coupling, and the two-body rf\ntransition rate takes the form\nΓ2(ω) =2πγ2\nrf\n/planckover2pi1×\n/summationdisplay\nk1k2/parenleftbigg/vextendsingle/vextendsingle/vextendsingle˜φ↑↓/vextendsingle/vextendsingle/vextendsingle2\n+/vextendsingle/vextendsingle/vextendsingle˜φ↓↑/vextendsingle/vextendsingle/vextendsingle2\n+2/vextendsingle/vextendsingle/vextendsingle˜φ↓↓/vextendsingle/vextendsingle/vextendsingle2/parenrightbigg\nδ(/planckover2pi1ω−∆E),(131)\nwhere\n˜φσ1σ2(k1,k2) =ˆ\ndr1dr2φσ1σ2(r1,r2)e−ik1·r1e−ik2·r2.\n(132)\nIf we are only interested in the high-frequency tail of the\ntransition rate, we can use the asymptotic behavior ofthe two-body wave function for a 2D system with 2D SO\ncoupling, i.e. Eq. (25). Combining with Eqs.(131) and\n(132), we obtain the two-body rf transition rate Γ2(ω)\nas\nΓ2(ω) =Mγ2\nrf\n4π/planckover2pi13/bracketleftBigg\nc(1)\na\nMω//planckover2pi1\n+c(0)\na/2+c(1)\nb/2−λp(1)\nλ\n(Mω//planckover2pi1)2/bracketrightBigg\n,(133)\nwherec(0)\na,c(1)\na,c(1)\nbandp(1)\nλare contacts for a two-body\nsystem with N= 1in the definitions (123)-(125).\nFor many-body systems, all N=N(N−1)/2pairs\ncontribute to the transition rate. Similarly, we can re-\ndefining the constant Ninto the contacts, and then ob-\ntain\nΓN(ω) =Mγ2\nrf\n4π/planckover2pi13/bracketleftBigg\nC(1)\na\nMω//planckover2pi1\n+C(0)\na/2+C(1)\nb/2−λP(1)\nλ\n(Mω//planckover2pi1)2/bracketrightBigg\n,(134)\nwhereC(0)\na,C(1)\na,C(1)\nbandP(1)\nλare corresponding con-\ntacts for many-body systems. In the absence of SO cou-\npling, Eq. (134) simply reduces to the ordinary results\nfors- andp-wave interactions, respectively [29, 64].\nD. Pair correlation function at short distances\nLet us then discuss the short-distance behavior of the\npair correlation function for a 2D Fermi gases with 2D\nSO coupling. Inserting the asymptotic form of the many-\nbody wave function at short distance, i.e. Eq. (25) into\nthe Eq. (78), we easily obtain spatially integrated pair\ncorrelation function G2(r). If we are only interested in\nthe dependence of G2(r)on the amplitude of r=|r|, we\nmay integrate over the direction of r, and obtain\nG2(r)≈1\n4π2\n/summationtext\nm=±1C(m)\na\nr2+C(0)\na/parenleftbigg\nlnr\n2a0/parenrightbigg2\n+/parenleftBigg\n2γC(0)\na+λu√\n2/summationdisplay\nm=±1P(m)\nλ/parenrightBigg\nlnr\n2a0\n+/summationdisplay\nm=±11\n2/parenleftBig\n−C(m)\nb+λP(m)\nλ/parenrightBig\nlnr\n2b1/bracketrightBigg\n.(135)\nIn the absence of SO coupling, Eq. (135) simply reduces\nto the ordinary results for s- andp-wave interactions,\nrespectively [24, 29].14\nE. Grand canonical potential and pressure relation\nSimilarly, according to the dimension analysis, we eas-\nily obtain\n−J −a0\n2∂E\n∂a0−a1∂E\n∂a1−b1\n2∂E\n∂b1=E. (136)\nFurther by using adiabatic energy relations, Eq.(136) be-\ncomes\nJ=−E−/planckover2pi12\n16π2M/parenleftBigg\nC(0)\na+λ\n2/summationdisplay\nm=±1P(m)\nλ/parenrightBigg\n−/planckover2pi12\n16πM/summationdisplay\nm=±1/bracketleftBigg\nC(m)\na\na1+1\n2π/parenleftBig\nC(m)\nb−λP(m)\nλ/parenrightBig/bracketrightBigg\n.(137)\nThe pressure relation can be obtained by dividing both\nsides of Eq.(137) by −V, which respectively reduces to\nthe results in the absence of SO coupling\nP=E\nV+/planckover2pi12C(0)\na\n16π2MV(138)\nfors-wave interactions, which is consistent with the result\nof Ref.[22], and\nP=E\nV+/summationdisplay\nm=±1/planckover2pi12\n16πMV/parenleftBigg\nC(m)\na\na1+C(m)\nb\n2π/parenrightBigg\n(139)\nforp-wave interactions, which is consistent with the re-\nsult of Ref.[29].\nV. CONCLUSIONS\nIn conclusion, we systematically study a set of univer-\nsal relations for spin-orbit-coupled Fermi gases in three o r\ntwo dimension, respectively. The universal short-range\nforms of two-body wave functions are analytically de-\nrived, by using a perturbation method, in the sub-Hilbert\nspace of zero center-of-mass momentum and zero total\nangular momentum of pairs. The obtained short-range\nbehaviors of two-body wave functions do not depend on\nthe short-range details of interatomic potentials. We find\nthat two new microscopic scattering parameters appear\nbecause of spin-orbit coupling, and then new contacts\nneed to be introduced in both three- and two-dimensional\nsystems. However, due to different short-range behav-\niors of two-body wave functions for three- and two-\ndimensional systems, the specific forms of universal re-\nlations are distinct in different dimensions. As we antic-\nipate, the universal relations for spin-orbit-coupled sys -\ntems, such as the adiabatic energy relations, the large-\nmomentum distributions, the high-frequency behavior ofthe radio-frequency responses, short-range behaviors of\nthe pair correlation functions, grand canonical poten-\ntials, and pressure relations, are fully captured by the\ncontacts defined. In general, more partial-wave scatter-\nings should be taken into account for nonzero center-of-\nmass momentum and nonzero total angular momentum\nof pairs. Consequently, we may expect more contacts to\nappear. Our results may shed some light for understand-\ning the profound properties of the few- and many-body\nspin-orbit-coupled quantum gases.\nACKNOWLEDGMENTS\nThis work has been supported by the NKRDP\n(National Key Research and Development Program)\nunder Grant No. 2016YFA0301503, NSFC (Grant\nNo.11674358, 11434015, 11474315) and CAS under Grant\nNo. YJKYYQ20170025.\nAPPENDIX A: NORMALIZATION OF THE\nWAVE FUNCTION FOR 3D SYSTEMS WITH 3D\nSO COUPLING\nIn this section of Appendix A, we are going to derive\n´\nDǫN/producttext\ni=1dri|Ψ|2for 3D many-body systems with 3D SO\ncoupling. Let us consider two many-body wave functions\nΨ′andΨ, corresponding to different energies /planckover2pi12k′2/M\nand/planckover2pi12k2/M, respectively. They should be orthogonal,\ni.e.,´\nDǫN/producttext\ni=1driΨ′∗Ψ = 0 , and therefore we have\nˆ\nr<ǫN/productdisplay\ni=1driΨ′∗Ψ =−ˆ\nr>ǫN/productdisplay\ni=1driΨ′∗Ψ. (140)\nFrom the Schrödinger equation satisfied by Ψ′andΨ\noutside the interaction potential, i.e., r > ǫ , we easily\nobtain\nˆ\nr>ǫN/productdisplay\ni=1driΨ′∗Ψ =ǫ2\nk2−k′2Nˆ\ndXˆ\nr=ǫdˆ r\n/bracketleftbigg/parenleftbigg\nΨ′∗∂\n∂rΨ−Ψ∂\n∂rΨ′∗/parenrightbigg\n+λ\n2π(ψ′∗\n0ψ1−ψ′∗\n1ψ0)/bracketrightbigg\n.(141)\nIn the presence of SO coupling, only s- andp-wave scat-\nterings are involved in the subspace K= 0andJ= 0,\nand the wave function at short distance takes the form\nof Eq. (17). Using the asymptotic behavior of the wave\nfunction, we easily evaluate15\nˆ\nr<ǫN/productdisplay\ni=1dri|Ψ|2\n=−lim\nk′→k1\n2/parenleftBiggˆ\nr>ǫN/productdisplay\ni=1driΨ′∗Ψ+ˆ\nr>ǫN/productdisplay\ni=1driΨ′Ψ∗/parenrightBigg\n=−Nˆ\ndX/braceleftBigg\n|α1|2\nǫ+|α1|2b1\n2/bracerightBigg\n=−/parenleftbigg1\nǫ+b1\n2/parenrightbigg\nI(1)\na, (142)\nwhich in turn yields\nˆ\nDǫN/productdisplay\ni=1dri|Ψ|2= 1 +/parenleftbigg1\nǫ+b1\n2/parenrightbigg\nI(1)\na.(143)\nAPPENDIX B: NORMALIZATION OF THE\nWAVE FUNCTION FOR 2D SYSTEM WITH 2D\nSO COUPLING\nIn this section of Appendix B, we are going to derive\n´\nDǫN/producttext\ni=1dri|Ψ|2for 2D many-body systems with 2D SO\ncoupling. Let us consider two many-body wave functions\nΨ′andΨ, corresponding to different energies /planckover2pi12k′2/M\nand/planckover2pi12k2/M, respectively. They should be orthogonal,\ni.e.,´\nDǫN/producttext\ni=1driΨ′∗Ψ = 0 , and therefore we have\nˆ\nr<ǫN/productdisplay\ni=1driΨ′∗Ψ =−ˆ\nr>ǫN/productdisplay\ni=1driΨ′∗Ψ. (144)\nFrom the Schrödinger equation satisfied by Ψ′andΨ\noutside the interaction potential, i.e., r > ǫ , we easilyobtain\nˆ\nr>ǫN/productdisplay\ni=1driΨ′∗Ψ =ǫ\nk2−k′2Nˆ\ndXˆ\nr=ǫdˆ r\n/bracketleftBigg/parenleftbigg\nΨ′∗∂\n∂rΨ−Ψ∂\n∂rΨ′∗/parenrightbigg\n+/summationdisplay\nm=±1λ√\n2π(ψ′∗\n0ψm−ψ′∗\nmψ0)/bracketrightBigg\n.\n(145)\nIn the presence of SO coupling, only s- andp-wave scat-\nterings are involved in the subspace K= 0andJ= 0,\nand the wave function at short distance takes the form\nof Eq. (25). Using the asymptotic behavior of the wave\nfunction, we easily evaluate\nˆ\nr<ǫN/productdisplay\ni=1dri|Ψ|2\n=−lim\nk′→k1\n2/parenleftBiggˆ\nr>ǫN/productdisplay\ni=1driΨ′∗Ψ+ˆ\nr>ǫN/productdisplay\ni=1driΨ′Ψ∗/parenrightBigg\n=Nˆ\ndX/summationdisplay\nm=±1/parenleftbigg\nlnǫ\n2b1+γ/parenrightbigg\n|αm|2\n=/summationdisplay\nm=±1/parenleftbigg\nlnǫ\n2b1+γ/parenrightbigg\nI(m)\na, (146)\nwhich in turn yields\nˆ\nDǫN/productdisplay\ni=1dri|Ψ|2= 1−/summationdisplay\nm=±1/parenleftbigg\nlnǫ\n2b1+γ/parenrightbigg\nI(m)\na.(147)\n[1] I. Bloch, J. Dalibard, and W. Zwerger, Many-body\nphysics with ultracold gases, Rev. Mod. Phys. 80, 885\n(2008).\n[2] S. Giorgini, L. P. Pitaevskii, and S. Stringari, Theory o f\nultracold atomic Fermi gases, Rev. Mod. Phys. 80, 1215\n(2008).\n[3] T. K ¨ ohler, K. G ´ oral, and P. S. Julienne, Production of\ncold molecules via magnetically tunable Feshbach reso-\nnances, Rev. Mod. Phys. 78, 1311 (2006).\n[4] C. Chin, R. Grimm, P. Julienne, and E. Tiesinga, Fesh-\nbach resonances in ultracold gases, Rev. Mod. Phys. 82,\n1225 (2010).\n[5] C. Gross and I. Bloch, Quantum simulations with ultra-\ncold atoms in optical lattices, Science 357, 995 (2017).\n[6] X. L. Qi and S. C. Zhang, Topological insulators and\nsuperconductors, Rev. Mod. Phys. 83, 1057 (2011).[7] J. Dalibard, F. Gerbier, G. Juzeli ¯ unas, and P. ¨Ohberg,\nColloquium: Artificial gauge potentials for neutral\natoms, Rev. Mod. Phys. 83, 1523 (2011).\n[8] D. A. Abanin, E. Altman, I. Bloch, M. Serbyn, Collo-\nquium: Many-body localization, thermalization, and en-\ntanglement, Rev. Mod. Phys. 91, 021001 (2019).\n[9] S. Tan, Energetics of a strongly correlated Fermi gas,\nAnn. Phys. 323, 2952 (2008).\n[10] S. Tan, Large momentum part of a strongly correlated\nFermi gas, Ann. Phys. 323, 2971 (2008).\n[11] S. Tan, Generalized virial theorem and pressure relati on\nfor a strongly correlated Fermi gas, Ann. Phys. 323, 2987\n(2008).\n[12] W. Zwerger, The BCS-BEC crossover and the unitary\nFermi gas , Lecture Notes in Physics Vol. 836 (Springer,\nBerlin, 2011), see Chap. 6 for a brief review.16\n[13] Z. H. Yu, J. H. Thywissen, and S. Z. Zhang, Universal\nrelations for a Fermi gas close to a p-wave interaction\nresonance, Phys. Rev. Lett. 115, 135304 (2015); Phys.\nRev. Lett. 117, 019901(E) (2016).\n[14] S. M. Yoshida and M. Ueda, Universal high-momentum\nasymptote and thermodynamic relations in a spinless\nFermi gas with a resonant p-wave interaction, Phys. Rev.\nLett.115, 135303 (2015).\n[15] M. Y. He, S. L. Zhang, H. M. Chan, and Q. Zhou, Con-\ncept of a contact spectrum and its applications in atomic\nquantum Hall states, Phys. Rev. Lett. 116, 045301\n(2016).\n[16] S. M. Yoshida and M. Ueda, p-wave contact tensor: Uni-\nversal properties of axisymmetry-broken p-wave Fermi\ngases, Phys. Rev. A 94, 033611 (2016).\n[17] S. G. Peng, X. J. Liu, and H. Hu, Large-momentum dis-\ntribution of a polarized Fermi gas and p-wave contacts,\nPhys. Rev. A 94, 063651 (2016).\n[18] F. Qin, X. L. Cui, and W. Yi, Universal relations and\nnormal phase of an ultracold Fermi gas with coexisting s-\nand p-wave interactions, Phys. Rev. A 94, 063616 (2016).\n[19] P. F. Zhang, S. Z. Zhang, and Z. H. Yu, Effective theory\nand universal relations for Fermi gases near a d-wave-\ninteraction resonance, Phys. Rev. A 95, 043609 (2017).\n[20] S. L. Zhang, M. Y. He, and Q. Zhou, Contact matrix in\ndilute quantum systems, Phys. Rev. A 95, 062702 (2017).\n[21] M. Barth and W. Zwerger, Tan relations in one dimen-\nsion, Ann. Phys. 326, 2544 (2011).\n[22] M. Valiente, N. T. Zinner, and K. M /diameterlmer, Universal\nrelations for the two-dimensional spin-1/2 Fermi gas with\ncontact interactions, Phys. Rev. A 84, 063626 (2011).\n[23] F. Werner and Y. Castin, General relations for quan-\ntum gases in two and three dimensions: Two-component\nfermions, Phys. Rev. A 86, 013626 (2012).\n[24] F. Werner and Y. Castin, General relations for quantum\ngases in two and three dimensions. II. Bosons and mix-\ntures, Phys. Rev. A 86, 053633 (2012).\n[25] X. L. Cui, Universal one-dimensional atomic gases near\nodd-wave resonance, Phys. Rev. A 94, 043636 (2016).\n[26] X. L. Cui and H. F. Dong, High-momentum distribu-\ntion with a subleading k−3tail in odd-wave interacting\none-dimensional Fermi gases, Phys. Rev. A 94, 063650\n(2016).\n[27] Y.-C. Zhang and S. Zhang, Strongly interacting p-wave\nFermi gas in two dimensions: Universal relations and\nbreathing mode, Phys. Rev. A 95, 023603 (2017).\n[28] X. G. Yin, X. W. Guan, Y. B. Zhang, H. B. Su, and S.\nZ. Zhang, Momentum distribution and contacts of one-\ndimensional spinless Fermi gases with an attractive p-\nwave interaction, Phys. Rev. A 98, 023605 (2018).\n[29] S. G. Peng, Universal relations for a spin-polarized Fe rmi\ngas in two dimensions, J. Phys. A: Math. Theor. 52,\n245302 (2019).\n[30] J. T. Stewart, J. P. Gaebler, T. E. Drake, and D. S. Jin,\nVerification of universal relations in a strongly interacti ng\nFermi gas, Phys. Rev. Lett. 104, 235301 (2010).\n[31] E. D. Kuhnle, H. Hu, X. J. Liu, P. Dyke, M. Mark, P. D.\nDrummond, P. Hannaford, and C. J. Vale, Universal be-\nhavior of pair correlations in a strongly interacting Fermi\ngas, Phys. Rev. Lett. 105, 070402 (2010).\n[32] S. Hoinka, M. Lingham, K. Fenech, H. Hu, C. J. Vale,\nJ. E. Drut, and S. Gandolfi, Precise determination of the\nstructure factor and contact in a unitary Fermi gas, Phys.\nRev. Lett. 110, 055305 (2013).[33] B. Mukherjee, P. B. Patel, Z. Yan, R. J. Fletcher, J.\nStruck, and M. W. Zwierlein, Spectral response and con-\ntact of the unitary Fermi gas, Phys. Rev. Lett. 122,\n203402 (2019).\n[34] C. Carcy, S. Hoinka, M. G. Lingham, P. Dyke, C. C. N.\nKuhn, H. Hu, and C. J. Vale, Contact and sum rules in\na near-uniform Fermi gas at unitarity, Phys. Rev. Lett.\n122, 203401 (2019).\n[35] C. Luciuk, S. Trotzky, S. Smale, Z. Yu, S. Zhang, and J.\nH. Thywissen, Evidence for universal relations describing\na gas with p-wave interactions, Nat. Phys. 12, 599 (2016).\n[36] S.-G. Peng, S. Tan, and K. Jiang, Manipulation of p-\nwave scattering of cold atoms in low dimensions using\nthe magnetic field vector, Phys. Rev. Lett. 112, 250401\n(2014).\n[37] Y. J. Lin, K. Jim ´ enez-Garcia, and I. B. Spielman, Spin-\norbit-coupled Bose-Einstein condensates, Nature 471, 83\n(2011).\n[38] L. W. Cheuk, A. T. Sommer, Z. Hadzibabic, T. Yefsah,\nW. S. Bakr, and M. W. Zwierlein, Spin-injection spec-\ntroscopy of a spin-orbit coupled Fermi gas, Phys. Rev.\nLett.109, 095302 (2012).\n[39] P. J. Wang, Z. Q. Yu, Z. K. Fu, J. Miao, L. H. Huang,\nS. J. Chai, H. Zhai, and J. Zhang, Spin-orbit coupled\ndegenerate Fermi gases, Phys. Rev. Lett. 109, 095301\n(2012).\n[40] Z. Wu, L. Zhang, W. Sun, X. T. Xu, B. Z. Wang, S. C. Ji,\nY. Deng, S. Chen, X. J. Liu, and J. W. Pan, Realization\nof two-dimensional spin-orbit coupling for Bose-Einstein\ncondensates, Science 354, 83 (2016).\n[41] L. H. Huang, Z. M. Meng, P. J. Wang, P. Peng, S. L.\nZhang, L. C. Chen, D. H. Li, Q. Zhou, and J. Zhang, Ex-\nperimental realization of two-dimensional synthetic spin -\norbit coupling in ultracold Fermi gases, Nat. Phys. 12,\n540 (2016).\n[42] H. R. Chen, K. Y. Lin, P. K. Chen, N. C. Chiu, J. B.\nWang, C. A. Chen, P. P. Huang, S. K. Yip, Y. Kawaguchi,\nand Y. J. Lin, Spin-orbital-angular-momentum coupled\nBose-Einstein condensates, Phys. Rev. Lett. 121, 113204\n(2018).\n[43] P. K. Chen, L. R. Liu, M. J. Tsai, N. C. Chiu, Y.\nKawaguchi, S. K. Yip, M. S. Chang, and Y. J. Lin,\nRotating atomic quantum gases with light-induced az-\nimuthal gauge potentials and the observation of the Hess-\nFairbank effect, Phys. Rev. Lett. 121, 250401 (2018).\n[44] D. Zhang, T. Gao, P. Zou, L. Kong, R. Li, X. Shen,\nX. L. Chen, S. G. Peng, M. Zhan, H. Pu, and K.\nJiang, Ground-state phase diagram of a spin-orbital-\nangular-momentum coupled Bose-Einstein condensate,\nPhys. Rev. Lett. 122, 110402 (2019).\n[45] S. L. Zhu, H. Fu, C. J. Wu, S. C. Zhang, and L. M. Duan,\nSpin Hall effects for cold atoms in a light-induced gauge\npotential, Phys. Rev. Lett. 97, 240401 (2006).\n[46] M. C. Beeler, R. A. Williams, K. Jim ´ enez-Garcia, L. J.\nLeBlanc, A. R. Perry, and I. B. Spielman, The spin Hall\neffect in a quantum gas, Nature 498, 201 (2013).\n[47] J. Y. Choi, S. Kang, S. W. Seo, W. J. Kwon, and Y. I.\nShin, Observation of a geometric Hall effect in a spinor\nBose-Einstein condensate with a skyrmion spin texture,\nPhys. Rev. Lett. 111, 245301 (2013).\n[48] P. Zhang, L. Zhang, and Y. J. Deng, Modified Bethe-\nPeierls boundary condition for ultracold atoms with spin-\norbit coupling, Phys. Rev. A 86, 053608 (2012).17\n[49] S.-G. Peng, C.-X. Zhang, S. Tan, and K. Jiang, Contact\ntheory for spin-orbit-coupled Fermi gases, Phys. Rev.\nLett.120, 060408 (2018).\n[50] P. Zhang and N. Sun, Universal relations for spin-orbit -\ncoupled Fermi gas near an s-wave resonance, Phys. Rev.\nA97, 040701(R) (2018).\n[51] J. Jie, R. Qi, and P. Zhang, Universal relations of an\nultracold Fermi gas with arbitrary spin-orbit coupling,\nPhys. Rev. A 97, 053602 (2018).\n[52] X. L. Cui, Mixed-partial-wave scattering with spin-or bit\ncoupling and validity of pseudopotentials, Phys. Rev. A\n85, 022705 (2012).\n[53] Y. X. Wu and Z. H. Yu, Short-range asymptotic behavior\nof the wave functions of interacting spin-1/2 fermionic\natoms with spin-orbit coupling: A model study, Phys.\nRev. A87, 032703 (2013).\n[54] Y. A. Bychkov and E. I. Rashba, Oscillatory effects and\nthe magnetic susceptibility of carriers in inversion layer s,\nJ. Phys. C: Solid State Phys. 17, 6039 (1984).\n[55] C.-X. Zhang, S.-G. Peng, and K. Jiang, High-\ntemperature thermodynamics of spin-polarized Fermi\ngases in two-dimensional harmonic traps, Phys. Rev. A\n98, 043619 (2018).\n[56] H. Hu, H. Pu, J. Zhang, S.-G. Peng, and X.-J. Liu,\nRadio-frequency spectroscopy of weakly bound moleculesin spin-orbit-coupled atomic Fermi gases, Phys. Rev. A\n86, 053627 (2012).\n[57] S.-G. Peng, X.-J. Liu, H. Hu, and K. Jiang, Momentum-\nresolved radio-frequency spectroscopy of a spin-orbit-\ncoupled atomic Fermi gas near a Feshbach resonance in\nharmonic traps, Phys. Rev. A 86, 063610 (2012).\n[58] E. Braaten, D. Kang, and L. Platter, Short-time operato r\nproduct expansion for rf spectroscopy of a strongly inter-\nacting Fermi gas, Phys. Rev. Lett. 104, 223004 (2010).\n[59] S. Gandolfi, K. E. Schmidt, and J. Carlson, BEC-BCS\ncrossover and universal relations in unitary Fermi gases,\nPhys. Rev. A 83, 041601(R) (2011).\n[60] N. Sakumichi, Y. Nishida, and M. Ueda, Lee-Yang cluster\nexpansion approach to the BCS-BEC crossover: BCS and\nBEC limits, Phys. Rev. A 89, 033622 (2014).\n[61] L. D. Landau and E. M. Lifshitz, Statistical Physics , 3rd\ned. (Elsevier, Singapore, 2007), Part I.\n[62] E. Braaten, D. Kang, and L. Platter, Universal relation s\nfor a strongly interacting Fermi gas near a Feshbach res-\nonance, Phys. Rev. A 78, 053606 (2008).\n[63] M. Iskin, Isothermal-sweep theorems for ultracold qua n-\ntum gases in a canonical ensemble, Phys. Rev. A 83,\n033613 (2011).\n[64] M. Barth, and J. Hofmann, Pairing effects in the nonde-\ngenerate limit of the two-dimensional Fermi gas, Phys.\nRev. A89, 013614 (2014)."    },    {        "title": "2211.04809v1.Giant_efficiency_of_long_range_orbital_torque_in_Co_Nb_bilayers.pdf",        "content": "Giant efficiency  of long-range orbital torque  in Co/Nb bilayers  \nFufu Liu1, Bokai Liang1, Jie Xu1, Cheng long Jia1,2†, Changjun Jiang1,2* \n1 Key Laboratory for Magnetism and Magnetic Materials, Ministry of Education, Lanzhou \nUniversity, Lanzhou 730000, China  \n2 Lanzhou Center for Theoretical Physics & Key Laboratory of Theoretical Physics of Gansu \nProvince, Lanzhou University, Lanzhou 730000, China  \n \nCorresponding author. E -mail address: †cljia@lzu.edu.cn, *jiangchj@lzu.edu.cn  \n \nABSTRACT  \nWe report unambiguous ly experimental evidence of a strong orbital current in Nb films \nwith weak spin -orbit coupling via the spin-torque ferromagnetic resonance (ST -FMR)  \nspectrum for Fe/Nb and Co/Nb bilayers.  The sign change of the damping -like torque in \nCo/Nb demonstrates a large spin -orbit correlation  and thus great efficiency of orbital \ntorque in Co/Nb.  By studying the efficiency as a function of the thickness of Nb \nsublayer, we reveal a long orbital diffusion length (~ 3.1 nm)  of Nb.  Further planar Hall \nresistance (PHE) measurements at positive and negative apply ing current  confirm the \nnonlocal orbital transport in ferromagnetic -metal/Nb heterostructures.  \n \n \n \n \n \n \n \n \n  Spin-related  torque s induc ed by spin current s have drawn considerable interest  in \nspintronic  over the decade  [1-3]. Especially, spin Hall effect (SHE)  prevail s in heavy \nmetal (HM) with strong spin -orbit coupling , where the resulting spin -orbit torque (SOT) \nhas served as an efficient manipulation of magnetization in HM/ferromagnet (HM/FM)  \nheter ostructures  [1, 4-7]. Recently, orbital current based on the flowing  orbital angular \nmomentum  (OAM)  [8] via orbital Hall effect (OHE) receive s quite attention  [9-12]. \nLike SHE, orbital current is generated along the perpendicular direction to the charge \ncurrent. However, there is distinctive  features  of OHE compared  to SHE. Theoretically, \nthe OHE roots in orbital textures  in momentum -space , which means  that the OHE \nallows for the absence of  spin-orbit coupling (SOC)  due to the direct action of charge \ncurrent on orbital degree of freedom  [13], where charge current directly acts on spin  \ndegree of freedom  through strong SOC that results in  SHE.  Moreover , orbital Hall \nconductivity  (OHC)  is much larger than  spin Hall conductivity  (SHC)  in many \ntransition metals , which could be estimated that the orbital torque (OT) efficiency can \nbe comparable to or larger than spin torque efficiency  [10]. Thus, the OT promote s the \nbirth of orbitronics  which holds great  potential for future highly efficient magnetic  \ndevices and a  complement to spintronics .  \nMany alternative materials can be regarded as the orbital source  to induce OT , and \ntheoretically  found  in multiorbital centrosymmetric systems  such as transition metals  \n[10], graphene  [14], semiconductors  [11] as well as two-dimensional  transition metal \ndichalcogenides  [15-17]. Experimentally, the OT was investigated in various systems, \nespecially in thulium iron garnet TmIG/Pt/CuO x [18], FM/Cu/Al 2O3 [19], Py/CuO x [20-\n21], etc. which all originates from the orbital Rashba effect (ORE) and modulates OT \nvia layer design  [22]. In these systems, Pt or oxidized Cu was utilized as a n orbital \nsource , it undoubtedly complicate s the discussion of OT.  Moreover, the intrinsic OHE \nwas observed in Ta/Ni by evaluating the sign of damping -like torque similar to the sign \nof OHC in Ta  [13]. However, the relatively larger SHC for Ta leads to slightly smaller \nOT efficiency in Ta/Ni system. Thus, materials selection without strong SOC for large \nOT efficiency is extremely necessary.  The transition metal  niobium (Nb) is \ncharacterized by weak SOC as well as large enough OHC  (𝜎𝑂𝐻𝑁𝑏) relative to SHC  (𝜎𝑆𝐻𝑁𝑏) and opposite sign of them  [10,], which implies highly possible to observe obvious OHE  \nvia sign relation  and generate expected  OT acting on magnetization of FM  [23]. \nMoreover, the large enough OHC is expected to be accompanied by a large OT \nefficiency .  \nHerein , we present experimental evidence confirm ing the existence of OHE in \nFM/Nb bilayers  by the sign relation of damping -like torque according with OHC  (or \nSHC)  in Nb via spin -torque ferromagnetic resonance (ST -FMR) . After fitting the FM \ndependence of ST -FMR spectra, the variety between the signs of damping -like torque  \nfor Fe/Nb and Co/Nb  bilayers unambiguous ly proves the emergence of orbital torque. \nFurthermore, we support our conclusion that the sign of damping -like torque is \nexamined by utilizing the method of planar Hall resistance  (PHE)  at positive and \nnegative measuring current, which result coincides wi th ST -FMR and the theoretical \ncalculations.  \nGenerally, it is difficult to disentangle  OT and ST due to identical properties. Thus, \nthe total torque usually is ascribed to the combined effect  of ST and OT,  which includes \nfollowing three cases: The first  case corresponds to the  same sign of 𝜎𝑆𝐻 and 𝜎𝑂𝐻 as \nshown in Figure 1(a). The total torque is enhanced by a synergistic effect of  the OT and \nST. In the second  case, the sign of 𝜎𝑆𝐻 varies from 𝜎𝑂𝐻 as shown in the top of Figure \n1(b), while the magnitude of spin Hall contribution ( ASH) is larger than orbital Hall \ncontribution ( AOH), the total torque is dominated by ST. In the final case, the sign of \n𝜎𝑆𝐻 varies from 𝜎𝑂𝐻 in the bottom of Figure 1(b), and the ASH is smaller than AOH that \nprovides a great chance to disentangl e the OT from the ST . Given all this, the 4d \ntransition metal Nb is  an excellent candidate  to observe OT  where 𝜎𝑂𝐻 is opposite sign \nand much larger than 𝜎𝑆𝐻 [10]. Specifically speaking , as shown in Figure 1(c), the \norbital current and spin current are generated by OHE and SHE and injected into  FM, \nrespectively. Next the injected orbital current  is converted to spin via SOC of the FM.  \nA crucial  conversion ratio 𝜂𝐹𝑀 (spin -orbit correlation) [ 24] is used to describe  how \nmuch spin in FM is induced by the  orbital current injected from the Nb.  For most FMs, \nthe sign of 𝜂𝐹𝑀 is positive, such as Fe, Co, Ni, etc. However, the small 𝜂𝐹𝑀 tends to \ninduce ST , while the large 𝜂𝐹𝑀 leads to OT , as demonstrated in Figure 1(c). Based on  the previous theoretical[13], we summary the total torque of FM/Nb systems in Figure \n1(d). Obviously, the Co/Nb bilayer  is a good candidate to clarify the OT, given that  the \nsign of the effective Hall conductivity ( 𝜎𝑆𝐻𝑁𝑏+𝜂𝐹𝑀𝜎𝑂𝐻𝑁𝑏) obtained from  torque coincides \nwith the sign of  𝜎𝑂𝐻𝑁𝑏 . The FM/Nb  thin films were deposited  on MgO substrate by \nmagnetron sputtering at room temperature.  In order to clarify the physical mechanism \nof OT in experiments, f our different types of bilayers were prepared: Fe(tFe)/Nb(6 nm), \nCo(tCo)/Nb(6 nm).  Fe(8 nm)/Nb( tNb) and Co(8 nm)/Nb( tNb). For comparison , the \nsamples Pt(6 nm)/Co( tCo) and Pt(6 nm)/Fe( tFe) were grown on MgO substrate  as well . \nMoreover, Fe(2 nm)/Nb(8 nm) and Co(2 nm)/Nb(8 nm) are prepared to conduct the \nPHE measurements.  \n To measure the OT efficiency, we exploit the spin -torque ferromagnetic resonance  \n(ST-FMR)  measurement  as shown in Figure 2(a), the ST -FMR technique is employed \nbecause it is a well-established  method to determine the charge -spin conversion \nefficiency  as well as  the type, number and efficiency of torques  [1, 25-28]. In this \nmeasurement,  a radio frequency (RF) microwave current is applied along the x -\ndirection , this RF current will induce the  resulting  torque and cause the FM \nmagnetization process, which contributes to the ST -FMR resonance signal. The typical \nST-FMR signal for Fe(8 nm)/Nb(6 nm) is shown in Figure 2(b), which could be fitted \nby symmetric  (Vs) and antisymmetric component  (Va). Generally, the Vs and Va \ncorresponds to different torques, where the Vs arises solely from the damping -like \ntorque ( 𝝉𝐃𝐋∝𝒎×(𝒚×𝒎)) and Va originates jointly from the  field-like torque ( 𝝉𝐅𝐋∝\n𝒎×𝒚   and current -induced Oersted field , respectively  [13]. Furthermore, the \nquantitative  conversion efficiency ξ can be determined by the ratio of Vs and Va based \non the following equation  [1, 13, 29-30]: \n                    𝜉=𝑉𝑠\n𝑉𝑎𝑒𝜇0𝑀𝑠𝑡𝐹𝑀𝑑𝑁𝑏\nℏ[1+(𝑀𝑠𝐻𝑟⁄ )]12⁄,              (1) \nwhere tFM and dNb are the thickness of FM and Nb sublayer s, respectively.  Hr is the \nresonance field . Ms is the effective saturation magnetization  that can be obtained by \nKittel equation:  (2𝜋𝑓)/𝛾=√[𝐻𝑟(𝐻𝑟+𝑀𝑠)] with γ being  the gyromagnetic ratio.   \nAnalogous to torque components of SOT, the 𝝉𝐃𝐋 and 𝝉𝐅𝐋 components of OT can be generated in adjacent magnetic layer  and separately determined  via ST -FMR. In the \npresent study, we are more interested in  the sign of 𝝉𝐃𝐋, which could reflect the sign of \nSHC and /or OHC. As shown in Figure 2(b), a negative  Vs is observed for Fe(8 nm)/Nb(6 \nnm), which mean s the existence of 𝝉𝐃𝐋. Obviously, the sign of Vs is consistent with the \nexpected sign of 𝜎𝑆𝐻𝑁𝑏 and opposite t o the 𝜎𝑂𝐻𝑁𝑏. The sign of 𝝉𝐃𝐋 for Fe/Nb could be \nfurther determined  by the FM layer thickness dependence of the ST-FMR resonance \nsignal Vdc [24] as shown in Figure 2( c). The experimentally measured conversion \nefficiency ξ as a function of tFM provides a route to separately evaluate the damping -\nlike ( 𝜉𝐷𝐿) and field-like (𝜉𝐹𝐿) torque efficiencies as  [13, 20 ] \n                          1\n𝜉=1\n𝜉𝐷𝐿(1+ℏ\n𝑒𝜉𝐹𝐿\n𝜇0𝑀𝑠𝑡𝐹𝑀𝑑𝑁𝑏)                   (2) \nwhere the intercept implies the sign of 𝜉𝐷𝐿. As shown in Figure 2(c), the sign of the \nintercept is in accord with the symmetric component  Vs in Figure 2(b). Hence , it is \napparent that the sign of 𝜉𝐷𝐿  of Fe/Nb bilayer is consistent with the sign of 𝜎𝑆𝐻𝑁𝑏 . \nHowever, we have a sign reversal of the symmetric component  Vs in ST-FMR signal \nfor Co(8 nm)/Nb(6 nm) , i.e., a positive Vs as shown in Figure 2( d). Consequently , the \nFM layer thickness dependence of efficiency for Co/Nb bilayers  [24] gives a positive \n𝜉𝐷𝐿  as plotted in Figure 2(e) , which coincides with  the expected  sign of 𝜎𝑂𝐻𝑁𝑏  but \nopposite to the sign of 𝜎𝑆𝐻𝑁𝑏. On the other hand , for the Pt/Fe and Pt/Co control sample \nas shown in Figures 3, the heavy metal Pt, is acknowledged as a strong spin source \nmaterial [1]. From the experimental result, the positive sign of Vs is consistent with the \nsign of 𝜎𝑆𝐻𝑃𝑡 , and further accords w ith the sign of 𝜉𝐷𝐿 . According to schematic \nillustration for the mechanism of the orbital torque  in Figure 1( c) and  theoretical \ncalculations  [24], the to tal torque in Co/Nb includes conventional SOT arising from \nspin Hall contribution ( 𝜎𝑆𝐻𝑁𝑏 ) and OT attributing to orbital Hall contribution ( 𝜎𝑂𝐻𝑁𝑏 ). \nWhen the spin Hall contribution is much less than orbital Hall contribution, the sign of \nthe total torque is same  as the sign of 𝜎𝑂𝐻𝑁𝑏. In this case, as for Co/Nb bilayer,  the sign \nof OT depends on the 𝜎𝑂𝐻𝑁𝑏  and spin -orbit correlation ˂L·S˃Co [13]. The fact from \ntheoretical calculation is that the orbital Hall effect contribution is larger than the spin \nHall one in magnitude  of Co/Nb , the sign of total torque is in accord with the expected sign of 𝜎𝑂𝐻𝑁𝑏, i.e., the orbital torque .  \nFurthermore, to verify the source of OT, we perform the ST -FMR measurement \nfor Co(8 nm)/Cu(t)/Nb(6 nm) , Figure S4 shows the typical ST -FMR spectra. The \npositive Vs means the existence of OT even if inserting the Cu layer. Cu layer thickness \ndependence of ef ficiency ξ is plotted in Figure 4(a). Enough efficiency exclude s the \ninterface effect at Co/Nb interface and bulk OHE is responsible for the OT . Moreover, \nto obtain the orbital  diffusion length of Nb, a series of ST-FMR samples with different \nNb thickness es of Co (and Fe, t = 8 nm) /Nb(d = 6-15 nm) was fabricated, and the \nconversion efficiency  ξ was characterized by Eq. (1) and summarized in Figure 3(a), \nthe Nb thickness dependence of ξ match with 𝜉=𝜉∞[1−sech  (𝑑/𝜆𝑠)] [31], where  ξ \nis the measured conversion efficiency at different Nb thickness and ξꝏ is the conversion \nefficiency at infinite Nb thickness. From the fitting, λs of Nb is quantitatively \ndetermined to be 3.1 nm. This point sets it apart from conventional heavy metal, such \nas Pt, where spin Hall contribution dominates and shows a slightly small spin diffusion \nlength (~ 1.5 nm) [31].  \nFurthermore, to verify the sign relation of damping -like torque in FM/Nb bila yers, \nwe perform the method of PHE at positive and negative measuring current, which has \nbeen regarded as a reliable and effective method to measure charge -to-spin conversion \n[32-34]. In this measurement, the damping -like torque induces the out -of-plane \neffective field Hp, while the field -like torque induces the in -plane effective field HT as \nshown in Figure 4(a). Here we are just concerned with the sign of Hp. Figure 4(b) and \n(c) exhibits the typical magnetic angle dependence of planar Hall resistance RH as well \nas the corresponding RDH curve at I = ± 0.9 mA  for Fe(2 nm)/N b (8 nm) , and I = ± 1.0 \nmA for Co (2 nm)/Nb(8 nm), respectively  (see Supplemental Material for other currents  \n[24]). Note  that the effective fields Hp and HT can be extracted from the resistance \ndifference RDH and charactered by following equation [ 32-34]: \n   𝑅𝐷𝐻(𝐼,𝜑)=2𝑅𝐻(𝐻𝑇+𝐻𝑂𝑒)\n𝐻𝑒𝑥𝑡(𝑐𝑜𝑠𝜑 +𝑐𝑜𝑠3𝜑)+2𝑑𝑅𝐴𝐻𝐸\n𝑑𝐻𝑝𝑒𝑟𝑝𝐻𝑃𝑐𝑜𝑠𝜑 +𝐶   (3) \nwhere C is the resistance offset, HOe is the Oersted effective field, φ is defined in Figure \n4(a), Hperp is the applied out -of-plane magnetic field, 𝑑𝑅𝐴𝐻𝐸\n𝑑𝐻𝑝𝑒𝑟𝑝 is the slope of  RAHE vs Hperp [24]. After determining relevant parameters in Eq. (3), the out -of-plane effective \nfield Hp is obtained by fitting the φ dependence of RDH. Figure 4(d) shows the linear \nfitting Hp relative to the current  for Fe/Nb and Co/Nb , respectively . The sign of effective \nconversion efficiency can be quantitatively determined by [ 32]:𝜉~𝐻𝑝\n𝐽𝑁𝑀, where JNM is \nthe density of current. Obviously, the sign of 𝜉  depends on the slope of Hp versus \ncurrent I. Any difference in sign of slope for Hp vs I between Fe/Nb and Co/Nb would \nsuggest an existence of orbital torque, which is consistent with ST -FMR result and the \ntheoretical calculations.  \n In conclusion, the generation of orbital current and orbital torque is experimentally \nconfirmed based on orbital Hall effect present in FM/Nb systems measured by spin -\ntorque ferromagnetic resonance. After fitting the FM thicknesses dependence of the ST -\nFMR reso nance sign al Vdc, we characterize the sign of 𝜉𝐷𝐿 for Fe/Nb and Co/Nb is \npositive, which is  consistent with the spin Hall conductivity and  orbital Hall \nconductivity  of Nb from theoretical calculations , respectively . This result can be further \nverif ied by the method of planar Hall resistance . This strategy of generating orbital \ncurrent and orbital torque by experimental method provide a route of enabling \napplications in spintronic and orbitronic  devices  by orbital engineering.  \n  REFERENCES  \n[1] L. Q. Liu, T. Moriyama, D. C. Ralph, and R. A. Buhrman, Phys. Rev. Lett. 106, \n036601  (2010).  \n[2] Y. K. Kato, R. C. Myers, A. C. Gossard, and D. D. Awschalom, Science 306, 1910 –\n1913  (2004).  \n[3] M. H. Nguyen, C. F. Pai, K. X. Nguyen, D. A. Muller, D. C. Ralph, and R. A. \nBuhrman, Appl. Phys. Lett. 106, 222402  (2015).  \n[4] W. Zhang, W. Han, X. Jiang, S .-H. Yang, and S. S. P. Parkin, Nat. Phys. 11, 496  \n(2015).  \n[5] L. Liu, C. F. Pai, Y . Li, H. W. Tseng, D. C. Ralph, and R. A. Buhrman, Science 336, \n555 (2012).  \n[6] F. F. Liu, C. Zhou, R. J. Tang, G. Zh. Chai, and Ch. J. Jiang, Phys. Status Solidi \nRRL.  15, 2100342  (2021).  \n[7] K. F. Huang, D. S. Wang, H. H. Lin, and C. H. Lai, Appl. Phys. Lett. 107, 232407  \n(2015).  \n[8] P. M. Haney and M. D. Stiles , Phys. Rev. Lett. 105, 126602  (2010).  \n[9] H. Kontani, T. Tanaka, D. S. Hirashima, K. Yamada, and J. Inoue , Phys. Rev. Lett. \n102, 016601  (2009).  \n[10] T. Tanaka, H. Kontani, M. Naito, T. Naito, D. S. Hirashima, K. Yamada, and J. \nInoue , Phys. Rev. B 77, 165117  (2008).  \n[11] B. A . Bernevig, T . L. Hughes, and Sh . Ch. Zhang , Phys. Rev. Lett. 95, 066601  \n(2005).  \n[12] D. Go, D . Jo, Ch . Y . Kim, and H . W. Lee, Phys. Rev. Lett. 121, 086602  (2018).  \n[13] D. Lee, D . Go, H . J Park, W . Jeong, H . W. Ko, D . Yun, D . Jo, S . Lee, G . Go, J . H. \nOh, K . J. Kim, B . G. Park, B . Ch. Min, H . Ch. Koo, H . W. Lee, O . Lee and K. J. Lee, \nNat. Commun . 12, 6710 (2021).  \n[14] S. Bhowal and G . Vignale , Phys. Rev. B 103, 195309  (2021).   \n[15] S. Bhowal and S. Satpathy , Phys. Rev. B 101, 121112(R)  (2020).  \n[16] L. M. Canonico, T . P. Cysne, A . M. Sanchez, R. B. Muniz, and T . G. Rappoport , \nPhys. Rev. B 101, 161409(R)  (2020).  [17] T. P. Cysne, M . Costa, L . M. Canonico, M. B . Nardelli, R.B. Muniz, and Tatiana \nG. Rappoport , Phys. Rev. Lett. 126, 056601  (2021).  \n[18] Sh. L. Ding, A . Ross, D . W. Go, L . Baldrati, Z . Y . Ren, F . Freimuth, S . Becker, F . \nKammerbauer, J . B. Yang, G . Jakob, Y . Mokrousov, and M . Kläui , Phys. Rev. Lett. 125, \n177201  (2020).  \n[19] J. Kim, D . W. Go, H . Sh.  Tsai, D . Jo, K . Kondou, H . W. Lee, and Y . Ch.  Otani , \nPhys. Rev. B 103, L020407  (2021).  \n[20] H. Y . An, Y . Kageyama, Y . Kanno, N . Enishi  and K. Ando , Nat. Commun. 7, 13069  \n(2016).  \n[21] Y . Kageyama, Y . Tazaki, H. An, T. Harumoto, T. Gao, J. Shi, and K. Ando, Sci. \nAdv. 5, eaax4278  (2019).  \n[22] D. W. Go, D . Jo, T . H. Gao, K . Ando, S . Blügel, H . W. Lee, and Y . Mokrousov , \nPhys. Rev. B 103, L121113  (2021).  \n[23] S. Dutta and A. A. Tulapurkar, Phys. Rev. B 106, 184406 (2022)  \n[24] See Supplemental Material for details of the theoretical analysis,  FM-thickness \ndependence of ST -FMR spectra , anomalous Hall resistance measurements  and p lanar \nHall resistance measurements . \n[25] A. R. Mellnik, J. S. Lee, A. Richardella, J. L. Grab, P. J. Mintun, M. H. Fischer, A. \nVaezi, A. Manchon, E. A. Kim, N. Samarth, D. C. Ralph, Nature 511, 449 -451 (2014).  \n[26] D. MacNeill, G. M. Stiehl, M. H. D. Guimaraes, R. A. Buhrman, J. Park and D. C. \nRalph, Nat. Phys.  13, 300 –305 (2017).  \n[27] F. F. Liu, Y . Jin, Y . B. Zhao, Ch . L. Jia and  Ch. J. Jiang , Appl. Phys. Lett. 121, \n022404 (2022) . \n[28] T. Nan, C. X. Quintela, J. Irwin, G. Gurung, D. F. Shao, J. Gibbons, N. Campbell, \nK. Song, S. Y . Choi, L. Guo, R. D. Johnson, P. Manuel, R. V . Chopdekar, I. Hallsteinsen, \nT. Tybell, P. J. Ryan, J. W. Kim, Y . Choi, P. G. Radaelli, D. C. Ralph, E. Y . Tsymb al, M. \nS. Rzchowski, C. B. Eom, Nat. Commun. 11, 4671  (2020).  \n[29] W. F. Zhang, W . Han, X . Jiang, S . H. Yang  and S . S. P. Parkin , Nat. Phys. 11, 496 –\n502 (2015) . \n[30] C. F. Pai, Y . X. Ou, L . H.e Vilela -Leão, D. C. Ralph, and R. A. Buhrman , Phys. Rev. B 92, 064426  (2015).  \n[31] Y. Wang, P . Deorani, X . P. Qiu, J . H. Kwon, and H . Yang , Appl. Phys. Lett. 105, \n152412 (2014) . \n[32] Q. Lu, P . Li, Zh . X. Guo, G . H. Dong, B . Peng, X . Zha, T . Min, Z . Y . Zhou and M. \nLiu, Nat. Commun . 13, 1650 (2022).  \n[33] M. DC, R . Grassi, J . Y. Chen, M . Jamali, D . R. Hickey, D . L. Zhang, Zh . Y . Zhao, \nH. Sh. Li, P. Quarterman, Y . Lv, Mo Li, A . Manchon, K. A . Mkhoyan, T . Low and J. P. \nWang , Nat. Mater . 17, 800 –807 (2018).  \n[34] M. Kawaguchi, K . Shimamura, S . Fukami, F . Matsukura, H . Ohno, T . Moriyama, \nD. Chiba and T . Ono, Appl. Phys. Express 6, 113002  (2013).  \n   \nFigure 1 . (a) The same sign for 𝜎𝑆𝐻 and 𝜎𝑂𝐻, where the total torque includes the sum of  OT and \nST. (b) The opposite sign of 𝜎𝑆𝐻 and 𝜎𝑂𝐻. The total torque  could be either ST or OT. (c) Schematic \nillustration for the mechanism of the orbital torque in FM/N b. The orbital current  generated by OHE  \nin the Nb is injected into FM. Next the injected orbital current  is converted to spin through SOC o f \nthe FM. The resulting torque exerted by t his spin exerts and acts on the magnetization of FM, which \nis referred to as orbital torque (OT). (d) Theoretical calculation for  𝜎𝑆𝐻𝑁𝑏+𝜂𝐹𝑀𝜎𝑂𝐻𝑁𝑏 of FM/Nb \nbilayers ( FM = Fe and Co ). Where 𝜎𝑆𝐻𝑁𝑏, 𝜎𝑂𝐻𝑁𝑏 and 𝜂𝐹𝑀 are spin Hall conductivity, orbital Hall \nconductivity of Nb and orbital -to-spin conversion efficiency of FM, respectively.  \n  \n-80-60-40-20020\n+5\nCo/NbNb\nSH + FMNb\nOH(ћ/e -1cm-1)\nFe/Nb-68(c) (d)(a) (b) \nFigure 2. (a) Schematic of the circuit for ST -FMR measurements.  ST-FMR voltage signal Vdc for \n(b) MgO/Fe (8 nm)/Nb(6 nm)  at 8 GHz , (d) MgO/Co (8 nm)/Nb(6 nm)  at 7 GHz . The inverse of the \nconversion efficiency 1/ ξ as a function of 1/ tFM for (c) Fe/Nb bilayer  and (e) Co/Nb bilayer. The \nsolid line s are the linear fit to the experimental data (grids) . \n0.00 0.05 0.10 0.15 0.20-6061218\nDL = - 0.22 Fe/Nb\n Linear1/\n1/t Fe (nm-1)\n0 200 400 600 800-24-120122436\n Data   Fitting     Va    Vs\n  Vdc (V)\nH (Oe)7 GHzCo(8nm)/Nb(6nm)\n0 300 600 900-2-1012\n Data   Fitting     Va    Vs\n  Vdc (V)\nH (Oe)8 GHzFe(8nm)/Nb(6nm)(b) (c)\n(d) (e)(a)\n0.00 0.07 0.14 0.21 0.28-12-606\nDL = + 0.16\n Co/Nb\n Linear1/\n1/t Co (nm-1) \nFigure 3. ST-FMR voltage signal Vdc for (a) MgO/Pt (6 nm)/Fe(8 nm)  at 7 GHz , (c) MgO/Pt (6 \nnm)/Co(8 nm)  at 7 GHz . The inverse of the conversion efficiency 1/ ξ as a function of 1/ tFM for (b) \nPt/Fe bilayer and (d) Pt/Co bilayer. The girds are the experimental data,  and the solid line is the \nlinear fit to the data.  \n0.00 0.07 0.14 0.21 0.28061218\nDL = + 0.06\n Pt/Co\n Linear1/\n1/t Co (nm-1)\n0.00 0.07 0.14 0.21 0.28071421DL = + 0.05\n Pt/Fe\n Linear1/\n1/t Fe (nm-1)\n0 300 600 900-12-6061218\n Data   Fitting     Va    Vs\n  Vdc (V)\nH (Oe)7 GHzPt(6nm)/Fe(6nm)\n0 300 600 900 1200-10-5051015  Data   Fitting     Va    Vs\n  Vdc (V)\nH (Oe)7 GHzPt(6nm)/Co(6nm)(a) (b)\n(c) (d) \nFigure 4. (a) Cu-layer thickness tCu dependence of efficiency for Co(8 nm)/Cu( tCu)/Nb(6 nm). (b) \nThe conversion efficiency ξ (orange squares) as a function of Nb thickness for Co (8 nm)/Nb(t) as \nwell as Fe  (8 nm)/Nb(t)  systems and a fit (solid orange line).  \n  \n0 1 2 3 4 50.000.060.120.180.240.30\n  \ntCu (nm)Co(8nm)/Cu (t\nCu)/Nb(6nm)\n0 4 8 12 16-0.26-0.130.000.130.26\n Co/Nb\n Fitting Fe/Nb\n Fitting\ntNb(nm)s= 3.1nm(b) (a) \nFigure 5. (a) Current -induced effective fields and schematic illustration of the planar Hall resistance  \nmeasurement srtup.  The external magnetic field angle φ dependence of planar Hall resistance RH of \n(b) Fe(2 nm)/Nb(8 nm) and ( c) Co(2 nm)/Nb(8 nm). The right axis  of (b) and ( c) represents the \ndifference of the Hall resistances RDH at positive and negative currents.  (d) The variety of Hp with \nthe applied current for Fe(2 nm)/Nb(8 nm) and Co(2 nm)/Nb(8 nm) . \n(a)\n0 90 180 270 360-0.2-0.10.00.10.2\nRDH ()\n (°)  +1 mA     -1 mA    DifferenceRH ()\n-0.18-0.090.000.090.18\nCo/Nb\n0 90 180 270 360-0.2-0.10.00.10.2 RH ()\n (°) Fe/Nb\n-0.14-0.070.000.070.14\n RDH () +0.9 mA      -0.9 mA  Difference\n0.0 0.3 0.6 0.9-4004080\n Co/Nb\n Fe/NbHP (Oe)\nCurrent (mA) (b)\n(c) (d)\nHP\nτDLτFL\nHTz\nyx\ny\nx zφIHext\nV"    },    {        "title": "1701.00786v2.Universal_Absence_of_Walker_Breakdown_and_Linear_Current_Velocity_Relation_via_Spin_Orbit_Torques_in_Coupled_and_Single_Domain_Wall_Motion.pdf",        "content": "arXiv:1701.00786v2  [cond-mat.mes-hall]  18 Apr 2017Universal Absence of Walker Breakdown and Linear Current–V elocity Relation\nvia Spin–Orbit Torques in Coupled and Single Domain Wall Mot ion\nVetle Risinggård∗and Jacob Linder\nDepartment of Physics, NTNU, Norwegian University of Scien ce and Technology, N-7491 Trondheim, Norway\n(Dated: June 29, 2018)\nWe consider theoretically domain wall motion driven by spin –orbit and spin Hall torques. We find that it\nis possible to achieve universal absence of Walker breakdow n for all spin–orbit torques using experimentally\nrelevant spin–orbit coupling strengths. For spin–orbit to rques other than the pure Rashba spin–orbit torque, this\ngives a linear current–velocity relation instead of a satur ation of the velocity at high current densities. The\neffect is very robust and is found in both soft and hard magnet ic materials, as well as in the presence of the\nDzyaloshinskii–Moriya interaction and in coupled domain w alls in synthetic antiferromagnets, where it leads\nto very high domain wall velocities. Moreover, recent exper iments have demonstrated that the switching of a\nsynthetic antiferromagnet does not obey the usual spin Hall angle-dependence, but that domain expansion and\ncontraction can be selectively controlled toggling only th e applied in-plane magnetic field magnitude and not\nits sign. We show for the first time that the combination of spi n Hall torques and interlayer exchange coupling\nproduces the necessary relative velocities for this switch ing to occur.\nI. INTRODUCTION\nDomain wall motion in ferromagnetic strips is a central\ntheme in magnetization dynamics and has recently been in-\nstrumental to the discovery of several new current-induced\neffects.1–6The attainable velocity of a domain wall driven by\nconventional spin-transfer torques (STTs)7–9is limited by the\nWalker breakdown,10upon which the domain wall deforms,\nresulting in a reduction of its velocity.\nCurrent-induced torques derived from spin–orbit effects\n(SOTs) such as the spin Hall effect4–6,11or an interfacial\nRashba spin–orbit coupling12–14have enabled large domain\nwall velocities. We here consider the dependence of the do-\nmain wall velocity on the current and find that regardless of\nthe relative importance of the reactive and dissipative com -\nponents of the torque it is possible to achieve universal ab-\nsence of Walker breakdown for all current densities for ex-\nperimentally relevant spin–orbit coupling strengths. For spin–\norbit torques other than the pure Rashba SOTs, such as the\nspin Hall torques, the velocity will not saturate as a func-\ntion of current, but will increase linearly as long as a conve n-\ntional spin-transfer torque is present. This behavior is ro bust\nagainst the presence of an interfacial Dzyaloshinskii–Mor iya\ninteraction15–17and is found both in perpendicular anisotropy\nferromagnets, in shape anisotropy-dominated strips and in\nsynthetic antiferromagnets (SAFs),18–22where it enables very\nhigh domain wall velocites for relatively small current den si-\nties. Moreover, the combination of SOTs with the interlayer\nexchange torque was recently shown experimentally to pro-\nduce novel switching behavior that circumvents the usual sp in\nHall angle-dependence.22We show that the combination of\nspin Hall torques and interlayer exchange produces the re-\nquired dependence of the domain wall velocity on the topolog -\nical charge to qualitatively reproduce the experimental da ta.II. UNIVERSAL ABSENCE OF WALKER BREAKDOWN\nWe consider an ultrathin ferromagnet with a heavy metal\nunderlayer as shown in Figure 1 . We describe the dynamics of\nthe magnetization m(r,t)using the Landau–Lifshitz–Gilbert\n(LLG) equation,23\n∂tm=γm×H−α\nmm×∂tm+τ, (1)\nwhereγ < 0is the gyromagnetic ratio, mis the saturation\nmagnetization,α<0is the Gilbert damping, H=−δF/δm\nis the effective field acting on the magnetization and τis the\ncurrent-induced torques. The free energy Fof the ferromag-\nnet is a sum,\nF=/uni222B.dsp\ndr(fZ+fex+fDM+fa), (2)\nof the Zeeman energy due to applied magnetic fields, the\nisotropic exchange, the interfacial Dzyaloshinskii–Mori ya in-\nteraction and the magnetic anisotropy.\nThe Zeeman energy and the isotropic exchange can be\nwritten respectively as fZ=−H0·m, whereH0is the ap-\nplied magnetic field, and fex=(A/m2)[(∇mx)2+(∇my)2+\n(∇mz)2], where Ais the exchange stiffness.23Inversion sym-\nmetry breaking at the interface between the heavy metal and\nthe ferromagnet gives rise to an anisotropic contribution t o\nthe exchange known as the Dzyaloshinskii–Moriya interac-\ntion, which favors a canting of the spins.15–17The resulting\nFigure 1. Ultrathin ferromagnet with a heavy metal underlay er. We\nconsider transverse domain wall motion along the xaxis. r,σland\nσsdenote the three nontrivial operations of the symmetry grou pC2v.2\ncontribution to the free energy is fDM=(D/m2)[mz(∇ ·m)−\n(m· ∇)mz], where Dis the magnitude of the Dzyaloshinskii–\nMoriya vector. Ultrathin magnetic films are prone to exhibit\nperpendicular magnetization due to interface contributio ns to\nthe magnetic anisotropy.24Consequently, we write the mag-\nnetic anisotropy energy as fa=−Kzm2\nz+Kym2\ny, correspond-\ning to an easy axis in the zdirection and a hard axis in the y\ndirection.\nA. Current-Induced Torques\nThe current-induced torques τare conventionally divided\ninto spin-transfer torques and spin–orbit torques. The spi n-\ntransfer torques can be written as7–9\nτSTT=u∂xm−βu\nmm×∂xm, (3)\nwhere u=µBP j/[em(1+β2)]andjis the electric current, Pis\nits spin polarization, µBis the Bohr magneton, eis the electric\ncharge andβis the nonadiabacity parameter. The spin–orbit\ntorques can be written as4–6,11–14\nτR=γm×HRey−γm×/parenleftbigg\nm×βHRey\nm/parenrightbigg\n, (4)\nτSH=γm×/parenleftbigg\nm×HSHey\nm/parenrightbigg\n+γm×βSHHSHey,(5)\nwhere HR=αRP j/[2µBm(1+β2)]andαRis the Rashba pa-\nrameter and where HSH=/planckover2pi1θSHj/(2emt)andθSHis the spin\nHall angle and tis the magnet thickness. Since the spin Hall\neffect changes sign upon time-reversal, the principal spin Hall\ntorque term is dissipative instead of reactive, in contrast to the\nprincipal term of the STTs and the Rashba SOTs.\nIn fact, assuming that the stack can be described using the\nC2vsymmetry group (see Figure 1 ) it can be shown that these\ntorques exhaust the number of possible torque components.\nHals and Brataas25describe spin–orbit torques and general-\nized spin-transfer torques in terms of a tensor expansion. A s-\nsuming the lowest orders are sufficient to describe the essen -\ntial dynamics, the reactive and dissipative spin–orbit tor ques\nare described by, respectively, an axial second-rank tenso r and\na polar third-rank tensor while the generalized spin-trans fer\ntorques are described using a polar fourth-rank tensor and a n\naxial fifth-rank tensor. The torques that arise in a given str uc-\nture are limited by the requirement that the tensors must be\ninvariant under the symmetry operations fulfilled by the str uc-\nture. We have assumed that the physical systems we con-\nsider are described by C2vsymmetry. Combined with the\nfact that the current is applied in the xdirection only and\nthat∂ym=0and∂zm=0, this implies that there is only\none relevant nonzero element in the axial second-rank tenso r,\ntwo elements in the polar third-rank tensor, three elements in\nthe polar fourth-rank tensor and six elements in the axial fif th-\nrank tensor.26\nThe three relevant nonzero elements of the second- and\nthird-rank tensors give rise to three spin–orbit torques. A de-\ntailed analysis shows that these torque components are cap-\ntured by the Rashba and spin Hall torques in equations ( 4) and(5). As an aside, we note that although the Rashba and spin\nHall effects may not necessarily capture all of the relevant mi-\ncroscopic physics27–29these torques can still be used to model\nthe dynamics because they contain three ‘free’ parameters, αR,\nθSHandβSH.\nAs has been shown in Ref. 25, the generalized spin-transfer\ntorques reduce to the ordinary STTs in the nonrelativistic l imit.\nThus, by using the ordinary STTs we neglect possible spin–\norbit coupling corrections to these higher-order terms.\nB. The Collective Coordinate Model\nThe magnetization is conveniently parametrized in spher-\nical coordinates as m/m=cosφsinθex+sinφsinθey+\ncosθez. Using the assumption that there is no magnetic tex-\nture along the yand the zaxes,∇=∂xex, we can find the\ndomain wall profile by minimizing the free energy. The re-\nsulting Euler–Lagrange equations are\nA(θ′′cscθsecθ−φ′2)−Dφ′sinφtanθ=(Kz+Kysin2φ)\nand\nA(φ′′+2θ′φ′cotθ)+Dθ′sinφ=Kycosφsinφ.\nOne solution of these differential equations is the Néel wal l\nsolutionφ=nπandθ=2 arctan exp [Q(x−X)/λ], where Q\nis the topological charge of the wall,30Xis the wall position\nandλ=/radicalbig\nA/Kzis the domain wall width. nis even if D<0\nandQ=+1, and nis odd if D<0andQ=−1. This domain\nwall profile is known as the Walker profile.10To be sure that\nφ=nπis really the global minimum, we solve the full LLG\nequation ( 1) for a single magnetic layer and let the solution re-\nlax without any applied currents or fields. The angle φ(x)can\nthen be calculated as φ(x)=arctan[my(x)/mx(x)]. However,\nφ(x)is ill defined in the domains where θ→0orπ. Conse-\nquently, we consider φonly inside the domain wall. As shown\ninFigure 2 (a), the solutionφ=0works very well.\nSubstitution of the Walker profile into the full LLG equa-\ntion ( 1) usingH0=HxexandQ=+1gives the collective\ncoordinate equations, for the wall position Xand tiltφ\nα∝dotaccX\nλ−∝dotaccφ=+π\n2γ/parenleftBig\nHSH−βHR/parenrightBig\ncosφ+βu\nλ, (6)\n(1+α2)∝dotaccφ=−αγKy\nmsin 2φ+παγ(D−Hxmλ)\n2mλsinφ(7)\n−u(α+β)\nλ−π\n2γ/bracketleftBig\nHSH(1−αβSH)−HR(α+β)/bracketrightBig\ncosφ.\nBy doing this substitution, we are assuming that the domain\nwall moves as a rigid object described by two collective coor -\ndinates X(t)andφ(t)(Ref. 30). In particular, we are neglect-\ning any position dependence in the domain wall tilt φ. The col-\nlective coordinate model, or one-dimensional model, has be en\nused previously to explain the qualitative behavior of both\nspin-transfer and spin–orbit torques.4,5,7,10,18–20,28,30,31How-\never, it is important to remember that the model will always3\n−2 0 2−0.02−0.010.000.010.02\npositionx/λtilt angle φ(x)[rad]\n−2 0 20.000.250.500.751.00\nmagnetization mx(x)/m(a)\n468 10−0.02−0.010.000.010.02\npositionx/λtilt angle φ(x)[rad]\n468 100.000.250.500.751.00\nmagnetization mx(x)/m(b)\n20 22 24 260.000.010.020.030.04\npositionx/λtilt angle φ(x)[rad]\n20 22 24 260.000.250.500.751.00\nmagnetization mx(x)/m(c)\n2 4 68−0.02−0.010.000.010.02\npositionx/λtilt angle φ(x)[rad]\n2 4 680.000.250.500.751.00\nmagnetization mx(x)/m(d)\nFigure 2. Position dependence of the domain wall tilt φ. In each\npanel, the orange curve mx(x)shows the extension of the domain\nwall while the black solid curve shows the domain wall tilt φ(x)ob-\ntained by solving the full LLG equation ( 1) and the black dashed line\nshows the prediction of the collective coordinate model. (a ) Equi-\nlibrium solution. (b) Spin-transfer torque dynamics. (c) S pin Hall\ntorque dynamics. (d) Rashba spin–orbit torque dynamics. (a )–(d) We\nuse the material parameters supplied in the first column of Table I\nwith j=5 MA/cm2except that J=0.\nbe an approximation, and we cannot necessarily expect quant i-\ntative agreement between experimental results and model pr e-\ndictions nor can we completely exclude the possibility of dy -\nnamics that is not captured by the one-dimensional model.31\nWe can nevertheless test the adequacy of the collective coor -\ndinate model by calculating φ(x)from a solution of the full\nLLG equation for a single magnetic layer, just as we did for\nthe static case. As shown in Figure 2 (b) the xdependence of\nφis negligible for spin-transfer torques. The xdependence of\nφis larger for spin Hall [ Figure 2 (c)] and Rashba spin–orbit\ntorques [ Figure 2 (d)]. Nonetheless, the ability of the collec-\ntive coordinate model to consistently qualitatively repro duce\nexperimental behavior indicates that it captures the gener ality,\nif not all, of the physics in the system.\nEquations ( 6) and ( 7) can be simplified by introducing a j=\nπ\n2γ(HSH−βHR),bj=βu/λ,c=−2αγKy/m,d=παγ(D−\nHxmλ)/(2mλ),ej=−π\n2γ[HSH(1−αβSH) −HR(α+β)]and\nf j=−u(α+β)/λ. Walker breakdown is absent when the time\nderivative ∝dotaccφvanishes, resulting in the condition\n0=csinφcosφ+dsinφ+j(ecosφ+f). (8)\nProvided that the transverse domain wall is not transformed\ninto for instance a vortex wall,31Walker breakdown will be\nuniversally absent if e>fbecause this equation always has\na solution forφregardless of the value of j. For increasing\nj,φwill level off to a value cosφ=−f/e. For realistic\nmaterial values e>fcorresponds to a Rashba parameter\nαR>4µ2\nB/(πeγλ)=1to6 meV nm (pure Rashba SOTs) or\na spin Hall angleθSH>4µBPt/(π/planckover2pi1γλ)=0.05to0.09(pure\nspin Hall torques). To the best of our knowledge, the absenceof Walker breakdown for spin Hall torques has not been noted\npreviously, whereas absence of Walker breakdown for suffi-\nciently strong Rashba spin–orbit coupling was pointed out i n\nRef. 32, and can also be noted in Refs 13and33–35.\nLet us writeξ=cosφandη=sinφ, so thatξ2+η2=1.\nSolving equation ( 8) forηto getη=−j(eξ+f)/(cξ+d), this\nrelation gives a quartic equation\nc2ξ4+2cdξ3+[(ej)2+d2−c2]ξ2+2(e f j2−cd)ξ=d2−(f j)2.\nThe exact solutions of the quartic are hopelessly complicat ed.\nHowever, they all have the same series expansion around j=\n0andj→ ∞ . We consider first the asymptotic expansion,\nξ=−f\ne+S1\nj+O/parenleftBig\nj−2/parenrightBig\n, (9)\nwhere S1represents the solutions of the quadratic equation\ne6ζ2=d2e4+c2f4+(c2−d2)f2e2+2cde f(f2−e2). Using\nequation ( 6), the wall velocity is then\nα∝dotaccX\nλ=/parenleftbigg\nb−a f\ne/parenrightbigg\nj+aS1+aO/parenleftBig\nj−1/parenrightBig\n. (10)\nBack substitution of the abbreviations a,b,eand fshows\nthat for pure Rashba SOTs the coefficient of the linear term\nreduces to zero because the ratio of the reactive to the dissi -\npative torque is the same for the STTs and the Rashba SOTs.\nThus, for large jthe domain wall velocity approaches a con-\nstant. For pure spin Hall torques we get instead the linear te rm\n−uα(1+ββSH)/[λ(1−αβSH)]. This means that for large j the\nvelocity is actually independent of the sign of the spin Hall\nangle and increases linearly with j. Note the importance of\nincluding the STTs—which are always present—in these con-\nsiderations: in the absence of STTs ( u→0) both band fgo\nto zero and the velocity levels off to a constant for large jfor\nany combination of SOTs.\nFor completeness, we also consider the series expansion\nabout j=0, which gives\nξ=−1+(e−f)2\n2(c−d)2j2+O/parenleftBig\nj4/parenrightBig\n(11)\nand\nα∝dotaccX\nλ=(b−a)j+a(e−f)2\n2(c−d)2j3+aO/parenleftBig\nj5/parenrightBig\n. (12)\nThe key observation here is that in this regime the velocity\ndoes depend on the sign of the spin Hall angle ( a∝θSHfor\npure spin Hall torques) and increases with the cube of j. Com-\nbined with the spin Hall angle-independence of the velocity\nin the j→ ∞ limit, this implies that even in the absence\nof Walker breakdown a nonmonotonic current–velocity rela-\ntion is possible. Figure 3 (a) shows a numerical solution of\nthe coupled equations ( 6) and ( 7) as a function of jfor pure\nRashba SOTs and for pure spin Hall torques both in the cases\nofθSH>0andθSH<0together with the analytical solutions\nclose to j=0and for large jfor parameters that are typical\nfor a standard cobalt–nickel multilayer. We see that our ana -\nlytical results successfully approximate the full solutio n in the4\n0 2 4 6\n·1013−2−10\ncurrentj/bracketleftBig\nA/m2/bracketrightBigvelocity ˙X[km/s](a) (b)\n0 3 69\n·1012−0.6−0.300.3\ncurrentj/bracketleftBig\nA/m2/bracketrightBigvelocity ˙X[km/s]\nspin Hall, θSH>0 spin Hall, θSH<0 Rashba SOTs\nFigure 3. Current–velocity relation for three different SO Ts in the ab-\nsence of Walker breakdown. The Rashba SOTs level off to a cons tant\nvelocity at large currents, whereas the spin Hall torques as ymptoti-\ncally approach a linear current–velocity relation. Dashed lines show\nthe asymptotic expansion and dotted curves show the series a bout\nj=0. We use the material parameters supplied in the (a) first and\n(b) second column of Table I except that J=0.\nexpected ranges of validity indicating the absence of Walke r\nbreakdown in the numerical solution.\nThe in-plane hard axis included in the magnetic anisotropy\nis appropriate for narrow ferromagnetic strips, which gene r-\nally host Néel walls. Wider strips give Bloch walls,24and\nby making the necessary modifications to the above calcula-\ntions, we find that in this case the domain wall velocity re-\ntains the qualitative features elucidated above. This is al so\ntrue for shape anisotropy-dominated strips, which host hea d-\nto-head walls. This shows that universal absence of Walker\nbreakdown is a robust effect that does not depend on the de-\ntails of the ferromagnetic material , unlike other SOT effects\nstudied previously.36This fact is also illustrated by the numer-\nics. In Figure 3 (b) we present numerical results obtained for a\nNéel wall in a PMA ferromagnet with anisotropies weaker byan order of magnitude, weaker magnetic damping and much\nlarger Rashba spin–orbit coupling and spin Hall angle in the\nadjacent heavy metal. The results are qualitatively simila r to\nthose obtained in Figure 3 (a).\nIII. COUPLED DOMAIN WALLS IN A SAF STRUCTURE\nWe consider next an asymmetric stack of two ultrathin\nferromagnets separated by an insulating spacer as shown in\nFigure 4 (a). We describe the dynamics of each of the ferro-\nmagnets using separate LLG equations, but add to the free\nenergy a coupling term,\nFIEC=/uni222B.dspdr1\nm(1)/uni222B.dspdr2\nm(2)J(r1−r2)/bracketleftBig\nm(1)(r1)·m(2)(r2)/bracketrightBig\n,(13)\nrepresenting the interlayer exchange (IEC). We assume that\nthe IEC is local in the plane, J(r1−r2)=Jδ(x1−x2)δ(y1−\ny2). Equation ( 13) then represent the lowest order coupling\nproposed by Bruno.37\nFollowing the same procedure as in the previous section\nwe may now derive four coupled collective coordinate equa-\ntions. With an antiferromagnetic coupling the walls will\nhave opposite topological charges, Q2=−Q1. Since a lo-\ncal IEC can only affect the chiralities, and not the profiles o f\nthe walls, we can use the static solution derived previously ,\nθ=2 arctan exp [Q(x−X)/λ], whereλ=/radicalbig\nA/Kzis the do-\nmain wall width and Qis the topological charge. For a single\nwall the azimuthal angle φis given byφ=nπ.nis even if\nD<0andQ=+1, and nis odd if D<0andQ=−1. To\nlimit the scope of the treatment, we consider only the case\nwhere D1andD2have the same sign, D1<0andD2<0.\nThen the DMI and the IEC cooperate to give the static solu-\ntionφ1=0(Q1=+1) andφ2=π(Q2=−1).\nSubstituting this static solution into the LLG equations us -\ningH0=Hxexgives the collective coordinate equations\n(1+α2)∝dotaccX1\nλ=−γKy\nmsin 2φ1+πγ(D1−Hxmλ)\n2mλsinφ1+γJt2\n2m/bracketleftBig\nαU(s)cos(φ1−φ2)+αW(s)+V(s)sin(φ1−φ2)/bracketrightBig\n−u(1−αβ)\nλ+π\n2γ/bracketleftBig\nH(1)\nSH/parenleftBig\nα+β(1)\nSH/parenrightBig\n+H(1)\nR(1−αβ)/bracketrightBig\ncosφ1,(14)\n(1+α2)∝dotaccX2\nλ=+γKy\nmsin 2φ2+πγ(D2+Hxmλ)\n2mλsinφ2−γJt1\n2m/bracketleftBig\nαU(s)cos(φ1−φ2)+αW(s)−V(s)sin(φ1−φ2)/bracketrightBig\n−u(1−αβ)\nλ+π\n2γ/bracketleftBig\nH(2)\nSH/parenleftBig\nα+β(2)\nSH/parenrightBig\n+H(2)\nR(1−αβ)/bracketrightBig\ncosφ2,(15)\n(1+α2)∝dotaccφ1=−αγKy\nmsin 2φ1+παγ(D1−Hxmλ)\n2mλsinφ1−γJt2\n2m/bracketleftBig\nU(s)cos(φ1−φ2)+W(s)−αV(s)sin(φ1−φ2)/bracketrightBig\n−u(α+β)\nλ−π\n2αγ/bracketleftBig\nH(1)\nSH/parenleftBig\n1−αβ(1)\nSH/parenrightBig\n−H(1)\nR(α+β)/bracketrightBig\ncosφ1,(16)\n(1+α2)∝dotaccφ2=−αγKy\nmsin 2φ2−παγ(D2+Hxmλ)\n2mλsinφ2−γJt1\n2m/bracketleftBig\nU(s)cos(φ1−φ2)+W(s)+αV(s)sin(φ1−φ2)/bracketrightBig\n+u(α+β)\nλ+π\n2αγ/bracketleftBig\nH(2)\nSH/parenleftBig\n1−αβ(2)\nSH/parenrightBig\n−H(2)\nR(α+β)/bracketrightBig\ncosφ2.(17)5\nFigure 4. (a) Two ultrathin ferromagnets separated by an ins ulating\nspacer with heavy metal over- and underlayers. The ferromag nets are\nidentical except for their thicknesses, but the different h eavy metals\ninduce different DMIs and SOTs. (b) Dependence of the IEC ter ms\nV(s),U(s)andW(s)on the wall separation.\nwhere we have assumed that the bulk parameters of the two\nferromagnets are equal and where sis the separation between\nthe two walls, s=(X1−X2)/λ. The IEC terms are expressed\nusing the three functions V(s),U(s)andW(s);\nV(s)=2scsch s,\nU(s)=2 csch s−2scoth scsch s,\nW(s)=2 coth s−2scsch2s.\nThese functions are plotted in Figure 4 (b).\nEquations ( 14) and ( 16) reduce to equations ( 6) and ( 7)\nwhen J→0. To solve equations ( 14)–(17) numerically, we\nrescale the equations to obtain dimensionless variables. T he\ndimension of equations ( 14)–(17) isHz. A convenient scal-\ning factor with the same dimensions is µ0γm. By dividing\nequations ( 14)–(17) byµ0γmwe get the rescaled variables\n˜t=tµ0γm,˜Xi=Xi/λ,˜Hx=Hx/µ0m,˜Ky=Ky/µ0m2,\n˜Di=Di/µ0m2λ,˜ti=ti/λ,˜J=Jλ/µ0m2and˜u=u/µ0γmλ.\nWe solve the equations using an explicit fourth order Runge–\nKutta scheme with adaptive stepsize control, implemented a s\na Dormand–Prince pair.38\nA. Universal Absence of Walker Breakdown in SAF structures\nFor parameter values representative of a standard cobalt–\nnickel multilayer we obtain the current–velocity and curre nt–\ntilt relations shown in Figure 5 (a) and (b) for t1/t2=1in the\ncase where only STTs are present and in the case where spin\nHall torques are additionally present. We see that the prese nce\nof the IEC delays Walker breakdown when the wall is driven\nby ordinary STTs, but the subcritical differential velocit y re-\nmains unaffected. This can also be shown analytically by sol v-\ning for the tilt angle of the wall as a function of current. Suc h a\ncalculation shows that the tilt angle is suppressed by the IE C00.61.2 1.8\n·1014−6−303\ncurrentj/bracketleftBig\nA/m2/bracketrightBigvelocity ˙X[km/s](a)\nSTT, single\nSTT, coupledSH, single\nSH, coupled\n00.61.2 1.8\n·1014−2−1012\ncurrentj/bracketleftBig\nA/m2/bracketrightBigtilt angle φ1[rad](b)\n0.5 1 1.5 20.511.522.5\nthickness ratio t1/t2velocity ˙X[km/s](c)\n0.5 1 1.5 2024\nthickness ratio t1/t2tiltφ[rad]\n12.22.42.62.8\ntiltφ2−φ1[rad]φ2\nφ1(d)\n0612 18\n·1013−20−10010\ncurrentj/bracketleftBig\nA/m2/bracketrightBigvelocity ˙X[km/s](e)\nSTT, single\nSTT, coupledSH, single\nSH, coupled\n0 0.30.60.9\n·1013−0.8−0.400.4\ncurrentj/bracketleftBig\nA/m2/bracketrightBigvelocity ˙X[km/s](f)\n0612 18\n·1013−2−1012\ncurrentj/bracketleftBig\nA/m2/bracketrightBigtilt angle φ1[rad](g)\n0.25 0.5 1 2 402468\nthickness ratio t1/t2velocity ˙X[km/s](h)\nFigure 5. Domain wall dynamics in interlayer exchange coupl ed\nferromagnets. (a) and (b) The IEC delays Walker breakdown fo r\nSTT driving, but the subcritical differential velocity rem ains unaf-\nfected. With spin Hall torques the tilt angle stabilizes at a finite\nvalue, indicating universal absence of Walker breakdown. T he tilt an-\ngle approaches its limiting value more slowly in the presenc e of IEC.\n(c) and (d) The IEC gives the velocity a nonmonotonic thickne ss-\ndependence resulting in a peak close to t1/t2=1. [j=3 GA/cm2,\ncorresponding to the dashed vertical line in (a).] We use the material\nparameters supplied in the first column of Table I . (e)–(h) These re-\nsults are robust against a change in parameters to those in th e second\ncolumn of Table I .\n(but the breakdown angle is still π/4). Back-substitution of\nthis angle into the torque acting on the wall shows that this\ntorque is independent of J, explaining why there is no change\nin the differential velocity.\nWhen spin Hall torques are included, the domain wall tilt\nlevels off to a finite value and the current–velocity relatio n is6\nTable I. Parameters used for the numerical solution of equat ions ( 14)–\n(17) and for analytical estimates in the text.\nparameter Co–Ni strong SOC Bi et al. unit\ngyromagnetic ratio γ−0.19−0.19−0.19 THz /T\ndomain wall width λ 4 16 2 nm\nhard axis anisotropy Ky200 20 2 kJ /m3\nsaturation magn. m 1 1 1 .1 MA/m\nDM constant D −1.4−1.0−0.1 mJ/m2\nGilbert dampingα −0.25−0.1−0.5\nspin-polarization P 0.5 0.5 0.5\nnonadiabacity param. β 0.5 0.4 2\nRashba parameterαR 6.3 75 meVnm\nspin Hall angleθSH 0.1 0.2 0.12\nspin Hallβ-termβSH 0.02 0.02 0.02\ninterlayer exchange Jt1t25 5 1 .5 mJ/m2\nthickness t1 1.2 1.2 0.6 nm\nthickness t2 1.2 1.2 1.7 nm\nlinear in the j→ ∞ limit. This shows that universal absence\nof Walker breakdown is also found in SAF structures. The\neffect of the IEC can be understood simply as a rescaling of\nthe constant S1and the higher order constants S2,S3,... in\nthe expansion ( 9), making the tilt angle approach its limiting\nvalue more slowly. Thus, the effect of the IEC on both the\nSTT and spin Hall results is to suppress the domain wall tilt,\nas shown in Figure 5 (b). We note that the combination of spin\nHall torques and IEC produces much higher domain wall ve-\nlocities than in single ferromagnets for comparatively sma ll\ncurrent densities.20\nIn a single ferromagnet the velocity of a wall driven by spin\nHall torques decreases with tas1/t. When changing t2from\nt2=t1/2tot2=2t1in a SAF structure, we find that the veloc-\nity peaks close to t1/t2≈1, which maximizes the IEC torque\n[seeFigure 5 (c); the deviation from 1 is due to the DMI]. This\ncan be understood by considering Figure 5 (d); at t1/t2≈1\nthe magnetizations in both layers are tilted in the ydirection.\nIncreasing (decreasing) t2tot2=2t1(t2=t1/2) reduces (in-\ncreases) H(2)\nSHand increases (reduces) H(1)\nIEC, thus(φ2−φ1)ap-\nproachesπand the IEC torque is reduced.\nJust as for the single ferromagnetic layer the results for th e\ncoupled walls are robust against a change of parameters, as\nshown in Figure 5 (e)–(h).\nB. Novel Switching Behavior in SAF Structures\nBiet al.22have very recently demonstrated completely\nnovel switching behavior in SAF structures. In single ferro -\nmagnets, domain walls with one topological charge will trav el\nfaster than those with the opposite topological charge if an\nin-plane magnetic field is applied.39If the relative velocity is\nlarge enough the favored domains can overcome the destabi-\nlizing action of the current (see Refs 40–43) and merge.44–46\nThe favored magnetization direction is uniquely determine d\nby the spin Hall angle and the applied magnetic field for a\nfixed direction of the current. Biet al. observed this behavior−2−1 0 1 21416\nvelocity ˙X[m/s]\n−2−1 0 1 2−16016\nin-plane magnetic field Hx[T]rel. vel.[m/s]\n(+,−)\n(−,+)\nFigure 6. Qualitative reproduction of the experimental res ults of Bi\net al.22The sign of the relative velocity of walls with (Q1,Q2)=\n(+1,−1)and(Q1,Q2)=(−1,+1)can be toggled only by changing\nthe magnitude of the applied field. We use the material parame ters\nsupplied in the third column of Table I .\nin SAF structures for small in-plane fields, but by toggling b e-\ntween large and small values of the in-plane field (same sign) ,\nthey were able to toggle the sign of the relative velocity of t he\nwalls and thereby the favored magnetization direction. Usi ng\nmaterial parameters that approximate the samples of Biet al. ,\nour model is the first to qualitatively reproduce this behavi or,\nas shown in Figure 6 . Under an in-plane field in the range\n0.3 Tto1.4 T, walls with (Q1,Q2)=(+1,−1)travel faster\nthan walls with (Q1,Q2)=(−1,+1)and ‘up’ magnetization\nis favored. If the field is increased beyond 1.4 T, the relative\nvelocity changes sign, and ‘down’ magnetization is favored .\n(The offset from zero is due to the DMI.)\nIV . CONCLUSION\nWe have shown that complete suppression of Walker break-\ndown is possible in a wide range of domain wall systems\ndriven by spin–orbit torques, including head-to-head wall s\nin soft magnets, Bloch and Néel walls in perpendicular\nanisotropy magnets, in the presence of the Dzyaloshinskii–\nMoriya interaction and in coupled domain walls in syn-\nthetic antiferromagnets. For spin–orbit torques other tha n\npure Rashba spin–orbit torques this leads to a linear curren t–\nvelocity relation instead of a saturation of the velocity fo r large\ncurrents. In combination with interlayer exchange couplin g,\nspin–orbit torque driven domain wall motion in synthetic an ti-\nferromagnets gives rise to novel switching behavior and ver y\nhigh domain wall velocities.\nACKNOWLEDGMENTS\nFunding via the “Outstanding Academic Fellows” program\nat NTNU, the COST Action MP-1201, the NV Faculty, and\nthe Research Council of Norway Grants No. 216700 and\nNo. 240806, is gratefully acknowledged. We thank Morten\nAmundsen for very useful discussions of the numerics.7\n∗vetle.k.risinggard@ntnu.no\n1I. M. Miron, P.-J. Zermatten, G. Gaudin, S. Auffret, B. Rodma cq,\nand A. Schuhl, Phys. Rev. Lett. 102, 137202 (2009) .\n2I. M. Miron, G. Gaudin, S. Auffret, B. Rodmacq,\nA. Schuhl, S. Pizzini, J. V ogel, and P. Gambardella,\nNat. Mater. 9, 230 (2010) .\n3I. M. Miron, T. Moore, H. Szambolics, L. D. Buda-Prejbeanu,\nS. Auffret, B. Rodmacq, S. Pizzini, J. V ogel, M. Bonfim,\nA. Schuhl, and G. Gaudin, Nat. Mater. 10, 419 (2011) .\n4S. Emori, U. Bauer, S.-M. Ahn, E. Martinez, and G. S. D. Beach,\nNat. Mater. 12, 611 (2013) .\n5K.-S. Ryu, L. Thomas, S.-H. Yang, and S. Parkin,\nNat. Nanotechnol. 8, 527 (2013) .\n6P. P. J. Haazen, E. Murè, J. H. Franken, R. Lavrijsen, H. J. M.\nSwagten, and B. Koopmans, Nat. Mater. 12, 299 (2013) .\n7S. Zhang and Z. Li, Phys. Rev. Lett. 93, 127204 (2004) .\n8D. C. Ralph and M. D. Stiles,\nJ. Magn. Magn. Mater. 320, 1190 (2008) .\n9A. Brataas, A. D. Kent, and H. Ohno, Nat. Mater. 11, 372 (2012) .\n10N. L. Schryer and L. R. Walker, J. Appl. Phys. 45, 5406 (1974) .\n11K. Ando, S. Takahashi, K. Harii, K. Sasage, J. Ieda, S. Maekaw a,\nand E. Saitoh, Phys. Rev. Lett. 101, 036601 (2008) .\n12A. Manchon and S. Zhang, Phys. Rev. B 78, 212405 (2008) .\n13K.-W. Kim, S.-M. Seo, J. Ryu, K.-J. Lee, and H.-W. Lee,\nPhys. Rev. B 85, 180404 (2012) .\n14A. Chernyshov, M. Overby, X. Liu, J. K. Furdyna, Y . Lyanda-\nGeller, and L. P. Rokhinson, Nat. Phys. 5, 656 (2009) .\n15I. E. Dzyaloshinskii, J. Exp. Theor. Phys. 5, 1259 (1957) .\n16T. Moriya, Phys. Rev. Lett. 4, 228 (1960) ;\nPhys. Rev. 120, 91 (1960) .\n17A. Fert, Mater. Sci. Forum 59–60 , 439 (1990) .\n18H. Saarikoski, H. Kohno, C. H. Marrows, and G. Tatara,\nPhys. Rev. B 90, 094411 (2014) .\n19S. Lepadatu, H. Saarikoski, R. Beacham, M. J. Benitez, T. A.\nMoore, G. Burnell, S. Sugimoto, D. Yesudas, M. C. Wheeler,\nJ. Miguel, S. S. Dhesi, D. McGrouther, S. McVitie, G. Tatara,\nand C. H. Marrows, arXiv:1604.07992 .\n20S.-H. Yang, K.-S. Ryu, and S. Parkin,\nNat. Nanotechnol. 10, 221 (2015) .\n21R. Tomasello, V . Puliafito, E. Martinez, A. Manchon, M. Ricci ,\nM. Carpentieri, and G. Finocchio, arXiv:1610.00894 .\n22C. Bi, H. Almasi, K. Price, T. Newhouse-Illige, M. Xu, S. R.\nAllen, X. Fan, and W. Wang, Phys. Rev. B 95, 104434 (2017) .\n23L. D. Landau and E. M. Lifshitz, Phys. Zeitschrift der Sowje-\ntunion 8, 153 (1935); Ukr. J. Phys. 53, 14 (2008) ; T. Gilbert,\nIEEE Trans. Magn. 40, 3443 (2004) .24R. C. O’Handley, Modern magnetic materials. Principles and ap-\nplications (John Wiley & Sons, 2000).\n25K. M. D. Hals and A. Brataas, Phys. Rev. B 88, 085423 (2013) ;\nPhys. Rev. B 91, 214401 (2015) .\n26R. R. Birss, Symmetry and Magnetism , 1st ed., edited by E. P.\nWohlfarth, Selected Topics in Solid State Physics, V ol. 3 (N orth-\nHolland Publising Company, 1964).\n27J. Kim, J. Sinha, M. Hayashi, M. Yamanouchi, S. Fukami,\nT. Suzuki, S. Mitani, and H. Ohno, Nat. Mater. 12, 240 (2012) .\n28K.-S. Ryu, S.-H. Yang, L. Thomas, and S. S. P. Parkin,\nNat. Commun. 5, 3910 (2014) .\n29X. Fan, H. Celik, J. Wu, C. Ni, K.-J. Lee, V . O. Lorenz, and J. Q.\nXiao, Nat. Commun. 5, 3042 (2014) .\n30J. Shibata, G. Tatara, and H. Kohno,\nJ. Phys. D. Appl. Phys. 44, 384004 (2011) .\n31G. Beach, M. Tsoi, and J. Erskine,\nJ. Magn. Magn. Mater. 320, 1272 (2008) .\n32J. Linder and M. Alidoust, Phys. Rev. B 88, 064420 (2013) .\n33M. Stier, M. Creutzburg, and M. Thorwart,\nPhys. Rev. B 90, 014433 (2014) .\n34O. Boulle, L. D. Buda-Prejbeanu, E. Jué, I. M. Miron, and\nG. Gaudin, J. Appl. Phys. 115, 17D502 (2014) .\n35P.-B. He, H. Yan, M.-Q. Cai, and Z.-D. Li,\nEurophys. Lett. 114, 67001 (2016) .\n36A. V . Khvalkovskiy, V . Cros, D. Apalkov, V . Nikitin, M. Kroun bi,\nK. A. Zvezdin, A. Anane, J. Grollier, and A. Fert,\nPhys. Rev. B 87, 020402 (2013) .\n37P. Bruno, Phys. Rev. B 52, 411 (1995) .\n38J. R. Dormand and P. J. Prince,\nJ. Comput. Appl. Math. 6, 19 (1980) .\n39S.-G. Je, D.-H. Kim, S.-C. Yoo, B.-C. Min, K.-J. Lee, and S.-B .\nChoe, Phys. Rev. B 88, 214401 (2013) .\n40J. Shibata, G. Tatara, and H. Kohno,\nPhys. Rev. Lett. 94, 076601 (2005) .\n41Y . Nakatani, J. Shibata, G. Tatara, H. Kohno, A. Thiaville, a nd\nJ. Miltat, Phys. Rev. B 77, 014439 (2008) .\n42J. Torrejon, F. Garcia-Sanchez, T. Taniguchi, J. Sinha, S. M itani,\nJ.-V . Kim, and M. Hayashi, Phys. Rev. B 91, 214434 (2015) .\n43T. Taniguchi, S. Mitani, and M. Hayashi,\nPhys. Rev. B 92, 024428 (2015) .\n44G. Yu, P. Upadhyaya, K. L. Wong, W. Jiang, J. G. Alzate, J. Tang ,\nP. K. Amiri, and K. L. Wang, Phys. Rev. B 89, 104421 (2014) .\n45K. Garello, C. O. Avci, I. M. Miron, M. Baumgartner, A. Ghosh,\nS. Auffret, O. Boulle, G. Gaudin, and P. Gambardella,\nAppl. Phys. Lett. 105, 212402 (2014) .\n46J.-C. Rojas-Sánchez, P. Laczkowski, J. Sampaio, S. Collin,\nK. Bouzehouane, N. Reyren, H. Jaffrès, A. Mougin, and J.-M.\nGeorge, Appl. Phys. Lett. 108, 082406 (2016) ."    },    {        "title": "1807.11806v1.Spin_absorption_at_ferromagnetic_metal_platinum_oxide_interface.pdf",        "content": "arXiv:1807.11806v1  [cond-mat.mes-hall]  31 Jul 2018Spin absorption at ferromagnetic-metal/platinum-oxide i nterface\nAkio Asami,1Hongyu An,1Akira Musha,1Makoto Kuroda,1and Kazuya Ando1,∗\n1Department of Applied Physics and Physico-Informatics, Ke io University, Yokohama 223-8522, Japan\n(Dated: August 1, 2018)\nWe investigate the absorption of a spin current at a ferromag netic-metal/Pt-oxide interface by\nmeasuring current-induced ferromagnetic resonance. The s pin absorption was characterized by the\nmagnetic damping of the heterostructure. We show that the ma gnetic damping of a Ni 81Fe19film\nis clearly enhanced by attaching Pt-oxide on the Ni 81Fe19film. The damping enhancement is\ndisappeared by inserting an ultrathin Cu layer between the N i81Fe19and Pt-oxide layers. These\nresults demonstrate an essential role of the direct contact between the Ni 81Fe19and Pt-oxide to\ninduce sizable interface spin-orbit coupling. Furthermor e, the spin-absorption parameter of the\nNi81Fe19/Pt-oxide interface is comparable to that of intensively st udied heterostructures with strong\nspin-orbit coupling, such as an oxide interface, topologic al insulators, metallic junctions with Rashba\nspin-orbit coupling. This result illustrates strong spin- orbit coupling at the ferromagnetic-metal/Pt-\noxide interface, providing an important piece of informati on for quantitative understanding the spin\nabsorption and spin-charge conversion at the ferromagneti c-metal/metallic-oxide interface.\nI. INTRODUCTION\nAn emerging direction in spintronics aims at discover-\ning novel phenomena and functionalities originatingfrom\nspin-orbit coupling (SOC)1. An important aspect of the\nSOC is the ability to convert between charge and spin\ncurrents. The charge-spin conversion results in the gen-\neration of spin-orbit torques in heterostructures with a\nferromagnetic layer, enabling manipulation of magneti-\nzation2–4. Recent studies have revealed that the oxida-\ntion of the heterostructure strongly influences the gen-\neration of the spin-orbit torques. The oxidation of the\nferromagnetic layer alters the spin-orbit torques, which\ncannot be attributed to the bulk spin Hall mechanism5–7.\nThe oxidation of a nonmagnetic layer in the heterostruc-\nture also offers a route to engineer the spin-orbit devices.\nDemasius et al. reported a significant enhancement of\nthe spin-torque generation by incorporating oxygen into\ntungsten, which is attributed to the interfacial effect8.\nThe spin-torque generation efficiency was found to be\nsignificantly enhanced by manipulating the oxidation of\nCu, enablingto turn the light metal into anefficient spin-\ntorque generator, comparable to Pt9. We also reported\nthat the oxidation of Pt turns the heavy metal into an\nelectrically insulating generatorof the spin-orbit torques,\nwhich enables the electrical switching of perpendicular\nmagnetization in a ferrimagnet sandwiched by insulating\noxides10. These studies have provided valuable insights\ninto the oxide spin-orbitronics and shown a promising\nway to develop energy-efficient spintronics devices based\non metal oxides.\nThe SOC in solids is responsible for the relaxation of\nspins, as well as the conversion between charge and spin\ncurrents. The spin relaxation due to the bulk SOC of\nmetals and semiconductors has been studied both ex-\nperimentally and theoretically11–14. The influence of the\nSOC at interfaces on spin-dependent transport has also\nbeen recognized in the study of giant magnetoresistance\n(GMR). The GMR in Cu/Pt multilayers in the current-perpendicular-to-plane geometry indicated that there\nmust be a significant spin-memory loss due to the SOC\nat the Cu/Pt interfaces15. The interface SOC also plays\na crucial role in recent experiments on spin pumping.\nThe spin pumping refers to the phenomenon in which\nprecessing magnetization emits a spin current to the sur-\nrounding nonmagnetic layers12. When the pumped spin\ncurrent is absorbed in the nonmagnetic layer due to the\nbulk SOC or the ferromagnetic/nonmagnetic interface\ndue to the interface SOC, the magnetization damping\nof the ferromagnetic layer is enhanced because the spin-\ncurrent absorption deprives the magnetization of the an-\ngularmomentum16. Althoughthedampingenhancement\ninduced by the spin pumping has been mainly associated\nwith the spin absorption in the bulk of the nonmagnetic\nlayer, recent experimental and theoretical studies have\ndemonstrated that the spin-current absorption at inter-\nfaces also provides a dominant contribution to the damp-\ning enhancement17. Since the absorption of a spin cur-\nrent at interfaces originates from the SOC, quantifying\nthe damping enhancement provides an important infor-\nmation for fundamental understanding of the spin-orbit\nphysics.\nIn this work, we investigate the absorption of a spin\ncurrent at a ferromagnetic-metal/Pt-oxide interface. We\nshow that the magnetic damping of a Ni 81Fe19(Py) film\nis clearly enhanced by attaching Pt-oxide, Pt(O), despite\nthe absence of the absorption of the spin current in the\nbulk of the Pt(O) layer. The damping enhancement dis-\nappears by inserting an ultrathin Cu layer between the\nPyand Pt(O)layers. This resultindicates that the direct\ncontact between the ferromagnetic metal and Pt oxide is\nessential to induce the sizable spin-current absorption, or\nthe interface SOC. We further show that the strength of\nthe damping enhancement observedfor the Py/Pt(O) bi-\nlayer is comparable with that reported for other systems\nwith strong SOC, such as two-dimensional electron gas\n(2DEG) at an oxide interface and topological insulators.2\nII. EXPERIMENTAL METHODS\nThree sets of samples, Au/SiO 2/Py,\nAu/SiO 2/Py/Pt(O) and Au/SiO 2/Py/Cu/Pt(O),\nwere deposited on thermally oxidized Si substrates\n(SiO2) by RF magnetron sputtering at room tempera-\nture. To avoid the oxidation of the Py or Cu layer, we\nfirst deposited the Pt(O) layer on the SiO 2substrate in\na mixed argon and oxygen atmosphere. After the Pt(O)\ndeposition, the chamber was evacuated to 1 ×10−6Pa,\nand then the Py or Cu layer was deposited on the top\nof the Pt(O) layer in a pure argon atmosphere. For\nthe Pt(O) deposition, the amount of oxygen gas in the\nmixture was fixed as 30%, in which the flow rates of\nargon and oxygen were set as 7.0 and 3.0 standard cubic\ncentimeters per minute (sccm), respectively. The SiO 2\nlayer was deposited from a SiO 2target in the pure argon\natmosphere. The film thickness was controlled by the\ndeposition time with a precalibrated deposition rate.\nWe measured the magnetic damping using current-\ninduced ferromagnetic resonance (FMR). For the fabri-\ncation of the devices used in the FMR experiment, the\nphotolithography and lift-off technique were used to pat-\nternthefilmsintoa10 µm×40µmrectangularshape. A\nblanket Pt(O) film on a 1 cm ×1 cm SiO 2substrate was\nfabricatedforthecompositionconfirmationbyx-raypho-\ntoelectron spectroscopy (XPS). We also fabricated Pt(O)\nsingle layer and SiO 2/Py/Pt(O) multilayer films with a\nHall bar shape to determine the resistivity of the Pt(O)\nand Py using the four-probe method. The resistivity of\nPt(O) (6.3 ×106µΩ cm) is much larger than that of Py\n(106µΩ cm). Because of the semi-insulating nature of\nthe Pt(O) layer, we neglect the injection of a spin cur-\nrent into the Pt(O) layer from the Py layer; only the\nPy/Pt(O) interface can absorb a spin current emitted\nfrom the Py layer. Transmission electron microscopy\n(TEM) was used to directly observe the interface and\nmultilayer structure of the SiO 2/Py/Pt(O) film. All the\nmeasurements were conducted at room temperature.\nIII. RESULTS AND DISCUSSION\nFigure 1(a) exhibits the XPS spectrum of the Pt(O)\nfilm. Previous XPS studies on Pt(O) show that bind-\ning energies of the Pt 4 f7/2peak for Pt, PtO and PtO 2\nare around 71.3, 72.3 and 74.0 eV, respectively18. Thus,\nthe Pt 4 f7/2peak at 72.3 eV in our Pt(O) film indi-\ncates the formation of PtO. By further fitting the XPS\nspectra, we confirm that the Pt(O) film is composed of\na dominant structure of PtO with a minor portion of\nPtO2. Figure 1(b) shows the cross-sectional TEM image\nof the SiO 2(4 nm)/Py(8 nm)/Pt(O)(10 nm) film. As can\nbe seen, continuous layer morphology with smooth and\ndistinct interfaces is formed in the multilayer film. Al-\nthough we deposited the Py layer on the Pt(O) layer to\navoid the oxidation of the Py, it might still be possible\nthat the Py layer is oxidized by the Pt(O) layer. There-\nFIG. 1. (a) The XPS spectrum of the Pt(O) film. The gray\ncurve is the experimental data, and the red fittingcurve is th e\nmerged PtO and PtO 2peaks. (b) The cross-sectional TEM\nimage of the SiO 2(4 nm)/Ni 81Fe19(8 nm)/Pt(O)(10 nm) film.\nfore, we measured the resistance of the Au/SiO 2/Py and\nAu/SiO 2/Py/Pt(O) samples used in the FMR experi-\nment. The resistance of both samples show the same\nvalue (60 Ω). Furthermore, as described in the follow-\ning section, the saturation magnetization for each device\nwas obtained by using Kittel formula (0.746 T and 0.753\nT for the Au/SiO 2/Py and Au/SiO 2/Py/Pt(O), respec-\ntively). The only 1% difference indicates that the minor\noxidationofthePylayerdue tothe presenceofthe Pt(O)\nlayer can be neglected.\nNext, we conduct the FMR experiment to investigate\nthe absorption and relaxation of spin currents induced\nby the spin pumping. Figure 2(a) shows a schematic\nof the experimental setup for the current-induced FMR.\nWe applied an RF current to the device, and an in-plane\nexternal magnetic field µ0Hwas swept with an angle\nof 45ofrom the longitudinal direction. The RF charge\ncurrentflowingintheAulayergeneratesanOerstedfield,\nwhich drives magnetization precession in the Py layer\nat the FMR condition. The magnetization precession\ninduces an oscillation of the resistance of the device due\nto the anisotropic magnetoresistance (AMR) of the Py\nlayer. We measured DC voltage generated by the mixing\nof the RF current and the oscillating resistance using a\nbias tee.\nFigures 2(b), 2(c) and 2(d) show the FMR spec-\ntra for the Au/SiO 2/Py, Au/SiO 2/Py/Pt(O) and\nAu/SiO 2/Py/Cu/Pt(O) films, respectively. For the\nFMRmeasurement, asmallRFcurrentpower P= 5mW\nwas applied. Around P= 5 mW, the FMR linewidth is\nindependent of the RF power as shown in the inset to\nFig. 3(a). This confirms that the measured linewidth\nis unaffected by additional linewidth broadening due to\nnonlinear damping mechanisms and Joule heating. As\nshown in Fig. 2, clear FMR signals with low noise are\nobtained, allowing us to precisely fit the spectra and ex-\ntract the magnetization damping for the three samples.\nHere, the mixing voltage due to the FMR, Vmix, is ex-3\nFIG. 2. Schematic illustration of the experimental setup\nfor the current-induced FMR. Mis the magnetization in\nthe Py layer. The FMR spectra of the (b) Py(9 nm),\n(c) Py(9 nm)/Pt(O)(7.3 nm), and (d) Py(9 nm)/Cu(3.6\nnm)/Pt(O)(7.3 nm) films by changing the RF current fre-\nquency from 4 to 10 GHz. All the films are capped with 3\nnm-thick SiO 2and 10 nm-thick Au layers. The RF current\npower was set as 5 mW. The schematic illustrations of the\ncorresponding films are also shown.\npressed as\nVmix=Vsym(µ0∆H)2\n(µ0H−µ0HR)2+(µ0∆H)2\n+Vasyµ0∆H(µ0H−µ0HR)\n(µ0H−µ0HR)2+(µ0∆H)2,(1)\nwhereµ0∆Handµ0HRare the spectral width and res-\nonance field, respectively19.VsymandVasymare the\nmagnitudes of the symmetric and antisymmetric com-\nponents. The symmetric and antisymmetric components\narise from the spin-orbit torques and Oersted field. In\nthe devices used in the present study, the Oersted field\ncreated by the top Au layer dominates the RF effective\nfields acting on the magnetization in the Py layer [see\nalso Fig. 2(a)]. The large Oersted field enables the elec-\ntric measurement of the FMR even in the absence of the\nspin-orbit torques in the Au/SiO 2/Py film.\nThe damping constant αof the Py layer\nin the Au/SiO 2/Py, Au/SiO 2/Py/Pt(O) and\nAu/SiO 2/Py/Cu/Pt(O) films can be quantified byfitting the RF current frequency fdependence of the\nFMR spectral width µ0∆Husing\nµ0∆H=µ0∆Hext+2πα\nγf, (2)\nwhere ∆ Hextandγare the inhomogeneous linewidth\nbroadening of the extrinsic contribution and gyromag-\nnetic ratio, respectively19,20. Figure 3(a) shows the f\ndependence of the FMR linewidth µ0∆H, determined by\nfitting the spectra shown in Fig. 2 using Eq. (1). As\nshown in Fig. 3(a), the frequency dependence of the\nlinewidth is well fitted by Eq. (2). Importantly, the slope\nof thefdependence of µ0∆Hfor the Py/Pt(O) film is\nclearly larger than that for the Py and Py/Cu/Pt(O)\nfilms. This indicates larger magnetic damping in the\nPy/Pt(O) film. By using Eq. (2), we determined the\ndamping constant αas 0.0126, 0.0169 and 0.0124 for the\nPy, Py/Pt(O) and Py/Cu/Pt(O)films, respectively. The\ndifference in αbetween the Py and Py/Cu/Pt(O)films is\nvanishingly small, which is within an experimental error.\nIn contrast, the damping of the Py/Pt(O) film is clearly\nlarger than that of the other films, indicating an essen-\ntial role of the Py/Pt(O) interface on the magnetization\ndamping.\nThe larger magnetic damping in the Py/Pt(O) film\ndemonstrates an important role of the direct contact be-\ntween the Py and Pt(O) layers in the spin-current ab-\nsorption. If the bottom layers influence the magnetic\nproperties of the Py layer, the difference in the mag-\nnetic properties can also result in the different magnetic\ndamping in the Au/SiO 2/Py, Au/SiO 2/Py/Pt(O) and\nAu/SiO 2/Py/Cu/Pt(O) films. However, we have con-\nfirmed that the difference in the magnetic damping is not\ncaused by different magnetic properties of the Py layer.\nIn Fig. 3(b), we plot the RF current frequency fdepen-\ndence of the resonance field µ0HR. As can be seen, the\nfdependence of µ0HRis almost identical for the differ-\nent samples, indicating the minor change of the magnetic\npropertiesofthe Pylayerdueto the different bottomlay-\ners. In fact, by fitting the experimental data using Kittel\nformula21, 2πf/γ=/radicalbig\nµ0HR(µ0HR+µ0Ms), the satura-\ntion magnetizationis obtainedto be µ0Ms= 0.746, 0.753\nand 0.777 T for the Py, Py/Pt(O) and Py/Cu/Pt(O)\nfilms, respectively. The minor difference ( <5%) in the\nsaturation magnetization indicates that the larger damp-\ning of the Py/Pt(O) film cannot be attributed to possi-\nble different magnetic properties of the Py layer. Thus,\nthe larger magnetic damping of the Py/Pt(O) film can\nonly be attributed to the efficient absorption of the spin\ncurrent at the interface. Notable is that the additional\ndamping due to the spin-current absorption disappears\nby inserting the 3.6 nm-thick Cu layer between the Py\nand Pt(O) layers. Here, the thickness of the Cu layer is\nmuchthinnerthanitsspin-diffusionlength( ∼500nm)22,\nallowing us to neglect the relaxation of the spin current\nin the Cu layer. This indicates that the directcontactbe-\ntween the Py and Pt(O) layersis essential for the absorp-\ntion of the spin current at the interface, or the interface\nSOC.4 0∆H (mT) \nµ\n3 6 9 12 0246\nf (GHz) (a) (b)\n8\n610 \n0\n 0HR (mT) µf (GHz) \n4\n100150 50 Py/Cu/Pt(O) Py/Pt(O)Py \nPy/Cu/Pt(O) Py/Pt(O) Py \n036\n10 010 110 2\nP (mW)  0∆H (mT) \nµ\nFIG. 3. (a) The RF current frequency fdependence of\nthe half-width at half-maximum µ0∆Hfor the Py, Py/Pt(O)\nand Py/Cu/Pt(O) samples. The solid lines are the linear fit\nto the experimental data. The inset shows RF current power\nPdependence of µ0∆Hfor the Py film at f= 7 GHz. (b)\nThe RF currentfrequency fdependenceof the resonance field\nµ0HRfor the three samples. The solid curves are the fitting\nresult using the Kittel formula.\nTABLE I. The summarized spin-absorption parameter Γ 0η\nin different material systems. In order to directly compare\nthis work with previous works, we used International Sys-\ntem of Units. We used the magnetic permeability in vac-\nuumµ0= 4π×10−7H/m. ∆ αand Γ 0ηfor the Sn 0.02-\nBi1.08Sb0.9Te2S/Ni81Fe19is the values at T <100 K.\nHeterostructure ∆ αΓ0η[1/m2] Ref.\nBi/Ag/Ni 80Fe20 0.015 8.7 ×1018[25]\nBi2O3/Cu/Ni 80Fe20 0.0045 1.5 ×1018[26]\nSrTiO 3/LaAlO 3/Ni81Fe19 0.0013 2.3 ×1018[27]\nPt(O)/Ni 81Fe19 0.0044 2.3 ×1018This work\nα-Sn/Ag/Fe 0.022 1.2 ×1019[28]\nSn0.02-Bi1.08Sb0.9Te2S/Ni81Fe190.013 1.4 ×1019[29]\nBi2Se3/Ni81Fe19 0.0013 2.5 ×1018[30]\nTo quantitatively discuss the spin absorption at the\nPy/Pt(O)interfaceand comparewith othermaterialsys-\ntems, we calculate the spin absorption parameters. In a\nmodel of the spin pumping where the interface SOC is\ntaken into account, the additional damping constant is\nexpressed as23\n∆α=gµBΓ0\nµ0Msd/parenleftbigg1+6ηξ\n1+ξ+η\n2(1+ξ)2/parenrightbigg\n.(3)\nHere,g= 2.11 is the gfactor24,µB= 9.27×10−24Am2\nis the Bohr magneton, dis the thickness of the Py layer,\nandΓ0isthemixingconductanceattheinterface. ξisthe\nback flow factor; no backflow refers to ξ= 0 and ξ=∞indicates that the entire spin current pumped into the\nbulk flows back across the interface. ηis the parameter\nthat characterizes the interface SOC. For the Py/Pt(O)\nfilm,ξcan be approximated to be ∞because of the spin\npumping into the bulk of the semi-insulating Pt(O) layer\ncan be neglected. Thus, Eq. (3) can be simplified as\n∆α=6gµBΓ0η\nµ0Msd. (4)\nHere, 6Γ 0ηcorresponds to the effective spin mixing\nconductance g↑↓\neff. From the enhancement of magnetic\ndamping ∆ α, we obtain Γ 0η= 2.3×1018m−2for the\nPy/Pt(O) film. We further compared this value with\nΓ0ηfor other systems where efficient interface charge-\nspin conversion has been reported. As shown in Table\nI, the spin-absorption parameter Γ 0ηof the Py/Pt(O)\nfilm is comparable with that of the heterostructures with\nthe strong SOC, such as the 2DEG at an oxide interface,\ntopological insulators, as well as metal/oxide and metal-\nlic junctions with the Rashba SOC. This result therefore\ndemonstrates the strong SOC at the Py/Pt(O) interface.\nIV. CONCLUSIONS\nIn summary, we have investigated the spin-current ab-\nsorption and relaxation at the ferromagnetic-metal/Pt-\noxide interface. By measuring the magnetic damping for\nthe Py, Py/Pt(O)and Py/Cu/Pt(O)structures, we show\nthat the direct contact between Py and Pt(O) is essential\nfor the absorption of the spin current, or the sizable in-\nterface SOC. Furthermore, we found that the strength of\nthe spin-absorption parameter at the Py/Pt(O) interface\nis comparable to the value for intensively studied het-\nerostructureswithstrongSOC,suchas2DEGatanoxide\ninterface, topological insulators, metallic junction with\nRashba SOC. The comparable value with these material\nsystems illustrates the strong SOC at the ferromagnetic-\nmetal/Pt-oxide interface. This indicates that the oxida-\ntion of heavy metals provides a novel approach for the\ndevelopment of the energy-efficient spintronics devices\nbased the SOC.\nACKNOWLEDGMENTS\nThis work was supported by JSPS KAKENHI Grant\nNumbers 26220604, 26103004, the Asahi Glass Founda-\ntion, JGC-SScholarshipFoundation, andSpintronicsRe-\nsearch Network of Japan (Spin-RNJ). H.A. is JSPS In-\nternational Research Fellow (No. P17066) and acknowl-\nedges the support from the JSPS Fellowship (Grant No.\n17F17066).\n∗ando@appi.keio.ac.jp1A. Soumyanarayanan, N. Reyren, A. Fert, and\nC. Panagopoulos, Nature (London) 539, 509 (2016).5\n2P. Gambardella and I. M. Miron, Philos. Trans. R. Soc.\nLondon A 369, 3175 (2011).\n3X.Qiu, Z.Shi, W.Fan, S.Zhou, andH.Yang,Adv.Mater.\n30, 1705699 (2018).\n4A. Brataas, A. D. Kent, and H. Ohno, Nat. Mater. 11,\n372 (2012).\n5X. Qiu, K. Narayanapillai, Y. Wu, P. Deorani, D.-H. Yang,\nW.-S.Noh, J.-H. Park, K.-J.Lee, H.-W.Lee, andH.Yang,\nNat. Nanotechnol. 10, 333 (2015).\n6Y. Hibino, T. Hirai, K. Hasegawa, T. Koyama, and\nD. Chiba, Appl. Phys. Lett. 111, 132404 (2017).\n7T. Gao, A. Qaiumzadeh, H. An, A. Musha,\nY. Kageyama, J. Shi, and K. Ando,\nPhys. Rev. Lett. 121, 017202 (2018).\n8K.-U. Demasius, T. Phung, W. Zhang, B. P. Hughes, S.-H.\nYang, A. Kellock, W. Han, A. Pushp, and S. S. Parkin,\nNat. Commun. 7, 10644 (2016).\n9H. An, Y. Kageyama, Y. Kanno, N. Enishi, and K. Ando,\nNat. Commun. 7, 13069 (2016).\n10H. An, T. Ohno, Y. Kanno, Y. Kageyama, Y. Monnai,\nH. Maki, J. Shi, and K. Ando, Sci. Adv. 4, eaar2250\n(2018).\n11S. Mizukami, Y. Ando, and T. Miyazaki, Phys. Rev. B\n66, 104413 (2002).\n12Y. Tserkovnyak, A. Brataas, and G. E. Bauer, Phys. Rev.\nB66, 224403 (2002).\n13L. Chen, F. Matsukura, and H. Ohno, Nat. Commun. 4,\n2055 (2013).\n14H. Nakayama, K. Ando, K. Harii, T. Yoshino, R. Taka-\nhashi, Y. Kajiwara, K.-i. Uchida, Y. Fujikawa, and\nE. Saitoh, Phys. Rev. B 85, 144408 (2012).\n15H. Kurt, R. Loloee, K. Eid, W. Pratt Jr, and J. Bass,\nAppl. Phys. Lett. 81, 4787 (2002).16Y. Tserkovnyak, A. Brataas, and G. E. Bauer, Phys. Rev.\nLett.88, 117601 (2002).\n17J. R. S´ anchez, L. Vila, G. Desfonds, S. Gambarelli, J. At-\ntan´ e, J. De Teresa, C. Mag´ en, and A. Fert, Nat. Commun.\n4, 2944 (2013).\n18Q. Wang, X. Lu, Q. Xin, and G. Sun, Chin. J. of Catal.\n35, 1394 (2014).\n19D.Fang, H.Kurebayashi, J. Wunderlich,K.V` yborn` y,L.P.\nZˆ arbo, R. Campion, A. Casiraghi, B. Gallagher, T. Jung-\nwirth, and A. Ferguson, Nat. Nanotechnol. 6, 413 (2011).\n20B. Heinrich, J. Cochran, and R. Hasegawa, J. Appl. Phys.\n57, 3690 (1985).\n21C. Kittel, Physical Review 73, 155 (1948).\n22H. Wang, C. Du, Y. Pu, R. Adur, P. C. Hammel, and\nF. Yang, Phys. Rev. Lett. 112, 197201 (2014).\n23K. Chen and S. Zhang, Phys. Rev. Lett. 114, 126602\n(2015).\n24J. M. Shaw, H. T. Nembach, T. J. Silva, and C. T. Boone,\nJ. Appl. Phys. 114, 243906 (2013).\n25M. Jungfleisch, W. Zhang, J. Sklenar, W. Jiang, J. Pear-\nson, J. Ketterson, and A. Hoffmann, Phys. Rev. B 93,\n224419 (2016).\n26S. Karube, K. Kondou, and Y. Otani, Appl. Phys. Express\n9, 033001 (2016).\n27E. Lesne, Y. Fu, S. Oyarzun, J. Rojas-S´ anchez, D. Vaz,\nH. Naganuma, G. Sicoli, J.-P. Attan´ e, M. Jamet,\nE. Jacquet, et al., Nat. Mater. 15, 1261 (2016).\n28J.-C. Rojas-S´ anchez, S. Oyarz´ un, Y. Fu, A. Marty,\nC. Vergnaud, S. Gambarelli, L. Vila, M. Jamet, Y. Oht-\nsubo, A. Taleb-Ibrahimi, et al., Phys. Rev. Lett. 116,\n096602 (2016).\n29A. Nomura, N. Nasaka, T. Tashiro, T. Sasagawa, and\nK. Ando, Phys. Rev. B 96, 214440 (2017).\n30P. Deorani, J. Son, K. Banerjee, N. Koirala, M. Brahlek,\nS. Oh, and H. Yang, Phys. Rev. B 90, 094403 (2014)."    },    {        "title": "0904.1185v2.Curvature_enhanced_spin_orbit_coupling_in_a_carbon_nanotube.pdf",        "content": "arXiv:0904.1185v2  [cond-mat.mes-hall]  3 Aug 2009Curvature-enhanced spin-orbit coupling in a carbon nanotu be\nJae-Seung Jeong and Hyun-Woo Lee\nPCTP and Department of Physics, Pohang University of Scienc e and Technology, Pohang, 790-784, Korea\n(Dated: November 1, 2018)\nStructure of the spin-orbit coupling varies from material t o material and thus finding the correct\nspin-orbit coupling structure is an important step towards advanced spintronic applications. We\nshow theoretically that the curvature in a carbon nanotube g enerates two types of the spin-orbit\ncoupling, one of which was not recognized before. In additio n to the topological phase-related\ncontribution of the spin-orbit coupling, which appears in t he off-diagonal part of the effective Dirac\nHamiltonian of carbon nanotubes, there is another contribu tion that appears in the diagonal part.\nThe existence of thediagonal term can modify spin-orbitcou pling effects qualitatively, an example of\nwhichistheelectron-hole asymmetricspinsplittingobser vedrecently, andgeneratefourqualitatively\ndifferent behavior of energy level dependence on parallel ma gnetic field. It is demonstrated that the\ndiagonal term applies to a curved graphene as well. This resu lt should be valuable for spintronic\napplications of graphitic materials.\nPACS numbers:\nI. INTRODUCTION\nGraphitic materials such as carbon nanotubes (CNTs)\nand graphenes are promising materials for spintronic ap-\nplications. Various types of spintronic devices are re-\nported such as CNT-based three terminal magnetic tun-\nnel junctions1, spin diodes2, and graphene-based spin\nvalves3. Graphitic materials are believed to be excellent\nspin conductors4. The hyperfine interaction of electron\nspins with nuclear spins is strongly suppressed since12C\natoms do not carry nuclear spins. It is estimated that\nthe spin relaxation time in a CNT5and a graphene6is\nlimited by the spin-orbit coupling (SOC).\nCarbon atoms are subject to the atomic SOC Hamil-\ntonianHso. In an ideal flat graphene, the energy shift\ncaused byHsois predicted to be ∼10−3meV7,8. Re-\ncently it is predicted8,9that the geometric curvature\ncan enhance the effective strength of the SOC by orders\nof magnitude. This mechanism applies to a CNT and\nalso to a graphene which, in many experimental situa-\ntions, exhibits nanometer-scale corrugations10. There is\nalso a suggestion?that artificial curved structures of a\ngraphene may facilitate device applications.\nA recent experiment12on ultra-clean CNTs measured\ndirectly the energy shifts caused by the SOC, which\nprovides an ideal opportunity to test theories of the\ncurvature-enhanced SOC in graphitic materials. The\nmeasured shifts are in order-of-magnitude agreement\nwith the theoretical predictions8,9, confirming that the\ncurvature indeed enhances the effective SOC strength.\nThe experiment revealed discrepancies as well; While ex-\nisting theories predict the same strength of the SOC for\nelectrons and holes, which is natural considering that\nboth the conduction and valence bands originate from\nthe sameπorbital, the experiment found considerable\nasymmetry in the SOC strength between electrons and\nholes. Thiselectron-holeasymmetryimpliesthatexisting\ntheoriesoftheSOCingraphiticmaterialsareincomplete.\nIn this paper, we show theoretically that in additionto effective SOC in the off-diagonal part of the effective\nDirac Hamiltonian, which was reported in the existing\ntheories8,9, there exists an additional type of the SOC\nthat appears in the diagonal part both in CNTs and\ncurved graphenes. It is demonstrated that the combined\naction ofthe twotypes ofthe SOC producesthe electron-\nhole asymmetry observed in the CNT experiment12and\ngives rise to four qualitatively different behavior of en-\nergy level dependence on magnetic field parallel to the\nCNT axis.\nThis paper is organized as follows. In Sec. II, we show\nanalytical expressions of two types of the effective SOC\nin a CNT and then explain how the electron-hole asym-\nmetric spin splitting can be generated in semiconducting\nCNTs generically. Section III describes the second-order\nperturbation theory that is used to calculate the effective\nSOC, and tight-binding models of the atomic SOC and\ngeometric curvature. Section IV reports four distinct en-\nergy level dependence on magnetic field parallel to the\nCNT axis. We conclude in Sec. V with implications of\nour theory on curved graphenes and a brief summary.\nII. EFFECTIVE SPIN-ORBIT COUPLING IN A\nCNT\nWe begin our discussion by presenting the first main\nresult for a CNT with the radius Rand the chiral angle\nθ(0≤θ≤π/6, 0(π/6) for zigzag (armchair) CNTs). We\nfind that when the two sublattices AandBof the CNT\nare used as bases, the curvature-enhanced effective SOC\nHamiltonian Hsocnear the K point with Bloch momen-\ntumKbecomes\nHK\nsoc=/parenleftbigg\n(δ′\nK/R)σy(δK/R)σy\n(δ∗\nK/R)σy(δ′\nK/R)σy/parenrightbigg\n, (1)\nwhereσyrepresents the real spin Pauli matrix along the\nCNT axis. The pseudospin is defined to be up (down)\nwhen an electron is in the sublattice A(B). Here the off-\ndiagonal term that can be described by a spin-dependent2\ntopological phase are reported in Refs.8,9but the diago-\nnal term was not recognized before. Expressions for the\nparameters δKandδ′\nKare given by13\nδK\nR=λsoa(εs−εp)(Vπ\npp+Vσ\npp)\n12√\n3Vσsp2e−iθ\nR(2)\nand\nδ′\nK\nR=λsoaVπ\npp\n2√\n3(Vσpp−Vπpp)cos3θ\nR, (3)\nwhereλso∼12meV14is the atomic SOC constant, ais\nthe lattice constant 2 .49˚A, andεs(p)is the atomic energy\nfor thes(p) orbital. Here, Vσ\nspandVπ(σ)\npprepresent the\ncoupling strengths in the absence of the curvature for the\nσcoupling between nearest neighbor sandporbitals and\ntheπ(σ) coupling between nearest neighbor porbitals,\nrespectively. Note that the |δ′\nK|has theθ-dependence,\nwhose implication on the CNT energy spectrum is ad-\ndressed in Sec. IV. For K′point with K′=−K,HK′\nsocis\ngiven by Eq. (1) with δKandδ′\nKreplaced by δK′=−δ∗\nK\nandδ′\nK′=−δ′\nK, respectively.\nImplications of the diagonal term of the SOC be-\ncome evident when Eq. (1) is combined with the two-\ndimensional Dirac Hamiltonian HDiracof the CNT. For\nastate nearthe Kpoint with the Blochmomentum K+k\n[k= (kx,ky),|k| ≪ |K|],HDiracbecomes15\nHK\nDirac=/planckover2pi1vF/parenleftbigg\n0e−iθ(kx−iky)\ne+iθ(kx+iky) 0/parenrightbigg\n,(4)\nwherevFis the Fermi velocity and the momentum com-\nponentkxalong the circumference direction satisfies the\nquantization condition kx= (1/3R)νfor a (n,m) CNT\nwithn−m= 3q+ν(q∈Zandν=±1,0) andθ=\ntan−1[√\n3m/(2n+m)]. For a semiconducting ( ν=±1)\nE E\nky ky ky0 0 0E (a) (b) (c)\n2δ′\nK/R−2νRe[δKeiθ]/R−2δ′\nK/R−2νRe[δKeiθ]/R −2νRe[δKeiθ]/R\n−2νRe[δKeiθ]/R\nFIG. 1: (Color online) Schematic diagram of the lowest con-\nduction (red, E >0) and highest valence (blue, E <0) band\npositions of asemiconducting CNT predictedby HK\nDirac+HK\nsoc\nfor (a)δK=δ′\nK=0, (b)δK/negationslash=0,δ′\nK=0, and (c) δK/negationslash=0,δ′\nK/negationslash=0.\nIn (c), the conduction or valence band has larger spin split-\nting depending on the sign of ν. Arrows (green) show the\nspin direction along the CNT. The expressions for the energy\nlevel spacing are also provided. When they are negative, the\npositions of the two spin-split bands should be swapped.CNT, thediagonalizationof HK\nDirac+HK\nsocresultsindiffer-\nent spin splittings [Fig. 1(c)] of −2δ′\nK/R−2νRe[δKeiθ]/R\nand 2δ′\nK/R−2νRe[δKeiθ]/Rfor the conduction and va-\nlence bands, respectively. This explains the electron-hole\nasymmetryobservedin the recent experiment12. Here we\nremark that neither the off-diagonal ( δK) nor the diago-\nnal(δ′\nK) termofthe SOCalonecangeneratetheelectron-\nhole asymmetrysince the twospin splittings can differ by\nsign at best, which actually implies the same magnitude\nof the spin splitting (see Fig. 1 for the sign convention).\nThus the interplay of the two types is crucial for the\nasymmetry.\nIII. THEORY AND MODEL\nWe calculate the δKandδ′\nKanalytically using degener-\natesecond-orderperturbationtheoryandtreatingatomic\nSOC and geometric curvature as perturbation. For sim-\nplicity, we evaluate δKandδ′\nKin the limit k= 0. Al-\nthough this limit is not strictly valid since k=0 does not\ngenerally satisfy the quantization condition on kx, one\nmay still take this limit since the dependence of δKand\nδ′\nKonkis weak. An electron at the K point is described\nby the total Hamiltonian HK,(0)+Hso+Hc, whereHc\ndescribes the curvature effects and HK,(0)describes the\nπandσbands in the absence of both HsoandHc. The\nπband eigenstates of HK,(0)are given by\n|ΨK,(0)\n↑(↓)/angbracketright=1√\n2/parenleftbig\nνe−iθ/vextendsingle/vextendsingleψK\nA/angbracketrightbig\n±/vextendsingle/vextendsingleψK\nB/angbracketrightbig/parenrightbig\nχ↑(↓)(5)\nwith the corresponding eigenvalues EK,(0)\n↑(↓)≡E(0)= 0.\nHere|ΨK,(0)\n↑(↓)/angbracketrightwith the upper (lower) sign amounts to the\nk= 0 limit of the eigenstate at the the conduction band\nbottom (valenceband top). χ↑(↓)denotes the eigenspinor\nofσy.|ψK\nA(B)/angbracketright=1√\nN/summationtext\nr=rA(B)eiK·r|pr\nz/angbracketrightis the orbital\nprojection of |ΨK,(0)\n↑(↓)/angbracketrightinto the sublattice A(B),|pr\nz/angbracketrightrep-\nresents the pzorbital at the atomic position r, and thez\naxis is perpendicular to the CNT surface.\nWhenHsoandHcare treated as weak perturbations,\nthe first order contribution Hsoto the effective SOC van-\nishes since it causes the inter-band transition (Fig. 3)\nto theσband8. The next leading order contribution to\nthe effective SOC comes from the following second order\nperturbation Hamiltonian HK,(2)16,\nHK,(2)=HcPK\nE(0)−HK,(0)Hso+H.c.,(6)\nwhere the projection operator PKis defined by PK≡1−/summationtext\nα=↑,↓|ΨK,(0)\nα/angbracketright/angbracketleftΨK,(0)\nα|. Another spin-dependent second\norder term Hso[PK/(E(0)−HK,(0))]Hso17is smaller than\nEq. (6) (by two orders of magnitude for a CNT with\nR∼2.5nm), and thus ignored. Then the second order3\nenergy shift EK,(2)\n↑(↓)is given by18\nEK,(2)\n↑=/angbracketleftbig\nψK\nA/vextendsingle/vextendsingleHK,(2)/vextendsingle/vextendsingleψK\nA/angbracketrightbig\n±νRe/bracketleftBig\n/angbracketleftψK\nA|HK,(2)|ψK\nB/angbracketrighteiθ/bracketrightBig\nEK,(2)\n↓=−EK,(2)\n↑, (7)\nwhere the upper (lower) sign applies to the energy shift\nof the conduction band bottom (valence band top) and\n/angbracketleftψK\nA|HK,(2)|ψK\nA/angbracketright=/angbracketleftψK\nB|HK,(2)|ψK\nB/angbracketrightis used. Then by com-\nparingEK,(2)\n↑(↓)with Fig. 1, one finds\nδK\nR=/angbracketleftbig\nψK\nA/vextendsingle/vextendsingleHK,(2)/vextendsingle/vextendsingleψK\nB/angbracketrightbig\n,δ′\nK\nR=/angbracketleftbig\nψK\nA/vextendsingle/vextendsingleHK,(2)/vextendsingle/vextendsingleψK\nA/angbracketrightbig\n.(8)\nNote thatδKandδ′\nKare related to pseudospin-flipping\nand pseudospin-conserving processes, respectively.\nTo evaluate Eq. (8), one needs explicit expressions for\nHso,Hc, andHK,(0).Hsois given by λso/summationtext\nrLr·Sr7,\nwhereLrandSrare respectively the atomic orbital and\nspin angular momentum of an electron at a carbon atom\nr. The tight-binding Hamiltonian of the Hsocan be\nwritten8asHso= (λso/2)/summationtext\nr=rA/B(cz†\nr−cx\nr+−cz†\nr+cx\nr−+\nicz†\nr+cy\nr−+icz†\nr−cy\nr++icy†\nr+cx\nr+−icy†\nr−cx\nr−) + H.c., where\ncx\nr+(−),cy\nr+(−), andcz\nr+(−)denote the annihilation opera-\ntors for|pr\nx/angbracketrightχ+(−),|pr\ny/angbracketrightχ+(−), and|pr\nz/angbracketrightχ+(−). Hereχ+(−)\ndenotes the eigenspinor of σz(+/−for outward/inward).\nFor later convenience, we express χ+(−)in term ofχ↑(↓)\nto obtain a expression for Hso,\nHso=λso\n2/summationdisplay\nr=rA/B/bracketleftBig\ni/parenleftBig\ncz†\nr↓cx\nr↓−cz†\nr↑cx\nr↑/parenrightBig\n+/parenleftBig\ne−iϕcz†\nr↑cy\nr↓−eiϕcz†\nr↓cy\nr↑/parenrightBig\n+i/parenleftBig\ne−iϕcy†\nr↑cx\nr↓+eiϕcy†\nr↓cx\nr↑/parenrightBig/bracketrightBig\n+H.c..(9)\nFor the curvature Hamiltonian Hc, we retain only the\nleading order term in the expansion in terms of a/R. Up\ny\nxy′\nω3 ω2ω1 a\nB3AB1\nB2a2\na1 θϕzxz′\nx\nx′\nFIG. 2: (Color online) Two-dimensional honeycomb lattice\nstructure. x(y) is the coordinate around (along) the CNT\nwith chiral vector na1+ma2≡(n,m) and chiral angle θ.\nωj(j=1,2,3), the length between yaxis passing Aatom and\nits parallel (red dashed) line is related with ξjbyξj≈ωj/(2R)\n[Eq. (10)]. The coordinates for the CNT is illustrated on the\nright. Here, x=ϕR.to the first order in a/R,Hcreduces toHπσ\nc,\nHπσ\nc=/summationdisplay\nrA3/summationdisplay\nj=1/summationdisplay\nα=↑,↓/bracketleftBig\nSj/parenleftBig\ncz†\nrAαcs\nBjα+cs†\nrAαcz\nBjα/parenrightBig\n+Xj/parenleftBig\ncz†\nrAαcx\nBjα−cx†\nrAαcz\nBjα/parenrightBig\n(10)\n+Yj/parenleftBig\ncz†\nrAαcy\nBjα−cy†\nrAαcz\nBjα/parenrightBig/bracketrightBig\n+H.c.,\nwhererAis a lattice site in the sublattice Aand its\nthree nearest neighbor sites in the sublattice Bare rep-\nresented by Bj(j=1,2,3) (Fig. 2). Here Sj,Xj,Yjare\nproportional to a/Rand denote the curvature-induced\ncoupling strengths of s,px,pyorbitals with a nearest\nneighborpzorbital. Their precise expressions that can\nbe determined purely from geometric considerations, are\ngiven bySj=ξj˜Sj,Xj=ξj˜Xj, andYj=ξj˜Yjwith\nξ1≈a/(2√\n3R)sinθ,ξ2≈a/(2√\n3R)sin(π/3−θ), and\nξ3≈a/(2√\n3R)sin(π/3+θ)(Fig. 2). Here\n˜S1=Vσ\nspsinθ,\n˜S2=Vσ\nspcos/parenleftBigπ\n6+θ/parenrightBig\n,\n˜S3=Vσ\nspcos/parenleftBigπ\n6−θ/parenrightBig\n,\n˜X1=−Vσ\nppsin2θ−Vπ\npp−Vπ\nppcos2θ,\n˜X2=−Vσ\nppsin2/parenleftBigπ\n3−θ/parenrightBig\n−Vπ\npp−Vπ\nppcos2/parenleftBigπ\n3−θ/parenrightBig\n,\n˜X3=Vσ\nppsin2/parenleftBigπ\n6−θ/parenrightBig\n+Vπ\npp+Vπ\nppcos2/parenleftBigπ\n6−θ/parenrightBig\n,\n˜Y1= sin(2θ)Vπ\npp−Vσ\npp\n2,\n˜Y2= sin/parenleftbigg\n2θ−2π\n3/parenrightbiggVπ\npp−Vσ\npp\n2,\n˜Y3= sin/parenleftBig\n2θ−π\n3/parenrightBigVπ\npp−Vσ\npp\n2. (11)\nLastly, for the factor HK,(0), we use the Slater-Koster\nparametrization19for nearest-neighbor hopping. In σ\nband calculation, s,px, andpyorbitals are used as basis.\nCombined effects of the three factors Hso,PK/(E(0)−\nHK,(0)),Hcare illustrated in Fig. 3. The real spin de-\npendence arises solely from Hso, which generates the\nfactorσy20. For the pseudospin, the combined effect\nofHsoandHcis to flip the pseudospin. When they\nare combined with the pseudospin conserving part of\nPK/(E(0)−HK,(0)), one obtains the pseudospin flip-\nping process [Eq. (8)] determining δK. In addition,\nPK/(E(0)−HK,(0)) contains the pseudospin flipping part,\nwhich is natural since states localized in one particular\nsublattice are not eigenstates of HK,(0). When the pseu-\ndospin flipping part of PK/(E(0)−HK,(0)) is combined\nwithHsoandHc, one obtains the pseudospin conserving\nprocess [Eq. (8)] determining δ′\nK.\nThe signs of δKeiθandδ′\nK/cos3θare negative. We\nfind|(δ′\nK/cos3θ)/δK|= 4.5 for tight-binding parame-\nters in Ref.21. Thusδ′\nKis of the same order as δK22,4\nHsoHπσ\nc\nHso\nπbandHπσ\ncσbandA\nB ABPK\nE(0)−HK,(0)PK\nE(0)−HK,(0)PK\nE(0)−HK,(0)\nFIG. 3: (Color online) Schematic diagram ofthe second order\ntransition process generated by HK,(2)[Eq. (6)]. Pseudospin\ntransitions (between the sublattices AandB) and interband\ntransitions (between πandσbands) are illustrated.\nwhich is understandable since pseudospin flipping terms\ninE(0)−HK,(0)(with amplitudes Vσ\npp,Vσ\nsp) are compara-\nble in magnitude to pseudospin conserving terms (with\namplitudes E(0)−εs(p)).\nIV. BEHAVIOR IN A MAGNETIC FIELD\nNext we examine further implications of our result in\nview of the experiment12, where the conduction band\nbottom and valence band top positions of semiconduct-\ning CNTs ( ν=±1) are measured as a function of the\nmagnetic field Bparallel to the CNT axis. We find that\ntheθdependence [Eq. (3)] of δ′\nKhas interesting impli-\ncations. When cos3 θis sufficiently close to 0 (close to\narmchair-type), |δ′\nK|is smaller than |δKeiθ|. The predic-\ntion of our theory in this situation is shown in Figs. 4(a)\nand (b). Note that the spin splitting of both the conduc-\ntion and valence bands becomes smaller as the energy\nEincreases. On the other hand, when cos3 θis suffi-\nciently close to 1 (close to zigzag-type), |δ′\nK|is larger\nthan|δKeiθ|. In this situation [Figs. 4(c) and (d)], the\nenergy dependence of either valence or conduction band\nis inverted; For ν=+1(−1), the spin splitting of the va-\nlence (conduction) band becomes largerasthe the energy\nincreases.\nCombined with the electron-prevailing [Figs. 4(a) and\n(c) forν=+1] vs. hole-prevailing [Figs. 4(b) and (d) for\nν=−1] asymmetries in the zero-field splitting, one then\nfinds that there exist four distinct patterns of Evs.B\ndiagram, which is the second main result of this paper.\nAmong these 4 patterns, only the pattern in Fig. 4(a) is\nobserved in the experiment12, which measured two CNT\nsamples. We propose further experiments to test the ex-\nistence of the other three patterns.\nHere we remark that although Eqs. (1), (2), (3) are\ndemonstrated so far for semiconducting CNTs, they hold\nfor metallic CNTs ( ν=0) as well. For armchair CNTs\nwith cos3θ=0,δ′\nKbecomes zero and the spin splitting isdetermined purely by δK. For metallic but non-armchair\nCNTs, finding implications of Eq. (1) is somewhat tech-\nnical since the curvature-induced minigap appears near\nthe Fermi level23. Our calculation for (37 ,34)(cos3θ≈0)\nand (60,0)(cos3θ=1) CNTs including the minigap effect\nindicates that they show behaviors similar to Fig. 4(b)\nand (d), respectively. Thus nominally metallic CNTs ex-\nhibit spin splitting patterns of ν=−1 CNTs.\n-0.0810-0.08050.08050.0810\n-0.5 0.0 0.5E ( eV )\nB ( T ) (a)\n∆so\n∆so= 0.09 meV\n= 0.03 meVν = +1\nK\nK′\nK\nK′\n-0.5 0.0 0.5-0.0820-0.08150.08150.0820\nE ( eV )\nB ( T ) (b)\n∆so\n∆so= 0.03 meV\n= 0.09 meVν = −1\n-0.083-0.0820.0820.083\n-1.0 0.0 1.0E ( eV )\nB ( T ) (c)\n∆so\n∆so= 0.34 meV\n= 0.22 meVν = +1\nK\nK′\nK\nK′\n-1.0 0.0 1.0-0.082-0.0810.0810.082\nE ( eV )\nB ( T ) (d)\n∆so\n∆so= 0.21 meV\n= 0.33 meVν = −1\nFIG. 4: (Color online) Calculated energy spectrum of the\nconduction band bottom (red, E >0) and valence band top\n(blue,E <0) near K (solid lines) and K′(dashed lines) points\nin semiconducting CNTs with R≈2.5nm as a function of\nmagnetic field Bparallel to the CNT axis. The chiral vectors\nfor each CNT are (a) (38,34), (b) (39,34), (c) (61,0), and (d)\n(62,0), respectively. Arrows (green) show spin direction a long\ntheCNTaxis and∆ sodenotesthezero-field splitting. Assum-\ningky=0, the energy Eincluding the SOC, the Aharonov-\nBohm flux15φAB=BπR2, and the Zeeman coupling effects is,\nE=±/planckover2pi1vFp\n(kx+(1/R)(φAB/φ0))2+EK(K′),(2)\n↑(↓)+(g/2)µBτ/bardblB,\nwith upper (lower) sign applying to the conduction (valence )\nband.φ0=hc/|e|,τ/bardbl=+1(−1) forχ↑(↓),vF=−aVπ\npp√\n3/2,\nandg= 212. For estimation of EK(K′),(2)\n↑(↓), we use tight-\nbinding parameters in Ref.21;Vσ\nss=−4.76eV,Vσ\nsp=4.33eV,\nVσ\npp=4.37eV,Vπ\npp=−2.77eV,εs=−6.0eV, andεp=022.5\nV. DISCUSSION AND SUMMARY\nLastly we discuss briefly the effective SOC in a curved\ngraphene10. Unlike CNTs, there can be both convex-\nshaped and concave-shaped curvatures in a graphene.\nWe first address the convex-shaped curvatures. When\nthe local structure of a curved graphene has two princi-\npal curvatures, 1 /R1and 1/R2with the corresponding\nbinormal unit vectors n1andn2, each principal curva-\nture 1/Ri(i=1,2) generates the effective SOC, Eq. (1),\nwithσyreplaced by σ·niandRbyRi. The correspond-\ningδiandδ′\nivalues are given by Eqs. (2) and (3) with θ\nreplaced by θi, whereθiis the chiral angle with respect\ntoni. Thus the diagonal term of the effective SOC is\nagain comparable in magnitude to the off-diagonal term.\nFor the concave-shaped curvatures, we find that the two\ntypes of the SOC become −δiand−δ′\niwithθi, respec-\ntively. We expect that this result may be relevant for the\nestimation of the spin relaxation length in graphenes6\nand may provide insights into unexplained experimen-\ntal data in graphene-based spintronic systems24. We\nalso remark that the effective SOC in a graphene may\nbe spatially inhomogeneous since the local curvature of\nthe nanometer-scale corrugations10is not homogeneous,\nwhose implications go beyond the scope of this paper.\nIn summary, we have demonstrated that the interplayofthe atomic SOC and the curvature generatestwotypes\nof the effective SOC in a CNT, one of which was not\nrecognized before. Combined effects of the two types of\nthe SOC in CNTs explain recently observedelectron-hole\nasymmetric spin splitting12and generates four qualita-\ntively different types of energy level dependence on the\nparallel magnetic field. Our result may have interesting\nimplications for graphenes as well.\nNote added.– While we werepreparingour manuscript,\nwe became aware of a related paper25. However the ef-\nfective Hamiltonian [Eq. (1)] for the SOC and the four\ndistinct types of the magnetic field dependence (Fig. 4)\nare not reported in the work.\nAcknowledgments\nWe appreciate Philp Kim for his comment for the\ncurved graphenes. We acknowledge the hospitality of\nHyunsoo Yang and Young Jun Shin at National Uni-\nversity of Singapore, where parts of this work were per-\nformed. We thank Seung-Hoon Jhi, Woojoo Sim, Seon-\nMyeong Choi and Dong-Keun Ki for helpful conversa-\ntions. This work was supported by the KOSEF (Ba-\nsic Research Program No. R01-2007-000-20281-0) and\nBK21.\n1S. Sahoo, T. Kontos, J. Furer, C. Hoffmann, M. Gr¨ aber,\nA. Cuttet, and C. Sch¨ onenberger, Nat. Phys. 1, 99 (2005).\n2C. A. Merchant and N. Markovi´ c, Phys. Rev. Lett. 100,\n156601 (2008).\n3N. Tombros, C. Jozsa, M. Popinciuc, H. T. Jonkman, and\nB. J. van Wees, Nature (London) 448, 571 (2007); S.\nCho, Y.-F. Chen, and M. S. Fuhrer, Appl. Phys. Lett. 91,\n123105 (2007).\n4K. Tsukagoshi, B. W. Alphenaar, and H. Ago, Nature\n(London) 401, 572 (1999).\n5D. V. Bulaev, B. Trauzettel, and D. Loss, Phys. Rev. B\n77, 235301 (2008).\n6D. Huertas-Hernando, F. Guinea, and A. Brataas,\narXiv:0812.1921 (unpublished).\n7H. Min, J. E. Hill, N. A.Sinitsyn, B. R.Sahu, L. Kleinman,\nand A. H. MacDonald, Phys. Rev. B 74, 165310 (2006);\nY. Yao, F. Ye, X.-L. Qi, S.-C. Zhang, and Z. Fang, ibid.\n75, 041401(R) (2007).\n8D. Huertas-Hernando, F. Guinea, and A. Brataas, Phys.\nRev. B74, 155426 (2006).\n9T. Ando, J. Phys. Soc. Jpn. 69, 1757 (2000); A. De-\nMartino, R. Egger, K. Hallberg, and C. A. Balseiro, Phys.\nRev. Lett. 88, 206402 (2002).\n10J. C. Meyer, A. K. Geim, M. I. Katsnelson, K. S.\nNovoselov, T. J. Booth, and S. Roth, Nature (London)\n446, 60 (2007); E. Stolyarova, K. T. Rim, S. Ryu, J.\nMaultzsch, P. Kim, L. E. Brus, T. F. Heinz, M. S. Hy-\nbertsen, and G. W. Flynn, Proc. Natl. Acad. Sci. U.S.A.\n104, 9209 (2007); V. Geringer, M. Liebmann, T. Echter-\nmeyer, S. Runte, M. Schmidt, R. R¨ uckamp, M. C. Lemme,\nand M. Morgenstern, Phys. Rev. Lett. 102, 076102 (2009);A. K. Geim, Science 324, 1530 (2009).\n11V. M. Pereira and A. H. Castro Neto, Phys. Rev. Lett.\n103, 046801 (2009).\n12F. Kuemmeth, S. Ilani, D. C. Ralph, and P. L. McEuen,\nNature (London) 452, 448 (2008).\n13The corresponding expression in Ref.8is slightly different\nfrom Eq. (2) since the σband is treated in different ways.\nWhen a few minor mistakes in Ref.8are corrected, the two\nexpressions result in similar numerical values.\n14J. Serrano, M. Cardona, and T. Ruf, Solid State Commun.\n113, 411 (2000).\n15J. Ajiki and T. Ando, J. Phys. Soc. Jpn. 62, 1255 (1993).\n16Leonard I. Schiff, Quantum Mechanics (McGraw-Hill, New\nYork, 1968).\n17Hc[PK/(E(0)−HK,(0))]Hcis spin-independent and thus ig-\nnored [see Eq. (10)].\n18For the K′point,EK′,(2)\n↑=/angbracketleftψK′\nA|HK′,(2)|ψK′\nA/angbracketright±νRe [/angbracketleftψK′\nA\n|HK′,(2)|ψK′\nB/angbracketrighte−iθ],EK′,(2)\n↓=−EK′,(2)\n↑with upper (lower)\nsign for the conduction band bottom (valence band top).\n19J. C. Slater and G. F. Koster, Phys. Rev. 94, 1498 (1954).\n20The last four terms of Hsoin Eq. (9), which do not com-\nmute withσy, do not contribute to the effective SOC near\nthe Fermi energy due to the factor e±iϕ8.\n21J. W. Mintmire and C. T. White, Carbon 33, 893 (1995).\n22Using other sets of tight-binding parameters [D. Tom´ anek\nand M. A. Schluter, Phys. Rev. Lett. 67, 2331 (1991); R.\nSaito, M. Fujita, G. Dresselhaus, and M. S. Dresselhaus,\nPhys. Rev. B 46, 1804 (1992)] does not change results\nqualitatively.\n23C. L. Kane and E. J. Mele, Phys. Rev. Lett. 78, 19326\n(1997); L.YangandJ.Han, ibid.85, 154(2000); A.Kleiner\nandS. Eggert, Phys. Rev.B 63, 073408 (2001); J.-C. Char-\nlier, X. Blase, and S. Roche, Rev. Mod. Phys. 79, 677\n(2007).\n24See for instance, W. Han, W. H. Wang, K. Pi, K. M. Mc-Creary, W. Bao, Y. Li, F. Miao, C. N. Lau, and R. K.\nKawakami, Phys. Rev. Lett. 102, 137205 (2009).\n25L. Chico, M. P. L´ opez-Sancho, and M. C. Mu˜ noz, Phys.\nRev. B79, 235423 (2009)."    },    {        "title": "1302.1063v3.Inertial_effect_on_spin_orbit_coupling_and_spin_transport.pdf",        "content": "Inertial e\u000bect on spin orbit coupling and spin transport\nB. Basu\u0003and Debashree Chowdhuryy1\n1Physics and Applied Mathematics Unit, Indian Statistical Institute, 203 B.T.Road, Kolkata 700 108, India\nWe theoretically study the renormalization of inertial e\u000bects on the spin dependent transport of\nconduction electrons in a semiconductor by taking into account the interband mixing on the basis of\n~k:~ pperturbation theory. In our analysis, for the generation of spin current we have used the extended\nDrude model where the spin orbit coupling plays an important role. We predict enhancement of\nthe spin current resulting from the rerormalized spin orbit coupling e\u000bective in our model in cubic\nand non cubic crystal. Attention has been paid to clarify the importance of gauge \felds in the\nspin transport of this inertial system. A theoretical proposition of a perfect spin \flter has been\ndone through the Aharonov-Casher like phase corresponding to this inertial system. For a time\ndependent acceleration, e\u000bect of ~k:~ pperturbation on the spin current and spin polarization has also\nbeen addressed. Furthermore, achievement of a tunable source of polarized spin current through\nthe non uniformity of the inertial spin orbit coupling strength has also been discussed.\nPACS numbers: 72.25.-b, 85.75.-d, 71.70.Ej\nI. INTRODUCTION\nIn semiconductor band structure spin-orbit coupling (SOC), which originates from the relativistic coupling of spin\nand orbital motion of electrons, plays a very important role from the perspective of spin Hall e\u000bect. Understanding the\ne\u000bect of SOI is indispensable in the study of spincurrent , a \row of spins. Although spin current can be induced easily,\ndetection and control of spin current is a challenging research area both for theoretical and experimental physicists and\nhas attracted a lot of attention in the \feld of spintronics [1{3]. Spintronics aims to use the spin properties of electrons\nalong with the charge degrees of freedom and has emerged as the most pursued area in condensed matter physics\nand nanotechnology. In this regard, the theoretical prediction of the spin Hall e\u000bect (SHE) [4] and its application to\nspintronics has seen considerable advancement. This e\u000bect is observed experimentally in semiconductors [5, 6] and\nmetals[7].\nThough studies on the inertial e\u000bect of electrons has a long standing history [8{11] but the contribution of the\nspin-orbit interaction (SOI) in accelerating frames has not much been addressed in the literature [12, 13]. Recently, a\ntheory has been proposed [14, 15] describing the direct coupling of mechanical rotation and spin where the generation\nof spin current arising from rotational motion has been predicted. Inclusion of the inertial e\u000bects in semiconductors\ncan open up some fascinating phenomena, yet not addressed. So it is appealing to investigate how the inertial e\u000bect\na\u000bects some aspects of spin transport in semiconductors. In addition, the role of SOI in connection to spin Hall e\u000bect\nmay inspire one to study the gauge theory of this inertial spin orbit Hamiltonian.\nThe spin dynamics of the semiconductor is in\ruenced by the ~k:~ pperturbation theory as the band structure of a\nsemiconductor in the vicinity of the band edges can be very well described by the ~k:~ pmethod. On the basis of ~k:~ p\nperturbation theory, by taking into account the interband mixing, one can reveal many characteristic features related\nto spin dynamics. In this paper, we theoretically investigate the generation of spin current in a solid on the basis of\n~k:~ pperturbation [6] with a generalized spin orbit Hamiltonian which includes the inertial e\u000bect due to acceleration.\nThe generation of spin current is studied in the extended Drude model framework, where the spin orbit coupling\nhas played an important role. It is shown in our present paper that spin current appearing due to the combined\naction of the external electric \feld, crystal \feld and the induced inertial electric \feld via the total e\u000bective spin-orbit\ninteraction is enhanced by the interband mixing of the conduction and valence band states. We have also studied the\nAharonov-Casher like phase which corresponds to the e\u000bective SOI present in the model. Through the interplay of\nAharonov-Bohm phase ( AB) and Aharonov-Casher ( AC) phases, we are able to propose a perfect spin \flter for the\naccelerating system. Also by taking into consideration of a special pro\fle of the acceleration in a trilayer system, we\ncan set up a tunable spin \flter. Renormalization of the spin current and spin polarization for the time dependent\nacceleration has also been investigated. Here we consider the ~k:~ pperturbation in the 8 \u00028 Kane model and write the\ntotal Hamiltonian including the inertial e\u000bect due to acceleration.\n\u0003Electronic address: sribbasu@gmail.com\nyElectronic address:debashreephys@gmail.comarXiv:1302.1063v3  [cond-mat.mes-hall]  20 May 20132\nThe paper is organized as follows. In Section II we write the total Hamiltonian of the 8 \u00028 Kane model with\n~k:~ pperturbation including the e\u000bect of acceleration. Section III deals with the generation of the spin current and\nconductivity in the semiconductors with cubic and non-cubic symmetry. The e\u000bect of time dependent acceleration on\nspin current and conductivity is discussed in section IV. The details on the gauge theory of our model, particularly\ntheACphase, perfect spin \flter and tunable spin \flter is narrated in section V. Finally we conclude with section VI.\nII. INERTIAL SPIN ORBIT HAMILTONIAN AND ~k:~ pMETHOD\nWe start with the Dirac Hamiltonian for a particle with charge eand massmin an arbitrary non-inertial frame\nwith constant linear acceleration ~ aand without rotation which is given by [11],\nHI=\fmc2+c \n\u000b:(~ p\u0000e~A\nc)!\n+1\n2c\"\n(~ a:~ r)((~ p\u0000e~A\nc):~ \u000b)((~ p\u0000e~A\nc):~ \u000b)(~ a:~ r)#\n+\fm(~ a:~ r) +eV(~ r); (1)\nwhere the subscript Iin Hamiltonian (1) is due to the e\u000bect of inertia. Applying a series of Foldy-Wouthuysen\n(FW) transformations [16, 17] on the Hamiltonian(1 ) we can write the Pauli-Schrodinger Hamiltonian for the two\ncomponent electron wave function in the low energy limit as\nHFW= \nmc2+(~ p\u0000e~A\nc)2\n2m!\n+eV(~ r) +m(~ a:~ r)\u0000e~\n2mc~ \u001b:~B\u0000e~\n4m2c2~ \u001b:(~E\u0002~ p) +\f~\n4mc2~ \u001b:(~ a\u0002~ p) (2)\nwhere~Eand~Bare the external electric and magnetic \feld respectively.\nIn the right hand side of Hamiltonian (2), the third term is an inertial potential term arising due to the acceleration\n~ a. This potential V~ a(~ r) =\u0000m\ne~ a:~ rinduces an electric \feld ~E~ a=m\ne~ a[12, 14]. The induced electric \feld ~E~ aproduces\nan inertial SOI term (sixth term in the right hand side of (2)) apart from the SOI term due to the external electric\n\feld (\ffth term in the right hand side of (2)). The Hamiltonian (2) thus can be rewritten in terms of ~E~ aandV~ a(~ r) as\nHFW= \nmc2+(~ p\u0000e~A\nc)2\n2m!\n+e(V(~ r)\u0000Va(~ r))\u0000e~\n2mc~ \u001b:~B\u0000e~\n4m2c2~ \u001b:((~E\u0000E~ a)\u0002~ p): (3)\nIn the above calculations we have neglected the1\nc4terms and the terms due to red shift e\u000bect of kinetic energy. The\ngeneralized spin-orbit interactione~\n4m2c2~ \u001b:\u0010\n(~E\u0000~E~ a)\u0002~ p\u0011\ne\u000bective in our inertial system plays a signi\fcant role in\nour analysis.\nWe are interested in an e\u000bective Hamiltonian describing the motion of electrons in a solid incorporating the inertial\ne\u000bect due to acceleration. It is known that the physical parameters present in any Hamiltonian in vacuum are\nrenormalized when considered in a solid. In an inertial frame, such renormalization e\u000bects in a crystalline solid is\ngenerally studied in the framework of ~k:~ pperturbation theory using the Bloch eigenstates. One can renormalize the\ne\u000bect of acceleration on the basis of ~k:~ pperturbation and the 8 \u00028 Kane model [18].\nThe basic idea of the Kane model is that the band edge eigenstates constitute a complete basis and to obtain the\neigenstates away from the band edge the wave function is expanded in the band edge states, which gives rise to an\n8\u00028 band Hamiltonian. Bands that are far away in energy can be neglected. In presence of magnetic \feld the crystal\nmomentum is given by ~~k=~ p\u0000q~A:The~k:~ pmethod leads to high-dimensional Hamiltonians, for example, an 8 \u00028\nmatrix for the Kane model [6].\nTo this end, we start with a Hamiltonian of the well known 8 \u00028 Kane model which takes into account the ~k:~ p\ncoupling between the \u0000 6conduction band and \u0000 8and \u0000 7valance bands which is given by\nH8\u00028=0\n@H6c6cH6c8vH6c7v\nH8v6cH8v8vH8v7v\nH7v6cH7v8vH7v7v1\nA (4)\n=0\nB@(Ec+eVtot)I2p\n3P~T:~k \u0000Pp\n3~ \u001b:~kp\n3P~Ty:~k (Ev+eVtot)I4 0\n\u0000Pp\n3~ \u001b:~k 0 (Ev\u00004 0+eVtot)I21\nCA (5)3\nHere,Vtot=V(~ r)\u0000Va(~ r); EcandEvare the energies at the conduction and valence band edges respectively. 40\nis the spin orbit gap, P is the Kane momentum matrix element which couples slike conduction bands with plike\nvalence bands. This Kane Momentum matrix is almost constant for group III to V semiconductors, whereas 40and\nEG=Ec\u0000Evvaries with materials. The ~Tmatrices are given as\nTx=1\n3p\n2\u0012\n\u0000p\n3 0 1 0\n0\u00001 0p\n3\u0013\n; Ty=\u0000i\n3p\n2\u0012p\n3 0 1 0\n0 1 0p\n3\u0013\n;Tz=p\n2\n3\u0012\n0 1 0 0\n0 0 1 0\u0013\n(6)\nandI2;I4are unit matrices of size 2 and 4 respectively.\nIt may be noted here that as the e\u000bect of rotation [14] is not considered, the crystal momentum used in ~k:~ p\nperturbation is not modi\fed in our model. The e\u000bect of acceleration changes the electric potential V(~ r) as well as\nthe electric \feld ~E. In our framework, the total potential and total electric \feld have been modi\fed as Vtot(~ r) and\n~Etotrespectively.\nThe Hamiltonian (5) can now be reduced to an e\u000bective Hamiltonian of the conduction band electron states [6] in\npresence of acceleration as\nHkp=P2\n3\u00122\nEG+1\nEG+40\u0013\n~k2+eVtot(~ r)\u0000P2\n3\u00121\nEG\u00001\n(EG+40)\u0013ie\n~~ \u001b:(~k\u0002~k)+eP2\n3\u00121\nE2\nG\u00001\n(EG+40)2\u0013\n~ \u001b:(~k\u0002~Etot)\n(7)\nIn our analysis the derivation of ~k:~ pperturbed Hamiltonian of the accelerated system is carried out by using ~Etotand\nVtot:The total Hamiltonian for the conduction band electrons including the e\u000bect of acceleration is then given by,\nHtot=~2~k2\n2m\u0003+eVtot(~ r) + (1 +\u000eg\n2)\u0016B~ \u001b:~B+e(\u0015+\u000e\u0015)~ \u001b:(~k\u0002~Etot); (8)\nwhere1\nm\u0003=1\nm+2P2\n3~2\u0010\n2\nEG+1\nEG+40\u0011\nis the e\u000bective mass and ~Etot=\u0000~rVtot(~ r) =~E\u0000~Ea;is the e\u000bective total electric\n\feld of the inertial system and \u0015=~2\n4m2c2is the spin orbit coupling strength as considered in vacuum. Furthermore,\nthe perturbation parameters \u000egand\u000e\u0015are given by\n\u000eg=\u00004m\n~2P2\n3\u00121\nEG\u00001\nEG+40\u0013\n\u000e\u0015= +P2\n3\u00121\nE2\nG\u00001\n(EG+40)2\u0013\n(9)\nSpeci\fcally, the parameter \u000egis related to the renormalized Zeeman coupling strength, whereas \u000e\u0015is responsible for\nthe renormalization of spin orbit coupling. Now one can rewrite the Hamiltonian as\nHtot=~2k2\n2m\u0003+eVtot+ (1 +\u000eg\n2)\u0016B~ \u001b:~B+e\u0015eff~ \u001b:(~k\u0002~Etot); (10)\nwhere\u0015eff=\u0015+\u000e\u0015is the e\u000bective SO coupling.\nWe shall note in due course that the parameter \u0015eff= (\u0015+\u000e\u0015), which comes into play due to the interband mixing\non the basis of ~k:~ pperturbation theory, is responsible for the enhancement of the spin current.\nIII. SPIN HALL CURRENT AND CONDUCTIVITIES FOR CUBIC AND NONCUBIC CRYSTAL\nWe are interested in the generation of spin current through the e\u000bective SOI and therefore consider only the relevant\npart of the Hamiltonian (for the positive energy solution) of spin1\n2electron for zero external magnetic \feld as\nH=~ p2\n2m\u0003+eVtot(~ r)\u0000\u0015effe\n~~ \u001b:(~Etot\u0002~ p) (11)\nThe semiclassical equation of motion of electron can be de\fned as\n~F=1\ni~h\nm\u0003~_r;Hi\n+m\u0003@~_r\n@t; (12)4\nwith~_r=1\ni~[~ r;H]:Thus from (11)\n~_r=~ p\nm\u0003\u0000\u0015effe\n~\u0010\n~ \u001b\u0002~Etot\u0011\n(13)\nFinally, the force\n~F=m\u0003~ r=\u0000e~rVtot(~ r) +\u0015effem\u0003\n~_~ r\u0002~r\u0002(~ \u001b\u0002~Etot) (14)\nis the spin Lorentz force with an e\u000bective magnetic \feld ~r\u0002(~ \u001b\u0002~Etot):Explicitly, the vector potential is given by\n~A(~ \u001b) =\u0015effm\u0003c\n~(~ \u001b\u0002~Etot) (15)\nLater we shall discuss about this spin dependent gauge ~A(~ \u001b) which is closely related to the ACphase and show\nhow the~k:~ pperturbation modi\fes the corresponding ACphase. The spin dependent e\u000bective Lorentz force noted in\neqn.(14) is responsible for the spin transport of the electrons in the system, and hence responsible for the spin Hall\ne\u000bect of this inertial system. It is clear from (9) that the expression in (14) i.e the Lorentz force is enhanced due to\n~k:~ pperturbation in comparison to the inertial spin force studied in [12]. From the expression of _~ rin (13) we can write\nthe linear velocity in a linearly accelerating frame with ~k:~ pperturbation as\n_~ r=~ p\nm+~ v~ \u001b; ~ a (16)\nwhere\n~ v~ \u001b; ~ a=\u0000\u0015effe\n~(~ \u001b\u0002~Etot) (17)\nis the spin dependent anomalous velocity term. The anomalous velocity term is related to the spin current as\nji\ns=enTr\u001bi~ v~ \u001b; ~ a:One should note that the velocity depends on \u000e\u0015i.e on the spin orbit gap and the band gap energy\nof the crystal considered. The expression shows for a non zero spin orbit gap, the spin dependent velocity changes\nwith the energy gap. For vanishing spin-orbit gap, there is no extra contribution to the anomalous velocity for the\n~k:~ pperturbation. The spin current and spin Hall conductivity in an accelerated frame of a semiconductor can now\nbe derived by taking resort to the method of averaging [12, 19]. We proceed with equation(14) as\n~F=~F0+~F~ \u001b (18)\nwhere~F0and~F~ \u001bare respectively the spin independent and the spin dependent parts of the total spin force. With\nthe help of eqn. (15) the Hamiltonian (11) can be written as\nH=1\n2m\u0003(~ p\u0000e\nc~A(~ \u001b))2+eVtot(~ r) (19)\nwhereVtot(~ r) =V(~ r)\u0000Va(~ r) andV(~ r) is the sum of the external electric potential V0(~ r) and the lattice electric\npotentialVl(~ r). In this calculation we have neglected the terms of O(~A2(~ \u001b)):Breaking into di\u000berent parts, the\nsolution of equation (14) can be written as _~ r=_~ r0+_~ r~ \u001b[19]. If the relaxation time \u001cis independent of ~ \u001band for the\nconstant total electric \feld ~Etot, following [12, 19] we can write,\nh_~ r0i=\u0000\u001c\nm\u0003\u001c@Vtot\n@r\u001d\n=e\u001c\nm\u0003~Eeff; (20)\nand\nD\n_~ r(~ \u001b)E\n=\u0000\u0015effe2\u001c2\nm\u0003~~Eeff\u0002\u001c@\n@r\u0002(~ \u001b\u0002@Vl\n@r)\u001d\n+\u0015effe2\u001c2\nm\u0003~~Eeff\u0002\u001c@\n@r\u0002(~ \u001b\u0002@V~ a\n@r)\u001d\n: (21)5\nwhere~Eeff=\u0000e~r(V0(~ r)\u0000V~ a(~ r)):We can now derive the spin current by the evaluation of the averages in equation\n(21) with di\u000berent symmetry.\nSemiconductors with cubic symmetry\nFor the case of semiconductors with cubic symmetry and constant acceleration [12]\nD\n_~ r(~ \u001b)E\n=2e2\u001c2\u0016\nm\u0003~\u0015eff(~ \u001b\u0002~Eeff) (22)\nas in this case the only non zero contribution permitted by symmetry [12, 19] is\n\u001c@2Vl\n@ri@rj\u001d\n=\u0016\u000eij; (23)\nwith\u0016being a system dependent constant. The total spin current of this inertial system with ~k:~ pperturbation can\nnow be obtained as\n~jkp=eD\n\u001as~_rE\n=~jo;~ a\nkp+~js;~ a\nkp(~ \u001b) (24)\nThe charge component of this current in our accelerated system is\n~jo;~ a\nkp=e2\u001c\u001a\nm\u0003(~E0\u0000~E~ a): (25)\nLet us introduce the density matrix for the charge carriers as\n\u001as=1\n2\u001a(1 +~ n:~ \u001b);\nwhere\u001ais the total charge concentration and ~ n=h~ \u001biis the spin polarization vector.\nWithin the ~k:~ pframework, due to the interband mixing the spin current of this inertial system is given by\n~js;~ a\nkp(~ \u001b) =\u00122e3\u001c2\u001a\u0016\nm\u0003~\u0013\u0014~2\n4(m)2c2+P2\n3\u00121\nE2\nG\u00001\n(EG+40)2\u0013\u0015\u0010\n~ n\u0002(~E0\u0000~E~ a\u0011\n(26)\nor\n~js;~ a\nkp(~ \u001b) =m\nm\u0003\u0014\n1 +4m2c2P2\n3~2\u00121\nE2\nG\u00001\n(EG+40)2\u0013\u0015\n~js;~ a(~ \u001b) (27)\n=m\nm\u0003(1 +\u000e\u0015\n\u0015)~js;~ a(~ \u001b) (28)\nwhere~js;~ a(~ \u001b) =~e3\u001c2\u001a\u0016\n2m3c2\u0010\n~ n\u0002(~E0\u0000~E~ a\u0011\nis the spin current in an accelerating frame[12] without ~k:~ pperturbation.\nWith~E0= (0;0;Ez^z) and~ a= (0;0;az^z) we can explicitly derive the spin currents in the xandydirections. The\nratio of spin current in an accelerating system with and without ~k:~ pperturbation is given by\nj~js;~ a(~ \u001b)kpj\nj~js;~ a(~ \u001b)j=m\nm\u0003(1 +\u000e\u0015\n\u0015): (29)\nThe coupling constant \u000e\u0015has di\u000berent values for di\u000berent materials and \u0015, the coupling parameter in the vacuum has a\nconstant value 3 :7\u000210\u00006\u0017A2:We tabulate the ratio of spin currents for di\u000berent semiconductors with cubic symmetry as\nEG(eV)40(eV)P(eV\u0017A)\u000e\u0015(\u0017A2)j~js;~ a(~ \u001b)kpj\nj~js;~ a(~ \u001b)j\nGaAs = 1.519 0.341 10.493 5.3 2:154\u0002107\nAlAs = 3.13 0.300 8.97 0.318 5:748\u0002105\nInSb = 0.237 0.810 9.641 523.33 1:0175\u00021010\nInAs = 0.418 0.380 9.197 120 1:41\u0002109\nThe table reveals that in a linearly accelerating frame, how ~k:~ pmethod is useful for the generation of large spin current6\nGaAsInAsInSb\n0 10 20 30 40az02/Multiply1094/Multiply1096/Multiply1098/Multiply1091/Multiply1010A/LBracketBar1jx/RBracketBar1\nFIG. 1: (Color online) Variation of spin current with acceleration for three di\u000berent semiconductors, where A =2m2c2\n~e2\u001c2\u001a\u0016.\nin semiconductors. In \fgure 1 we have plotted the variation of spin current( xdirection) with acceleration( zdirection)\nusing equ. (31) and the values of the Kane model parameters in the table, for three di\u000berent semiconductors.\nNow if we switch o\u000b the external electric \feld, we have the following expression of spin current in our system\n~js;~ a\nkp(~ \u001b) =\u0000\u001bs;a\nH;kp\u0010\n~ n\u0002~E~ a\u0011\n; (30)\nwhere\u001bs;a\nH;kp=2e3\u001c2\u001a\u0016\nm\u0003~\u0015effis the spin Hall conductivity. If we now consider the acceleration along zdirection, the\nspin current in the xdirection becomes\nj~js;~ a\nx;kp(~ \u001b)j=\u001bs;a\nH;kp(nyEa;z) (31)\nOne can notice that even if the external electric \feld is zero, still we can achieve huge spin current by the application\nof acceleration only. For di\u000berent semiconductors we get di\u000berent spin current for non zero spin orbit gap. It\nis interesting to point out that for a critical value ~E0=~Eawe see no spin current in the system. Though the\nacceleration under which we get the result is very high [12], still it elicits that we can control spin current by adjusting\nacceleration.\nThe corresponding charge and spin Hall conductivities in a linearly accelerating frame from the expressions of the\ncurrents (25), (26) can be readily obtained as\n\u001b~ a\nH;kp =e2\u001c\u001a\nm\u0003\n\u001bs;~ a\nH;kp=2e3\u001c2\u001a\u0016\nm\u0003~\u0015eff (32)\nAs expected [12], both the charge and spin conductivities are not a\u000bected by the inertial e\u000bect of acceleration but the\nspin conductivity is renormalized by the ~k:~ pperturbation. The ratio of spin and charge Hall conductivity is\n\u001bs;~ a\nH;kp\n\u001b~ a\nH;kp=2e\u001c\u0016\n~\u0015eff (33)\n=2e\u001c\u0016\n~\u0014~2\n4(m)2c2+P2\n3\u00121\nE2\nG\u00001\n(EG+40)2\u0013\u0015\n(34)\nThis spin to charge ratio is independent of the concentration of charge carriers but depends on the relaxation time,\nthe constant term \u0016, and also on the Kane model parameters P,EGand40, i.e. the ratio varies with the material\nconsidered. For a non accelerating system with zero spin orbit gap parameter, the ratio in (33) is the same as found in\n[19]. The condition under which the spin conductivity becomes exactly equal to charge conductivity is \u0015eff=~\n2e\u001c\u0016:\nThe comparison of the spin Hall conductivity in our system with that as obtained in [12] without ~k:~ pperturbation\nshows an enhancement due to the presence of the term \u000e\u0015which can be observed from the relation\nj\u001bs;~ a\nH;kpj\nj\u001bs;~ a\nHj=m\nm\u0003(1 +\u000e\u0015\n\u0015) (35)\nin any crystalline solid with a non zero spin orbit gap 40:\nSemiconductors with non-cubic symmetry7\nThere are semiconductors which do not have cubic symmetry, but are examples of producing spin Hall e\u000bect. Our\nobjective now is to study those systems and derive the expressions for the spin current. We can consider orthorhombic\ncrystals and can choose the axes of the coordinate frame along the crystal axes [20]. Instead of eqn (23), for the non\ncubic symmetry we can now write\n\u001c@2V\n@ri@rj\u001d\n=\u0016\u001fi\u000eij; (36)\nwhere\u001fx6=\u001fy6=\u001fzare the factors of order unity. Using this, eqn (22) becomes,\nD\n_~ r(~ \u001b)iE\n=\u0015effe2\u001c2\nm\u0003~\u0016[\u001fx+\u001fy+\u001fz\u0000\u001fi](~ \u001b\u0002~Eeff)i (37)\nwhich shows that the spin dependent velocity is not uniform in all directions. Following the same procedure [19], the\nspin current in ~ xdirection is attained as\n~js;~ a\nx;kp(~ \u001b) =\u0015eff\u0012e3\u001c2\u001a\u0016\nm\u0003~\u0013\n(\u001fy+\u001fz)\u0010\n~ n\u0002(~E0\u0000~E~ a)\u0011\nx(38)\nHence, the spin Hall conductivity is\n\u001bs;~ a\nx;kp=\u0015eff\u0012e3\u001c2\u001a\u0016\nm\u0003~\u0013\n(\u001fy+\u001fz) (39)\nThe charge conductivity remains the same as in the cubic case i.e\n\u001b~ a\nH;kp =e2\u001c\u001a\nm\u0003(40)\nFor an orthorhombic crystal in the ~ xdirection we can \fnd out the ratio of the spin to charge conductivity as\n\u001bs;~ a\nH;kp\n\u001b~ a\nH;kp=\u0015effe\u001c\u0016(\u001fy+\u001fz)\n~(41)\n=e\u001c\u0016(\u001fy+\u001fz)\n~(42)\n\u0014~2\n4m2c2+P2\n3\u00121\nE2\nG\u00001\n(EG+40)2\u0013\u0015\n: (43)\nThe charge to spin conductivity ratio does not depend upon the concentration of charge carriers, rather it depends\non the Kane model parameters and the values of \u001fxand\u001fy:It can be readily observed that we return back to the\nresult of the cubic case for \u001fx=\u001fy=\u001fz.\nIV. SPIN CURRENT AND SPIN POLARIZATION WITH TIME DEPENDENT ACCELERATION\nLet us now analyze the case for a time dependent acceleration. As an example of time dependent acceleration we\nconsider [12, 14]\n~ a=u!2\naexp(i!~ at)~ ex; (44)\nwhere the acceleration is induced by harmonic oscillation with frequency !~ aand amplitude u:The time dependent\nacceleration ~ ainduces a time dependent electric \feld ~E~ aas\n~E~ a=mu!2\na\neexp(i!~ at)~ ex: (45)\nFor the external electric \feld ~E= 0, from (26), the spin current for a semiconductor with cubic symmetry is then\ngiven by\n~js;~ a\nkp(~ \u001b;t) =\u0000mu!2\na\u00122e2\u001c2\u001a\u0016\nm\u0003~\u0013\u0014~2\n4(m)2c2+P2\n3\u00121\nE2\nG\u00001\n(EG+40)2\u0013\u0015\n(~ n\u0002ex)exp(i!~ at) (46)8\n Ω = 10 GHz\nGaAs\n2 4 6 8 10u2.0/Multiply1084.0/Multiply1086.0/Multiply1088.0/Multiply1081.0/Multiply1091.2/Multiply1091.4/Multiply109A/LBracketBar1jz/RBracketBar1\nu = 10 nm\nGaAs\n2 4 6 8 10Ωa2.0/Multiply1084.0/Multiply1086.0/Multiply1088.0/Multiply1081.0/Multiply1091.2/Multiply1091.4/Multiply109A/LBracketBar1jz/RBracketBar1\nFIG. 2: (Color online) Left: Variation of Ajjzjwith ufor!a= 10GHz . Right: Variation of Ajjzjwith !aforu= 10 nm for\nGaAs semiconductor, where A=2mm\u0003c2\ne2\u001c2\u001a\u0016~.\nThezpolarized spin current along y direction is\n~js;~ a\nz;kp(~ \u001b;t) =\u0000mu!2\na\u00122e2\u001c2\u001a\u0016\nm\u0003~\u0013\u0014~2\n4(m)2c2+P2\n3\u00121\nE2\nG\u00001\n(EG+40)2\u0013\u0015\nexp(i!~ at)~ ey (47)\nThe absolute value of the current is then given by jjs;~ a\nz;kp(~ \u001b)j=1\nAu!2\na(1 +\u000e\u0015\n\u0015);whereA=2mm\u0003c2\ne2\u001c2\u001a\u0016~. Equation (47)\ndemonstrates that if the semiconductor sample is attached to a mechanical resonator and vibrated in xdirection with\n!a= 10GHz andu= 10nm, we get the zpolarized ac spin current along ydirection. Application of ~k:~ pperturbation\nenhances the ac spin current in semiconductor [14]. In \fgure 2 we plot the variation of spin current with amplitude\nuand frequency !afor theGaAs semiconductor.\nNow we move towards the evaluation of the out of plane spin polarization. The constant acceleration of the inertial\nsystem cannot explain the out-of-plane transverse spin current and in what follows we consider a time dependent\nacceleration within the ~k:~ pperturbation formalism. From (11) we write the time dependent Hamiltonian for the time\ndependent acceleration with the choice of ~ a(t) = (0;0;az^z(t)), which subsequently results ~E~ a(t) = (0;0;Ea;z^z(t));[12]\nand\nH(t) =~2~k2\n2m\u0003+\u000beff(kx;t\u001by\u0000ky;t\u001bx); (48)\nwhere we use the fact that, for electrons moving through a lattice, the electric \feld ~Eis Lorentz transformed to an\ne\u000bective magnetic \feld ( ~k\u0002~E)\u0019~B(~k) in the rest frame of the electron. Hamiltonian (48) resembles to the well known\nRashba Hamiltonian and \u000beff;the spin orbit coupling strength depends on the acceleration of the system as well as on\nthe material parameters. This Rashba like coupling parameter has signi\fcant importance in the understanding of the\nspin transport with inertial e\u000bects. The SOC in semiconductor causes electron to experience an e\u000bective momentum\ndependent magnetic \feld ~B~ a(~k);which breaks the spin degeneracy of electron. Time dependence of the spin orbit\nHamiltonian will generate an additional component [21] ~B?= (_~n~ a\u0002~ n~ a);in addition to the e\u000bective magnetic \feld\n~Ba(~k);where the unit vector ~ na=~Ba(~k)\nj~Ba(~k)j:Let us now assume the e\u000bective electric \feld due to acceleration is in the x\ndirection such that ~E~ a=E~ a;x^x[12, 21]. As ~B~ a(~k) is in the x-y plane, the term ~B?completely represents an e\u000bective\nout of plane magnetic \feld component along zdirection. With ~B\u0006=~B~ a(~k) +~B?;the total contribution of magnetic\n\feld, the classical spin vector in the zdirection can be written as\nsz=\u00061\nj~B\u0006j~\n2(_~n~ a\u0002~ n~ a):^z (49)\nNow if we make a choice for the unit vector along ~Ba(~k) as~ n~ a=p\u00001(py;\u0000px;0);we get _~ na=p\u00001(0;e~E~ a;x;0):Here\n\u0006represents the spin aligned parallel and anti-parallel to ~B\u0006:In the adiabatic limit, where ~B~ a(~k)\u001d~B?[12],~B\u00069\n 0 5 10 15 20 0 0.5 1 1.5 2\n 0 250 500 750 1000A|sz|\naz py/p3A|sz|\n 0 20 40 60 80 100 0 2 4 6 8 10\n 0 25 50 75 100A|sz|\naz uA|sz|\n 0 5 10 15 20 0 2 4 6 8 10\n 0 10 20 30A|sz|\nazωaA|sz|\nFIG. 3: (Color online) (i) Variation of Ajszjwith azandpy=p3for!a= 10 GHz, u= 10 nm. (ii) Variation of Ajszjwith az\nandufor!a= 10 GHz andpy\np3=const . (iii) Variation of Ajszjwith azand!aforu= 10 nm andpy\np3=const where A=4\u0015eff\ne~3\n.\napproaches ~B~ a(~k) and the out of plane spin polarization can be derived as\nsz;kp\u0019 \u00061\nj~B~ a(~k)j~\n2(_~n~ a\u0002~ n~ a):^z\n=\u0006~2\n2\u000beffp~\n2\u0012\n\u00001\np2eEa;xpy\u0013\n=\u0007e~3pyEa;x\n4\u000beffp3: (50)\nSubstituting the value of Ea;xfrom (45) in (50), we obtain the absolute value of sz;kpas\njsz;kpj=\u0007e~3pyu!2\na\n4\u0015effazp3: (51)\nThis shows how the Kane model parameters modify the spin polarization vector in an accelerated system [12]. The\ndependence of spin polarization on acceleration and Kane model parameters is clear from (51).\nIn \fgure 3 we show the variation of the spin polarization with respect to acceleration azand any one of the\nparameters, u;!a;py\np3keeping the two other parameters \fxed. Here A=4\u0015eff\ne~3, depends on the solid considered. As\nspin polarization is a measurable quantity, from the experimentally obtained values of szusing (51) we can have an\ninsight for the experimental veri\fcation of the parameters of the Kane model.\nV. GAUGE FIELD THEORY OF INERTIAL SOC AND SPIN FILTER\nThe study of gauge \felds in spintronics has become a topic of recent interest [22]. In the context of SOI, the\nimportance of Berry phase [23] was realized following the discovery of the intrinsic spin Hall e\u000bect[24]. The Berry\nphase results from cyclic, adiabatic transport of quantum states with respect to parameter space (e.g. real space\n~ r;momentum space ~k):In this regard, analysis of the Aharonov-Casher phase through the spin dependent gauge\npotential also has remarkable importance. Our next goal is to explore the conditions of the Berry curvature and study\ntheir consequences in spin transport. In this section we consider the Dirac Hamiltonian in a linearly accelerating frame\nwithout any external electric \feld and consider the physical consequences appearing as a result of the the induced\ninertial electric \feld due to acceleration.\nA. Spin orbit coupling, spin dependent phase and perfect spin \flter\nThere are lots of attempt to describe spin \flter in di\u000berent systems [25], but as far as our knowledge goes, proposal\nof a perfect \flter through an inertial system is not noted in the literature. As the name suggests, the function of a\nspin \flter is to spin polarize the injected charge current.\nIn this subsection we consider the induced spin orbit Hamiltonian (11) in the presence of the external magnetic\n\feld and study the gauge theory of the inertial spin orbit interaction.10\nIn this regard, let us consider the SO Hamiltonian in (48)(time independent) in the presence of external magnetic\n\feld~B, as\nH=~\u00052\n2m\u0003+\u000beff\n~(\u0005x\u001by\u0000\u0005y\u001bx); (52)\nwhere~\u0005 =~ p\u0000e~A(~ r);and~B=r\u0002~A(~ r):The Hamiltonian in (52) can also be rewritten in the following form\nH=1\n2m\u0003\u0010\n~ p\u0000e\nc~A(~ r)\u0000q\nc~A0(~ r;~ \u001b)\u00112\n; (53)\nwhere the spin dependent real space gauge \feld, ~A0(~ r;\u001b) is\n~A0(~ r;\u001b) =c\n2(\u0000\u001by;\u001bx;0): (54)\nHere we neglect the second order of ~A0in deriving eqn. (53). The new constant term q=2m\u0003\u000beff\n~can be regarded as\ncharge and \u000beffrepresents a Rashba [5] like spin orbit coupling strength [12]. The expression of the spin dependent\ngauge indicates that it is non-Abelian in nature, whereas the gauge due to external magnetic \feld provides an Abelian\ncontribution. The equation (52) can be written in terms of the total gauge \feld~~A0(~ r;\u001b) acting on the system as\nH=1\n2m\u0003\u0012\n~ p\u0000~e\nc~~A\u00132\n(55)\nwhere~~A=e~A(~ r) +q~A(~ r;~ \u001b);is the total gauge e\u000bective in the system and ~ eis a coupling constant which is set to be\n1 for future convenience. From the \feld theoretical point of view, the physical \feld generated due to the presence of\nthe total gauge~~Ais given by\n\n\u0015= \n\u0016\u0017=@\u0016~A\u0017\u0000@\u0017~A\u0016\u0000i~e\nc~h\n~A\u0016;~A\u0017i\n(56)\nThe \feld in the zdirection is then\n\nz=\u0010\n@x~Ay\u0000@y~Ax\u0011\n\u0000i~e\nc~h\n~Ax;~Ayi\n: (57)\nAs the commutators of di\u000berent components of the spin gauge ~A(~ r;~ \u001b) exists, \n zin our case boils down to the following\nform\n\nz=eBz+q2c\n2~\u001bz: (58)\nThe equn (58) can be expressed in terms of the \rux generated through area Sas\n\nz=e\u001eB\nS+q\u001eI\nS; (59)\nwhere\u001eB=SBz;\rux due to external magnetic \feld and \u001eI=Sqc\n2~\u001bzis the physical \feld generated due to inertial\nspin orbit coupling e\u000bect. The \frst term on the right hand side(rhs) of (59) is the contribution due to the external\nmagnetic \feld, which causes the ABphase, whereas the second term on the rhs of (59) is the \rux due to the physical\n\feld, is actually responsible for a AClike phase. This AClike phase is generated when a spin circulates an electric\n\rux. The second term in the expression of \n zin (58), actually represents a magnetic \feld in zdirection, with opposite\nsign for spin polarized along + zdirection or\u0000zdirection. As the spin up and spin down electrons experience equal\nbut opposite vertical magnetic \felds, they will subsequently carry equal and opposite AClike phase [26] which can\nbe obtained from\n\u001eAC=I\nd~ r:~A(~ r;\u001b); (60)\nwhereas the ABphase appears due to the \frst term in (58) is the same for both up and down electrons.\nInterestingly, one can take advantage of this acceleration induced spin orbit Hamiltonian for the proposition of a\nperfect spin \flter. In a semiconductor the interplay between this ACphase due to the induced spin dependent gauge11\nin presence of acceleration and the ABphase due to an external magnetic \feld can be used to achieve a spin \flter\n[27]. If a spatial circuit can be realized such that the up (down) spin electrons acquire an ACphase of\u0019=2 (-\u0019=2)\nand \fnite magnetic vector potential in the interior of the circuit makes both the up and down electrons attain an AB\nphase of\u0019=2, then the output consists of only spin down electrons and a perfect spin \flter is set up. The reversal of\nthe direction of the applied magnetic \feld may switch the polarity of the \flter and the output may consist of only\nspin down electrons. Thus we can propose theoretically a perfect spin \flter without any external electric \feld. The\nbeauty of our result is that without the application of any external electric \feld, only through the acceleration of the\ncarriers and external magnetic \feld we can at least theoretically propose a perfect spin \flter for our system.\nB. Spatially non uniform SOC and tunable spin \flter\nSpin transport through the magnetic barriers is a topic of recent interest, which naturally gives us an idea of spin\n\flter [28]. Theoretically, spin \fltering through the formation of magnetic barrier was \frst investigated in a trilayer\nsystem constructed using the FM stripes and 2DEG [29] .\nIn this subsection we consider a trilayer structure, within which a semiconducting (SC) channel is sandwiched\nbetween two metallic contacts. The SC channel is assumed to be in an accelerated frame with SO coupling strength\n\u000beff, while within metallic parts have \u000beff= 0 i.e we are considering a spatial discontinuity of the inertial spin orbit\ncoupling strength. This sharp discontinuity in the SO coupling, in turn gives a highly localized e\u000bective magnetic\n\feld barriers at the interfaces. The Hamiltonian with spatially non uniform spin orbit coupling can be obtained as\nH=~ p2\n2m\u0003+\u000beff(~ r)(ky\u001bx\u0000kx\u001by); (61)\nwhich can be written in the following form\nH=1\n2m\u0003\u0010\n~ p\u0000e\nc~A(~ r;\u001b)\u00112\n(62)\nwhere\n~A(~ r;\u001b) =\u000beff(~ r)m\u0003c\n~(\u0000\u001by;\u001bx;0); (63)\nis the spin gauge. The trilayer structure consists of metals at x= 0 andx=Land in between there exists\nsemiconducting channel [30]. Let us now consider the spatial pro\fle of \u000beffto be a step function as\n\u000beff(x) =\u000b0[\u0002(x)\u0000\u0002(x\u0000L)]; (64)\nwhere \u0002(x) is the unit step function and Lis the length of the semiconductor channel. Thus we can write the\ncurvature \feld using (64) as\n\nz(~ r) =m\u0003c\n~(@x\u000beff(x)\u001bx\u0000@y\u000beff(y)\u001by) +\u000b2\neff2(m\u0003)2ec\n~3\u001bz: (65)\nAs the spin orbit coupling is non uniform, the \frst term in (65), which is vanishing for an uniform coupling, exists in\nour case. Finally we can write the curvature in the zdirection as\n\nz(~ r) =\u000b0m\u0003c\n~[\u000e(x)\u0000\u000e(x\u0000L)]\u001bx+ (\u000beff)22(m\u0003)2ec\n~3\u001bz; (66)\nwhere\u000b0and\u000e(x) are the inertial spin orbit coupling at the barrier of the sample and the Dirac delta function\nrespectively. The \frst term on the rhs of eqn. (66) appears as the consequence of the spatial discontinuity of \u000beffand\nactually gives narrow spikes of magnetic \felds at the interfaces. This term is interesting as it gives a \u000eDirac function\ncentered at the interfaces of the trilayer structure. The second term is a physical \feld di\u000berent from the e\u000bective\nmagnetic \feld, generated due to the SO coupling. This narrow magnetic \felds are spin dependent as \u001bx=\u00061:One\nshould notice here that if the mixing of spin states are not considered, i.e if we take the length of the channel large\ncompared to the spin precession length, we can write the non- Abelian gauge in (63) as an Abelian gauge as\nA\u0006= (0;A\u0006;y;0) = (0;\u0006\u000beffm\u0003c\n~;0); (67)\nwhere\u0006denotes two states corresponding to \u001bx=\u00061:This structure is useful as a tunable source of spin current,\nwhich is very important concept in spintronics applications.12\nVI. CONCLUSION\nIn this paper we have theoretically investigated the generation of spin Hall current in a linearly accelerating semi-\nconductor system in presence of electromagnetic \felds with the help of well known Kane model by taking into account\nthe interband mixing on the basis of ~k:~ pperturbation theory. The explicit form of inertial spin Hall current and\nconductivity is derived for both cubic and noncubic crystals. We have shown how the interband mixing explains the\nspin current in an inertial system and also show the dependence of conductivity on the ~k:~ pperturbation parameters.\nIn the case of time dependent acceleration with ~k:~ pmethod we show the explicit expression of spin current and spin\npolarization. From the gauge theoretical point of view, next we have investigated the real space Berry curvature\nappearing in an inertial system with ~k:~ pmethod. Lastly based on the gauge theoretical aspects we have discussed a\nperfect spin \flter and a tunable spin \flter in an inertial frame.\n[1] S. A. Wolf, D. D. Awschalom, R. A. Buhrman, J. M. Daughton, S. von Molnar, M. L. Roukes, A.Y. Chtchelkanova, and\nD. M. Treger, Science 294, 1488 (2001).\n[2] I. Zutic, J. Fabian, and S. D. Sarma, Rev. Mod. Phys. 76, 323 (2004).\n[3] J. E. Hirsch, Phys. Rev. Lett. 83, 1834 (1999).\n[4] M. I. Dyakonov and V. I. Perel, Sov. Phys. JETP Lett. 13: 467 (1971).\n[5] E.I Rashba Sov. Phys. Solid State 2, 1224 (1960); Y. Bychkov and E.I Rashba JETP Lett. 39, 78 (1984).\n[6] R. Winkler, Spin-orbit Coupling E\u000bects in Two-Dimentional Electron and Hole Systems (Springer-Verlag, Berlin, 2003)\n[7] Y. K. Kato, R. C. Myers, A. C. Gossard, and D. D. Awschalom, Science 306, 1910 (2004); J. Wunderlich, B. Kaestner, J.\nSinova, T. Jungwirth, Phys. Rev. Lett. 94, 047204 (2005); S. O. Valenzuela and M. Tinkham, Nature 442, 176 (2006), D.\nCulcer, J. Sinova, N. A. Sinitsyn, T. Jungwirth and A. H. MacDonald and Q. Niu, Phys. Rev. Lett. 93, 046602 (2004); J.\nShi, P. Zhang, D. Xiao and Q. Niu, Phys. Rev. Lett. 96, 076604 (2006).\n[8] S. J. Barnett, Phys. Rev. 6, 239 (1915).\n[9] A. Einstein and W. J. de Haas, Verh. Dtsch. Phys. Ges. 17, 152 (1915).\n[10] R. T. Tolman and T. Stewart, Phys. Rev. 8, 97 (1916).\n[11] F. W. Hehl and Wei Tou Ni, Phys. Rev. D 42, 2045 (1990).\n[12] Debashree Chowdhury and B. Basu, Annals of Physics 329166 (2013).\n[13] B Basu, D Chowdhury, S Ghosh, arXiv:1212.4625.\n[14] M. Matsuo et.al., Physical Review B 84, 104410 (2011).\n[15] M. Matsuo , J Ieda, S Maekawa, Phys. Rev. B 87, 115301 (2013).\n[16] L.L Foldy and S.Wouthuysen, Phys Rev 78, 29 (1950).\n[17] W. Greiner, Relativistic Quantum Mechanics:Wave Equation (Springer-Verlag, Berlin, 2000), p.277.\n[18] E. O. Kane, J. Phys. Chem. Solids 1, 249 (1957).\n[19] E. M. Chudnovsky, Phys. Rev. Lett. 99, 206601 (2007).\n[20] E. M. Chudnovsky, Phys. Rev. B 80.153105 (2009).\n[21] T Fujita, M B A Jalil and S G Tan, New Journal of Physics 12, 013016 (2010).\n[22] K.Yu. Bliokh, Yu.P. Bliokh, Annals of Physics 319, 13 (2005).\n[23] M.V. Berry, Proc. R. Soc. London A 392, 45 (1984).\n[24] S. Q. Shen, Phys. Rev. B 70, 081311(R)(2004).\n[25] Y. Z. Xu and Z. Shi, Appl. Phys. Lett., 81, 691 (2002), Y. Jiang, M. B. A. Jalil, and T. S. Low, Appl. Phys. Lett., 80,\n1673, (2002), M. B. A. Jalil, S. G. Tan, T. Liew, K. L. Teo, and T. C. Chong, J. Appl. Phys., 95, 7321, (2004).\n[26] Y. Aharonov and A. Casher, Phys. Rev. lett. 53, 319 (1984).\n[27] N. Hatano, R. Shirasaki and H. Nakamura, Phys. Rev. A 75, 032107 (2007).\n[28] X. Hao, J. Moodera and R. Meservey, Phys. Rev. B 42, 8235 (1990).\n[29] A. Majumdar, Phys. Rev. B 54, 11911, (1996).\n[30] T. Fujita, M. B. A. Jalil and S. G. Tan, IEEE Transactions of Magnetics, 46, 6 (2010)."    },    {        "title": "1709.07042v1.Superfluid_transition_temperature_of_spin_orbit_and_Rabi_coupled_fermions_with_tunable_interactions.pdf",        "content": "Super\ruid transition temperature of spin-orbit and\nRabi coupled fermions with tunable interactions\nPhilip D. Powell,1, 2Gordon Baym,2and C. A. R. S\u0013 a de Melo3\n1Lawrence Livermore National Laboratory, 7000 East Ave., Livermore, CA 94550, USA\n2Department of Physics, University of Illinois at Urbana-Champaign,\n1110 W. Green Street, Urbana, Illinois 61801, USA\n3School of Physics, Georgia Institute of Technology, 837 State Street, Atlanta, Georgia 30332, USA\n(Dated: October 12, 2021)\nWe obtain the super\ruid transition temperature of equal Rashba-Dresselhaus spin-orbit and Rabi\ncoupled Fermi super\ruids, from the Bardeen-Cooper-Schrie\u000ber (BCS) to Bose-Einstein condensate\n(BEC) regimes in three dimensions. Spin-orbit coupling enhances the critical temperature in the\nBEC limit, and can convert a \frst order phase transition in the presence of Rabi coupling into second\norder, as a function of the Rabi coupling for \fxed interactions. We derive the Ginzburg-Landau\nequation to sixth power in the super\ruid order parameter to describe both \frst and second order\ntransitions as a function of spin-orbit and Rabi couplings.\nPACS numbers: 67.85.Lm, 03.75.Ss, 47.37.+q, 74.25.Uv, 75.30.Kz\nThe ability to simulate magnetic \felds in cold atoms\nsystems opens the possibility of exploring new physics\nunachievable elsewhere. In addition to arti\fcial Abelian\nmagnetic \felds [1{3], one can also generate non-Abelian\n\felds in both bosonic and fermionic systems [4{12]. The\nlatter will eventually lead to the possibility of simulat-\ning quantum chromodynamics lattice gauge theory [13{\n16]. Present experiments on three dimensional spin-orbit\ncoupled Fermi gases are still at too high a temperature\nfor these systems to undergo Bardeen-Cooper-Schrie\u000ber\n(BCS) pairing, because the current Raman scheme causes\nheating. In contrast, theory has concentrated at zero\ntemperature [17{22]. Once such fermionic systems can be\ncooled below the super\ruid transition temperature, the\nspin-orbit coupling is expected to reveal new states with\nnon-conventional pairing. Even weak spin orbit coupling\nwill produce an admixture of s-wave and p-wave pairing.\nIn this Letter, we investigate the transition tempera-\nture of Fermi super\ruids with an equal mixture of Rashba\nand Dresselhaus spin-orbit coupling as a function of the\nRabi coupling, throughout the entire BCS (Bardeen-\nCooper-Schrie\u000ber)-to-BEC (Bose-Einstein condensation)\nevolution in three dimensions; the single particle Hamil-\ntonian matrix is\nHso(^p) =(^px\u0000\u0014\u001by)2\n2m+^p2\ny\n2m+^p2\nz\n2m\u0000\n2\u001bz; (1)\nthe Pauli sigma matrices operate in the two-level space,\n^pis the momentum, \n is the Rabi frequency, and \u0014is\nthe momentum transfer to the atoms in a two-photon Ra-\nman process [7]. This problem bears a close relation to\nspin-orbit coupling in solids, where the coupling \u0018pi\u001bj\nis intrinsic, and where the role of the Rabi frequency is\nplayed by an external Zeeman magnetic \feld. While a\nmean \feld treatment describes well the evolution from\nthe BCS to the BEC regime at zero temperature [23],\nthis order of approximation fails to describe the correctcritical temperature of the system in the BEC regime,\nbecause the physics of two-body bound states (Feshbach\nmolecules) [24] is not captured when the pairing order pa-\nrameter goes to zero. To remedy this problem, we include\ne\u000bects of order-parameter \ructuations in the thermody-\nnamic potential.\nWe stress that our present results are applicable to\nboth neutral cold atomic and charged condensed matter\nsystems. We \fnd that the spin-orbit coupling can en-\nhance the critical temperature of the super\ruid in the\nBEC regime and that it can convert a discontinuous \frst\norder phase transition in the presence of Rabi coupling\ninto a continuous second order transition, as a function of\nthe Rabi frequency (or Zeeman \feld in solids) for \fxed in-\nteractions. We analyze the nature of the phase transition\nin terms of the Ginzburg-Landau free energy, calculating\nit to six powers of the super\ruid order parameter to al-\nlow for the description of continuous and discontinuous\ntransitions as a function of the spin-orbit coupling, Rabi\nfrequency, and interactions.\nTo describe three dimensional Fermi super\ruids in the\npresence of spin-orbit and Zeeman \felds, we start from\nthe Hamiltonian density\nH(r) =Hso(r) +HI(r); (2)\nand use units ~=kB= 1. The \frst term in Eq. (2)\nis the independent-particle contribution including spin-\norbit coupling,\nHso(r) =X\nss0 y\ns(r) [Hso(^p)]ss0 s0(r); (3)\nThe second term describes the two-body s-wave contact\ninteraction\nHI(r) =\u0000g y\n\"(r) y\n#(r) #(r) \"(r); (4)\nwhere the arrows indicate the pseudospins of the\nfermions, which we refer to simply as \\spins\". Here g>0arXiv:1709.07042v1  [cond-mat.quant-gas]  20 Sep 20172\ncorresponds to a constant attraction between opposite\nspins.\nThe pairing \feld \u0001( r;\u001c) =\u0000gh #(r;\u001c) \"(r;\u001c)ide-\nscribes the formation of pairs of two fermions with oppo-\nsite spins, where \u001c=itis the imaginary time. Standard\nmanipulations lead to the Lagrangian density\nL(r;\u001c) =1\n2\ty(r;\u001c)G\u00001(r;\u001c)\t(r;\u001c) +j\u0001(r;\u001c)j2\ng\n+K(r)\u000e(r\u0000r0); (5)\nwhere \t = (  \" # y\n\" y\n#)Tis the Nambu spinor, and\nK\u0011\u0000r2=2m\u0000\u0016is the kinetic energy operator measured\nwith respect to the fermion chemical potential \u0016. We\nnote that the de\fnition of \u0016already includes the overall\npositive shift \u00142=2min the single particle kinetic energies\ndue to spin-orbit coupling, that is, \u0016is measured with\nrespect to\u00142=2m.\nThe inverse Green's function appearing in Eq. (5) is\nG\u00001\nk(\u001c) =0\nBB@@\u001c\u0000K\"\u0000i\u0014kx=m 0\u0000\u0001\ni\u0014kx=m @\u001c\u0000K# \u0001 0\n0 \u0001\u0003@\u001c+K\"\u0000i\u0014kx=m\n\u0000\u0001\u00030i\u0014kx=m @\u001c+K#1\nCCA;\n(6)\nwhereK\"=K\u0000\n=2;andK#=K+ \n=2 are the ki-\nnetic energy terms shifted by the Rabi coupling. As men-\ntioned above, the mean \feld treatment fails to describe\nthe correct critical temperature of the system in the BEC\nregime. To incorporate the physics of two-body bound\nstates, we must include e\u000bects of order-parameter \ructu-\nations in the thermodynamic potential.\nTo obtain the transition temperature to the super\ruid\nstate, we analyze the partition function Zas a functional\nintegralR\nD\u0001D\u0001\u0003R\nD\tD\tyeSfor the Fermi super\ruid,\nwhereS=R\f\n0R\nd3rL(r;\u001c) is the full action of the sys-\ntem. Upon integration over the fermion \felds, the ther-\nmodynamic potential \n = \u0000TlnZcontains two terms\n\n = \n 0+ \nF, where \n 0=\u0000TlnZ0=\u0000TS0is the sad-\ndle point contribution, at which point \u0001( r;\u001c) = \u0001 0, and\n\nF=\u0000TlnZFis the \ructuation part. The subscript 0\ndenotes quantities calculated in mean \feld.\nThe mean-\feld (saddle-point) term in the thermody-\nnamic potential is\n\n0=Vj\u00010j2\ng\u0000T\n2X\nk;jlnh\n1 +e\u0000\fEj(k)i\n+X\nk\u0018k;(7)\nwhere\u0018k=\"k\u0000\u0016,\"k=k2=2m, and theEj(k) are the\neigenvalues of the Nambu Hamiltonian matrix H0(k) =\n1@\u001c\u0000G\u00001\nk(\u001c), withj=f1;2;3;4g. The \frst set of eigen-\nvalues\nE1;2(k) =r\nE2\n0;k+h2\nk\u00062q\nE2\n0;k+h2\nk\u0000j\u00010j2(\u0014kx=m)2;\n(8)describe quasiparticle excitations, and the second set\nof eigenvalues E3;4(k) =\u0000E1;2(k) correspond to\nquasiholes. Here E0;k=p\n\u00182\nk+j\u00010j2;andhk\u0011p\n(\u0014kx=m)2+ \n2=4 is the magnitude of the combined\nspin-orbit and Rabi couplings.\nThe order parameter equation is found from the saddle\npoint condition \u000e\n0=\u000e\u0001\u0003\n0jT;V;\u0016 = 0, leading to\nm\n4\u0019as=1\n2VX\nk\u00141\n\"k\u0000A+\u0000h2\nz\n\u0018khkA\u0000\u0015\n: (9)\nHere, we write the interaction gin terms of the renor-\nmalizeds-wave scattering length asvia the relation\n1=g=\u0000m=4\u0019as+(1=V)P\nk1=2\"k[25{27], and for short\nwriteA\u0006= (1\u00002nk;1)=2E1\u0006(1\u00002nk;2)=2E2;with\nnk;i= 1=(e\fEk;i+1) the Fermi function. In addition, the\nparticle number at the saddle point N0=\u0000@\n0=@\u0016jT;V;\nis given by\nN0=X\nk\u001a\n1\u0000\u0018k\u0014\nA++(\u0014kx=m)2\n\u0018khkA\u0000\u0015\u001b\n: (10)\nThe saddle point transition temperature T0is deter-\nmined by solving Eq. (9) for given \u0016. The correspond-\ning number of particles is given by Eq. (10). This mean\n\feld treatment leads to a transition temperature grow-\ning ase1=kFasforkFas!0+. To \fnd the physically\ncorrect transition temperature we must, in construct-\ning the thermodynamic potential, include the physics of\ntwo-body bound states near the transition via the two-\nparticle t-matrix [28, 29]. With all the two particle chan-\nnels taken into account, the t-matrix calculation leads to\na two-particle scattering amplitude, \u0000, where\n\u0000\u00001(q;z) =m\n4\u0019as\u00001\n2VX\nk\u00141\n\"k+2X\ni;j=1\u000bijWij\u0015\n; (11)\nherezis the (complex) frequency, Wij= (1\u0000nk;i\u0000\nnk+q;j)=(z\u0000Ei(k)\u0000Ej(k+q)):In the limit that the\norder parameter goes to 0, the single particle eigenvalues\nreduce toE1;2(k) =\u0018k\u0006hk[30]. The coe\u000ecients \u000b11=\n\u000b22=jukuk+q\u0000vkv\u0003\nk+qj2and\u000b12=\u000b21=jukvk+q+\nuk+qvkj2are weighting functions of the amplitudes\nuk=s\n1\n2\u0012\n1 +\n2hk\u0013\n; v k=is\n1\n2\u0012\n1\u0000\n2hk\u0013\n:(12)\nAs the fermion chemical potential becomes large\nand negative, the system becomes non-degenerate and\n\u0000\u00001(q;z) = 0 becomes the exact eigenvalue equation for\nthe two-body bound state in the presence of spin-orbit\nand Rabi coupling [31]. The solution is z=Ebs(q)\u00002\u0016,\nwhereEbs(q) is the two-body bound state energy. The\n\ructuation correction to the thermodynamic potential is\nthen \nF=\u0000TP\nq;iqnln [\f\u0000(q;iqn)=V]:3\nFrom \nFwe obtain the \ructuation contribution to the\nparticle number NF=\u0000@\nF=@\u0016jT;V=Nsc+Nb. Here,\nNsc=X\nqZ1\n!tp(q)d!\n\u0019nB(!)\u0014@\u000e(q;!)\n@\u0016\u0000@\u000e(q;0)\n@\u0016\u0015\nV;T\n(13)\nis the number of particles in scattering states, where the\nphase shift \u000e(q;!) is de\fned via the relation \u0000( q;!\u0006\ni\u000f) =j\u0000(q;!)je\u0006i\u000e(q;!)and!tp(q) is the two-particle\ncontinuum threshold corresponding to the branch point\nof \u0000\u00001(q;z) [28, 32]. Also,\nNb= 2X\nqnB(Ebs(q)); (14)\nis the number of fermions in bound states with nB(!) =\n1=(e\f!\u00001) the Bose distribution function. The total\nnumber of fermions, as a function of \u0016, becomes\nN=N0+NF; (15)\nwhereN0is given in Eq. (10) and NFis the sum the two\ncontributions NscandNbdiscussed above [24, 28].\nDe\fningkFto be the Fermi momentum of the atomic\ngas with total density n=k3\nF=3\u00192, we obtain Tcas a\nfunction of the scattering parameter 1 =kFasby solving\nsimultaneously the order parameter and number equa-\ntions (9) and (15). Figure 1 shows the e\u000bects of spin-\norbit and Rabi couplings on the transition temperature\nTc. The solutions correspond to minima of the free en-\nergyF= \n +\u0016N. In Fig. 1, we scale energies and\ntemperatures by the Fermi energy \"F=k2\nF=2m.\nThe solid (black) line in Fig. 1a shows the transi-\ntion temperature Tcbetween the normal and super\ruid\nstate versus the scattering parameter 1 =kFasfor zero\nRabi coupling (\n = 0) and zero one-dimensional Rashba-\nDresselhaus (ERD) [33{35] spin-orbit coupling ( \u0014= 0).\nIf \n = 0, the spin-orbit coupling \u0014can be removed by a\nsimple gauge transformation, and thus plays no role. In\nthis situation, the pairing is purely s-wave. The dashed\n(blue) line shows Tcfor \n6= 0, with vanishing ERD spin-\norbit coupling. We see that for \fxed interaction strength,\nthe pair-breaking e\u000bect of the Rabi coupling (as a Zee-\nman \feld breaks pairs in a superconductor) suppresses\nsuper\ruidity, compared with \n = 0. With both ERD\nspin-orbit and Rabi coupling present, the pairing is no\nlonger pure s-wave, but has a triplet p-wave component\n(and higher) mixed into the super\ruid order parameter;\nthe admixture stabilizes the super\ruid phase, as shown\nby the dotted (green) line. The latter curve shows that\nin the BEC regime with large positive 1 =kFas, the tran-\nsition temperature Tcis larger with spin-orbit and Rabi\ncouplings than in their absence, as a consequence of the\nreduction of the bosonic e\u000bective mass in the x-direction\nbelow 2m. However, with su\u000eciently large \n, the ge-\nometric mean bosonic mass MBincreases and Tcde-\ncreases [36].\nTemperature\n-1/k f a     0No SOC\nNo SOC/uni03A9 = 0NORMAL\nPAIRED\nBEC molecules BCS pairs/uni03A9 = fSOC\nε/uni03A9 = fεa)\nFIG. 1: (Color online) a) The transition temperature Tcfor ERD\nspin-orbit coupling for two Rabi coupling strengths, \n = 0 and\n\"F. For \n = 0, solid (black) curve, Tcis that for zero spin-orbit\ncoupling, since the ERD \feld can be gauged away. The dashed\n(blue) line shows Tcfor zero spin-orbit coupling, with \n = \"F,\nwhile the dotted (green) line shows Tcfor \n = 0 and \"F, and\n\u0014= 0:5kF. In b)Tcis drawn at unitarity, 1 =kFas= 0, and in\nthe inset at 1 =kFas=\u00002:0, as a function of e\n = \n=\"F. The\nsolid (red) curves are for ~ \u0014= 0, and the dashed (blue) curves are\nfor ~\u0014= 0:5. Across the dotted (red) curves below the solid (red)\ncurves, the phase transition is \frst order.\nFigure 1b shows Tcversus \n for \fxed 1 =kFas, without\nand with ERD spin-orbit coupling at \u0014= 0:5kF. When\u0014\nand the temperature are zero, super\ruidity is destroyed\nat a critical value of \n corresponding to the Clogston\nlimit [37]. At low temperature the phase transition to\nthe normal state is \frst order, because the Rabi coupling\n(Zeeman \feld) is su\u000eciently large to break singlet Cooper\npairs. However, at higher temperatures the singlet s-wave\nsuper\ruid starts to become polarized due to thermally\nexcited quasiparticles that produce a paramagnetic re-\nsponse. Therefore above the characteristic temperature\nindicated by the large (red) dots, the transition becomes\nsecond order, as pointed out by Sarma [38]. The critical\ntemperature for \u00146= 0 vanishes only asymptotically in\nthe limit of large \n. We note that for \n = EFand\u0014= 0\nthe transition from the super\ruid to the normal state is\ncontinuous at unitarity, but very close to a discontinuous\ntransition. In the range 1 :05.\n=EF.1:10 numerical4\nuncertainties as \u0014!0 prevent us from predicting ex-\nactly whether the transition at unitarity is continuous or\ndiscontinuous.\nTo understand further the e\u000bects of \ructuations on the\norder of the transition to the super\ruid phase and to as-\nsess the impact of spin-orbit and Rabi couplings near the\ncritical temperature, we now derive the Ginzburg-Landau\ndescription of the free energy near the transition, where\nthe actionSFcan be expanded in powers of the order\nparameter \u0001( q), beyond Gaussian order. The expansion\nofSFto quartic power is su\u000ecient to describe the con-\ntinuous (second order) transition in Tcversus 1=kFasin\nthe absence of an external Zeeman \feld [24]. However,\nto describe correctly the \frst order transition [37, 38] at\nlow temperature (Fig. 1), it is necessary to expand the\nfree energy to sixth order in \u0001.\nThe quadratic (Gaussian order) term in the action is\nSG=\fVX\nqj\u0001qj2\n\u0000(q;z): (16)\nFor an order parameter varying slowly in space and time,\nwe may expand\n\u0000\u00001(q;z) =a+ciq2\ni\n2m\u0000d0z+\u0001\u0001\u0001; (17)\nwith the sum over i=x;y;z implicit. The full result, as\na functional of \u0001( r;\u001c), has the form\nSF=Z\f\n0d\u001cZ\nd3r\u0010\nd0\u0001\u0003@\n@\u001c\u0001 +aj\u0001j2\n+cijri\u0001j2\n2m+b\n2j\u0001j4+f\n3j\u0001j6\u0011\n: (18)\nThe full time-dependent Ginzburg-Landau action de-\nscribes systems in and near equilibrium, e.g., with col-\nlective modes. The imaginary part of d0measures the\nnon-conservation of j\u0001j2in time.\nWe are interested here in systems at thermodynamic\nequilibrium where the order parameter is independent of\ntime. Then minimizing the free energy TSFwith respect\nto \u0001\u0003, we obtain the Ginzburg-Landau equation\n\u0010\n\u0000ci\n2mr2\ni+bj\u0001j2+fj\u0001j4+a\u0011\n\u0001 = 0: (19)\nForbpositive the system undergoes a continuous phase\ntransition when achanges sign. However, when bis nega-\ntive the system is unstable in the absence of f. Forb<0\nanda > 0, a \frst order phase transition occurs when\n3b2= 16af. Positivefstabilizes the system even when\nb<0.\nIn the BEC regime, we de\fne an e\u000bective bosonic wave-\nfunction \t =pd0\u0001 to recast Eq. (19) in the form of the\nGross-Pitaevskii equation for a dilute Bose gas\n\u0012\n\u0000r2\ni\n2Mi+U2j\t(r)j2+U3j\t(r)j4\u0000\u0016B\u0013\n\t(r) = 0:\n(20)Here,\u0016B=\u0000a=d 0is the bosonic chemical potential, the\nMi=m(d0=ci) are the anisotropic bosonic masses, and\nU2=b=d2\n0andU3=f=d3\n0represent contact interactions\nof two and three bosons. In the BEC regime these terms\nare always positive, thus leading to a system consisting\nof a dilute gas of stable bosons. The boson chemical\npotential\u0016Bis\u00192\u0016+Eb<0, whereEbis the two-\nbody bound state energy in the presence of spin-orbit\ncoupling and Rabi frequency, obtained from the condition\n\u0000\u00001(q;E\u00002\u0016) = 0, discussed earlier.\nThe anisotropy of the e\u000bective bosonic masses, Mx6=\nMy=Mz\u0011M?stems from the anisotropy of the ERD\nspin-orbit coupling, which together with the Rabi cou-\npling modi\fes the dispersion of the constituent fermions\nalong thexdirection. In the limit kFas\u001c1 the many-\nbody e\u000bective masses reduce to those obtained by ex-\npanding the two-body binding energy Ebs(q)\u0019\u0000Eb+\nq2\ni=2Mi;and agree with known results [31]. However,\nfor 1=kFas<\u00182, many-body and thermal e\u000bects produce\ndeviations from the two-body result.\nIn the absence of two and three-body boson-boson in-\nteractions ( U2andU3), we directly obtain the analytic\nexpression for Tcin the Bose limit from Eq. (14),\nTc=2\u0019\nMB\u0012nB\n\u0010(3=2)\u00132=3\n; (21)\nwithMB= (MxM2\n?)1=3, by noting that !p(q) =Ebs(q)\nand using the condition that nB'n=2 (with corrections\nexponentially small in (1 =kFas)2), wherenBis the den-\nsity of bosons and nis the density of fermions. In the\nBEC regime, the results shown in Fig. 1 include the ef-\nfects of the mass anisotropy, but do not include e\u000bects of\nboson-boson interactions.\nTo account for boson-boson interactions, we use the\nHamiltonian of Eq. (20) with U26= 0, but with U3= 0,\nand apply the method developed in Ref. [39] to show that\nthese interactions further increase TBEC to\nTc(aB) = (1 +\r)TBEC; (22)\nwhere\r=\u0015n1=3\nBaB. Here,aBis the s-wave boson-\nboson scattering length, \u0015is a dimensionless constant\n\u00181, and we used the relation U2= 4\u0019aB=MB. Since\nnB=k3\nF=6\u00192and the boson-boson scattering length is\naB=U2MB=4\u0019, we have\r=~\u0015fMBeU2;wherefMB=\nMB=2m;eU2=U2k3\nF=\"F;and~\u0015=\u0015=4(6\u00195)1=3\u0019\u0015=50:\nFor \fxed 1=kFas,Tcis enhanced both by a spin-orbit and\nthe \n dependent decrease in the e\u000bective boson mass MB\n(\u001810-15%), as well as a stabilizing boson-boson repulsion\nU2(\u00182-3%), for the parameters used in Fig. 1.\nIn summary, we have analysed the \fnite temperature\nphase diagram of three dimensional Fermi super\ruids in\nthe presence ERD spin-orbit coupling, Rabi coupling,\nand tunable s-wave interactions. Furthermore, we de-\nveloped the Ginzburg-Landau theory up to sixth power5\nin the amplitude of the order parameter to show the ori-\ngin of discontinuous (\frst order) phase transitions when\nthe Rabi frequency is su\u000eciently large for vanishing spin-\norbit coupling.\nThe research of author PDP was supported in part\nby NSF Grant PHY1305891 and that of GB by NSF\nGrants PHY1305891 and PHY1714042. Both GB and\nCARSdM thank the Aspen Center for Physics, supported\nby NSF Grants PHY1066292 and PHY1607611, where\npart of this work was done. This work was performed\nunder the auspices of the U.S. Department of Energy by\nLawrence Livermore National Laboratory under contract\nDE- AC52- 07NA27344.\n[1] Y. Lin, R. Compton, K. Jimin\u0013 ez-Garc\u0013 \u0010a, J. Porto, and I.\nSpielman, Nature (London) 462, 628 (2009).\n[2] Y. Lin, R. Compton, A. Perry, W. Phillips, J. Porto, and\nI. Spielman, Phys. Rev. Lett. 102, 130401 (2009).\n[3] C. J. Kennedy, W. C. Burton, W. C. Chung and W.\nKetterle, Nature Physics, 11, 859 (2015).\n[4] Y.-J. Lin, K. Jim\u0013 enez-Garc\u0013 \u0010a and I. B. Spielman, Nature\n471, 83 (2011).\n[5] L. W. Cheuk, A. T. Sommer, Z. Hadzibabic, T. Yefsah,\nW. S. Bakr, and M. W. Zwierlein, Phys. Rev. Lett. 109,\n095302 (2012).\n[6] P. Wang, Z.-Q. Yu, Z, Fu, J, Miao, L, Huang, S. Chai, H,\nZhai, and J. Zhang Phys. Rev. Lett. 109, 095301 (2012).\n[7] R. A. Williams, M. C. Beeler, L. J. LeBlanc, K. Jimin\u0013 ez-\nGarc\u0013 \u0010a and I. B. Spielman, Phys. Rev. Lett. 111, 095301\n(2013).\n[8] V. Galitski and I. B. Spielman, Nature 494, 49 (2013).\n[9] Z. Fu, L. Huang, Z. Meng, P. Wang, L. Zhang, S. Zhang,\nH. Zhai, P. Zhang, and J. Zhang, Nat. Phys. 10, 110\n(2014).\n[10] M. Mancini, G. Pagano, G. Cappellini, L. Livi, M. Rider,\nJ. Catani, C. Sias P. Zoller, M. Inguscio, M. Dalmonte,\nand L. Fallani, Science 349, 1510 (2015).\n[11] L. Huang, Z. Meng, P. Wang, P. Peng, S.-L. Zhang, L.\nChen, D. Li, Q. Zhou, and J. Zhang, Nature Physics 12,\n540 (2016).\n[12] Z. Wu, L.Zhang, W. Sun, X.-T. Xu, B.-Z. Wang, S.-C.\nJi, Y. Deng, S. Chen, X.-J. Liu, and J.-W. Pan, Science\n354, 83-88 (2016).\n[13] J. I. Cirac and P. Zoller, Nature Phys. 8, 264 (2012).\n[14] U.-J. Wiese, Ultracold Quantum Gases and Lattice Sys-\ntems: Quantum Simulation of Lattice Gauge Theories,\nAnnalen der Physik 525, 777 (2013).\n[15] E. Zohar, J. I. Cirac, and B. Reznik, Rep. Prog. Phys.\n79, 1 (2015).\n[16] M. Dalmonte and S. Montangero, Contemp. Phys. 57388\n(2016); arXiv:1602.03776.\n[17] M. Gong, S. Tewari, and C. Zhang, Phys. Rev. Lett. 107,\n195303 (2011).\n[18] Z.-Q. Yu and H. Zhai, Phys. Rev. Lett. 107, 195305\n(2011).[19] H. Hu, L. Jiang, X.-J. Liu, and H. Pu, Phys. Rev. Lett.\n107, 195304 (2011).\n[20] L. Han and C. A. R. S\u0013 a de Melo, Phys. Rev. A 85,\n011606(R) (2012).\n[21] K. Seo, L. Han, and C. A. R. S\u0013 a de Melo, Phys. Rev.\nLett. 109, 105303 (2012).\n[22] K. Seo, L. Han, and C. A. R. S\u0013 a de Melo, Phys. Rev. A\n85, 033601 (2012).\n[23] A. J. Leggett in Modern Trends in the Theory of Con-\ndensed Matter edited by A. Pekalski and R. Przystawa,\nSpringer-Verlag, Berlin (1980).\n[24] C. A. R. S\u0013 a de Melo, M. Randeria, and J. Engelbrecht,\nPhys. Rev. Lett. 71, 3202 (1993).\n[25] Note that asis thes-wave scattering length in the ab-\nsence of spin-orbit and Zeeman \felds. It is, of course,\npossible to express all relations obtained in terms of a\nscattering length which is renormalized by the presence\nof the spin-orbit and Zeeman \felds [26, 27]. However,\nin addition to complicating our already cumbersome ex-\npressions, it would make reference to a quantity that is\nmore di\u000ecult to measure experimentally and that would\nhide the explicit dependence of the properties that we\nanalyzed in terms of the spin-orbit and Zeeman \felds, so\nwe do not consider such complications here.\n[26] S. Gopalakrishnan, A. Lamacraft, and P. M. Goldbart,\nPhys. Rev. A 84, 061604 (2011).\n[27] T. Ozawa, Ph.D. Thesis, University of Illinois (2012).\n[28] P. Nozi\u0012 eres and S. Schmitt-Rink, J. Low Temp. Phys.,\n59, 195 (1985).\n[29] Z. Yu and G. Baym, Phys. Rev. A 73, 063601 (2006).\n[30] Note that setting \u0001 0= 0 in the general eigenvalue ex-\npressions yields E1;2=jj\u0018kj\u0006hkj. It is straightforward\nto show that neglecting the absolute values does not re-\nsult in any change in either the mean \feld order pa-\nrameter or number equation. Thus, for simplicity we let\nE1;2=\u0018k\u0006hk.\n[31] D. M. Kurkcuoglu and C. A. R. S\u0013 a de Melo, Phys. Rev.\nA93, 023611 (2016). See also arXiv:1306.1964v1 (2013).\n[32] Z. Yu, G. Baym, and C. J. Pethick, J. Phys. B: At. Mol.\nOpt. Phys. 44, 195207 (2011).\n[33] G. Dresselhaus, Phys. Rev. 100, 580 (1955).\n[34] E. Rashba, Fizika Tverdogo Tela 2, 1244 (1960).\n[35] This form is equivalent to another common form of\nthe Rashba-Dresselhaus coupling found in the litera-\nture: hso=hR+hDwhere hR=vR(kx^y\u0000ky^x)\nandhD=vD(kx^y+ky^x). The two forms are related\nvia a momentum-space rotation and the correspondences\n\u0014=m(vR+vD) and\u0011= (vR\u0000vD)=(vR+vD). The\nEqual-Rashba-Dresselhaus limit (ERD) corresponds to\nvR=vD=v, leading to \u0011= 0 and\u0014= 2mv.\n[36] This renormalization of the mass of the bosons can be\ntraced back to the actual change in the energy disper-\nsion of the fermions when both spin-orbit coupling and\nZeeman \felds are present.\n[37] A. M. Clogston, Phys. Rev. Lett. 9, 266 (1962).\n[38] G. Sarma, J. Phys. Chem. Sol. 24, 1029 (1963).\n[39] G. Baym, J.-P. Blaizot, M. Holzmann, F. Lalo e and D.\nVautherin, Phys. Rev. Lett. 83, 1703 (1999)."    },    {        "title": "2401.16738v1.A_novel_non_adiabatic_spin_relaxation_mechanism_in_molecular_qubits.pdf",        "content": "A novel non-adiabatic spin relaxation mechanism in molecular qubits\nPhilip Shushkova)\nDepartment of Chemistry, Indiana University, Bloomington, Indiana, 47405\n(Dated: 31 January 2024)\nThe interaction of electronic spin and molecular vibrations mediated by spin-orbit coupling governs spin relaxation in\nmolecular qubits. I derive an extended molecular spin Hamiltonian that includes both adiabatic and non-adiabatic spin-\ndependent interactions, and I implement the computation of its matrix elements using state-of-the-art density functional\ntheory. The new molecular spin Hamiltonian contains a novel spin-vibrational orbit interaction with non-adiabatic\norigin together with the traditional molecular Zeeman and zero-field splitting interactions with adiabatic origin. The\nspin-vibrational orbit interaction represents a non-Abelian Berry curvature on the ground-state electronic manifold and\ncorresponds to an effective magnetic field in the electronic spin dynamics. I further develop a spin relaxation rate\nmodel that estimates the spin relaxation time via the two-phonon Raman process. An application of the extended\nmolecular spin Hamiltonian together with the spin relaxation rate model to Cu(II) porphyrin, a prototypical S=1/2\nmolecular qubit, demonstrates that the spin relaxation time at elevated temperatures is dominated by the non-adiabatic\nspin-vibrational orbit interaction. The computed spin relaxation rate and its magnetic field orientation dependence are\nin excellent agreement with experimental measurements.\nI. INTRODUCTION\nMolecular qubits, paramagnetic systems that exhibit long\nspin-coherence times, are emerging as a promising platform\nfor the implementation of quantum information processing.1–4\nThe molecular electronic spin is an excellent candidate to\nencode quantum information because of protection by time-\nreversal symmetry. Taken together with the unprecedented\naccuracy of chemical methods to synthesize and assem-\nble molecular systems, there has been a recent surge in\nefforts to engineer molecular qubits that are suitable to\nadvance the much sought-after room-temperature quantum\ntechnologies.5–18\nA major limiting factor to the room-temperature quantum\ninformation storage in molecular qubits is the spin relax-\nation time T1, called also spin-lattice or longitudinal relax-\nation time,19that characterizes the timescale of thermal equi-\nlibration of the electronic spin by the molecular vibrational\nmotion.5–7,9The spin relaxation theory dates back to the pio-\nneering work of Van Vleck,20Mattuck and Strandberg,21and\nOrbach,22who focused on the relaxation dynamics of para-\nmagnetic impurities in crystals. The physical picture that\nemerged from their seminal contributions is that both adia-\nbatic and non-adiabatic processes can dominate the spin re-\nlaxation time depending on the specifics of the electronic\nstructure of the paramagnetic impurities. Furthermore, they\nshowed that one-phonon processes contribute to the the spin\nrelaxation dynamics only at very low temperatures and at el-\nevated temperatures spin relaxation is driven by two-phonon\nabsorption-emission processes.\nDensity functional theory and multi-reference wavefunc-\ntion approaches have provided a firm theoretical basis to pre-\ndict molecular spin interactions.23–28Fundamental to this suc-\ncess is the concept of the static spin Hamiltonian, an effective\nHamiltonian defined at the equilibrium nuclear geometry that\na)phgshush@iu.eduincorporates the influence of the molecular electronic struc-\nture in spin-dependent interactions.29,30The dynamical exten-\nsion of the static spin Hamiltonian by accounting for the nu-\nclear geometry-dependence of the traditional spin-dependent\ninteractions has underscored recent efforts to simulate the spin\nrelaxation dynamics of molecular qubits.31–38This approach\nto spin relaxation, however, does not account for the contribu-\ntion of the non-adiabatic interactions.\nThe goal of the paper is to derive a molecular spin Hamil-\ntonian that includes both adiabatic and non-adiabatic spin-\ndependent interactions and to implement the computation of\nthe matrix elements of this Hamiltonian using density func-\ntional theory. I achieve this goal by applying the Born-\nOppenheimer approximation in the adiabatic representation\nfor the electronic wavefunctions,39,40circumventing the need\nfor diabatization, and allowing seamless integration with\nlinear response density functional theory. Similar to the\nstatic spin Hamiltonian, I use unitary degenerate perturbation\ntheory41,42to derive the molecular spin Hamiltonian, which\nlimits its applicability to molecular systems with weak spin-\norbit interaction and orbitally non-degenerate ground elec-\ntronic states.30The derived molecular spin Hamiltonian con-\ntains a novel, non-adiabatic spin-vibrational orbit interaction\ntogether with the traditional molecular Zeeman and zero-field\nsplitting interactions29that have an adiabatic origin. I fur-\nther develop a rate model to estimate the contribution to the\nspin relaxation time of the interactions in the molecular spin\nHamiltonian. The rate model is specialized to elevated tem-\nperatures and evaluates the spin relaxation time via the two-\nphonon Raman process20using density functional calcula-\ntions on the isolated paramagnetic molecule. A pilot appli-\ncation of the molecular spin Hamiltonian together with the\nspin relaxation rate model to Cu(II) porphyrin,43a prototyp-\nicalS=1/2 molecular qubit,44,45demonstrates that the two-\nphonon spin relaxation time is dominated by the non-adiabatic\nspin-vibrational orbit interaction with doubly degenerate nor-\nmal modes being the major vibrational relaxation channel.\nThe paper is organized as follows: I present the derivation\nof the molecular spin Hamiltonian and of the spin relaxationarXiv:2401.16738v1  [physics.chem-ph]  30 Jan 20242\nrate model in Sec. II. I outline the numerical evaluation of\nthe molecular spin Hamiltonian matrix elements using density\nfunctional theory in Sec. III. I present the results of the appli-\ncation of the new approach to Cu(II) porphyrin in Sec. IV,\nand I conclude the paper with an outline of future directions\nin Sec. V.\nII. THEORY\nSpin-vibrational interactions involve the coupling of the\nelectronic spin with the molecular vibrations that is mediated\nby the electronic orbital motion. I start Sec. II by specifying\nthe molecular Hamiltonian and establishing the notation used\nin the paper. I then present an overview of unitary degenerate\nperturbation theory42that I apply to derive the molecular spin\nHamiltonian in Sec. II A and proceed with the application of\nthe Born-Oppenheimer approximation. In Sec. II A, I derive\nthe expression for the Berry connection46–48on the ground-\nstate spin manifold and arrive at a molecular spin Hamilto-\nnian that contains a novel spin-vibronic vector potential. In\nSec. II B, I apply a unitary gauge transformation40to a sym-\nmetrized gauge for the spin-vibronic vector potential and de-\nrive the spin-vibrational orbit interaction. Harmonic expan-\nsion of the resulting molecular spin Hamiltonian in Sec. II C\ngives one- and two-phonon spin-vibrational interactions with\nnon-adiabatic and adiabatic origins. In Sec. II D, I present a\nrate model for the computation of the spin relaxation time of\nS=1/2 molecular qubits at elevated temperature and at weak\nto moderate magnetic field intensities using the two-phonon\nspin-vibrational interactions of Sec. II C.\nA. Molecular Spin Hamiltonian\nI write the molecular Hamiltonian,29including relativistic\ninteractions and the interaction of the molecule with an exter-\nnal magnetic field as:\nˆHmolec=ˆTR+ˆHNR\nR+ˆHSOZ\nR. (1)\nˆTRin Eq. (1) is the nuclear kinetic energy, ˆHNR\nR- the non-\nrelativistic electronic Hamiltonian together with the spin-\nindependent, scalar relativistic corrections, and ˆHSOZ\nR- the\nspin-dependent relativistic interactions that include the spin-\norbit, the electronic Zeeman, and the electronic spin-spin in-\nteractions. Rstands for the collection of Cartesian nuclear co-\nordinates, and the subscript denotes the explicit dependence\nof the Hamiltonian terms on the nuclear configuration. In this\npaper, I do not include the interactions of the electronic spin\nwith the nuclear magnetic moments, which give rise to hyper-\nfine splitting of the electronic energy levels.\nI define the zero-order adiabatic electronic wavefunctions,\f\f\fK(0)\nRE\n, and the zero-order adiabatic electronic potential en-\nergy surfaces, E(0)\nK,R, as the eigenvectors and the eigenvalues\nof the non-relativistic electronic Hamiltonian:\nˆHNR\nR\f\f\fK(0)\nRE\n=E(0)\nK,R\f\f\fK(0)\nRE\n. (2)The eigenfunctions and eigenvalues are zero order with re-\nspect to the spin-dependent interactions, in which case the\nzero-order electronic states form multiplets of degenerate\nstates that correspond to different projection of the electronic\nspin angular momentum. To account for the degeneracy of the\nzero-order electronic spectrum, the multi-index Kstands for a\ncollection of quantum numbers K={kSM S}that specify the\norbital quantum number k, the spin quantum number S, and\nthe associated spin projection MS. I assume that the state de-\ngeneracy in the multiplets is only of spin origin and exclude\ndegeneracy in the multiplets that results from spatial symme-\ntry. The adiabatic eigenfunctions in Eq. (2) are parametrically\ndependent on the nuclear configuration, R, and form an or-\nthonormal set at each nuclear geometry.\nThe spin-dependent interactions in Eq. (1) couple the elec-\ntronic states of the non-relativistic Hamiltonian and remove\npartially or fully the degeneracy of the electronic spectrum.\nI apply unitary degenerate perturbation theory42to diagonal-\nize the inter-multiplet interactions due to the spin-dependent\nHamiltonian ˆHSOZ\nR. The perturbative treatment is justified for\nmolecules that consist of atoms of light elements, including\nearly transition metal and main group elements, because for\nthem the spin-dependent multiplet splittings are significantly\nsmaller than the energy differences between the zero-order\nelectronic multiplets. The perturbed wavefunctions, |KR⟩, in\nunitary perturbation theory are obtained by a unitary transfor-\nmation of the zero-order wavefunctions:\n|KR⟩=eˆGR\f\f\fK(0)\nRE\n, (3)\nwhere ˆGis the generator of the transformation and satisfies\nˆG†=−ˆGto ensure the unitarity of the transformation. The\ngenerator is obtained perturbatively from the requirement of\nvanishing inter-multiplet interactions in the unitarily trans-\nformed Hamiltonian, and the perturbed eigenvalues derive\nfrom the intra-multiplet blocks of the transformed Hamilto-\nnian:\nD\nJ(0)\nR\f\f\fe−ˆGR\u0000ˆHNR\nR+ˆHSOZ\nR\u0001\neˆGR\f\f\fK(0)\nRE\n=δJKEK′K,R.(4)\nMulti-indeces J,K,..refer to general multiplets, and I reserve\nthe multi-index Ifor the ground-state multiplet. HIJstands\nfor the matrix block between two different multiplets IandJ,\nwhereas HI′Istands for the matrix block of the multiplet I.δJK\nstands for a multi-index Kronecker delta symbol. Use of the\nBaker-Campbell-Hausdorff formula in Eq. (4) and collection\nof terms according to the order of perturbation provides the\nequation for the first-order generator:\nh\nˆHNR\nR,ˆG(1)\nRi\nJK+HSOZ\nJK,R=0, (5)\nwhich is solved for the inter-multiplet blocks of the genera-\ntorG(1)\nJKand is supplemented with the additional condition of\nvanishing intra-multiplet blocks G(1)\nK′K=0. The perturbation\nexpansion in Eq. (4) gives also the perturbed eigenvalues up\nto second order in the perturbation:\nEK′K,R=E(0)\nK,RδK′K+HSOZ\nK′K,R+1\n2h\nˆHSOZ\nR,ˆG(1)\nRi\nK′K.(6)3\nEq. (5) gives the ground-state electronic wavefunction to first\norder in the spin-orbit perturbation as:\n|IR⟩=\f\f\fI(0)\nRE\n+ˆG(1)\nR\f\f\fI(0)\nRE\n=\f\f\fI(0)\nRE\n−∑\nJ̸=I\f\f\fJ(0)\nRE HSOI\nJI,R\nE(0)\nJ,R−E(0)\nI,R.(7)\nComponents of multiplets with different projections of spin\nangular momentum are non-interacting at zero order; coupling\nbetween them arises in the first-order contribution to the elec-\ntronic wavefunction, the second term in Eq. (7), from the spin-\norbit interaction (SOI) in ˆHSOZ\nR. The perturbative expansion of\nthe ground-state multiplet energies from Eq. (6) is given by:\nEI′I,R=E(0)\nI,RδI′I+E(1)\nI′I,R+E(2)\nI′I,R\n=E(0)\nI,RδI′I+HSOZ\nI′I,R+∑\nJ̸=IHSOZ\nI′J,RHSOZ\nJI,R\nE(0)\nI,R−E(0)\nJ,R,(8)\nwhere EI′I,Ris a matrix within the ground-state manifold,\nEM′M,R. The first-order and second-order adiabatic energy\ncontributions, the second and third term in Eq. (8), give rise\nto the traditional molecular spin Hamiltonian with the molec-\nular Zeeman interaction, µB⃗BgR⃗S, characterized by the elec-\ntronic g-tensor and µBthe Bohr magneton, and the zero-field\nsplitting interaction, ⃗SDR⃗S, characterized by the electronic D-\ntensor. Both the g-tensor and D-tensor depend parametrically\non the nuclear positions, and the existence of the molecular\nspin Hamiltonian requires an orbitally-non-degenerate ground\nstate, which is the case for all systems that I consider in the\npaper. I establish the connection with the traditional spin\nHamiltonian and the spin-component structure of the adia-\nbatic wavefunctions in Eq. (7) using the Wigner-Eckart the-\norem in Appendix A.\nI apply the Born-Oppenheimer approximation39,40toun-\ncouple the electron-nuclear dynamics on different electronic\nmultiplets and preserve the coupled electron-nuclear dynam-\nicswithin an electronic multiplet. This treatment of the\nelectron-nuclear dynamics is justified when the energy dif-\nferences between electronic multiplets are significantly larger\nthan the vibrational quanta involved in the spin relaxation dy-\nnamics, whereas the energy differences within electronic mul-\ntiplets are of similar magnitude or smaller than the vibrational\nquanta of the nuclear dynamics. This regime holds for the\nS=1/2 systems considered in this work. The case where\ntwo or more electronic multiplets come close in energy and\ninteract non-adiabatically as it occurs in conical intersections\nis outside the scope of the present treatment. In the Born-\nOppenheimer approximation, the vibronic wavefunction for\nthe ground-state multiplet is expressed as the sum of prod-\nucts of the electronic wavefunction of a multiplet component,\n|iSM,R⟩, and the associated nuclear wavefunction, χiSM(R):\n\f\fΨBO\nI\u000b\n=∑\nM,M′\f\fiSM′,R\u000b\ncM′M,R0χiSM(R) =|IR⟩χI(R).(9)\nThe electronic wavefunctions in Eq. (9) are the perturbed\nwavefunctions of Eqs. (3) and (7), and the last equality impliesa sum over the multiplet components. I choose a diabatic rep-\nresentation within the multiplet manifold that conserves the\ncharacter of the multiplet components, the dominant projec-\ntion of spin angular momentum M, and the matrix cM′M,R0\nin Eq. (9) allows for a specific choice of the spin quantiza-\ntion axis at a fixed nuclear configuration. I use this matrix to\nspecify the reference spin quantization axis for the spin re-\nlaxation dynamics at the equilibrium nuclear configuration.\nIf the matrix cM′M,Rwere allowed to vary with nuclear po-\nsition and at each nuclear configuration were chosen as the\neigenvectors of the matrix, EI′I,R, in Eq. (8), then the result-\ning wavefunctions, ∑M,M′|iSM′,R⟩cM′M,R, would be the adi-\nabatic electronic wavefunctions of the multiplet components\nand the associated eigenvalues would be the adiabatic poten-\ntial energy surfaces of the multiplet components.\nTo derive the Born-Oppenheimer Hamiltonian for the\nground-state multiplet, I apply the nuclear kinetic energy op-\nerator to the Born-Oppenheimer wavefunction in Eq. (9) and\nobtain:\nˆTR\f\fΨBO\nI\u000b\n=|IR⟩ˆTRχI(R)\n−i∂µ|IR⟩ˆpµχI(R)+χI(R)ˆTR|IR⟩.(10)\nThe notation in Eq. (10) implies a sum over the components\nof the electronic multiplet. I use the Einstein convention for\nan unrestricted summation over repeated indeces, and I ex-\nplicitly include the sum symbol when there are restrictions on\nthe sum indeces and when there is a summation without in-\ndex repetition. µis the index of the mass-weighted nuclear\nCartesian coordinates, and ˆ pµis the nuclear momentum oper-\nator of coordinate µ.iis the imaginary unit, and I use atomic\nunits, such that ¯h=1. The action of the nuclear kinetic en-\nergy operator gives terms with nuclear derivatives acting only\non the nuclear, first term, and the electronic wavefunctions,\nthird term, as well as a mixed term with nuclear derivatives\nacting on both the electronic and nuclear wavefunctions.\nI project the molecular Hamiltonian, Eq. (1), in the ground-\nstate multiplet, Eq. (9), and use Eqs. (4) and (10) to derive the\nBorn-Oppenheimer molecular Hamiltonian:\nˆHBO\nI′I=δI′IˆTR−iAI′Iµ,Rˆpµ+DI′I,R+EI′I,R, (11)\nThe Hamiltonian in Eq. (11) is a matrix within the ground-\nstate electronic multiplet space and an operator in nuclear\nCartesian coordinate space. The first term is the nuclear ki-\nnetic energy, and the last term is the adiabatic potential en-\nergy matrix of the ground-state multiplet, Eq. (8). The second\nand third terms are non-adiabatic interactions and contain the\nfirst-derivative non-adiabatic coupling matrix,\nAI′Iµ,R=\nI′\nR\f\f∂µ|IR⟩, (12)\nand the second-derivative non-adiabatic coupling matrix,\nDI′I,R=\nI′\nR\f\fˆTR|IR⟩. (13)\nAI′Iµ,Ris the Berry connection,40,46,48,49which in this case\nhas a non-Abelian algebraic structure and arises as a result of\nnon-adiabatic interactions with excited electronic multiplets.4\nAll inter-multiplet non-adiabatic coupling matrices vanish be-\ncause of the application of the Born-Oppenheimer approxima-\ntion.\nI expand the Berry connection in the spin-orbit interaction\nto derive explicit expressions for the non-adiabatic coupling\nmatrix elements between multiplet components. I use the per-\nturbative expansion of the electronic wavefunctions, Eq. (3),\nand the Baker-Campbell-Hausdorff formula in Eq. (12) and\nobtain to first order in the SOI:\nAI′Iµ,R=D\nI′(0)\nR\f\f\fe−ˆGR∂µeˆGR\f\f\fI(0)\nRE\n=D\nI′(0)\nR\f\f\f∂µI(0)\nRE\n+D\nI′(0)\nR\f\f\fh\n∂µ,ˆG(1)\nRi\f\f\fI(0)\nRE\n.(14)\nThe zero-order term of the Berry connection A(0)\nI′Iµ,R, the first\nterm in the last equality in Eq. (14), is diagonal in the ground-\nstate multiplet space and is the generalization of the traditional\nBerry connection that arises from conical intersections.46,47\nBecause I consider systems away from conical intersections in\nthis paper, A(0)\nI′Iµ,Rvanishes. The first-order term in the spin-\norbit interaction, A(1)\nI′Iµ,R, the second term in Eq. (14), is the\nleading contribution to the Berry connection on the ground-\nstate manifold and it couples different multiplet components\nas a result of the interplay of spin-orbit and non-adiabatic\ninteractions with excited electronic states. Expansion of the\ncommutator in Eq. (14) gives an explicit formula for the first-\norder Berry connection in terms of zero-order adiabatic elec-\ntronic wavefunctions:\nA(1)\nI′Iµ,R=∑\nJ̸=IHSOI\nI′J,R\nE(0)\nI′,R−E(0)\nJ,RD\nJ(0)\nR\f\f\f∂µI(0)\nRE\n−∑\nJ̸=ID\nI′(0)\nR\f\f\f∂µJ(0)\nRE HSOI\nJI,R\nE(0)\nJ,R−E(0)\nI,R.(15)\nThe non-Abelian algebraic structure of A(1)\nI′Iµ,Rfollows from\nan application of the Wigner-Eckart theorem50to Eq. (15),\nwhich gives an expression for A(1)\nI′Iµ,Rin terms of the set of\nnon-commuting spin matrices, SM′M,α, with dimension equal\nto the dimensionality of the ground-state multiplet space:\nA(1)\nI′Iµ,R=iSM′M,αa(1)\nαµ,R,\na(1)\nαµ,R=2∑\nj̸=ihSOI\ni jα,R\nE(0)\ni,R−E(0)\nj,RD\nj(0)\nR\f\f\f∂µi(0)\nRE\n.(16)\nThe sum over the Cartesian index α=x,y,zis implicit\nin Eq. (16), and hSOI\ni jα,Ris the imaginary part of the one-\nelectron, mean-field spin-orbit coupling operator as given in\nAppendix A. The functions a(1)\nαµ,Rare the three Cartesian com-\nponents of a vector potential associated with each nuclear de-\ngree of freedom µ.a(1)\nαµ,Rare real functions of the nuclear\npositions, which ensures that A(1)\nI′Iµ,Ris an anti-Hermitian ma-\ntrix. Note that with the definition of the Berry connection in\nEq. (12), the product −i¯hA(1)\nI′Iµ,Ris a Hermitian matrix. Thelast equality of Eq. (16) expresses the first-order vector po-\ntential, a(1)\nαµ,R, in terms of matrix elements of zero-order adia-\nbatic electronic wavefunctions,\f\f\f{jSS}(0)\nRE\n=\f\f\fj(0)\nRE\n, with spin\nquantum number Sequal to the spin quantum number of the\nground-state multiplet and maximum projection of the spin\nangular momentum MS=S.\f\f\fi(0)\nRE\nis the maximum-spin-\nprojection ground-state electronic wavefunction and the sum\nin Eq. (16) runs over the maximum-spin-projection excited-\nstate electronic wavefunctions. The benefit of the Wigner-\nEckart theorem is to separate the exact spin-component struc-\nture from the electronic matrix elements and to express all rel-\nevant quantities in terms of electronic wavefunctions of fixed\nprojection of spin angular momentum. I implement Eq. (16)\ntogether with state-of-the-art density functional theory ap-\nproximations to evaluate the first-order Berry connection in\nSec. III.\nSimilar expansion of the second-derivative non-adiabatic\ncoupling matrix, Eq. (13), in the spin-orbit interaction gives:\nDI′I,R=D\nI′(0)\nR\f\f\fˆTR\f\f\fI(0)\nRE\n+D\nI′(0)\nR\f\f\fh\nˆTR,ˆG(1)\nRi\f\f\fI(0)\nRE\n=D(0)\nI′I,R+D(1)\nI′I,R.(17)\nThe zero-order term of the second-derivative non-adiabatic\ncoupling matrix, D(0)\nI′I,R, is similarly diagonal in the ground-\nstate multiplet space and vanishes away from conical inter-\nsections. The first-order contribution, D(1)\nI′I,R, can be entirely\nexpressed in terms of the first-order Berry connection:\nD(1)\nI′I,R=−∑\nµ1\n2D\nI′(0)\nR\f\f\f∂2\nµˆG(1)\nR\f\f\fI(0)\nRE\n−∑\nµD\nI′(0)\nR\f\f\f∂µˆG(1)\nR\f\f\f∂µI(0)\nRE\n=−1\n2∑\nµ∂µA(1)\nI′Iµ,R.(18)\nThe first line of Eq. (18) follows from the expansion of the\ncommutator with the nuclear kinetic energy in Eq. (17), and\nin the second line I apply an identity proved in Appendix B\ntogether with the definition of the first-order Berry connection\nin Eq. (14).\nCollecting the results from the perturbative treatment of the\nnon-adiabatic coupling matrices, Eqs. (14) and (18), and the\nadiabatic potential energy matrix, Eq. (8), gives an expansion\nof the Born-Oppenheimer molecular Hamiltonian, Eq.(11), in\nthe spin-orbit interaction:\nˆHBO\nI′I=δI′IˆTR−iA(1)\nI′Iµ,Rˆpµ−1\n2∑\nµ∂µA(1)\nI′Iµ,R\n+δI′IE(0)\nI,R+E(1)\nI′I,R+E(2)\nI′I,R.(19)\nUniting the Berry connection terms with the nuclear kinetic\nenergy gives a covariant form of this Hamiltonian:\nˆHBO\nI′I=−1\n2∑\nµ\u0010\n∂µ+A(1)\nI′Iµ,R\u00112\n+δI′IE(0)\nI,R+E(1)\nI′I,R+E(2)\nI′I,R.(20)5\nEq. (20) and Eq. (19) agree to first-order in the Berry connec-\ntion upon expansion of the kinetic energy term. I use Eq. (16)\nand Eq. (8) in Eq. (20) to derive the final form of the new\nmolecular spin Hamiltonian in terms of spin operators, ˆSα,\nacting in the ground-state multiplet space:\nˆHspin=−1\n2∑\nµ\u0010\n∂µ+iˆSαa(1)\nαµ,R\u00112\n+E(0)\nR+µBBαg(2)\nαβ,RˆSβ+ˆSαD(2)\nαβ,RˆSβ.(21)\nEq. (21), the extended molecular spin Hamiltonian, is a main\nresult of this paper. It contains a novel, spin-vibronic vec-\ntor potential that derives from the first-order Berry connec-\ntion, as well as the traditional spin Hamiltonian terms that\nderive from the adiabatic potential energy matrix. The spin-\nvibronic vector potential gives rise to an effective magnetic\nfield on the ground-state multiplet, and I explore its effects on\nthe spin-vibrational dynamics in the rest of this paper. The\nspin-orbit interaction makes a leading first-order contribution\nto the vector potential, unlike the adiabatic molecular Zeeman\nand zero-field splitting terms, which are of second order in the\nspin-orbit and orbital Zeeman interactions. I expect that the\nspin-vibronic vector potential dominates the spin-vibrational\ndynamics in weak external magnetic fields, and I demonstrate\nits fundamental role for the spin relaxation dynamics of a pro-\ntotypical molecular qubit. The external magnetic field also\nmakes a contribution to the Berry connection: at first order\nthe contribution to the Berry connection is diagonal in the\nground-state multiplet space and does not couple to the spin-\nvibrational dynamics.\nB. Gauge transformation\nI showed in Sec. II A that the spin-vibrational dynamics in\nthe ground-state multiplet is governed by the molecular spin\nHamiltonian in Eq. (21), which contains a novel, spin-vibronic\nvector potential together with the traditional spin Hamiltonian\ninteractions. In Sec. II B, I transform the representation of\nthe Hamiltonian to bring the molecular spin Hamiltonian to a\nsymmetrized form that is convenient for application in time-\ndependent perturbation theory.\nGauge transformations40,48are nuclear configuration-\ndependent single-valued unitary transformations of the elec-\ntronic multiplet wavefunctions:\n\f\f˜IR\u000b\n=|IR⟩UI′I,R=∑\nM′\f\fiSM′,R\u000b\nUM′M,R, (22)\nwhich preserve the dynamics generated by the Born-\nOppenheimer molecular Hamiltonian, Eq. (11). As a result of\nthe gauge transformation in Eq. (22), the spin-vibronic vec-\ntor potential, Eq. (12), transforms as a non-Abelian gauge\npotential,40,48\n˜AI′Iµ,R=U†\nI′I′′,RAI′′I′′′µ,RUI′′′I,R+U†\nI′I′′,R∂µUI′′I,R,(23)\nand the Born-Oppenheimer Hamiltonian undergoes a unitary\ntransformation,\n˜HBO\nI′I,R=U†\nI′I′′,RHBO\nI′′I′′′,RUI′′′I,R. (24)Physical observables are independent of the specific choice of\nelectronic representation; they are gauge-covariant quantities,\nwhich transform upon gauge transformations like the Hamil-\ntonian in Eq. (24). For instance, the gauge-covariant nuclear\nmomentum, called kinematic momentum ˆΠI′Iµ, differs from\nthe canonical momentum ˆ pµby the vector potential:\nˆΠI′Iµ=δI′Iˆpµ−iAI′Iµ,R, (25)\nand neither ˆ pµnorAI′Iµ,Rare separately gauge-covariant. The\nvector potential in Eq. (25) results in the non-commutativity\nof the kinematic momenta:\n\u0002ˆΠµ,ˆΠν\u0003\nI′I=−FI′Iµν,R, (26)\nunlike the well-known commutation relation of the canonical\nmomenta, [ˆpµ,ˆpν] =δµν. In Eq. (26), bold symbols denote\nmatrices in the ground-state multiplet space, and FI′Iµν,Ris\nthe field tensor (also called Berry curvature46):\nFI′Iµν,R=∂µAI′Iν,R−∂νAI′Iµ,R+\u0002\nAµ,R,Aν,R\u0003\nI′I,(27)\nwhich is a gauge-covariant measure of the strength of the in-\nduced effective magnetic field on the ground-state manifold.\nThe gauge-covariance of the field tensor requires that UI′IR\nis both unitary and single-valued. Expansion of the spin-\nvibronic vector potential to first order in the spin-orbit interac-\ntion as in Sec. II A gives the first-order spin-vibronic magnetic\nfield tensor:\nF(1)\nI′Iµν,R=∂µA(1)\nI′Iν,R−∂νA(1)\nI′Iµ,R=iSM′M,αf(1)\nαµν,R,(28)\nwith\nf(1)\nαµν,R=∂µa(1)\nαν,R−∂νa(1)\nαµ,R=∂µ∧a(1)\nαν,R. (29)\nIn Eq. (29), ∧denotes the antisymmetric product, the gener-\nalization of the cross-product ×to multiple dimensions. With\nthe first-order spin-vibronic vector potential, the kinematic\nmomentum becomes:\nˆΠM′Mµ=δM′Mˆpµ+SM′M,αa(1)\nαµν,R. (30)\nThe nuclear dynamics depends only on gauge-covariant quan-\ntities, and I derive the Heisenberg equation of motion for the\nkinematic momentum Eq. (30) with the Hamiltonian Eq. (21)\nto leading order in the spin-orbit interaction:\nd2ˆRµ\ndt2=ˆSα1\n2\u001adˆRν\ndtf(1)\nανµ,R−f(1)\nαµν,RdˆRν\ndt\u001b\n−∂µE(0)\nR−µBBα∂µg(2)\nαβ,RˆSβ−ˆSα∂µD(2)\nαβ,RˆSβ.(31)\nThe derivation of Eq. (31) uses the equation of motion for\nthe nuclear position δM′MdˆRµ/dt=ˆΠM′Mµwith the veloc-\nity operator dˆRµ/dt. The first term on the right-hand side of\nEq. (31) is a quantum Lorentz force, the quantum equivalent to\nthe classical Lorentz force51FL=v×Bfor a unit-charge par-\nticle moving in three dimensions in a magnetic field B, where\nBrelates to the electromagnetic tensor Fi jasεi jkBk=Fjiwith6\nthe completely antisymmetric tensor εi jk. Because of the non-\nAbelian algebraic structure of the spin-vibronic magnetic field\ntensor, there is a separate magnetic field for each generator of\nthe associated non-Abelian group SO(3) or SU(2). The rest of\nthe terms on the right-hand side are the forces that originate\nfrom the adiabatic potential energy matrix.\nI use the freedom of gauge transformation to explicitly de-\nrive the spin-vibronic magnetic field tensor contribution to the\nspin Hamiltonian in Eq. (21). With the hindsight of the har-\nmonic approximation, I first expand the spin-vibronic vector\npotential to linear order in the deviations uµfrom a fixed nu-\nclear configuration R0, which is equivalent to the multipole\nexpansion of the electromagnetic vector potential50:\nA(1)\nI′Iµ,R=A(1)\nI′Iµ,R0+∂νA(1)\nI′Iµ,R0uν. (32)\nI construct a single-valued unitary gauge transformation using\nthe generator approach UI′I,R=eΛI′I,Rwith the anti-Hermitian\ngenerator function:\nΛI′I,R=A(1)\nI′Iµ,R0uν+1\n2∂νA(1)\nI′Iµ,R0uνuµ, (33)\nthat transforms the molecular spin Hamiltonian to a symmet-\nric gauge. I use this gauge transformation in Eq. (23) to obtain\nthe first-order, symmetrized spin-vibronic vector potential:\n˜A(1)\nI′Iµ,R=1\n2\u0010\n∂νA(1)\nI′Iµ,R0uν−∂µA(1)\nI′Iν,R0uν\u0011\n. (34)\nGauge transforming Eq. (21) together with the vector potential\nin Eq. (34) gives to leading order in the spin-orbit interaction\nthe symmetrized molecular spin Hamiltonian:\nˆHspin=ˆH(0)+1\n4f(1)\nανµ,R0ˆSα\u0000\nuν∧ˆpµ\u0001\n+µBBαg(2)\nαβ,RˆSβ+ˆSαD(2)\nαβ,RˆSβ,(35)\nwith\nˆH(0)=−1\n2∑\nµ∂2\nµ+E(0)\nR\nand summation over repeated indeces. The significance of\nthe spin-vibronic vector potential is particularly revealing in\nthe symmetric gauge where it gives rise to a spin-vibrational\norbit interaction, second term in Eq. (35), which involves\nthe effective magnetic field f(1)\nανµ,R0ˆlνµinduced by the vi-\nbrational angular motion with vibrational angular momen-\ntum ˆlνµ=uν∧ˆpµ. The spin-vibrational orbit interaction is\nanalogous to the spin-electronic orbit interaction (SOI) and\nit is similarly symmetric upon both time-reversal and par-\nity transformations. Note that the time-reversal invariance\nrequires that the first-order magnetic field tensor f(1)\nανµ,Ris\nreal. This implies that in the absence of an external magnetic\nfield, the spin-vibrational orbit interaction preserves Kramers’\ntheorem50and guarantees that half-integer spin systems have\ndoubly degenerate electronic states.C. Harmonic approximation\nI carried out a multipole expansion of the spin-vibronic vec-\ntor potential followed by a gauge transformation to a sym-\nmetric gauge in Sec. II B to derive the spin-vibrational orbit\ninteraction in the molecular spin Hamiltonian Eq. (35). In\nSec. II C, I derive a harmonic approximation to the molec-\nular spin Hamiltonian in Eq. (35), which I use to obtain an\nexpression for the spin relaxation time in Sec. II D using time-\ndependent perturbation theory. I start with the case of har-\nmonic vibrational dynamics of molecules in Sec. II C 1, which\nI extend to molecular crystals in Sec. II C 2.\n1. Vibrations of molecules\nThe harmonic approximation to the vibrational dynamics\nof molecules52is based on the second-order expansion of the\nground-state potential energy surface E(0)\nRinˆH(0)around the\nequilibrium nuclear geometry R0:\nˆHvib=−1\n2∑\nµ∂2\nµ+1\n2∂µ∂νE(0)\nR0uµuν\n=1\n4∑\niωi\u0000\nP2\ni+Q2\ni\u0001(36)\nI introduce in Eq. (36) the dimensionless normal mode coor-\ndinates Qiand momenta Piof vibrations with frequencies ωi.\nThe mass-weighted Cartesian displacements and momenta re-\nlate to the normal mode coordinates and momenta by the lin-\near transformation:\nuµ=∑\niCµ,ir\n1\n2ωiQi,Qi=b†\ni+bi\nˆpµ=∑\niCµ,ir\nωi\n2Pi,Pi=i\u0010\nb†\ni−bi\u0011\nwith Cµ,ithe eigenvectors of the mass-weighted Hessian ma-\ntrix. The equations also give the expressions for the quanti-\nzation of QiandPiin terms of the ladder operators biandb†\ni.\nThe second-order expansion of the spin-dependent terms in\nthe Hamiltonian Eq. (35) gives: (i) the traditional static spin\nHamiltonian at the equilibrium nuclear configuration, which I\ncall the reference spin Hamiltonian:\nˆHspin-ref=µBBαg(2)\nαβ,R0ˆSβ+ˆSαD(2)\nαβ,R0ˆSβ; (37)\n(ii) one-phonon (1P) spin-vibrational coupling Hamiltonian\nwith adiabatic (AD) origin, deriving from the adiabatic po-\ntential energy matrix in Eq. (35):\nˆH1P-AD=µB∂νg(2)\nαβ,R0BαˆSβuν+∂νD(2)\nαβ,R0ˆSαˆSβuν;(38)\n(iii) two-phonon (2P) spin-vibrational coupling Hamiltonian\nwith non-adiabatic (NA) origin, deriving from the spin-\nvibrational orbit interaction in Eq. (35), as well as the adia-7\nbatic two-phonon spin-vibrational coupling Hamiltonian:\nˆH2P-NA=1\n4f(1)\nανµ,R0ˆSα\u0000\nuν∧ˆpµ\u0001\n,\nˆH2P-AD=µB\n2∂ν∂µg(2)\nαβ,R0BαˆSβuνuµ+\n1\n2∂ν∂µD(2)\nαβ,R0ˆSαˆSβuνuµ.(39)\nTransforming the spin-vibrational coupling Hamiltonians in\nEqs. (38) and (39) to normal mode representation gives: (i)\nthe one-phonon spin-vibrational interactions:\nˆH1P-AD=µB∂ig(2)\nαβBαˆSref\nβQi+∂iD(2)\nαβˆSref\nαˆSref\nβQi; (40)\n(ii) the two-phonon spin-vibrational interactions:\nˆH2P-NA=1\n4f(1)\nαjiˆSref\nα(Qj∧Pi),\nˆH2P-AD=µB\n2∂i∂jg(2)\nαβBαˆSref\nβQiQj+\n1\n2∂i∂jD(2)\nαβˆSref\nαˆSref\nβQiQj.(41)\nIn Eqs. (40) and (41), indeces iandjrefer to the dimension-\nless normal modes, and fαji=ωi∂jaαi−ωj∂iaαj.\n2. Vibrations of molecular crystals\nThe generalization of the harmonic approximation in\nSec. II C 1 to molecular crystals requires a straightforward\nchange of notation to account for the wavevector qdepen-\ndence of the normal mode coordinates Qiq, momenta Piq, and\nfrequencies ωiq:53\nˆHvib=1\n4∑\ni,qωiq\u0000\nP2\niq+Q2\niq\u0001\n. (42)\nThe sum in Eq. (42) goes over all normal mode branches\niand all values of the wavevector qin the first Brillouin\nzone. The mass-weighted Cartesian displacements uµLand\nmomenta ˆ pµLof atomic coordinate µin the cell centered at\nLin terms of the normal mode coordinates and momenta be-\ncome:\nuµL=∑\ni,qCµ,iqs\n1\n2Nωiqeiq·LQiq,Qiq=b†\niq+bi−q\nˆpµL=∑\ni,qCµ,iqr\nωiq\n2Neiq·LPiq,Piq=i\u0010\nb†\niq−bi−q\u0011\nwith Cµ,iqthe eigenvectors of the mass-weighted dynamical\nmatrix. The equations give also the quantization of the crys-\ntal vibrations in terms of the ladder operators biqandb†\niq. The\nspin relaxation rate model in Sec. II D 3 assumes that the para-\nmagnetic molecular center, a paramagnetic molecule in a crys-\ntal environment of diamagnetic molecular counterparts, is lo-\ncalized in the unit cell at the origin L=0 and relies only on thedynamics of the optical vibrational modes, for which I adopt\nan approximation that neglects the dependence of the opti-\ncal mode eigenvectors Cµ,iqon the wavevector q. Within this\napproximation, the paramagnetic molecular center displace-\nments and momenta in terms of the optical normal modes are\ngiven by:\nuµ=∑\ni,qr\n1\n2NωiCµ,iQiq,\nˆpµ=∑\ni,qr\nωi\n2NCµ,iPiq.(43)\nSubstituting Eq. (43) in Eq. (39) gives the expression for\nthe two-phonon spin-vibrational coupling Hamiltonian of the\nparamagnetic molecular center,\nˆH2P-NA=1\nN∑\niq,jk1\n4f(1)\nαjiˆSref\nα\u0000\nQjk∧Piq\u0001\nˆH2P-AD=1\nN∑\niq,jkµB\n2∂i∂jg(2)\nαβBαˆSref\nβQjkQiq+\n1\nN∑\niq,jk1\n2∂i∂jD(2)\nαβˆSref\nαˆSref\nβQjkQiq.(44)\nEq. (44) is the extension of the isolated molecule two-phonon\nspin-vibrational interactions Eq. (41) to a single paramagnetic\nmolecular spin center in a diamagnetic crystal environment. I\napply Eq. (44) in Sec. II D to compute the two-phonon Raman\nprocess contribution to the spin relaxation time.\nD. Spin relaxation rate model\n1. Rate model assumptions\nThe rate model in this paper focuses on the simulation of\nthe spin relaxation dynamics of S=1/2 molecular qubits near\nroom temperature and at weak to moderate magnetic field in-\ntensities. The rate model is motivated by experimental mea-\nsurements of the temperature and magnetic field dependence\nof the spin-lattice relaxation time ( T1time) of molecular qubit\ncrystals under diamagnetic dilution.5–9,11,14,43,44The assump-\ntions of the rate model are:\n• the model applies to single paramagnetic molecular\ncenters which are embedded in the crystal environment\nof diamagnetic molecular homologs.\n• the model targets early-transition-metal-based molec-\nular qubits, allowing the perturbative treatment of the\nspin-orbit interaction of Sec. II A. It further focuses\nonS=1/2 molecular qubits, for which the zero-field\ninteraction is unnecessary, leaving the spin-vibronic\nmagnetic-field and molecular Zeeman interactions as\nthe only spin-dependent contributions to the Hamilto-\nnian in Eq. (35).8\n• the model assumes that the spin relaxation dynam-\nics near room temperature is determined by the two-\nphonon spin-vibrational interactions in Eq.(44), which\ninvolve a broad range of vibrational modes in the re-\nlaxation dynamics by the virtual absorption-emission of\nphonons, the two-phonon Raman process.\n• the model includes only the intra-molecular, optical\nvibrational modes in the approximation of Eq. (43)\nas they are more efficient in modulating the intra-\nmolecular spin-dependent interactions.\n• the model approximates the density-of-states of an op-\ntical vibrational mode by a Gaussian function centered\nat the isolated molecule normal mode frequency. The\nwidth of the Gaussian function is treated as an empiri-\ncal parameter.\nSubsections II D 2 and II D 3 develop the rate model based on\ntime-dependent perturbation theory,20–22the molecular spin\nHamiltonian of Sec. II C, and the model assumptions.\n2. Reference state\nThe application of time-dependent perturbation theory to\nevaluate the two-phonon Raman process rate of the model of\nSec. II D 1 requires a definition of the initial and final states of\nthe relaxation dynamics.21I assume that as a result of thermal\nequilibration the initial and final states are the eigenvectors\nof the reference spin Hamiltonian Eq. (37) at the equilibrium\nnuclear configuration:\nˆU†ˆHspin-ref ˆU=EM\nˆSref\nα=ˆU†ˆSαˆU.(45)\nThis definition of reference states can be straightforwardly im-\nplemented in the molecular spin Hamiltonian by rotating the\nquantization axis of the reference spin operators ˆSref\nαas written\nin the second line of Eq. (45), where ˆUis the unitary matrix\ndiagonalizing the reference spin Hamiltonian. With this defi-\nnition, the quantization axis of ˆSref\nαclosely follows the direc-\ntion of the external magnetic field for S=1/2 paramagnetic\nmolecular centers with a near isotropic g-tensor.\n3. Two-phonon Raman process rate\nThe rate kb→afor the spin-flip transition from the initial\nstate bto the final state avia the two-phonon Raman process,\nwhere a phonon in mode jkis absorbed and a phonon in mode\niqis emitted, is given by time-dependent perturbation theory\nas:\nkb→a=2π\n¯h∑\niq,niq∑\njk,njkρniqρnjkδ\u0000\n¯hωiq−¯hωjk−∆ba\u0001\n\f\f\na,niq+1,njk−1\f\fˆH2P\f\fb,niq,njk\u000b\f\f2.(46)In Eq. (46), the sum ranges over all optical vibrational modes\naccording to the rate model assumptions in Sec. II D 1, their\nassociated wavevectors and occupation numbers niq.ρniqis\nthe thermal weight of mode iq,ωiq- the frequency of the\nmode, and ∆ba- the energy gap between the initial and final\nstates. δis the Dirac delta function that imposes energy con-\nservation. The virtual absorption-emission process in Eq. (46)\nallows optical modes with close frequencies to match ener-\ngetically the small energy gap in the spin-flip transition, jus-\ntifying the leading role of the two-phonon Raman process in\nthe near-room-temperature spin relaxation dynamics. Evalu-\nation of the two-phonon spin-vibrational Hamiltonian matrix\nelements using Eq. (44) gives: (i) for the non-adiabatic two-\nphonon coupling matrix elements\n\na,niq+1,njk−1\f\fˆH2P-NA\f\fb,niq,njk\u000b\n=\n=i f(1)\nαjiSref\nα,ab1\nN√njkp\nniq+1\n=H2P-NA\nji,ab1\nN√njkp\nniq+1,(47)\nwith the definition of H2P-NA\nji,abin the last line of Eq. (47); (ii)\nfor the adiabatic two-phonon coupling matrix elements\n\na,niq+1,njk−1\f\fˆH2P-AD\f\fb,niq,njk\u000b\n=\n=µB∂i∂jg(2)\nαβBαSref\nβ,ab1\nN√njkp\nniq+1\n=H2P-AD\nji,ab1\nN√njkp\nniq+1(48)\nwith the definition of H2P-AD\nji,abin the last line of Eq. (48). I\ninclude only the g-tensor contribution in Eq. (48) because I\nconsider only S=1/2 systems, for which the D-tensor term\nvanishes. With the model for the optical vibrational modes\nin Eq. (43), the Hamiltonian matrix elements are independent\nof the mode wavevector, resulting in the simplified expression\nfor the Raman process rate:\nkb→a=2π\n¯h∑\ni,j\f\f\fH2P-NA\nji,ab+H2P-AD\nji,ab\f\f\f2\n1\nN2∑\nq,kδ\u0000\n¯hωiq−¯hωjk−∆ba\u0001\n(¯n(ωiq)+1)¯n(ωjk).(49)\nI carry out the sum over the thermal weight in Eq. (49) to ob-\ntain the average thermal occupation of the vibrational mode\n¯n(ωiq). Furthermore, I carry out the sum over the mode\nwavevectors in Eq. (49) using the vibrational mode density of\nstates gi(ω) =1\nN∑qδ(ω−ωiq)to obtain the final expression\nfor the Raman process rate:\nkb→a=2π\n¯h2∑\ni,j\f\f\fH2P-NA\nji,ab+H2P-AD\nji,ab\f\f\f2\nZ\ndωgi(ω−ωba)gj(ω)(¯n(ω−ωba)+1)¯n(ω).(50)\nThe spin-lattice ( T1) relaxation time characterizes the\ntimescale of thermal equilibration of the spin system as a re-\nsult of the spin-vibrational interactions with the phonon bath.9\nFor the case of an S=1/2 spin system that interacts with a\nphonon bath of reciprocal temperature βwith rate constants\nkb→aandka→bsatisfying detailed balance, the T1time is given\nby:\n1\nT1=kb→a+ka→b≈2k. (51)\nThe last equality in Eq. (51) follows from the smallness of\nωbacompared to the vibrational mode frequencies, and kis\ndefined as the expression obtained from Eq. (50) for vanishing\nωba. Substitution of Eq. (50) in Eq. (51) gives the expression\nfor the contribution of the two-phonon Raman process to the\nT1time:\n1\nT1=4π\n¯h2∑\ni,j\f\f\fH2P-NA\nji,ab+H2P-AD\nji,ab\f\f\f2\nZ\ndωgi(ω)gj(ω)(¯n(ω)+1)¯n(ω).(52)\nThe implementation of Eq. (52) requires the vibrational mode\ndensity-of-states, for which I adopt the Gaussian approxima-\ntion from Sec. II D 1 with mode frequency ωiand width σi.\nWith the Gaussian approximation, I arrive at the final ex-\npression for the T1spin relaxation time of the rate model of\nSec. II D 1:\n1\nT1=4π\n¯h2∑\ni,j\f\f\fH2P-NA\nji,ab+H2P-AD\nji,ab\f\f\f2\n1q\n2π(σ2\ni+σ2\nj)e−1\n2(ωi−ωj)2\nσ2\ni+σ2\nje−β¯hωji\n\u0010\n1−e−β¯hωji\u00112,(53)\nwhere ωji=σ2\njωi+σ2\niωj\nσ2\ni+σ2\njis the center of the product of the mode\ndensity-of-states Gaussians. To obtain Eq. (53), I assumed\nthat the thermal probabilities change slowly at the scale of the\nmode density-of-states, allowing their evaluation at the center\nof the Gaussian product. Section III presents the computa-\ntion of the spin-vibrational Hamiltonian matrix elements in\nEq. (53) based on density functional theory calculations on\nthe isolated paramagnetic molecule. The only empirical pa-\nrameters that enter in the implementation of Eq. (53) are the\nwidths of the mode density-of-states, for which I use a simple\nmodel with mode widths: σi=10cm−1forωi<100cm−1,\nσi=5cm−1forωi<200cm−1, and σi=1cm−1forωi>\n200cm−1. The ab initio estimation of the mode density-of-\nstates requires a computational model of the molecular crystal,\nwhich is outside of the scope of the present work. I present re-\nsults in Sec. IV for the two-phonon Raman contribution to the\nT1time as a function of temperature and magnetic field orien-\ntation for the prototypical molecular qubit Cu(II) porphyrin.\nIII. IMPLEMENTATION\nI derived in Sec. II an extended molecular spin Hamilto-\nnian that contains a novel, non-adiabatic contribution fromthe spin-vibronic vector potential. In Sec. III, I present the\nnumerical evaluation of the spin-vibronic vector potential via\nstate-of-the-art density functional theory.\nThe starting point of the derivation is Eq. (16), which I eval-\nuate using linear response unrestricted density functional the-\nory. As derived in Refs.23,54, the first-order perturbed Kohn-\nSham electronic wavefunctions\f\f\fψ(1)\n0,αE\nwith respect to the\nspin-orbit interaction are given by:\n\f\f\fψ(1)\n0,αE\n=∑\na,i,σUSOI\naiσ,α|ψaσ\niσ⟩,(54)\nwhere I use the convention that i,j,..denote occupied molec-\nular orbitals (MOs) and a,b,..- unoccupied (virtual) MOs in\nthe ground-state determinant, and σis the MO spin projec-\ntion. The sum in Eq. (54) ranges over all single excitations\nfrom the occupied to the unoccupied MOs of both spin pro-\njections. The coefficients USOI\naiσsatisfy the coupled-perturbed\nself-consistent-field Kohn-Sham equations:\n∑\na,iMσσ\nb j,aiUSOI\naiσ,α=−hSOI\nbσjσ,α,(55)\nwith the magnetic electronic Hessian Mσ′σ\nb j,aigiven by:\nMσ′σ\nb j,ai=(εaσ−εiσ)δb j,aiδσσ′+\ncHF(\naσj′\nσ\f\fiσb′\nσ\u000b\n−\naσb′\nσ\f\fiσj′\nσ\u000b\n).(56)\nIn Eq. (56), εiσare the Kohn-Sham orbital energies, cHFis\nthe percentage of exact Hartree-Fock exchange in the density\nfunctional approximation, and ⟨aσj′\nσ|iσb′\nσ⟩are the electron\nrepulsion integrals. hSOI\nbσjσ,αin Eq. (55) are the matrix ele-\nments of the spin-orbit coupling operator as defined in Ap-\npendix A.\nWith Eq. (54) the spin-vibronic vector potential in Eq. (16)\nbecomes:\na(1)\nαµ,R=−2∑\na,i,σUSOC\niaσ,αR\naσR\f\f∂µiσR\u000b\n.(57)\nWith generalized-gradient density functional approximations,\nthe coupled-perturbed Kohn-Sham equations have a non-\niterative solution, and Eq. (57) gives:\na(1)\nαµ,R=2∑\na,i,σhSOI\niσaσ,αR\nεiσ,R−εaσ,R\naσR\f\f∂µiσR\u000b\n, (58)\nwhich is the result of Eq. (57) for electronic states repre-\nsented as single Kohn-Sham Slater determinants. To com-\nplete the evaluation of the spin-vibronic vector potential, I ob-\ntain the last term in the sum in Eq. (57) from the solution\nof the coupled-perturbed Kohn-Sham equations for the nu-\nclear displacements55,56. These equations are conventionally\nsolved when computing analytical second derivatives of the\nground-state energy, and using the results of analytical gra-\ndient theory,55,56I write the nuclear derivatives of the Kohn-\nSham MOs as:\n\f\f∂µiσR\u000b\n=∑\np,iUG\npiσ,µ|pσR⟩+\f\f∂µ˜iσR\u000b\n,(59)10\ni j ωi ωj|fi j|\nPBE/SV|fi j|\nPBE/TZ|fi j|\nPBE0/SV|∂i jg|\nPBE/SV|∂i jg|\nPBE/TZ|∂i jg|\nPBE0/SV\n10 10 126.8 126.8 0.000 0.000 0.000 0.042 0.043 0.061\n10 11 126.8 126.8 13.500 16.189 7.995 0.018 0.018 0.027\n11 11 126.8 126.8 0.000 0.000 0.000 0.042 0.043 0.061\n13 13 204.3 204.3 0.000 0.000 0.000 0.029 0.030 0.039\n13 14 204.3 204.4 3.356 2.885 4.824 0.007 0.007 0.009\n14 14 204.4 204.4 0.000 0.000 0.000 0.029 0.030 0.039\n12 16 196.5 233.6 0.000 0.000 0.000 0.071 0.076 0.117\n15 17 212.9 240.6 4.211 4.727 5.345 0.002 0.002 0.002\n18 19 288.9 288.9 13.961 14.213 15.205 0.003 0.003 0.012\n22 23 387.1 389.1 3.532 3.306 3.092 0.001 0.001 0.002\n23 25 389.1 433.4 8.129 8.826 9.726 0.001 0.001 0.002\n25 26 433.4 437.8 4.292 4.033 4.351 0.001 0.001 0.002\n28 29 454.3 454.3 7.686 8.606 2.158 0.000 0.000 0.000\nTABLE I. Two-phonon spin-vibrational Hamiltonian matrix elements for the leading pairs of normal modes. iandjare the indeces of the\nmodes, and ωiandωjare the mode frequencies in cm−1.|fi j|2=∑αf2\nαi jdenotes the magnitude of the spin-vibronic magnetic field tensor for\nmodes iandjin 10−3cm−1.|∂i jg|2=µB∑α̸=β∂i∂jg2\nαβi jdenotes the magnitude of the off-diagonal portion of the second derivative g-tensor\nfor modes iandjin 10−3cm−1/T. The matrix elements are computed by: PBE density functional and def2-SVP basis set (PBE/SV), PBE\ndensity functional and def2-TZVP basis set (PBE/TZ), and PBE0 density functional and def2-SVP basis set (PBE0/SV).\nFIG. 1. Major normal modes contributing to the two-phonon spin\nrelaxation time T1for the prototypical Cu(II) porphyrin S=1/2\nmolecular qubit. The four normal modes are doubly degenerate with\nnormal mode frequencies 126 .8cm−1(upper pair) and 288 .9cm−1\n(lower pair). The mode at 126 .8cm−1brings the molecule out of the\nsymmetry plane, whereas the mode at 288 .9cm−1develops within\nthe plane of Cu(II) porphyrin. Arrows portray the relative magnitude\nand direction of the atomic displacements. Color code: Carbon is\ndark gray, Nitrogen - blue, Copper - orange, and Hydrogen - light\ngray.\nwhere UG\npiσ,µare the solutions of the coupled-perturbed Kohn-\nSham equations for the nuclear displacement µ, and\f\f˜iσR\u000b\nde-\nnotes MOs with frozen orbital coefficients, such that the nu-\nclear derivative operates only on the atom-centered basis func-\ntions. Using Eq. (59) in Eq. (57), I arrive at the final formulafor the evaluation of the spin-vibronic vector potential:\na(1)\nαµ,R=−2∑\na,i,σUSOC\niaσ,αR\u0010\nUG\naiσ,µR+Saiσ,µR\u0011\n,(60)\nwhere Saiσ,µRis the right-hand derivative of the basis overlap\nmatrix in the MO basis. The numerical evaluation of Eq. (60)\nrequires two coupled-perturbed Kohn-Sham response calcu-\nlations: one response calculation for the spin-orbit interac-\ntion perturbation USOI\niaσ,αR, and a second response calculation\nfor the nuclear displacement perturbations UG\naiσ,µR. I imple-\nment Eq. (60) in an in-house version of the quantum chem-\nistry software package ORCA57, for which I adapted already\nexisting highly optimized algorithms for molecular g-tensor\ncalculations23and analytical second derivatives55. I applied\nin all calculations a widely-used mean-field approximation for\nthe spin-orbit coupling operator26that is already available in\nthe ORCA suite.\nThe implementation of the spin-vibronic vector potential\nEq. (60) is consistent with the numerical evaluation of the\ng-tensor23and the D-tensor54in linear response unrestricted\ndensity functional theory. The molecular g-tensor, for in-\nstance, is expressed as:\ng(2)\nαβ,R=−∑\na,i,σUSOI\niaσ,αRIm(Laiσ,βR),(61)\nwhere Im (Laiσ,βR)is the imaginary part of the electronic or-\nbital angular momentum operator in the MO basis. Both the\nspin-vibronic vector potential Eq. (60) and the molecular g-\ntensor Eq. (61) expressions contain the self-consistent pertur-\nbation of the Kohn-Sham orbitals with respect to the spin-orbit\ninteraction; the crucial difference is the second perturbing in-\nteraction: for the spin-vibronic vector potential, these are the\nnuclear displacement perturbations, whereas for the molecu-\nlar g-tensor this is the orbital Zeeman interaction. This dif-\nference determines the different orders of the two effective11\n103105107\nPBE/SVP PBE/TZVP PBE0/SVP1031051071/T1, s-1Non-adiabatic rate\n10-2100102\n 50  150  250\nT, KTotal rate\nAdiabatic rateCuPc \nCuTTP\nFIG. 2. Two-phonon spin relaxation rate 1 /T1as a function of tem-\nperature Tin K for Cu(II) porphyrin. Rate is in s−1and the y-axis is a\nlogarithmic scale (base 10). The rate is uniformly averaged over the\nmagnetic field orientation. Magnetic field intensity is B=330mT.\nUpper panel is the total rate, middle panel - the non-adiabatic con-\ntribution to the rate, and lower panel - the adiabatic contribution to\nthe rate. Panels share the same temperature range. Black circles rep-\nresent experimentally measured T1times for Cu(II)-phthalocyanine\n(CuPc) from Ref.14. Black diamond is an experimetally measured\nT1time for Cu(II) tetratolylporphyrin (CuTPP) from Ref.43. Color\ncode: orange - PBE/def2-SVP, red - PBE/def2-TZVP, and blue -\nPBE0/def2-SVP. Note the difference in scale of the adiabatic and\nthe non-adiabatic rate.i j ωi ωj DoS Th. Pr.1\nT1\n10 11 126.8 126.8 0.056 2.588 1.758\n13 14 204.4 204.4 0.282 0.949 1.172\n18 19 288.9 289.0 0.282 0.438 5.385\n22 23 387.1 389.2 0.102 0.215 0.039\n28 29 454.3 454.3 0.282 0.141 0.035\n33 34 662.9 662.9 0.282 0.044 0.036\nTABLE II. Two-phonon spin relaxation rate contributions for room\ntemperature T=298K, magnetic field intensity B=330mT, and\nmagnetic field orientation θ=90orelative to the molecular axis.\niand jare the indeces of the modes, and ωiandωjare the mode\nfrequencies in cm−1. DoS denotes the mode density-of-states contri-\nbution to the rate using the Gaussian model in Sec. II D. Th. Pr. is the\nthermal probability contribution to the rate. 1 /T1is the two-phonon\nspin relaxation rate in MHz. The spin-vibronic matrix elements are\ncalculated at PBE0/def2-SVP level of theory.\ninteractions: the spin-vibronic vector potential is first order in\nthe spin-orbit interaction, whereas the molecular g-tensor is\nfirst order in the spin-orbit interaction but also first order in\nthe small orbital Zeeman interaction, resulting in an overall\nsecond order expression.\nThe matrix elements that enter in the calculation of the\nspin relaxation time Eq. (44) require geometric derivatives\nof both the spin-vibronic vector potential and the molecular\ng-tensor. I calculate these derivatives by numerical differen-\ntiation using a central difference approximation and normal\nmode displacements. I apply a dimensionless mode displace-\nment (displacement in units of the zero-point amplitude) of\n0.5. The calculation protocol starts with a gas-phase opti-\nmization of the molecular geometry using the selected ba-\nsis set and density functional together with: resolution-of-\nidentity approximation for the Coulomb portion of the Fock\nmatrix,58fine grid for the exchange-correlation functional in-\ntegration, very tight criteria for the self-consistent-field con-\nvergence, and tight criteria for the geometry convergence, all\nper ORCA 3.0.0 definitions.57A calculation of the analyti-\ncal second derivatives follows using the same convergence\ncriteria as applicable. In this work, I computed the molec-\nular Hessian using the Perdew-Burke-Ernzerhof (PBE) den-\nsity functional59and the Ahlrichs def2-TZVP basis set.60–62\nI calculated the spin-vibronic magnetic field tensor and the\nmolecular g-tensor second geometrical derivatives using three\ndensity functional and basis set combinations: PBE functional\nand def2-SVP basis set, PBE functional and def2-TZVP basis\nset, and the hybrid PBE0 functional63,64and def2-SVP ba-\nsis set. The results of these calculations and the associated\nspin-lattice relaxation times for the Cu(II) porphyrin molecu-\nlar qubit are presented in Sec. IV.\nIV. RESULTS\nI apply the new molecular spin Hamiltonian to the spin re-\nlaxation dynamics of Cu(II) porphyrin, a prototypical S=1/2\nmolecular qubit that represents the common core of a homol-12\ni j f jix fjiy fjiz1\nT1\n10 11 0.000 0.000 1.999 1.758\n13 14 0.000 -0.003 -1.206 1.172\n18 19 0.000 0.000 3.801 5.385\n22 23 0.000 0.000 -0.773 0.039\n28 29 0.004 0.000 0.539 0.035\n33 34 0.000 0.000 0.971 0.036\nTABLE III. Spin-vibronic magnetic-field tensor components for the\nleading contributions to the two-phonon spin relaxation rate for room\ntemperature T=298K, magnetic field intensity B=330mT, and\nmagnetic field orientation θ=90orelative to the molecular symme-\ntry axis. iandjare the indeces of the modes. fjixdenotes the value\nof the spin-vibronic tensor x-component in 10−3cm−1. 1/T1is the\ntwo-phonon spin relaxation rate in MHz. The spin-vibronic matrix\nelements are calculated at PBE0/def2-SVP level of theory.\nogous series of Cu(II)-based molecular qubits. I present the\nresults for the two-phonon spin relaxation time using the spin\nrate model of Sec. II D, and I calculate the two-phonon spin-\nvibrational Hamiltonian matrix elements according to the im-\nplementation in Sec. III using three levels of density func-\ntional theory to investigate the density functional and basis\nset dependence of the computed matrix elements.\nTable I presents the magnitudes of the spin-vibronic mag-\nnetic field tensor and the off-diagonal second derivative g-\ntensor for the leading pairs of normal modes. The striking\nresult is that the spin-vibronic magnetic field matrix elements\nare two to three orders of magnitude larger than the deriva-\ntive g-tensor matrix elements, demonstrating that the domi-\nnating spin-vibrational coupling mechanism in this class of\nmolecular qubits is of non-adiabatic character. The results in\nTable I affirm that this conclusion is independent of both the\nbasis set size and the density functional approximation. The\nbasis set dependence of the matrix elements is mild, show-\ning that computationally affordable basis sets of double zeta\nsize can be employed to compute spin-vibronic coupling ma-\ntrix elements. The amount of Hartree-Fock exchange in the\ndensity functional (the PBE0 density functional has 25% ex-\nact exchange compared to 0% for the PBE functional) leads\nto larger deviations in the magnitudes of the matrix elements,\nbut the values are in quantitative agreement with one another,\nallowing the use of generalized gradient density functionals\nfor initial investigations.\nThe data in Table I further demonstrate that degenerate\nnormal modes have a disproportionately large contribution to\nthe spin-vibrational coupling mechanism in Cu(II) porphyrin.\nFigure 1 depicts the two normal modes that are the major con-\ntributors to the spin-vibrational coupling in the qubit. The\nmode at 126 .8cm−1brings the molecule out of the molec-\nular plane, whereas the mode at 288 .9cm−1develops in the\nmolecular plane. Table I shows that the spin-vibronic matrix\nelements of the 288 .9cm−1in-plane mode increase with the\namount of Hartree-Fock exchange in the density functional\nand this mode becomes the major spin-vibrational coupling\nchannel at PBE0/def2-SVP level of theory. The emerging\nphysical picture from the results in Table I and Fig. 1 is that\n fitNon-adiabatic rate\n 0  45  90\nθ, deg1/T1, s-1\nAdiabatic rate\nPBE/SVP PBE/TZVP PBE0/SVP10-410-2100102104106FIG. 3. Two-phonon spin relaxation rate 1 /T1as a function of the\nangle θin degrees relative to the C4axis of Cu(II) porphyrin. Rate is\nin s−1and the y-axis is a logarithmic scale (base 10). The rate is uni-\nformly averaged over the polar angle φin the plane of the molecule.\nTemperature is 100K and magnetic field intensity is 330 mT. Upper\npanel is the the non-adiabatic contribution to the rate and lower panel\n- the adiabatic contribution to the rate. Panels share the same angle\nrange. The non-adiabatic rate is practically equal to the total rate -\nnote the difference in the scales of the two panels. Color codes the\nlevel of theory: orange - PBE/def2-SVP, red - PBE/def2-TZVP, and\nblue - PBE0/def2-SVP. The dashed black line is a Asin2(θ)fit to the\nPBE0/def2-SVP data. The fit to the non-adiabatic contribution coin-\ncides with the computed results.\nnearly degenerate normal modes linearly superpose to give vi-\nbrational rotational modes, which strongly interact with the\nmolecular spin via the spin-vibrational orbit interaction (see\nend of Sec. II B). The spin-vibrational orbit interaction is of\nnon-adiabatic origin and results from the non-Abelian Berry\ncurvature on the ground-state electronic spin multiplet.\nFigure 2 presents the magnetic-field-orientation averaged\ntwo-phonon spin relaxation rate 1 /T1, resulting from the cou-\npling to the optical normal modes of the molecular qubit com-\nputed by the spin relaxation rate model of Sec. II D. The data\nin Figure 2 confirm that the total two-phonon relaxation rate is\ndetermined by the non-adiabatic spin-vibrational orbit interac-\ntion and the adiabatic contribution to the relaxation rate is five13\norders of magnitude smaller. Similar to the spin-vibronic cou-\npling, this conclusion holds true independently of the basis set\nsize and the density functional approximation. The calculated\nrates are in excellent agreement with experimentally measured\nT1times for both a powder of Cu(II) phthalocyanine14and sin-\ngle crystals of Cu(II) tetratolylporphirin43. The theory con-\nfirms that the leading spin relaxation mechanism in Cu(II)-\nbased molecular qubits is the spin-vibrational orbit interac-\ntion.\nThe rates in Fig. 2 display a characteristic activated dynam-\nics with increasing temperature, which results from the suc-\ncessive thermal population of the doubly degenerate normal\nmodes. Table II presents the major normal mode contribu-\ntions to the two-phonon spin relaxation rate for T=298K and\nB=330mT. The data demonstrate that the doubly degener-\nate normal modes at 288 .9cm−1, 126.8cm−1, and 204 .4cm−1\nare the major channels for two-phonon spin relaxation in the\nsystem, and the three normal mode pairs fully determine the\norder of magnitude of the rate. This conclusion suggests an\nappealing route to control the spin relaxation rate of S=1/2\nmolecular qubits by decreasing the molecular symmetry via\nremoval of symmetry axes greater than second order.\nFigure 3 plots the two-phonon spin relaxation rate as a\nfunction of the angle θbetween the molecular C4axis and\nthe external magnetic field for T=100K and B=330mT.\nThe relaxation rates are averaged over the angle φthat de-\nscribes the rotation of the magnetic field vector around the C4\naxis. The results in Figure 3 demonstrate the strong magnetic-\nfield orientation dependence of the two-phonon relaxation rate\nwith distinct functional dependence of the rate contributions\nonθ: the non-adiabatic contribution, which entirely domi-\nnates the total rate, shows a clear sin2(θ)dependence (dashed\nline), whereas the adiabatic contribution goes through a max-\nimum at 45oand is described poorly by the sin2(θ)func-\ntion. This orientational dependence of the two-phonon rate\nallows to distinguish theoretically and experimentally the non-\nadiabatic and the adiabatic mechanisms of spin relaxation.\nThe computed sin2(θ)dependence of the total rate is in com-\nplete agreement with the observed sin2(θ)trend of the exper-\nimentally measured T1times43, which presents an indepen-\ndent confirmation of the non-adiabatic spin relaxation mech-\nanism. Furthermore, as a consequence of the orientational\ndependence, the two-phonon spin relaxation rate for in-plane\norientation θ=90oof the magnetic field is several orders of\nmagnitude larger than the rate for perpendicular orientation\nθ=0o. This result leads to a greater than one T1anisotropy,\ndefined as the ratio of the in-plane to the perpendicular relax-\nation rate, that similarly parallels experimental observations.\nThe dependence of the two-phonon rate on the orienta-\ntion of the external magnetic field can be rationalized based\non the data in Table III, which presents the components of\nthe spin-vibronic magnetic field tensor for the leading normal\nmode pairs in the spin relaxation dynamics at T=298Kand\nB=330mT. Table III shows that the largest component of\nthe spin-vibronic magnetic field tensor is along the C4axis of\nCu(II) porphyrin and the in-plane components are smaller by\nseveral orders of magnitude. As a result, the induced effective\nmagnetic field fαi jˆli jby the spin-vibrational orbit interactionis entirely directed along the C4axis of the molecule, and the\nmost efficient relaxation occurs when the molecular spin is\noriented perpendicularly to the C4axis in the molecular plane.\nProvided that the molecular g-tensor at the equilibrium geom-\netry is very nearly axial, the molecular spin closely follows\nthe orientation of the external magnetic field, such that the in-\nplane molecular spin component varies as sin (θ)withθthe\nangle between the magnetic field vector and the C4axis. The\nrelaxation rate depends on the squared modulus of the spin-\nvibrational orbit interaction, resulting in the observed sin2(θ)\ndependence in Fig. 3.\nV. CONCLUSIONS\nI derive in the present paper an extended molecular spin\nHamiltonian that takes into account both the traditional adia-\nbatic spin-dependent interactions and includes a novel, non-\nadiabatic spin-vibrational interaction. The derivation employs\nthe Born-Oppenheimer approximation to decouple the dy-\nnamics on different electronic spin manifolds, which induces a\nnon-Abelian Berry connection on the ground-state electronic\nmultiplet. The resulting non-Abelian Berry curvature is the\nspin-vibrational orbit interaction, which couples the vibra-\ntional angular momentum to the electronic spin in complete\nanalogy to the spin-electronic orbit interaction. Thus, the vi-\nbrational angular motion induces an effective magnetic field\nin the spin dynamics on the electronic ground state and vice\nversa the electronic spin exerts a quantum Lorentz force on the\nnuclear motion. The spin-vibronic magnetic field tensor that\nquantifies the strength of coupling between the vibrational an-\ngular motion and the electronic spin is first order in the spin-\norbit interaction and first order in the non-adiabatic interac-\ntions between the electronic multiplets. As such, this previ-\nously unappreciated interaction is expected to dominate the\nspin relaxation dynamics at weak magnetic field intensities\nand elevated temperatures.\nI implement the computation of the matrix elements of\nthe molecular spin Hamiltonian using linear response density\nfunctional theory, which allows the ab initio prediction of the\nstrength of the different interactions and the unbiased compar-\nison between them. The density-functional implementation\nrequires two different kinds of coupled-perturbed calculations\nfor computing the response of the electronic wavefunctions to\nthe spin-orbit interaction and to the nuclear displacements. I\nfurther develop a rate model to estimate the spin relaxation\ntime using the molecular spin Hamiltonian computed for the\nisolated paramagnetic molecule. The rate model is special-\nized to isolated paramagnetic molecular centers in molecular\ncrystals and to elevated temperatures and estimates the spin\nrelaxation time via the two-phonon Raman process with the\nsole participation of the optical crystal vibrations. The only\nempirical parameters of the model are the widths of the mode\ndensity-of-states, for which I employ a Gaussian approxima-\ntion.\nI apply the molecular spin Hamiltonian together with the\nspin relaxation rate model to the prototypical S=1/2 molec-\nular qubit Cu(II) porphyrin. The striking result is that the14\nspin relaxation rate near room temperature is determined by\nthe new spin-vibrational orbit interaction, and that the com-\nputed spin relaxation rate is in remarkable agreement with\nexperimental measurements. This result is independent of\nboth basis set size and density functional approximation. The\nphysical picture that emerges for spin relaxation in Cu(II)-\nbased molecular qubits is that the vibrational angular motion\nof nearly degenerate normal modes strongly couples to the\nelectronic spin dynamics and is responsible for spin relaxation\nin the system. Furthermore, the spin relaxation time shows a\nstrong dependence on the orientation of the magnetic field to\ntheC4symmetry axis of the molecule and is distinct for the\nnon-adiabatic and the adiabatic relaxation mechanisms, which\nallows to distinguish them both theoretically and experimen-\ntally.\nThe current work lays the foundation for a further investiga-\ntion of a much broader set of S=1/2 qubits. Furthermore, the\napproach can be successfully applied to S>1/2 systems that\ninclude the emerging class of optically addressable molecular\nqubits. Taken together, the new molecular spin Hamiltonian\nformalism together with density functional theory is expected\nto provide a much-needed theoretical approach to simulate\nand understand spin relaxation dynamics in molecular qubits.\nACKNOWLEDGMENTS\nI would like to acknowledge start-up funds from Indiana\nUniversity, Bloomington and Tufts University.\nDATA AVAILABILITY\nThe data that support the findings of this study are available\nfrom the corresponding author upon reasonable request.\nAppendix A SPIN STRUCTURE VIA THE\nWIGNER-ECKART THEOREM\nIn Appendix A, I apply the Wigner-Eckart theorem to estab-\nlish the spin-operator equivalents of the effective interactions\nin the molecular spin Hamiltonian. The spin-dependent rel-\nativistic interactions ˆHSOZ, including the interaction with the\nweak external magnetic field, consists of: (i) the spin-orbit\ninteraction:\nˆHSOI=∑\ni,m(−1)mˆF(1)\n−m(i)ˆS(1)\nm(i), (62)\nwhich is a one-electron operator in the mean-field approxi-\nmation and contains an orbital-dependent part ˆF(i), a func-\ntion of the position and momentum of electron i, and a spin-\ndependent part ˆS(i), the spin operator of electron i; (ii) the\nZeeman interaction:\nˆHZ=µB∑\ni,m(−1)m\u0010\ngeB(1)\n−mˆS(1)\nm(i)+B(1)\n−mˆL(1)\nm(i)\u0011\n,(63)which includes the interaction of the magnetic field Bwith the\nelectron spin, first term, and the interaction with the orbital\nmotion of the electrons, second term. The sum over iruns\nover all electrons, and the sum over mruns over all spher-\nical components of the spherical tensors of rank (1).geis\nthe anomalous g-factor of the electron. Application of the\nWigner-Eckart theorem to the matrix elements of the spin-\norbit interaction gives:\nHSOI\nJI=\njSM′\f\fˆHSOI|jSM⟩\n=∑\nm(−1)m\nS1;Mm\f\fS1;SM′\u000b\n⟨jS||∑iˆF(1)\n−m(i)ˆS(1)(i)||iS⟩√\n2S+1\n=∑\nm(−1)mS(1)\nM′Mm⟨jSS|∑iˆF(1)\n−m(i)ˆS(1)\n0(i)|iSS⟩\nS\n=i∑\nαSM′MαIm⟨jSS|∑iˆFα(i)ˆSz(i)|iSS⟩\nS\n=i∑\nαSM′MαhSOI\njiα.(64)\nThe second line of Eq. (64) results directly from the Wigner-\nEckart theorem, where ⟨S1;Mm|S1;SM′⟩is a Clebsch-Gordan\ncoefficient in the nomenclature of Sakurai50and⟨jS||·||iS⟩\ndenotes a reduced matrix element. In the third line, I express\nthe Clebsch-Gordan coefficient using the matrix elements of\nthe electron spin operator, and in the fourth line, I account for\nthe purely imaginary nature of the resulting orbital-dependent\noperator. The last line of Eq. (64) provides the definition of\nhSOI\njiα. The sum over mis a sum over the spherical compo-\nnents of the rank-1 spherical tensors as before, and the sum\noverαis the equivalent sum over the Cartesian components of\nthe vector operators. Similar application of the Wigner-Eckart\ntheorem to the Zeeman interaction reveals the following spin-\ncomponent structure:\nˆHZ\nJI=\njSM′\f\fˆHZ|jSM⟩\n=µBge∑\nm(−1)mB(1)\n−mS(1)\nM′Mmδji\n+µB∑\nm(−1)mB(1)\n−m⟨jSS|ˆL(1)\nm|iSS⟩δM′M.(65)\nUse of Eqs. (64) and (65) in the second order term of Eq. (8),\nand keeping only the combination terms between the orbital\nZeeman and spin-orbit interactions gives the second order\ncontribution to the molecular g-tensor:\n∆g(2)\nαβ=1\nS2∑\nj̸=0Im⟨0SS|ˆLα|jSS⟩Im⟨jSS|ˆFβˆSz|0SS⟩\nE(0)\nj−E(0)\ni\n+1\nS2∑\nj̸=0Im⟨0SS|ˆFαˆSz|jSS⟩Im⟨jSS|ˆLβ|0SS⟩\nE(0)\nj−E(0)\ni.(66)\nEquation (66) is the sum-over-states expression of the sec-\nond order molecular g-tensor and demonstrates the relation\nbetween adiabatic terms in the molecular spin Hamiltonian15\nand the traditional static spin Hamiltonian interactions. Simi-\nlar account of the pure spin-orbit interaction terms in Eq. (8)\ngives the second-order sum-over-states expression for the\nmolecular D-tensor, which also involves matrix elements of\nthe spin-orbit interaction between spin manifolds that differ\nby one unit of angular momentum.\nFinally, the same-spin-manifold matrix elements of the\nfirst-order spin-orbit perturbing operator can be written using\nthe Wigner-Eckart theorem as:\nG(1)\nJI=\njSM′\f\fˆG(1)|iSM⟩=−∑\nJ̸=IHSOI\nJI\nE(0)\nJ−E(0)\nI\n=−iSM′Mα∑\nj̸=ihSOI\njiα\nE(0)\nj−E(0)\ni=iSM′Mαg(1)\njiα,(67)\nwhere in the last line I define the Cartesian components of the\nfirst-order perturbing operator gjiα.gjiαare purely real func-\ntions and are symmetric matrices with respect to the orbital\nindeces.\nAppendix B PROOF OF IDENTITY IN EQ. 18\nI prove in Appendix B an identity that is needed to trans-\nform the second derivative non-adiabatic coupling matrix in\nEq. (18):\nD\nI′(0)\nR\f\f\f∂µˆG(1)\nR\f\f\f∂µI(0)\nRE\n=\n=∑\nJ̸=I∂µGI′J\nJ\f\f∂µI\u000b\n+∑\nJ̸=IGI′J\n∂µJ\f\f∂µI\u000b\n+\n∑\nK,J,J̸=K\nI′\f\f∂µK\u000b\nGKJ\nJ\f\f∂µI\u000b\n=iSM′Mα(∑\nj̸=i∂µgi jα\nj\f\f∂µi\u000b\n+∑\nj̸=igi jα\n∂µj\f\f∂µi\u000b\n+\n∑\nk,j,j̸=k\ni\f\f∂µk\u000b\ngk jα\nj\f\f∂µi\u000b\n)\n=iSM′Mα(∑\nj̸=i\n∂µi\f\fj\u000b\n∂µgjiα+∑\nj̸=i\n∂µi\f\f∂µj\u000b\ngjiα+\n∑\nk,j,j̸=k\n∂µi\f\fj\u000b\ngjkα\n∂µk\f\fi\u000b\n)\n=D\n∂µI′(0)\nR\f\f\f∂µˆG(1)\nR\f\f\fI(0)\nRE(68)\nThe first line follows from the expansion of the first-order per-\nturbing operator, the second line employs Eq. (67), and the\nthird line uses the symmetry of gjiαand of the derivative ma-\ntrix elements in the case of real zero-order wavefunctions.\n1A. Gaita-Ariño, F. Luis, S. Hill, and E. Coronado, “Molecular spins for\nquantum computation,” Nature chemistry 11, 301–309 (2019).\n2M. S. Fataftah and D. E. Freedman, “Progress towards creating optically ad-\ndressable molecular qubits,” Chemical Communications 54, 13773–13781\n(2018).\n3M. Atzori and R. Sessoli, “The second quantum revolution: role and chal-\nlenges of molecular chemistry,” Journal of the American Chemical Society\n141, 11339–11352 (2019).\n4M. J. Graham, J. M. Zadrozny, M. S. Fataftah, and D. E. Freedman, “Forg-\ning solid-state qubit design principles in a molecular furnace,” Chemistry\nof Materials 29, 1885–1897 (2017).5N. P. Kazmierczak, K. M. Luedecke, E. T. Gallmeier, and R. G. Hadt, “T 1\nanisotropy elucidates spin relaxation mechanisms in an s= 1 cr (iv) optically\naddressable molecular qubit,” The Journal of Physical Chemistry Letters\n14, 7658–7664 (2023).\n6N. P. Kazmierczak and R. G. Hadt, “Illuminating ligand field contributions\nto molecular qubit spin relaxation via t 1 anisotropy,” Journal of the Amer-\nican Chemical Society 144, 20804–20814 (2022).\n7N. P. Kazmierczak, R. Mirzoyan, and R. G. Hadt, “The impact of ligand\nfield symmetry on molecular qubit coherence,” Journal of the American\nChemical Society 143, 17305–17315 (2021).\n8J. M. Zadrozny, J. Niklas, O. G. Poluektov, and D. E. Freedman, “Mil-\nlisecond coherence time in a tunable molecular electronic spin qubit,” ACS\ncentral science 1, 488–492 (2015).\n9M. J. Amdur, K. R. Mullin, M. J. Waters, D. Puggioni, M. K. Wojnar,\nM. Gu, L. Sun, P. H. Oyala, J. M. Rondinelli, and D. E. Freedman, “Chemi-\ncal control of spin–lattice relaxation to discover a room temperature molec-\nular qubit,” Chemical Science 13, 7034–7045 (2022).\n10C.-J. Yu, S. V on Kugelgen, M. D. Krzyaniak, W. Ji, W. R. Dichtel, M. R.\nWasielewski, and D. E. Freedman, “Spin and phonon design in modular ar-\nrays of molecular qubits,” Chemistry of Materials 32, 10200–10206 (2020).\n11M. S. Fataftah, M. D. Krzyaniak, B. Vlaisavljevich, M. R. Wasielewski,\nJ. M. Zadrozny, and D. E. Freedman, “Metal–ligand covalency enables\nroom temperature molecular qubit candidates,” Chemical Science 10,\n6707–6714 (2019).\n12M. S. Fataftah, S. L. Bayliss, D. W. Laorenza, X. Wang, B. T. Phelan, C. B.\nWilson, P. J. Mintun, B. D. Kovos, M. R. Wasielewski, S. Han, et al. , “Trig-\nonal bipyramidal v3+ complex as an optically addressable molecular qubit\ncandidate,” Journal of the American Chemical Society 142, 20400–20408\n(2020).\n13M. J. Graham, J. M. Zadrozny, M. Shiddiq, J. S. Anderson, M. S. Fataftah,\nS. Hill, and D. E. Freedman, “Influence of electronic spin and spin–orbit\ncoupling on decoherence in mononuclear transition metal complexes,” Jour-\nnal of the American Chemical Society 136, 7623–7626 (2014).\n14A. H. Follmer, R. D. Ribson, P. H. Oyala, G. Y . Chen, and R. G.\nHadt, “Understanding covalent versus spin–orbit coupling contributions\nto temperature-dependent electron spin relaxation in cupric and vanadyl\nphthalocyanines,” The Journal of Physical Chemistry A 124, 9252–9260\n(2020).\n15L. Tesi, E. Lucaccini, I. Cimatti, M. Perfetti, M. Mannini, M. Atzori,\nE. Morra, M. Chiesa, A. Caneschi, L. Sorace, et al. , “Quantum coherence\nin a processable vanadyl complex: new tools for the search of molecular\nspin qubits,” Chemical Science 7, 2074–2083 (2016).\n16M. Atzori, L. Tesi, E. Morra, M. Chiesa, L. Sorace, and R. Sessoli, “Room-\ntemperature quantum coherence and rabi oscillations in vanadyl phthalo-\ncyanine: toward multifunctional molecular spin qubits,” Journal of the\nAmerican Chemical Society 138, 2154–2157 (2016).\n17M. Atzori, S. Benci, E. Morra, L. Tesi, M. Chiesa, R. Torre, L. Sorace, and\nR. Sessoli, “Structural effects on the spin dynamics of potential molecular\nqubits,” Inorganic chemistry 57, 731–740 (2018).\n18M. Atzori, L. Tesi, S. Benci, A. Lunghi, R. Righini, A. Taschin, R. Torre,\nL. Sorace, and R. Sessoli, “Spin dynamics and low energy vibrations: in-\nsights from vanadyl-based potential molecular qubits,” Journal of the Amer-\nican Chemical Society 139, 4338–4341 (2017).\n19M. Goldman, “Formal theory of spin–lattice relaxation,” Journal of Mag-\nnetic Resonance 149, 160–187 (2001).\n20J. Van Vleck, “Paramagnetic relaxation times for titanium and chrome\nalum,” Physical Review 57, 426 (1940).\n21R. D. Mattuck and M. Strandberg, “Spin-phonon interaction in paramag-\nnetic crystals,” Physical Review 119, 1204 (1960).\n22R. Orbach, “Spin-lattice relaxation in rare-earth salts,” Proceedings of the\nRoyal Society of London. Series A. Mathematical and Physical Sciences\n264, 458–484 (1961).\n23F. Neese, “Prediction of electron paramagnetic resonance g values using\ncoupled perturbed hartree–fock and kohn–sham theory,” The Journal of\nChemical Physics 115, 11080–11096 (2001).\n24F. Neese and E. I. Solomon, “Calculation of zero-field splittings, g-values,\nand the relativistic nephelauxetic effect in transition metal complexes. ap-\nplication to high-spin ferric complexes,” Inorganic chemistry 37, 6568–\n6582 (1998).16\n25F. Neese, “Importance of direct spin- spin coupling and spin-flip excitations\nfor the zero-field splittings of transition metal complexes: A case study,”\nJournal of the American Chemical Society 128, 10213–10222 (2006).\n26F. Neese, “Efficient and accurate approximations to the molecular spin-orbit\ncoupling operator and their use in molecular g-tensor calculations,” The\nJournal of chemical physics 122(2005).\n27S. K. Singh, M. Atanasov, and F. Neese, “Challenges in multireference per-\nturbation theory for the calculations of the g-tensor of first-row transition-\nmetal complexes,” Journal of Chemical Theory and Computation 14, 4662–\n4677 (2018).\n28J. Gauss, M. Kallay, and F. Neese, “Calculation of electronic g-tensors\nusing coupled cluster theory,” The Journal of Physical Chemistry A 113,\n11541–11549 (2009).\n29R. McWeeny, “On the origin of spin-hamiltonian parameters,” The Journal\nof Chemical Physics 42, 1717–1725 (1965).\n30M. Pryce, “A modified perturbation procedure for a problem in paramag-\nnetism,” Proceedings of the Physical Society. Section A 63, 25 (1950).\n31A. Lunghi and S. Sanvito, “The limit of spin lifetime in solid-state elec-\ntronic spins,” The Journal of Physical Chemistry Letters 11, 6273–6278\n(2020).\n32A. Lunghi and S. Sanvito, “Multiple spin–phonon relaxation pathways in a\nkramer single-ion magnet,” The Journal of Chemical Physics 153(2020).\n33A. Lunghi, “Toward exact predictions of spin-phonon relaxation times: An\nab initio implementation of open quantum systems theory,” Science Ad-\nvances 8, eabn7880 (2022).\n34M. Briganti, F. Santanni, L. Tesi, F. Totti, R. Sessoli, and A. Lunghi, “A\ncomplete ab initio view of orbach and raman spin–lattice relaxation in a\ndysprosium coordination compound,” Journal of the American Chemical\nSociety 143, 13633–13645 (2021).\n35E. Garlatti, A. Albino, S. Chicco, V . Nguyen, F. Santanni, L. Paolasini,\nC. Mazzoli, R. Caciuffo, F. Totti, P. Santini, et al. , “The critical role of\nultra-low-energy vibrations in the relaxation dynamics of molecular qubits,”\nNature Communications 14, 1653 (2023).\n36A. Albino, S. Benci, L. Tesi, M. Atzori, R. Torre, S. Sanvito, R. Ses-\nsoli, and A. Lunghi, “First-principles investigation of spin–phonon coupling\nin vanadium-based molecular spin quantum bits,” Inorganic chemistry 58,\n10260–10268 (2019).\n37D. Reta, J. G. Kragskow, and N. F. Chilton, “Ab initio prediction of high-\ntemperature magnetic relaxation rates in single-molecule magnets,” Journal\nof the American Chemical Society 143, 5943–5950 (2021).\n38J. G. Kragskow, A. Mattioni, J. K. Staab, D. Reta, J. M. Skelton, and\nN. F. Chilton, “Spin–phonon coupling and magnetic relaxation in single-\nmolecule magnets,” Chemical Society Reviews (2023).\n39D. R. Yarkony, “Diabolical conical intersections,” Reviews of Modern\nPhysics 68, 985 (1996).\n40C. A. Mead, “The geometric phase in molecular systems,” Reviews of mod-\nern physics 64, 51 (1992).\n41J. Van Vleck, “On σ-type doubling and electron spin in the spectra of di-\natomic molecules,” Physical Review 33, 467 (1929).\n42H. Primas, “Generalized perturbation theory in operator form,” Reviews of\nModern Physics 35, 710 (1963).\n43J.-L. Du, G. R. Eaton, and S. S. Eaton, “Electron spin relaxation in vanadyl,\ncopper (ii), and silver (ii) porphyrins in glassy solvents and doped solids,”\nJournal of Magnetic Resonance, Series A 119, 240–246 (1996).\n44C.-J. Yu, M. D. Krzyaniak, M. S. Fataftah, M. R. Wasielewski, and D. E.\nFreedman, “A concentrated array of copper porphyrin candidate qubits,”Chemical science 10, 1702–1708 (2019).\n45C. M. Moisanu, R. M. Jacobberger, L. P. Skala, C. L. Stern, M. R.\nWasielewski, and W. R. Dichtel, “Crystalline arrays of copper porphyrin\nqubits based on ion-paired frameworks,” Journal of the American Chemi-\ncal Society 145, 18447–18454 (2023).\n46M. V . Berry, “Quantal phase factors accompanying adiabatic changes,” Pro-\nceedings of the Royal Society of London. A. Mathematical and Physical\nSciences 392, 45–57 (1984).\n47C. A. Mead and D. G. Truhlar, “On the determination of born–oppenheimer\nnuclear motion wave functions including complications due to conical inter-\nsections and identical nuclei,” The Journal of Chemical Physics 70, 2284–\n2296 (1979).\n48A. Shapere and F. Wilczek, Geometric phases in physics , V ol. 5 (World\nscientific, 1989).\n49C. A. Mead, “Molecular kramers degeneracy and non-abelian adiabatic\nphase factors,” Physical review letters 59, 161 (1987).\n50J. Sakurai and J. Napolitano, “Modern quantum mechanics. 2-nd edition,”\nPerson New International edition , 39 (2014).\n51J. D. Jackson, “Classical electrodynamics,” (1999).\n52E. B. Wilson, J. C. Decius, and P. C. Cross, Molecular vibrations: the theory\nof infrared and Raman vibrational spectra (Courier Corporation, 1980).\n53N. W. Ashcroft and N. D. Mermin, Solid state physics (Cengage Learning,\n2022).\n54F. Neese, “Calculation of the zero-field splitting tensor on the basis of hy-\nbrid density functional and hartree-fock theory,” The Journal of chemical\nphysics 127(2007).\n55D. Bykov, T. Petrenko, R. Izsák, S. Kossmann, U. Becker, E. Valeev, and\nF. Neese, “Efficient implementation of the analytic second derivatives of\nhartree–fock and hybrid dft energies: a detailed analysis of different ap-\nproximations,” Molecular Physics 113, 1961–1977 (2015).\n56J. Pople, R. Krishnan, H. Schlegel, and J. S. Binkley, “Derivative studies in\nhartree-fock and møller-plesset theories,” International Journal of Quantum\nChemistry 16, 225–241 (1979).\n57F. Neese, “The orca program system,” Wiley Interdisciplinary Reviews:\nComputational Molecular Science 2, 73–78 (2012).\n58K. Eichkorn, F. Weigend, O. Treutler, and R. Ahlrichs, “Auxiliary basis\nsets for main row atoms and transition metals and their use to approximate\ncoulomb potentials,” Theoretical Chemistry Accounts 97, 119–124 (1997).\n59J. P. Perdew, K. Burke, and M. Ernzerhof, “Generalized gradient approxi-\nmation made simple,” Physical review letters 77, 3865 (1996).\n60A. Schäfer, H. Horn, and R. Ahlrichs, “Fully optimized contracted gaussian\nbasis sets for atoms li to kr,” The Journal of chemical physics 97, 2571–\n2577 (1992).\n61A. Schäfer, C. Huber, and R. Ahlrichs, “Fully optimized contracted gaus-\nsian basis sets of triple zeta valence quality for atoms li to kr,” The Journal\nof chemical physics 100, 5829–5835 (1994).\n62F. Weigend and R. Ahlrichs, “Balanced basis sets of split valence, triple\nzeta valence and quadruple zeta valence quality for h to rn: Design and\nassessment of accuracy,” Physical Chemistry Chemical Physics 7, 3297–\n3305 (2005).\n63C. Adamo and V . Barone, “Toward reliable density functional methods\nwithout adjustable parameters: The pbe0 model,” The Journal of chemi-\ncal physics 110, 6158–6170 (1999).\n64M. Ernzerhof and G. E. Scuseria, “Assessment of the perdew–burke–\nernzerhof exchange-correlation functional,” The Journal of chemical\nphysics 110, 5029–5036 (1999)."    },    {        "title": "2403.14408v1.Spin_orbit_interaction_with_large_spin_in_the_semi_classical_regime.pdf",        "content": "arXiv:2403.14408v1  [math-ph]  21 Mar 2024Spin-orbit interaction with large spin\nin the semi-classical regime\nDidier Robert∗\nAbstract\nWe consider the time dependent Schr¨ odinger equation with a cou-\npling spin-orbit in the semi-classical regime /planckover2pi1ց0 and large spin num-\nbers→+∞suchthat /planckover2pi1δs=cwherec>0andδ>0areconstant. The\ninitial state Ψ(0) is a product of an orbital coherent state in L2(Rd)\nand a spin coherent state in a spin irreducible representation space\nH2s+1. Forδ<1, at the leading order in /planckover2pi1, the time evolution Ψ( t) of\nΨ(0) is well approximated by the product of an orbital and a spin co-\nherent state. Nevertheless for 1 /2<δ <1 the quantum orbital leaves\nthe classical orbital. For δ= 1 we prove that this last claim is no more\ntrue when the interaction depends on the orbital variables. For th e\nDicke model, we prove that the orbital partial trace of the projec tor\non Ψ(t) is a mixed state in L2(R) for smallt>0.\n1 Introduction\nWe consider here the Schr¨ odinger equation for a system of pa rticles with\nlarge spin number when the spin and the position variables ar e coupled:\ni/planckover2pi1∂tΨ(t) =ˆH(t)Ψ(t),\nΨ(0) =ϕz0⊗ψn0,inL2(Rd)⊗H2s+1(1.1)\nwhereϕz0,ψn0are respectively Schr¨ odinger, spin coherent states and\nˆH(t) =ˆH0(t)+/planckover2pi1/hatwideC(t)·S. (1.2)\nˆH0(t) has a scalar /planckover2pi1-Weyl symbol as well as ˆCj(t) andS= (S1,S2,S3)\nare spin matrices representation in an irreducible space H2s+1of dimension\n2s+1, wheresis the total spin number.\nA particular case is the Pauli equation for the electron ( s=1/2).\nˆH=1\n2(/planckover2pi1D−A)2+ˆV+/planckover2pi1B·σ,\n∗Laboratoire de Math´ ematiques Jean Leray, Universit´ e de N antes, 2 rue de la\nHoussini` ere, BP 92208, 44322 Nantes Cedex 3, France\nEmail:didier.robert@univ-nantes.fr\n1whereD=i−1∇x,x∈R3,A= (A1,A2,A3) is a magnetic potential, Van\nelectric potential, B= (B1,B1,B1) the magnetic field in R3;σ= (σ1,σ2,σ3)\nare the Pauli matrices:\nσ1=/parenleftbigg0 1\n1 0/parenrightbigg\n, σ2=/parenleftbigg0−i\ni0/parenrightbigg\n, σ3=/parenleftbigg1 0\n0−1/parenrightbigg\n.\nFors=1/2we haveSk=σk\n2.\nMore generally the Weyl symbol of the interaction is /planckover2pi1H1(t,X) where\nH1(t,X) =/summationdisplay\n1≤k≤3Ck(t,X)Sk=C(t,X)·S.\nIf the spin sis fixed, in the semi-classical regime, /planckover2pi1ց0,H1(t,X) is the sub-\nprincipal symbol of ˆH(t). Hence we can get a semiclassical approximation\nat any order for Ψ( t) using generalized coherent states as it will be recalled\nlater. For details see [3], Chapter 14 and [4] for matrix prin cipal symbol\nH0(t).\nThe spin matrices Skare realized as hermitian matrices in the Hermitian\nspaceH2s+1such thatSk=dD(s)(σk/2) are defined by the derivative at I\nof the irreducible representation D(s)inH2s+1. In particular from the Lie\nalgebrasu(2) we have the commutation relations\n[Sk,Sℓ] =iǫk,ℓ,mSm. (1.3)\nCk(t,X) are real scalar symbols in X∈R2d. So the full Weyl symbol of\nˆH(t) is the matrix H(t,X) =H0(t,X)I+/planckover2pi1C(t,X)·S, with the spin opera-\ntorS= (S1,S2,S3).\nFor simplicity we shall assume in all this paper that the Weyl symbols\nH0(t) andC(t,X) are subquadratic [3](p.409). It follows that the quan-\ntum Hamiltonian ˆH(t) generates a propagator U(t,t0) in the Hilbert space\nL2(Rd,H2s+1). (see Proposition 123 of [3] easily extended to systems).\nIt is known (see [3] ( Ch.1 and Chap.7) that the Schr¨ odinger c oherent\nstates are labelled by the phase space T∗(Rd) and the spin (or atomic)\ncoherent states are labelled by the sphere S2. So it is natural to study the\nsemi-classical limit of the propagator with the phase space T∗(Rd)×S2. In\nparticular we can reformulate the propagation for coherent statesϕz⊗ψn\nlabelled by ( z,n)∈R2d×S2whereϕzis a Schr¨ odinger coherent state and\nψ(s)\nn=ψnis a spin coherent state.\nLet us recall our notations.\n•Heisenberg translations in L2(Rd):\nˆT(z) = exp/parenleftbiggi\n/planckover2pi1p·ˆx−q·ˆp/parenrightbigg\nwherez= (q,p)∈Rd×Rd, ˆxis mutiplication by xand ˆp=/planckover2pi1\ni∇x.\n2•Schr¨ odinger coherent states :\nϕz=ˆT(z)ϕ0, ϕ0(x) = (π/planckover2pi1)−d/4exp/parenleftbigg|x|2\n2/planckover2pi1/parenrightbigg\n•SU(2) and the sphere S2: let\nn= (sinθcosϕ,sinθsinϕ,cosθ),0≤θ<π,0≤ϕ<2π;\nTo anyn∈S2we associate the transformation in SU(2),\ng=gn= exp/parenleftbigg\niθ\n2((sinϕ)σ1−(cosϕ)σ2)/parenrightbigg\n(1.4)\n•irreducible representations and spin coherent states : letsbe an half\ninteger and Dsthe irreducible representation in the Hilbert space\nH2s+1of dimension 2 s+1. Ton∈S2is associated the spin coherent\nstates1\nψn=Ds(gn)ψ0,\nwhereψ0is a unit eigenvector of S3inH2s+1with minimal eigenvalue\n−s.\nLet us consider uncorrelated initial states Ψ z0,n0=ϕz0⊗ψ(s)\nn0, where\nϕz0is a Schr¨ odinger (orbital) coherent state and ψ(s)\nn0a spin coherent state.\nWhen the spin number sis fixed we have the following result proved in [2]\nand revisited in [3].\nTheorem 1.1. [2] For the initial state Ψz0,n0=ϕΓ0z0⊗ψ(s)\nn0we have\nU(t,t0)Ψz0,n0= ei\n/planckover2pi1S(t,t0)+isα(t)ϕΓ(t)\nz(t)⊗ψ(s)\nn(t)+O(√\n/planckover2pi1) (1.5)\nwherez0/ma√sto→z(t)is the classical flow for the Hamiltonian for H0(t),S(t,t0)\nis the classical action, the covariance matrix Γ(t)is computed from the dy-\nnamics generated by the linearized flow of H0(t)andα(t)is a real phase\ncomputed from the spin motion n(t)which satisfies the Landau-Lifshitz [7]\nequation\n∂tn(t) =C(t,z(t))∧n(t). (1.6)\nRemark 1.2. In Theorem 1.1 the motion of the spin depends on the motion\nalong the orbit but the orbit motion is independent of the spi n.\nRemark 1.3. Theorem 1.1 is a particular case of propagation of coherent\nstates for systems with a subprincipal term of order /planckover2pi1and a principal term\nH0(t)(here scalar) without crossing eigenvalues. Moreover one c an get a\ncomplete asymptotic expansion in power of√\n/planckover2pi1modO(/planckover2pi1∞).\nFor smooth crossings some non-adiabatic results are proved in [4].\n1this construction follows from the computation of the isotr opy subgroups of the SU(2)\naction: we get roughly SU(2)/ISO≈S2(see [3] for details).\n3The first new result in this paper is a generalization of Theor em 1.1 for\ns→+∞and/planckover2pi1ց0 such that s/planckover2pi1δ=cwhere 0< δ <1 andc >0 are\nconstants. Let us denote κ:=/planckover2pi1swhich is here a small positive parameter\nforδ<1.\nTheorem 1.4. Let us assume that 0<δ<1. For the initial state Ψz0,n0=\nϕz0⊗ψ(s)\nn0, the solution of (1.1)for the Hamiltonian (1.2)satisfies:\nU(t,t0)Ψz0,n0= ei\n/planckover2pi1S(t)+isα(t)ϕΓ(t)\nz(t)⊗ψ(s)\nn(t)+O(√\n/planckover2pi1+κ)),(1.7)\nwhere the dynamics of the coherent states satisfies the follo wing system of\nequations:\n˙q=∂pH0(t,q,p)+κ∂pC(t,q,p)·n(t)\n˙p=−∂qH0(t,q,p)−κ∂qC(t,q,p)·n(t)\n˙n(t) =C(t,q,p)∧n(t).(1.8)\nMoreoverS(t)is the action along the trajectory z(t)andα(t)is the action\nalong the trajectory n(t)of the classical spin, in the time interval [t0,t].\nLet be (zκ\nt,nκ\nt) the flow satisfying (1.8) with zκ\n0=z0,nκ\n0=n0.\nCorollary 1.5. Let be1/2>ε>0small. Then we have\n•ifs≤c/planckover2pi1−1/2+εthen the Theorem 1.4 is valid by taking κ= 0in(1.8),\nwith the error term O(/planckover2pi1ε). So in this case the quantum orbital follows\nthe classical trajectory for H0(t)at the leading order in /planckover2pi1.\n•Ifs≈c/planckover2pi1−1/2−εthen the quantum orbital motion depends on the spin\nmotion: in general we cannot take κ= 0in(1.8). If∇XC(t0,z0)/\\e}atio\\slash= 0\nthe orbital Gaussian ϕΓ(t)\nz(t)follows the new trajectory zκ\ntand notzκ\nt.\nMoreover there exist τ >0,c>0, such that\n|zκ\nt−z0\nt| ≥c|t−t0|/planckover2pi11/2−ε,for|t−t0| ≤τ. (1.9)\n(see Lemma A.1)\nRemark 1.6. Notice that the coherent state ϕΓ\nz(x)is localized in any disc\n{|X−z| ≤c/planckover2pi11/2−ε},c>0,ε>0, in the phase space R2d. More precisely its\nWigner function WϕΓz(X)satisfies for some µ>0,C >0,\n|WϕΓz(X)| ≤(π/planckover2pi1)−de−µ\n/planckover2pi1|X−z|2.\nAnd its Husimi function HϕΓz(X)satisfies, for some C′>0andµ′>0,\n0≤ HϕΓz(X)≤C′/planckover2pi1−de−µ′\n/planckover2pi1|X−z|2,\n(for more properties of the Husimi functions see [3], sectio n 2.5).\nHence from (1.9)we get that HϕΓ(t)\nz(t)(X)isO(/planckover2pi1∞)in a neighborhood of size\n/planckover2pi11/2−ε/2ofz0\ntfor0<t1<t<t+τ.\n4Let us consider now the case κ>0 fixed and /planckover2pi1ց0. We shall see that\nTheorem1.4 cannot besatisfied ingeneral. Inparticular for theDicke model\nconsidered by Hepp-Lieb (for /planckover2pi1= 1) [5] (See also [11] for a recent review).\nˆHDic=/planckover2pi1ωca†a+ω3ˆS3+λ√\nN(a†+a)ˆS1,\nIn quantum optics this model describes the interaction betw een light and\nmatter where the light (photons) is a field operator. Here we c onsider a toy\nmodel with an quantum harmonic oscillator for photons and Natoms of\nmatter being in two spin levels.\nωc>0,ω3≥0,N= 2s+1is the dimension of the spin states space, λ>0\na coupling constant, a=1√\n2/planckover2pi1(x+/planckover2pi1∂x),a†=1√\n2/planckover2pi1(x−/planckover2pi1∂x) are the usual\ncreation and annihilation operators (satisfying [ a,a†] = 1) and ˆSj=/planckover2pi1Sj.\nHere the Fock space is simply L2(R).\nA more explicite expression for ˆHDicconsidered here is the following\nˆHDic=ωc\n2(−/planckover2pi12∂2\nx+x2)+ω3ˆS3+2λ√\nN/planckover2pi1xˆS1. (1.10)\nWith this elementary model we get a contradiction with Theor em 4.7 of [2].\nIn the regime /planckover2pi1≈s−1,sր ∞the orbital trajectory blows up into a mixed\nstate fort>0 (”decoherence of the orbital state”).\nRemark 1.7. Why to consider large spin quantum systems?\nNotice that for Natoms of spin 1/2 the spin Hilbert space should be of dimen-\nsion2Ncorresponding to the tensor product ⊗ND1/2whereD1/2is the rep-\nresentation of degree 2 of SU(2)for the spin 1/2. Using the Clebsch-Gordon\nformula we see that the representation ⊗ND1/2contains the irreducible rep-\nresentation DN/2ofSU(2)of maximal degree N+ 1corresponding to an\neffective spin s=N\n2. So that in this settting the large spin limit is also the\nlarge number of atoms limit (thermodynamic limit).\nTheorem 1.8. Let be0<κ=/planckover2pi1s,Ψ(0) =ϕz0(x)ψn0andΨ(t) = e−it\n/planckover2pi1ˆHDicΨ(0)\nwithΨ(0) =ϕz0⊗ψn0.\nThen there exists c0>0such that for any 0<µ<1/2and/planckover2pi1ց0, we have\ntr[trH2s+1ΠΨ(t)]2≤(1+c0κt2)−1/2+O(/planckover2pi1µ), (1.11)\nwhereΠΨis the projection on the pure state Ψ∈L2(R)⊗H2s+1,trH2s+1ΠΨ\nis the partial trace of ΠΨinH2s+1and the last trace in (6.7)is computed\nfor operators in the Hilbert space L2(R,C).\nRemark 1.9. The previous result shows that for a spin sof order /planckover2pi1−1the\norbital evolution is transformed into a mixed state as the sp in interaction is\n5switched on and /planckover2pi1is small enough.\nFor a density matrix ˆρ,P[ˆρ] := tr[ˆρ2]is called the purity of the density\nmatrixˆρ. Here it is applied for ˆρs(t) := tr H2s+1ΠΨ(t). This is related with\nthe von-Neumann entropy which is defined as SvN[ˆρ] =−tr[ˆρlog ˆρ]. So we\nhave easily SvN[ˆρ]≥1−P[ˆρ]. For a pure state SvN[ˆρ] = 0andP[ˆρ] = 1.\nThen from (6.7)we get that the density matrix trH2s+1ΠΨ(t)has a positive\nvon Neumann entropy for t>0and/planckover2pi1>0small enough.\nIn the analysis of the Dicke model one consider that the orbita l part is an\nopen sub-system of the closed total system (orbital+spin). For other open\nsystems and time evolution of coherent states (like the ”Sch r¨ odinger cat”)\nwe refer to [3], Chapter 13 for more details.\n2 Preliminaries\n2.1 Reduction to the interaction propagator\nIt is convenient to annihilate the scalar part H0(X) by considering the inter-\naction representation for the propagator U(t,t0) of the Hamiltonian ˆH(t).\nHence we have U(t,t0) =U0(t−t0)V(t,t0), where U0(t−t0) = e−it−t0\n/planckover2pi1ˆH0and\nthe propagator V(t,t0) must satisfy\ni/planckover2pi1∂tV(t,t0) = (/hatwideCI(t)·S)V(t,t0).\nFor 1≤k≤3 the Weyl symbols CI,k(t,X) is computed by the Egorov\nTheorem. In particular its principal term is given by\nCI,k(t,X) =Ck(t,Φt−t0\n0(X)).\nSo for simplicity, in what follows we shall assume that H0≡0 and consider\nthe simpler interaction spin-orbit Hamiltonian ˆHint(t) =/planckover2pi1/hatwideC(t)·S.\nFor the Dicke model H0is the harmonic oscillator so we have:\nΦt\n0(x,ξ) = (cos(ωct)x(0)+sinωctξ(0),−sin(ωct)x(0)+cos(ωct)ξ(0),\nhenceHint(t,x,ξ) = (cos(ωct)x+sin(ωct)ξ)S1+/planckover2pi1ω3S3.\n2.2 A realization of spin- srepresentation\nBecause our aim is to perform explicit computations for the s pin side we\nchoose to work with a concrete representation (see for examp le [3], chap.7).\nWe assume here that V(s)=H2s+1, the complex linear space of homogenous\npolynomials of degree 2 sin two variables ( z1,z2)∈C2.H2s+1is an Hermi-\ntian space for the scalar product defined such that the monomi als (named\nDicke states):\nD(s)\nm=zs+m\n1zs−m\n2/radicalbig\n(s+m)!(s−m)!,\n6where−s≤m≤sandmis an integer or half an integer according sis.\nInH2sthe spin operators are realized as\nS3=1\n2(z1∂z1−z2∂z2), S+=z1∂z2, S−=z2∂z1\nS1=S++S−\n2, S2=S+−S−\n2i, (2.1)\nwith the commutation relations of the Lie algebra su(2) of the group SU(2):\n[S3,S±] =±S±.\nRecall that g∈SU(2) ifg=/parenleftbiggα−¯β\nβ α/parenrightbigg\n,α,β∈C,|α|2+|β|2= 1.\nSpin coherent states are defined by the action of SU(2) on the Dicke state of\nminimal weight: D(s)\n−s. As known, to get a good parametrization of coherent\nstates we choose a family of transformations in SU(2) indices by the sphere\nS2. This is obtained by computing the isotropy subgroup ISO of t heSU(2)\naction:g∈ISO if and only if g=/parenleftbiggα0\n0 ¯α/parenrightbigg\n,α= eiψ,ψ∈R. Then we get\nthat the space of orbits SU(2)\\ISO can be identified with the sphere S2(for\ndetails see for example [3]).\nLet us denote\nn= (sinθcosϕ,sinθsinϕ,cosθ),0≤θ<π,0≤ϕ<2π;\nTo anyn∈S2we associate the transformation in SU(2),\ng=gn= exp/parenleftbigg\niθ\n2(sinϕσ1−cosϕσ2)/parenrightbigg\n(2.2)\nwhereσ1, σ2are the Pauli matrices which satisfy the commutation relati ons\n[σk,σl] = 2iǫk,ℓ,mσm (2.3)\nFors=1\n2we haveSk=σk\n2. So ifg/ma√sto→ D(s)gis the representation of SU(2)\ninH2s+1we have\nD(s)gn= exp/parenleftbiggθ\n2(eiϕS−−e−iϕS+)/parenrightbigg\n(2.4)\nDefinition 2.1. The coherent states of SU(2)are defined in the represen-\ntation space H2s+1as follows:\n|n/a\\}b∇acket∇i}ht=Ds(gn)D(s)\n−s:=ψ(s)\nn. (2.5)\n7It is some time convenient to consider a complex parametriza tion of the\nsphereS2using the stereographic projection n/ma√sto→η, on the complex plane\nand an identification of the coherent states |n/a\\}b∇acket∇i}htwith the state |η/a\\}b∇acket∇i}ht:=ψη\ndefined as follows:\n|η/a\\}b∇acket∇i}ht= (1+|η|2)−jexp(ηS+)|s,−s/a\\}b∇acket∇i}ht,\nMore precisely we denote |η/a\\}b∇acket∇i}ht=|n/a\\}b∇acket∇i}htwith the following correspondence:\nn= (sinθcosϕ,sinθsinϕ,cosθ), η=−tanθ\n2e−iϕ\nThe geometrical interpretation is that −¯ηis the stereographic projection of\nn.\nRecall the following expression of gn\ngn=/parenleftbiggcosθ\n2−sinθ\n2e−iϕ\nsinθ\n2eiϕcosθ\n2/parenrightbigg\n(2.6)\nFor further investigations we shall need two results:\n1) compute the derivative of t/ma√sto→T(gnt) for aC1path onS2.\n2) compute the adjoint action of T(gn) onSk, 1≤k≤3.\nFrom (2.4) we get\n∂ϕT(gn) =i\n2T(gn)/parenleftbig\nsinθ(e−iϕS++eiϕS−)+(1−cosθ)S3/parenrightbig\n(2.7)\ni∂θT(gn) =i\n2T(gn)(eiϕS−−e−iϕS+) (2.8)\nLemma 2.2. Let be aC1path onS2:t/ma√sto→(θt,ϕt). Then we have\n∂tT(gn) =i\n2T(gn)/parenleftbig\nsinθ(e−iϕS++eiϕS−)+(1−cosθ)S3/parenrightbig\n˙ϕt\n+1\n2T(gn)/parenleftbig\neiϕS−−e−iϕS+/parenrightbig˙θt. (2.9)\nTo compute ˙θ, ˙ϕwe use the Riemann model for S2. Let be −¯η∈Cthe\nstereographic projection of non the equatorial plane, from the south pole\nnso. The coordinates of nare given by\nn1=−η+ ¯η\n1+|η|2, n2=η−¯η\ni(1+|η|2), n3=1−|η|2\n1+|η|2.\nSo we have\nη=−tan/parenleftbiggθ\n2/parenrightbigg\ne−iϕ,cosθ=1−|η|2\n1+|η|2,sinθeiϕ=−2¯η\n1+|η|2.(2.10)\nThen we get easily\n˙θ=η˙¯η+ ¯η˙η\n|η|(1+|η|2),˙ϕ=i\n2/parenleftbigg˙η\nη−˙¯η\n¯η/parenrightbigg\n. (2.11)\n8Lemma 2.3.\n−i∂tT(gnt) =iT(gnt)(A˙θ+B˙ϕ)\nwith\n(A˙θ+B˙ϕ) =−1\n1+|η|2(˙ηS+−˙¯ηS−)+|η|2\n1+|η|2/parenleftbigg˙η\nη−˙¯η\n¯η/parenrightbigg\nS3\nProof. Standard computations. /square\nLet us denote Sk(n) =Ds(gn)∗SkDs(gn),k= +,−,3 ork= 1,2,3.\nIn the following Lemma similar results are stated in [6], sec tion 2.6, and\nproved by a different method.\nLemma 2.4. If(θ,ϕ)are the coordinates of nonS2, we have,\nS3(θ,ϕ) = cosθ.S3−sinθ/parenleftbiggeiϕS−+e−iϕS+\n2/parenrightbigg\n(2.12)\nS+(θ,ϕ) = eiϕsinθ.S3+cosθ+1\n2S++cosθ−1\n2e2iϕS−(2.13)\nS−(θ,ϕ) = e−iϕsinθ.S3+cosθ+1\n2S−+cosθ−1\n2e−2iϕS+(2.14)\nS1(θ,ϕ) =/parenleftbigcosθ+1\n2+cosθ−1\n2cos2ϕ/parenrightbig\nS1+cosθ−1\n2sin2ϕS2\n+cosϕsinθS3\n(2.15)\nS2(θ,ϕ) =cosθ−1\n2sin2ϕS1+/parenleftbigcosθ+1\n2−cosθ−1\n2cos2ϕ/parenrightbig\nS2\n+sinθsinϕS3. (2.16)\nProof. We write Ds(gn) = eθ\n2LwhereL= eiϕS−−e−iϕS+. Let beSone\nof the spin operator, we have\n∂θS(θ,ϕ) =1\n2e−θ\n2L[S,L]eθ\n2L\nand the commutation relations\n[L,S3] = eiϕS++e−iϕS+,[L,S−] =−2e−iϕS3,[L,S+] =−2eiϕS3.\nSo we find\n∂θS3(θ,ϕ) =−1\n2(eiϕS−(θ,ϕ)+e−iϕS+(θ,ϕ)) (2.17)\n∂θS+(θ,ϕ) = eiϕS3(θ,ϕ) (2.18)\n∂θS−(θ,ϕ) = e−iϕS3(θ,ϕ). (2.19)\n9hence\n∂2\nθS3(θ,ϕ) =−S3(θ,ϕ). (2.20)\nSolving the differential equation (2.20) we get (2.12). /square\nNow we come to the spin-coherent states. Le be n0the north pole on\nS2corresponding to the Dicke state D(s)\n−s:=ψn0. The following Lemma is\nbasic for our next computations.\nLemma 2.5. For any n∈S2we have\nSψn=−snψn+/radicalbiggs\n2v(n)ψ1,n\nwhereψ1,n=D(s)(gn)D(s)\n1−sandv(n) = (v1,v2,v3)∈C3is defined as\nv1(n) =cosθ+1\n2+cosθ−1\n2e−2iϕ(2.21)\nv2(n) =cosθ+1\n2i−cosθ−1\n2ie−2iϕ(2.22)\nv3(n) =−sinθe−iϕ(2.23)\nIn particular we have n·v(n) = 0.\nProof. The formulas are proved from elementary computations usin g\nLemma 2.4 and that S3ψn0=−sψn0,S+ψn0=√\n2sψ1,n0,S−ψn0= 0. In\nparticular we recover here a well known property of the spin c oherent states:\nn·Sψn=−sψn.\n/square\nRemark 2.6. In Lemma 2.5 the formulas have only two terms, a leading\nterm of order sand a second term of order√s. In Lemma 4.5 of [2] the\nauthors claims that the expansion is an asymptotic power ser ie ins−1. This\ncontradicts our computations.\nIt is also convenient to write down the evolution of the spin m atrices on\nthe Riemann sphere, denoting Sk(η) =Sk(n) whereη∈Cis the complex\ncoordinate of n∈S2.\nLemma 2.7.\nS3(η) =1−|η|2\n1+|η|2S3+ηS++ ¯ηS−\n1+|η|2,\nS+(η) =−2¯η\n1+|η|2S3+1\n1+|η|2S+−η2\n1+|η|2S−\nS−(η) =−2η\n1+|η|2S3+1\n1+|η|2S−−¯η2\n1+|η|2S+.\n102.3 The classical spin space\nLet us recall that the sphere S2has a natural symplectic form:\nσn(u,v) = (u∧v)·n= det(u,v,n), where n∈S2,u,v∈Tn(S2).\nLet beH:R3→Ra smooth function. Its resriction to S2defined an\nHamiltonian vector field XHonS2satisfyingdH(Y) =σn(Y,XH),∀Y∈\nTn(S2). So we get the Hamilton equation (named in this context the L andau\nequation):\n˙n=∇H∧n.\nIncomplexcoordinates(2.10)thecovariant symbol Hc(t,η,¯η) =/a\\}b∇acketle{tψη,ˆH(t)ψη/a\\}b∇acket∇i}ht\nof the Hamiltonian ˆH(t) =/planckover2pi1C(t)·S, becomes\nHc(t,η,¯η) = (1+ |η|2)−1(C3(t)(1−|η|2)−(C−(t)−¯η+C−(t)+η))\nwhereC±=C1±iC2. Recall that the symplectic form on the Riemann\nsphereˆCisσc= 2i(1 +|η|2)−2dη∧d¯η. So the Hamilton equation in ˆC\nbecomes\n˙η=(1+|η|2)2\n2i∂¯ηHc(t,η,¯η). (2.24)\nThis is the Landau-Lifschitz equation (1.6) in complex coor dinates.\nFollowing [10] the symplectic two form satisfies dθc=σcwhere the one-form\nisθc=i−1(∂¯ηKd¯η−∂ηKdη) andK(η,¯η) = 2log(1 + η¯η) is the K¨ ahler\npotential for ˆC.\nIn particular the action Γ for Hcis the one form in ˆCη×Rtsatisfying\ndΓc=θc−Hcdt. So, along an Hamiltonian path in time interval [0 ,T], the\naction is given by (see Appendix):\nγ(T) =/integraldisplayT\n0/parenleftbiggℑ(ηt˙¯ηt)\n2(1+|ηt|2)−Hc(ηt,¯ηt)/parenrightbigg\ndt (2.25)\nIn the same Appendix it is proved that α(t) =γ(t) whereα(t) is the\nphase given in Theorem 1.4.\n2.4 The Schr¨ odinger coherent states\nWe shall use some well known formulas concerning the Heisenb erg transla-\ntions operators ˆT(z) and coherent states.\nLemma 2.8. Le bet/ma√sto→zt= (qt,pt)aC1path in the phase space R2d. Then\nwe have\ni/planckover2pi1∂tˆT(zt) =ˆT(zt)/parenleftbigg1\n2σ(zt,˙zt)+ ˙qt·/planckover2pi1∇x−˙pt·x/parenrightbigg\n.(2.26)\nLemma 2.9. [3, chap.2] Assume that Aa sub-polynomial symbol. Then for\neveryN≥1, we have\nˆAϕz=/summationdisplay\n|γ|≤N/planckover2pi1|γ|\n2∂γA(z)\nγ!Ψγ,z+O(/planckover2pi1(N+1)/2), (2.27)\n11the estimate of the remainder is uniform in L2(Rn)forzin every bounded\nset of the phase space and\nΨγ,z=ˆT(z)Λ/planckover2pi1Opw\n1(zγ)g. (2.28)\nwhereg(x) =π−n/4e−|x|2/2,Opw\n1(zγ)is the 1-Weyl quantization of the\nmonomial :\n(x,ξ)γ=xγ′ξγ′′,γ= (γ′,γ′′)∈N2d. In particular Opw\n1(zγ)g=Pγgwhere\nPγis a polynomial of the same parity as |γ|.\n3 The Schr¨ odinger equation for the spin-orbit in-\nteraction\nRecall our reduced Hamiltonian ˆH(t) =/planckover2pi1/hatwideC(t)·Sand the Schr¨ odinger equa-\ntion\n(i/planckover2pi1∂t−ˆH(t))Ψ(t) = 0,Ψ(0) =ϕz0⊗ψn0. (3.1)\nLet us consider the following ansatz\nΨapp(t) = ei\n/planckover2pi1γ(t)ϕΓ(t)\nz(t)⊗ψ(s)\nn(t)\nsuch that for /planckover2pi1δs=κ>0and someµ>0 we have:\n(i/planckover2pi1∂t−ˆH(t))Ψapp(t) =O(/planckover2pi11+µ) for/planckover2pi1→0 (3.2)\nhence from Duhamel formula we should have\n/ba∇dblΨ(t)−Ψapp(t)/ba∇dbl=O(/planckover2pi1µ).\nIfC(t) depends only on time tthen the ansatz (3.2) gives an exact solution\nfor someγ(t),nt[2].\nProposition 3.1. Let us assume that ˆH(t) =/planckover2pi1C(t)·S, whereC(t)depends\nonly on time. Then the ansatz (3.2)is exact for any s. More explicitly we\nhave\nΨapp(t) = eisα(t)ϕz0⊗ψ(s)\nn(t)(3.3)\nThe classical spin motion n(t)follows the Landau-Lifshitz equation\n˙nt=C(t)∧nt. (3.4)\nand the phase α(t)is the classical spin action computed in (2.25).\nProof.This is well known (see a proof in Appendix B).\n124 The regime s=c/planckover2pi1−δ,0< δ <1\nInthissectionwegiveaproofofTheorem1.4. Wehavetwosmal lparameters\n/planckover2pi1andκ:=s/planckover2pi1=c/planckover2pi11−δ.\nIn this regime we shall prove that the formula (1.5) is still v alid with the\nerror estimate O(√\n/planckover2pi1+κ).\nLet us consider the ansatz\nΨ♯(t) = ei\n/planckover2pi1S(t,t0)+isα(t)ϕΓ(t)\nz(t)⊗ψ(s)\nn(t). (4.1)\nAs for the scalar Schr¨ odinger equation (see [3], Chap. 4) we shall compute\na patht/ma√sto→(z(t),n(t)), a covariance matrix Γ( t) and phases S(t,t0),α(t,t0)\nsuch that (1.5) is satisfied.\nIt is enough to prove the result for the interaction propagat orV(t,t0) such\nthat the orbital Hamiltonian H0(t) is absent. The full result is obtained by\napplying the scalar propagator of the orbital motion.\nIt is enough to prove, for the norm in L2(Rd,H2s+1), we have\ni/planckover2pi1∂tΨ♯(t) =ˆH(t)Ψ♯(t)+O(/planckover2pi1(√\n/planckover2pi1+κ)).\nBy standard computations we get asymptotic expansions in /planckover2pi1for each term\ni/planckover2pi1∂tΨ♯(t) andˆH(t)Ψ♯(t) mod an error O(/planckover2pi1(√\n/planckover2pi1+κ)).\nRecall the notations\nϕΓ\nz=ˆT(z)Λ/planckover2pi1gΓ\n0, g0(x) = (π)−d/4cΓei<x,Γx>,Λ/planckover2pi1f(x) =/planckover2pi1−d/4f(x/√\n/planckover2pi1) and\nψn=T(gn)ψn0. (4.2)\nUsing (2.26), (2.2), Lemma 2.3, Lemma 2.9, and Taylor formul a aroundzt\nat the order 3, the Schr¨ odinger equation for the ansatz (3.2 ) is transformed\nas follows.\n/parenleftbigg1\n2σ(zt,˙zt)−∂tS(t)−s/planckover2pi1˙α(t)+√\n/planckover2pi1(˙ qt·∇x−˙ pt·x)/parenrightbigg\ng0⊗ψn0+ (4.3)\ng0⊗/planckover2pi1(A˙θ+B˙ϕ)ψn0\n=/summationdisplay\n1≤k≤3/parenleftBig\nCk(t,zt)+√\n/planckover2pi1∇XCk(t,zt)Opw\n1(X)/parenrightBig\ng0⊗/planckover2pi1Sk(nt)ψn0+O(/planckover2pi1(√\n/planckover2pi1+κ)).\nBut we have\n(˙qt·∇x−˙pt·x)g0= (i˙q−˙p)·xg0,and Opw\n1(a·X)g0= (α+iβ)·xg0\nwherea·X=α·x+β·ξ. From Lemma 2.3 we have\nA˙θ+B˙ϕ) =−√\n2s˙ηt\n1+|ηt|2ψ1,n0+s|ηt|2\n1+|ηt|2/parenleftbigg˙ηt\nηt−˙¯ηt\n¯ηt/parenrightbigg\nψn0.(4.4)\n13From Lemma 2.7 we have also\nS3(η)ψn0=−s1−|η|2\n1+|η|2ψn0+√\n2sη\n1+|η|2ψ1,n0,\nS+(η)ψn0= 2s¯η\n1+|η|2ψn0+√\n2s1\n1+|η|2ψ1,n0\nS−(η)ψn0= 2sη\n1+|η|2ψn0−√\n2s¯η2\n1+|η|2ψ1,n0\nNow we get the equations to compute the ansatz by identifying the co-\nefficients of ( /planckover2pi1,κ) in (4.3). Easy computations give the following results.\n•Projection on g0⊗ψn0: the coefficient of /planckover2pi10gives the classical action\nS(t,t0) andthecoefficient of κgives thespinaction α(t). Inparticular.\nwe get\n˙α(t) =ℑ(ηt˙¯ηt)\n2(1+|ηt|2)−Hc(ηt,¯ηt), (4.5)\nwhereHc(η,¯η) =/a\\}b∇acketle{tψη,C(t)·Sψη/a\\}b∇acket∇i}ht.\n•Projection on xg0⊗ψn0determinestheHamilton equation fortheorbit\nzt.\n•projection on xg0⊗ψ1,n0determines the Landau-Lifshitz equation for\nn(t).\nNotice that for the interaction dynamics the covariant matr ix Γ is con-\nstant, itdependsonlyontheorbitaldynamicsfor H0(t)notintheinteraction\nwith the spin, contrary to the orbital motion when κ≫√\n/planckover2pi1.\nUsing the same method as in the scalar case considered in [3], Chapter 4 we\ncan complete the proof of Theorem 1.4.\n5 The regime /planckover2pi1s=κ=constant\nThe ansatz 3.2 is very natural when considering the classica l analogue of\n(3.1). We assume /planckover2pi1s=κ>0and we keep /planckover2pi1as our semi-classical parameter.\nFor simplicity assume that κ=1\n2.\nThe symbol of ˆH(t) =/hatwideC(t)·ˆS, isH(t,q,p;n) =−κC(t,q,p)·n, defined on\nT∗(R)×S2, (recall that the covariant symbol of Sis−sn, see [3], prop. 90)\nwe get the classical system of equations\n˙q=−1/2∂pC(t,q,p)·n,\n˙p= 1/2∂qC(t,q,p)·n,\n˙n=C(t,q,p)∧n. (5.1)\nWhenC(t) depends only on time there is no orbit interaction with the s pin\nand using section 2.2 we can compute the phase α(t) such that (3.2) is the\n14exact solution of (3.1). /square\nBut in the following computations we shall see that (3.2)is not possible if\n∇XC(t,X)/\\e}atio\\slash= 0.\nFor simplicity we assume here that for 1 ≤k≤3,Ck(t) is a linear form on\nR2d,Ck(t,X) =ak(t)·X=αk(t)·x+βk(t)·ξ,X= (x,ξ).\nLet us revisit the computations of Section 4 in the particula r case. Denote\nC±=C1±iC2.\nLet us denote (0) Land (0)Rthe coefficient of /planckover2pi10on left and right side of\n(4.3). In the same way we introduce (1 /2)L,Rand (1)L,R. for the coefficients\nof/planckover2pi11/2and/planckover2pi11. Just compute to get\n(0)L=1\n2σ(zt,˙zt)−˙γ(t)−1\n2|ηt|2\n1+|ηt|2/parenleftbigg˙ηt\nη−˙¯ηt\n¯ηt/parenrightbigg\n(5.2)\n(0)R=−1−|ηt|2\n1+|ηt|2C3(t,zt)−¯η2\n1+|η|2C−(t,zt)−¯η2\n1+|η|2C+(t,zt)\nThe equation (0) L= (0)Rdetermineγ(t) whenzt,ηtare known.\nIn the linear case consider here ∇XC(t) is independent on X. So we have:\n(1/2)L= (−i˙q−˙p)xg0⊗ψn0−˙ηt\n1+|ηt|2g0⊗ψ1,n0\n(1/2)R=−/parenleftbigg1−|ηt|2\n1+|ηt|2(α3(t)+iβ3(t))+ηt\n1+|ηt|2(α+(t)+iβ+(t))\n+¯ηt\n1+|ηt|2(α−(t)+iβ−(t))/parenrightbigg\n·xg0⊗ψ0\n+/parenleftbigg\nC3(t)ηt\n1+|ηt|2+C−(t)1\n1+|ηt|2\n−C+(t)¯η2\nt\n1+|ηt|2/parenrightbigg\ng0⊗ψ1,n0 (5.3)\nWe denote C(t) :=C(t,zt). From the equation (1 /2)L= (1/2)R, using that\nthe statesxg0⊗ψ0andg0⊗ψ1,n0are orthogonal in H2s, we obtain a system\nof coupled equations which determines a trajectory ( zt,ηt) inR2d×S2. In\nparticular we get again for ntthe Landau equation (3.4) for C(t,zt) but here\nthe time dependent equation for ztdepends on nt.\nNow let us consider (1) L,R. First we have (1) L= 0. Let us compute (1) R\nwhich is a term supported by the the mod xg0⊗ψ1,n0.\n(1)R=−/parenleftbiggηt\n1+|ηt|2(α3(t)+iβ3(t))+1\n1+|ηt|2(α+(t)+iβ+(t))\n+1\n1+|ηt|2(α−(t)+iβ−(t))/parenrightbigg\n·xg0⊗ψ1,n0.\nThen we get that (1) R/\\e}atio\\slash= 0 if∇XC(t)/\\e}atio\\slash= 0.\n15Remark 5.1. The section 4.2 of [2] use in a fondamental way their Lemma\n4.5 concerning the action of the spin operators on spin coher ent states. But\nthe conclusion of this Lemma is false as we have shown in (4.5). In particu-\nlar the√sterm in our Lemma 2.5 is at the origin of the wrong ansatz (3.2)\nin the regime /planckover2pi1s=κ.\n6 More on the Dicke model\n6.1 Preliminary computations and reductions\nA simpler form of the interaction Hamiltonian for the Dicke m odel is\nˆHDint(t) = ((cost)x+(sint)/planckover2pi1Dx)/planckover2pi1S1. (6.1)\nMore generally consider the Hamiltonian ˆH(t) = (α(t)x+β(t)Dx))A, where\nAis a bounded Hermitian operator in the Hilbert space H. Let be U(t) the\npropagator for the Schr¨ odinger equation with initial data att= 0.\ni∂tΨ(t) =ˆH(t)Ψ(t)\nThenwehavethefollowinglemmarelatedwithCampbell-Haus dorffformula.\nLemma 6.1. There exist two scalar functions c(t)and˜c(t)such that\nU(t) = e−i\n2c(t)A2e−ib(t)DxAe−ia(t)xA(6.2)\nwherec(t) =/integraltextt\n0α(τ)b(τ)dτ,a(t) =/integraltextt\n0α(τ)dτ,b(t) =/integraltextt\n0β(τ)dτ, and\nU(t) = e−i\n2˜c(t)A2e−i(b(t)Dx+a(t)x)A(6.3)\nwhere˜c(t) =c(t)−a(t)b(t).\nProof.Letusfirstremarkthat (6.3)isadirectconsequenceof theCa mpbell-\nHausdorffformula. Soitisenoughtoprove(6.2). Let V(t) = e−ib(t)DxAe−ia(t)xA.\nBy a direct computation and a commutation we get that that\ni∂tV(t) = (˙a(t)xA+˙b(t)Dx)V(t)+a(t)b(t)A2.\nSo we get (6.2) with ˙ c(t) =α(t)b(t). Now from the Campbell-Hausdorff\nformula, we get (6.3) with ˜ c(t) =c(t)−a(t)b(t).\n/square.\nLet us assume that Ψ(0 ,x) =ϕz0(x)ψn0, the product of a Schr¨ odinger\ncoherent state and a spin coherent state. Let us assume first t hatβ= 0.\nThen the time evolution is given by Ψ( t,x) =ϕz0(x)e−ia(t)xS1ψn0. We have\nseen (Proposition 3.1) that\ne−ia(t)xS1ψn0= eisγ(t,x)ψn(t,x).\n16The classical spin n(t,x) is given here by\nn1(t,x) =n0,1= cosθ0,\nn2(t,x) = sinθ0cos(θ(t,x)),\nn3(t,x) = sinθ0sin(θ(t,x)),\nwhereθ(t,x) =θ0−a(t)x. The phase α(t,x) is given by\n˙α(t,x) =α(t)x\n1+|η|2/parenleftbiggn1(t,x)\n1+n3(t,x)+|η|2n2(t,x)\n1+n3(t,x)/parenrightbigg\n(6.4)\nwithγ(t0,x) = 0 and |η|2=1−n2\n3\n(1+n3)2. So we have the formula\nΨ(t,x) =ϕz0(x)eisγ(t,x)ψn(t,x).\nIfsis frozen ( /planckover2pi1-independent ) we can use the Taylor formula and we get\n/ba∇dblΨ(t)−ϕz0eisγ(t,q0)ψn(t,q0)/ba∇dblL2(R,Hs)=O(√\n/planckover2pi1)\nwhich is a compatible with the ansatz (3.2) for µ= 1/2.\nFors/planckover2pi1=κ>0we also consider Taylor expansion.\nγ(t,x) =γ(t,q0)+∂xγ(t,q0)(x−q0)+∂2\nxγ(t,q0)(x−q0)2\n2+O(|x−q0|3)\nHence\nϕz0(x)eisγ(t,x)=ϕz0(x)eis(γ(t,q0)+∂xγ(t,q0)(x−q0)+∂2\nxγ(t,q0)(x−q0)2\n2)+O(κ√\n/planckover2pi1)\nin this formula we see that the spin motion add a momentum to th e orbit\nmotion (the linear term in ( x−q0) and a quadratic contribution to the\nGaussian.\nBut the Taylor argument to eliminate the x-dependence does not work for\nthe spin part ψn(t,x).\nThe reason is the following. From explicit formulas [3], we k now that\n|/a\\}b∇acketle{tψn,ψm/a\\}b∇acket∇i}ht|= eslog(1−|n−m|2\n4), (6.5)\nhence we have\n/ba∇dblψn−ψm/ba∇dbl2≈2(1−e−s|n−m|2\n4)\nWith the localization by ϕz0forxin a neighborhood of q0we have|n(t,x)−\nn(t,q0)|of order/planckover2pi1. Assume that 0 <θ0<π/2 andf= 1. Then there exist\nc1>,c2>0 such that for t>0 we have\n/ba∇dblψn(t,x)−ψn(t,q0)/ba∇dbl2≥c1(1−e−c2st2|x−q0|2).\n17Using that s=κ\n/planckover2pi1, we get with c3>0, fort0>0 small enough,\n/ba∇dblϕz0(x)(ψn(t,q0)−ψn(t,x))/ba∇dbl2\nL2(Rx,H2s+1)≥c3κt2,∀t∈]0,t0].\nThis shows that the classical spin n(t,x) has to depend not only on the\norbit but also on the position xon the orbit which is not compatible with\nthe ansatz (3.2).\nAnother and more accurate way to understand the last computa tions is\nrelated with an entanglement-decoherence phenomenon for t he time evolu-\ntion of the initial state Ψ(0) = ϕz0⊗ψn0, when the interaction with a large\nspin system is switched on.\nWe shall see that for t∈[0,t0],κ >0 ands=κ\n/planckover2pi1, then Ψ(t,x) is anentan-\ngled state which means that it is not possible to have a decomposition li ke\nΨ(t) =ϕ(t)⊗ψ(t) withϕ(t)∈L2(R) andψ(t)∈ H2s+1.\nWeusethepartialtracesintheHilbertspace L2(R,C)⊗H2s+1=L2(R,H2s+1).\nRecall that in a tensor product of Hilbert spaces H=H1⊗H2, for a trace\nclass operator AinH, the partial trace of AonH2is the unique trace class\noperator in H1, denoted tr H2(A), such that the following Fubini identity is\nsatisfied for any bounded operator BonH1,\ntrH1(trH2(A)B) = trH(A(B⊗IH2)).\nWe shall use the following invariance property: if Ukare invertible operators\ninHk, andU=U1⊗U2, then we have\ntrH2(U−1AU) =U−1\n1trH2(A)U1.\nNotice that if H2=Cthen we have H1=H1⊗Cand trC(A) =A.\nLet Ψ =ψ1⊗ψ2∈ H1⊗H2. and denote by Π Ψthe orthogonal projector on\nΨ. Then we have tr H2ΠΨ= Πψ1.\nPhysical interpretation : ifAis a density matrix in H(non negative operator\nwith trace 1) then tr H2(A) is also a density matrix in H1which represents\nthe state of the sub-system H1.\nSupposethat thetotal system satisfies aSchr¨ odingerequat ion withaninitial\nstate Ψ 0=ψ1⊗ψ2(pure state in H). For time t >0 the density matrix\nΠΨ(t)of the total system is pure but the density matrix tr H2(ΠΨ(t)) of the\nsubsystem in H1is not necessary a pure state because it is not isolated\n(decoherence ). When the rank of tr H2(ΠΨ(t)) is≥2 the sub-system (1) can\noccupy at least two orthogonal pure states with probabiliti es in ]0,1[, like\nfor the Schr¨ odinger cat.\nLet be Ψ ∈L2(R,C)⊗ H2s+1)≃L2(R,H2s+1). A simple computation\ngives\ntrH2s+1(ΠΨ)f(x) =/integraldisplay\nRf(y)/a\\}b∇acketle{tΨ(y),Ψ(x)/a\\}b∇acket∇i}htH2s+1dy,∀f∈L2(R).\n18In the same way we have also\ntrL2(R)u=/integraldisplay\nR/a\\}b∇acketle{tΨ(x),u/a\\}b∇acket∇i}htH2s+1Ψ(x)dx,∀u∈ H2s+1.\nThe orbital density matrix ρO(t) := tr H2s+1(ΠΨ(t)) has the integral kernel\nK(t,x,y) = eis(γ(t,x)−γ(t,y)˜K(t,x,y) where\n˜K(t,x,y) =ϕzt(x)ϕzt(y)/a\\}b∇acketle{tψn(t,x),ψn(t,y)/a\\}b∇acket∇i}htH2s+1. (6.6)\nSo we have\ntr(ρO(t)2) =/integraldisplay\nR2|˜K(t,x,y)|2dxdy.\nRecall that ˆH(t) =α(t)xS1.\nLemma 6.2. There exists c0>0such that for any 0<µ<1/2we have\n/integraldisplay\nR2|˜K(t,x,y)|2dxdy≤(1+c0κt2)−1/2+O(/planckover2pi1µ) (6.7)\nNotice that ˆK(t) is a non negative operator of trace 1. So let be λj(t)\nthe eigenvalues of ρO(t) (in decreasing order with multiplicities). So if t>0,\nκ>0 and/planckover2pi1small enough, we get from the Lemma that\n/summationdisplay\njλ2\nj(t)</summationdisplay\njλj(t) = 1\nhenceˆK(t) has at least two eigenvalues <1 when the spin interaction is\nswitched on (recall that ˆK(0) is the projector on a pure state).\nProofof Lemma 6.2.\nWe use (6.5) to compute the Schwartz kernel (6.6) and Lemma 6. 3\nSo we get for any µ>0,\n/integraldisplay\nR2|˜K(t,x,y)|2dxdy≤(π/planckover2pi1)−1/integraldisplay\n{u2+v2≤r0}e−1\n/planckover2pi1(u2+v2+c1κt2(u−v)2)dudv\nThen a direct computation gives the estimate:\n/integraldisplay\nR2|˜K(t,x,y)|2dxdy≤(1+2c1κt2)−1/2+O(/planckover2pi1µ)\n/square\nLet usnow consider thefull interaction Hamiltonian for the Dicke model:\nˆHDint(t) = ((cost)x+(sint)/planckover2pi1Dx)/planckover2pi1S1. By a symplectic transformthis Hamil-\ntonian is conjugate to the previous one. So Lemma 6.2 is also t rue in this\ncase.\n196.2 Proof of Theorem 1.8\nStrategy:\n1) Reduce to the interaction picture with the propagator V(t,t0):\nDenote Ψ I(t) =V(t,t0)Ψ0. Then we have Π Ψ(t)=U0(t−t0)ΠΨI(t)U0(t0−t).\nHence we have\ntrH2s+1ΠΨ(t)=U0(t−t0)trH2s+1ΠΨI(t)U0(t0−t)\nButU0(t−t0) is a unitary operator in L2(R,C) so it is enough to prove the\nresult for Ψ I(t).\n2) Proof of the result if ω3= 0\nLetusconsiderthetimedependentHamiltonian ˆH(t) = (α(t)x+β(t)Dx))S1.\nUsing (6.3), to compute partial trace on H2s+1it is enough the consider the\npropagator:\nW(t) := e−i(b(t)/planckover2pi1Dx+a(t)x)S1\nFinally by a symplectic rotation R(t) we get a metaplectic transformation\nˆR(t) inL2(R,C) such that\nˆR(t)W(t)ˆR∗(t) = e˜α(t)xS1:=˜W(t).\nWe havealready proved abovethedecoherencefor theevoluti on of theinitial\nstate Ψ(0,x) =ϕz0(x)ψn0by˜W(t). So the proof for ω3= 0 is achieved.\n3) The case ω3/\\e}atio\\slash= 0.\nNow the interaction is computed from the decoupled Hamilton ian\nˆK0=ˆH0+/planckover2pi1ω3S3. So we have U(t) =U0(t)V(t), where U0(t) = e−it\n/planckover2pi1ˆK0and\nthe propagator V(t) must satisfy\ni/planckover2pi1∂tV(t) =ˆKI(t)V(t),V(0) =I,\nwhere\nˆKI(t) = (α(t)x+β(t)/planckover2pi1Dx)/parenleftBig\ncos(ω3t)ˆS1+sin(ω3t)ˆS2/parenrightBig\n.\nLike in the case ω3= 0, to compute the partial trace it is enough to consider\nthe caseβ(t) = 0. Hence we can conclude, like for ω3= 0, using Proposition\n3.1, (6.5) and the following Lemma\nLemma 6.3. Let us consider the Landau equation depending on the param-\neterx∈R,\n∂tn(t) =C(t,x)∧n(t).\nAssume that C3≡0,∇xC1(t,x0)/\\e}atio\\slash= 0andn3(0)/\\e}atio\\slash= 0. Then there exists\nc0>0,t0>0,r0>0such that\n|n(t,x)−n(t,y)| ≥c0t|x−y|,if|x0−y|+|x0−x| ≤r0,0≤t≤t0.(6.8)\n20Proof.From the Landau-Lifshitz equation we get |∂s∂xn2(s,u)| ≥c1>0\nfromsandu−x0small enough. Then (6.8) follows.\nRemark 6.4. It seems possible that these results coud be extended to more\ngeneral spin-orbit interaction like /hatwideC1S1such that ∇XC1(X)/\\e}atio\\slash= 0(principal\ntype condition) by constructing a unitary Fourier-integra l operatorUsuch\nthatU/hatwideC1U∗≈ˆx.\nA Classical perturbations of Hamiltonians\nOur aim here is to analyze the consequence on the trajectorie s of the pertur-\nbation of order κin (1.8) for the critical regime s≈/planckover2pi1−1/2. More generally\nlet us consider times dependent smooth vector fields in Rm,F(t,X),G(t,X)\nand the differential equations\n˙Y(t) =F(t,Y(t)),˙X(t) =F(t,X(t))+κG(t,X(t)), X(0 =Y(0) =X0.\nLemma A.1. There exists t0>0,κ0>0such that for we have\n/ba∇dblX(t)−Y(t)/ba∇dbl=κt/ba∇dblG(0,X0)/ba∇dbl+O(κt2),for 0<t<t 0,0<κ<κ 0.\nIn particular if G(0,X0)/\\e}atio\\slash= 0we have, for c2>0,\n/ba∇dblX(t)−Y(t)/ba∇dbl ≥c2κt,for 0<t<t 0.\nProof.The following argument is standard to compare solutions of d ifferen-\ntial equations.\nIn a first step, we get for some c1>0,t0>0, that we have\n/ba∇dblX(t)−Y(t)/ba∇dbl ≤c1κ,for 0<t<t 0.\nThenwe usethevariation equation with the linearized equat ion around Y(t)\nto get\nX(t)−Y(t) =κ/integraldisplayt\n0R(s,t)G(s,Y(s))ds+O(κ2t2).\nSo we get the Lemma with t0>0 small enough.\nB Proof of Proposition 3.1\nHere we can assume /planckover2pi1= 1.\nWeareusingcomplexcoordinatesonthespherefortheHamilt onian(Section\n2.3):\nHc(η,¯η) = (1+ |η|2)−1(C3(1−|η|2)−(C−¯η+C+η))\nRecall that η=in2−n1\n1+n3forn= (n1,n2,n3) (η(0,0,−1) =∞)\nLet us compute the time derivative of the ansatz (3.3), like i n Section 5.\n21The classical dynamics of the spin is given by (2.24).\nBy a straightforward computation we get the spin phase α:\nα(t) =/integraldisplayt\n0/parenleftbiggℑ(˙η¯η)\n2(1+|η|2)−Hc(η,¯η)/parenrightbigg\ndτ=γ(t).\nReferences\n[1] F.T Arecchi and E. Courtens and R. Gilmore and H. Thomas. Atomic\nCoherent States in Quantum Optics Physical Review A, Vol.6, No.6,\np.2211-2237, December 1972.\n[2] J. Bolte and R. Glaser. Semi-classical propagation of coherent states\nwith spin-orbit interaction Annales Henri Poincar´ e 6, p. 625-656, 2004.\n[3] M. Combescure and D. Robert. Coherent states and applications\nin mathematical physics .Theoretical and Mathematical Physics .\nSpringer, 2nd edition 2021.\n[4] C. Fermanian-Kammerer, C. Lasser and D. Robert. Propagation\nof wave packets for systems presenting codimension one cros sings.\nComm. Math. Phys., 385(3), p. 1685-1739 (2021).\n[5] K. Hepp and E. Lieb. On the superradiant phase transition for\nmolecules in a quantized radiation field: the Dicke maser mod el. An-\nnals of Physics. 76(2):p.360-404 (1973).\n[6] J. Fr¨ ohlich, A. Knowles and E. Lenzmann. Semi-Classical Dynamics\nin Quantum Spin Systems . Letters in Mathematical Physics, 82(2), p.\n275-296 (2007).\n[7] L.LandauandE.Lifshitz. On the theory of dispersion of magnetic per-\nmeability in ferromagnetic bodies . Phys. Zeitsch. der Sow., 8:153–169,\n(1935).\n[8] Y. Kosmann-Schwarzbach. Groupes et sym´ etries ´Editions de l’ ´Ecole\nPolytechnique (2006)\n[9] E. Lieb. A proof of an entropy conjecture of Wehrl . Communications\nin Mathematical Physics, 62p. 1-13, (1978).\n[10] E. Onofri A note on coherent state representations of Lie groups Jour-\nnal of Mathematical Physics. Vol. 16, No. 5 p. 1087-1089. (19 75)\n[11] M. Roses and G. Dalla Torre Dicke model PLOS,\nhttps://doi.org/10.1371/journal.pone.0235197 Published:\nSeptember 4, ’2020).\n22"    },    {        "title": "2202.04797v1.Local_breaking_of_the_spin_degeneracy_in_the_vortex_states_of_Ising_superconductors__Induced_antiphase_ferromagnetic_order.pdf",        "content": "arXiv:2202.04797v1  [cond-mat.supr-con]  10 Feb 2022Local breaking of the spin degeneracy in the vortex states of Ising superconductors:\nInduced antiphase ferromagnetic order\nHong-Min Jiang1and Xiao-Yin Pan2\n1School of Science, Zhejiang University of Science and Techno logy, Hangzhou 310023, China\n2Department of Physics, Ningbo University, Ningbo 315211,C hina\n(Dated: February 11, 2022)\nIsing spin-orbital coupling is usually easy to identify in t he Ising superconductors via an in-plane\ncritical field enhancement, but we show that the Ising spin-o rbital coupling also manifests in the\nvortex physics for perpendicular magnetic fields. By self-c onsistently solving the Bogoliubov-de\nGennes equations of a model Hamiltonian built on the honeyco mb lattice with the Ising spin-orbital\ncoupling pertinent to the transition metal dichalcogenide s, we numerically investigate the local\nbreaking of the spin and sublattice degeneracies in the pres ence of a perpendicular magnetic field.\nItisrevealedthattheferromagnetic ordersareinducedins idethevortexcoreregion bytheIsingspin-\norbital coupling. The induced magnetic orders are antiphas e in terms of their opposite polarizations\ninside the two nearest-neighbor vortices with one of the two polarizations coming dominantly from\none sublattice sites, implying the local breaking of the spi n and sublattice degeneracies. The finite-\nenergy peaks of the local-density-of-states for spin-up an d spin-down in-gap states are split and\nshifted oppositely by the Ising spin-orbital coupling, and the relative shifts of them on sublattices\nAandBare also of opposite algebraic sign. The calculated results and the proposed scenario may\nnot only serve as experimental signatures for identifying t he Ising spin-orbital coupling in the Ising\nsuperconductors, but also be prospective in manipulation o f electron spins in motion through the\norbital effect in the superconducting vortex states.\nPACS numbers: 74.20.Mn, 74.25.Ha, 74.62.En, 74.25.nj\nI. INTRODUCTION\nThe superconductivity uncovered in atomically thin\ntwo-dimensional (2D) forms of layered transition metal\ndichalcogenides (TMDs) have recently attracted remark-\nable scientific and technical interests1–14. Although these\nsuperconductors belong to the conventional s-wave su-\nperconductivity with low transition temperature1–14, the\nuniqueness of the TMDs makes them alluring to the re-\nsearchers. On one hand, similar to graphene, these ma-\nterials have a honeycomb lattice structure, and exhibit\na valley degree of freedom with minima/maxima of con-\nduction/valence bands at the corners Kand−Kof the\nBrillouin zone. On the other hand, unlike graphene, the\nin-plane mirror symmetry is broken in the TMDs, lead-\ning to a strong atomic Ising type spin-orbital coupling\n(ISOC)3,4,7–9. The ISOC strongly pins the electron spins\nto the out-of-plane directions and have opposite direc-\ntions in opposite valleys ( Kand−K)3,4,7–9,12,14–16, so\nthat it preserves time-reversal symmetry and is compat-\nible with superconductivity. Due to the strong pinning\nof electron spins in the out-of-plane directions, external\nin-plane magnetic fields are much less effective in align-\ning electron spins, and lead to the in-plane upper critical\nfieldHc2ofthe system severaltimes largerthan the Pauli\nlimit10,12.\nNevertheless, an out-of-plane magnetic field will gen-\nerate the magnetic flux in conductors due to the domi-\nnating orbital effect over the Zeeman splitting. It is well\nknown that the superconductors expel the magnetic flux\nfrom their interior, the so called Meissner effect. While\nsome superconductors expel the magnetic field globally(they are called type I superconductors), a type II super-\nconductorwill only keep the whole magnetic field out un-\ntil a first critical field Hc1is reached. Then vortices start\ntoappear. Avortexisalocalmagneticfluxquantumthat\npenetrates the superconductor, where the superconduct-\ning (SC) order parameter drops to zero to save the rest\nof the SC state in metal from being destroyed. While the\nISOC exemplifies itself as the spin-valley locking in the\nmomentum space, it acts as coupling between spins and\nthe orbital derived effectively periodic spin and sublat-\ntice dependent fluxes in real space with the quantization\naxis along the out-of-plane direction. This is to say the\nspins, sublattices and the effectively periodic fluxes are\nbound together by the ISOC in real space. Thus, the lo-\ncal breaking of the spin and sublattice degeneracies may\nbe expected if the fluxes are altered locally, and the spin\norders in real space may also be expected to emerge.\nIn this paper, we numerically demonstrate that the\nspin and sublattice degeneracies break locally with an\ninduced ferromagnetic order inside the vortex core of\nthe Ising superconductors, as a result of the contrast-\ning variation of the effectively periodic fluxes for sublat-\nticesAandBcaused by the out-of-plane magnetic field.\nBy self-consistently solving the Bogoliubov-de Gennes\n(BdG) equations of the Hamiltonian, it is shown that\nthere is no magnetic order induced inside the vortex core\nwhen the ISOC is zero. Accordingly, the curves of the\nlocal-density-of-states (LDOS) for the spin-up and spin-\ndown in-gap states are almost identical, forming a series\nof discrete energy peaks inside the core region. The in-\nclusion of the ISOC induces a ferromagnetic order inside\nthe vortex core, where the SC order parameter is sup-2\npressed. The induced magnetic orders are antiphase in\nterms of their opposite polarizations inside two nearest-\nneighbor (NN) vortices with one of the two polarizations\ncoming dominantly from one sublattice sites. The finite-\nenergy peaks of the LDOS for spin-up and spin-down\nin-gap states are shifted oppositely by the ISOC, and\nthe sign of the relative shifts of them depends on which\nsublattices the site is belonging to. Based on a scenario\nof local breaking of the spin and sublattice degeneracies\ndue to the interactionofthe ISOCderived effective fluxes\nwith the local magnetic flux inside the vortex core, we\ngive an explanation to the unusual phenomena regarding\nthe polarization of the induced magnetic orders and the\nenergy shifts of the finite-energy in-gap peaks. The cal-\nculated results may not only serve as experimental sig-\nnatures for identifying the ISOC proposed in the Ising\nsuperconductors, but also put forward effective thinking-\nways in manipulation of electron spins in motion through\nthe orbital effect in the SC vortex states.\nThe remainder of the paper is organized as follows. In\nSec. II, we introduce the model Hamiltonian and carry\nout analytical calculations. In Sec. III, we present nu-\nmerical calculations and discuss the results. In Sec. IV,\nwe make a conclusion.\nII. THEORY AND METHOD\nThe effective electron hoppings between the NN sites\niandi+τjon a honeycomb lattice can be described by\nthe following tight-binding Hamiltonian,\nH0=−/summationdisplay\ni,τj,σ(ti,i+τja†\ni,σbi+τj,σ+h.c.)−µ(/summationdisplay\ni∈A,σa†\ni,σai,σ\n+/summationdisplay\ni∈B,σb†\ni,σbi,σ), (1)\nwhereti,i+τjisthehoppingintegralbetweentheNNsites.\nτjdenotes the three NN vectors with τ0=a(√\n3\n2,1\n2),τ1=\na(−√\n3\n2,1\n2) andτ2=a(0,−1) as defined in Fig. 1(a) with\nabeing the lattice constant. a†\ni,σ(b†\ni,σ) is the electron\ncreation operator in sublattice A(B) ifi∈sublattice A\n(B), andµthe chemical potential. For the free hopping\ncase with ti,i+τj=t, the Hamiltonian H0can be written\nin the momentum space,\nH0(k) =/summationdisplay\nk,σ[ξka†\nk,σbk,σ+ξ∗\nkb†\nk,σak,σ−µ(a†\nk,σak,σ\n+b†\nk,σbk,σ)], (2)\nwhere\nξk=−t2/summationdisplay\nj=0eik·τj. (3)One can readily find the energy bands for this Hamilto-\nnian as17,\nε±\nk=±t[3+2cos(√\n3kx)+4cos(√\n3kx/2)cos(3ky/2)]1\n2\n−µ. (4)\nwith + ( −) indexing the conduction (valence) band. We\nfocus on systems which have been doped such that the\nchemical potential µlies in the upper conduction bands,\nand produce six spin degenerate pockets at the corners\nof the hexagonal Brillouin zone when ε+\nk= 0, as shown\nin Fig. 1(b).\nFIG. 1: (a) Honeycomb lattice structure of the Ising super-\nconductor, made out of two sublattices A(blue dots) and B\n(red dots). τ0,τ1andτ2are the nearest-neighbor vectors,\nandτ′\n1-τ′\n6the next-nearest-neighbor vectors. (b) The Bril-\nlouin zone (dashed line) and the six spin degenerate Fermi\npockets (solid lines) of the Ising superconductor. The red a nd\nblue colors indicate the opposite sign of the effective Zeema n\nfields between adjacent Fermi pockets located at Kand−K.\nThe positive phase hopping directions for spin-up electron s\ndepicted by HISOCin Eq. (6) (c1), and by HKMin Eq. (15)\n(c2), respectively. The arrows in bothfiguresindicate thep os-\nitive phase hopping directions. (d) The ISOC dependencies o f\nthe maximum of the absolute value for the induced magnetic\norder|S|maxand the magnitude of the relative energy shifts\n|δ|between the spin-up and spin-down in-gap state peaks on\nthe vortex core center [reference to text and Fig. 4(b)].\nThe ISOC acts as strong effective Zeeman fields, which\npolarize electron spins oppositely to the out-of-plane di-\nrection at opposite valleys, that is, at the Kand−K\npoints in Fig. 1(b). If we choose the out-of-plane direc-\ntion as the z-axis, the ISOC term has the form18\nHISOC(k) =β/summationdisplay\nk,σ,σ′gk·ˆσσσ′(a†\nk,σak,σ′+b†\nk,σbk,σ′),(5)\nwhereβis the ISOC strength, and ˆ σdenotes the Pauli\nmatricesactinginthespinspace. TheISOCrequiresthat3\nthe form factor gkalternates its sign between adjacent\nFermipocketslocatedat Kand−K[seeFig.1(b)], which\nshould be the form gk= ˆzFkwithFk= 2sin(√\n3kx)−\n4cos(3ky/2)sin(√\n3kx/2) =−F−ksatisfying the time-\nreversal symmetry. In this way, the spins are bound\nto the orbitals in the momentum space and accordingly\nexhibit various valley dependent behaviors such as val-\nley spintronics in these materials19–23. By making the\nFouriertransformationof Fk, theISOCterminrealspace\ncan be reached as24,\nHISOC=iβ/summationdisplay\ni,τ′\nj,σ,σ′ˆσz\nσσ′(−1)j(a†\ni,σai+τ′\nj,σ′+b†\ni,σbi+τ′\nj,σ′),(6)\nwhere the vectors τ′\njconnecting the six next-nearest-\nneighbor (NNN) sites are located at τ′\n1=−τ′\n4=√\n3a(1,0),τ′\n2=−τ′\n5=√\n3a(1\n2,√\n3\n2) andτ′\n3=−τ′\n6=√\n3a(−1\n2,√\n3\n2), as indicated by the dashed arrows in\nFig. 1(a). We will see later that the ISOC in real space\ndepicted by Eq. (6) plays the role of the coupling be-\ntween spins and the effectively periodic fluxes with the\nquantization axis along the out-of-plane direction. Then,\nthe Hamiltonian including both the free hoppings and\nthe ISOC term is reached, in real space as,\nHTMD=H0+HISOC. (7)\nThe SC pairing is assumed to be derived from the ef-\nfective attraction between electrons,\nHP=V0\n2/summationdisplay\ni,σni,σni,¯σ. (8)\nHere, we consider the on-site interactions with V0denot-\ning the effective interaction potential15,16. By making\nthe mean-field decoupling, HPcan be rewritten in terms\nof the SC pairings as,\nHP=/summationdisplay\ni∈A(∆Aa†\ni,↑a†\ni,↓+h.c.)+/summationdisplay\ni∈B(∆Bb†\ni,↑b†\ni,↓\n+h.c.), (9)\nwhere ∆ A=−V0/angbracketleftai,↑ai,↓/angbracketright(∆B=−V0/angbracketleftbi,↑bi,↓/angbracketright) defines\nthe on-site spin-singlet s-wave SC pairing.\nThen the total Hamiltonian is arrived as follows,\nH=HTMD+Hpair. (10)\nBased on the Bogoliubov transformation, the diagonal-\nization of the Hamiltonian Hcan be achieved by solving\nthe following discrete BdG equations,\n/summationdisplay\nj\n−µδijHij,↑↑∆Aδij0\nH∗\nij,↑↑−µδij0 ∆ Bδij\n∆∗\nAδij0µδij−H∗\nij,↓↓\n0 ∆∗\nBδij−Hij,↓↓µδij\n\nuA,n,j,↑\nuB,n,j,↑\nvA,n,j,↓\nvB,n,j,↓\n=\nEn\nuA,n,i,↑\nuB,n,i,↑\nvA,n,i,↓\nvB,n,i,↓\n,(11)where,\nHij,↑↑=−tijδi+τj,j+iβσz\n↑↑(−1)jδi+τ′\nj,j,\nHij,↓↓=−tijδi+τj,j+iβσz\n↓↓(−1)jδi+τ′\nj,j,(12)\nwithuA,n,j,↑(uB,n,j,↑) andvA,n,j,↓(vB,n,j,↓) being the\nBogoliubovquasiparticleamplitudeson the j-th site with\ncorresponding eigenvalues En. The SC pairing ampli-\ntudes satisfy the following self-consistent conditions,\n∆A=−V0\n2/summationdisplay\nnuA,n,i,↑v∗\nA,n,i,↓tanh(En\n2kBT),\n∆B=−V0\n2/summationdisplay\nnuB,n,i,↑v∗\nB,n,i,↓tanh(En\n2kBT).(13)\nThe spin dependent electron density nA(B),i,σand the\nlocal magnetic orders SA(B),i,zare determined respec-\ntively by,\nnA(B),i,↑=/summationdisplay\nn|uA(B),n,i,↑|2f(En),\nnA(B),i,↓=/summationdisplay\nn|vA(B),n,i,↓|2f(En),\nSA(B),i,z=1\n2[nA(B),i,↑−nA(B),i,↓]. (14)\nIII. RESULTS AND DISCUSSION\nIn numerical calculations, we choosethe zero field hop-\nping integral t= 200meVasthe energyunit, andfix tem-\nperature T= 1×10−5, unless otherwise specified. The\nfilling factor n=/summationtext\ni,σni,σ/N= 1.08 (Ndenotes the\nnumber of total lattice sites) such that the chemical po-\ntentialµlies in the upper conduction band and gives rise\nto the Fermi surfaces in Fig. 1(b). In the presence of a\nperpendicularmagneticfield, the orbitaleffect dominates\nover the the Zeeman splitting, so we neglect the Zeeman\nterm of the external magnetic field in the following cal-\nculations. In this case, the hopping terms are described\nby the Peierls substitution. For the NN hopping between\nsitesiandi+τj, one has ti,i+τj=teiϕi,i+τj, and for\nthe NNN hopping between iandi+τ′\njone should have\nβ→βeiϕi,i+τ′\nj, whereϕi,i+τj(τ′\nj)=π\nΦ0/integraltextri\nri+τj(τ′\nj)A(r)·dr\nwith Φ 0=hc\n2ebeing the SC flux quanta. We consider\na system with a parallelogram vortex unit cell as shown\nin Fig. 1(a), where two vortices are accommodated. The\nvortex unit cell with size of 24 a1×48a2is adopted in the\ncalculations, unless otherwise stated. The vector poten-\ntialA(r) = (0,Bx,0) is chosen in the Landau gauge to\ngive rise to the magnetic field Balong the z-direction.\nIn this study, we have no ambition to explore the SC\nmechanism underlying the Ising superconductors. In-\nstead, we assume a phenomenological pairing potential\nV0to give rise to the SC pairing. Within the BCS the-\nory,the coherencelengthisgivenby ξ0=/planckover2pi1vF/π∆, where4\nvFis the Fermi velocity, linking the coherence length\nto the inverse size of the SC gap ∆. The coherence\nlength of NbSe 2is about 10nm as obtained from Hc2(T)\nmeasurement25,26. The estimated vortex core size is of\nξV∼30nm26. A system contains two such vortex cores\nwould be larger than the size of 60nm ×120nm, which\nroughly amounts to a parallelogram sample with the size\nlargerthan 200 a1×400a2. Such a large size is far beyond\nthe computational capability. However, it is still capable\nof mimicking the vortex physics on a relative small size\nof sample by artificially enlarging the SC gap ∆. In the\nself-consistent calculations, the length scale of the sam-\nple with size 24 a1×48a2is about one order smaller than\nthe actual size. Thus, we need to choose a large V0= 1.6\nin the self-consistent calculations to give rise to a bulk\nvalue of ∆ ≈0.09∼18meV, a value about one order\nlarger than the actual measurements26, so as to meet the\nrequirement.\nFIG. 2: The spatial distributions of the SC and magnetic\norder parameters in the vortex states for β= 0.04 are shown\nin (a) and (b), respectively. The spatial distributions of t he\nmagnetic order in the vortex states for β= 0.04 on sublattice\nA(c), and on sublattice B(d), respectively.\nA. The induced antiphase magnetic orders inside\nvortex cores\nUnder a perpendicular magnetic field, the vanishment\nof the screening current density at the vortex center\ndrives the system into the vortexstates with the suppres-\nsionoftheSCorderparameteraroundthevortexcore. In\nthe absence of the ISOC interaction, we find that except\nfor the suppression of the SC order around the vortex\ncore region there is no other order to be induced. On the\nother hand, when the ISOC is present, a ferromagnetic\norder can emerge inside the vortex core region with its\nmaximum appearing at the vortex core center. The max-\nimum of the absolute value for the magnetic order |S|max\nexhibits roughly linear increasing trend with βin a widerange of ISOC, and finally reaches a saturated value at\nlarge ISOC, as displayed in Fig. 1(d). Typical results on\nthevortexstructurewith β= 0.04areshowninFigs.2(a)\nand 2(b) for the spatial distributions of SC and magnetic\norders, respectively. As shown in Fig. 2(a), each vortex\nunit cell accommodates two SC vortices each carrying a\nflux quantum Φ 0. The SC order parameter |∆A/B|van-\nishes at the vortex corecenter where the maximum of the\ninduced magnetic order appears. It is interesting to note\nthat the magnetic order parameters have opposite polar\ndirections around two NN vortices along the long side of\nthe parallelogram vortex unit cell, as shown in Fig. 2(b).\nThe most unusual aspect of the spatial distribution of\nthe magnetic order parameters SA(B),i,zappears when\nwe replot in Figs. 2(c) and 2(d) the magnetic orders sep-\narately on the sublattices AandB. Specifically, the pos-\nitive magnetic order alone z-axis inside one vortex comes\ndominantly from the Asublattice while the negative one\ninside another vortex comes dominantly from the Bsub-\nlattice.\nIn ordertounderstandthe originaswellasthe unusual\ndistributions of the induced magnetic order, we should\nnote the fact that there is no magnetic order induced\nwhen ISOC is zero. In real space, the ISOC depicted\nby Eq. (6) plays the role of coupling between spins and\nthe effectively periodic fluxes with the quantization axis\nalong the out-of-plane direction. Following the ISOC\nterm in Eq. (6), we display the positive phase [noting\nthati=eiπ/2] hopping directions of ISOC in Fig. 1(c1)\nby arrows on NNN bonds for spin-up electrons at sub-\nlatticesAandB, from which the effective spin fluxes\nare generated. If the positive phase hoppings on NNN\nbonds for spin-up electrons on sublattice Agenerate spin\nflux pointing to z-direction, then they generate spin flux\npointing to −z-direction on sublattice B, and contrary is\ntrue for spin-down electrons. That is, the NNN hoppings\nhave opposite chirality, for sublattices AandB. Since\nthe spins, sublattices and the effectively periodic fluxes\nare bound together in real space, local breaking of the\nspin and sublattice degeneracies may be expected if the\neffective fluxes for sublattices AandBare contrastively\naltered by an out-of-plane magnetic field, and thus the\nspin ordersin realspacemay alsobe expected. Neverthe-\nless, we can not expect the appearance of magnetic order\nin the normal state under an out-of-plane magnetic field.\nThis is due to the fact that the energy scale of the hop-\nping integral toverwhelms the ISOC strength β, inter-\nchanging the electrons between sites of sublattices Aand\nBleading to the suppression of the local orders. How-\never, the situation is totally different in the vortex state,\nwhere the localized electrons in the vortex core, which\ncome from the breaking of the Cooper pairs, contribute\nto the magnetic order. If one vortex core resides on the\nAsublattice site, the blue site shown in Fig. 1(a), the\npositive phase hoppings on NNN bonds bound to spin-\nup electrons on sublattice Agenerate effective spin flux\npointing to z-direction [noting the negative charge of the\nelectrons], which is in the same direction as the mag-5\nnetic field. On the contrary, the spin-down electrons on\nsublattice Agenerate effective spin flux in the opposite\ndirection of the magnetic field. Thus, the spin degener-\nacy breaks locally to two branches with a lower energy\nfor the spin-up electrons, leading to the positive mag-\nnetic order around one vortex as shown in Fig. 2(c) on\nsublattice A. In principle, the pairing breaking from the\nspin-singlet SC pairings due to the orbital effect of the\nmagnetic field results in equal numbers of spin-up and\nspin-down electrons, so the total spins should be zero as\na global. The excess of spin-down electrons accumulate\ninto the NN vortex to give rise to the negative magnetic\nordershowninFig.2(d)onsublattice B, wherebyitsaves\nthe energy as the effective spin flux generated by spin-\ndown electrons being in compliance with the direction of\nthe magnetic field.\nTwo situations could lend support to the above sce-\nnario. Firstly, we consider the case with a reversal of the\ndirectionofthemagneticfield, i.e., amagneticfield inthe\n−z-direction. From the above argument, the polariza-\ntions of the induced magnetic orders should be reversed\nif the magnetic field reverses its direction. It is exactly\nthe case as evidenced in Figs. 3(a) and 3(b), where the\nresults are obtained with an out-of-plane magnetic field\nin the−z-directionwhile keepother parametersthe same\nas that in Fig. 2. Secondly, we should make a comparison\nwith the spin-orbital coupling (SOC) term in Kane-Mele\nmodel27, which has the form\nHKM=iβ/summationdisplay\ni,τ′\nj,σ,σ′ˆσz\nσσ′(−1)j(a†\ni,σai+τ′\nj,σ′−b†\ni,σbi+τ′\nj,σ′).(15)\nBothHISOCandHKMpreserve time-reversal symme-\ntry, so the spins remain degenerate in both cases. The\nonly difference lies that HISOCpreserves the sublattice\nsymmetry but HKMbreaks it. As a result, the NNN\nhopping phases carried by the same spins in HKMwould\nhave same chirality for sublattices AandB, as denoted\nby arrowsin Fig. 2(c2). According to the above scenario,\nwe deduce that the induced magnetic orders should be\nin the same direction for the two adjacent vortices. This\nis also verified in Figs. 3(c) and 3(d), where the results\nfor the spatial distribution of the induced magnetic or-\nders are calculated by replacing HISOCwithHKMwhile\nother parameters remain unchanged.\nB. The splitting and shift of the finite-energy\npeaks for the spin-resolved LDOS\nNext, we examine the energy dependence of the LDOS\nin the vortex states on the honeycomb lattice. The\nLDOS is defined as N(Ri,E) =N↑(Ri,E)+N↓(Ri,E)\nwithN↑(Ri,E) =−/summationtext\nn|uA(B),n,i,↑|2f′(En−E) and\nN↓(Ri,E) =−|vA(B),n,i,↓|2f′(En+E) being the spin-\nresolved LDOS for spin-up and spin-down states, respec-\ntively. In order to reduce the finite size effect, the calcu-\nlations of the LDOS are carried out on a periodic latticeFIG. 3: The spatial distributions of the induced magnetic or -\nders in the vortex states with the magnetic field along the −z\ndirection for β= 0.04 on sublattice A(a), and on sublattice\nB(b), respectively. (c) and (d) show the calculated results\nfot the spatial distributions of the induced magnetic order s\nby replacing HISOCwithHKM[see text for details].\nwhich consists of 16 ×8 parallelogram vortex unit cells,\nwith each vortex unit cell being the size of 24 a1×48a2.\nIn Fig. 4, we plot a series of the spin-resolved LDOS as\na function of energy at sites along the zigzag direction\nmoving away from the vortex center for β= 0.0 and\nβ= 0.04, respectively. For comparison, we have also\ndisplayed the LDOS at the midpoint between the two\nNN vortices, which resembles the U-shaped full gap fea-\nture for the bulk system. In the absence of ISOC, the\nstates of spin-up and spin-down are nearly equal occupa-\ntion and empty in the vortex core, as shown in Fig. 4(a),\nin accordance with the empty cores without the induced\nmagnetic orders. Besides the almost identical LDOS line\nshapes for the spin-up and spin-down states, the LDOS\nshown in Fig. 4(a) exhibits another two prominent fea-\ntures within the SC gap edges. On one hand, the LDOS\nshows the pronounced discrete energy peaks inside the\ncoreregionwithonelocatednearthezeroenergyandoth-\ners located at finite energies, as indicated by the dashed\nvertical lines in the figure. Here, the asymmetric line\nshape of the LDOS with respect to zero energy reflects\nthe lack of particle-hole symmetry as the chemical po-\ntentialµdeviates from zero for the filling factor nbeing\ngreater than the half filling ( n >1). Due to the particle-\nhole asymmetry, the finite-energy bound states at the\ncore site only appear on the E >0 side28(There are also\nweak peaks at finite energies on the E <0 side when\nmoving away from the core center.). The existence of\nthe zero-energy vortex core sates in the Dirac fermion\nsystem have been predicted analytically by Jackiw and\nRossi in terms of the zero-energy solutions of relativistic\nfield theory29. Although these zero-energysolutions were6\nsubsequently demonstrated that the existence of these\nzero-energy solutions is connected to an index theorem30\nand the zero modes were shown to exist in the Dirac con-\ntinuum theory of the honeycomb lattice at half filling31,\nthe zero-energy levels split when adopting a honeycomb\nlattice model description by setting the size of the vortex\ncoretobezero32. Itisalsofoundthattheenergysplitting\ndecreases with the vortex size and leads to the near-zero-\nenergy states in the circumstance of finite core size32.\nWhile the notion of the zero-energy vortex core states\npresents an important subject of study being worthy of\nfurther research, we identify the near-zero-energy vortex\ncore states here in a self-consistent manner by employing\nthe honeycomb lattice model, where the band structure\nhasthe Dirac-typedispersionnearthe halffilling. Onthe\nother hand, though the peaks’ intensities are suppressed\nas the site departing from the core center, the energy lev-\nels of these peaks are almost independent of positions. It\nis worth while to note that a dispersionless zero-energy\nconductance peak has been recently observed inside the\nSC vortex core by Chen’s group33in the kagome super-\nconductor CsV 3Sb5, which shares the lattice structure\nwith component of hexagonal honeycomb and the elec-\ntronic structure with Dirac points in a manner similar to\nthose in honeycomb lattices. How the calculated results\nwithnear-zero-energypeaksinthepresentstudyrelateto\nthe experimental observations, and whether these near-\nzero-energy vortex core states have a common underly-\ning symmetrical cause, constituting another fascinating\nquestions deserving further studies.\nIn the presence of the ISOC, the local breaking of the\nspin and sublattice degeneracies in the vortex states is\nalso reflected in the energy dependence of the LDOS.\nFigs. 4(b), 4(c) and 4(d) present the typical results of\nthe spin-resolved LDOS for β= 0.04. As can be seen\nfrom Fig. 4(b), while the energy level of the near-zero-\nenergy peaks remain virtually unchanged for both spins,\nthe energy levels of the finite-energy peaks are shifted\ndifferently by the ISOC for different spins and at differ-\nent sublattice sites, as compared with the case of β= 0.\nSpecifically, for the LDOS on the same site within the\ncore region, the finite-energy peaks for the spin-up and\nspin-down bound states shift oppositely, as indicated by\nthe arrows in the figures, depicting a picture of local\nbreaking of the spin degeneracy. At the same time, for\nthe bound states with the same spin, the finite-energy\npeaks on the sites belonging to different sublattices also\nhave the opposite shifts, indicating the local breaking of\nthe sublattice degeneracy. Since there are induced mag-\nnetic orders in the vortex cores as well as the similar\nISOC dependencies of the magnitudes of the magnetic\norders|S|maxand the relative energy shifts |δ|as shown\nin Fig. 1(d), it is natural to suspect whether the spin\nsplitting for the LDOS is derived from the Zeeman effect\nof the induced local magnetic order interacting with the\nelectrons34, or from the above scenario where the spin\ndegree of freedom is manipulated by the orbital effect of\nmagnetic field via the ISOC. Several aspects render theFIG. 4: The energy dependence of the spin-resolved LDOS on\na series of sites for β= 0.0 (a), and for β= 0.04 (b), (c) and\n(d). (a), (b) and (d) are the results for a magnetic field along\nthez-direction, while (c) the results for a magnetic field along\nthe−z-direction. (b) and (c) show the LDOS inside the same\nvortex core, and (d) the LDOS inside another vortex core.\nIn each panel from top to bottom, the curves stand for the\nLDOS at sites along the zigzag direction moving away from\nthe core center. The curves are vertically shifted for clari ty.\nThe three dashed vertical lines in each panel denote the thre e\nlow energy peak positions for β= 0. The arrows in (b), (c)\nand (d) indicate the peak position shift with respect to that\nofβ= 0. The magnitude of the relative energy shifts |δ|is\nshown in (b).\nZeemaneffect mechanism impossible. As hasbeen shown\nin Fig. 2(b), the magnetic orders polarize oppositely in-\nside two NN vortices. If the Zeeman effect mechanism\nruns, the energy level shifts of the peaks should behave\nthe opposite way on the sites located respectively at the\ntwo NN vortices. Nevertheless, as displayed in Figs. 4(b)\nand 4(d), the consistency of the peaks’ shifts on the sites\nlocated at different vortices while belonging to the same\nsublattice rules out the Zeeman effect mechanism. The\nsecond thing we notice about the energy level shifts of\nthe peaks is that they occur only for the ones with finite\nenergy, while the near-zero-energy peaks almost stay the\nsame, being at odds with the Zeeman effect mechanism.\nFinally, if we reverse the direction of the out-of-plane\nmagnetic field, as shown in Fig. 4(c), the peaks’ shifts\nbehave exactly the opposite way as compared with that\nin Fig. 4(b). It is thus confirmed that the local spin split-\nting and the local breaking of sublattice degeneracy are\nconformed with the above scenario where the spin de-\ngree of freedom is manipulated by the orbital effect of7\nmagnetic field via the ISOC in the SC vortex states.\nFIG. 5: Temperature (a), and magnetic field (b) evolutions of\nthe maximum of the absolute value for the induced magnetic\norders at vortex cores with β= 0.04.\nC. The effects of temperature and magnetic field\nstrength on the induced orders\nDue to the 2D nature of the Ising superconductors,\nthe thermal effect on the induced magnetic orders con-\nstitutes an inevitable issue from both theoretical per-\nspective and experimental realization. Although the sys-\ntem under study is 2D, the induced magnetic orders are\nformed under the combined actions of the magnetic field,\nthe ISOC and the SC order, so they are not spontaneous\nones. Meanwhile, the induced magnetic orders are local-\nized inside the vortex core regions, and thus they are lo-\ncal ones. As a result, one may expect a different manner\nof the thermal effect on the induced magnetic orders as\ncompared with the Mermin-Wagner theorem35. To see\nthe thermal effect on the induced magnetic orders, we\ncalculate the temperature dependence of the magnitude\nof the magnetic orders. Fig. 5(a) shows the temperature\ndependence of the maximum of the absolute value for\nthe induced magnetic order at the vortex core, where T\nis rescaled by Tc≈0.05. As can be seen from the figure,\nthe magnitude of the magnetic orders remains approxi-\nmately constantat lowtemperature T≤0.02Tc[seeinset\nof Fig. 5(a)] as a result of the small thermal excitations\nandthealmostunchangedvortexcoresizeatthistemper-\nature regime36. After then it shows a steady decreasing\ntrend with increasing temperature, and finally reaches a\ntiny value at T∼0.5Tc. The decreasing trend is mainly\nascribed to the enlarging vortex core size with temper-\nature36, the so-called Kramer-Pesch Effect37. The en-\nlargedvortexcorewould involvemoredifferent sublattice\nsites into the vortex core center, resulting in the reduc-\ntion of the induced magnetic orders. Though the mag-\nnetic orders reduce their magnitude upon the increasing\nof the temperature, they sustain to a finite temperature.\nTherefore, one may expect to observe the induced mag-\nnetic orders under temperatures well below the SC criti-\ncal temperature.\nAnotherimportantfactortobeconsideredinobserving\ntheinducedmagneticordersishowthestrengthoftheex-\nternal magnetic field affects the induced magnetic orders.Since one vortex unit cell accommodates two vortices in\nthe calculations, we have B= 2Φ0/A∼1/NwithAand\nNbeing the area and the site number of the vortex unit\ncell. Fig. 5(b) displays the variation of the maximum of\nthe absolute value for the induced magnetic orders with\nrespect to different strengths of the magnetic field, which\nare realized in the calculations by varying the size of the\nparallelogram vortex unit cell. In the weak to moderate\nmagnetic field region, there is little interference between\nthe vortex cores due to the large inter-vortex spacing d.\nThe increase of the magnetic field leads to more broken\nCooper pairs inside the vortex cores to contribute to the\nformation of the magnetic orders, so the magnitude of\nthe induced magnetic orders increases with the magnetic\nfield strength, as evidenced in Fig. 5(b). However, as\nthe magnetic field increasing further, the adjacent vortex\ncores with opposite polarizations of the induced mag-\nnetic order would get close enough (with a length scale\nbeing less than two times of the penetration depth λ) to\ninterfere with one another, leading to the reduction of\nthe magnitude of the magnetic order. This suggests the\ninduced magnetic orders will be altered in an Abrikosov\nvortexlattice38. Ononehand, the formationofthe Bloch\nwave39or the interactions among vortices40in the vortex\nlattice will suppress the induced magnetic orders. On the\nother hand, since there are many vortices in the sample\ninstead of just two, the polarizationofthe induced orders\nis not necessarily opposite for two adjacent vortex cores.\nNevertheless, the result also means the induced magnetic\norders would survive in the vortex lattice under a weak\nto moderate magnetic field as long as d≫λ, i.e., the\ninter-vortex spacing is much larger than the penetration\ndepth.\nIV. REMARKS AND CONCLUSION\nThe local magnetic orders induced in the SC vor-\ntex states have been extensively investigated on the\ncuprates superconductors34,41–45, where the emergence\nof the magnetic orders inside the core region was gener-\nally believed to be originated from the electrons’ correla-\ntions. These correlationsusually comefrom the Coulomb\ninteractionsbetween electronsandthat the induced mag-\nnetic orders have nothing to do with the chirality of\nthe electrons. However, the induced local magnetic or-\nders inside the SC vortex core by the ISOC has a di-\nrect bearing on what the electrons’ chirality is. As has\nbeen demonstrated that the amplitudes of the induced\nmagnetic orders and the unusual energy shifts of the in-\ngap state peaks present here are related to the ISOC\nstrength β, while their directions are determined by the\ndirection of the magnetic field. The amplitude and the\ndifferent polar direction of the induced local magnetic or-\nders could be measured by the muon spin rotation ( µSR)\nspectroscopy and the nuclear magnetic resonance experi-\nments, and the energy shifts of the in-gap state peaks on\ndifferent sublattice for different spins could be observed8\nin the spin-polarized scanning tunneling microscopy ex-\nperiments. Both of these observations may be served\nas signatures to characterize the ISOC proposed for the\nIsing superconductors. In the meantime, since the in-\nduced magnetic orders are derived from the ISOC, the\nbreaking of the spin degeneracy and the energy shifts of\nthe in-gap state peaks are selectively occurred for the\nelectrons which possess finite momentum with respect to\nthe vortex center. The scenario proposed here may also\nprovide a possibility in manipulation of electron spins in\nmotion via the orbital effect in the SC vortex states.\nIn conclusion, we have numerically investigated the\nvortex states of the Ising superconductors, with the em-\nphasis on the local breaking of the spin and sublattice\ndegeneracies as a result of the interaction between the\nISOC derived effective fluxes and the local magnetic flux\ninside the vortex core. In the absence of the ISOC, there\nwas no magnetic order induced inside the vortex core,\nand the almost identical line shapes of the LDOS for the\nspin-upandspin-downin-gapstateswereshownupinside\nthe core region, forming a series of discrete energy peaks\nwithin the gap edges. The inclusion of the ISOC induced\nthe ferromagnetic orders inside the vortex core region,where the magnetic orders polarized oppositely for the\ntwo NN vortices with one of the two polarizations com-\ning dominantly from one specie of the two sublattices.\nAccordingly, the finite-energy peaks of the LDOS on the\nsame site for spin-up and spin-down in-gap states were\nshifted oppositely by the ISOC, and the relative shifts of\nthem on sublattices AandBwere also of opposite alge-\nbraic sign. The calculated results might serve as exper-\nimental signatures for identifying the ISOC in the Ising\nsuperconductors, and the scenario proposed here might\nalso be prospective in manipulation of electron spins in\nmotion through the orbital effect in the SC vortex states.\nV. ACKNOWLEDGEMENT\nThis work was supported by the National Natural Sci-\nence Foundation of China (Grant Nos. 11574069 and\n61504035) and the Natural Science Foundation of Zhe-\njiang Province (No. LY16A040010). This work was also\nsupported by K. C. Wong Magna Foundation in Ningbo\nUniversity.\n1R. A. Bromley, R. B. Murray, and A. D. Yoffe, J. Phys. C\n5, 759 (1972).\n2Th. B¨ oker, R. Severin, A. M¨ uller, C. Janowitz, R. Manzke,\nD. Voß, P. Kr¨ uger, A. Mazur, and J. Pollmann, Phys. Rev.\nB64, 235305 (2001).\n3Z. Y. Zhu, Y. C. Cheng, and U. Schwingenschl¨ ogl, Phys.\nRev. B84, 153402 (2011).\n4D. Xiao, G.-B. Liu, W. Feng, X. Xu, and W. Yao, Phys.\nRev. Lett. 108, 196802 (2012).\n5J. T. Ye, Y. J. Zhang, R. Akashi, M. S. Bahramy, R. Arita,\nand Y. Iwasa, Science 338, 1193 (2012).\n6K.Taniguchi, A.Matsumoto, H.Shimotani, andH.Takagi,\nAppl. Phys. Lett. 101, 042603 (2012).\n7A. Korm´ anyos, V. Z´ olyomi, N. D. Drummond, P. Rakyta,\nG. Burkard, and V. I. Fal’ko, Phys. Rev. B 88, 045416\n(2013).\n8F. Zahid, L. Liu, Y. Zhu, J. Wang, and H. Guo, AIP Adv.\n3, 052111 (2013).\n9E. Cappelluti, R. Rold´ an, J. A. Silva-Guill´ en, P. Ordej´ o n,\nand F. Guinea, Phys. Rev. B 88, 075409 (2013).\n10X. Xi, L. Zhao, Z. Wang, H. Berger, L. Forr´ o, J. Shan, and\nK. F. Mak, Nat. Nanotechnol. 10, 765 (2015).\n11W. Shi, J. T. Ye, Y. Zhang, R. Suzuki, M. Yoshida, J.\nMiyazaki, N. Inoue, Y. Saito, and Y. Iwasa, Sci. Rep. 5,\n12534 (2015).\n12J. M. Lu, O. Zheliuk, I. Leermakers, N. F. Q. Yuan, U.\nZeitler, K. T. Law, and J. T. Ye, Science 350, 1353 (2015).\n13Y. Saito, Y. Nakamura, M. S. Bahramy, Y. Kohama, J.\nYe, Y. Kasahara, Y. Nakagawa, M. Onga, M. Tokunaga,\nT. Nojima, Y. Yanase, and Y. Iwasa, Nat. Phys. 12, 144\n(2016).\n14X. Xi, Z. Wang, W. Zhao, J.-H. Park, K. T. Law, H.\nBerger, L. Forr´ o, J. Shan, and K. F. Mak, Nat. Phys. 12,\n139 (2016).15B. T. Zhou, N. F. Q. Yuan, H.-L. Jiang, and K. T. Law,\nPhys. Rev. B 93, 180501(R) (2016).\n16G. Sharma and S. Tewari, Phys. Rev. B 94, 094515 (2016).\n17A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S.\nNovoselov, and A. K. Geim, Rev. Mod. Phys. 81, 109\n(2009).\n18P. A. Frigeri, D. F. Agterberg, A. Koga, and M. Sigrist,\nPhys. Rev. Lett. 92097001 (2004).\n19B. Radisavljevic, A. Radenovic, J. Brivio, V. Giacometti,\nand A. Kis, Nat. Nanotechnol. 6, 147 (2011).\n20Y. J. Zhang, J. T. Ye, Y. Matsuhashi, and Y. Iwasa, Nano\nLett.12, 1136 (2012).\n21Q. H. Wang, K. Kalantar-Zadeh, A. Kis, J. N. Coleman,\nand M. S. Strano, Nat. Nanotechnol. 7, 699 (2012).\n22W. Bao, X. Cai, D. Kim, K. Sridhara, and M. S. Fuhrer,\nAppl. Phys. Lett. 102, 042104 (2013).\n23J. Lee, K. F. Mak, and J. Shan, Nat. Nanotechnol. 11, 421\n(2016).\n24H.-M. Jiang, J.-L. Shang, andL.-Z.Hu, J. Phys.: Condens.\nMatter31, 295602 (2019).\n25V. G. Kogan and N. V. Zhelezina, Phys. Rev. B 71, 134505\n(2005).\n26A. Fente, E. Herrera, I. Guillam´ on, H. Suderow, S. Ma˜ nas-\nValero, M. Galbiati, E. Coronado, and V. G. Kogan, Phys.\nRev. B94, 014517 (2016).\n27C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 226801\n(2005).\n28N. Hayashi, T. Isoshima, M. Ichioka, and K. Machida,\nPhys. Rev. Lett. 80, 2921 (1998).\n29R. Jackiw and P. Rossi, Nucl. Phys. B 190, 681 (1981).\n30E. J. Weinberg, Phys. Rev. D 24, 2669 (1981).\n31P. Ghaemi and F. Wilczek, Phys. Scr. T146, 014019\n(2012).\n32D. L. Bergman and K. Le Hur, Phys. Rev. B, 79, 1845209\n(2009).\n33Z. Liang, X. Hou, F. Zhang, W. Ma, P. Wu, Z. Zhang,\nF. Yu, J.-J. Ying, K. Jiang, L. Shan, Z. Wang, and X.-H.\nChen, Phys. Rev. X 11, 031026 (2021).\n34J.-X. Zhu and C. S. Ting, Phys. Rev. Lett. 87, 147002\n(2001).\n35N. D. Mermin and H. Wagner, Phys. Rev. Lett. 17, 1133\n(1966).\n36R. I. Miller, R. F. Kiefl, J. H. Brewer, J. Chakhalian, S.\nDunsiger, G. D. Morris, J. E. Sonier, and W. A. MacFar-\nlane, Phys. Rev. Lett. 85, 1540 (2000).\n37L. Kramer and W. Pesch, Z. Phys. 269, 59 (1974).\n38A. A. Abrikosov, Rev. Mod. Phys, 76, 975 (2004).\n39M. Franz and Z. Teˇ sanovi´ c, Phys. Rev. Lett. 84, 554(2000).\n40G. Blatter and V. Geshkenbein, Phys. Rev. Lett. 77, 4958\n(1996).\n41M. Ogata, Int. J. Mod. Phys. B 13, 3560 (1999).\n42J.-X. Zhu, I. Martin, and A. R. Bishop, Phys. Rev. Lett.\n89, 067003 (2002).\n43Y. Chen, Z. D. Wang, J.-X. Zhu, and C. S. Ting, Phys.\nRev. Lett. 89, 217001 (2002).\n44M. Takigawa, M. Ichioka, and K. Machida, Phys. Rev.\nLett.90, 047001 (2003).\n45H. Tsuchiura, M. Ogata, Y. Tanaka, and S. Kashiwaya,\nPhys. Rev. B 68, 012509 (2003)."    },    {        "title": "2304.12928v1.Designing_Valley_Dependent_Spin_Orbit_Interaction_by_Curvature.pdf",        "content": "Designing Valley-Dependent Spin-Orbit Interaction by Curvature\nAi Yamakage,1,∗T. Sato,2,†R. Okuyama,3T. Funato,4, 5W. Izumida,6K. Sato,7T. Kato,2and M. Matsuo5, 8, 9, 10\n1Department of Physics, Nagoya University, Nagoya 464-8602, Japan\n2Institute for Solid State Physics, University of Tokyo, Kashiwa 277-8581, Japan\n3Department of Physics, Meiji University, Kawasaki 214-8571, Japan\n4Center for Spintronics Research Network, Keio University, Yokohama 223-8522, Japan\n5Kavli Institute for Theoretical Sciences, University of Chinese Academy of Sciences, Beijing, 100190, China.\n6Department of Physics, Tohoku University, Sendai 980-8578, Japan\n7National Institute of Technology, Sendai College, Sendai 989-3128, Japan\n8CAS Center for Excellence in Topological Quantum Computation,\nUniversity of Chinese Academy of Sciences, Beijing 100190, China\n9Advanced Science Research Center, Japan Atomic Energy Agency, Tokai, 319-1195, Japan\n10RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan\nWe construct a general theoretical framework for describing curvature-induced spin-orbit interac-\ntions on the basis of group theory. Our theory can systematically determine the emergence of spin\nsplitting in the band structure according to symmetry in the wavenumber space and the bending\ndirection of the material. As illustrative examples, we derive the curvature-induced spin-orbit cou-\npling for carbon and silicon nanotubes. Our theory offers a strategy for designing valley-contrasting\nspin-orbit coupled materials by tuning their curvatures.\nIntroduction.— The spin-orbit interaction (SOI) in\nsolids lies at the heart of the the ability to manipu-\nlate electron spins in spintronics. It has been utilized\nthrough the spin Hall effect, a charge-to-spin conversion\nphenomenon, to generate, manipulate, and detect spin\ncurrents by various electrical means [1]. It also enables\ncontrol of the magnetization by transferring spin as a\ntorque from materials such as heavy metals, antiferro-\nmagnets, as well as oxide, topological [2, 3], and chiral\nmaterials [4]. In addition, valleytronics, which utilizes\nthe valley degree of freedom in materials, has stimulated\nresearchers’ interest because the interplay between spin\nand valley plays a crucial role in quantum transport phe-\nnomena [5–7].\nWhile the design of materials for a suitable SOI is an\nimportant subject in modern spintronics, the choices of\nelements and crystal structures are limited. As another\nstrategy, researchers have extensively studied tuning the\nstrength of the SOI by using the mechanical proper-\nties of the materials, such as strain. Strain-induced SOI\nhas been studied in semiconductor quantum wells with\nstatic [8–14] and dynamical [15, 16] strains. However,\nthe application of strain engineering has been restricted\nto inherently strong SOI materials, because strain does\nnot directly couple to spins. To overcome the previous\nlimitations, we propose an alternate route for designing\nvalley-dependent SOI by using the curvature of the ma-\nterial.\nIn particular, we propose a novel strategy to realize\nvalley-dependent SOI for electrons by breaking the crys-\ntalline symmetry through the application of a finite cur-\nvature to a material as shown in Fig. 1. Our theoreti-\ncal framework provides a powerful method for determin-\ning the emergence of spin splitting in the band structure\ndepending on high-symmetry points in the wavenumber\nFIG. 1. Schematic diagram of valley-dependent spin splitting\ninduced by the curvature of the material.\nspace and the bending direction of the material. Our pro-\ncedure based on group theory generally describes valley-\ndependent spin splitting induced by the curvature. We\nconsider a single-wall carbon nanotubes (CNTs) and sil-\nicon as prominent examples. For CNTs, we derive an\neffective spin- and valley-dependent Hamiltonian, with\nan SOI allowed by the group theoretic approach, that is\nconsistent with microscopic theory [17]. For silicon, we\nderive an effective model for conduction electrons near six\nvalleys that clarifies the valley-dependent SOI induced by\nthe curvature. Our theory shares the group-theoretic ap-\nproach of Ref. [18], in which the curvature-induced spin-\nphonon coupling in graphene was discussed. It provides\na general strategy for inducing SOI in materials and de-\nsigning valley contrasting spintronics via tuning of the\nbending directions of material.\nGeneral procedure.— By using group theory, we con-\nstruct an effective Hamiltonian with curvature as follows.\nFirst, we identify the space group of the crystal struc-arXiv:2304.12928v1  [cond-mat.mes-hall]  25 Apr 20232\nture without curvature. In particular, the magnetic little\ngroupGis determined for a given valley. We can find a\nsymmetry of wavefunctions for a target band as an irre-\nducible representation (irrep) of G. Then, an irreducible\ndecomposition is performed for significant physical quan-\ntities, such as momentum k, spins, and pseudospin σ,\nas well as the curvature which is the second derivative\nof the displacement field. The products of these quanti-\nties appear in the effective Hamiltonian if they belong to\nthe totally symmetric irrep, as the Hamiltonian is invari-\nant to any symmetry operation in G. The specific forms\nand product rules of irreps are provided online [19–22].\nWith this procedure, we can systematically construct an\neffective Hamiltonian with a valley-dependent SOI.\nCarbon nanotube.— A carbon nanotube (CNT) is\nrolled-up graphene forming a cylindrical surface. We will\nstart by considering flat graphene with the space group\nP6/mmm (No. 191). The electronic states nearby the\nFermi energy are around the Kpoint. The Kpoint has\nthe point symmetry of ¯62m(D3h) and the corresponding\nmagnetic little group is 6/prime/mmm/prime(D3h×{I,PT}with\nPTbeing the parity-time transformation). We consider\ntheπbands, which are doubly degenerate at the Fermi\nenergy and belong to the K6(E/prime/prime) irrep ofD3h(defined\nin the Supplemental Material [23]). The theory contains\nthe degrees of freedom of the momentum kmeasured\nfrom theKpoint, spins, and pseudospin for sublat-\nticeσ. For the irreducible decomposition, we examine\nrepresentations for the generators of the magnetic little\ngroup: the threefold rotation along the z-axisC3, twofold\nrotation along the y-axisC/prime\n2(y), and the horizontal mir-\nrorσh, are represented by DK(C3) =ei2πσz/3e−iπsz/3,\nDK(C/prime\n2(y)) =−iσysy, andDK(σh) =−isz, respectively.\nDK(PT) =σxsyK, whereKdenotes complex conjuga-\ntion. An operator O(=σiorsi,i=x,y,z ) transforms\nby the symmetry operation gasO→DK(g)−1ODK(g),\nwhereask→gk. Then, we find the conversion rules for\nk,s, andσand decompose these quantities (as well as\nthe possible products of sandσ) as shown in Table I.\nAn effective Hamiltonian for πelectrons is constructed\nfrom this table as a totally symmetric irrep, namely, a\nPT-even irrep of A/prime\n1. For example, such an irrep can be\nobtained as a product of the two PT-even E/primeirreps:\nH(0)\nK(k) =/planckover2pi1vF(kxσx+kyσy). (1)\nThis is nothing but the well-known Dirac Hamiltonian\nfor graphene.\nThe curvature is expressed in terms of the displacement\nfieldu(x), wherex= (x,y,0) is the coordinates of the\ntube surface. For the tube geometry shown in Fig. 2, the\ndisplacement field is given by\nu(x) = (0,0,uz(x·n)), (2)\nusing the unit vector of the bending direction n=\n(cosϕ,sinϕ,0), whereϕ=π/6−θandθis the chiral\n(c) (b)\nB1B2\nB3A(a)\nKK'\nAB2\nB3B1\nFIG. 2. (a) Atomic structure of a cylindrical surface. (b)\nCoordinates of CNT in real space. (c) Coordinates of CNT\nin momentum space.\nangle (see Fig. 2 (b)). The second derivative ∂i∂juzis\nproportional to the first order of the curvature 1 /R:\n(∂2\nx+∂2\ny)uz∝1\nR,\n(∂2\nx−∂2\ny)uz∝cos 2ϕ\nR,2∂x∂yuz∝sin 2ϕ\nR. (3)\nThe irreducible decomposition of the first- and second-\norder curvatures is also shown in Table I. The cou-\npling terms in the Hamiltonian between the curva-\nture and other quantities are obtained as PT-even ir-\nreps ofA/prime\n1: (σxsy−σysx)/R, (sxcos 2ϕ−sysin 2ϕ)/R,\n[cos 2ϕ(σxsy+σysx)−sin 2ϕ(σxsx−σysy)]/R, (σxsin 2ϕ+\nσycos 2ϕ)/R2, and (σxsin 4ϕ−σycos 4ϕ)/R2. To sim-\nplify the Hamiltonian, we introduce the tube coordinates\n(kc,kt)T=ˆR(−ϕ)(kx,ky)T, withkcandktbeing mo-\nmenta along the circumference and axis directions, re-\nspectively. ˆR(ϑ) is a two-dimensional rotation matrix,\nˆR(ϑ) =/parenleftBigg\ncosϑ−sinϑ\nsinϑcosϑ/parenrightBigg\n. (4)\nσandsare also represented in this basis as ( σc,σt)T=\nˆR(−ϕ)(σx,σy)Tand (sc,st)T=ˆR(−ϕ)(sx,sy)T. Then,\nwe can construct the effective Hamiltonian in the pres-\nence of the curvature-induced SOI as HK(k) by collect-\ning the terms with PT-even irrep of A/prime\n1. The effective\nHamiltonian for the opposite valley, K/prime, is obtained by\nrequiring the Hamiltonian to be even under time re-\nversal:THK(k)T−1=HK/prime(−k). Here, we have used\nT=iτxσzsyK, withτbeing the valley-pseudospin opera-\ntor [24]. Finally, we obtain the valley-dependent effective\nHamiltonian as,\nHτzK(k) =/planckover2pi1vF(kcσc+τzktσt)\n−/epsilon1soτz(sccos 3ϕ−stsin 3ϕ)\n−/planckover2pi1vF∆kso(σcst−τzσtsc)−/planckover2pi1vF∆k/prime\nso(σcst+τzσtsc)\n−/planckover2pi1vF(τzσc∆kcsin 3ϕ+σt∆ktcos 3ϕ). (5)\nτzKrepresentsKandK/primeforτz=±1. In the curvature-\ninduced SOI terms, /epsilon1so, ∆kso, and ∆k/prime\nsoare proportional3\nTABLE I. Irreducible decomposition under D3hof momentum k, spins, pseudospin for sublattice σ, and∂i∂juzof the order\nof 1/R(see Eq. (3)). The twofold axis of C/prime\n2is set toy. The momentum kis PT-even and the curvature ∂i∂jukis PT-odd.\nIrrep PT-even PT-odd 1 /R(PT-odd) 1 /R2(PT-even)\nA/prime\n1σzsz 1/R2\nA/prime\n2 σz,sz\nE/prime(kx,ky), (σx,σy) (σysx,−σxsz) (sin 2 ϕ,cos 2ϕ)/R2, (sin 4ϕ,−cos 4ϕ)/R2\nA/prime/prime\n1 σxsx+σysy\nA/prime/prime\n2 σxsy−σysx 1/R\nE/prime/prime(σzsy,−σzsx) (sx,sy), (cos 2 ϕ,−sin 2ϕ)/R\n(σxsy+σysx,σxsx−σysy)\nto 1/R. The spin-independent shift (∆ kc, ∆kt) of Dirac\npoints is proportional to 1 /R2. The obtained Hamil-\ntonian is consistent with the previous study [25]. We\nshould note that the chiral-angle dependence of the CNT\nis also fully reproduced. This means that the 3 ϕdepen-\ndence is ascribable to crystalline symmetry. In conclu-\nsion, the curvature breaks the spatial-inversion symme-\ntry of graphene and induces the antisymmetric SOI which\nleads to valley-dependent spin-split energy bands.\nSilicon.— Next, we consider curvature-induced SOI\nin silicon, which is a fundamental material used in semi-\nconductor technology. We focus on the lowest conduc-\ntion bands, whose edges are located at ∆ points near\nthe zone boundary along the Γ-X symmetry lines and\ndiscuss the expected effect of curvature on, e.g., silicon\nnanotubes [26, 27].\nThree-dimensional silicon crystallizes into a diamond\nstructure whose symmetry is characterized by the\nspace group Fd¯3m(No. 227). The conduction elec-\ntrons are located at the six valleys at ∆ points,\n(±k0,0,0),(0,±k0,0), and (0,0,±k0). Their magnetic\nlittle group is 4 /m/primemm (C4v×{I,PT}). The orbital\nwavefunctions for the conduction band minima belong\nto ∆ 1irrep [20, 28]. For ( ±k0,0,0)-valley, the mo-\nmentumk(measured from the band bottom), spin s,\nand curvature ∂i∂jukdecompose into the irreps summa-\nrized in Table II. The irreducible decomposition for the\nother valleys can be obtained by the threefold rotation,\n(x,y,z )→(y,z,x )→(z,x,y ).\nAn effective Hamiltonian for the valley located at k0\nis a totally symmetric irrep of the little group (PT-even\nirrep ofA1). In the absence of curvature, we obtain\nquadratic kinetic terms with anisotropic effective masses,\nH(0)\nk0(k) =/planckover2pi12k2\n/lscript\n2m/lscript+/planckover2pi12k2\nt\n2mt, (6)\nwherek/lscriptandktare longitudinal (parallel to k0) and\ntransversal (perpendicular to k0) momenta, respectively.\nHereafter, we consider curved silicon expressed with\nthe displacement vector u= (0,0,uz(x·n)) withn=\n(cosϕ,sinϕ,0), for simplicity. The curvature is repre-\nsented by the second-order derivative u/prime/prime\nz(x·n)∝1/RTABLE II. Irreducible decomposition of momentum k, spin\ns, and∂i∂jukof the order of 1 /Runder the point group C4v,\nwhich is the little group of ( ±k0,0,0) from the space group\nFd¯3m.\nIrrep PT-even PT-odd ∇∇u(PT-odd)\nA1kx/parenleftbig\n∂2\ny+∂2\nz/parenrightbig\nux,∂2\nxux,∂x(∂yuy+∂zuz)\nA2 sx∂x(∂yuz−∂zuy)\nB1/parenleftbig\n∂2\ny−∂2\nz/parenrightbig\nux,∂x(∂yuy−∂zuz)\nB2 ∂y∂zux,∂x(∂yuz+∂zuy)\nE(ky,kz) (sz,−sy)∂2\nx(uy,uz),\n∂y∂z(uy,uz), (∂y,∂z)∂xux,/parenleftbig\n∂2\nyuy,∂2\nzuz/parenrightbig\n,/parenleftbig\n∂2\nzuy,∂2\nyuz/parenrightbig\nas in the discussion on the CNT (see Eq. (3)). For\n(±k0,0,0)-valley, as shown in Table II [29], we have a\nnon-vanishing term of order 1 /R,∂x(∂yuz−∂zuy)∝\nsin 2ϕ/R, as anA2irrep. This can be coupled to the\nsame irrep, sx, to be a totally symmetric irrep. Similarly,\nanEirrep (sz,−sy) can be coupled to ( ∂2\nxuy,∂2\nxuz)∝\n(0,cos2ϕ)/Rand (∂2\nzuy,∂2\nyuz)∝(0,sin2ϕ)/R. Accord-\ningly, the curvature-induced SOI yields\nH/prime\n(±k0,0,0)\n=±/planckover2pi1v1\nRsxsin 2ϕ±/planckover2pi1v2\nRsycos2ϕ±/planckover2pi1v3\nRsysin2ϕ. (7)\nHere, the double-sign is in the same order, and the\nsigns are determined to satisfy time-reversal symme-\ntry,syH(k0,0,0)(k)∗sy=H(−k0,0,0)(−k). For (0,±k0,0)-\nvalley, the irreducible decomposition can be carried out\nby making the threefold rotation for momentum, spin,\nand curvature in Table II. The SOI reads\nH/prime\n(0,±k0,0)\n=∓/planckover2pi1v1\nRsysin 2ϕ∓/planckover2pi1v2\nRsxsin2ϕ∓/planckover2pi1v3\nRsxcos2ϕ.(8)\nFor the (0 ,0,±k0)-valley, in contrast, no curvature-\ninduced term appears up to 1 /R. We have a nonzero\nterm, (∂2\nx+∂2\ny)uz∝1/R, as anA1irrep, but this can-\nnot be coupled to a same irrep, e.g., kz, due to the PT\nsymmetry.4\nIn order to estimate the magnitude of the spin split-\nting, we calculated the band structure of the curved sil-\nicon by using the tight binding model, taking the ef-\nfect of the curvature up to order 1 /Rinto account [23].\nLet us consider curved silicon as schematically shown in\nFig. 3 (a), where the [010] axis is kept unchanged, cor-\nresponding to ϕ= 0. Figure 3 (b) shows the calculated\nconduction bands near ( k0,0,0), (0,k0,0), and (0,0,k0)\nwith a curvature of R= 50 nm. At the valleys of ( k0,0,0)\nand (0,k0,0), the electron spin is fully polarized along the\nyandxdirections, respectively. The spin splitting of the\nconduction band is estimated to be ∼2µeV at the val-\nley of (k0,0,0) and∼90µeV at the valley of (0 ,k0,0).\nNote that the spin splitting at the valley of (0 ,0,±k0)\nis negligibly small. These results agree with the above\ngroup-theoretic prediction. We can also calculate the\nband structures when ϕ=π/4 [23]. By carefully compar-\ning these numerical results with Eqs. (7) and (8), the val-\nues ofv1,v2,v3can be estimated to be /planckover2pi1v1/R/similarequal30µeV,\n/planckover2pi1v2/R/similarequal1µeV, and /planckover2pi1v3/R/similarequal50µeV forR= 50 nm. We\ncan show that v2is induced by interband mixing, while v1\nandv3are induced by mixing between subbands formed\nby the circumferential boundary condition [23]. The spin\nsplitting obtained here should be able to be detected with\nresonance microwave measurements [30].\nDiscussion.— Our theory can be utilized as a conve-\nnient method to produce valley-dependent SOI in various\nsystems. Nanotubes produced by rolled-up atomic layer\nmaterials such as transition-metal dichalcogenides would\nbe an interesting example [31]. Our method can also be\nused to control the SOI in rolled-up semiconductor struc-\ntures with different elastic properties [32, 33].\nWhile our theory was applied to homogeneous systems,\nit is also applicable to curvature-induced SOI by using\nspatially inhomogeneous structures such as corrugations\nof atomic layers [34]. Moreover, the curvature effect de-\nscribed by the second derivative of the displacement has\na similar aspect to the spin-vorticity coupling [35–37],\nHsv=−1\n2s·ω, whereωi=/epsilon1ijk∂j∂tukmeans vorticity,\ndynamical anti-symmetric lattice distortion. The vor-\nticity can be regarded as a kind of curvature that ex-\ntends the second-order derivative of the lattice displace-\nments in space to those in space-time. In this sense,\nour group-theoretic approach can be extended to include\nspin-vorticity theory in a unified way.\nWe should note that the SOI induced in curved mate-\nrials has been discussed in the literature in terms of the\ntechnique of thin-film quantization [38, 39]. We clarified\nthat this method cannot be applied to curved materi-\nals in usual situations because the confinement potential\nhas to be made unphysically large (see the Supplemental\nMaterial [23] for details).\nConclusion.— We proposed a group-theoretical\nmethod to describe the valley-dependent spin-orbit in-\nteraction induced in curved materials. This method sys-\ntematically determines an effective Hamiltonian from the\n(eV)\n(/μm)\n(a)\n(b)\n(eV)\n(/μm)\n(eV)\n(/μm)\nFIG. 3. (a) Crystal structure of silicon and schematic diagram\nof curved silicon whose bending direction is n= (1,0,0). (b)\nSchematic diagram of the valleys of silicon in the wavenumber\nspace (lower left panel) and the conduction bands at the val-\nleys of [100] (lower right panel), [010] (upper right panel) and\n[001] (upper left panel) along the kydirection. The red and\nblue lines indicate band dispersions for different spin polariza-\ntions. The solid lines indicate the lowest subbands, while the\ndashed line indicates the next lowest subband. Here, the sub-\nbands are formed by imposing the boundary condition along\nthe circumferential direction. The curvature radius is taken\nto beR= 50 nm and the energy is measured from the con-\nduction band bottom for unbent silicon.\nsymmetries of the crystals, momenta, and spins. The\nmethod succeeded in reproducing the effective Hamilto-\nnian of CNTs with sublattice- and valley-dependent SOI\ninduced by curvature. Furthermore, we derived an ef-\nfective Hamiltonian for curved silicon and revealed that\nthe curvature activates SOI in a valley-selective man-\nner. Combining this method with a tight-binding calcu-\nlation, we demonstrated significant SOI splittings with\nnontrivial bending-direction dependences. Our method\nwill provide a general strategy for curvature engineering\nof valley-dependent SOI in nanomaterials.\nWe acknowledge JSPS KAKENHI for providing Grants\n(No. JP18H04282, No. JP19K14637, No. JP20K03831,\nNo. JP20H01863, No. JP20K03835, No. JP20K05258,\nNo. JP21K20356, and No. JP21K03414) and the Sumit-5\nomo Foundation (190228). M. M. is partially supported\nby the Priority Program of the Chinese Academy of Sci-\nences, Grant No. XDB28000000. T.S. was supported by\nthe Japan Society for the Promotion of Science through\nthe Program for Leading Graduate Schools (MERIT).\n∗ai@st.phys.nagoya-u.ac.jp\n†sato-tetsuya163@g.ecc.u-tokyo.ac.jp\n[1] J. Sinova, S. O. Valenzuela, J. Wunderlich, C. Back, and\nT. Jungwirth, Spin Hall effects, Rev. Mod. Phys. 87, 1213\n(2015).\n[2] A. Manchon, H. C. Koo, J. Nitta, S. Frolov, and\nR. Duine, New perspectives for rashba spin–orbit cou-\npling, Nat. Mater. 14, 871 (2015).\n[3] A. Manchon, J. ˇZelezn` y, I. M. Miron, T. Jungwirth,\nJ. Sinova, A. Thiaville, K. Garello, and P. Gambardella,\nCurrent-induced spin-orbit torques in ferromagnetic and\nantiferromagnetic systems, Rev. Mod. Phys. 91, 035004\n(2019).\n[4] S.-H. Yang, R. Naaman, Y. Paltiel, and S. S. Parkin,\nChiral spintronics, Nature Reviews Physics 3, 328 (2021).\n[5] S. A. Vitale, D. Nezich, J. O. Varghese, P. Kim, N. Gedik,\nP. Jarillo-Herrero, D. Xiao, and M. Rothschild, Val-\nleytronics: Opportunities, challenges, and paths forward,\nSmall 14, 1801483 (2018).\n[6] A. Ciarrocchi, F. Tagarelli, A. Avsar, and A. Kis, Exci-\ntonic devices with van der waals heterostructures: val-\nleytronics meets twistronics, Nat. Rev. Mater. 7, 449\n(2022).\n[7] J. F. Sierra, J. Fabian, R. K. Kawakami, S. Roche, and\nS. O. Valenzuela, Van der waals heterostructures for spin-\ntronics and opto-spintronics, Nature Nanotechnology 16,\n856 (2021).\n[8] B. A. Bernevig and S.-C. Zhang, Spin splitting and spin\ncurrent in strained bulk semiconductors, Physical Review\nB72, 115204 (2005).\n[9] M. Hruˇ ska, ˇS. Kos, S. A. Crooker, A. Saxena, and D. L.\nSmith, Effects of strain, electric, and magnetic fields on\nlateral electron-spin transport in semiconductor epilay-\ners, Phys. Rev. B 73, 075306 (2006).\n[10] Y. Li and Y. Q. Li, Strain-assisted spin manipulation in\na quantum well, Eur. Phys. J. B 63, 493 (2008).\n[11] B. Norman, C. Trowbridge, J. Stephens, A. Gossard,\nD. Awschalom, and V. Sih, Mapping spin-orbit splitting\nin strained (in, ga) as epilayers, Physical Review B 82,\n081304 (2010).\n[12] G. Liu, Y. Chen, C. Jia, G.-D. Hao, and Z. Wang, Spin\nsplitting modulated by uniaxial stress in InAs nanowires,\nJ. Phys.: Condens. Matter 23, 015801 (2011).\n[13] T. Matsuda and K. Yoh, Effect of strain on the Dres-\nselhaus effect of InAs-based heterostructures, Journal of\nCrystal Growth 323, 52 (2011).\n[14] B. M. Norman, C. J. Trowbridge, D. D. Awschalom, and\nV. Sih, Current-Induced Spin Polarization in Anisotropic\nSpin-Orbit Fields, Phys. Rev. Lett. 112, 056601 (2014).\n[15] T. Sogawa, P. V. Santos, S. K. Zhang, S. Eshlaghi, A. D.\nWieck, and K. H. Ploog, Transport and Lifetime En-\nhancement of Photoexcited Spins in GaAs by Surface\nAcoustic Waves, Phys. Rev. Lett. 87, 276601 (2001).\n[16] H. Sanada, T. Sogawa, H. Gotoh, K. Onomitsu, M. Ko-hda, J. Nitta, and P. Santos, Acoustically induced spin-\norbit interactions revealed by two-dimensional imaging\nof spin transport in gaas, Physical review letters 106,\n216602 (2011).\n[17] W. Izumida, K. Sato, and R. Saito, Spin–Orbit Inter-\naction in Single Wall Carbon Nanotubes: Symmetry\nAdapted Tight-Binding Calculation and Effective Model\nAnalysis, J. Phys. Soc. Jpn. 78, 074707 (2009).\n[18] H. Ochoa, A. H. Castro Neto, V. I. Fal’ko, and F. Guinea,\nSpin-orbit coupling assisted by flexural phonons in\ngraphene, Phys. Rev. B 86, 245411 (2012).\n[19] M. I. Aroyo, J. M. Perez-Mato, C. Capillas, E. Kroumova,\nS. Ivantchev, G. Madariaga, A. Kirov, and H. Won-\ndratschek, Bilbao Crystallographic Server: I. Databases\nand crystallographic computing programs, Z. Kristallogr.\nCryst. Mater. 221, 15 (2006).\n[20] M. I. Aroyo, A. Kirov, C. Capillasm, J. M. Perez-Mato,\nand H. Wondratschek, Bilbao Crystallographic Server.\nII. Representations of crystallographic point groups and\nspace groups, Acta Cryst. A62, 115 (2006).\n[21] Y. Xu, L. Elcoro, Z.-D. Song, B. J. Wieder, M. G.\nVergniory, N. Regnault, Y. Chen, C. Felser, and\nB. A. Bernevig, High-throughput calculations of mag-\nnetic topological materials, Nature 586, 702 (2020).\n[22] L. Elcoro, B. J. Wieder, Z. Song, Y. Xu, B. Bradlyn, and\nB. A. Bernevig, Magnetic topological quantum chemistry,\nNat. Commun. 12, 5965 (2021).\n[23] See Supplemental Material at\nhttp://link.aps.org/supplemental/10.1103/PhysRevLett.***.******\nfor detailed theoretical formulation,...\n[24] The time reversal operator Tfor our basis (same as the\none used in Ref. [17]) is derived from the well-known\nexpression, T=−isyK, for the position basis.\n[25] For the perturbative calculation given in Ref. [17], we\nhave ∆k/prime\nSO= 0. This is due to the oversimplification in\nthe model, i.e., the artificial electron-hole symmetry for π\nandσorbitals. ∆k/prime\nSO/negationslash= 0 is obtained from more realistic\nanalyses.\n[26] X. Huang, R. Gonzalez-Rodriguez, R. Rich, Z. Gryczyn-\nski, and J. L. Coffer, Fabrication and size dependent\nproperties of porous silicon nanotube arrays, Chem.\nCommun. 49, 5760 (2013).\n[27] M. Taghinejad, H. Taghinejad, M. Abdolahad, and\nS. Mohajerzadeh, A nickel–gold bilayer catalyst engineer-\ning technique for self-assembled growth of highly ordered\nsilicon nanotubes, Nano Lett. 13, 889 (2013).\n[28] Unlike K6irrep (πelectrons) for graphene, ∆ 1irrep is\nnon-degenerate, and therefore orbital degrees of freedom,\nsuch asσfor graphene, do not appear in this case.\n[29] Non-vanishing irreps of the curvature as functions of 1 /R\nandϕfor each valley are listed in Table IV of the Sup-\nplemental Material [23].\n[30] J. Sichau, M. Prada, T. Anlauf, T. J. Lyon, B. Bosn-\njak, L. Tiemann, and R. H. Blick, Resonance microwave\nmeasurements of an intrinsic spin-orbit coupling gap in\ngraphene: A possible indication of a topological state,\nPhys. Rev. Lett. 122, 046403 (2019).\n[31] J. L. Musfeldt, Y. Iwasa, and R. Tenne, Nanotubes from\nlayered transition metal dichalcogenides, Physics Today\n73, 42 (2020).\n[32] V. Y. Prinz, V. A. Seleznev, A. K. Gutakovsky, A. V.\nChehovskiy, V. V. Preobrazhenskii, M. A. Putyato,\nand T. A. Gavrilova, Free-standing and overgrown in-\ngaas/gaas nanotubes, nanohelices and their arrays, Phys-6\nica E: Low-dimensional Systems and Nanostructures 6,\n828 (2000).\n[33] Y. Zhang, F. Zhang, Z. Yan, Q. Ma, X. Li, Y. Huang,\nand J. A. Rogers, Printing, folding and assembly meth-\nods for forming 3D mesostructures in advanced materials,\nNature Reviews Materials 2, 17019 (2017).\n[34] A. Avsar, H. Ochoa, F. Guinea, B. ¨Ozyilmaz, B. van\nWees, and I. Vera-Marun, Colloquium: Spintronics in\ngraphene and other two-dimensional materials, Reviews\nof Modern Physics 92, 021003 (2020).\n[35] M. Matsuo, J. Ieda, K. Harii, E. Saitoh, and S. Maekawa,\nMechanical generation of spin current by spin-rotation\ncoupling, Phys. Rev. B 87, 180402 (2013).\n[36] D. Kobayashi, T. Yoshikawa, M. Matsuo, R. Iguchi,\nS. Maekawa, E. Saitoh, and Y. Nozaki, Spin Current Gen-\neration Using a Surface Acoustic Wave Generated via\nSpin-Rotation Coupling, Phys. Rev. Lett. 119, 077202\n(2017).\n[37] S. Tateno, G. Okano, M. Matsuo, and Y. Nozaki, Electri-\ncal evaluation of the alternating spin current generated\nvia spin-vorticity coupling, Phys. Rev. B 102, 104406\n(2020).\n[38] R. C. T. da Costa, Quantum mechanics of a constrained\nparticle, Phys. Rev. A 23, 1982 (1981).\n[39] S. Matsutani, Berry phase of dirac particle in thin rod,\nJ. Phys. Soc. Jpn 61, 3825 (1992).Supplementary Information:\nDesigning Valley-Dependent Spin-Orbit Interaction by Curvature\nAi Yamakage,1,∗T. Sato,2,†R. Okuyama,3T. Funato,4, 5W. Izumida,6K. Sato,7T. Kato,2and M. Matsuo5, 8, 9, 10\n1Department of Physics, Nagoya University, Nagoya 464-8602, Japan\n2Institute for Solid State Physics, University of Tokyo, Kashiwa 277-8581, Japan\n3Department of Physics, Meiji University, Kawasaki 214-8571, Japan\n4Center for Spintronics Research Network, Keio University, Yokohama 223-8522, Japan\n5Kavli Institute for Theoretical Sciences, University of Chinese Academy of Sciences, Beijing, 100190, China.\n6Department of Physics, Tohoku University, Sendai 980-8578, Japan\n7National Institute of Technology, Sendai College, Sendai 989-3128, Japan\n8CAS Center for Excellence in Topological Quantum Computation,\nUniversity of Chinese Academy of Sciences, Beijing 100190, China\n9Advanced Science Research Center, Japan Atomic Energy Agency, Tokai, 319-1195, Japan\n10RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan\nI. EFFECTIVE HAMILTONIAN IN THE\nPRESENCE OF CURVATURE\nThis section shows a procedure to generate an effective\nHamiltonian induced by curvature that can be applied to\narbitrary systems including ones with multiple internal\ndegrees of freedom (spin, orbital, and sublattice). The\noutline of the procedure is:\n1. Find the space group, kpoints, and irreducible rep-\nresentations (irreps) of the low-lying excitations in\nthe flat (uncurved) system.\n2. Set the irreps of the symmetry operations.\n3. Compute the irreducible decomposition of the ma-\ntrices (operators) Oi.\n4. Compute the irreducible decomposition of the mo-\nmentumki, strains∂iujand curvatures ∂i∂juk.\n5. Construct the products of the matrix, momentum,\nstrains, and curvatures. Decompose them into ir-\nreps. The totally symmetric representations can\nappear in the effective Hamiltonian. The other\nirreps correspond to physical observables such as\nelectric/spin currents.\nAs a representative example, we consider a carbon nan-\notube (CNT).\n1. Irrep of low-energy states\nFor nonmagnetic graphene, which is the flat (unrolled)\nsystem of CNT, characterized by the magnetic space\ngroupP6/mmm 1/prime(No. 191.234), the low-lying excita-\ntions are governed by the Dirac cones around the Kand\n∗ai@st.phys.nagoya-u.ac.jp\n†sato-tetsuya163@g.ecc.u-tokyo.ac.jpK/primepoints, on which the magnetic little cogroup is given\nby 6/prime/mmm/primeand its unitary maximum subgroup ¯62m\n(D3h) [1, 2]. They come from the pzorbitals to form\na double degeneracy, single-valued two-dimensional odd-\nparity irrep K6.\n2. Symmetry operation\nTheK6irreps of the generators, i.e., the 3+threefold\nrotation along the /angbracketleft001/angbracketright(z) direction, 2 100twofold rota-\ntion along the/angbracketleft100/angbracketright(y) direction, and m001horizontal\nmirror, are listed as follows [1–4]:\nDK(3+) =eiσz2π/3, (1)\nDK(2100) =σy, (2)\nDK(m001) =−σ0, (3)\nwhereσ0andσi(i=x,y,z ) denote the identity and Pauli\nmatrices acting in the sublattice space, respectively. Ad-\nditionally, we have the parity-time (PT) symmetry oper-\nation,\nO→σxO∗σx. (4)\nTaking the spin degrees of freedom into account, the ir-\nreps are given by\n¯DK(3+) =eiσz2π/3e−iszπ/3, (5)\n¯DK(2100) =σy(−isy), (6)\n¯DK(m001) =−σ0isz, (7)\nand the PT symmetry operation by\nO→σxsyO∗syσx. (8)\n3. Irreps of matrices\nThe theory on the Kpoint has the sublattice σand\nspinsdegrees of freedom, represented by 4 ×4 matrixarXiv:2304.12928v1  [cond-mat.mes-hall]  25 Apr 20232\nσµsν. This matrix transforms as O→D†OD. They\nare decomposed into irreps under the point group ¯62m\n(D3h). Furthermore, they are decomposed into PT–even\nand odd irreps. The results are summarized in Table I.\nNote that we will use the following convention for\nthe components of the two-dimensional irreps, E/primeand\nE/prime/prime. Their components, ( E(1),E(2)), are transformed for\ng∈D3has if they are components of in-plane geometric\nvectors:\n¯DK(3+)†/parenleftbigg\nE(1)\nE(2)/parenrightbigg\n¯DK(3+) =ˆR(2π/3)/parenleftbigg\nE(1)\nE(2)/parenrightbigg\n,(9)\n¯DK(2100)†/parenleftbigg\nE(1)\nE(2)/parenrightbigg\n¯DK(2100) =/parenleftbigg\n−E(1)\nE(2)/parenrightbigg\n, (10)\nwhere ˆR(ϑ) is a two-dimensional rotation matrix,\nˆR(ϑ) =/parenleftbigg\ncosϑ−sinϑ\nsinϑcosϑ/parenrightbigg\n. (11)\nBy using this convention, the product rules involving E/prime\nandE/prime/primeirreps are simplified as shown in Table III. These\nresults can be easily interpreted geometrically: A/prime\n1and\nA/prime\n2components of E/prime×E/primeare inner and outer products\nof the two in-plane vectors, respectively.\n4. Irreps of momentum, strain, and curvature\nA three-dimensional displacement u= (ux,uy,uz) is\ntransformed using a symmetry operation gof the little\ngroup as\n3+:\nux\nuy\nuz\n→\n−1/2−√\n3/2 0√\n3/2−1/2 0\n0 0 1\n\nux\nuy\nuz\n,(12)\n2100:\nux\nuy\nuz\n→\n−1 0 0\n0 1 0\n0 0−1\n\nux\nuy\nuz\n, (13)\nm001:\nux\nuy\nuz\n→\n1 0 0\n0 1 0\n0 0−1\n\nux\nuy\nuz\n. (14)\nTwo-dimensional vectors, momentum k= (kx,ky) and\ngradient ∇= (∂x,∂y), transform in a similar way. Note\nthat momentum is PT–even, while the gradient and dis-\nplacement are PT–odd. The irreducible decomposition\nof strains and linear and quadratic polynomials of mo-\nmentum is shown in Table I.\n5. Hamiltonian\nNow we are in a position to obtain any observable oper-\nator, including the Hamiltonian. The Hamiltonian must\nbelong to the totally symmetric irrep, A/prime\n1. Of the orderofk0and on the flat space, the Hamiltonian is given by a\nlinear combination of σ0s0andσzsz, the latter of which\ncorresponds to spin-orbit coupling in graphene leading to\na quantum spin Hall insulator [5] and is ignored in the\nmain text as it is quite small. Since both ( kx,ky) and\n(σx,σy) belong to the PT–even E/primeirrep, the Hamilto-\nnian of the first order of kis given by∝kxσx+kyσy,\nwhich is nothing but a Dirac cone.\nStrain-induced chiral gauge field. Similarly, strain\n∂iujcan appear in the Hamiltonian when the product of\n∂iujand matrices are in the PT–even A/prime\n1irrep. The A/prime\n2\nstrain,∂xuy−∂yux, is not allowed in the Hamiltonian\nbecause there is no PT–even A/prime\n2matrix. The E/primestrains,\non the other hand, can appear as\n∝(∂xuy+∂yux)σx+ (∂xux−∂yuy)σy, (15)\nreproducing the chiral gauge field in strained\ngraphene [6].\nTheE/prime/primestrain, (∂y,−∂x)uz, also appears in the Hamil-\ntonian to couple with PT–even spin-dependent E/prime/primema-\ntrixσz(sy,−sx):\n∝∂yuzσzsy+∂xuzσzsx, (16)\nwhich is the mass term for the Dirac cone with spin de-\npendence.\nCurvature-induced spin-orbit interaction. The\nnext-order terms, ∂i∂juk, including curvature, are cou-\npled with PT–odd matrices in the Hamiltonian, result-\ning in spin-split eigenstates. For instance, the A/prime/prime\n2andE/prime/prime\nterms are included in the Hamiltonian in the form of\n∝∇2uz(σxsy−σysx),/parenleftbig\n∂2\ny−∂2\nx/parenrightbig\nuzsx+ 2∂x∂yuzsy,\n/parenleftbig\n∂2\ny−∂2\nx/parenrightbig\nuz(σxsy+σysx) + 2∂x∂yuz(σxsx−σysy),\n(17)\nwhich is a kind of Rashba-like antisymmetric spin-orbit\ninteraction (SOI) induced by curvature ∂i∂juz, as dis-\ncussed in the main text.\nValley-asymmetric velocities. Similarly, we can re-\nproduce the curvature-induced valley-asymmetric Fermi\nvelocities of electrons that are microscopically derived in\nRef. [7]. These are determined by terms proportional to k\nin the effective Hamiltonian. For the Kvalley, we decom-\npose the products of sublattice pseudospin, momentum,\nand derivatives of uby neglecting tiny spin dependence.\nWe obtain the effective Hamiltonian as the PT–even ir-\nreps ofA/prime\n1, where the leading order in curvature is 1 /R2.\nBy requiring the time-reversal symmetry, the additional\nterms for the Hamiltonian for τzKvalley yield,\nH/prime\nτzK(k) =1\nR2/bracketleftBig\ng1σckc+g2τzσtkt+g3τzkcsin 3ϕ\n+g4τzktcos 3ϕ+g5(σc,−τzσt)ˆR(−6ϕ)/parenleftBigg\nkc\nkt/parenrightBigg/bracketrightBig\n.(18)\nHere,gj’s are constants independent of both Randϕ.\nkcandktare momenta along the circumference and3\nTABLE I. Irreps of matrices under ¯62m(D3h) point group. Irreps of momenta and strains are also shown.\nIrrep 1 3+2100m001 PT–even PT–odd k kk ∇u\nA/prime\n1 1 1 1 1 1, σzsz k2\nx+k2\ny ∂xux+∂yuy\nA/prime\n2 1 1−1 1 σz,sz ∂xuy−∂yux\nE/prime2−1 0 2 ( σx,σy) ( σy,−σx)sz (kx,ky) (2kxky,k2\nx−k2\ny) (∂xuy+∂yux,∂xux−∂yuy)\nA/prime/prime\n1 1 1 1−1 σxsx+σysy\nA/prime/prime\n2 1 1−1−1 σxsy−σysx\nE/prime/prime2−1 0−2σz(sy,−sx) ( sx,sy), ( ∂y,−∂x)uz\n(σxsy+σysx,σxsx−σysy)\nTABLE II. Irreps of curvatures.\nIrrep ∇∇u\nA/prime\n1 (∂2\nx−∂2\ny)uy+ 2∂x∂yux\nA/prime\n2 (∂2\nx−∂2\ny)ux−2∂x∂yuy\nE/prime∇2(ux,uy),/parenleftbig\n2∂x∂yuy+ (∂2\nx−∂2\ny)ux,2∂x∂yux−(∂2\nx−∂2\ny)uy/parenrightbig\nA/prime/prime\n1\nA/prime/prime\n2 ∇2uz\nE/prime/prime(∂2\ny−∂2\nx,2∂x∂y)uz\nTABLE III. Product rules for two-dimensional irreps of D3h.\nXindicatesX/primeorX/prime/primein the table ( X=A1,A2,orE).X/prime×\nY/prime=X/prime/prime×Y/prime/prime=Z/primeandX/prime×Y/prime/prime=X/prime/prime×Y/prime=Z/prime/prime.\nProduct Components\nA1×E=E A 1(E(1),E(2))\nA2×E=E A 2(E(2),−E(1))\nE×E=A1E(1)E(1) +E(2)E(2)\n+A2E(1)E(2)−E(2)E(1)\n+E(E(1)E(2) +E(2)E(1),E(1)E(1)−E(2)E(2))\naxis directions, respectively: ( kc,kt)T=ˆR(−ϕ)(kx,ky)T.\nEquation (18) is nothing but the Hamiltonian derived in\nRef. [7]. Note that some coupling constants, which are\nindependent in Ref. [7], satisfy the relation −c3=c4=\n4g5//planckover2pi1in this analysis. This should be because the sym-\nmetries of the system may not be fully taken into account\nin the previous calculation based on perturbation theory.\nActually,c3andc4obtained from a fitting with the nu-\nmerical calculations satisfy −c3/similarequalc4in Ref. [7].\nII. IRREDUCIBLE DECOMPOSITION FOR\nCURVED SILICON\nIn this section, we provide the irreducible decomposi-\ntion for curved silicon, for each conduction band minima.\nThe curvature is expressed in terms of the displacementfield as,\nu= (0,0,uz(x·n)), (19)\nn= (cosϕ,sinϕ,0). (20)\nAs shown in Eq. (3) in the main text, the second deriva-\ntive ofuis proportional to 1 /R. While the irreducible\ndecomposition is only shown for valleys along [100]-\ndirection in the forms of derivative in the main material,\nwe provide them for all the six valleys in Table IV as a\nfunction of 1 /Randϕ.\nIII. NUMERICAL ESTIMATE OF\nCURVATURE-INDUCED SPIN SPLITTING\nIn this section, we summarize how to estimate the\ncurvature-induced spin splitting for curved thin films.\nIn the theoretical work for CNTs [8], the spin splitting\ninduced by the curvature was calculated by considering\ninclinations of the atomic orbitals at each site, which en-\nables hopping between, e.g., pzandsorbitals that are\nforbidden in graphene. Here, we employ an alternative\nmethod to simplify the implementation of the numerical\nsimulation. We consider a general coordinate transforma-\ntion to make a curved thin film flat in new coordinates. In\nthe new coordinates, the hopping energy between neigh-\nboring sites is modified due to the curvature of the co-\nordinates. This method can treat not only atomic-layer\nmaterials such as CNTs but also thin curved films of\nthree-dimensional materials.\nThe coordinate transformation is schematically illus-\ntrated in Fig. 1. The gray area in the left panel indi-\ncates a curved thin material with curvature R, which\nis assumed to be much larger than the thickness of the\nmaterial. The Cartesian coordinates at an arbitrary\nposition are expressed in the cylindrical coordinates as\n(x,y,z ) = (rsinφ,y,r cosφ−R), whereris the dis-\ntance from the center of the curved material and φis\nthe azimuth angle measured from the z-axis (see Fig. 1).\nWe consider a general transformation from the origi-\nnal Cartesian coordinates ( x,y,z ) to curved coordinates\n(x/prime,y/prime,z/prime), in which the curved thin material is mapped to\na flat thin material, keeping the ycoordinate unchanged\n(y=y/prime). The new coordinates are written with the cylin-4\nTABLE IV. Irreducible decomposition of momentum k, spins, and curvature of order 1 /R, under the point group C4v, which\nis the little group of six conduction valley minima, (0 ,0,±k0), (±k0,0,0), and (0,±k0,0), from the space group Fd¯3m. We\nassume that curvature is expressed as u= (0,0,uz(x·n)) withn= (cosϕ,sinϕ,0).\n(0,0,±k0) (±k0,0,0) (0,±k0,0)\ns k 1/Rs k 1/R s k 1/R\nIrrep (PT-odd) (PT-even) (PT-odd)\nA1 kz 1/R kx ky\nA2sz sx sin 2ϕ/R sy −sin 2ϕ/R\nB1 cos 2ϕ/R\nB2 sin 2ϕ/R sin 2ϕ/R sin 2ϕ/R\nE(sy,−sx) (kx,ky) (sz,−sy) (ky,kz) (0,cos2ϕ/R),(0,sin2ϕ/R)(sx,−sz) (kz,kx) (sin2ϕ/R, 0),(cos2ϕ/R, 0)\nφ φ\n φ\nφ\nFIG. 1. Schematic diagram of the coordinate transformation.\nThe gray region indicates a thin material. Through this coor-\ndinate transformation, the curved material in the laboratory\ncoordinates ( x,y,z ) is mapped to a flat one in the new coor-\ndinates (x/prime,y/prime,z/prime).\ndrical coordinates as ( x/prime,y/prime,z/prime) = (Rφ,y,r−R), which\nleads to the relation,\n\n\nx= (z/prime+R) sin(x/prime/R),\ny=y/prime,\nz= (z/prime+R) cos(x/prime/R)−R.(21)\nFor this coordinate transformation, the Jacobian is r/R\n(= (R+z/prime)/R). Therefore, the volume integral in the old\ncoordinates can be rewritten as\n/integraldisplay\ndxdydz (···) =/integraldisplay\ndx/primedy/primedz/primer\nR(···). (22)\nNext, we consider a modulation of a potential energy\nby this coordinate transformation. For simplicity, we as-\nsume that the potential of an ion is the Coulomb poten-\ntial, i.e.,V(x,y,z ) =k/(x2+y2+z2)1/2, wherekis the\nCoulomb constant. Here, we take the Coulomb constant\nto bek=Z/planckover2pi1c/137/epsilon1r, wherecis the velocity of light, Zis\nthe effective ionic charge, and /epsilon1ris the relative electrical\npermittivity. We set /epsilon1r= 3 for carbon nanotubes [9] and\n/epsilon1r= 11.7 for silicon [10], while we set Z= 4 for both\nsystems. If the curvature Ris much larger than r, the\npotential energy of an ion located at the origin can bewritten in the new coordinates V(x/prime,y/prime,z/prime) as\nV(x/prime,y/prime,z/prime) =k/radicalbig\nR2+r2−2rRcosφ+y2\n=k/radicalbig\nx/prime2+y/prime2+z/prime2\n×/parenleftbigg\n1−x/prime2z/prime\n2R(x/prime2+y/prime2+z/prime2)+O(1/R2)/parenrightbigg\n≡V0(x/prime,y/prime,z/prime) +1\nRVcur(x/prime,y/prime,z/prime) +O(1/R2).(23)\nIn addition to the spherical Coulomb potential\nV0(x/prime,y/prime,z/prime), we also obtain the curvature-induced\nanisotropic correction Vcur(x/prime,y/prime,z/prime)/R.\nThe present coordinate transformation also modifies\nthe kinetic energy of electrons. The Laplace operator is\nexpressed in cylindrical coordinates as\n∆=1\nr∂r(r∂r) +1\nr2∂2\nφ+∂2\ny. (24)\nUsing (x/prime,y/prime,z/prime) = (Rφ,y,r−R), the spatial derivatives\nin the new coordinates are written in terms of r,φ, and\nyas\n∂x/prime=1\nR∂φ, ∂y/prime=∂y, ∂z/prime=∂r. (25)\nCombining Eqs. (24) and (25), the Laplace operator in\nthe new coordinates becomes\n∆/similarequal∂2\nx/prime+∂2\ny/prime+∂2\nz/prime+1\nR∂z/prime−2z/prime\nR∂2\nx/prime\n≡∆0+1\nR∆cur, (26)\nwhere∆0=∂2\nx/prime+∂2\ny/prime+∂2\nz/primeis the Laplace operator in the\nnew coordinates and ∆curis a correction term induced\nby the coordinate transformation.\nThe wave functions are also modified by the coordinate\ntransformation. For instance, the wave function of the pz5\norbital in the original frame is transformed as\nψpz(r) =rcosφ−R\n4/radicalbig\n2πa5ze−√\n(rcosφ−R)2+r2sin2φ+y2/2az\n/similarequal1\n4/radicalbig\n2πa5z/bracketleftBigg\nz/prime+1\nR/parenleftBigg\n−x/prime2\n2−x/prime2z/prime2\n4az/radicalbig\nx/prime2+y/prime2+z/prime2/parenrightBigg/bracketrightBigg\n×e−√\nx/prime2+y/prime2+z/prime2/2az\n≡ψ(0)\npz(r/prime) +1\nRψ(1)\npz(r/prime), (27)\nwhereaz=a0/Z,a0is the Bohr radius, ψ(0)\npz(r/prime) =\nψpz(r/prime) is the original wavefunction, and ψ(1)\npz(r/prime)/Risa correction term induced by the coordinate transforma-\ntion.\nIn the new coordinates, the crystal structure has no\ncurvature and its electronic structure can be obtained\nusing the standard tight-binding calculation. This helps\nto reduce the computational cost compared with direct\ncalculation for the curved crystal structure. In return, we\nhave to consider the correction terms in the kinetic and\npotential energies induced by curvature carefully. We can\nevaluate these corrections using perturbation theory.\nIn our tight-binding calculation, we employ the non-\northogonal Slater-Koster two-center parameters [11, 12]\nfor silicon and CNTs and take the atomic SOIs into ac-\ncount through one-site terms. The correction of the hop-\nping integral, which is proportional to 1 /R, is written\nas\n/integraldisplay\nd3r/prime/braceleftbigg\nψ(0)∗\nn(r/prime+di/2)/parenleftbigg\n−/planckover2pi12\n2m∆cur+Vcur(r/prime+di/2) +Vcur(r/prime−di/2)/parenrightbigg\nψ(0)\nm(r/prime−di/2) (28)\n+ψ(1)∗\nn(r/prime+di/2)/parenleftbigg\n−/planckover2pi12\n2m∆0+V0(r/prime+di/2) +V0(r/prime−di/2)/parenrightbigg\nψ(0)\nm(r/prime−di/2)\n+ψ(0)∗\nn(r/prime+di/2)/parenleftbigg\n−/planckover2pi12\n2m∆0+V0(r/prime+di/2) +V0(r/prime−di/2)/parenrightbigg\nψ(1)\nm(r/prime−di/2)\n+z/primeψ(0)∗\nn(r/prime+di/2)/parenleftbigg\n−/planckover2pi12\n2m∆0+V0(r/prime+di/2) +V0(r/prime−di/2)/parenrightbigg\nψ(0)\nm(r/prime−di/2)/bracerightbigg\n,\nFIG. 2. Cylindrical coordinates on a nanotube.\nwhere the displacement vectors between two atomic or-\nbitals are denoted with di(i= 1,···,M). Note that the\nlast term is due to the correction in the Jacobian.\nThe atomic SOI is defined in cylindrical coordinates,\nwhich is (xc,xt,xn) in Fig. 2 and is given as λs·l, where\ns= (sc,st,sn) is the spin operator and l= (lc,lt,ln)\nis the orbital angular momentum operator in cylindrical\ncoordinates. Since the spin operator is related to the\nspin of an itinerant electron, shas to be rewritten in the\nlaboratory frame ( x,y,z ) (see Fig. 2). When the axis ofthe nanotube is taken to be in the y(=y/prime) direction,\nthe spin operator in cylindrical coordinates, s, can be\nrewritten as [8]\nsc= (˜s+µ−−˜s−µ+)/i, (29)\nst= ˜sy, (30)\nsn= ˜s+µ−+ ˜s−µ+, (31)\nwhere ˜s= (˜sx,˜sy,˜sz) is the spin operator in the labo-\nratory frame, ˜ s±= ˜sz±i˜sxis the spin ladder operator,\nandµ±is the operator which changes the momentum of\nthe circumference direction kcby±δkc=±1/R. Using\nthese relations, the atomic SOI takes the form of\nλs·l=λ(˜syly+ ˜s+µ−l−+ ˜s−µ+l+). (32)\nThis spin-orbit interaction hybridizes a up-spin state\nwith wavenumber ( kc,kt,kn) with a down-spin state with\nwavenumber ( kc±δkc,kt,kn). This transfer between kc\nand ˜syis ascribable to the conservation of the total angu-\nlar momentum in the ydirection in the laboratory frame,\nas we will discuss soon. For example, let us consider a\nband calculation of silicon using eight atomic orbitals,\ni.e., four atomic orbitals (3 s, 3px, 3py, and 3pz) per sub-\nlattice. Here, we need to consider a 16 ×16 matrix for the\nHamiltonianH(k) by taking the spin degree of freedom\ninto account, in which the eight up-spin orbitals with6\n(b)\n(d)2.0\n-2.00\n2.0\n-2.00(meV)\n000(eV)\n(meV) (eV)-1 1246\n-246248\n-248-1 1\n0-2 21\n0\n-12.0\n-2.00 -1 1(eV)\n2.0\n-2.00\n0(eV)\n-2 20(a)\n(c)\n5 -5(/nm) (/nm)\n(/nm)(/nm)\n(/nm) (/nm)\nFIG. 3. Band structures for (a-b) (9 ,0) zigzag nanotube and (c-d) (6 ,6) armchair nanotube. k0/similarequal1.3225 nm−1andktis\na momentum along the axis direction. (a)(c) Band structures in the absence of curvature, which are obtained from that of\ngraphene on the cutting lines, reflecting the boundary condition in the circumferential direction. (b)(d) Band structures in the\npresence of curvature. A band gap opens up at the Dirac point in both the zigzag and armchair nanotubes. The gap for the\nzigzag nanotube ( ∼500 meV) is much larger than that for the armchair ( ∼0.4 meV). For the zigzag nanotube, spin splitting\nis observed in both the conduction band ( ∼0.2 meV) and the valence band ( ∼1.1 meV). On the other hand, for the armchair\nnanotube, spin splitting is negligibly small in both bands. These results are consistent with Ref. [8].\nwavenumber ( kc,kt,kn) couple to the eight downspin or-\nbitals with shifted wavenumber ( kc+δkc,kt,kn). Since\nscandsnchange the (quantized) momentum in the cir-\ncumference direction, kc, it is no longer a good quantum\nnumber. Instead, the total angular momentum in the di-\nrection of the y-axis,Jy=Rkc+sy/2, is conserved, due\nto the axial symmetry in the laboratory frame.\nThus, the effect of the curvature is taken into account\nthrough the modification of the hopping integral and the\nSOI. We can obtain the band structure by numerically\ndiagonalizing the Hamiltonian H(k) for fixedk.\nFor nanotube structures, we should also take into ac-\ncount the boundary condition for the circumference di-\nrection. In the absence of the SOI, kcis discretized in\norder to satisfy the boundary condition. Therefore, the\noriginal Brillouin zone for an unbent material is quan-\ntized into line segments [13], called cutting lines. A cut-\nting line is nothing but a quasi-one-dimensional subband,\nlabeled bykc. In the presence of the SOI, Jyshould be\nused instead of kcto specify a subband [8], as kcis no\nlonger a good quantum number.A. Carbon nanotubes\nTo discuss the effect of curvature on graphene, we\nwill focus on (9 ,0) (zigzag) and (6 ,6) (armchair) nan-\notubes. Fig. 3 (a) and (c) show the band structures for\nthe low-lying cutting lines in the absence of curvature for\nzigzag and armchair nanotubes, respectively. When the\ncurvature is non-zero, the band structures of the zigzag\nand armchair nanotubes change into those in Fig. 3 (b)\nand (d), respectively. Here, the radii of the (9 ,0) and\n(6,6) nanotubes are R/similarequal0.35 nm and R/similarequal0.41 nm.\nThe band gap between the conduction and valence bands\nis estimated to be 500 meV for the (9 ,0) nanotube and\n0.4 meV for the (6 ,6) nanotube. The spin splitting ener-\ngies for the conduction and valence bands are estimated\nto be 0.2 meV and 1 .1 meV for the (9 ,0) nanotube, while\nthey become much smaller for the (6 ,6) nanotube. The\nband gap opens exactly at the Kpoint for (9 ,0) nan-\notube, while its position is shifted from the Kpoint by\nktT/2π∼0.08 for (6,6) nanotube, where ktis the mo-\nmentum along the axis direction, T= 2√\n3πR/dRis the\nlength of the unit cell of the nanotube, and dRis the\ngreatest common divisor of 2 n+mand 2m+nfor the\n(n,m) nanotube [13].7\n(eV)\nFIG. 4. Energy dispersion at the valley of (0 ,k0,0) forϕ= 0\nas a function of angular momentum along the axis direction,\nJy.\nThese results semi-quantitatively agree with the previ-\nous theoretical work. In fact, the band gap is opened by\nthe curvature in the same way as in Ref. [8]. The spin\nsplitting of the (9 ,0) nanotube and the gap of the (6 ,6)\nnanotube, which are proportional to 1 /R, have similar\nvalues to those in Ref. [8]. However, the gap of the (9 ,0)\nnanotube, which is proportional to 1 /R2, is several times\nlarger than the value reported in Ref. [8]. This deviation\nis consistent with the fact that our calculation neglected\nthe contribution of 1 /R2; the present tight-binding cal-\nculation leads to a reasonable estimate only up to 1 /R.\nB. Silicon\nNext, we consider spin-orbit coupling in curved silicon.\nWe assume that the ydirection is unchanged, while the\nxdirection is maximally curved. This situation corre-\nsponds toϕ= 0 in Eq. (7) in the main text. Here, the\nspinsyand the momentum kcare not conserved, because\nthey do not commute with the effective Hamiltonian near\nthe valley. Instead, the total angular momentum along\nthe axis direction, Jy=Rkc+sy/2, becomes a good\nquantum number (see Eq. (32) and the subsequent ex-\nplanation), and therefore the cutting lines are labeled by\nJy. Figure 4 depicts the energy dispersion at the valley\nof (0,k0,0) as a continuous function of Jy. Note that Jy\nis actually quantized by the boundary condition; Jy= 0\n(Jy=±1) yields the lowest (next-lowest) cutting line,\nfor example. All the band structures in Fig. 3(b) in the\nmain text were obtained with this procedure.\nThe band structure can be obtained in the case of ϕ=\nπ/4 by changing the directions of xt,xc, andxnwith\ncarefully treating the direction of a crystal structure.IV. THIN-FILM QUANTIZATION\nThe thin-film quantization is a method to describe a\nparticle confined in a one- or two-dimensional curved\nspace embedded in three-dimensional space [14, 15]. In\nthis method, the spatial constraint on a particle can be\nrepresented by the geometric potentials depending on the\ncurvatures of the low-dimensional space. In this section,\nwe briefly explain that the thin-film quantization proce-\ndure does not lead to an appropriate curvature-induced\nSOI by considering the CNT as an illustrative example.\nLet us derive an effective Hamiltonian for the CNT by\nusing the method given in Refs. [14, 15]. For an electron\nconfined in a curved thin film such as the CNT, one starts\nwith a Dirac equation in 3+1 dimensional curved space-\ntime with the geometric potentials of a cylindrical surface\nwhich is considered to be the CNT. Following Ref. [15],\nwe use the cylindrical coordinates shown in Fig. 2. Note\nthatxn= 0 when an electron is on the nanotube. We\nstart with the Dirac equation:\ni/planckover2pi1∂\n∂tψ(x)/radicalbig\n1 +xn/R\n= [−i/planckover2pi1cαˆiej\nˆi∂j+βmc2+V(xn)]ψ(x)/radicalbig\n1 +xn/R,(33)\nwhereαˆiis the alpha matrix, V(xn) =vcx2\nnis the con-\nfinement potential to the nanotube with strength param-\netervc, andej\nˆiis the triad field, whose indices ˆiandj\nrespectively label the Cartesian coordinates ( x,y,z ) and\ncylindrical coordinates ( xc,xt,xn):\nec\nˆi=1/radicalbig\n1 +xn/R/parenleftBig\n−sinxc\nRδˆiz+ cosxc\nRδˆix/parenrightBig\n,(34)\net\nˆi=δˆiy, (35)\nen\nˆi= cosxc\nRδˆiz+ sinxc\nRδˆix, (36)\nwhereδˆiˆjis Kronecker’s delta. By taking the confinement\npotential to be infinitely large, i.e., vc→∞ , we obtain\nthe SOI modulated by the geometric potential in the non-\nrelativistic limit as follows:\nHtfq\nso=/planckover2pi1stpc\n2mR. (37)\nNext, we estimate the SOI modulation derived by the\nthin-film quantization on a CNT (see Fig. 5). The ma-\ntrix elements of the SOI modulation between the nearest-\nneighbor sites on the CNT are given by\n/angbracketleftRA|Htfq\nso|RA+dj/angbracketright=2/planckover2pi1vF\n3ist∆ktfq\nsocosφj (38)\nwherej= 1,2,3 are integers, vF/similarequal8.32×105m/s is the\nFermi velocity, and φj=−ϕ+2π\n3(j−1) is the angle be-\ntween the displacement vector djand thexc-axis.∆ktfq\nso\ncharacterizes the energy gap between the conduction and8\nFIG. 5. Crystalline structure of a CNT. a1anda2are primi-\ntive lattice vectors, and ∆j(j= 1,2,3) are vectors connecting\nthe nearest-neighbor sites. θis the angle between a1and the\nxc-axis.\nvalence bands around KandK/primepoints due to the SOI\nmodulation: it is estimated as\n∆ktfq\nso=−v\n2vFR/similarequal−0.82×1\n2R, (39)where we have defined v=/planckover2pi1κ/m, and 2κ/3 denotes the\nmatrix elements of the momentum operator, given by [16,\n17]\n2\n3κ=/angbracketleftRA|px|RA+dj/angbracketright/similarequal0.22 a.u. (40)\nOur estimate of the curvature-induced SOI by the thin-\nfilm quantization is three orders of magnitude larger than\nthe previous results, ∆kso/similarequal5.3×10−4/2R, and its sign is\nopposite to that of the SOI estimated by a first-principles\ncalculation [8]. This overestimation is caused by the con-\nstraint imposed by the infinitely large confinement poten-\ntialV(xn), which is larger than the energy of the electron\nmass. This situation is not suitable to the consideration\nof electronic states confined in a CNT.\n[1] M. I. Aroyo, J. M. Perez-Mato, C. Capillas, E. Kroumova,\nS. Ivantchev, G. Madariaga, A. Kirov, and H. Won-\ndratschek, Bilbao Crystallographic Server: I. Databases\nand crystallographic computing programs, Z. Kristallogr.\nCryst. Mater. 221, 15 (2006).\n[2] M. I. Aroyo, A. Kirov, C. Capillasm, J. M. Perez-Mato,\nand H. Wondratschek, Bilbao Crystallographic Server.\nII. Representations of crystallographic point groups and\nspace groups, Acta Cryst. A62, 115 (2006).\n[3] Y. Xu, L. Elcoro, Z.-D. Song, B. J. Wieder, M. G.\nVergniory, N. Regnault, Y. Chen, C. Felser, and\nB. A. Bernevig, High-throughput calculations of mag-\nnetic topological materials, Nature 586, 702 (2020).\n[4] L. Elcoro, B. J. Wieder, Z. Song, Y. Xu, B. Bradlyn, and\nB. A. Bernevig, Magnetic topological quantum chemistry,\nNat. Commun. 12, 5965 (2021).\n[5] C. L. Kane and E. J. Mele, Quantum spin Hall effect in\ngraphene, Phys. Rev. Lett. 95, 226801 (2005).\n[6] M. A. H. Vozmediano, M. I. Katsnelson, and F. Guinea,\nGauge fields in graphene, Phys. Rep. 496, 109 (2010).\n[7] W. Izumida, A. Vikstr¨ om, and R. Saito, Asymmetric ve-\nlocities of dirac particles and vernier spectrum in metallic\nsingle-wall carbon nanotubes, Phys. Rev. B 85, 165430\n(2012).\n[8] W. Izumida, K. Sato, and R. Saito, Spin–Orbit Inter-\naction in Single Wall Carbon Nanotubes: Symmetry\nAdapted Tight-Binding Calculation and Effective Model\nAnalysis, J. Phys. Soc. Jpn. 78, 074707 (2009).\n[9] Y. Wang, V. W. Brar, A. V. Shytov, Q. Wu, W. Regan,\nH.-Z. Tsai, A. Zettl, L. S. Levitov, and M. F. Crommie,\nMapping Dirac quasiparticles near a single Coulomb im-\npurity on graphene, Nature Physics 8, 653 (2012).\n[10] W. C. Dunlap and R. L. Watters, Direct measurement of\nthe dielectric constants of silicon and germanium, Phys.\nRev.92, 1396 (1953).[11] H. Min, J. E. Hill, N. A. Sinitsyn, B. R. Sahu, L. Klein-\nman, and A. H. MacDonald, Intrinsic and rashba spin-\norbit interactions in graphene sheets, Phys. Rev. B 74,\n165310 (2006).\n[12] D. A. Papaconstantopoulos, Handbook of the Band Struc-\nture of Elemental Solids (Springer New York, NY, 2015)\np. 344.\n[13] R. Saito, G. Dresselhaus, and M. S. Dresselhaus, Physical\nProperties of Carbon Nanotubes (Imperial College Press,\nLondon, 1998).\n[14] R. C. T. da Costa, Quantum mechanics of a constrained\nparticle, Phys. Rev. A 23, 1982 (1981).\n[15] S. Matsutani, Berry phase of dirac particle in thin rod,\nJ. Phys. Soc. Jpn 61, 3825 (1992).\n[16] A. Gr¨ uneis, R. Saito, G. G. Samsonidze, T. Kimura,\nM. A. Pimenta, A. Jorio, A. G. S. Filho, G. Dressel-\nhaus, and M. S. Dresselhaus, Inhomogeneous optical ab-\nsorption around the k point in graphite and carbon nan-\notubes, Phys. Rev. B 67, 165402 (2003).\n[17] A. Gr¨ uneis, Resonance Raman spectroscopy of single\nwall carbon nanotubes , Ph.D. thesis, Tohoku University\n(2004)."    },    {        "title": "1503.06835v2.Critical_Temperature_and_Tunneling_Spectroscopy_of_Superconductor_Ferromagnet_Hybrids_with_Intrinsic_Rashba_Dresselhaus_Spin_Orbit_Coupling.pdf",        "content": "Critical Temperature and Tunneling Spectroscopy of Superconductor/Ferromagnet\nHybrids with Intrinsic Rashba–Dresselhaus Spin-Orbit Coupling\nSol H. Jacobsen,1\u0003Jabir Ali Ouassou,1\u0003and Jacob Linder1\n1Department of Physics, Norwegian University of Science and Technology, N-7491 Trondheim, Norway\n\u0003These authors contributed equally to this work.\nWe investigate theoretically how the proximity effect in superconductor/ferromagnet hybrid structures with in-\ntrinsic spin-orbit coupling manifests in two measurable quantities, namely the density of states and critical tem-\nperature. To describe a general scenario, we allow for both Rashba and Dresselhaus type spin-orbit coupling.\nOur results are obtained via the quasiclassical theory of superconductivity, extended to include spin-orbit cou-\npling in the Usadel equation and Kupriyanov–Lukichev boundary conditions. Unlike previous works, we have\nderived a Riccati parametrization of the Usadel equation with spin-orbit coupling which allows us to address\nthe full proximity regime and not only the linearized weak proximity regime. First, we consider the density of\nstates in both SF bilayers and SFS trilayers, where the spectroscopic features in the latter case are sensitive to the\nphase difference between the two superconductors. We find that the presence of spin-orbit coupling leaves clear\nspectroscopic fingerprints in the density of states due to its role in creating spin-triplet Cooper pairs. Unlike\nSF and SFS structures without spin-orbit coupling, the density of states in the present case depends strongly\non the direction of magnetization. Moreover, we show that the spin-orbit coupling can stabilize spin-singlet\nsuperconductivity even in the presence of a strong exchange field h\u001dD. This leads to the possibility of a mag-\nnetically tunable minigap: changing the direction of the exchange field opens and closes the minigap. We also\ndetermine how the critical temperature Tcof an SF bilayer is affected by spin-orbit coupling and, interestingly,\ndemonstrate that one can achieve a spin-valve effect with a single ferromagnet. We find that Tcdisplays highly\nnon-monotonic behavior both as a function of the magnetization direction and the type and direction of the\nspin-orbit coupling, offering a new way to exert control over the superconductivity of proximity structures.\nI. INTRODUCTION\nMaterial interfaces in hybrid structures give rise to proxim-\nity effects, whereby the properties of one material can “leak”\ninto the adjacent material, creating a region with properties\nderived from both materials. In superconductor/ferromagnet\n(SF) hybrid structures1, the proximity effect causes supercon-\nducting correlations to penetrate into the ferromagnetic re-\ngion and vice versa. These correlations typically decay over\nshort distances, which in diffusive systems is of the orderp\nD=h, where Dis the diffusion coefficient of the ferromag-\nnet and his the strength of the exchange field. However, for\ncertain field configurations, the singlet correlations from the\nsuperconductor may be converted into so-called long-range\ntriplets (LRTs)2. These triplet components have spin projec-\ntion parallel to the exchange field, and decay over much longer\ndistances. This results in physical quantities like supercur-\nrents decaying over the length scale xN=p\nD=T, which is\nusually much larger than the ferromagnetic coherence length\nxF=p\nD=h, where Tis the temperature. This distance is\nindependent of h, and at low temperatures it becomes increas-\ningly large, which allows the condensate to penetrate deep into\nthe ferromagnet. The isolation and enhancement of this fea-\nture has attracted much attention in recent years as it gives rise\nto novel physics and possible low-temperature applications by\nmerging spintronics and superconductivity3.\nIt is by now well-known that the conversion from singlet\nto long-range triplet components of the superconducting state\ncan happen in the presence of magnetic inhomogeneities4,5,\ni.e.a spatially varying exchange field, and until recently such\ninhomogeneities were believed to be the primary source of\nthis conversion6–15, although other proposals using e.g.non-\nequilibrium distribution functions and intrinsic triplet super-conductors also exist16–19. However, it has recently been es-\ntablished that another possible source of LRT correlations is\nthe presence of a finite spin-orbit (SO) coupling, either in the\nsuperconducting region20or on the ferromagnetic side21,22. In\nfact, it can be shown that an SF structure where the magnetic\ninhomogeneity is due to a Bloch domain wall, as considered\nine.g. Refs. 23–25, is gauge equivalent to one where the\nferromagnet has a homogeneous exchange field and intrinsic\nSO coupling21. It is known that SO scattering can be caused\nby impurities26, but this type of scattering results in purely\nisotropic spin-relaxation, and so does not permit the desired\nsinglet-LRT conversion. To achieve such a conversion, one\nneeds a rotation of the spin pair into the direction of the ex-\nchange field27. This can be achieved by using materials with\nan intrinsic SO coupling, either due to the crystal structure in\nthe case of noncentrosymmetric materials28, or due to inter-\nfaces in thin-film hybrids29, where the latter also modifies the\nfundamental process of Andreev reflection30,31. The role of\nSO coupling with respect to the supercurrent in ballistic hy-\nbrid structures has also been studied recently32.\nIn this paper, we establish how the presence of spin-orbit\ncoupling in SF structures manifests in two important exper-\nimental observables: the density of states D(e)probed via\ntunneling spectroscopy (or conductance measurements), and\nthe critical temperature Tc. A common consequence for both\nof these quantities is that neither becomes independent of the\nmagnetization direction. This is in contrast to the case with-\nout SO coupling in conventional monodomain ferromagnets,\nwhere the results are invariant with respect to rotations of the\nmagnetic exchange field. This symmetry is now lifted due to\nSO coupling: depending on the magnetization direction, LRT\nCooper pairs are created in the system which leave clear fin-\ngerprints both spectroscopically and in terms of the Tcbehav-arXiv:1503.06835v2  [cond-mat.supr-con]  6 Jul 20152\nior. On the technical side, we will present in this work for the\nfirst time a Riccati parametrization of the Usadel equation and\nits corresponding boundary conditions that include SO cou-\npling. This is an important advance in terms of exploring the\nfull physics of triplet pairing due to SO coupling as it allows\nfor a solution of the quasiclassical equations without any as-\nsumption of a weak proximity effect. We will also demon-\nstrate that the SO coupling can actually protect the singlet su-\nperconducting correlations even in the presence of a strong\nexchange field, leading to the possibility of a minigap that is\nmagnetically tunable via the orientation of the exchange field.\nThe remainder of the article will be organised as follows:\nIn Section II, we introduce the relevant theory and notation,\nstarting from the quasiclassical Usadel equation, which de-\nscribes the diffusion of the superconducting condensate into\nthe ferromagnet. We also motivate our choice of intrinsic SO\ncoupling in this section, and propose a new notation for de-\nscribing Rashba–Dresselhaus couplings. The section goes on\nto discuss key analytic features of the equations in the limit\nof weak proximity, symmetries of the density of states at zero\nenergy, and analytical results needed to calculate the critical\ntemperature of hybrid systems. We then present detailed nu-\nmerical results in Section III: we analyze the density of states\nof an SF bilayer in III A [see Fig. 1(a)], with the case of pure\nRashba coupling considered in Section III B, and we study\nthe SFS Josephson junction in III C [see Fig. 1(b)]. We con-\nsider different orientations and strengths of the exchange field\nand SO coupling, and in the case of the Josephson junction,\nthe effect of altering the phase difference between the con-\ndensates. Then, in Section III D, we continue our treatment\nof the SF bilayer in the full proximity regime by including a\nself-consistent solution in the superconducting layer, and fo-\ncus on how the presence of SO coupling affects the critical\ntemperature of the system. We discover that the SO coupling\nallows for spin-valve functionality with a single ferromagnetic\nlayer, meaning that rotating the magnetic field by p=2 induces\na large change in Tc. Finally, we conclude in Section IV with a\nsummary of the main results, a discussion of some additional\nconsequences of the choices made in-text, as well as possibil-\nities for further work.\nII. THEORY\nA. Fundamental concepts\nThe diffusion of the superconducting condensate into the\nferromagnet can be described by the Usadel equation, which\nis a second-order partial differential equation for the Green’s\nfunction of the system33. Together with appropriate bound-\nary conditions, the Usadel equation establishes a system of\ncoupled differential equations that can be solved in one di-\nmension. We will consider the case of diffusive equilibrium,\nwhere the retarded component ˆ gRof the Green’s function is\nsufficient to describe the behaviour of the system34,35. We\nstart by examining the superconducting correlations in the fer-\nromagnet, and use the standard Bardeen–Cooper–Schrieffer\n(BCS) bulk solution for the superconductors. In particular,we will clarify the spectroscopic consequences of having SO\ncoupling in the ferromagnetic layer.\nIn the absence of SO coupling, the Usadel equation33in the\nferromagnet reads\nDFÑ(ˆgRшgR)+i\u0002\neˆr3+ˆM;ˆgR\u0003\n=0; (1)\nwhere the matrix ˆr3=diag(1;\u00001), andeis the quasiparticle\nenergy. The magnetization matrix ˆMin the above equation is\nˆM=\u0012\nh\u0001s 0\n0(h\u0001s)\u0003\u0013\n;\nwhere h= (hx;hy;hz)is the ferromagnetic exchange field, (\u0003)\ndenotes complex conjugation, s= (sx;sy;sz)is the Pauli\nvector, and skare the usual Pauli matrices. The corresponding\nKupriyanov–Lukichev boundary conditions are36\n2LjzjˆgR\njшgR\nj= [ˆgR\n1;ˆgR\n2]; (2)\nwhere the subscripts refer to the different regions of the hybrid\nstructure; in the case of an SF bilayer as depicted in Fig. 1(a),\nj=1 denotes the superconductor, and j=2 the ferromagnet,\nwhile Ñdenotes the derivative along the junction 1 !2. The\nrespective lengths of the materials are denoted Lj, and the in-\nterface parameters zj=RB=Rjdescribe the ratio of the barrier\nresistance RBto the bulk resistance Rjof each material.\nz = 0(a) Bilayer\n(b) Josephson junctionS\nF\nFS\nSSO\nSOz = ‒L S\nz = L F\nz = ‒L F/2\nz = L F/2\nFIG. 1: (Color online) (a) The SF bilayer in III A, III B and\nIII D. We take the thin-film layering direction along the z-\naxis, and assume an xy-plane Rashba–Dresselhaus coupling\nin the ferromagnetic layer. (b) The SFS trilayer in III C.\nWe will use the Riccati parameterisation37for the quasi-\nclassical Green’s function ˆ gR,\nˆgR=\u0012\nN(1+g˜g) 2Ng\n\u00002˜N˜g\u0000˜N(1+˜gg)\u0013\n; (3)3\nwhere the normalisation matrices are N= (1\u0000g˜g)\u00001and\n˜N= (1\u0000˜gg)\u00001. The tilde operation denotes a combination of\ncomplex conjugation i!\u0000 iand energy e!\u0000e, with g!˜g,\nN!˜N. The Riccati parameterisation is particularly useful for\nnumerical computation because the parameters are bounded\n[0;1], contrary to the multi-valued q-parameterisation34. In\npractice, this means that for certain parameter choices the nu-\nmerical routines will only converge in the Riccati formulation.\nAppendix A contains some further details on this parameteri-\nsation.\nTo include intrinsic SO coupling in the Usadel equation, we\nsimply have to replace all the derivatives in Eq. (1) with their\ngauge covariant counterparts:21,38\nÑ(\u0001)7!˜Ñ(\u0001)\u0011Ñ(\u0001)\u0000i[ˆA;\u0001]: (4)\nThis is valid for any SO coupling linear in momentum. We\nconsider the leading contribution; higher order terms, e.g.\nthose responsible for the SU(2) Lorentz force, are neglected\nhere. Such higher order terms are required to produce so-\ncalled j0junctions which have attracted interest of late39, and\nconsequently we will see no signature of the j0effect in the\nsystems considered herein. The object ˆAhas both a vector\nstructure in geometric space, and a 4 \u00024 matrix structure in\nSpin–Nambu space, and can be written as ˆA=diag(A;\u0000A\u0003)\nin terms of the SO field A= (Ax;Ay;Az), which will be dis-\ncussed in more detail in the next subsection. SO coupling in\nthe context of quasiclassical theory has also been discussed\nin Refs. 38,40. When we include the SO coupling as shown\nabove, we derive the following form for the Usadel equation\n(see Appendix A):\nDF\u0000\n¶2\nkg+2(¶kg)˜N˜g(¶kg)\u0001\n=\u00002ieg\u0000ih\u0001(sg\u0000gs\u0003)\n+DF\u0002\nAAg\u0000gA\u0003A\u0003+2(Ag+gA\u0003)˜N(A\u0003+˜gAg)\u0003\n+2iDF\u0002\n(¶kg)˜N(A\u0003\nk+˜gAkg)+(Ak+gA\u0003\nk˜g)N(¶kg)\u0003\n;(5)\nwhere the index kindicates an arbitrary choice of direction\nin Cartesian coordinates. The corresponding equation for ˜gis\nfound by taking the tilde conjugate of Eq. (5). Similarly, the\nboundary conditions in Eq. (2) become:\n¶kg1=1\nL1z1(1\u0000g1˜g2)N2(g2\u0000g1)+iAkg1+ig1A\u0003\nk;\n¶kg2=1\nL2z2(1\u0000g2˜g1)N1(g2\u0000g1)+iAkg2+ig2A\u0003\nk;(6)\nand the ˜gcounterparts are found in the same way as before.\nFor the details of these derivations, see Appendix A.\nWe will now discuss the definition of current in the presence\nof spin-orbit interactions. Since the Hamiltonian including SO\ncoupling contains terms linear in momentum (see below), the\nvelocity operator vj=¶H=¶kjis affected. We stated above\nthat the Kupriyanov-Lukichev boundary conditions are sim-\nply modified by replacing the derivative with its gauge covari-\nant counterpart including the SO interaction. To make sure\nthat current conservation is still satisfied, we must carefully\nexamine the Usadel equation. In the absence of SO coupling,the quasiclassical expression for electric current is given by\nIe=I0Z¥\n\u0000¥deTrfr3(ˇgÑˇg)Kg; (7)\nwhere ˇ gis the 8\u00028 Green’s function matrix in Keldysh space\nˇg=\u0012ˆgRˆgK\nˆ0 ˆgA\u0013\n; (8)\nandI0is a constant that is not important for this discussion.\nCurrent conservation can now be proven from the Usadel\nequation itself. We show this for the case of equilibrium,\nwhich is relevant for the case of supercurrents in Josephson\njunctions. In this case ˆ gK= (ˆgR\u0000ˆgA)tanh(e=2T)and we get\nIe=I0Z¥\n\u0000¥deTrfr3(ˆgRшgR\u0000ˆgAшgA)gtanh(e=2T):(9)\nPerforming the operation Tr fr3\u0001\u0001\u0001g on the Usadel equation,\nwe obtain\nDÑ\u0001Trfr3(ˆgRшgR)+iTrfr3[er3+ˆM;ˆgR]g=0: (10)\nNow, inserting the most general definition of the Green’s func-\ntion ˆgR, one finds that the second term in the above equation\nis always zero. Thus, we are left with\nÑ\u0001Trfr3(ˆgRшgR)g=0; (11)\nwhich expresses precisely current conservation since the same\nanalysis can be done for ˆ gA. Now, let us include the SO cou-\npling. The current should then be given by\nIe=I0Z¥\n\u0000¥deTrfr3(ˇg˜Ñˇg)Kg; (12)\nso that the expression for the charge current is modified by the\npresence of SO coupling, as is known. The question is now if\nthis current is conserved, as it has to be physically. We can\nprove that it is from the Usadel equation by rewriting it as\nDÑ\u0001(ˆgR˜ÑˆgR)\n=D[A;ˆgRшgR]+D[A;[A;ˆgR]]\u0000i[er3+ˆM;ˆgR];(13)\nand then performing the operation Tr fr3\u0001\u0001\u0001g, one finds:\nDÑ\u0001Trfr3(ˆgR˜ÑˆgR)g=0; (14)\nso we recover the standard current conservation law Ñ\u0001Ie=0.\nB. Spin-orbit field\nThe precise form of the generic SO field Ais imposed by\nthe experimental requirements and limitations. As the name\nsuggests, spin-orbit coupling couples a particle’s spin with its\nmotion, and more specifically its momentum. As mentioned\nin the Introduction, the SO coupling in solids can originate\nfrom a lack of inversion symmetry in the crystal structure.4\nSuch spin-orbit coupling can be of both Rashba and Dressel-\nhaus type and is determined by the point group symmetry of\nthe crystal41,42. It is also known that the lack of inversion sym-\nmetry due to surfaces, either in the form of interfaces to other\nmaterials or to vacuum, will give rise to antisymmetric spin-\norbit coupling of the Rashba type. For sufficiently thin struc-\ntures, the SO coupling generated in this way can permeate the\nentire structure, but the question of precisely how far into ad-\njacent materials such surface-SO coupling may penetrate ap-\npears to be an open question in general. Intrinsic inversion\nasymmetry arises naturally due to interfaces between materi-\nals in thin-film hybrid structures such as the ones considered\nherein. Noncentrosymmetric crystalline structures provide an\nalternative source for intrinsic asymmetry, and are considered\nin Ref. 43. In thin-film hybrids, the Rashba spin splitting de-\nrives from the cross product of the Pauli vector swith the\nmomentum k,\nHR=\u0000a\nm(s\u0002k)\u0001ˆz; (15)\nwhere ais called the Rashba coefficient, and we have chosen\na coordinate system with ˆ zas the layering direction. Another\nwell-known type of SO coupling is the Dresselhaus spin split-\nting, which can occur when the crystal structure lacks an in-\nversion centre. For a two-dimensional electron gas (quantum\nwell) confined in the ˆ z-direction, then to first order hkzi=0,\nso the Dresselhaus splitting becomes\nHD=b\nm(syky\u0000sxkx); (16)\nwhere bis called the Dresselhaus coefficient. In our struc-\nture, we consider a thin-film geometry with the confinement\nbeing strongest in the z-direction. Although there may cer-\ntainly be other terms contributing to the Dresselhaus SO cou-\npling in such a structure, since real thin-film structures will\nhave three-dimensional quasiparticle diffusion and we use a\n2Dform of the SO coupling here, we consider the standard\nform Eq. (16) as an approximation that captures the main\nphysics in the problem. This is a commonly used model in\nthe literature to explore the effects originating from SO cou-\npling in a system. When we combine both interactions, we\nobtain the Hamiltonian for a general Rashba–Dresselhaus SO\ncoupling,\nHRD=kx\nm(asy\u0000bsx)\u0000ky\nm(asx\u0000bsy): (17)\nIn this work, we will restrict ourselves to this form of SO cou-\npling. It should be noted that our setup may also be viewed as\na simplified model for a scenario where the SO coupling and\nferromagnetism exist in separate, thin layers, in which case\nwe expect qualitatively similar results to the ones reported in\nthis manuscript.\nAs explained in Ref. 21, the SO coupling acts as a back-\nground SU(2) field, i.e.an object with both a vector structure\nin geometric space, and a 2 \u00022 matrix structure in spin space.\nWe can therefore identify the interaction above with an effec-\ntive vector potential Awhich we will call the SO field ,\nHRD\u0011\u0000k\u0001A=m; (18)from which we derive that\nA= (bsx\u0000asy;asx\u0000bsy;0): (19)\nAt this point, it is convenient to introduce a new notation\nfor describing Rashba–Dresselhaus couplings, which will let\nus distinguish between the physical effects that derive from\nthe strength of the coupling, and those that derive from the\ngeometry. For this purpose, we employ polar notation defined\nby the relations\na\u0011\u0000asinc;\nb\u0011acosc; (20)\nwhere we will refer to aas the SO strength , andcas the SO\nangle . Rewritten in the polar notation, Eq. (19) takes the form:\nA=a(sxcosc+sysinc)ˆx\u0000a(sxsinc+sycosc)ˆy:(21)\nFrom the definition, we can immediately conclude that c=0\ncorresponds to a pure Dresselhaus coupling, while c=\u0006p=2\nresults in a pure Rashba coupling, with the geometric interpre-\ntation of cillustrated in Fig. 2. Note that A2\nx=A2\ny=a2, which\nmeans that A2=2a2. Another useful property is that we can\nswitch the components Ax$Ayby letting c!3p=2\u0000c.\nsxcosc+sysinc\nkxcky\nsxsinc+sycosc\nc\nFIG. 2: Geometric interpretation of the SO field (21) in polar\ncoordinates: the Hamiltonian couples the momentum com-\nponent kxto the spin component (sxcosc+sysinc)with a\ncoefficient +a=m, and the momentum component kyto the\nspin component ( sxsinc+sycosc) with a coefficient \u0000a=m.\nThus, adetermines the magnitude of the coupling, and cthe\nangle between the coupled momentum and spin components.\nThe appearance of LRTs in the system depends on the inter-\nplay between SO coupling and the direction of the exchange\nfield. Recall that the LRT components are defined as hav-\ning spin projections parallel to the exchange field, as opposed\nto the short-ranged triplet (SRT) component which appears\nas long as there is exchange splitting44but has spin projec-\ntion perpendicular to the field and is therefore subject to the\nsame pair-breaking effect as the singlets3,27, penetrating only\na very short distance into strong ferromagnets. Thus if we\nhave an SO field component along the layering direction, e.g.\nif we had Az6=0 in Figs. 1(a) and 1(b), achievable with a non-\ncentrosymmetric crystal or in a nanowire setup, then a non-\nvanishing commutator [A;h\u0001s]creates the LRT. However, we\nwill from now only consider systems where Az=0, in which5\ncase the criterion for LRT is21that[A;[A;h\u0001s]]must not be\nparallel to the exchange field h\u0001s. Expanding, we have\n[A;[A;h\u0001s]] =4a2(h\u0001s+hzsz)\n\u00004a2(hxsy+hysx)sin2c; (22)\nfrom which it is clear that no LRTs can be generated for a pure\nDresselhaus coupling c=0 or Rashba coupling c=\u0006p=2\nwhen the exchange field is in-plane. However, the effect of SO\ncoupling becomes increasingly significant for angles close to\n\u0006p=4 (see Fig. 4 in Section III A). We also see that no LRTs\ncan be generated for in-plane magnetization in the special case\nhx=hyandhz=0, since hxsy+hysxcan then be rewritten as\nhxsx+hysy, which is parallel to h. There is no LRT genera-\ntion for the case hx=hy=0 and hz6=0 for similar reasons. In\ngeneral however, the LRT will appear for an in-plane magne-\ntization as long as hx6=hyand the SO coupling is not of pure\nDresselhaus or pure Rashba type. It is also important to note\nthat the LRT can be created even for pure Rashba type SO\ncoupling if the magnetization has both in- and out-of-plane\nmagnetization components. We will discuss precisely this sit-\nuation in Sec. III B.\nOnce the condition for long-range triplet generation is sat-\nisfied, increasing the corresponding exchange field will also\nincrease the proportion of long-range triplets compared with\nshort-range triplets. Whether or not the presence of long-\nrange triplets can be observed in the system, i.e.if they retain a\nclear signature in measurable quantities such as the density of\nstates when the criteria for their existence is fulfilled, depends\non other aspects such as the strength of the spin-orbit coupling\nand will be discussed later in this paper. Thus, a main moti-\nvation for this work is to take a step further than discussing\ntheir existence21and instead make predictions for when long-\nranged triplet Cooper pairs can actually be observed via spec-\ntroscopic or T cmeasurements in SF structures with spin-orbit\ncoupling. However, we will also demonstrate that the pres-\nence of SO coupling offers additional opportunities besides\nthe creation of LRT Cooper pairs. We will show both ana-\nlytically and numerically that the SO coupling can protect the\nsinglet component even in the presence of an exchange field,\nwhich normally would suppress it. This provides the possi-\nbility of tuning the well-known minigap magnetically , both in\nbilayer and Josephson junctions, simply by altering the direc-\ntion of the magnetization.\nC. Weak proximity effect\nIn order to establish a better analytical understanding of the\nrole played by SO coupling in the system before presenting\nthe spectroscopy and Tcresults, we will now consider the limit\nof weak proximity effect, which means that jgi jj\u001c1,N\u00191\nin the ferromagnet. The anomalous Green’s function in gen-\neral is given by the upper-right block of Eq. (3), f=2Ng,\nwhich we see reduces to f=2gin this limit. It will also prove\nprudent to express the anomalous Green’s function using a\nsinglet/triplet decomposition, where the singlet component is\ndescribed by a scalar function fs, and the triplet componentsencapsulated in the so-called d-vector45,46,\nf= (fs+d\u0001s)isy: (23)\nCombining the above with the weak proximity identity f=2g,\nwe see that the components of gcan be rewritten as:\ng=1\n2 \nidy\u0000dxdz+fs\ndz\u0000fsidy+dx!\n: (24)\nUnder spin rotations, the singlet component fswill then\ntransform as a scalar, while the triplet component d=\n(dx;dy;dz)transforms as an ordinary vector. Another useful\nfeature of this notation is that it becomes almost trivial to dis-\ntinguish between short-range and long-range triplet compo-\nnents; the projection d=d\u0001ˆhalong the exchange field corre-\nsponds to the SRTs, while the perpendicular part d?=jd\u0002ˆhj\ndescribes the LRTs, where ˆhhere denotes the unit vector of\nthe exchange field. For a concrete example, if the exchange\nfield is oriented along the z-axis, then dzwill be the short-\nrange component, while both dxanddyare long-ranged com-\nponents. In the coming sections, we will demonstrate that the\nLRT component can be identified from its density of states\nsignature, as measurable by tunneling spectroscopy.\nIn the limit of weak proximity effect, we may linearize both\nthe Usadel equation and Kupriyanov–Lukichev boundary con-\nditions. Using the singlet/triplet decomposition in Eq. (24),\nand the Rashba–Dresselhaus coupling in Eq. (19), the lin-\nearized version of the Usadel equation can be written:\ni\n2DF¶2\nzfs=efs+h\u0001d; (25)\ni\n2DF¶2\nzd=ed+hfs+2iDFa2W(c)d; (26)\nwhere we for brevity have defined an SO interaction matrix\nW(c) =0\n@1\u0000sin2c 0\n\u0000sin2c 1 0\n0 0 21\nA: (27)\nWe have now condensed the Usadel equation down to two\ncoupled differential equations for fsandd, where the cou-\npling is proportional to the exchange field and the SO interac-\ntion term. The latter has been written as a product of a factor\n2iDFa2, depending on the strength a, and a factor W(c)d, de-\npending on the angle cin the polar notation. The matrix W(c)\nbecomes diagonal for a Dresselhaus coupling with c=0 or\na Rashba coupling with c=\u0006p=2, which implies that there\nis no triplet mixing for such systems. In contrast, the off-\ndiagonal terms are maximal for c=\u0006p=4, which suggests\nthat the triplet mixing is maximal when the Rashba and Dres-\nselhaus coefficients have the same magnitude. In addition to\nthe off-diagonal triplet mixing terms, we see that the diagonal\nterms of W(c)essentially result in imaginary energy contri-\nbutions 2 iDFa2. As we will see later, this can in some cases\nresult in a suppression of all the triplet components in the fer-\nromagnet.\nWe will now consider exchange fields in the xy-plane,\nh=hcosqˆx+hsinqˆy: (28)6\nSince the linearized Usadel equations show that the presence\nof a singlet component fsonly results in the generation of\ntriplet components along h, and the SO interaction term only\nmixes the triplet components in the xy-plane, the only nonzero\ntriplet components will in this case be dxanddy. The SRT\namplitude dand LRT amplitude d?can therefore be written:\nd=dxcosq+dysinq; (29)\nd?=\u0000dxsinq+dycosq: (30)\nBy projecting the linearized Usadel equation for dalong the\nunit vectors (cosq;sinq;0)and(\u0000sinq;cosq;0), respectively,\nthen we obtain coupled equations for the SRTs and LRTs:\ni\n2DF¶2\nzfs=efs+hd; (31)\ni\n2DF¶2\nzd=[e+2iDFa2(1\u0000sin2qsin2c)]d\n\u00002iDFa2cos2qsin2cd?+h fs; (32)\ni\n2DF¶2\nzd?=[e+2iDFa2(1+sin2qsin2c)]d?\n\u00002iDFa2cos2qsin2cd: (33)\nThese equations clearly show the interplay between the singlet\ncomponent fs, SRT component d, and LRT component d?.\nIf we start with only a singlet component fs, then the presence\nof an exchange field hresults in the generation of the SRT\ncomponent d. The presence of an SO field can then result in\nthe generation of the LRT component d?, where the mixing\nterm is proportional to a2cos2qsin2c. This implies that in the\nweak proximity limit, LRT mixing is absent for an exchange\nfield direction q=p=4, corresponding to hx=hy, while it is\nmaximized if q=f0;p=2;pgand at the same time c=\u0006p=4.\nIn other words, the requirement for maximal LRT mixing is\ntherefore that the exchange field is aligned along either the x-\naxis or y-axis, while the Rashba and Dresselhaus coefficients\nshould have the same magnitude. It is important to note here\nthat although the mixing between the triplet components is\nmaximal at q=f0;p=2;pg, this does not necessarily mean\nthat the signature of the triplets in physical quantities is most\nclearly seen for these angles, as we shall discuss in detail later.\nMoreover, these equations show another interesting conse-\nquence of having an SO field in the ferromagnet, which is\nunrelated to the LRT generation. Note that the effective quasi-\nparticle energies coupling to the SRTs and LRTs become\nE=e+2iDFa2(1\u0000sin2qsin2c); (34)\nE?=e+2iDFa2(1+sin2qsin2c): (35)\nWhen q=c=\u0006p=4, then the SRTs are entirely unaffected\nby the presence of SO coupling; the triplet mixing term van-\nishes for these parameters, and Eis also clearly independent\nofa. However, when q=\u0000c=\u0006p=4, the situation is dras-\ntically different. There is still no possibility for LRT genera-\ntion, however the SRT energy E=e+4iDFa2will now ob-\ntain an imaginary energy contribution which destabilizes the\nSRTs. In fact, numerical simulations show that this energy\nshift destroys the SRT components as aincreases. As we willsee in Section III D, this effect results in an increase in the\ncritical temperature of the bilayer. Thus, switching between\nq=\u0006p=4 in a system with c'\u0006p=4 may suggest a novel\nmethod for creating a triplet spin valve.\nWhen c=\u0006p=4 but q6=\u0006p=4, the triplet mixing term\nproportional to cos2 qsin2cwill no longer vanish, so we get\nLRT generation in the system. We can then see from the ef-\nfective triplet energies that as q!sgn(c)p=4, the imaginary\npart of Evanishes, while the imaginary part of E?increases.\nThis leads to a relative increase in the amount of SRTs com-\npared to the amount of LRTs in the system. In contrast, as\nq!\u0000 sgn(c)p=4, the imaginary part of E?vanishes, and the\nimaginary part of Eincreases. This means that we would\nexpect a larger LRT generation for these parameters, up until\nthe point where the triplet mixing term cos2 qsin2cbecomes\nso small that almost no LRTs are generated at all. The ratio\nof effective energies coupling to the triplet component at the\nFermi level e=0 can be written as\nE?(0)\nE(0)=1+sin2qsin2c\n1\u0000sin2qsin2c: (36)\nD. Density of states\nThe density of states D(e)containing all spin components\ncan be written in terms of the Riccati matrices as\nD(e) =Tr[N(1+g˜g)]=2; (37)\nwhich for the case of zero energy can be written concisely in\nterms of the singlet component fsand triplet components d,\nD(0) =1\u0000jfs(0)j2=2+jd(0)j2=2: (38)\nThe singlet and triplet components are therefore directly com-\npeting to lower and raise the density of states47. Furthermore,\nsince we are primarily interested in the proximity effect in the\nferromagnetic film, we will begin by using the known BCS\nbulk solution in the superconductor,\nˆgBCS=\u0012\ncosh(q) sinh(q)isyeif\nsinh(q)isye\u0000if\u0000cosh(q)\u0013\n; (39)\nwhere q=atanh(D=e), and fis the superconducting phase.\nUsing Eq. (24) and the definition of the tilde operation, and\ncomparing ˆ gRin Eq. (3) with its standard expression in a bulk\nsuperconductor Eq. (39), we can see that at zero energy the\nsinglet component fs(0)must be purely imaginary and the\nasymmetric triplet dz(0)must be purely real if the supercon-\nducting phase is f=0.\nBy inspection of Eq. (26), we can see that a transformation\nhx$hyalong with dx$dyleaves the equations invariant.\nThe density of states will therefore be unaffected by such per-\nmutations,\nD[h= (a;b;0)] = D[h= (b;a;0)]; (40)\nwhile in general\nD[h= (a;0;b)]6=D[h= (b;0;a)]: (41)7\nHowever, whenever one component of the planar field is ex-\nactly twice the value of the other component, one can confirm\nthat the linearized equations remain invariant under a rotation\nof the exchange field\nh= (a;2a;0)!h= (a;0;2a); (42)\nwith associated invariance in the density of states.\nE. Critical temperature\nWhen superconducting correlations leak from a supercon-\nductor and into a ferromagnet in a hybrid structure, there will\nalso be an inverse effect, where the ferromagnet effectively\ndrains the superconductor of its superconducting properties\ndue to tunneling of Cooper pairs. Physically, this effect is\nobservable in the form of a reduction in the superconducting\ngapD(z)near the interface at all temperatures. Furthermore,\nif the temperature of the hybrid structure is somewhat close\nto the bulk critical temperature Tcsof the superconductor, this\ninverse proximity effect can be strong enough to make the su-\nperconducting correlations vanish entirely throughout the sys-\ntem. Thus, proximity-coupled hybrid structures will in prac-\ntice always have a critical temperature Tcthat is lower than the\ncritical temperature Tcsof a bulk superconductor. Depending\non the exact parameters of the hybrid system, Tccan some-\ntimes be significantly smaller than Tcs, and in some cases it\nmay even vanish ( Tc!0).\nTo quantify this effect, it is no longer sufficient to solve the\nUsadel equation in the ferromagnet only. We will now also\nhave to solve the Usadel equation in the superconductor,\nDS¶2\nzg=\u00002ieg\u0000D(sy\u0000gsyg)\u00002(¶zg)˜N˜g(¶zg); (43)\nalong with a self-consistency equation for the gap D(z),\nD(z) =N0lD0cosh(1=N0l)Z\n0deReffs(z;e)gtanh\u0012p\n2ege=D0\nT=Tcs\u0013\n;(44)\nwhere N0is the density of states per spin at the Fermi level,\nandl>0 is the electron-electron coupling constant in the\nBCS theory of superconductivity. For a derivation of the gap\nequation, see Appendix B.\nTo study the effects of the SO coupling on the critical tem-\nperature of an SF structure, we therefore have to find a self-\nconsistent solution to Eq. (5) in the ferromagnet, Eq. (6) at\nthe interface, and Eqs. (43) and (44) in the superconductor.\nIn practice, this is done by successively solving one of the\nequations at a time numerically, and continuing the procedure\nuntil the system converges towards a self-consistent solution.\nTo obtain accurate results, we typically have to solve the Us-\nadel equation for 100–150 positions in each material, around\n500 energies in the range (0;2D0), and 100 more energies in\nthe range (2D0;wc), where the Debye cutoff wc\u001976D0for\nthe superconductors considered herein. This procedure will\nthen have to be repeated up to several hundred times before\nwe obtain a self-consistent solution for any given temperatureof the system. Furthermore, if we perform a conventional lin-\near search for the critical temperature Tc=Tcsin the range (0;1)\nwith a precision of 0.0001, it may require up to 10,000 such it-\nerations to complete, which may take several days depending\non the available hardware and efficiency of the implementa-\ntion. The speed of this procedure may, however, be signifi-\ncantly increased by performing a binary search instead. Using\nthis strategy, the critical temperature can be determined to a\nprecision of 1 =212+1\u00190:0001 after only 12 iterations, which\nis a significant improvement. The convergence can be fur-\nther accelerated by exploiting the fact that D(z)from iteration\nto iteration should decrease monotonically to zero if T>Tc;\nhowever, the details will not be further discussed in this paper.\nIII. RESULTS\nWe consider the proximity effect in an SF bilayer in III A,\nusing the BCS bulk solution for the superconductors. The\ncase of pure Rashba coupling is discussed in III B, and the\nSFS Josephson junction is treated in III C. We take the thin-\nfilm layering direction to be oriented in the z-direction and\nfix the spin-orbit coupling to Rashba–Dresselhaus type in the\nxy-plane as given by Eq. (19). We set LF=xS=0:5. The co-\nherence length for a diffusive bulk superconductor typically\nlies in the range 10 \u000030 nm. We solve the equations using\nMATLAB with the boundary value differential equation pack-\nagebvp6c and examine the density of states D(e)for en-\nergies normalised to the superconducting gap D. For brevity\nof notation, we include the normalization factor in the coeffi-\ncients aandbin these sections. This normalization is taken\nto be the length of the ferromagnetic region LF, so that for in-\nstance a=1 in the figure legends means aLF=1. Finally, in\nSection III D, we calculate the dependence of the critical tem-\nperature of an SF bilayer as a function of the different system\nparameters.\nA. SF Bilayer\nConsider the SF bilayer depicted in Fig. 1(a). In section\nII B we introduced the conditions for the LRT component to\nappear, and from Eq. (22) it is clear that no LRTs will be gen-\nerated if the exchange field is aligned with the layering direc-\ntion, i.e. hkˆz, since Eq. (22) will be parallel to the exchange\nfield. Conversely, the general condition for LRT generation\nwith in-plane magnetisation is both that hx6=hyand that the\nSO coupling is not of pure Rashba or pure Dresselhaus form.\nHowever, it became clear in Section II C that the triplet mix-\ning was maximal for equal Rashba and Dresselhaus coupling\nstrengths, and in fact the spectroscopic signature is quite sen-\nsitive to deviations from this.\nIn Ref. 50, the density of states for an SF bilayer was shown\nto display oscillatory behavior as a function of distance pene-\ntrated into the ferromagnet. The physical origin of this stems\nfrom the non-monotonic dependence of the superconducting\norder parameter inside the F layer, which oscillates and leads\nto an alternation of dominant singlet and dominant triplet cor-8\nrelations as a function of distance from the interface. When\nthe triplet ones dominate, the proximity-induced change in the\ndensity of states is inverted compared to SN structures, giving\nrise to an enhancement of the density of states at low-energies\nin this so-called p-phase where the proximity-induced super-\nconducting order parameter is negative.\nFor SF bilayers without SO coupling and a homogeneous\nexchange field, one expects to see a spectroscopic mini-\ngap whenever the Thouless energy is much greater than the\nstrength of the exchange field. The minigap in SF structures\ncloses when the resonant condition h\u0018Egis fulfilled, where\nEgis the minigap occuring without an exchange field, and a\nzero-energy peak emerges instead48. The minigap Egdepends\non both the Thouless energy and the resistance of the junc-\ntion. For stronger fields we will have an essentially feature-\nless density of states (see e.g.Ref. 49 and references therein).\nThis is indeed what we observe for a=b=0 in Fig. 5. With\npurely out-of-plane magnetisation hkˆz, the effect of SO cou-\npling is irrespective of type: Rashba, Dresselhaus or both will\nalways create a minigap. With in-plane magnetisation how-\never, the observation of a minigap above the SO-free resonant\ncondition h>Egindicates that dominant Rashba or domi-\nnant Dresselhaus coupling is present. The same is true for\nSFS trilayers, and thus to observe a signature of long-range\ntriplets the Rashba–Dresselhaus coefficients must be similar\nin magnitude, and in the following we shall primarily focus\non this regime. To clarify quantitatively how much the Rashba\nand Dresselhaus coefficients can deviate from each other be-\nfore destroying the low-energy enhancement of the density of\nstates, which is the signature of triplet Cooper pairs in this sys-\ntem, we have plotted in Fig. 3 the density of states at the Fermi\nlevel ( e=0) as a function of the spin-orbit angle cand the\nmagnetization direction q. For purely Rashba or Dresselhaus\ncoupling (c=f0;\u0006p=2g), the deviation from the normal-\nstate value is small. However, as soon as both components are\npresent a highly non-monotonic behavior is observed. This\nis particularly pronounced for c!\u0006p=4, although the con-\nversion from dominant triplets to dominant singlets as one ro-\ntates the field by changing qis seen to occur even away from\nc=\u0006p=4.\nWith either h=hˆx6=0, or equivalently h=hˆy6=0, LRTs\nare generated provided ab6=0, and in Fig. 6 we can see that\nthe addition of SO coupling introduces a peak in the density\nof states at zero energy, which saturates for a certain cou-\npling strength. This peak manifests as sharper around e=0\nthan the zero-energy peak associated with weak field strengths\nof the order of the gap ( i.e. as evident from a=b=0 in\nFig. 6), which occurs regardless of magnetisation direction or\ntexture48,49. By analysing the real components of the triplets,\nfor a gauge where the superconducting phase is zero, we can\nconfirm that this zero-energy peak is due to the LRT compo-\nnent, in this case dx, also depicted in Fig. 6, in agreement\nwith the predictions for textured magnetisation without SO\ncoupling49. However, it is also evident from Fig. 6 that in-\ncreasing the field strength rapidly suppresses the density of\nstates towards that of the normal metal, making the effect\nmore difficult to detect experimentally. The way to amelio-\nrate this situation is to remember that the introduction of SOcoupling means the direction of the exchange field is crucially\nimportant, as we see in Fig. 4, and this allows for a dramatic\nspectroscopic signature for fields without full alignment with\nthex- ory-axes.\nFIG. 3: Zero-energy density of states D(0)as a function of the\nspin-orbit angle cand magnetization angle q. We have used\na ferromagnet of length LF=xS=0:5 with an exchange field\nh=D=3 and a spin-orbit magnitude axS=2.\nFig. 4 shows how the density of states at zero energy varies\nwith the angle qbetween hxand hyat zero energy; with\nq=0 the field is aligned with hx, and with q=p=2 it is\naligned with hy. We see that the inclusion of SO coupling\nintroduces a nonmonotonic angular dependance in the den-\nsity of states, with increasingly sharp features as the SO cou-\npling strength increases, although the optimal angle at ap-\nproximately q=7p=32 and q=9p=32 varies minimally with\nincreasing SO coupling. Clearly the ability to extract max-\nimum LRT conversion from the inclusion of SO coupling is\nhighly sensitive to the rotation angle, with near step-function\nbehaviour delineating the regions of optimal peak in the den-\nsity of states and an energy gap for strong SO coupling. It is\nremarkable to see how D(0)vs.qformally bears a strong re-\nsemblance to the evolution of a fully gapped BCS64density of\nstates D(e)vs.eto a flat density of states as the SO coupling\ndecreases.\nThese results can again be explained physically by the lin-\nearized equations (31)–(33). Since the case a=bcorresponds\ntoc=\u0000p=4 in the notation developed in the preceding sec-\ntions, Eq. (36) implies that E?(0)>E(0)when q<0, while\nE?(0)<E(0)when q>0. In other words, for negative q,\nthe SO coupling suppresses the LRT components, and the ex-\nchange field suppresses the other components. Since the sin-\nglet and SRT components have opposite sign in Eq. (38), this\nrenders the density of states essentially featureless. However,\nfor positive q, both the SO coupling and the exchange field\nsuppress the SRT components, meaning that LRT generation\nis energetically favoured. Note that E?=E!¥asq!+p=4,\nwhich explains why the LRT generation is maximized in this\nregime. Since the triplet mixing term in Eq. (33) is propor-9\ntional to (cos2qsin2c), the LRT component vanishes when\nthe value of qgets too close to +p=4. Furthermore, since\nEhas a large imaginary energy contribution in this case, the\nSRTs are also suppressed at q= +p=4. Thus, despite LRTs\nbeing most energetically favored at this exact point, we end up\nwith a system dominated by singlets due to the SRT suppres-\nsion and lack of LRT production pathway. Nevertheless, one\nwould conventionally expect that exchange fields of a mag-\nnitude h\u001dDas depicted in Fig.4 would suppress any fea-\ntures in the density of states, while we observe an obvious\nminigap. Thus, the singlet correlations become much more\nresilient against the pair-breaking effect of the exchange field\nwhen spin-orbit coupling is present.\nTo identify the physical origin of this effect, we solve the\nlinearized equations (31)–(33) along with their corresponding\nboundary conditions for the specific case e=0,q=\u0000c=\np=4. We consider a bulk superconductor occupying the space\nx<0 while the ferromagnet length LFis so large that one in\npractice only needs to keep the decaying parts of the anoma-\nlous Green’s function. We then find the following expression\nfor the singlet component at the SF interface in the absence of\nSO coupling:\nf0\ns=sinh(arctanh (D=e))\n2zLFr\nDF\nh: (45)\nWith increasing h, the singlet correlations are suppressed in\nthe conventional manner. However, we now incorporate SO\ncoupling in the problem. For more transparent analytical re-\nsults, we focus on the case 2 (ax)2\u001dh=D. This condition can\nbe rewritten as 2 DFa2\u001dh. In this case, a similar calculation\ngives the singlet component at the SF interface in the presence\nof SO coupling:\nfs=f0\nsr\nDFa2\n2h: (46)\nClearly, the SO coupling enhances the singlet component in\nspite the presence of an exchange field sincep\nDFa2=h\u001d1.\nThis explains the presence of the conventional zero energy\ngap for large SO coupling even with a strong exchange field.\nA consequence of this observation is that SO coupling in fact\nprovides a route to a magnetically tunable minigap . Fig. 4\nshows that when both an exchange field and SO coupling is\npresent, the direction of the field determines when a minigap\nappears. This holds even for strong exchange fields h\u001dDas\nlong as the SO coupling is sufficiently large as well.\nWe recall that the LRT Cooper pairs, defined as the com-\nponents of dperpendicular to h, may be characterized by a\nquantity d?which is defined by the cross product of the two\nvectors: d?=jd\u0002ˆhj. We saw above that the spectroscopic\nsignature of LRT generation is strongly dependent on the an-\ngle of the field, and this angle is a tunable parameter for suffi-\nciently weak magnetic anisotropy. In Fig. 7 we see an example\nof the effect this rotation can have on the spectroscopic signa-\nture of LRT generation: when the exchange field is changed\nfrom h= (6D;3D;0)!(6D;5D;0),i.e.changing the direction\nof the field, we see that a strong zero-energy peak emerges\ndue to the presence of LRT in the system. This large peakemerges despite the stronger exchange field that would ordi-\nnarily reduce the density of states towards the normal state,\ni.e.as in Fig. 6 for h=Dˆy!3Dˆy. If one were to remove\nthe SO coupling, the low-energy density of states would thus\nhave no trace of any superconducting proximity effect, which\ndemonstrates the important role played by the SO interactions\nhere. Finally, for completeness we include an example of the\neffect of rotating the field to have a component along the junc-\ntion in Fig. 8. Comparing the case of h= (0;3D;6D)in Fig. 8\nwith h= (6D;3D;0)in Fig. 7, we see that the two cases are\nidentical, as predicted in the limit of weak proximity effect,\nand increasing the magnitude of the out-of-plane zcomponent\nof the field has no effect on the height of the zero-energy peak,\nwhich is instead governed by the in-plane ycomponent.\nπ/4 π/2 0D(0)\nθα = β = 0\nα = β = 0.5\nα = β = 1\nα = β = 5α = β = 2\nh=6Δ(cos(θ), sin(θ), 0)\n00.250.50.7511.251.51.75\n-π/4 -π/2\nFIG. 4: The dependence of the density of states of the SF\nbilayer at zero energy on the angle qbetween the xand\nycomponents of the magnetisation exchange field h=D=\n6(cos(q);sin(q);0)for increasing SO coupling. As the\nstrength of the SO coupling increases we see increasingly\nsharp variations in the density of states from an optimal peak\nat around q\u00197p=32 and q\u00199p=32 to a gap around q=p=4.10\n-1.5 -1 -0.5 0 0.5 1 1.500.511.522.5\n-1.5 -1 -0.5 0 0.5 1 1.500.511.522.5D(ε)\nε/Δ ε/Δh = (0, 0, 3Δ)\n-1.5 -1 -0.5 0 0.5 1 1.500.511.522.5\nε/Δh = (0, 3Δ, 0)\nα = β = 0 α =β = 0.5\nα = β = 1 α = β = 2α = β = 0 α = 0.1, β = 0.5 α = 0.1, β = 1 α = 0.1, β = 2\nFIG. 5: Density of states D(e)for the SF bilayer with energies normalised to the superconducting gap Dand SO coupling\nnormalised to the inverse ferromagnet length 1 =LF. The table shows the spectroscopic effect of increasing SO coupling\nwitha=bwhen the magnetisation h=3Dˆz,i.e.with the field perpendicular to the interface, and the effect of increasing\ndifference between the Rashba and Dresselhaus coefficients for both h=3Dˆzandh=3Dˆy. Although the conditions for\nLRT generation are fulfilled in the latter case, it is clear that no spectroscopic signature of this is present.\n0.511.522.53\n-1.5 -1 -0.5 0 0.5 1 1.50.60.811.2\nε/Δε/ΔD(ε)(0,Δ,0)\n(0,3Δ,0)|Re(d y)| \nα = β = 0 α = β = 0.1 α = β = 0.5 α = β = 2D(ε)h D(ε) |Re(d x)|\n00.511.5|Re(d y)| |Re(d y)| |Re(d x)| |Re(d x)|\n-1.5 -1 -0.5 0 0.5 1 1.50.050.20.350.5\nε/Δε/Δ ε/Δ\n00.40.81.21.4\n-1.5 -1 -0.5 0 0.5 1 1.500.10.20.30.4\nFIG. 6: Density of states D(e)for the SF bilayer with energies normalised to the superconducting gap Dand SO coupling\nnormalised to the inverse ferromagnet length 1 =LF. The table shows the spectroscopic effect of equal Rashba–Dresselhaus\ncoefficients when the magnetisation is oriented entirely in the y-direction, and also the correlation between the SO-induced\nzero-energy peak with the long-range triplet component jRe(dx)j\u0011Re(d?). It is clear that the predominant effect of the\nLRT component, which appears only when the SO coupling is included, is to increase the peak at zero energies. Increasing\nthe field strength rapidly suppresses the density of states towards that of the normal metal.11\n-1.5 -1 -0.5 0 0.5 1 1.50.60.811.2\n-1.5 -1 -0.5 0 0.5 1 1.50.511.522.5\n-1.5 -1 -0.5 0 0.5 1 1.500.511.52\nε/Δ ε/Δ ε/ΔD(ε)h = (6Δ, 3Δ, 0) h = (6Δ, 5Δ, 0) h = (6Δ, 5Δ, 0) Re(d ⊥) for\nα = β = 0α = β = 0.5 α = β = 2 α = β = 1 α = β = 5Re(d ⊥)\nFIG. 7: Density of states D(e)in the SF bilayer for energies normalised to the superconducting gap Dand SO coupling\nnormalised to the inverse ferromagnet length 1 =LF. The table shows the spectroscopic features of the SF bilayer with\nrotated exchange field in the xy-plane. Again we see a peak in the density of states at zero energy due to the LRT\ncomponent, i.e.the component of dperpendicular to h,d?. The height of this zero-energy peak is strongly dependent\non the angle of the field vector in the plane, as shown in Fig. 4. For near-optimal field orientations increasing the SO\ncoupling leads to a dramatic increase in the peak of the density of states at zero energy.\n-1.5 -1 -0.5 0 0.5 1 1.50.60.811.2\n-1.5 -1 -0.5 0 0.5 1 1.50.60.811.2\n-1.5 -1 -0.5 0 0.5 1 1.50.60.811.2\nε/Δ ε/Δ ε/ΔD(ε)h = (0, 3Δ, 3Δ) h= (0, 3Δ, 6Δ) h = (0, 6Δ, 3Δ)\nα = β = 0 α = β = 0.5 α = β = 2 α = β = 1 α = β = 0.1\nFIG. 8: Density of states D(e)in the SF bilayer for energies normalised to the superconducting gap Dand SO coupling\nnormalised to the inverse ferromagnet length 1 =LF. The table shows the spectroscopic features of the SF bilayer with a\nrotated exchange field in the xz\u0011yz-plane. Note that when the field component along the junction is twice the component\nin the y-direction, here h= (0;3D;6D), the density of states is equivalent to the case h= (6D;3D;0)illustrated in Fig. 7,\nas predicted in the limit of weak proximity effect.12\nB. SF bilayer with pure Rashba coupling\nThere exists another experimentally viable setup where the\nLRT can be created. In the case where pure Rashba SO cou-\npling is present, originating e.g. from interfacial asymmetry,\nthe condition for the existence of LRT is that the exchange\nfield has a component both in-plane and out-of-plane. Al-\nthough the LRT formally is non-zero, it is desirable to clarify\nif and how it can be detected through spectroscopic signatures.\nFrom an experimental point of view, it is known that PdNi\nand CuNi11can in general feature a canted magnetization\norientation relative to the film-plane due to the competition\nbetween shape anisotropy and magnetocrystalline anisotropy.\nThis is precisely the situation required in order to have an ex-\nchange field with both an in-plane ( xy-plane in our notation)\nand out-of-plane ( z-direction) component. In our model, the\nferromagnetism coexists with the Rashba SO coupling, which\nmay be taken as a simplified model of two separate layers\nwhere the SO coupling is induced e.g. by a very thin heavy\nmetal and PdNi or CuNi is deposited on top of it.\nTo determine how the low-energy density of states is influ-\nenced by the triplet pairing, we plot in Fig. 9(a) D(0)as a\nfunction of the misalignment angle jbetween the film-plane\nand its perpendicular axis [see inset of Fig. 9(b) for junc-\ntion geometry]. In order to correlate the spectroscopic features\nwith the LRT, we plot in Fig. 9(b) the LRT Green’s function\njd?j. It is clear that the LRT vanishes when j=0 orj=p=2.\nThis is consistent with the fact that for pure Rashba coupling,\npurely in-plane or out-of-plane direction of the exchange field\ngives d?=0 according to our previous analysis. However,\nforj2(0;p=2)the LRT exists. Its influence on D(0)is seen\nin Fig. 9(a): an enhancement of the zero-energy density of\nstates. For any particular set of junction parameters there is\nan optimal value of the SO coupling, and in approaching this\nvalue the density of states is correlated with Re fd?g. Beyond\nthis optimal value, they are anticorrelated, as evident from Fig.\n9 as the SO coupling increases, but the angular correlation re-\nmains. We note that the magnitude of the enhancement of\nthe density of states is substantially smaller than what we ob-\ntained with both Rashba and Dresselhaus coupling. At the\nsame time, the magnitude of the enhancement is of precisely\nthe same order as previous experimental works that have mea-\nsured the density of states in S/F structures50,51.\nNote that it is only the angle between the plane and the tun-\nneling direction which is of importance: the density of states\nis invariant under a rotation in the film-plane of the exchange\nfield. The SO-induced enhancement of the zero-energy den-\nsity of states reaches an optimal peak before further increases\nin the magnitude of the Rashba coupling results in a suppres-\nsion of both the short- and long-ranged triplet components,\ncausing the low-energy density of states enhancement to van-\nish. The correlation with the LRT component jd?jcorre-\nspondingly changes to anticorrelation, evident in Fig. 9. Nev-\nertheless, the strong angular variation with D(0)remains al-\nthough D(0)<1 for all j[see inset of Fig. 9(a)]. Increasing\nthe exchange field hfurther suppressed the proximity effect\noverall.\nThe main effect of the SO coupling is that D(0)depends onthe exchange field direction. As seen for the case of a=0\nin Fig. 9(a), there is no directional dependence without SO\ncoupling. Thus, depending on the exchange field angle be-\ntween the in-plane and out-of plane direction, measuring an\nenhanced D(0)at low-energies is a signature of the presence\nof LRT Cooper pairs in the ferromagnet. More generally, mea-\nsuring a dependence on the exchange field direction jwould\nbe a direct consequence of the presence of SO coupling in the\nsystem, even in the regime of e.g.moderate to strong Rashba\ncoupling where the triplets are suppressed.\n(a) (b)\nFIG. 9: (Color online) (a) Plot of the zero-energy density\nof states D(0)in an S/F structure with pure Rashba spin-\norbit coupling. We have set h=D=4 and L=xS=0:5. Inset:\nstronger SO coupling a=1:5, demonstrating that the angu-\nlar variation of D(0)remains, although the enhancement due\nto triplets is absent. (b) Plot of the magnitude of the LRT\nanomalous Green’s function jd?jate=0. As seen, its en-\nhancement correlates with an accompanying increase in the\ndensity of states for the same angle j, and beyond an optimal\nSO coupling value there is anticorrelation between the density\nof states peak and jd?j. The only angle of importance is the\nangle jbetween the out-of-plane and in-plane component of\nthe exchange field, shown in the inset.\nC. Josephson junction\nBy adding a superconducting region to the right interface\nof the SF bilayer we form an SFS Josephson junction. It is\nwell known that the phase difference between the supercon-\nducting regions governs how much current can flow through\nthe junction52, and the density of states for a diffusive SNS\njunction has been measured experimentally with extremely\nhigh precision53. Here we consider such a transversal junc-\ntion structure as depicted in Fig.1(b), again with intrinsic SO\ncoupling in the xy-plane (Eq. 19) in the ferromagnet and with\nBCS bulk values for each superconductor. In III C 1 we con-\nsider single orientations along the principal axes of the system\n(x;y;z)of the uniform exchange field and in III C 2 we con-\nsider a rotated field. Experimentally, the density of states can\nbe probed at the superconductor/ferromagnet interface if one13\nof the superconductors is a superconducting island, and the\nscanning tunneling microscope approaches from the top, next\nto this superconductor island.\nLet us first recapitulate some known results. We saw in Sec-\ntion II that the spin-singlet, SRT and LRT components com-\npete to raise and lower the density of states at low energies.\nTheir relative magnitude is affected by the magnitude and di-\nrection of both the exchange field and SO coupling and results\nin three distinctive qualitative profiles: the zero-energy peak\nfrom the LRTs, the singlet-dominated regime with a minigap,\nand the flat, featureless profile in the absence of superconduct-\ning correlations. In the Josephson junction, the spectroscopic\nfeatures are in addition sensitive to the phase difference fbe-\ntween the superconductors. In junctions with an interstitial\nnormal metal, the gap decreases as f=0!p, closing entirely\natf=psuch that the density of states is that of the isolated\nnormal metal; identically one53,54. Without an exchange field\nthe density of states is unaffected by the SO coupling. This\nis because without an exchange field the equations governing\nthe singlet and triplet components are decoupled and thus no\nsinglet-triplet conversion can occur. From a symmetry point\nof view, it is reasonable that the time-reversal invariant spin-\norbit coupling does not alter the singlet correlations.\nWithout SO coupling and as long as the exchange field is\nnot too large, changing the phase difference can qualitatively\nalter the density of states from minigap to peak at zero en-\nergy (see Fig. 10), a useful feature permitting external con-\ntrol of the quasiparticle current flowing through the junction.\nThe underlying reason is that the phase difference controls the\nrelative ratio of the singlet and triplet correlations: when the\nsinglets dominate, a minigap is induced which mirrors their\norigin in the bulk superconductor.\nAs in the bilayer case, there is a resonant\ncondition48,49indicating an exchange field strength be-\nyond which the minigap can no longer be sustained and\nincreasing the phase difference simply lowers the density of\nstates towards that of the normal metal. Amongst the features\nwe outline in the following subsections, one of the effects of\nadding SO coupling is to make this useful gap-to-peak effect\naccessible with stronger exchange fields, i.e. for a greater\nrange of materials. At the same time, the SO coupling cannot\nbetoo strong since the triplet correlations are suppressed\nin this regime leaving only the minigap and destroying the\ncapability for qualitative change in the spectroscopic features.\n1. Josephson junction with uniform exchange field in single\ndirection\nConsider first the case in which the exchange field is\naligned in a single direction, meaning that we only consider an\nexchange field purely along the principal fx;y;zgaxes of thesystem. If we again restrict the form of the SO-vector to (19),\naligning hin the z-direction will not result in any LRTs. In this\ncase the spectroscopic effect of the SO coupling is dictated by\nthe singlet and short-range triplet features, much as in the SF\nbilayer case (Fig. 5). This is demonstrated in Fig. 10, where\nagain we see a qualitative change in the density of states as\nthe exchange field increases, with the regions of minigap and\nzero-energy-peak separated by the resonant condition h\u0018Eg\nwithout SO coupling.\nWe will now examine the effect of increasing the exchange\nfield aligned in the x- or, equivalently, the y-direction. In this\ncase, we have generation of LRT Cooper pairs. If his suffi-\nciently weak to sustain a gap independently of SO coupling,\nintroducing weak SO coupling will increase the gap at zero\nphase difference while maintaining a peak at zero energy for\na phase difference of 0 :75p(see Fig. 10). Increasing the SO\ncoupling increases this peak at zero energy up to a saturation\npoint. As the exchange field increases sufficiently beyond the\nresonant condition to keep the gap closed, increasing the SO\ncoupling increases the zero-energy peak at all phases, again\ndue to the LRT component, eventually also reaching a satu-\nration point. As the phase difference f=0!p, the density\nof states reduces towards that of the normal metal, closing en-\ntirely at f=pas expected43,54,55. As the value of the density\nof states at zero energy saturates for increasing SO coupling,\nfixed phase differences yield the same drop at zero energy re-\ngardless of the strength of SO coupling.\nWe note in passing that when the SO coupling field has a\ncomponent along the junction direction (z), it can qualitatively\ninfluence the nature of the superconducting proximity effect.\nAs very recently shown in Ref. 43, a giant triplet proximity\neffect develops at f=pin this case, in complete contrast to the\nnormal scenario of a vanishing proximity effect in p-biased\njunctions.\n2. Josephson junction with rotated exchange field\nWith two components of the field h,e.g.from rotation, it is\nagain useful to separate the cases with and without a compo-\nnent along the junction direction. When the exchange field lies\nin-plane (the xy-plane), and provided we satisfy the conditions\nhx6=hyandab6=0, increasing the SO coupling drastically in-\ncreases the zero energy peak as shown in Fig. 11, again due to\nthe LRT component. This is consistent with the bilayer behav-\nior, where the maximal generation of LRT Cooper pairs occurs\nat an angle 0 <q<p=4. As the phase difference approaches\np, the proximity-induced features are suppressed in the centre\nof the junction. This can be understood intuitively as a con-\nsequence of the order parameter averaging to zero since it is\npositive in one superconductor and negative in the other.14\n01234\n01234\n01234\n01234\n01234\n01234\n01234\n-1.5 -1 -0.5 0 0.5 1 1.501234\n01234\n-1.5 -1 -0.5 0 0.5 1 1.501234\n-1.5 -1 -0.5 0 0.5 1 1.500.511.522.2\n00.511.52\n00.511.52\n00.511.52\n00.511.52\n-1.5 -1 -0.5 0 0.5 1 1.500.511.52\n00.511.52\n00.511.52\n00.511.52\n00.511.52\nε/Δ ε/Δ ε/Δ ε/ΔD(ε) D(ε) D(ε) D(ε) D(ε)h = (0, 0, Δ) h = (0, Δ, 0) h = (0, 0, 3Δ) h = (0, 3Δ, 0)  α, β\n0\n0.1\n0.5\n1\n2\nφ = 0 φ =0.25π φ = 0.5π φ = 0.75π φ = π\nFIG. 10: The table shows the density of states D(e)in the SFS junction with increasing SO coupling and exchange field\nin a single direction, with D(e)normalised to the superconducting gap Dand SO coupling normalised to the inverse\nferromagnet length 1 =LF. With no SO coupling and very weak exchange field we see a phase-dictated gap-to-peak\nqualitative change in the density of states at zero energy. When the field is strong enough to destroy this gap, i.e.above\nthe resonant condition, increasing the phase difference simply lowers the density of states towards that of the normal\nmetal, which is achieved at a phase difference of f=p. With the addition of SO coupling we see a clear difference in\nthe density of states due to the long range triplet component, which is present when the field is oriented in ybut not in z.\nWhen LRTs are present with weak exchange fields, a phase-dictated gap-to-peak feature is retained and increased as the\nstrength of SO coupling increases the gap, with the peak shown here at a phase difference of 0 :75p. For stronger exchange\nfields, increasing the SO coupling produces the minigap when there is no LRT component, whereas the existence of an\nLRT component again introduces an increasing peak at zero energy when no minigap is present.15\n0.50.7511.251.5\n0.50.7511.251.5\n0.7511.251.51.75\n0.511.52\n-1.5 -1 -0.5 0 0.5 1 1.500.511.522.5\n-1.5 -1 -0.5 0 0.5 1 1.500.511.522.5\n0.511.52\n0.7511.251.51.75\n0.50.7511.251.5\n0.50.7511.251.5\n-1.5 -1 -0.5 0 0.5 1 1.50.50.7511.251.5\n0.50.7511.251.5\n0.50.7511.251.5\n0.50.7511.251.5\n0.50.7511.251.5h = (1.5Δ, 3.5Δ, 0) h = (0, 1.5Δ, 3.5Δ) h = (0, 3.5Δ, 5.5Δ)  α, β\n0\n0.1\n0.5\n1\n2\nD(ε) D(ε) D(ε) D(ε) D(ε)\nε/Δ ε/Δ ε/Δ\nφ = 0 φ =0.25π φ = 0.5π φ = 0.75π φ = π\nFIG. 11: Density of states D(e)in the SFS junction for energies normalised to the superconducting gap Dand SO coupling\nnormalised to the inverse ferromagnet length 1 =LF. The table shows the spectroscopic effects of increasing SO coupling\nin SFS with rotated exchange field. In the absence of SO coupling, the density of states is flat and featureless at low\nenergies. Increasing the SO coupling again leads to a strong increase in the peak of the density of states at zero energy,\nwhile increasing the phase difference reduces the peak and shifts the density of states weight toward the gap edge for\nhigher SO coupling strengths. With a component of the field in the junction direction a qualitative change in the density\nof states from strongly suppressed to enhanced at zero energy can be achieved by altering the phase difference between\nthe superconductors. This change can occur in the presence of stronger exchange fields when SO coupling is included.\nIncreasing the exchange field destroys the ability to maintain a gap in the density of states and the LRT component of the\nSO coupling increases the zero-energy peak as it did in the bilayer case.16\nThe 2D plots in this paper of the local density of states are\ngiven for the centre of the junction ( z=0), where one natu-\nrally expects the relative proportion of LRTs to be greatest.\nHowever, it is interesting to note that the large peak at zero\nenergy – the signature of the LRTs – is maintained through-\nout the ferromagnet. This is shown in Fig. 12, for the case\na=b=1 and h= (1:5D;3:5D;0), where the maximal peak\nforf=0 is almost twice the normal-state value. In compar-\nison, the depletion of this peak is surprisingly small at the\nsuperconductor interfaces.\nFIG. 12: Spatial distribution of the density of states D(e)\nthroughout the ferromagnet of an SFS junction with phase dif-\nference f=0, spin-orbit coupling a=b=1 and magnetisa-\ntionh= (1:5D;3:5D;0).\nWith one component of the exchange field along the junc-\ntion and another along either xory, a phase-dictated gap-to-\npeak transition at zero energy is possible with stronger fields\nthan with the field aligned in a single direction, as shown in\nFig. 11. Notice that in this case increasing the phase differ-\nencef=0!0:5pgives an increase in the peak at zero energy\nbefore reducing towards the normal metal state. For higher\nfield strengths we find once again that increasing the SO cou-\npling increases the peak at zero energy, up to a system-specific\nthreshold, and increasing phase difference reduces the density\nof states towards that of the normal metal.\nIt is also useful to consider how the zero-energy density\nof states depends simultaneously on the phase-difference and\nmagnetization orientation. To this end, we show in Fig. 13 a\ncontour plot of the density of states at the Fermi level (e=0)\nas a function of the superconducting phase difference facross\nthe junction and the magnetization direction q. The proxim-\nity effect vanishes in the centre of the junction at f=pfor\nany value of the exchange field orientation, giving the normal-\nstate value. Just as in the bilayer case (Fig. 3), we see that the\nproximity effect is strongly suppressed for the range of an-\nglesq>0. When rotating the field in the opposite direction,\nq<0, strongly non-monotonic behavior emerges. For zero\nphase-difference, the physics is qualitatively similar to the bi-\nlayer situation. In this case, we proved analytically that the\nLRT is not produced at all when q=\u0000p=4. Accordingly, Fig.\n13 shows a full minigap there.Whether or not a clear zero-energy peak can be seen due to\nthe LRT depends on the relative strength of the Rashba and\nDresselhaus coupling. In the top panel, we have dominant\nDresselhaus coupling in which case the low-energy density\nof states show either normal-state behavior or a minigap. In-\nterestingly, we see that the same opportunity appears in the\npresent case of a Josephson setup as in the bilayer case: a\nmagnetically tunable minigap appears. This effect exists as\nlong as the phase difference is not too close to p, in which\ncase the minigap closes. In the bottom panel correspond-\ning to equal magnitude of Rashba and Dresselhaus, however,\na strong zero-energy enhancement due to long-range triplets\nemerges as one moves away from q=\u0000p=4. With increasing\nphase difference, the singlets are seen to be more strongly sup-\npressed than the triplet correlations since the minigap region\n(dark blue) vanishes shortly after f=p'0:6 while the peaks\ndue to triplets remain for larger phase differences.\nFIG. 13: Zero-energy density of states D(0)as a function of\nthe phase-difference fand magnetization angle q, both tun-\nable parameters experimentally. The other parameters used\nareLF=xS=0:5,h=D0=3,axS=2. In the top panel, we\nhave dominant Dresselhaus strength ( c=0:15p) while in the\nbottom panel we have equal magnitude of Rashba and Dres-\nselhaus ( c=p=4).17\nD. Critical temperature\nIn this section, we present numerical results for the criti-\ncal temperature Tcof an SF bilayer. The theory behind these\ninvestigations is summarized in Section II E, and discussed\nin more detail in Appendix B. An overview of the physical\nsystem is given in Fig. 1(a). In all of the simulations we per-\nformed, we used the material parameter N0l=0:2 for the su-\nperconductor, the exchange field h=10D0for the ferromag-\nnet, and the interface parameter z=3 for both materials. The\nother physical parameters are expressed in a dimensionless\nform, with lengths measured relative to the superconducting\ncorrelation length xS, energies measured relative to the bulk\nzero-temperature gap D0, and temperatures measured relative\nto the bulk critical temperature Tcs. This includes the SO cou-\npling strength a, which is expressed in the dimensionless form\naxS. The plots presented in this subsection were generated\nfrom 12–36 data points per curve, where each data point has\na numerical precision of 0.0001 in Tc=Tcs. The results were\nsmoothed with a LOESS algorithm.\nBefore we present the results with SO coupling, we will\nbriefly investigate the effects of the ferromagnet length LF\nand superconductor length LSon the critical temperature, in\norder to identify the interesting parameter regimes. The criti-\ncal temperature as a function of the size of the superconductor\nis shown in Fig. 14.\nLF/ξS= LF/ξS= 0.25 LF/ξS= 0.50 = 1.00\n0.000.250.500.751.00\n0.5 0.6 0.7 0.8 0.9 1.0\nLSξSTcTcs\nFIG. 14: Plot of the critical temperature Tc=Tcsas a function\nof the length LS=xSof the superconductor for axS=0. Be-\nlow a critical length LS, superconductivity can no longer be\nsustained and Tcbecomes zero. For larger thicknesses of the\nsuperconducting layer, Tcreverts back to its bulk value.\nFirst of all, we see that the critical temperature drops to zero\nwhen LS=xS\u00190:5. This observation is hardly surprising;\nsince the superconducting correlation length is xS, the criti-\ncal temperature is rapidly suppressed once the length of the\njunction goes below xS. After this, the critical temperature in-\ncreases quickly, already reaching nearly 50% of the bulk value\nwhen LS=xS=0:6, demonstrating that the superconductivity\nof the system is clearly very sensitive to small changes in pa-\nrameters for this region.The next step is then to observe how the behaviour of the\nsystem varies with the size of the ferromagnet, and these re-\nsults are presented in Fig. 15.\nLS/ξS = 0.575 LS/ξS = 0.525 LS/ξS = 0.550\n0.000.250.500.751.00\n0.00 0.25 0.50 0.75 1.00\nLFξSTcTcs\nFIG. 15: Plot of the critical temperature Tc=Tcsas a function\nof the ferromagnet length LF=xSforaxS=0. Increasing the\nthickness of the ferromagnet gradually suppresses the Tcof the\nsuperconductor, causing a stronger inverse proximity effect.\nWe again observe that the critical temperature increases with\nthe size of the superconductor, and decreases with the size of\nthe ferromagnet. The critical temperature for a superconduc-\ntor with LS=xS=0:525 drops to zero at LF=xS\u00190:6, and stays\nthat way as the size of the ferromagnet increases. Thus we\ndo not observe any strongly nonmonotonic behaviour, such\nas reentrant superconductivity, for our choice of parameters.\nThis is consistent with the results of Fominov et al. , who only\nreported such behaviour for systems where either the interface\nparameter or the exchange field is drastically smaller than for\nthe bilayers considered herein56.\nWe now turn to the effects of the antisymmetric SO cou-\npling on the critical temperature, which has not been studied\nbefore. Figs. 16 and 17 show plots of the critical temperature\nas a function of the SO angle cfor an exchange field in the z-\ndirection. The critical temperature is here independent of the\nSO angle c. This result is reasonable, since the SO coupling is\nin the xy-plane, which is perpendicular to the exchange field\nfor this geometry. We also observe a noticeable increase in\ncritical temperature for larger values of a. This behaviour can\nbe explained using the linearized Usadel equation. Accord-\ning to Eq. (26), the effective energy Ezcoupling to the triplet\ncomponent in the z-direction becomes\nEz=e+4iDFa2; (47)\nso in other words, the SRTs obtain an imaginary energy shift\nproportional to a2. However, as shown in Eq. (25), there is\nno corresponding shift in the energy of the singlet component.\nThis effect reduces the triplet components relative to the sin-\nglet component in the ferromagnet, and as the triplet prox-\nimity channel is suppressed the critical temperature becomes\nrestored to higher values.18\naξS= 0 aξS= 2 aξS= 6 \n  0.8250.8500.8750.9000.925\n-0.50 -0.25 0.00 0.25 0.50\nχπTcTcs\nFIG. 16: Plot of the critical temperature Tc=Tcsas a function of\nthe SO angle c, when LS=xS=1:00,LF=xS=0:2, and hkˆz.\nIncreasing the SO coupling causes Tcto move closer to its\nbulk value, since the triplet proximity effect channel becomes\nsuppressed.\naξS= 0 aξS= 2 aξS= 6 \n                                  0.30.40.50.60.70.8\n-0.50 -0.25 0.00 0.25 0.50\nχπTcTcs\nFIG. 17: Plot of the critical temperature Tc=Tcsas a function\nof the SO angle c, when LS=xS=0:55,LF=xS=0:2, and hkˆz.\nThe same situation for an exchange field along the x-axis\nis shown in Figs. 18 and 19. For this geometry, we observe a\nsomewhat smaller critical temperature for all a>0 and all c\ncompared to Figs. 16 and 17. This can again be explained by\nconsidering the linearized Usadel equation in the ferromagnet,\nwhich suggests that the effective energy Excoupling to the x-\ncomponent of the triplet vector should be\nEx=e+2iDFa2; (48)\nwhich has a smaller imaginary part than the corresponding\nequation for Ez. Furthermore, note the drop in critical temper-\nature as c!\u0006p=4. Since the linearized equations contain a\ntriplet mixing term proportional to sin2 c, which is maximal\nprecisely when c=\u0006p=4, these are also the geometries for\nwhich we expect a maximal LRT generation. Thus, this de-\ncrease in critical temperature near c=\u0006p=4 can be explained\nby a net conversion of singlet components to LRTs in the sys-\ntem, which has an adverse effect on the singlet amplitude in\nthe superconductor, and therefore the critical temperature.\naξS= 0 aξS= 2 aξS= 6 \n                                   0.8250.8500.8750.9000.925\n-0.50 -0.25 0.00 0.25 0.50\nχπTcTcsFIG. 18: Plot of the critical temperature Tc=Tcsas a function\nof the SO angle c, when LS=xS=1:00,LF=xS=0:2, and hkˆx.\nThe critical temperature depends on the relative weight of the\nRashba and Dresselhaus coefficients.\naξS= 0 aξS= 2 aξS= 6 \n                                0.30.40.50.60.70.8\n-0.50 -0.25 0.00 0.25 0.50\nχπTcTcs\nFIG. 19: Plot of the critical temperature Tc=Tcsas a function\nof the SO angle c, when LS=xS=0:55,LF=xS=0:2, and hkˆx.\nIn Figs. 20 and 21 we present the results for a varying\nexchange field h\u0018cosqˆx+sinqˆyin the xy-plane. In this\ncase, we observe particularly interesting behaviour: the crit-\nical temperature has extrema at jcj=jqj=p=4, where the\nextremum is a maximum if qandchave the same sign, and\na minimum if they have opposite signs. Since q=\u0006p=4\nis precisely the geometries for which we do not expect any\nLRT generation, triplet mixing cannot be the source of this\nbehaviour. For the choice of physical parameters chosen in\nFig. 21, this effect results in a difference between the minimal\nand maximal critical temperature of nearly 60% as the mag-\nnetization direction is varied. As shown in Fig. 20, the effect\npersists qualitatively in larger structures as well, but is then\nweaker.19\nχ = 0 χ = +π/4 χ = -π/4\n                      \n0.830.840.850.860.87\n-0.50 -0.25 0.00 0.25 0.50\nθπTcTcs\nFIG. 20: Plot of critical temperature Tc=Tcsas a function of\nthe exchange field angle q, when LS=xS=1:00,LF=xS=0:2,\nandaxS=2. In contrast to ferromagnets without SO coupling,\nTcnow depends strongly on the magnetization direction. This\ngives rise to a spin-valve like functionality with a single fer-\nromagnet featuring SO coupling.\nχ = 0 χ = +π/4 χ = -π/4\n                                  \n0.350.400.450.500.55\n-0.50 -0.25 0.00 0.25 0.50\nθπTcTcs\nFIG. 21: Plot of critical temperature Tc=Tcsas a function of\nthe exchange field angle q, when LS=xS=0:55,LF=xS=0:2,\nandaxS=2.\nInstead, these observations may be explained using the theory\ndeveloped in Section II. When we have a general exchange\nfield and SO field in the xy-plane, Eq. (34) reveals that the\neffective energy of the SRT component is\nE=e+2iDFa2(1\u0000sin2qsin2c): (49)\nSince the factor (1\u0000sin2qsin2c)vanishes for q=c=\u0006p=4,\nwe get E=efor this case. This geometry is also one where\nwe do not expect any LRT generation, since the triplet mix-\ning factor cos2 qsin2c=0, so the conclusion is that the SO\ncoupling has no effect on the behaviour of SRTs for these\nparameters—at least according to the linearized equations.\nHowever, since 1\u0000sin2qsin2c=2 forq=\u0000c=\u0006p=4, the\nsituation is now dramatically different. The SRT effective en-\nergy is now E=e+4iDFa2, with an imaginary contribution\nwhich again destabilizes the SRTs, and increases the critical\ntemperature of the system. We emphasize that the variation ofTcwith the magnetization direction is present when c6=p=4\nas well, albeit with a magnitude of the variation that gradually\ndecreases as one approaches pure Rashba or pure Dresselhais\ncoupling.\nE. Triplet spin-valve effect with a single ferromagnet\nThe results discussed in the previous section show that the\ncritical temperature can be controlled via the magnetization\ndirection of one single ferromagnetic layer. This is a new re-\nsult originating from the presence of SO coupling. In conven-\ntional SF structures, Tcis independent of the magnetization\norientation of the F layer. By using a spin-valve setup such\nas FSF57–61, it has been shown that the relative magnetization\nconfiguration between the ferromagnetic layers will tune the\nTcof the system. In contrast, in our case such a spin-valve\neffect can be obtained with a single ferromagnet (see Figs. 20\nand 21): by rotating the magnetization an angle p=2,Tcgoes\nfrom a maximum to a minimum. The fact that only a single\nferromagnet is required to achieve this effect is of practical\nimportance since it can be challenging to control the relative\nmagnetization orientation in magnetic multilayered structures.\nIV . SUMMARY AND DISCUSSION\nIt was pointed out in Ref. 21 that for the case of transver-\nsal structures as depicted in Fig. 1(b), pure Rashba or pure\nDresselhaus coupling and arbitrary magnetisation direction\nare insufficient for long range triplets to exist. However, al-\nthough these layered structures are more restrictive in their\nconditions for LRT generation than lateral junctions they are\nnevertheless one of the most relevant for current experimental\nsetups10,11,50, and herein we consider the corresponding ex-\nperimentally accessible effects of SO coupling as a comple-\nment to the findings of Ref. 21. We have provided a detailed\nexposition of the density of states and critical temperature for\nboth the SF bilayer and SFS junction with SO coupling, high-\nlighting in particular the signature of long range triplets.\nWe saw that the spectroscopic signature depends nonmono-\ntonically on the angle of the magnetic exchange field, and that\nthe LRT component can induce a strong peak in the density\nof states at zero energy for a range of magnetization direc-\ntions. In addition to the large enhancement at zero energy, we\nsee that by carefully choosing the SO coupling and exchange\nfield strengths in the Josephson junction it is again possible to\ncontrol the qualitative features of the density of states by al-\ntering the phase difference between the two superconductors\ne.g.with a loop geometry53.\nThe intrinsic SO coupling present in the structures con-\nsidered herein derives from their lack of inversion symme-\ntry due to the e.g. junction interfaces, so-called interfacial\nasymmetry, and we restricted the form of this coupling to the\nexperimentally common and, in some cases, tunable Rashba-\nDresselhaus form. A lack of inversion symmetry can also de-\nrive from intrinsic noncentrosymmmetry of a crystal. This\ncould in principle be utilised to provide a component of the20\nSO-field in the junction direction, but to date we are not\naware of such materials having been explored in experiments\nwith SF hybrid materials. However, analytic and numerical\ndata suggest that such materials could have significant impor-\ntance for spintronic applications making use of a large triplet\nCooper pair population43.\nIt is also worth considering the possibility of separating the\nspin-orbit coupling and the ferromagnetic layer, which would\narguably be easier to fabricate, and we are currently pursuing\nthis line of investigation. In this case, we would expect similar\nconclusions regarding when the long-range triplets leave clear\nspectroscopic signatures and also regarding the spin-valve ef-\nfect with a single ferromagnet, as found when the SO cou-\npling and exchange field coexist in the same material. One\nway to practically achieve such a setup would be to deposit\na very thin layer of a heavy normal metal such as Au or Pt\nbetween a superconductor and a conventional homogeneous\nferromagnet. The combination of the large atomic number Z\nand the broken structural inversion symmetry at the interface\nregion would then provide the required SO coupling. With a\nvery thin normal metal layer (of the order of a couple of nm),\nthe proximity effect would be significantly stronger, and thus\nanalysis of this regime is only possible with the full Usadel\nequations in the Riccati parameterisation developed herein.\nThe current analysis pertains to thin film ferromagnets.\nUpon increasing the length of ferromagnetic film one will in-\ncrease the relative proportions of long-range to short-range\ntriplets in the middle of the ferromagnet. For strong ferro-\nmagnets where the exchange field is a significant fraction of\nthe Fermi energy, the quasiclassical Usadel formalism may no\nlonger describe the system behaviour appropriately, since it\nassumes that the impurity scattering rate is much larger than\nthe other energy scales involved, and the Eilenberger equation\nshould be used instead62.\nIn the previous section, we also observed that the presence\nof SO coupling will in many cases increase the critical tem-\nperature of a hybrid structure. This effect is explained through\nan increase in the effective energy coupled to the triplet com-\nponent in the Usadel equation, which destabilizes the triplet\npairs and closes that proximity channel. However, for the\nspecial geometry q=\u0000c=\u0006p=4, the linearized equations\nsuggest that the SRTs are unaffected by the presence of SO\ncoupling, and this is consistent with the numerical results. We\nalso note that for the geometries with a large LRT generation,\nsuch as q=0 and c=\u0006p=4, the LRT generation reduces the\ncritical temperature again. Thus, for the physical parameters\nconsidered herein, we see that there is a very slight increase\nin critical temperature for these geometries, but not as large as\nfor the geometries without LRT generation.\nOne particularly striking result from the critical temper-\nature calculations is that when the Rashba and Dresselhaus\ncontribution to the SO coupling is of similar magnitude, one\nobserves that the critical temperature can change by as much\nas 60% upon changing q=\u0000p=4 toq= +p=4,i.e.by a 90\u000e\nrotation of the magnetic field. This implies that it is possible\nto create a novel kind of triplet spin valve using an SF bilayer,\nwhere the ferromagnet has a homogeneous exchange field and\nRashba–Dresselhaus coupling. This is in contrast to previoussuggestions for triplet spin valves, such as the one described\nby Fominov et al. , which have required trilayers with differ-\nent homogeneous ferromagnets63. The construction of such a\ndevice is likely to have possible applications in the emerging\nfield of superconducting spintronics3.\nAcknowledgments\nThe authors thank Angelo di Bernardo, Matthias Eschrig,\nCamilla Espedal, and Iryna Kulagina for useful discussions\nand gratefully acknowledge support from the ‘Outstanding\nAcademic Fellows’ programme at NTNU and COST Action\nMP-1201’ Novel Functionalities through Optimized Confine-\nment of Condensate and Fields’. J.L. was supported by the\nResearch Council of Norway, Grant No. 205591 (FRINAT)\nand Grant No. 216700.\nAppendix A: Riccati parametrization of the Usadel equation\nand Kupriyanov–Lukichev boundary conditions\nThe 4\u00024 components of the retarded Green’s function ˆ g\nare not entirely independent, but can be expressed as\nˆg(z;e) =\u0012\ng(z;+e) f(z;+e)\n\u0000f\u0003(z;\u0000e)\u0000g\u0003(z;\u0000e)\u0013\n; (A1)\nwhich suggests that the notation can be simplified by intro-\nducing the tilde conjugation\n˜g(z;+e)\u0011g\u0003(z;\u0000e): (A2)\nMoreover, the normalization condition ˆ g2=1 further con-\nstrains the possible form of ˆ gby relating the gcomponents\nto the fcomponents,\ngg\u0000f˜f=1; g f\u0000f˜g=0: (A3)\nRemarkably, if we pick a suitable parametrization of ˆ g, which\nautomatically satisfies the symmetry and normalization re-\nquirements above, then both the Usadel equation and the\nKupriyanov–Lukichev boundary conditions can be reduced\nfrom 4\u00024 to 2\u00022 matrix equations. In this paper, we em-\nploy the so-called Riccati parametrization for this purpose,\nwhich is defined by\nˆg=\u0012\nN0\n0\u0000˜N\u0013\u0012\n1+g˜g2g\n2˜g1+˜gg\u0013\n; (A4)\nwhere the normalization matrices are N\u0011(1\u0000g˜g)\u00001and ˜N\u0011\n(1\u0000˜gg)\u00001. Solving the Riccati parametrized equations for\nthe function g(z;e)in spin space is then sufficient to uniquely\nconstruct the whole Green’s function ˆ g(z;e). It is noteworthy\nthat ˆg!1 when g!0, while the elements of ˆ gdiverge to\ninfinity when g!1; so we see that a finite range of variation\ningparametrizes an infinite range of variation in ˆ g.\nWe begin by deriving some basic identities, starting with\nthe inverses of the two matrix products Ngandg˜N:\n(Ng)\u00001=g\u00001N\u00001=g\u00001(1\u0000g˜g) =g\u00001\u0000˜g; (A5)\n(g˜N)\u00001=˜N\u00001g\u00001= (1\u0000˜gg)g\u00001=g\u00001\u0000˜g: (A6)21\nBy comparison of the results above, we see that Ng=g˜N.\nSimilar calculations for other combinations of the Riccati ma-\ntrices reveal that we can always move normalization matrices\npast gamma matrices if we also perform a tilde conjugation in\nthe process:\nNg=g˜N;˜Ng=gN;N˜g=˜g˜N;˜N˜g=˜gN: (A7)\nSince we intend to parametrize a differential equation, we\nshould also try to relate the derivatives of the Riccati matri-\nces. This can be done by differentiating the definition of N\nusing the matrix version of the chain rule:\n¶zN=¶z(1\u0000g˜g)\u00001\n=\u0000(1\u0000g˜g)\u00001[¶z(1\u0000g˜g)](1\u0000g˜g)\u00001\n= (1\u0000g˜g)\u00001[(¶zg)˜g+g(¶z˜g)](1\u0000g˜g)\u00001\n=N[(¶zg)˜g+g(¶z˜g)]N: (A8)\nPerforming a tilde conjugation of the equation above, we get\na similar result for ¶z˜N. Thus, the derivatives of the normal-\nization matrices satisfy the following identities:\n¶zN=N[(¶zg)˜g+g(¶z˜g)]N; (A9)\n¶z˜N=˜N[(¶z˜g)g+˜g(¶zg)]˜N: (A10)\nIn addition to the identities derived above, one should note that\nthe definition of the normalization matrix N= (1\u0000g˜g)\u00001can\nbe rewritten in many forms which may be of use when sim-\nplifying Riccati parametrized expressions; examples of this\ninclude g˜g=1\u0000N\u00001and 1 =N\u0000Ng˜g.\nNow that the basic identities are in place, it is time to\nparametrize the Usadel equation in the ferromagnet,\nDF˜Ñ(ˆg˜Ñˆg)+i\u0002\neˆr3+ˆM;ˆg\u0003\n=0; (A11)\nwhere we for simplicity will let DF=1 in this appendix. We\nbegin by expanding the gauge covariant derivative ˜Ñ(ˆg˜Ñˆg),\nand then simplify the result using the normalization condition\nˆg2=1 and its derivative fˆg;¶zˆgg=0, which yields the result\n˜Ñ\u0001(ˆg˜Ñˆg) =¶z(ˆg¶zˆg)\u0000i¶z(ˆgˆAzˆg)\n\u0000i[ˆAz;ˆg¶zˆg]\u0000[ˆA;ˆgˆAˆg]:(A12)\nWe then write ˆ gin component form using Eq. (A1), and also\nwrite ˆAin the same form using ˆA=diag(A;\u0000A\u0003). In the rest\nof this appendix, we will for simplicity assume that Ais real,\nso that ˆA=diag(A;\u0000A); in practice, this implies that Acan\nonly depend on the spin projections sxandsz. The derivation\nfor the more general case of a complex ˆAis almost identical.The four terms in Eq. (A12) may then be written as follows:\n¶z(ˆg¶zˆg)\n=\u0014\n¶z(g¶zg\u0000f¶z˜f)¶z(g¶zf\u0000f¶z˜g)\n¶z(˜g¶z˜f\u0000˜f¶zg)¶z(˜g¶z˜g\u0000˜f¶zf)\u0015\n; (A13)\n¶z(ˆgˆAˆg)\n=\u0014\n¶z(gAg+f A˜f)¶z(gA f+f A˜g)\n\u0000¶z(˜gA˜f+˜f Ag)\u0000¶z(˜gA˜g+˜f A f)\u0015\n; (A14)\n[ˆA;ˆg¶zˆg]\n=\u0014\n[A;g¶zg\u0000f¶z˜f]fA;g¶zf\u0000f¶z˜gg\n\u0000fA;˜g¶z˜f\u0000˜f¶zgg \u0000[A;˜g¶z˜g\u0000˜f¶zf]\u0015\n; (A15)\n[ˆA;ˆgˆAˆg]\n=\u0014\n[A;gAg+f A˜f]fA;gA f+f A˜gg\nfA;˜gA˜f+˜f Agg[A;˜gA˜g+˜f A f]\u0015\n: (A16)\nSubstituting these results back into Eq. (A12), we can find the\nupper blocks of the covariant derivative ˜Ñ\u0001(ˆg˜Ñˆg),\n[˜Ñ\u0001(ˆg˜Ñˆg)](1;1)\n=¶z(g¶zg\u0000f¶z˜f)\u0000i¶z(gAzg+f Az˜f)\n\u0000i[Az;g¶zg\u0000f¶z˜f]\u0000[A;gAg+f A˜f]; (A17)\n[˜Ñ\u0001(ˆg˜Ñˆg)](1;2)\n=¶z(g¶zf\u0000f¶z˜g)\u0000i¶z(gAzf+f Az˜g)\n\u0000ifAz;g¶zf\u0000f¶z˜gg\u0000f A;gAf+f A˜gg: (A18)\nIn this context, the notation ˆM(n;m)refers to the n’th row and\nm’th column in Nambu space. Since the Green’s function ˆ g\nand background field ˆAalso have a structure in spin space, the\n(1;1)element in Nambu space is the upper-left 2 \u00022 block of\nthe matrix, and the (1;2)element is the upper-right one.\nThere are two kinds of expressions that recur in the equa-\ntions above, namely the components of ˆ g¶zˆg, and the compo-\nnents of ˆ gˆAˆg. After we substitute in the Riccati parametriza-\ntiong=2N\u00001 and f=2Ng, these components take the form:\n[ˆg¶zˆg](1;1)=g¶zg\u0000f¶z˜f\n=2N[(¶zg)˜g\u0000g(¶z˜g)]N; (A19)\n[ˆg¶zˆg](1;2)=g¶zf\u0000f¶z˜g\n=2N[(¶zg)\u0000g(¶z˜g)g]˜N; (A20)\n[ˆgˆAˆg](1;1)=gAg+f A˜f\n=4N(A+gA˜g)N\u00002fA;Ng+A; (A21)\n[ˆgˆAˆg](1;2)=gAf+f A˜g\n=4N(Ag+gA)˜N\u00002fA;Ngg: (A22)\nIf we explicitly calculate the commutators of ˆAwith the two22\nmatrices ˆ g¶zˆgand ˆgˆAˆg, then we find:\n[ˆA;ˆg¶zˆg](1;1)= [A;g¶zg\u0000f¶z˜f]\n=2N(1\u0000g˜g)AN[(¶zg)˜g\u0000g(¶z˜g)]N\n\u00002N[(¶zg)˜g\u0000g(¶z˜g)]NA(1\u0000g˜g)N; (A23)\n[ˆA;ˆg¶zˆg](1;2)=fA;g¶zf\u0000f¶z˜gg\n=2N(1\u0000g˜g)AN[(¶zg)\u0000g(¶z˜g)g]˜N\n+2N[(¶zg)\u0000g(¶z˜g)g]˜NA(1\u0000˜gg)˜N; (A24)\n[ˆA;ˆgˆAˆg](1;1)= [A;gAg+f A˜f]\n=4AN(A+gA˜g)N\n\u00004N(A+gA˜g)NA\n\u00002[A2;N]; (A25)\n[ˆA;ˆgˆAˆg](1;2)=fA;gAf+f A˜gg\n=4AN(Ag+gA)˜N\n+4N(Ag+gA)˜NA\n\u00004ANgA\u00002fA2;Ngg: (A26)\nIf we instead differentiate the aforementioned matrices with\nrespect to z, we obtain:\n[¶z(ˆg¶zˆg)](1;1)=¶z(g¶zg\u0000f¶z˜f)\n=2N[(¶2\nzg)+2(¶zg)˜N˜g(¶zg)]˜gN\n\u00002Ng[(¶2\nz˜g)+2(¶z˜g)Ng(¶z˜g)]N; (A27)\n[¶z(ˆg¶zˆg)](1;2)=¶z(g¶zf\u0000f¶z˜g)\n=2N[(¶2\nzg)+2(¶zg)˜N˜g(¶zg)]˜N\n\u00002Ng[(¶2\nz˜g)+2(¶z˜g)Ng(¶z˜g)]g˜N; (A28)\n[¶z(ˆgAˆg)](1;1)=¶z(gAg+f A˜f)\n=2N(1+g˜g)AN[g(¶z˜g)+(¶zg)˜g]N\n+2N[g(¶z˜g)+(¶zg)˜g]NA(1+g˜g)N\n+4NgA˜N[(¶z˜g)+˜g(¶zg)˜g]N\n+4N[(¶zg)+g(¶z˜g)g]˜NA˜gN; (A29)\n[¶z(ˆgAˆg)](1;2)=¶z(gAf+f A˜g)\n=2N(1+g˜g)AN[(¶zg)+g(¶z˜g)g]˜N\n+2N[(¶zg)+g(¶z˜g)g]˜NA(1+˜gg)˜N\n+4NgA˜N[˜g(¶zg)+(¶z˜g)g]˜N\n+4N[g(¶z˜g)+(¶zg)˜g]NA˜g˜N: (A30)\nCombining all of the equations above, we can express\nEqs. (A17) and (A18) using Riccati matrices. In order to iso-\nlate the second-order derivative ¶2\nzgfrom these, the trick is\nto multiply Eq. (A17) by gfrom the right, and subsequentlysubtract the result from Eq. (A18):\n1\n2N\u00001\b\n[˜Ñ\u0001(ˆg˜Ñˆg)](1;2)\u0000[˜Ñ\u0001(ˆg˜Ñˆg)](1;1)g\t\n=¶2\nzg+2(¶zg)˜N˜g(¶zg)\n\u00002i(Az+gAz˜g)N(¶zg)\u00002i(¶zg)˜N(Az+˜gAzg)\n\u00002(Ag+gA)˜N(A+˜gAg)\u0000A2g+gA2: (A31)\nIf we finally rewrite [˜Ñ\u0001(ˆg˜Ñˆg)](1;1)and[˜Ñ\u0001(ˆg˜Ñˆg)](1;2)in the\nequation above by substituting in the Usadel equation (A11),\nthen we obtain the following equation for the Riccati matrix g:\n¶2\nzg=\u00002ieg\u0000ih\u0001(sg\u0000gs\u0003)\u00002(¶zg)˜N˜g(¶zg)\n+2i(Az+gAz˜g)N(¶zg)+2i(¶zg)˜N(Az+˜gAzg)\n+2(Ag+gA)˜N(A+˜gAg)+A2g\u0000gA2: (A32)\nThe corresponding equation for ˜gcan be found by tilde con-\njugation of the above. After restoring the diffusion coeffi-\ncient DF, and generalizing the derivation to a complex SO\nfield A, the above result takes the form shown in Eq. (5).\nAfter parametrizing the Usadel equation, the next step is\nto do the same to the Kupriyanov–Lukichev boundary condi-\ntions. The gauge covariant version of Eq. (2) may be written\n2Lnznˆgn˜Ñˆgn= [ˆg1;ˆg2]; (A33)\nwhich upon expanding the covariant derivative ˆ g˜Ñˆgbecomes\nˆgn¶zˆgn=1\n2Wn[ˆg1;ˆg2]+iˆgn[ˆAz;ˆgn]; (A34)\nwhere we have introduced the notation Wn\u00111=Lnznfor the\ninterface parameter. We will now restrict our attention to the\n(1,1) and (1,2) components of the above,\ngn¶zgn\u0000fn¶z˜fn=1\n2Wn(g1g2\u0000g2g1\u0000f1˜f2+f2˜f1)\n+ign[Az;gn]+i fnfAz;˜fng; (A35)\ngn¶zfn\u0000fn¶z˜gn=1\n2Wn(g1f2\u0000g2f1\u0000f1˜g2+f2˜g1)\n+ignfAz;fng+i fn[Az;˜gn]: (A36)\nSubstituting the Riccati parametrizations gn=2Nn\u00001 and\nfn=2Nngnin the above, we then obtain:\nNn[(¶zgn)˜gn\u0000gn(¶z˜gn)]Nn=WnN1(1\u0000g1˜g2)N2\n\u0000WnN2(1\u0000g2˜g1)N1\n\u0000iNn(1\u0000gn˜gn)ANn\n\u0000iNnA(1\u0000gn˜gn)Nn\n+2iNn(A+gnA˜gn)Nn;(A37)\nNn[(¶zgn)\u0000gn(¶z˜gn)gn]˜Nn=WnN1(1\u0000g1˜g2)g2˜N2\n\u0000WnN2(1\u0000g2˜g1)g1˜N1\n+iNn(1+gn˜gn)Agn˜Nn\n+iNngnA(1+˜gngn)˜Nn:(A38)23\nIf we multiply Eq. (A37) by gnfrom the right, subtract this\nfrom Eq. (A38), and divide by Nnfrom the left, then we obtain\nthe following boundary condition for gn:\n¶zgn=Wn(1\u0000g1˜g2)N2(g2\u0000gn)\n+Wn(1\u0000g2˜g1)N1(gn\u0000g1)\n+ifAz;gng: (A39)\nWhen we evaluate the above for n=1 and n=2, then it sim-\nplifies to the following:\n¶zg1=W1(1\u0000g1˜g2)N2(g2\u0000g1)+ifAz;g1g; (A40)\n¶zg2=W2(1\u0000g2˜g1)N1(g2\u0000g1)+ifAz;g2g: (A41)\nThe boundary conditions for ¶z˜g1and¶z˜g2are found by tilde\nconjugating the above. If we generalize the derivation to a\ncomplex SO field A, and substitute back Wn\u00111=Lnznin the\nresult, then we arrive at Eq. (6).\nAppendix B: Derivation of the self-consistency equation for D\nFor completeness, we present here a detailed derivation of\nthe self-consistency equation for the BCS order parameter64\nin a quasiclassical framework. Similar derivations can also be\nfound in Refs. 52,65–68. In this paper, we follow the conven-\ntion where the Keldysh component of the anomalous Green’s\nfunction is defined as\nFK\nss0(r;t;r0;t0)\u0011\u0000ih[ys(r;t);ys0(r;t)]i; (B1)\nwhere ys(r;t)is the spin-dependent fermion annihilation op-\nerator, and the superconducting gap is defined as\nD(r;t)\u0011lhy\"(r;t)y#(r;t)i; (B2)\nwhere l>0 is the electron–electron coupling constant in the\nBCS theory. For the rest of this appendix, we will also assume\nthat we work in an electromagnetic gauge where Dis a purely\nreal quantity. Comparing Eqs. (B1) and (B2), and using the\nfermionic anticommutation relation\ny\"(r;t)y#(r;t) =\u0000y#(r;t)y\"(r;t); (B3)\nwe see that the superconducting gap D(r;t)can be expressed\nin terms of the Green’s functions in two different ways,\nD(r;t) =il\n2FK\n\"#(r;t;r;t); (B4)\nD(r;t) =\u0000il\n2FK\n#\"(r;t;r;t): (B5)\nWe may then perform a quasiclassical approximation by first\nswitching to Wigner mixed coordinates, then Fourier trans-\nforming the relative coordinates, then integrating out the en-\nergy dependence, and finally averaging the result over the\nFermi surface to obtain the isotropic part. The resulting equa-\ntions for the superconducting gap are\nD(r;t) =1\n4N0lZ\ndefK\n\"#(r;t;e); (B6)\nD(r;t) =\u00001\n4N0lZ\ndefK\n#\"(r;t;e); (B7)where fK\nss0is the quasiclassical counterpart to FK\nss0,eis the\nquasiparticle energy, and N0is the density of states per spin at\nthe Fermi level.\nIn the equilibrium case, the Keldysh component ˆ gKcan be\nexpressed in terms of the retarded and advanced components\nof the Green’s function,\nˆgK= (ˆgR\u0000ˆgA)tanh(e=2T); (B8)\nand the advanced Green’s function may again be expressed in\nterms of the retarded one,\nˆgA=\u0000r3ˆgR†r3; (B9)\nwhich implies that the Keldysh component can be expressed\nentirely in terms of the retarded component,\nˆgK= (ˆgR\u0000r3ˆgR†r3)tanh(e=2T): (B10)\nIf we extract the relevant anomalous components fK\n\"#and fK\n#\"\nfrom the above, we obtain the results\nfK\n\"#= [fR\n\"#(r;+e)+fR\n#\"(r;\u0000e)]tanh(e=2T); (B11)\nfK\n#\"= [fR\n#\"(r;+e)+fR\n\"#(r;\u0000e)]tanh(e=2T): (B12)\nWe then switch to a singlet/triplet-decomposition of the\nretarded component fR, where the singlet component is de-\nscribed by a scalar function fs, and the triplet component by\nthe so-called d-vector (dx;dy;dz). This parametrization is de-\nfined by the matrix equation\nfR= (fs+d\u0001s)isy; (B13)\nor in component form,\n \nfR\n\"\"fR\n\"#\nfR\n#\"fR\n##!\n= \nidy\u0000dxdz+fs\ndz\u0000fsidy+dx!\n: (B14)\nParametrizing Eqs. (B11) and (B12) according to Eq. (B14),\nwe obtain\nfK\n\"#(r;e) = [ dz(r;+e)+fs(r;+e)\n+dz(r;\u0000e)\u0000fs(r;\u0000e)]tanh(e=2T); (B15)\nfK\n\"#(r;e) = [ dz(r;+e)\u0000fs(r;+e)\n+dz(r;\u0000e)+fs(r;\u0000e)]tanh(e=2T): (B16)\nThe triplet component dzcan clearly be eliminated from the\nabove equations by subtracting Eq. (B15) from Eq. (B16),\nfK\n\"#\u0000fK\n#\"=2[fs(r;e)\u0000fs(r;\u0000e)]tanh(e=2T); (B17)\nand a matching expression for the superconducting gap can be\nacquired by adding Eqs. (B6) and (B7),\n2D(r) =1\n4N0lZ\nde[fK\n\"#(r;e)\u0000fK\n\"#(r;e)]tanh(e=2T):(B18)\nBy comparing the two results above, we finally arrive at an\nequation for the superconducting gap which only depends on\nthe singlet component of the quasiclassical Green’s function:\nD(r) =1\n4N0lZ\nde[fs(r;e)\u0000fs(r;\u0000e)]tanh(e=2T):(B19)24\nIf the integral above is performed for all real values of e,\nit turns out to be logarithmically divergent e.g.for a bulk su-\nperconductor. However, physically, the range of energies that\nshould be integrated over is restricted by the energy spectra\nof the phonons that mediate the attractive electron–electron\ninteractions in the superconductor. This issue may therefore\nbe resolved by introducing a Debye cutoff wc, such that we\nonly integrate over the region where jej<wc. Including the\nintegration range, the gap equation is therefore\nD(r) =1\n4N0lwcZ\n\u0000wcde[fs(r;e)\u0000fs(r;\u0000e)]tanh(e=2T):(B20)\nThe equation above can however be simplified even further.\nFirst of all, both fs(e)\u0000fs(\u0000e)and tanh (e=2T)are clearly\nantisymmetric functions of e, which means that the product\nis a symmetric function, and so it is sufficient to perform an\nintegral over positive values of e,\nD(r) =1\n2N0lwcZ\n0de[fs(r;e)\u0000fs(r;\u0000e)]tanh(e=2T):(B21)\nHowever, because of the term fs(r;\u0000e), we still need to know\nthe Green’s function for negative values of ebefore we can\ncalculate the gap. On the other hand, the singlet component\nof the quasiclassical Green’s functions also has a symmetry\nwhen the superconducting gauge is chosen as real\nfs(r;e) =\u0000f\u0003\ns(r;\u0000e); (B22)\nwhich implies that\nfs(r;e)\u0000fs(r;\u0000e) =2Reffs(r;e)g: (B23)\nSubstituting Eq. (B23) into Eq. (B21), the gap equation takes\na particularly simple form, which only depends on the real\npart of the singlet component fs(r;e)for positive energies e:\nD(r) =N0lwcZ\n0deReffs(r;e)gtanh(e=2T): (B24)\nLet us now consider the case of a BCS bulk superconductor,\nwhich has a singlet component given by the equation\nfs(e) =Dp\ne2\u0000D2; (B25)so that the gap equation may be written as\nD=N0lwcZ\n0deRe\u001aDp\ne2\u0000D2\u001b\ntanh(e=2T): (B26)\nThe part in the curly braces is only real when jej\u0015D, which\nmeans that the equation can be simplified by changing the\nlower integration limit to D. After also dividing the equation\nbyDN0l, we then obtain the self-consistency equation\n1\nN0l=wcZ\nDdetanh(e=2T)p\ne2\u0000D2: (B27)\nFor the zero-temperature case, where T!0 and D!D0, per-\nforming the above integral and reordering the result yields\nwc=D0cosh(1=N0l): (B28)\nUsing the above equation for wc, and the well-known result\nD0\nTc=p\neg; (B29)\nwhere g\u00190:57722 is the Euler–Mascheroni constant, we can\nfinally rewrite Eq. (B24) as:\nD(r) =N0lD0cosh(1=N0l)Z\n0deReffs(r;e)gtanh\u0012p\n2ege=D0\nT=Tc\u0013\n:(B30)\nThis version of the gap equation is particularly well-suited for\nnumerical simulations. One advantage is that we only need to\nknow the Green’s function for positive energies, which halves\nthe number of energies that we need to solve the Usadel equa-\ntion for. The equation also takes a particularly simple form\nif we use energy units where D0=1 and temperature units\nwhere Tc=1, which is common practice in such simulations.\n1A. I. Buzdin, Rev. Mod. Phys. 77, 935 (2005).\n2F.S. Bergeret, A.F. V olkov and K.B. Efetov, Rev. Mod. Phys. 77,\n1321 (2005).\n3J. Linder and J. W. A. Robinson, Nature Physics 11, 307 (2015).\n4F. Bergeret, A. V olkov, and K. Efetov, Phys. Rev. Lett. 86, 4096\n(2001).\n5M. Eschrig, J. Kopu, J. C. Cuevas, and Gerd Sch ¨on, Phys. Rev.\nLett. 90, 137003 (2003).6K. Halterman, P. H. Barsic, and O. T. Valls, Phys. Rev. Lett. 99,\n127002 (2007).\n7A. Cottet, Phys. Rev. Lett. 107, 177001 (2011).\n8Y . Asano, Y . Sawa, Y . Tanaka, and A. A. Golubov, Phys. Rev. B\n76, 224525 (2007).\n9Y . Kalcheim et al. , Phys. Rev. B 89, 180506(R) (2014).\n10J. W. A. Robinson, J. D. Witt, and M. Blamire, Science 329, 59\n(2010).25\n11T. S. Khaire et al. , Phys. Rev. Lett. 104, 137002 (2010).\n12M. Alidoust and J. Linder, Phys. Rev. B 82, 224504 (2010); I. B.\nSperstad et al. , Phys. Rev. B 78, 104509 (2008).\n13L. Trifunovic and Z. Radovi ´c, Phys. Rev. B 82, 020505(R) (2010).\n14I. Sosnin, H. Cho, V . T. Petrashov, and A. F. V olkov, Phys. Rev.\nLett. 96, 157002 (2006).\n15Z. Shomali, M. Zareyan, W. Belzig, New J. Phys. 13, 083033\n(2012).\n16G. Annunziata et al. , Phys. Rev. B 83, 060508(R) (2011).\n17I. V . Bobkova and A. M. Bobkov, Phys. Rev. Lett. 108, 197002\n(2012).\n18F. Konschelle, J. Cayssol, and A. Buzdin, Phys. Rev. B 82,\n180509(R) (2010).\n19M. Houzet, Phys. Rev. Lett. 101, 057009 (2008).\n20G. Annunziata, D. Manske, and J. Linder, Phys. Rev. B 86,\n174514 (2012).\n21F.S. Bergeret and I.V . Tokatly, Phys. Rev. B. 89, 134517 (2014).\n22F.S. Bergeret and I.V . Tokatly, Phys. Rev. Lett. 110, 117003\n(2013).\n23M. Eschrig, J. Kopu, A. Konstandin, J.C. Cuevas, M. Fogelstr ¨om,\nG. Sch ¨on, Advances in Solid State Physics V ol. 44, pp. 533-546\n(2004).\n24I. S. Burmistrov and N. M. Chtchelkatchev, Phys. Rev. B 72,\n144520 (2005).\n25J. Linder and K. Halterman, Phys. Rev. B 90, 104502 (2014).\n26A.A. Abrikosov and L.P. Gor’kov, Sov. Phys. JETP 15, 752\n(1962).\n27M. Eschrig, Physics Today 64(1), 43 (2011).\n28K.V . Samokhin, Ann. Phys. 324, 2385 (2009).\n29V .M. Edelstein, Phys. Rev. B. 67, 020505 (2003).\n30J. Linder and T. Yokoyama, Phys. Rev. Lett. 106, 237201 (2011).\n31K. Sun and N. Shah, Phys. Rev. B 91, 144508 (2015).\n32X. Liu, J. K. Jain, and C.-X. Liu, Phys. Rev. Lett. 113, 227002\n(2014).\n33K. Usadel, Phys. Rev. Lett. 25, 507 (1970).\n34W. Belzig et al. , Superlattices and Microstructures, 5, 25 (1999).\n35V . Chandrasekhar, arXiv:0312507.\n36M.Y . Kupriyanov and V .F. Lukichev, Sov. Phys. JETP 67, 1163\n(1988).\n37N. Schopohl and K. Maki, Phys. Rev. B. 52, 490 (1995); N.\nSchopohl, arXiv:cond-mat/9804064.\n38C. Gorini, P. Schwab, R. Raimondi, and A. L. Shelankov, Phys.\nRev. B 82, 195316 (2010).\n39A.I. Buzdin, Phys. Rev. Lett. 101, 107005 (2008); A.A. Raynoso\net al. , Phys. Rev. B. 86, 214519 (2012); I. Kulagina and J. Linder,\nPhys. Rev. B 90, 054504 (2014); F.S. Bergeret and I.V . Tokatly,preprint, arXiv:1409.4563.\n40F. Konschelle, Eur. Phys. J. B 87, 119 (2014).\n41G. Dresselhaus, Phys. Rev. 100, 580 (1955); E. Rashba, Fiz.\nTverd. Tela (Leningrad) 2, 1224 (1960), [Sov. Phys. Solid State\n2, 1109 (1960)].\n42E. Bauer, M. Sigrist. Non-Centrosymmetric Superconductors: In-\ntroduction and overview. Springer (2012).\n43S.H. Jacobsen and J. Linder, Phys. Rev. B 92, 024501 (2015)\n44P. Fulde and R.A. Ferrell, Phys. Rev. 135, A550 (1964); A.I.\nLarkin and Y .N. Ovchinnikov, Sov. Phys. JETP 20762 (1965).\n45R. Balian and N.R. Werthamer, Phys. Rev. 131, 1553 (1963).\n46A.P. Mackenzie and Y . Maeno, Rev. Mod. Phys. 75, 657 (2003).\n47Y . Tanaka and A. Golubov, Phys. Rev. Lett. 98, 037003 (2007).\n48T. Yokoyama, Y . Tanaka and A.A. Golubov, Phys. Rev. B 72,\n052512 (2005);T. Yokoyama, Y . Tanaka and A.A. Golubov, Phys.\nRev. B 73, 094501 (2006).\n49S. Kawabata et al. , J. Phys. Soc. Jpn. 82, 124702 (2013).\n50T. Kontos et al. , Phys. Rev. Lett. 86, 304 (2001); V .V . Ryazanov\net al. , Phys. Rev. Lett. 86, 2427 (2001).\n51P. SanGiorgio et al. , Phys. Rev. Lett. 100, 237002 (2008).\n52M. Tinkham, Introduction to Superconductivity , (McGraw-Hill,\n1996), 2nd Edition.\n53H. le Sueur et al, Phys. Rev. Lett. 100, 197002 (2008).\n54F. Zhou, P. Charlat, and B. Pannetier, J. Low Temp. Phys. 110,\n841 (1998).\n55J. C. Hammer, J. C. Cuevas, F. S. Bergeret, and W. Belzig, Phys.\nRev. B 76, 064514 (2007).\n56Y . Fominov et al. , Phys. Rev. B. 66, 014507 (2002).\n57J. Y . Gu et al. , Phys. Rev. Lett. 89, 267001 (2002).\n58I. Moraru, W. P. Pratt Jr., and N. Birge, Phys. Rev. Lett. 96,\n037004 (2006).\n59J. Zhu, I. N. Krivorotov, K. Halterman, and O. T. Valls, Phys. Rev.\nLett. 105, 207002 (2010).\n60P. V . Leksin et al. , Phys. Rev. Lett. 109, 057005 (2012).\n61N. Banerjee et al. , Nature Commun. 5, 3048 (2014).\n62G. Eilenberger, Sov. Phys. JETP 214, 195 (1968).\n63Y . Fominov et al. , JETP Lett., 91, 308 (2010).\n64J. Bardeen, L. N. Cooper, and J. R. Schrieffer, Phys. Rev. 108,\n1175 (1957).\n65N. Kopnin. Theory of Nonequilibrium Superconductivity (2001).\n66A. A. Abrikosov, L.P. Gorkov, I.E. Dzyaloshinski. Quantum Field\nTheoretical Methods in Statistical Physics (V olume 4 of Interna-\ntional Series of Monographs in Natural Philosophy) (1965).\n67J. W. Serene and D. Rainer, Phys. Rep. 101, 4 (1983).\n68P. G. deGennes. Superconductivity Of Metals And Alloys . (1999)."    },    {        "title": "1401.0575v1.Spin_Orbit_Coupled_Bose_Gases_at_Finite_Temperatures.pdf",        "content": "arXiv:1401.0575v1  [cond-mat.quant-gas]  3 Jan 2014Spin-Orbit Coupled Bose Gases at Finite Temperatures\nRenyuan Liao1, Oleksandr Fialko2\n1College of Physics and Energy, Fujian Normal University, Fu zhou 350108, China and\n2Centre for Theoretical Chemistry and Physics, Massey Unive rsity, Auckland 0632, New Zealand\n(Dated: June 24, 2021)\nSpin-orbit couplingis predictedto havedramatic effects on thermal properties of a two-component\natomic Bose gas. We show that in three spatial dimensions it l owers the critical temperature of\ncondensation and enhances thermal depletion of the condens ate fraction. In two dimensions we show\nthat spin-orbit coupling destroys superfluidity at any finit e temperature, modifying dramatically the\ncerebrated Berezinskii-Kosterlitz-Thouless scenario. W e explain this by the increase of the number\nof low energy states induced by spin-orbit coupling, enhanc ing the role of quantum fluctuations.\nPACS numbers: 67.85.Fg, 03.75.Mn, 05.30.Jp, 67.85.Jk\nThere are numerous phenomena in a wide range of\nquantum systems, ranging from condensed matter to\natomic and nuclear physics, where spin-orbit coupling\n(SOC) plays an important role. Recently discovered\nnew class of topological insulators, quantum spin Hall\neffect [1] and Majorana fermions [2] rely on SOC and are\nexpected to retain their quantum properties up to room\ntemperature. However, the electronic systems can not\nbe easily controlled and the details of SOC are usually\nnot known. Therefore, it is a difficult task to manip-\nulate such systems. In contrast, ultracold atoms have\nbeen demonstrated to be a remarkable platform for em-\nulation of various condensed matter phenomena due to\ntheir ability to be easily manipulated at will [3]. The pio-\nneering experimental realization of synthetic gauge fields\nand SOC [4–8] is defining a new dimension in explor-\ning quantum many-body systems with ultracold atomic\ngases. The engineered SOC (with equal Rashba and\nDresselhaus strength) in a neutral atomic Bose-Einstein\ncondensate was achieved by dressing two atomic spin\nstates with a pair of lasers. It allows to study the rich\nphysics of SOC effects in bosonic systems [9–13], which\nhavenotbeenexploredbefore. Recently, methodstogen-\nerate pure Rashba type SOC have been suggested [14].\nIts realization will make it possible to study rich ground\nstate physics proposed in fermionic [15] and bosonic sys-\ntems [10, 16], of which many properties have no con-\ndensed matter analogues.\nSOC leads to a huge degeneracy of the ground state\nof a single particle [17]. This enhances the role of quan-\ntum fluctuations making condensation of non-interacting\nbosons not possible [17, 18]. However, it has been\nshown that interactions among atoms stabilize conden-\nsation [19–21]. The role of quantum fluctuations is espe-\ncially essential in two dimensions destroying condensa-\ntion but not necessary superfluidity. This yields, in par-\nticular, the celebrated Berezinskii-Kosterlitz-Thouless\n(BKT) phase transition in two dimensions with a criti-\ncaltemperatureseparatingsuperfluidandnormalphases.\nHow do the quantum fluctuations in the presence of SOC\naffect the BKT phenomenon? What is the effect of SOCon thermal properties of a Bose condensate? These ques-\ntions shall be addressed in this Letter. Previous theo-\nretical studies have been focused mainly on the ground\nstate properties of interacting SOC quantum gases, leav-\ning the experimentally relevant physics at finite tempera-\ntures intact. In the light of the recent experiment [22] on\na finite-temperature phase diagram of SOC Bose gases,\nour present study is an interesting and urgent task.\nAccording to the Mermin-Wagner theorem long-range\norder at finite temperature does not exist in one spatial\ndimension. In this Letter, we shall present studies of a\nSOC two-component atomic Bose gas at finite temper-\nature in two and three spatial dimensions. The inter-\nplay between quantum and thermal fluctuations in the\npresence of SOC is shown to yield dramatic modifica-\ntions of the familiar physics. First, by resorting to the\nPopov approximation, we develop a formalism suitable\nfor treating the system at finite temperature in three di-\nmensions. Within this formalism, we find that the SOC\ngreatly suppresses the critical temperature of condensa-\ntion and enhances thermal depletion of the condensate.\nWe then derive an effective theory suitable to study the\ncelebrated BKT phase transition in two dimensions. The\nBKT transition temperature, in contrast to the previous\ncase, is shown to drop to zero in the presence of SOC.\nWe consider a three-dimensional homogeneous two-\ncomponent Bose gas with an isotropic in-plane (x-y\nplane) Rashba spin-orbit coupling, described by the fol-\nlowing grand canonical Hamiltonian in real space:\nH=/integraldisplay\ndr/bracketleftigg/summationdisplay\nσψ†\nσ/parenleftbigg\n−/planckover2pi12∇2\n2m−µ/parenrightbigg\nψσ+(ψ†\n↑ˆRψ↓+h.c.)\n+/summationdisplay\nσgσσ\n2(ψ†\nσψσ)2+g↑↓ψ†\n↑ψ↑ψ†\n↓ψ↓/bracketrightigg\n. (1)\nHere,ψσis a Bose field satisfying the usual commuta-\ntion relation [ ψσ(r),ψ†\nσ′(r′)] =i/planckover2pi1δσσ′δ(r−r′), the spin\nindexσ=↑,↓denotes two pseudo-spin states of the Bose\ngas with atomic mass m. The inter-particle interac-\ntiongσσ′is related to the two-body scattering length\naσσ′asgσσ′= 4π/planckover2pi12aσσ′/m. For simplicity, we shall2\nassume that the interactions between like-spin particles\nare the same, g↑↑=g↓↓=g. The chemical potential\nµis introduced to fix the total particle number den-\nsity. The Rashba spin-orbit coupling is described by\nthe operator ˆR=λ(ˆpx−iˆpy) withλbeing the cou-\npling strength. Throughout the rest of this paper, we\nset/planckover2pi1= 2m=kB= 1 and define n1/3as a momentum\nscale, while n2/3as an energy scale. For the system to\nbe weakly interacting, we set g= 0.1n−1/3.\nThe Hamiltonian of a non-interacting system is diag-\nonalized in the helicity basis after the Fourier transform\nof the fields ψσ(r) = 1/√\nL3/summationtext\nqψσ(q)exp(−iqr). The\ngas is assumed to be in a box with size L. This results\nin two branches of spectrum E±\nq=q2−µ±λq⊥, where\nq⊥is the magnitude of the in-plane momentum. The\nlowest energy state is therefore infinitely degenerate, sit-\nting on the circle q⊥=λ/2 in the plane qz= 0. This\nincreases the low energy density of states with dramatic\nimplications on the thermal properties of the Bose gas\nto be explored below. For an interacting system, earlier\nmean-field study [10] found that there exist the plane\nwave phase (PW) for g≥g↑↓and the striped phase for\ng < g ↑↓. The PW phase is the result of condensation\nat a single finite momentum state breaking explicitly the\nrotational symmetry, while the striped phase represents\na coherent superposition of two condensates at two op-\nposite momenta.\nWithin the framework of the functional field in-\ntegral, the partition function of the system is [23]\nZ=/integraltext\nD[ψ∗,ψ]exp(−S[ψ∗,ψ]) with the action S=/integraltextβ\n0dτ[/summationtext\nσψ∗\nσ∂τψσ+H(ψ∗,ψ)], whereβ= 1/Tis the\ninverse temperature. Here for simplicity we restrict our-\nself to study the PW phase, as analogous treatment of\nthe striped phase is more involved and will be addressed\nelsewhere. Our choice of the PW phase is also motivated\nby the fact that the striped phase has not been realized\nexperimentally yet. We further assume that the conden-\nsation occurs at momentum κ= (λ/2,0,0). Without\nloss of generality, the condensate wavefunction can bechosen as ( φ0↑,φ0↓) =√n0(1,−1)eiλx/2, withn0being\nthe condensate density for either species. We split the\nBose field into the mean-field part φ0σand the fluctuat-\ning partφqσasψqσ=φ0σδqκ+φqσ. After substitution,\nthe action can be formally written as S=S0+Sf, where\nS0=βL3/bracketleftbig\n−2(κ2+µ)n0+(g+g↑↓)n2\n0/bracketrightbig\nis the mean-field\ncontribution and Sfdenotes a contribution from fluctu-\nating fields. At this point, the action is exact. However,\nit contains terms of cubic and quartic orders in fluctu-\nating fields. To deal with such an action, one must re-\nsort to some sort of approximation. The celebrated Bo-\ngoliubov approximation for BEC is valid strictly at zero\ntemperature. At finite temperatures, the self-consistent\nHartree-Fock-Bogoliubov (HFB) approximation gives a\ngapped spectrum [24], violating the Hugenholtz-Pines\ntheorem [25] and the Goldstone theorem [26], which re-\nsults from the spontaneous symmetry breaking of U(1)\ngauge symmetry. We choose the Popov theory [27] yield-\ning a gapless spectrum and therefore it is more suitable\nfor treating finite-temperature Bose gases.\nUnder the Popov approximation, which takes into ac-\ncount interactions between excitations [28], the terms\nwith three and four fluctuating fields in the action are\napproximated as follows (neglecting anomalous average):\nφ∗\nσφσφσ≈2∝angb∇acketleftφ∗\nσφσ∝angb∇acket∇ightφσ, (φ∗\nσφσ)2≈4∝angb∇acketleftφ∗\nσφσ∝angb∇acket∇ightφ∗\nσφσand\n|φ↑φ↓|2≈ ∝angb∇acketleftφ∗\n↑φ↑∝angb∇acket∇ightφ∗\n↓φ↓+∝angb∇acketleftφ∗\n↓φ↓∝angb∇acket∇ightφ∗\n↑φ↑. Within the Popov\napproximation we also require the first order term to\nvanish, which fixes the chemical potential as µ=µp=\n−κ2+ (g+g↑↓)n0+ (2g+g↑↓)nf, wherenf≡ ∝angb∇acketleftφ∗\nσφσ∝angb∇acket∇ight\nis the density of particles of either component excited\nout of the condensate. We define a four-dimensional col-\numn vector Φ qn= (φκ+q,n↑,φκ+q,n↓,φ∗\nκ−q,n↑,φ∗\nκ−q,n↓),\nwhose components are defined through the Fouriertrans-\nformφσ(r,τ) = 1//radicalbig\nL3β/summationtext\nq,nφq,nσexp[i(q·r−wnτ)],\nwherewn= 2nπ/βis the bosonic Matsubara frequen-\ncies. Retaining terms of zeroth and quadratic orders\nin the fluctuating fields we can then bring the fluctu-\nating part of the action into the compact form Sf≈/summationtext\nq,n1\n2Φ∗\nqnG−1(q,iwn)ΦT\nqn−β/summationtext\nqǫ−q, whereǫq= (q+\nκ)2−µ+ (2g+g↑↓)(n0+nf) and the inverse Green’s\nfunction G−1(q,iwn) reads\nG−1(q,iwn) =\n−iwn+ǫqRκ+q−g↑↓n0gn0 −g↑↓n0\nR∗\nκ+q−g↑↓n0−iwn+ǫq−g↑↓n0gn0\ngn0 −g↑↓n0iwn+ǫ−qR∗\nκ−q−g↑↓n0\n−g↑↓n0gn0Rκ−q−g↑↓n0iwn+ǫ−q\n, (2)\nwhereRq=λ(qx−iqy) is the Fourier transformed\nRashba operator. The poles of the Green’s function give\nthe spectrum of elementary excitations ωqs. It can be\nfound by replacing iωnwithωqsand solving the secu-\nlar equation det G−1(q,ωqs) = 0. The index sdenotesdifferent solutions of the secular equation. We retain\nonly two positive solutions in the following. The spec-\ntrum is used to study thermodynamic properties of the\nsystem. The Gaussian action permits direct integration\nover the fluctuating fields to yield the grand potential3\n/s48 /s49 /s50 /s51 /s52/s48/s46/s48/s48/s48/s46/s50/s53/s48/s46/s53/s48/s48/s46/s55/s53/s49/s46/s48/s48/s84\n/s67/s47/s84/s48 /s67\n/s47/s110/s49/s47/s51\nFIG. 1. (color online). The critical temperature of conden-\nsation as a function of the SOC strength. SOC decreases the\ncritical temperature due to increase of the low energy densi ty\nof states on the circle q⊥=λ/2. HereT0\ncis the critical tem-\nperature for a non-interacting Bose system without SOC. We\nhave set g↑↓=g.\n−lnZ/β=S0/β+Ωf, where the second term\nΩf=/summationdisplay\nq,s/bracketleftigg\nln(1−e−βωqs)\n2β+ωqs−ǫq\n2+(g2+g2\n↑↓)n2\n0\n2q2/bracketrightigg\n(3)\nis the result of the Gaussian integration. The density of\nparticles excited out of the condensates for either species\ncan be easily evaluated as nf= 1/(2L3)(∂Ωf/∂µ)µ=µp.\nAt the critical temperature the condensed density n0= 0\nandnfis equal to the density of the total number of\nparticlesn. In this special case, it is straightforward to\ncalculate analytically the poles of the Green’s function\nand obtain the following secular equation determining\nthe critical temperature\nn=1\n2L3/summationdisplay\nq,s=±1\ne[q2z+(q⊥+sλ/2)2]/Tc−1.(4)\nIntheabsenceofSOC,thetwotermsareidenticalandwe\nget the usual number equation to determine the critical\ntemperature of condensation. The biggest contribution\nto the sum over the momentum is from the vicinity of\nthe point |q|= 0. SOC changes the situation. Now the\nbiggest contribution to the sum comes from the circle\nqz= 0,q⊥=λ/2 with much higher weight than in the\nprevious case. The critical temperature Tcmust be de-\ncreasedforfixeddensity. Morequantitatively, the critical\ntemperature in the thermodynamic limit is estimated as\nTc≈T0\ncg3/2(1)\ng3/2(z). (5)/s48/s46/s48/s48 /s48/s46/s50/s53 /s48/s46/s53/s48 /s48/s46/s55/s53 /s49/s46/s48/s48/s48/s46/s48/s48/s48/s46/s50/s53/s48/s46/s53/s48/s48/s46/s55/s53/s49/s46/s48/s48/s110\n/s48/s47/s110\n/s84/s47/s84\n/s67/s32 /s61/s48\n/s32 /s110/s49/s47/s51\n/s32 /s110/s49/s47/s51\nFIG. 2. (color online). The condensate fraction as a func-\ntion of temperature for various spin-orbit coupling streng ths\nλ. The effect of SOC on the thermal depletion is more pro-\nnouncedathighertemperatures, sincethethermalcomponen t\nbenefits more from the low energy states than the condensed\none. We have set g↑↓=g.\nHere,T0\nc= 4π[n/g3/2(1)]2/3is the critical tempera-\nture for a non-interacting Bose gas without SOC, z=\nexp[−λ2/(4Tc)], andgp(z) =/summationtext∞\nl=1zl/lp[29]. As shown\nin Fig. 1, the critical temperature Tcindeed decreases\nrapidly as the SOC strength increases.\nThe situation is more complicated for temperatures\nsmaller than the critical temperature. The secular equa-\ntionshouldbesupplementedwiththeequationfortheto-\ntal densityn0+nf=n. By solving the two equations we\nobtain the condensate density n0for a fixed total density\nofatomsn. BeinganintrinsicpropertyofaBose-Einstein\ncondensate, the condensate fraction n0/nprovides key\ninformation about the robustness of the superfluid state.\nIt is shown in Fig. 2, where three typical SOC couplings\n(λ= 0,λ/n1/3= 0.2, andλ/n1/3= 0.4) are chosen\nfor comparison. In the absence of SOC, the condensate\nfraction decreases gradually to zero as the temperature\nreachesT0\nc, since under the Popov approximation, the\ncritical temperature of a weakly interacting Bose gas is\nidentical to that of a non-interacting one [30, 31]. The\neffect of SOC on the condensate fraction is more pro-\nnounced when the temperature gets closer to the tran-\nsition temperature Tc. We can explain this by a similar\nargument leading to the reduction of the critical temper-\nature: At a fixed temperature the thermal component\nbenefits mainly from the low energy states living on the\ncircleq⊥=λ/2, while the condensed component mainly\nfrom a single momentum at qx=λ/2. Therefore, SOC\naffects the thermal component more than the condensed\none, leading to the more pronounced thermal depletion\nat higher temperatures.4\nWe turn now to study the effects of SOC on the sys-\ntem in two spatial dimensions. In the absence of SOC\nthere is a quasi-long range order in two dimensions at\nsufficiently low temperatures with correlation function\ndecaying according to some power law. At higher tem-\nperatures the correlation function decays exponentially,\nwhere the algebraic order is destroyed by proliferation\nof vortices. The BKT separates these two regimes [32].\nSince there is no condensation now, the Popov approx-\nimation is not applicable. Here we derive a low energy\neffective theory suitable to probe the correlation func-\ntion. Anticipating that phase degrees of freedom play an\nessential role now, we adopt the density-phase represen-\ntation of the field operators, ψσ=ρσeiθσ. The fields are\nthen separated again into a mean-field part and a fluctu-\nating part as ρσ=ρ0+δρσandθσ=θ0σ+δθσ. Here,\nρ0is usually called a quasi-condensate density and the\nexplicit breaking of the rotational symmetry by SOC re-\nsults inθ0↑=θ0↓−π=λx/2. Retaining terms of zeroth\nand quadratic order in the fluctuating fields, the action\nis split into two parts S≈S0+Sf, where similarly to the\nprevious case S0=βL2/bracketleftbig\n−2(λ2/4+µ)ρ0+(g+g↑↓)ρ2\n0/bracketrightbig\nis the mean-field contribution. The Gaussian action Sf\ncontains the fluctuating fields up to the second order.\nThis allows us to integrate the density fluctuations δρσ\nout yieding\nSf=/integraldisplay\ndτ/integraldisplay\ndr/braceleftbigg(∂τθ+)2\ng+g↑↓+(∂τθ−+i2λ∂xθ−)2\ng−g↑↓+λ2/(2ρ0)\n+ρ0\n2/bracketleftbig\n(∇θ+)2+(∇θ−)2/bracketrightbig\n+ρ0λθ−∂yθ++ρ0\n2λ2θ2\n−/bracerightig\n,(6)\nwhereθ±=δθ↑±δθ↓. In the absence of SOC, the\nabove action describes two decoupled quantum XY mod-\nels representing two branches of phase fluctuations with\nlinear spectrum in momentum space yielding the quasi-\nlong range order. The corresponding correlation func-\ntionsC±(r) =∝angb∇acketlefte[θ±(r)−θ±(0)]∝angb∇acket∇ight ∝r−T/ρ0(g±g↑↓)decay ac-\ncording to the power law rather than exponentially [32].\nIn the presence of SOC, we notice that the field θ−de-\nscribes a massive fluctuation as the action Sgcontains\nthe termρ0λ2θ2\n−/2. At low energy, we can integrate out\nthe massive field θ−by assuming homogeneous temporal\nfluctuations to get an effective action solely in terms of\nthe collective variable θ+(for simplicity we set g≈g↑↓):\nSf≈1\n2/integraldisplay\ndτ/integraldisplay\ndr/bracketleftbigg1\ng(∂τθ+)2+ρ0(∂xθ+)2+ρ0\nλ2(∂2\nyθ+)2/bracketrightbigg\n.\n(7)\nThe first two terms in the action are the part of the fa-\nmiliar XY model. The third term is the manifestation of\nthe broken rotational symmetry and it adds a twist. To\nsee this we calculate the spectrum of elementary excita-\ntions. We first Fourier transform the action for it to be-\ncomeSf= 1/2/summationtext\nq,n(w2\nn/g+ρ0q2\nx+ρ0q4\ny/λ2)|θ+(q,wn)|2.\nThe phase fluctuations in the Fourier space can be eas-\nily evaluated, ∝angb∇acketleft|θ+(q,wn)|2∝angb∇acket∇ight=g(w2\nn+w2\nq)−1. The polesof this expression gives the low energy spectrum of the\nsymmetric phase wq=/radicalig\ngρ0(q2x+q4y/λ2). It is clear\nthat now the energy carried by elementary excitations of\nthe system does not scale linearly as in the XY model\ncase. As a result, less energy is required to excite low\nenergy modes as compared to the XY model case, en-\nhancing the role of quantum fluctuations. The Lan-\ndau criterion for the superfluid critical velocity gives\nminq(wq/q) = 0, implying SOC destroys superfluidity\nin two dimensions. This is substantiated by the form of\nthe correlation function, which reads C+(r)≈e−Iwith\nI=/summationtext\nq,n∝angb∇acketleft|θ+(q,wn)|2∝angb∇acket∇ight(1−eiqr)/2β. The asymptotic\nbehavior of it for large separations along the x-direction\nisI≈T/radicalbig\nλ|x|/(2πgρ0) and along the y-direction is\nI≈Tλ|y|/(2gρ0). Hence, at any finite temperature, the\ncorrelation function decays exponentially along both x-\nandy- directions, in stark contrast to the power law ex-\nhibited by the conventional XY model in the low tem-\nperature phase. The superfluid order in two dimensions\nis predicted to be disordered by SOC.\nSummary. – We predict dramatic implications of SOC\non the thermal properties of atomic Bose gases: In three\ndimensions SOC reduces the critical temperature of con-\ndensation and enhances the thermal depletion of the con-\ndensate, while in two dimensions it destroys superfluid\norder at any finite temperature. This is explained by the\nrole of quantum fluctuations amplified by SOC. We hope\nthatcurrentworkwillstimulateexperimentstoverifyour\npredictions and will add to the overall understanding of\nthe thermal quantum fluids.\nWe would like to thank Zhi-Gao Huang, Xiu-Min Lin\nand Joachim Brand for useful discussions. RL was sup-\nported by the NBPRC under Grant No. 2011CBA00200,\nthe NSFC under Grants No. 11274064, and by NCET.\nOF was supported by the Marsden Fund Council from\nGovernment funding (contract No. MAU1205), adminis-\ntrated by the Royal Society of New Zealand.\n[1] D. Xiao, M.-C. Chang, and Q. Niu, Rev. Mod. Phys. 82,\n1959 (2010), M. Z. Hasan and C. L. Kane, ibid. 82, 3045\n(2010); X. L. Qi and S. C. Zhang, ibid. 83, 1057 (2011).\n[2] V. Mourik, K. Zuo, S. M. Frlov, S. R. Plissard,\nE.P.A.M.Bakkers, and L. P. Kouwenhoven, Science 336,\n1003 (2012)\n[3] D. Jaksch and P. Zoller, Annals of Physics 315, 52 (2005)\n[4] Y.-J. Lin, R. L. Compton, A. R. Perry, W. D. Phillips,\nJ. V. Porto, and I. B. Spielman, Phys. Rev. Lett. 102,\n130401 (2009); Y.-J. Lin and R. L. Compton and K.\nJim´ enez-Garcia and J. V. Porto and I. B. Spielman, Na-\nture462, 628 (2009); Y.-J. Lin and K. Jim´ enez-Garcia\nand I. B. Spielman, Nature 471, 83 (2011).\n[5] M. Aidelsburger, M. Atala, S. Nascimb´ ene, S. Trotzky,\nY.-A. Chen, and I. Bloch, Phys. Rev. Lett. 107, 255301\n(2011)\n[6] J.Zhang, S.Ji, Z.Chen, L.Zhang, Z.Du,B.Yan, G.Pan,5\nB. Zhao, Y. Deng, H. Zhai, S. Chen, and J. Pan, arxiv:\n1201.6018\n[7] P. Wang, Z.-Q. Yu, Z. Fu, J. Miao, L. Huang, S. Chai,\nH. Zhai, and J. Zhang, Phys. Rev. Lett. 109, 095301\n(2012)\n[8] L. W. Cheuk, A. T. Sommer, Z. Hadzibabic, T. Yefsah,\nW. S. Bakr, and M. W. Zwierlein, Phys. Rev. Lett. 109,\n095302 (2012)\n[9] T. D. Stanescu, B. Anderson, and V. Galitski, Phys. Rev.\nA78, 023616 (2008)\n[10] C. Wang, C. Gao, C. M. Jian, and H. Zhai, Phys. Rev.\nLett.105, 160403 (2010)\n[11] C.-M. Jian and H. Zhai, Phys. Rev. B 84, 060508 (2011)\n[12] T.-L. Ho and S. Zhang, Phys. Rev. Lett. 107, 150403\n(2011)\n[13] Y. Li, L. P. Pitaevskii, and S. Stringari, Phys. Rev. Let t.\n108,225301(2012); Y.Li, G.I.Martone, L.P.Pitaevskii,\nand S. Stringari, ibid. 110, 235302 (2013).\n[14] B. M. Anderson, G. Juzeli¯ unas, V. M. Galitski, and I. B.\nSpielman, Phys. Rev. Lett. 108, 235301 (2012); B. M.\nAnderson, I. B. Spielman and G. Juzeli¯ unas, Phys. Rev.\nLett.111, 125301 (2013).\n[15] M. Iskin and A. L. Subasi, Phys. Rev. Lett. 107, 050402\n(2011), M. Gong, S. Tewari, and C. Zhang, ibid. 107,\n195303 (2011); H. Hu, L. Jiang, X.-J. Liu, and H. Pu,\nibid.107, 195304 (2011); Z.-Q. Yu and H. Zhai, ibid.\n107,195305 (2011); R. Liao, Y. Yi-Xiang and W-M. Liu,\nibid.108, 080406(2012).\n[16] S. Sinha, R. Nath, and L. Santos, Phys. Rev. Lett.\n107, 270401 (2011); H. Hu, B. Ramachandhran, H. Pu,\nand X.-J. Liu, Phys. Rev. Lett. 108,010402 (2012); T.Kawakami, T. Mizushima, M. Nitta, and K. Machida,\nPhys. Rev. Lett. 109,015301 (2012); Z. F. Xu, Y.\nKawaguchi, L. You, and M. Ueda, Phys. Rev. A 86,\n033628 (2012).\n[17] H. Hu and X.-J. Liu, Phys. Rev. A 85, 013619 (2012)\n[18] Q. Zhou and X. Cui, Phys. Rev. Lett. 110, 140407 (2013)\n[19] R. Barnett, S. Powell, T. Grass, M. Lewenstein, and\nS. DasSarma, Phys. Rev. A 85, 023615 (2012)\n[20] X. Cui and Q. Zhou, Phys. Rev. A 87, 031604 (2013)\n[21] T. Ozawa and G. Baym, Phys. Rev. Lett. 109, 025301\n(2012)\n[22] J.-Y. Zhang, S.-C. Ji, L. Zhang, Z.-D. Du, W. Zheng, Y.-\nJ. Deng, H.Zhai, S.Chen,andJ.-W. Panarxiv:1305.7054\n[23] A. Altland and B. Simons, Condensed Matter Field The-\nory(Cambridge University Press, Cambridge, 2006)\n[24] A. Griffin, Phys. Rev. B 53, 9341 (1996)\n[25] N. Hugenholtz and D. Pines, Phys. Rev. 116, 489 (1959)\n[26] J. Goldstone, Phys. Rev. 127, 2 (1962)\n[27] V. Popov, Functional integrals in quantum field theory\nand stastical physics (Reidel, Dordrecht, 2001)\n[28] J. O. Anderson, Rev. Mod. Phys. 76, 599 (2004)\n[29] L. Pitaevskii and S. Stringari, Bose-Einstein Condensa-\ntion(Oxford, New York, 2003) page 17\n[30] F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. String ari,\nRev. Mod. Phys. 71, 463 (1999)\n[31] C. Pethick and H. Smith, Bose-Einstein Condensation in\nDilute Gases (Cambridge University Press, Cambridge,\n2008)\n[32] N. Nagaosa, Quantum Field Theory in Condensed Matter\nPhysics (Springer, Berlin, 1999)"    },    {        "title": "1812.07949v2.Manifestations_of_spin_orbit_coupling_in_a_cuprate_superconductor.pdf",        "content": "arXiv:1812.07949v2  [cond-mat.supr-con]  12 Dec 2019Manifestations of spin-orbit coupling in a cuprate superco nductor\nZachary M. Raines,∗Andrew A. Allocca, and Victor M. Galitski\nJoint Quantum Institute and Condensed Matter Theory Center , Department of Physics,\nUniversity of Maryland, College Park, Maryland 20742-4111 , U.S.A.\nExciting new work on Bi2212 shows the presence of non-trivia l spin-orbit coupling effects as seen\nin spin resolved ARPES data [Gotlieb et al.,Science,362, 1271-1275 (2018)]. Motivated by these\nobservations we consider how the picture of spin-orbit coup ling through local inversion symmetry\nbreaking might be observed in cuprate superconductors. Fur thermore, we examine two spin-orbit\ndriveneffects, thespin-Hall effect andtheEdelstein effect, focusing onthe details of theirrealizations\nwithin both the normal and superconducting states.\nI. INTRODUCTION\nSince their discovery three decades ago, the cuprate\nfamily of superconductors has been a focus of intense\nresearch interest.1To this day they maintain the high-\nest superconducting transition temperature at ambient\npressure.2Despite many years of active investigation\nthe cuprates continue to generate new discoveries and\nnew puzzles. Discussion continues on phenomena rang-\ning from the exact nature of the pairing mechanism, to\nthe origin of the pseudogap,3–5and the various charge-\nordered states now being observed.6–9\nWhile there has been some work on the consequences\nof spin-orbit coupling in the cuprates, it is generally be-\nlieved that such effects are weak10–14and they are often\nignored. However, recent spin-resolved ARPES measure-\nments have shown striking evidence of spin textures in\nthe Brillouin zone.15In particular, the observed behav-\nior can be explained by a model of spin-orbit coupling\nwhich is opposite on the two layers of the BSCCO unit\ncell. Such a model preserves the inversion symmetry of\nsystem but can still host non-trivial effects arising from\nthe spin-orbit coupling. There is precedent for supercon-\nductors with such a staggered noncentrosymmetry,16but\nthe consequences for the cuprate superconductors have\nnot yet been investigated.\nIt is well established both theoretically17–21and\nexperimentally22–24that systems with spin-orbit cou-\npling may display novel transport properties linking spin\nand chargedegrees of freedom. These are typically called\nspintronic effects, and provide a potential means to ma-\nnipulate spins with charges and vice versa.25,26One of\nthemostcommonlyconsideredoftheseeffectsisthe spin-\nHall effect, in which a net spin polarization accumulates\nat the boundaries of a sample in response to an electric\ncurrent. As in the caseof the Hall effect, the spin-Hall ef-\nfect can also be related to a transversecurrent associated\nwith the accumulated quantity, although in the case of a\nsuperconductor the relation between the two viewpoints\nis moresubtle. Another notable phenomenon is the Edel-\nstein or inverse spin-galvanic effect, which relates a spin\npolarization throughout the bulk of a sample to the flow\nof a charge current.10In light of the observations of spin-\norbit coupling in cuprate superconductors, the question\nnaturally arises as to how such effects manifest in thesematerials,particularlywithinthesuperconductingphase.\nThe structure of this paper is as follows. In Section II\nwe introduce the model of spin-orbit coupling in BSCCO\nand discuss some of its properties. In Section III we re-\nview the theory of superconductivity in spin-orbit cou-\npled materials and construct a general Bogoliubov-de\nGennes Hamiltonian describing d-wave superconductiv-\nity in this model. In Sections IV and V we then use this\nmodel to calculate spintronic effects in both the normal\nand superconducting states. In Section VI we review and\ndiscuss our results.\nII. MODEL\nIn this work we use a model which was introduced\nto explain the experimentally observed spin-texture in\nBi2212 under spin-resolved ARPES.27The model is a\ntight-binding description of the copper sites in a bilayer\nof CuO 2planes. It is given by\nH0=/summationdisplay\nkc†\nk/parenleftbig\nξ(k)+t⊥(k)τx+λ(k)·στz/parenrightbig\nck,(1)\nwhereξ(k) includes hopping terms up to third-nearest\nneighbor, leading to a hole-like Fermi surface, λ(k) =\nλ(sinky,−sinkx,0) is the spin-orbit coupling vector, and\nt⊥(k) =t⊥(coskx−cosky)2is the interlayer hopping\nterm.28,29Hereτandσrepresent two sets of Pauli ma-\ntrices, with the τmatrices operating in the layer space\nand theσmatrices operating in spin space, and ckis the\n4-component vector of electron annihilation operators in\nthe spin and layerspaces. The form ofthe spin-orbitcou-\npling can be ascribed to local inversion symmetry break-\ning between the layers; field effects in between the layers\nlead to inversion symmetry breaking with opposite sense\nin the top and bottom layer, such that the system as a\nwhole retains inversion symmetry.\nThis system contains two Kramers degenerate bands\nwith energies\nǫb(k) =ξ(k)+bA(k), (2)\nwhereb=±andA(k) =/radicalBig/vextendsingle/vextendsingleλ(k)/vextendsingle/vextendsingle2+t⊥(k)2. It should\nbe noted that at the level of the electronic dispersion,2\nFIG. 1. The Fermi surfaces for the two bands with spin\naligned along the spin-orbit coupling vector λ(k). For the\nopposite helicity the two bands are exchanged. The spin-\ntexture for the band parallel to the spin-orbit vector is sho wn\nby the arrows.\nspin-orbit coupling enters in the same manner as inter-\nlayer coupling and it is difficult to disentangle the two.\nThe eigenstates of this model can be expressed as tensor\nproducts of spin and layer space states as\n|b↑/angb∇acket∇ight=|↑/angb∇acket∇ightσ⊗|b/angb∇acket∇ightτ\n|b↓/angb∇acket∇ight=|↓/angb∇acket∇ightσ⊗τx|b/angb∇acket∇ightτ. (3)\nThe states |↑/angb∇acket∇ightσand|↓/angb∇acket∇ightσare defined as the spin states\npointing parallel or anti-parallel to λ(k), i.e. helicity\nstates, while |b=±/angb∇acket∇ightτare the states with layer pseudo-\nspin parallel or anti-parallel to ( t⊥,0,|λ|). They can be\nexpressed as\n/angb∇acketleftσ|h/angb∇acket∇ightσ=1√\n2/parenleftBig\n1he−iφ(k)/parenrightBigT\n/angb∇acketleftτ|+/angb∇acket∇ightτ=/parenleftbigw(k)z(k)/parenrightbigT\n/angb∇acketleftτ|−/angb∇acket∇ightτ=/parenleftbig−z(k)w(k)/parenrightbigT(4)\nwhereλcos/parenleftbig\nφ(k)/parenrightbig\n=λ(k)·ˆxand\nw(k), z(k) =/radicalBigg\n1\n2/parenleftbigg\n1±λ(k)\nA(k)/parenrightbigg\n(5)\nwhich implicitly defines the change of basis ψk=ˇUck\nto eigenstate operators. The structure of the eigenstates\nleads to two Kramers degenerate Fermi surfaces, split\nfrom each other by the spin-orbit and interlayer coupling\nas depicted in Fig. 1. Each Kramers doublet consists of\nstates with helical winding of the electronic spin about\nthe Brillouin zone center in opposite senses.\nFor the purposes of calculating response functions in\nSections IV and V below it is convenient to re-expressthe Hamiltonian in the following manner. We define the\nmatrices\nˇΣ0=σzτx,ˇΣ1=σxτy,\nˇΣ2=σyτy,ˇΣ3=σzτ0.(6)\nThese matrices along with the products ˇΣ0ˇΣiare closed\nunder the commutators\n/bracketleftBig\nˇΣ0,ˇΣi/bracketrightBig\n= 0,/bracketleftBig\nˇΣi,ˇΣj/bracketrightBig\n= 2iǫijkˇΣk(7)\nand generatethe algebra so(4)⊕u(1) with the ˇΣiforming\nansu(2) sub-algebra. In terms of these matrices along\nwith the the modified adjoint ¯ c≡c†ˇΣ0, the Hamiltonian\nis simply\nH0=/summationdisplay\nk¯ck/parenleftBig\nξkˇΣ0+d(k)·ˇΣ/parenrightBig\nck (8)\nwhere we have defined the vector\nd(k)≡A(k)n(k)≡(λsinkx,λsinky,t⊥(k))T,(9)\nwith/vextendsingle/vextendsingled(k)/vextendsingle/vextendsingle≡A(k) and unit vector n(k).\nFor all calculations in this work we use intralayer hop-\nping strengths t1= 1,t2=−0.32,t3= 0.16, inter-\nlayer hopping strength t⊥= 0.08, and chemical potential\nµ=−1.18.\nIII. BOGOLIUBOV-DE GENNES\nHAMILTONIAN\nWhen writing the Bogoliubov-de Gennes Hamiltonian\nfor the superconducting state of this model, we impose\nseveral constraints in order to match empirical details of\nsuperconductivity in this system. The order parameter\nin cuprates is known to belong to the B1grepresentation\n(dx2−y2), which we enforce by hand. We additionally re-\nstrict pairing to be between degenerate bands; pairing\nbetween bands with different energies would either re-\nquire pairing at finite momentum, which is not observed,\nor pairing of excitations away from the Fermi surface,\nwhich is energeticallydisfavored. Finally, we impose that\nthe system remains inversion symmetric.\nWith these restrictions we can now write the BdG\nHamiltonian for this system as\nHBdG=/summationdisplay\nkbh′Ψ†\nkhb/bracketleftbigg\nǫb(k) ∆bf(k)\n∆∗\nbf(k)−ǫb(k)/bracketrightbigg\nΨkhb.(10)\nHere ∆ bis the order parameter for pairing in each band\nandf(k) = coskx−coskyis thed-wave form factor.\nThe Nambu spinor Ψ =/parenleftBig\nψk˜ψ†\nk/parenrightBigT\nis defined in terms of\nthe eigenstate operators ψbhassociated with the states in\nEq. (3) and their time-reverse ˜ψ= ΘψΘ−1, where Θ is\nthe time-reversal operator. We do this because this sys-\ntem has non-trivial properties under time-reversal owing3\nto the presence of spin-orbit coupling. The notation/summationtext′\nindicates that we sum overonly half the Brillouin zone to\navoid double counting states that would naturally arise\nin this framework.\nWe have written our Nambu spinors in this form be-\ncause the usual procedure of considering pairing be-\ntween states of opposite spin and momentum would\nhaveunfavorableconsequencesin systemswith inversion-\nsymmetrybreakingorspin-orbitcoupling,namelythesu-\nperconducting gap function would no longer transform-\ning as a representation of the point group of the system.\nThe order parameter would acquire an extra phase under\ngroup operations that could not be removed by a gauge\ntransformation, and so provides an obstruction to clas-\nsifying its symmetry. By writing the BdG Hamiltonian\nexplicitly in terms of operators and their time-reversal,\nhowever, this problem is avoided.30–32Additionally, one\nrecovers the notion of separation into singlet and triplet\ngaps, where now this distinction is with respect to helic-\nity instead of spin along a fixed quantization axis.\nThe global inversion symmetry of this system enforces\nthat the gap is singlet in helicity space, analogously to\nthe case of a system without spin-orbit coupling. Note\nthat unlikein the caseofan inversionsymmetry-breaking\nsuperconductor, our quasiparticle bands remain doubly\ndegenerate. The difference of the gap magnitude in the\ntwo bands depends on the strength of the interaction in\nthe respective channels, which we do not focus on here.\nOnecan diagonalizethe Hamiltonian Eq.(10) asasum\nof normal BdG Hamiltonians, and the Bogoliubov quasi-\nparticle dispersions are found to be\nEb(k) =/radicalBig\nǫb(k)2+|∆b|2f(k)2. (11)\nBecause of the singlet nature of the gap, Bogoliubov\nquasiparticles are superpositions of a quasi-electron state\nandaquasi-holeinthecorrespondingtime-reversedstate.\nThese two states will have the same spin and so as a di-\nrect consequence, the Bogoliubov quasiparticles inherit\nthe spin-texture of the normal state bands. This means\nthat near the nodes, there are gapless spin-orbit-coupled\nexcitations.\nFinally, we note that the parametrization introduced\nin Eq. (9) can be straightforwardly extended to the BdG\nHamiltonian Eq. (10), allowing the response functions\nto be neatly expressed in terms of the vector dand its\nderivatives (for more details see Appendix A).\nIV. SPIN-HALL EFFECT\nOne of the most commonly studied spin transport ef-\nfects in spin-orbit coupled materials is the spin-Hall ef-\nfect (SHE), in which spin accumulates on the boundaries\nof a material parallel to the electrical current flowing\nthrough it, with the projection of the spin being opposite\non opposing boundaries, as depicted in Fig. 2. A quan-\ntity often considered in the context of the SHE is the\nFIG. 2. (Color online) A schematic depiction of the spin-Hal l\neffect. The charge current jis split into spin-up (red) and\nspin-down (blue) components, which accumulate on oppos-\ning boundaries. This can be interpreted as a charge current\ninducing a transverse spin current, jz\nS.\nspin-hall conductivity, which describes the flow of a spin-\ncurrent perpendicular to an applied electric current, with\nthe projection of the carried spin being perpendicular to\nthe plane defined by the currents themselves.26Here we\nfocus on the spin-hall conductivity as a hallmark of the\nSHE, but note that the two are not necessarily simply\nrelated, as will be discussed further below.\nIn particular, we are interested in the dc intrinsic spin-\nHall conductivity\nσz\nxy= lim\nω→0lim\nq→0Πint\nxy(q,iωm→ω+i0+)\niω,(12)\nwritten here in terms of the intrinsic contribution to the\nspin-Hall response,\nΠint\nxy(q,ωm) =/angbracketleftBig\njz\nS,x(q,ωm)jy(−q,−ωm)/angbracketrightBig\n,(13)\njz\nS,x(q,ωm) =/summationdisplay\nk,ǫnc†\nk−q\n21\n2/braceleftbig\nvx(k),σz/bracerightbig\nck+q\n2,(14)\njy(q,ωm) =e/summationdisplay\nk,ǫnc†\nk−q\n2vy(k)ck+q\n2.(15)\nHerek= (k,ǫn) stands for the momentum and Mat-\nsubara frequency, respectively, e=−|e|is the electron\ncharge,vi(k) =∂H0(k)/∂kiis the velocity operator,\nandc†\nk,ckare the electron creation and annihilation op-\nerators. For our analysis these operators create quasi-\nparticles of definite spin and layer index. Equation (14)\ngives a common convention for the spin-current, and\nthe superscript zindicates the polarization of the spin-\ncurrent.33Explicit calculation leads to\nσz\nxy= 4e/summationdisplay\nk′∂ξ\n∂kx/parenleftBigg\n∂d\n∂ky×n/parenrightBigg\n·ˆz\n×/bracketleftBigg/parenleftbiggǫ+\nE++ǫ−\nE−/parenrightbiggπN\nE+−E−\n+/parenleftbiggǫ+\nE+−ǫ−\nE−/parenrightbiggπS\nE++E−/bracketrightBigg\n(16)4\n0.0 0.5 1.0 1.5 2.0\nT/T c0.860.880.900.920.940.960.981.00σz\nxy/σz\nxy ;N\n0.01000.01880.02750.03620.04500.05380.06250.07130.0800\nFIG. 3. (Color online) The total DC spin-Hall conductivity\nσz\nxyas a function of temperature for different values of the\nspin-orbit coupling strength λ. The conductivity is roughly\nconstant in the normal phase. We therefore normalize all\nresults by the value of the spin-hall conductivity in the nor -\nmal phase for the corresponding choice of λ. The spin hall\nconductivity however exhibits a decrease with the onset of s u-\nperconductivity. This can be attributed to a finite energy fo r\nreorienting spins associated with the formation of singlet -like\nCooper pairs in the superconducting phase. In these calcula -\ntions, we have chosen Tc= 0.016t1.\nwhere we have defines the normal- and superfluid-like\nbubbles\nπN/S(k) =n(E−(k))−n(±E+(k))\nE+(k)∓E−(k)(17)\nwithnthe quasiparticle occupation function, and nand\ndthe vectors introduced in the parametrization Eq. (9).\nWe have suppressed the dependence on the momentum\nk.\nSpin-orbit coupling will in general lead to a non-zero\nspin-Hall conductivity. There are notable exceptions,\nhowever, where the spin-Hall conductivity exactly van-\nishes, and the exact conditions under which it remains\nfinite, particularly for Rashba spin-orbit coupling, have\nbeen a subject of lively discussion.20,34–41These argu-\nments for a vanishing of the spin-Hall conductivity do\nnot hold in our model, however, and we find a non-zero\nresult.\nOne of the main issues regarding the consideration of\nthe spin-Hall conductivityis the difficulty in relatingit to\nexperimental measurements of the spin-Hall effect. Since\nspin is not a conserved quantity in a system with spin-\norbit coupling, it does not obey a continuity equation,\nso a spin-current cannot be consistently and rigorously\ndefined; Eq. (14), which we use in our calculation, is a\ncommon choice, but is not uniquely determined. There-\nfore, the accumulation of spin at the boundaries of the\nsystem is not directly related to the spin-Hall conductiv-\nityinthe samewaythataccumulationofchargeisrelated\nto electrical Hall conductivity.\nThis can be most easily demonstrated by considering\nFIG. 4. (Color online) A schematic depiction of the Edelstei n\neffect. The charge current jinduces a uniform transverse spin\npolarization s= ˆαEEjthroughout the bulk of the material.\nthe transformation properties of spin and spin-current\nundertimereversal. AspointedoutbyRashba,42thedef-\ninition of the spin current used in defining the spin-Hall\nconductivity is even under time reversal, while magneti-\nzation measured at sample boundaries is odd under the\nsame operation. Consequently, there must be some addi-\ntional time-reversal-symmetry breaking process that re-\nlates spin-current to spin accumulation at sample bound-\naries, which is not included in calculations of the spin-\nHall conductivity. Indeed, we note that we find a total\nspin-Hall conductivity, despite the fact that the system\ndoes not break global inversion symmetry. Furthermore,\nthe sense of the spin-Hall conductivity does not depend\non the sign of the spin-orbit coupling. This is a strongin-\ndication that the spin-Hall conductivity as traditionally\ncalculated is not directly related to an observable.\nNonetheless, we have included this calculation of the\nspin-Hall conductivity for completeness. For the afore-\nmentioned reasons it is not clear how to relate this result\nto a precise experimental prediction, except to note that\na non-zerospin-Hall conductivity typically indicates that\naspin-Halleffect canbe observed. Becausethe spin-orbit\ncoupling is seen to change sense between layers, we may\nexpectwhateverspin-Hallresponsethereisinexperiment\nto be layer-staggered as well.\nV. EDELSTEIN EFFECT\nAnother frequently considered spintronic effect is the\nEdelstein effect, also called the inverse spin-galvanic ef-\nfect (ISGE). In the Edelstein effect a charge current gen-\nerates a uniform spin polarizationthroughout the bulk of\na SOC material.10Traditionally the effect is discussed in\nthe context ofapplyingan electricfield andobservingthe\nresultant spin polarization, and such behavior has been\nexperimentally observed.22,43\nThe effect can typically be quantified through the dc\nEdelstein conductivity σEEgiven by\nχEE(q,iωm) =/angbracketleftbig\nsx(q,iωm)jy(−q,−iωm)/angbracketrightbig\nσEE= lim\nω→0lim\nq→0χEE(q,iωm→ω+i0+)\niω,(18)5\nwhich defines the linear response relationship sx=\nσEEEy. Thisrelationis, however,problematicinthe con-\ntext of a superconductor. As stated above, the Edelstein\neffect is, properly, the relationship between spin polar-\nization and current flow. In the normal state the current\nis directly related to an applied electric field through the\ndc electrical conductivity ji=σEi, and soσEEis an\naccurate proxy for the strength of the effect. However,\nin a superconductor (or the pathological case of a metal\nwithout disorder), the electrical conductivity σis infi-\nnite, so application of an electric field does not result in\na steady state current, and the dc Edelstein conductivity\nin Eq. (18) is not a meaningful quantity.\nInstead, in order to consider the Edelstein effect in a\nsuperconductor we need to directly relate the spin po-\nlarization and the supercurrent as sx=αEEjy, with\nαEEnow being the quantity of primary interest. Con-\nsidering the finite frequency response of the system, we\ncan relateαEE(ω) to the Edelstein susceptibility χEE(ω)\nabove and the electromagnetic response function Π( ω)\nfrom which one obtains the dc electrical conductivity as\nσ= limω→0Π(ω)/(iω). We have\nsx(ω) =αEE(ω)Π(ω)\niωEy(ω)\n/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\n=jy(ω)=χEE(ω)\niωEy(ω)\n=⇒αEE(ω) =χEE(ω)\nΠ(ω),(19)\ngivingαEE(ω), forwhichthedclimit ω→0isfinite. This\nresult will also be useful in evaluating the normal state\nresponse since our model contains no disorder effects.\nWe can understand, at a heuristic level, how the cur-\nrent and spin can be related through the following argu-\nment. Let us consider the case where this model contains\na supercurrent. The Cooper pairs then acquire finite mo-\nmentum Q. We can absorb this momentum by making\nthe gauge transformation A→A+Q/(2e). To lowest\norder in Qthe action is shifted to S−j·Q/(2e). Com-\nputing the spin expectation value of the system, we find\nto lowest order in Q\n/angb∇acketleftsx/angb∇acket∇ight=−Q\n2e/angbracketleftbig\nsxjy/angbracketrightbig\n=−Q\n2eχEE(0,ω→0).(20)\nFrom Ginzburg-Landau theory we have that the super-\ncurrent is\njy=−e\nmnsQ, (21)\nwherensis the density of superfluid electrons, and we\nassumethat quasiparticlesdonot significantlycontribute\ntothecurrent,sothisapproximatelyrepresentstheentire\ncharge current of the system. We thus have that in the\nsuperconducting state\n/angb∇acketleftsx/angb∇acket∇ight=mχEE\n2e2nsjy. (22)\nSo in the case of a uniform supercurrent the non-\nvanishing of the Edelstein susceptibility χEEimpliesa relationship between the supercurrent and spin-\npolarization, as expected from Eq. (19). Such a relation\nwas studied in the case of an s-wave pairing previously.12\nWe note, however, that because our model is inversion\nsymmetric, there can be no total spin polarization in the\nsystem,sx= 0. However, the layer-staggered analogs of\nEqs. (19) and (22), corresponding to the response of the\nlayer-staggered spin ˜sx≡sxτz, behave in much the same\nmanner.\nThis response ˜ sx=αEEjS\nycan obtained be via the\nanalog of (19), replacing the normal spin with layer-\nstaggered spin. Some calculation allows us to find the\ngeneral formulae\nχ(ω→0) = 8e/summationdisplay\nk′A(k)∂ny(k)\n∂ky\n×/parenleftBig\nl(k)2πN(k)+p(k)2πS(k)/parenrightBig\n(23)\nand\nΠ(ω→0) = 2e2/summationdisplay\nk′\n×\n2A(k)2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂n(k)\n∂ky/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2/parenleftBig\nl(k)2πN(k)+p(k)2πS(k)/parenrightBig\n+/summationdisplay\nb\n∂2ǫb(k)\n∂k2y−bA(k)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂n(k)\n∂ky/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\nǫb(k)\n2Eb(k)tanhEb(k)\n2T\n,\n(24)\nwhere we have defined the coherence functions\nl(k)2,p(k)2= 1±ǫ+(k)ǫ−(k)+∆+∆−f2\nk\nE+(k)E−(k),(25)\nnis again the unit vector introduced in Eq. (9), and\nπN/Sare the quantities defined in Eq. (17). The cor-\nresponding expressions for the normal state are found\nas the ∆ →0 limit of these. The quantity αEE, ob-\ntained as lim ω→0χ(ω)/Π(ω), is plotted in Fig. 5 as a\nfunction oftemperature acrossthe superconductingtran-\nsition. Whereas the effect is only weakly dependent on\ntemperature above Tc, in the superconducting phase the\nmagnitude of the effect can change dramatically with\ntemperature, especially for weaker spin-orbit coupling.\nTheresultbecomesmorecomplicatedinthecasewhere\nthe current is non-uniform or if the two gaps have dif-\nferent phase structures. To obtain insight into this\ncase, as well as the above linear response calculation,\nwe now derive a Ginzburg-Landau-like generating func-\ntional for the layer-staggeredspin-density. We start with\nthe Bogoliubov-deGennes (BdG) action SBdGalongwith\nthe associated Hubbard-Stratonovich terms. The first\nmodification is to give the order parameter a spatially\nvarying phase in order to describe a supercurrent carry-\ning state. The ∆ terms now connect states of momentum6\n0.0 0.5 1.0 1.5 2.0\nT/T c0.8250.8500.8750.9000.9250.9500.9751.000αEE/αEE ;N\n0.01000.02750.04500.06250.0800\nFIG. 5. (Color online) The Edelstein susceptibility αEErelat-\ning the layer-staggered polarization to the transverse cur rent\nvia ˜sx=αEEjyas depicted schematically in Fig. 4. The coef-\nficientαis plotted for a range of spin-orbit coupling strengths\nλwith each line normalized by the (roughly) constant normal\nstate Edelstein response. There is a marked change in the\nmagnitude of the effect coinciding with the onset of super-\nconductivity.\nk+Q/2 andk−Q/2 where Qis the Cooper pair mo-\nmentum. Secondly we will add a source field for layer\nstaggered spin-density, which takes the form B¯ψ˜sxψ. In-\ntegrating out the Fermions we obtain a generating func-\ntional forthe staggeredspin density Z[∆,Q,B]such that\n/angb∇acketleft˜sx/angb∇acket∇ight=−Td\ndBlnZ|B=0.\nOur next step is to approximate the generating func-\ntional within a Ginzburg-Landau approximation\nZ=e−βF≈e−βFGL(26)\nwhere\nFGL=/integraldisplay\ndr/parenleftBig\nαb|∆b|2+Kbb′D∆∗\nbD∆b′\n+βb|∆b|4+BKxy\nbb′∆∗\nb(−iD)∆b′/parenrightBig\n(27)\nandD=∇−2ieAis the covariant derivative, with sums\nover repeated indices. To do so we start with the stan-\ndard Hubbard-Stratonovichdecoupled form of the mean-\nfield problem, including now the layer-staggered source\nterm\nS=/summationdisplay\nb1\ngb|∆b|2+SBdG[∆b]+/summationdisplay\nk′B¯ψ˜sxψ.(28)\nWe then integrate out the gaussian fermionic theory to\nobtain\nS=/summationdisplay\nb1\ngb|∆b|2−Trln/bracketleftBig\nˆG−1\n0−∆bˆV∆;b−BˆVB/bracketrightBig\n.(29)\nPerforminga4thordergradientexpansionin∆andkeep-\ning only to lowest order in Bwe obtain the explicit formof the Ginzburg-Landau coefficients44\nαb=1\ngb−/summationdisplay\nk′f(k)2tanh/parenleftbigǫb\n2T/parenrightbig\nǫb\nβb=/summationdisplay\nk′f(k)4\n2ǫ2\nbc(ǫb)\nKxy\nbb=b/summationdisplay\nk′f(k)2\nǫb/bracketleftBigg\nny∂ǫb\n∂ky/parenleftbigg\nn′′(ǫb)+c(ǫb)\nǫb/parenrightbigg\n+∂ny\n∂ky/parenleftbig\nc(ǫb)+ǫ+ǫ−S(ǫ+,ǫ−)/parenrightbig/bracketrightBigg\nKxy\nb,−b=−2/summationdisplay\nk′f(k)2A∂ny\n∂kyS(ǫ+,ǫ−)\nKbb=/summationdisplay\nk′f(k)2\n4ǫb/bracketleftBigg\n∂2ǫb\n∂k2xc(ǫb)+/parenleftbigg∂ǫb\n∂kx/parenrightbigg2\nn′′(ǫb)\n+2A2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂n\n∂ky/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\nǫ+ǫ−\nǫb−ǫ−bS(ǫ+ǫ−)\n\nKb,−b=−1\n4/summationdisplay\nk′f(k)2A2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂n\n∂ky/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\nS(ǫ+,ǫ−)(30)\nwhere\nc(x) =tanh/parenleftbigx\n2T/parenrightbig\n2x+n′(x),\nS(x,y) =1\nxtanhx\n2T−1\nytanhy\n2T\nx2−y2,(31)\nnis the Fermi function, and we recall that/summationtext′indicates\nsummation over half the Brillouin zone. The interest-\ning magneto-electro effects are due to the the presence of\ntheKxyterms, sometimes called Lifshitz invariants, al-\nlowed by the breaking of inversion symmetry within each\nlayer45, which arise from the Tr/bracketleftBig\n(ˆGˆV∆)3ˆGˆVB/bracketrightBig\nterm of\nthe above expansion. In our case, instead of coupling to\nthe magnetic field, these terms couple to the generating\nfield of layer-staggered spin.\nWith this we can now see how the Edelstein effect\narises from Ginzburg-Landau theory. Suppose we have\na uniform supercurrent in the ydirection. From the\nGinzburg-Landau theory, we have that\njy= 2e/summationdisplay\nbb′Kbb′∆∗\nb∆b′Qy. (32)\nRecall that since Fis a generating functional, we also\nhave\n/angb∇acketleft˜sx/angb∇acket∇ight=/summationdisplay\nbb′Kxy\nbb′∆∗\nb∆b′Qy, (33)\nand we can then write\n/angb∇acketleft˜sx/angb∇acket∇ight=/summationtext\nbb′Kxy\nbb′∆∗\nb∆b′\n2e/summationtext\nbb′Kbb′∆∗\nb∆b′jy, (34)7\ngiving the Ginzburg-Landau theory equivalent of the\nquantityαEEcalculated above from linear response. In\nthe more general case, the expression becomes\n/angb∇acketleft˜sx/angb∇acket∇ight=/summationtext\nbb′Kxy\nbb′∆∗\nb∂y∆b′\n2e/summationtext\nbb′Kbb′∆∗\nb∂y∆b′jy, (35)\nSimilar effects have been derived from the Ginzburg-\nLandau free energies of inversion-symmetry breaking\nsuperconductors12,46, but the layer-staggered polariza-\ntion predicted here is novel in the cuprates.\nOne might expect from the above that since an ana-\nlog of the ISGE exists in bilayer cuprates, so might an\nanalog of the spin Galvanic effect, which would allow a\nsupercurrent to be driven by application of a static mag-\nnetic field. In inversion asymmetric systems such behav-\nior is generally preempted by the transition to a helical\nsuperconducting phase with zero net supercurrent.47–50\nIt remains to be investigated whether similar reasoning\nrules out the spin Galvanic effect in this system.\nVI. DISCUSSION AND CONCLUSIONS\nIn this work we have considered a model of spin-orbit\ncoupling the superconducting state of a bilayer cuprate\nsuperconductor. We have shown that the inversion-\nsymmetry-preserving spin-orbit coupling posited to be\npresent in Bi221215should lead to non-trivial layer po-\nlarized spin-orbit effects.\nIn particular, wecalculate the layerpolarizedspin-Hall\nconductivity, which was found to be non-trivial. While\nthis quantity cannot be directly related to a measurable\nquantity, such as the accumulation of spin at sample\nboundaries, we nonetheless expect that a layer-staggered\nspin-Hall effect should be present.\nMore interestingly, the presence of the new coupling\nterm leads to a layer-staggered analog of the Edelstein\n(or inverse spin-galvanic) effect, an in-plane spin polar-\nization in the presence of an applied supercurrent. This\nshould be visible through optical methods such as Fara-\nday rotation22,51or by measuring the degree of circular\npolarization in photoluminescence.43Furthermore, one\ncould attempt ARPES measurements in a supercurrent-\ncarrying state to directly see that canting of the in-plane\nspins due to this effect.52\nThere are still interesting effects to consider beyond\nwhat we have looked at in this work. In particular, spin-\nresolved ARPES observes a non-trivial variation of spin\ntexture within the Brillouin zone, most notably including\na reversal of the spin texture as a function from the zone\ncenter.15This suggests a more complicated form of the\nspin orbit coupling which could lead to further effects.\nAdditionally, for dx2−y2superconductivity a gradient in\nthed-wave order parameter can admix an s-wave pair-\ning term through a coupling of the gradients.53–55Re-\ngardless, the presence of spin-orbit coupling in BSCCO\nshould lead to the presence of a multitude of fascinatingspin-orbit driven effects, including the spin-Hall effect,\nEdelstein effect, and more.\nACKNOWLEDGMENTS\nWe would like to thank A. Lanzara, K. Gotlieb, C.-Y\nLin, M. Serbyn, I. Appelbaum, V. Yakovenko, and V.\nStanev for enlightening discussions. This work was sup-\nported by DOE-BES (DESC0001911) and the Simons\nFoundation (V.G. and A.A.) and NSF DMR-1613029\n(Z.R.).\nAppendix A: Parametrization of the BdG\nHamiltonian\nFor purposes of calculation, it is convenient to\nparametrize the BdG Hamiltonian, and associated\nNambu Green’s function in terms of the vector dand\nˇΣ matrices introduced in Eq. (9). To do so we note that\nthe matrices ˇP±≡/parenleftBig\nˇΣ0±n·ˇΣ/parenrightBig\n/2 satisfy the the rela-\ntion\nˇPbn·ˇΣˇPb′=bδbb′ˇPb, (A1)\nmeaning that they act as projectors on the degenerate\neigenspaces of the normal state Hamiltonian, with the\nsomewhat non-standard properties\nˇPbˇPb′=ˇΣ0ˇPbδbb′,\nn·ˇΣˇPb=bˇΣ0ˇPb.(A2)\nUsing these matrices we can compactly express the BdG\nHamiltonian\nHBdG=/summationdisplay\nk′¯Ψk/summationdisplay\nb/bracketleftbig\nǫb(k)ρ3+∆bf(k)ρ1/bracketrightbigˇPbΨk(A3)\nwhere we have introduced the adjoint Nambu spinor\n¯Ψ = Ψ†ˇΣ0and theρiare Pauli matrices in Nambu\nspace. The corresponding Nambu Green’s function is\nfound, completely analogous to the usual expresion, to\nbe\nˇG(iǫn,k) =/summationdisplay\nbˇPbiǫnρ0+ǫb(k)ρ3+∆bf(k)ρ1\n(iǫn)2−E2\nb(A4)\nwith the energies Ebas defined in Eq. (11).\nAppendix B: Spin Susceptibility and Knight Shift\nAs the presence of spin-orbit coupling induces a triplet\ncomponent to the Gor’kov anomalous Green’s function\none might wonder why this is not in general seen in ex-\nperimental signatures such as the Knight shift, where\nthe observed behavior is seen to be consistent with sin-\nglet pairing.56,57In general, the telltale sign of triplet8\n0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4\nT/T c0.00.20.40.60.81.0χij(T)/χN\nxx\nxx\nyy\nzzxy\nFIG. 6. (Color online) The components of the static spin sus-\nceptiblity χij(0,0) as a function of temperature for the layer\nspin-orbit coupled model. The diagonal in-plane component s\nhave a noticeable decrease below Tcbut do not go exactly to 0\nat zero temperature. This behavior is consistent with the de -\ncrease normally attributed to spin-singlet superconducti vity\nin the cuprates.pairing in the Knight shift is that the spin-susceptibility\nremains constant across the superconducting transition.\nOn the other hand, for the case of singlet pairing there\nis a noticeable decrease.58,59This is, however, consistent\nwith this model as the singlet component of the order pa-\nrameter still leads to qualitative behavior similar to the\ntypical singlet case i.e. the Knight shift should rapidly\ndecrease below Tc. However, unlike the pure singlet case\nas can be seen in Fig. 6, the spin-susceptibility does not\ngo exactly to zero at T= 0, a fact which was already\nappreciated by Gor’kov and Rashba in 2001.60\n∗raineszm@gmail.com\n1J. G. Bednorz and K. A. Mller,\nZ. Fr Phys. B Condens. Matter 64, 189 (1986).\n2A. Schilling, M. Cantoni, J. D. Guo, and H. R. Ott,\nNature363, 56 (1993).\n3P. A. Lee, Phys. Rev. X 4, 031017 (2014).\n4S.Badoux, W. Tabis, F. Lalibert, G.Grissonnanche, B. Vi-\ngnolle, D. Vignolles, J. Bard, D. A. Bonn, W. N. Hardy,\nR. Liang, N. Doiron-Leyraud, L. Taillefer, and C. Proust,\nNature531, 210 (2016).\n5S. Chatterjee and S. Sachdev,\nPhys. Rev. B 94, 205117 (2016).\n6J. Chang, E. Blackburn, A. T. Holmes, N. B. Christensen,\nJ. Larsen, J. Mesot, R. Liang, D. A. Bonn, W. N. Hardy,\nA. Watenphul, M. V. Zimmermann, E. M. Forgan, and\nS. M. Hayden, Nat Phys 8, 871 (2012).\n7G. Ghiringhelli, M. Le Tacon, M. Minola, S. Blanco-\nCanosa, C. Mazzoli, N. B. Brookes, G. M. De Luca,\nA. Frano, D. G. Hawthorn, F. He, T. Loew, M. M. Sala,\nD. C. Peets, M. Salluzzo, E. Schierle, R. Sutarto, G. A.\nSawatzky, E. Weschke, B. Keimer, and L. Braicovich,\nScience337, 821 (2012).\n8K. Fujita, M. H. Hamidian, S. D. Edkins, C. K.\nKim, Y. Kohsaka, M. Azuma, M. Takano, H. Tak-\nagi, H. Eisaki, S.-i. Uchida, A. Allais, M. J. Lawler,\nE.-A. Kim, S. Sachdev, and J. C. S. Davis,\nProc. Natl. Acad. Sci. 111, E3026 (2014).\n9R. Comin, R. Sutarto, F. He, E. H. da Silva Neto,\nL. Chauviere, A. Frao, R. Liang, W. N. Hardy, D. A.\nBonn, Y. Yoshida, H. Eisaki, A. J. Achkar, D. G.\nHawthorn, B. Keimer, G. A. Sawatzky, and A. Damas-\ncelli, Nat Mater 14, 796 (2015).\n10V.M.Edelstein,Solid State Communications 73, 233 (1990).\n11W. Koshibae, Y. Ohta, and S. Maekawa,\nPhys. Rev. B 47, 3391 (1993).12V. M. Edelstein, Phys. Rev. Lett. 75, 2004 (1995).\n13N. Harrison, B. J. Ramshaw, and A. Shekhter,\nSci Rep 5(2015), 10.1038/srep10914.\n14C. Wu, J. Zaanen, and S.-C. Zhang,\nPhys. Rev. Lett. 95, 247007 (2005).\n15K. Gotlieb, C.-Y. Lin, M. Serbyn, W. Zhang, C. L. Small-\nwood, C. Jozwiak, H. Eisaki, Z. Hussain, A. Vishwanath,\nand A. Lanzara, Science 362, 12711275 (2018).\n16M. Sigrist, D. F. Agterberg, M. H. Fischer,\nJ. Goryo, F. Loder, S.-H. Rhim, D. Maruyama,\nY. Yanase, T. Yoshida, and S. J. Youn,\nJ. Phys. Soc. Jpn. 83, 061014 (2014).\n17M. Dyakonov and V. Perel,\nPhysics Letters A 35, 459 (1971).\n18J. E. Hirsch, Phys. Rev. Lett. 83, 1834 (1999).\n19S. Murakami, N. Nagaosa, and S.-\nC. Zhang, Science 301, 1348 (2003),\nhttp://science.sciencemag.org/content/301/5638/1348 .full.pdf.\n20J. Sinova, D. Culcer, Q. Niu, N. A. Sinit-\nsyn, T. Jungwirth, and A. H. MacDonald,\nPhys. Rev. Lett. 92(2004), 10.1103/PhysRevLett.92.126603.\n21V. V. Bel’kov and S. D. Ganichev,\nSemiconductor Science and Technology 23, 114003 (2008).\n22Y. K. Kato, R. C. Myers, A. C. Gossard, and\nD. D. Awschalom, Science 306, 1910 (2004),\nhttp://science.sciencemag.org/content/306/5703/1910 .full.pdf.\n23J. Wunderlich, B. Kaestner, J. Sinova, and T. Jungwirth,\nPhys. Rev. Lett. 94, 047204 (2005).\n24H. Zhao, E. J. Loren, H. M. van Driel, and A. L. Smirl,\nPhys. Rev. Lett. 96, 246601 (2006).\n25A.Hoffmann,IEEE Transactions on Magnetics 49, 5172 (2013).\n26J. Sinova, S. O. Valenzuela, J. Wunderlich, C. Back, and\nT. Jungwirth, Rev. Mod. Phys. 87, 1213 (2015).\n27See the Supplemental material for Ref. 15.9\n28T. Xiang and W. N. Hardy,\nPhys. Rev. B 63, 024506 (2000).\n29R. S. Markiewicz, S. Sahrakorpi, M. Lindroos, H. Lin, and\nA. Bansil, Phys. Rev. B 72, 054519 (2005).\n30I. A. Sergienko and S. H. Curnoe,\nPhys. Rev. B 70, 214510 (2004).\n31K. V. Samokhin, Phys. Rev. B 92, 174517 (2015).\n32X. Gong, M. Kargarian, A. Stern, D. Yue, H. Zhou,\nX. Jin, V. M. Galitski, V. M. Yakovenko, and J. Xia,\nSci. Adv. 3, e1602579 (2017).\n33The presence of spin-orbit coupling means that there is no\nlonger a natural definition of spin-current in terms of a\nconserved Noether charge.\n34J.-i. Inoue, G. E. W. Bauer, and L. W. Molenkamp,\nPhys. Rev. B 70, 041303 (2004).\n35E. G. Mishchenko, A. V. Shytov, and B. I. Halperin,\nPhys. Rev. Lett. 93, 226602 (2004).\n36O. V. Dimitrova, ArXivcond-Mat0405339 (2004),\narXiv:cond-mat/0405339.\n37O. V. Dimitrova, Phys. Rev. B 71, 245327 (2005).\n38P. L. Krotkov and S. Das Sarma,\nPhys. Rev. B 73, 195307 (2006).\n39W.-K. Tse and S. Das Sarma,\nPhys. Rev. Lett. 96, 056601 (2006).\n40V. M. Galitski, A. A. Burkov, and S. Das Sarma,\nPhys. Rev. B 74, 115331 (2006).\n41O.Bleibaum, Phys. Rev. B 74(2006), 10.1103/PhysRevB.74.113309.\n42E.I.Rashba,Phys. E Low-Dimens. Syst. Nanostructures 34, 31 (2006).\n43A. Y. Silov, P. A. Blajnov, J. H. Wolter,\nR. Hey, K. H. Ploog, and N. S. Averkiev,\nAppl. Phys. Lett. 85, 5929 (2004).\n44Wenotethatthecoefficientsobtainedheredifferfromwhat\none would obtain by expanding Eqs. (23) and (24). This\nis due to different order of limits between the two expres-\nsions. Here we take the limit ω→0 before q→0 reflecting\nthe fact that we are considering a supercurrent carrying\nsystem in thermal equilibrium. In contrast, the general ex-pressions derived above consider the case where the system\nis responding to the presence of a supercurrent which has\nbeen “switched on”. The general expressions above can be\nderived in the opposite limit without much extra complica-\ntion leading toadditional intra-bandterms in theresponse .\n45V. P. Mineev and K. V. Samokhin,\nPhys. Rev. B 78, 144503 (2008).\n46F. Konschelle, I. V. Tokatly, and F. S. Bergeret,\nPhys. Rev. B 92, 125443 (2015).\n47D.F.Agterberg,Physica C: Superconductivity Proceedings of the 3rd Polish-US Workshop on Superconductivity and Magn etism of Advanced Materials, 387, 13 (2003).\n48O. Dimitrova and M. V. Feigelman,\nPhys. Rev. B 76, 014522 (2007).\n49D.F.Agterberg,in Non-Centrosymmetric Superconductors ,\nVol. 847, edited by E. Bauer and M. Sigrist (Springer\nBerlin Heidelberg, Berlin, Heidelberg, 2012) pp. 155–170.\n50M. Smidman, M. B. Salamon, H. Q. Yuan, and D. F.\nAgterberg, Rep. Prog. Phys. 80, 036501 (2017).\n51Y. K. Kato, R. C. Myers, A. C. Gossard, and D. D.\nAwschalom, Phys. Rev. Lett. 93, 176601 (2004).\n52G. Boyd, S. Takei, and V. Galitski, Solid State Commu-\nnications 189, 6367 (2014).\n53Q. P. Li, B. E. C. Koltenbah, and R. Joynt,\nPhys. Rev. B 48, 437 (1993).\n54A. J. Berlinsky, A. L. Fetter, M. Franz, C. Kallin, and\nP. I. Soininen, Phys. Rev. Lett. 75, 2200 (1995).\n55D. L. Feder and C. Kallin, Phys. Rev. B 55, 559 (1997).\n56M. Takigawa, P. C. Hammel, R. H. Heffner, and Z. Fisk,\nPhysical Review B 39, 7371 (1989).\n57S. E. Barrett, D. J. Durand, C. H. Pennington, C. P.\nSlichter, T. A. Friedmann, J. P. Rice, and D. M. Gins-\nberg, Physical Review B 41, 6283 (1990).\n58K. Yosida, Physical Review 110, 769 (1958).\n59J. R. Schrieffer, Theory of Superconductivity , Advanced\nbook classics (Advanced Book Program, Perseus Books,\nReading, Mass, 1999).\n60L. P. Gor’kov and E. I. Rashba,\nPhys. Rev. Lett. 87, 037004 (2001)."    },    {        "title": "2106.01046v1.Enhanced_spin_orbit_coupling_and_orbital_moment_in_ferromagnets_by_electron_correlations.pdf",        "content": "arXiv:2106.01046v1  [cond-mat.str-el]  2 Jun 2021Enhanced spin-orbit coupling and orbital moment in ferroma gnets by electron\ncorrelations\nZe Liu,1Jing-Yang You,2Bo Gu,1,3,∗Sadamichi Maekawa,4,1and Gang Su1,3,5,†\n1Kavli Institute for Theoretical Sciences, and CAS Center fo r Excellence in Topological Quantum Computation,\nUniversity of Chinese Academy of Sciences, Beijing 100190, China\n2Department of Physics, National University of Singapore, 2 Science Drive 3, Singapore 117551\n3Physical Science Laboratory, Huairou National Comprehens ive Science Center, Beijing 101400, China\n4Center for Emergent Matter Science, RIKEN, Walo 351-0198, J apan\n5School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China\nIn atomic physics, the Hund rule says that the largest spin an d orbital state is realized due to\nthe interplay of the spin-orbit coupling (SOC) and the Coulo mb interactions. Here, we show that\nin ferromagnetic solids the effective SOC and the orbital mag netic moment can be dramatically\nenhanced bya factor of 1 /[1−(2U′−U−JH)ρ0], whereUandU′are the on-site Coulomb interaction\nwithin the same oribtals and between different orbitals, res pectively, JHis the Hund coupling, and\nρ0is the average density of states. This factor is obtained by u sing the two-orbital as well as\nfive-orbital Hubbard models with SOC. We also find that the spi n polarization is more favorable\nthan the orbital polarization, being consistent with exper imental observations. This present work\nprovides a fundamental basis for understanding the enhance ments of SOC and orbital moment by\nCoulomb interactions in ferromagnets, which would have wid e applications in spintronics.\nIntroduction —The Hund’s rule in atomic physics says\nthat the state with both the largest spin moment and the\nlargest orbital moment is realized in an atom, required\nby the minimum of the Coulomb repulsive energy. The\nsimilar picture was obtained in the magnetic impurity\nsystems. In the Anderson impurity model, the spin mag-\nnetic moment of impurities is developed due to the large\non-site Coulomb interaction U [ 1]. In 1964, the extended\nAnderson impurity model with degenerate orbitals has\nbeen studied, where the role of Uand the Hund coupling\nJHhas been addressed [ 2,3]. Forty years ago, Yafet also\nstudied the Anderson impurity model within Hartree-\nFock approximation and found that the on-site Coulomb\ninteraction of impurities can enhance the effective SOC\nin the spin-flip cross section [ 4]. Later, Fert and Jaoul\napplied this result to study the anomalousHall effect due\nto magnetic impurities [ 5]. The relation between the on-\nsite Coulomb interaction Uand the effective spin-orbit\ncoupling (SOC) in magnetic impurity systems has also\nbeen discussed by the density functional theory (DFT)\ncalculations [ 6] and the quantum Monte Carlo simula-\ntions [7].\nIn these years, one of the fast developing areas in con-\ndensed matter physics is spintronics [ 8,9]. It aims to\nmanipulate the spin rather than the charge degree of\nfreedom of electrons to design the next-generation elec-\ntronic devices with small size, faster calculating abil-\nity, and lower energy consumption. SOC, as one of the\nkey ingredients in spintronics, is related to many sig-\nnificant physical phenomena and novel matter [ 10]. In\naddition to the magnetic anisotropy [ 8,11], SOC plays\nan important role in the phenomena such as anomalous\nHall effect [ 12], spin Hall effect associated with the spin-\ncharge conversion [ 13–16], topological insulators [ 17–21],\nskymions [ 22–24] and so on. To design better spintronicdevices, a large SOC is usually required. As SOC is a\nrelativistic effect in quantum mechanics, it is often small\nin many materials. A key issue is what factors can affect\nthe magnitude of the SOC in solids.\nOn the other hand, the orbital moment in the FeCo\nnanogranules was experimentally shown to be about\nthree times larger than that in bulk FeCo, as a result of\nthe enhanced Coulomb interaction in the FeCo/insulator\ninterface [ 25], because the Coulomb interaction in the\nFeCo/insulator interface is expected to be larger than\nthat in the ferromagnetic FeCo bulk. In addition, a large\nCoulomb interaction up to 10 eV was discussed in Fe\nthin films in the experiment [ 26]. The spin polarization\nin the Hubbard model with Rashba SOC can also be\nenhanced by the on-site Coulomb interaction U [ 27]. Re-\ncently, in the two-dimensional magnetic topological insu-\nlators PdBr 3and PtBr 3, the DFT calculations show that\nthe band gap and the SOC can be strongly enhanced by\nthe Coulomb interaction [ 28].\nInspired by recent experimental and numerical results\non the enhanced SOC due to the Coulomb interaction in\nstrongly correlated electronic systems, here we develop a\ntheory on the relation between SOC and Coulomb inter-\naction in ferromagnets. By a two-orbital Hubbard model\nwith SOC, we find that the effective SOC and orbital\nmagnetic moment in ferromagnets can be enhanced by a\nfactorof1 /[1−(2U′−U−JH)ρ0], whereUandU′arethe\non-site Coulomb interaction within the same oribtalsand\nbetween different orbitals, respectively, JHis the Hund\ncoupling, and ρ0is the average density of states. The\nsame factor has also been obtained for the five-orbital\nHubbard model with degenerate bands. Our theory can\nbe viewed as the realization of Hund’s rule in ferromag-\nnets.\nTwo-orbital Hubbard model with SOC —Let us consider2\na two-orbital Hubbard model, where only a pair of or-\nbitals with opposite orbital magnetic quantum numbers\nm(-1 and 1, or -2 and 2) are considered. Thus, the\nHamiltonian can be written as\nH=/summationdisplay\nk,m,σǫkmσnkmσ+U/summationdisplay\ni,mnim↑nim↓\n+U′/summationdisplay\ni,σ,σ′nimσni¯mσ′−JH/summationdisplay\ni,σnimσni¯mσ,(1)\nwhereǫkmσis the energy of electron with wave vector k,\norbitalm, and spin σ(↑,↓) [29],UandU′are the on-\nsite Cuolomb repulsion within the orbital mand between\ndifferent orbitals mandm′, respectively, JHis the Hund\ncoupling, and nkmσ(nimσ)representstheparticlenumber\nwith wave vector k(site index i), orbital mand spin σ.\nFor simplicity, we consider four degenerate energy bands,\nwhich are lifted by an external magnetic field hand the\nIsing-type SOC [ 5]:\nǫkmσ=ǫk−σµBh−1\n2σλsom, (2)\nwhereλsois the SOC constant, ǫkis the electron en-\nergy without external magnetic filed and SOC. Using\nthe Hartree-Fock approximation, we have nimσnim′σ′≈\n/angbracketleftnimσ/angbracketrightnim′σ′+/angbracketleftnim′σ′/angbracketrightnimσ−/angbracketleftnimσ/angbracketright/angbracketleftnim′σ′/angbracketright. Assum-\ning the system is homogeneous, the occupation number\nnimσisindependentoflatticesite i:/angbracketleftnimσ/angbracketright ≈ /angbracketleftnmσ/angbracketright, and\nthrough Fourier transformation/summationtext\ninimσ=/summationtext\nknkmσ,\nthe Halmiltonian in Eq.( 1) can be diagonalized as:\nH≈/summationdisplay\nk,m,σ˜ǫkmσnkmσ, (3)\nwith\n˜ǫkmσ=ǫk−σµBh−1\n2σλsom+U/angbracketleftnm¯σ/angbracketright\n+U′(/angbracketleftn¯mσ/angbracketright+/angbracketleftn¯m¯σ/angbracketright)−JH/angbracketleftn¯mσ/angbracketright.(4)\nWe define the spin polarization per site as sz=\nµB(/angbracketleftnm↑/angbracketright−/angbracketleftnm↓/angbracketright+/angbracketleftn¯m↑/angbracketright−/angbracketleftn¯m↓/angbracketright), and the orbital po-\nlarization per site as lz=mµB(/angbracketleftnm↑/angbracketright−/angbracketleftn¯m↑/angbracketright+/angbracketleftnm↓/angbracketright−\n/angbracketleftn¯m↓/angbracketright). Here we should remark that the so-defined or-\nbital polarization from itinerant electrons on different or-\nbitals with SOC differs from the conventionalorbital mo-\nments of atoms that are usually quenched owing to the\npresence of the crystal fields in transition metal ferro-\nmagnets. Introduce the particle numbers of the parallel\n(np) and antiparallel ( nap) states of the spin σand or-\nbitalm:np=/angbracketleftnm↑/angbracketright+/angbracketleftn¯m↓/angbracketright,nap=/angbracketleftn¯m↑/angbracketright+/angbracketleftnm↓/angbracketright. Then\nthe energy ˜Ekmσcan be written as\n˜ǫkmσ=¯ǫ−σµB/parenleftbigg\nh+U+JH\n4µ2\nBsz/parenrightbigg\n−1\n2m/parenleftbigg\nσλso−U−2U′+JH\n2µBm2lz/parenrightbigg\n.(5)When spin σand orbital mare antiparallel (parallel) the\nenergy ¯ǫ= (ǫk+1\n2Unap(p)+1\n2U′nap(p)+1\n2U′np(ap)−\n1\n2JHnap(ap)).\nSpin polarization — It is noted that without external\nmagneticfield handSOC λso, thefourenergybandswith\nspinσ(↑and↓) and orbital m(for example 1 and −1)\naredegenerate,andtheoccupationnumbers nap=np. In\ntermsofthe translationalsymmetryofthelatticesystem:\n/angbracketleftnmσ/angbracketright=1\nN/summationtext\ni/angbracketleftnimσ/angbracketright=1\nN/summationtext\nk/angbracketleftnkmσ/angbracketright=1\nN/summationtext\nkf(˜ǫkmσ),\nwherefistheFermidistributionfunction, thespinpolar-\nizationcanbewrittenas sz=µB\nN/summationtext\nk[f(˜ǫkm↑)−f(˜ǫkm↓)+\nf(˜ǫk¯m↑)−f(˜ǫk¯m↓)]. For the system with a paramagnetic\n(PM) state ( h= 0),f(˜ǫkmσ) can be expanded according\ntoh, which is a small value compared to Fermi energy,\nandnap=np,sz=µB/summationtext\nk[f(˜ǫPM,km ↑)−f(˜ǫPM,km ↓) +\nf(˜ǫPM,k¯m↑)−f(˜ǫPM,k¯m↓)] = 0. Up to the linear order of\nh, the spin polarization becomes\nsz=4µ2\nBρ0\n1−(U+JH)ρ0h, (6)\nwhereρ0=1\n4/integraltext∞\n0[−∂f(E)\n∂E][ρm↑(E) +ρ¯m↑(E) +ρm↓(E) +\nρ¯m↓(E)]dEis the average density of states of the four\nenergy bands. The instability condition of the spin po-\nlarization is\n(U+JH)ρ0>1. (7)\nThis condition canbe takenas an extensionofStoner cri-\nterion in the presence of SOC in itinerant ferromagnets.\nOrbital polarization — Similarly, the orbital polariza-\ntion can be expressed as lz=µBm(/angbracketleftnm↑/angbracketright − /angbracketleftn¯m↑/angbracketright+\n/angbracketleftnm↓/angbracketright−/angbracketleftn¯m↓/angbracketright) =µBm\nN/summationtext\nk[f(˜ǫkm↑)−f(˜ǫk¯m↑)+f(˜ǫkm↓)−\nf(˜ǫk¯m↓)]. Forthe ferromagnetic(FM) state, the SOCcan\nbe regarded as a small value [ 5], sof(˜ǫkmσ) can be ex-\npanded according to λso, and when λso= 0,nap=np,\nthe zero-order term is zero. To the linear order of λso,\nthe orbital polarization gives\nlz=m2µBρs\n1−(2U′−U−JH)ρ0λso, (8)\nwhereρs=1\n2/integraltext∞\n0[−∂f(E)\n∂E][ρm↑(E) +ρ¯m↑(E)−ρm↓(E)−\nρ¯m↓(E)]dEisthe averagespin polarizeddensityofstates.\nThen Eq.( 8) can be rewritten as lz=µBm2ρsλeff\nso, where\nthe effective SOC λeff\nsois\nλeff\nso=λso\n1−(2U′−U−JH)ρ0. (9)\nOnemaynotethattheorbitalpolarizationdiscussedhere\n[Eq. (8)] is totally induced by the SOC, which can be\nenhanced by increasing U′or decreasing UandJH, we\nwilldiscussthisindetail. IntheabsenceoftheSOC,such\nan orbital polarization is absent according to Eq. ( 8).\nThe instability condition of orbital polarization would\nbe:\n(2U′−U−JH)ρ0>1. (10)3\nTABLE I. Comparison of the theoretical results among the And erson impurity model, the one-orbital Hubbard model (Stone r\nmodel), and our two- and five-orbital Hubbard models with the spin-orbit coupling (SOC). szandlzare the spin and orbital\npolarization, respectively. The instability conditions ( IC) ofszandlzin these models are listed. λeff\nsois the effective SOC\naffected by atomic SOC λso, the electron correlations U,U′andJH, and the electron density of state ρ. The equations of\nfive-orbital Hubbard model can be found in the Supplementary Information.\nAnderson impurity\nmodelOne-orbital Hubbard\nmodel (Stoner)Two-orbital Hubbard model\nwith SOC( m=±1 orm=±2)Five-orbital Hubbard model\nwith SOC ( m= 0,±1,±2)\nsz –2µ2\nBρ(EF)\n1−Uρ(EF)h[30]4µ2\nBρ0\n1−(U+JH)ρ0h[Eq.(6)]10µ2\nBρ0\n1−(U+4JH)ρ0h[Eq.(63)]\nlz – –m2µBρs\n1−(2U′−U−JH)ρ0λso[Eq.(8)]µB(ρ1s+4ρ2s)\n1−(2U′−U−JH)ρ0λso[Eq.(78)]\nIC ofsz(U+4JH)ρ(EF)>1[2,3]Uρ(EF)>1 [30] ( U+JH)ρ0>1 [Eq.(7)] ( U+4JH)ρ0>1 [Eq.(65)]\nIC oflz – – (2 U′−U−JH)ρ0>1 [Eq.(10)]\nλeff\nsoλat\n1−(U−JH)ρ(EF)[4] –λso\n1−(2U′−U−JH)ρ0[Eq.(9)]\nThe detailed derivation is given in the Supplementary\nInformation.\nFive-orbital Hubbard model with SOC —Ourtheorycan\nbeeasilyextendedtothefive-orbitalHubbardmodelwith\ndegenerate bands, and the detailed derivation is given in\ntheSupplementaryInformation. Forthefive-orbitalcase,\nthe instability condition of the spin polarization becomes\nas (U+4JH)ρ0>1. The same expression has been ob-\ntained for the presence of localized spin moment in the\nAnderson impurity model with degenerate orbitals [ 2,3].\nThe obtained instability condition of the orbital polar-\nization is (2 U′−U−JH)ρ0>1, which is the same as\nEq.(10) for the two-orbital case. In the five-orbital case,\nthe effective SOC and the orbital magnetic moment can\nalsobe enhanced by a factorof1 /[(2U′−U−JH)ρ0], that\nisthesameenhancementfactorasin thetwo-orbitalcase.\nDiscussion —The comparison between our theory, the\nStoner model and the Anderson impurity model is shown\nin Table I. It is interesting to note that the instabil-\nity conditions of szbetween our five-orbital Hubbard\nmodel with SOC and the Anderson impurity model are\nthe same, while the obtained effective SOC λeff\nsobetween\nthe two models are different. Comparing Eqs.( 7) and\n(10), which are the spin and orbital instability condi-\ntions of the two-orbital model in Table I, one may note\nthat the condition of the orbital spontaneous polariza-\ntion is more stringent than that of the spin spontaneous\npolarization. The phase diagram of the spin and orbital\nspontaneous polarizations as a function of the inverse of\naverage density of state 1 /ρ0and the Coulomb interac-\ntion U obtained with Eqs. ( 7) and (10) is depicted in\nFig.1. Considering the relation U=U′+2JHand the\nreasonable values of U= 4∼7 eV in the 3d transitional\nmetal oxides [ 31], for 3d electrons, JH= 1,U′= 5,\nU= 7 eV are a set of reasonable values, for simplicity\nwe keep the ratio U:U′:JH= 7 : 5 : 1 in Eq.( 9),\nand the shaded area with blue (red) solid lines indicatesUnpolarized\nStoner\n1\u0010ρ/g3674 (eV)(eV) Spin polarization (Eq. 7)Orbital polarization\n (Eq.10)\nFIG. 1. The phase diagram of spin and orbital spontaneous\npolarization as a function of the inverse average density of\nstates and the Coulomb interaction U. The shaded area with\nblue solid lines represents the spin spontaneous polarizat ion\ndetermined by Eq.( 7). The shaded area with red solid lines\nrepresentsthe orbital spontaneous polarization determin edby\nEq.(10). The black dotted line indicates the Stoner criterion\nof the spin spontaneous polarization, which is obtained by t he\nsingle orbital Hubbard model.\nthe spin (orbital) spontaneous polarization. The Stoner\ncriterion of the spin spontaneous polarization based on\nthe single orbital Hubbard model is also plotted in Fig.\n1for a comparison. The results show that the area of or-\nbital spontaneous polarization is enclosed in the area of\nspin spontaneous polarization. In other words, it is more\nstringent to have the orbital spontaneous polarization,\nwhich is consistent with the fact that the orbital sponta-\nneous polarization is rarely observed in experiments.\nThe relation between the electron correlations U,U′\nandJHand the spin polarization szin Eq. (6) and the4\norbital polarization lzin Eq. (8) can be understood by\ntheenergytermsinEq. ( 4). Foragivenstatewithorbital\nmand spin ↑, according to the principle of minimum\nenergy, in order to compensate the Coulomb interaction\nU, the occupancy number /angbracketleftnm↓/angbracketrightwill decrease, which will\nincreaseszand decrease lz. To compensate the Coulomb\ninteraction U′, the occupancy numbers /angbracketleftn¯m↑/angbracketrightand/angbracketleftn¯m↓/angbracketright\nwill equally decrease, which will have no effect on szand\nincrease lz. To compensate the Hund interaction JH,\nthe occupancy number /angbracketleftn¯m↑/angbracketrightwill increase, which will\nincrease szand decrease lz. The above argument by Eq.\n(4) is consistent with the obtained enhancement factor\n1/[1−(2U′−U−JH)]ρ0forlzin Eq. ( 8), where U\nandJHwill decrease lzandU′will increase lz. The\nsame argument by Eq. ( 4) is also consistent with the\ncalculated enhancement factor 1 /[1−(U+JH)]ρ0forsz\nin Eq. (6), where UandJHwill increase szandU′has\nno effect on sz.\nApplication —Equation ( 9) shows that Coulomb inter-\nactions can enhance the effective SOC. Recently, for\nmagnetic topological insulators PdBr 3and PtBr 3, it is\nfound that the energy gap increases with the increase of\nCoulomb interaction [ 28]. In these topological materi-\nals, the energy gap is opened due to the SOC, and the\nenhancement of SOC by the Coulomb interaction can\nbe naturally obtained by Eq.( 9). In addition, since the\nmagnetic optical Kerr effect (MOKE) and the Faraday\neffect aredetermined by the SOC, the experimentally ob-\nservedlargeFaradayeffectinmetalfluoridenanogranular\nfilms [32] and the predicted large MOKE at Fe/insulator\ninterfaces [ 33] can also be understood by the effect of\nCoulomb interaction as revealed by Eq.( 9), because the\nCoulomb interaction becomes important with the de-\ncreased screening effect at the interfaces. It is noted that\nthe Hubbard model with SOC has been extensively stud-\nied, where the SOC can induce the Dzyaloshinski-Moriya\ninteraction and the pesudo-dipolar interaction [ 29,34–\n45].\nThe orbital magnetic moment can also be enhanced\nby Coulomb interaction, as given by Eq.( 8). In the re-\ncent experiment, the orbital magnetic moment in FeCo\nnanogranules is observed to be three times larger than\nthat of FeCo bulk [ 25]. Using Eq.( 8), the ratio of the or-\nbital magnetic moment without the Coulomb interaction\nlz(U= 0) to the orbital magnetic moment with finite U\nlz(U) can be approximately written as:\nlz(U)\nlz(0)=1\n1−(2U′−U−JH)ρ0. (11)\nAs the Coulomb interactions can be approximately ne-\nglected in the metal bulk, and become important in the\nmetal/insulator interfaces, lz(0) andlz(U) can represent\nthe orbital moment of FeCo bulk and nanogranules, re-\nspectively. Toreproducethe experimentalratiooforbital\nmagnetic moment between FeCo nanogranules and bulk\nwemaytake, lz(U)/lz(0) = 3, thefitted value U= 4.4eVis obtained by Eq.( 11), which is reasonable for 3d tran-\nsition metals. In the fitting, we use the approximation\nin the DFT calculation, to keep JH= 0 eV, U=U′.\nρ0= 0.15 (1/eV) is obtained by DFT calculation for the\nFeCo interface, where ρ0is approximately estimated as\nthe density of states at Fermi level. Therefore, Eq.( 11)\ncan be used to qualitatively explain the enhancement of\norbital magnetic moment for the FeCo nanogranules in\nthe experiment.\nConclusion — Atwo-orbitalHubbardmodelwithSOC,\nwe show that the orbital polarization and the effec-\ntive SOC in ferromagnets are enhanced by a factor of\n1/[1−(2U′−U−JH)ρ0], whereUandU′are the on-site\nCoulomb interaction within the same orbitals and be-\ntweendifferentorbitals,respectively, JHistheHund cou-\npling, and ρ0is the average density of states. The same\nfactor is obtained for the five-orbital Hubbard model\nwith degenerate bands. Our theory can be viewed as\nthe realization of Hund’s rule in ferromagnets. The the-\nory can be applied to understand the enhanced band\ngap due to SOC in magnetic topological insulators, and\nthe enhanced orbital magnetic moment in ferromagnetic\nnanogranules in a recent experiment. In addition, our\nresults reveal that it is more stringent to have the or-\nbital spontaneouspolarizationthan the spin spontaneous\npolarization, which is consistent with experimental ob-\nservations. As the electronic interaction in some two-\ndimensional (2D) systems can be controlled experimen-\ntally [46], according to our theory, the enhanced SOC,\nspin and orbital magnetic moments are highly expected\nto be observed in these 2D systems. This present work\nnot only provides a fundamental basis for understanding\nthe enhancements of SOC in some magnetic materials,\nbut also sheds light on how to get a large SOC through\nhybrid spintronic structures.\nThe authors acknowledge Q. B. Yan, Z. G. Zhu, and\nZ. C. Wang for many valuable discussions. This work is\nsupported in part by the National Key R&D Program of\nChina (Grant No. 2018YFA0305800), the Strategic Pri-\nority Research Program of the Chinese Academy of Sci-\nences (Grant No. XDB28000000), the National Natural\nScience Foundation of China (Grant No. 11834014), and\nBeijing Municipal Science and Technology Commission\n(Grant No. Z191100007219013). B.G. is also supported\nby the National Natural Science Foundation of China\n(Grants No. Y81Z01A1A9 and No. 12074378), the Chi-\nnese Academy of Sciences (Grants No. Y929013EA2and\nNo. E0EG4301X2), the University of Chinese Academy\nof Sciences (Grant No. 110200M208), the Strategic Pri-\nority Research Program of Chinese Academy of Sciences\n(Grant No. XDB33000000), and the Beijing Natural\nScience Foundation (Grant No. Z190011). SM is sup-\nported by JST CREST Grant (No. JPMJCR19J4 No.\nJPMJCR1874 and No. JPMJCR20C1) and JSPS KAK-\nENHI (Nos. 17H02927 and 20H01865) from MEXT,\nJapan.5\n∗gubo@ucas.ac.cn\n†gsu@ucas.ac.cn\n[1] P. W. Anderson, Localized magnetic states in metals,\nPhys. Rev. 124, 41 (1961) .\n[2] T. Moriya, Ferro- and antiferromagnetism of transition\nmetals and alloys, Prog. Theor. Phys. 33, 157 (1965) .\n[3] K. Yosida, A. Okiji, and S. Chikazumi, Mag-\nnetic anisotropy of localized state in metals,\nProg. Theor. Phys. 33, 559 (1965) .\n[4] Y.Yafet,Spin-orbitinducedspin-flipscatteringbyalo cal\nmoment, J. Appl. Phys. 42, 1564 (1971) .\n[5] A. Fert and O. Jaoul, Left-right asymmetry in the scat-\ntering of electrons by magnetic impurities, and a hall\neffect,Phys. Rev. Lett. 28, 303 (1972) .\n[6] G.-Y. Guo, S. Maekawa, and N. Nagaosa, En-\nhanced spin hall effect by resonant skew scat-\ntering in the orbital-dependent kondo effect,\nPhys. Rev. Lett. 102, 036401 (2009) .\n[7] B. Gu, J.-Y. Gan, N. Bulut, T. Ziman, G.-Y. Guo, N. Na-\ngaosa, and S. Maekawa, Quantum renormalization of the\nspin hall effect, Phys. Rev. Lett. 105, 086401 (2010) .\n[8] I.ˇZuti´ c, J. Fabian, and S. D. Sarma, Spin-\ntronics: Fundamentals and applications,\nRev. Mod. Phys. 76, 323 (2004) .\n[9] S. Maekawa, ed., Concepts in Spin Electronics (Oxford\nUniversity Press, 2006).\n[10] A. Soumyanarayanan, N. Reyren, A. Fert, and\nC. Panagopoulos, Emergent phenomena induced\nby spin–orbit coupling at surfaces and interfaces,\nNature539, 509 (2016) .\n[11] J.-Y. You, Z. Zhang, X.-J. Dong, B. Gu, and G. Su, Two-\ndimensional magnetic semiconductors with room curie\ntemperatures, Phys. Rev. Research 2, 013002 (2020) .\n[12] N. Nagaosa, J. Sinova, S. Onoda, A. H. Mac-\nDonald, and N. P. Ong, Anomalous hall effect,\nRev. Mod. Phys. 82, 1539 (2010) .\n[13] M. Dyakonov and V. Perel, Current-induced\nspin orientation of electrons in semiconductors,\nPhys. Lett. A 35, 459 (1971) .\n[14] J. E. Hirsch, Spin hall effect,\nPhys. Rev. Lett. 83, 1834 (1999) .\n[15] Y. K. Kato, Observation of the spin hall effect in semi-\nconductors, Science306, 1910 (2004) .\n[16] J. Sinova, S. O. Valenzuela, J. Wunderlich, C.\nH. Back, and T. Jungwirth, Spin hall effects,\nRev. Mod. Phys. 87, 1213 (2015) .\n[17] L. Fu, C. L. Kane, and E. J. Mele, Topological insulators\nin three dimensions, Phys. Rev. Lett. 98, 106803 (2007) .\n[18] J. E. Moore and L. Balents, Topological invari-\nants of time-reversal-invariant band structures,\nPhys. Rev. B 75, 121306(R) (2007) .\n[19] M. Z. Hasan and C. L. Kane, Colloquium: Topological\ninsulators, Rev. Mod. Phys. 82, 3045 (2010) .\n[20] X.-L. Qi and S.-C. Zhang, Topological insulators and su -\nperconductors, Rev. Mod. Phys. 83, 1057 (2011) .\n[21] J.-Y. You, C. Chen, Z. Zhang, X.-L. Sheng,\nS. A. Yang, and G. Su, Two-dimensional weyl half-\nsemimetal and tunable quantum anomalous hall effect,\nPhys. Rev. B 100, 064408 (2019) .\n[22] S. Muhlbauer, B. Binz, F. Jonietz, C. Pflei-\nderer, A. Rosch, A. Neubauer, R. Georgii, andP. Boni, Skyrmion lattice in a chiral magnet,\nScience323, 915 (2009) .\n[23] X. Z. Yu, Y. Onose, N. Kanazawa, J. H. Park, J. H.\nHan, Y. Matsui, N. Nagaosa, and Y. Tokura, Real-\nspace observation of a two-dimensional skyrmion crystal,\nNature465, 901 (2010) .\n[24] N. Nagaosa and Y. Tokura, Topological prop-\nerties and dynamics of magnetic skyrmions,\nNat. Nanotechnol. 8, 899 (2013) .\n[25] Y. Ogata, H. Chudo, B. Gu, N. Kobayashi,\nM. Ono, K. Harii, M. Matsuo, E. Saitoh, and\nS. Maekawa, Enhanced orbital magnetic moment\nin FeCo nanogranules observed by barnett effect,\nJ. Magn. Magn. Mater. 442, 329 (2017) .\n[26] R. Gotter, A. Verna, M. Sbroscia, R. Moroni, F. Bi-\nsio, S. Iacobucci, F. Offi, S. R. Vaidya, A. Ruocco,\nand G. Stefani, Unexpectedly large electron correlation\nmeasured in auger spectra of ferromagnetic iron thin\nfilms: Orbital-selected coulomb and exchange contribu-\ntions,Phys. Rev. Lett. 125, 067202 (2020) .\n[27] J. A. Riera, Spin polarization in the hubbard\nmodel with rashba spin-orbit coupling on a ladder,\nPhys. Rev. B 88, 045102 (2013) .\n[28] J.-Y. You, Z. Zhang, B. Gu, and G. Su, Two-\ndimensional room-temperature ferromagnetic semi-\nconductors with quantum anomalous hall effect,\nPhys. Rev. Appl 12, 024063 (2019) .\n[29] T.A.Kaplan,Single-bandhubbardmodelwithspin-orbi t\ncoupling, Z. Phys. B 49, 313 (1983) .\n[30] Collective electron ferronmagnetism,\nProc. Math. Phys. Eng. Sci. 165, 372 (1938) .\n[31] S. Maekawa, T. Tohyama, S. E. Barnes,\nS. Ishihara, W. Koshibae, and G. Khaliullin,\nPhysics of Transition Metal Oxides (Springer Berlin\nHeidelberg, 2004).\n[32] N. Kobayashi, K. Ikeda, B. Gu, S. Takahashi,\nH. Masumoto, and S. Maekawa, Giant fara-\nday rotation in metal-fluoride nanogranular films,\nSci. Rep. 8, 4978 (2018) .\n[33] B. Gu, S. Takahashi, and S. Maekawa, Enhanced\nmagneto-optical kerr effect at fe/insulator interfaces,\nPhys. Rev. B 96, 214423 (2017) .\n[34] D. Coffey, T. M. Rice, and F. C. Zhang,\nDzyaloshinskii-moriya interaction in the cuprates,\nPhys. Rev. B 44, 10112 (1991) .\n[35] L. Shekhtman, O. Entin-Wohlman, and A. Aharony,\nMoriya’s anisotropic superexchange interaction, frus-\ntration, and dzyaloshinsky’s weak ferromagnetism,\nPhys. Rev. Lett. 69, 836 (1992) .\n[36] L. Shekhtman, A. Aharony, and O. Entin-\nWohlman, Bond-dependent symmetric and anti-\nsymmetric superexchange interactions in La 2CuO4,\nPhys. Rev. B 47, 174 (1993) .\n[37] N. E. Bonesteel, Theory of anisotropic superexchange i n\ninsulating cuprates, Phys. Rev. B 47, 11302 (1993) .\n[38] W. Koshibae, Y. Ohta, and S. Maekawa, Electronic and\nmagnetic structures of cuprates with spin-orbit interac-\ntion,Phys. Rev. B 47, 3391 (1993) .\n[39] W. Koshibae, Y. Ohta, and S. Maekawa, Comment on\n“moriya’s anisotropic superexchange interaction, frus-\ntration, and dzyaloshinsky’s weak ferromagnetism”,\nPhys. Rev. Lett. 71, 467 (1993) .\n[40] H. E. Vierti¨ o and N. E. Bonesteel, Interplanar couplin g\nandtheweakferromagnetic transitioninLa 2−xNdxCuO4,6\nPhys. Rev. B 49, 6088 (1994) .\n[41] T. Yildirim, A. B. Harris, A. Aharony, and O. Entin-\nWohlman, Anisotropic spin hamiltonians due to\nspin-orbit and coulomb exchange interactions,\nPhys. Rev. B 52, 10239 (1995) .\n[42] S. G. Tabrizi, A. V. Arbuznikov, and M. Kaupp, Hub-\nbard trimer with spin–orbit coupling: Hartree–fock so-\nlutions, (non)collinearity, and anisotropic spin hamilto -\nnian,J. Phys. Chem. A 123, 2361 (2019) .\n[43] A. N. Kocharian, G. W. Fernando, K. Fang,\nK. Palandage, and A. V. Balatsky, Spin-orbit\ncoupling, electron transport and pairing insta-bilities in two-dimensional square structures,\nAIP Advances 6, 055711 (2016) .\n[44] M. Laubach, J. Reuther, R. Thomale, and S. Rachel,\nRashba spin-orbit coupling in the kane-mele-hubbard\nmodel,Phys. Rev. B 90, 165136 (2014) .\n[45] W. Koshibae, Y. Ohta, and S. Maekawa, Theory of\ndzyaloshinski-moriya antiferromagnetism in distorted\nCuO2and NiO 2planes,Phys. Rev. B 50, 3767 (1994) .\n[46] X. Liu, Z. Wang, K. Watanabe, T. Taniguchi, O. Vafek,\nand J. I. A. Li, Tuning electron correlation in magic-\nangle twisted bilayer graphene using coulomb screening,\nScience371, 1261 (2021) ."    },    {        "title": "2401.11065v1.Extremely_strong_spin_orbit_coupling_effect_in_light_element_altermagnetic_materials.pdf",        "content": "Extremely strong spin-orbit coupling effect in light element altermagnetic materials\nShuai Qu1,3, Ze-Feng Gao1,2,3, Hao Sun2, Kai Liu1,3, Peng-Jie Guo1,3,∗and Zhong-Yi Lu1,3†\n1. Department of Physics and Beijing Key Laboratory of Opto-electronic Functional\nMaterials &Micro-nano Devices. Renmin University of China, Beijing 100872, China\n2. Gaoling School of Artificial Intelligence, Renmin University of China, Beijing, China and\n3. Key Laboratory of Quantum State Construction and Manipulation (Ministry of Education),\nRenmin University of China, Beijing 100872, China\n(Dated: January 23, 2024)\nSpin-orbit coupling is a key to realize many novel physical effects in condensed matter physics,\nbut the mechanism to achieve strong spin-orbit coupling effect in light element antiferromagnetic\ncompounds has not been explored. In this work, based on symmetry analysis and the first-principles\nelectronic structure calculations, we demonstrate that strong spin-orbit coupling effect can be real-\nized in light element altermagnetic materials, and propose a mechanism for realizing the correspond-\ning effective spin-orbit coupling. This mechanism reveals the cooperative effect of crystal symmetry,\nelectron occupation, electronegativity, electron correlation, and intrinsic spin-orbit coupling. Our\nwork not only promotes the understanding of light element compounds with strong spin-orbit cou-\npling effect, but also provides an alternative for realizing light element compounds with an effective\nstrong spin-orbit coupling.\nIntroduction. Spin-orbit coupling (SOC) is ubiquitous\nin realistic materials and crucial for many novel physical\nphenomena emerging in condensed matter physics, in-\ncluding topological physics [1–3], anomalous Hall effect\n[4], spin Hall effect [5, 6], magnetocrystalline anisotropy\n[7] and so on. For instance, quantum anomalous Hall\n(QAH) insulators are characterized by non-zero Chern\nnumbers [8]. The Chern number is derived from the inte-\ngration of Berry curvature over the occupied state of the\nBrillouin zone (BZ). For collinear ferromagnetic and an-\ntiferromagnetic systems, the integral of Berry curvature\nover the occupied state of the Brillouin zone must be zero\nwithout SOC due to the spin symmetry {C⊥\n2T||T}. Here\ntheC⊥\n2andTrepresent the 180 degrees rotation perpen-\ndicular to the spin direction and time-reversal operation,\nrespectively. Therefore, QAH effect can only be real-\nized in collinear magnetic systems when SOC is included\n[9, 10]. On the other hand, strong SOC may open up a\nlarge nontrivial bandgap, which is very important to re-\nalize QAH effect at high temperatures. In general, strong\nSOC exists in heavy element compounds. Unfortunately,\nthe chemical bonds of heavy element compounds are\nweaker than those of light element compounds, which\nleads to more defects in heavy element compounds. Thus,\nthe stability to realize exotic functionalities in heavy el-\nement compounds is relatively weak.\nAn interesting question is whether the strong SOC ef-\nfect can be achieved in light element compounds. Very\nrecently, Li et al. demonstrated that the SOC can be en-\nhanced in light element ferromagnetic materials, which\nderives from the cooperative effects of crystal symme-\ntry, electron occupancy, electron correlation, and intrin-\nsic SOC [11]. This provides a new direction for the design\nof light element materials with strong effective SOC.\n∗guopengjie@ruc.edu.cn\n†zlu@ruc.edu.cnVery recently, based on spin group theory, altermag-\nnetism is proposed as a new magnetic phase distinct from\nferromagnetism and conventional collinear antiferromag-\nnetism [12, 13]. Moreover, altermagnetic materials have\na wide range of electronic properties, which cover met-\nals, semi-metals, semiconductors, and insulators [13, 14].\nDifferent from ferromagnetic materials with s-wave spin\npolarization, altermagnetic materials have k-dependent\nspin polarization, which results in many exotic physical\neffects [12, 13, 15–21]. With spin-orbit coupling, similar\nto the case of ferromagnetic materials, the time-reversal\nsymmetry-breaking macroscopic phenomena can be also\nrealized in altermagnetic materials [10, 22–24]. Never-\ntheless, altermagnetism is proposed based on spin group\ntheory and the predicted altermagnetic materials basi-\ncally are light element compounds [13, 14]. Therefore,\nit is very important to propose a mechanism to enhance\nSOC in light element compounds with altermagnetism\nand predict the corresponding compounds with strong\nSOC effect.\nIn this work, based on symmetry analysis and the first-\nprinciples electronic structure calculations, we predict\nthat the light element compound NiF 3is an i-wave al-\ntermagnetic material with extremely strong SOC effect.\nThen, we propose a mechanism to enhance SOC effect\nin light element compounds with altermagnetism, which\nreveals the cooperative effects of crystal symmetry, elec-\ntron occupation, electronegativity, electron correlation,\nand intrinsic SOC. We also explain the weak SOC effect\nin altermagnetic materials VF 3, CrF 3, FeF 3, CoF 3.\nResults and discussion. The NiF 3takes rhombohedral\nstructure with nonsymmorphic R −3c (167) space group\nsymmetry, as shown in Fig. 1 (a)and(b). The corre-\nsponding elementary symmetry operations are C 3z, C1\n2t\nand I, which yield the point group D 3d. The t repre-\nsents (1/2, 1/2, 1/2) fractional translation. To confirm\nthe magnetic ground state of NiF 3, we consider six dif-\nferent collinear magnetic structures, including one ferro-arXiv:2401.11065v1  [cond-mat.mtrl-sci]  19 Jan 20242\nFM AFM2 AFM4\nAFM3 AFM5\nAFM1\na\nb\nc\nNi\nF\na\nb\nc\n140\n°(a) (c)(e)(g)\n(b) (d)\n(f)\n(h)\na\nb\nc\nFIG. 1. The crystal structure and six collinear magnetic structures of NiF 3.(a)and(b)are side and top views of the crystal\nstructure, respectively. The cyan arrow represents the direction of easy magnetization axis. (c)-(h)are six different collinear\nmagnetic structures including one ferromagnetic and five different collinear antiferromagnetic structures. The bond angle of\nNi−F−Ni for the nearest neighbour Ni ions is 140 degrees. The primitive cell of NiF 3is shown in (d). The red and blue\narrows represent spin-up and spin-down magnetic moments, respectively.\nmagnetic and five collinear antiferromagnetic structures\nwhich are shown in Fig. 1 (c)-(h) . Then we calculate\nrelative energies of six magnetic states with the varia-\ntion of correlation interaction U. With the increase of\ncorrelation interaction U, the NiF 3changes from the fer-\nromagnetic state to the collinear antiferromagnetic state\nAFM1 (Fig. 2 (a)). The AFM1 is intralayer ferromag-\nnetism and interlayer antiferromagnetism (Fig. 1 (d)). In\nprevious works, the correlation interaction U was selected\nas 6.7eV for Ni 3dorbitals [25, 26]. Thus, the magnetic\nground state of NiF 3is the AFM1 state, which is consis-\ntent with previous works[14]. On the other hand, since\nthe bond angle of Ni −F−Ni for the nearest neighbour\nNi ions is 140 degrees, the spins of the nearest neigh-\nbour and next nearest neighbour Ni ions are in antipar-\nallel and parallel arrangement according to Goodenough-\nKanamori rules [27], respectively. This will result in NiF 3\nbeing the collinear antiferromagnetic state AFM1. Thus,\nthe results of theoretical analysis are in agreement with\nthose of theoretical calculation.\nIndeed, the structure of AFM1 is very simple and the\ncorresponding magnetic primitive cell only contains twomagnetic atoms with opposite spin arrangement which\nis shown in Fig. 2 (b). From Fig. 2 (b), the two Ni\natoms with opposite spin arrangement are surrounded by\nF-atom octahedrons with different orientations, respec-\ntively. Thus, the two opposite spin Ni sublattices cannot\nbe connected by a fractional translation. Due to two Ni\nions located at space-inversion invariant points, the two\nopposite spin Ni sublattices cannot be either connected\nby space-inversion symmetry. However, the two opposite\nspin Ni sublattices can be connected by C1\n2t symmetry.\nThus, the NiF 3is an altermagnetic material. The BZ of\naltermagnetic NiF 3is shown in Fig. 2 (c)and both the\nhigh-symmetry and non-high-symmetry lines and points\nare marked. In order to display the altermagnetic prop-\nerties more intuitively, we calculate polarization charge\ndensity of altermagnetic NiF 3, which is shown in Fig.\n2(d). From Fig. 2 (d), the polarization charge densi-\nties of two Ni ions with opposite spin arrangement are\nanisotropic and their orientations are different, result-\ning from F-atom octahedrons with different orientations.\nThe anisotropic polarization charge densities can result in\nk-dependent spin polarization in reciprocal space. More-3\nFM\nAFM1\nAFM2\nAFM3\nAFM4\nAFM5\nΓFT\nL\nSHSଶmSଶHଶ(a)\n(c) (d)(b)\nabc\nFIG. 2. The magnetic ground state of NiF 3and the cor-\nresponding properties. (a)Relative energies of six differ-\nent magnetic states with the variation of correlation inter-\naction U. (b)and(c)are the magnetic primitive cell of NiF 3\nand the corresponding Brillouin zone, respectively. The red\nand blue arrows represent spin-up and spin-down magnetic\nmoments, respectively. The high-symmetry and non-high-\nsymmetry lines and points are marked in the BZ. (d)The\nanisotropic polarization charge densities. The red and blue\nrepresent spin-up and spin-down polarization charge density,\nrespectively.\nover, according to different spin group symmetries, the\nk-dependent spin polarization can form d-wave, g-wave,\nori-wave magnetism [12].\nWithout SOC, the nontrivial elementary spin sym-\nmetry operations in altermagnetic NiF 3have{E||C3z},\n{C⊥\n2||M1t},{E||I}, and {C⊥\n2T||T}. The spin symme-\ntries{C⊥\n2||M1t},{T||TM 1t}, and {E||C3z}make alter-\nmagnetic NiF 3being an i-wave magnetic material, as\nshown in Fig. 3 (a). Moreover, the spins of bands are\nopposite along non-high-symmetry S 2−Γ and Γ −m(S 2)\ndirections, reflecting features of i-wave magnetism (Fig.\n3(b)).\nIn order to well understand the electronic properties,\nwe also calculate the electronic band structures of alter-\nmagnetic NiF 3along the high-symmetry directions. Ig-\nnoring SOC, the NiF 3is an altermagnetic metal. There\nare four bands crossing the Fermi level due to spin de-\ngeneracy on the high-symmetry directions (Fig. 3 (c)).\nEspecially, these four bands are degenerate on the Γ −T\naxis. In fact, any kpoint on the Γ −T axis has nontriv-\nial elementary spin symmetry operations {E||C3z}and\n{C⊥\n2||M1t}. And the spin symmetry {E||C3z}has one\none-dimensional irreducible real representation and two\none-dimensional irreducible complex representations. Al-\nthough the time-reversal symmetry is broken, altermag-\n(d)Cଷ\nCଶୄMଵt\nTT Mଵt(c)\n(b)(a)\nNi-3d\nF-2p\nΓTHଶ|HLΓS |SଶFΓ-0.50-0.250.000.250.50Energy  (eV)\nNi-3d\nF-2p\nΓTHଶ|HLΓS|SଶFΓ-2-1012Energy  (eV)\nΓ Sଶ mSଶ-0.50-0.250.000.250.50Energy  (eV)FIG. 3. Schematic diagram of the i-wave magnetism\nand electronic band structures of altermagnetic NiF 3.(a)\nSchematic diagram of the i-wave magnetism. The red and\nblue parts represent spin up and down, respectively. (b)The\nelectronic band structure without SOC along the non-high-\nsymmetry directions. The red and blue lines represent spin-\nup and spin-down bands, respectively. (c)and(d)are the\nelectronic band structures without and with SOC along the\nhigh-symmetry directions.\nnetic materials have equivalent time-reversal spin sym-\nmetry {C⊥\n2T||T}. The spin symmetry {C⊥\n2T||T}will\nresult in two one-dimensional irreducible complex repre-\nsentations to form a Kramers degeneracy. Meanwhile,\nthe spin symmetry {C⊥\n2||M1t}protects the spin degen-\neracy. Therefore, there is one four-dimensional and one\ntwo-dimensional irreducible representations on the Γ −T\naxis. The quadruple degenerate band crossing the Fermi\nlevel is thus protected by the spin group symmetry. Fur-\nthermore, the orbital weight analysis shows that these\nfour bands are contributed by both the 3dorbitals of Ni\nand the porbitals of F (Fig. 3 (c)). As is known to all,\nthe F atom has the strongest electronegativity among all\nchemical elements, but the 2porbitals of F do not fully\nacquire the 3d-orbital electrons of Ni, which is very in-\nteresting.\nIn our calculations, the number of valence electrons of\nNiF 3is 74, which makes the quadruple band only half-\nfilled. This is the reason why the porbitals of F do not\nfully acquire the d-orbital electrons of Ni. When SOC is\nincluded, the spin group symmetry breaks down to mag-\nnetic group symmetry. The reduction of symmetry will\nresult in the quadruple band to split into multiple bands.\nSince the F atom has the strongest electronegativity, the\n2porbitals of F will completely acquire the 3d-orbital\nelectrons of Ni. This will result in altermagnetic NiF 3to\ntransform from metal phase to insulator phase. In order\nto prove our theoretical analysis, we calculate the elec-\ntronic band structure of altermagnetic NiF 3with SOC.\nThe calculation of the easy magnetization axis and sym-\nmetry analysis based on magnetic point group are shown\nin Supplementary Material[28]. Just like our theoretical\nanalysis, the 2porbitals of F indeed fully acquire the4\n3d-orbital electrons of Ni and altermagnetic NiF 3trans-\nforms into an insulator with a bandgap of 2 .31eV (Fig.\n3(d)). In general, the SOC strength of Ni is in the order\nof 10meV, so the SOC strength of altermagnetic NiF 3is\ntwo orders of magnitude higher than that of Ni. Thus,\nthe SOC effect of altermagnetic NiF 3is extremely strong.\nOn the other hand, we also examine the effect of cor-\nrelation interaction in altermagnetic NiF 3. We calcu-\nlate the electronic band structures of altermagnetic NiF 3\nalong the high-symmetry directions without SOC under\ncorrelation interaction U = 3 ,5,7eV, which are shown in\nFig. 4 (a),(b), and (c), respectively. From Fig. 4 (a),\n(b)and(c), the correlation interaction has a slight ef-\nfect on the band structure around the Fermi level with-\nout SOC, due to the constraints of spin symmetry and\nelectron occupancy being 74. When including SOC, al-\ntermagnetic NiF 3transforms from a metal phase to an\ninsulator phase under different correlation interaction\nU. Moreover, the bandgap of altermagnetic NiF 3in-\ncreases linearly with the correlation interaction U. Thus,\nthe correlation interaction can substantially enhance the\nbandgap opened by the SOC of altermagnetic materials.\n(a) (c)\n(b)(d)ΓTL ΓFΓ-0.50-0.250.000.250.50Energy  (eV)\nHଶHSSଶ ΓTL ΓFΓ-0.50-0.250.000.250.50Energy  (eV)\nHଶHSSଶ\nΓTLΓ FΓ-0.50-0.250.000.250.50Energy  (eV)\nHଶHSSଶ3 4 5 6 7\nU  (eV)1.01.52.02.5Bandgap  (eV)\nFIG. 4. The electronic properties of altermagnetic NiF 3\nunder different correlation interaction U. (a),(b)and(c)\nare the electronic band structures along the high-symmetry\ndirections without SOC under correlation interaction U =\n3,5,7eV, respectively. (d)The bandgap as a function of cor-\nrelation interaction U under SOC.\nNow we well understand the reason for the extremely\nstrong SOC effect in altermagnetic NiF 3. A natural ques-\ntion is whether such a strong SOC effect can be realized\nin other altermagnetic materials. According to the above\nanalysis, we propose four conditions for realizing such an\neffective strong SOC in light element altermagnetic ma-\nterials: First, the spin group of altermagnetic material\nhas high-dimensional (greater than four dimensions) irre-\nducible representation (crystal symmetry groups are pre-\nsented in the Supplementary Material[28]); Second, the\nband with high-dimensional representation crossing the\nFermi level is half-filled by valence electrons; Third, non-\nmetallic elements have strong electronegativity; Fourth,the altermagnetic material has strong electron correla-\ntion. To verify these four conditions, we also calcu-\nlate the electronic band structures of four i-wave alter-\nmagnetic materials (VF 3, CrF 3, FeF 3and CoF 3), which\nhave the same crystal structure and spin group sym-\nmetry as NiF 3[14]. The calculations show that none of\nthe four altermagnetic materials meets the second con-\ndition, and the SOC effect is very weak (Detailed calcu-\nlations and analysis are presented in the Supplementary\nMaterial[28]). On the other hand, since high-dimensional\nirreducible representations can be protected by spin space\ngroup in two-dimensional altermagnetic systems, the pro-\nposed mechanism is also applicable to two-dimensional\nlight element altermagnetic materials, which may be ad-\nvantage for realizing quantum anomalous Hall effect at\nhigh temperatures [10].\nThe mechanism for enhancing the SOC effect that\nwe propose in altermagnetic materials is different from\nthat in ferromagnetic materials [11]. First, the high-\ndimensional representation of the symmetry group is 2 or\n3 dimensions in ferromagnetism, while in altermagnetism\nthe high-dimension representation is 4 or 6 dimensions, so\ntheir symmetry requirements are entirely different. Sec-\nond, the band with high-dimensional representation in\nferromagnetism comes from dorbitals, while the band\nwith high-dimensional representation in altermagnetism\ncan come from the combination of porbitals and dor-\nbitals. Third, the enhancement of SOC effect derives\nfrom correlation interaction for ferromagnetic materials,\nbut from both correlation interaction and electronegativ-\nity of nonmetallic element for altermagnetic materials.\nDue to one more degree of freedom to enhance the SOC\neffect, a stronger SOC effect can be achieved in the al-\ntermagnetic materials. Moreover, if electronegativity of\nnonmetallic element is weak, different topological phases\nmay be realized in altermagnetic materials when includ-\ning SOC. On the other hand, the mechanism for enhanc-\ning SOC effect in altermagnetic materials can be also\ngeneralized to conventional antiferromagnetic materials.\nDue to the equivalent time-reversal symmetry, more spin\ngroups with conventional antiferromagnetism have high-\ndimensional irreducible representations. Moreover, con-\nventional antiferromagnetic materials are more abundant\nthan altermagnetic materials, thus conventional antifer-\nromagnetic materials of light elements with strong SOC\neffect remain to be discovered.\nSummary. Based on spin symmetry analysis and\nthe first-principles electronic structure calculations, we\ndemonstrate that extremely strong SOC effect can be re-\nalized in altermagnetic material NiF 3. Then, we propose\na mechanism to enhance SOC effect in altermagnetic ma-\nterials. This mechanism reveals the cooperative effect\nof crystal symmetry, electron occupation, electronega-\ntivity, electron correlation, and intrinsic spin-orbit cou-\npling. The mechanism can explain not only the extremely\nstrong SOC effect in altermagnetic NiF 3, but also the\nweak SOC in altermagnetic VF 3, CrF 3, FeF 3, CoF 3.\nMoreover, the mechanism for enhancing SOC effect can5\nbe also generalized to two-dimensional altermagnetic ma-\nterials.ACKNOWLEDGMENTS\nThis work was financially supported by the Na-\ntional Key R&D Program of China (Grant No.\n2019YFA0308603), the National Natural Science Foun-\ndation of China (Grant No.11934020, No.12204533,\nNo.62206299 and No.12174443) and the Beijing Natural\nScience Foundation (Grant No.Z200005). Computational\nresources have been provided by the Physical Laboratory\nof High Performance Computing at Renmin University of\nChina.\n[1] M. Z. Hasan and C. L. Kane, Colloquium: Topological\ninsulators, Rev. Mod. Phys. 82, 3045 (2010).\n[2] X.-L. Qi and S.-C. Zhang, Topological insulators and su-\nperconductors, Rev. Mod. Phys. 83, 1057 (2011).\n[3] A. Bansil, H. Lin, and T. Das, Colloquium: Topological\nband theory, Rev. Mod. Phys. 88, 021004 (2016).\n[4] N. Nagaosa, J. Sinova, S. Onoda, A. H. MacDonald, and\nN. P. Ong, Anomalous hall effect, Rev. Mod. Phys. 82,\n1539 (2010).\n[5] J. Sinova, S. O. Valenzuela, J. Wunderlich, C. H. Back,\nand T. Jungwirth, Spin hall effects, Rev. Mod. Phys. 87,\n1213 (2015).\n[6] J. E. Hirsch, Spin hall effect, Phys. Rev. Lett. 83, 1834\n(1999).\n[7] M. Cinal, D. M. Edwards, and J. Mathon, Magnetocrys-\ntalline anisotropy in ferromagnetic films, Phys. Rev. B\n50, 3754 (1994).\n[8] F. D. M. Haldane, Model for a quantum hall effect with-\nout landau levels: Condensed-matter realization of the\n”parity anomaly”, Phys. Rev. Lett. 61, 2015 (1988).\n[9] C.-Z. Chang, C.-X. Liu, and A. H. MacDonald, Collo-\nquium: Quantum anomalous hall effect, Rev. Mod. Phys.\n95, 011002 (2023).\n[10] P.-J. Guo, Z.-X. Liu, and Z.-Y. Lu, Quantum anomalous\nhall effect in collinear antiferromagnetism, npj Comput.\nMater. 9, 70 (2023).\n[11] J. Li, Q. Yao, L. Wu, Z. Hu, B. Gao, X. Wan, and\nQ. Liu, Designing light-element materials with large effec-\ntive spin-orbit coupling, Nat. Commun. 13, 919 (2022).\n[12] L. ˇSmejkal, J. Sinova, and T. Jungwirth, Beyond conven-\ntional ferromagnetism and antiferromagnetism: A phase\nwith nonrelativistic spin and crystal rotation symmetry,\nPhys. Rev. X 12, 031042 (2022).\n[13] L. ˇSmejkal, J. Sinova, and T. Jungwirth, Emerging re-\nsearch landscape of altermagnetism, Phys. Rev. X 12,\n040501 (2022).\n[14] Z.-F. Gao, S. Qu, B. Zeng, Y. Liu, J.-R. Wen, H. Sun,\nP.-J. Guo, and Z.-Y. Lu, Ai-accelerated discovery of al-\ntermagnetic materials (2023), arXiv:2311.04418 [cond-\nmat.mtrl-sci].\n[15] H.-Y. Ma, M. Hu, N. Li, J. Liu, W. Yao, J.-F. Jia, and\nJ. Liu, Multifunctional antiferromagnetic materials with\ngiant piezomagnetism and noncollinear spin current, Nat.\nCommun. 12, 2846 (2021).\n[16] L. ˇSmejkal, A. B. Hellenes, R. Gonz´ alez-Hern´ andez,\nJ. Sinova, and T. Jungwirth, Giant and tunneling mag-netoresistance in unconventional collinear antiferromag-\nnets with nonrelativistic spin-momentum coupling, Phys.\nRev. X 12, 011028 (2022).\n[17] R. Gonz´ alez-Hern´ andez, L. ˇSmejkal, K. V´ yborn´ y, Y. Ya-\nhagi, J. Sinova, T. c. v. Jungwirth, and J. ˇZelezn´ y,\nEfficient electrical spin splitter based on nonrelativis-\ntic collinear antiferromagnetism, Phys. Rev. Lett. 126,\n127701 (2021).\n[18] H. Bai, L. Han, X. Y. Feng, Y. J. Zhou, R. X. Su,\nQ. Wang, L. Y. Liao, W. X. Zhu, X. Z. Chen, F. Pan,\nX. L. Fan, and C. Song, Observation of spin splitting\ntorque in a collinear antiferromagnet RuO 2, Phys. Rev.\nLett. 128, 197202 (2022).\n[19] S. Karube, T. Tanaka, D. Sugawara, N. Kadoguchi,\nM. Kohda, and J. Nitta, Observation of spin-splitter\ntorque in collinear antiferromagnetic RuO 2, Phys. Rev.\nLett. 129, 137201 (2022).\n[20] D. Zhu, Z.-Y. Zhuang, Z. Wu, and Z. Yan, Topological\nsuperconductivity in two-dimensional altermagnetic met-\nals, Phys. Rev. B 108, 184505 (2023).\n[21] P.-J. Guo, Y. Gu, Z.-F. Gao, and Z.-Y. Lu, Altermagnetic\nferroelectric LiFe 2F6and spin-triplet excitonic insulator\nphase (2023), arXiv:2312.13911 [cond-mat.mtrl-sci].\n[22] L. ˇSmejkal, R. Gonzalez-Hernandez, T. Jungwirth, and\nJ. Sinova, Crystal time-reversal symmetry breaking and\nspontaneous hall effect in collinear antiferromagnets, Sci.\nAdv. 6, eaaz8809 (2020).\n[23] X. Zhou, W. Feng, X. Yang, G.-Y. Guo, and Y. Yao,\nCrystal chirality magneto-optical effects in collinear an-\ntiferromagnets, Phys. Rev. B 104, 024401 (2021).\n[24] X.-Y. Hou, H.-C. Yang, Z.-X. Liu, P.-J. Guo, and Z.-Y.\nLu, Large intrinsic anomalous hall effect in both Nb 2FeB 2\nand Ta 2FeB 2with collinear antiferromagnetism, Phys.\nRev. B 107, L161109 (2023).\n[25] B. ˇZemva, K. Lutar, L. Chacon, M. Fele-Beuermann,\nJ. A. Allman, C. J. Shen, and N. Bartlett, Thermody-\nnamically unstable fluorides of nickel: NiF 4and NiF 3\nsyntheses and some properties, J. Am. Chem. Soc. 117,\n10025 (1995).\n[26] S. Mattsson and B. Paulus, Density functional theory\ncalculations of structural, electronic, and magnetic prop-\nerties of the 3d metal trifluorides MF 3(M = Ti-Ni) in\nthe solid state, J. Comput. Chem. 40, 1190 (2019).\n[27] J. B. Goodenough, Goodenough-Kanamori rule, Scholar-\npedia 3, 7382 (2008).\n[28] Supplemental material, ."    },    {        "title": "1101.1268v3.Quantum_phase_transitions_in_a_strongly_entangled_spin_orbital_chain__A_field_theoretical_approach.pdf",        "content": "arXiv:1101.1268v3  [cond-mat.str-el]  31 May 2011Quantum phase transitions in a strongly entangled spin-orb ital chain:\nA field-theoretical approach\nAlexander Nersesyan\nThe Abdus Salam International Centre for Theoretical Physi cs, 34100, Trieste, Italy\nAndronikashvili Institute of Physics, Tamarashvili 6, 017 7, Tbilisi, Georgia\nCenter of Condensed Mater Physics, ITP, Ilia State Universi ty, 0162, Tbilisi, Georgia\nGia-Wei Chern and Natalia B. Perkins\nDepartment of Physics, University of Wisconsin, Madison, W isconsin 53706, USA\nMotivated by recent experiments on quasi-1D vanadium oxide s, we study quantum phase transi-\ntions in a one-dimensional spin-orbital model describing a Haldane chain and a classical Ising chain\nlocally coupled by the relativistic spin-orbit interactio n. By employing a field-theoretical approach,\nwe analyze the topology of the ground-state phase diagram an d identify the nature of the phase\ntransitions. In the strong coupling limit, a long-range N´ e el order of entangled spin and orbital\nangular momentum appears in the ground state. We find that, de pending on the relative scales\nof the spin and orbital gaps, the linear chain follows two dis tinct routes to reach the N´ eel state.\nFirst, when the orbital exchange is the dominating energy sc ale, a two-stage ordering takes place\nin which the magnetic transition is followed by melting of th e orbital Ising order; both transitions\nbelong to the two-dimensional Ising universality class. In the opposite limit, the low-energy orbital\nmodes undergo a continuous reordering transition which rep resents a line of Gaussian critical points.\nOn this line the orbital degrees of freedom form a Tomonaga-L uttinger liquid. We argue that the\nemergence of the Gaussian criticality results from merging of the two Ising transitions in the strong\nhybridization region where the characteristic spin and orb ital energy scales become comparable.\nFinally, we show that, due to the spin-orbit coupling, an ext ernal magnetic field acting on the spins\ncan induce an orbital Ising transition.\nI. INTRODUCTION\nOver the past decades, one-dimensional spin-orbital\nmodels have been a subject of intensive theoretical stud-\nies. The interest is to a large extent motivated by exper-\nimental discovery of unusual magnetic properties in vari-\nous quasi-one-dimensional Mott insulators.1,2The inter-\ndependence of spin and orbital degrees of freedom is usu-\nally described by the so-called Kugel-Khomskii Hamil-\ntonian in which the effective spin exchange constant de-\npends on the orbital configuration and vice versa.3,4An-\nother mechanism of coupling spin and orbital degrees of\nfreedom is the on-site relativistic spin-orbit (SO) interac-\ntionλL·S, whereListhe orbitalangularmomentumand\nλis the coupling constant. In compounds with quenched\norbital degrees of freedom, the presence of the SO term\nusuallyleadstothesingle-ionspinanisotropy DS2\nzwhere\nD∼λ2/∆ and ∆ denotes the energy scale of the crystal\nfield which lifts the degenerate orbital states.\nFor systems with residual orbital degeneracy, on the\nother hand, the effect of the SO term is much less\nexplored compared with the Kugel-Khomskii-type cou-\npling. Due to the directional dependence of the orbital\nwave functions, the SU(2) symmetry of the Heisenberg\nspin exchange is expected to be broken in the presence of\ntheSOinteraction. Theresultantspinanisotropyislikely\nto induce a long-range magnetic order in the spin sector.\nAmoreintriguingquestioniswhathappenstotheorbital\nsector. To answer this question, one needs to consider\nthe details of the interplay between the orbital exchange\nand the SO coupling. Here we consider the simplestcase of a two-fold orbital degeneracy per site. Specifi-\ncally, the two degenerate states could be the dyzanddzx\norbitals in a tetragonal crystal field observed in several\ntransition-metal compounds. We introduce pseudospin-\n1/2 operators τa(a=x,y,z) to describe the doublet\norbital degrees of freedom assuming that τz=±1 corre-\nspond to the states |yz/an}bracketri}htandi|zx/an}bracketri}ht, respectively. Alterna-\ntively, one can also realize the double orbital degeneracy\nin the Mott-insulating phase of a 1D fermionic optical\nlattice where the eigenvectors of τzrefers to pxandpy\norbitals in an anisotropic potential.5,6Restricted to this\ndoublet space, the orbital angular momentum operator\nL= (0,0,τx). This can be easily seen by noting that the\neigenstatesof τxcarryanangularmomentum /an}bracketle{tLz/an}bracketri}ht=±1.\nThe exchange interaction between localized orbital de-\ngrees of freedom is characterized by its highly direc-\ntional dependence: the interaction energy only depends\non whether the relevant orbital is occupied for bonds of\na given orientation. This is particularly true for inter-\nactions dominated by direct exchange mechanism. De-\nnoting the relevant orbital projectors on a given bond as\nP= (1+τβ)/2, where τβ/2isanappropriatepseudospin-\n1/2 operator ( τβbeing a Pauli matrix), the orbital inter-\naction is thus described by an Ising-type term τβ\niτβ\nj. The\nwell studied orbital compass model and Kitaev model\nboth belong to this category.7,8The quantum nature of\nthese models comes from the fact that different operators\nτβ, which do not commute with each other, are used for\nbonds of different types. To avoid unnecessary compli-\ncations coming from the details of orbital interactions,\nwe assume that there is only one type of bond in our 1D2\nsystem and the orbital interaction is thus governed by a\nclassical Ising Hamiltonian.\nWe incorporate these features into the following toy\nmodel of spin-orbital chain ( Js,Jτ>0):\nH=HS+Hτ+HSτ (1)\n=JS/summationdisplay\nnSn·Sn+1+Jτ/summationdisplay\nnτz\nnτz\nn+1+λ/summationdisplay\nnτx\nnSz\nn.\nMotivated by the recent experimental characterizations\nof quasi-1D vanadium oxides,14–19here we focus on the\ncase of quantum spin with length S= 1. The above\nmodel thusdescribesaHaldanechainlocallycoupled toa\nclassical Ising chain by the SO interaction HSτ. The role\nof theλ-term is two-fold: firstly it introduces anisotropy\ntothespin-1subsystem, andsecondlyit endowsquantum\ndynamics to the otherwise classical Ising chain.\nBefore turning to a detailed study of the phase dia-\ngram of model (1), we first discuss its connections to\nreal compounds. As mentioned above, the interest in\nthe toy model is partly motivated by the recent experi-\nmental progresson vanadium oxides which include spinel\nZnV2O414–17and quasi-1D CaV 2O4.18,19In both types\nof vanadates, the two delectrons of V3+ions have a\nspinS= 1 in accordance with Hund’s rule. In the low-\ntemperature phase of both vanadates, the vanadium site\nembeddedinaflattenedVO 6octahedronhasatetragonal\nsymmetry. This tetragonal crystal field splits the degen-\neratet2gtriplet into a singlet and a doublet. As one of\nthe twodelectrons occupies the lower-energy dxystate,\na double orbital degeneracy arises as the second electron\ncould occupy either dzxordyzorbitals. The fact that\nthedxyorbital is occupied everywherealso contributes to\ntheformationofweaklycoupledquasi-1Dspin-1chainsin\nthese compounds.20On the other hand, the details of the\norbital exchange depends on the geometry of the lattice\nand in the case of vanadium spinel the orbital interaction\nis of three-dimensional nature. The Ising orbital Hamil-\ntonian in Eq. (1) thus should be regarded as an effective\ninteraction in the mean-field sense. Nonetheless, the toy\nmodel provides a first step towardsunderstanding the es-\nsentialphysics introducedby the SO coupling. Moreover,\nmany conclusions of this paper can be applied to the case\nof quasi-1Dcompound CaV 2O4where the vanadium ions\nform a zigzag chain.\nIt is instructive to first establish regions of stable mas-\nsive phases. In the decoupling limit, λ→0, our model\ndescribes two gapped systems: a quantum spin-1 Heisen-\nbergchainand aclassicalorbitalIsingchain. Theground\nstate of the spin sector is a disordered quantum spin\nliquid with a finite spectral gap21∆S, whereas the or-\nbital ground state is characterized by a classical N´ eel\norder along the chain: /an}bracketle{tτz\nn/an}bracketri}ht= (−1)nηz. Quantum ef-\nfects in the orbital sector induced by the SO coupling\nplay a minor role. Obviously, just because of being\ngapped, both the spin-liquid phase and the orbital or-\ndered state are stable as long as λremains small. Con-\nsider now the opposite limit, λ≫JS,Jτ. In the ze-\nroth order approximation, the model is dominated by thesingle-ionterm HSτwhose doubly degenerateeigenstates\n|±/an}bracketri}ht=|Sz=±1/an}bracketri}ht⊗|τx=±1/an}bracketri}htrepresent locally entangled\nspin and orbital degrees of freedom. Switching on small\nJSandJτleads to a staggered ordering of the |+/an}bracketri}htand\n|−/an}bracketri}htstates alongthe chain. Physically, the large- λground\nstate can be viewed as a simultaneous N´ eel ordering of\nspin and orbital angular momentum characterized by or-\nder parameters ζandηxsuch that /an}bracketle{tSz\nn/an}bracketri}ht= (−1)nζand\n/an}bracketle{tLz\nn/an}bracketri}ht=/an}bracketle{tτx\nn/an}bracketri}ht= (−1)nηx. The Ising order parameter ηz\nvanishes identically in this phase.\nThese observations naturally lead to the following\nquestions. How is the magnetically ordered N´ eel state\nat large λconnected to the disordered Haldane phase\nasλ→0 ? What is the scenario for the orbital re-\norientation transition ηz→ηx, which is of essentially\nquantum nature ? In this paper we employ the field-\ntheoretical approach to address these questions. We first\nnote that the one-dimensional model (1) is not exactly\nintegrable. As a consequence, the regime of strong hy-\nbridization of the spin and orbital excitations, which is\nthe case when Jτ,JSandλare all of the same order,\nstays beyond the reach of approximate analytical meth-\nods. We thus will be mainly dealing with limiting cases\nJτ≫JSandJτ≪JS, in which one can integrate out\nthe “fast” variables to obtain an effective action for the\n“slow” modes. Following this approach, we establish the\ntopology and main features of the ground-state phase\ndiagram in the accessible parts of the parameter space\nof the model. We were able to unambiguously identify\nthe universality classes of quantum criticalities separat-\ning different massive phases. Using plausible arguments\nwe comment on some features of the model in the regime\nof strong spin-orbital hybridization.\nWe demonstrate that the aforementioned reorientation\ntransition ηz→ηxcan be realized in one of two possi-\nble ways. In the limit of large Jτ, we find a sequence\nof two quantum Ising transitions and an intermediate\nmassive phase, sandwiched between these critical lines,\nin which both ηzandηxare nonzero. This is consis-\ntent with the recent findings22based on DMRG calcu-\nlations and some analytical estimations. In the oppo-\nsite limit, when the Haldane gap ∆ Sis the largest en-\nergy scale, integrating out the spin excitations yields an\neffective lowest-energy action for the orbital degrees of\nfreedom, which shows that the ηz→ηxcrossover takes\nplace as a single Gaussian quantum criticality. At this\ncritical point, the orbital degrees of freedom display an\nextremelyquantumbehaviour: they aregaplessand form\na Tomonaga-Luttinger liquid. This is the main result of\nthis paper. We bring about arguments suggesting that\nthe emergence of the Gaussian critical line is the result\nof merging of the two Ising criticalities in the region of\nstrong spin-orbital hybridization.\nAny field-theoretical treatment of the model (1) must\nbe based on a properly chosen contiuum description of\nthe spin-1 antiferromagnetic Heisenberg chain. Its prop-\nerties have been thoroughly studied, both analytically\nand numerically (see for a recent review Ref. 23). In3\nwhat follows, the spin sector of the model (1) will be\ntreated within the O(3)-symmetric Majorana field the-\nory, proposed by Tsvelik:24\nHM=/summationdisplay\na=1,2,3/bracketleftbiggiv\n2(ξa\nL∂xξa\nL−ξa\nR∂xξa\nR)−imξa\nRξa\nL/bracketrightbigg\n+Hint.\n(2)\nHereξa\nR,L(x) is a degenerate triplet of real (Majorana)\nFermi fields with a mass m, the indices RandLlabel\nthe chirality of the particles, and\nHint=1\n2g/summationdisplay\na(ξa\nRξa\nL)2\nis a weak four-fermion interaction which can be treated\nperturbatively. The continuum theory (2) adequately de-\nscribesthelow-energypropertiesofthegeneralizedspin-1\nbilinear-biquadratic chain\nHS→¯HS=JS/summationdisplay\nn/bracketleftBig\nSn·Sn+1−β(Sn·Sn+1)2/bracketrightBig\n.(3)\nin the vicinityofthe criticalpoint β= 1.25Thisquantum\ncriticality belongs to the universality class of the SU(2) 2\nWess-Zumino-Novikov-Witten(WZNW)modelwithcen-\ntral charge c= 3/2.\nAt small deviations from criticality the Majoranamass\nm∼JS|β−1|determines the magnitude of the triplet\ngap, ∆ S=|m| ≪JS. The theory of a massive\ntriplet of Majorana fermions is equivalent to a system\nof three degenerate noncritical 2D Ising models, with\nm∼(T−Tc)/Tc. This is one of the most appealing fea-\ntures of the theory because the most strongly fluctuating\nphysical fields of the S= 1 chain, namely the staggered\nmagnetization and dimerization operators, have a simple\nlocal representation in terms of the Ising order and dis-\norder parameters.24,26,27It is this fact that greatly sim-\nplifies the analysis of the spin-orbital model (1). While\nthe correspondence between the models (2) and (3) is\nwell justified at |β−1| ≪1, it is believed that the Ma-\njorana model (2) captures generic properties of the Hal-\ndanespin-liquidphaseofthe spin-1chain, eventhoughat\nlarge deviations from criticality ( |β−1| ∼1,∆S∼Js)\nall parameters of the model should be treated as phe-\nnomenological.\nThe remainderofthe paperis organizedasfollows. We\nstart our discussion with Sec. II which contains a brief\nsummary of known facts about the Majorana model24\nthat will be used in the rest of the paper. In Sec. III\nwe consider the limit Jτ/∆S≫1 and by integrating\nout the ‘fast’ orbital modes, show that on increasing the\nSO coupling λthe system undergoes a sequence of two\nconsecutive quantum Ising transitions in the spin and or-\nbital sectors, respectively. In section IV we analyze the\nopposite limiting case, Jτ/∆S≪1, and, by integrating\nover the ‘fast’ spin modes, show that there exists a sin-\ngle Gaussian transition in the orbital sector accompaniedNeel\nζ=0/\nOrbital\nIsing orderIIII\nIIPath 1Orbital Ising order\nxτxS Path 2\nζ=0 Haldane spin liquid\nxzη =0, η =0//η =0, η =0z x\nxz//η =0, η =0\nFIG. 1: Schematic phase diagram of the model on the ( xS,\nxτ)-plane, where xS= ∆S/λandxτ=Jτ/λ.\nby a Neel ordering of the spins. We then conjecture on\nthe topology of the ground-state phase diagram of the\nmodel. In Sec. V we show that spin-orbital hybridiza-\ntion effects near the orbital Gaussian transition lead to\nthe appearance of a non-zero spectral weight well below\nthe Haldane gap which can be detected by inelastic neu-\ntron scattering experiments and NMR measurements. In\nSec. VI we comment on the role of an external magnetic\nfield. We show that, through the SO interaction, a suf-\nficiently strong magnetic field affects the orbital degrees\nof freedom and can lead to a quantum Ising transition\nin the orbital sector. Sec. VII contains a summary of\nthe obtained results and conclusions. The paper has two\nappendices containing certain technical details.\nII. SOME FACTS ABOUT MAJORANA\nTHEORY OF SPIN-1 CHAIN\nIn this Section, we provide some details about the\nO(3)-symmetric Majorana field theory,24Eq. (2), which\nrepresents the continuum limit of the biquadratic spin-1\nmodel (3) at |β−1| ≪1.\nIn the continuum description, the local spin density of\nthe spin model (3) has contributions from the low-energy\nmodes centered in momentum space at q= 0 and q=π:\nS(x) =IR(x)+IL(x)+(−1)x/a0N(x) (4)\nThe smooth part of the local magnetization, I=IR+\nIL, is a sum of the level-2 chiral vector currents. The\nSU(2)2Kac-Moody algebra of these currents is faithfully\nreproduced in terms of a triplet of massless Majorana\nfields28ξ= (ξ1,ξ2,ξ3):\nIν=−i\n2(ξν×ξν),(ν=R,L) (5)4\nThis fact is not surprising because, as already men-\ntioned, the central charge of the SU(2) 2WZNW theory\nisc= 3/2, whereas that of the theory of a massless Ma-\njorana fermion (equivalently, critical 2D Ising model) is\nc= 1/2. At smalldeviationsfromcriticality( |β−1| ≪1)\nthefermionsacquireamass. Stronglyfluctuatingfieldsof\nthe spin-1 chain, the staggered magnetization N(x) and\ndimerization operator ǫ(x) = (−1)nSn·Sn+1, are nonlo-\ncal in terms of the Majorana fields but admit a simple\nrepresentation in terms of the order, σ, and disorder, µ,\noperators of the related noncritical Ising models:\nN∼(1/α)(σ1µ2µ3, µ1σ2µ3, µ1µ2σ3),\nǫ∼(1/α)σ1σ2σ3, (6)\nwhereα∼a0is a short-distance cutoff of the contin-\nuum theory. These expressions together with their duals\n(i.e. their counterparts obtained by the duality trans-\nformation in all Ising copies, σa↔µa) determine the\nvector and scalar parts of the WZNW 2 ×2 matrix field\nˆgwhich is a primary scalar field with scaling dimension\n3/8. It has been demonstrated in Ref. 28 that using\nthe representation (6) and the short-distance operator\nproduct expansions for the Ising fields, one correctly re-\nproduces all fusion rules of the SU(2) 2WZNW model.\nAn equivalent way to make sure that this is indeed the\ncase is to consider the four-Majorana representation of\nthe weakly coupled spin-1/2 Heisenberg ladder26,27and\ntake the limit of a infinite singlet Majorana mass to map\nthe low-energy sector of the model on the O(3) theory\n(2).\nIn the spin-liquid phase of the spin chain (3), which\nis the case β <1, the Majorana mass mis positive,\nimplying that the degenerate triplet of2D Isingmodels is\ninadisorderedphase: /an}bracketle{tσa/an}bracketri}ht= 0,/an}bracketle{tµa/an}bracketri}ht /ne}ationslash= 0 (a= 1,2,3). In\nparticular, this implies that the O(3) symmetry remains\nunbroken, /an}bracketle{tN/an}bracketri}ht= 0, and the ground state of the system\nis not spontaneously dimerized, /an}bracketle{tǫ/an}bracketri}ht= 0.\nThe representation (6) proves to be very useful for cal-\nculatingthe dynamicalspincorrelationfunctionsbecause\nthe asymptotics of the Ising correlators /an}bracketle{tσ(x,τ)σ(0,0)/an}bracketri}ht\nand/an}bracketle{tµ(x,τ)µ(0,0)/an}bracketri}htarewellknownbothatcriticalityand\nin a noncritical regime. In the disordered phase ( m >0),\nthe leading asymptotics of the Ising correlators are:\n/an}bracketle{tµ(r)µ(0)/an}bracketri}ht ∼(a/ξS)1/4/bracketleftBig\n1+O(e−2r/ξS)/bracketrightBig\n,\n/an}bracketle{tσ(r)σ(0)/an}bracketri}ht ∼(a/ξS)1/4/radicalbig\nξS/r e−r/ξS(7)\nwhereξS=v/mis the correlation length, and r=√\nx2+v2τ2. (By duality, in the ordered phase ( m <0)\nthe asymptotics of the correlators in (7) must be inter-\nchanged.) Correspondingly, the dynamical correlation\nfunction\n/an}bracketle{tN(r)N(0)/an}bracketri}ht ∼(a/ξS)3/4/radicalbig\nξS/r e−r/ξS.(8)\nIts Fourier transform at q∼πand small ωdescribes a\ncoherent excitation – a triplet magnon with the mass gapm:\nℑm χ(q,ω)∼m\n|ω|δ/parenleftBig\nω−/radicalbig\n(q−π)2v2+m2/parenrightBig\n.(9)\nSince the single-ion anisotropy Hanis=D/summationtext\nn(Sz\nn)2\nlowers the original O(3) symmetry down to O(2) ×Z2,\none expects24that in the continuum theory it will induce\nanisotropy in the Majorana masses\nm1=m2/ne}ationslash=m3,\nas well as in the coupling constants parametrizing the\nfour-fermion interaction:\nHint→1\n2/summationdisplay\na/negationslash=bgab(ξa\nRξa\nL)/parenleftbig\nξb\nRξb\nL/parenrightbig\n, g13=g23/ne}ationslash=g12.\nThis can be checked by using the correspondence (4)\nand short-distance operator product expansions (OPE)\nfor the physical fields. There will also appear anisotropy\nin the velocities, v1=v2/ne}ationslash=v3, but we will systematically\nneglect this effect. Thus, we have Hanis=/integraltext\ndxHanis,\nwith\nHanis=Dα/integraldisplay\ndx/bracketleftbig\nI3(x)I3(x+α)+N3(x)N3(x+α)/bracketrightbig\n,(10)\nwhereα∼ais a short-distance cutoff of the continuum\ntheory. Using (5) and keeping only the Lorentz invariant\nterms (i.e. neglecting renormalization of the velocities)\nwe can replace ( I3)2by 2I3\nRI3\nL. To treat the second term\nin the r.h.s. of (10), we need OPEs for the products of\nIsing operators:29\nσ(z,¯z)σ(w,¯w)\n=1√\n2/parenleftbiggα\n|z−w|/parenrightbigg1/4/bracketleftbig\n1−π|z−w|ε(w,¯w)/bracketrightbig\n,(11)\nµ(z,¯z)µ(w,¯w)\n=1√\n2/parenleftbiggα\n|z−w|/parenrightbigg1/4/bracketleftbig\n1+π|z−w|ε(w,¯w)/bracketrightbig\n.(12)\nHereε=iξRξLis the energy density (mass bilinear) of\nthe Ising model, z=vτ+ ixandw=vτ′+ ix′are\ntwo-dimensional complex coordinates, ¯ zand ¯ware their\nconjugates. From the above OPEs it follows that\nN3(x)N3(x+α) = i(π/α)/parenleftbig\nξ1\nRξ1\nL+ξ2\nRξ2\nL−ξ3\nRξ3\nL/parenrightbig\n−(π2C)[(ξ1\nRξ1\nL)(ξ2\nRξ2\nL)−(ξ1\nRξ1\nL)(ξ3\nRξ3\nL)−(ξ2\nRξ2\nL)(ξ3\nRξ3\nL)],\nwhereC∼1 is a nonuniversal constant. As a result,\nHanis=−i/summationdisplay\na=1,2,3δmaξa\nRξa\nL+1\n2/summationdisplay\na/negationslash=bδgij(ξa\nRξa\nL)/parenleftbig\nξb\nRξb\nL/parenrightbig\n,(13)\nwhere\nδm1=δm2=−δm3=−(πC)D (14)5\nare corrections to the single-fermion masses, and δg12=\n(2−π2C)Dα, δg 13=δg23=π2CDαare cou-\npling constants of the induced interaction between the\nfermions. Smallness of the Majorana masses ( |m|α/v≪\n1) implies that the additional mass renormalizations\ncaused by the interaction in (13) are relatively small,\nm(Dα/v)ln(v/|m|α)≪D, so that the main effect of\nthe single-ion anisotropy is the additive renormalization\nof the fermionic masses, ma=m+δma, withδmagiven\nby Eq.(14).\nThe cases D >0 andD <0 correspond to an easy-\nplane and easy-axis anisotropy, respectively. The spin\nanisotropy (18) induced by the spin-orbit coupling is of\nthe easy-axis type. At D <0 the singlet Majorana\nfermion, ξ3, is the lightest, m3< m1=m2. Increas-\ning anisotropy drives the system towards an Ising crit-\nicality at D=−D∗, where m3= 0. At D <−D∗\nthe system occurs in a new phase where the Ising dou-\nblet remains disordered while the singlet Ising system\nbecomes ordered. It then immediately follows from the\nrepresentation (6) that the new phase is characterized by\na N´ eel long-range order with /an}bracketle{tN3/an}bracketri}ht /ne}ationslash= 0. Transverse spin\nfluctuations, as well as fluctuations of dimerization, are\nincoherent in this phase.\nIII. TWO ISING TRANSITIONS IN THE\n∆S≪JτLIMIT\nNowwe turn to ourmodel (1). Let us considerthe case\nwhen, in the absence of spin-orbit coupling, the orbital\ngapisthelargest: Jτ≫Js. Theorbitalpseudospinsthen\nrepresent the ‘fast’ subsystem and can be integrated out.\nAssuming that λ≪Jτ, we treat the spin-orbit coupling\nperturbatively. In this case, the zero order Hamiltonian\nH0=HS+Hτdescribes decoupled spin and orbital sys-\ntems, while the spin-orbit interaction HSτdenotes per-\nturbation. Defining the interaction representation for all\noperators according to A(τ) =eτH0Ae−τH0(hereτde-\nnotes imaginary time), the interaction term in the Eu-\nclidian action is given by\nSSτ=λ/summationdisplay\nn/integraldisplay\ndτ τx\nn(τ)Sz\nn(τ). (15)\nThe first nonvanishing correction to the effective action\nin the spin sector is of the second order in λ:\n∆Ss=−λ2\n2/summationdisplay\nnm/integraldisplay\ndτ1dτ2/angbracketleftbig\nτx\nn(τ1)τx\nm(τ2)/angbracketrightbig\nτSz\nn(τ1)Sz\nm(τ2).\n(16)\nAveraging in the right-hand side of (16) goes over config-\nurations of the classical Ising chain Hτ. The correlation\nfunction /an}bracketle{tτx\nn(τ1)τx\nm(τ2)/an}bracketri}htτis calculated in Appendix A. It\nis spatially ultralocal (because there are no propagating\nexcitations in the classical Ising model) and rapidly de-\ncaying at the characteristic time ∼1/Jτ, which is muchshorter than the spin correlation time ∼1/∆0:\n/an}bracketle{tτx\nn(τ1)τx\nm(τ2)/an}bracketri}htτ=δnmexp(−4Jτ|τ1−τ2|).(17)\nPassing to new variables, τ= (τ1+τ2)/2 andρ=τ1−τ2,\nand integrating over ρyields a correction to the effec-\ntive spin action which has the form of a single-ion spin\nanisotropy. Thus in the second order in λ, the spin\nHamiltonian acquires an additional term\nHani=−λ2\n4Jτ/summationdisplay\nn(Sz\nn)2. (18)\nThe anisotropy splits the Majorana triplet into a doublet\n(ξ1,ξ2) and singlet ( ξ3), with masses\nm1=m2=m+πCλ2\n4Jτ, m3=m−πCλ2\n4Jτ,(19)\nwhereC∼1 is a nonuniversal positive constant. The\nanisotropy is of the easy-axis type, so that the singlet\nmode has a smaller mass gap.\nAs long as all the masses maremain positive, the sys-\ntem maintains the properties of an anisotropic Haldane’s\nspin-liquid. The dynamical spin susceptibilities calcu-\nlated at small ωandq∼π(see Sec. II),\nℑm χxx(q,ω) =ℑm χyy(q,ω) (20)\n∼m1\n|ω|δ/parenleftbigg\nω−/radicalBig\n(q−π)2v2+m2\n1/parenrightbigg\n,\nℑm χzz(q,ω)∼m3\n|ω|δ/parenleftbigg\nω−/radicalBig\n(q−π)2v2+m2\n3/parenrightbigg\n,\nindicate the existence of the Sz=±1 andSz= 0 optical\nmagnons with mass gaps m1andm3, respectively. In-\ncreasing the spin-orbital coupling leads eventually to an\nIsingcriticalityat λ=λc1= 2/radicalbig\nJτm/πC, wherem3= 0.\nAtm3<0thesystemoccursinalong-rangeorderedN´ eel\nphase with staggered magnetization /an}bracketle{tSz\nn/an}bracketri}ht= (−1)nζ(λ),\nin whichthe Z2-symmetryofmodel (18) is spontaneously\nbroken. Using the Ising-model representation (6) of the\nstaggered magnetization of the spin-1 chain, we find that\nat 0< λ−λc1≪λc1the order parameter ζ(λ) follows a\npower-law increase:\nζ(λ)∼/parenleftbiggλ−λc1\nλc1/parenrightbigg1/8\n. (21)\nThe transverse spin fluctuations become incoherent in\nthis phase. The situation here is entirely similar to that\nin the spontaneously dimerized massive phase of a two-\nchain spin-1/2 ladder27,30, where the dimerization kinks\nmake spin fluctuations incoherent. In the present case,\nthe spontaneouslybroken Z2symmetry ofthe Neel phase\nleadstotheexistenceofpairsofmassivetopologicalkinks\ncontributing to a broad continuum with a threshold at\nω=m1+|m3|(the details of calculation can be found in\nRef.27):\nℑm χxx(q,ω) (22)\n∼1/radicalbig\nm1|m3|θ(ω2−(q−π)2v2−(m1+|m3|)2)/radicalbig\nω2−(q−π)2v2−(m1+|m3|)2.6\nIn the N´ eel phase, the orbital sector acquires quantum\ndynamicsbecauseantiferromagneticorderingofthe spins\ngenerates an effective transverse magnetic field which\ntransforms the classical Ising model Hτto a quantum\nIsing chain. At λ > λ c1the spin-orbit term takes the\nform\nHSτ=−h/summationdisplay\nn(−1)nτx\nn+H′\nSτ, (23)\nwhereh=λζ(λ) andH′\nSτ=−λ/summationtext\nn(Sz\nn−/an}bracketle{tSz\nn/an}bracketri}ht)τx\nnac-\ncounts for fluctuations. Since both the orbital and spin\nsectors are gapped, the main effect of this term is a\nrenormalization of the mass gaps and group velocities.\nThe transverse field hgives rise to quantum fluctuations\nwhich decrease the classical value of ηzand, at the same\ntime, lead to a staggered ordering of the orbital pseu-\ndospins in the transverse direction. Since the orbital sec-\ntor has a finite susceptibility with respect to a transverse\nstaggered field, in the right vicinity of the critical point\nηxfollows the same power-law increase as ζbut with a\nsmaller amplitude:\nηx∼/parenleftbiggh\nJτ/parenrightbigg\n∼/radicalbigg\n∆S\nJτ/parenleftbiggλ−λc1\nλc1/parenrightbigg1/8\n.(24)\nThis result is in a good agreement with previously\nobtained numerical results for order parameters (See\nFig. 4(a) in Ref. 22).\nPerforming an inhomogeneous π-rotation of the pseu-\ndospins around the y-axis,τx,z\nn→(−1)nτx,z\nn,τy\nn→τy\nn,\nwe find that at λ > λc1the effective model in the orbital\nsectorreducesto a ferromagneticIsing chainin auniform\ntransverse (pseudo)magnetic field:\nHτ;eff=−Jτ/summationdisplay\nnτz\nnτz\nn+1−h/summationdisplay\nnτx\nn.(25)\nNotice that the restriction λ≪Jτ, which was imposed\nin the derivation of the effective Hamiltonian in the spin\nsector, now can be released because the spin sector is\nassumed to be in the N´ eel phase.\nAth=Jτ, i.e. at λ=λc2whereλc2satisfies the\nequation\nλc2ζ(λc2) =Jτ, (26)\nthe model (25) undergoes a 2D Ising transition27,31to a\nmassive disordered phase with /an}bracketle{tτz\nn/an}bracketri}ht= 0. This quantum\ncritical point can be reached when λis further increased\nin the region λ > λc1. It is clear from (26) that λc2is of\ntheorderoforgreaterthan Jτ. Itisreasonabletoassume\nthat for such values of λthe N´ eel magnetization is close\nto its nominal value, ζ∼1, implying that λc2∼Jτ. We\nsee that the two Ising transitions are well separated:\nλc2/λc1∼(Jτ/∆S)1/2≫1. (27)\nThus, in the limit Jτ≫∆S, the ground-state phase\ndiagram of the model (1) consists of three gapped phasesFIG. 2: Schematic diagram of order parameters as functions\nof the SO coupling constant λ. (a) Two Ising transitions in\ntheJτ≫∆Slimit. (b) A single Gaussian transition in the\n∆S≫Jτlimit. These two scenarios correspond to path-1\nand path-2 in the phase diagram (Fig. 1), respectively.\nseparated by two Ising criticalities, one in the spin sector\n(λ=λc1) and the other in the orbital sector ( λ=λc2).\nAt 0< λ < λ c1the spin sector represents an anisotropic\nspin-liquid while in the orbital sector there is a N´ eel-like\nordering of the pseudospins: ( −1)n/an}bracketle{tτz\nn/an}bracketri}ht ≡ηz(λ)/ne}ationslash= 0.\nAtλc1< λ < λ c2the orbital degrees of freedom reveal\ntheir quantum nature: the onset of the spin N´ eel order\n(ζ/ne}ationslash= 0) is accompanied by the emergence of the trans-\nversecomponent of the staggered pseudospin density:\n(−1)n/an}bracketle{tτx\nn/an}bracketri}ht ≡ηx(λ)/ne}ationslash= 0. Upon increasing λ, the stag-\ngered orbital order parameter ηundergoes a continuous\nrotationfrom the z-directionto x-direction. At λ=λc2a\nquantum Ising transition takes place in the orbital sector\nwhereηzvanishes. At λ > λ c2both sectors are long-\nrange ordered, with order parameters ζ,ηx/ne}ationslash= 0. The de-\npendenceoforderparameterson λisschematicallyshown\nin Fig. 2(a); this picture is in full qualitative agreement\nwith the results of the recent numerical studies.22\nThe crossover between the small and large λlimits\nstudied in this sectioncorrespondsto path 1 onthe phase\ndiagram shown in Fig. 1. The path is located in the re-\ngionJτ≫∆S. Starting from the massive phase I and\nmoving along this path we first observe the spin-Ising\ntransition (I →II) to the N´ eel phase. Long-range or-\ndering of the spins induces quantum reconstruction of\nthe initialy classical orbital sector (i.e. generation of a\nnonzeroηx). Theorbital-Ising transition (II →III) takes\nplace inside the spin N´ eel phase. Of course, feedback ef-\nfects (that is, orbitaffecting spin) become inreasinglyim-\nportant upon deviating from the critical curve ∆ SJτ∼1\ninto phases II and III, especially in the vicinity of the\norbital transition where the spin-orbit coupling is very\nstrong,λ∼Jτ. In this region the behavior of the spin\ndegrees of freedom is not expected to follow that of an\nisolated anisotropic spin-1 chain in the N´ eel phase since\nthe effect of an “explicit” staggered magnetic field ∼ληx\nbecomes important. We will see a pattern of such be-\nhavior in the opposite limit of “heavy” spins, which is\ndiscussed in the next section.7\nIV. GAUSSIAN CRITICALITY AT J τ≪∆S\nIn this section we turn to the opposite limiting case:\n∆S≫Jτ. Now the spin degrees of freedom constitute\nthe “fast” subsystem and can be integrated out to gen-\nerate an effective action in the orbital sector. We will\nshowthat, in this regime, the intermediate massivephase\nwherethe orbitalorderparameter ηundergoesacontinu-\nousrotationfrom η= (0,0,ηz) toη= (ηx,0,0)nolonger\nexists. Going along path 2, Fig. 1, which is located in the\nregion ∆ S≫Jτ, we find that the two massive phases, I\nand III, are separated by a single Gaussian critical line\ncharacterized by central charge c= 1. On this line the\nvectorηvanishes, the orbital degrees of freedom become\ngapless and represent a spinless Tomonaga-Luttinger liq-\nuid characterized by power-law orbital correlations.\nAtλ= 0 the spin-1 subsystem represents a disordered,\nisotropic spin liquid. Therefore the first nonzero correc-\ntiontothelow-energyeffectiveactionintheorbitalsector\nappears in the second order in λ:\n∆S(2)\nτ=−1\n6/an}bracketle{tS2\nSτ/an}bracketri}htS (28)\n=−1\n2λ2/summationdisplay\nnm/integraldisplay\ndτ1/integraldisplay\ndτ2/an}bracketle{tSn(τ1)Sm(τ2)/an}bracketri}htSτx\nn(τ1)τx\nm(τ2),\nwhere/an}bracketle{t···/an}bracketri}htSmeans averaging over the massive spin de-\ngrees of freedom. According to the decomposition of the\nspin density, Eq. (4), the correlation function in (29) has\nthe structure:\n/an}bracketle{tSl(τ)S0(0)/an}bracketri}ht= (−1)lf1(r/ξS)+f2(r/ξS).(29)\nHereξs=vs/∆Sis the spin correlation length and\nr= (vsτ,x) is the Euclidian two-dimensional radius-\nvector.f1andf2aresmooth functions with the following\nasymptotic behaviour27\nf1(x) =C1x−1/2e−x, f2(x) =C2x−1e−2x(x≫1),(30)\nwhereC1andC2are nonuniversal constants. DMRG\ncalculations show32thatC2≪C1; for this reason the\ncontribution of the smooth part of the spin correlation\nfunction can be neglected in (29).\nIntegrating over the relative time τ−=τ1−τ2we find\nthat the spin-orbit coupling generates a pseudospin xx-\nexchange with the following structure:\nH′\nτ=/summationdisplay\nn/summationdisplay\nl≥1(−1)l+1J′\nτ(l)τx\nnτx\nn+l (31)\nHere the exchange couplings exponentially decay with\nthe separation l,J′\nτ(l)∼(λ2/∆S)exp(−la0/ξS), so the\nsummation in (31) actually extends up to l∼ξS/a0. In\nthe Heisenberg model ξSis of the order of a few lattice\nspacings, so for a qualitative understanding it would be\nsufficient to consider the l= 1 term as the leading one\nand treat the l= 2 term as a correction. Making a π/2rotation in the pseudospin space, τz\nn→τy\nn,τy\nn→ −τz\nn,\nwe passto the conventionalnotationsand write down the\neffective Hamiltonian for the orbital degrees of freedom\nas a perturbed XY spin-1/2 chain:\nHeff\nτ=/summationdisplay\nn/parenleftbig\nJxτx\nnτx\nn+1+Jyτy\nnτy\nn+1/parenrightbig\n+H′\nτ.(32)\nwhere\nH′\nτ=−J′\nx/summationdisplay\nnτx\nnτx\nn+2+···. (33)\nHereJy=Jτ,Jx=J′\nτ(1)>0 andJ′\nx=J′\nτ(2)>0. By\norder of magnitude J′\nx< Jx∼λ2/∆S.\nIn the absence of the perturbation H′\nτ, the model (32)\nrepresents a spin-1/2 XY chain which for any nonzero\nanisotropy in the basal plane ( Jx/ne}ationslash=Jy) has a N´ eel long-\nrange order in the ground state and a massive excitation\nspectrum. This follows from the Jordan-Wigner trans-\nformation\nτz\nn= 2a†\nnan−1, τ+\nn=τx\nn+iτy\nn= 2a†\nneiπ/summationtext\nj<na†\njaj(34)\nwhich maps the XY chain onto a model of complex spin-\nless fermions with a Cooper pairing:33\nHeff\nτ= (Jx+Jy)/summationdisplay\nn/parenleftbig\na†\nnan+1+h.c./parenrightbig\n+ (Jx−Jy)/summationdisplay\nn/parenleftBig\na†\nna†\nn+1+h.c./parenrightBig\n.(35)\nBy increasing λ(equivalently, decreasing Jτ) the model\n(35) can be driven to a XX quantum critical point, Jx=\nJy(1), i.e. λ=λc∼√Jτ∆S, where the the system\nacquires a continuous U(1) symmetry. At this point the\nJordan-Wignerfermionsbecomemasslessand thesystem\nundergoes a continuous quantum transition.\nThe transition is associated with reorientation of the\npseudospins. Away from the Gaussian criticality the\neffective orbital Hamiltoian is invariant under Z2×Z2\ntransformations: τx\nn→ −τx\nn,τz\nn→ −τz\nn. In massive\nphasesthis symmetry is spontaneouslybroken. Making a\nback rotation from τytoτzwe conclude that at Jy> Jx\n(λ < λ c)ηz/ne}ationslash= 0,ηx= 0, while at Jy< Jx(λ > λ c)\nηz= 0,ηx/ne}ationslash= 0. Both ηzandηxvanish at the critical\npoint, so contrary to the case Jτ≫∆S, here there is no\nregion of their coexistence.\nThe passage to the continuum limit for the model (32)\nbased on Abelian bosonization is discussed in Appendix\nB. There we show that the perturbation H′\nτadds a\nmarginal four-fermion interaction g=J′\nx(2)/πv≪1\nto the free-fermion model (B3). In the spin-chain lan-\nguage, this is equivalent to adding a weak ferromag-\nneticzz-coupling. In the limit of weak XY anisotropy,\n|λ−λc|/λc≪1, the low-energy properties of the orbital\nsector are described by a quantum sine-Gordon model\n(all notations are explained in Appendix B)\nH=u\n2/bracketleftbigg\nKΠ2+1\nK(∂xΦ)2/bracketrightbigg\n+2γ\nπαcos√\n4πΘ,(36)8\nwhere\nγ∼Jτ/parenleftbiggλ−λc\nλc/parenrightbigg\n, K= 1+2g+O(g2).(37)\nThe U(1) criticality is reached at λ=λcwhere, due to a\nfinite value of g, the orbital degrees of freedom represent\na Tomonaga-Luttingerliquid. Close to the criticality, the\nspectral gap in the orbital sector scales as the renormal-\nized mass of the sine-Gordon model (36):\nMorb∼/vextendsingle/vextendsingle/vextendsingleλ−λc\nλc/vextendsingle/vextendsingle/vextendsingleK\n2K−1. (38)\nStrongly fluctuating physical fields acquire coupling\ndependentscalingdimensions. Inparticular,accordingto\nthe bosonization rules,27the staggered pseudospin den-\nsities are expressed in terms of the vertex operators,\n(−1)nτx\nn≡nx(x)∼sin√πΘ(x),\n(−1)nτz\nn≡nz(x)∼cos√πΘ(x),(39)\nboth with scaling dimension d= 1/4K. This anomalous\ndimension determines the power-lawbehaviourof the av-\nerage staggered densities close to the criticality:\nηz(λ)∼(λc−λ)1/4K, λ < λ c\nηx(λ)∼(λ−λc)1/4K, λ > λ c. (40)\nA finite staggered pseudospin magnetization ηxat\nλ > λ cgenerates an effective external staggered mag-\nnetic field in the spin sector:\nHS→¯H=HS+H′\nS, H′\nS=−hS/summationdisplay\nn(−1)nSz\nn,(41)\nwherehS=−ληx. The spectrum of the Hamiltonian ¯H\nis always massive. This can be easily understood within\nthe Majorana model (2). According to (6), in the con-\ntinuum limit, the sign-alternating component of the spin\nmagnetization, N3∼(−1)nSz\nn, canbeexpressedinterms\nof the order and disorder fields of the degenerate triplet\nof 2D disordered Ising models: N3∼µ1µ2σ3. In the\nleading order, the magnetic interaction H′\nSgives rise to\nan effective magnetic field h3=hS/an}bracketle{tµ1µ2/an}bracketri}htapplied to the\nthird Ising system: h3σ3. The latter always stays off-\ncritical.\nSince in the Haldane phase the spin correlations are\nshort-ranged, close to the transition point the induced\nstaggered magnetization ζcan be estimated using linear\nresponse theory. Therefore, at 0 < λ−λc≪λc,ζfollows\nthe same power-law increase as that of ηxbut with a\nsmaller amplitude:\nζ∼hS\n∆S∼/parenleftbiggJτ\n∆S/parenrightbigg1/2/parenleftbiggλ−λc\nλc/parenrightbigg1/4K\n(42)\nSo, in the part of the phase C, Fig. 1, where ∆ S≫Jτ,\ntheηx-orbital order, being the result of a spontaneousbreakdown of a Z2symmetry τx\nn→ −τx\nn, acts as an ef-\nfective staggered magnetic field applied to the spins and\ninducestheir N´ eel alignment. This fact is reflected in a\ncoupling dependent, nonuniversal exponent 1 /4Kchar-\nacterizing the increase of the staggered magnetization at\nλ > λ c. The order parameters as functions of λin the\n∆S≫Jτlimit is schematically shown in Fig. 2(b).\nAs already mentioned, the absence of a small parame-\nter in the regime of strong hybridization, Jτ∼JS∼λ,\nmakestheanalysisofthephasediagraminthisregionnot\neasily accessible by analytical tools. Nevertheless some\nplausible arguments can be put forward to comment on\nthe topology of the phase diagram. It is tempting to\ntreat the curve Jτ∆S/λ2∼1 as a single critical line go-\ning throughout the whole phase plane ( Jτ/λ,∆S/λ). If\nso, we then can expect that there exists a special sin-\ngular point located in the region Jτ∆S/λ2∼1. This\nexpectation is based on the fact that at Jτ≫∆slimit\nthe transition is of the Ising type and the spontaneous\nspin magnetization below the critical curve follows the\nlawζ∼(λ−λc1)1/8with auniversal critical exponent,\nwhereas at Jτ≪∆sthe spin magnetization has a differ-\nent,nonuniversal exponent, ζ∼(λ−λc)1/4K. Continu-\nity considerations make it very appealing to suggest that\nat the special point the Tomonaga-Luttinger liquid pa-\nrameter takes the value K= 2, and the two power laws\nmatch. Since the central charges of two Ising and one\nGaussian criticalities satisfy the relation 1 /2+1/2 = 1,\nthe singular point must be a point where the two Ising\ncritical curves merge into a single Gaussian one.\nV. DYNAMICAL SIN SUSCEPTIBILITY AND\nNMR RELAXATION RATE IN THE VICINITY\nOF GAUSSIAN CRITICALITY\nIt may seem at the first sight that, in the regime\n∆S≫J, the spin degrees of freedom which have been\nintegrated out remain massive across the orbital Gaus-\nsian transition, and the spectral weight of the staggered\nspin fluctuations is only nonzero in the high-energy re-\ngionω∼∆S. However, this conclusion is only correct\nfor the zeroth-order definition of the spin field N0(x),\ngiven by Eq. (6), with respect to the spin-orbit inter-\naction. In fact, the staggered magnetization hybridizes\nwith low-energyorbitalmodes viaSO couplingalreadyin\nthe first order in λand thus acquires a low-energy pro-\njection which contributes to a nonzero spectral weight\ndisplayed by the dynamical spin susceptibility at ener-\ngies well below the Haldane gap.\nTofind the low-energyprojectionofthe field Nz(r), we\nmust fuse the local operator Nz\n0(r) with the perturbative\npart of the total action. Keeping in mind that close to\nand at the Gaussian criticality most strongly fluctuating\nfields are the staggered components of the orbital po-\nlarization, we approximate the SO part of the Euclidian9\naction by the expression\nSSτ≃λa0\nvS/integraldisplay\nd2rNz(r)nx(r), (43)\nwherer= (vSτ,x) is the two-dimensional radius vector\n(hereτis the imaginary time). We thus construct\nNz\nP(r) =/an}bracketle{te−SSτNz(r)/an}bracketri}ht\n=Nz\n0(r)−λa0\nvS/integraldisplay\nd2r1/an}bracketle{tNz\n0(r)Nz\n0(r1)/an}bracketri}htSnx(r1)\n+O(λ2), (44)\nwhere averaging is done over the unperturbed, high-\nenergy spin modes. For simplicity, here we neglect the\nanisotropy of the spin-liquid phase of the S=1 chain and\nuse formula (8). The spin correlation function is short-\nranged. Treating the spin correlation length ξS∼vS/∆S\nas a new lattice constant (new ultraviolet cutoff) and be-\ning interested in the infrared asymptotics |r| ≫ξS, we\ncan replace in (44) nx(r1) bynx(r). The integral\n/integraldisplay\nd2ρ/an}bracketle{tNz\n0(ρ)Nz\n0(0)/an}bracketri}htS (45)\n∼1\na2\n0(a/ξS)3/4/integraldisplay∞\n0dρ ρ/radicalbig\nξs/ρ e−ρ/ξS∼(ξS/a0)5/4.\nSo the first-order low-energy projection of the staggered\nmagnetization is proportional to\nNz\nP(r)∼λ\n∆S/parenleftbiggξS\na0/parenrightbigg1/4\nnx(r). (46)\nThisresultclarifiestheessenceofthehybridizationeffect:\nclose to the Gaussian criticality the spin fluctuations ac-\nquire a finite spectral weight in the low-energy region,\nω≪∆S,q∼π, which is contributed by orbital fluctu-\nations and can be probed in magnetic inelastic neutron\nscattering experiments and NMR measurements.\nAway from but close to the Gaussian criticality the\nbehavior of the dynamical spin susceptibility ℑmχ(q,ω)\nis determined by the excitation spectrum of the sine-\nGordon model for the dual field, Eq.(36). Since K >1, it\nconsists of kinks, antikinks carrying the mass Morb, and\ntheir bound states (breathers) with masses (see e.g. Ref.\n27)\nMj= 2Morbsin(πj/2ν),\nj= 1,2,...ν−1, ν= 2K−1 (47)\nSinceK−1 = 2gis small, there will be only the first\nbreather in the spectrum, with mass M1= 2Morb(1−\n2π2g2). The sine-Gordon model is integrable, and the\nasymptotics of its correlation functions in the massive\nregime have been calculated using the form-factor ap-\nproach (see for a recent review 35). Here we utilize\nsome of the known results. At λ < λ cthe operator\nnx∼sin√πΘ has a nonzero matrix element betweenthe vacuum and the first breather state. This form-\nfactor contributes to a coherent peak in the dynamical\nspin susceptibility at frequencies much smaller than than\nthe Haldane gap:\nℑmχ(q,ω,T= 0) = A(λ/∆S)2δ[ω2−(q−π)2v2−M2\n1]\n+ℑmχcont(q,ω,T= 0). (48)\nHereAis a constant and the second term is the contribu-\ntion of a multi-kink continuum of states with a threshold\natω= 2Morb. Atλ > λ cthe spectral properties of\nthe operator cos√πΘ coincide with those of the opera-\ntor sin√πΘ atλ < λ c. For symmetry reasons35, this\noperator does not couple to the first breather, so that\natλ > λcℑmχ(q,ω) will only display the kink-antikink\nscattering continuum.\nWeseethat, due tospin-orbithybridizationeffects, the\nspin sector of our model loses the properties of a spin liq-\nuid alreadyin anoncriticalorbitalregime. Thistendency\ngets strongly enhanced at the orbital Gaussian criticality\n(Morb→0) where all multi-particle processesmerge, and\nthe spin correlationfunction exhibits an algebraicallyde-\ncaying asymptotics\n/an}bracketle{tNz(r)Nz(0)/an}bracketri}ht ≃ /an}bracketle{tNz\nP(r)Nz\nP(0)/an}bracketri}ht ∼/parenleftbiggλ\n∆S/parenrightbigg2/parenleftBiga\nr/parenrightBig1\n2K,\n(49)\nimplying that the spin sector of the model becomes remi-\nniscent of Tomonaga-Luttinger liquid. In this limit (here\nfor simplicity we consider the T= 0 case) the dynamical\nspin susceptibility is given by34\nℑmχ(q,ω,T= 0)∼(λ/∆S)2/bracketleftbig\nω2−v2(q−π)2/bracketrightbig1\n4K−1.\n(50)\nThe NMR relaxation rate probes the spectrum of local\nspin fluctuations\n1\nT1=A2Tlim\nω→01\nω/summationdisplay\nqℑmχzz(q,ω,T)\nwhereAis an effective hyperfine constant. In spin-\nliquid regime of an isolated spin-1 chain, the existence of\na Haldane gap makes 1 /T1exponentially suppressed36:\n1/T1∼exp(−2∆S/T). The admixture of low-energy or-\nbital states in the spin-fluctuation spectrum drastically\nchangedthisresult. Asimplepowercountingargument37\nleads to a power-law temperature dependence of the\nNMR relaxation rate:\n1\nT1∼A2/parenleftbiggλ\n∆s/parenrightbigg2\nT1\n2K−1(51)\nThis result is valid not only exactly at the Gaussian crit-\nicality but also in its vicinity provided that the tempera-\nture is larger than the orbital mass gap. By construction\n(see the preceding section) K≥1. This means that the\nexponent 1 /2K−1 isnegative and the NMR relaxation\nrateincreases on lowering the temperature. It is worth10\nnoticingthatsuchregimesarenotunusualforTomonaga-\nLuttingerphasesoffrustratedspin-1/2ladders.38Forour\nmodel, such behavior of 1 /T1would be a strong indica-\ntion of an extremely quantum nature of the collective\norbital excitations.39\nVI. BEHAVIOR IN A MAGNETIC FIELD:\nQUANTUM ISING TRANSITION IN ORBITAL\nSECTOR\nWe have seen in Sec.III that, due to spin-orbit cou-\npling, the N´ eel ordering of the spins is accompanied by\nthe emergence of quantum effects in the orbital sector:\nthe classical orbital Ising chain transforms to a quan-\ntum one. In this section we briefly comment on a similar\nsituation that can arise upon application of a uniform\nexternal magnetic field h.\nSince the spin-1 chain is massive, it will acquire a finite\nground-statemagnetization /an}bracketle{tSz/an}bracketri}htonly when the magnetic\nfield,h, is higher than the critical value hc1∼∆S, corre-\nsponding to the commensurate-incommensurate (C-IC)\ntransition. According to the definition (5), a uniform\nmagnetic field along the z-axis,Hmag=−hIz, mixes\nup a pair of Majorana fields, ξ1andξ2, and splits the\nspectrum of Sz=±1 excitations (the Sz= 0 modes are\nunaffected by the field). At h=hc1the gap in the spec-\ntrum of the Sz= 1 excitations closes, and at h > h c1\nthese modes condense giving rise to a finite magnetiza-\ntion. Once /an}bracketle{tSz/an}bracketri}ht /ne}ationslash= 0, the effective Hamiltonian of the\nτ-chain becomes\n¯Hτ=Jτ/summationdisplay\nnτz\nnτz\nn+1−∆τ/summationdisplay\nnτx\nn,∆τ=λ/an}bracketle{tSz/an}bracketri}ht.(52)\nHere we ignore the fluctuation term that couples τx\nnto\n∆Sz\nn=Sz\nn−/an}bracketle{tSz\nn/an}bracketri}ht.\nOne should keep in mind that there exists the sec-\nond C-IC transition at a higher field hc2associated with\nfull polarization of the spin-1 chain. To simplify further\nanalysis, let us assume that the range of magnetic fields\nhc1< h < h c2, where an isolated spin-1 chain has an in-\ncommensurate,gaplessgroundstate,issufficientlybroad.\nThis can be easily achieved in the biquadratic model (3)\nwithβ∼1, in which case the Haldane gap – and hence\nhc1– is small, and the effects associated with the second\nC-IC transition can be neglected.\nNow, by increasing the magnetic field hin the region\nh > h c1, the effective orbital chain (52) can be driven\nto an Ising criticality. The induced transverse “magnetic\nfield” ∆ τis proportional to a nonzero magnetization of\nthe spin-1 chain. If λ/Jτis large enough, then upon in-\ncreasing the field the effective quantum Ising chain (52)\ncan reach the point ∆ τ(h∗) =Jτwhere the Ising transi-\ntion occurs. This will happen at some field h=h∗> hc1.\nIn the region |h−h∗|/h∗≪1 the quantum Ising τ-chain\nwill be slightly off-critical. Due to the SO coupling, thesemassive orbital excitations will interact with the gap-\nlessSz=±1 spin modes. However, this interaction can\nonly give rise to the orbital mass renormalization (i.e. a\nsmall shift of the Ising critical point) and a group veloc-\nity renormalization of the spin-doublet modes. For this\nreason we do not expect the aforementioned spin-orbital\nfluctuation term to cause any qualitative changes.\nThe above discussion reveals an interesting fact: a suf-\nficiently strong magnetic field acting on the spin degrees\nof freedom can affect the orbital structure of the chain\nanddriveit to aquantum Isingtransition. Thedifference\nwith the situation discussedin Sec.III is that the external\nmagnetic field induces a uniform spin polarization which,\nin turn, gives rise to a uniform transverse orbital order-\ning/an}bracketle{tτx\nn/an}bracketri}ht /ne}ationslash= 0. Thus, the classical long-range orbital order\n/an}bracketle{tτz\nn/an}bracketri}ht= (−1)nηz, present at h < h∗, disappears in the\nregionh > h∗, where the orbital degrees of freedom are\ncharacterized by a transverse ferromagnetic polarization,\n/an}bracketle{tτx/an}bracketri}ht /ne}ationslash= 0.\nVII. CONCLUSION AND DISCUSSION\nIn this paper, we have proposed and analyzed a 1D\nspin-orbital model in which a spin-1 Haldane chain is lo-\ncally coupled to an orbital Ising chain by an on-site term\nλτxSzoriginating from relativistic spin-orbit (SO) in-\nteraction. The SO term not only introduces anisotropy\nto the spin sector, but also gives quantum dynamics to\nthe orbital degrees of freedom. We approach this prob-\nlem from well defined limits where either the spin or the\norbital sector is strongly gapped and becomes a ‘fast’\nsubsystem which can be integrated out. By analyzing\nthe resultant effective action of the remaining ‘slow’ de-\ngrees of freedom, we have identified the stable massive\nand critical phases of the model which are summarized\nin a schematic phase diagram shown in Fig. 1.\nInthe limit dominatedbyalargeorbitalgap, i.e. Jτ≫\n∆S, integrating out the orbital variables gives rise to an\neasy-axis spin anisotropy D(Sz)2whereD∼ −λ2/Jτ.\nAsλincreases, the disordered Haldane spin liquid un-\ndergoes an Ising transition into a magnetically ordered\nN´ eel state. The presence of antiferromagnetic spin order\nζin the N´ eel phase in turn generates an effective trans-\nverse field h∼λζacting on the orbital Ising variables.\nThe orbital sector which is described by the Hamiltonian\nof a quantum Ising chain reaches criticality when h=Jτ.\nIn between the two Ising critical points lies an interme-\ndiate phase (phase II in Fig. 1) where both Ising order\nparameters ηxandηzare nonzero. Such a two-stage or-\nderingscenarioillustratedbypath1inthephasediagram\n(Fig.1)hasbeenconfirmednumericallybyrecentDMRG\ncalculations.22Interestingly, the orbital Ising transition\ncan also be induced by applying a magnetic field to the\nspin sector. As the field strength is greater than the Hal-\ndane gap, a field-induced magnon condensation results\nin a finite magnetization density /an}bracketle{tSz/an}bracketri}htin the linear chain.\nThanks to the SO coupling, the orbital sector again ac-11\nquires a transverse field h∼λ/an}bracketle{tSz/an}bracketri}htand becomes critical\nwhenh=Jτ.\nA distinct scenario of the orbital reorientation tran-\nsitionηz→ηxoccurs in the opposite limit ∆ S≫Jτ.\nThis time we integrate out the fast spin subsystem and\nobtain a perturbed spin-1/2 XY Hamiltonian for the or-\nbital sector. The effective exchange constants are given\nbyJx∼λ2/∆SandJy=Jτ. Asλis varied, the orbital\nsector reaches a Gaussian critical point when Jx=Jy,\nat which the system acquires an emergent U(1) sym-\nmetry. The orbital order parameter goes directly from\nη= (0,0,ηz)to(ηx,0,0)inthissingle-transitionscenario\n(illustrated by path 2 in Fig. 1). Both order parameters\nηxandηzvanish at the critical point. We have shown\nthat spin-orbital hybridization effects near the Gaussian\ntransition lead to the appearance of a non-zero spectral\nweight of the staggered spin density well below the Hal-\ndane gap – the effect which can be detected by inelastic\nneutronscatteringexperimentsandNMRmeasurements.\nThe stability analysis of the orbital Gaussian criti-\ncality in the original lattice model (1), done in Ap-\npendix B, has shown that this critical regime is pro-\ntected by the τz→ −τzsymmetry of the underlying\nmicroscopic model. This symmetry will be broken in\nthe presence of an orbital field δ/summationtext\nnτz\nnwhich removes\ndegeneracy between the local orbitals dzxanddyzand\nadds a ”magnetic” field along the y-axis in the effective\nXY model (32). Such perturbation will drive the orbital\nsector away from the Gaussian criticality. The same\nargument applies to a perturbation with the structure\nβ/summationtext\nnSz\nnτz\nnwhich also breaks the aforementioned sym-\nmetry. Integrating over the spins will generate an extra\nterm∼λβ/summationtext\nn(τx\nnτy\nn+1+τy\nnτz\nn+1) which, in the contin-\nuum limit, translates to λβsin√\n4πΘ. As explained in\nAppendix B, such perturbation will keep the orbital sec-\ntor gapped with coexisting ηxandηzorderings.\nSince the analysispresentedin thispaper isdonein the\nlimiting cases, precise predictions on the detailed shape\nof the phase diagram or on the behavior of correlation\nfunctions in the regime of strong hybridization of spin\nand orbital degrees of freedom, where all interactions in-\ncluded inthe modelareofthe sameorder, arebeyondour\nreach and require further numerical calculations. On the\nother hand, the continuity and scaling analysis allow us\nto believe that the global topology of the phase diagram\nand character of critical lines are given correctly. Finally\nthe spin-orbital model Eq. (1) can be generalized to the\nzigzag geometrical where two parallel spin-1 chains are\ncoupled to a zigzag Ising orbital chain via on-site SO in-\nteraction. The zigzag case is closely related to the quasi-\n1D compound CaV 2O4. While the two-Ising-transitions\nscenario is expected to hold in the Jτ≫∆Sregime, the\ncounterpart of Gaussian criticality in the zigzag chain\nremains to be explored and will be left for future study.Acknowledgements\nThe authors are grateful to Andrey Chubukov, Fabian\nEssler, Vladimir Gritsev, Philippe Lecheminant and\nAlexei Tsvelik for stimulating discussions. A.N. grate-\nfully acknowledges hospitality of the Abdus Salam Inter-\nnational Centre for Theoretical Physics, Trieste, where\npart of this work has been done. He is also supported\nby the grants GNSF-ST09/4-447 and IZ73Z0-128058/1.\nG.W.C. acknowledges the support of ICAM and NSF\ngrant DMR-0844115. N.P. acknowledges the support\nfrom NSF grant DMR-1005932 and ASG ”Unconven-\ntional magnetism”. G.W.C. and N.P. also thank the hos-\npitality of the visitors program at MPIPKS, where the\npart of the work on this manuscript has been done.\nAppendix A: Ising correlation function\nIn this Appendix we estimate the correlation func-\ntion Γxx\nnm(τ) =/an}bracketle{tτx\nn(τ)τx\nm(0)/an}bracketri}ht, where the averaging is per-\nformed over the ground state of the Ising Hamiltonian\nHτ=Jτ/summationtext\nnτz\nnτz\nn+1, andτx\nn(τ) =eτHττx\nne−τHτ.\nIt proves useful to make a duality transformation:\nτz\nnτz\nn+1=µx\nn, τx\nn=µz\nnµz\nn+1.\nThe new set of Pauli matrices µa\nnrepresents disorder op-\nerators. The Hamiltonian and correlation function be-\ncome:\nH→Jτ/summationdisplay\nnµx\nn, (A1)\nΓzz\nnm(τ)→ /an}bracketle{tµz\nn(τ)µz\nn+1(τ)µz\nm(0)µz\nm+1(0)/an}bracketri}ht.(A2)\nThe most important fact about the dual representation\nis the additive, single-spin structure of the Hamiltonian:\nthe latter describes noninteracting spins in an external\n“magnetic field” Jτ. Notice that by symmetry /an}bracketle{tµz\nn/an}bracketri}ht= 0.\nTherefore the correlationfunction in (A2) has an ultralo-\ncal structure:\nΓxx\nnm(τ) =δnmY2(τ), Y(τ) =/an}bracketle{tµz\nn(τ)µz\nn(0)/an}bracketri}ht.(A3)\nThe time-dependence of the disorder operator can be ex-\nplicitly computed,\nµz\nn(τ) =eτJτµx\nnµz\nne−τJτµx\nn=µz\nncosh(2Jττ)−iµy\nnsinh(2Jττ).\nTherefore (below we assume that τ > τ′)\nY(τ−τ′) = cosh2 Jτ(τ−τ′)+/an}bracketle{tµx/an}bracketri}htsinh2Jτ(τ−τ′)\n= exp[−2|Jτ|(τ−τ′)]. (A4)\nHereweusedthefactthat, inthegroundstatetheHamil-\ntonianHτ,/an}bracketle{tµx/an}bracketri}ht=−sgnJτ. Thus, as expected for the\n1D Ising model, the correlation function Γxx\nnm(τ) is local\nin real space and decays exponentially with τ:\nΓxx\nnm(τ) =δnmexp(−4J⊥|τ|). (A5)12\nAppendix B: Perturbed XY chain, Eq. (32)\nIn this Appendix we analyze the perturbation (33) to\nthe XY spin chain (32) and show that at the XX point\nit represents a marginal perturbation which transforms\nthe free-fermion regime to a Gaussian criticality describ-\ning a Luttinger-liquid behavior of the orbital degrees of\nfreedom.\nUsing the Jordan-Wigner transformation (34) we\nrewrite (33) as H′\nτ=H′\n1+H′\n2, where\nH′\n1=J′\nx(2)\n2/summationdisplay\nn(a†\nnan+2+h.c.)(a†\nn+1an+1−1\n2),(B1)\nH′\n2=J′\nx(2)\n2/summationdisplay\nn(a†\nna†\nn+2+h.c.)(a†\nn+1an+1−1\n2).(B2)\nAssuming that |Jx−Jy|,J′\nx≪Jx+Jy, we pass to a con-\ntinuum description of the XY chain in terms of chiral,\nright (R) and left (L), fermionic fields based on the de-\ncomposition (to simplify notations we set here a0= 1):\nan→(−i)nR(x)+inL(x).Then the Hamiltonian density\nof the XY model takes the form:\nHXY(x) =−iv/parenleftbig\nR†∂xR−L†∂xL/parenrightbig\n−2iγ/parenleftbig\nR†L†−h.c./parenrightbig\n,\n(B3)\nwhereγ=Jx−Jy. Standard rules of Abelian\nbosonization27transform(B3)toaquantumsine-Gordon\nmodel:\nHXY(x) =v\n2/bracketleftBig\nΠ2+(∂xΦ)2/bracketrightBig\n+2γ\nπαcos√\n4πΘ,(B4)\nwherev= 2(Jx+Jy)a0is the Fermi velocity, Π( x) =\n∂xΘ(x) is the momentum conjugate to the scalar field\nΦ(x) = Φ R(x) + ΦL(x), and Θ( x) =−ΦR(x) + ΦL(x)\nis the field dual to Φ( x). Here Φ R,L(x) are chiral\ncomponents of the scalar field. Using the fact that\nthe fermions are spinless, one can impose the condi-\ntion [Φ R(x),ΦL(x′)] =i/4 and thus make sure that the\nbosonization rules correctly reproduce the anticommuta-\ntion relations {R(x),L(x′)}={R(x),L†(x′)}= 0. An\nexplicit introduction of the so-called Klein factors be-\ncomes necessary when bosonizing fermions with an inter-\nnal degree of freedom, such as spin 1/2, chain index etc,\nwhich is not the case here.\nLet is find the structure of the perturbation (33) in the\ncontinuum limit. First of all we notice that\na†\nn+1an+1−1/2≡:a†\nn+1an+1:\n→(:R†R: + :L†L:)+(−1)n+1(R†L+L†R)\n=1√π∂xΦ+(−1)n\nπαsin√\n4πΦ. (B5)\nSimilarly\na†\nnan+2+h.c.\n→ −2/bracketleftbig\n(:R†R: + :L†L:)+(−1)n(R†L+L†R)/bracketrightbig\n=−2/bracketleftbigg1√π∂xΦ−(−1)n\nπαsin√\n4πΦ/bracketrightbigg\n. (B6)Dropping Umklapp processes R†(x)R†(x+α)L(x+\nα)L(x) +h.c.∼cos√\n16πΦ as strongly irrelevant (with\nscaling dimension 4) at the XXcriticality and ignoring\ninteraction of the fermions in the vicinity of the same\nFermi point, we find that\n(a†\nnan+2+h.c.)(a†\nn+1an+1−1/2)/vextendsingle/vextendsingle/vextendsingle\nsmooth\n→ −8 :R†R::L†L:= 2/bracketleftbig\nΠ2−(∂xΦ)2/bracketrightbig\n.(B7)\nWe see that the perturbation H′\n1generates a marginal\nfour-fermion interaction to the free-fermion model (B3),\nthus transforming the model (32) to an XYZ model with\na weak ferromagnetic( zz)-coupling. This interaction can\nbe incorporated into the Gaussian part of the bosonic\ntheory (B4) by changing the compactification radius of\nthe field Φ:\nH=HXY+H′\n1\n=u\n2/bracketleftbigg\nKΠ2+1\nK(∂xΦ)2/bracketrightbigg\n−2γ\nπαcos√\n4πΘ.(B8)\nHereuis the renormalized velocity and Kis the inter-\naction constant which at J′\nx≪(Jx+Jy) is given by\nK= 1+2g+O(g2),whereg=J′\nx(2)a0/πv≪1.\nNow we turn to H′\n2. We have:\na†\nna†\nn+2+h.c. (B9)\n→ −/bracketleftbig\nR†(x)L†(x+α)+L†(x)R†(x+α)+h.c./bracketrightbig\n+(−1)n/bracketleftbig\nR†(x)R†(x+α)+L†(x)L†(x+α)+h.c./bracketrightbig\n.\nBosonizing the smooth term in the r.h.s. of (B10) one\nobtains∂xΦcos√\n4πΘ. Bosonizing the staggered term\nyields sin√\n4πΦcos√\n4πΘ. Using the OPE\nsin√\n4πΦ(x)sin√\n4πΦ(x+α)\n= const−πα2(∂xΦ)2−1\n2cos√\n16πΦ,\nwe find that, in the continuum limit, the Hamiltonian\ndensityH′\n2is contributed by the operatorscos√\n4πΘ and\n(∂xΦ)2cos√\n4πΘ (as before, we drop corrections related\nto Umklapp processes). The former leads to a small ad-\nditive renormalization of the fermionic mass γand thus\nproduces a shift of the critical point. The latter repre-\nsents an irrelevant perturbation (with scaling dimension\n3) at the XX criticality. In a noncritical regime it renor-\nmalizes the mass and four-fermion coupling constant g.\nConsidering the structure of the remaining terms in\nthe expansion (31) one arrives at similar conclusions.\nHere a remark is in order. The only dangerous perturba-\ntion which would dramatically affect the above picture\nis sin√\n4πΘ. The presence of two nonlinear terms in\nthe Hamiltonian, γcos√\n4πΘ+δsin√\n4πΘ, would make\nthe fermionic mass equal to/radicalbig\n(λ−λc)2+δ2. The Gaus-\nsian criticality in this case would never be reached, the13\nmodel would always remain massive, and nonzero stag-\ngered pseudospin densities, ηzandηx, would coexist in\nthe whole parameter range of the model.\nFortunately, the appearance of the operator sin√\n4πΘ\nis forbidden by symmetry. The initial Hamiltonian (1)\nis invariant under global pseudospin inversion In the z-\ncomponent only: τz\nn→ −τz\nn. After rotation τz→τythis translates to τy\nn→ −τy\nn. Using the bosonized ex-\npressions (39) for the staggered pseudospin densities we\nfind that the corresponding transformation of the dual\nfield is Θ →√π−Θ and so the bosonized Hamiltonian\ndensity must be invariant under this transfomation. This\nexplainswhythe operatorsin√\n4πΘcannotappearin the\neffective continuum theory.\n1E. Axtell, T. Ozawa, S. Kauzlarich, and R. R. P. Singh, J.\nSolid State Chem. 134, 423 (1997).\n2M. Isobe and Y. Ueda, J. Phys. Soc. Jpn. 65, 1178 (1996)\n3K. I. Kugel andD. I. Khomsky, JETP Lett. 15, 446 (1971).\n4Y. Tokura and N. Nagaosa, Science, 288, 462 (2000).\n5E. Zhao and W. V. Liu, Phys. Rev. Lett. 100, 160403\n(2008).\n6C. Wu, Phys. Rev. Lett. 100, 200406 (2008).\n7Z. Nussinov and E.Fradkin, Phys. Rev. B 71, 195120\n(2005).\n8A. Kitaev, Ann. Phys. (N.Y.) 321, 2 (2006).\n9P. Azaria, A.O. Gogolin., P. Lecheminant, and A.A. Ners-\nesyan, Phys. Rev. Lett. 83, 624 (1999).\n10C. Itoi, S. Qin, and I. Affleck, Phys. Rev. B 61, 6747\n(2000).\n11Y. Yamashita, N. Shibata, and K. Ueda, Phys. Rev. B 58,\n9114 (1998).\n12B. Frischmuth, F. Mila, and M. Troyer, Phys. Rev. Lett.\n82, 835 (1999).\n13C. Itoi, S. Qin, andI. Affleck, Phys. Rev.B 61, 6747 (2000)\n14H. Mamiya, M. Onoda, T. Furubayashi, J. Tang, and I.\nNakatani, J. Appl. Phys. 81, 5289 (1997).\n15M. Reehuis, A. Krimmel, N. B´ uttgen, A. Loidl, A.\nProkofiev, Eur. Phys. J. B 35, 311 (2003).\n16M. Onoda and J. Hasegawa, J. Phys.: Condens. Matter\n15, 95 (2003).\n17S.-H. Lee, D. Louca, H. Ueda, S. Park, T. J. Sato, M.\nIsobe, Y. Ueda, S. Rosenkranz, P. Zschack, J. Iniguez, Y.\nQiu, R. Osborn, Phys. Rev. Lett. 93, 15640 (2004).\n18A. Niazi et al, Phys. Rev. B 79, 104432 (2009).\n19O. Pieper et al, Phys. Rev. B 79,180409(R) (2009).\n20G.-W. Chern and N. Perkins, Phys. Rev. B 80, 220405(R)\n(2009).\n21F. D. M. Haldane, Phys. Lett. 93A, 464 (1983).\n22G.-W. Chern, N. Perkins and G. I. Japaridze, Phys. Rev.\nB82, 172408 (2010).\n23H.-J. Mikeska and A. K. Kolezhuk, Lect. Notes Phys. 645,\n1 (2004).24A. M. Tsvelik, Phys. Rev. B 42, 10 499 (1990).\n25L. Takhtajian, Phys. Lett. 87 A, 479 (1982); H. Babujian,\nibid.90 A, 464 (1982).\n26D. Shelton, A. A. Nersesyan and A. M. Tsvelik, Phys. Rev.\nB53, 8521 (1996).\n27A. O. Gogolin, A. A. Nersesyan and A. M. Tsvelik,\nBosonization andStronglyCorrelated Systems , Cambridge\nUniversity Press, 1998.\n28A. B. Zamolodchikov and V. A. Fateev, Sov. J. Nucl. Phys.\n43, 657 (1986).\n29P. Di Francesco, P. Mathieu and D. Senechal, Conformal\nField Theory , (Springer-Verlag, 1997).\n30A. A. Nersesyan and A. M. Tsvelik, Phys. Rev. Lett. 78,\n3939 (1997).\n31J. Kogut, Rev. Mod. Phys. 51, 659 (1979).\n32E. S. Sorensen and I. Affleck, Phys. Rev. B 49, 15771\n(1994).\n33E. Lieb, T. D. Schultz, and D. C. Mattis, Ann. Phys. (NY)\n16, 407 (1961).\n34H. J. Schulz and C. Bourbonnais, Phys. Rev. B 27, 5856\n(1983); H.J. Schulz, Phys. Rev. B 34, 6372 (1986).\n35F.H.L. Essler and R.M. Konik, Applications of Massive\nIntegrable Quantum Field Theories to Problems in Con-\ndensed Matter Physics , in the Ian Kogan Memorial Vol-\nume ”From Fields to Strings: Circumnavigating Theoret-\nical Physics”, eds. M. Shifman, A. Vainshtein and J.F.\nWheater, World Scientific, 2005, p. 684-831.\n36J. Sagi and I. Affleck, Phys. Rev. B 539188 (1996).\n37T. Giamarchi, Quantum Physics in One Domension , Ox-\nford University Press, 2003.\n38M. Sato, T. Momoi, and A. Furusaki, Phys. Rev. B 79,\n060406(R) (2009).\n39In the most general setting, there will be a contribution to\nthe NMR relaxation rate coming directly from the orbital\nsector. However, since the coupling to orbitals is non-loca l,\nthis contribution is thus less singular and is usually negli -\ngible compared with contribution from spins."    },    {        "title": "1106.5667v2.Phase_separation_in_a_polarized_Fermi_gas_with_spin_orbit_coupling.pdf",        "content": "Phase separation in a polarized Fermi gas with spin-orbit coupling\nW. Yi and G.-C. Guo\nKey Laboratory of Quantum Information, University of Science and Technology of China,\nCAS, Hefei, Anhui, 230026, People's Republic of China\n(Dated: November 20, 2018)\nWe study the phase separation of a spin polarized Fermi gas with spin-orbit coupling near a wide\nFeshbach resonance. As a result of the competition between spin-orbit coupling and population\nimbalance, the phase diagram for a uniform gas develops a rich structure of phase separated states\ninvolving topologically non-trivial gapless super\ruid states. We then demonstrate the phase sepa-\nration induced by an external trapping potential and discuss the optimal parameter region for the\nexperimental observation of the gapless super\ruid phases.\nPACS numbers: 03.75.Ss, 03.75.Lm, 05.30.Fk\nSpin-orbit coupling (SOC), common in condensed mat-\nter systems for electrons, has been considered a key in-\ngredient for many interesting phenomena such as topo-\nlogical insulators [1], quantum spin Hall e\u000bects [2], etc.\nThe recent realization of synthetic gauge \feld and hence\nspin-orbit couplings in ultracold atomic systems opens\nup exciting new routes in the study of these phenomena\n[3, 4], allowing us to take advantage of the features of the\nultracold atoms, e.g. clean environment and highly con-\ntrollable parameters. In particular, with the Feshbach\nresonance technique, the e\u000bective interaction strength\nbetween atoms can be tuned [5, 6]. This technique has\nbeen applied to study various interesting topics, e.g. the\nBCS-BEC crossover [7], polarize Fermi gases [8], itinerant\nferromagnetism [9], etc. The introduction of spin-orbit\ncoupling may shed new light on these strongly correlated\nsystems.\nSpin-orbit coupled Fermi gas near a Feshbach reso-\nnance has recently attracted much theoretical attention\n[10{16]. The SOC has been shown to enhance pairing\non the BCS side of the Feshbach resonance [12, 14, 15].\nFurthermore, for a polarized Fermi gas, the SOC intro-\nduces competition against population imbalance, which\ncan lead to topologically non-trivial phases [13, 16]. Re-\ncently, the phase diagrams for a polarized Fermi gas with\nspin-orbit coupling near a Feshbach resonance have been\nreported for a uniform gas [16]. The phase boundaries\nhave been calculated by solving the gap equation and\nthe number equations self-consistently. However, simi-\nlar to the case of a polarized Fermi gas near Feshbach\nresonance [17], due to the competition between di\u000ber-\nent phases, the solutions of the gap equation may corre-\nspond to metastable or unstable states. By considering\nthe compressibility criterion [16], the unstable solutions\nare correctly discarded, while the metastable solutions\nmay survive, rendering the resulting phase boundaries,\nin particular those representing \frst order phase transi-\ntions, unreliable.\nIn this paper, we examine in detail the zero temper-\nature phase diagrams for a polarized Fermi gas with\nRashba spin-orbit coupling near a wide Feshbach reso-nance for both the uniform and the trapped cases. To\navoid getting metastable or unstable solutions, instead\nof solving the gap equation, we minimize the thermody-\nnamic potential directly as in Ref. [17]. For the uniform\ngas, we \fnd larger stability regions for the phase sepa-\nrated state at unitarity as compared to the results in Ref.\n[16]. More interestingly, we \fnd that SOC may induce\nmore complicated phase separated states involving gap-\nless super\ruid phases that are topologically non-trivial,\nin addition to the typical phase separated state composed\nof normal (N) and gapped super\ruid (SF) phases. We\ncalculate the stability region for the various phase sepa-\nrated states as well as for the gapless super\ruid states, SF\nstate and normal state. We show that there are two dis-\ntinct gapless phases that di\u000ber by the number of crossings\ntheir excitation spectra have with the zero energy in mo-\nmentum space, consistent with previous results [13, 16].\nThese novel gapless phases are stabilized by intermedi-\nate SOC strengths; whereas for large enough SOC, the\nsystem always becomes a gapped super\ruid of `rashbons'\n[12]. We show how these phases can be characterized by\ntheir di\u000berent excitation spectra and momentum space\ndensity distributions. We then discuss the phase separa-\ntion in an external trapping potential, where the various\nphases naturally phase separate in real space. By exam-\nining their respective stability regions, we demonstrate\nthe optimal parameter region to observe the gapless su-\nper\ruid states in the presence of a trapping potential.\nFor all of our calculations in the paper, we adopt the\nBCS-type mean \feld treatment. Although the mean \feld\ntheory does not give quantitatively accurate results near\na wide Feshbach resonance, it is a natural \frst step for\nus to qualitatively estimate what phases may be stable,\nas well as to understand their respective properties. We\nalso note that we have neglected the Fulde-Ferrell-Larkin-\nOvchinnikov (FFLO) phase in our calculations. This is\nmotivated by the fact that the FFLO phase is stable only\nin a narrow parameter region in the absence of SOC due\nto competition against other phases [8]. As SOC intro-\nduces new gapless phases into this competition, we do\nnot expect a signi\fcant increase in its stability region.arXiv:1106.5667v2  [cond-mat.quant-gas]  30 Sep 20112\nWe \frst consider a uniform three dimensional polarized\nFermi gas with Rashba spin-orbit coupling in the plane\nperpendicular to the quantization axis z. The model\nHamiltonian takes the form [13, 14, 16]\nH\u0000X\n\u001b\u0016\u001bN\u001b=X\nk;\u001b\u0018ka†\nk;\u001bak;\u001b\n+h\n2X\nk\u0010\na†\nk;#ak;#\u0000a†\nk;\"ak;\"\u0011\n+U\nVX\nk;k0a†\nk;\"a†\n\u0000k;#a\u0000k0;#ak0;\"\n+X\nk\u000bk?\u0010\ne\u0000i'ka†\nk;\"ak;#+h:c:\u0011\n; (1)\nwhere\u0018k=\u000fk\u0000\u0016, with the kinetic energy \u000fk=~2k2\n2m;\n\u001b=f\";#gare the atomic spins; N\u001bdenotes the to-\ntal number of particles with spin \u001b;ak;\u001b(a†\nk;\u001b) annihi-\nlates (creates) a fermion with momentum kand spin\u001b;\n\u0016\u001b=\u0016\u0006h=2 is the chemical potential of the correspond-\ning spin species, and Vis the quantization volume. The\nRashba spin-orbit coupling strength \u000bcan be tuned via\nparameters of the gauge-\feld generating lasers [4], while\nk?=q\nk2x+k2yand'k= arg (kx+iky). In writing\nHamiltonian (1), we assume s-wave contact interaction\nbetween the two fermion species, with the bare interac-\ntion rateUrenormalized following the standard relation\n1\nU=1\nUp\u00001\nVP\nk1\n2\u000fk[7]. The physical interaction rate is\ngiven asUp=4\u0019~2as\nm, whereasis the s-wave scattering\nlength between the two fermionic spin species.\nTo diagonalize the Hamiltonian, we make the trans-\nformation: ak;\"=1p\n2ei'k(ak;++ak;\u0000),ak;#=\n1p\n2(ak;+\u0000ak;\u0000), whereak;\u0006are the annihilation op-\nerators for the dressed spin states with di\u000berent helic-\nities (\u0006) [12{16]. Taking the pairing mean \feld \u0001 =\nU\nVP\nkha\u0000k;#ak;\"ias in the standard BCS-type theory,\nwe may diagonalize the mean \feld Hamiltonian in the\nbasis of the dressed spins:n\nak;+;a†\n\u0000k;+;ak;\u0000;a†\n\u0000k;\u0000oT\n.\nThe thermodynamic potential is then evaluated from\n\n =\u00001\n\fln tr\u0002\ne\u0000\f(H\u0000P\n\u001b\u0016\u001bN\u001b)\u0003\n, with\f= 1=kBT. In\nthis paper, we will focus on the zero temperature case,\nfor which the thermodynamic potential has the form\n\n =1\n2X\nk;\u0015=\u0006(\u0018\u0015\u0000Ek;\u0015)\u0000Vj\u0001j2\nU; (2)\nwith the quasi-particle excitation spectrum Ek;\u0006=r\n\u00182\nk+\u000b2k2\n?+j\u0001j2+h2\n4\u00062q\n(h2\n4+\u000b2k2\n?)\u00182\nk+h2\n4j\u0001j2.\nBefore proceeding, let us examine the quasi-particle\nexcitations \frst and study the conditions for possible\ngapless phases. We see that at the points in the\nmomentum space where Ek;\u0000crosses zero, the quasi-\nparticle excitation becomes gapless while the pairing\ngap \u0001 remains \fnite. The SOC, together with the\npopulation imbalance re-arranges the topology of the\n0 0.5 1−0.2−0.15−0.1Ω/h\n∆/h0 0.5 1−0.4−0.35−0.3\n∆/hΩ/h\n0 0.5 1−0.23−0.22−0.21−0.2\n∆/hΩ/h\n0 0.5 1−0.32−0.3−0.28−0.26\n∆/hΩ/h(a)\n(c)(b)\n(d)N\nGP2SF\nSFFIG. 1. Illustration of typical shapes of the thermodynamic\npotential \n =has a function of order parameter \u0001 =hfor var-\nious phases at unitarity: (a) \u0016=h = 0:52,\u000bkh=h= 0:1; (b)\n\u0016=h= 0:7,\u000bkh=h= 0:1; (c)\u0016=h= 0:52,\u000bkh=h= 0:3, (d)\n\u0016=h= 0:52,\u000bkF=h= 0:6. The chemical potential his taken\nto be the energy unit, while the unit of momentum khis de-\n\fned through~2k2\nh\n2m=h.\nFermi surfaces of the spin species [13, 16]. The points\nof gapless excitations lie on the kzaxis withk?= 0,\nand are symmetric with respect to the kz= 0 plane.\nMore speci\fcally, for \u0016\u00140, the excitation spectrum\nhas two gapless points \u00062m\n~2\u0012\n\u0016+q\nh2\n4\u0000j\u0001j2\u00131\n2\n,\nso long asjhj\n2>p\n\u00162+j\u0001j2. For\u0016 > 0,\nthe excitation spectrum has four gapless points(\n\u00062m\n~2\u0012\n\u0016+q\nh2\n4\u0000j\u0001j2\u00131\n2\n;\u00062m\n~2\u0012\n\u0016\u0000q\nh2\n4\u0000j\u0001j2\u00131\n2)\n,\nwithj\u0001j<jhj\n2<p\n\u00162+j\u0001j2; two gapless points\n\u00062m\n~2\u0012\n\u0016+q\nh2\n4\u0000j\u0001j2\u00131\n2\n, withjhj\n2>p\n\u00162+j\u0001j2. We\nidentify the super\ruid states with two excitation points\n(GP1) and those with four excitation points (GP2) as\ndi\u000berent topological phases [13, 16].\nWe illustrate in Fig. 1 typical shapes of the thermo-\ndynamic potential as a function of \u0001 with di\u000berent pa-\nrameters. Notably, due to the competition between dif-\nferent phases, a double-well structure appears (see Fig.\n1(a-c)). Hence the solutions to the gap equation may\ncorrespond to the metastable states (local minimum) or\nthe unstable states (local maximum). To make sure that\nthe ground state is achieved, we directly minimize the\nthermodynamic potential [17].\nAnother complication comes from the existence of the\nphase separated state, which must be considered explic-\nitly for a uniform gas. As in the case of polarized Fermi\ngases without SOC [8], we introduce the mixing coe\u000e-\ncientx(0\u0014x\u00141), and the thermodynamic potential\nbecomes\n\n =x\n(\u0001 1) + (1\u0000x)\n(\u0001 2); (3)3\n00.2 0.4 0.6 0.8 11.200.20.40.60.81\nα kF/EFPGP1\nN+SF GP2+SF\nSFGP2N\nGP2+GP2\nFIG. 2. Zero temperature phase diagram for a uniform Fermi\ngas with population imbalance at ( kFas)\u00001= 0. Within the\nbold phase boundaries are the various phase separated states\n(see text). These phase separated states can be connected\nwith the non-phase separated states by \frst order phase tran-\nsitions (solid bold curve). The thin curves represent various\nsecond order phase transitions (see text). Here kF= (3\u00192n)1\n3,\nEF=~2k2\nF\n2m, andnis the total density of the system.\nwhere \u0001i(i= 1;2) is the pairing gap for the ith com-\nponent state. Note that due to SOC, we now have the\npossibility of a phase separated state of two distinct su-\nper\ruid states (see Fig. 1(c)). The number equations of\nthe phase separated state become\nN\u001b=x@\n@\u0016\u001b\f\f\f\f\n\u0001=\u0001 1+ (1\u0000x)@\n@\u0016\u001b\f\f\f\f\n\u0001=\u0001 2: (4)\nMinimizing the thermodynamic potential Eq. (3) with\nrespect to \u0001 iandxwhile implementing the number con-\nstraints Eq. (4), we map out the phase diagram for a uni-\nform polarized Fermi gas with SOC at ( kFas)\u00001= 0. Fig.\n2 illustrates the resulting phase boundaries in the plane of\n(P;\u000bkF=EF), where the polarization P=N\"\u0000N#\nN\"+N#. When\nthe SOC is o\u000b ( \u000b= 0), the system remains in a phase\nseparated state of normal and gapped super\ruid (PS1)\nup toP\u00180:93 before it becomes a normal state via a\n\frst order phase transition. This is consistent with pre-\nvious mean \feld calculations for a polarized Fermi gas\n[8, 19], while di\u000berent from the result in Ref. [16]. As\nthe SOC strength \u000bincreases, a rich structure of dif-\nferent phases shows up, e.g. gapped super\ruid phase\n(SF), gapless super\ruid phases with di\u000berent Fermi sur-\nface topology (GP1 and GP2), and notably, various phase\nseparated states. These phase separated states are con-\n\fned by a phase boundary of \frst order phase transition\n(bold curve in Fig. 2). In addition to the typical PS1\nphase, we now have a phase separated state with GP2\nand SF phases (PS2), and a phase separated state of two\ndistinct GP2 phases (PS3). As \u000bincreases, the system\ncan undergo second order phase transitions from PS1 to\nPS2 and then to PS3 for intermediate Pand\u000b. As-\nsumingj\u00011j<j\u00012j, the phase boundaries between them\n−1 0 101\nkz/khEk,−/h\n  \n0 1 200.51\nkz/khDensity\n0 200.51\nkz/khDensity\n0 200.51\nkz/khDensity0121\nk⊥/kh0120.51\nk⊥/kh\n0121\nk⊥/khSF\nGP1\nGP2(a) (b)\n(d) (c)FIG. 3. Typical excitation spectrum and momentum space\ndensity distribution for di\u000berent phases. (a) Lower branch of\nthe excitation spectra for GP1 (solid), GP2 (dashed) and SF\n(dash-dotted) phases; (b-d) Density distribution in momen-\ntum space for spin-up (solid) and spin-down (dashed) species\nalongk?= 0 andkz= 0 (inset), for (b) \u000bkh=h= 0:35,\n\u0016=h = 0:5 (SF); (c) \u000bkh=h= 0:35,\u0016=h = 0:45 (GP1); (d)\n\u000bkh=h= 0:45,\u0016=h= 0:43 (GP2), respectively.\ncan be determined by imposing \u0001 1= 0 (PS1 and PS2)\nandh\n2=j\u00012j(PS2 and PS3), respectively. These phase\nseparated states \fnally become unstable and give way to\nsingle component super\ruid phases as \u000bbecomes large.\nThe phase boundaries between these single component\nstates are determined by settingh\n2=j\u0001j(SF and GP2),\nh\n2=p\n\u00162+j\u0001j2(GP2 and GP1), and \u0001 = 0 (N and\nGP1), respectively. When \u000bis large enough, the stabil-\nity region of the GP2 phase decreases and \fnally vanishes\nat a tri-critical point ( \u0016= 0), beyond which only GP1,\nSF and normal phase may exist. Note that beyond the\ntri-critical point, the chemical potential \u0016becomes neg-\native, and the phase boundary between GP1 and SF will\nbend upwards so that in the large \u000blimit the SF phase\nbecomes dominant in the phase diagram.\nTo characterize the properties of the di\u000berent\nphases, we calculate the excitation spectrum and\nnumber distribution in momentum space for SF,\nGP1 and GP2 states (see Fig. 3). Several interest-\ning observations are in order. Firstly, the gapless\nphases leave their signatures in the momentum\nspace density distribution. For k?= 0 andjkzj 2\"\nmin \n0;\u0000r\n\u0016\u0000q\nh2\n4\u0000j\u0001j2!\n;r\n\u0016+q\nh2\n4\u0000j\u0001j2#\n,\nthe minority spin component vanishes, and pairing does\nnot occur in this region. This is reminiscent of the\nmomentum space phase separation of a breached pairing\nphase in the polarized Fermi gas [18], though now the\nunpaired region lies only on the kzaxis. Away from\nkzaxis, the occupation of the minority spin recovers\nfrom zero gradually, leaving a signature which may be\ndetected in the time of \right imaging experiment [16].\nSecondly, for \fnite \u000b, both the gapless and the gapped4\n0 0.5 1 1.5−0.500.5\nαkh/hµ/h0.42 0.450.430.46\nαkh/hµ/h\nSF GP1 NGP2\nV\nFIG. 4. Phase diagram in the ( \u0016=h;\u000bk h=h) plane at\n(khas)\u00001= 0. While the second order phase transitions are\nin dashed thin curves, the \frst order phase transitions are\nshown in solid bold curves, which end at the point where the\ndouble-well structure in the thermodynamic potential disap-\npears (inset). The boundary for vacuum (V) is determined\nby setting the chemical potential of the majority spin species\nto vanish in the normal phase.\nsuper\ruid phases can support population imbalance,\nwhich can be seen from the density distribution along\nk?(see Fig. 3 insets). Indeed as we will see later, for\nlarge enough \u000b, we may expect no phase separation even\nin the presence of a harmonic trapping potential. The\natoms in the trap will all be in the super\ruid phase\ninduced by SOC.\nTo understand the spatial distribution of the various\nphases in a trapping potential, we calculate the phase\ndiagram as a function of ( \u0016=h;\u000bkh=h) at unitarity (Fig.\n4), wherekhis de\fned in the caption of Fig. 1. Under\nthe Local Density Approximation (LDA) while assuming\nboth spin species experience the same harmonic poten-\ntial, the local chemical potential \u0016(r) can be related to\nthat at the center of the trap \u0016as\u0016(r) =\u0016\u0000V(r),\nwhereV(r) gives the trapping potential. Thus a down-\nward vertical line in Fig. 4 represents a trajectory from\na trap center to its edge, with the chemical potential at\nthe trap center \fxed by that at the starting point of the\nline. In Fig. 4, consistent with Fig. 2, the GP2 phase\nonly exists in a small parameter region in the trap, while\nthere appears to be considerable stability regions for the\nGP1 phase. When \u000bis small, the Fermi gas in the trap\nwill phase separate into two regions, SF at the core, nor-\nmal phase (N) towards the edge. At intermediate \u000b, the\ngapless phases GP2 and GP1 may appear either near the\ncenter of the trap or as a ring between the SF core and\nthe normal edge, depending on the chemical potentials.\nNote that the boundary of the \frst order phase transi-\ntion between PS3 and GP2 (dotted thin curve in Fig. 2)\ncorresponds to a small scale structure here (Fig. 4 inset),\nwhere a \frst order phase transition (bold black curve) ex-\n0 1 2012\nαkh/hµ/h\n00.20.40.6−0.5−0.4−0.3−0.2\nαkh/hµ/hGP2\nGP1SF\nGP1SF(b) (a)\nN\nVN\nVFIG. 5. Phase diagram in the ( \u0016=h;\u000bk h=h) plane at (a)\n(khas)\u00001=\u00001 and (b) ( khas)\u00001= 0:5. First order phase\ntransitions are shown in solid bold curves, while second order\nphase transitions are in dashed thin curves.\nists between two distinct gapless super\ruids, both in the\nGP2 phase. However, this region is found to be small at\nunitarity and only increases slightly towards the BCS side\nof the resonance. It is therefore di\u000ecult to observe this\nphase transition in the trapped Fermi gas in the parame-\nter region that we considered. For large SOC beyond the\ntip of the GP1-SF phase boundary, there is only the SF\nphase in the phase diagram, and hence we will have only\nthe SOC induced SF phase in the trap for large enough\nSOC.\nWe have also calculated the phase diagram in\n(\u0016=h;\u000bkh=h) plane away from the resonance point. On\nthe BCS side (Fig. 5(a)), the stability region for the GP2\nphase increases considerably. It is therefore desirable to\nprepare the system on the BCS side of the resonance for\nthe observation of GP2 phases. On the BEC side (Fig.\n5(b)), the GP2 phase vanishes from the phase diagram\naltogether for \u0016<0, consistent with our previous discus-\nsion.\nIn summary, we have calculated in detail the phase di-\nagrams near a wide Feshbach resonance for a polarized\nFermi gas with Rashba spin-orbit coupling. We \fnd that\nthe competition among pairing, polarization and SOC\ngives rise to a rich structure of phases and phase separa-\ntions involving topologically non-trivial phases. From the\nphase diagrams for both uniform and trapped systems,\nwe \fnd that the interesting gapless super\ruid phases are\nmost likely to be observed in an experiment with moder-\nate polarization and SOC strength.\nWe would like to thank L.-M. Duan for helpful discus-\nsions. This work was supported by NFRP 2011CB921200\nand 2011CBA00200, NNSF 60921091, and The Fun-\ndamental Research Funds for the Central Universities\nWK2470000001.\n[1] M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045\n(2010).5\n[2] D. Xiao, M.-C. Chang, Q. Niu, Rev. Mod. Phys. 82, 1959\n(2010).\n[3] Y.-J. Lin, R. L. Compton, A. R. Perry, W. D. Phillips,\nJ. V. Porto, and I. B. Spielman, Phys. Rev. Lett. 102,\n130401 (2009).\n[4] Y.-J. Lin, K. Jim\u0013 enez-Garc\u0013 \u0010a and I. B. Spielman, Nature\n(London) 471, 83 (2011).\n[5] R. A. Duine and H. T. C. Stoof, Phys. Rep. 396, 115\n(2004).\n[6] S. Inouye, M. R. Andrews, J. Stenger, H.-J. Miesner,\nD. M. Stamper-Kurn and W. Ketterle, Nature (London)\n392, 15 (1998).\n[7] Q.-J. Chen, J. Stajic, S. Tan and K. Levin, Phys. Rep.\n412, 1 (2005).\n[8] D. E Sheehy and L. Radzihovsky, Ann. Phys. 322, 1790\n(2007).\n[9] G.-B. Jo, Y.-R. Lee, J.-H. Choi, C. A. Christensen, T. H.Kim, J. H. Thywiseen, D. E. Pritchard and W. Ketterle,\nScience 325, 1521 (2009).\n[10] S. Tewari et al. , New J. Phys. 1, 065004 (2011).\n[11] J. P. Vyasanakere and V. B. Shenoy, Phys. Rev. B 83,\n094515 (2011).\n[12] J. P. Vyasanankere, S. Zhang, and V. B. Shenoy, Phys.\nRev. B 84, 014512 (2011).\n[13] M. Gong, S. Tewari and C. Zhang, arXiv:1105.1796.\n[14] Z.-Q. Yu and H. Zhai, arXiv:1105.2250.\n[15] H. Hu, L. Jiang, X.-J. Liu, H. Pu, arXiv:1105.2488.\n[16] M. Iskin and A. L. Subasi, Phys. Rev. Lett. 107, 050402\n(2011).\n[17] W. Yi and L.-M. Duan, Phys. Rev. A 73, 031604(R)\n(2006).\n[18] W. V. Liu and F. Wilczek, Phys. Rev. Lett. 90, 047002\n(2003).\n[19] M. M. Parish, F. M. Marchetti, A. Lamacraft and B. D.\nSimons, Nat. Phys. 3, 124 (2007)."    },    {        "title": "2012.13914v1.Microwave_spectroscopy_of_spin_orbit_coupled_states__energy_detuning_versus_interdot_coupling_modulation.pdf",        "content": "arXiv:2012.13914v1  [cond-mat.mes-hall]  27 Dec 2020Microwave spectroscopy of spin-orbit coupled states: ener gy detuning versus interdot\ncoupling modulation\nG. Giavaras1and Yasuhiro Tokura1,2\n1Faculty of Pure and Applied Sciences, University of Tsukuba , Tsukuba 305-8571, Japan\n2Tsukuba Research Center for Energy Materials Science (TREM S), Tsukuba 305-8571, Japan\nWe study the AC field induced current peaks of a spin blockaded double quantum dot with\nspin-orbit interaction. The AC field modulates either the in terdot tunnel coupling or the energy\ndetuning, and we choose the AC field frequency range to induce two singlet-triplet transitions\ngiving rise to two current peaks. We show that for a large detu ning the two current peaks can be\nsignificantly stronger when the AC field modulates the tunnel coupling, thus making the detection\nof the spin-orbit gap more efficient. We also demonstrate the i mportance of the time dependence of\nthe spin-orbit interaction.\nI. INTRODUCTION\nThe singlet-triplet states of two electron spins in\ntunnel-coupled quantum dots can be used to define spin-\nqubits in semiconductor devices.1,2In the presence of a\nstrongspin-orbitinteraction(SOI)anappliedACelectric\nfield can give rise to singlet-triplet transitions and spin\nresonance can be achieved.3,4In a double quantum dot\nwhich is tuned to the spin blockade regime,5the transi-\ntions can be probed by the AC induced current peaks.\nIt has been experimentally demonstrated that the two-\nspin energy spectra can be extracted by examining the\nmagneticfielddependent positionofthecurrentpeaks.3,4\nThe exchange energy, the strength of the SOI, as well\nas theg-factors of the quantum dots can then be esti-\nmated. Microwavespectroscopy has also been performed\nfor the investigation of charge qubits,6,7as well as other\nhybrid spin systems.8–10Charge localization in quantum\ndot systems can be controlled with AC fields,11–13while\nvarious important parameters of the spin and/or charge\ndynamics can be extracted from AC induced interference\npatterns.14–17\nIn this work, we study the current through a double\ndot(DD) fortwodifferentcasesoftheACelectricfield; in\nthe first case the AC field modulates the interdot tunnel\ncoupling of the DD, and in the second case the AC field\nmodulates the energy detuning of the DD. We consider\na specific energy configuration and AC frequency range\nwhich involve two SOI coupled singlet-triplet states, and\na third state with (mostly) triplet character. The two\nSOI-coupled singlet-triplet states form an anticrossing\npoint,3,4,18,19and in this work we focus on this point.\nSpecifically, we perform electronic transport calcula-\ntions and demonstrate that for a large energy detuning\nthe tunnel coupling modulation results in stronger AC-\ninduced current peaks than the corresponding peaks in-\nduced by the detuning modulation. The stronger peaks\ncanofferasignificantadvantagewhenthespectroscopyof\nthe coupled spin system is performed by monitoring the\nmagnetic field dependence of the position of the peaks.\nWhen the peaks are suppressed no reliable information\ncan be extracted.\nThe tunnel coupling modulation offers a similar ad-vantage when the transitions involve only the two states\nforming the anticrossing point.20This finding together\nwith the results of the present work demonstrate that\nmodulating the tunnel coupling of a DD with an AC\nelectric field is a robust method to perform spectroscopy\nof spin-orbit coupled states. Furthermore, in the present\nwork, we explore the time dependent role of the SOI, and\nspecify the regime in which this time dependence should\nbe taken into accountbecause it can drasticallyaffect the\nAC-induced current peaks.\nIn some experimental works the interdot tunnel cou-\npling has been accurately controlled and transport\nmeasurements have been performed.21,22For instance,\nBertrand et al[Ref. 22] have demonstrated that the tun-\nnel coupling can be tuned by orders of magnitude on the\nnanosecond time scale. Therefore, our theoretical find-\nings could be tested with existing semiconductor technol-\nogy.\nII. DOUBLE QUANTUM DOT MODEL\nWe focus on the spin blockade regime5for two serially\ntunnel-coupled quantum dots. In this regime the quan-\ntum dot 1 (dot 2) is coupled to the left (right) metallic\nlead and with the appropriate bias voltage current can\nflow through the DD when the blockade is partially or\ncompletely lifted. Each quantum dot has a single orbital\nlevel and dot 2 is lower in energy by an amount equal to\nthe charging energy which is assumed to be much larger\nthanthetunnelcoupling. Consequently,fortheappropri-\nate bias voltage a single electron can be localized in dot\n2 during the electronic transport process.5If we use the\nnotation (n,m) to indicate nelectrons on the dot 1 and\nmelectronsonthedot2thenelectronictransportprocess\nthroughthe DDtakesplaceviathe chargecycle: (0 ,1)→\n(1,1)→(0,2)→(0,1). For the DD system there are\nin total six two-electron states but in the spin blockade\nregime the double occupation on dot 1 can be ignored\nbecause it lies much higher in energy and does not affect\nthe dynamics. Therefore, the relevanttwo-electronstates\nare the triplet states |T+/angbracketright=c†\n1↑c†\n2↑|0/angbracketright,|T−/angbracketright=c†\n1↓c†\n2↓|0/angbracketright,\n|T0/angbracketright= (c†\n1↑c†\n2↓+c†\n1↓c†\n2↑)|0/angbracketright/√\n2 and the two singlet states2\n|S02/angbracketright=c†\n2↑c†\n2↓|0/angbracketright,|S11/angbracketright= (c†\n1↑c†\n2↓−c†\n1↓c†\n2↑)|0/angbracketright/√\n2. The\nfermionic operator c†\niσcreates an electron on dot i= 1,\n2 with spin σ=↑,↓, and|0/angbracketrightdenotes the vacuum state.\nIn this singlet-triplet basis the DD Hamiltonian20is\nHDD= ∆[|T−/angbracketright/angbracketleftT−|−|T+/angbracketright/angbracketleftT+|]−δ|S02/angbracketright/angbracketleftS02|\n−√\n2Tc|S11/angbracketright/angbracketleftS02|+∆−|S11/angbracketright/angbracketleftT0|+H.c.\n−Tso[|T+/angbracketright/angbracketleftS02|+|T−/angbracketright/angbracketleftS02|]+H.c.(1)\nHere,δis the energy detuning, Tcis the tunnel coupling\nbetween the two dots, and Tsois the SOI-induced tunnel\ncoupling causing a spin-flip.23,24The magnetic field is\ndenoted by Bwhich gives rise to the Zeeman splitting\n∆i=giµBB(i= 1, 2) in each quantum dot. Then\n∆ = (∆ 2+∆1)/2, and the Zeeman asymmetry is ∆−=\n(∆2−∆1)/2. To a good approximation, in the transport\nprocess of a spin-blockaded DD only the c†\n2↑|0/angbracketright,c†\n2↓|0/angbracketright\nsingle electron states are important, and HDDcan be\nalso derived using a standard two-site Hubbard model.25\nIn the present work, we consider two cases for the AC\nfield. Specifically, in the first case the AC field modu-\nlates the energy detuning of the DD, thus we consider\nthe following time dependence:\nδ(t) =ε+Adsin(2πft), (2)\nwhereAdis the AC amplitude and fis the AC frequency.\nTheconstantvalueofthedetuningisdenotedby ε. Inthe\nsecond case, the AC field modulates the interdot tunnel\ncoupling, thus we assume the time dependent terms\nTc(t) =tc+Absin(2πft),\nTso(t) =tso+xsoAbsin(2πft).(3)\nThe AC amplitude is Aband in general Ab/negationslash=Ad. For\nmost calculations we assume that xso=tso/tc, so at any\ntime theratio Tso(t)/Tc(t) isatimeindependent constant\nequaltoxso. We alsoaddressthe casewhere xso/negationslash=tso/tc,\nbut for simplicity we assume no phase difference between\nthe tunnel couplings Tc(t) andTso(t). For the numerical\ncalculations the DD parameters are taken to be tc=\n0.2 meV,tso= 0.02 meV,g1= 2 andg2= 2.4. The\nbasic conclusions of this work are general enough and\nnot specific to these numbers.\nIII. RESULTS\nIn this section we present the basic results of our work.\nWedeterminethe ACinducedcurrentforeachcaseofthe\ntwo AC fields Eq. (2) and Eq. (3). The DD eigenenergies\nEisatisfyHDD|ψi/angbracketright=Ei|ψi/angbracketrightwithAd=Ab= 0, and are\nshown in Fig. 1(a) at B= 1 T. When tso= 0 andg1=g2\nsinglet and triplet states are uncoupled. The energy lev-\nelsE2,E3andE4correspond to the pure triplet states\n|T+/angbracketright,|T0/angbracketrightand|T−/angbracketrightrespectively. These levels are detun-\ning independent and are Zeeman-split due to the applied\nmagnetic field. The energy levels E1,E5correspond to-0.6-0.4-0.2 0 0.2\n 0  0.5  1  1.5E (meV)\nε (meV)E1E2E3E4E5(a)\n 0 0.01 0.02\n 0 0.5 1 1.5 2 2.5 3 0 2 4 6∆so (meV)\n∆so (GHz)\nε (meV)*\n*(b)\nFIG. 1: (a) Two-electron eigenenergies as a function of the\nenergydetuningfor themagnetic field B= 1T. The levels E4,\nE5anticross at ε= 0.5 meV due to the spin-orbit interaction.\nThe two vertical arrows indicate possible transitions whic h\ncan be induced by the AC fields defined in the main text\nEq. (2) and Eq. (3). (b) Spin-orbit gap (∆ so=E5−E4at the\nanticrossing point) as a function of the detuning. In this ca se\nthe magnetic field is detuning dependent. The AC induced\ncurrent is computed for the marked points.\npure singlet states which are |S11/angbracketright,|S02/angbracketrighthybridized due\nto the tunnel coupling tcand are independent of the field\nas can be seen from HDD. Importantly, the singlet levels\nE1,E5define a two-level system and for a fixed tcthe\nhybridization is controlled by the energy detuning. The\ntwo levelsE1,E5anticross at ε= 0 where the hybridiza-\ntion is maximum. This is the only anticrossing point in\nthe energy spectrum for tso= 0. However, according to\nHDDwhentso/negationslash= 0 the polarized triplets |T±/angbracketrightcouple to\nthe singlet state |S02/angbracketright. Therefore, as seen in Fig. 1, at\nε≈0.5 meV the levels E4andE5form an anticrossing\npoint due to the SOI. Another SOI-induced anticross-\ning point is formed at ε <0 between the energy levels\nE1andE2, but here we consider ε >0 and as in the\nexperiments3,26–28we taketso<tc.\nBecauseoftheSOI( tso/negationslash= 0)andthedifferenceinthe g-\nfactors (g1/negationslash=g2) the DD eigenstates |ψi/angbracketrightare hybridized\nsinglet-triplet states and can be written in the general\nform\n|ψi/angbracketright=αi|S11/angbracketright+βi|T+/angbracketright+γi|S02/angbracketright+ζi|T−/angbracketright+ηi|T0/angbracketright.(4)3\nThe coefficients denoted by Greek letters determine the\ncharacter of the states, and are sensitive to the detuning.\nOne method to probe the SOI anticrossing point is to\nfocus on the AC frequency range 0 <hf/lessorsimilarE5−E4and\ndetermine the position of the AC induced current peak.\nThis method has been theoretically studied in Ref. 20.\nAnother method to probe the anticrossing point is to fo-\ncus on the AC frequency range E4−E2/lessorsimilarhf/lessorsimilarE5−E2,\nand determine the positions of the two AC induced cur-\nrentpeaks. The presentworkisconcernedwith the latter\nmethod and the main subject of the present work is to\ncompare the current peaks induced separately by the two\nAC fields; the tunnel barrier modulation and energy de-\ntuningmodulation. InRef.3bothmethodshavebeenex-\nperimentally investigated under the assumption that the\nAC field modulates the energy detuning of the DD. The\ncase where the AC field modulates simultaneouslythe in-\nterdot tunnel coupling and the energy detuning might be\nexperimentally relevant,29but this case is not pursued in\nthe present work.\nIn Fig. 1 the SOI anticrossing is formed at ε≈0.5\nmeV forB= 1 T. A lower magnetic field shifts the SOI\nanticrossing point at larger detuning, and the degree of\nhybridization due to the SOI decreases. The reason is\nthat asεincreases the |S02/angbracketrightcharacter in the original sin-\nglet state ( tso= 0) is gradually replaced by the |S11/angbracketright\ncharacter. As a result, the SOI gap ∆ so=E5−E4,\ndefined at the anticrossing point, decreases with εas\nshown in Fig. 1(b). For the parameters considered in\nthis work, the SOI gap can be analytically determined\nfrom the expression30∆so= 2tso/radicalbig\n(1−cosθ)/2, with\nθ= arctan(2√\n2tc/ε).\nIn ourpreviouswork20weexaminedthe transitionsbe-\ntween the two singlet-triplet states |ψ4/angbracketrightand|ψ5/angbracketright, whose\nenergy levels form the SOI anticrossing point (Fig. 1).\nThese transitions give rise to one current peak which is\nsuppressedneartheanticrossingpoint, inagreementwith\nan experimental study.3In the present work, we focus on\nthe transitions between the two pairs of states |ψ5/angbracketrightand\n|ψ2/angbracketrightas well as |ψ4/angbracketrightand|ψ2/angbracketright. Here, |ψ4/angbracketrightand|ψ5/angbracketrightare\nstrongly hybridized singlet-triplet states, whereas |ψ2/angbracketright\nhas mostly triplet character provided the detuning is\nlarge.\nWe compute the AC-induced current flowing through\nthe double dot within the Floquet-Markov density ma-\ntrix equation of motion.31,32In this approach we treat\nthe time dependence of the AC field exactly, taking ad-\nvantage of the fact that the DD Hamiltonian is time pe-\nriodic and thus it can be expanded in a Fourier series.\nThe model uses for the basis states of the DD density\nmatrix the periodic Floquet modes,33and consequently\nit is applicable for any amplitude of the AC field. In\nmost calculations we take the parameter xso= 0.1 unless\notherwise specified.\nTo study the AC current spectra we choose two values\nfortheenergydetuning ε(2, 0.5meV),anddeterminethe\nmagnetic field at which |ψ4/angbracketrightand|ψ5/angbracketrightanticross. At this\nspecific field we plot in Fig. 2 the AC-induced current as 0.05 0.1 0.15 0.2 0.25\n 17 17.5 18 18.5 19 19.5 20Ι (pA)\nf (GHz)Ab = 10 µeV \nAd = 10 µeV (a)\n 0.4 0.8 1.2 1.6\n 58 59 60 61 62 63 64 65Ι (pA)\nf (GHz)Ab = 10 µeV \nAd = 10 µeV (b)\nFIG. 2: Current as a function of AC frequency, when the\nAC field modulates the tunnel barrier with the AC amplitude\nAb= 10µeV, and the energy detuning with Ad= 10µeV.\nThe constant value of the detuning is ε= 2, 0.5 meV and the\ncorresponding magnetic field is B= 0.3, 1 T for (a) and (b)\nrespectively. These fields define the singlet-triplet antic ross-\ning point for each value of the detuning.\na function of the AC frequency. As the energy detuning\ndecreases the magnetic field defining the corresponding\nanticrossing point increases. This in turn means that the\nAC frequencyhasto increaseto satisfythe corresponding\nresonance condition hf=E5−E2(orhf=E4−E2).\nThis increase in the frequency explains the different fre-\nquency range in Fig. 2. Furthermore, the off-resonant\ncurrent is larger for ε= 0.5 meV due to the stronger SOI\nhybridization.25\nIn the two cases shown in Fig. 2 two peaks are formed;\none peak is due to the transition between the eigenstates\n|ψ2/angbracketrightand|ψ4/angbracketright, and the second peak is due to the transi-\ntion between |ψ2/angbracketrightand|ψ5/angbracketright. Therefore, the distance be-\ntween the centres of the two peaks is equal to the singlet-\ntriplet energy splitting E5−E4. For the specific choice\nof magnetic field this energy splitting is equal to the SOI\ngap of the anticrossing point. For example, for ε= 0.5\nmeV the gap is ∆ so≈3.5 GHz, and for ε= 2 meV the\ngap is ∆ so≈1.1 GHz. These numbers are in agreement\nwith those derived from the exact energies of the time\nindependent part of the Hamiltonian HDD. According to\nFig. 2, for a given energy detuning and driving field the\ntwo peaks are almost identical. This is due to the fact,\nthat at the anticrossing point the states |ψ4/angbracketright,|ψ5/angbracketrighthave\nidentical characters when the driving field is off, and the\nrelevant transition rates are almost equal. In contrast,4\n 0 0.5  1 1.5  2\nB (T) 0.5 1 1.5 2ε (meV)\n 0   0.3   0.6\nqb (µeV)(b) 0 0.5  1 1.5  2 0.5 1 1.5 2ε (meV)\n 0     0.25     0.5\nqd (µeV)(a)\nFIG. 3: (a) Absolute value of the coupling parameter qdas\na function of the energy detuning and magnetic field for the\nAC amplitude Ad= 10µeV. The dotted curve defines the\nanticrossing point for each εandB. (b) The same as (a) but\nforqbwithAb= 10µeV.\n 0 10 20 30 40\n 0 0.5 1 1.5 2 2.5 3 3.5qb/qd\nε (meV)0.00.020.040.060.080.1xso = 0.12\nFIG. 4: The ratio qb/qddefined in Eq. (9) as a function of\ndetuning for different values of xsoandAd=Ab.\naway from the anticrossing point the two peaks can be\nvery different.33\nThe results in Fig. 2 demonstrate that the two driving\nfields Eq. (2) and Eq. (3) induce different peak magni-\ntudes. Specifically, the peaks due to the tunnel barrier\nmodulation are stronger than those due to the detuningmodulation. As an example, for ε= 0.5 meV [Fig. 2(b)]\nthe tunnel barrier modulation induces a relative peak\nheight of about 1 pA, whereas the relative peak height is\nonly 0.1 pA for the energy detuning modulation. Some\ninsight into this interesting behavior can be obtained by\ninspecting the time-scale (“Rabi” frequency) of the co-\nherent transitions between the eigenstates |ψ2/angbracketrightand|ψ4/angbracketright.\nWhen the AC field modulates the tunnel coupling, the\ntransitionscanbestudiedwithintheexactFloqueteigen-\nvalue problem, but for simplicity we here employ an ap-\nproximate approach.20This two-level approach gives the\ntransition frequency qb/h, with\nqb=hfhb\n24\n(hb\n22−hb\n44)J1/parenleftbiggAb(hb\n22−hb\n44)\nhf/parenrightbigg\n,(5)\nwhereJ1(r) is a Bessel function of the first kind, and the\nargument is r=Ab(hb\n22−hb\n44)/hf, withhf=E4−E2\nand\nhb\nij=−γj(√\n2αi+xsoβi+xsoζi)\n−γi(√\n2αj+xsoβj+xsoζj), i,j= 2,4(6)\nWhen the AC field modulates the energy detuning the\ntime-scale of the coherent transitions between the eigen-\nstates|ψ2/angbracketrightand|ψ4/angbracketrightis approximately qd/h. The cou-\npling parameter qdis found by qbwith the replacements\nAb→Adandhb\nij→hd\nij, where\nhd\nij=−γiγj, i,j= 2,4 (7)\nIn general, qbcan be very different from qd, even when\nAb=Ad. Therefore, the two driving fields are expected\nto induce current peaks with different width and height.\nTo quantify the two parameters qb,qdwe plot in Fig. 3\nqb,qd, as a function of the energy detuning and the mag-\nnetic field. Here, Ad=Ab= 10µeV, andhf=E4−E2\n[in Eq. (5)] is magnetic field as well as detuning depen-\ndent, and is determined by the energies of HDD. If we\ndenote byBanthe field at which the anticrossing point is\nformed, then as seen in Fig. 3 both qbandqdare largefor\nB > B an, but vanishingly small for B≪Ban. The rea-\nson is that the state |ψ4/angbracketrightis singlet-like for B >B an, but\ntriplet-like for B <B an, whereas |ψ2/angbracketrighthas mostly triplet\ncharacterindependent of B, providedεisawayfromzero.\nTransitions between triplet-like states are in general slow\nleading to vanishingly small qb,qdforB≪Ban. In\ncontrast, if we choose hf=E5−E2, then both qband\nqdare large for B < B an. For large enough detuning\nwhere the two spins are in the Heisenberg regime, the\nexchange energy is approximately 2 t2\nc/εthereforeBan\nsatisfies (g1+g2)µBBan/2≈2t2\nc/ε.\nMost importantly Fig. 3 demonstrates that qb> qd\nwhenε/greaterorsimilar0.2 meV. To understand this result we focus\non the anticrossing point where r<1, then from Eq. (5)\nqb≈hb\n24Ab/2 becauseJ1(r)≈r/2, and similarly qd≈\nhd\n24Ad/2. Moreover, away from zero detuning the state\n|ψ2/angbracketrighthas mostly triplet character, therefore\nhb\n24≈ −γ4(√\n2α2+xsoβ2)−γ2(√\n2α4+xsoζ4),(8)5\nand the ratio qb/qdis\nqb\nqd=Ab\nAd/parenleftbigg√\n2α2\nγ2+xsoβ2\nγ2+√\n2α4\nγ4+xsoζ4\nγ4/parenrightbigg\n.(9)\nAsεincreasesβ2→1,γ2≪1 and, considering absolute\nvalues, the second term in Eq. (9) dominates\nβ2\nγ2≫α2\nγ2,α4\nγ4,ζ4\nγ4. (10)\nConsequently, qbcan be much greater than qd, especially\nat largeε, and for a fixed tunnel coupling tcthe exact\nvalue of the ratio qb/qddepends sensitively on xso. This\ndemonstrates the importance of the time dependence of\nthe spin-orbit coupling. The conclusions derived from\nthe parameters qb,qdassume that there is no ‘multi-\nlevel’ interference and only the levels Ei,Ejsatisfying\nhf=|Ei−Ej|are responsible for the current peaks.\nThe approximate results are more accurate when the ar-\ngumentrof the Bessel function is kept small.\nTo examine the xso-dependence, we consider Ab=Ad\nand plot in Fig. 4 the ratio qb/qdversus the detuning\nat the anticrossing point, and for different values of xso.\nBy increasing εand for large values of xsothe coupling\nparameters qb,qdcan differ by over an order of mag-\nnitude;qb/qd>10. This leads to (very) different cur-\nrent peaks with the tunnel barrier modulation inducing\nstronger peaks. The special value xso= 0 corresponds to\natimeindependent SOItunnelcoupling[seeEq.(3)], and\nthespecialvalue xso= 0.1correspondstoatimeindepen-\ndent ratioTso/Tc= 0.1. Although, the ratio qb/qdcan\nbe computed at any ε, the regime of small ε(<0.2 meV)\nis not particularly interesting in this work. The reason\nis that with decreasing εthe character of the state |ψ2/angbracketright\nchanges from triplet-like to singlet-triplet, which even-\ntually becomes approximately equally populated to |ψ4/angbracketright\nand|ψ5/angbracketright. Therefore, the current peaks induced by both\ndriving fields are suppressed even when qborqdis large.\nIn Fig. 4 the maximum value of the detuning is chosen\nto giveε/tc≈17.5 which can be easily achieved in dou-\nble quantum dots. Some experiments1,3,4have reported\nvalues greater than ε/tc≈100, thusqbcan be even two\norders of magnitude greater than qd.\nAccording to the above analysis if qb/qd≈1 then the\ncurrent peaks induced by the two driving fields should\napproximately display the same characteristics. As an\nexample, consider the two sets of current peaks shown\nin Fig. 2 both for xso= 0.1 andε= 2 meV, ε= 0.5\nmeV respectively. Focusing on xso= 0.1 in Fig. 4, we\nsee that at ε= 2 meVqb/qd≈19 and atε= 0.5 meV\nqb/qd≈4.9. These numbers suggestthat if at ε= 2 meV\nwe choose for the AC amplitudes the ratio Ab/Ad≈1/19\nthen the detuning and the barrier modulation should in-\nduce approximately the same peak characteristics. Like-\nwise atε= 0.5 meV the ratio should be Ab/Ad≈1/4.9.\nThese arguments are quantified in Fig. 5 where we plot\nthe current peaks for the two driving fields for different\nAC amplitudes satisfying the condition qb/qd≈1. The 0.05 0.1 0.15 0.2 0.25\n 17 17.5 18 18.5 19 19.5 20Ι (pA)\nf (GHz)Ad = 190 µeV \nAb = 10 µeV (a)\n 0.4 0.8 1.2 1.6\n 58 59 60 61 62 63 64 65Ι (pA)\nf (GHz)Ad = 49 µeV \nAb = 10 µeV (b)\nFIG. 5: As in Fig. 2, but (a) Ab= 10µeV andAd= 19Ab,\n(b)Ab= 10µeV andAd= 4.9Ab. The value of Adis chosen\nso that to approximately induce the same current peaks as\nthose induced by Ab.\n 0.05 0.1 0.15 0.2 0.25\n 17.5  18  18.5  19  19.5Ι (pA)\nf (GHz)\nFIG. 6: Current as a function of AC frequency, when the AC\nfield modulates the tunnel barrier, with the AC amplitude\nAb= 10µeV. The detuning is ε= 2 meV and from the upper\nto the lower curve the parameter xso= 0.1, 0.04, 0.02, 0.\nresults confirm that the induced current peaks display\napproximately the same characteristics.\nInducing strong current peaks can be advantageous in\norder to perform spectroscopy of the singlet-triplet levels\nand extract the SOI anticrossing gap. However, an im-\nportant aspect is that the SOI gap cannot be extracted\nfrom the positions of the current peaks at arbitrary large\nAC amplitudes. In particular, by increasing the AC am-\nplitude the two peaks start to overlap and eventually6\nthe resonant pattern of the current changes drastically.33\nTherefore, the distance between the two peaks cannot\naccurately predict the SOI gap. This effect has been\ntheoretically studied for the case of a time dependent\nenergy detuning,33and it can be readily shown that sim-\nilar trends occur for a time dependent tunnel coupling.\nThe driving regime where the two current peaks strongly\noverlap is not considered in the present work, since it is\nnot appropriate for the spectroscopy of the SOI gap.\nFinally, in Fig. 6 we plot the current peaks when the\nAC field modulatesthe tunnel barrierwith the amplitude\nAb= 10µeVandtheconstantdetuning ε= 2meV.With\ndecreasing xsothe two peaks gradually weaken and for\nxso= 0 the peaks arevanishingly small; for this value the\npeaks are of the same order as the peaks induced by the\ndetuning modulation with the same amplitude Ad= 10\nµeV (for clarity these peaks are not shown). The small\ndifference between the left and the right peaks, for exam-\nple whenxso= 0.02,can be understood byinspecting the\ndifferent values of qb[Eq. (5)] which involvedifferent ma-\ntrixelementsandfrequencies. Theoveralltrendsindicate\nthe important role of the time dependent spin-orbit term\nandareconsistentwiththeresultsshowninFig.4. As xso\ndecreases the coupling parameter qbdecreases too, thus\nthe time scale of the singlet-triplet transitions becomes\nlonger leading to smaller peaks. Moreover, by decreasing\nxso,qbbecomes approximately equal to qd, therefore the\ntunnel barrier modulation and the detuning modulationresult in approximately the same current peaks.\nIV. SUMMARY\nIn summary, we considered a double quantum dot in\nthe spin blockade regime and studied the AC induced\ncurrent peaks for a specific energy configuration which\ninvolves two hybridized singlet-triplet states as well as\na third state with mostly triplet character. The two\nAC induced transitions which rely on the spin-orbit in-\nteraction, result in two current peaks. We found that\nfor a large energy detuning the two peaks are stronger\nwhen the time periodic field modulates the interdot tun-\nnel coupling (barrier)instead ofthe energy detuning. We\ndemonstrated that a time dependence in the spin-orbit\ncoupling can significantly modify the peak characteris-\ntics, and should be taken into account even when the\nactual spin-orbit coupling is small. Our work suggests\nan efficient way of probing the spin-orbit energy gap in\ntwo-spin states based on transport measurements.\nACKNOWLEDGEMENT\nPart of this work was supported by CREST JST (JP-\nMJCR15N2), and by JSPS KAKENHI (18K03479).\n1J. R. Petta, A. C. Johnson, J. M. Taylor, E. A. Laird, A.\nYacoby, M. D. Lukin, C. M. Marcus, M. P. Hanson, and\nA. C. Gossard, Science 309, 2180 (2005).\n2J. J. L. Morton, and B. W. Lovett, Annu. Rev. Condens.\nMatter Phys. 2, 189 (2011).\n3K. Ono, G. Giavaras, T. Tanamoto, T. Ohguro, X. Hu,\nand F. Nori, Phys. Rev. Lett. 119, 156802 (2017).\n4S. Nadj-Perge, V. S. Pribiag, J. W. G. van den Berg, K.\nZuo, S. R. Plissard, E. P. A. M. Bakkers, S. M. Frolov, and\nL. P. Kouwenhoven, Phys. Rev. Lett. 108, 166801 (2012).\n5K. Ono, D.G. Austing, Y. Tokura, andS. Tarucha, Science\n297, 1313 (2002).\n6T. Hayashi, T. Fujisawa, H. D. Cheong, Y. H. Jeong, and\nY. Hirayama, Phys. Rev. Lett. 91, 226804 (2003).\n7Z.V.Penfold-Fitch, F.Sfigakis, andM. R.Buitelaar, Phys.\nRev. Applied 7, 054017 (2017).\n8B. C. Wang, G. Cao, H. O. Li, M. Xiao, G. C. Guo, X.\nHu, H. W. Jiang, and G. P. Guo, Phys. Rev. Applied 8,\n064035 (2017).\n9X. C. Yang, G. X. Chan, and X. Wang, Phys. Rev. A 98,\n032334 (2018).\n10W. Song, T. Du, H. Liu, M. B. Plenio, and J. Cai, Phys.\nRev. Applied 12, 054025 (2019).\n11E. Paspalakis and A. Terzis, J. Appl. Phys. 95, 1603\n(2004).\n12A. Terzis and E. Paspalakis, J. Appl. Phys. 97, 023523\n(2005).\n13C. E. Creffield and G. Platero, Phys. Rev. B 65, 113304\n(2002).14S. N. Shevchenko, S. Ashhab, and F. Nori, Phys. Rep. 492,\n1 (2010).\n15S. N. Shevchenko, S. Ashhab, and F. Nori, Phys. Rev. B\n85, 094502 (2012).\n16F. Gallego-Marcos, R. Sanchez, and G. Platero J. Appl.\nPhys.117, 112808 (2015).\n17A. Chatterjee, S. N. Shevchenko, S. Barraud, R. M. Otxoa,\nF. Nori, J. J. L. Morton, and M. F. Gonzalez-Zalba, Phys.\nRev. B97, 045405 (2018).\n18S. Takahashi, R. S. Deacon, K. Yoshida, A. Oiwa, K. Shi-\nbata, K. Hirakawa, Y. Tokura, and S. Tarucha, Phys. Rev.\nLett.104, 246801 (2010).\n19Y. Kanai, R. S. Deacon, S. Takahashi, A. Oiwa, K.\nYoshida, K. Shibata, K. Hirakawa, Y. Tokura, and S.\nTarucha, Nature Nanotechnology 6, 511 (2011).\n20G. Giavaras and Y. Tokura, Phys. Rev. B 99, 075412\n(2019).\n21T. H. Oosterkamp, T. Fujisawa, W. G. van der Wiel, K.\nIshibashi, R. V. Hijman, S. Tarucha, and L. P. Kouwen-\nhoven, Nature (London) 395, 873 (1998).\n22B. Bertrand, H. Flentje, S. Takada, M. Yamamoto, S.\nTarucha, A. Ludwig, A. D. Wieck, C. Buerle, and T. Me-\nunier, Phys. Rev. Lett. 115, 096801 (2015).\n23F. Mireles and G. Kirczenow, Phys. Rev. B 64, 024426\n(2001).\n24H. Pan and Y. Zhao, J. Appl. Phys. 111, 083703 (2012).\n25G. Giavaras, N. Lambert, and F. Nori, Phys. Rev. B 87,\n115416 (2013).\n26S. J. Chorley, G. Giavaras, J. Wabnig, G. A. C. Jones, C.7\nG. Smith, G. A. D. Briggs, and M. R. Buitelaar, Phys.\nRev. Lett. 106, 206801 (2011).\n27G. Xuet al., Appl. Phys. Express 13, 065002 (2020).\n28M. Marx , J. Yoneda, T. Otsuka, K. Takeda, Y. Yamaoka,\nT. Nakajima, S. Li, A. Noiri, T. Kodera, and S. Tarucha,\nJpn. J. Appl. Phys. 58, SBBI07 (2019).\n29T. Nakajima, M. R. Delbecq, T. Otsuka, S. Amaha, J.\nYoneda, A. Noiri, K. Takeda, G. Allison, A. Ludwig, A. D.\nWieck, X. Hu, F. Nori, and S. Tarucha, Nature Communi-cations9, 2133 (2018).\n30D. Stepanenko, M. Rudner, B. I. Halperin, and D. Loss,\nPhys. Rev. B 85, 075416 (2012).\n31M. Grifoni and P. H¨ anggi, Phys. Rep. 304, 229 (1998).\n32S. Kohler, J. Lehmann, and P. H¨ anggi, Phys. Rep. 406,\n379 (2005).\n33G. Giavaras and Y. Tokura, Phys. Rev. B 100, 195421\n(2019)."    },    {        "title": "2003.13181v1.Intrinsic_orbital_moment_and_prediction_of_a_large_orbital_Hall_effect_in_the_2D_transition_metal_dichalcogenides.pdf",        "content": "Intrinsic orbital moment and prediction of a large orbital Hall e\u000bect in the 2D\ntransition metal dichalcogenides\nSayantika Bhowal\u0003and S. Satpathy\nDepartment of Physics & Astronomy, University of Missouri, Columbia, MO 65211, USA\nCarrying information using generation and detection of the orbital current, instead of the spin\ncurrent, is an emerging \feld of research, where the orbital Hall e\u000bect (OHE) is an important ingredi-\nent. Here, we propose a new mechanism of the OHE that occurs in non-centrosymmetric materials.\nWe show that the broken inversion symmetry in the 2D transition metal dichalcogenides (TMDCs)\ncauses a robust orbital moment, which \row in di\u000berent directions due to the opposite Berry cur-\nvatures under an applied electric \feld, leading to a large OHE. This is in complete contrast to the\ninversion-symmetric systems, where the orbital moment is induced only by the external electric \feld.\nWe show that the valley-orbital locking as well as the OHE both appear even in the absence of the\nspin-orbit coupling. The non-zero spin-orbit coupling leads to the well-known valley-spin locking\nand the spin Hall e\u000bect, which we \fnd to be weak, making the TMDCs particularly suitable for\ndirect observation of the OHE, with potential application in orbitronics .\nOrbital Hall e\u000bect (OHE) is the phenomenon of trans-\nverse \row of orbital angular momentum in response to\nan applied electric \feld, similar to the \row of spin angu-\nlar momentum in the spin Hall e\u000bect (SHE). The OHE\nis more fundamental in the sense that it occurs with or\nwithout the presence of the spin-orbit coupling (SOC),\nwhile in presence of the SOC, OHE leads to the addi-\ntional \row of the spin angular momentum resulting in\nthe SHE. In fact, the idea of OHE has already been in-\nvoked to explain the origin of a large anomalous and spin\nHall e\u000bect in several materials [1{3]. Because of this and\nthe fact that OHE is expected to have a larger magnitude\nthan its spin counterpart, there is a noticeable interest\nin developing the OHE [4{7], with an eye towards future\n\\orbitronics\" device applications.\nIn this work, we propose a new mechanism of the OHE\nthat occurs in non-centrosymmetric materials and explic-\nitly illustrate the ideas for monolayer transition metal\ndichalcogenides (TMDCs) which constitute the classic\nexample of 2D materials with broken inversion symme-\ntry. In complete constrast to the centrosymmetric mate-\nrials [4, 6], where orbital moments are quenched due to\nsymmetry and a non-zero moment develops only due to\nthe symmetry-breaking applied electric \feld, here an in-\ntrinsic orbital moment is already present in the Brillouin\nzone (BZ) even without the applied electric \feld. Unlike\nthe centrosymmetric systems, the physics here is dom-\ninated by the non-zero Berry curvatures, which deter-\nmines the magnitude of the OHE. Our work emphasizes\nthe intrinsic nature of orbital transport in contrast to the\nvalley Hall e\u000bect [8{12], for example, which can only be\nachieved by extrinsic means (doping, light illumination,\netc.).\nWe develop the key physics of the underlying mecha-\nnism of the OHE using a tight-binding (TB) model as\nwell as from density-functional calculations. The e\u000bect\nis demonstrated for the selected members of the family\n\u0003bhowals@missouri.edu\n  \nMXxy\nX(a) (b)\nx2-y2+i xy\nx2-y2-i xy\n3z2-1\n3z2-1\nK (t = +1) K' (t = -1)\nMX2(d)\nK K'\nEky\nkxkz=0Mz < 0\nMz >0E\nv =(e/ħ) E X Ωv \nv \nMz < 0\nMz >0a\n10\n5\n0\n-5\n-10(c)\nFIG. 1. Illustration of OHE in monolayer MX 2. (a) Crystal\nstructure of MX 2, showing the triangular network of transi-\ntion metal M atoms as viewed from top. The two out-of-plane\nchalcogen atoms X occur above and below the plane. (b) The\nband structure near K(\u00004\u0019=3a;0) andK0(4\u0019=3a;0), show-\ning the valley dependent spin and orbital characters. (c) The\norbital moment Mz(~k) in the BZ and the anomalous veloci-\ntiesv, indicated by the blue and the red arrows. (d) Orbital\nmoments \row in the transverse direction leading to the OHE.\nof monolayer TMDCs, viz., 2H-Mo X2(X= S, Se, Te),\nwhere we \fnd a large OHE and at the same time a negli-\ngible intrinsic spin Hall e\u000bect, making these materials an\nexcellent platform for the direct observation of the OHE.\nThe basic physics is illustrated in Fig. 1, where we have\nshown the computed intrinsic orbital moments in the BZ\nas well as the electron \\anomalous\" velocities at the K,\nK0valleys. Symmetry demands that in the presence\nof inversion (I), orbital moments satisfy the condition\n~M(~k) =~M(\u0000~k), while if time-reversal ( T) symmetry is\npresent, we have ~M(~k) =\u0000~M(\u0000~k). Thus for a non-zero\n~M(~k), at least one of the two symmetries must be broken.arXiv:2003.13181v1  [cond-mat.mtrl-sci]  30 Mar 20202\nIn the present case, broken Ileads to a nonzero ~M(~k),\nwhile its sign changes between the KandK0points due\nto the presence of T. The Berry curvatures ~\n(~k) follow\nthe same symmetry properties leading to the non-zero\nanomalous velocity ~ v= (e=~)~E\u0002~\n~k[13] which has op-\nposite directions at the two valleys, and thus leads to\nthe OHE. These arguments are only suggestive, and one\nmust evaluate the magnitude of the e\u000bect from the cal-\nculation of the orbital Berry curvatures [13] as discussed\nbelow.\nTight Binding results near the valley points { The val-\nley points ( K/K0) have the major contributions to the\nOHE in the TMDCs and this can be studied analytically\nusing a TB model. Due to the broken I[see Fig. 1 (a)],\nthe chalcogen atoms must be kept along with the transi-\ntion metal atom (M) in the TB basis set; However, their\ne\u000bect may be incorporated via the L owdin downfolding\n[14] producing an e\u000bective TB Hamiltonian for the M-\ndorbitals with modi\fed Slater-Koster matrix elements\n[15]. The e\u000bective Hamiltonian, valid near the KandK0\nvalley points reads\nH(~ q) = (~d\u0001~ \u001b)\nIs+\u001c\u0015\n2(\u001bz+ 1)\nsz; (1)\nwhere only terms linear in ~ q=~k\u0000~Khave been kept,\nignoring thereby the higher-order trigonal warping [16],\nwhich are unimportant for the present study. Here ~ sand\n~ \u001bare respectively the Pauli matrices for the electron spin\nand the orbital pseudo-spins, jui= (p\n2)\u00001(jx2\u0000y2i+\ni\u001cjxyi) andjdi=j3z2\u0000r2i.Isis the 2\u00022 identity\noperator in the electron spin space, \u0015is the SOC con-\nstant, and the valley index \u001c=\u00061 for theKandK0\nvalleys, respectively. The TB hopping integrals appear\nin the parameter ~d, withdx=\u001ctqxa;dy=\u0000tqya;and\ndz=\u0000\u0001=2, whereais the lattice constant, \u0001 is the\nenergy gap at the K(K0) point, and tis an e\u000bective\ninter-band hopping, determined by certain d\u0000dhopping\nmatrix elements. We note that Eq. (1) is consistent with\nthe Hamiltonian derived earlier [10] using the k\u0001pthe-\nory. The TB derivation has the bene\ft that it directly\nexpresses the parameters of the Hamiltonian in terms of\nthe speci\fc hopping integrals.\nThe magnitude of the orbital moment ~M(~k) can be\ncomputed for a speci\fc band of the Hamiltonian (1) using\nthe modern theory of orbital moment [13, 17], viz.,\n~M(~k) =\u00002\u00001Im[h~rku~kj\u0002(H\u0000\"~k)j~rku~ki]\n+ Im[h~rku~kj\u0002(\u000fF\u0000\"~k)j~rku~ki]; (2)\nwhere\"~kandu~kare the band energy and the Bloch wave\nfunction, and the two terms in (2) are, respectively, the\nangular momentum ( ~ r\u0002~ v) contribution due to the self-\nrotation and due to the motion of the center-of-mass of\nthe Bloch electron wave packet. Diagonalizing the 4 \u00024\nHamiltonian (1), we \fnd the energy eigenvalues: \"\u0017\n\u0006=\n2\u00001[\u001c\u0017\u0015\u0006((\u0001\u0000\u001c\u0017\u0015)2+ 4t2a2q2)1=2], where\u0017=\u00061 are\nthe two spin-split states within the conduction or valence\nband manifold, denoted by the subscript \u0006. The wave\n  \nK'(a)\n(b)3z2-r2 x2-y2 - i xyxz + i yz\nxz - i yz\nx2-y2 + i xyM\nzDFTTB ModelE\nF\nx2-y2 - i xy3z2-r2 DEnergy (eV)M\nz (eV Å2\n)FIG. 2. (a) Density-functional band structure together with\nthe orbital characters near the valley points and (b) the com-\nputed sum of the orbital moments ( Mz) over all occupied\nbands along selected symmetry lines.\nfunctions in the basis set ( ju\"i;jd\"i;ju#i;andjd#i)\nare\nju\u0017=1\n\u0006(q)i=Nh\n1 (D\u0017\u0007p\n(D\u0017)2+d2)=d\u00170 0iT\n;\nju\u0017=\u00001\n\u0006 (q)i=Nh\n0 0 1 (D\u0017\u0007p\n(D\u0017)2+d2)=d\u0017iT\n(3)\nwhereD\u0017= (\u0001\u0000\u0017\u001c\u0015)=2,d\u0017=ta(\u001cqx\u0006i\u0017qy),d2=\nt2a2(q2\nx+q2\ny), andNis the appropriate normalization\nfactor. With these wave functions, the orbital moments\ncan be evaluated exactly within the TB model from Eq.\n2. For the two valence bands ( \u0017=\u00061), the result is\nMz(~ q) =\u001cm0D\u0017(D\u0000\u0017\u0000\u0015)\u0001\n2[(D\u0017)2+t2q2a2]3=2(4a)\n\u0019\u001cm0[1 +\u0015(3\u0017\u001c\u00002)=\u0001](1\u00006m0q2=\u0001); (4b)\nwherem0= \u0001\u00001t2a2, only the out-of-plane ^ zcomponent\nof the orbital moment is non-zero, and the second line is\nthe expansion for small qand\u0015, both\u001c\u0001.\nNote the important result (4) that a large orbital mo-\nmentMzexists at the valley points ( ~ q= 0) and its sign al-\nternates between the two valleys ( \u001c=\u00061) (valley-orbital\nlocking ). Furthermore, it exists even in absence of the\nSOC (\u0015= 0). For typical parameters, t= 1:22 eV, \u0001 =\n1.66 eV, and \u0015= 0:08 eV, relevant for the monolayer\nMoS 2,m0\u00199:1 eV. \u0017A2\u00192:4\u0016B\u0002(~=e). As seen from\nEq. 4 (b), there is only a weak dependence on \u0015.\nIn fact it is interesting to note that the valley-\ndependent spin splitting [Fig. 1 (b)] directly follows from\nthe valley orbital moments due to the h~L\u0001~Siterm, which\nfavors anti-alignment of spin with the orbital moment [3].\nThus for the valence bands, the spin- #band is lower in\nenergy atK, while the spin-\"band is lower at K0, with\na spin splitting of about 2 \u0015. Therefore, the well-known\nspin polarization of the bands at the valley points can be\nthought of to be driven by the robust orbital moments\nvia the perturbative SOC.3\nThe orbital moment is the largest at the valley points\nK;K0, as seen from Eq. (4), falling o\u000b quadratically with\nmomentum q. This is also validated by the DFT results\nshown in Fig. 2. The orbital moment at the center of the\nBZ (\u0000) vanishes exactly due to symmetry reasons, and\ntherefore is expected to be small in the neighborhood of\n\u0000 as seen from Fig. 2 (b) as well.\nIt is easy to argue that under an applied electric \feld,\nthe electrons in the two valleys move in opposite direc-\ntions, so that a net orbital Hall current is produced. To\nsee this, we \frst realize that only the Berry curvature\nterm in the semi-classical expression [13] for the electron\nvelocity _~ rc=~\u00001[~rk\"k+e~E\u0002~\n(~k)]~kcis non-zero for\nthe two valleys. Furthermore, only the ^ zcomponent of\nthe Berry curvature survives, which we evaluate near the\nK;K0valleys within the TB model using the Kubo for-\nmula below. The result is\n\nz\nn(~ q) =\u00002~2X\nn06=nIm\u0002\nhun~ qjvxjun0~ qihun0~ qjvyjun~ qi\u0003\n(\"n0~ q\u0000\"n~ q)2\n=2Mz(~ q)\n\u0001 +\u0015(\u0017\u001c\u00002)\u00192\u001cm0\n\u00012(\u0001 + 2\u0017\u001c\u0015\u00006m0q2):(5)\nClearly, \nzhas opposite signs for the two valleys, so that\n~ v/~E\u0002~\n is in opposite directions for the Kand the\nK0valley electrons. Thus the positive orbital moment of\ntheKvalley moves in one direction, while the negative\norbital moment of K0moves in the opposite direction,\nleading to a net orbital Hall current.\nThe magnitude of the orbital Hall conductivity (OHC)\nmay be calculated using the Kubo formula by the mo-\nmentum sum of the orbital Berry curvatures [4, 6], viz.,\n\u001b\r;orb\n\u000b\f=\u0000e\nNkVcoccX\nn~k\n\r;orb\nn;\u000b\f(~k); (6)\nwhere\u000b;\f;\r are the cartesian components, jorb;\r\n\u000b =\n\u001b\r;orb\n\u000b\fE\fis the orbital current density along the \u000bdirec-\ntion with the orbital moment along \r, generated by the\nelectric \feld along the \fdirection. In the 2D systems, Vc\nis the surface unit cell area, so that the conductivity has\nthe dimensions of ( ~=e) Ohm\u00001.\nThe orbital Berry curvature \n\r;orb\nn;\u000b\fin Eq. 6 can be\nevaluated as\n\n\r;orb\nn;\u000b\f(~k) = 2 ~X\nn06=nIm[hun~kjJ\r;orb\n\u000bjun0~kihun0~kjv\fjun~ki]\n(\"n0~k\u0000\"n~k)2;\n(7)\nwhere the orbital current operator is J\r;orb\n\u000b =1\n2fv\u000b;L\rg,\nwithv\u000b=1\n~@H\n@k\u000bis the velocity operator and L\ris the\norbital angular momentum operator.\nIt turns out that due to the simplicity of the TB Hamil-\ntonian (1), valid near the valley points, the orbital and\nthe standard Berry curvatures are the same, apart from\na valley-dependent sign, viz.,\n\nz;orb\nn;yx(~ q) =\u001c\u0002\nz\nn(~ q): (8)To see this, we take the momentum derivative of (1) to\nget\n~vx(~ q) =2\n640\u001cta 0 0\n\u001cta 0 0 0\n0 0 0 \u001cta\n0 0\u001cta 03\n75=ta\u001c\u001bx\nIs;(9)\nand, similarly, ~vy(~ q) =\u0000ta\u001by\nIsandvz(~ q) = 0.\nFurthermore, in the subspace of the TB Hamiltonian,\nLx=Ly= 0, andLz=\u001c~(\u001bz+ 1)\nIs. By matrix mul-\ntiplication, we immediately \fnd that Jz;orb\n\u000b =\u001c~v\u000band\nJx;orb\n\u000b =Jy;orb\n\u000b = 0, which leads to the result (8). The\nexpression for the orbital Berry curvature then follows\nfrom Eqs. (5) and (8), viz.,\n\nz;orb\n\u0017;yx(~ q) =2\u001cMz(~ q)\n\u0001 +\u0015(\u0017\u001c\u00002); (10)\nwhereMz(~ q) is the orbital moment in Eq. (4). At a\ngeneralkpoint, the full expression (7) must be evaluated\nto obtain the OHC.\nThis is a key result of the paper, which shows that the\norbital Berry curvatures near the KandK0points are\ndirectly proportional to the respective orbital moments,\nand, more importantly, they have the same sign at the\ntwo valleys as both \u001candMzchange signs simultane-\nously. Thus, the contributions from these two valleys\nadd up, leading to a non-zero OHC. Another important\npoint is that \nz;orb\n\u0017;yx exists even without the SOC, and\nit has only a weak dependence on \u0015as seen from Eq.\n(10). Neglecting the \u0015dependence, we see that at both\nvalley points, the contribution to the OHC is given by\n\nz;orb\n\u0017;yx = 2t2a2=\u00012. In fact, the momentum sum in OHC\ncan be performed analytically in this limit by integrating\nup to the radius qc(\u0019q2\nc= \nBZ) to yield the result\n\u001bz;orb\nyx =\u00002e\n(2\u0019)2X\n\u0017=\u00061Zqc\n0d2q\u0002\nz;orb\n\u0017;yx(~ q)\n=\u0000e\n\u0019\u0002h\n1\u0000\u0001q\n\u00012+ (32\u0019t2=p\n3)i\n+O(\u00152=\u00012);\n(11)\nwhich is consistent with the anticipated result that the\nlarger the parameter t2=\u00012, the larger is the OHC, pri-\nmarily because the orbital moment Mzincreases.\nWe pause here to compare the OHE with the related\nphenomenon of the valley Hall e\u000bect, which has been pro-\nposed in the gapped graphene as well as in the TMDCs\n[11, 12]. In the valley Hall e\u000bect, electrons in the two\nvalleys \row in opposite directions, leading to a charge\ncurrent and additionally to an orbital current (the valley\norbital Hall e\u000bect [12]), if there is a valley population im-\nbalance (e.g., created by shining light). This is in com-\nplete contrast to the OHE, which is an intrinsic e\u000bect\nwithout any need for population imbalance between the\nvalleys. More interestingly, unlike the valley Hall e\u000bect,\nthe OHE described here does not have any net charge4\n  \nMK K'(a)\nK K'\nGM\n1.00.50-0.5\n 1.5G\n-505101520(b)\n00\nFIG. 3. (a) Orbital and (b) spin Berry curvatures (in units of\n\u0017A2), summed over the occupied states, on the kz= 0 plane for\n2H-MoS 2. The contours correspond to the tick values on the\ncolor bar and the zero contours have been indicated explicitly.\ncurrent but there exists only a pure orbital current. Fur-\nthermore, in the valley Hall e\u000bect, the non-zero valley\norbital magnetization [11] explicitly breaks the Tsym-\nmetry, which is preserved in the present case. In this\nsense the OHE studied here is completely di\u000berent from\nthe valley Hall e\u000bect proposed earlier.\nDensity functional results { We now turn to the DFT\nresults for the monolayer TMDCs. Orbital moments were\ncomputed using pseudopotential methods [18] and the\nWannier functions as implemented in the Wannier90 code\n[19, 20] [see Supplementary Materials [21] for details].\nThe complementary mu\u000en-tin orbitals based method\n(NMTO) [22] was used to compute the orbital moment\nas well as the orbital and the spin Hall conductivities.\nIn the latter method, e\u000bective TB hopping matrix ele-\nments between the M- dorbitals are obtained for several\nneighbors, which yields the full TB Hamiltonian valid\neverywhere in the BZ, using which all quantities of inter-\nest are computed. The BZ sums for the OHC and spin\nHall conductivity (SHC) were computed with 400 \u0002400\nkpoints in the 2D zone. The computed orbital moments\nusing the Wannier90 or the NMTO method agree quite\nwell.\nThe DFT band structure and the corresponding or-\nbital moments are shown in Fig. 1 (c) and Fig. 2 for\nTABLE I. DFT results for the OHC of the monolayer TMDCs,\nincluding the partial contributions ( \u001bz;orb\nyx =\u001bK+\u001b\u0000+\u001brest),\n\u001bK,\u001b\u0000, and\u001brestbeing the contributions, respectively, from\nthe valley, \u0000-point, and the remaining regions of the BZ. OHC\nare in units of 103\u0002(~=e)\n\u00001, while the SHC are in units of\n(~=e)\n\u00001.\nMaterials \u001bK\u001b\u0000\u001brest\u001bz;orb\nyx\u001bz;spin\nyx\nMoS 2 -9.1 1.7 -3.2 -10.6 1.0\nMoSe 2 -8.0 1.7 -3 -9.3 1.8\nMoTe 2 -9.1 1.1 -2.5 -10.5 3.0\nWTe 2 -8.6 1.0 -2.6 -10.2 9.4MoS 2. As shown in Fig. 2 (b), the orbital moments com-\nputed from the Hamiltonian (1) near the valley points\nagree quite well with the DFT results. Note that the to-\ntal orbital moment (summed over the BZ) vanishes due\nto the presence of T, though it is non-zero at individual\nkpoints. From the TB model (1), we had studied the or-\nbital moment and the OHE near the valley points. From\nthe DFT calculations, we can compute the same over the\nentire BZ, the result of which is shown in Fig. 3 (a). As\nseen from the \fgure, the dominant contribution comes\nfrom thekspace near the valley points K,K0. Since\nthe intrinsic orbital moment near the \u0000 point is absent,\nthe orbital Berry curvature in this region takes a non-\nzero value only due to the orbital moments induced by\nthe applied electric \feld in the Hall measurement, similar\nto the centrosymmetric case [4]. This results in a small\ncontribution \u001b\u0000to the net OHC, as seen from Table I,\nwhich lists the partial contributions to the OHC coming\nfrom di\u000berent parts of the BZ. Note that there is only one\nindependent component of OHC, viz., \u001bz;orb\nyx = -\u001bz;orb\nxy.\nSpin Hall E\u000bect { For a material to be a good can-\ndidate for the detection of the OHE, the SHC must be\nsmall, as both carry angular momentum. To this end, we\ncompute the SHC, \frst from the model Hamiltonian and\nthen from the full DFT calculations. Analogous to the\ncalculation of the OHC, the SHC can be obtained by the\nsum of the spin Berry curvatures, \nz;spin\n\u0017;yx (~k), evaluated\nby replacing the orbital current operator with the spin\ncurrent operator J\r;spin\n\u000b =1\n4fv\u000b;s\rgin Eq. 7. For the\ntwo spin-split valence bands near the valley points in the\nTB model, we \fnd\n\nz;spin\n\u0017;yx (~ q) =\u0017Mz(~ q)\n\u0001 +\u0015(\u0017\u001c\u00002)=\u0017\u001c\n2\nz;orb\nyx(~ q):(12)\nNote that \nz;spin\n\u0017;yx (~ q) has opposite signs for the two spin-\nsplit bands and in the limit of \u0015= 0, they exactly cancel\neverywhere producing a net zero SHC. For a non-zero\n\u0015, these two contributions add up to produce a small\nnet SHC. Calculating the contributions from the valley\npoints with a similar procedure as Eq. (11), we obtain the\nresult\u001bz;spin\nyx\u0018\u0000e\u0015(\u0019\u0001)\u00001;in the limit \u0015\u001c\u0001. This is\nclearly much smaller than the OHC (11), by a factor of\n\u0015=\u0001. From the DFT results (see Table I), we do indeed\n\fnd that the SHC is about three orders of magnitude\nsmaller than the OHC. Even in doped samples though\nthe SHC is expected to be higher than the undoped sam-\nple, the typical values [8] are nevertheless still an order of\nmagnitude smaller than the computed OHC. These ar-\nguments suggest the TMDCs to be excellent candidates\nfor the observation of OHE, since the intrinsic SHC is\nnegligible in comparison.\nIn conclusion, we examined the intrinsic OHE in non-\ncentrosymmetric materials and illustrated the ideas for\nthe monolayer TMDCs. The broken Iin TMDCs pro-\nduces a robust momentum-space intrinsic orbital moment\n~M(~k), present even in the absence of \u0015. Due to the op-\nposite Berry curvatures at the valley points KandK0,\nthese orbital moments \row in opposite directions, leading5\nto a large OHC (\u0019104~=e\n\u00001). The vanishingly small\nintrinsic SHC in these materials make them particularly\nsuitable for the direct observation of the OHE, which can\nbe measured by detecting the orbital torque generated by\nthe orbital Hall current [5]. Magneto-optical Kerr e\u000bect\nmay also be used to detect the orbital moments accumu-\nlated at the edges of the sample due to the OHE [23].\nFurthermore, the valley-orbital locking can be probed in\nphoton polarized angle-resolved photoemission measure-ments [24]. In addition, it may be possible to tune the\nOHC by applying a transverse electric \feld [25, 26]. Ex-\nperimental con\frmation of the OHE in the TMDC's may\nopen up new avenues for the realization of orbitronics de-\nvices.\nWe thank the U.S. Department of Energy, O\u000ece of\nBasic Energy Sciences, Division of Materials Sciences\nand Engineering for \fnancial support under Grant No.\nDEFG02-00ER45818.\n[1] T. Tanaka, H. Kontani, M. Naito, T. Naito, D. S. Hi-\nrashima, K. Yamada and J. Inoue, Intrinsic spin Hall\ne\u000bect and orbital Hall e\u000bect in 4d and 5d transition met-\nals, Phys. Rev. B 77, 165117 (2008).\n[2] H. Kontani, T. Tanaka, D. S. Hirashima, K. Yamada and\nJ. Inoue, Giant Intrinsic Spin and Orbital Hall E\u000bects\nin Sr 2MO4(M= Ru, Rh, Mo), Phys. Rev. Lett. 100,\n096601 (2008).\n[3] H. Kontani, T. Tanaka, D. S. Hirashima, K. Yamada and\nJ. Inoue, Giant Orbital Hall E\u000bect in Transition Metals:\nOrigin of Large Spin and Anomalous Hall E\u000bects, Phys.\nRev. Lett. 102, 016601 (2009).\n[4] D. Go, D. Jo, C. Kim, and H.-W. Lee, Intrinsic Spin and\nOrbital Hall E\u000bects from Orbital Texture, Phys. Rev.\nLett. 121, 086602 (2018).\n[5] D. Go and H.-W. Lee, Orbital Torque: Torque Gen-\neration by Orbital Current Injection, arXiv:1903.01085\n(2019).\n[6] D. Jo, D. Go and H.-W. Lee, Gigantic intrinsic orbital\nHall e\u000bects in weakly spin-orbit coupled metals, Phys.\nRev. B 98, 214405 (2018).\n[7] V. T. Phong, Z. Addison, S. Ahn, H. Min, R. Agarwal,\nand E. J. Mele, Optically Controlled Orbitronics on a\nTriangular Lattice, Phys. Rev. Lett. 123, 236403 (2019).\n[8] W. Feng, Y. Yao, W. Zhu, J. Zhou, W. Yao, and D.\nXiao, Intrinsic spin Hall e\u000bect in monolayers of group-VI\ndichalcogenides: A \frst-principles study, Phys. Rev. B\n86, 165108 (2012).\n[9] B. T. Zhou, K. Taguchi, Y. Kawaguchi, Y. Tanaka, and\nK.T. Law, Spin-orbit coupling induced valley Hall ef-\nfects in transition-metal dichalcogenides, Communica-\ntions Physics 2, 26 (2019).\n[10] D. Xiao, G.-B. Liu, W. Feng, X. Xu, and W. Yao, Cou-\npled Spin and Valley Physics in Monolayers of MoS 2and\nOther Group-VI Dichalcogenides, Phys. Rev. Lett. 108,\n196802 (2012).\n[11] D. Xiao, W. Yao, and Q. Niu, Valley-Contrasting Physics\nin Graphene: Magnetic Moment and Topological Trans-\nport, Phys. Rev. Lett. 99, 236809 (2007).\n[12] Z. Song, R. Quhe, S. Liu, Y. Li, J. Feng, Y. Yang, J.\nLu, and J. Yang, Tunable Valley Polarization and Val-\nley Orbital Magnetic Moment Hall E\u000bect in Honeycomb\nSystems with Broken Inversion Symmetry, Sc. Rep. 5,\n13906 (2015).\n[13] D. Xiao, J. Shi and Q. Niu, Berry Phase Correction to\nElectron Density of States in Solids, Phys. Rev. Lett. 95,\n137204 (2005); D. Xiao, M.-C. Chang, and Q. Niu, Berry\nphase e\u000bects on electronic properties, Rev. Mod. Phys.\n82, 1959 (2010).[14] P.-O. L owdin, A Note on the QuantumMechanical Per-\nturbation Theory, J. Chem. Phys. 19, 1396 (1951).\n[15] J. C. Slater and G. F. Koster, Simpli\fed LCAO Method\nfor the Periodic Potential Problem, Phys. Rev. 94, 1498\n(1954).\n[16] A. Korm\u0013 anyos, V. Z\u0013 olyomi, N. D. Drummond, P. Rakyta,\nG. Burkard, and V. I. Fal'ko, Monolayer MoS 2: Trigo-\nnal warping, the \u0000 valley, and spin-orbit coupling e\u000bects,\nPhys. Rev. B 88, 045416 (2013).\n[17] D. Ceresoli, T. Thonhauser, D. Vanderbilt, and R. Resta,\nOrbital magnetization in crystalline solids: Multi-band\ninsulators, Chern insulators, and metals, Phys. Rev. B\n74, 024408 (2006).\n[18] Giannozzi, P. et al. QUANTUM ESPRESSO: a modular\nand open-source software project for quantum simula-\ntions of materials, J. Phys. Condens. Matter 21, 395502\n(2009).\n[19] Marzari, N. & Vanderbilt, D. Maximally localized gen-\neralized Wannier functions for composite energy bands,\nPhys. Rev. B 56, 12847 (1997); I. Souza, N. Marzari and\nD. Vanderbilt, Maximally Localized Wannier Functions\nfor Entangled Energy Bands, Phys. Rev. B 65, 035109\n(2001).\n[20] M. G. Lopez, D. Vanderbilt, T. Thonhauser, and I. Souza\nWannier-based calculation of the orbital magnetization in\ncrystals, Phys. Rev. B 85, 014435 (2012); A. A. Mosto\f,\nJ. R. Yates, G. Pizzi, Y. S. Lee, I. Souza, D. Vanderbilt\nand N. Marzari, An updated version of Wannier90: A\nTool for Obtaining Maximally Localised Wannier Func-\ntions, Comput. Phys. Commun. 185, 2309 (2014).\n[21] Supplementary Materials describing the density func-\ntional methods and additional DFT results.\n[22] O. K. Andersen and T. Saha-Dasgupta, Mu\u000en-tin or-\nbitals of arbitrary order, Phys. Rev. B 62, R16219 (2000).\n[23] Y. K. Kato, R. C. Myers, A. C. Gossard, and D. D.\nAwschalom, Intrinsic Spin and Orbital Hall E\u000bects from\nOrbital Texture, Science 306, 1910 (2004).\n[24] S. R. Park, J. Han, C. Kim, Y. Y. Koh, C. Kim, H.\nLee, H. J. Choi, J. H. Han, K. D. Lee, N. J. Hur, M.\nArita, K. Shimada, H. Namatame, and M. Taniguchi,\nChiral Orbital-Angular Momentum in the Surface States\nof Bi 2Se3, Phys. Rev. Lett. 108, 046805 (2012).\n[25] S. Bhowal and S. Satpathy, Electric \feld tuning of the\nanomalous Hall e\u000bect at oxide interfaces, npj Computa-\ntional Materials 5(1), 61 (2019).\n[26] D. Go, J.-P. Hanke, P. M. Buhl, F. Freimuth, G.\nBihlmayer, H.-W. Lee, Y. Mokrousov and Stefan Bl ugel,\nToward surface orbitronics: giant orbital magnetism from\nthe orbital Rashba e\u000bect at the surface of spmetals, Sci.6\nRep.7, 46742 (2017)."    },    {        "title": "2105.04172v1.Self_Bound_Quantum_Droplet_with_Internal_Stripe_Structure_in_1D_Spin_Orbit_Coupled_Bose_Gas.pdf",        "content": "arXiv:2105.04172v1  [cond-mat.quant-gas]  10 May 2021Self-Bound Quantum Droplet with Internal Stripe Structure in 1D\nSpin-Orbit-Coupled Bose Gas\nYuncheng Xiong and Lan Yin∗\nPeking University\n(Dated: May 11, 2021)\nWe study the quantum-droplet state in a 3-dimensional (3D) B ose gas in the presence of 1D\nspin-orbit-coupling and Raman coupling, especially the st ripe phase with density modulation, by\nnumerically computing the ground state energy including th e mean-field energy and Lee-Huang-\nYang correction. In this droplet state, the stripe can exist in a wider range of Raman coupling,\ncompared with the BEC-gas state. More intriguingly, both sp in-orbit-coupling and Raman coupling\nstrengths can be used to tune the droplet density.\nPACS numbers: 03.75.Hh; 03.75.Mn; 05.30.Jp; 31.15.Md;\nIntroduction. Ultracold atoms have been excellent\nplatforms for investigating many-body quantum phe-\nnomena since the experimental realization of Bose-\nEinstein condensation(BEC) [ 1–3]. In most cases, Gross-\nPitaevskii (GP) equations, derived by minimizing mean-\nfield energy functional with respect to condensation\nwavefunction, provide a good description for the BEC\nstate of trapped Bose gases [ 4]. The next-order cor-\nrection to the gound state energy, i.e. the Lee-Huang-\nYang(LHY) energy, is usually negligible in the dilute\nlimit. However in a binary boson mixture, it was found\nthat when the attractive inter-species coupling constant\ng↑↓is a little larger in magnitude than the geometric av-\nerageofthe repulsiveintra-speciescoupling constants g↑↑\nandg↓↓, the repulsive LHY energy overtakes the attrac-\ntive MF ground-state energy, and the system becomes\na self-bound quantum droplet [ 5]. The quantum droplet\nwasfirstobservedindipolarBosegases[ 6–10]andlaterin\nbinary boson mixtures [ 11–13]. The self-binding mech-\nanism of a single-component dipolar Bose gas is simi-\nlar to that of a boson mixture except that the residual\nMF attraction arises from the counterbalance between\nattractive dipole-dipole interaction and repulsive contact\ninteraction. Theoretically, the quantum droplet has been\ninvestigated with various methods, including variational\nHNC-EL method [ 14], ab initial diffusion Monte-Carlo\n[15], and extended GPEs with the LHY correction in-\ncluded [16].\nOn the other hand, Raman-induced spin-orbit-\ncoupling(SOC)hasbeenrealizedexperimentallyinrecent\nyears both in bosonic [ 17,18] and fermionic [ 19,20] sys-\ntems. Alternative scheme of SOC which is immune from\nheating problem has been theoretically [ 21–23] and ex-\nperimentally [ 24] investigated. In a two-component Bose\ngas with a one-dimensional (1D) SOC, a stripe struc-\nture appears when the inter-species coupling constant\nis smaller than geometric average of intra-species cou-\npling constants below a critical Raman coupling (RC)\n[17,25–30]. In previous studies, the stripe state has\nbeen investigated in the BEC-gas region with repulsive\nMF ground-state energy. Recently, theoretically stud-ies [31,32] reveal that the stripe state can also exist in\nthe quantum droplet regime. In Ref. [ 31], the quantum\ndroplet was found in a two-dimensional Bose gas with\nvery weak SOC, where the LHY energy density was ap-\nproximated by that of a uniform system without SOC. In\nRef. [32] the LHY energy of a three-dimensional system\nwith SOC was calculated numerically, and its fitted form\nwasusedintheextended(GP)equation. Thephasetran-\nsition between a stripe gas and a stripe liquid was found\nby tuning coupling constants and RC. In their calcula-\ntion, the ultraviolet divergence in the expression of LHY\nenergy was removed by dimensional regularization. In\nthis work, we apply the standard regularization scheme\nto treat the ultraviolet divergence in the LHY energy of\na three-dimensional system with SOC. We found that\nthe droplet density can be easily tuned by RC and SOC,\neven to the zero limit. Compared to the case with a re-\npulsive inter-species interaction, in the quantum droplet\nthe stripe phase can exist in a bigger regime of RC and\nSOC.\nBoson mixture with SOC. Westudyatwo-component\nBose gas system with total particle number Nand vol-\numeV. In momentum space, its Hamiltonian is given\nby\nH=/summationdisplay\nk/summationdisplay\nρρ′ˆφ†\nρk/parenleftbigg(k−krexσz)2\n2+Ω\n2σx/parenrightbigg\nρρ′ˆφρ′k\n+1\n2V/summationdisplay\nk1,k2,q/summationdisplay\nρρ′gρρ′ˆφ†\nρk1+qˆφ†\nρ′k2−qˆφρ′k2ˆφρk1,(1)\nwhereˆφρkandˆφ†\nρkare the annihilation and creation op-\neratorsofthe ρ-component boson with the momentum k,\n{ρ,ρ′}={↑,↓},krand Ω are the strengths of SOC and\nRCrespectively, existhe unit vectorin x-directionwhich\nis the SOC direction. For convenience, we set ¯ hand the\nboson mass to be one. In this paper, we focus on the\nΩ<4Erregime where the lower excitation spectrum of\nthe single-particle Hamiltonian has two degenerate min-\nima [17,28]. For simplicity, the interactions are chosen\nto be symmetric g↑↑=g↓↓=g. The droplet regime is\nset by the condition g↑↓<∼−g[5].2\nTo implement Bogoliubov approximation to obtain ex-\ncitation spectra, and thereby LHY correction, we need\nto know ground state(GS) wavefunction. To this end, we\ndetermine GS by variationally minimizing MF energy.\nWe choose GS ansatz to be superposition of plane waves\n[27,28],\nφ(r) =/parenleftbiggφ↑\nφ↓/parenrightbigg\n=/radicalbigg\nN0\nV/summationdisplay\nm/parenleftbiggφ↑m\n−φ↓m/parenrightbigg\neimk1·r,(2)\nwhereN0is the particle number of the condensate, k1≡\n(γkr,0,0),γis a variational parameter to be determined.\nIn the lowest order, only m=±1 components corre-\nsponding to the two minima of the lower single-particle\nspectrum are relevent [ 27]. However, the periodic stripes\ninduced by the condensation of ±k1will lead to the cou-\nplings between the momenta differing from each other by\nreciprocal lattice vectors. Therefore, it is necessary to\ninclude all the components with momenta K±k1in the\nhigher order approximations [ 28], where K= 2sk1withs= 0,±1,±2,..., arereciprocallattice vectors. In short,\nthe summation is over all the odd integer m= 2s±1 in\nthe region −C1≤m≤C1where the cutoff C1is a posi-\ntive odd number. The normalization relation is given by/summationtext\nρ,m|φρm|2= 1.\nIn the BEC state, following the Bogoliubov prescrip-\ntion, we replace ˆφ(†)\nρmk1by√N0φ(∗)\nρm+ˆφ(†)\nρmk1and keep\nterms up to the quadratic order. The first-order terms\nvanish due to the minimization of MF energy. The\nnumber of atoms in the condensation is given by N0=\nN−/summationtext′′\nρ,kˆφ†\nρkˆφρkwhich can be used to rewrite N0in\nterms of the total atom number N. The MF energy\nper particle εMFand the Bogoliubov Hamiltonian HBare\ngiven by\nεMF=/summationdisplay\nm/summationdisplay\nρρ′/parenleftbiggk2\nr\n2(mγ−σz)2−Ω\n2σx/parenrightbigg\nρρ′φ∗\nρmφρ′m\n+/summationdisplay\nm+l=i+j/summationdisplay\nρρ′gρρ′n\n2φ∗\nρmφ∗\nρ′lφρ′iφρj,(3)\nHB=EMF+/summationdisplay\nρρ′/summationdisplay\nk/parenleftbigg(k−krexσz)2\n2+Ω\n2σx−µˆI/parenrightbigg\nρρ′ˆφ†\nρkˆφρ′k\n+/summationdisplay\nρ,q/summationdisplay\nm+l=α+βgρρl\n2/bracketleftbigg\n2φ∗\nρmφ∗\nρlˆφραk1−qˆφρβk1+q+2φ∗\nρmφρ−l(ˆφ†\nρ−αk1+qˆφρβk1+q+ˆφ†\nρ−αk1−qˆφρβk1−q)+H.c./bracketrightbigg\n+/summationdisplay\nρ/negationslash=ρ′,q/summationdisplay\nm+l=α+βgρρ′l\n2/bracketleftbigg\n(−φ∗\nρmφ∗\nρ′l)(ˆφρ′αk1−qˆφρβk1+q+ˆφρ′αk1+qˆφρβk1−q)+(−φ∗\nρmφρ′−l)×\n(ˆφ†\nρ′−αk1+qˆφρβk1+q+ˆφ†\nρ′−αk1−qˆφρβk1−q)+φ∗\nρmφρ−l(ˆφ†\nρ′−αk1+qˆφρ′βk1+q+ˆφ†\nρ′−αk1−qˆφρ′βk1−q)+H.c./bracketrightbigg\n,(4)\nwherem,l,i,j,α,βare all odd integers with\n−C1≤m,n,i,j ≤C1and−C2≤α,β≤C2,C2\nis another cutoff necessary for numerical diagonaliza-\ntion of Bogoliubov Hamiltonian, qxis in the first Bril-\nlioun zone, 0 < qx< k1,nis total particle density,\nandµ=/summationtext\nm/summationtext\nρ,ρ′/parenleftbigg\nk2\nr\n2(mγ−σz)2−Ω\n2σx/parenrightbigg\nρρ′φ∗\nρmφρ′m+\n/summationtext\nm+l=i+j/summationtext\nρ,ρ′gρρ′nφ∗\nρmφ∗\nρ′lφρ′iφρjis, in nature, the MF\nchemical potential, satisfying µ=∂EMF/∂N.\nIn Hamiltonian Eq.( 4) there are not only terms equiv-\nalent to periodic potentials, but also off-diagonal terms\nsuch as ˆφραk1+qˆφραk1−q. The quasiparticle spectra are\ncharacterized by band index and quasimomentum.\nWe define a column operator\nˆAq≡/parenleftbig\n···,ˆφ†\n↑αk1−q,ˆφ†\n↓αk1−q,ˆφ↑αk1+q,ˆφ↓αk1+q,···/parenrightbigT\nwithα=±1,±3,...,±C2, and rewrite Eq.( 4) in a com-pact form\nHB=EMF+E1+/summationdisplay\nqˆA†\nqHqˆAq,\nwhere\nE1=−/summationdisplay\nm,q,±/bracketleftbigg(mk1−q±kr)2\n2−µ+gn+g↑↓n\n2/bracketrightbigg\n.(5)\nThematrix Hqcanbe obtainedfromEq.( 4)andsubse-\nquentlydiagonalizedtoobtainquasiparticlespectra. The\ndiagonalized Bogoliubov Hamiltonian is given by\nHB=EMF+E1+/summationdisplay\nα,q(E↑−\nα(q)+E↓−\nα(q))\n+/summationdisplay\nα,q,±/summationdisplay\nρEρ±\nα(q)ˆ˜φ†\nραk1±qˆ˜φραk1±q, (6)\nwhere/parenleftbig\n···,ˆ˜φ†\n↑αk1−q,ˆ˜φ†\n↓αk1−q,ˆ˜φ↑αk1+q,ˆ˜φ↓αk1+q,···/parenrightbigT=\nMqˆAq,Mqthe Bogoliubov transformation matrix sat-\nisfyingMqΣM†\nq= Σ, and Σ is a diagonal matrix with3\neveryfourdiagonalmatrixelementsgivenby −1,−1,1,1.\nThe quasi-particle energy Eρ±\nα(q) can be solved from\nthe generalized secular equation |Hq−λΣ|= 0.\nBefore we write down the expression of LHY en-\nergy, we need to rewrite gρρ′in terms of scattering\nlengthaρρ′through regularization relation gρρ′=Uρρ′+\n(U2\nρρ′/V)/summationtext\nk1/k2[33] whereUρρ′= 4πaρρ′. The LHY\nenergy is therefore given by\nELHY=E1+/summationdisplay\nα,q/bracketleftbigg\n(E↑−\nα(q)+E↓−\nα(q))+/parenleftbigg/summationdisplay\nm+l=i+j\n/summationdisplay\nρ,ρ′(Uρρ′n)2\n2φ∗\nρmφ∗\nρ′lφρ′iφρj/parenrightbigg/parenleftbigg/summationdisplay\n±1\n(αk1±q)2/parenrightbigg/bracketrightbigg\n,(7)\nwherethe summation convergesquicklyfor largemomen-\ntum due to regularization. In contrast, the divergence of\nLHY energy was removed by dimensional regularization\nin Ref. [32].\nSelf-bound quantum droplet with stripe With the ex-\nplicit expressions of MF energy Eq.( 3) and LHY cor-\nrection Eq.( 7), we are ready to investigate the inter-\nplay between SOC, RC and interactions in the forma-\ntion of droplet. We take Er≡k2\nr/2 andkras en-\nergy and momentum units respectively. The dimension-\nless version of MF and LHY energies are given in ap-\npendix. In the following numerical calculations, we use\nthe parameters a↑↑=a↓↓≡a= 89.08a0,a↑↓=−1.1a\nwherea0is the Bohr radius. Correspondingly, we de-\nfineU≡4πa=U↑↑=U↓↓. And before the effect\nof SOC is considered, the recoil momentum is fixed at\nkr= 2π×106m−1(or equivalently akr≈0.0296).\nWhen implementing the numerical calculations, we\nintroduce two cutoffs: C1is for the ground-state\nansatz, Eq.( 2), andC2is for the diagonalization of the\notherwise infinite-dimensional Bogoliubov Hamiltonian,\nEq.(4). We have numerically verified the convergence of\nwavefunctionsandLHYenergies,andfindthatthechoice\nofC1= 9 and C2= 39 can produce sufficiently accurate\nresults. The condensate fraction at m= 9 is about 10−14\nof the total density. In our calculations, the two charac-\nteristic momenta,√gnand√\nΩ, are at most of the same\norder of the recoil momentum kr. The momentum cut-\noffC2kris much larger than any of them. Therefore,\nthroughout our calculations, we set C1= 9 and C2= 39.\nWe first minimize the mean-field energy at fixed to-\ntal density nto determine variational parameters φρm\nandγ. It shows that, at low densities, the mean-field\nenergy per particle εMFshows linear dependence on den-\nsity, and can be fitted by εMF=c0+c1(Un/E r). The\ndensity-independent background energy appears due to\nRaman energy in Eq.( 3), and thus c0depends strongly\non the strength of RC, while the proportionality coef-\nficientc1, as shown in Fig. 1, weakly relies on the RC\nstrength in the considered regime. Both c0andc1are\nirrelevant to the strength of SOC, kr, since the MF en-\nergy has been rescaled in the unit of Er. Moreover, c1isnegative throughout the considered regime, which indi-\ncates the tendency to collapse in the mean field and thus\nhigher-order correction is necessary to stabilize such a\nsystem.\nIn the low density region, the mean-field density dis-\ntribution exhibits stripes as in the low RC limit in re-\npulsive BEC-gas [ 27]. As has been studied [ 17,27,29],\nin experiments on87Rb atoms, stripe phase exists only\nfor very small Ω ( <∼0.2Er), and stripes cannot be de-\ntected directly in the absoption imaging. In contrast,\nthe stripe phase of the quantum droplet can survive in\na much larger range of RC, of the order of several Er.\nThis result can be obtained in the variational theory [ 27]\nof a Bose gas with SOC, where the stripe phase and the\nplane-wave phase can all be described. In the low den-\nsity limit with strong SOC, there is a transition between\nthese two phases at a critical RC, ΩI-II= 4Er/radicalBig\n2γ\n1+2γ\nwhereγ= (g−g↑↓)/(g+g↑↓) (see Eq.(12) in [ 27]). Con-\nsequently, we can reach a conclusion that in a quantum\ndroplet with Ω <4Erand strong SOC, stripe phase is\nfavored. Compared to the BEC case with repulsive iner-\nspecies interaction where ΩI-II<∼0.2Er, one can draw\nthe conclusion that it is the strong inter-species attrac-\ntion that significantly enlarges the region of stripe phase.\nWe compute the LHY energy by solving excitation\nspectra and numerically performing the integration in\nEq.(7). As in the case without SOC [ 5], the lowest exci-\ntation spectrum at qx≈0 and 2krhas small imaginary\npart. When performing the numerical integration, we\nkeep all the real and imaginary contributions in the exci-\ntation energies. Even though the resulting LHY energy\nis complex, the imaginary part is at least three orders\nsmaller in magnitude than the real part in the parameter\nregion that we are considering, i.e. with low density and\nstrong SOC. Consequently, the imaginary parts can be\nsafely omitted in the ensuing calculations as in the case\nwithout SOC [ 5].\nNotice that LHY energy is a function of three di-\nmensionless parameters, Un/E r, Ω/Erandakr(see\nEq.(A.9)) whereas the MF energy depends only on the\nfirst two parameters as can be seen in Eq.( A.8). In the\ndilute limit, in terms of Un/E r, the LHY energy per par-\nticle can be fitted by the formula εLHY=c2(Un/E r) +\nc3(Un/E r)3/2, wherec2andc3arepositivefittingparam-\neters. Compared to the case without SOC, in addition\nto the usual term proportional to n3/2, a linear repulsive\nterm appears in the LHY energy which will be discussed\nlater.\nThe coefficient c2, as shown in Fig. 1, has a roughly\nquadratic dependence on RC strength while c3remains\nconstant in the range of 0 <Ω<3Er, in agreement with\nthe preceding work [ 32]. Also, such a behavior occurs in\nthe Rabi-coupled case [ 34] where the coefficient of usual\nn3/2term in LHY energy is free of Ω. For Ω >3Er,\nonly low density region can be sampled, and while the4\n0 1 2 3 40.000.010.020.030.04\nΩ/Er|c1|\nc2\nc3\nFIG. 1. Three fitting coefficients c1,c2andc3in MF en-\nergy and LHY energy vs the strength of Raman coupling Ω\natakr≈0.0296.c1is negative and has been shown by its\nmagnitude for convenient comparison with c2.\nfittedc1andc2remains reliable, the value of c3, which\ndetermines the behavior of the LHY energy in the higher\ndensity region, can not be trusted due to the deficiency\nof sampling.\n0.001 0.010 0.100 1-0.008-0.006-0.004-0.0020.0000.002\nUn/Er\n\u0001/Er\n\u0002=1/2\u0000=3/2\u0003=5/2\u0004=7/2\nFIG. 2. Sampled points (dots) and fitting functions (solid\nlines) of total energy per particle ε=εMF+εLHYfor several\nRaman coupling strengths at akr≈0.0296. The constant\nenergy background from the MF energy has been subtracted\nfor comparison.\nIncluding both MF and LHY energies, the total en-\nergy per particle in the droplet regime is shown in Fig. 2.\nNear the collapse point of the MF energy, contrary to\nthe monotonously decreasing tendency of the MF energy\nwith density, the total energy has a minimum, because\nthe repulsive LHY energy overcomes the attractive MF\nenergy at larger densities. As discussed above, the MF\nwavefunction shows density modulation. Therefore, in\nthe dropletregime, the self-bound stripe phase exists and\ncan survive for even larger Raman coupling compared to\nthe BEC gas regime [ 17,27,29].\nAlthough the general analytic total energy per particle0.050.100.150.200.250.301017101810191020\nakrnd/m-3\nΩ=1/2\nΩ=3/2\nΩ=5/2\nFIG. 3. Droplet density ndversus SOC strength krfor several\nRC strengths. With RC fixed, the stronger the SOC is, the\nsmaller the equilibrium density of droplet becomes. When\nSOC is strong enough, for example at akr≈0.32 for Ω =\n1/2Er, the droplet state disappears and the system expands\nwithout an external trap.\nis inaccessible in the presence of SOC and RC, it can be\napproximatedbyaddingtogetherthe fitted MF andLHY\nenergies, ε=c0+(c2−|c1|)(Un/E r)+c3(Un/E r)3/2with\nc0,1,2,3all fitted numerically. The equilibrium density of\nself-bound droplet can obtained by solving zero-pressure\ncondition, i.e. P=∂ε/∂V= 0, yielding Und/Er=\n4(|c1|−c2)2/9c2\n3for|c1|−c2≥0.\nWe now discuss the role played by RC strength on\ndroplet formation. The dependence of droplet on RC\nis similar to the case in uniform Rabi-coupled binary\nmixture which has been reported in 3D [ 34] and lower\ndimensions [ 35]. The similarity stems from the gapped\nsingle-particlespectrum. Without RC,thesingle-particle\nspectra of both components are gapless. The finite RC\ninduces coupling between the two components leading to\nthe new quasi-particle spectra, with the lower one gap-\nlessandthe higheronegapped. Due tothe gappedmode,\nLHY correction per particle acquires a positive term lin-\near in density nin addition to the n3/2term, as men-\ntioned above.\nAs Ω increases, the rapid increase of c2results in the\ndecreasing of |c1| −c2in magnitude, which is mani-\nfest in Fig. 1, withc3remaining almost constant, mak-\ning it easier to counterbalance the attractive MF energy.\nThus the equilibrium density of droplet is smaller for\nlarger Ω as shown in Fig. 2. A critical point is reached\nat|c1|=c2, as shown in Fig. 1where Ω c≈3.5Erat\nakr≈0.0296. Above this point, the total energy in-\ncreases monotonously with density since both c2− |c1|\nandc3are positive, and thereby no self-bound droplet\ncan exist. A similar droplet-gas transition has been re-\nported theoretically in Rabi-coupled binary mixture in\nboth 3D [ 34] and lower dimensions [ 35].\nItiseasiertoconsiderthedependenceonSOCstrength\naskr(Er≡k2\nr/2) serves as the momentum (energy)\nunit. From the dimensionless expression of MF and LHY\nenergies Eq.( A.8) and Eq.( A.9), it’s easy to see that the\nMF energy only depends on Un/E rand Ω/Er, and so5\ndoes the excitation energy Eρ±\nα(q). With fixed reduced\ninteraction Un/E rand Raman coupling Ω /Er, the LHY\nenergy is proportional to kr, and as a consequence, the\nfitting parameters in LHY energy, c2andc3are both\nlinearly proportional to kr. Since c1is independent of\nkr, increasing SOC strength krhas the same effect as\nincreasing RC strength Ω. Although the energy unit Er\nis also increased in the same process, the overall effect of\nincreasing SOC strength is, as shown in Fig. 3, decreasing\nequilibrium density of droplet. And finally above some\nspecific value, no droplet can exist any more.\nConclusion and Discussion In current experiment on\n39K, the droplet state has been realized in the mixture of\nhyperfine states |1,−1/an}bracketri}htand|1,0/an}bracketri}htby tunning scattering\nlengths [5,11–13], but the artificial SOC has not been re-\nalizedinthissystem. Incontrast,in87Rbsystems[ 17,18]\nthe SOC has been realized and the stripe state has been\nobserved, but tunning the interactions in this system has\nnot been achieved. Our results could be tested experi-\nmentallyiftheinteractionscanbetunedandtheartificialSOC can be generated in the same system.\nIn conclusion, we have studied the quantum droplet\nstate of a uniform binary Bose gas in the presence of 1D\nspin-orbit-coupling and Raman coupling, and find that\ngroundstate can display density modulation of the stripe\nphase in the low Ω regime [ 27,28]. The density modula-\ntion can survive for much larger Ω than in the BEC gas\nstate with inter-species interaction. Compared to the\ncase without SOC, the droplet density can be tuned by\nchanging the strength of SOC and RC. With the increase\nof SOC and RC, the droplet density can be reduced by\nseveral orders of magnitude, and eventually to the zero\nlimit at a critical kror Ω. We plan to study the finite-\nsize effect of the quantum droplet with SOC in the future\nwork.\nAppendix\nDimensionless MF and LHY energies per particle are\ngiven by\nEMF/(NEr) =/summationdisplay\nm/summationdisplay\nρρ′/parenleftbigg\n(mγ−σz)2−Ω\n2σx/parenrightbigg\nρρ′φ∗\nρmφρ′m+/summationdisplay\nm+l=i+j/summationdisplay\nρρ′Uρρ′n\n2φ∗\nρmφ∗\nρ′lφρ′iφρj,(A.8)\nELHY/(NEr) = (akr)1\nπ2(Un)/summationdisplay\nα/integraldisplayγ\n0/integraldisplayγC2\n−γC2/integraldisplayγC2\n−γC2dqxdqydqz/bracketleftbigg\n(E↑−\nα(q)+E↓−\nα(q))−/summationdisplay\n±/parenleftbigg\n(αγ−qx±1)2+q2\ny\n+q2\nz−µ+Un+U↑↓n\n2/parenrightbigg\n+/parenleftbigg/summationdisplay\nm+l=i+j/summationdisplay\nρρ′Uρρ′n\n2φ∗\nρmφ∗\nρ′lφρ′iφρj/parenrightbigg/parenleftbigg1/2\n(αγ±qx)2+q2y+q2z/parenrightbigg/bracketrightbigg\n, (A.9)\nwhere Ω, Uρρ′nand excitation spectra Eρ−\nαare in the\nunit ofEr.\n∗yinlan@pku.edu.cn\n[1] M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E.\nWieman, and E. A. Cornell, Science269, 198 (1995) .\n[2] C. C. Bradley, C. A. Sackett, J. J. Tollett, and R. G.\nHulet,Phys. Rev. Lett. 75, 1687 (1995) .\n[3] K. B. Davis, M. O. Mewes, M. R. Andrews, N. J. van\nDruten, D. S. Durfee, D. M. Kurn, and W. Ketterle,\nPhys. Rev. Lett. 75, 3969 (1995) .\n[4] F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S.Stringa ri,\nRev. Mod. Phys. 71, 463 (1999) .\n[5] D. S. Petrov, Phys. Rev. Lett. 115, 155302 (2015) .\n[6] H. Kadau, M. Schmitt, M. Wenzel, C. Wink, T. Maier,\nI. Ferrier-Barbut, and T. Pfau, Nature530, 194 (2016) .\n[7] I. Ferrier-Barbut, H. Kadau, M. Schmitt, M. Wenzel,\nand T. Pfau, Phys. Rev. Lett. 116, 215301 (2016) .\n[8] L. Chomaz, S. Baier, D. Petter, M. J. Mark, F. W¨ achtler,L. Santos, and F. Ferlaino, Phys. Rev. X 6, 041039\n(2016).\n[9] M. Schmitt, M. Wenzel, F. Bottcher, I. Ferrier-Barbut,\nand T. Pfau, Nature539, 259 (2016) .\n[10] A. Trautmann, P. Ilzh¨ ofer, G. Durastante, C. Politi,\nM. Sohmen, M. J. Mark, and F. Ferlaino, Phys. Rev.\nLett.121, 213601 (2018) .\n[11] C. R. Cabrera, L. Tanzi, J. Sanz, B. Naylor, P. Thomas,\nP. Cheiney, and L. Tarruell, Science359, 301 (2018) .\n[12] P. Cheiney, C. R. Cabrera, J. Sanz, B. Naylor, L. Tanzi,\nand L. Tarruell, Phys. Rev. Lett. 120, 135301 (2018) .\n[13] G. Semeghini, G. Ferioli, L. Masi, C. Mazzinghi, L. Wol-\nswijk, F. Minardi, M. Modugno, G. Modugno, M. In-\nguscio, and M. Fattori, Phys. Rev. Lett. 120, 235301\n(2018).\n[14] C. Staudinger, F. Mazzanti, andR.E. Zillich, Phys. Rev.\nA98, 023633 (2018) .\n[15] V. Cikojevi´ c, K. Dˇ zelalija, P. Stipanovi´ c, L. Vranj eˇ s\nMarki´ c, and J. Boronat, Phys.Rev. B 97, 140502 (2018) .\n[16] F. W¨ achtler and L. Santos, Phys. Rev. A 93, 061603\n(2016).\n[17] Y. J. Lin, K.Jim´ enez-Garc´ ıa, andI.B. Spielman, Nature6\n471, 83 (2011) .\n[18] J.-Y. Zhang, S.-C. Ji, Z. Chen, L. Zhang, Z.-D. Du,\nB.Yan, G.-S.Pan, B.Zhao, Y.-J.Deng, H.Zhai, S.Chen,\nand J.-W. Pan, Phys. Rev. Lett. 109, 115301 (2012) .\n[19] L. W. Cheuk, A. T. Sommer, Z. Hadzibabic, T. Yefsah,\nW. S. Bakr, and M. W. Zwierlein, Phys. Rev. Lett. 109,\n095302 (2012) .\n[20] P. Wang, Z.-Q. Yu, Z. Fu, J. Miao, L. Huang, S. Chai,\nH. Zhai, and J. Zhang, Phys. Rev. Lett. 109, 095301\n(2012).\n[21] Y. Zhang, G. Chen, and C. Zhang, Sci. Rep. 3, 1937\n(2013).\n[22] Z.-F. Xu, L. You, and M. Ueda, Phys. Rev. A 87, 063634\n(2013).\n[23] B. M. Anderson, I. B. Spielman, and G. Juzeli¯ unas,\nPhys. Rev. Lett. 111, 125301 (2013) .\n[24] X. Luo, L. Wu, J. Chen, Q. Guan, K. Gao, Z. F. Xu,\nL. You, and R. Wang, Sci. Rep. 6, 18983 (2016) .\n[25] C. Wang, C. Gao, C.-M. Jian, and H. Zhai, Phys. Rev.\nLett.105, 160403 (2010) .\n[26] T.-L. Ho and S. Zhang, Phys. Rev. Lett. 107, 150403\n(2011).[27] Y. Li, L. P. Pitaevskii, and S. Stringari, Phys. Rev. Lett.\n108, 225301 (2012) .\n[28] Y. Li, G. I. Martone, L. P. Pitaevskii, and S. Stringari,\nPhys. Rev. Lett. 110, 235302 (2013) .\n[29] S.-C. Ji, J.-Y. Zhang, L. Zhang, Z.-D. Du, W. Zheng,\nY.-J. Deng, H. Zhai, S. Chen, and J.-W. Pan, Nature\nPhysics10, 314 (2014) .\n[30] J.-R. Li, J. Lee, W. Huang, S. Burchesky, B. Shteynas,\nF. a. Top, A. O. Jamison, and W. Ketterle, Nature543,\n91 (2017) .\n[31] R. Sachdeva, M. N. Tengstrand, and S. M. Reimann,\nPhys. Rev. A 102, 043304 (2020) .\n[32] J. S´ anchez-Baena, J. Boronat, and F. Mazzanti, Phys.\nRev. A102, 053308 (2020) .\n[33] L. Pitaevskii and S. Stringari, Bose-Einstein condensa-\ntion and superfluidity , Vol. 164 (Oxford University Press,\n2016).\n[34] A. Cappellaro, T. Macri, G. F. Bertacco, and L. Salas-\nnich,Sci. Rep. 7, 13358 (2017) .\n[35] E. Chiquillo, Phys. Rev. A 99, 051601 (2019) ."    },    {        "title": "2203.12263v1.Effects_of_a_single_impurity_in_a_Luttinger_liquid_with_spin_orbit_coupling.pdf",        "content": "E\u000bects of a single impurity in a Luttinger liquid with spin-orbit coupling\nM. S. Bahovadinov1, 3and S. I. Matveenko2, 3\n1Physics Department, National Research University Higher School of Economics, Moscow, 101000, Russia\n2L. D. Landau Institute for Theoretical Physics, Chernogolovka, Moscow region 142432, Russia\n3Russian Quantum Center, Skolkovo, Moscow 143025, Russia\n(Dated: March 24, 2022)\nIn quasi-1D conducting nanowires spin-orbit coupling destructs spin-charge separation, intrinsic\nto Tomonaga-Luttinger liquid (TLL). We study renormalization of a single scattering impurity\nin a such liquid. Performing bosonization of low-energy excitations and exploiting perturbative\nrenormalization analysis we extend the phase portrait in K\u001b\u0000K\u001aspace, obtained previously for\nTLL with decoupled spin-charge channels.\nI. INTRODUCTION\nLow-energy excitations of the one-dimensional con-\nductors have collective phonon-like character, formalized\nwithin the Tomonaga-Luttinger liquid (TLL) theory [1{\n4]. The TLL formalism correctly predicts algebraic decay\nof single-particle correlations with exponents depending\non the strength of the (short-range) electron-electron (e-\ne) interactions. Due to the collective nature of these ex-\ncitations the interparticle interactions drastically change\nthe low-energy physics leading to fractionalization of con-\nstituent carriers [5{7], vanishing density of states and\npower law singularity of the zero-temperature momen-\ntum distribution at the Fermi level [8{10]. Another sur-\nprising prediction of the TLL theory is related to the\ne\u000bects of a single scattering impurity on transport prop-\nerties at low temperatures, presented in prominent se-\nries of works in Refs. [11{13]. Perturbative renormaliza-\ntion group (RG) studies have shown that if the electron-\nelectron interactions of the single-channel TLL is attrac-\ntive (with the TLL parameter K > 1), impurity is irrele-\nvant in the RG sense due to the pinned superconducting\n\ructuations. On the other hand, if the e-e interactions\nhave a repulsive character ( K < 1), the backscattering\ne\u000bects are relevant and grow upon the RG integration\nof short-distant degrees of freedom. At T= 0 the sys-\ntem is e\u000bectively decoupled into two disjoint TLLs for\narbitrarily weak impurity potentials. The latter implies\na metal-insulator transition with vanishing two-terminal\ndc-conductance G= 0 atT= 0, whereas at \fnite\ntemperatures it exhibits power-law dependence on T, i.e\nG\u0018T2=K\u00002. These results are in sharp constrast with\nthe result in the non-interacting limit ( K= 1), where\nthe impurity is marginal in the RG sense (in all orders of\nRG) and the transmission coe\u000ecient Tdepends on the\nimpurity strength with Landauer conductance G=Te2\nh.\nExperimental con\frmation of these e\u000bects was performed\nrecently, using remarkable quantum simulator based on\na hybrid circuit [14].\nIn TLLs composed of spin-1/2 electrons [15{18] spin\nand charge degrees of freedom are decoupled [19{23], in\ncontrast to higher dimensional counterparts. Each chan-\nnel has individual bosonic excitations with the proper\nvelocity and carry the corresponding quantum number\n0.5 2\nKS 0.52KC\n 0.5 2\nKS 0.52KCweak barrier strong barrier (a) (b)\nIIII\nIIIV\nIIIII\nIIVFIG. 1. Phase portrait at T= 0 for TLL with separated spin\nand charge channels. Results are obtained from perturbative\nRG studies of (a) a weak scattering barrier or (b) a weak link.\nIn region I the backscattering of carriers from a weak barrier\n(tunneling through a link) is relevant (irrelavant) and \rows\nto the strong-coupling (to the weak-coupling) \fxed point in\nboth channels. The opposite occurs in the regions IV. In the\nremained regions the mixed phase is realized, where one of\nthe channels is insulating and the other is conducting.\nseparately. However, this decoupling of modes is not\ncharacteristic for all TLLs. Particularly, the earlier stud-\nies have shown that spin-charge coupling (SCC) occurs\nin the TLL subjected to a strong Zeeman \feld [24] or\nwith strong spin-orbit interactions [25{27]. The latter is\ntypically expected in nanowires, where electrons in trans-\nverse directions are con\fned, whereas in the other direc-\ntion they move freely. Spin-orbit coupling (SOC) in these\nsystems plays an important role on the realization of spin-\ntronic devices [28, 29]. Thus, an interplay of (large) SOC\nand (large) e-e interactions in the TLL regime is an in-\nteresting question and has gained both theoretical and\nexperimental attention in recent years [30{32].\nImpurity e\u000bects in conventional TLLs with decoupled\nchannels were studied in the original works [12, 13], and\ncan be summarized by phase portrait at T= 0, presented\nin Fig. 1. RG studies imply that electrons scattering on a\nweak potential barrier [Fig. 1(a)] are fully re\rected when\nKc+Ks<2 (region I). In region II (III) charge (spin)\nquanta are fully re\rected, while the spin (charge) ones\nare transmitted. The backscattering terms scale to thearXiv:2203.12263v1  [cond-mat.str-el]  23 Mar 20222\nξ(k) ξ(k)\nkυ1\nkF,1υ2\n-kF,2-kF,1\nkF,2υ1 υ2υ\nυ\nkF (a) Polarized TLL \n-kF (b) TLL with SOC\nkυ\nυ\nFIG. 2. Linearized electronic excitation spectrum corre-\nsponding to the (a) TLL in a strong Zeeman \feld and (b)\nTLL with SOC. In both models spin-charge separation is vi-\nolated by the mixing terms given in Eq. (11) and Eq. (30).\nweak-coupling \fxed point in region IV, so electrons fully\ntransmit through the barrier. The same physical picture\nis obtained from the perturbative analysis of the tun-\nneling events between disconnected wires [Fig. 1(b)]. In\nregion I (1\nKc+1\nKs<2;Kc<2;Ks<2) all hopping events\nare irrelevant, so the system renormalizes into two dis-\nconnected wires. In region II (region III) only tunneling\nof spin (charge) is relevant. Numerical con\frmation of\nthese results was presented using path-integral Monte-\nCarlo methods [33].\nThe modi\fcation of the phase portrait in the presence\nof SCC has not yet been considered, although there are\nseveral studies which partially addressed this question in\ndi\u000berent aspects [34, 35]. Particularly, impurity e\u000bects\non transport properties in the case of the SCC caused by\nSOC were recently studied [31, 32]. As our main moti-\nvation of this work we consider modi\fcation of the phase\nportrait in the presence of SOC. We also show that the\nspin-\fltering e\u000bect conjectured for this model in Ref. [35],\nis not exhibited.\nThe paper is organized as follows: In Section II we\npresent the model of our study and set the necessary for-\nmalism and conventions. We next present a generalized\napproach to tackle the impurity problem for a \fnite SOC\nin Section III. In Section IV we present our main results\nand discussions. Concluding remarks are given in Section\nV.\nII. MODEL AND METHODS\nIn this section we set up our used conventions and ter-\nminology and present the model of our study. We ap-\nproach the problem using the standard Abelian bosoniza-\ntion technique with the consequent perturbative RG\nanalysis of impurity terms.\nBosonization of low-lying modes relies on the assump-\ntion of linear electronic spectrum with the corresponding\nright (R) and left (L) movers for carriers with both spin\nprojections (see Fig. 2). Fermionic \feld operators foreach branch and spin component can be expressed via\nthe bosonic displacement and phase \felds:\n \u001b;\u0016=^F\u001bp\n2\u0019aei\u0016kF;\u001bxei\u0016p\u0019(\u001e\u001b(x)\u0000\u0016\u0012\u001b(x))(1)\nwith\u001b2f\"\u0011 +1;#\u0011\u0000 1gand\u00162fR\u0011+1;L\u0011\u00001g.\nThe dual \felds satisfy the commutation relations:\n[\u001e\u001b(x);\u0005(x0)\u001b0] =i\u000e\u001b;\u001b0\u000e(x\u0000x0); (2)\nwhere \u0005\u001b(x) =r\u0012\u001b(x) is a conjugate to \u001e\u001b(x) momen-\ntum. The UV cuto\u000b aof the theory is on the order\nof inter-atomic distances. As in the standard litera-\nture, the Klein factors ^F\u001bguarantee anti-commutation\nof fermionic \felds with di\u000berent spin orientations.\nThe second exponent can also be expressed in terms of\nconstituent bosonic ladder operators in momentum space\n(lis the length of the system)\ni\u0016p\u0019(\u001e\u001b(x)\u0000\u0016\u0012\u001b(x)) =X\n\u0016q>0Aq\u0000\nbq;\u001beiqx\u0000by\nq;\u001be\u0000iqx\u0001\n(3)\nwithAq=q\n2\u0019\nljqje\u0000ajqj=2and [bq;by\nq0] =\u000eq;q0,q=2\u0019\nl.\nWe hereafter use the spin and the charge basis as the\ncanonical basis, i.e we work with the \felds \u001ec;s=\u001e\"\u0006\u001e#p\n2\nand \u0005c;s=\u0005\"\u0006\u0005#p\n2, which also satisfy Eq. (2). Hereafter,\nthe subbscripts \"s\" and \"c\" stand for spin and charge\ndegrees of freedom. The \ructuations of the spin and the\ncharge density are given by,\n\u001as(x) =r\n2\n\u0019X\n\u001b\u001b@x\u001e\u001b (4)\nand\n\u001ac(x) =r\n2\n\u0019X\n\u001b@x\u001e\u001b: (5)\nTo get similar expressions for the current, one uses the\ntransformation @x\u001e\u001b!@x\u0012\u001b. In the following, we imply\nnormal ordering with respect to Dirac sea, whenever it is\nneeded and avoid : () : symbol.\nModel\nThe important e\u000bect of SOC in quantum nanowires is\nband distortion, which is usually considered within the\ntwo-band model [36]. This distortion causes the veloc-\nity di\u000berence \u0001 = v1\u0000v2, pronounced in Fig. 2(b) in\nthe approximation of linearized spectrum. Initially pro-\nposed in Refs.[25, 26], the model can be realized by tun-\ning chemical potential, \flling only the lowest subband.\nIt should be noted that the spin orientation of carriers\nmoving in the same direction can be tuned also from the\nparallel [37] to anti-parallel [25] by band-\flling [32]. In3\nthis work we consider anti-parallel spin orientation, as\nshown in Fig. 2(b).\nThe non-interacting Hamiltonian of the model with lin-\nearized excitation spectrum is\nH0=\u0000iv1Z\ndx\u0010\n y\nR;#(x)@x R;#(x)\u0000 y\nL;\"(x)@x L;\"(x)\u0011\n\u0000iv2Z\ndx\u0010\n y\nR;\"(x)@x R;\"(x)\u0000 y\nL;#(x)@x L;#(x)\u0011\n;\n(6)\nwith distinct Fermi velocities v1andv2. As it is clear\nfrom the electronic spectrum, the model has broken chiral\nsymmetry, but the time-reversal symmetry is preserved.\nCorresponding excitations can be rewritten equiva-\nlently in the bosonic language bq(neglecting zero-energy\nmodes)\nH0=v1 X\nq>0jqjby\nq;#bq;#+X\nq<0jqjby\nq;\"bq;\"!\n+v2 X\nq>0jqjby\nq;\"bq;\"+X\nq<0jqjby\nq;#bq;#!\n:(7)\nCollecting the common terms, we obtain\nH0=vF\n2X\nq6=0;\u001bjqjby\nq;\u001bbq;\u001b+\u0001\n2X\nq6=0;\u001bq\u001bby\nq;\u001bbq;\u001b (8)\nwith \u0001 =v1\u0000v2andvF=v1+v2. The second term\nvanishes for vanishing SOC and has the following form\nin the spin-charge basis:\nHmix=\u0000\u0001\n2X\nq6=0q\u0002\nby\nq;sbq;c+by\nq;cbq;s\u0003\n: (9)\nThis term with \fnite \u0001 6= 0 violates the spin-charge sep-\naration and demands more general approach.\nWe consider the interacting theory within the general-\nized g-ology approach and do not impose any constraint\non TLL parameters K\u0017,\u00172(s;c). Within this general-\nization, the coordinate space representation of the inter-\nacting Hamiltonian in the spin-charge basis has quadratic\nGaussian form:\nHSOC =X\n\u0017=s;cv\u0017\n2Z\u00141\nK\u0017(@x\u001e\u0017)2+K\u0017(@x\u0012\u0017)2\u0015\ndx+Hmix:\n(10)\nThe SCC term is expressed as follows:\nHmix=\u0001\n2Z\n[@x\u001es@x\u0012c+@x\u001ec@x\u0012s]dx: (11)\nFor vanishing SOC, the chiral symmetry is restored, with\nfullSU(2) symmetry and Ks= 1.\nIn principle, one has to include also the backscattering\nterm to the Hamiltonian,\nHBS=gs\n2(\u0019a)2Z\ndxcos(p\n8\u0019\u001es): (12)In the parameter space of its relevancy, this term opens\na gap in the spin channel via the Berezinskii-Kosterlitz-\nThouless mechanism. There are several works which have\naddressed the relevancy of this term for repulsive e-e in-\nteractions [25, 38{40]. However, in a recent experiment\non InAs nanowires with a strong SOC, no sign of spin\ngap is observed [31]. Based on the phenomenology, we\nneglect all such terms within the whole parameter space.\nWe also emphasize that impurity e\u000bects in the system\nwith gapped spin channel were also previously studied\n[40, 41].\nFor perturbative analysis of impurity e\u000bects, the\nimaginary-time Euclidean actions for the displacement\n\felds can be obtained by integrating out the quadratic\nphase \felds. In the ~ x= (x;\u001c) space it takes the following\nform:\nS\u001e\n\u0017=1\n2vZ\ndxd\u001c\u0000\n(@\u001c\u001e\u0017)2+ ~v2\n\u0017(@x\u001e\u0017)2\u0001\n(13)\nwithv=vsKs=vcKc, guaranteed by Galilean invari-\nance of the model with \u0001 = 0. The renormalized veloci-\nties are:\n~v2\n\u0017=v2\n\u0017d\u0017; (14)\nd\u0017=\u0012\n1\u0000\u00012\n4v2\u0017\u0013\n: (15)\nThe contribution from the mixing term is,\nS\u001e\nmix=i\u0001\n2vZ\ndxd\u001c (@x\u001es@\u001c\u001ec+@x\u001ec@\u001c\u001es): (16)\nEuclidean actions for the phase \feld \ructuations can\nbe obtained in a similar fashion,\nS\u0012\n\u0017=v\n2Z\ndxd\u001c\u00121\nv2\u0017(@\u001c\u0012\u0017)2+d\u0000\u0017(@x\u0012\u0017)2\u0013\n(17)\nalong with the mixing \u0012-term\nS\u0012\nmix=iv\u0001\n2Z\ndxd\u001c\u00121\nv2c(@x\u0012s@\u001c\u0012c) +1\nv2s(@x\u0012c@\u001c\u0012s)\u0013\n:\n(18)\nFor \u0001 = 0, one correctly obtains the standard expres-\nsions for actions S\u001e\n\u0017andS\u0012\n\u0017.\nImpurity bosonization\nWe consider a single impurity embedded at the origin\nx= 0. This impurity causes backscattering of carriers\nHbs=VbsX\n\u001b\u0010\n y\nR;\u001b L;\u001b+h:c:\u0011\n; (19)\nwhereVbsis the Fourier component of the backscattering\npotentialV(kF;1+kF;2). The forward-scattering term\ncan be always gauged out [3].4\nThe bosonized expression of the backscattering term\nleads to the boundary sine-Gordon model in the spin-\ncharge basis:\nHbs=2Vbs\n\u0019acos(\fI\u001es(x= 0)) cos(\fI\u001ec(x= 0)) (20)\nwith\f2\nI= 2\u0019. This term also couples spin and charge\ndegrees of freedom at the impurity point. We note that in\nthe bosonization process of this term we do not take into\naccount Klein factors ^F\u001b, since their e\u000bect is irrelevant\nwithin our model with all terms included terms in this\nwork. The corresponding backscattering action takes the\nfollowing form:\nSb=2Vbs\n\u0019aZ\f\n0d\u001ccos(\fI\u001es(0;\u001c)) cos(\fI\u001ec(0;\u001c)) (21)\n\f=1\nTis the inverse temperature and should not be\nconfused with \fI. We hereafter assume Vbs=\u0003\u001c1, where\n\u0003 in the e\u000bective bandwidth for both channels.\nFor a complete analysis, pertubative study of the\nbackscatteting action Eq. (21) should be accompanied\nwith the RG study in the strong barrier limit. For this\npurpose, we consider two disjointed wires and treat in-\nterwire tunneling term perturbatively. The most relevant\ntunneling term has the following contribution to the total\naction,\nSt=t\n\u0019aZ\f\n0d\u001ccos(\fI\u0012s(0;\u001c)) cos(\fI\u0012c(0;\u001c));(22)\nwheretis the bare tunneling amplitude through the weak\nlink.\nIII. EFFECTIVE BATHS ACTIONS\nAt this point it is worth noting that the previous\nbosonization studies of TLL with SCC [24, 32, 35] were\nperformed using basis rotation approach, where one\ntransforms the initial basis to the new spin-charge basis\nwith decoupled channels and new renormalized velocities\nandK\u0017parameters. This approach usually results on\nredundant expressions for model parameters and com-\nplicates further analysis. Instead, we follow a generic\napproach [3] without performing a basis rotation. As\nwe show below, this procedure is not required. We next\ntrace out all space degrees of freedom except for the im-\npurity point. The low-lying excitations of the \felds away\nfrom the impurity act as a dissipative bath reducing the\nproblem to an e\u000bective 0D \feld theory of a single Brow-\nnian quantum particle in a 2D harmonic impurity poten-\ntial [42].\nWe follow the standard procedure [3, 12] to get e\u000bective\nbath actions for the displacement \ructuation. The same\nprocedure applies for the phase \felds.The integration of the bulk degrees of freedom is done\nusing the standard trick of introducing Lagrange multi-\npliers with the real auxilliary \felds \u0015\u0017\nZ\u001e=Z\nD\u001esD\u001ecD\bcD\bsD\u0015sD\u0015ce\u0000ST(23)\nwith\nST=S[\u001ec;\u001es] +iZ\f\n0d\u001cX\n\u0017[\b\u0017(\u001c)\u0000\u001e\u0017(0;\u001c)]\u0015\u0017(\u001c):\n(24)\nOne needs consequently integrate out the \u001e\u0017and\u0015\u0017to\nget the \fnal baths actions:\nS\u001e\n\u0017=X\n!nj!nj ^F\u0000\u0017\ndet(F)!\nj\b\u0017(!n)j2(25)\nwith the bosonic Matsubara frequencies !n=2\u0019n\n\f;(n2\nZ). The mixing term is:\nS\u001e\nmix=\u0000X\n!nj!nj ^Fmix\ndet(F)!\n\bs(\u0000!n)\bc(!n);(26)\nwhere theFmatrix and ^F\u0017are given in Eqs. (A.1)-(A.3).\nImportantly, the mixing action S\u001e\nmixvanishes (see Ap-\npendix) and one is left with the decoupled set:\nS\b\n\u0017=X\n!nj!nj\n~K\u001e\n\u0017j\b\u0017(!n)j2(27)\nwith ~K\u001e\n\u0017given by Eq. (A.5) and \u00172(s;c). The resulted\nCaldeira-Legett type actions are common and describe\ndynamics of a single quantum Brownian particle in the\nregime of Ohmic dissipation [42, 43]. These actions rep-\nresent the weak-coupling \fxed point of pure Luttinger\nliquid.\nSimilarly, the e\u000bective baths actions for \u0012\u0017\felds can\nbe obtained as\nS\u0002\n\u0017=X\n!nj!nj~K\u0012\n\u0017j\u0002\u0017(!n)j2(28)\nwith the new TLL parameters ~K\u0012\n\u0017given by Eq. (A.6).\nThe mixing action also vanishes in this case.\nVanishing mixing terms and decoupled baths map the\nproblem onto the one with conventional TLL baths,\nbut with the new parameters ~K\u0017andv\u0000;+given by\nEqs. (A.5)-(A.7). Hence, the usual basis-rotation proce-\ndure is excessive here. Obviously, for the vanishing spin-\ncharge mixing term \u0001 = 0, the standard TLL parameters\nare correctly recovered, ~K\u0017!K\u0017. The characteristic\nvelocitiesv+!max(vs;vc) andv\u0000!min(vs;vc). This\nimplies that in the presence of SOC, one has new modes,\nwith carriers composed of spin and charge quanta. These\nnew modes have excitation velocities v\u0000;+and posses new\nTLL parameters ~K\u0017. Prior to the discussion of our main5\nresults, we emphasize two important features arisen al-\nready at this stage of analysis.\nMode freezing . As the strength of SOC is increased, v+\nmonotonically increases, whereas v\u0000similarly decreases\nand vanishes at the critical \u0001c\n\u0017. As it was mentioned\nin previous works [25, 27], at the critical SOC strength\n\"freezing\" of the corresponding mode (phase separation)\nwith diverging spin (or charge) susceptibility is exhibited.\nFor repulsive e-e interactions, it was shown [25] that it is\nthe spin susceptibility which diverges at the critical SOC\nas,\n\u001f=\u001f0 \n1\u0000\u0012\u0001\n\u0001cs\u00132!\u00001\n(29)\nwith spin-susceptibility \u001f0at zero SOC and \u0001c\ns= 2=Ks.\nSimilar divergence can be observed in the charge channel,\nif one considers full Kc\u0000Ksspace. These divergences in-\ndicate a phase transition, which occurs in the spin/charge\nchannel (see Ref. [27] and references therein). In this\nwork, we consider SOC strengths causing the velocity dif-\nference \u0001=v < 0:8, which restricts the parameter space\ntoK\u0017<5=2.\nAbsence of spin-\fltering e\u000bect . In the case of the\nspin-charge mixing caused by the strong Zeeman \feld\n[Fig. 2(a)] the mixing bath actions Smixdoes not vanish,\ndue to the di\u000berent type of SCC mechanism. The mix-\ning term Eq. (11) in the Hamiltonian takes the following\nform:\nHmix=\u0001B\n2Z\n[@x\u001es@x\u001ec+@x\u0012s@x\u0012c]dx (30)\nwith \u0001B=v\"\u0000v#. As it was demonstrated earlier in\nRefs. [24, 35], a single impurity as in Eq. (19) embedded\nto such TLL causes polarization of the spin current with\na ratio of tunneling amplitudes,\nt\"\nt#=\u0012T\n\u0003\u0013\u0011\n; (31)\nand with a \fnite exponent \u0011(\u0001B) for \fnite \u0001 B.\nObservation of this e\u000bect in our model was also pro-\nposed earlier in Ref. [35]. However, an important conse-\nquence of vanishing Smixaction terms of Eq. (26) (and\nsimilar term for \u0002 \feld) is the absence of such spin-\n\fltering e\u000bect, albeit the spin-charge separations is vi-\nolated.\nIV. RESULTS AND DISCUSSIONS\nWeak potential barrier\nOnce the e\u000bective bath actions are obtained as in\nEq. (27) and Eq. (28) , the standard pRG analysis in\nthe limit of weak and strong impurity potential can be\nperformed.We \frst consider the scattering of electrons on a weak\nbarrier. In this limit, the partition function is,\nZ=Z\nD\bsD\bce\u0000ST; (32)\nwith the total action,\nST=S\b\nc+S\b\ns+Vm;n\n\u0019aZ\f\n0d\u001ccos(m\fI\bc) cos(n\fI\bs):\n(33)\nTo analyze the relevance of the last term within the Kc\u0000\nKsparameter space, we generalize the impurity term to\ntake di\u000berent values of mandnwith the amplitudes\nVm;n\u001c\u0003, since such terms are necessarily generated\nduring the RG process.\nTreating the backscattering terms perturbatively, one\nobtains the standard set of \frst-order RG equations [12,\n13],\ndV1;1(l)\ndl= \n1\u0000~K\u001e\ns+~K\u001e\nc\n2!\nV1;1(l); (34)\ndV0;2(l)\ndl=\u0010\n1\u00002~K\u001e\ns\u0011\nV0;2(l); (35)\ndV2;0(l)\ndl=\u0010\n1\u00002~K\u001e\nc\u0011\nV2;0(l) (36)\nwithdl=d\u0003\n\u0003.\nThe \frst equation for V1;1\u0011Vbscorresponds to the\nkF;1+kF;2backscattering of a single electron, whereas\nthe second and the third equations correspond to the\nbackscattering of spin or charge (electron pair) degree of\nfreedom. The last processes have the fermionic expres-\nsions y\nR;\" L;\" y\nL;# R;#+h:c:and y\nR;\" L;\" y\nR;# L;#+\nh:c:, respectively. The sketches of these scattering pro-\ncesses in the momentum space are presented in Fig. 3(b)-\n(c). All higher-order decendent terms are neglected, since\nthe regions of their relevancy are covered with the ones\nof the last two equations, even for the \fnite \u0001. Thus,\nthe main low-temperature processes are dictated by these\nthree equations Eqs.(34)-(36).\nFor vanishing SOC \u0001 = 0 ( ~K\u0017!K\u0017), the marginal\n\fxed line de\fned in Eq. (34) is determined by the con-\nditionKs+Kc= 2. The region I in Fig. 3(a) corre-\nsponds to a parameter space where the backscattering of\na single electron in a weak potential becomes a relevant\nprocess and the backscattering amplitude V1;1\rows to\nthe strong-coupling regime. This leads to the blocked\ntransport in both charge and spin channels and the \felds\n\bs;care pinned on the minima of cosine functions. Qual-\nitatively, in this region either single electron ( \"or#) is\nfractionalized by e-e interactions, which is responsible for\nfull backscattering of single carrier and hence, charge and\nspin carriers, too. Thus, one has vanishing conductance\nG= 0 atT= 0.6\n(b)\n(c)charge backscattering\nspin backscattering(a)\nIV\nI\nIIIII\nKF,1\nKF,2\nKF,2K\nK\nKF,1ξ(k)\nξ(k)\n0 0.5 1 1.5 2 2.5\nKs00.511.522.5Kc∆=0\n∆=0.4\n∆=0.8\nFIG. 3. (a) The modi\fcation of the phase portrait for the\nTLL with \fnite SOC. For \u0001 >0 the phase boundary of region\nI is modi\fed. In (b) and (c) the backscattering processes of\ncharge and spin are sketched, which correspond to Eqs. (35)-\n(36), respectively.\nAs dictated by Eqs. (35)-(36), the backscattering pro-\ncesses of the charge/spin carriers also become relevant\nwhenKc=s<0:5. For the charge backscattering, this im-\nplies the charge quanta to carry smaller than 0 :5 unit, to\nmake it a relevant process. In the region of relevance II\n(III for spin channel), the charge (spin) channel is insu-\nlating since the the \b c(\bs) is pinned. The spin (charge)\nfully transmit the barrier in this region, implying the re-\nalization of mixed phases in these regions. In region IV\nboth channels have dominating superconducting \ructu-\nations [27]. We emphasize again that in this region with\nattractive e-e interactions, other type of instabilities in\nthe bulk may arise, which usually open a gap in one of\nthe channels. Here, we neglect all such processes and\npresent the simplest picture.\nTo investigate the e\u000bect of SOC on backscattering of\ncarriers, we represent the new TLL parameters in a sym-\nmetric way:\n~K\u001e\n\u0017=K\u0017\u0012v\u0017\n~v\u0017\u0013\u0012~vc+ ~vs\nv++v\u0000\u0013\n; (37)\nwhere ~v\u0017andv\u0006are given in Eq. (14) and Eq. (A.7),\nrespectively. In the presence of SOC the \frst e\u000bect is\nthe renormalization of spin and charge velocities, which\nis manifested in the \frst factor of Eq. (37). The second\nfactor is due to the emerged modes with velocities v+;\u0000.\nAs it was mentioned in the previous section, we limit\nconsidiration of the parameter space K\u0017<5\n2and \u0001=v<\n0:8.\nFor non-interacting electrons, the e\u000bect of SOC is lim-\nited to the breaking of chiral symmetry and the renor-\nmalization of excitation velocities. Indeed, one has ~K\u001e\n\u0017=\nK\u001e\n\u0017= 1, since renormalizing factors in Eq. (37) cancel\neach other. As shown in Fig. 3a this is also valid for\nweakly-interacting electrons, K\u0017\u00181.\nFor the given \fnite SOC strengths, where 0 <\u0001=v <\n0:8, the second factor\u0010\n~vs+~vc\nv\u0000+v+\u0011\n\u00191 is \fxed for both\nK\u0017, whereas the \frst one de\fnes scattering of carriersand determines the boundary of region I. The strongest\ne\u000bect of SOC is exhibited when in one of the channels\nK\u0017\u001c1. In this limit one can lock the corresponding\n\b\u0017\feld to the minimum of cosine function and reexpress\nthe scaling dimension of the impurity term for m= 1\nandn= 1 as follows:\n2\u000e1;1=K\u0017+K\u0000\u0017(1 +\u00012\n8v2\n\u0000\u0017): (38)\nThe resulting marginality line is presented in Fig. 3(a)\nfor \u0001=v= 0:4 and \u0001=v= 0:8. At the critical K\u0017=5\n2,\nthe excitations in the spin/charge channel become frozen\n(v\u0000= 0) and the bulk of the channel becomes insulating.\nThe e\u000bect of \fnite SOC on the boundaries of regions\nII/III with the region IV is negligibally small. The largest\ncorrection to the scaling dimension \u000e0;1(\u000e1;0) is of the\norder of 10\u00002for the largest \u0001 =v= 0:8. Thereby it can\nbe safely neglected.\nFinally, one can straightforwardly generalize the ex-\npressions for corrections to (bulk) conductances obtained\nin Refs. [12][13] to the case with \fnite SOC by K\u0017!~K\u0017:\n\u000eG=e2\nhX\nm;ncm;njVm;nj2T(m2~Kc+n2~Ks)=2\u00002(39)\nwith dimensionless coe\u000ecients cm;n.\nStrong barrier\nFor analysis of tunneling term in Eq. (22) one considers\nthe following total action ST,\nST=S\u0002\nc+S\u0002\ns+tm;n\n\u0019aZ\f\n0d\u001ccos(m\fI\u0002c) cos(n\fI\u0002s):\n(40)\nSimilar to the previous case, the impurity term is gener-\nalized for di\u000berent mandn.\nThe RG transformation of the impurity term leads to\nthe following set of equations,\ndt1;1(l)\ndl=\u0012\n1\u00001\n2\u00121\n~K\u0012c+1\n~K\u0012s\u0013\u0013\nt1;1(l); (41)\ndt2;0(l)\ndl=\u0012\n1\u00002\n~K\u0012c\u0013\nt2;0(l); (42)\ndt0;2(l)\ndl=\u0012\n1\u00002\n~K\u0012s\u0013\nt0;2(l) (43)\nwith the TLL prameters rewritten as:\n1\n~K\u0012\u0017=1\nK\u0017\u0012v\u0017\n^v\u0017\u0013\u0012^vc+ ^vs\nv\u0000+v+\u0013\n; (44)7\nKSKC\n0 0.5 1 1.5 2 2.5IIVIII\nII∆=0\n∆=0.4\n∆=0.8\n00.511.522.5\nFIG. 4. (a) The modi\fcation of the phase portrait obtained\nfrom the renormalization analysis of tunneling events. The\ne\u000bect of SOC is manifested on phase boundaries between the\nregions I-II and I-III. In the vicinity of mode freezing, hopping\nevents of a single ( \")=(#) carrier become irrelevant. All shaded\nregions correspond to disconnected wires with no transport of\ncarriers.\nand\n^v\u0017=v\u0017q\n1\u0000\u00012\n4v2\u0017: (45)\nAn amplidute t1;1corresponds to the interwire hopping\nevent of a single ( \")=(#) electron, while t2;0andt0;2are\ntunneling amplitude of charge and spin, respectively.\nBased on the set of equations, one obtains phase por-\ntrait atT= 0. It consists of four regions, exhibited also\nin the limit of weak scattering potential and it is shown\nin Fig. 4(a). For vanishing SOC ~K\u0017!K\u0017, the bound-\naries of regions are de\fned by1\nKc+1\nKs= 2;Kc= 2,\nandKs= 2. In region I (IV), hopping event of a single\n(\")=(#) electron is irrelevant (relevant) and one eventu-\nally renormalizes onto the \fxed point of disjoined (con-\nnected) wires in the RG process. In region II, the tunnel-\ning amplitude t0;2for spin is relevant and grows upon RG\ntransformation, however charge carriers can not tunnel.\nAn opposite situation with a conducting charge channel\nand an insulating spin channel occurs in the region III.\nWe note that the results obtained in the opposite limits\nof weak and strong barrier are consistent.\nFor relatively weak SOC (\u0001 = 0 :4) the only pro-\nnounced e\u000bect is modi\fcation of boundaries between re-\ngions I-II and I-III, as shown in Fig. 4(a). These e\u000bects\nare dictated by Eqs. (42)-(43). The regions II and III\nare extended towards the region I with the new bound-ariesK\u0017\u00191:8<2. The area of the extended region is\nlarger for the larger value of \u0001 = 0 :8 with boundaries\nde\fned via K\u0017\u00191:6. Remarkably, these e\u000bects are also\nexhibited in the weak barrier analysis, in Fig. 3(a).\nAt the largest \u0001 = 0 :8, the tunneling of single carri-\ners with amplitude t1;1becomes irrelevant near the phase\nseparation point K\u0017= 5=2. This is consistent with the\npicture of mode freezing, discussed in the previous sec-\ntions. On the other hand, for weakly-interacting elec-\ntrons withK\u0017\u00191 the e\u000bect of SOC is negligibely small,\neven for large values of \u0001, also compatible with the re-\nsults of the previous section.\nConductance for insulating regions at \fnite tempera-\ntureTcan be also generalized as follows:\nG=e2\nhX\nm;ndm;nt2\nm;nT2(m2=~Kc+n2=~Ks\u00002)(46)\nwith dimensionless coe\u000ecients dm;n.\nExperimentally relevant values of \u0001 =v\u00180:2 [25, 26,\n31] are smaller than the maximum value considered in\nthis work. For this range of \u0001 values, one can neglect\nmode-freezing e\u000bects safely. The results of the RG anal-\nysis on the last sections a\u000erm that for weak and mod-\nerate e-e interactions e\u000bects of SOC are negligible small.\nThis results are in accrodance with recent theoretical and\nexpertimental studies [31, 32].\nV. CONCLUSIONS\nWe studied carrier scattering e\u000bects upon a single im-\npurity embedded to a Luttinger liquid with spin-orbit\ncoupling using Abelian bosonization and pertubative\nrenormalization techniques. Spin-orbit interaction de-\ngrades spin-charge separation and renormalizes the TLL\nparameters Ks;cand the excitation velocities vs;c. We\ndemonstrated that the scaling dimension of impurity op-\nerator is identical for both ( \")=(#) carriers. This im-\nplies the absence of conjectured spin-\fltering e\u000bect in\nRef.[35]. The strongest e\u000bects of spin-orbit coupling are\npronounced for strong e-e interactions, whereas these ef-\nfects are negligibly small for moderate e-e interactions.\nOur main results are summarized by the phase portrait\nmodi\fcations presented in Figs (3)-(4).\nACKNOWLEDGMENTS\nWe thank F. Yilmaz and V. I. Yudson for useful com-\nments and discussions.8\nAppendix: Expressions for new TLL parameters\nand excitations velocities\nTheFmatrix has the following form,\nF=\u0012\n2^Fc^Fmix\n^Fmix 2^Fs\u0013\n: (A.1)\nThe matrix elements ^F\u0017=j!jF\u0017are expressed via the\nfollowing integrals,\nF\u0017=Z+1\n\u00001dk\n2\u0019 \n[G\u001e\n\u0000\u0017]\u00001\ndet(Q)!\n(A.2)\nand for the mixing element,\nFmix=\u0000Z+1\n\u00001dk\n2\u0019 \n[G\u001e\nmix]\u00001\ndet(Q)!\n(A.3)\nwhere [G\u001e\n\u0017]\u00001are the inverse propogators of Eqs. (13)-\n(16) in~ q= (k;!) space, and the Qmatrix is de\fned\nas,\nQ=\u0012\n2[G\u001e\nc]\u00001[G\u001e\nmix]\u00001\n[G\u001e\nmix]\u000012[G\u001e\ns]\u00001\u0013\n: (A.4)For the mixing parameter Fmixthe numerator of the\nkernel is odd function of k, i.e [G\u001e\nmix]\u00001=k!, whereas\nthe denominator is an even function of k. This leads to\nthe vanishing action S\u001e\nmix. Similarly, S\u0012\nmixvanishes and\none is left with fully decoupled actions [Eqs. (27)-(28)].\nEvaluation of integrals leads to the following results for\nnew TLL parameters for displacement \ructuation \felds,\n^K\u001e\n\u0017=v\n(v++v\u0000)\u0014~v2\n\u0000\u0017\nv+v\u0000+ 1\u0015\n; (A.5)\nand for the phase \felds,\n1\n^K\u0012\u0017=v2\n\u0017\nv(v\u0000+v+)\u0014v2\n\u0000\u0017~v\u0017\nv2\u0017~v\u0000\u0017+ 1\u0015\n: (A.6)\nThe sound velocities for the newly emerged modes are\ngiven by,\nv2\n\u0006=1\n2h\n~v2\nc+ ~v2\ns+ \u00012\u0006p\n(~v2c\u0000~v2s+ \u00012)2+ 4~v2s\u00012i\n:\n(A.7)\nFor vanishing SOC \u0001 = 0, the sound velocities in the\nspin/charge channel are recovered.\n[1] S.-i. Tomonaga, Progress of Theoretical Physics 5, 544\n(1950).\n[2] J. Luttinger, Journal of mathematical physics 4, 1154\n(1963).\n[3] A. O. Gogolin, A. A. Nersesyan, and A. M. Tsvelik,\nBosonization and strongly correlated systems (Cambridge\nuniversity press, 2004).\n[4] F. D. M. Haldane, 14, 2585 (1981).\n[5] M. P. Fisher and L. I. Glazman, in Mesoscopic Electron\nTransport (Springer, 1997) pp. 331{373.\n[6] H. Steinberg, G. Barak, A. Yacoby, L. N. Pfei\u000ber, K. W.\nWest, B. I. Halperin, and K. Le Hur, Nature Physics 4,\n116 (2008).\n[7] H. Kamata, N. Kumada, M. Hashisaka, K. Muraki, and\nT. Fujisawa, Nature Nanotechnology 9, 177 (2014).\n[8] J. Voit, Reports on Progress in Physics 58, 977 (1995).\n[9] T. Giamarchi, Quantum physics in one dimension , Vol.\n121 (Clarendon press, 2003).\n[10] E. Jeckelmann, Journal of Physics: Condensed Matter\n25, 014002 (2012).\n[11] C. Kane and M. P. Fisher, Phys. Rev. Lett. 68, 1220\n(1992).\n[12] A. Furusaki and N. Nagaosa, Phys. Rev. B 47, 4631\n(1993).\n[13] C. Kane and M. P. Fisher, Phys. Rev. B 46, 15233 (1992).\n[14] A. Anthore, Z. Iftikhar, E. Boulat, F. D. Parmentier,\nA. Cavanna, A. Ouerghi, U. Gennser, and F. Pierre,\nPhys. Rev. X 8, 031075 (2018).\n[15] A. M. Chang, Rev. Mod. Phys. 75, 1449 (2003).\n[16] X. G. Wen, Phys. Rev. B 41, 12838 (1990).\n[17] M. Bockrath, D. H. Cobden, J. Lu, A. G. Rinzler, R. E.\nSmalley, L. Balents, and P. L. McEuen, Nature 397, 598(1999).\n[18] C. Blumenstein, J. Sch afer, S. Mietke, S. Meyer,\nA. Dollinger, M. Lochner, X. Y. Cui, L. Patthey,\nR. Matzdorf, and R. Claessen, Nature Physics 7, 776\n(2011).\n[19] Y. Jompol, C. J. B. Ford, J. P. Gri\u000eths, I. Farrer,\nG. A. C. Jones, D. Anderson, D. A. Ritchie, T. W. Silk,\nand A. J. Scho\feld, Science 325, 597 (2009).\n[20] E. Bocquillon, V. Freulon, J.-. M. Berroir, P. Degiovanni,\nB. Pla\u0018 cais, A. Cavanna, Y. Jin, and G. F\u0012 eve, Nature\nCommunications 4, 1839 (2013).\n[21] M. Hashisaka, N. Hiyama, T. Akiho, K. Muraki, and\nT. Fujisawa, Nature Physics 13, 559 (2017).\n[22] T. Vekua, S. I. Matveenko, and G. V. Shlyapnikov, JETP\nLetters 90, 289 (2009).\n[23] S. Brazovskii, S. Matveenko, and P. Nozieres, Journal\nde Physique I 4, 571 (1994).\n[24] T. Kimura, K. Kuroki, and H. Aoki, Phys. Rev. B 53,\n9572 (1996).\n[25] A. V. Moroz, K. V. Samokhin, and C. H. W. Barnes,\nPhys. Rev. B 62, 16900 (2000).\n[26] A. V. Moroz, K. V. Samokhin, and C. H. W. Barnes,\nPhys. Rev. Lett. 84, 4164 (2000).\n[27] A. Iucci, Phys. Rev. B 68, 075107 (2003).\n[28] S. Datta and B. Das, Applied Physics Letters 56, 665\n(1990), https://doi.org/10.1063/1.102730.\n[29] S. Nadj-Perge, S. M. Frolov, E. P. A. M. Bakkers, and\nL. P. Kouwenhoven, Nature 468, 1084 (2010).\n[30] C. Fasth, A. Fuhrer, L. Samuelson, V. N. Golovach, and\nD. Loss, Phys. Rev. Lett. 98, 266801 (2007).\n[31] Y. Sato, S. Matsuo, C.-H. Hsu, P. Stano, K. Ueda,\nY. Takeshige, H. Kamata, J. S. Lee, B. Shojaei, K. Wick-9\nramasinghe, J. Shabani, C. Palmstr\u001cm, Y. Tokura,\nD. Loss, and S. Tarucha, Phys. Rev. B 99, 155304 (2019).\n[32] C.-H. Hsu, P. Stano, Y. Sato, S. Matsuo, S. Tarucha,\nand D. Loss, Phys. Rev. B 100, 195423 (2019).\n[33] Y. Hamamoto, K.-I. Imura, and T. Kato, Phys. Rev. B\n77, 165402 (2008).\n[34] T. Hikihara, A. Furusaki, and K. A. Matveev, Phys.\nRev. B 72, 035301 (2005).\n[35] K. Kamide, Y. Tsukada, and S. Kurihara, Phys. Rev. B\n73, 235326 (2006).\n[36] M. Governale and U. Z ulicke, Solid State Communica-\ntions 131, 581 (2004), new advances on collective phe-\nnomena in one-dimensional systems.\n[37] M. Governale and U. Z ulicke, Phys. Rev. B 66, 073311(2002).\n[38] V. Gritsev, G. Japaridze, M. Pletyukhov, and\nD. Baeriswyl, Phys. Rev. Lett. 94, 137207 (2005).\n[39] A. Schulz, A. De Martino, P. Ingenhoven, and R. Egger,\nPhys. Rev. B 79, 205432 (2009).\n[40] N. Kainaris and S. T. Carr, Phys. Rev. B 92, 035139\n(2015).\n[41] N. Kainaris, S. T. Carr, and A. D. Mirlin, Phys. Rev. B\n97, 115107 (2018).\n[42] M. P. A. Fisher and W. Zwerger, Phys. Rev. B 32, 6190\n(1985).\n[43] A. Caldeira and A. Leggett, Annals of Physics 149, 374\n(1983)."    },    {        "title": "0809.5119v1.Multi_terminal_Spin_Transport__Non_applicability_of_linear_response_and_Equilibrium_spin_currents.pdf",        "content": "arXiv:0809.5119v1  [cond-mat.mes-hall]  30 Sep 2008Multi-terminal Spin Transport: Non applicability of linea r response and Equilibrium\nspin currents\nT. P. Pareek\nHarish Chandra Research Institute\nChhatnag Road, Jhusi, Allahabad - 211019, India\nWe present generalized scattering theory for multi-termin al spin transport in systems with broken\nSU(2) symmetry either due to spin-orbit interaction,magne tic impurities or magnetic leads. We\nderive equation for spin current consistent with charge con servation. It is shown that resulting spin\ncurrent equations can not be expressed as difference of poten tial pointing to non applicability of\nlinear response for spin currents and as a consequence equilibrium spin currents(ESC) in the l eads\nare non zero. We illustrate the theory by calculating ESC in t wo terminal normal system in presence\nof Rashba spin orbit coupling and show that it leads to spin re ctification consistent with the non\nlinear nature of spin transport.\nPACS numbers: 75.60.Jk, 72.25-b,72.25.Dc, 72.25.Mk\nSpin transport has emerged as an important subfield\nof research in bulk condensed matter system as well in\nmesoscopic and nano system[1]. In macroscopic systems,\nthe very definition of spin currents is still debated due to\nnon conservation of spin in presence of SO interaction[2]\n. On the other hand in mesoscopic hybrid system since\ncurrent is defined in the leads where SO coupling is\nabsent,therefore, it has been assumed that Landauer-\nB¨ uttiker formula for charge current[3] (Eq. (9) in this\nmanuscript), which determines current in leads in terms\nofapplied voltagedifference multiplied by totaltransmis-\nsionprobability, can be straightawaygeneralizedforspin\ncurrents by replacing total transmission probability with\nsome particular combination of spin resolved transmis-\nsion probabilities[4, 5, 6]. This simple generalization has\nbeen widely used in the literature to study spin depen-\ndent phenomena in nanosystems[4, 5, 6]. The Landauer-\nB¨ uttiker formula in its widely used form has inbuilt cur-\nrent conservation , i.e., total current is divergence less\n(divj=0) which follows from basic Maxwell equations of\nelectrodynamics. Physically this implies that total cur-\nrent has neither sources nor sinks this would be true for\nspin currents as well if spin is conserved. However, in\npresence of spin-orbit interaction, magnetic impurities or\nnon-collinear magnetization in leads, spin is no longer a\nconserved quantity, hence the spin currents can not be\ndivergence less. Therfore a straight forward generaliza-\ntion of Landauer-B¨ uttiker charge current formula to spin\ncurrents can not be correct for spin non-conserving sys-\ntems.\nIn view of the above discussion in this work we de-\nvelopaconsistentscatteringtheoreticformulationofcou-\npled spin and charge transport in multiterminal systems\nwith broken SU(2) symmetry in spin space following\nB¨ uttiker’s work on charge transport [3]. The SU(2) sym-\nmetry in spin space can be broken due to either SO inter-\naction, magnetic impurities or non-collinear magnetiza-\ntion in leads[7, 8]. Our analysis provides a correct gener-\nalization of Landauer-B¨ uttiker theory for spin transport.\nIn particular we derive a spin currents equation (Eq. (8)inthismanuscript)consistentwithchargecurrentconser-\nvation. However, the resulting spin current equation can\nnot be cast in terms of spin resolved transmission and re-\nflectionprobabilitiesmultiplied byvoltagedifferenceasis\nthe casefor chargecurrent(Eq. (9) in this manuscript)[3].\nTherefore, equilibrium spin currents are generically non\nzero.Moreover, it implies that linear response theory\nwith respect to electric field is not applicable to the spin\ncurrents (equilibrium as well non-equilibrium).Thus spin\ncurrents are intrinsically non-linear in electrical circu its.\nHowever, this is not surprising since linear response is\nvalid for thermodynamically conjugate variable. In an\nelectrical circuit thermodynamically conjugate variable\nto electric field is charge current not the spin currents.\nWe illustrate the theory by calculating ESC analytically\nfor two-dimensional electron system with Rashba SO in-\nteraction in contact with two unpolarized metallic con-\ntacts. Our analytical formula for ESC clearly demon-\nstrates that it is transfer of angular momentum per unit\ntime from SO coupled sample to leads where SO inter-\naction is zero. Therefore,it is truly a transport current\nin contrast to equilibrium spin currents in macroscopic\nRashba medium(see ref.[2, 9]).\nTo formulate scattering theory for spin transport we\nconsider a mesoscopicconductor with brokenSU(2) sym-\nmetry in spin space connected to a number of ideal mag-\nnetic and non-magnetic leads (without SO interaction)\nwhich in turn are connected to electron reservoirs. To\ninclude the effect of broken SU(2) symmetry, it is neces-\nsaryto write spin scattering state in eachlead along local\nspin quantizationaxis. For magnetic leadslocal magneti-\nzation direction provides a natural spin quantization axis\nwhich we denote in a particular lead αbyˆmα(ϑα,ϕα)\nwhereϑαandϕαis polar and azimuthal angle respec-\ntively. For nonmagnetic leads since there is no preferred\nspin quantization axis, hence we choose an arbitrary spin\nquantization axis ˆu(θ,φ) which is same for all nonmag-\nnetic leads. Thus the most general spin scattering state\nin leadαwhich can be either magnetic or nonmagnetic2\nis given by,\nˆΨσ\nα(r,t) =/integraldisplay\ndENσ\nα(E)/summationdisplay\nn=1Φαn(r⊥)χα(σ)/radicalbig\n2π/planckover2pi1vσαn(E)\n(aσ\nαn(E)eikσ\nαn(E)x+bσ\nαn(E)e−ikσ\nαn(E)x) (1)\nwhere Φ αn(r⊥) is transverse wavefunction of channel n\nandχα(σ) is corresponding spin wave function along\nchosen spin quantization axis, ˆuorˆmαsuch that S·\nˆuχ(σ)=(σ/planckover2pi1/2)χ(σ) orS·ˆmαϕ(σ)=(σ/planckover2pi1/2)ϕ(σ) withσ=\nσ(σ=±1,representing local up or down spin compo-\nnents) for nonmagnetic and magnetic leads respectively.\nHereS= (/planckover2pi1/2)σis a vector of Pauli spin matrices and\nNσ\nαisnumberofchannelswith spin σinleadα. The rela-\ntion between spin dependent wavevector kσ\nαn(E) and en-\nergyEis specified by, E=/bracketleftbig\n/planckover2pi12k2\nαnσ/2m+εαn+σ∆α/bracketrightbig\n,\nwhereεαnis energy due to transverse motion, ∆ αis\nstoner exchange splitting in the magnetic lead α. The\nstoner exchange splitting is zero for nonmagnetic leads.\nThe operators aσ\nαnandbσ\nαnare annihilation operator for\nincoming and outgoing spin channels in lead αand are\nrelated via the scattering matrix,\nbσ\nαm=/summationdisplay\nβnσ′Sσσ′\nαm;βnaσ′\nβn (2)\nThe scattering matrix elements Sσσ′\nαm;βnprovides scatter-\ning amplitude between spin channel nσ′in leadβto spin\nchannelmσin leadα. These scattering matrix elements\nwill be function of energy E as well angles, ϑandϕ.\nThe angular dependence of scattering matrix elements\non polar and azimuthal angle arise due to broken SU(2)\nsymmetry. Note that for noncollinear magnetization in\nleads and in absence of SO interaction and magnetic im-\npurities, the angular dependence is purely of geometric\norigin and is related to the angular variation of various\nmagnetoresistance phenomena[10].\nThe current in spin channel σalong longitudinal direc-\ntionˆx(through a cross section of lead α) and the local\nspin quantization axis ˆuis defined as,\nˆIˆxσ\nαˆu(t) =/planckover2pi1\n2mi/integraldisplay/bracketleftBig\nˆΨ†σ\nα(S·ˆu)∇xˆΨσ\nα−∇xˆΨ†σ\nα(S·ˆu)ˆΨσ\nα/bracketrightBig\ndr⊥.\n(3)\nSubstituting for ˆΨσ\nαfrom Eq.(1) into Eq.(3, we get an ex-\npression for spin current in terms of creation and annihi-\nlation operators. On the resulting expression we perform\nquantum statistical averages and after a lengthy algebra\nwe obtain followingexpression for averagecurrentin spin\nchannelσ(for brevity of notation we suppress the super-\nscriptˆxwritten in Eq. (3),\n/an}b∇acketle{tIσ\nαˆu/an}b∇acket∇i}ht=g\nh/integraldisplay∞\n0dE/summationdisplay\nβfβ(E)/bracketleftBigg\nNσ\nαδαβ−/summationdisplay\nσ′mnS†σ′σ\nβm;αnSσσ′\nαn;βm/bracketrightBigg\n(4)\nWherefβ= 1/exp[(E−µβ)/kT]+1 is Fermi distribu-\ntion function with chemical potential µβand the pre-\nfactorgequalsσ/planckover2pi1/2. The summation over σ′in Eq. (4)can take on values ±σcorresponding to two spin pro-\njections along local spin quantization axis. The second\nterm of Eq.(4) can be written explicitly in terms of spin\nresolved reflection and transmission probabilities as,\n/summationdisplay\nβσ′;mnS†σ′σ\nβm;αnSσσ′\nαn;βm=/summationdisplay\nσ′;mnS†σ′σ\nαm;αnSσσ′\nαn;αm\n+/summationdisplay\nβ/negationslash=ασ′;mnS†σ′σ\nβm;αnSσσ′\nαn;βm\n≡/summationdisplay\nσ′Rσσ′\nαα+/summationdisplay\nβ/negationslash=ασ′Tσσ′\nαβ(5)\nWhereRσσ′\nααandTσσ′\nαβare spin resolved reflection and\ntransmission probability in the same probe and between\ndifferent probes respectively. In Eq.5 on right hand side\nspin resolvedreflectionandtransmissionprobabilitiesare\nsummed over all possible input modes for a fixed output\nspin mode σin leadα. Because partial scattering matrix\nin spin subspaceis not unitary due to non conservationof\nspin hence this summation need not to be equal to num-\nber of spin σchannels in lead α,i.e.Nσ\nα, rather it can\nhave any value lying between zero and Nσ\nα. To determine\nNσ\nαin terms of spin resolved reflection and transmission\nprobabilities, consider a situation where current is in-\njected from reservoir only in spin channels σin leadα.\nIn this casechargeconservationrequiresthat this current\nshould leavethe spin channel σthrough all other possible\nchannels in the same lead as well in differing leads, which\nimplies,\nNσ\nα=/summationdisplay\nσ′Rσ′σ\nαα+/summationdisplay\nβ/negationslash=ασ′Tσ′σ\nβα. (6)\nAs we can see that Eq.(6) differs from Eq.(5) in a sub-\ntle way and are not equal because in general spin re-\nsolved transmission or reflection probabilities can not be\nrelated among themselves by interchanging spin indices,\ni.e.,Tσ′σ\nαβ/ne}ationslash=Tσσ′\nβαandRσ−σ\nαα/ne}ationslash=R−σσ\nαα( we will discuss\nconstraints due to time reversal symmetry below). If we\ndemand that sum in Eq.(5) also equals to Nσ\nαthen it\nwould imply spin conservation which is incorrect in pres-\nence of spin flip scattering or broken SU(2) symmetry.\nThe inadvertent use of this charge conservation sum rule\nfor spin degrees of freedom in Ref. [5, 6, 11] has led to\nincorrect spin current equation. Though the partial scat-\ntering matrix in spin subspace is not unitary,however,\nthe full scattering matrix is unitary,i.e., SS†=S†S=I,\ntherefore, if we sum over σalso in Eq.(5) or Eq.(6) then\nit should give total number of channels in leads α,i.e.\nN=Nσ\nα+N−σ\nα, and as a result we get the following sum\nrule for total transmission probability,\nTβα=/summationdisplay\nσ′σTσ′σ\nβα=/summationdisplay\nσσ′Tσσ′\nαβ=Tαβ (7)\nwhereTαβis total tranmission probability.\nThe net spin current flowing in lead αis defined as\nIS\nˆuα=/an}b∇acketle{tIσ\nˆuα/an}b∇acket∇i}ht+/an}b∇acketle{tI−σ\nˆuα/an}b∇acket∇i}htwhile the net charge current flowing3\nis given by sum of absolute values, i.e., Iq\nˆuα=| /an}b∇acketle{tIσ\nˆuα/an}b∇acket∇i}ht |+|\n/an}b∇acketle{tI−σ\nˆuα/an}b∇acket∇i}ht |with pre-factor greplace by the electronic charge\nein Eq. (4). Using Eqs.( 5),(6) in Eq.(4) we obtain net\nspin and charge current as,\nIs\nαˆu= (/planckover2pi1\n2h)/integraldisplay∞\n0dE2fα(E)(R−σσ\nαα−Rσ−σ\nαα)+\n/summationdisplay\nβ/negationslash=ασ′/bracketleftBig\nfα(E)(Tσ′σ\nβα−Tσ′−σ\nβα)−fβ(E)(Tσσ′\nαβ−T−σσ′\nαβ)/bracketrightBig\n(8)\nIq\nαˆu= (e\nh)/integraldisplay∞\n0dE/summationdisplay\nβ/negationslash=ασ′[fα(E)−fβ(E)]Tαβ(9)\nEquation (8) is the central result of this work. We stress\nthat Eqs. (8) and (9) are valid under most general condi-\ntions as we have not made any assumptions about sym-\nmetries of the scattering region . It is instructive to note\nthat in general Tσ′σ\nαβ/ne}ationslash=Tσ′−σ\nαβandRσ−σ\nαα/ne}ationslash=R−σσ\nααthere-\nfore, spin current equation can not be simplified further\nand written in terms of difference of Fermi function mul-\ntiplied by transmission or reflection probabilities as is\nthe case for charge current in Eq. (9) which is stan-\ndard Landauer-B¨ uttikerresult[3]. Hence the spin current\ngiven by Eq. (8) will be nonzero even when all the leads\nareatequilibrium, i.e., fα(E,µα) =f(E,µ),∀α, whereµ\nis equilibrium chemical potential. For sake of complete-\nness we mention that equilibrium chargecurrentvanishes\nas is evident from Eq. (9). The preceding discussion im-\nplies that linear response for spin currents is not appli-\ncable in an electrical circuit where external perturbation\nis applied voltages which is conjugate to charge currents\nand not to the spin currents as discussed in introduction.\nTherefore,the most widely used equation for spin cur-\nrent,see Ref.[4, 6] obtained by a generalization of charge\ncurrent Eq. (9) has to regarded as incorrect. In view of\nthis the theoretical study of spin dependent phenomena\nin mesoscopic systems needs to be re-investigated.\nWe can gain further insight into spin current by con-\nsidering non-equilibrium situation such that the chemi-\ncal potential at the different leads differ only by a small\namount so that we can expand the Fermi distribution\nfunction around equilibrium chemical potential µas,\nfβ(E,µβ) =f(E,µ)+(−df/dE)(µβ−µ). In this case we\ncan immediately notice from Eq. (8) that total spin cur-\nrent in non-equilibrium situation will have equilibrium as\nwell non equilibrium parts of spin current. For ESC the\nfull Fermi sea of occupied levels will contribute. There-\nfore even in non-equilibrium situation ESC cannot be\nneglected.\nEquilibrium spin currents in time reversal symmetric\ntwo terminal system: In time reversalsymmetric systems\nspin resolved transmission and reflection probabilities in\nEq. (8) obey following relations i.e.,Rσσ′\nαα=R−σ′−σ\nααand\nTσσ′\nαβ=T−σ′−σ\nβα[7]. In this case the spin currents Eq. (8)\nfurther simplifies to (here we denote left and right termi-nals byLandRrespectively),\nIs,eq\nL,ˆu= (/planckover2pi1\n2h)/integraldisplay∞\n0dE2f(E,µ)/bracketleftbig\n(R−σσ\nLL−Rσ−σ\nLL)\n+(Tσ−σ\nRL−T−σσ\nRL)+(T−σ−σ\nRL−Tσσ\nRL)/bracketrightbig\n,(10)\nabove equation gives spin current in Left terminal. Spin\ncurrent in right terminal are obtained from the same\nequation by interchanging L↔R. On right hand side\nin Eq. (10), σand−σrefers to up and down spin states\nalongˆu. From the above equation and previous discus-\nsion it is evident that even in time reversal symmetric\ntwo terminal systems ESC are non zero. Incase SU(2)\nsymmetry in spin space is preserved, the spin resolved\ntransmission and reflection probabilities obey a further\nrotational symmetry in spin space, i.e, Tσσ′\nαβ=T−σ−σ′\nαβ,\nRσσ′\nαα=R−σ−σ′\nααand spin flip components are zero, which\nimplies that spin currents are identically zero for all ter-\nminals as is evident from Eq. (10). This conclusion re-\nmains valid even for systems without time reversal sym-\nmetry as can be seen easily from Eq. (8).\nThe expression in Eq. (10) can be cast in a more useful\nform as (the details will be provided in Ref.[14]),\nIs,eq\nαˆu=1\n2π/integraldisplay\nTrσ[{ΓαGrΓβGa+\n(ΓαgrΓαga)}(σ·u)]f(E,µ)dEdk/bardbl.(11)\nIn the above equation all symbols represents 2 ×2 matri-\nces in spin space( in σ·ubasis) and trace is taken over\nspin space. Where Γ α,βrepresents broadening matrices\ndue to contacts, Gr(a)are retarded and advanced Green\nfunction and gr,ais a off diagonal matrix in spin space\ndefined as gr,a= [{0,Gr,a\nσ−σ},{Gr,a\n−σσ,0}]. First term in\nEq. (11) corresponds to spin resolved transmission while\nthe second and third term give spin resolved reflection\nprobabilities as required by Eq. (10). Notice that the\nabove formula can not be simply written in terms of\ntransmission matrix and it is reminiscent of the charge\ncurrentformulaforinteractingsystemderivedinRef.[12].\nIn our case this happens for spin current because in pres-\nence of SO interaction spin can not be described as a\nnon-interacting object.\nEquilibrium spin currents in two terminal Rashba sys-\ntem:We now apply Eq.(11) to study ESC(zero tempera-\nture)inafinite sizeRashbasampleoflength L, contacted\nby two ideal and identical unpolarized leads. The Hamil-\ntonian for two-dimensional electron system with Rashba\nSO interaction and short-range spin independent disor-\nder isH=/planckover2pi12k2/(2m∗)I+λso(σxky−σykx) +U(x,y),\nwhereλsois Rashba SO coupling strength, U(x,y) is the\nrandom disorder potential and Iis 2×2 identity ma-\ntrix. Neglecting weak localization effects, the disorder\naveraged retarded Green function including the effect of\nleads is given by[13],\nGr,a(E,k) =E−/planckover2pi12k2\n2m+iη(k)+λso(σxky−σykx)\n[(E−/planckover2pi12k2\n2m∗+iη(k))2−(λsok)2]\n(12)4\nwithη(k) = (2γ(k) +/planckover2pi1\nτ(k)) where τ(k) is momentum\nrelaxation time due to elastic scattering caused by im-\npurities and γ(k) broadening due to leads. The contact\nbroadeningmatricesinEq.(11)arediagonalinspinspace\nand defined as Γ 1,2= [{γ(k),0},{0,γ(k)}]. Physically\nsignificance of γ(k) is that it represents /planckover2pi1/2 times the\nrate at which an electron placed in a momentum state\nkwill escape into left lead or right lead, hence as a first\napproximation we can write, γ(k) =/planckover2pi1vx(k)\nL≡/planckover2pi12kcos(φ\nmL,\nwhereφis angle with respect to xaxis. The impurity\nscattering time can be approximated as1\nτ(k)≈/planckover2pi1k\nlel, where\nlelis elasticmean free path. With these inputs we can in-\ntegrate Eq. (11) over transverse momentum (multichan-\nnel case) and energy to obtain an analytical expression\nfor equilibrium spin current. We find that ESC with\nspin parallel(antiparallel) to the ˆzorˆzaxis and flowing\nto thexdirection vanishes in both leads( Iˆxs\nˆz≡σzvx=0,\nIˆxs\nˆx≡σxvx=0). The ESC with spin parallel(antiparallel)\nto theˆyaxis are nonzero and given by,\nI−ˆx,s\nL,+ˆy=I+ˆx,s\nR,−ˆy≅m∗λsoEFL\n32π/planckover2pi12(2+(L\nlel)2).(13)\nIn left lead ESC is polarizedalong+ ˆydirectionand flows\noutwards from sample to lead,i.e.,along −ˆxdirection,\nwhile in the right lead ESC is polarized along −ˆyand\nflows outwards from sample to lead,i.e.,along + ˆxdirec-\ntion. Physically this implies that spin angular momen-\ntum is generated in sample with SO coupling which then\nflows outwards in the regions where SO coupling is zero,\ni.e., the left and right leads. This implies a spin rectifi-\ncation effect which can only occur if the transport is non\nlinear and we see that this consistent with nonlinear na-\nture of spin currents as remarked earlier. It is important\nto note that due to ESC there is no net magnetization in\nthe total system (sample+leads) which is consistent with\nthe Kramer’s degeneracy.\nWe can gain a deeper understanding of the above ex-\npression if we analyze the systems using additional sym-\nmetries. The disorder averaging establishes reflection\nsymmetry with respect to x(y)axis and the system has\na symmetry related with the operator σyRy. (σxRx).\nAs a result the total symmetry operator(time rever-\nsal+reflection) for the system is UT R=ItσxRxσyRy=\nIt(iσz)RxRy, whereItis time reversal operator. Under\nthis symmetry operation the disorder averaged system\nis invariant and the spin current operators σxvx,σyvx\nare even while σzvxis odd. Therefore the spin current\nalongˆzdirectionvanishes while in-planespin currentcanbe non zero. However as we have seen above that only\ntheycomponent of spin current survives after integrat-\ning over momentum. Fig.(1) illustrate the conservation\nofσyvxunder these symmetry operation. The depen-\ndence of ESC on λsocan also be inferred from symmetry\nconsideration. As we can check easily that the Rashba\nSO interaction changes sign under reflection along ˆz\naxis(λso(σxky−σykx)/mapsto→ −λso(σxky−σykx)). Physi-\ncally it corresponds to reversing the asymmetry of con-\nfining potential along ˆzaxis. Therefore the spin currents\nxy\nzspin parallel to\n+ y direction\nflow parallel to \n−ve x directionspin parallel to\n−ve y directionflow parallel to \n+ve x direction−\nFIG. 1: Fig 1: The figure illustrate the conservation of spin\ncurrentσyvxif th system is rotated by πalongˆzaxis, which\nrepresents reflection along and y axis respectively. Under t his\ntransformation configuration in left and right goes over int o\neach other hence the spin current remains invariant.\n(equilibrium as wellnon-equilibrium) can only depend on\nthe odd powers of spin orbit coupling constants λso. Ac-\ncording to Eq. (13) ESC are proportional to λsowhich is\nconsistent with these symmetry consideration and ESC\nvanishes if λsozero as expected on physical grounds. It\nis worth noting that even the local ESC in macroscopic\nRashba medium, discussed in Ref.[2] are proportional to\nλ3\nsoandaswehaveseenthisisaconsequenceofsymmetry\nconsideration. Moreover ESC are proportional to length\nof SO region because SO region acts as source of these\ncurrents. Note that the right hand-side of Eq. (13) has\ndimension of angular momentum per unit time signifying\nthat these currents are truly transport current.\nTo conclude, we have derived spin current formula\nfor multiterminal spin transport for system with broken\nSU(2) symmetry in spin space. We have demonstrated\nthat spin currents are fundamentally different from the\ncharge currents and the ESC are generically non zero. In\nview of this the spin transport phenomena in mesoscopic\nsystem needs a fresh look. Further it will also be inter-\nesting and desirable to study different magnetoresistance\nphenomena from the perspective of spin currents.\nI acknowledge helpful discussion with M. B¨ uttiker and\nA. M. Jayannavar.\n[1] I. Zutic, J. Fabian, and S. Das Sarma, Rev. Mod. Phys.\n76, 323 (2004).\n[2] E. I. Rashba, Phys. Rev. B 68, 241315(R) (2003). J. Shi,\nP. Zhang,D. Xiao, and Q. Niu, Phys. Rev. Lett. 96,\n076604 (2004).\n[3] M. B¨ uttiker, Phys. Rev. B. 46, 12485 (1992).,M. B¨ uttiker, Phys. Rev. Lett. 57, 1761 (1986).\n[4] J. H. Bardarson, I. Adagideli and P. Jacquod, Phy. Rev.\nLett.98196601, (2006). W. Ren,Z. Qiao,S. Q. Wang\nand H. Guo,Phy. Rev. Lett. 97066603, (2006).\nE. M. Hankiewicz, L. W. Molenkamp,T. Jungwirth and\nJ. Sinova,Phy. Rev. B. 70241301 (2004). B. K. Nikolic,5\nL. P. Zrbo and S. Souma, Phys. Rev. B 72075361(2005).\n[5] Y. Jiang and L. Hu, Phys. Rev. B 75, 195343 (2007).\n[6] M. Scheid, D. Bercioux, and K. Ritcher, New. J. Phys.\n9, 401 (2007).\n[7] T. P. Pareek, Phys. Rev. Lett 92, 076601 (2004).\n[8] T. P. Pareek, Phys. Rev. B. 66, 193301 (2002). T. P. Pa-\nreek, Phys. Rev. B. 70, 033310 (2004).\n[9] R. H. Silsbee, J. Phys:Cond. Matter 16, R179-R207\n(2004).[10] A. Brataas,Yu. V. Nazarov, and G. E. W. Bauer, Phys.\nRev. Lett. 842481 (2000).\n[11] A. A. Kiselev and K. W. Kim, Phys. Rev. B 71, 153315\n(2005).\n[12] Y. Meir and N. S. Wingreen, Phys. Rev. Lett. 682512\n(1992).\n[13] M. A. Skvortsov, JETP Lett. 67, 133 (1998).\n[14] T. P. Pareek manuscripr under preparation."    },    {        "title": "1705.04427v1.Analytical_slave_spin_mean_field_approach_to_orbital_selective_Mott_insulators.pdf",        "content": "Analytical slave-spin mean-\feld approach to orbital selective Mott insulators\nYashar Komijani1;\u0003, and Gabriel Kotliar1,\n1Department of Physics and Astronomy, Rutgers University, Piscataway, New Jersey, 08854, USA\n(Dated: October 29, 2021)\nWe use the slave-spin mean-\feld approach to study particle-hole symmetric one- and two-band\nHubbard models in presence of Hund's coupling interaction. By analytical analysis of Hamiltonian,\nwe show that the locking of the two orbitals vs. orbital-selective Mott transition can be formulated\nwithin a Landau-Ginzburg framework. By applying the slave-spin mean-\feld to impurity problem,\nwe are able to make a correspondence between impurity and lattice. We also consider the stability of\nthe orbital selective Mott phase to the hybridization between the orbitals and study the limitations\nof the slave-spin method for treating inter-orbital tunnellings in the case of multi-orbital Bethe\nlattices with particle-hole symmetry.\nINTRODUCTION\nIron-based superconductors are the subject of inten-\nsive study in the pursuit of high-temperature super-\nconductivity [1{7] . These systems are interacting via\nCoulomb repulsion and Hund's rule coupling and they\nrequire the consideration of multiple bands with crys-\ntal \feld and inter-orbital tunnelling [8, 9]. Early DMFT\nstudies, pointed out the importance of the corrlations [10]\nand Hund's rule coupling [11], and reported a notice-\nable tendency towards orbital di\u000berentiation, with the\ndxyorbital more localized than the rest [12]. They also\ndemonstrated orbital-spin separation [13{15]. Note that\nthe orbital di\u000berentiations has been recently observed in\nexperiments [16].\nAnother perspective on the electron correlations in\nthese materials is that the combination of Hubbard in-\nteraction and Hunds coupling place them in proximity to\na Mott insulator [17] and, correspondingly, the role of the\norbital physics is provided by the orbital selective Mott\npicture [18, 19]. Ref. [18] demonstrated an orbital selec-\ntive Mott phase in the multi-orbital Hubbard models for\nsuch materials, in the presence of the inter-orbital kinetic\ntunneling. In such a phase, the wavefunction renormal-\nization for some of the orbitals vanishes. Such a phase\nhas been observed in angle-resolved photo-emission spec-\ntroscopy (ARPES) experiments [20, 21]. Although desir-\nable, these e\u000bects have not been understood analytically\nin the past, partly due to the fact that an analytical\nstudy is di\u000ecult for realistic models. However, there are\nsimpler models, capable of capturing part of the relevant\nphysics, which are amenable to such analytical under-\nstanding, and this is what we study in this paper.\nThe mean-\feld approaches to study these problems\nrely on various parton constructions or slave-particle\ntechniques. The latter include slave-bosons [22, 23],\nKotliar-Ruckenstein four-boson method [24] and its rota-\ntionally invariant version [25], slave-rotor [26], Z2slave-\nspin [27{30] and its U(1) version [18, 31], slave spin-1\nmethod [32] and the Z2mod-2 slave-spin method [33, 34].\nFor a comparison of some of these methods see Appendix\nA. While these methods are all equivalent in the sensethat they are exact representation of the partition func-\ntion if the degrees of freedom are taken into account ex-\nactly, di\u000berent approximation schemes required for an-\nalytical tractability, lead to di\u000berent \fnal results and\ntherefore they have to be tested against an unbiased\nmethod like the dynamical mean-\feld theory (DMFT)\n[36{44] in large dimensions or density function renormal-\nization group (DMRG) [45] in one dimension.\nWe use the Z2slave-spin [27{30] in the following to\nstudy the orbital selectivity with and without Hund's\ncoupling. We brie\ry go through the method for the sake\nof completeness and setting the notations. By study-\ning the free energy analytically we develop a Landau-\nGinzburg theory for the orbital selectivity. A Landau-like\npicture has been useful in understanding the Mott tran-\nsition in in\fnite dimensions. Using a Landau-Ginzburg\napproach, we show how the interaction in the slave-spin\nsector tend to lock the two bands together in absence of\nHund's coupling and that the Hund's coupling promotes\norbital selectivity. We also apply the method to an im-\npurity problem (\fnite- UAnderson impurity) and its use\nas an impurity show that the slave-spin mean-\feld re-\nsult can be understood as the DMFT solution with an\nslave-spin impurity solver. This puts the method in per-\nspective by showing that the mean-\feld result is a subset\nof DMFT. Additionally, we study the e\u000bect on the orbital\nselective Mott phase produced by inter-orbital kinetic\ntunnelling and point out to some of the limitations of\nthe slave-spin for treating such inter-orbital tunnelling in\nparticle-hole symmetric Bethe lattices. Finally, we study\nstudy the instability of the orbital selective Mott phase\nby including hybridization between the two orbitals.\nZ2Slave-spin method\nWe consider the Hamiltonian H=H0+Hint, where\nH0=X\nhiji\u000b\ft\u000b\f\nijdy\ni\u000bdj\f(1)\nWe must demand t\u000b\f\nij= [t\f\u000b\nji]\u0003for this Hamiltonian to\nbe Hermitian. Unless mentioned explicitly, \u000bis a super-arXiv:1705.04427v1  [cond-mat.str-el]  12 May 20172\nindex that contains both spin and orbital degrees of free-\ndom. We replace the d-fermions with the parton con-\nstruction [27]\ndy\ni\u000b= ^zi\u000bfy\ni\u000b; ^zi\u000b=\u001cx\ni\u000b: (2)\n\u001c\u0016\ni\u000b,\u0016=x;y;z are SU(2) Pauli matrices acting on an\nslave-spin subspace per site/spin/\ravour, that is intro-\nduced to capture the occupancy of the levels. Slave-spin\nstatesj*i\u000biandj+i\u000bicorrespond to occupied/unoccupied\nstates of orbital/spin \u000bat sitei, respectively. Away from\nhalf-\flling, [28] has shown that \u001cx\ni\u000bhas to be replaced\nwith\u001c+\ni\u000b=2+c\u000b\u001c\u0000\ni\u000b=2 wherecis a gauge degree of freedom\nand is determined to give the correct non-interacting re-\nsult. Here, for simplicity we assume p\u0000h(particle-hole)\nsymmetry and thus maintain the form of Eq. (2). Note\nthat this parton construction has a Z2gauge degree of\nfreedom\u001cx;y!\u0000\u001cx;yandf!\u0000f, thus the name Z2\nslave-spin. The representation (2) increases the size of\nthe Hilbert space. Therefore, the constraint\n2fy\ni\u000bfi\u000b=\u001cz\ni\u000b+ 1; (3)\nto imposed to remove the redundancy and restrict the\nevolution to the physical subspace. Using Eqs. (2,3) it\ncan be shown that the standard anti-commutation rela-\ntions ofd-electron are preserved.\nPlugging Eq. (2) in H0, and imposing the constraint\n(on average) via a Lagrange multiplier, we have\nH0=X\nhiji\u000b\ft\u000b\f\nijfy\ni\u000bfj\f^zy\ni\u000b^zj\f\u0000\u0015i\u000b[fy\ni\u000bfi\u000b\u0000(\u001cz\ni\u000b+ 1)=2]\nOn a mean-\feld level, the transverse Ising model of slave-\nspins can be decoupled from fermions. The decoupling\nis harmless in large dimensions [46] as the leading op-\nerator introduced by integrating over the fermions be-\ncomes irrelevant at the critical point of the transverse\nIsing model. Therefore, writing H0\u0019Hf+H0S, we have\nH0S=X\nhiji\u000b\fJ\u000b\f\nijh\n^zy\ni\u000b^zj\f\u0000Q\u000b\f\niji\n+X\n\u000b\u0015i\u000b\u001cz\ni\u000b=2;\nHf=X\nhiji\u000b\f~t\u000b\f\nijfy\ni\u000bfj\f\u0000\u0015i\u000b(fy\ni\u000bfi\u000b\u00001=2) (4)\nwhere ~t\u000b\f\nij=t\u000b\f\nijQ\u000b\f\nijwithQ\u000b\f\nij=h^zy\ni\u000b^zj\fiis the renor-\nmalized tunnelling and J\u000b\f\nij=t\u000b\f\nijhfy\ni\u000bfj\fiis an Ising\ncoupling between slave-spins. The advantage of the par-\nton construction (2) is that the interaction Hintf\u001cgcan\nbe often written only in terms of the slave-spin variables,\nso thatH=Hf+HSandHS=H0S+Hint.\nParticle-hole symmetry - p\u0000hsymmetry on the orig-\ninal Hamiltonian is de\fned as ( nis a site index)\ndn\u000b!(\u00001)ndy\nn\u000b; dy\nn\u000b!(\u00001)ndn\u000b (5)\nOn a bipartite lattice, the nearest neighbor tunnelling\nterm preserves p\u0000hsymmetry, even in presence of inter-\norbital tunnelling. So, if the system is at half-\flling theHamitonian is invariant under p\u0000hsymmetry. We have\nto decide what p\u0000hsymmetry does to our slave-spin\n\felds. We choose\nfn\u000b!(\u00001)nfy\nn\u000b; \u001cx\nn\u000b!\u001cx\nn\u000b; \u001cz\nn\u000b!\u0000\u001cz\nn\u000b (6)\nSo, we see that if the original Hamiltonian had p\u0000h\nsymmetry, we necessarily have \u0015i\u000b= 0.\nSingle-site approximation - The Hamiltonian HSis a\nmulti-\ravour transverse Ising model which is non-trivial\nin general. Following [27{34] we do a further single-site\nmean-\feld for the Ising model, exact in the limit of large\ndimensions:\n^zy\ni\u000b^zj\f\u0019h^zy\ni\u000bi^zy\nj\f+ ^zy\ni\u000bh^zj\fi\u0000h^zy\ni\u000bih^zj\fi;(7)\nThe last term together with the second term of Eq. (8)\ncontributes a\u00002P\nhiji\u000b\fJ\u000b\f\nijQ\u000b\f\nij. We de\fne zi\u000b=h^zi\u000bi\nandZi\u000b=jzi\u000bj2as the wavefunction renormalization of\norbital\u000bat sitei. The slave-spin Hamiltonian becomes\n(using the symmetry of J\u000b\f\nij)\nH0S=X\ni\u000b(h\u0003\ni\u000b^zi\u000b+h:c:); hi\u000b=X\nj\fJ\u000b\f\nijzj\f(8)\nIn translationally invariant cases hi\u000bandzi\u000bbecome in-\ndependent of the site index and J\u000b\f\nijdepends on the dis-\ntance between sites iandj. Therefore, we can simply\nwriteh\u000b=P\n\fJ\u000b\fz\fwhere\nJ\u000b\f\u0011X\n(i\u0000j)J\u000b\f\n(i\u0000j)=X\njt\u000b\f\nijD\nfy\ni\u000bfj\fE\n:\nIn absence of inter-orbital tunnelling, Jis a diagonal ma-\ntrix, corresponding to individual orbitals, where for each\norbitalJ\u000b=RD\u000b\n\u0000D\u000bd\u000f\u001a\u000b(\u000f)f(\u000f)\u000fis the average kinetic en-\nergy and depends only on bare parameters, una\u000bected\nby the renormalization factor z. For semicircular band\n(Bethe lattice),J=\u00000:2122D, while for a 1 Dtight-\nbinding modelJ1D=\u00000:318DwithD= 2t. Since the\noperator ^z\u000b=\u001cx\n\u000bis Hermitian, we can write the slave-\nspin Hamiltonian (for each site) as [47]\nHS=X\n\u000ba\u000b\u001cx\n\u000b+Hint (9)\nwherea\u000b= 2P\n\fJ\u000b\fz\f(at half-\flling). The only non-\ntrivial part of computation is the diagonalization of HS.\nThis is a 4Mdimensional matrix where Mis the number\nof orbitals. The free energy (per site) is\nF=\u00001\n\fX\nnkTr log[\u0000G\u00001\nf(k;i!n)]\u00002X\nnJ\u000b\fz\u0003\n\u000bz\f\n\u00001\n\flogn\nTrh\ne\u0000\fHSio\n: (10)\nHere\f= 1=Tis the inverse temperature and the second\npart comes from two constants introduced in Eqs. (4) and3\n(7). At zero temperature, the \frst term is just J\u000b\fz\u0003\n\u000bz\f\nand the last term is ESwhich depends on zviaa. Hence,\nF=\u0000X\n\u000b\fJ\u000b\fz\u0003\n\u000bz\f+ES(fag): (11)\nONE-BAND MODEL\nIn the one-band case the interaction is Hint=\nUP\ni~ni\"~ni#where ~ni\u001b\u0011ni\u001b\u00001=2. Representing the\nlatter with \u001cz\ni\u001b=2 and using translational symmetry we\nobtainHint!(U=4)\u001cz\n\"\u001cz\n#. Since we are in the para-\nmagnetic phase ( a\"=a#), only sum of the two spins\n2~T=~ \u001c\"+~ \u001c#enter (the singlet decouples) and the Hamil-\ntonain can be written as HS= 2aTx+U\n2(Tz)2\u0000U=4,\ncreating a connection to the spin-1 representation of [32].\nFurthermore, we can form even and odd linear combina-\ntions of the empty and \flled states and at the half-\flling,\nonly the even linear super-positions enters the the Hamil-\ntonian. Thus, choosing atomic states of HSas\nj \u00060i=j*i\u0006j+ip\n2;j \u00061i=j*+i\u0006jOip\n2(12)\nwithE\u00060=\u0000U=4 andE\u00061=U=4, the Hamiltonian\ncan be written as HS= 2a\u001cx+ (U=4)\u001czwhere~ \u001care\nPauli matrices acting between j +0iandj +1i, i.e. it\nreduces to the Z2mod-2 slave-spin method [33, 34]. In\nwriting the states in Eq. (12) we have used a short-hand\nnotation (also used in the next section) j*\"+#i ! j*i\nandj+\"*#i!j+i ,j*\"*#i!j*+i and so on. The in-\nset of Fig. (1b) shows a diagrammatic representation of\nthe slave-spin Hamiltonian and two states decouple. The\nground state of HSis that of a two-level system\nES=\u0000U\n4p\n1 + (4\u000b=U)2 (13)\nwith the level-repulsion \u000b= 2aand the zero-temperature\n(free) energy is given by [factor of 2 sdue to spin]\nF= 2sjJjz2+ES(z) (14)\nThe free energy is plotted in Fig. (1a) and it shows a\nsecond-order phase transition as Uis varied. Close to\nthe the transition \u000b!0, we can approximate ES\u0019\n\u00002\u000b2=U+ 8\u000b4=U3. Writing the \frst term of the free en-\nergy as +\u000b2=8jJj, we can read o\u000b the critical interaction\nUC= 16jJj. Minimization of the free energy gives the\nGutzwiller projecion fomrula of Brinkman and Rice [48]\nZ=\u001a1\u0000u2u<1\n0u>1(15)\nwithu=U=UCand is plotted in Fig. (1b). At \fnite\ntemperature this procedure gives a \frst order transition\nterminating at a critical point [34].\nFIG. 1: (color online) (a) Free energy (at T= 0) as a\nfunction of zshowing a second-order phase transition as\nU=UCis varied. (b) Wavefunction renormalization\nZ=jzj2as a function of Uhas the Brinkman-Rice\nform. Inset: Diagrammatic representation of the\nslave-spin Hamiltonian. Each dot denotes on atomic\nstate. Two states decouple and HSis equivalent to that\nofZ2mod-2 slave-spin.\nSpectral function - The Green's functions of the d-\nfermionsGd(\u001c)\u0011\n\u0000Td\u001b(\u001c)dy\n\u001b(0)\u000b\nfactorizes\nGd;\u001b(\u001c)\u0019\n\u0000Tf\u001b(\u001c)fy\n\u001b(0)\u000b\nhT\u001c\u001cx\n\u001b(\u001c)\u001cx\n\u001b(0)i (16)\nto thef-electron and the slave-spin susceptibility and\nthus the spectral function is obtained from a convolu-\ntion with the slave-spin function Ad(!) =Af(!)\u0003AS(!),\nin whichAfis a semicircular density of states with the\nwidthZand within single-site approximation ASis\nAS(!) =Z\u000e(!) +1\u0000Z\n2[\u000e(!+ 2ES) +\u000e(!\u00002ES)] (17)\nThe spectral density has the correct sum-rule (in contrast\nto the usual slave-bosons [22, 23]) since the commuta-\ntion relations of the slave-spins are preserved. However,\nthe single-site approximation does not capture incoherent\nprocesses, and this re\rects in sharp Hubbard peaks in the\nMott phase ( Z= 0) where Af=\u000e(!). Also, the spatial\nindependence of the self-energy implies that the inverse\ne\u000bective-mass of \\spinons\" m=~m=Z[1 + (m=kF)@k\u0006]\nis zero in the Mott phase. This is again an artifact of\nthe single-site approximation. Both of these problems\nare remedied, e.g. by doing a cluster mean-\feld calcu-\nlation [28, 33] or including quantum \ructuations around\nthe mean-\feld value within a spin-wave approximation\nto the slave-spins [33].\nThe fact that (beyond single-site approximation)\nspinons disperse in spite of h\u001cxi! 0 and they carry a\nU(1) charge as seen by Eq. (2), implies that vanishing of\nh\u001cxidoes not generally correspond to the Mott phase in\n\fnite dimensions. However, in large dimensions, this is\ncorrect [34] and that is what we refer to in the following.4\nTWO-BAND MODEL\nIn absence of inter-orbital tunnellings, the free-energy\nis\nF=a2\n1=2jJ1j+a2\n2=2jJ2j+ES(a1;a2) (18)\nwhereESis the ground state of the slave-spin Hamilto-\nnian. For two bands we have the interaction\nHint=U(~n1\"~n1#+ ~n2\"~n2#)\n+U0(~n1\"~n2#+ ~n1#~n2\")\n+ (U0\u0000J)(~n1\"~n2\"+ ~n1#~n2#) +HXP (19)\nwhere ~n\u000b\u0011nf\u000b\u00001=2 =\u001cz\n\u000b=2. The spin-\rip and pair-\ntunnelling terms are\nHXP=\u0000JX[dy\n1\"d1#dy\n2#d2\"+dy\n1#d1\"dy\n2\"d2#]\n+JP[dy\n1\"dy\n1#d2#d2\"+dy\n2\"dy\n2#d1#d1\"]:(20)\nThis term mixes the Hilbert space of f-electron with that\nof slave-spins. Following [27{29] we include this term ap-\nproximately by dy\n\u000b\u001b!\u001c+\n\u000b\u001bandd\u000b\u001b!\u001c\u0000\n\u000b\u001bsubstitution so\nthat it acts only in the slave-spin sector. The justi\fcation\nis that such a term captures the physics of spin-\rip and\npair-hopping. Using the spherical symmetry U0=U\u0000J\nthis can be written as\nHint=U\n2(~n1\"+ ~n1#+ ~n2\"+ ~n2#)2\u0000U\n2+HXP\n\u0000J[~n1\"~n2#+ ~n1#~n2\"+ 2~n1\"~n2\"+ 2~n1#~n2#] (21)\nForJX=JandJP= 0 it has a rotational symmetry\n[49]. Alternatively, U0=U\u00002JandJX=JP=J\nhas rotational symmetry. The choice does not a\u000bect the\ndiscussion qualitatively. We keep the former values in\nthe following.\nAtomic orbitals - We start by diagonalizing the atomic\nHamiltonian in absence of the hybridizations. Close to\nhalf-\flling the doubly-occupied states have the lowest en-\nergy and are given by\nj \u00060i=j*1*2i\u0006j+ 1+2ip\n2; E\u00060=\u0000U\u0000J=2;\nj \u00061i=j*1+2i\u0006j+ 1*2ip\n2; E\u00061=\u0000U+J=2\u0007JX;\nj \u00062i=j*+ 1;O2i\u0006jO1*+2ip\n2; E\u00062=\u0000U+ 3J=2\u0007JP;\nThese 3 doublets become the 6-fold degenerate ground\nstate when J!0. The 1;3-particle states are then next\nj \u00063i=j*+ 1ij*2i\u0006j+ 2ip\n2; E\u00063=\u00151;\nj \u00064i=jOi1j*2i\u0006j+ 2ip\n2; E\u00064=\u0000\u00151\nj \u00065i=j*1i\u0006j+ 1ip\n2j*+ 2i; E\u00065=\u00152;\nj \u00066i=j*1i\u0006j+ 1ip\n2jOi2; E\u00065=\u0000\u00152;and \fnally, there are two (empty and quadruple occu-\npancy) states at the top of the ladder\nj 7i=j*+i1j*+i2; E 7=\u00151+\u00152+ 3U\u00003J=2;\nj 8i=jOi1jOi2; E 8=\u0000\u00151\u0000\u00152+ 3U\u00003J=2:\nNo Hund's rule coupling - The hybridization causes\ntransition among atomic states. In the case of no Hund's\ncoupling we can block diagonalize HSinto several sectors\nand diagrammatically represent it as shown in Fig. (2).\nTherefore, the calculation can be reduced from 16 \u000216 to\n5\u00025. The larger the level-repulsion, the lower the ground\nstate energy in each sector. The fact that the slave-spins\ndecouple into several sectors brings about the possibility\nof possible ground-state crossings between various sectors\nas the parameters a1anda2are varied. Here, however,\nit can be shown that the sector Chas the lowest ground\nstate energy for arbitrary parameters.\nFIG. 2: Diagrammatic representation of the slave-spin\nHamiltonian HSin the two-band model with J= 0 and\n\u00151=\u00152= 0. Each dot represents an atomic state with a\ncertain energy, denoted on the left, whereas the connecting\nlines represent o\u000b-diagonal elements of the Hamiltonian\nmatrix, all assumed to be real. We have used the short-hand\nnotationp\n2 S=D\na;b\u0011 a\u0006 b. Also note that ai= 2Jizi. The\nHamiltonian factorizes into several sectors.\nNumerical minimization of the free-energy leads to\nFig. (3) which reproduces the results of [27]. For t2=t1>\n0:2 the metal-insulator transition happens at the same\ncriticalUfor the two bands and we refer to it as the\nlocking phase , whereas for t2=t1<0:2 the critical Ufor\nthe bands are di\u000berent U2< U 1and we refer to it as\norbital selective Mott (OSM) phase .\nIn order to have the result analytically tractable we do\none further simpli\fcation and that is to project out the\nzero and quartic occupancies per site, by dropping the\nhigh energy site at the apex of sector C. We expect such\nan approximation to be valid close to the Mott transition\nof the wider band, but invalid at low U. As a result5\nFIG. 3: (color online Wavefunction renormalizations Z1\n(blue) andZ2(green) as a function of U=UC1in absence of\nHund's rule coupling J= 0. The states at the bottom row\ncorrespond to doubly occupied sites. The middle-row states\nhave occupancy of 1 or 3 and the states at the top row\ncorrespond to zero or four-electron \fllings. (a) Moderate\nbandwidth anisotropy t2=t1= 0:5 shows locking. (b) Large\nbandwidth anisotropy t2=t1= 0:15 can unlock the bands and\ncause OSM transition (OSMT). We also reproduce the kink\nin the wider-bandwidth (blue) band as the narrow band\ntransitions to the Mott phase [27], marked with an arrow. In\nthe OSM phase, the wavefunction renormalization of the\nwider band follows the Brinkman-Rice formula (solid line).\nthe sectorCdecouples into two smaller sectors C\u0006, each\nequivalent to a two-level system with the level-repulsions\n\u000b\u0006=q\na2\n1(3=2 +p\n2) +a2\n2(3=2\u0000p\n2)\n\u0006q\na2\n1(3=2\u0000p\n2) +a2\n2(3=2 +p\n2):(22)\nThe ground state energy of the slave-spin sector is deter-\nmined with \u000b+inserted in the ESexpression (13) (after\nan inert\u0000U=4 energy shift). Note that this ground state\nhas theZ2symmetrya1$a2of the Hamiltonian HS.\nES(\u000b+) as a function of ( a2\n1\u0000a2\n2)=(a2\n1+a2\n2), is mini-\nmized fora1=a2. Discarding empty and \flled states\ncorresponds to truncating part of the Hilbert space and\nthus leads to reduced wavefunction renormalization at\nU\u00180. In Fig. (4) we have compared our analytical so-\nlution to that of the exact result. When a2= 0, Eq. (22)\ngives\u000b!2a1as in the single-band case and there-\nfore, same critical interaction UC1= 16jJ1jis obtained.\nBut for symmetric bands a1=a2, it gives\u000b= 2p\n3a.\nFollowing similar analysis as before, the free energy is\na2=jJj\u0000 2\u000b2=Uand we obtain UC= 24jJj= 1:5UC1in\nagreement with [27, 29].\nLocking vs. OSM phase - We formulate the locking vs.\nOSM question as the following. Under what condition,\na1>0 buta2= 0 can be a minima of the Free energy. As\nmentioned before, setting a2= 0,\u000bin Eq. (22) reduces to\nthe one-band \u000b!2a1. Therefore, the Mott transition for\nthe wide band happens at the same critical Uas before.\nTo have a non-zero a1solution, we must have U < Uc1.\nThe pointa2= 0 always satis\fes dF=da 2= 0. To ensure\nthat it is the energy minima we need to check the second\nFIG. 4: (color online) A comparison of numerical\nminimization of the free energy vs. the analytical two-level\nsystem. Discarding the empty and full occupancy states\nleads to underestimation of ZasU!0 but close to the\nMott transition the approximation is accurate.\nderivative\nd2F\nda2\n2\f\f\f\na2=0=1\njJ2j\u00005\njJ1j>0; (23)\nwhich gives the condition jJ2=J1j<0:2.\nWe can better understand the transition by using an\norder parameter. The trouble with the expression of \u000bis\nthat it cannot be Taylor expanded when a1anda2are\nboth small. However, we may assume a2=ra1, with\nras an order parameter replacing a2, and write down\n\u000b(a1;a2) =a1\u000b(r) where\u000b(r) =\u000b+(a1!1;a2!r). A\n\fniterclose to the transition implies locking whereas r=\n0 orr=1implies OSM phase. Close to the transition\nof both bands \u000b\u00190 and we can write ES\u0019\u00002\u000b2=U+\n8\u000b4=U3and Eq. (18) becomes\nF(a1;r) =a2\n1W\n2jJ1j+O(a4); Wx(r;u) = 1 +xr2\u0000\u000b2(r)\n4u\nHere,x=jJ1=J2j, andu=U=UC1. The metal-insulator\ntransition for a1happens when the mass coe\u000ecient W\nchanges sign. For negative W,a2\n1>0 and we still have to\nminimize the free energy with respect to r. At smallr, we\ncan expand \u000b(r)\u00192 + 5r2. To zeroth order in r, theW-\nsign-change happen at u= 1. Another transition from\nr= 0 tor > 0 happens when the corresponding mass\nterm (x\u00005=u)r2changes sign, giving the same critical\nbandwidth ratio xc= 5 as we had before. So we have two\nequationsW(r;u) = 0 and@rW(r;u) = 0. The function6\nWis plotted in the \fgure and the transition from locking\nr>0 to OSM phase r= 0 are shown.\nFIG. 5: The coe\u000ecient W(r;u) is shown for various uas\nfunction of r=a2=a1. Equations W= 0 and@rW= 0 are\nsatis\fed at the minimum of the red curve, which is (a) at a\n\fniter= 1 in the Locking phase, jJ1j=jJ2j. (b) and zero\nr= 0 in the OSM phase, jJ1j\u00155jJ2j.\nLarge Hund's coupling - In presence of Hund's cou-\npling the slave-spin Hamiltonian is modi\fed to the dia-\ngram shown in Fig. (6).\nFIG. 6: Diagrammatic representation of the slave-spin\nHamiltonian HSin the two-band model at half-\flling with\nin presence of Hund's rule coupling J. Various degeneracies\nare lifted by J-interaction. In the limit of large Hund's\ncouplingJ=U!1=4 we may only keep sector Cand neglect\nall the gray lines.\nThe ground state still belongs to the sector C. In the\nlimit of large J=U!1=4, we may ignore all the gray lines\non the block Cand \fnd that the ground state is that of\na two-level system, Eq. (13) with the level-repulsion\n\u000b= 2q\na2\n1+a2\n2 (24)\nIt is remarkable that the (orbital) rotational invariance\nof the model (even though absent in HS) is recovered in\nthis ground state. When the two bands have the same\nbandwidth, this formula predicts UC=UC1. SinceESno longer depends on a2\n1\u0000a2\n2, there is no more compe-\ntition between the two terms and an slight bandwidth\nasymmetry lead to OSM phase. This can be formulated\nagain, following previous section, in terms of stability of\naa16= 0 buta2= 0 solution. We can check that\nd2F\nda2\n2\f\f\f\na2=0=1\njJ2j\u00001\njJ1j>0; (25)\nwhich givesjJ2j<jJ1jas the su\u000ecient condition for\nOSMT, i.e., any di\u000berence in bandwidth drives the sys-\ntem to the OSM phasse. Alternatively, by expanding the\nlevel-repulsion in this case \u000b(r)\u00192 +r2and plugging\nit intoW(r;u), we \fnd that the critical bandwidth ratio\nxc=jJ1=J2jis equal to one.\nTUNNELLING BETWEEN THE ORBITALS\nA very interesting question is about the fate of or-\nbital selective Mott phase upon turning on an inter-\norbital tunnelling. The band in Mott insulating phase\nhas one electron per site forming localized magnetic mo-\nment. There is a large entropy associated with this phase\nand it is natural to expect that it would be unstable to-\nward possible ordering. A possible mechanism that can\ncompete with magnetic ordering, is the Kondo screening\nof the insulating band by the itinerant band, leading to\nconduction in the former and opening a hybridization gap\nin the latter band (e\u000bectively a new locking e\u000bect com-\ning from Kondo screening). Within single-site approxi-\nmation, however, the form of the renormalized coupling\n~t\u000b\f\nij=z\u0003\ni\u000bt\u000b\f\nijzj\fimplies that once an orbital goes to the\nMott phase, it automatically shuts down its coupling to\nall the other orbitals. We speculate that this e\u000bect might\nbe responsible for the orbital selective Mott transition so-\nlution found in [18]. However, it is still a valid question\nwhether or not the critical interactions UCfor a Mott\ntransition are modi\fed by inter-orbital tunnelling, which\nwe explore in the following.\nBefore treating inter-orbital tunnelling, we discuss\nhow the slave-spin method can be applied to the impurity\nproblem, and its relation to the lattice.\nImpurity vs. Lattice and the DMFT loop\nWe can also apply the slave-spin method to an im-\npurity problem. In particular, we can use the slave-spin\n(as well as any other slave-particle) method as an im-\npurity solver for the DMFT. We show in the following\nthat the slave-spin mean-\feld result corresponds to such\na DMFT solution with the corresponding slave-spin im-\npurity solver. This puts the method on \frm ground and\nallows comparison between various methods.\nFirst, consider a generic p\u0000hsymmetric impurity\nmodel described by the Hamiltonian H=H0+Hint7\nwhere\nH0=\u0000X\nk\u000b\ft\u000b\f\nk(dy\n\u000bck\f+h:c:) +X\nk\f\u000f\f\nkcy\nk\fck\f:(26)\nAgain\u000b;\f are superindices that include both orbital\nand spin. We have assumed that the bath is diagonal\nand discarded any local `crystal \feld' dy\n1d2for simplic-\nity. In the simple case of single-orbital impurity Hint=\nU~nd\"~nd#. Via a substitution of Eq. (2), the hybridiza-\ntion term becomes H0=\u0000P\nk\u000b\ft\u000b\f\nk(fy\n\u000b\u001cx\n\u000bck\f+h:c:).\nThis problem can be written in a similar way as be-\nforeH\u0019Hf+HSwhereHSis exactly what we had\nin single-band lattice case. However, since the fy\n\u000b\u001cx\n\u000bck\f\ninteraction happens only on the impurity site, we do not\nneed the second single-site approximation here, and ob-\ntaina\u000b=\u00002P\u000b\f\nk\ft\u000b\f\nkhfy\nk\u000bck\fi. In order to have a gen-\neral formalism that applies to both impurity and lattice,\nas well as scenarios with inter-orbital tunnelling for which\nJ\u000b\frenormalizes and is di\u000ecult to compute, we regard a\nandzas independent variables and write the free energy\nof Eq. (12) as [47]\nF(fz;ag) =Ff(fzg) +FS(fag)\u0000X\n\u000ba\u000bz\u000b: (27)\nThe saddle-point of Fwith respect to aandzgives the\ncorrect mean-\feld equations. Ffis the free energy of\nthef-electron given by Ff=\u0000TP\nnTr log[\u0000G\u00001\nf(i!n)]\nwhereG\u00001\nf(i!n) =i!n1\u0000zy\u0001(i!n)zwith\u0001(i!n) =P\nktyGc(k;i!n)t, the hybridization function. The slave-\nspin part is given by FS= Tr[e\u0000\fHS] where for a single-\norbital Anderson impurity, HS= 2a\u001cx+U\u001cz=4, as we\nhad in the single-band case before.\nThe mean-\feld equations w.r.t zandaare, respec-\ntively\na\u000b=1\nz\u000bZd!\n\u0019f(!)!Im\u0002\nG\u000b\u000b\nf(!+i\u0011)\u0003\n; (28)\nz\u000b=dFS\nda\u000b: (29)\nThe \frst equation provides a relation between aand\nzthat generalizes a\u000b= 2P\n\fJ\u000b\fz\f(see appendix D).\nHaving expressions for ES(a) we can eliminate ain fa-\nvor ofz, or vice versa, which is equivalent to a Legendre\ntransformation. In the appendix C, we apply these equa-\ntions to the (single-orbital) \fnite- UAnderson impurity\nproblem and show the `transition' to the Kondo phase as\nthe temperature is lowered.\nIn a lattice, the free energy has the same\nform as Eq. (27) with the di\u000berence that Ff=\n\u0000TP\nk;nTr log[\u0000G\u00001\nf(k;i!n)] where the Green's func-\ntion isGf(k;!+i\u0011) = [(!+i\u0011)1\u0000zyEkz]\u00001. It can\nbe shown that exactly same mean-\feld equations are ob-\ntained ifGfin Eq. (28) is replaced with G\u000b\u000b\nf(!+i\u0011)!P\nkG\u000b\u000b\nf(k;!+i\u0011). Therefore, we conclude that the twoproblems (lattice and impurity) are equivalent provided\nthat the hybridization function in the impurity problem\nis chosen such that the impurity Green's function and the\nlocal Green's function of the lattice are equal, i.e.\n[i!n1\u0000zy\u0001(i!n)z]\u00001=X\nk[(!+i\u0011)1\u0000zyEkz]\u00001:(30)\nwhich is the DMFT consistency equation. Therefore,\nslave-spin mean-\feld is equivalent to a DMFT solution\nusing the slave-spin method as the impurity solver. Also,\nnote that a lattice problem in the OSM phase, corre-\nsponds to an impurity problem in which the hybridiza-\ntion of one of the orbitals to the bath has been turned\no\u000b [50]. See also Appendix B.\nInter-orbital tunnelling\nSlave spins have been used to study Iron-based su-\nperconductors [18] where the inter-orbital tunnelling are\nimportant. We study this tunnelling e\u000bect in the speci\fc\ncase withp\u0000hsymmetry and without orbital-splitting\n(which allows for analytic calculations). The cases that\ngo beyond such conditions, as arising in the models for\nthe iron-based superconductors [18], remain to be ex-\nplored and are left for future work.\nA troublesome feature of the slave-spins is that\nthey break the rotational symmetry among the orbitals.\nWithin the p\u0000hsymmetric Bethe lattices that we study\nhere, this rotational variation leads to ambiguities in\npresence of inter-orbital tunnellings, as we point out here.\nLet us consider a 1D chain with two orbitals H0=\n\u0000P\nn\u001b(Dy\nn\u001bTDn+1;\u001b+h:c:), no Hund's coupling in Hint,\nand a dispersion\nEk=\u00002Tcosk;T=\u0012t11t12\nt12t22\u0013\n(31)\nWe have chosen t12=t21and all the elements real (and\npositive)to preserve the p\u0000hsymmetry. Strictly speak-\ning, in 1D the mean-\feld factorization that led to Eq. (4)\nand the consequent single-site approximation are both\nunjusti\fed. The choice of dimensionality, here, is only\nfor the ease of discussion and not essential to the con-\nclusions. As long as the dispersion matrix can be diago-\nnalized with a momentum-independent unitary transfor-\nmation (as well as any Bethe lattice, see the appendix\nD), the following discussion applies. Diagonalizing the\ntunnelling matrix gives E\u0006\nk=\u00002t\u0006coskwith\nt\u0006=t11+t22\n2\u0006r\u0010t11+t22\n2\u00112\n\u0000detT (32)\nIncluding renormalization just changes t\u000b\f!~t\u000b\f. We\ncan simply use the diagonalized form of the tunnelling8\nmatrix to calculate F0atT= 0. Assuming det T>0,\nFf=X\n\r=\u0006Zdk\n2\u0019E\u0006\nkf(E\r\nk)\n!\u0000(~t++~t\u0000)Z\u0019=2\n\u0000\u0019=2dk\n\u0019cos(k) =\u00002(~t11+~t22)=\u0019\nNote thatt12does not enter the free energy. Inserting\nthis expression into Eq. (27) and setting dF=dzi= 0, we\ncan remove aiin favor of zi. This seems to imply that\nthere is a \fnite threshold (topological stability) for inter-\norbital tunnelling: as long as det T>0, introducing t12\ndoes not change anything in the problem and it simply\ndrops out and OSM phase is stable against inter-orbtital\ntunnelling. For large t12eventually det T<0. So, we get\nt+>0 andt\u0000<0 and second band is inverted and F0\nbecomes\nFf!\u00002(~t+\u0000~t\u0000)=\u0019=\u00004\n\u0019s\n\u0010~t11\u0000~t22\n2\u00112\n+\f\f~t12\f\f2(33)\nHence,t12has non-trivial e\u000bects on renormalization.\nOn the other hand, we could have used the rotational\ninvariance of Hintand done a rotation in d1\u0000d2ba-\nsis to band-diagonalize H0with the bandwidths T!\ndiagft+;t\u0000g, before using slave-spins to treat the inter-\nactions. It is clear then that t12always has non-trivial\ne\u000bects by modifying t\u0006. For example we could start in\nthe locking phase where t\u0000=t+>0:2, and by increasing\nt12slightly get to the OSMT phase t\u0000=t+<0:2, without\nchanging the sign of det T. This paradox exist for any\np\u0000hsymmetric lattice with diagonalizable tunnelling\nmatrix. The root of the problem is that our expression\nin Eq. (9) is not invariant under rotations between vari-\nous orbitals. Therefore, the critical value where the OSM\nphase persists, is basis-dependent. This ambiguity calls\nfor the use of unbiased techniques to understand the role\nof inter-orbital tunnelling on OSMT. It might be that\nthe model we studied analytically here is a singular limit\nwhich can be avoided by breaking p\u0000hsymmetry and\ninclusion of crystal \feld in more realistic settings [18].\nThis remains to be explored in a future work.\nAs discussed in [25], the way to achieve rotational-\ninvariance is to liberate the f-electrons that describe\nquasi-particles from the physical d-electrons. This is\nachieved by a d\u000b!P\n\f^z\u000b\ff\frepresentation which leads\nto a wavefunction-renormalization matrix z\u000b\f=h^z\u000b\fi\nwith o\u000b-diagonal elements. So far, we have not been\nable to generalize the slave-spin to a rotationally invari-\nant form and we leave it as a future project.\nON-SITE INTER-ORBITAL HYBRIDIZATION\nEven though models for the Iron-based superconduc-\ntors have \fnite crystal level splitting and no on-site hy-\nbridization, it is interesting to introduce a hybridizationbetween the two orbitals within the current formalism\n[27]. This is interesting, because the on-site hybridiza-\ntion, does not su\u000ber from the singe-site approximation\nh^zi\u000b^zi\fi 6=h^zi\u000bih^zi\fi, as opposed to the inter-orbital\ntunnelling andh^zi\u000b^zi\fiappears as an independent or-\nder parameter, which leads to the emergence of Kondo\nscreening as we show in this section.\nWe can include a termP\nn;\u001b(v12dy\nn;1\u001bdn;2\u001b+h:c:) to\nthe Hamiltonian. In order to preserve the p\u0000hsymmetry,\nv12has to be purely imaginary. The modi\fcations to the\nmean-\feld Hamiltonians are\n\u0001Hf=X\nn;\u001b(~v12fy\nn;1\u001bfn;2\u001b+h:c:)\u00002sA12Z12(34)\n\u0001HS=X\n\u001bA12\u001cx\n1\u001b\u001cx\n2\u001b (35)\nwhere ~v12=v12Z12withZ12=h\u001cx\n1\u001b\u001cx\n2\u001biandA12=\nv12P\nnhfy\nn;1\u001bf2\u001bi+h:c:.Z12andA12are are related to\neach other via the Hamiltonian above and they are inde-\npendent of \u001bin the paramagnetic regime. Alternatively,\nwe can regard them as independent and impose the mean-\n\feld equation Z12=@FS=@A 12to eliminate A12by a\nLagrange multiplier. Assuming a small A12we can com-\npute the change in slave-spin energy using second-order\nperturbation theory. The result is of the form \u0001 ES=\n\r(A12)2where\ris (in absence of Hund's coupling) a\npositive constant which contains all the matrix elements\nand the inverse gaps \r=P\nj\u001b\u001b0h 0j\u001cx\n1\u001b\u001cx\n2\u001bj ji(Ej\u0000\nE0)\u00001h jj\u001cx\n1\u001b0\u001cx\n2\u001b0j 0iwhereEjandj jiare the eigen-\nvalue/states of the HSsolved in the previous section.\nEliminating A12in favor ofZ12we \fnd that the free en-\nergy of the system is\nF(z1;z2;Z12) =\u00002s\n\fX\nknTr log\u0012~\u000fk1\u0000i!niZ0\n12\n\u0000iZ0\n12 ~\u000fk2\u0000i!n\u0013\n+E0\nS(z1;z2) +(Z0\n12)2\n\r0(36)\nHereE0\nSis the value of ES(a1;a2)\u0000P\niaiziin absence\nof hybridization v12in whicha1anda2are eliminated in\nfavor ofz1andz2. Also, we have rede\fned jv12jZ12!\nZ0\n12and\rjv12j2!\r0.\nEq. (36) is nothing but the free energy of a Kondo\nlattice at half-\flling [51] with renormalized dispersions\n~\u000fk1and ~\u000fk2. In a Kondo lattice, this form of the free\nenergy appears using Z0\n12as the Hubbard-Stratonovitch\n\feld that decouples the Kondo coupling \r0~S2\u0001dy\n1~ \u001bd1. Here\nS2=dy\n2~ \u001bd2is the spin of the Mott-localized band and\n\r0plays the role of the Kondo coupling. As a result of\nthis coupling, a new energy scale TK\u0018Dexp[\u00001=\r0] ap-\npears, with D\u00182~t11the bandwidth of the wider band,\nbelow which the Kondo screening takes place which in\nthep\u0000hsymmetric case gaps out both bands but away\nfromp\u0000hsymmetry mobilizes the Mott localized band.\nEither way, we conclude that orbital selective Mott in-\nsulating phase is unstable against hybridization between9\nthe two orbitals in agreement with [27]. However, even\nthough a true selective Mottness is unstable, orbital dif-\nferentiation, re\rected as large di\u000berence in e\u000bective mass\ncan exist [16].\nCONCLUSION\nIn conclusion, we have used slave-spin mean-\feld\nmethod to study two-band Hubbard systems in presence\nof Hund's rule coupling. We have developed a Landau-\nGinzburg theory of the locking vs. OSMT. We discussed\nthe relation between slave-spins and the KR boson\nmethods (Appendix). We have also applied the method\nto impurity problems and shown a correspondence\nbetween the latter and the single-site approximation\nof the lattice using the DMFT loop. Finally, we have\ndiscussed the limitations of the slave-spin method for\nmulti-orbital models with both particle-hole symmetry\nand inter-orbital tunnelling and shown that the orbital\nselective Mott phase is unstable against on-site hy-\nbridization between the two orbitals.\nWe appreciate valuable discussions with P. Coleman,\nT. Ayral, M. Metlitski, L. de'Medici, K. Haule and\nC.-H. Yee, and in particular, a detailed reading of the\nmanuscript and constructive comments by Q. Si. The\nauthors acknowledge \fnancial support from NSF-ONR.\nAfter completion of this manuscript, we became aware\nof another work [52] which contains a Landau-Ginzburg\ntheory of OSMT in presence of the inter-orbital tun-\nnelling. The conclusions of the two work agrees wherever\nthere is an overlap.\nAPPENDIX\nA. Various slave-particle methods\nFor a one band model, KR introduces four bosons and\nuses the representation ^ zy\n\u001b=P+[py\n\u001be+dyp\u0000\u001b]P\u0000, where\npy\n\u001b,eyanddyare (hardcore) bosonic creation operators\nfor\u001b-spinon, holon and doublon, respectively and P\u0006are\nprojectors that depend on the occupations of the bosons\nand are introduced to normalize the probability ampli-\ntudes over the restricted set of physical states. On the\nother hand, a SU(2) spin-variable ~ \u001c\u000bcan be represented\nby two Schwinger bosons a\u000bandb\u000bsatisfying the con-\nstraintay\n\u000ba\u000b+by\n\u000bb\u000b= 1 (hardcore-ness), via\n\u001cz\n\u000b=by\n\u000bb\u000b\u0000ay\n\u000ba\u000b; \u001cx\n\u000b=ay\n\u000bb\u000b+by\n\u000ba\u000b (37)\nOn an operator level, the two methods have the same\nHilbert space as depicted in Table (I) for the case of oneay\n\"a\"by\n\"b\"ay\n#a#by\n#b#\n1 0 1 0\n0 1 1 0\n1 0 0 1\n0 1 0 1eyepy\n\"p\"py\n#p#dyd\n10 0 0\n01 0 0\n00 1 0\n00 0 1\nTABLE I: Comparison of the Schwinger boson\nrepresentation of the slave-spin (left) and\nKotliar-Ruckenstein slave-bosons (right).\norbital. Average polarization of the spin along various di-\nrection in the Bloch sphere corresponds to condensation\nofaandbbosons.\nA trouble with the slave-spin representation is that\nthef-quasi-particles carry the charge of the d-electron\nand thus the disordered phase of the slave-spins (in which\nthef-electrons still disperse beyond single-site approxi-\nmation) is not a proper description of the Mott phase.\nAs a remedy, it has been suggested [31] to replace \u001cxin\nEq. (2) with \u001c+and \fxing the problem of non-unity Z\nin the non-interacting case by applying \fne-tuned pro-\njectors ^zy=P+\u001c+P\u0000. We note that this looks quite\nsimilar to KR.\nForMspinful orbitals, KR requires introducing 4M\nbosons (only one of them occupied at a time) whereas\nonly 2Mslave-spins are required (each with the Hilbert\nspace of 2). Thus the size of the two Hilbert spaces are\nthe same 22M= 4M.\nB. General low-energy considerations\nGenerally for a lattice we can expand the self-energy\nGd(k;!) = [!1\u0000Ek\u0000\u0006d(k;!)]\u00001; (38)\nExpanding the self-energy\n\u0006d;lat(k;!) =\u0006(0;0) +~k\u0001@~k\u0006(0;0) +!@!\u0006(0;0) +\u0001\u0001\u0001\nWithin single-site approximation, the second term is\nzero. Denoting the third term as @!\u0006d\u00191\u0000Z\u00001and\nassuming Z=zzywe can write\nGd(k;!)\u0019z[!1\u0000zyEkz]\u00001zy; (39)\nwhich simply means Gf(!) = [!1\u0000~Ek]\u00001and the corre-\nlation functions of the slave-particles are just decoupled\nh^zi\u000b(\u001c)^zy\nj\fi!z\u0003\n\u000bz\fwithin single-site approximation also\ndiscarding any time dynamics. For the tunnelling ma-\ntrix, we simply have ~t=zytz. Similarly, for an impurity\nwe have\ni!n1\u0000\u0006d;imp (i!n) =G\u00001\nd;imp(i!n); (40)\nGd;loc(i!n) =X\nk[i!n\u0000Ek\u0000\u0006d;lat(k;i!n)]\u00001(41)10\nDenoting the interaction part of the self-energy\n\u0006d;imp (i!n) =\u0001(i!n) +\u0006d;I(i!n), the DMFT approxi-\nmation identi\fes \u0006d;I(i!n) =\u0006d;lat(k;i!n). Again ex-\npanding\u0006d;I(!)\u0019(1\u0000Z\u00001)!we have\nGd;imp (i!n) =z[i!n1\u0000~\u0001(i!n)]\u00001zy; (42)\nwith ~\u0001(i!n) = zy\u0001(i!n)zin agreement with\nGf;imp (i!n) = [i!n1\u0000~\u0001(i!n)]\u00001. Using the same\napproximation for Gd;locleads to\nG\u00001\nd;loc(i!n)!zX\nk[i!n\u0000~Ek]\u00001z (43)\nthe DMFT self-consistency loop equation is Gf;loc(i!n) =\nGf;imp (i!n) or\nX\nk[i!n\u0000~Ek]\u00001= [i!n1\u0000~\u0001(i!n)]\u00001: (44)\nWithin the slave-spin approach there are no interactions\n\u0006f;imp =zy\u0001(i!n)z;\u0006f;I= 0;\u0006f;lat= 0 (45)\nand Eq. (44) is satis\fed as it does for any non-interacting\nproblem.\nRotation - Using the vector Dfor thed-electrons, in\npresence of inter-orbital tunnelling we may sometimes\nbe able to eliminate such inter-orbital tunnelling by a\nrotation to D=UD\u0006. SinceD=zF, we assume the\nsame rotation in the F-spaceF=UF\u0006(otherwise they\nwould contain inter-orbital tunnelling) and the two z-s\nare related by z=Uyz\u0006U. Assuming that Uis a SO(2)\nmatrix, and z\u0006is diagonal, we \fnd\nz=z++z\u0000\n21\u0000z+\u0000z\u0000\n2 \n\u0000cos 2\u000bsin 2\u000b\nsin 2\u000bcos 2\u000b!\n(46)\nwhich has o\u000b-diagonal elements. Note that if one of the\nz\u0006elements vanishes, e.g. z\u0000= 0, we can factorize z\nz=z+ \ncos\u000b\n\u0000sin\u000b!\u0010\ncos\u000b\u0000sin\u000b\u0011\n: (47)\nThen, it can be seen that Z=zzy!z+zhas the same\nform. This basically means one linear combination of f\nelectrons is decoupled (localized) and the itinerant spinon\nband carries characters of both d1andd2bands. This\nbasis-dependence of the orbital Mott selectivity is again\nan artefact due to lack of rotational invariance.\nC. Finite- UAnderson model\nThe slave-spin part of the Hamiltonian is as we had\nin the one band case. We can use Eq. (27) to eliminate a\nin favour of z. In the wide band limit for the conductionband, we have Gf(i!n) = [i!n\u0000i\u0001Ksign(!n)]\u00001where\n\u0001K=\u0019\u001at2z2, and the free energy is\nF(z) =\u00002sZD\n\u0000Dd!\n\u0019f(!)Im [log (i\u0001K\u0000!)] +E0\nS(z):(48)\nE0\nSis obtained by eliminating afromES(a)\u00002sazpart\nof the free energy in Eqs. (13) and (27) and is equal to\nE0\nS=\u0000U\n4p\n1\u0000z2. Here, we have done a simpli\fcation to\nreplaceFSwith its zero temperature value (ground state\nenergy) while maintaining the temperature dependence\nof theFf. We expect this approximation to be valid in\nthe large-Ulimit especially close to the transition. The\nmean-\feld equation w.r.t zis\nzZD\n\u0000Dd!f(!)Re\u00141\n!\u0000i\u0001K\u0015\n+U\n4\u001at2zp\n1\u0000z2= 0 (49)\nClose to the transition, the second term is e\u000bectively like\naz=\u001aJ withJ(z)\u0011(4t2=U)p\n1\u0000z2. At zero tempera-\nture the left-side simpli\fes\nzlog\u0001K\nD+z\n\u001aJp\n1\u0000z2=zlog\u0001K\nTK(z)= 0; (50)\nwhereTK(z) =De\u00001=\u001aJ(z). So to have non-zero zwe\nmust have \u0001 K=TKwhich determines z. Also, we can\ngo to non-zero temperature. We just replace the log-term\nin above expression with its \fnite-temperature expression\nfrom Eq. (49)\nzReh\n~ (i\u0001K)\u0000~ (D)i\n+z\n\u001aJ(z)= 0 (51)\nThis is solved numerically and the result shown in\nFig. (7). It shows a Kondo phase z>0 forT <T0\nK.\nFIG. 7: (color online) The order parameter zfor the\nAnderson model calculated from numerical evaluation of\nEq.(53).T0\nK=De\u00001=\u001aJ, whereJ= 4t2=U. Note that this is\no\u000b with a factor of 4, an artifact of slave-spin method. We\nhave usedD= 100Uwhereast=Uis varied.11\nD. Stability of OSMT against interorbital tunnelling\nin a Bethe lattice\nUsing equations of motion, the coe\u000ecient J\u000b\fde\fned\nEq. (4) can be related to the correlation function of the\nelectrons at the same site, The result is\nJ\u000b\f=t\u000b\fX\n\u0011[~t\u00001]\f\u0011Zd!\n2\u0019f(!)!A\u0011\u000b\nii(!) (52)\nThis together with a\u000b= 2P\n\fJ\u000b\fz\fleads to Eq. (27).\nIn a Bethe lattice we can use recursive methods [53] to\ncomputeA\u000b\f\nii. When the tunnelling matrix is hermitian\nand there is no chemical potential or crystal \feld, the\nprocedure is especially simple. We diagonalize the renor-\nmalized tunnelling matrix ~t=U~tDU\u00001. Then the re-\ntarded and the spectral functions are\nGR(!) =~t\u00001U\u0003(!)U\u00001;A(!) =UAD(!)U\u00001(53)\nwhere diagonal matirix \u0003contains\u0015-elements that sat-\nisfy\u0015i+\u0015\u00001\ni=!=ti\nDwith the retarded boundary condi-\ntion.ADis diagonal matrix of semicircular density states\nwhose width are given by the eigenvalues of ~t. By plug-\nging this into Eq. (53) and (52) and using\nZd!\n2\u0019f(!)!AD\nii(!) =\u00000:2122\u00022\f\f\f~tD\f\f\f\nwe see that if the eigenvalues of the matrix ~tall have\nthe same sign, then U(j~tDj=~tD)U\u00001=~t. This is the\ngeneralization of the protection of OSM phase against\ninter-orbital tunnelling, discussed in the 1D case in the\npaper. For the case of two bands,\ndet~t>0) J\f\u000b=\u00000:2122\u00022t\f\u000b\u000e\f\u000b;\ndet~t<0) J\f\u000b=\u00000:2122\u00022t\f\u000bR\u000b\f(54)\ni.e. for det ~t>0, theJ-matrix does not have any o\u000b-\ndiagonal elements and the diagonal elements are propor-\ntional to the bare diagonal hoppings (as before), but but\nif det ~t<0, there is a matrix R=U\u001czU\u00001multiplying\nelement-by-elements of the J-matrix which does depend\non renormalization.\nAgain in this problem, one could have done the ro-\ntation ind\u000b-sector before using the slave-spins, in which\ncase, inter-orbital tunnelling would have an e\u000bect and\ncould cause OSM transition. Therefore, the stability\nfound above is basis-dependent. This ambiguity is ab-\nsent whenp\u0000hsymmetry is broken and the tunnelling\nmatrix cannot be diagonalized independent of the mo-\nmentum [18].\n[1] Y. Kamihara, T. Watanabe, M. Hirano, H. Hosono, J.\nAm. Chem. Soc. 130, 3296 (2008).[2] D. C. Johnston, Adv. Phys. 59, 803-1061 (2010).\n[3] G. R. Stewart, Rev. Mod. Phys. 83, 1589 (2011).\n[4] F. Wang, D.-H. Lee, Science 332, 200-204 (2011).\n[5] P. Dai, Rev. Mod. Phys. 87, 855-896 (2015).\n[6] Q. Si, R. Yu, E. Abrahams, Nature Reviews 1, 1 (2016).\n[7] P. J. Hirschfeld, Comptes Rendus Physique 17, 197\n(2016).\n[8] M. Daghofer, A. Moreo, J. A. Riera, E. Arrigoni,\nD. J. Scalapino, E. Dagotto, Phys. Rev. Lett. 101,\n237004 (2008).\n[9] M. J. Calderon, B. Valenzuela, E. Bascones, Phys. Rev.\nB80, 094531 (2009).\n[10] K. Haule, J. H. Shim, and G. Kotliar, Phys. Rev. Lett.\n100, 226402 (2008).\n[11] K. Haule, G. Kotliar, New J. Phys. 11, 025021 (2009).\n[12] Z. P. Yin, K. Haule, G. Kotliar, Nature Materials 10,\n932-935 (2011).\n[13] Z. P. Yin, K. Haule, G. Kotliar, Phys. Rev. B 86, 195141\n(2012).\n[14] C. Aron, G. Kotliar Phys. Rev. B 91, 041110(R) (2015).\n[15] K. Stadler, Z. P. Yin, J. von Delft, G. Kotliar, A. Weich-\nselbaum, Phys. Rev. Lett. 115, 136401 (2015)\n[16] H. Miao, Z. P. Yin, F. Wu, J. M. Li, J. Ma, B.-\nQ. Lv, X. P. Wang, T. Qian, P. Richard, L.-Y. Xing,\nX.-C. Wang, C. Q. Jin, K. Haule, G. Kotliar, H. Ding,\nPhys. Rev. B 94, 201109(R) (2016).\n[17] Q. Si, E. Abrahams, Phys. Rev. Lett. 101, 076401 (2008).\n[18] R. Yu, Q. Si, Phys. Rev. Lett. 110, 146402 (2013).\n[19] L. de'Medici, G. Giovannetti, M. Capone, Phys. Rev.\nLett. 112, 177001 (2014).\n[20] M. Yi, D. H. Lu, R. Yu, S. C. Riggs, J.-H. Chu, B. Lv,\nZ. K Liu, M. Lu, Y.-T. Cui, M. Hashimoto, S. -K. Mo,\nZ. Hussain, C. W. Chu, I. R. Fisher, Q. Si, Z.-X. Shen,\nPhys. Rev. Lett. 110, 067003 (2013).\n[21] M. Yi, Z-K Liu, Y. Zhang, R. Yu, J.-X. Zhu, J. J. Lee,\nR. G. Moore, F. T. Schmitt, W. Li, S. C. Riggs, J.-\nH. Chu, B. Lv, J. Hu, M. Hashimoto, S.-K. Mo, Z. Hus-\nsain, Z. Q. Mao, C. W. Chu, I. R. Fisher, Q. Si, Z.-\nX. Shen, D. H. Lu, Nature Commun. 6, 7777 (2015).\n[22] S. E. Barnes, J. Phys. F 6, 1375 (1976) ibid.7, 2637\n(1977).\n[23] P. Coleman, Phys. Rev. B 29, 3035 (1984).\n[24] G. Kotliar, A. E. Ruckenstein, Phys. Rev. Lett. 57, 1362\n(1986).\n[25] F. Lechermann, A. Goerges, G. Kotliar, O. Parcollet,\nPhys. Rev. B 76, 155102 (2007).\n[26] S. Florens, A. Georges, Phys. Rev. B 70, 035114 (2004).\n[27] L. de'Medici, A. Georges, S. Biermann, Phys. Rev. B 72,\n205124 (2005).\n[28] S. R. Hassan, L. de' Medici, Phys. Rev. B 81, 035106\n(2010).\n[29] L. de'Medici, M. Capone, arXiv:1607.08468v1 (2016).\n[30] A. B. Georgescu, S. Ismail-Beigi, Phys. Rev. B 92,\n235117 (2015).\n[31] R. Yu, Q. Si, Phys. Rev. B 86, 085104 (2012).\n[32] M. Mardani, M.-S. Vaezi, A. Vaezi, arXiv:1111.5980\n(2011).\n[33] A. R uegg, S. D. Huber, M. Sigrist, Phys. Rev. B 81,\n155118 (2010).\n[34] R. \u0015Zitko, M. Fabrizio, Phys. Rev. B 91, 245130 (2015).\n[35] TheZ2mod-2 slave-spin introduces a spin variable\nthat corresponds to charge per site moduli 2 (appli-\ncable to single-orbital systems), whereas the slave-spin12\nmethod introduces separate slave-spin variables for each\nspin/orbital. To distinguish the two, we refer to the for-\nmer as mod-2 slave-spin method.\n[36] A. Georges, G. Kotliar, W. Krauth, M. J. Rozenberg,\nRev. Mod. Phys. 68, 13 (1996).\n[37] A. Koga, N. Kawakami, T. M. Rice, M. Sigrist, Phys.\nRev. B 72, 045128 (2005).\n[38] Y. Song, L.-J. Zou, Phys. Rev. B 72, 085114 (2005).\n[39] J. B unemann, D. Rasch, F. Gebhard, J. Phys.: Cond.\nMatt. 19, 436206 (2007).\n[40] A. I. Poteryaev, M. Ferrero, A. Georges, O. Parcollet,\nPhys. Rev. B 78, 045115 (2008).\n[41] Y. Song, L.-J. Zou, Eur. Phys. J. 72, 59 (2009).\n[42] K. S. D. Beach, F. F. Assaad, Phys. Rev. B 83, 045103\n(2011).\n[43] E. A. Winograd, L. de'Medici, Phys. Rev. B 89, 085127\n(2014).\n[44] Z.-Y. Song, H. Lee, Y.-Z. Zhang, New J. of Phys. 17,\n033034 (2015).[45] S. R. White, Phys. Rev. Lett. 692863 (1992).\n[46] R. Nandkishore, M. A. Metlitski, T. Senthil, Phys. Rev.\nB86, 045128 (2012).\n[47] When dealing with complex z\u000b, the \frst term of the\nHamiltonian (9) and the last term of the free energy\n(27) have to be modi\fed to a\u000bz\u000b!(h\u0003\n\u000b^z\u000b+h:c:) where\nh\u000b=P\n\fJ\u000b\fz\fand note that a\u000b=h\u000bc\u000b+h\u0003\n\u000b. Since\n^zy\n\u000b= ^z\u000bhere, it is more convenient to work with a\u000b.\n[48] W. F. Brinkman, T. M. Rice, Phys. Rev. B 2, 4302\n(1970).\n[49] A. Georges, L. de' Medici, J. Mravlje, Annu. Rev. Con-\ndens. Matter, 4, 137 (2013).\n[50] L. de'Medici, A. Georges, G. Kotliar, S. Biermann, Phys.\nRev. Lett. 95066402 (2005).\n[51] P. Coleman, Introduction to Many-Body Physics , Cam-\nbridge Univ. Press (2016).\n[52] R. Yu, Q. Si, private communication.\n[53] Y. Komijani, I. A\u000feck, Phys. Rev. B 90, 115107 (2014)."    },    {        "title": "1705.05826v3.Theory_of_electron_spin_resonance_in_one_dimensional_topological_insulators_with_spin_orbit_couplings.pdf",        "content": "Theory of electron spin resonance in one-dimensional topological insulators with\nspin-orbit couplings: Detection of edge states\nYuan Yao,1,\u0003Masahiro Sato,2, 3Tetsuya Nakamura,4Nobuo Furukawa,4and Masaki Oshikawa1\n1Institute for Solid State Physics, University of Tokyo, Kashiwa, Chiba 277-8581, Japan\n2Department of Physics, Ibaraki University, Mito, Ibaraki 310-8512, Japan\n3Spin Quantum Recti\fcation Project, ERATO, Japan Science and Technology Agency, Sendai 980-8577, Japan\n4Department of Physics and Mathematics, Aoyama-Gakuin University, Sagamihara, Kanagawa 229-8558, Japan\nEdge/surface states often appear in a topologically nontrivial phase when the system has a bound-\nary. The edge state of a one-dimensional topological insulator is one of the simplest examples.\nElectron spin resonance (ESR) is an ideal probe to detect and analyze the edge state for its high\nsensitivity and precision. We consider ESR of the edge state of a generalized Su-Schrie\u000ber-Heeger\nmodel with a next-nearest neighbor (NNN) hopping and a staggered spin-orbit coupling. The\nspin-orbit coupling is generally expected to bring about nontrivial changes on the ESR spectrum.\nNevertheless, in the absence of the NNN hoppings, we \fnd that the ESR spectrum is una\u000bected by\nthe spin-orbit coupling thanks to the chiral symmetry. In the presence of both the NNN hopping and\nthe spin-orbit coupling, on the other hand, the edge ESR spectrum exhibits a nontrivial frequency\nshift. We derive an explicit analytical formula for the ESR shift in the second-order perturbation\ntheory, which agrees very well with a non-perturbative numerical calculation.\nI. INTRODUCTION\nIn recent decades, topological phases have become a\ncentral issue in condensed matter physics. An important\nclass of topological phases is topological insulators and\ntopological superconductors1{4.\nIn condensed matter and statistical physics, one-\ndimensional (1-D) systems, which are amenable to\nseveral powerful analytical and numerical methods, of-\nten provide useful insights. 1-D topological phases are no\nexceptions. One of the simplest 1-D models possessing\nnontrivial topological nature is the Su-Schrie\u000ber-Heeger\n(SSH) model5, which has been used to describe the lattice\nstructure of polyacetylene [C 2H2]n. The SSH model can\nbe also applied to the 1-D charge density wave systems,\nsuch as quasi-one-dimensional conductors like TTF-\nTCNQ (tetrathiofulvalinium-tetracyanoquinodime-\nthanide) and KCP (potassium-tetracyanoplatinate)6.\nWhile the SSH model had been studied intensively\nmuch earlier than the notion of topological phases\nwas conceived, there is a renewed interest from the\nviewpoint of topology. In fact, distinct phases of the\nSSH model are classi\fed by the Zak phase7which is\na topological invariant, and the bulk winding number\nof the momentum-space Hamiltonian8. In this sense,\nthe SSH model can be regarded as a 1-D topological\ninsulator.\nAn important nontrivial signature of many topologi-\ncal phases is edge states. The SSH model indeed pos-\nsesses zero-energy edge states that are protected by a\nchiral symmetry8. The number of edge states at a do-\nmain wall is equal to the bulk winding number. This is\nknown as the bulk-boundary correspondence in the spin-\nless inversion-symmetric SSH model8. Experimentally,\n1-D systems with boundaries or edges can be realized by\nadding impurities to the material so that the system is\nbroken to many \fnite chains. However, the edge statesare often experimentally di\u000ecult to observe, since they\nare localized near the boundaries or the impurities and\ntheir contribution to bulk physical quantities is small.\nGiven this challenge, electron spin resonance (ESR) pro-\nvides one of the best methods to probe the edge states,\nthanks to its high sensitivity. In fact, the edge states\nof theS= 1 Haldane chain were created by doping im-\npurities and then successfully detected by ESR9,10. Fur-\nthermore, combined with near-edge x-ray absorption \fne-\nstructure experiments, ESR was applied successfully to\nprobe the magnetic edge state in a graphene nanoribbon\nsample11,12. Such a strategy could also be applied to 1-\nD topological insulators, which are described by the SSH\nmodel.\nAnother intriguing nature of ESR is that it is highly\nsensitive to magnetic anisotropies, such as the anisotropic\nexchange interaction, single-spin anisotropy, and the\nDzyaloshinskii-Moriya (DM) interaction. The e\u000bect of\nmagnetic anisotropies on ESR is well understood only\nin limited circumstances, and there remain many open\nissues13,14. These magnetic anisotropies are often con-\nsequences of spin-orbit (SO) coupling which generally\nbreaks spin-rotation symmetry. The e\u000bects of magnetic\nanisotropies and SO couplings also play important roles\nin magnetic dynamics in higher-dimensional topological\nphases1{4,15,16. Thus it is of great interest to study\nthe e\u000bect of SO coupling on ESR directly. However, this\nquestion has not been explored in much detail so far.\nAn obstacle for the potential experimental ESR study of\nSO coupling is the electromagnetic screening in metallic\nsystems. This problem does not exist in insulators. Un-\nfortunately, band insulators are generally non-magnetic\nand we cannot expect interesting ESR properties. On\nthe other hand, Mott insulators can have interesting\nmagnetic properties. However, strong correlation e\u000bects,\nwhich are essential in Mott insulators, make theoretical\nanalysis di\u000ecult.\nIn this context, the 1-D topological insulator providesarXiv:1705.05826v3  [cond-mat.str-el]  18 Nov 20172\na unique opportunity to study the e\u000bects of SO coupling\non ESR. This would be of signi\fcant interest in several\naspects. Experimentally, the insulating nature makes\nthe observation of edge states by ESR easier. Theoret-\nically, the interesting e\u000bects of anisotropic SO coupling\non ESR can be studied accurately for the SSH model\nof non-interacting electrons. Moreover, the chiral sym-\nmetry, which is essential for the well-de\fned topological\ninsulator phase, is often broken explicitly in realistic sys-\ntems. When we introduce a chiral-symmetry breaking\nperturbation to the 1-D SSH model, the energy eigen-\nvalues of the edge states generally deviate from zero en-\nergy. However, the edge states are expected to still sur-\nvive and be localized near the edge if the perturbation is\nsmall enough. As we will demonstrate, the ESR of the\nedge state can detect the breaking of the chiral symme-\ntry. The purpose of this paper is to present a theoretical\nanalysis on ESR of the edge states in 1-D topological\ninsulators, based on a generalized SSH model with SO\ncouplings. We demonstrate several interesting aspects of\nESR, which will hopefully stimulate corresponding ex-\nperimental studies.\nThe paper is organized as follows. In Sec. II, we present\nthe model of interest and review the basic topological na-\nture of the SSH model. The next three sections are the\nmain part of this paper. The properties of edge states\nare discussed in detail in Sec. III. In Sec. IV, we obtain\na compact analytical formula of the ESR frequency shift\nin perturbation theory with respect to SO coupling. Sec-\ntion V is devoted to a direct numerical calculation of the\nESR frequency shift, which is independent of the pertur-\nbative approach in Sec. IV. We \fnd that the perturba-\ntion theory agrees with the numerical results very well.\nFinally, we present conclusions and future problems in\nSec. VI.\nII. THE MODEL\nA. A generalized SSH model\nFirst let us consider a generalized SSH model with SO\ncoupling\nH0=\u0000+1X\nj=1\u001a\nt\u0002\n1 + (\u00001)j\u000e0\u0003\ncy\nj+1exp\u0014\ni(\u00001)j\u001e\n2~ n\u0001~ \u001b\u0015\ncj\n+h.c.g (1)\nwherecjis the two-component electron annihilation op-\neratorcj\u0011[cj\";cj#]Tat thej-th site,t > 0 is the\nnearest-neighbor (NN) electron hopping amplitude, and\n\u00001\u0014\u000e0\u00141 is the bond-alternation parameter. The\nangle\u001eand axis~ n(which is a unit vector) parametrizes\nthe SO coupling on the NN bond. The angle \u001edenotes\nthe ratio of the SO coupling to the hopping amplitude on\nthe bond. In this paper, we assume that \u001eis su\u000eciently\nsmall (j\u001ej\u001c1), which is the case in many real materials.Expanding \u001eto \frst order, we obtain a standard form\nwith so-called intrinsic and Rashba SO couplings17.\nIn our model Eq. (1), SO coupling is assumed to be\nstaggered along the chain. This would be required, in\nthe limit of \u000e0= 0, if the system had site-centered inver-\nsion symmetry. In general, other forms of SO coupling\nincluding the uniform one along the chain are also pos-\nsible. In this paper, however, we focus on the particular\ncase of the staggered SO coupling to demonstrate its in-\nteresting e\u000bects on the ESR spectrum.\nB. The SSH model and its topological properties\nIn the limit \u001e= 0, our model is reduced to the standard\nSSH model\nHSSH=\u0000Xn\nt\u0002\n1 + (\u00001)j\u000e0\u0003\ncy\nj+1cj+ h.c.o\n:(2)\nLet us \frst consider a system of 2 Nsites (Nunit cells)\nwith the periodic boundary condition (PBC). It is then\nnatural to take the momentum representation\nc2j;\u001b=1p\nNX\nkak;\u001bexp (ik(2j)); (3)\nc2j+1;\u001b=1p\nNX\nkbk;\u001bexp (ik(2j+ 1)); (4)\nwhere the summation of kis in the reduced Brillouin zone\n[0;\u0019) withk=n\u0019=N andn= 0;\u0001\u0001\u0001;N\u00001. Correspond-\ning to the each sublattice (even and odd), there are two\n\ravors of fermions, aandb. The Hamiltonian can then\nbe written as\nHSSH=X\nk(ay\nk;by\nk)hSSH(k)\u0012\nak\nbk\u0013\n; (5)\nwith\nhSSH(k)\u0011dx(k)\u001cx+dy(k)\u001cy; (6)\nwhere\u001cx;y;zare Pauli matrices acting on the \ravor space,\nanddx;y(k) are real numbers\ndx(k) =\u00002tcos(k);dy(k) = 2tsin(k)\u000e0: (7)\nThe spin indices are again suppressed in the Hamiltonian.\nFrom this expression, the single-electron energy reads\n\u000f(k) =\u0006q\ndx(k)2+dy(k)2=\u00062tq\ncos2k+\u000e02sin2k:\n(8)\nThe gap is closed at k=\u0019=2 when\u000e0= 0, while the sys-\ntem has a gap 4 tj\u000e0jwhenever the bond alternation does\nnot vanish ( \u000e06= 0). The gapless point can be regarded\nas a quantum critical point separating the two gapped\nphases,\u000e0<0 and\u000e0>0. Throughout this paper, we\nconsider the half-\flled case with 2 Nelectrons. The bulk\nmode near this gap-closing point has a linear dispersion\nrelation indicated in Fig. 1, and it can be described by a\none-dimensional Dirac fermion3.3\nFIG. 1. Band structure of the SSH model in Eq. (2) with\na periodic boundary condition.1The solid and dashed lines\nrespectively represent band structures of the insulating case\nwith\u000e06= 0 and the gapless point at \u000e0= 0. The low-energy\nphysics around k=\u0019=2 can be described by one-dimensional\nDirac fermion model.\nIt is evident from the Hamiltonian that each of the\ngapped phases is simply a dimerized phase. Neverthe-\nless, we can identify them as a trivial insulator phase\nand a \\topological insulator\" phase. This can be under-\nstood by considering the system with the open boundary\ncondition. Let us consider the chain of 2 Nsites labeled\nwithj= 1;2;:::; 2N, and the open ends at sites j= 1\nand 2N. For\u000e0>0 (\u000e0<0), sitesj= 2nandj= 2n+1\n(j= 2n\u00001 andj= 2n) form a dimerized pair, re-\nspectively. As a consequence, for \u000e0>0 the end sites\nj= 1 andj= 2Nremain unpaired. The electrons at\nthese unpaired sites give rise to S= 1=2 edge states.\nIn contrast, for \u000e0<0, there are no unpaired sites and\nthus no edge states. In this sense, \u000e0>0 is a topo-\nlogical insulator phase and \u000e0<0 is a trivial insulator\nphase. Of course, considering the equivalence of the two\nphases in the bulk, such a distinction involves an arbi-\ntrariness. That is, if we consider the an open chain of\nNsites withj= 0;1;:::; 2N\u00001, the edge states appear\nonly for\u000e0<0. It is still useful to identify the gapless\npoint at\u000e0= 0 as a quantum critical point separating\nthe topological insulator and the trivial insulator phase.\nThe particular shape of the Hamiltonian also implies\nthe existence of a chiral symmetry:\nfhSSH;\u0000g= 0; (9)\nwhere \u0000\u0011\u001cz. The chiral symmetry turns out to be im-\nportant for the distinction of the two phases. In the con-\ntext of the general classi\fcation of topological insulators,\nthe present system corresponds to the \\AIII\" class with\nparticle number conservation and the chiral symmetry in\none spatial dimension18,19.\nIn a general one dimensional free fermion system, we\ncan de\fne a topological invariant called the Zak phase7\nfor each band as follows:\n\rZak=iI\nBZh\t(k)jOkj\t(k)i; (10)\nFIG. 2. Amplitude j jof the edge-state wave function of the\nSSH model in Eq. (2) under an open boundary condition.8\nBlue and red colors respectively represent the spatial distri-\nbution of the existing probability for left and right localized\nedge states. The total site number is set to be even, the\ndimerization parameter \u000e0>0, and symbols vandwdenote\nhopping amplitudes t(1\u0000\u000e0) andt(1 +\u000e0), respectively. The\nwave-function amplitude decays exponentially into the bulk4.\nwherej\t(k)iis the Bloch wavefunction of the band with\nthe momentum k. In the presence of the chiral symmetry,\n\rZakis quantized to integral multiples of \u0019, if the band\nis separated from others by gaps20.\nFor the present two-band SSH model in Eq. (5) with\nthe chiral symmetry, we can compute the Zak phase using\nthe explicit Bloch wavefunction. For the lower band,\nj\t(k)i=1p\n2\u0012\nexp (\u0000i\u001ek)\n1\u0013\n; (11)\nwith\u001ek\u0011arctan[dy(k)=dx(k)]. As a result, we \fnd\n\rZak=\u0019= 1 for 0 < \u000e 0\u00141 in which the system is a\ntopological insulator. In the other case \u00001\u0014\u000e0<0,\nwhere the system is a trivial insulator, \rZak=\u0019= 0. In\nthis case, we can see that the Zak phase can be also iden-\ntifed20with a winding number of the Hamiltonian as\n\rZak\n\u0019=i\n\u0019I\nBZdkOkln [dx(k)\u0000idy(k)]: (12)\nWhen\rZak=\u0019= 1, there is an edge state localized at each\nend of an open \fnite chain as shown in Fig. 2. This is\nthe bulk-boundary correspondence21in the SSH model.\nIn general, this topological number can take arbitrary\ninteger values, corresponding to Zclassi\fcation of BDI\nor AIII class in d= 1 dimension. However, in the present\nSSH model, its value is restricted to 0 or 1.\nThe existence of the edge states in the SSH model can\nbe demonstrated by an explicit calculation for a \fnite-\nsize chain. In Fig. 3, we can see that, when \u000e0decreases\nfrom 1 to -1, the edge states merge into the bulk spectrum\nas\u000e0!0+. In addition, when the thermodynamic limit,\nN!+1, is taken in the open-end SSH model, the edge\nstates are strictly at zero energy and topologically stable\nagainst any local adiabatic deformation that respects the\nchiral symmetry8.4\n−1 −0.5 0 0.5 1−2−1012\nδ0/tEnergy spectrum EEnergy spectrum with di fferentδ0\nFIG. 3.\u000e0dependence of the energy spectrum of the SSH\nmodel with t= 1 andN= 40 under an open boundary con-\ndition. As \u000e0!0+, two localized edge states at the ends\nmerge into the bulk. For the limit N!+1, the edge states\nare strictly at zero energy as 0 < \u000e\u0014twhich are protected\nby the chiral symmetry.8\nC. ESR of edge states\nLet us consider ESR of the 1-d half-\flled topological in-\nsulator phases at the low-temperature and low-frequency\nlimitj!j;T\u001cj\u000e0jton which we focus in this paper. The\nESR contribution from bulk excitations is negligible in\nthis limit since there is a large bond-alternation driven\nband gap 4j\u000e0jt. On the other hand, spin-1/2 edge states\nare located at the (nearly) zero energy point in the band-\ngap regime. Therefore, ESR is dominated by the edge\nstate contribution.\nWhen the chiral symmetry is preserved and SO cou-\npling is absent, the edge spin is precisely equivalent to a\nfreeS= 1=2. In this case, the edge ESR spectrum is triv-\nial, which means that it just consists of the delta function\nat the Zeeman energy. However, breaking of the chiral\nsymmetry and introduction of SO couplings can bring\na nontrivial change on the edge ESR spectrum. In the\nfollowing, we shall analyze this e\u000bect theoretically.\nIn ESR, absorption of an incoming electromagnetic\nwave is measured under a static magnetic \feld. Thus\nwe introduce the Zeeman term for the static, uniform\nmagnetic \feld\nHZ=\u0000H\n2X\nj=1cy\nj(~ \u001b\u0001~ nH)cj; (13)\nwhere~ nHis a unit vector representing the direction of\nthe magnetic \feld, H > 0 is its magnitude, and ~ \u001b=\n(\u001bx;\u001by;\u001bz).\nIn the paramagnetic resonance of independent electron\nspins, absorption occurs for the oscillating magnetic \feld\nperpendicular to the static magnetic \feld, which is mea-sured in the standard Faraday con\fguration. Therefore,\nin this paper, we assume that the oscillating magnetic\n\feld is perpendicular to the static magnetic \feld ~ nH. The\nfrequency of the electromagnetic wave is denoted by !.\nIn an electron system with the SO interaction, the\nelectric current operator contains a \\SO current\" that\ninvolves the spin operator. Since the electric current\ncouples to the oscillating electric \feld, in the actual\nsetting of the ESR experiment, the optical conductiv-\nity due to the SO current also contributes to the ab-\nsorption of the electromagnetic wave with a spin \rip.\nThis e\u000bect is called Electron Dipole Spin Resonance\n(EDSR)22{25. The EDSR contribution is generically\nlarger than ESR if SO coupling is at the same order\nas the Zeeman splitting26,27, as their relative contribu-\ntions are of ( a=\u0015C)2\u0019106, wherea\u001910\u000010m is the\nlattice spacing and \u0015C\u001910\u000013m the Compton length\nof the electron25. In general, EDSR requires a sepa-\nrate consideration from ESR as they involve di\u000berent\noperators25. Nevertheless, in the low-temperature/low-\nfrequency regime, only the two spin states of the edge\nstate are involved. Thus, although EDSR contributes to\nthe absorption intensity di\u000berently from ESR, the reso-\nnance frequency is identical between the ESR and EDSR.\nWith this in mind, we do not consider EDSR explicitly\nin the rest of the paper. It should be noted that, for a\nhigher temperature or a higher frequency, the EDSR con-\ntribution to the absorption spectrum is rather di\u000berent\nfrom the ESR one, as the absorption spectrum involves\nbulk excitations.\nWe now consider the ESR in the system with the\nHamiltonian H0+Hz. Within the linear response the-\nory28, the ESR spectrum is generally given by the dynam-\nical susceptibility function in the limit of zero-momentum\ntransfer\n\u001f00\n+\u0000(q= 0;!> 0) =\u0000ImGR\n+\u0000(q= 0;!);(14)\nwhere\nGR\n+\u0000(0;!) =\u0000iZ1\n0dtX\nr;r0h[s+(r;t);s\u0000(r0;0)]iei!t\n=\u0000iZ+1\n0dth[S+(t);S\u0000(0)]iei!t; (15)\nwhereS\u0006means the ladder operator de\fned with respect\nto the direction ~ nHof the static \feld, h\u0001\u0001\u0001i denotes the\nquantum and ensemble average at the given temperature\nT, and\nS\u0006(t)\u0011X\nrexp(iHt)s\u0006(r) exp(\u0000iHt) =X\nrs\u0006(r;t);\n(16)\nwhereHis the static Hamiltonian we consider (e.g.,\nH=H0+Hz).\nIn the absence of SO coupling ( \u001e= 0),H0has the\nexact SU(2) spin rotation symmetry which is broken\nonly \\weakly\"14by the Zeeman term HZ. As a gen-\neral principle of ESR, in this case, the ESR spectrum (if5\nany) remains paramagnetic, namely a single \u000e-function\nat!=H. As we will demonstrate later in Sec. III, in\nthe low-temperature limit, this paramagnetic ESR can\nbe attributed to the edge states of the SSH model. Once\nthe SO coupling is introduced ( \u001e6= 0), the SU(2) sym-\nmetry is broken and we would expect a nontrivial ESR\nlineshape. However, somewhat surprisingly, (as we will\nshow later), the ESR spectrum attributed to the edge\nstates remains a \u000e-function at !=Heven when \u001e6= 0.\nThus, in order to investigate possible nontrivial e\u000bects\nof the SO coupling on ESR, we further consider the NNN\nhoppings\n\u0001H=X\nj=1\u0001tcy\nj+2exp[i\r~ n\r\u0001\u001b=2]cj+ h.c.; (17)\nwhere \u0001tis the NNN electron hopping amplitude. The\nangle\rand~ n\rare the SO turn angle and the axis for\nthe NNN hopping, respectively.\nThe Hamiltonian of the system to be considered is then\nHESR=H0+HZ+ \u0001H: (18)\nOnce we include the NNN hoppings, the chiral symme-\ntry is broken and the edge states are not protected to be\nat zero energy. Nevertheless, when the chiral symmetry\nbreaking perturbations are weak, we may still identify\n\\edge states\" localized near the ends although they are\nno longer at exact zero energy even when H= 0. Un-\nder a magnetic \feld H, contributions from these edge\nstates dominate the ESR in the low-energy limit. Now,\nthe ESR spectrum can be nontrivially modi\fed by the\nSO couplings \u001eand\r. It is the main purpose of the\npresent paper to elucidate this e\u000bect. In real materials,\nthe NNN hoppings might be small but they are generally\nnon-vanishing. Thus it is important to develop a theory\nof ESR in the presence of the NNN hoppings, especially\nbecause we can detect the NNN hoppings with ESR even\nwhen they are small.\nWe will treat the NNN hopping \u0001 H, which would be\nsmaller than the NN hopping H0in many experimental\nrealizations, as a perturbation. This also turns out to\nbe convenient for our theoretical analysis. We also as-\nsume thatj\u001ej;j\rj\u001c 1 since SO couplings are weak in\nmost of the realistic systems, and they will formulate a\nperturbation expansion in \u0001 t,\u001e, and\r.\nIII. EDGE STATES OF H0ANDU(1)~S\u0001~ nHSYMMETRY\nAs we discussed earlier, ESR would be an ideal probe\nto detect the edge state of the 1D topological insulator\nand various perturbations. In this section, we discuss\nand explicitly solve the edge states of the unperturbed\nHamiltonian H0to demonstrate the robustness of the\nedge states against SO coupling. As a consequence, in the\nmodelH0only with NN hoppings, there is no nontrivial\nchange in the edge ESR spectrum.Since our Hamiltonian H0is bilinear in fermion op-\nerators, we can focus on single-electron states and rep-\nresent them with the ket notation. For a half-in\fnite\nchain with sites j= 1;2;:::, wherej= 1 corresponds to\nthe end of the chain, we \fnd a single-electron eigenstate\njEdge;\u001bilocalized near the edge in the \\topological in-\nsulator\" phase \u000e0>0. Here\u001b=\u00061 represents the spin\ncomponent in the direction of the magnetic \feld. Namely,\n(H0+HZ)jEdge,\u001bi=E(0)\n\u001bjEdge,\u001bi (19)\n~S\u0001~ nHjEdge,\u001bi=\u001b\n2jEdge,\u001bi; (20)\nwith the energy eigenvalues E(0)\n\u001b=\u0000\u001bH=2 and~Sis the\ntotal spin of the system. The wave function of the edge\nstates is exactly calculated as\nhj;\u001bjEdge,\u001b0i=8\n<\n:\u000e\u001b\u001b 0p\nN\u0010\n\u00001\u0000\u000e0\n1+\u000e0\u0011(j\u00001)=2\n(j22N0+ 1 );\n0 (otherwise) ;\n(21)\nwhere 2 N0is the set of non-negative even integers, and\nNis the normalization constant. The energy eigenvalue\nof the edge state for H0is, independently of the spin\ncomponent, exactly zero, re\recting its topological nature.\nIt is also remarkable that the edge state wavefunction\nis independent of \u001eand~ n. This is a consequence of a\ncanonical \\gauge transformation\"29\n~c2k+1=c2k+1;\n~c2k= exp (i\u001e~ n\u0001~ \u001b=2)c2k;(22)\nwhich eliminates the SO coupling from H0. In this sense,\nH0still has a hidden SU(2) symmetry30,31even though\nthe SO coupling breaks the apparent spin SU(2) sym-\nmetry. However, since the gauge transformation involves\nthe local rotation of spins, the uniform magnetic \feld\nHZgives rise to a staggered \feld after the gauge trans-\nformation. This staggered \feld completely breaks the\nSU(2) symmetry. This is similar to the situation in a\nspin chain with a staggered Dzyaloshinskii-Moriya inter-\naction13. Thus, following the general principle of ESR,\nwe would expect a nontrivial ESR spectrum in the pres-\nence of the staggered SO coupling as in H0.\nNevertheless, somewhat unexpectedly, we \fnd that the\nedge ESR spectrum for the model H0remains the \u000e-\nfunction\u000e(!\u0000H), as if there is no anisotropy at all.\nThis is due to the fact that the edge-state wavefunction\nEq. (21) is non-vanishing only on the even sites. Since\nthe gauge transformation can be de\fned so that it only\nacts on the even sites where the edge-state wavefunction\nEq. (21) vanishes, the edge state is completely insensi-\ntive to the SO coupling. Therefore, the spectral shape of\nthe edge ESR remains unchanged by the staggered SO\ncoupling. In addition, since the edge wave functions are\neigenstates of the total spin component along ~ nHaccord-\ning to Eq. (20), the edge states have U(1)~S\u0001~ nHsymmetry\ngenerated by ~S\u0001~ nH.\nIn fact, the robustness of the edge ESR spectrum is\nvalid for a wider class of models. The edge ESR only6\nprobes a transition between two states with opposite po-\nlarization of the spin, which form a Kramers pair in the\nabsence of the magnetic \feld. Thus, at zero magnetic\n\feld, the time-reversal invariance of the model requires\nthese two states to be exactly degenerate. For a \fnite\nmagnetic \feld, if the system still has U(1)~S\u0001~ nHsymme-\ntry of rotation about the magnetic \feld axis, the two\nstates can be labelled by the eigenvalues of Sz=\u00061=2,\nand their energy splitting is exactly\n!ESR=H: (23)\nThus, as far as the edge ESR involving only the Kramers\npair is concerned, the U(1)~S\u0001~ nHsymmetry is su\u000ecient to\nprotect the \u000e-function peak at !=H. We note that,\nmore generally, when more than two states contribute to\nESR, these symmetries are not su\u000ecient to protect the\nsingle-peak ESR spectrum, as there can be transitions\nbetween states not related by time reversal. In fact, this\nwould be the case for the absorption due to bulk excita-\ntions which we do not discuss in this paper.\nIV. PERTURBATION THEORY OF THE EDGE\nESR FREQUENCY SHIFT\nAs we have shown in the previous Section, even in the\npresence of SO coupling, there is no frequency shift for\nedge ESR in the NN hopping model H0. This is a con-\nsequence of the chiral symmetry, which stems from the\nbipartite nature of the NN hopping model.\nAs we will discuss below, the introduction of NNN hop-\npings breaks the chiral symmetry and generally causes a\nnontrivial frequency shift of the edge ESR. In this Sec-\ntion, we develop a perturbation theory of ESR for the\nedge states, \frst by regarding H0+HZas the unper-\nturbed Hamiltonian and \u0001 Has a perturbation.\nIn the presence of the perturbation with SO couplings,\nthe eigenstate of the Hamiltonian is generally no longer\nan eigenstate of the spin component ~S\u0001~ nHas in Eq. (20).\nNevertheless, as long as the perturbation theory is valid,\ntwo edge states can still be identi\fed by using spin \u001b=\n+ and\u0000. Namely, for the full Hamiltonian (18), we\ncan de\fne the edge state labeled by \u001b=\u0006as the state\nadiabatically connected to jEdge;\u0006ias \u0001H!0.\nHere let us introduce new symbols E+(\u0000)andE(n)\n+(\u0000)as\nthe energy eigenvalue of the almost spin up (down) edge\nstate in Eq. (18) and its n-th order correction in the\nperturbation theory, respectively. With these symbols,\nthe ESR frequency is given by\n!ESR=E\u0000\u0000E+=H+ \u0001!; (24)\nwhere the ESR trivial peak position is given by E(0)\n\u0000\u0000\nE(0)\n+=H. The ESR frequency shift \u0001 !, driven by the\nperturbation \u0001 H, is expanded in the perturbation the-\nory as\n\u0001!= \u0001!(1)+ \u0001!(2)+:::; (25)where then-th order term is\n\u0001!(n)=E(n)\n\u0000\u0000E(n)\n+; (26)\nforn2N. In the following parts, we perturbatively solve\nthe single-electron problem to compute the eigen-energy\ndi\u000berence of two edge states E+andE\u0000.\nA. First order in \u0001t\nThe NNN hopping \u0001 Hbreaks the chiral symmetry,\nand thus it can change the edge ESR spectrum. In fact,\nthe energy of the edge states is already shifted in the \frst\norder of \u0001 H. However, the energy shift is the same for\nthe two edge states with di\u000berent spin polarizations. This\nis a consequence of the time-reversal (TR) symmetry of\nthe SO coupling, as demonstrated below:\nE(1)\n\u001b=hEdge,\u001bj\u0001HjEdge,\u001bi\n=h\u0002 (Edge,\u0000\u001b)j\u0002\u0001H\u0002\u00001j\u0002 (Edge,\u0000\u001b)i\n=E(1)\n\u0000\u001b\n=\u00002\u0001t1\u0000\u000e0\n1 +\u000e0cos\u0010\r\n2\u0011\n(27)\nwhere \u0002 is the TR operator and \u0002\u0001 H\u0002\u00001= \u0001His\nused. Therefore, the edge ESR spectrum remains un-\nchanged in the \frst order of \u0001 tas the frequency shift\nvanishes in this order:\n\u0001!(1)=E(1)\n\u0000\u0000E(1)\n+= 0: (28)\nB. Second order in \u0001t\nWe can formally write down the second-order pertur-\nbation correction\nE(2)\n\u001b=hEdge,\u001bj\u0001H\u0001(E(0)\n\u001b\u0000H0)\u00001\u0001P\u001b\u0001\u0001HjEdge,\u001bi\nwhere the projection operator P\u001b\u0011 1\u0000\njEdge,\u001bihEdge,\u001bj. It is rather di\u000ecult to evaluate\nthis formula directly, since the intermediate states in the\nperturbation term include the bulk eigenstates of H0in\nwhich the hidden symmetry is generally broken. Assum-\ning thatj\u001ej;j\rj\u001c1 (i.e., the SO coupling is su\u000eciently\nsmall), we can develop a perturbative expansion in \u001e\nand\rin addition to \u0001 t. In this framework, we expand\nE\u001bas a Taylor series of \u001e,\rand \u0001t. The quantity\nof interest is the energy splitting E\u0000\u0000E+, since it\ncorresponds to the ESR frequency.\nThe edge state energy splitting in the second order in\n\u0001t, and up to the second order in \u001e,\ris required to take\nthe form\nE(2)\n\u001b\u0000E(2)\n\u0000\u001b\u00191\n2\u001bH\u0001t2\b\na(~ n\r\u0002~ nH)2\r2+b(~ n\u0002~ nH)2\u001e2\n+c(~ n\u0002~ nH)\u0001(~ n\r\u0002~ nH)\u001e\rg; (29)\nbased on the following symmetry considerations, and a,\nbandcare constants to be determined.7\nFirst of all, ( E\u001b\u0000E\u0000\u001b) will not change under\n(H;~ nH;\u001b)!(\u0000H;\u0000~ nH;\u0000\u001b); (30)\nbecause this corresponds to a trivial rede\fnition of co-\nordinate system. Therefore, we attach the factor \u001bHin\nthe r.h.s. of Eq. (29).\nNext we notice that \u001ealways appears with ~ n, and\r\nwith~ n\r, which leads to further constraints as we will\nsee below. Let us consider the limiting case \u001e= 0 with\nnonzero\r. The splitting can only depend on the relative\nangle between ~ n\rand~ nH. Furthermore, if ~ n\rk~ nH,\nthe hidden symmetry of the edge state implies that the\nenergy splitting is exactly given by the Zeeman energy\nand there is no perturbative correction. Thus, for \u001e= 0,\nthe energy splitting can only depend on ( ~ n\r\u0002~ nH)2\r2up\ntoO(\r2) since the energy split is a scalar and it must\nbe written in terms of inner and vector products of ~ nH,\n~ nand~ n\r. Then, similarly, if \r= 0 with nonzero \u001e,\nthe energy splitting can only depend on ( ~ n\u0002~ nH)2\u001e2.\nFinally, the O(\u001e\r) term should be linear in ~ nand~ n\r,\nand it vanishes when ~ nk~ n\rk~ nHbecause of the U(1)~S\u0001~ n\nsymmetry. These requirements uniquely determine the\nform of (~ n\u0002~ nH)\u0001(~ n\r\u0002~ nH). Thus, the symmetries\nreduce the possible forms of the second-order corrections\nto Eq. (29) with only the three parameters a,b, andc.\nTo obtain these parameters, we note that the expan-\nsion in\u001eand\rintroduced above can be naturally done\nby regarding the SSH model without SO coupling\n~H0=\u0000+1X\nj=1n\nt\u0002\n1 + (\u00001)j\u000e0\u0003\ncy\nj+1cj+ h.c.o\n\u0000HX\nj=1cy\nj(~ \u001b\u0001~ nH)cj=2 (31)\nas the unperturbed Hamiltonian, and\nHpert= \u0001H0+ \u0001H; (32)\nas the perturbation, where\n\u0001H0\u0011\u0000+1X\nj=1n\nt\u0002\n1 + (\u00001)j\u000e0\u0003\ncy\nj+1\n\b\nexp\u0002\n(\u00001)ji\u001e~ n\u0001~ \u001b=2\u0003\n\u00001\t\ncj+ h.c.\t\n(33)\nand \u0001Has de\fned in Eq. (17).\nAs the result of the perturbation calculation given in\nthe Appendix, we \fnd that the second-order term of fre-\nquency shift is non-positive given in the form\n\u0001!(2)=\u0000H\n2X\nm3;j3(~M+~N)y\u0001(~M+~N)\u00140;(34)\nwhere\n~M\u0011X\nm2;j2hm3;j3j\u0001H0\n0jm2;j2ihm2;j2j\u0001H00jEdgei\nm2Ej2Ej3\n(~ n\u0002~ nH)\u001e\n~N\u0011hm3;j3j\u0001H0jEdgei\nEj3(~ n\r\u0002~ nH)\r: (35)Herejm;ji's are single-particle (bulk) energy eigenstates\nof~H0in Eq. (31) where m=\u0006labels the positive or neg-\native energy sector (i.e., band indices) and jlabels other\npossible quantum numbers, which is not the wave vector\nsince we have the open-ended boundary condition. The\nenergy\u0006Ejstands for the energy eigenvalue of j\u0006;ji.\nThe perturbation terms \u0001 H0, \u0001H00, and \u0001 H0\n0are de-\n\fned as\n\u0001H0\u0011X\nj=1\u0001tcy\nj+2cj\u0000h.c.; (36)\n\u0001H00\u0011X\nj=1\u0001tcy\nj+2cj+ h.c.; (37)\n\u0001H0\n0\u0011\u0000+1X\nj=1n\nt\u0002\n1 + (\u00001)j\u000e0\u0003\ncy\nj+1cj\u0000h.c.o\n:(38)\nWe note that Eqs. (36) and (38) are anti-Hermitian.\nAfter putting the de\fnitions of ~Mand~Ninto Eq. (34),\nwe \fnd the result consistent with the general form\nEq. (29) required by symmetries. The parameters are\nthen identi\fed as\na=b=\u0014\u0001t\nt(1 +\u000e0)\u00152\n; c =\u00002\u0014\u0001t\nt(1 +\u000e0)\u00152\n:\nThe detailed derivation of a,bandcis given in Appendix.\nThe \fnal result can be given in a compact form as\n\u0001!(2)=\u0000H\n2\f\f\f\f\u0001t\nt(1 +\u000e0)(\u001e~ n\u0002~ nH\u0000\r~ n\r\u0002~ nH)\f\f\f\f2\n;(39)\nwhich is non-positive. Since the \frst-order correction\nvanishes as we have already seen, the second-order term\nEq. (39) gives the leading term for the frequency shift\nof the edge ESR. It also implies that, although \ris the\nSO-coupling turn angle for the NNN hopping terms, it\nis equally important in \u0001 !as the NN SO coupling turn\nangle\u001eeven if the NNN hopping itself is small (\u0001 t\u001ct).\nOne of the most remarkable features of our result is\nthat, the shift (up to the second order in the perturba-\ntion) vanishes when \u001e~ n\u0002~ nH=\r~ n\r\u0002~ nH. This corre-\nsponds to the zeros of curves in Fig. 4 where the direction\nof the magnetic \feld is in the plane spanned by ~ n= ^y\nand~ n\r=\u0000^z, and\u0012is the angle between ~ nHand ^yin the\n^y-^zplane, as shown in Fig. 5. Therefore, we predict two\ndirections of the magnetic \feld for which the ESR shift\nvanishes, when the magnetic \feld direction ~ nHsweeps\nthe plane spanned by ~ nand~ n\r.\nV. NON-PERTURBATIVE CALCULATION OF\nTHE EDGE ESR SPECTRUM\nHere, in order to see the validity of our perturba-\ntion theory in the preceding section, let us re-compute\nthe edge ESR frequency with a more direct numerical\nmethod. As already mentioned, when the magnetic \feld8\nFIG. 4. Angle \u0012dependence of the ESR frequency shift \u0001 !.\nWe set the parameters t= 1:0, \u0001t=\u000e0= 0:2, andH=\n0:05. Red and blue curves are obtained by the second-order\nperturbation calculation in Sec. IV B. Square and circle points\nare the results of direct numerical diagonalization for a system\n(of 100 sites) with an open boundary condition. The de\fnition\nof angle\u0012is indicated in Fig. 5. The zeros of \u0001 !occur at\n\u001e~ n\u0002~ nH=\r~ n\r\u0002~ nH.\nH(and thus the ESR frequency !) is much smaller com-\npared to the bulk excitation gap 4 tj\u000e0j, we may ignore\nthe e\u000bects of the bulk excitations in the ESR spectrum.\nThen the edge ESR spectrum is of the \u000e-function form\n\u001f00\n+\u0000edge(q= 0;!)/\u000e[!\u0000(E\u0000\u0000E+)]; (40)\nwhereE+(\u0000)is the energy eigenvalue of the \\almost\"\nspin-up (spin-down) edge state. These energies can be\naccurately computed by numerical diagonalization of a \f-\nnite (but long) size full Hamiltonian with an open bound-\nary condition. Then we obtain the spectrum peak shift\n\u0001!\u0011E\u0000\u0000E+\u0000Hfrom the numerical results of E\u0006. In\nthe present work, we calculated E\u0006using a \fnite open\nchain of 100 sites.\nIn Fig. 4, we compare the numerical results of the peak\nshift with the analytical perturbation theory of \u0001 !(2)\nESRin\nEq. (39) when the magnetic \feld is in the plane spanned\nby~ nand~ n\r. The \fgure clearly shows that our perturba-\ntion theory agrees with the numerical results quite well.\nVI. CONCLUSIONS AND DISCUSSION\nWe have analyzed ESR of edge states in a general-\nized SSH model with staggered SO couplings and with\nan open end. In this paper, we assume that the en-\nergy scales of the magnetic \feld, the frequency, and the\ntemperature are su\u000eciently small compared to the bulk\ngap. Then the ESR spectrum only consists of a single\n\u000e-function spectrum corresponding to the transition be-\ntween two spin states at the edge, but is expected to show\na nontrivial frequency shift in general as the SO coupling\nFIG. 5. Geometric relation among some vectors in Fig. 4,\nwhere the direction of magnetic \feld is in the plane spanned\nby~ n= ^yand~ n\r=\u0000^z. The parameter \u0012is de\fned as the\nangle between ~ nHand ^yin the ^y-^zplane.\nbreaks the SU(2) symmetry strongly under the applied\nmagnetic \feld. Nevertheless, there is no ESR frequency\nshift in the model with only NN hoppings, thanks to its\nchiral symmetry.\nThe chiral symmetry is broken by NNN hoppings,\nwhich should be generally present in any realistic materi-\nals even if they are small. This NNN hoppings, together\nwith the SO coupling, can induce a nontrivial frequency\nshift on the edge ESR. Thus we have developed a pertur-\nbation theory of the frequency shift, regarding the NNN\nhoppings and the SO couplings as perturbations. Our\nmain result, the ESR frequency shift up to second order\nin the perturbation theory, is found in Eq. (29). It is\nnon-positive in this order. (The resonance \feld shift for\na \fxed frequency, which is usually measured in experi-\nments, is always positive.) In the presence of the NNN\nhoppings, the SO couplings in the NN hoppings, which\ndid not cause a frequency shift by themselves, also con-\ntribute to the frequency shift. We \fnd an interesting\ndependence of the ESR frequency shift on the direction\nof the static magnetic \feld, relative to the SO couplings\non the NN and the NNN hoppings. In particular, the\nESR frequency shift is predicted to vanish when the static\nmagnetic \feld points to a certain direction on the plane\nspanned by the two SO coupling axes (see Fig. 4). Fur-\nthermore, we performed a direct estimate of the ESR\nfrequency shift by a numerical calculation of the edge\nstate spectrum, without relying on the perturbation the-\nory. The result agrees very well with the perturbation\ntheory, establishing its validity.\nOur results indicate that, the chiral symmetry break-\ning by the NNN hoppings in the \\SSH\"-type topological\ninsulators in one dimension may be detected by ESR, in9\nthe presence of the SO couplings. If the NNN hoppings\nare small, which would be the case in many realistic ma-\nterials, it might be di\u000ecult to detect their e\u000bects with\nother experimental techniques. ESR has been success-\nful in detecting even very small magnetic anisotropies,\nthanks to its high sensitivity and accuracy. We hope that\nthe present work will pave the way for a new application\nof ESR in detecting (small) chiral symmetry breaking.\nWhile we do not discuss any particular material in this\npaper, let us discuss here the prospect of experimentally\nobserving the e\u000bects we predict. The maximal frequency\nshift is given by the order of\nH\u0012\u0001t\ntmaxf\u001e;\rg\u00132\n: (41)\nThe ratio of NNN to NN hoppings, \u0001 t=t, of course\nstrongly depends on each material. It is even possible\nthatj\u0001tj=t\u001d1, in which case the system may be re-\ngarded as two chains coupled weakly by zigzag hopping.\nIt should be however noted that our theory is valid only\nwhenj\u0001tj=tis su\u000eciently small. We still expect that\nour theory works reasonably well for (\u0001 t=t)2\u00180:1. In\ncarbon-based systems, such as polyacetylene, the SO in-\nteraction is known to be weak. For example, even with\nthe enhanced SO interaction due to a curvature32,\u001e;\r\nis of order of 10\u00005. This would give an ESR shift that\nis too small to be observed in experiments. However,\nthe SO interaction is stronger in heavier atoms. In fact,\neven in carbon-based systems, the SO interaction can be\nsigni\fcantly enhanced by heavy adatoms. For example,\nplacing Pb as adatoms can enhance \u001e;\rup to 0:1 or more\nin graphene33. This would give the edge ESR shift cor-\nresponding to the g-shift up to the order of 10\u00003(1,000\nppm), which should be observable. In particular, even\nif the absolute value of the shift is di\u000ecult to be deter-\nmined, the angular dependence of the ESR shift would be\nmore evident in experiments. Furthermore, it is known\nthat an SO interaction generally becomes larger when\nthe electron system we consider is located in the vicinity\nof an interface between two bulk systems or is under a\nstrong, static electric \feld34{36. Therefore, if we set up an\nSSH chain system under such an environment, it would\nbecome easier to detect an ESR frequency shift due to a\nstrong SO coupling.\nThroughout this paper, we have taken the NN SO cou-\npling in the model Eq. (1) to be staggered. However,\nthere are other possibilities. In particular, in a transla-\ntionally symmetric system, the NN SO coupling is uni-\nform. Although the only di\u000berence is the signs in the\nHamiltonian, ESR spectra should be signi\fcantly di\u000ber-\nent between these two cases. This is clear if we consider\nthe limit of the zero NNN hopping. In the staggered SO\ncoupling case, there is no ESR frequency shift. This is\nbecause the NN SO coupling can be gauged out by the\ncanonical transformation Eq. (22), without changing the\nodd site amplitude and thus leaving the SSH edge state\nwavefunction Eq. (21) unchanged. On the other hand,\nin the uniform SO coupling case, gauging out the NNSO coupling a\u000bects any wavefunction including the SSH\nedge state wavefunction, resulting in the change of the\nESR spectrum. A similar di\u000berence has been recognized\nbetween the ESR spectum in the presence of a staggered\nDM interaction13and that with a uniform DM interac-\ntion along the chain29. The analysis of the edge ESR\nspectrum in the presence of a uniform SO coupling is left\nfor future studies.\nACKNOWLEDGMENTS\nM. S. was supported by Grant-in-Aid for Scienti\fc\nResearch on Innovative Area, \\Nano Spin Conversion\nScience\" (Grant No.17H05174), and JSPS KAKENHI\nGrants (No. 17K05513 and No. 15H02117), and M. O.\nby KAKENHI Grant No. 15H02113. This work was also\nsupported in part by U.S. National Science Foundation\nunder Grant No. NSF PHY-1125915 through Kavli In-\nstitute for Theoretical Physics, UC Santa Barbara where\na part of this work was performed by Y. Y. and M. O.\nAppendix: Perturbation theory for the frequency\nshift \u0001!\nHere we explain how to determine the parameters a,b\nandcin the general form of Eq. (29), based on a pertur-\nbation theory. As we have discussed earlier, in order to\nexpand the edge-state energy eigenstates with respect to\nthe SO coupling parameters \u001eand\r, which are assumed\nto be small, we regard ~H0of Eq. (31) as the unperturbed\npart, and ~Hpertof Eq. (32) as the perturbation. More-\nover, we have already shown in Sections III and IV A that\nthe ESR shift vanishes exactly up to O(\u0001t). Therefore,\nthe leading terms with coe\u000ecients a,bandcstem from\nthe second-order perturbation proportional to \u0001 t2. Be-\nlow, we will determine three parameters a,bandcfrom\nthe perturbative expansion of ~Hpert.\n1. Second order in Hpert\nThe shift of the edge energy eigenvalue in the second\norder of Hpertis given by\nE[2]\n\u001b=X\nm=\u0006;j;~\u001b=\u0006hEdge,\u001bjHpertjm;j; ~\u001bihm;j; ~\u001bjHpertjEdge,\u001bi\n\u0000\u001bH\n2\u0000Em;j;~\u001b\n\u0019X\nm=\u0006;jhEdgej(\u0001H0)yjm;jihm;jj\u0001H0jEdgei\nE2\nj\u001bH\n\u0001jn\r\u0002nHj2\r2=4; (A.1)\nwhere the bulk single-particle eigen-energy Em;j;~\u001bis\ngiven by\nEm;j;~\u001b\u0011mEj\u0000~\u001bH=2 (A.2)10\nwithmEjthe energy eigenvalue of spinless single-particle\neigenstatesjm;ji, and we have used the facts that in the\nunperturbed sector, the orbital and spin parts of single-\nparticle eigenstates can be decomposed, e.g. jm;j;\u001bi=\njm;jij\u001bisince ~H0commutes with the spin operator of\neach site. Here we de\fne E[n]\n\u001bas the energy eigenvalue\nof the edge state with spin \u001bin then-th order of pertur-\nbation in Hpert. This is to be distinguished from E(n)\n\u001b\nintroduced in Eq. (26), where nrefers to the order in \u0001 t.\nIn the second line of Eq. (A.1), we assume jEm;j;~\u001bj\u001dH\nand perform the Taylor expansion of E[2]\n\u001bwith respect\ntoH. The matrixhm;jj\u0001H0jEdgeican be calculated\nby using the anti-Hermitian operator \u0001 H0of Eq. (36).\nFrom Eq. (A.1), the ESR frequency shift driven by E(2)\n\u001bis expressed as\nE[2]\n\u0000\u0000E[2]\n+=\u0000jn\r\u0002nHj2\r2=2\n\u0001X\nm=\u0006;jhEdgej(\u0001H0)yjm;jihm;jj\u0001H0jEdgei\nE2\njH:(A.3)\nThis indeed corresponds to the parameter a. To obtain\nbandc, we should proceed to higher orders of Hpert.\n2. Third order in Hpert\nIn the third order perturbation theory within \u0001 t2, the\nenergy correction takes the form as\nE[3]\n\u001b=X\nm;j;~\u001bhEdge,\u001bjHpertjm3;j3;~\u001b3ihm3;j3;~\u001b3jHpertjm2;j2;~\u001b2ihm2;j2;~\u001b2jHpertjEdge,\u001bi\u0002\n\u0000(\u001b\u0000~\u001b2)H\n2\u0000m2Ej2\u0003\u0002\n\u0000(\u001b\u0000~\u001b3)H\n2\u0000m3Ej3\u0003\n+hEdge,\u001bjHpertjEdge,\u001bijhEdge,\u001bjHpertjm3;j3;~\u001b3ij2\n\u0002\n\u0000(\u001b\u0000~\u001b3)H\n2\u0000m3Ej3\u00032\n\u0019X\nm;jRe\"\nhEdgej\u0001H00jm3;j3ihm3;j3j\u0001H0\n0jm2;j2ihm2;j2j\u0001H0jEdgei\nm3E2\nj2Ej3\u001bH#\n\u0001(~ n\u0002~ nH)\u0001(~ n\r\u0002~ nH)\u001e\r\n2;(A.4)\nwhere \u0001 H00is de\fned by Eq. (37).\nTherefore,\nE[3]\n\u0000\u0000E[3]\n+\n=X\nm;j\u0000Re\"\nhEdgej\u0001H00jm3;j3ihm3;j3j(\u0001H0\n0)yjm2;j2i\nm3E2\nj2Ej3\n\u0001hm2;j2j\u0001H0jEdgeiH] (~ n\u0002~ nH)\u0001(~ n\r\u0002~ nH)\u001e\r:\n(A.5)\nWe see that this correction term corresponds to the pa-\nrametercin Eq. (29).\n3. Fourth order in Hpert\nIn order to derive the leading term of the parameter b,\nwe have to calculate the fourth-order term. For conve-nience of the fourth-order calculation, we introduce the\nabbreviated notation of the matrix elements:\nAqr\u0011hmr;jr;~\u001brjAjmq;jq;~\u001bqi; (A.6)\nEEq\u0011E[0]\n\u001b\u0000Emq;jq;~\u001bq (A.7)\nfor any operator A, and denote the zeroth order edge\nstate by \\E\". In this notation, the fourth order edge-\nenergy correction up till (\u0001 t\u001e)2-order is given by\nE[4]\n\u001b=X\nm;j;~\u001b\u0001HE4H4;3\npertH3;2\npert\u0001H2E\nEE2EE3EE4\u0000E[2]\n\u001b\u0000\n\u0001HE4\u00012\n(EE4)2\u00002\u0001HEE\u0001HE4H43\npert\u0001H3E\nE2\nE3EE4+\u0000\n\u0001HEE\u00012\u0000\n\u0001HE4\u00012\n(EE3)3\n\u0019X\nm;j;hEdgej\u0001H00jm2;j2ihm2;j2j(\u0001H0\n0)yjm3;j3ihm3;j3j\u0001H0\n0jm4;j4ihm4;j4j\u0001H00jEdgei\nm2m4Ej2E2\nj3Ej4\u001bHj~ n\u0002~ nHj2\u001e2\n4:(A.8)11\nThen we can arrive at\nE[4]\n\u0000\u0000E[4]\n+\n\u0019\u0000X\nm;j;hEdgej\u0001H00jm2;j2ihm2;j2j(\u0001H0\n0)yjm3;j3ihm3;j3j\u0001H0\n0jm4;j4ihm4;j4j\u0001H00jEdgei\nm2m4Ej2E2\nj3Ej4Hj~ n\u0002~ nHj2\u001e2\n2:(A.9)\nThis correspond to the term coe\u000eciented by the param-\neterbin Eq. (29).4. Summation over the perturbation orders\nThrough some algebra, we \fnd that terms of the \ffth\nand higher orders in Hpertdo not contribute to the order\nofO\u0000\n\u0001t2\u001e2\u0001\n. Summing up Eqs. (28), (A.3), (A.5), and\n(A.9), we arrive at the energy shift of the energy di\u000ber-\nence between down- and up-spin edge states in an elegant\nform as in Eq. (34).\n\u0003smartyao@issp.u-tokyo.ac.jp\n1M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045\n(2010).\n2X.-L. Qi and S.-C. Zhang, Rev. Mod. Phys. 83, 1057\n(2011).\n3B. A. Bernevig and T. L. . X. Hughes, Topological insula-\ntors and topological superconductors (Princeton University\nPress, 2013).\n4S.-Q. Shen, Topological Insulators: Dirac Equation in Con-\ndensed Matters (Springer Series in Solid-State Sciences)\n(Springer-Verlag, Berlin, 2013).\n5W. P. Su, J. R. Schrie\u000ber, and A. J. Heeger, Phys. Rev.\nLett. 42, 1698 (1979).\n6H. J. Schulz, Phys. Rev. B 18, 5756 (1978).\n7J. Zak, Phys. Rev. Lett. 62, 2747 (1989).\n8J. K. Asb\u0013 oth, L. Oroszl\u0013 any, and A. P\u0013 alyi, Lecture Notes\nin Physics, Berlin Springer Verlag , Vol. 919 0075-8450\n(Springer, 2016).\n9M. Hagiwara, K. Katsumata, I. A\u000feck, B. I. Halperin,\nand J. P. Renard, Phys. Rev. Lett. 65, 3181 (1990).\n10S. H. Glarum, S. Geschwind, K. M. Lee, M. L. Kaplan,\nand J. Michel, Phys. Rev. Lett. 67, 1614 (1991).\n11V. L. J. Joly, M. Kiguchi, S.-J. Hao, K. Takai, T. Enoki,\nR. Sumii, K. Amemiya, H. Muramatsu, T. Hayashi,\nY. A. Kim, M. Endo, J. Campos-Delgado, F. L\u0013 opez-Ur\u0013 \u0010as,\nA. Botello-M\u0013 endez, H. Terrones, M. Terrones, and M. S.\nDresselhaus, Phys. Rev. B 81, 245428 (2010).\n12J. Campos-Delgado, J. M. Romo-Herrera, X. Jia, D. A.\nCullen, H. Muramatsu, Y. A. Kim, T. Hayashi, Z. Ren,\nD. J. Smith, and Y. Okuno, Nano letters 8, 2773 (2008).\n13M. Oshikawa and I. A\u000feck, Phys. Rev. Lett. 82, 5136\n(1999).\n14M. Oshikawa and I. A\u000feck, Phys. Rev. B 65, 134410\n(2002).\n15R. Shindou, A. Furusaki, and N. Nagaosa, Phys. Rev. B\n82, 180505 (2010).\n16M. Nakamura and A. Tokuno, Phys. Rev. B 94, 081411\n(2016).17S. Konschuh, M. Gmitra, and J. Fabian, Phys. Rev. B 82,\n245412 (2010).\n18A. P. Schnyder, S. Ryu, A. Furusaki, and A. W. W. Lud-\nwig, Phys. Rev. B 78, 195125 (2008).\n19C.-K. Chiu, J. C. Y. Teo, A. P. Schnyder, and S. Ryu,\nRev. Mod. Phys. 88, 035005 (2016).\n20P. Delplace, D. Ullmo, and G. Montambaux, Phys. Rev.\nB84, 195452 (2011).\n21S. Ryu and Y. Hatsugai, Phys. Rev. Lett. 89, 077002\n(2002).\n22E. I. Rashba, Sov. Phys. Solid State 2, 1109 (1960).\n23E. I. Rashba and A. L. Efros, Phys. Rev. Lett. 91, 126405\n(2003).\n24A. L. Efros and E. I. Rashba, Phys. Rev. B 73, 165325\n(2006).\n25A. Bolens, H. Katsura, M. Ogata, and S. Miyashita, Phys.\nRev. B 95, 235115 (2017).\n26A. Shekhter, M. Khodas, and A. M. Finkel'stein, Phys.\nRev. B 71, 165329 (2005).\n27S. Maiti, M. Imran, and D. L. Maslov, Phys. Rev. B 93,\n045134 (2016).\n28R. Kubo and K. Tomita, JPSJ 9, 888 (1954).\n29S. Gangadharaiah, J. Sun, and O. A. Starykh, Phys. Rev.\nB78, 054436 (2008).\n30T. A. Kaplan, Zeitschrift f ur Physik B Condensed Matter\n49, 313 (1983).\n31L. Shekhtman, O. Entin-Wohlman, and A. Aharony, Phys.\nRev. Lett. 69, 836 (1992).\n32D. Huertas-Hernando, F. Guinea, and A. Brataas, Phys.\nRev. B 74, 155426 (2006).\n33L. Brey, Phys. Rev. B 92, 235444 (2015).\n34J. Nitta, T. Akazaki, H. Takayanagi, and T. Enoki, Phys.\nRev. Lett. 78, 1335 (1997).\n35A. D. Caviglia, M. Gabay, S. Gariglio, N. Reyren, C. Can-\ncellieri, and J. M. Triscone, Phys. Rev. Lett. 104, 126803\n(2010).\n36A. Soumyanarayanan, N. Reyren, A. Fert, and\nC. Panagopoulos, Nature 539, 509 (2016)."    },    {        "title": "1809.10852v2.Spin_orbit_crossed_susceptibility_in_topological_Dirac_semimetals.pdf",        "content": "arXiv:1809.10852v2  [cond-mat.mes-hall]  20 Feb 2019Spin-orbit crossed susceptibility in topological Dirac se mimetals\nYuya Ominato1, Shuta Tatsumi1, and Kentaro Nomura1,2\n1Institute for Materials Research, Tohoku University, Send ai 980-8577, Japan and\n2Center for Spintronics Research Network, Tohoku Universit y, Sendai 980-8577,Japan\n(Dated: February 21, 2019)\nWe theoretically study the spin-orbit crossed susceptibil ity of topological Dirac semimetals. Be-\ncause of strong spin-orbit coupling, theorbital motion of e lectrons is modulated byZeeman coupling,\nwhich contributes to orbital magnetization. We find that the spin-orbit crossed susceptibility is pro-\nportional totheseparation oftheDirac points anditis high lyanisotropic. The orbital magnetization\nis induced only along the rotational symmetry axis. We also s tudy the conventional spin susceptibil-\nity. The spin susceptibility exhibits anisotropy and the sp in magnetization is induced only along the\nperpendicular to the rotational symmetry axis in contrast t o the spin-orbit crossed susceptibility.\nWe quantitatively compare the two susceptibilities and find that they can be comparable.\nI. INTRODUCTION\nIn the presence of an external magnetic field, magne-\ntization is induced by both the orbital motion and spin\nmagnetic moment ofelectrons. When spin-orbit coupling\nisnegligible, the magnetizationiscomposedoftheorbital\nand spin magnetization, which are induced by the mini-\nmal substitution, p→p+eA, and the Zeeman coupling,\nrespectively. Additionally, spin-orbit coupling gives rise\nto the spin-orbitcrossedresponse, in whichthe spinmag-\nnetization is induced by the minimal substitution, and\nthe orbital magnetization is induced by the Zeeman cou-\npling. In the strongly spin-orbit coupled systems, the\nspin-orbit crossed response can give comparable contri-\nbution to the conventional spin and orbital magnetic re-\nsponses.\nSpin-orbit coupling plays a key role to realize a topo-\nlogical phase of matter, such as topological insulators [1]\nand topological semimetals [2]. A natural question aris-\ning is what kind of the spin-orbit crossed response occurs\nin the topological materials. Because of the topologically\nnontrivial electronic structure and the existence of the\ntopological surface states, the topological materials ex-\nhibit the spin-orbit crossed response as a topological re-\nsponse [3–7]. The spin-orbit crossed response has been\ninvestigated in several systems. In the literature the con-\nnection between the spin-orbit crossed susceptibility and\nthe spin Hall conductivity was pointed out [3, 4]. In\nrecent theoretical work, the spin-orbit crossed response\nhas been investigated also in Rashba spin-orbit coupled\nsystems [8, 9].\nThe topological Dirac semimetal is one of the topo-\nlogical semimetals [10–15] and experimentally observed\nin Na3Bi and Cd 3As2[16–18]. The topological Dirac\nsemimetals have an inverted band structure originating\nfrom strong spin-orbit coupling. They are characterized\nby a pair of Dirac points in the bulk and Fermi arcs on\nthe surface [10, 11]. The Dirac points are protected by\nrotational symmetry along the axis perpendicular to the\n(001) surface in the case of Na 3Bi and Cd 3As2[10, 11].\nThis is an important difference from the Dirac semimet-\nals appearing at the phase boundary of topological in-\nsulators and ordinary insulators [19–22], in which thereis no Fermi arc. A remarkable feature of the topological\nDirac semimetals is the conservation of the spin angular\nmomentum along the rotation axis within a low energy\napproximation [23]. The topological Dirac semimetals\nare regarded as layers of two-dimensional (2D) quantum\nspin Hall insulators (QSHI) stacked in momentum space\nand exhibit the intrinsic semi-quantized spin Hall effect.\nThe magnetic responses of the generic Dirac electrons\nhave been investigated in several theoretical papers. The\norbital susceptibility logarithmically diverges and ex-\nhibits strong diamagnetism at the Dirac point [6, 24–26].\nWhen spin-orbit coupling is not negligible, the spin sus-\nceptibility becomes finite even at the Dirac point where\nthe density of states vanishes [6, 27–29]. This is contrast\nto the conventional Pauli paramagnetism and known as\nthe Van Vleck paramagnetism [29–32].\nIn this paper, we study the spin-orbit crossed suscep-\ntibility of the topological Dirac semimetals. We find that\nthe spin-orbitcrossedsusceptibility isproportionalto the\nseparation of the Dirac points and independent of the\nother microscopic parameters of the materials. We also\ninclude the spin conservation breaking term which mixes\nup and down spins [10, 11]. We confirm that the spin-\norbit crossedsusceptibility is approximatelyproportional\nto the separation of the Dirac points even in the absence\nof the spin conservation as long as the separation is suf-\nficiently small. We also calculate the spin susceptibility\nand quantitatively compare the two susceptibilities. Us-\ning the material parameters for Na 3Bi and Cd 3As2, we\nshow that the contribution of the spin-orbit crossed sus-\nceptibilityisimportantinordertoappropriatelyestimate\nthe total susceptibility.\nThe paper is organized as follows. In Sec. II, we in-\ntroduce a model Hamiltonian and define the spin-orbit\ncrossed susceptibility. In Secs. III and IV, we calculate\nthe spin-orbit crossed susceptibility and the spin suscep-\ntibility. In Secs. V and VI, the discussion and conclusion\nare given.2\nII. MODEL HAMILTONIAN\nWe consider a model Hamiltonian on the cubic lattice\nHk=HTDS+Hxy+HZeeman, (1)\nwhich is composed of three terms. The first and sec-\nond terms describe the electronic states in the topolog-\nical Dirac semimetals, which reduces to the low energy\neffective Hamiltonian around the Γ point [10–12, 14, 15].\nThe first term is given by\nHTDS=εk+τxσztsin(kxa)−τytsin(kya)+τzmk,(2)\nwhere\nεk=C0−C1cos(kzc)−C2[cos(kxa)+cos(kya)],\nmk=m0+m1cos(kzc)+m2[cos(kxa)+cos(kya)].\n(3)\nPauli matrices σandτact on real and pseudo spin (or-\nbital) degrees of freedom. aandcare the lattice con-\nstants.t,C1, andC2are hopping parameters. C0gives\nconstant energy shift. m0,m1, andm2are related to\nstrength of spin-orbit coupling and lead band inversion.\nThere are Dirac points at (0 ,0,±kD),\nkD=1\ncarccos/parenleftbigg\n−m0+2m2\nm1/parenrightbigg\n. (4)\nThe separation of the Dirac points is tuned by chang-\ning the parameters, m0,m1, andm2. The first term,\nHTDS, commutes with the spin operator σz, andHTDS\nis regarded as the Bernevig-Hughes-Zhangmodel [12, 33]\nextended to three-dimension. The second term is given\nby\nHxy=τxσxγ[cos(kya)−cos(kxa)]sin(kzc)\n+τxσyγsin(kxa)sin(kya)sin(kzc),(5)\nwhich mixes up and down spins. When Hxyis expanded\naround the Γ point, leading order terms are third or-\nder terms, which are related to the rotational symmetry\nalongtheaxisperpendiculartothe(001)surfaceinNa 3Bi\nand Cd 3As2. In the currentsystem, this axiscorresponds\nto thez-axis and we call it the rotational symmetry axis\nin the following. γcorresponds to the coefficient of the\nthird order terms in the effective model [10, 11]. When γ\niszero, the z-componentofspin conserves. At finite γ, on\nthe otherhand, the z-componentofspin isnot conserved.\nAswementionedintheintroduction,theexternalmag-\nnetic field enters the Hamiltonian via the minimal substi-\ntution,p→p+eA, and the Zeeman coupling. We for-\nmally distinguish the magnetic field by the way it enters\nthe Hamiltonianinordertoextractthespin-orbitcrossed\nresponse. BorbitandBspinrepresent the magnetic field\nin the minimal substitution and in the Zeeman coupling\nrespectively. They are the same quantities so that we\nhave to set Borbit=Bspinat the end of the calculation.In the following, the subscripts α,β,γ,δ refer tox,y,z.\nWe define the orbital magnetization Morbit\nαand the spin\nmagnetization Mspin\nαas follows\nMorbit\nα=−1\nV∂Ω\n∂Borbitα, (6)\nMspin\nα=−1\nV∂Ω\n∂Bspin\nα, (7)\nwhere Ω is the thermodynamic potential and Vis the\nsystemvolume. Thesequantitiesarewritten, up tolinear\norder in BorbitandBspin, as\nMorbit\nα=χorbit\nαβBorbit\nβ+χSO\nαβBspin\nβ, (8)\nMspin\nα=χspin\nαβBspin\nβ+χSO\nαβBorbit\nβ, (9)\nwhere\nχorbit\nαβ=∂Morbit\nα\n∂Borbit\nβ, (10)\nχspin\nαβ=∂Mspin\nα\n∂Bspin\nβ, (11)\nχSO\nαβ=∂Morbit\nα\n∂Bspin\nβ=∂Mspin\nα\n∂Borbit\nβ. (12)\nSpin-orbit coupling can give the spin-orbit crossed sus-\nceptibility χSO\nαβ, in addition to the conventional spin and\norbital susceptibilities, χspin\nαβandχorbit\nαβ[6, 7].\nIn the rest of the paper, we focus on the Zeeman cou-\npling, which can induce both of the orbital and spin mag-\nnetization as we see in Eqs. (8) and (9). The Zeeman\ncoupling is given by\nHZeeman=−µB\n2/parenleftbigg\ngsσ0\n0gpσ/parenrightbigg\n·Bspin,\n=−g+µBτ0σ·Bspin−g−µBτzσ·Bspin,(13)\nwhereµBis the Bohr magneton and gs,gpcorrespond to\ntheg-factors of electrons in sandporbitals, respectively.\nWe define g+= (gs+gp)/4 andg−= (gs−gp)/4, so that\nthe Zeeman coupling contains two terms, the symmetric\ntermτ0σand the antisymmetric term τzσ[7, 34, 35].\nIII. SPIN-ORBIT CROSSED SUSCEPTIBILITY\nA. Formulation\nThe orbital magnetization is calculated by the formula\n[36–40],\nMorbit\nα=e\n2/planckover2pi1/summationdisplay\nn/integraldisplay\nBZd3k\n(2π)3fnkǫαβγ\n×Im/angbracketleft∂βn,k|(εnk+Hk−2µ)|∂γn,k/angbracketright,(14)\nwherefnk=/bracketleftbig\n1+e(εnk−µ)/kBT/bracketrightbig−1is the Fermi distribu-\ntion function, ∂α=∂\n∂kα, and|n,k/angbracketrightis a eigenstate of Hk3\nand its eigenenergy is εnk. The derivative of the eigen-\nstates|∂αn,k/angbracketrightis expanded as [39]\n|∂αn,k/angbracketright=cn|n,k/angbracketright+/summationdisplay\nm/negationslash=n/angbracketleftm,k|/planckover2pi1vα|n,k/angbracketright\nεmk−εnk|m,k/angbracketright,(15)\nwhere the velocity operator vαis given by vα=∂αHk//planckover2pi1\nandcnis a pure imaginary number. Using Eq. (15), the\nformula, Eq. (14), is written as\nMorbit\nα=e\n2/planckover2pi1/summationdisplay\nn/integraldisplay\nBZd3k\n(2π)3fnkǫαβγ\n×Im/summationdisplay\nm/negationslash=n/angbracketleftn,k|/planckover2pi1vβ|m,k/angbracketright/angbracketleftm,k|/planckover2pi1vγ|n,k/angbracketright\n(εmk−εnk)2(εnk+εmk−2µ).\n(16)\nWeusetheaboveformulainnumericalcalculation. Using\nthe 2D orbital magnetization Morbit(2D)\nz(kz) at fixed kz,\nMorbit\nzis expressed as\nMorbit\nz=/integraldisplayπ/c\n−π/cdkz\n2πMorbit(2D)\nz(kz).(17)\nThe aboveexpressionis useful when wediscussnumerical\nresults for χSO\nzz. We can relate χSO\nαβto the Kubo formula\nfor the Hall conductivity,\nσαβ=e2\n/planckover2pi1/summationdisplay\nn/integraldisplay\nBZd3k\n(2π)3fnkǫαβγ\n×Im/summationdisplay\nm/negationslash=n/angbracketleftn,k|/planckover2pi1vβ|m,k/angbracketright/angbracketleftm,k|/planckover2pi1vγ|n,k/angbracketright\n(εmk−εnk)2.(18)\nWhen the density of states at the Fermi level vanishes,\nthe intrinsic anomalous Hall conductivity is derived by\nthe Streda formula [3, 4, 41],\nσαβ=−eǫαβγ∂Morbit\nγ\n∂µ,\n=−eǫαβγ∂χSO\nγδ\n∂µBspin\nδ. (19)\nThe topological Dirac semimetals possess time reversal\nsymmetry, so that the Hall conductivity is zero in the\nabsence of the magnetic field. On the other hand, in the\npresence of the magnetic field, this formula suggests that\nthe anomalous Hall conductivity at the Dirac point be-\ncomes finite beside the ordinary Hall conductivity, if χSO\nγδ\nis not symmetric as a function of the Fermi energy εF.\nIn the following section, we only consider χSO\nαα, because\nχSO\nαβ(α/negationslash=β) becomes zero from the view point of the\ncrystalline symmetry in Na 3Bi and Cd 3As2.\nB. Numerical results\nNumerically differentiating Eq. (16) with respect to\nBspin\nα, we obtain χSO\nαα. In Sec. III and IV, we omit εkin Eq. (2) for simplicity. This simplification does not\nchange essential results in the following calculations. In\nSec. V, we incorporate εkin order to compare the spin-\norbit crossed susceptibility and the spin susceptibility\nquantitatively in Na 3Bi and Cd 3As2. Figure 1 shows\nthe spin-orbit crossed susceptibility χSO\nzzatεF= 0 as\na function of the separation of the Dirac points kD. In\nthe present model, there are several parameters, such as\nt,a,m 0,and so on. We systematically change them and\nfind which parameter affect the value of χSO\nzz. Figure 1\n(a), (b), and (c) show that χSO\nzzincreases linearly with\nkDand satisfy following relation,\nχSO\nzz=g+µB2e\nhkD\nπ. (20)\nχSO\nzzis proportional to the separation of the Dirac points\nkDand the coupling constant g+µB.\nEq. (20) is given by numerical calculation. This result\nis understood as follows. χSO\nzzis obtained as\nχSO\nzz=/integraldisplayπ/c\n−π/cdkz\n2πχSO\nzz(2D)(kz), (21)\nwhereχSO\nzz(2D)(kz) is the 2D spin-orbit crossed suscepti-\nbility at fixed kz, which is defined in the same way as\nEq. (12). χSO\nzz(2D)is quantized as 2 g+µBe/hin the 2D-\nQSHI and vanishes in the ordinary insulators [4, 7]. The\ntopological Dirac semimetal is regarded as layers of the\n2D-QSHI stacked in the momentum space and the spin\nChern number on the kx-kyplane with fixed kzbecomes\nfinite only between the Dirac points. As a result, we\nobtain Eq. (20). The sign of χSO\nzzdepends on the spin\nChern number on the kx-kyplane with fixed kzbetween\ntheDiracpoints. ThisisanalogoustotheanomalousHall\nconductivity in the Weyl semimetals [2, 23, 42]. In Fig. 1\n(d),χSO\nzzincreases linearly at small kDbut deviates from\nEq. (20) for finite γ. This is because the z-component\nof spin is not conserved in the presence of Hxy, Eq. (5),\nand the above argument for 2D-QSHI is not applicable\nto the present system. In the following calculation, we\nsetm0=−2m2,m1=m2,m1/t= 1 and c/a= 1.\nFigure 2 shows χSO\nααatεF= 0 as a function of γ. At\nγ= 0,χSO\nzzis finite as we mentioned above. On the other\nhand,χSO\nxxandχSO\nyyare zero. This means that the orbital\nmagnetization is induced only along z-axis, which is the\nrotational symmetry axis. As a function of γ,χSO\nzzis an\neven function and χSO\nxx(yy)is an odd function.\nFigure 3 (a) shows χSO\nzzaround the Dirac point as a\nfunction of εF. Wheng−/g+= 0,χSO\nzzisan evenfunction\naround the Dirac point. At εF= 0,χSO\nzzis independent\nofg−/g+as we see it in Fig. 1 (b). When g−/g+/negationslash= 0,\nhowever, χSO\nzzis asymmetric and the derivative of χSO\nzzis\nfinite. This suggests that the Hall conductivity is finite\nwheng−/g+/negationslash= 0. Calculating Eq. (18) numerically, We\nconfirm that the Hall conductivity is finite at εF= 0.\nFigure 3 (b) shows σxyas a function g−/g+.σxylinearly\nincreases with g−/g+. The topological Dirac semimetal4\n0.0 0.5\nkD [ π/c]1.00.0 0.5 1.0(a)εF=0 \n0.0 0.5\nkD [ π/c]1.00.0 0.5 1.0(c)\n0.0 0.5\nkD [ π/c]1.00.0 0.5 1.0(d)0.0 0.5\nkD [ π/c]1.00.0 0.5 1.0(b)\nm1/t=0.5\n1.0 \n2.0 g-/g+=0.0 \n0.5 \n1.0 \nc/a=1.0 \n2.0 \n3.0 γ/t=0.0 \n0.5 \n1.0 χzz   [2g+μBe/(ha)]SO \nχzz   [2g+μBe/(ha)]SO χzz   [2g+μBe/(ha)]SO \nχzz   [2g+μBe/(ha)]SO \nFIG. 1: The spin-orbit crossed susceptibility χSO\nzzatεF= 0\nas a function of kD. We set the parameters m1=m2,m1/t=\n1,g−/g+= 1,c/a= 1,andγ= 0, if the parameters are\nnot indicated in each figure. The panels (a), (b), and (c)\nshow that χSO\nzzis proportional to kD, which means that χSO\nzz\nreflects the topological property of the electronic structu re.\nFrom these numerical results, we obtain analytical express ion\nforχSO\nzz, Eq. (20), which is independent of model parameters\nexcept for kDandg+. The panel (d) show that Hxyreduces\nχSO\nzzbut it is negligible for sufficiently small kD.\nis viewed as a time reversal pair of the Weyl semimetal\nwith up and down spin. Therefore, the Hall conductivity\ncompletely cancel with each other. Even in the presence\nofg+Zeeman term (the symmetric term), the cancella-\ntion is retained. In the presence of g−Zeeman term (the\nantisymmetric term), on the other hand, the cancella-\ntion is broken. This is because g−Zeeman term changes\nthe separation of the Dirac points and the direction of\nthe change is opposite for the up and down spin Weyl\nsemimetals. As a result, the Hall conductivity is finite in\ng−/g+/negationslash= 0 and given by\nσxy=2\nπe2\nhag−µBBspin\nt. (22)\nThis expression is quantitatively consistent with the nu-\nmerical result in Fig. 3 (b).γ [t] 1.5\n1.0\n0.5\n0.0\n-0.5\n0.0 1.0 0.5 -0.5 -1.0χzz \nχyy χxx εF=0 \nSO \nSO \nSO χ    [g+μB(2e/h)k D]SO \nFIG. 2: The spin-orbit crossed susceptibility as a function of\nγ. The solid black curve is χSO\nzz, the blue dashed curve is\nχSO\nxx, and the red dashed curve is χSO\nyy. We set the parameters\nm0=−2m2,m1=m2,m1/t= 1,g−/g+= 1, and c/a= 1.\nBreaking the conservation of σz, i.e., with the increase of γ,\nχSO\nzzis reduced, while χSO\nxxandχSO\nyybecome finite.\nIV. SPIN SUSCEPTIBILITY\nIn this section, we calculate the spin susceptibility us-\ning the Kubo formula,\nχspin\nαα(q,εF) =1\nV/summationdisplay\nnmk−fnk+fmk−q\nεnk−εmk−q\n×µ2\nB|/angbracketleftn,k|g+τ0σα+g−τzσα|m,k−q/angbracketright|2,\n(23)\nwhereVis the system volume, fnkis the Fermi distribu-\ntion function, εnkis energy of n-th band and |n,k/angbracketrightis a\nBloch state of the unperturbed Hamiltonian. Taking the\nlong wavelength limit |q| →0, we obtain\nlim\n|q|→0χspin\nαα(q,εF) =χintra\nαα(εF)+χinter\nαα(εF),(24)\nwhereχintra\nαα(εF) is an intraband contribution,\nχintra\nαα(εF) =1\nV/summationdisplay\nnk/parenleftbigg\n−∂fnk\n∂εnk/parenrightbigg\n×µ2\nB|/angbracketleftn,k|g+τ0σα+g−τzσα|n,k/angbracketright|2,(25)\nandχinter\nαα(εF) is an interband contribution,\nχinter\nαα(εF) =1\nV/summationdisplay\nn/negationslash=m,k−fnk+fmk\nεnk−εmk\n×µ2\nB|/angbracketleftn,k|g+τ0σα+g−τzσα|m,k/angbracketright|2.(26)\nAt the zero temperature, only electronic states on the\nFermi surface contribute to χintra\nαα. On the other hand, all\nelectronic states below the Fermi energy can contribute\ntoχinter\nαα[29]. From the above expression, we see that\nχinter\nααbecomes finite, when the matrix elements of the5\n-0.02-0.010.000.02\n0.01\n0 2 1 -1 -2 \ng-/g +σxy [e2/(ha)] εF [t]-1.0-0.50.0\n0.0 1.0 0.5 -0.5 -1.01.5\n1.0\n0.5\n-1.5g-/g +=1 g-/g +=-1 \ng-/g +=0 (a) \n(b) \nεF=0 \nBspin =0.01t/(g +μB)χzz   [g+μB(2e/h)k D]SO \nFIG.3: Thespin-orbitcrossed susceptibility χSO\nzzasafunction\nofεFand the Hall conductivity as a function of g−/g+. We\nset the parameters m0=−2m2,m1=m2,m1/t= 1,c/a= 1,\nandγ= 0. At εF= 0, the value of χSO\nzzis independent of\ng−but itsεFdependence changes at finite g−. Consequently,\nthe Hall conductivity becomes finite in accordance with Eq.\n(19).\nspin magnetization operatorbetween the conduction and\nvalence bands is non-zero, i.e. the commutation relation\nbetween the Hamiltonian and the spin magnetization op-\nerator is non-zero. If the Hamiltonian and the spin mag-\nnetization operator commute,\n/angbracketleftn,k|[Hk,g+τ0σα+g−τzσα]|m,k/angbracketright= 0,(27)\nthe interband matrix element satisfies\n(εnk−εmk)/angbracketleftn,k|g+τ0σα+g−τzσα|m,k/angbracketright= 0.(28)\nThis equation means that there is no interband matrix\nelement and χinter\nαα= 0, because εnk−εmk/negationslash= 0.\nIn the following, we set εF= 0, where the density of\nstates vanishes. Therefore, there is no intraband contri-\nbution and we only consider the interband contribution.\nWe numerically calculate Eq. (26). Figure 4 shows the\nspin susceptibility χspin\nααas a function of (a) γand (b)\ng−/g+. In the following, we explain the qualitative be-\nhavior of χspin\nααusing the commutation relation between\nthe Hamiltonianand the spin magnetizationoperator. In\nFig. (4) (a), χspin\nzzvanishes at γ= 0, because the Hamil-\ntonian,HTDS, and the spin magnetization operator of\nz-component, g+µBτ0σz, commute,\n[HTDS,g+µBτ0σz] = 0. (29)For finite γ, on the other hand, χspin\nzzincreases with |γ|.\nThis is because the commutation relation between Hxy\nandg+µBτ0σzis non-zero,\n[Hxy,g+µBτ0σz]/negationslash= 0, (30)\nandχinter\nzzgives finite contribution. χspin\nxxandχspin\nyyare\nfinite even in the absence of Hxy, i.e.γ= 0, because\nHTDSandg+µBτ0σα(α=x,y) do not commute,\n[HTDS,g+µBτ0σx]/negationslash= 0,\n[HTDS,g+µBτ0σy]/negationslash= 0. (31)\nAtγ= 0,χspin\nxxis equal to χspin\nyy. For finite γ, however,\nthey deviate from each other. This is because HTDSpos-\nsessesfour-foldrotationalsymmetryalong z-axisbut Hxy\nbreaks the four-fold rotational symmetry. Figure (4) (b)\nshows that χSO\nzzbecomes finite when g−/g+/negationslash= 0. The\nantisymmetric term, g−µBτzσz, andHTDSdo not com-\nmute,\n[HTDS,g−µBτzσz]/negationslash= 0. (32)\nConsequently, χinter\nzzgives finite contribution, though the\nz-componentof spin is a good quantum number. The an-\ntisymmetric term does not break the four-fold rotational\nsymmetry along z-axis, so that χspin\nxxis equal to χspin\nyyin\nFig. (4) (b).\nThe spin susceptibility χspin\nααis also anisotropic but\ncontrasts with the spin-orbit crossed susceptivity χSO\nαα.\nχspin\nxxandχspin\nyyare larger than χspin\nzz, in contrast χSO\nzzis\nlarger than χSO\nxxandχSO\nyy. Therefore, the angle depen-\ndence measurement of magnetization will be useful to\nseparate the contribution from the each susceptibility.\nV. DISCUSSION\nIn this section, we quantitatively compare the spin-\norbit crossed susceptibility χSO\nzzand the spin susceptibil-\nityχspin\nzzat the Dirac points as a function of g−/g+. In\nthe following calculation, we set the parameters to re-\nproduce the energy band structure around the Γ point\ncalculated by the first principle calculation for Cd 2As3\nand Na 3Bi [10, 15]. The parameters are listed in the\ntable and we omit Hxy, i.e.γ= 0.\nFigure 5 shows the two susceptibilities as a function of\ng−/g+. We find that the two susceptibilities are approx-\nimately written as\nχspin\nzz∼/parenleftbiggg−\ng+/parenrightbigg2\n, (33)\nand\nχSO\nzz∼ −1\ng+/parenleftbigg\nχ0+g−\ng+/parenrightbigg\n, (34)\nby numerical fitting. In the present parameters, χSO\nzzis\nnegative and depends on g−/g+. The dependence on6\nεF=0 χxx \nχyy \nχzz γ [t] \ng-/g +εF=0 \n0.6\n0.4\n0.2\n0.0\n0.0 1.0 0.5 -0.5 -1.00.0 1.0 0.5 -0.5 -1.00.30.4\n0.2\n0.00.1\n(b) (a) \nχxx \nχyy \nχzz g-/g +=0 \nγ=0 spin\nspin\nspin\nspin\nspin\nspinχαα     [(g +μB/2) 2/(ta 3)]spin χαα    [(g +μB/2) 2/(ta 3)]spin \nFIG. 4: The spin susceptibility χspin\nααatεF= 0 as a function\nof (a)γand (b) g−/g+. We set m0=−2m2,m1=m2,\nm1/t= 1, and c/a= 1. At γ= 0 and g−/g+= 0,χspin\nzz= 0\nwhileχspin\nxx,χspin\nyy>0. These behaviors are explained by the\ncommutation relation between the Hamiltonian and the spin\nmagnetization operators as discussed in the main text.\ng−/g+originates from the existence of εk, which breaks\ntheparticle-holesymmetry. The g-factorsareexperimen-\ntally estimated as gs= 18.6 for Cd 2As3[43] andg−= 20\nfor Na 3Bi [44]. Unfortunately, there is no experimental\ndatawhichdeterminesbothof gs,gporg+,g−. FromFig.\n5, we see that χSO\nzzcan dominate over χspin\nzzifg−/g+≃0.\nAs far as we know, there is no experimental observation\nof the magnetic susceptibility in these materials. We ex-\npect the experimental observation in near future and our\nestimation of χSO\nzzwill be useful to appropriately analyze\nexperimental data.\nMaterial parameters\nCd3As2Na3Bi\nC00.306[eV] -1.183[eV]\nC10.033[eV] 0.188[eV]\nC20.144[eV] -0.654[eV]\nm00.376[eV] 1.754[eV]\nm1-0.058[eV] -0.228[eV]\nm2-0.169[eV] -0.806[eV]\nt0.070[eV] 0.485[eV]\na12.64[˚A]5.07[˚A]\nc25.43[˚A]9.66[˚A]\nχ [(g +μB/2) 2/(ta 3)]0.030.04\n0.02\n0.000.01χ [(g +μB/2) 2/(ta 3)]\n0.030.05\n0.02\n0.000.01\ng-/g +0.0 0.4 0.2 -0.2 -0.40.04g+=5 \ng+=10 \ng+=15 \ng+=5 \ng+=10 \ng+=15 −χ zz SO −χ zz SO \nχzz spinχzz spin\nNa 3Bi Cd 2As 3\nFIG. 5: The spin-orbit crossed susceptibility χSO\nzzand the spin\nsusceptibility χspin\nzzat the Dirac points as a function of g−/g+.\nThe dashed curve is χspin\nzzand the solid lines are χSO\nzz. The\nupper (lower) panel shows Cd 2As3(Na3Bi). When g−/g+are\nsufficiently small, χSO\nzzbecomes comparable to χspin\nzz.\nVI. CONCLUSION\nWe theoretically study the spin-orbit crossed suscepti-\nbility of topological Dirac semimetals. We find that the\nspin-orbit crossed susceptibility along rotational symme-\ntry axis is proportional to the separation of the Dirac\npoints and is independent of the microscopic model pa-\nrameters. This means that χSO\nzzreflects topological prop-\nerty of the electronic structure. The spin-orbit crossed\nsusceptibility is induced only along the rotational sym-\nmetryaxis. We alsocalculatethe spinsusceptibility. The\nspin susceptibility is anisotropic and vanishingly small\nalong the rotational symmetry axis, in contrast to the\nspin-orbit crossed susceptibility. The two susceptibilities\nare quantitatively compared for material parameters of\nCd2As3and Na 3Bi. At the Dirac point, the orbital sus-\nceptibility logarithmically diverges and gives dominant\ncontribution to the total susceptibility. Off the Dirac\npoint, on the other hand, the orbital susceptibility de-\ncreases [6, 24, 25] and the contribution from the spin\nsusceptibility and the spin-orbit crossed susceptibility is\nimportant for appropriate estimation of the total suscep-\ntibility.7\nACNOWLEDGEMENT\nThis work was supported by JSPS KAKENHI Grant\nNumbers JP15H05854 and JP17K05485, and JSTCREST Grant Number JPMJCR18T2.\n[1] M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045\n(2010).\n[2] N. P. Armitage, E. J. Mele, and A. Vishwanath, Rev.\nMod. Phys. 90, 015001 (2018).\n[3] M.-F. Yang and M.-C. Chang, Phys. Rev. B 73, 073304\n(2006).\n[4] S. Murakami, Phys. Rev. Lett. 97, 236805 (2006).\n[5] Y. Tserkovnyak, D. A. Pesin, and D. Loss, Phys. Rev. B\n91, 041121 (2015).\n[6] M. Koshino and I. F. Hizbullah, Phys. Rev. B 93, 045201\n(2016).\n[7] R. Nakai and K. Nomura, Phys. Rev. B 93, 214434\n(2016).\n[8] H. Suzuura and T. Ando, Phys. Rev. B 94, 085303\n(2016).\n[9] T. Ando and H. Suzuura, Journal of the Physical Society\nof Japan 86, 014701 (2017).\n[10] Z. Wang, Y. Sun, X.-Q. Chen, C. Franchini, G. Xu,\nH. Weng, X. Dai, and Z. Fang, Physical Review B 85,\n195320 (2012).\n[11] Z. Wang, H. Weng, Q. Wu, X. Dai, and Z. Fang, Physical\nReview B 88, 125427 (2013).\n[12] T. Morimoto and A. Furusaki, Phys. Rev. B 89, 235127\n(2014).\n[13] B.-J. Yang and N. Nagaosa, Nature Communications 5,\n4898 EP (2014), article.\n[14] D. Pikulin, A. Chen, and M. Franz, Physical Review X\n6, 041021 (2016).\n[15] J. Cano, B. Bradlyn, Z. Wang, M. Hirschberger, N. P.\nOng, and B. A. Bernevig, Phys. Rev. B 95, 161306\n(2017).\n[16] Z. Liu, B. Zhou, Y. Zhang, Z. Wang, H. Weng, D. Prab-\nhakaran, S.-K. Mo, Z. Shen, Z. Fang, X. Dai, et al., Sci-\nence343, 864 (2014).\n[17] M. Neupane, S.-Y. Xu, R. Sankar, N. Alidoust, G. Bian,\nC. Liu, I. Belopolski, T.-R. Chang, H.-T. Jeng, H. Lin,\net al., Nature communications 5(2014).\n[18] S. Borisenko, Q. Gibson, D. Evtushinsky, V. Zabolotnyy ,\nB. B¨ uchner, and R. J. Cava, Physical review letters 113,\n027603 (2014).\n[19] S. Murakami, New Journal of Physics 9, 356 (2007).\n[20] H. Guo, K. Sugawara, A. Takayama, S. Souma, T. Sato,\nN. Satoh, A. Ohnishi, M. Kitaura, M. Sasaki, Q.-K. Xue,\net al., Physical Review B 83, 201104 (2011).\n[21] S.-Y. Xu, Y. Xia, L. Wray, S. Jia, F. Meier, J. Dil, J. Os-\nterwalder, B. Slomski, A. Bansil, H. Lin, et al., Science\n332, 560 (2011).\n[22] T. Sato, K. Segawa, K. Kosaka, S. Souma, K. Nakayama,\nK. Eto, T. Minami, Y. Ando, and T. Takahashi, NaturePhysics7, 840 (2011).\n[23] A. A. Burkov and Y. B. Kim, Phys. Rev. Lett. 117,\n136602 (2016).\n[24] H. Fukuyama and R. Kubo, Journal of the Physical So-\nciety of Japan 28, 570 (1970).\n[25] M. Koshino and T. Ando, Phys. Rev. B 81, 195431\n(2010).\n[26] G. P. Mikitik and Y. V. Sharlai, Phys. Rev. B 94, 195123\n(2016).\n[27] A. Thakur, K. Sadhukhan, and A. Agarwal, Phys. Rev.\nB97, 035403 (2018).\n[28] J. Zhou and H.-R. Chang, Phys. Rev. B 97, 075202\n(2018).\n[29] Y. Ominato and K. Nomura, Phys. Rev. B 97, 245207\n(2018).\n[30] J. Van Vleck, The theory of electronic and magnetic sus-\nceptibility (1932).\n[31] R. Yu, W. Zhang, H.-J. Zhang, S.-C. Zhang, X. Dai, and\nZ. Fang, Science 329, 61 (2010).\n[32] J. Zhang, C.-Z.Chang, P.Tang, Z.Zhang, X.Feng, K.Li,\nL.-l. Wang, X. Chen, C. Liu, W. Duan, et al., Science\n339, 1582 (2013).\n[33] B. A. Bernevig, T. L. Hughes, and S.-C. Zhang, Science\n314, 1757 (2006).\n[34] C.-X. Liu, X.-L. Qi, H. Zhang, X. Dai, Z. Fang, and S.-C.\nZhang, Physical Review B 82, 045122 (2010).\n[35] R. Wakatsuki, M. Ezawa, and N. Nagaosa, Scientific Re-\nports5, 13638 EP (2015).\n[36] G. Sundaramand Q.Niu, Phys.Rev. B 59, 14915 (1999).\n[37] D. Xiao, J. Shi, and Q. Niu, Phys. Rev. Lett. 95, 137204\n(2005).\n[38] T. Thonhauser, D. Ceresoli, D. Vanderbilt, and R. Resta ,\nPhys. Rev. Lett. 95, 137205 (2005).\n[39] D. Ceresoli, T. Thonhauser, D. Vanderbilt, and R. Resta ,\nPhys. Rev. B 74, 024408 (2006).\n[40] J. Shi, G. Vignale, D. Xiao, and Q. Niu, Phys. Rev. Lett.\n99, 197202 (2007).\n[41] P. Streda, Journal of Physics C: Solid State Physics 15,\nL717 (1982).\n[42] A. A. Burkov and L. Balents, Phys. Rev. Lett. 107,\n127205 (2011).\n[43] S. Jeon, B. B. Zhou, A.Gyenis, B. E. Feldman, I. Kimchi,\nA. C. Potter, Q. D. Gibson, R. J. Cava, A. Vishwanath,\nand A. Yazdani, Nature Materials 13, 851 EP (2014).\n[44] J. Xiong, S. K. Kushwaha, T. Liang, J. W. Krizan,\nM. Hirschberger, W.Wang, R.Cava, andN.Ong, Science\n350, 413 (2015)."    },    {        "title": "2003.12221v1.Generalized_magnetoelectronic_circuit_theory_and_spin_relaxation_at_interfaces_in_magnetic_multilayers.pdf",        "content": "Generalized magnetoelectronic circuit theory and spin relaxation at interfaces in\nmagnetic multilayers\nG. G. Baez Flores,1Alexey A. Kovalev,1M. van Schilfgaarde,2and K. D. Belashchenko1\n1Department of Physics and Astronomy and Nebraska Center for Materials and Nanoscience,\nUniversity of Nebraska-Lincoln, Lincoln, Nebraska 68588, USA\n2Department of Physics, King’s College London, Strand, London WC2R 2LS, United Kingdom\n(Dated: July 6, 2021)\nSpin transport at metallic interfaces is an essential ingredient of various spintronic device con-\ncepts, such as giant magnetoresistance, spin-transfer torque, and spin pumping. Spin-orbit coupling\nplays an important role in many such devices. In particular, spin current is partially absorbed at\nthe interface due to spin-orbit coupling. We develop a general magnetoelectronic circuit theory and\ngeneralize the concept of the spin mixing conductance, accounting for various mechanisms respon-\nsible for spin-flip scattering. For the special case when exchange interactions dominate, we give a\nsimple expression for the spin mixing conductance in terms of the contributions responsible for spin\nrelaxation (i.e., spin memory loss), spin torque, and spin precession. The spin-memory loss param-\neter\u000eis related to spin-flip transmission and reflection probabilities. There is no straightforward\nrelation between spin torque and spin memory loss. We calculate the spin-flip scattering rates for\nN|N, F|N, F|F interfaces using the Landauer-Büttiker method within the linear muffin-tin orbital\nmethod and determine the values of \u000eusing circuit theory.\nI. INTRODUCTION\nSpin-orbit coupling (SOC) plays an essential role at\nmetallicinterfaces, especiallyinthecontextofspintrans-\nport related phenomena such as giant magnetoresistance\n(GMR),1,2spin injection and spin accumulation,3spin\ntransfer torque,4spin pumping,5–7spin-orbit torque,8,9\nspin Hall magnetoresistance (SMR),10and spin Seebeck\neffect (SSE).11–13The concept of the spin mixing conduc-\ntance,originallyintroducedwithinthemagnetoelectronic\ncircuit theory,14plays a very important role in describing\nthe spin transport at magnetic interfaces.15\nNevertheless, the spin mixing conductance in its orig-\ninal form cannot account for various important con-\ntributions associated with spin-flip processes,16–22cou-\npling to the lattice,23,24and other effects associated with\nmagnons.25–27One can generalize the concept of spin\nmixing conductance by considering spin pumping in the\npresence of spin-flip processes28or by considering the\nmagnetoelectronic circuit theory in the presence of spin-\nflipscattering.29Sofarsuchgeneralizationswerenotable\nto clarify the role of interfacial spin relaxation (usually\nreferred to as spin memory loss or spin loss) in processes\nresponsible for spin pumping and spin-transfer torque.\nRecent progress in first-principles calculations of interfa-\ncial spin loss29suggests that an approach fully account-\ning for spin-nonconserving processes can be developed.\nExperimentally, a great deal of data is available on the\nrelationbetweenspin-orbitinteractionsandtheefficiency\nof spin-orbit torque.30–33This data is often interpreted\nintuitively in terms of the spin memory loss parameter,1\nwhile lacking careful theoretical justification.\nIn this work, we develop the most general form of the\nmagnetoelectronic circuit theory and apply it to studies\nof spin transport, concentrating on such phenomena as\nspin-orbit torque and interfacial spin relaxation in mul-tilayers. We introduce a tensor form for the generalized\nspin mixing conductance describing spin-nonconserving\nprocesses, such as spin dephasing, spin memory loss, and\nspin precession. We numerically calculate parts of the\nspinmixingconductanceresponsibleforthespinmemory\nloss in N|N, F|N, F|F interfaces in the presence of spin-\norbit interactions using the Landauer-Büttiker method\nbased on linear muffin-tin orbital (LMTO) method. We\nshow that the generalized spin mixing conductance can\nbe also used to describe spin-orbit torque when exchange\ninteractions dominate and the torque on the lattice can\nbe disregarded. Our results for the generalized spin mix-\ning conductance suggest that two distinct combinations\nof scattering amplitudes are responsible for spin mem-\nory loss and torque, and in general there is no simple\nconnection between the two.\nThe paper is organized as follows. In Sec. II, we de-\nvelop a general formulation of the magnetoelectronic cir-\ncuit theory in the presence of spin-flip scattering. In Sec.\nIII, we apply the magnetoelectronic circuit theory to cal-\nculations of spin loss in (N 1N2)N, (N 1F2)N, or (F 1F2)N\nmultilayers connected to ferromagnetic leads. In Sec. IV,\nwe apply the magnetoelectronic circuit theory to spin-\norbit torque calculations. Computational details are de-\nscribed in Sec. V, and the technicalities of the adiabatic\nembedding approach are detailed in Sec. VI. Section VII\npresents numerical results for the spin-flip transmission\nand reflection rates and area-resistance products for N|N,\nF|N, F|F interfaces. Section VIII concludes the paper.arXiv:2003.12221v1  [cond-mat.mes-hall]  27 Mar 20202\nII. GENERALIZED CIRCUIT THEORY\nA. Formalism\nThe magnetoelectronic circuit theory follows from\nthe boundary conditions linking pairs of nodes in a\ncircuit.14Here we consider the general case, allowing\nspin-nonconserving scattering at interfaces between mag-\nnetic or non-magnetic metals due to the presence of spin-\norbit interaction or non-uniform magnetization. The\nboundary condition at an interface between nodes 1 and\n2, witharbitrarydistributionfunctions ^fa(a= 1;2labels\nthe node), is:\n^I2=G0X\nnmh\n^t0\nmn^f1(^t0\nmn)y\u0000\u0010\nM2^f2\u0000^rmn^f2(^rmn)y\u0011i\n;\n(1)\nwhereG0=e2=h,^rmnis the spin-dependent reflection\namplitude for electrons reflected from channel ninto\nchannelmin node 2, ^t0\nmnis the spin-dependent transmis-\nsion amplitude for electrons transmitted from channel n\nin node 1 into channel min node 2, and the Hermitian\nconjugate is taken only in spin space. Equation (1) can\nbe easily rewritten for the current ^I1in node 1. For a\nferromagnetic node, the spin accumulation is taken to be\nparallel to its magnetization. The matrices ^rmnand^t0\nmn\nare generally off-diagonal in spin space.\nIt is customary to assume that the distribution func-\ntions in the nodes, ^fa= ^\u001b0f0\na+^\u001b\u0001fs\na, are isotropic, i.e.,\nindependent of k. In this case Eq. (1) reduces to gener-\nalized Kirchhoff relations:29\nI0\n2=Gcc\n2\u0001f0+Gcs\n2\u0001\u0001fs\u0000Gm\n2\u0001fs\n2;(2)\nIs\n2=Gsc\n2\u0001f0+^Gss\n2\u0001\u0001fs\u0000^Gm\n2\u0001fs\n2;(3)\nwhere \u0001f0=f0\n1\u0000f0\n2and\u0001fs=fs\n1\u0000fs\n2are interfacial\ndrops of charge and spin components of the distribution\nfunction, and ^Ia= (^\u001b0I0\na+^\u001b\u0001Is\na)=2. The conductances in\nEq.(2-3)carryasubscript2emphasizingthattheygener-\nally differ from their counterparts describing the currents\nin node 1; this subscript will be dropped where it doesn’t\nlead to confusion. The conductances are related through\nGcs=Gsc\u0000Gt,^Gss=Gcc^\u001b0\u0000^Gt,Gm=Gt+Gr,\n^Gm=^Gt+^Grto the following scalar, vector, and tensorquantities:\nGcc= 2G0X\nmnT\u0017\u0017\nmn; (4)\nGt\ni= 4G0X\nmni\"ijkTjk\nmn; (5)\nGr\ni= 4G0X\nmni\"ijkRjk\nmn; (6)\nGsc\ni= 2G0X\nmn(Ti0\nmn+T0i\nmn+i\"ijkTjk\nmn); (7)\nGt\nij= 2G0\u000ekl\nijX\nmn(Tkl\nmn+Tlk\nmn+i\"klp[T0p\nmn\u0000Tp0\nmn]);(8)\nGr\nij= 2G0\u000ekl\nijX\nmn(Rkl\nmn+Rlk\nmn+i\"klp[R0p\nmn\u0000Rp0\nmn]);\n(9)\nwhere\u000ekl\nij=\u000eij\u000ekl\u0000\u000eik\u000ejl, Latin indices i;:::;ldenote\nCartesian coordinates and m,nthe conduction channels,\nandrepeatedCartesianindicesaresummedoverhereand\nbelow. In the above expressions, we defined the following\ncombinations of scattering matrix elements:\nR\u0016\u0017\nmn=Tr[(^rmn\n^r\u0003\nmn)\u0001(^\u001b\u0016\n^\u001b\u0017)]=4;(10)\nT\u0016\u0017\nmn=Tr[(^t0\nmn\n^t0\u0003\nmn)\u0001(^\u001b\u0016\n^\u001b\u0017)]=4;(11)\nwhere Greek indices can take values from 0 to 3.\nIn order to obtain the circuit theory equations (2) and\n(3) from Eq. (1), we used the trace relations for Pauli\nmatrices, Tr (^\u001bi^\u001bj) = 2\u000eij, Tr(^\u001bi^\u001bj^\u001bk) = 2i\"ijk, and\nTr(^\u001bi^\u001bj^\u001bk^\u001bl) = 2(\u000eij\u000ekl+\u000eil\u000ejk\u0000\u000eik\u000ejl). The unitar-\nity condition gives the following identities:\nX\nmn^rmn^ry\nmn+^t0\nmn(^t0\nmn)y=M2^\u001b0; (12)\nX\nmn^r0\nmn(^r0\nmn)y+^tmn(^tmn)y=M1^\u001b0;(13)\nX\nmn^rmn(^rmn)y+^tmn(^tmn)y=M2^\u001b0;(14)\nX\nmn^r0\nmn(^r0\nmn)y+^t0\nmn(^t0\nmn)y=M1^\u001b0;(15)\nwhich relate the conductances defined for the two nodes\nseparated by the interface as Gcc\n1=Gcc\n2,Gcs\n1=Gcs\n2+\nGm\n2, and Gcs\n2=Gcs\n1+Gm\n1.\nThe interface conductances in the magnetoelectronic\ncircuit theory have to be renormalized by the Sharvin re-\nsistancefortransparentOhmiccontacts34,35whichallows\ncomparison between ab initio studies and experiment.36\nThe circuit theory in Eqs. (2) and (3) can be general-\nized to account for the drift contributions in the nodes\nby renormalizing the conductances Gcc,Gcs,Gsc,Gm,\n^Gss, and ^Gm. This can be done by connecting nodes 1\nand2to proper reservoirs with spin-dependent distribu-\ntion functions ^fLand ^fRvia transparent contacts. The\ncurrents in the nodes then become ^I1= 2G0^M1(^fL\u0000^f1)\nand^I2= 2G0^M2(^f2\u0000^fR), where ^M1(2)describethenum-\nber of channels (in general spin-dependent) in the nodes.3\nEffectively, this leads to substitutions f\"(#)\n1!f\"(#)\n1+\nI\"(#)\n1=(2G0M\"(#)\n1)andf\"(#)\n2!f\"(#)\n2\u0000I\"(#)\n2=(2G0M\"(#)\n2)\nin Eqs. (2) and (3).\nFinally, we note that the conductance ^Gmdescribes\nvarious spin-nonconserving processes, such as spin de-\nphasing, spin loss, and spin precession. Therefore, it can\nbe interpreted as a tensor generalization of the spin mix-\ning conductance14,37,38to systems with spin-flip scatter-\ning. In the limiting case described in Ref. 28, our defi-\nnition reduces to the generalized tensor expression sug-\ngested there. However, our definition is more general as\nit can account for processes corresponding to spin pre-\ncession and spin memory loss. Spin-nonconserving pro-\ncesses can also result in spin-charge conversion (i.e., spin\ngalvanic effect), which is described by GmandGcscon-\nductances. Furthermore, Gscdescribes the conversion of\ncharge imbalance into spin current (inverse spin galvanic\neffect), and ^Gssis the tensor spin conductance.\nB. Spin-conserving F|N interface\nWe now apply the generalized circuit theory to an\nF|N interface. In the special case of a spin-conserving\ninterface, Eqs. (2) and (3) should be invariant under\nSO(3)rotationsinspinspace, whichreproducesthespin-\nconserving circuit theory:14,37,38\nGm= 0; (16)\nGcs=Gsc=Gscm; (17)\n^Gss=Gccm\nm; (18)\n^Gm= 2G\"#\nr(^1\u0000m\nm) + 2G\"#\nim\u0002;(19)\nwhere the tensor m\nmimplements a projection onto\nthe magnetization direction, and G\"#\nrandG\"#\niare the\nreal and imaginary parts of the spin-mixing conductance\nG\"#=G0P\nmn(\u000enm\u0000r\"\"\nmnr##\u0003\nmn\u0000t\"\"\nmnt##\u0003\nmn).\nC. General F|N interface\nTo understand further the structure of current re-\nsponses, we expand the vector and tensor conductances\nin powers of magnetization:\nG\u000b\ni=G\u000b(0)\ni+G\u000b(1)\ni;kmk+G\u000b(2)\ni;klmkml+\u0001\u0001\u0001;(20)\nG\f\nij=G\f(0)\nij+G\f(1)\nij;kmk+G\f(2)\nij;klmkml+\u0001\u0001\u0001;(21)\nwhere\u000bstands forsc,cs,t,r, orm,\fstands forss,\nt,r, orm, and the tensors G\u000b(0)\ni,G\u000b(1)\ni;k,G\u000b(2)\ni;kl,G\f(0)\nij,\nG\f(1)\nij;k,G\f(2)\nij;kl, etc. are invariant under the nonmagnetic\npoint group of the system.\nThe circuit theory substantially simplifies for axially\nsymmetric interfaces, which are common in polycrys-\ntalline heterostructures. Choosing the zaxis to be nor-\nmal to the interface and applying the constraints corre-\nsponding to the C1vsymmetry, we obtain the expansionof vector conductances Gsc,GscandGmto second order\ninm:\n~G\u000b=0\nB@mxx\u000b(1)\n1+mymzx\u000b(2)\n1\nmyx\u000b(1)\n1\u0000mxmzx\u000b(2)\n1\nmzx\u000b(1)\n21\nCA;(22)\nwherex\u000b(1)\n1,x\u000b(1)\n2, andx\u000b(2)\n1are arbitrary coefficients.\nFor the tensor conductances ^Gssand ^Gmwe obtain\n^G\f=0\nB@x\f(0)\n1 0 0\n0x\f(0)\n1 0\n0 0x\f(0)\n21\nCA (23)\n+0\nB@0\u0000mzx\f(1)\n1myx\f(1)\n2\nmzx\f(1)\n1 0\u0000mxx\f(1)\n2\n\u0000myx\f(1)\n3mxx\f(1)\n3 01\nCA (24)\n+0\nB@m2\nxx\f(2)\n1+m2\nzx\f(2)\n2mxmyx\f(2)\n1mxmzx\f(2)\n4\nmxmyx\f(2)\n1m2\nyx\f(2)\n1+m2\nzx\f(2)\n2mymzx\f(2)\n4\nmxmzx\f(2)\n5 mymzx\f(2)\n5m2\nzx\f(2)\n31\nCA\n(25)\nwherex\f(0)\n1,x\f(0)\n2,x\f(1)\n1,x\f(1)\n2,x\f(1)\n3,x\f(2)\n1,x\f(2)\n2,x\f(2)\n3,\nx\f(2)\n4, andx\f(2)\n5are arbitrary coefficients.\nTheroleofspin-flipscatteringbecomesthemosttrans-\nparent if both the magnetization and the spin accumu-\nlation are either parallel or perpendicular to the inter-\nface. In this case, the tensor and vector conductances\nin Eqs. (2) and (3) can be simplified, and we arrive at\nthefollowingrelationsforrelevantcomponentsassociated\nwith the in-plane and perpendicular directions:\nGcc=G0(T\"\"+T##+T\"#+T#\"); (26)\nGsc=G0(T\"\"\u0000T##+T\"#\u0000T#\"); (27)\nGt= 2G0(T\"#\u0000T#\"); Gr= 2G0(R\"#\u0000R#\");(28)\nGt= 2G0(T\"#+T#\");Gr= 2G0(R\"#+R#\");(29)\nalongwithGcs=Gsc\u0000Gt,Gss=Gcc\u0000Gt,Gm=Gt+Gr,\nandGm=Gt+Gr. Of course, all quantities in these ex-\npressions are different for the in-plane and perpendicular\norientations of the magnetization; the corresponding in-\ndex has been dropped to avoid clutter. The spin-resolved\ndimensionless transmittances and reflectances\nT\u001b\u001b0=X\nmnt\u001b\u001b0\nmn(t\u001b\u001b0\nmn)\u0003; (30)\nR\u001b\u001b0=X\nmnr\u001b\u001b0\nmn(r\u001b\u001b0\nmn)\u0003(31)\nare defined in the reference frame with the spin quanti-\nzation axis aligned with the magnetization.\nEqs. (26)-(29), together with Eqs. (2) and (3), are\nalso valid for axially symmetric F|F interfaces, as long as\nthe magnetizations of the two ferromagnets are collinear.\nThese expressions generalize the result given in Ref. 29\nfor axially symmetric N|N junctions to include F|N and\nF|F interfaces.4\nD. Relation to Valet-Fert theory\nThe Valet-Fert model39incorporates spin relaxation\nin diffusive bulk regions but makes restrictive approx-\nimations for the interfaces, treating them as transpar-\nent, spin-conserving, and prohibiting transverse spin\naccumulation.2,16,40–43When spin relaxation at inter-\nfaces is of interest, the treatment based on the Valet-Fert\nmodel is forced to replace the interfaces by fictitious bulk\nregions,1,2which is restrictive even for N|N interfaces.29\nHere we show how diffusive bulk regions can be incor-\nporated in the generalized circuit theory. By introduc-\ning nodes near the interfaces and treating both interfaces\nand bulk regions as junctions, the generalized Kirchhoff’s\nrules2,16,40–43can be used to analyze entire devices with\nspin relaxation in the diffusive bulk regions and arbitrary\nspin-nonconserving scattering at interfaces.\nThe Valet-Fert model employs the following equations\nto describe spin and charge diffusion in a normal metal:\n@2\nx(DfN\n0) = 0; (32)\n@2\n@x2(DfN\ns) =fN\ns\n\u001cN\nsf; (33)\nand in a ferromagnet:\n@2\n@x2(D\"f\"+D#f#) = 0; (34)\n@2\n@x2(D\"f\"\u0000D#f#) =f\"\u0000f#\n\u001cF\nsf: (35)\nHere fF\ns=m(f\"\u0000f#)=2is the spin accumulation in\nthe ferromagnet, and the spin-flip relaxation times \u001cN\nsf=\n(lN\nsf)2=Dand\u001cF\nsf= (lF\nsf)2(1=D\"+ 1=D#)=2are given in\nterms of the spin-diffusion lengths lN\nsf,lF\nsfand diffusion\ncoefficients D,D\u001b. We now consider three basic circuit\nelements.\n1. Diffusive N region\nFor a diffusive N layer, the solution of Eqs. (32) and\n(33) leads to a simplified version of Eqs. (2) and (3)\nwith vanishing vector conductances Gsc,Gsc,Gm, and\nall tensor conductances reduced to scalars:\nGcc\nN=2D\ntN; (36)\nGss\nN=Gcc\nN\u000eN\nsinh\u000eN; (37)\nGm\nN=Gcc\nN\u000eNtanh\u000eN\n2; (38)\nwheretNisthethicknessoftheNlayer, and \u000eN=tN=lN\nsf.2. Diffusive F region\nFor a diffusive F layer with spin accumulation that is\nparallel to the magnetization, the solution of Eqs. (34)\nand (35) leads to vanishing Gmand the other conduc-\ntances defined as follows:\nGcc\nF= (D\"+D#)=tF; (39)\nGsc\nF=Gcs\nF=m(D\"\u0000D#)=tF; (40)\nGss\nF=G\u0003\nF\u000eF\nsinh\u000eF+(Gsc\nF)2\nGcc\nF; (41)\nGm\nF=G\u0003\nF\u000eFtanh\u000eF\n2; (42)\nwhereG\u0003\nF= [(Gcc\nF)2\u0000(Gsc\nF)2]=Gcc\nFis the effective con-\nductance and tFthe thickness of the F layer, and \u000eF=\ntF=lF\nsf.\n3. Diffusive F|N junction\nAs a simple application, consider a composite junc-\ntion consisting of F and N diffusive layers separated by\na transparent interface. Such an idealized junction can\nbe used to model an interface with spin-flip scattering\nbetween F and N layers.2,16,40–43Combining the results\nfor F and N regions with boundary conditions, we find\nGm= 0and the following effective conductances:\nGcc= (1=Gcc\nF+ 1=Gcc\nN)\u00001; (43)\nGcs=Gsc=Gsc\nF; (44)\n^Gss=Gss\nNGss\nF\nGss\nN+Gss\nF+Gm\nN+Gm\nFm\nm; (45)\n^Gm=Gss\nN+Gm\nN\u0000Gss\nN(Gss\nN+Gss\nF)\nGss\nN+Gss\nF+Gm\nN+Gm\nFm\nm;(46)\nwhere the conductances for the F and N layers should\nbe taken from the previous subsections. If spin-flip\nscattering is negligible, we recover the known result:16\n^Gm=Gss\nN(1\u0000m\nm).\nIII. SPIN LOSS AT INTERFACES\nThe experimental data on interfacial spin relaxation\ncomes primarily from the measurements of magnetore-\nsistance in (N 1N2)N, (N 1F2)N, or (F 1F2)Nmultilayers\nconnected to ferromagnetic leads,1,2whereNis the num-\nber of repetitions. The results have been reported1,2in\nterms of the effective spin memory loss parameter \u000eNor\n\u000eFobtained by treating the interface as a fictitious bulk\nlayer and fitting the data to the Valet-Fert model. Here\nwe relate the experimentally measured parameter \u000eNor\n\u000eFto the generalized conductances appearing in Eqs. (2)\nand (3). We assume that the interfaces are axially sym-\nmetricandthatthemagnetizationandspinaccumulation\nare either parallel or perpendicular to the interface.5\nA. N|N multilayer\nWe first consider a multilayer with repeated interfaces\nbetween normal metals N 1and N 2. We would like to as-\nsess the decay of spin current which may include the spin\nrelaxationbothatinterfacesandinthebulk. Tothisend,\nwe place nodes in both N 1and N 2layers and consider\nthe case of axially symmetric interfaces corresponding to\nrelations, Gsc=Gsc=Gm= 0. The relevant conduc-\ntancesGcc,Gss,Gm\n1, andGm\n2account for the scattering\nin the bulk and/or at the interfaces. Using Eq. (3), we\narrive at the following equations for the spin current in\nsome arbitrary node iin the superlattice:\nIs\ni=Gss(fs\ni\u00001\u0000fs\ni)\u0000Gm\nifs\ni; (47)\nIs\ni=Gss(fs\ni\u0000fs\ni+1) +Gm\nifs\ni; (48)\nwhich results in the recursive formula:\n2Gm\ni\nGssfs\ni=fs\ni\u00001\u00002fs\ni+fs\ni+1: (49)\nThis equation has analytical solutions:\nfi\ns=C1e\u000ei+C2e\u0000\u000ei; (50)\nwhere the constants C1andC2are determined by the\nboundary conditions. In the limit of weak spin-flip scat-\ntering, we obtain the leading term for the decay rate:\n\u000e2\u0019Gm\n1+Gm\n2\nGcc; (51)\nwhere the constants C1andC2are defined by the bound-\nary conditions. Note that to the lowest order in the\nspin-flip processes, only denominator in Eq. (51) needs\nto be renormalized by the Sharvin resistance for trans-\nparent Ohmic contacts, i.e., 1=~Gcc= 1=Gcc\u0000(1=M 1+\n1=M 2)=(4G0). It is clear that the constant \u000edescribes\nhow the spin current decays as we increase the num-\nber of layers in the superlattice. The conductances in\nEq. (51) may also include scattering in the bulk where\nthe total conductances can be calculated by concatenat-\ning the corresponding bulk and interface conductances\nusing Eqs. (2) and (3). When obtaining \u000efrom experi-\nmental data, one typically considers only interfacial con-\ntributions in Eq. (51), while the bulk contributions are\nsimply removed.1This does not cause any problem when\nspin-orbit interaction is weak as in this limit the total\nGmis a simple sum of contributions from interface and\nbulk.\nB. F|N and F|F multilayers\nBy considering F|N and F|F multilayers connected to\nferromagnetic leads one can also quantify spin relaxation\nat magnetic interfaces.1In this case, a parameter \u000ede-\nscribing the decay of spin current can also be relatedto the scattering matrix elements and to the general-\nized conductances in Eq. (2) and (3). We assume that\nwe have a superlattice with repeated interfaces between\nnormal (N 1) and ferromagnetic (F 2) layers. Normal can\nbe considered a special case of F in this section, equa-\ntions derived below also apply to F|F multilayers with-\nout any modifications. We would like to assess the decay\nof spin current due to spin relaxation at interfaces and\nin the bulk. We take nodes in F and N layers and con-\nsider the case of axially symmetric interfaces. We also\nassume collinear spin transport with the magnetization\nbeing in-plane or perpendicular to interfaces. The gen-\neralized conductances may include scattering both in the\nbulk and at the interfaces. Using Eqs. (2) and (3), we\narrive at the following equations for the spin and charge\ncurrents in node i:\nI0\ni=Gcc(f0\ni\u00001\u0000f0\ni) +Gcs\ni\u00001(fs\ni\u00001\u0000fs\ni)\u0000Gm\nifs\ni;(52)\nI0\ni=Gcc(f0\ni\u0000f0\ni+1) +Gcs\ni+1(fs\ni\u0000fs\ni+1) +Gm\nifs\ni;(53)\nIs\ni=Gsc\ni\u00001(f0\ni\u00001\u0000f0\ni) +Gss(fs\ni\u00001\u0000fs\ni)\u0000Gm\nifs\ni;(54)\nIs\ni=Gsc\ni+1(f0\ni\u0000f0\ni+1) +Gss(fs\ni\u0000fs\ni+1) +Gm\nifs\ni;(55)\nwhich results in the recursive formula:\n2Gm\ni=Gsc\ni\u00001\u00002Gm\ni=Gcc\nGss=Gsc\ni\u00001\u0000Gcs\ni\u00001=Gccfs\ni=fs\ni\u00001\u00002fs\ni+fs\ni+1;(56)\nSimilar to non-magnetic case, the above equation has an-\nalytical solutions:\nfi\ns=C1e\u000ei+C2e\u0000\u000ei: (57)\nIn the limit of weak spin-flip scattering, we obtain the\nleading term for the decay rate:\n\u000e2\u0019Gm\nF+Gm\nN\nG\u0003; (58)\nwhereG\u0003= [(Gcc)2\u0000(Gsc)2]=Gccis the effective con-\nductance of the scattering region. Note that to the low-\nest order in the spin-flip processes, only denominator in\nEq. (58) needs to be renormalized by the Sharvin re-\nsistance for transparent Ohmic contacts, i.e., 1=~G\u0003=\n1=G\u0003\u0000(1=M\"\n1+ 1=M#\n1+ 1=M\"\n2+ 1=M#\n2)=(8G0). The\nconstant\u000edescribes how the spin current decays as we\nincrease the number of layers in the multilayers. The\nconductances in Eq. (58) may also include scattering in\nthe bulk. The bulk and interface conductances can be\nconcatenated using Eqs. (2) and (3).\nIV. SPIN-ORBIT TORQUE\nThe discontinuity of spin-current at the interface fol-\nlowing from the circuit theory in Eqs. (2) and (3) can\nbe used to calculate the total torque transferred to both\nthe magnetization and the lattice. In general, separating\nthese two contributions is not possible without consider-\nations beyond the circuit theory. When exchange inter-\nactions dominate and the torque on the lattice can be6\ndisregarded, we can use the circuit theory to calculate\nthe spin torque on magnetization. Note that spin-flip\nscattering and spin memory loss can still be present even\nin the absence of the lattice torque, e.g., due to magnetic\ndisorder at the interface.\nIn the absence of angular momentum transfer to the\nlattice, it is natural to assume axial symmetry with re-\nspect to magnetization direction which results in simpli-\nfications in Eqs. (22), (23), (24), and (25), i.e., x\u000b(2)\n1= 0,\nx\f(0)\n1=x\f(0)\n2,x\f(1)\n1=x\f(1)\n2=x\f(1)\n3,x\f(2)\n2= 0,x\f(2)\n3=\nx\f(2)\n4=x\f(2)\n5=x\f(2)\n1. This leads to the following gener-\nalization of Eq. (19) for the spin mixing conductance:\n^Gm= 2G\"#\nr(^1\u0000m\nm) + 2Gm\nkm\nm+ 2G\"#\nim\u0002;(59)\nwhereG\"#\nr=G0P\nmnRe(\u000enm\u0000r\"\"\nmnr##\u0003\nmn\u0000t\"\"\nmnt##\u0003\nmn)de-\nscribes the absorption of transverse spin current and\nGm\nk=G0(T\"#+T#\"+R\"#+R#\")the absorption of lon-\ngitudinal spin current (i.e., spin memory loss); G\"#\ni=\nG0P\nmnIm(\u000enm\u0000r\"\"\nmnr##\u0003\nmn\u0000t\"\"\nmnt##\u0003\nmn)describes the pre-\ncession of spins. Even though the formal expressions for\nG\"#\nrandG\"#\nidid not change compared to Eq. (19), their\nvalues can still be affected by the presence of spin-flip\nscattering due to unitarity of the scattering matrix. The\neffect of the unitarity constraint, however, does not have\na direct relation to the spin memory loss parameter \u000e.29\nUsing a typical spin-orbit torque geometry10and Eq.\n(3), we can write a boundary condition determining the\ntorque:\n2e2\n~~ \u001cF=e(^1\u0000m\nm)js= (^1\u0000m\nm)^Gm\u0001\u0016s;(60)\nwhere \u0016sis the spin accumulation and ~ \u001cFis the magneti-\nzation torque. The spin current can be further calculated\nfrom the diffusion equation:\nr2\u0016s=\u0016s=l2\nsf; (61)\nand\njs=\u0000\u001b\n2e@z\u0016s+jSH^y; (62)\nwhere the interface is orthogonal to zaxis andjSHis the\nspin Hall current. We recover conventional antidamping\nand field like torques:\n~ \u001cF= (~jSH=2e)\u0014g\"#\nrtanh\u000e=2\n1 + 2g\"#\nrcoth\u000em\u0002(m\u0002^y)(63)\n+g\"#\nitanh\u000e=2\n1 + 2g\"#\nicoth\u000em\u0002^y#\n;\nwhereg\"#\nr(i)= (lsf=\u001b)G\"#\nr(i)and\u001bistheconductivityofthe\nnormal metal. The results of this section are inconsistent\nwith the notion that spin memory loss should directly af-\nfect spin-orbit torque.30–33As can be seen from Eq. (59),\ntwo separate parameters are responsible for spin mem-\nory loss and spin-orbit torque, and in general there is nodirect connection between the two. In the presence of\nspin-orbit interactions, only the total torque acting on\nthe lattice and magnetization can be obtained from the\ncircuit theory. However, it seems that a similar conclu-\nsion can be reached about the absence of direct relation\nbetween spin memory loss and torque.\nV. COMPUTATIONAL DETAILS AND\nINTERFACE GEOMETRY\nThe transmittances and reflectances (30)-(31) were\ncalculated using the Landauer-Büttiker approach im-\nplemented in the tight-binding linear muffin-tin orbital\n(LMTO) method.44Spin-orbit coupling (SOC) was in-\ntroduced as a perturbation to the LMTO potential\nparameters.44,45Local density approximation (LDA) was\nused for exchange and correlation.46\nWe have considered a number of interfaces between\nmetals with the face-centered cubic lattice. The inter-\nfaceswereassumedtobeepitaxialwiththe(111)or(001)\ncrystallographic orientation. Lattice relaxations were ne-\nglected, and the average lattice parameter for the two\nlead metals was used for the given interface. The po-\nlarization of the spin current and the magnetization (in\nF|N and F|F systems) were taken to be either parallel or\nperpendicular to the interface.\nSelf-consistent charge and spin densities were obtained\nusing periodic supercells with at least 12 monolayers of\neach metal. The surface Brillouin zone integration in\ntransport calculations was performed with a 512\u0002512\nmesh for magnetic and 128\u0002128for non-magnetic sys-\ntems.\nWe also studied the influence of interfacial intermix-\ning on spin-memory loss at Pt jPd and AujPd interfaces.\nOne layer on each side of the interface was intermixed\nwith the metal on the other side. The mixing concentra-\ntions were varied from 11% to 50%. For example, an A|B\ninterface with 25% intermixing had two disordered lay-\ners with compositions A 0:75B0:25and A 0:25B0:75between\npure A and pure B leads. The transverse size of the su-\npercell was 2\u00022for 25% and 50% intermixing and 3\u00023\nfor 11% intermixing. The conductances were averaged\nover all possible configurations in the 2\u00022supercell and\nover 18 randomly generated configurations in 3\u00023. In\naddition, a model with long-range intermixing (LRI) was\nconsidered where the transition from pure A to pure B\noccurs over 8 intermixed monolayers with compositions\nA8=9B1=9, A 7=9B2=9,..., A 1=9B8=9. This model was im-\nplemented using 3 \u00023 supercells.\nVI. ADIABATIC EMBEDDING\nIn the Landauer-Büttiker approach, the active region\nwhere scattering takes place is embedded between ideal\nsemi-infinite leads. In the circuit theory, the leads are\nimagined to be built into the nodes of the circuit on7\nboth sides of the given interface. In order to define spin-\ndependent scattering matrices with respect to the well-\ndefined spin bases, we turn off SOC in the leads.\nToavoidspuriousscatteringattheboundarieswiththe\nSOC-free leads, we introduce “ramp-up” regions between\nthe interface and the leads, wherein the SOC is gradually\nincreased from zero at the edges of the active region to\nits actual magnitude near the interface. Specifically, for\nan atom at a distance xfrom the interface ( jxj>l0), the\nSOC parameters are scaled by (L\u00002jxj)=(L\u00002l0), where\nLis the total length of the active region and l0the length\nof the region on each side of the interface where SOC is\nretained at full strength. In our calculations we set l0to\n2 monolayers.\nBecause a slowly varying potential only allows scat-\ntering with a correspondingly small momentum transfer,\nsuch adiabatic embedding29allows a generic pure spin\nstate from the lead to evolve without scattering into the\nbulk eigenstate of the metal before being scattered at the\ninterface.\nIn a non-magnetic metal, as explained in Ref. 29, adia-\nbatic embedding leads to strong reflection near the lines\non the Fermi surface where the group velocity is paral-\nlel to the interface. Geometrically, when projected or-\nthographically onto the plane of the interface, these lines\nformtheboundariesoftheprojectedFermisurface. Elec-\ntrons with such wave vectors can backscatter from the\nSOC ramp-up region both with and without a spin flip.\nThecontributionofthisbackscatteringtothespin-flipre-\nflectance is an artefact of adiabatic embedding and needs\nto be subtracted out.29In a magnetic lead such backscat-\nteringconservesspinandis,therefore,inconsequentialfor\nspin-memory loss calculations.\nAdiabatic embedding can also produce strong scatter-\ning near the intersections of different sheets of the Fermi\nsurface, where an electron can scatter from one sheet to\nanother with a small momentum transfer. Such intersec-\ntions do not exist in non-magnetic metals considered in\nthis paper (Cu, Ag, Au, Pd, Pt), but they are present in\nall ferromagnetic transition metals. When the two inter-\nsecting sheets correspond to states of opposite spin, scat-\ntering from one sheet to the other is a spin-flip process.\nDepending on the signs of the normal (to the interface)\ncomponents v?of the group velocities at the intersection,\nthis scattering may or may not change the propagation\ndirection with respect to the interface and thereby show\nup in spin-flip reflection or transmission. These two sit-\nuations are illustrated in Fig. 1. If v?has opposite signs\non the two intersecting sheets [see Fig. 1(a-b)], then SOC\nopens a gap at the avoided crossing, and incident elec-\ntrons with quasi-momenta close to the intersection are\nfully reflected from the ramp-up region with a spin flip.\nOn the other hand, if v?has the same sign on the two\nsheets [see Fig 1(c-d)], then, instead of backscattering,\nthere is a large probability of forward spin-flip scattering\nas the electron passes through the ramp-up region.\nBecause we are interested in the spin-flip scattering\nprocesses introduced by the interface, the contribution\nFIG. 1. Crossing of the electronic bands in a ferromagnetic\nlead near an intersection of two Fermi surface sheets of op-\nposite spin. The parallel component of the quasi-momentum,\nkk, is fixed. (a-b) and (c-d): Cases where the normal compo-\nnent of the group velocity v?has the same or opposite sign\non the two sheets, resulting in resonant spin-flip reflection or\ntransmission, respectively. (a) and (c): no SOC; (b) and (d):\navoided crossings induced by SOC.\nof spin-flip scattering due to the presence of the ramp-\nup regions in the leads should be subtracted out. Un-\nfortunately, this can only be done approximately. The\napproach used for N 1|N2interfaces in Ref. 29 was to sub-\ntract the spin-flip reflectances of auxiliary systems N 1|N1\nand N 2|N2where the same lead material is used on both\nsidesofanimaginaryinterfacewithadiabaticembedding.\nThis method is reasonable because the electrons incident\nfrom one of the leads and backscattered by the ramp-up\nregion never reach the interface in the real N 1|N2system.\nIn an F|N system, the same is true for the backscattering\non Fermi sheet crossings in F [the case of Fig. 1(a-b)], but\nnot for the forward scattering [the case of Fig. 1(c-d)].\nNevertheless, as a simple approximation, we extend\nthe approach of Ref. 29 to the F|N interfaces, subtract-\ning both the spin-flip reflectances in auxiliary F|F and\nN|N systems and the spin-flip transmittance in auxiliary\nF|F.Likewise,foranF 1|F2interface,wesubtractbothre-\nflectances and transmittances in F 1|F1and F 2|F2. Thus,\nfor any kind of interface, we define\nT0\n\"#=T1j2\n\"#\u0000T1j1\n\"#\u0000T2j2\n\"#(64)\nR0\na;\"#=R1j2\na;\"#\u0000Raja\n\"#; (65)\nwherea=Lora=Rdenotes one of the leads, and the\nprimed quantities are used in Eq. (58). In the follow-\ning, we refer to this as the subtraction method, and the\nparameter\u000ecalculated in this way is denoted \u000es.8\nA.k-point filtering\nA more fine-grained approach is to identify the loca-\ntions in the surface Brillouin zone where spurious reflec-\ntion or transmission occurs and filter out the contribu-\ntions to spin-flip scattering probabilities from those lo-\ncations. This filtering requires care, because some spin-\nflip scattering processes near the Fermi surface crossings\nare, in fact, physical, rather than merely being artefacts\nof adiabatic embedding. This can be seen from Fig. 2,\nwhich shows possible spin-flip scattering processes facili-\ntated by the crossing of the Fermi sheets of opposite spin.\nFigure 2(a) shows a spin-flip backscattering process in\nthe left lead, which can occur near a Fermi projection\nboundary in a normal metal or near a Fermi crossing of\nthe type shown in Fig. 1(b). The processes shown in\nFigs. 2(b) and 2(c) result from the forward scattering\nnear a Fermi crossing of the type shown in Fig. 1(d) in\nthe left lead, where the electron is then either transmit-\nted through or reflected from the interface, respectively.\nEach process has a reciprocal version. The three pro-\ncesses shown in Figs. 2(a-c) exist solely due to the pres-\nence of a ramp-up region, which provides the small mo-\nmentum transfer needed to scatter from one Fermi sheet\nto another.\nIn contrast, Figs. 2(d) and 2(e) show physical scatter-\ning processes. Here, the momentum of an electron inci-\ndent from the left lead lies inside the spin-orbit gap of\nthe type shown in Fig. 1(b) in the right lead. As a re-\nsult, the electron experiences a resonant spin-flip trans-\nmission [Fig. 2(d)] or reflection [Fig. 2(e)] at the inter-\nface. Resonant spin-flip transmission shown in Fig. 2(d)\nis possible because an electron can scatter to a different\nFermi sheet with a large momentum transfer acquired\nfrom the interface. Illustrations in Fig. 2(d-e) are highly\nschematic because the wavefunction inside the spin-orbit\ngap is evanescent in the right lead.\nLetusfirstexaminethespin-flipscatteringprocessesin\nsystems without a physical interface, where all scattering\nis due to adiabatic embedding alone. Spin-flip reflection\nat the Fermi projection boundaries can be seen in Figs.\n3(a) and 3(d) for adiabatically embedded Pt and Pd, re-\nspectively, denoted in the figure caption as a fictitious\n“interface” of a material with itself (e.g., Pd|Pd).29The\nareas with strong spin-flip reflection are notably broader\nin Pt, which has a larger spin-orbit constant compared\nto Pd. Spin-flip reflection at Fermi crossings can be seen\nin Figs. 4(a) and 4(b) for adiabatically embedded Ni and\nCo, respectively. These two cases correspond to the dia-\ngram in Fig. 2(a). Spin-flip transmission at Fermi cross-\nings in Ni and Co is seen, in turn, in Figs. 4(c) and 4(d);\nthisistheprocessshowninFig.2(b)withoutthephysical\ninterface.\nNowconsiderphysicalinterfaces. Contourswithstrong\nspin-flip reflection in, say, Fig. 3(d) for Pd|Pd are also\nseen in Fig. 3(c) for electrons incident from the Pd lead\nin Pt|Pd; the same comparison can be made for con-\ntours with strong spin-flip reflection in, say, Fig. 4(a) forNi|Ni and 4(g) for Ni|Co. These processes correspond to\nFig. 1(a). Furthermore, the contours with strong spin-\nflip transmission in Fig. 4(c) for Ni|Ni show up in both\nFig. 4(e) and 4(g) for spin-flip transmission and reflection\nin Ni|Co, respectively. These processes correspond to\nFig. 2(b) and 2(c). The contours with resonant spin-flip\ntransmission in Co|Co [Fig. 4(d)] also show up in spin-\nflip transmission for Ni|Co [Fig. 4(e)]; this corresponds\nto Fig. 2(b) with the two leads interchanged.\nAll of the spin-flip scattering processes mentioned so\nfar and corresponding to Fig. 2(a-c) are artefacts of adia-\nbatic embedding and need to be filtered out in the calcu-\nlation of the interfacial spin loss parameter. On the other\nhand, the spin-flip transmission [Fig. 4(e)] and reflection\n[Fig. 4(g)] functions for the Ni|Co interface also show the\nspin-flip resonances of the types shown in Fig. 2(d-e).\nConsider the spin-flip reflection function for electrons in-\ncident from the Ni lead for the Ni|Co interface, which\nis shown in Fig. 4(g). Apart from the resonant contours\nappearing in Fig. 4(a) and 4(c) for spin-flip reflection and\ntransmission in Ni|Ni, there are also resonant contours in\nFig. 4(g) that correspond to the spin-flip reflection reso-\nnances in Co|Co, which are seen in Fig. 4(b). The same\nresonant contours appearing in Fig. 4(e) for the spin-flip\ntransmission in Ni|Co correspond to the process shown\nin Fig. 2(d). These resonances correspond to the physical\nprocess depicted in Fig. 2(e) and should notbe filtered\nout in the calculation of the spin loss parameter.\nThis analysis shows that both artefacts of adiabatic\nembedding [Fig. 2(a-c)] and physical resonant spin-flip\nscattering processes [Fig. 2(d-e)] can be located in k-\nspaceusingspin-fliptransmissionfunctionscalculatedfor\nauxiliarysystems. Thus, asanalternativetothesubtrac-\ntion method discussed above, the artefacts of adiabatic\nembedding can be removed using k-point filtering.\nFor nonmagnetic (N 1|N2) interfaces, we first identify\nthek-points where the spin-flip reflectance in an auxil-\niary system (N 1|N1or N 2|N2) exceeds a certain threshold\nvalue, which is chosen so that the spin-flip reflectance in\nthe auxiliary system becomes less than 0:001G0if the\ncontributions from the identified k-points are excluded.\nThen the contributions from those k-points are excluded\nin the calculation of the spin-flip reflectance for electrons\nincident from the corresponding lead. To ensure that\nthe artefacts are fully removed, the excluded regions are\nslightly enlarged.\nFerromagnetic leads induce resonant scattering near\nthe crossings of the Fermi surfaces for opposite spins.\nProcesses of the types shown in Fig. 2(a-c) should be\nfiltered out, as explained above. We found that the spin-\nflip reflectances and transmittances for all ferromagnetic\ninterfacesconsideredherearedominatedbyresonantpro-\ncessesdepictedinFig.2(d-e)ratherthanbycontributions\nfrom generic k-points. Indeed, the spin-loss parameters\nobtained by excluding the processes of Fig. 2(a-c) or by\nincluding only those in Fig. 2(d-e) are almost identical.\nFigures 4(i-l) show the spin-flip scattering functions ob-\ntained by starting from Figs. 4(e-h) and filtering out ev-9\nFIG. 2. Spin-flip scattering mechanisms induced by a crossing of two Fermi sheets of opposite spin in an adiabatically embedded\ninterface with no disorder. Dashed vertical lines show the interface; the label Fspecifies that the given metal must be\nferromagnetic. Blue and red lines schematically show the trajectory of an electron before and after the spin flip. Crosses show\nphysical spin-flip scattering processes, while circles denote those that occurs solely due to adiabatic embedding.\nFIG. 3.k-resolved spin-flip reflection functions for adiabatically embedded Pt|Pt, Pd|Pd, and Pt|Pd interfaces with and without\nk-point filtering. (a) R#\"in PtjPt; (b)RL#\"in PtjPd; (c)RR#\"in PtjPd; (d)R#\"in PdjPd; (e)R#\"in PtjPt, filtered; (f) RL#\"\nin PtjPd, filtered; (g) RR#\"in PtjPd, filtered; (h) R#\"in PdjPd, filtered.\nerything other than the processes of Fig. 2(d-e). By per-\nformingk-point filtering in this way we obtain a lower\nbound on the spin-flip scattering functions and the spin-\nloss parameter, ensuring that the artefacts of adiabatic\nembedding are completely removed. The values \u000eflisted\nin Table III were obtained in this way.\nVII. RESULTS\nA. Non-magnetic interfaces\nTable I lists the area-resistance products ARand the\nspin-loss parameters for nonmagnetic interfaces. The\nsubtraction and k-point filtering methods result in simi-\nlar values of \u000e. For all material combinations, \u000eis quite\nsimilar for (001) and (111) interfaces, suggesting that the\ncrystallographic structure of the interface does not havea strong effect on interfacial spin relaxation. In all cases,\nthe spin-loss parameter is slightly lower for the parallel\norientation of the spin accumulation relative to the inter-\nface.\nThe calculated ARproducts and \u000eparameters are in\ngoodagreementwithexperimentalmeasurements1insys-\ntems without Pd, but both are strongly overestimated\nfor (Au,Ag,Cu,Pd)|Pd interfaces. However, the results\nfor the Au|Pd (111) interface with the spin accumulation\nparallel to the interface are in good agreement with re-\ncent calculations of Gupta et al.47(AR= 0:81f\n\u0001m2and\n\u000e= 0:43) based on the analysis of the local spin currents\nnear the interface.\nThe large discrepancy in ARfor interfaces with Pd\nsuggests that the idealized interface model is inadequate\nfor these interfaces. Therefore, Pt jPd and AujPd with\ninterfacial intermixing were also constructed as described\nin Section V. The results for intermixed interfaces are\nlisted in Table II. It is notable that intermixing increases10\nFIG. 4.k-resolved spin-flip transmission and reflection functions for Ni|Ni, Co|Co, and Ni|Co, and an illustration of k-point\nfiltering. (a) R#\"in NijNi; (b)R#\"in CojCo; (c)T#\"in NijNi; (d)T#\"in Co|Co; (e) T#\"in Ni|Co; (f) T\"#in Ni|Co; (g) RL\n#\"in\nNijCo; (h)RR\n#\"in NijCo; (i)T#\"in NijCo, filtered; (j) T\"#in NijCo, filtered; (k) RL\n#\"in NijCo, filtered; (l) RR\n#\"in NijCo, filtered.\ntheARproduct, while its values for ideal interfaces with\nPd are already too large compared with experimental\nreports. The spin-loss parameter \u000eis also significantly\nincreased by intermixing, which moves it further away\nfrom experimental data.\nThe disagreement with experiment in the values of AR\nand\u000eforinterfaceswithPdislikelyduetothelackofun-\nderstanding of the interfacial structure in the sputtered\nmultilayers, for which no structural characterization is\navailable, to out knowledge. It seems somewhat implau-\nsiblethattherealsputteredinterfacesaremuchlessresis-\ntive compared to both ideal or intermixed interfaces con-\nsidered here. It is possible that nominally bulk regions\nin sputtered multilayers containing Pd are more disor-\ndered and thereby have a higher resistivity and shorter\nspin-diffusion length compared to pure Pd films. The fit-\nting procedure used to extract the ARand\u000eparameters\nfor the interface1would then ascribe this additional bulk\nresistance and spin relaxation to the interfaces.B. Ferromagnetic interfaces\nTable III lists the results for interfaces with one or two\nferromagnetic leads. The ARproducts for all interfaces\nare in excellent agreement with experimental data.1The\nvalues of the spin-loss parameter obtained using the sub-\ntraction method ( \u000es) tend to be larger, by up to a factor\nof2, comparedtothe k-pointfilteringmethod( \u000ef), which\nisexpectedtobemoreaccurate. ForPt|Cotheresultsfor\nARand\u000eare in good agreement both with experiment\nand with calculations using the discontinuity of the spin\ncurrent.47In other systems ARagrees very well with ex-\nperiment but \u000eis underestimated, which may be due to\nthe neglect of interfacial disorder and to the limitations\nof the adiabatic embedding method.\nVIII. CONCLUSIONS\nWe have developed a general formalism for analyzing\nmagnetoelectronic circuits with spin-nonconserving N|N,\nF|N, or F|F interfaces between diffusive bulk regions. A\ntensor generalization of the spin mixing conductance en-11\nTABLE I. Area-resistance products AR(f\n\u0001m2) and spin-\nloss parameters obtained using the subtraction method ( \u000es)\nand the filtering method ( \u000ef) for nonmagnetic interfaces. M\ndenotes the orientation of the spin accumulation relative to\nthe interface.\nNjNPlaneMARARexp\u000es\u000ef\u000eexp\nPtjPd001k0.42\n0.14\u00060.030.600.57\n0.13\u00060.08?0.44 0.710.65\n111k0.28 0.410.36\n?0.29 0.450.38\nAujPd001k0.96\n0.23\u00060.080.710.68\n0.08\u00060.08?0.96 0.860.82\n111k0.83 0.530.54\n?0.87 0.730.69\nAgjPd001k0.92\n0.35\u00060.080.410.47\n0.15\u00060.08?1.12 0.500.54\n111k0.89 0.410.47\n?0.92 0.500.55\nCujPd001k0.81\n0.45\u00060.0050.410.47\n0.24\u00060.05?0.81 0.470.52\n111k0.80 0.430.40\n?0.81 0.530.48\nCujAu001k0.13\n0.15\u00060.0050.080.08\n0.13\u00060.07?0.13 0.110.11\n111k0.11 0.080.07\n?0.12 0.110.10\nCujPt001k0.90\n0.75\u00060.051.000.87\n0.9\u00060.1?0.89 1.070.9\n111k0.75 0.880.72\n?0.82 1.110.83\nCujAg001k0.03\n0.045\u00060.0050.020.2\n0?0.03 0.030.02\n111k0.13 0.030.03\n?0.13 0.040.04\ncodes all possible spin-nonconserving processes, such as\nspin dephasing, spin loss, and spin precession. In the\nspecial case when exchange interactions dominate, those\ncontributions can be clearly separated into terms respon-\nsible for spin memory loss, spin-orbit torque, and spin\nprecession. Surprisingly, there is no direct relation be-\ntween spin-orbit torque and spin memory loss; the two\neffects are described by different combinations of scat-\ntering amplitudes responsible for the absorption of the\ntransverse and longitudinal components of spin current\nat the interface.\nThe spin relaxation (i.e., spin memory loss) param-\neter\u000ehas been numerically calculated using Eqs. (51)\nand (58) for a number of N|N, F|N, and F|F interfaces.\nFirst-principles calculations, aided by adiabatic embed-\nding, show reasonable agreement with experiment for \u000e\nand the area-resistance products with the exception of\nN|N interfaces including a Pd lead. For such interfacesTABLE II. Same as in Table I but for non-magnetic interfaces\nwith intermixing. The percentage indicates the composition\nin the two intermixed layers. LRI refers to the long-range\nintermixing model; see Section V for details.\nNjN (mix %) PlaneMARARexp\u000es\u000ef\u000eexp\nPtjPd (11%) 111k0.29\n0.14\u00060.030.450.38\n0.13\u00060.08?0.30 0.560.40\nPtjPd (25%) 111k0.32 0.520.46\n?0.34 0.650.52\nPtjPd (50%) 111k0.36 0.580.51\n?0.38 0.720.57\nPtjPd (LRI) 111k0.82 1.200.91\n?0.85 1.340.96\nAujPd (11%) 111k0.86\n0.23\u00060.080.560.46\n0.08\u00060.08?0.90 0.760.58\nAujPd (25%) 111k0.96 0.600.58\n?1.01 0.810.73\nAujPd (50%) 111k0.95 0.600.58\n?0.99 0.820.73\nAujPd (LRI) 111k1.24 0.790.65\n?1.29 0.980.76\nTABLEIII.SameasinTableIbutforF|NandF|Finterfaces.\nF(N)jFPlaneMAR\"AR#ARARexp\u000es\u000ef\u000eexp\nCujCo001k0.292.060.59\n0.51\n\u00060.050.220.12\n0.33\n\u00060.05?0.312.050.59 0.240.14\n111k0.361.540.48 0.180.11\n?0.361.520.47 0.190.12\nPtjCo001k0.464.671.28\n0.85\n\u00060.121.120.91\n0.9\n\u00060.4?0.444.601.26 1.170.96\n111k1.701.360.76 0.810.72\n?1.821.380.80 0.910.80\nAgjCo001k0.401.870.57\n0.56\n\u00060.060.330.21\n0.33\n\u00060.1?0.431.840.57 0.380.29\n111k0.221.580.45 0.200.12\n?0.221.570.45 0.210.13\nNijCo001k0.221.040.32\n0.255\n\u00060.0250.320.15\n0.35\n\u00060.05?0.241.020.32 0.340.16\n111k0.210.730.23 0.270.17\n?0.250.720.24 0.290.16\nboth\u000eandARare strongly overestimated, which can\nnot be explained by short or long-range interfacial in-\ntermixing. The analysis of spin-flip scattering probabil-\nities for F|N and F|F interfaces suggests that interfacial\nspin relaxation is dominated by electronic states near the\ncrossings of the Fermi surfaces for opposite spins in fer-\nromagnets.\nThe generalized magnetoelectronic circuit theory pro-\nvides a convenient framework for analyzing spin trans-\nport in magnetic nanostructures with strong spin-orbit12\ncoupling at interfaces.\nACKNOWLEDGMENTS\nA. K. is much indebted to Gerrit Bauer for stimulat-\ning discussions on circuit theory with spin-flip scatter-\ning. This work was supported by the National ScienceFoundation through Grant No. DMR-1609776 and the\nNebraska MRSEC, Grant No. DMR-1420645, as well as\nby the DOE Early Career Award DE-SC0014189 (AK)\nand the EPSRC CCP9 Flagship project, EP/M011631/1\n(MvS). Computations were performed utilizing the Hol-\nland Computing Center of the University of Nebraska,\nwhich receives support from the Nebraska Research Ini-\ntiative.\n1J. Bass, J. Magn. Magn. Mater. 408, 244 (2016).\n2J. Bass and W. P. Pratt, Jr., J. Phys.: Condens. Matter\n19, 183201 (2007).\n3M. Johnson and R. H. Silsbee, Phys. Rev. Lett. 55, 1790\n(1985).\n4D. Ralph and M. Stiles, J. Magn. Magn. Mater. 320, 1190\n(2008).\n5O. Mosendz, J. E. Pearson, F. Y. Fradin, G. E. W. Bauer,\nS. D. Bader, and A. Hoffmann, Phys. Rev. Lett. 104,\n046601 (2010).\n6B.Heinrich,C.Burrowes,E.Montoya,B.Kardasz,E.Girt,\nY.-Y. Song, Y. Sun, and M. Wu, Physical Review Letters\n107, 066604 (2011).\n7Y. Tserkovnyak, A. Brataas, G. E. W. Bauer, and B. I.\nHalperin, Rev. Mod. Phys. 77, 1375 (2005).\n8L. Liu, C.-F. Pai, Y. Li, H. W. Tseng, D. C. Ralph, and\nR. A. Buhrman, Science 336, 555 (2012).\n9V. E. Demidov, S. Urazhdin, H. Ulrichs, V. Tiberkevich,\nA. Slavin, D. Baither, G. Schmitz, and S. O. Demokritov,\nNature Materials 11, 1028 (2012).\n10Y.-T. Chen, S. Takahashi, H. Nakayama, M. Althammer,\nS. T. B. Goennenwein, E. Saitoh, and G. E. W. Bauer,\nPhys. Rev. B 87, 144411 (2013).\n11G.E.W.Bauer, E.Saitoh, andB.J.vanWees, Nat.Mater.\n11, 391 (2012).\n12H. Adachi, K.-I. Uchida, E. Saitoh, and S. Maekawa, Rep.\nProg. Phys. 76, 036501 (2013).\n13D. Meier, D. Reinhardt, M. van Straaten, C. Klewe, M. Al-\nthammer, M. Schreier, S. T. B. Goennenwein, A. Gupta,\nM. Schmid, C. H. Back, et al., Nat. Commun. 6, 8211\n(2015).\n14A. Brataas, G. E. W. Bauer, and P. J. Kelly, Phys. Rep.\n427, 157 (2006).\n15M. Weiler, M. Althammer, M. Schreier, J. Lotze, M. Pern-\npeintner, S.Meyer, H.Huebl, R.Gross, A.Kamra, J.Xiao,\net al., Phys. Rev. Lett. 111, 176601 (2013).\n16A. A. Kovalev, A. Brataas, and G. E. Bauer, Phys. Rev.\nB66, 224424 (2002).\n17P.M.Haney, H.-W.Lee, K.-J.Lee, A.Manchon, andM.D.\nStiles, Phys. Rev. B 87, 174411 (2013).\n18J.-C.Rojas-Sánchez, N.Reyren, P.Laczkowski, W.Savero,\nJ.-P.Attané, C.Deranlot,M.Jamet, J.-M.George,L.Vila,\nand H. Jaffrès, Physical Review Letters 112, 106602\n(2014).\n19K.ChenandS.Zhang, Phy.Rev.Lett. 114, 126602(2015).\n20V. P. Amin and M. D. Stiles, Phys. Rev. B 94, 104420\n(2016).\n21V. P. Amin and M. D. Stiles, Phys. Rev. B 94, 104419\n(2016).\n22X. Tao, Q. Liu, B. Miao, R. Yu, Z. Feng, L. Sun, B. You,J. Du, K. Chen, S. Zhang, et al., Sci. Adv. 4, eaat1670\n(2018).\n23A. A. Kovalev, G. E. W. Bauer, and A. Brataas, Phys.\nRev. B75, 014430 (2007).\n24P. M. Haney and M. D. Stiles, Phys. Rev. Lett. 105,\n126602 (2010).\n25R. Arias and D. L. Mills, Phys. Rev. B 60, 7395 (1999).\n26A. Azevedo, A. B. Oliveira, F. M. de Aguiar, and S. M.\nRezende, Phys. Rev. B 62, 5331 (2000).\n27M. Beens, J. P. Heremans, Y. Tserkovnyak, and R. A.\nDuine, J. Phys. D: Appl. Phys. 51, 394002 (2018),\n1804.02172.\n28Y. Tserkovnyak and H. Ochoa, Phys. Rev. B 96, 100402\n(2017).\n29K. D. Belashchenko, A. A. Kovalev, and M. van Schilf-\ngaarde, Phys. Rev. Lett. 117, 207204 (2016).\n30M.-H. Nguyen, D. C. Ralph, and R. A. Buhrman, Phys.\nRev. Lett. 116, 126601 (2016).\n31C.-F.Pai, Y.Ou, L.H.Vilela-Leão, D.C.Ralph, andR.A.\nBuhrman, Phys. Rev. B 92, 064426 (2015).\n32J.-C.Rojas-Sánchez, N.Reyren, P.Laczkowski, W.Savero,\nJ.-P.Attané,C.Deranlot,M.Jamet, J.-M.George,L.Vila,\nand H. Jaffrès, Phys. Rev. Lett. 112, 106602 (2014).\n33L. Zhu, D. C. Ralph, and R. A. Buhrman, Phys. Rev. Lett.\n122, 077201 (2019).\n34K. M. Schep, J. B. A. N. van Hoof, P. J. Kelly, G. E. W.\nBauer, andJ.E.Inglesfield, Phys.Rev.B 56, 10805(1997).\n35G. E. W. Bauer, Y. Tserkovnyak, D. Huertas-Hernando,\nand A. Brataas, Phys. Rev. B 67, 094421 (2003).\n36A. A. Kovalev, G. E. W. Bauer, and A. Brataas, Phys.\nRev. B73, 054407 (2006).\n37A. Brataas, Y. V. Nazarov, and G. E. W. Bauer, Phys.\nRev. Lett. 84, 2481 (2000).\n38A. Brataas, Y. V. Nazarov, and G. E. Bauer, Eur. Phys.\nJ. B22, 99 (2001).\n39T. Valet and A. Fert, Phys. Rev. B 48, 7099 (1993).\n40K. Eid, D. Portner, J. A. Borchers, R. Loloee, M. Al-\nHaj Darwish, M. Tsoi, R. D. Slater, K. V. O’Donovan,\nH. Kurt, W. P. Pratt, et al., Phys. Rev. B 65, 054424\n(2002).\n41J. Barnaś, A. Fert, M. Gmitra, I. Weymann, and V. K.\nDugaev, Phys. Rev. B 72, 024426 (2005).\n42S. Urazhdin, R. Loloee, and W. P. Pratt, Jr., Phys. Rev.\nB71, 100401 (2005).\n43Y. Liu, Z. Yuan, R. J. H. Wesselink, A. A. Starikov, and\nP. J. Kelly, Phys. Rev. Lett. 113, 207202 (2014).\n44D.Pashov, S.Acharya, W.R.Lambrecht, J.Jackson, K.D.\nBelashchenko, A. Chantis, F. Jamet, and M. van Schilf-\ngaarde, Comput. Phys. Commun. 249, 107065 (2020).\n45I. Turek, V. Drchal, and J. Kudrnovská, Philos. Mag. 88,13\n2787 (2008).\n46U. von Barth and L. Hedin, J. Phys. C: Solid State Physics\n5, 1629 (1972).47K. Gupta, R. J. H. Wesselink, R. Liu, Z. Yuan, and P. J.\nKelly, arXiv:2001.11520 (2020)."    },    {        "title": "1708.07247v2.Strong_influence_of_spin_orbit_coupling_on_magnetotransport_in_two_dimensional_hole_systems.pdf",        "content": "arXiv:1708.07247v2  [cond-mat.mes-hall]  25 Aug 2017Strong influence of spin-orbit coupling on magnetotranspor t in two-dimensional hole\nsystems\nHong Liu, E. Marcellina, A. R. Hamilton and Dimitrie Culcer\nSchool of Physics and Australian Research Council Centre of Excellence in Low-Energy Electronics Technologies,\nUNSW Node, The University of New South Wales, Sydney 2052, Au stralia\n(Dated: August 28, 2017)\nWith a view to electrical spin manipulation and quantum comp uting applications, recent signifi-\ncant attention has been devoted to semiconductor hole syste ms, which have very strong spin-orbit\ninteractions. However, experimentally measuring, identi fying, and quantifying spin-orbit coupling\neffects in transport, such as electrically-induced spin pol arizations and spin-Hall currents, are chal-\nlenging. Here we show that the magnetotransport properties of two-dimensional (2D) hole systems\ndisplay strong signatures of the spin-orbit interaction. S pecifically, the low-magnetic field Hall co-\nefficient and longitudinal conductivity contain a contribut ion that is second order in the spin-orbit\ninteraction coefficient and is non-linear in the carrier numb er density. We propose an appropriate\nexperimental setup to probe these spin-orbit dependent mag netotransport properties, which will\npermit one to extract the spin-orbit coefficient directly fro m the magnetotransport.\nLow-dimensional hole systems have attracted consid-\nerable recent attention in the context of nanoelectron-\nics and quantum information [ 1–9]. They exhibit strong\nspin-orbit coupling but a weak hyperfine interaction,\nwhich allows fast, low-power electrical spin manipula-\ntion [10,11] and potentially increased coherence times\n[12–15] while their effective spin-3/2 is responsible for\nphysics inaccessible in electron systems [ 16–20]. Struc-\ntures with strong spin-orbit interactions coupled to su-\nperconductors may enable topological superconductivity\nhosting Majorana bound states and non-Abelian particle\nstatistics relevant for topological quantum computation\n[21–24]. In the past fabricating high-quality hole struc-\ntures was challenging, but recent years have witnessed\nextraordinary experimental progress [ 12,25–43].\nA full quantitative understanding of spin-orbit cou-\npling mechanisms is vital for the realization of spin-\ntronics devices and quantum computation architectures\n[44,45]. At the same time experimental measurement\nof spin-orbit parameters is difficult [ 46]. Spin-orbit con-\nstants can be estimated from weak antilocalization [ 47–\n50], Shubnikov-de Haas oscillations and spin precession\nin large magnetic fields (up to 2 T) [ 51–53], and state-of-\nthe-art optical measurements [ 54,55]. Many techniques\nyield only the ratio between the Rashba and Dresselhaus\nterms or allow the determination of only one type of spin\nsplitting. Likewise,experimentallyquantifyingspin-orbit\ninduced effects, such as via spin-to-charge conversion or\nvice versa, is difficult. For instance, current-induced spin\npolarizations in spin-orbit coupled systems are small and\ntheir relationship to theoretical estimates is ambiguous\n[56–58], while spin-Hall currents [ 59] can only be identi-\nfied via an edge spin accumulation [ 60–62].\nHere we show that the spin-orbit interaction can have\na sizeable effect on low magnetic-field Hall transport in a\n2D hole system, which is density-dependent and experi-\nmentally visible. Our central result, shown in Fig. 1, is a(a)\n1.2\n1.1\n1.0RH/R0\n10 8 6 4 2 0\nFz(MV/m)p = 1 x 1011cm-2\np = 1.5 x 1011cm-2\np = 2 x 1011cm-2GaAs QW1.4\n1.3\n1.2\n1.1\n1.0RH/R0GaAs\nInAs\nInSbp = 2 x 1011cm-2\n(b)\nFigure 1. Spin-orbit correction to the Hall coefficient RHof\n2D holes in various 15 nm quantum wells as a function of the\nelectric field Fzacross the well, where R0≡1\npeis the bare\nHall coefficient. Panel shows results for (a) different quantu m\nwell materials at p= 1×1011cm−2and (b) GaAs quantum\nwells at different densities.\ncorrection to the low-field Hall coefficient\nRH=1\npe/bracketleftbigg\n1+/parenleftbigg64πm∗2α2\n/planckover2pi14/parenrightbigg\np/bracketrightbigg\n, (1)\nwhereαis the coefficient of the cubic Rashba spin-orbit\nterm, which arisesfromthe applicationofan electricfield\nFzacrossthequantumwell, m∗istheheavy-holeeffective\nmass atα= 0,pis the hole density, and eis the elemen-2\ntary charge. Note that here we have chosen the z−axis\nas the quantization direction. In hole systems, where the\nspin-orbit coupling can account for as much as 40% of\nthe Fermi energy [ 63], effects of second-order in the spin-\norbit strength can be sizable in charge transport. These\nreflect spin-orbit corrections to the occupation probabili-\nties, densityofstates, andscatteringprobabilities, aswell\nas the feedback of the current-induced spin polarization\non the charge current. Quantitative evaluation shows\nthat the spin-orbit corrections can reach more than 10%\nin GaAs quantum wells, and are of the order ∼20−30%\nin InAs and InSb quantum wells (Fig. 1a). The magni-\ntude of the spin-orbit corrections also increase with den-\nsity, which is consistent with the expectation that the\nstrength of spin-orbit interaction increases with density\n(Fig.1b). It is worth noting that the correction due to\nspin-orbit coupling has already taken into account the\nfact that the spin-split subbands may have different hole\nmobilities.\nIn the following we derive the formalism and show\nhow spin-orbit coupling can give rise to corrections in\nthe magnetotransport. We consider a 2D hole system in\nthe presence of a constant electric field Fand a perpen-\ndicular magnetic field B=Bzˆz. The full Hamiltonian is\nˆH=ˆH0+ˆHE+ˆU+ˆHZ, where the band Hamiltonian ˆH0\nis defined below in Eq. ( 2),ˆHE=−eF·ˆrrepresents the\ninteractionwith the externalelectricfield ofholes ˆristhe\nposition operator, and ˆUis the impurity potential, dis-\ncussed below. The Zeeman term HZ= 3κµBσ·Bwith\nκis a material-specific parameter [ 16],µBthe Bohr mag-\nneton and σthe vector of Pauli spin matrices. Rashba\nspin-orbit coupling is expected to dominate greatly over\nthe Dresselhaus term in 2D hole gases, even in materi-\nals such as InSb in which the bulk Dresselhaus term is\nvery large [ 63]. With this in mind, the band Hamiltonian\nused in our analysis in the absence of a magnetic field is\nwritten as [ 64]\nH0k=/planckover2pi12k2\n2m∗+iα(k3\n−σ+−k3\n+σ+)≡/planckover2pi12k2\n2m∗+σ·Ωk,(2)\nwherem∗=m0\nγ1+γ2, the Pauli matrix σ±=1\n2(σx±iσy),\nk±=kx±iky. ForB= 0 the eigenvalues of the band\nHamiltonian are εk±=/planckover2pi12k2/(2m∗)±αk3. In an exter-\nnal magnetic field we replace kby the gauge-invariant\ncrystal momentum ˜k=k−eAwith the vector potential\nA=1\n2(−y,x,0). The magnetic field is assumed small\nenough that Landau quantization can be neglected, in\nother words ωcτp≪1, where ωc=eBz/m∗is the cy-\nclotron frequency and τpthe momentum relaxation time.\nTo set up our transport formalism, in the spirit of\nRef. [65], we begin with a set of time-independent states\n{ks}, where srepresents the twofold heavy-hole pseu-\ndospin. We work in terms of the canonical momentum\n/planckover2pi1k. The terms ˆH0,ˆHEandˆHZare diagonal in wave vec-\ntor but off-diagonal in band index while for elastic scat-\ntering in the first Born approximation Uss′\nkk′=Ukk′δss′.\nWithout loss of generality, here we consider short-range\nimpurity scattering. The impurities are assumed uncor-related and the averageof ∝an}bracketle{tks|ˆU|k′s′∝an}bracketri}ht∝an}bracketle{tk′s′ˆU|ks∝an}bracketri}htoverim-\npurityconfigurationsis( ni|¯Uk′k|2δss′)/V, whereniis the\nimpurity density, Vthe crystalvolume, and ¯Uk′kthe ma-\ntrix element of the potential of a single impurity.\nThe central quantity in our theory is the density oper-\nator ˆρ, which satisfies the quantum Liouville equation,\ndˆρ\ndt+i\n/planckover2pi1[ˆH,ˆρ] = 0. (3)\nThe matrix elements of ˆ ρare ˆρkk′≡ˆρss′\nkk′=∝an}bracketle{tks|ˆρ|k′s′∝an}bracketri}ht\nwith understanding that ˆ ρkk′is a matrix in heavy hole\nsubspace. The density matrix ρkk′is written as ρkk′=\nfkδkk′+gkk′, wherefkis diagonal in wave vector, while\ngkk′is off-diagonal in wave vector. The quantity of in-\nterest in determining the charge current is fksince the\ncurrent operator is diagonal in wave vector. We there-\nfore derive an effective equation for this quantity by first\nbreaking down the quantum Liouville equation into the\nkinetic equations of fkandgkk′separately, and fkobeys\ndfk\ndt+i\n/planckover2pi1[H0k+HZ,fk]+ˆJ(fk) =DE,k+DL,k,(4)\nwhere the scattering term in the Born approximation\nˆJ(fk)=1\n/planckover2pi12/integraldisplay∞\n0dt′[ˆU,e−iH0t′\n/planckover2pi1[ˆU,ˆf(t)]eiH0t′\n/planckover2pi1]kk,(5)\nand the driving terms\nDE,k=−eE\n/planckover2pi1·∂fk\n∂k, (6a)\nDL,k=1\n2e\n/planckover2pi1{ˆv×B,∂fk\n∂k}, (6b)\nstem from the applied electric field and Lorentz force re-\nspectively [ 65]. In external electric and magnetic fields\none may decompose fk=f0k+fEk+fEBk, wheref0kis\ntheequilibriumdensitymatrix, fEkisacorrectiontofirst\norderin the electricfield (but atzeromagnetic field), and\nfEBkis an additional correction that isfirst order in the\nelectricand magnetic fields. The equilibrium density ma-\ntrix is written as f0k= (1/2)[(fk++fk−)11+σ·ˆΩ(fk+−\nfk−)], where ˆΩis a unit vector and Ωwas defined in\nEq. (2), andfk±represent the Fermi-Dirac distribution\nfunctions corresponding to the two band energies εk±.\nIn linear response one may replace fk→f0kin Eq. (6a).\nOn the other hand it is trivial to check that the driving\ntermDL,kvanishes when the equilibrium density matrix\nis substituted, so in Eq. ( 6b) one may replace fk→fEk.\nHence, in this work we perform a perturbation expansion\nup to first order in the electric and magnetic fields, and\nup to second orderin the spin-orbit interaction, retaining\ntermsuptoorder α2. ThedetailedsolutionofEq.( 4)and\nthe explicit evaluation of the scattering term Eq. ( 5) are\ngiven in the Supplement. We briefly summarize the pro-\ncedure here. Firstly, with f0kknown and only DE,kon\nthe right-hand side of Eq. ( 4), we obtain fEk. Next, with3\nonlyDL,kon the right-hand side of Eq. ( 4), we obtain\nfEBk. By taking the trace with current operator the lon-\ngitudinal and transverse components of the current are\nfound as jx,y=eTr/bracketleftbig\nˆvx,yfk/bracketrightbig\n, withvi= (1//planckover2pi1)∂H0k/∂k.\nFinally, with σxxandσxythe longitudinal and Hall con-\nductivities respectively, the Hall coefficient appearing in\nEq. (1) is found through RH=σxy\nBz(σ2xx+σ2xy). For the\nHall conductivity on the other hand one needs fEBk. We\nnote that the topological Berrycurvature terms that give\ncontributions analogous to the anomalous Hall effect in\nRashba systems (with the magnetization replaced by the\nmagnetic field Bz) vanish identically when both the band\nstructure and the disorder terms are taken into account.\nTable I. The maximal hole densities for which the current\ntheory is applicable for 15 nm-wide GaAs, InAs, and InSb\nquantum wells. Densities in units of 1011cm−2.\nGaAs InAs InSb\n6.55 8.08 8.60\nThe limits of applicability of our approach are as fol-\nlows. We assume that the magnetotransport considered\nhere occurs in the weak disorder regime, i.e. εFτp//planckover2pi1≫1,\nwhereεFis Fermi energy. Furthermore, we assume that\nthe scatteringdoes not changeappreciablywhen the gate\nfield is changed at low density [ 40], so the condition\nεFτp//planckover2pi1≫1 is still valid when the gate field is changed.\nWe assume αk3\nF/ǫkin≪1 where ǫkin=/planckover2pi12k2\nF\n2m∗is kinetic\nenergy, for example in Ref. [ 48], the spin-orbit-induced\nsplitting of the heavy hole sub-band at the Fermi level is\ndetermined to be around 30% of the total Fermi energy.\nIn addition, Eq. ( 2) withαindependent of wave vector\nis a result of the Schrieffer-Wolff transformation applied\nto the Luttinger Hamiltonian, and its use requires the\nSchrieffer-Wolff method to be applicable. Furthermore,\nthroughout this paper we consider cases where only the\nHH1 band is occupied. We have calculated the exact\nwindow of applicability of our theory in Table I.\nPhysically, the terms ∝α2entering the Hall coefficient\naretracedbacktoseveralmechanisms. Firstly, spin-orbit\ncoupling gives rise to corrections to: (i) the occupation\nprobabilities, through fk±; (ii) the band energies and\ndensity of states, through dεk±/dk; and (iii) the scatter-\ning term, whichincludes intra-andinter-bandscattering,\nas well as scattering between the charge and spin distri-\nbutions. Secondly, Rashba spin-orbit coupling gives rise\nto a current-induced spin polarization [ 56], which is of\nfirst order in α, and this in turn gives rise to a feedback\neffect on the charge current, which is thenresponsible for\napproximately a quarter of the overall spin-orbit contri-\nbution to the Hall coefficient.\nAs a concrete example, a 2D hole system confined to\nGaAs/AlGaAs heterostructures is particularly promising\nsince it has not only a very high mobility, but also a spin\nsplitting that hasbeen shownto be electricallytunable in\nboth square and triangular wells [ 66]. The spin splitting\ncan be tuned from large values to nearly zero in a square3500\n3000\n2500\n2000\n1500\n1000\n500\n0α (meV nm3)\n108 6 4 2 0\nFz (MV/m)360\n340\n320\n300α (meV nm3)\n10864\nFz (MV/m) GaAs\n InAs\n InSb\n GaAs\n \nFigure 2. The Rashba coefficient αof as a function of the\nnet perpendicular electric field Fzfor 15 nm GaAs, InAs,\nand InSb quantum wells. The inset shows that αfor GaAs\ndecreases by ∼20% asFzis increased from 4 MV/m to 10\nMV/m, due to the fact the well becomes quasi-triangular at\nFz/greaterorsimilar4 MV/m.\nquantumwellwhosechargedistributioncanbecontrolled\nfrom being asymmetric to symmetric via the application\nof a surface-gate bias. Whereas thus far the theoretical\nformalismhas been general, to makeconcrete experimen-\ntal predictions we first specialize to a two-dimensional\nhole gas (2DHG) in a 15 nm-wide GaAs quantum well\nsubjected to an electric field in the ˆ zdirection, so that\nthe symmetry ofthe quantum well can be tuned arbitrar-\nily. In the simplest approximation, taking into account\nonly the lowest heavy-hole and light-hole sub-bands, in a\n2DHG the Rashba coefficient αmay be estimated as\nα=3/planckover2pi14\nm2\n0∆Eγ2∝an}bracketle{tφL|φH∝an}bracketri}ht∝an}bracketle{tφH|(−id/dz)|φL∝an}bracketri}ht.(7)\nwhere∆ Eis energysplitting ofthe lowestheavy-holeand\nlight-hole sub-bands and γ=γ2+γ3\n2, andφH,L≡φH,L(z)\nrepresents the orbitalcomponent of the heavy-hole and\nlight-hole wave functions respectively in the direction ˆz\nperpendiculartotheinterface. Forasystemwith topand\nback gates, where the electric field Fzacross the well can\nbe turned on or off, we use a modified infinite square well\nwave function in which Fzis already encoded [ 67].\nTheRashbacoefficient α, asafunctionof Fz, for15nm\nhole quantum wells is shown Fig. 2. For GaAs, at low Fz\n(Fz≪4 MV/m), the Rashba coefficient increases with\nF, whichisinaccordancewiththetrendsreportedinRef.\n[68]. AsFzis increased, αthen saturates, and, at larger\nelectricfields( Fz>4MV/m), thequantumwellbecomes\nquasi-triangular and the Rashba coefficient αdecreases\nwith increasing electric field Fz. The decrease of αas\na function of Fzin quasi-triangular wells is consistent\nwith the experimental findings of Ref. [ 69]. Note that for\ndifferent materials, αsaturates at different values of Fz,4\n(a)\n(b)1.00\n0.98\n0.96\n0.94\n0.92\n0.90\n0.88σxx/σ0\n10 8 6 4 2 0\nFz(MV/m)p = 1 x 1011cm-2\np = 1.5 x 1011cm-2\np = 2 x 1011cm-2GaAs QW1.00\n0.95\n0.90\n0.85\n0.80\n0.75\n0.70σxx/σ0GaAs\nInAs\nInSbp = 2 x 1011cm-2\nFigure 3. Ratio of Drude conductivity at finite electric field s\nto its zero electric field value, with the bare Drude conduc-\ntivityσ0≡peµ, for (a) different quantum well materials at\np= 1×1011cm−2and (b) GaAs quantum wells at different\ndensities. Here, the well width is 15 nm.\nand that the αis larger in materials with a higher atomic\nnumber [ 63].\nGiventhe dependence of α(Fig.2), andhence theHall\ncoefficient RH(Fig.1), onFz, we now outline how αcan\nbe deduced experimentally. Using a top- and backgated\nquantum well, the quantum well is initially tuned to be\nsymmetric so that αwill be zero and the hole density\ncan be measured accurately. One subsequently increases\nFz, for example to ∼4 MV/m for the GaAs quantum\nwelldiscussedabove,whilst keepingthe densityconstant.\nThis in turn results in an appreciable increase in α, and\nhence a large change in RHas a function of Fz.\nThe non-monotonic change in αas a function of Fz\nlikewiseaffectsthelongitudinalconductivity σxx(Fig.3),\nwhich reads\nσxx=σ0/bracketleftbigg\n1−/parenleftbigg60πm∗2α2\n/planckover2pi14/parenrightbigg\np/bracketrightbigg\n. (8)\nThe spin-orbit corrections are larger in InAs and InSb\n(Fig.3a) rather than GaAs. Furthermore, as the density\nincreases, σxxdecreases faster with Fz(Fig.3b). How-\never, although the spin-orbit corrections to σxxhave a\nsimilar functional form as and a similar magnitude to\nthe corrections to RH, it is difficult to single out the de-\npendence of σxxonαexperimentally. As the shape of\nthe wave functions changes with Fz, the spin-orbit in-dependent scattering properties are also altered, which\nmay then introduce a larger correction to σxxthan the\nspin-orbitinduced corrections[ 70]. In fact, the spin-orbit\nindependent corrections can alter the carrier mobility by\n∼20% even in electron quantum wells [ 40].\nVarious possibilities exist to extend the scope of the\ncalculations presented in this paper. Here we have re-\nstricted ourselves, for the sake of gaining physical insight\nand without loss of generality, to hole systems in which\nthe Schrieffer-Wolff approximation is applicable so that\nαcan be approximated as constant. In a general 2D hole\nsystemα(k) is a function of wave vector, and decreases\nwithkatlargerwavevectors. Itsbehaviourisinprinciple\nnottractableanalyticallythoughitcanstraightforwardly\nbe calculated numerically. The results we have found re-\nmain true in their general closed form for hole systems\nat arbitrary densities provided αis replaced by α(k). An\nalternative approach would be to start directly with the\n4×4 Luttinger Hamiltonian and determine the charge\nconductivity using a spin-3/2 model. However, calculat-\ning the conductivity as a function of Fzcan quickly be-\ncomeverycomplicated analytically, limiting the utility of\nsuch an approach. Finally, the kinetic equation approach\nwehavediscussedcanstraightforwardlybe generalizedto\narbitrary band structures in a way that makes it suitable\nfor fully numerical approaches relying on maximally lo-\ncalized Wannier functions [ 71].\nIt is worth mentioning how the corrections in the mag-\nnetotransport properties of 2D electrons will differ from\nthose of 2D holes. In 2D electrons, to lowest order the\nspin-orbit coupling stems from k.pcoupling with the\ntopmost valence band, and the leading contribution to\nspin-orbit interaction in 2D electrons is linear in k[16].\nAs a result, the spin-orbit dependent corrections to the\nmagnetotransport in 2D electrons will be much smaller\ncompared to 2D holes, and thus may not be detectable\nwithin experimental resolution.\nIn summary, we have presented a quantum kinetic the-\nory of magneto-transport in 2D heavy-hole systems in\na weak perpendicular magnetic field and demonstrated\nthat the Hall coefficient, as well as the longitudinal con-\nductivity, display strong signatures of the spin-orbit in-\nteraction. We have also shown that our theory provides\nan excellent qualitative agreementto existing experimen-\ntal trends for α, although to the best of our knowledge,\nthere has not been a demonstration of RHchanging as a\nfunction of α. An appropriate experimental setup with\ntop and back gates can lead to a direct electrical mea-\nsurement of the Rashba spin-orbit constant via the Hall\ncoefficient.\nACKNOWLEDGMENTS\nThis research was supported by the Australian Re-\nsearch Council Centre of Excellence in Future Low-\nEnergy Electronics Technologies (project CE170100039)\nand funded by the Australian Government.5\n[1] R. Cuan and L. Diago-Cisneros,\nEurophys. Lett. 110, 67001 (2015) .\n[2] T. Biswas, S. Chowdhury, and T. K. Ghosh,\nEur. Phys. J. B 88, 220 (2015) .\n[3] F. A. Zwanenburg, A. S. Dzurak, A. Morello,\nM. Y. Simmons, L. C. L. Hollenberg, G. Klimeck,\nS. Rogge, S. N. Coppersmith, and M. A. Eriksson,\nRev. Mod. Phys. 85, 961 (2013) .\n[4] S. Conesa-Boj, A. Li, S. Koelling, M. Brauns, J. Rid-\nderbos, T. T. Nguyen, M. A. Verheijen, P. M. Koen-\nraad, F. A. Zwanenburg, and E. P. A. M. Bakkers,\nNano Letters 17, 2259 (2017) .\n[5] M. Brauns, J. Ridderbos, A. Li, W. G. van der\nWiel, E. P. A. M. Bakkers, and F. A. Zwanenburg,\nAppl. Phys. Lett. 109, 143113 (2016) .\n[6] M. Brauns, J. Ridderbos, A. Li, E. P. A. M. Bakkers,\nW. G. van der Wiel, and F. A. Zwanenburg,\nPhys. Rev. B 94, 041411 (2016) .\n[7] F. Mueller, G. Konstantaras, P. C. Spruijtenburg,\nW. G. van der Wiel, and F. A. Zwanenburg,\nNano Lett. 15, 5336 (2015) .\n[8] F. Qu, J. van Veen, F. K. de Vries, A. J. A. Beuk-\nman, M. Wimmer, W. Yi, A. A. Kiselev, B.-M. Nguyen,\nM. Sokolich, M. J. Manfra, F. Nichele, C. M. Marcus,\nand L. P. Kouwenhoven, Nano Lett. 16, 7509 (2016) .\n[9] J.-T. Hung, E. Marcellina, B. Wang, A. R. Hamilton,\nand D. Culcer, Phys. Rev. B 95, 195316 (2017) .\n[10] D. V. Bulaev and D. Loss,\nPhys. Rev. Lett. 95, 076805 (2005) .\n[11] F. Nichele, M. Kjaergaard, H. J. Suominen, R. Sko-\nlasinski, M. Wimmer, B.-M. Nguyen, A. A. Kiselev,\nW. Yi, M. Sokolich, M. J. Manfra, F. Qu, A. J. A.\nBeukman, L. P. Kouwenhoven, and C. M. Marcus,\nPhys. Rev. Lett. 118, 016801 (2017) .\n[12] T. Korn, M. Kugler, M. Griesbeck, R.Schulz, A. Wagner,\nM. Hirmer, C. Gerl, D. Schuh, W. Wegscheider, and\nC. Schller, New J. Phys. 12, 043003 (2010) .\n[13] J. Salfi, J. A. Mol, D. Culcer, and S. Rogge,\nPhys. Rev. Lett. 116, 246801 (2016) .\n[14] J. Salfi, M. Tong, S. Rogge, and D. Culcer,\nNanotechnology 27, 244001 (2016) .\n[15] J. R. Petta, A. C. Johnson, J. M. Taylor, E. A. Laird,\nA. Yacoby, M. D. Lukin, C. M. Marcus, M. P. Hanson,\nand A. C. Gossard, Science309, 2180 (2005) .\n[16] R. Winkler, Spin-Orbit Coupling Effects in Two-\nDimensional Electron and Hole systems (Springer,\nBerlin, 2003).\n[17] R. Moriya, K. Sawano, Y. Hoshi, S. Masubuchi,\nY. Shiraki, A. Wild, C. Neumann, G. Abstre-\niter, D. Bougeard, T. Koga, and T. Machida,\nPhys. Rev. Lett. 113, 086601 (2014) .\n[18] T. Biswas and T. K. Ghosh,\nJ. Appl. Phys. 115, 213701 (2014) .\n[19] K. V. Shanavas, Phys. Rev. B 93, 045108 (2016) .\n[20] G. Akhgar, O. Klochan, L. H. Willems van Beveren,\nM. T. Edmonds, F. Maier, B. J. Spencer, J. C. Mc-\nCallum, L. Ley, A. R. Hamilton, and C. I. Pakes,\nNano Lett. 16, 3768 (2016) .\n[21] R. M. Lutchyn, J. D. Sau, and S. Das Sarma,\nPhys. Rev. Lett. 105, 077001 (2010) .[22] J.Alicea, Y.Oreg, G.Refael, F.vonOppen, andM.P.A.\nFisher,Nat. Phys. 7, 412 (2011) .\n[23] S. T. Gill, J. Damasco, D. Car, E. P. A. M. Bakkers, and\nN. Mason, Appl. Phys. Lett. 109, 233502 (2016) .\n[24] S. V. Alestin Mawrie and T. K. Ghosh,\narXiv:1705.02483v1 (unpublished).\n[25] M. J. Manfra, L. N. Pfeiffer, K. W. West,\nR. de Picciotto, and K. W. Baldwin,\nAppl. Phys. Lett. 86, 162106 (2005) .\n[26] B. Habib, M. Shayegan, and R. Winkler,\nSemicond. Sci. Technol. 23, 064002 (2009) .\n[27] X.-J. Hao, T. Tu, G. Cao, C. Zhou, H.-O. Li, G.-\nC. Guo, W. Y. Fung, Z. Ji, G.-P. Guo, and W. Lu,\nNano Letters 10, 2956 (2010) , pMID: 20698609.\n[28] S. Chesi, G. F. Giuliani, L. P. Rokhinson, L. N. Pfeiffer,\nand K. W. West, Phys. Rev. Lett. 106, 236601 (2011) .\n[29] D. Q. Wang, O. Klochan, J.-T. Hung, D. Culcer,\nI. Farrer, D. A. Ritchie, and A. R. Hamilton,\nNano Lett. 16, 7685 (2016) .\n[30] A. Srinivasan, D. S. Miserev, K. L. Hudson,\nO. Klochan, K. Muraki, Y. Hirayama, D. Reuter,\nA. D. Wieck, O. P. Sushkov, and A. R. Hamilton,\nPhys. Rev. Lett. 118, 146801 (2017) .\n[31] J. D. Watson, S. Mondal, G. A. Cs´ athy, M. J. Manfra,\nE. H. Hwang, S. Das Sarma, L. N. Pfeiffer, and K. W.\nWest,Phys. Rev. B 83, 241305 (2011) .\n[32] A. Srinivasan, K. L. Hudson, D. Miserev, L. A. Yeoh,\nO. Klochan, K. Muraki, Y. Hirayama, O. P. Sushkov,\nand A. R. Hamilton, Phys. Rev. B 94, 041406 (2016) .\n[33] S. J. Papadakis, E. P. De Poortere, M. Shayegan, and\nR. Winkler, Phys. Rev. Lett. 84, 5592 (2000) .\n[34] R. Winkler, H. Noh, E. Tutuc, and M. Shayegan,\nPhys. Rev. B 65, 155303 (2002) .\n[35] J. van der Heijden, J. Salfi, J. A. Mol, J. Verduijn, G. C.\nTettamanzi, A. R. Hamilton, N. Collaert, and S. Rogge,\nNano Letters 14, 1492 (2014) .\n[36] F. Nichele, A. N. Pal, R. Winkler, C. Gerl,\nW. Wegscheider, T. Ihn, and K. Ensslin,\nPhys. Rev. B 89, 081306 (2014) .\n[37] T. Ota, K. Ono, M. Stopa, T. Hatano, S. Tarucha,\nH. Z. Song, Y. Nakata, T. Miyazawa, T. Ohshima, and\nN. Yokoyama, Phys. Rev. Lett. 93, 066801 (2004) .\n[38] K. Ono, D. G. Austing, Y. Tokura, and S. Tarucha,\nScience297, 1313 (2002) .\n[39] R. Hanson, L. P. Kouwenhoven, J. R. Petta,\nS. Tarucha, and L. M. K. Vandersypen,\nRev. Mod. Phys. 79, 1217 (2007) .\n[40] A. F. Croxall, B. Zheng, F. Sfigakis, K. Das Gupta,\nI. Farrer, C. A. Nicoll, H. E. Beere, and D. A. Ritchie,\nAppl. Phys. Lett. 102, 082105 (2013) .\n[41] H. Watzinger, C. Kloeffel, L. Vukui, M. D. Rossell,\nV. Sessi, J. Kukuka, R. Kirchschlager, E. Lausecker,\nA. Truhlar, M. Glaser, A. Rastelli, A. Fuhrer, D. Loss,\nand G. Katsaros, Nano Letters 16, 6879 (2016) .\n[42] C. Gerl, S. Schmultz, H.-P. Tranitz, C. Mitzkus, and\nW. Wegscheider, Appl. Phys. Lett. 86, 252105 (2005) .\n[43] W. R. Clarke, C. E. Yasin, A. R. Hamilton, A. P. Micol-\nich, M. Y. Simmons, K. Muraki, Y. Hirayama, M. Pep-\nper, and D. A. Ritchie, Nat. Phys. 4, 55 (2007) .\n[44] I. ˇZuti´ c, J. Fabian, and S. Das Sarma,\nRev. Mod. Phys. 76, 323 (2004) .6\n[45] D. D. Awschalom and M. E. Flatte,\nNat. Phys. 3, 153159 (2007) .\n[46] A. Sasaki, S. Nonaka, Y. Kunihashi, M. Kohda,\nT. Bauernfeind, T. Dollinger, K. Richter, and J. Nitta,\nNat. Nanotechnol. 9, 703 (2014) .\n[47] R. L. Kallaher, J. J. Heremans, N. Goel, S. J. Chung,\nand M. B. Santos, Phys. Rev. B 81, 075303 (2010) .\n[48] B. Grbi´ c, R. Leturcq, T. Ihn, K. Ensslin, D. Reuter, and\nA. D. Wieck, Phys. Rev. B 77, 125312 (2008) .\n[49] H. Nakamura, T. Koga, and T. Kimura,\nPhys. Rev. Lett. 108, 206601 (2012) .\n[50] T. Koga, J. Nitta, T. Akazaki, and H. Takayanagi,\nPhys. Rev. Lett. 89, 046801 (2002) .\n[51] J. Nitta, T. Akazaki, H. Takayanagi, and T. Enoki,\nPhys. Rev. Lett. 78, 1335 (1997) .\n[52] Y. H. Park, H. jun Kim, J. Chang, S. H.\nHan, J. Eom, H.-J. Choi, and H. C. Koo,\nAppl. Phys. Lett. 103, 252407 (2013) .\n[53] T. Li, L. A. Yeoh, A. Srinivasan, O. Klochan, D. A.\nRitchie, M. Y. Simmons, O. P. Sushkov, and A. R.\nHamilton, Phys. Rev. B 93, 205424 (2016) .\n[54] P. S. Eldridge, W. J. H. Leyland, P. G. Lagoudakis, O. Z.\nKarimov, M. Henini, D. Taylor, R. T. Phillips, and R. T.\nHarley,AIP Conference Proceedings 1199, 395 (2010) .\n[55] G. Wang, B. L. Liu, A. Balocchi, P. Renucci,\nC. R. Zhu, T. Amand, C. Fontaine, and X. Marie,\nNat. Commun. 4, 2372 (2013) .\n[56] C.-X. Liu, B. Zhou, S.-Q. Shen, and B.-f. Zhu,\nPhys. Rev. B 77, 125345 (2008) .\n[57] I. V. Tokatly, E. E. Krasovskii, and G. Vignale,\nPhys. Rev. B 91, 035403 (2015) .[58] C. H. Li, O. M. J. van‘t Erve, S. Rajput, L. Li, and B. T.\nJonker,Nat. Commun. 7, 13518EP (2016) .\n[59] A. Wong and F. Mireles,\nPhys. Rev. B 81, 085304 (2010) .\n[60] E. B. Sonin, Phys. Rev. B 81, 113304 (2010) .\n[61] K. Nomura, J. Wunderlich, J. Sinova, B. Kaest-\nner, A. H. MacDonald, and T. Jungwirth,\nPhys. Rev. B 72, 245330 (2005) .\n[62] T. Jungwirth, J. Wunderlich, and K. Olejnik,\nNat. Mater. 11, 382 (2012) .\n[63] E. Marcellina, A. R. Hamilton, R. Winkler, and D. Cul-\ncer,Phys. Rev. B 95, 075305 (2017) .\n[64] R. Winkler, Phys. Rev. B 62, 4245 (2000) .\n[65] F. T. Vasko and O. E. Raichev, Quantum Kinetic Theory\nand Applications (Springer, New York, 2005).\n[66] J. P. Lu, J. B. Yau, S. P. Shukla, M. Shayegan,\nL. Wissinger, U. R¨ ossler, and R. Winkler,\nPhys. Rev. Lett. 81, 1282 (1998) .\n[67] G. Bastard, E. E. Mendez, L. L. Chang, and L. Esaki,\nPhys. Rev. B 28, 3241 (1983) .\n[68] S. J. Papadakis, E. P. De Poortere, H. C. Manoharan,\nM. Shayegan, and R. Winkler, Science 283, 2056 (1999).\n[69] B. Habib, E. Tutuc, S. Melinte, M. Shayegan,\nD. Wasserman, S. A. Lyon, and R. Winkler,\nAppl. Phys. Lett. 85, 3151 (2004) .\n[70] M. Y. Simmons, A. R. Hamilton, S. J. Stevens,\nD. A. Ritchie, M. Pepper, and A. Kurobe,\nAppl. Phys. Lett. 70, 2750 (1997) .\n[71] D. Culcer, A. Sekine, and A. H. MacDonald,\nPhys. Rev. B 96, 035106 (2017) .arXiv:1708.07247v2  [cond-mat.mes-hall]  25 Aug 2017Supplement of “Strong influence of spin-orbit coupling on ma gnetotransport in\ntwo-dimensional hole systems”\nHong Liu, E. Marcellina, A. R. Hamilton and Dimitrie Culcer1\n1School of Physics and Australian Research Council Centre of Excellence in Low-Energy Electronics Technologies,\nUNSW Node, The University of New South Wales, Sydney 2052, Au stralia\nI. LUTTINGER HAMILTONIAN\nWe start from the bulk 4 ×4 Luttinger Hamiltonian [ 1]HL(k2,kz) describing holes in the uppermost valence band\nwith an effective spin J= 3/2. So the hole system with top and back gate in z-direction can be simplified as the\nisotropic Luttinger-Kohn Hamiltonian plus a confining asymmetrical t riangular potential.\nˆH=HL(k2,kz)−eFzz, z > 0, (1)\nwhereFzis the gate electric field and Fz≥0. The 4 ×4 Luttinger Hamiltonian, which is expressed in the basis of Jz\neigenstates {|+3\n2∝an}bracketri}ht,|−3\n2∝an}bracketri}ht,|+1\n2∝an}bracketri}ht,|−1\n2∝an}bracketri}ht}, reads\nHL(k2,kz)=\nP+Q0L M\n0P+Q M∗−L∗\nL∗M P−Q0\nM∗−L0P−Q\n, (2)\nwhere\nP=µ\n2γ1(k2+k2\nz), Q=−µ\n2γ2(2k2\nz−k2),\nL=−√\n3µγ3k−kz, M=−√\n3µ\n2(γk2\n−+δk2\n+).(3)\nwithµ=/planckover2pi12\nm0,γ1,γ2,γ3are the Luttinger parameters (Table I),γ=γ2+γ3\n2,δ=γ2−γ3\n2, andk2=/radicalig\nk2x+k2y,k±=\nkx±ikyandθ= arctanky\nkx. To obtain the spectrum of our system, we use modified infinite squa re well wave functions\n[3] for the heavy hole (HH) and light hole (LH) states\nφv=sin/bracketleftbigπ\nd/parenleftbig\nz+d\n2/parenrightbig/bracketrightbig\nexp/bracketleftbig\n−βv/parenleftbigz\nd+1\n2/parenrightbig/bracketrightbig\nπ/radicalig\ne−βvdsinh(βv)\n2π2βv+2β3v, (4)\nwherev=h,ldenote the HH and LH states and dis the width of the quantum well. The eigenvalues of the heavy\nhole and light hole as well as the corresponding kdependent expansion coefficients are then obtained by diagonalizing\nthe matrix ˜H, whose elements are given as\n˜H=∝an}bracketle{tν|HL(k2,ˆkz)+V(z)|ν′∝an}bracketri}ht, (5)\nwhere|ν∝an}bracketri}htdenotes the wave function Eq. ( 4) andˆkzstands for the operator −i∂\n∂z. The two lowest eigenenergies of\nthe 4×4 matrix Eq. ( 5) correspond to the dispersion of the spin-split HH1 ±subbands. Usually, only the lowest\nHH-subspace is taken into account at low hole densities. Accordingly , we perform a Schrieffer-Wolff transformation\non Eq.5to restrict our attention to the lowest HH subspace. Therefore, the effective Hamiltonian describing the two\ndimensional hole gas is [ 4]\nH0k=/planckover2pi12k2\n2m∗+iα(k3\n−σ+−k3\n+σ+), (6)\nTable I. Luttinger parameters used in this work [ 2].\nGaAs InAs InSb\nγ1 6.85 20.40 37.10\nγ2 2.10 8.30 16.50\nγ3 2.90 9.10 17.702\nwherem∗≡m∗\nhh=m0\nγ1+γ2and the Pauli matrix σ±=1\n2(σx±iσy). The eigenvalues of Eq. ( 6) areεk,±=ǫ0±αk3,\nwhereǫ0=/planckover2pi12k2\n2m∗. The Rashba coefficient αis expressed as\nα=3µ2\n∆Eγ2∝an}bracketle{tφL|φH∝an}bracketri}ht∝an}bracketle{tφH|kz|φL∝an}bracketri}ht. (7)\nwhere ∆ Eis energy splitting of heavy hole and light hole.\nII. SCATTERING TERM\nThek-diagonal part of density matrix fkis a 2×2 Hermitian matrix, which is decomposed into fk=nk11+Sk,\nwherenkrepresents the scalar part and 11 is the identity matrix into two dimensions. The component Skis written\npurely in terms of the Pauli σmatrices Sk=1\n2Sk·σ≡1\n2Skiσi. With this notation, the scattering term is in turn\ndecomposed into\nˆJ(fk) =ni\n/planckover2pi12/integraldisplayd2k′\n(2π)2|Ukk′|2(nk−nk′) lim\nη→0/integraldisplay∞\n0dt′e−ηt′e−iH0k′t′//planckover2pi1eiH0kt′//planckover2pi1+H.c.\n+ni\n2/planckover2pi12/integraldisplayd2k′\n(2π)2|Ukk′|2(Sk−Sk′)·lim\nη→0/integraldisplay∞\n0dt′e−ηt′e−iH0k′t′//planckover2pi1σeiH0kt′//planckover2pi1+H.c..(8)\nWe use perturbation theory solving the kinetic equation up to α2. In the process, we decompose the matrix Sk=\nSk/bardbl+Sk⊥and write those two parts as Sk/bardbl= (1/2)sk/bardblσk/bardblandSk⊥= (1/2)sk⊥σk⊥. The terms sk/bardblandsk⊥are\nscalars and given by sk/bardbl=Sk·ˆΩkandsk⊥=Sk·ˆΘkwithˆΩk=−sin3θˆx+cos3θˆyandˆΘk=−cos3θˆx−sin3θˆy.\nWithγ=θ′−θ, the scattering term becomes\nˆJ(n) =πni\n2/planckover2pi1/integraldisplayd2k′\n(2π)2|Ukk′|2(nk−nk′)·(1+ˆΩk′·ˆΩk)/bracketleftig\nδ(ǫ+−ǫ′\n+)+δ(ǫ−−ǫ′\n−)/bracketrightig\n+πni\n2/planckover2pi1/integraldisplayd2k′\n(2π)2|Ukk′|2(nk−nk′)·σ·(ˆΩk′+ˆΩk)/bracketleftig\nδ(ǫ′\n+−ǫ+)−δ(ǫ′\n−−ǫ−)/bracketrightig\n+πni\n2/planckover2pi1/integraldisplayd2k′\n(2π)2|Ukk′|2(nk−nk′)·(1−ˆΩk′·ˆΩk)/bracketleftig\nδ(ǫ+−ǫ′\n−)+δ(ǫ−−ǫ′\n+)/bracketrightig\n+πni\n2/planckover2pi1/integraldisplayd2k′\n(2π)2|Ukk′|2(nk−nk′)·σ·/bracketleftig\n(ˆΩk−ˆΩk′)/bracketrightig/bracketleftig\nδ(ǫ′\n−−ǫ+)−δ(ǫ′\n+−ǫ−)/bracketrightig\n,(9)\nand\nˆJ(S) =πni\n4/planckover2pi1/integraldisplayd2k′\n(2π)2|Ukk′|2(Sk−Sk′)·/bracketleftig\nσ(1−ˆΩk·ˆΩk′)+(ˆΩk·σ)ˆΩk′+ˆΩk(ˆΩk′·σ)/bracketrightig/bracketleftig\nδ(ǫ+−ǫ′\n+)+δ(ǫ−−ǫ′\n−)/bracketrightig\n+πni\n4/planckover2pi1/integraldisplayd2k′\n(2π)2|Ukk′|2(Sk−Sk′)·(ˆΩk+ˆΩk′)/bracketleftig\nδ(ǫ′\n+−ǫ+)−δ(ǫ′\n−−ǫ−)/bracketrightig\n+πni\n4/planckover2pi1/integraldisplayd2k′\n(2π)2|Ukk′|2(Sk−Sk′)·/bracketleftig\nσ(1+ˆΩk·ˆΩk′)−(ˆΩk·σ)ˆΩk′−ˆΩk(ˆΩk′·σ)/bracketrightig/bracketleftig\nδ(ǫ+−ǫ′\n−)+δ(ǫ−−ǫ′\n+)/bracketrightig\n+πni\n4/planckover2pi1/integraldisplayd2k′\n(2π)2|Ukk′|2(Sk−Sk′)·/bracketleftig\n(ˆΩk−ˆΩk′)/bracketrightig/bracketleftig\nδ(ǫ+−ǫ′\n−)−δ(ǫ−−ǫ′\n+)/bracketrightig\n.\n(10)\nWe now separate these terms according to the contributions from intra-band and inter-band scatterings\nˆJ(n) =πni\n2/planckover2pi1/integraldisplayd2k′\n(2π)2|Ukk′|2(nk−nk′)(1+cos3 γ)/bracketleftig\nδ(ǫ+−ǫ′\n+)+δ(ǫ−−ǫ′\n−)/bracketrightig\n+πni\n2/planckover2pi1/integraldisplayd2k′\n(2π)2|Ukk′|2(nk−nk′)(1−cos3γ)/bracketleftig\nδ(ǫ+−ǫ′\n−)+δ(ǫ−−ǫ′\n+)/bracketrightig\n,(11)3\nˆJ(S) =πni\n4/planckover2pi1/integraldisplayd2k′\n(2π)2|Ukk′|2/bracketleftig\n(sk/bardbl−sk′/bardbl)(1+cos3 γ)σk/bardbl+(sk/bardbl−sk′/bardbl)sin3γσk⊥\n+(sk⊥+sk′⊥)σk/bardblsin3γ+(sk⊥+sk′⊥)(1−cos3γ)σk⊥/bracketrightig/bracketleftig\nδ(ǫ+−ǫ′\n+)+δ(ǫ−−ǫ′\n−)/bracketrightig\n+πni\n4/planckover2pi1/integraldisplayd2k′\n(2π)2|Ukk′|2/bracketleftig\n(sk/bardbl+sk′/bardbl)(1−cos3γ)σk/bardbl−(sk/bardbl+sk′/bardbl)sin3γσk⊥\n−(sk⊥−sk′⊥)σk/bardblsin3γ+(sk⊥−sk′⊥)(1−cos3γ)σk⊥/bracketrightig/bracketleftig\nδ(ǫ+−ǫ′\n−)+δ(ǫ−−ǫ′\n+)/bracketrightig\n,(12)\nand\nˆJS→n(S) =πni\n4/planckover2pi1/integraldisplayd2k′\n(2π)2|Ukk′|2/bracketleftig\n(sk/bardbl−sk′/bardbl)(1+cos3 γ)+(sk⊥+sk′⊥)sin3γ/bracketrightig/bracketleftig\nδ(ǫ′\n+−ǫ+)−δ(ǫ′\n−−ǫ−)/bracketrightig\n=πni\n4/planckover2pi1/integraldisplayd2k′\n(2π)2|Ukk′|2/bracketleftig\n(sk/bardbl+sk′/bardbl)(1−cos3γ)−(sk⊥−sk′⊥)sin3γ/bracketrightig/bracketleftig\nδ(ǫ+−ǫ′\n−)−δ(ǫ−−ǫ′\n+)/bracketrightig\n,(13)\nˆJn→S(n) =πni\n2/planckover2pi1/integraldisplayd2k′\n(2π)2|Ukk′|2(nk−nk′)/bracketleftig\nσk/bardbl(1+cos3 γ)+σk⊥sin3γ/bracketrightig/bracketleftig\nδ(ǫ′\n+−ǫ+)−δ(ǫ′\n−−ǫ−)/bracketrightig\n+πni\n2/planckover2pi1/integraldisplayd2k′\n(2π)2|Ukk′|2(nk−nk′)/bracketleftig\nσk/bardbl(1−cos3γ)−σk⊥sin3γ/bracketrightig/bracketleftig\nδ(ǫ′\n−−ǫ+)−δ(ǫ′\n+−ǫ−)/bracketrightig\n.(14)\nWe next decompose the kinetic equation as follows:\ndnk\ndt+ˆJn→n(nk) =Dkn,\ndSk/bardbl\ndt+P/bardblˆJS→S(Sk/bardbl) =Dk/bardbl,\ndSk⊥\ndt+i\n/planckover2pi1/bracketleftbig\nH0k,Sk⊥/bracketrightbig\n=Dk⊥.(15)\nNote that the projection operator P/bardblabove acts on a matrix Mas Tr(Mσk/bardbl), where Tr refers to the matrix (spin)\ntrace.\nIII. SOLUTION FOR THE LONGITUDINAL CONDUCTIVITY\nHere we derive the longitudinal conductivity at zero magnetic field. E xpanding the δfunctions in Sec. IIup to\n∝α2, we get the following\nδ(ǫ+−ǫ′\n+)≈δ(ǫ0−ǫ′\n0)+α(k3−k′3)∂\n∂ǫ0δ(ǫ0−ǫ′\n0)+α2(k3−k′3)2\n2∂2δ(ǫ0−ǫ′\n0)\n∂ǫ2\n0\nδ(ǫ−−ǫ′\n−)≈δ(ǫ0−ǫ′\n0)−α(k3−k′3)∂\n∂ǫ0δ(ǫ0−ǫ′\n0)+α2(k3−k′3)2\n2∂2δ(ǫ0−ǫ′\n0)\n∂ǫ2\n0\nδ(ǫ+−ǫ′\n−)≈δ(ǫ0−ǫ′\n0)+α(k3+k′3)∂\n∂ǫ0δ(ǫ0−ǫ′\n0)+α2(k3+k′3)2\n2∂2δ(ǫ0−ǫ′\n0)\n∂ǫ2\n0\nδ(ǫ−−ǫ′\n+)≈δ(ǫ0−ǫ′\n0)−α(k3+k′3)∂\n∂ǫ0δ(ǫ0−ǫ′\n0)+α2(k3+k′3)2\n2∂2δ(ǫ0−ǫ′\n0)\n∂ǫ2\n0.(16)\nWe now insert Eq. ( 16) into the electric driving term DE,kand scattering term ˆJ(fk). With ρ0k=f0++f0−\n211 +\nf0+−f0−\n2σk/bardblandf0+,f0−equilibrium Fermi distribution function, the driving term DE,kbecomes,\nDE,kn=−eE·ˆk\n2/planckover2pi1(∂f0+\n∂k+∂f0−\n∂k)≈eE·ˆk\n2/planckover2pi1/bracketleftig\n2/planckover2pi12k\nm∗δ(ǫ0−ǫF)+6α2k5∂δ(ǫ0−ǫF)\n∂ǫ0/bracketrightig\n,\nDE,k/bardbl=−eE·ˆk\n2/planckover2pi1(∂f0+\n∂k−∂f0−\n∂k)σk/bardbl≈eE·ˆk\n2/planckover2pi1/bracketleftig\n6αk2δ(ǫ0−ǫF)+2/planckover2pi12k\nm∗αk3∂δ(ǫ0−ǫF)\n∂ǫ0)/bracketrightig\n.(17)4\nSolving Eqs. ( 15), we obtain the density matrices\nn(0)\nEk=τpeE·ˆk\n/planckover2pi1/bracketleftig/planckover2pi12k\nm∗δ(ǫ0−ǫF)/bracketrightig\n, (18a)\nS(1)\nEk/bardbl=τsαeE·ˆk\n/planckover2pi1/bracketleftig/planckover2pi12k4\nm∗∂δ(ǫ0−ǫF)\n∂ǫ0+3k2δ(ǫ0−ǫF)/bracketrightig\nσk/bardbl=s(1)\nEk/bardblσk/bardbl, (18b)\nn(2)\nEk=τpα2/braceleftigeE·ˆk\n/planckover2pi1/bracketleftig\n3k5∂δ(ǫ0−ǫF)\n∂ǫ0/bracketrightig\n−3km∗2ni\n4απ/planckover2pi15s(1)\nEk/bardblζ(γ)−n(0)\nEk6nim∗3\nπ/planckover2pi17k2ξ(γ)/bracerightig\n. (18c)\nwhereǫF=/planckover2pi12k2\nF\n2m∗,τp=2π/planckover2pi13\nm∗niξ(γ),τs=4π/planckover2pi13\nm∗niβ(γ), and\nζ(γ)=/integraldisplay\ndγ|Ukk′|2(cosγ−cos3γ), ξ(γ)=/integraldisplay\ndγ|Ukk′|2(1−cosγ), β(γ)=/integraldisplay\ndγ|Ukk′|2(1−cosγcos3γ).(19)\nIn the low temperature limit, the Thomas-Fermi wave-vector of a t wo-dimensional hole gas without spin-orbit\ncoupling is kTF=2\naB, withaB=/planckover2pi12ǫr\nm∗e2. The screened Coulomb potential between plane waves is given by\n|Ukk′|2=Z2e4\n4ǫ2\n0ǫ2r/parenleftbigg1\n|k−k′|+kTF/parenrightbigg2\n. (20)\nWith Eq. ( 20), we obtainζ(γ)\nξ(γ)≈2 andξ(γ)\nβ(γ)=1\n3. Using the velocity operator\nˆvx=/planckover2pi1kx\nm∗+α\n/planckover2pi13k2[−sin2θσx+cos2θσy],ˆvy=/planckover2pi1ky\nm∗+α\n/planckover2pi13k2[−sin2θσy−cos2θσx], (21)\nthe longitudinal current is jx=eTr/bracketleftbig\nˆvxρEk/bracketrightbig\n, where ρEk= (n(0)\nEk+n(2)\nEk)11 +S(1)\nEk/bardbl. Therefore, the longitudinal\nconductivity with Rashba spin orbit coupling up to second order in αis\nσxx=e2τp\n2πm∗k2\nF/bracketleftig\n1−15\n2/parenleftbiggαk3\nF\nǫkin/parenrightbigg2/bracketrightig\n, (22)\nwhereǫkin=/planckover2pi12k2\nF\n2m∗.\nIV. SOLUTION FOR THE HALL COEFFICIENT\nNow we consider the case of Bz>0. Firstly, we find that the Zeeman terms have no contribution to th e Hall\ncoefficient. With Eqs. ( 21), the Lorentz driving term DL,kbecomes\nDL,k=1\n2e\n/planckover2pi1/braceleftig\nˆv×B,∂ρEk\n∂k/bracerightig\n=1\n2eBz\n/planckover2pi1/braceleftig/braceleftbig\nˆvy,∂ρEk\n∂kx/bracerightbig\n−/braceleftbig\nˆvx,∂ρEk\n∂ky/bracerightbig/bracerightig\n. (23)\nWe separate DL,kinto the scalar and spin parts with DL,k=DL,n+DL,S, and, switching from the rectangular\ncoordinates to polar coordinates with∂D\n∂kx=∂D\n∂kcosθ−∂D\n∂θsinθ\nk;∂D\n∂ky=∂D\n∂ksinθ+∂D\n∂θcosθ\nk, we obtain\nDL,n=−eBz\nm∗/bracketleftbig\nn(0)\nk+n(2)\nk/bracketrightbig\n(−sinθ)+eBz\n/planckover2pi13αk\n/planckover2pi1s(1)\nk,/bardbl(−sinθ),\nDL,S/bardbl=−/braceleftigeBz\nm∗s(1)\nk,/bardbl(−sinθ)+eBz\n/planckover2pi13αk\n/planckover2pi1/bracketleftbig\nn(0)\nk+n(2)\nk/bracketrightbig\n(−sinθ)/bracerightig\nσk/bardbl,\nDL,S⊥= cosθeBz\n/planckover2pi13αk2\n/planckover2pi1∂/bracketleftbig\nn(0)\nk+n(2)\nk/bracketrightbig\n∂kσk⊥,(24)5\nwithn(0)\nEk=n(0)\nkcosθ,n(2)\nEk=n(2)\nkcosθands(1)\nEk,/bardbl=s(1)\nk,/bardblcosθ. Solving Eqs. ( 15), we obtain the following density\nmatrices in presence both electric and magnetic fields\nnBz,k=−sinθτpeBz/braceleftign(0)\nk+n(2)\nk\nm∗+3αk\n/planckover2pi12s(1)\nk,/bardbl/bracerightig\n,\nSBz,k/bardbl=−sinθτseBz/braceleftigs(1)\nk,/bardbl\nm∗+3αk\n/planckover2pi12/bracketleftbig\nn(0)\nk+n(2)\nk/bracketrightbig/bracerightig\nσk/bardbl,\nSBz,k⊥= cosθ3eBz\n2/planckover2pi1k∂/bracketleftbig\nn(0)\nk+n(2)\nk/bracketrightbig\n∂kσz.(25)\nThe Hall current is jy=eTr/bracketleftbig\nˆvyρEB\nk/bracketrightbig\n, whereρEBz\nk=nBz,k11 +SBz,k/bardbl+SBz,k⊥. The Hall coefficient, up to the\nsecond order in α, is thus given as\nRH=σxy\nBz(σ2xx+σ2xy)≈1\npe/bracketleftig\n1+8/parenleftbiggαk3\nF\nǫkin/parenrightbigg2/bracketrightig\n, (26)\nwhereωc=eBz\nm∗.\n[1] J. M. Luttinger, Phys. Rev. 102, 1030 (1956) .\n[2] R. Winkler, Spin-Orbit Coupling Effects in Two-Dimensional Electron and Hole systems (Springer, Berlin, 2003).\n[3] G. Bastard, E. E. Mendez, L. L. Chang, and L. Esaki, Phys. Rev. B 28, 3241 (1983) .\n[4] R. Winkler, Phys. Rev. B 62, 4245 (2000) ."    },    {        "title": "2304.12632v1.Magnetization_Switching_in_van_der_Waals_Systems_by_Spin_Orbit_Torque.pdf",        "content": "1 \n Magnetization Switching in van der Waals Systems by Spin-Orbit Torque  \n \nXin Lin1,2, Lijun Zhu1,2* \n1. State Key Laboratory of Superlattices and Microstructures, Institute of Semiconductors, Chinese Academy of \nSciences, Beijing 100083, China  \n2. College of Materials Science and Opto -Electronic Technology, University of Chinese Academy of Sciences, Beijing \n100049, China  \n*ljzhu@semi.ac.cn  \n \nAbstract : Electrical switching of magnetization via spin -orbit torque  (SOT) is of great potential in fast, dense, energy -\nefficient nonvolatile magnetic memory and logic technologies. Recently, enormous efforts have been stimulated to \ninvestigate switching of perpendicular magnetization in van der Waals systems that have unique, strong tunability and \nspin-orbit coupling effect  compared to conventional metals. In this review , we first give a brief, generalized introduction \nto the spin -orbit torque  and van der Waals materials. We will then discuss  the recent advances in magnetization switching \nby the spin current generated from van der Waals materials  and summary the progress in  the switching of Van der Waals \nmagnetization  by the spin current . \n \n1. Introduction  \n1.1 Spin -orbit torque  \n \nSpin-orbit torque s (SOT s) are a powerful tool to \nmanipulat e magnetization at the nanoscale for spintronic  \ndevices, such as magnetic random access  memory (MRAM) \nand logic  [1-5]. SOTs  are exerted on a magnetization when \nangular momentum is transferred from spin accumulation \nor spin currents carried by a flow of electrons or magnons  \n(Fig. 1 ). A spin current with spin polarization vector σ, can \nexert two types of SOTs on a magnetization M, i.e., a \ndamping -like (DL) torque [ τDL ~ M × (M × σ)] due to the \nabsorption of the spin current component transverse to M \nand a field-like (FL) torque [ τFL ~ M × σ] due to the \nreflection of the spin current with some spin rotation . In the \nsimple case of the spin-current generator /magnet bilayer, \nthe efficienc y of the damping -like SOT per unit bias \ncurrent density,  𝜉DL𝑗\n, can be estimated as [6] \n𝜉DL𝑗 ≈ Tint θSH τM-1/(τM-1+τso-1)            (1) \nwhere τM-1/(τM-1+τso-1) is the percentage of the spin \ncurrent relaxed via the spin-magnetization exchange \ninteraction  (with spin relaxation rate τM-1) within the \nmagnetic layer  and is less than 1 in presence of non -\nnegligible spin relaxation via the spin-orbit scattering  \n(with spin relaxation rate τM-1) [6], Tint is the  interfacial  \nspin transparency which determines what fraction of the \nspin current enters the magnet  (less than 1 in presence \nof spin backflow [7-11] and spin memory loss [12-15]), \nand θSH is the charge -to-spin conversion efficiency of \nthe spin current generator (e.g., the spin Hall ratio  in the \ncase of spin Hall materials  [2-4]). The quantitative \nunderstanding of the efficiency of the field -like torque , \n𝜉FL𝑗, remains an open question.  \nThe same SOT physics can be expressed in terms of \neffective SOT fields: a damping -like effective SOT field \n(HDL) parallel to M × σ and a field -like effective SOT field \n(HFL) parallel to σ. The magnitudes of the damping -like \nand field -like SOT  fields correlate  to their SOT efficiencies \nper unit bias current density via  \n  HDL = (ℏ/2e) j𝜉DL𝑗Ms-1t -1      （2） \n  HFL = (ℏ/2e) j𝜉FL𝑗Ms-1t -1      （3） where e is the elementary charge, ℏ reduced Plank’s \nconstant, t the magnetic layer thickness, Ms the saturation \nmagnetization of the magnetic layer, and j the charge  \ncurrent density in the spin current generating layer.  \nThe damping -like SOT is technologically more \nimport ant because it can excite dynamics and switching of \nmagnetization (even for low currents for which HDL is \nmuch less than the anisotropy field of a perpendicular \nmagnetization). Field -like SOTs by themselves can \ndestabilize magnets only if HFL is greater th an the \nanisotropy field, but they can still strongly affect the \ndynamics in combination with a damping -like SOT  [16-18].  \nThe g eneration of spin currents is central to the SOT \nphenomena. The spin polarization vector σ can have \nlongitudinal, transverse, and perpendicular components, \ni.e., σx, σy, and σz. Transversely polarized spin current can \nbe generated by a longitudinal charge current flow in either \nmagnetic or non -magnetic materials via a variety of \npossible spin -orbit -coupling (SOC) effects. The latter \nincludes the bulk spin Hall effect (SHE)  [19-22], \ntopological surface states [23,24], interfacial SOC effects \n[16,25-28], orbit -spin conversion [29], the anomalous Hall \neffect [30,31], the planar Hall effect [32,33], the magnetic \nSHE [34-36], Dresselhaus effect [37], Dirac nodal lines \n[38,39], etc. The bulk S HE has been widely observed in \nthin-film heavy metal (HM) [40-42], Bi-Sb [43], Bi xTe1-x \n[44], CoPt [45], FePt [46], Fe xTb1-x [47], and Co -Ni-B [48]. \n \n \nFig. 1 Schematic of damping -like and field -like spin-orbit \ntorque s exerted on a magnetic layer by an incident spin \ncurrent . \n2 \n Since the transverse spins, in principle, cannot switch \na uniform perpendicular magnetization without the \nassistance of an in -plane longitudinal magnetic field either \nvia coherent rotation of macros pin or via domain wall \ndepinning  [2-4,49], generation of perpendicular  and \nlongitudinal  spins  [50-53] are of great interests.  While \nperpendi cular and longitudinal spins are not allowed in \nnonmagnetic materials that are cubic crystal or \npolycrystalline/amorphous, additional crystal or magnetic \nsymmetry breaking can be introduced to make \nperpendicular and longitudinal spins permissive.  \n Generat ion of perpendicular spins has been  argued \nfrom  low-symmetry crystals (e.g., WTe 2 [50], MoTe 2 \n[54,55], and CuPt [56]), non -collinear antiferromagnetic \ncrystals  with magnetic asymmetry (e.g. , IrMn 3 [57], \nMn 3GaN [58,59], Mn 3Sn [34]), some collinear \nantiferromagnets with spin conversions (e.g. , Mn 2Au \n[60,61] and RuO 2 [62-64]), and also some magnetic \ninterfaces  [65]. Longitudinal spins  might be generated by \nlow-symmetry crystals ( e.g., MoTe 2 [54], (Ga, Mn)As [66], \nNiMnSb [67], and Fe/GaAs [68]) or by a non -zero \nperpendicular magnetization  [69,70]. \n \n1.2 van der Waals materials  \nSo far, the most widely used spin -source materials are \nheavy metals with strong spin Hall effect  (e.g., Pt with a \ngiant spin Hall conductivity of 1.6×106 (ℏ/2e) Ω-1 m-1 in the \nclean limit)  [71], while the 3 d ferromagnets (e.g., Co, Fe, \nNi81Fe19, CoFeB ) and ferrimagnets (e.g., FeTb, CoTb , and \nGdFeCo)  are well studied a s the spin -detect ors. However, \nit is believed that the energy efficiency of SOT devices may \nbe improved  by developing new materials and new \nmechanisms that generate spin currents.  \nVan der Waals materials have attracted enormous \nattention in the field of material science and condensed \nmatter physics since the  discovery of single monolayer \ngraphene  in ref. [72] and is increasingly investigated in the \nfield of spintronics . This is particularly due to the diversity \nof materials, the flexibility of preparation (e.g., by \nmechanical exfoliation), and strong tunability by interface \neffects. In the field of spintronics, the van der Waals \nmaterials are also interesting for intriguing SOC effects. \nFor example, the non-magnetic  van der Waals  materials  of \ntransition metal dichalcogenides (TMDs)  and topological \ninsulators (TIs) [73,74] exhibit a strong ability in \ngenerating transversely polarized spin current via the spin \nHall effect and/or topological surface states  [75-77], in \nsome cases also in generating currents of perpendicular and \nlongitudinal spins due to low  crystal symmetry  [50,51]. On \nother hand, van der Waals materials that are magnetic are \nalso interesting for spin-orbitronics due to their highly \ntunable magneti sm and low magnetization at small \nthicknesses  [78-83]. As indicated by Eq. (2)  and Eq. (3), \nthe damping -like and field -like effective SOT fields  \nexerted on the magnetic layer by a given spin current scale \ninversely with the thickness and magnetization of the \nmagnetic layer . \nThis review is intended to focus on highlighting t he \nrecent advances of magnetization switching in van der \nWaals systems  by spin -orbit torque , including switching of \nconventional magnetic metals by spin current from non -\nmagnetic van der Waals  TMDs and TIs and switching of  \nvan der Waals magnets by incident spin currents . 2. Magnetization s witching by spin current from \nTransition Metal Dichalcogenide  semimetals  \nTMDs typically consist of transition metals (e.g., Mo, \nW, Pt, Ta, Zr) and chalcogenide elements (e.g., S, Te, Se). \nFrom point of view of SOT applications, it is  most  \ninteresting to develop the TMD semi -metals with relatively \nhigh conductivity, strong spin -orbit coupling, high θSH, and \nreduced structural symmetry. However, TMD \nsemiconductors (e.g., MoS 2 [84,85], WS 2 [86,87], WSe 2 \n[85]) are merely studied in spin -orbitronics due to their \nvery poor conductivity that is detrimental to the energy \nTMD metal s (e.g., TaS 2 [88], NbSe 2 [89]) are highly \nconductive but typically not efficien t in generating spin \ncurrents. Therefore, b elow we mainly discuss the progress \nin spin-orbit torque stud ies of TMD semimetals that are \ninteresting for SOT applications (e.g., WTe 2 [50,90,91], \nPtTe 2 [92], MoTe 2 [51,93], and ZrTe 2 [94]).  \n \n2.1 Exfoliated Transition Metal Dichalcogenide  \nsemimetals   \nThe pioneering s tudies of TMD  semimetal s in the \nfield of spintronics [50] mainly focused on the \ncharacterization of damping -like and field -like SOTs of \ntransvers e, perpendicular, and longitudinal  spins generated \nin bilayers of mechanically exfoliated  TMD  semimetal s \nand 3d ferromagnets (e.g., Ni81Fe19). These studies have \nopened a new subject field that unitizes van der Waals \nmaterials for the possible generation of transverse, \nperpendicular, and longitudinal spin polarizations.  \nWTe 2 is a semimetal with a low inversion symmetry \nalong  the a axis of the crystal  (the space group Pmn21) [Fig. \n2(a)] and the Weyl points at the crossing of the oblique \nconduction and the valence bands only at low temperatures \n(typically below 100 K  [95]). In a WTe 2/ferromagnet \nbilayer, the screw -axis and glide -plane symmetries of this \nspace group are broken at the interface, so that \nWTe 2/ferromagnet bilayers have only one symmetry, a \nmirror symmetry relative to the bc plane (depicted in Fig. \n2(a)). There is no mirror symmetry in the ac plane, and \ntherefore no 180◦ rotational symmetry about the c axis \n(perpendicular to the sample plane). MacNeill et al. [50] \nfirst observe d in mechanically exfoliated WTe 2/Ni81Fe19 \nbilayers  damping -like (≈ 8×103 (ℏ/2e) Ω-1 m-1) and field -\nlike spin -orbit torques of transverse spins  at room \ntemperature  as well as t he exotic damping -like SOT of \nperpendicular spins  (≈ 3.6×103 (ℏ/2e) Ω-1 m-1). The  \ndamping -like SOT of perpendicular spins manifests as  an \nadditional sin2 𝜑 term in the antisymmetric signal of spin -\ntorque ferromagnetic resonance ( ST-FMR )[Fig. 2(b)] and \nwas attributed to the symmetry breaking at the interface of \nthe WTe 2 crystal.  The damping -like SOT of perpendicular \nspins is found to maximize when current is applied along  \nthe low-crystal -symmetry a axis and vanishes when  current \nis applied along the high-crystal -symmetry b axis [50]. \nThis is in  contrast  to the torques of the transverse spins that \nare independent of the c rystal orientation. The damping -\nlike SOT of perpendicular spins in WTe 2/Ni 81Fe19 was also \nfound to vary little with the WTe 2 thickness, which was \nsuggest ed as an indication that the spin current is mainly \ngenerated near the interface of the WTe 2 [90,91,96]. Xie et \nal. [52] reported  that in -plane direct current along the a axis \nof WTe 2 can induce partial switching of magnetization in \nabsence of an external magnetic field [Fig. 2(c)] and shift 3 \n of the anomalous Hall resistance loop in SrRuO 3/exfoliated \nWTe 2 bilayers , which was speculated as an indication of \ndamping -like torque  of perpendicular spins on the \nperpendicular magnetization  (macrospin ). However, the re \ncan be  longitudinal and perpendicular  Oersted field s due to \ncurrent spreading in  the WTe 2 layer  [90,91,96], which can \nalso induce “field -free” switching of perpendicular \nmagnetization and anomalous Hall loop shifts via adding to or subtracting from the domain wall depinning  field \n(coercivity) . The SOTs of the WTe 2/Ni 81Fe19 have also \nbeen reported to switch the  Ni81Fe19 layer with weak in -\nplane magnetic anisotropy at a current density of ≈ 3×105 \nA cm-2 [91]. An in-plane current along the a axis of WTe 2 \nhas also been reported to enable partial switching of \nperpendicular magnetization in WTe 2/Fe2.78GeTe 2 without \nan external magnetic field  [53]. \n \n \nFIG. 2. (a) Crystal structure near the surface of WTe 2, displaying a  mirror symmetry relative to the bc plane but not to the \nac plane.  (b) Symmetric (VS) and antisymmetric (VA) components of ST -FMR signal for WTe 2 (5.5)/Py  (6) device as a \nfunction of the angle of the in-plane magnetic  field [50]. Reprinted with permission from Mac Neill et al., Nat. Phys. 13, \n300 (2017).  (c) Current -induced magnetization switching of WTe 2(15)/SrRuO3 when the current is along the low -\nsymmetry a axis where the magnetization can be switched without an external magnetic field [52]. Reprinted with \npermission from  Xie et al., APL Mater. 9, 051114 (2021). (d) Structure of the MoTe 2 crystal in the monoclinic ( β or 1T′) \nphase depicted in the a-c plane for which the mirror plane is within the page and the Mo chains run into the page. (e) \nSymmetric and antisymmetric ST -FMR resonance components for the MoTe 2(0.7)/Py(6) device with a current applied \nperpendicular to the MoTe 2 mirror plane as a function of the orientation of the in-plane magnetic  field. (f) The \nconductivities of damping -like torque  of perpendicular spins  (blue)  and transverse spins  (red) as a function of the MoTe 2 \nthickness for devices with current aligned perpendicular to the MoTe 2 mirror plane  [51]. Reprinted with permission from  \nStiehl  et al ., Phys. Rev. B 100, 184402 (2019). (g) Crystal structure of the monoclinic 1T′ phase of MoTe 2. (h) \nAntisymmetric ST -FMR components for MoTe 2 (83.1)/Py(6) as a function of the orientation of the in-plane magnetic  field. \n(i) MOKE images implying switching  of Py by current [93]. Reprinted with permission from  Liang  et al., Adv. Mater. 32, \n2002799 (2020).  \n4 \n  \nFIG. 3. (a) Current -induced magnetization switching in sputter -deposited WTe 2(10)/CoTb(6)/Ta(2) Hall bar (Hx= ± 900 \nOe) [97]. Reprinted with permission from  Peng  et al., ACS Appl. Mater. Interfaces 13, 15950 (2021).  (b) Second harmonic \nlongitudinal resistance  (𝑅2𝜔𝑥𝑥) of WTe x(5)/Mo(2)/CoFeB(1)  measured as a function of pulse current amplitude Ipulse under \nzero external field  [98]. (c) Current -induced switching loops of WTe x(5)/Ti(2)/CoFeB (1.5) Hall bar  under  different in-\nplane magnetic fields at 200 K [99]. Reprinted with  permission from  Xie et al., Appl. Phys. Lett. 118, 042401 (2021). (d) \nDependences on the WTe x thickness of damping -like SOT efficiency  (𝜉DL𝑗) and the WTe x resistivity (𝜌𝑥𝑥) for WTe 2/CoFeB . \n(e) Apparent  spin Hall conductivity  as a function of the longitudinal conductivity  for WTe 2/CoFeB  [98]. Data in (b), (d), \nand (e) are r eprinted with permission from Li et al., Matter 4, 1639 (2021).  \n \nFIG. 4. (a) Schematic of the CVD growth process  for PtTe 2. (b) High -resolution transmission electron microscopy image \nof a 5 nm  PtTe 2 thin film. (c) Current -induced magnetization switching  in the PtTe 2(5)/Au(2.5) /CoTb (6) Hall bar under \ndifferent in -plane field s [92]. Reprinted with permission from  Xu et al., Adv. Mater. 32, 2000513 (2020).  (d) Spin torque \nferromagnetic resonance  spectrum of a ZrTe 2/Py bilayer at room temperature. (e) Current -induced magnetization \nswitching in ZrTe 2(8 u.c.)/CrTe 2(3 u.c.) Hall bar  under a 700 Oe in -plane field at 50 K [94]. Reprinted with permission \nfrom Ou et al., Nat. Commun. 13, 2972 (2022).  \n \nβ-MoTe 2 is a semimetal  that retains inversion \nsymmetry in bulk but has a low-symmetry  interface  (the \ngroup space is Pmn21 in bulk but Pm11 in few -layer \nstructures  [Fig. 2(d)]). Stiehl et al. [51] observed damping -\nlike SOT of both transverse spins (≈  8×103 (ℏ/2e) Ω-1 m-1) \nand perpendicular spins (≈  1×103 (ℏ/2e) Ω-1 m-1) in \nmechanically exfoliated  β-MoTe 2/Ni81Fe19 bilayers [Fig. 2(e)]. This torque of perpendicular spins is one -third strong \nthan that of WTe 2/Ni 81Fe19 and was attributed to \nperpendicularly polarized spin current from the surface of \nthe low -symmetry β-MoTe 2 [Fig. 2(f)]. This appears to \nsuggest that the breaking of bulk inversion symmetry is not \nan essential  requirement for producing perpendicular spins. \nHowever, 1T′ -MoTe 2 [Fig. 2(g)] was reported to generate \n5 \n no damping -like SOT of perpendicular spins in contact \nwith Ni 81Fe19 [Fig. 2(h)][93]. Instead, 1T′ -MoTe 2 only \ngenerates a nonzero damping -like SOT of transverse spins \nthat switches the in -plane magnetized Ni 81Fe19 layer at a \ncurrent density of 6.7×105 A cm-2 [Fig.  2(i)]. NbSe 2 with \nresistivity anisotropy was reported to generate a \nperpendicular Oersted field  but no perpendicular  or \nlongitudinal  spins when interfaced with Ni 81Fe19 [89]. The \ndamping -like toque of transverse spins in mechanically \nexfoliated NbSe 2/Ni81Fe19 is very weak and corresponds to \na spin Hall conductivity of ≈  103 (ℏ/2e) Ω-1 m-1 [89]. \nHere it is important to note that, while the presence \nof perpendicular spins has been widely concluded in the \nliterature from a sin2 φ-dependent contribution in \nsymmetric  spin-torque ferromagnetic  resonance signal of \nin-plane magnetization ( φ is the angle of the external \nmagnetic field relative  to the current), or a φ-independent \nbut field -dependent contribution in the second harmonic \nHall voltage of in -plane magnetization, or field -free \nswitching,  none of the three characteristics can simply \n“signify” the presence of a flow of perpendicular spins. \nThis is because non -uniform current effects that can \ngenerally exist and generates out -of-plane Oersted field in \nnominally uniform, symmetric Hall bars a nd ST -FMR \nstrips [90,91,96,100] also exhibit all three characteristics. \nAs demonstrated by Liu and Zhu [100], these \ncharacteristics can be considerable especially when the \ndevices have strong current spreading, e.g., in presence of \nnon-symmetric electric contact s. \n \n2.2 Large -area Transition Metal Dichalcogenides   \nSo far, most TMD studies have been based on \nmechanical exfoliation, which is unsuitable for the mass \nproduction of spintronic applications. Recently, efforts \nhave been made in large -area growth of thin-film TMDs  \ntowards the goal of SOT applications  [97,98]. For example, \nsputter -deposited WTe x has also developed to drive low -\ncurrent -density switching of CoTb ( jc ≈ 7.05×105 A cm-2 \nunder in -plane assisting field of 900 Oe) [Fig. 3(a)] [97] \nand in WTe x/Mo/CoFeB ( jc ≈ 7×106 A cm-2, under  no in-\nplane assisting field, Fig. 3(b))[99] and in WTe x/Ti/CoFeB \n(jc ≈ 2.0×106 A cm-2 under in -plane assisting field of ±  30 \nOe, Fig. 3(c))[98]. It has become a consensus that the spin -\norbit torque in these sputter -deposited WTe x/FM samples \narises from the bulk spin Hall effect of the WTe x [97,98]. \nAs indicated in F igs. 3(d) and 3(e), the measured spin -orbit \ntorque efficiency increases but the apparent spin Hall \nconductivity decreases as the resistivity increases in the \ndirty limit [41] due to increasing layer thickness.  \nLarge -area PtTe 2 films with relatively high electrical \nconducti vity (≈  3.3×106 Ω-1 m−1 at room temperature) and \nspin Hall conductivity (2×105 ℏ/2e Ω −1m−1) have also been \nreported by annealing a Pt thin film in tellurium vapor at ≈ \n460 °C  [Figs. 4(a) and 4(b)][92]. PtTe 2 is predicted to be \na type -II Dirac semimetal with spin -polarized surface \nstates. However, there is no indication of the generation of \ntorques of out -of-plane spins. Partial switching of \nmagnetization by in -plane current has also been reported in \na PtTe 2(10)/Au(2.5)/CoTb(6) Hall bar ( jc ≈ 9.9×106 A cm−2 \nunder in -plane assisting field of 2 kOe) [Fig. 4(c)].  \nGrowth of ZrTe 2 by molecular beam epitaxy (MBE) \nhas also been reported. A ST -FMR study has measured a \nsmall damping -like torque of transverse spins for MBE -grown ZrTe 2/Ni 81Fe19 bilayer at room temperature [Fig. \n4(d)][94]. This is consistent with the theoretical prediction \nthat ZrTe 2 is a Dirac semimetal with massless Dirac \nfermions in its band dispersion [101] but vanishing spin \nHall conductivity. Even so, a ZrTe 2(8 u.c.)/CrTe 2(3 u.c.) \nbilayer has been reported to be partially switched at 50 K \nby an in -plane current of 1.8×107 A cm-2 in density under \nan in-plane assisting field of 700 Oe [Fig. 4(e)]. \n \n3. Magnetization s witching by spin current from Bi-\nbased topological insulators   \nAnother kind of layered strong -SOC material is Bi -\nbased topological insulators [102-104]. As displayed in Fig. \n5 (a) , TIs are insulating in the bulk but conducting at the \nsurface. The initial interest of TIs for spin-orbit torque \nstudies is the topological surface states ( Fig. 5 (b) ). In the \nwavevector space, the spin and momentum of electrons are \none-to-one locked to each other at the Fermi level. With a \nflow of charge current, the shift in the electron distribut ion \nin the wavevector space induces non -equilibrium spin \naccumulation  (Fig. 5 ( c)). \n \nFig. 5. Topological surface states and spin -accumulation in \ntopological insulators. (a) Real -space picture of the \nconducting surface states in an ideal topological insulator  \n[103]. Reprinted with permission from Han an d Liu, APL \nMater. 9, 060901 (2021). (b) Angle -resolved \nphotoemission spectrum that indicates the bulk and surface \nbands of a six -quintuple -layer -thick Bi 2Se3 film [102]. \nReprinted with permission from Zhang et al., Nat. Phys. 6, \n584 (2010). Copyright 2010 Springer Nature Limited. (c) \nCurrent -induced spin accumulation in a topological \ninsulator  [104]. The arrows denote the directions of spin \nmagnetic moments, which are opposite to the \ncorresponding spin angular momenta. Reprinted with \npermission from He et al., Nat. Mater. 21, 15 –23 (2 022).  \n \n3.1 MBE -grown  and exfoliated  Topological insulators  \nTopological insulators were first introduced in the \nfield of spin -orbit torque i n 2014 . From ST -FMR \nmeasurement , Mellnik  et al . measured  a giant damping -\nlike spin-orbit torque efficiency  (𝜉DL𝑗 = 3.5) at room \ntemperature in Bi 2Se3/Py bil ayers grown by MBE  [Fig. \n6(a)][23]. In the same year, Fan et al . reported from \nharmonic Hall measurement a damping -like torque \nefficiency of 4 25 and the spin–orbit torque switching in the \n6 \n (Bi 0.5Sb0.5)2Te3/(Cr 0.08Bi0.54Sb0.38)2Te3 bilayers [24] at 1.9 \nK [Fig. 6(b)].  \nAs shown in Fig. 7 , room -temperature magnetization \nswitching by spin current from TIs (e.g., Bi 2Se3, Bi 2Te3, \nand BiSb ) has been demonstrated in Hall -bar samples \n[99,105]. Han et al. [76,106] first reported magnetization \nswitching in Hall bars of Bi 2Se3(7.4)/Co 0.77Tb0.23(4.6) \nbilayer ( Hx = 1000Oe, Hc ≈ 200Oe, Jc ≈ 2.8×106 A cm-2, \nswitching ratio=85%)[Fig. 7(a)]. The damping -like SOT \nefficiency was determined to be 0.16 ± 0.02 for the \nBi2Se3/Co 0.77Tb0.23. Similar results have been also reported \nby Wu et al. [77] in Bi 2Se3/Gd x(FeCo) 1−x Hall bars ( 𝜉DL𝑗= \n0.13, Jc ≈ 2.2×106 A cm-2)[Fig. 7(b)]. These values of spin -\norbit torque efficiency are significantly low compared to \nthose from Bi 2Se3/Py samples, which may be understood \npartly by the increased spin current relaxation via spin -\norbit scattering in the ferrimagnets  [6]. Khang et al. have \nreported a spin -orbit torque efficiency of 52 (as determined \nfrom a coercivity change measurement) and resistivity of \n400 μΩ cm for MBE -grown Bi 1-xSbx [107]. Switching of \nMBE -grown fully epitaxial Mn 0.45Ga0.55/Bi0.9Sb0.1 has also \nbeen demonstrated at a current density of 1.1×106Acm–2 \n(Hx = 3.5 kOe) in [Fig. 7(c)]. Non -epitaxial BiSb films (10 \n– 20 nm) grown by MBE were also reported to have a high \nspin-orbit torque efficiency of up to 3.2 and to enable \nmagnetization switching at a current density of 2.2×106 A \ncm-2. There have also been reports of SOT switching of \nexfoliated van-der-Waals  magnets at low temperatures, \nsuch as in Fe 3GeTe 2 [108,109] and Cr 2Ge2Te6 [110,111], by \nthe spin current from TIs. Liu et al . [112] reported a \nstrongly temperature -dependent damping -like torque \nefficiency of up to 70 from a field -dependent harmonic \nHall response measurement[Fig. 8(a)], and current \nswitching of MBE -grown Bi 2Te3/MnTe  Hall bar at a \ncritical current density of down to 6.6×106 Acm–2 (Hx = ± \n400 Oe, T = 90 K)  [Fig. 8(b)]. \n \nFIG. 6. (a) Spin torque ferromagnetic resonance spectrum \nfor Bi 2Se3(8)/Py(16) bilayer at room temperature [23]. \nReprinted with permission from  Mellnik et al., Nature 511, \n449 (2014). (b) Second  harmonic Hall resistance for \n(Bi 0.5Sb0.5)2Te3(3 QL)/(Cr 0.08Bi0.54Sb0.38)2Te3(6 QL) bilayer \nas a function of the in -plane field angle for different applied \na.c. current [24]. Reprinted with permission from  Fan et al., \nNat. Mater. 13, 699 (2014).  \n \n3.2 Sputter -deposited Topological Insulators  \nSince exfoliation and molecular -beam epitaxy are less realistic methods for the preparation of large -area TI thin \nfilms for practical SOT devices, s puttering has been \nintroduced to grow amorphous or polycrystalline \n“topological insulators”. The first report of sputter -\ndeposited “topological insulators” is BixSe(1–x) [113] with \nrelatively high electrical conductivity ( 0.78×105 Ω-1m-1 for \n4 nm thickness) . Such sputter -deposited Bi xSe(1–x) exhibits \na very high  damping -like spin-torque efficiency of 18 and \nenabled magnetization switching in a BixSe(1–\nx)(4)/Ta(0.5)/CoFeB(0.6)/Gd(1.2)/CoFeB(1.1) at a low \ncurrent density of ≈ 4.3×105 A cm-2 [Fig. 8(c)]. Wu et al. \n[105] also reported  room -temperature  witching of \nBi2Te3/Ti/CoFeB  at a current density of 2 .4×106Acm–2 (Hx \n= 100 Oe). In the Hall bar of  sputter -deposited PMA \nBi2Te3(8)/CoTb(6) bilayer, current -induced magnetization \nswitching was reported at  a low critical current density of \n9.7×105 A cm-2 [Fig. 8(d)]. Sputter -deposited BiSb films \n(10 nm) were reported to provide a spin -torque efficiency  \nof 1.2  and to drive switching of CoTb at 4×105 A/cm-2 \n[106].  \n \n3.3 Practical impact  \nAs we have discussed above, some TIs and their \nsputter -deposited counterparts are reported to have much \nhigher damping -like torque efficiency  than heavy metals . \nMeanwhile, the sputter -deposited TIs are typically several \ntimes more resistive than the MBE -grown ones since \ndisordered films typically have stronger electron scattering \nthan crystalline films. However, for practical SOT \napplications, the spin -source ma terials are required to have \nlow resistivity and large damping -like spin -orbit torque  \nefficiency . Despite their amazingly high damping -like \nspin-orbit torque  efficiency , most TIs are highly resistive (>  \n1×103 μΩ cm), much more resistive than ferromagnetic \nmetals in metallic spintronic devices (e.g., 110 μΩ cm for \nCoFeB). Current shunting into the adjacent metallic layers \nwould be considerably more than that flow s within the \ntopological insulator layer, resulting in increases in the \ntotal switching current a nd power consumption of devices.  \n \n3.4 Mechanism of the spin current generation   \nDespite the debate, t he two main mechanism s via \nwhich the TIs and their disordered counterparts generate \nspin current or spin accumulation are the spin Hall effect \nand the surface states. As suggested by Khang et al. [82], \nChi et al. [43], Tian et al. [44], the bulk spin Hall effect is \nthe dominant source of the spin current for the spin -orbit \ntorque in Bi 0.9Sb0.1, Bi0.53Sb0.47, and BixTe1–x. As shown in \nFigs. 9(a) and 9(b), in disordered Bi0.53Sb0.47 the apparent \nspin Hall conductivity increases non -linearly with \nincreasing layer thickness, which is a typical spin diffusion \nbehavior and in good consistent with a bulk spin Hall effect \nbeing the mechanism of the spin current generation. In \ncontrast, the surface states of the TIs have been suggested \nto be the main spin current source in MBE -grown (Bi 1-\nxSbx)2Te3 [105] and Bi 2Te3[112]. This suggestion is \nconsistent with the strong dependence of the damping -like \nspin-orbit  torque on the composition [105], the temperature  \n[112], and thus the location of the Fermi level relative  to \nthe Dirac point [Figs. 9(c) - 9(e)][105,114]. In addition, DC \net al . suggested that the quantum confinement effect of \nsmall grains should account for the high spin -torque \nefficiency  in the sputter -deposited Bi xSe(1–x) [113].\n7 \n  \nFIG. 7. Current -induced magnetization switching  at room temperature  in (a) Bi2Se3(7.4)/CoTb(4.6)  bilayer ( the in-plane \nmagnetic field is 1000  Oe) [76], (b) Bi2Se3(6)/Gd x(FeCo) 1−x(15) bilayer ( an in -plane magnetic field is 1000  Oe) [77], and  \n(c) Mn 0.45Ga0.55(3)/Bi0.9Sb0.1(5) (3.5 kOe)  [115]. Data in (a) is reprinted with permission from  Han et al., Phys. Rev. Lett. \n119, 077702 (2017).  Data in (b) is reprinted with permission from  Wu et al., Adv. Mater. 31, 1901681 (2019) ; Data in ( c) \nis reprinted with permission from  Khang et al., Nat. Mater. 17, 808 (2018).  \n \n \nFIG. 8. (a) Variation of the spin Hall ratio of Bi 2Te3 with temperature. ( b) Current -induced magnetization switching of \nBi2Te3(8)/MnTe(20) at 90 K under  different in -plane magnetic field s [112]. Reprinted with permission from Liu et al., \nAppl. Phys.  Lett. 118, 112406 (2021). (c) Bi xSe(1–x)(4)/Ta(0.5)/CoFeB(0.6)/Gd(1.2)/CoFeB(1.1)  (the in-plane magnetic \nfield is 80 Oe) [113], reprinted with  permission from  DC et al., Nat. Mater. 17, 800 (2018) . (d) Current switching of \nsputter -deposited Bi2Te3(8)/CoTb (6) under  an in -plane magnetic field of 400 Oe at room temperature [116]. Reprinted \nwith permission from  Zheng et al., Chin. Phys. B 29, 078505 (2020).   \n8 \n  \nFIG. 9. (a) Scanning transmission electron  microscopy  image of the 0.5 Ta/[0.35 Bi|0.35 Sb] N/0.3 Bi/2 CoFeB/2 MgO/1 \nTa structure with N = 8 . (b) thickness dependence of the apparent  spin Hall conductiv ity σSH of Bi0.53Sb0.47 [43]. Reprinted \nwith permission from  Chi et al., Sci. Adv. 6, eaay2324 (2020). (c) Fermi level , (d) Resistivity  ρxx and 2D carrier density \n|n2D| for of (Bi 1-xSbx)2Te3 with different Sb  percentage s. (e) Switching current density |Jjc| and effective damping -like spin -\norbit torque  field vs the Sb ratio  of (Bi 1-xSbx)2Te3 [105]. Reprinted with permission from  Wu et al., Phys. Rev. Lett. 123, \n207205 (2019).  \n \n4. Magnetization s witching of Van der Waals magnet  \n4.1 Van der Waals magnet  \nThe r ecent discovery of v an der Waals magnets (e.g., \nCr2Ge2Te6 [82], CrI 3 [83], etc.) has attached remarkable \nattention in the field of magnetism and spintronics. While \nthe origin of the  long-range magnetic order is still under \ndebate, it has been suggested to have a close correlation \nwith the suppression of thermal fluctuations by magnetic \nanisotropy. Note that in absence of magnetic anisotropy, no \nlong-range magnetic order is expected by the Mermin –\nWagner theorem [117] at finite temperature in a two-\ndimensional system . Van-der-Waals  magnets provide  a \nunique, highly tunable platform for spintronics.  Most \nstrikingly, the  properties of  van der Waals FMs, such as \nCurie  temperature  (TC) [78,80], coercivity [79,80], and \nmagnetic domain structure  [81], can be tuned significantly  \nby a variety of techniques ( e.g., layer thickness,  ionic liquid \ngating [78], proton doping [79], strain [80,81], exchange \nbias [118,119], interfacial proximity -effect [120], etc.). An \ninteresting example is CrI 3, whose magnetic ordering \ndepends on the number of layers and can be tuned by an \nexternal magnetic field. As shown in Fig. 10(a) , the CrI 3 is \nferromagnetic at 1 monolayer thickness, antiferromagnetic \nat 2 monolayer thickness , and ferromagnetic at 3 \nmonolayer thickness. Ferromagnetic CrI 3 also shows a \nrelatively square perpendicular magnetization loop [83]. \nFollowing the long -range ordering of magnetic \nlattices, magnetic materials can be grouped into \nferromagnets, ferrimagnets, and antiferromagnets . In  \ngeneral, ferromagnets and ferrimagnets are considered \nmore friendly than antiferromagnet s to be integrated into \nelectric circuits because their magnetization states can be \nelectrically detected by anomal ous Hall effect or tunnel \nmagnetoresistance and efficiently switched by SOTs. In \ncontrast, electrical detection and switching of collinear \nantiferromagnets  [121-123] are generally much more \nchallenging  [124], despite the recent discovery of magnetoresistance and anomalous Hall in non -collinear \nantiferromagnets Mn 3Sn [34,125,126]. For this reason, \nspin-torque switching of magnetization is mostly studied \nand better understood in ferromagnetic and ferrimagnetic \nsystems than in antiferromagnets. Our discussion below \nwill be focused on van der Waals ferromagnet s [116-134]. \nThe van-der-Waals magnet CrBr 3 (TC = 34 \nK)[127,128] (TC = 34 K), CrI 3 (TC = 45 K) [83], Cr 5Te8 \n[129] and VI 3 (TC = 60 K) [130] have  perpendicular \nmagnetic anisotropy but low Curie temperature . So far, \nroom -temperature ferromagnetism and low-temperature \nperpendicular magnetic anisotropy ha ve been reported for \nvan der Waals  materials FeTe [131], Fe 4GeTe 2 (Fig. \n10(b))[132], Fe 5GeTe 2 [133], CrTe [134], CrTe 2 (Fig. \n10(c))[135-137], Cr 1+δTe2 [138], Cr 2Te3 [139], Cr 3Te4 \n[140]), CrSe [141], and Fe 3GaTe 2 (Fig. 10( d))[142]. In Fig. \n11, we summar ize the representative results of the Curie \ntemperature and magnetization of relatively thin van der \nWaals magnets (note that TC of van der Waals magnets is \nstrongly thickness  dependent ). While FenGeTe 2 can have  \ngood PMA  at low temperatures  and CrnTem and CrSe  are \nrelatively stable in air, they lose square hysteresis loops at \nroom temperature  [Fig. 10(b) and 10(c)]. The recently \ndiscovered Fe 3GaTe 2 [142] is an outstanding van der Waals \nferromagnet that can have both a high Curie temperature  \n(TC ≈ 350 - 380 K) and large PMA energy density ( Ku ≈ \n4.8×105 J m-3) [Fig. 10(d)]. Search ing for Van der Waals \nmagnets  with room -temperature ferromagnetism, strong \nperpendicular magnetic anisotropy, a nd high stability  in the \nair at the same is expected to be an active topic in the field.  \n \n4.2 Magnetization s witching of v an der Waals \nferromagnets  \n \nSpin-orbit torque switching of v an der Waals \nferromagnets was first demonstrated in perpendicularly \nmagnetized Fe3GeTe 2/Pt bilayer s [108,109], where the spin \n9 \n current generated by the SHE in the Pt exerts a damping -\nlike spin torque on the Fe 3GeTe 2 [Fig. 12(a)]. Interestingly , \ndespite the small layer thicknesses and small magnetization \nof the Fe 3GeTe 2, the Fe 3GeTe 2/Pt samples have a high \ndepinning field (coercivity)  and strong perpendicular \nmagnetic anisotropy  such that they typically require a large \ncurrent density of ~  107 A cm-2 [108,109] as well as an in -\nplane magnetic field [Fig. 12(a)]. As indicated by the \nanomalous Hall resistance, the Fe 3GeTe 2 was also only \npartially switched , with the switching ratio of 20%-30% in \n[108] and 62% in  [118], probably due to the non-\nuniformity of the magnetic domains with the van der Waals \nlayer.  \nCr2Ge2Te6 is another well -studied van der Waals \nferromagnet . Spin-orbit torque switching of Cr2Ge2Te6 has \nbeen demonstrated in Ta/Cr 2Ge2Te6 bilayers  (Curie \ntemperature < 65 K)  at a low current density of 5×105 A cm-2 at 4 K, with an in -plane assisting magnetic field of 200 \nOe [143]. Zhu et al . [110] reported SOT switching of \nCr2Ge2Te6/W with interface -enhanced Curie temperature  \nof up to 150 K [Fig. 12(b)].  \nCurrent -induced magnetization switching has also \nbeen realized in all van der Waals heterojunction s. Nearly \nfull magnetization switching  (88%) has been reported in \nMBE -grown Cr2Ge2Te6/(Bi 1-xSbx)2Te3 bilayer s [144][Fig. \n13(a)]. In the  (Bi0.7Sb0.3)2Te3/Fe 3GeTe 2 bilayer  [145], the \nthreshold current density for the magnetization switching \nis 5. 8×106 A cm-2 at 100 K [Fig. 13(b)]. Field-free \nswitching magnetization has been reported in the \nexfoliation -fabricated WTe 2/Fe 2.78GeTe 2 bilayers  by a \ncurrent along the low symmetry axis (9.8×106 A cm-2, T \n=170K) [53]. In the same bilayer structure, Shin et al. also \nrealized magnetization switching at a current density of \n3.9×106 A cm-2 (T = 150 K, Hx = 300 Oe , Fig. 13(c)) [146].  \n \n \nFIG. 1 0. (a) Kerr rotation vs perpendicular magnetic field for monolayer (1L), bilayer (2L) , and trilayer (3L) CrI 3 flake \n[83]. Reprinted with permission from  Huang et al., Nature 546, 270 (2017). ( b) Hall conductivity hysteresis loop of a 11 -\nmono layer -thick Fe 4GeTe 2 crystal at various temperature s [132]. Reprinted with permission from  Seo et al., Sci. Adv. 6, \neaay8912 (2019). (c) Out-of-plane magnetization hysteresis loop of 7 monolayer CrTe 2 at different temperatures along \nthe out -of-plane direction [137]. Reprinted with permission from  Zhang et al ., Nat. Commun. 12, 2492 (2021). ( d) \nAnomalous Hall resistance  hysteresis  (of Fe3GaTe 2 with different thickness es at 3 K and 300 K [142]. Reprinted with \npermission from Zhang et al., Nat. Commun. 13, 5067 (2022).  \n \nFIG. 1 1 Saturation magnetization Ms vs Curie temperature TC of representative van der Wa als ferromagnets .  \n10 \n  \n \nFIG. 1 2. (a) Current -driven perpendicular magnetization switching for Fe 3GeTe 2(4)/Pt(6)  bilayer  under an in-plane \nmagnetic field of 50 0 Oe  at 100 K [108]. Reprinted with permission from  Wang et al., Sci. Adv. 5, eaaw8904 (2019).  (b) \nCurrent -driven perpendicular magnetization switching for  Cr2Ge2Te6(10)/W(7)  bilayer  under in -plane magnetic fields of \n± 1 kOe at 150 K  [110]. Reprinted with permission from  Zhu et al., Adv. Funct. Mater. 32, 2108953 (2022).  \n \n \nFIG. 1 3. Current induced switching of van der Waals magnets. (a) Normalized anomalous Hall resistance vs current \ndensity for (Bi 1-xSbx)2Te3(6)/Cr 2Ge2Te6(t) (x = 0.5) with different Cr 2Ge2Te6 thicknesses under an in-plane magnetic field \nof 1 kOe at 2 K [144]. Reprinted with permission from  Mogi et al., Nat. Commun. 12, 1404 (2021).  (b) Anomalous Hall \nconductivity vs current density for (Bi 1-xSbx)2Te3(8)/Fe3GeTe 2(6) at different temperatures under an in-plane magnetic \nfield of 1 kOe [145]. Reprinted with permission from  Fujimura et al., Appl. Phys. Lett. 119, 032402 (2021).  (c) Hall \nresistance vs current density  for Fe 3GeTe 2(7.3)/WTe 2(12.6)  under in -plane magnetic field  Hx = 300 Oe . The Hall resistance \nvaries during three consecutive current scans due to temperature rise towards the Curie temperature [146]. Reprinted with \npermission from  Shin et al., Adv. Mater. 34, 2101730 (2022).  \n \n5. Simplifying models of switching current density  \n \nIn this section, we provide a quantitative \nunderstanding of the switching current densities in the van \nder Waals system by considering the simplifying models. \nthe transverse spin damping -like SOT efficiency per unit \ncurrent density ( 𝜉𝐷𝐿𝑗 ) of a heterostructure with PMA \ninversely cor relates to the critical switching current density \n(jc) in the spin -current -generating layer via Eq. (4) in \nmacrospin limit  [149,150] and via Eq. (5)  in the domain \nwall depinning limit [151-153], i.e.,  jc ≈ eμ0MstFM (Hk-√2|𝐻𝑥|)/ћ𝜉DL𝑗,           (4)  \njc = (4e/πћ) μ0MstFMHc/𝜉DL𝑗,               (5)  \nwhere e is the elementary charge, ℏ  is the reduced Planck \nconstant, μ0 is the permeability of vacuum, Hx is the applied \nfield along the current direction, and tFM, Ms, Hk, and Hc \nare the thickness,  the saturation magnetization, the \neffective perpendicular anisotropy field, and the \nperpendicular coercivity  of the driven magnetic layer FM, \nrespectively.  \n11 \n However, r ecent experiments [154] on heavy \nmetal/m agnet bilayers have shown that neither Eq. (4) nor \nEq. (5) can provide a reliable prediction for the switching \ncurrent and 𝜉𝐷𝐿𝑗  and that there is no simple correlation \nbetween 𝜉𝐷𝐿𝑗 and the critical switching current density of \nrealistic perpendicularly magnetized spin -current \ngenerator/ferromagnet heterostructures. As shown in Table \nI, the same is true for the van der Waals systems. The \nmacrospin analysis does not seem to apply to the switching \ndynamics of micrometer -scale samples so that the values \nof 𝜉𝐷𝐿𝑗 determined using the switching current density and \nEq. ( 4) can produce overestimates by up to hundreds of \ntimes (𝜉𝐷𝐿,𝑚𝑎𝑐𝑟𝑜𝑗 and 𝜉𝐷𝐿,𝑚𝑎𝑐𝑟𝑜𝑗/𝜉𝐷𝐿𝑗 in Table I). A domain -\nwall depinning analysis [ Eq. ( 5)] can either under - or over -estimated 𝜉𝐷𝐿𝑗  by up to tens of times (𝜉𝐷𝐿,𝐷𝑊𝑗  and \n𝜉𝐷𝐿,𝐷𝑊𝑗/𝜉𝐷𝐿𝑗 in Table I). These observations consistently \nsuggest that the switching current or “switching efficiency” \nof perpendicular heterostructures in the micrometer or sub -\nmicrometer scales cannot provide a quantitative estimation \nof 𝜉𝐷𝐿𝑗. \nWhile the underlying mechanis m of the failure of the \nsimplifying models remains an open question, it is obvious \nthat Joule heating during current switching of the resistive \nor low Curie -temperature van der Waals systems can have \na rather significant influence on the apparent switching  \ncurrent density. As shown in Fig. 13(c), the anomalous Hall \nresistance hysteresis loop drifts for three consecutive \ncurrent scans because of Joule heating [146].  \n \nTABLE I. Comparison of spin -torque efficiencies determined from the harmonic response or ST -FMR (𝜉𝐷𝐿𝑗 ) and \nmagnetization switching ( 𝜉𝐷𝐿,𝐷𝑊𝑗, 𝜉𝐷𝐿,𝑚𝑎𝑐𝑟𝑜𝑗) of PMA samples, which is calculated from Eq. (5)  and Eq. (4)  using the \napplied external magnetic field ( Hx), saturation magnetization ( Ms), the perpendicular coercivity ( Hc), and the effective \nperpendicular anisotropy field ( Hk) of the driven magnetic layer , and the critical magnetization switching current density \n(jc). The value of 𝜉𝐷𝐿,𝑚𝑎𝑐𝑟𝑜𝑗 is estimated to be negative for Te2(10)/CoTb(6) [97] and ZrTe 2(7.2) /CrTe 2(1.8) [94] because \naccording to th e original reports an in -plane field greater that the effective perpendicular anisotropy field was applied.  \n \nSample  Technique  Ms \n(emu cm-3) Hc  \n(Oe) Hk  \n(kOe)  Hx  \n(Oe) jc  \n(MA cm-2) 𝜉𝐷𝐿𝑗 𝜉𝐷𝐿,𝐷𝑊𝑗 𝜉𝐷𝐿,𝑚𝑎𝑐𝑟𝑜𝑗 𝜉𝐷𝐿,𝐷𝑊𝑗/𝜉𝐷𝐿𝑗 𝜉𝐷𝐿,𝑚𝑎𝑐𝑟𝑜𝑗/𝜉𝐷𝐿𝑗 \nTe2(10)/CoTb(6) [97] sputtering  48 60 0.33 900 0.7 0.2 0.47 -5.8 2.4 -29 \nZrTe 2(7.2) /CrTe 2(1.8) [94] MBE  100 -- 0.2 700 18 0.014  -- -0.12 -- -8.5 \nBi2Se3(7.4)/CoTb(4.6)  [76] MBE  280 300 -- 1000  2.8 0.16 2.7 -- 17 -- \nBi2Se3(6)/Gd x(FeCo) 1−x(15) [77] MBE  46 160 0.35 1000  2.2 0.13 0.98 10 7.6 78 \nBixSe(1–x) (4)/Ta(0.5)/  CoFeB  \n(0.6)/Gd(1.2)/CoFeB(1.1) [113] sputtering  300 30 6 80 0.43 8.67 1.2 180 0.13 21 \n(BiSb) 2Te3(6QL)/Ti \n(2)/CoFeB(1.5) [105] MBE  868 30 2.24 100 0.12 2.5 6.3 345 2.5 138 \nBi2Te3/Ti(2) (6QL)/CoFeB(1.5) \n[105] MBE  868 27 2.06 100 2.4 0.08 0.287  16 3.5 194 \nBi2Te3(8)/MnTe(20)  [112] MBE  175 100 ≈50 400 6.6 10 1.0 397 0.10 40 \nBi0.9Sb0.1(5) /Mn 0.45Ga0.55(3) \n[115] MBE  240 4500  10 3500  1.1 52 57 50 1.1 0.96 \nSnTe(6QL)/Ti(2)/CoFeB(1.5) \n[105] MBE  868 53 2.18 100 1.5 1.41 0.91 27.5 0.65 20 \nFe3GeTe 2(4)/Pt(6)  [108] exfoliation  16 125 11 500 11.6 0.12 0.013  0.86 0.11 7.2 \nFe3GeTe 2 (15)/Pt(5)  [109] exfoliation  170 750 30 3000  20 0.14 1.8 50 13 355 \n \n6. Conclusion and outlook  \nWe have reviewed recent advances in spin -orbit \ntorque and resultant magnetization switching in van der \nWaals systems. Van der Waals materials provided unique \nopportuni ties for spintronics  because of their diversity of \nmaterials, the flexibility of preparat ion (e.g., by mechanical \nexfoliation), and strong tunability by interface effects.  \nVan der Waals TMDs  such as WTe 2 also exhibit potential \nas a spin current source of both transverse spins and exotic \nperpendicular and lo ngitudinal spins . Bismuth -based \ntopological insulators and their sputter -deposited \ndisordered counterparts are shown to generate giant \ndamping -like SOT with the efficiency  of up to 1-3 orders \nof magnitudes greater than 5 d metals. Moreover, van der \nWaals materials that are m agnetic are also interesting for \nspin-orbitronics due to their highly tunable magnetism and \nlow magnetization at small thicknesses  since the damping -like and field -like effective SOT fields exerted on the \nmagnetic layer by a given spin current scale invers ely with \nthe thickness and magnetization of the magnetic layer.  \nEfficient  switching of several  van der Waals magnet s by \nspin current  has been demonstrated .  \nDespite the se exciting progres s, essential challenges \nhave remained to overcome for spin-orbit torque switching \nof van der Waals systems : \n(i) While the generation of different spin components \nhas been demonstrated, the efficiencies are typically quite \nlow. It has remained unclear as to whether and how the \nefficiency of generating exotic perpendicular and \nlongitudinal spins by the low -symmetry TMDs can be \nimproved  significantly  to be compelling for practical SOT \napplications .  \n(ii) Some Bi -based t opological i nsulators and alloys \nhave both giant effective spin Hall ratio and resistivity at 12 \n the same time, the latter is undesirable for metallic SOT \ndevices that require energy efficiency, high endurance, and \nlow impedance. It would be interesting if new uniform, \nstable spin -orbit materials can be developed to provide \ndamping -like SOT efficiency of greater than 1 but \nsubstantially less resistive than the yet -know n topological \ninsulators.  \n(iii) So far, large -area growth of van der Waals \nmagnets that have high Cu rie temperature, strong \nperpendicular magnetic anisotropy, and high stabilities \nagainst heating and atmosphere  at the same time  has \nremained a key obstacle that prevents van der Waals \nmagnets from applications in spintronic technologie s. \nBreakthroughs in th e fabrication of such application -\nfriendly van der Waals magnets are in urgent need.   \n(iv) While spin -orbit torque switching of \nmagnetization has been demonstrated in Hall -bar devices \ncontaining van der Waals magnets, TMDs, or/and \ntopological insulators, t he simplifying models of \nmacrospin rotation and domain wall depinning cannot \nprovide an accurate prediction for the switching current \ndensity. So far, the quantitative understanding of the \nswitching current remains a fundamental problem . \n(v) From  the point  of view of magnetic memory and \nlogic application, s witching of thermally stable nanodots of \nvan der Waals magnet s, such as  the free layers of \nnanopillars of  magnetic tunnel junctions, by current pulse s \nof picosecond and nanosecond durations. Efforts are a lso \nrequired on the demonstration and optimization of t he key \nperformance , e.g., the endurance, the write error rates, the \nretention, and the tunnel magnetoresistance . \n \nAcknowledgments  \nThis work was supported partly by the National Key \nResearch and Develop ment Program of China (Grant No. \n2022YFA12 0400 4) and by the National Natural Science  \nFoundation of China  (Grant No. 12274405 ). \n \nReference  \n[1] A. Manchon, J. Železný, I. M. Miron, T. Jungwirth, J. \nSinova, A. Thiaville, K. Garello, and P. Gambardella, \nCurrent -induced spin -orbit torques in ferromagnetic and \nantiferromagnetic systems. Rev. Mod. Phys. 91, 035004 \n(2019).  \n[2] I. Mihai Miron, K. Garello, G. Gaudin, P. -J. \nZermatten, M. V . Costache, S. Auffret, S. Bandiera, B. \nRodmacq, A. Schuhl, a nd P. Gambardella, Perpendicular \nswitching of a single ferromagnetic layer induced by in -\nplane current injection. Nature 476, 189 (2011).  \n[3] L. Q. Liu, C. F. Pai, Y . Li, H. W. Tseng, D. C. Ralph, \nand R. A. Buhrman, Spin -Torque Switching with the Giant \nSpin Hall Effect of Tantalum. Science 336, 555 (2012).  \n[4] L. Liu, O. J. Lee, T. J. Gudmundsen, D. C. Ralph, and \nR. A. Buhrman, Current -Induced Switching of \nPerpendicularly Magnetized Magnetic Layers Using Spin \nTorque from the Spin Hall Effect. Phys. Rev. Let t. 109, \n096602 (2012).  \n[5] L. Zhu, Switching of Perpendicular Magnetization by \nSpin-Orbit Torque. Adv. Mater. n/a, 2300853 (2023).  \n[6] L. Zhu and D. C. Ralph, Strong variation of spin -orbit \ntorques with relative spin relaxation rates in ferrimagnets. \nNat. Commun. 14, 1778 (2023).  [7] P. M. Haney, H. -W. Lee, K. -J. Lee, A. Manchon, and \nM. D. Stiles, Current induced torques and interfacial spin -\norbit coupling: Semiclassical modeling. Phys. Rev. B 87, \n174411 (2013).  \n[8] Y.-T. Chen, S. Takahashi, H. Nakayama, M.  \nAlthammer, S. T. B. Goennenwein, E. Saitoh, and G. E. W. \nBauer, Theory of spin Hall magnetoresistance. Phys. Rev. \nB 87 (2013).  \n[9] V . P. Amin and M. D. Stiles, Spin transport at \ninterfaces with spin -orbit coupling: Phenomenology. Phys. \nRev. B 94, 104420 ( 2016).  \n[10] L. J. Zhu, D. C. Ralph, and R. A. Buhrman, Effective \nSpin-Mixing Conductance of Heavy -Metal -Ferromagnet \nInterfaces. Phys. Rev. Lett. 123, 7, 057203 (2019).  \n[11] L. Zhu, L. Zhu, and R. A. Buhrman, Fully Spin -\nTransparent Magnetic Interfaces Enabl ed by the Insertion \nof a Thin Paramagnetic NiO Layer. Phys. Rev. Lett. 126, \n107204 (2021).  \n[12] Y . Liu, Z. Yuan, R. J. H. Wesselink, A. A. Starikov, \nand P. J. Kelly, Interface Enhancement of Gilbert Damping \nfrom First Principles. Phys. Rev. Lett. 113, 2072 02 (2014).  \n[13] J. C. Rojas -Sánchez, N. Reyren, P. Laczkowski, W. \nSavero, J. P. Attané, C. Deranlot, M. Jamet, J. M. George, \nL. Vila, and H. Jaffrès, Spin Pumping and Inverse Spin Hall \nEffect in Platinum: The Essential Role of Spin -Memory \nLoss at Metallic Interfaces. Phys. Rev. Lett. 112, 106602 \n(2014).  \n[14] L. Zhu, D. C. Ralph, and R. A. Buhrman, Spin -Orbit \nTorques in Heavy -Metal –Ferromagnet Bilayers with \nVarying Strengths of Interfacial Spin -Orbit Coupling. Phys. \nRev. Lett. 122, 077201 (2019).  \n[15] L. J. Zhu and R. A. Buhrman, Maximizing Spin -\nOrbit -Torque Efficiency of Pt/Ti Multilayers: Trade -Off \nBetween Intrinsic Spin Hall Conductivity and Carrier \nLifetime. Phys. Rev. Appl. 12, 6, 051002 (2019).  \n[16] S. Li, K. Shen, and K. Xia, Interfacial spin Hall effect \nand spin swapping in Fe -Au bilayers from first principles. \nPhys. Rev. B 99, 134427 (2019).  \n[17] M. Baumgartner, K. Garello, J. Mendil, C. O. Avci, E. \nGrimaldi, C. Murer, J. Feng, M. Gabureac, C. Stamm, Y . \nAcremann, S. Finizio, S. Wintz, J. Raab e, and P. \nGambardella, Spatially and time -resolved magnetization \ndynamics driven by spin –orbit torques. Nat. Nanotechnol. \n12, 980 (2017).  \n[18] S. Dutta, A. Bose, A. A. Tulapurkar, R. A. Buhrman, \nand D. C. Ralph, Interfacial and bulk spin Hall \ncontributions  to fieldlike spin -orbit torque generated by \niridium. Phys. Rev. B 103, 184416 (2021).  \n[19] M. I. Dyakonov and V . I. Perel, Current -induced spin \norientation of electrons in semiconductors. Phys. Lett. A 35, \n459 (1971).  \n[20] J. E. Hirsch, Spin Hall Effect. Phys. Rev. Lett. 83, \n1834 (1999).  \n[21] T. Tanaka, H. Kontani, M. Naito, T. Naito, D. S. \nHirashima, K. Yamada, and J. Inoue, Intrinsic spin Hall \neffect and orbital Hall effect in 4d and 5d transition metals. \nPhys. Rev. B 77, 165117 (2008).  \n[22] G.-Y . Guo, S . Murakami, T. -W. Chen, and N. Nagaosa, \nIntrinsic spin Hall effect in platinum: First -principles \ncalculations. Phys. Rev. Lett. 100, 096401 (2008).  \n[23] A. R. Mellnik, J. S. Lee, A. Richardella, J. L. Grab, P. \nJ. Mintun, M. H. Fischer, A. Vaezi, A. Manchon , E. A. Kim, 13 \n N. Samarth, and D. C. Ralph, Spin -transfer torque \ngenerated by a topological insulator. Nature 511, 449 \n(2014).  \n[24] Y . Fan, P. Upadhyaya, X. Kou, M. Lang, S. Takei, Z. \nWang, J. Tang, L. He, L. -T. Chang, M. Montazeri, G. Yu, \nW. Jiang, T. Nie, R. N. Schwartz, Y . Tserkovnyak, and K. \nL. Wang, Magnetization switching through giant spin –orbit \ntorque in a magnetically doped topological insulator \nheterostructure. Nat. Mater. 13, 699 (2014).  \n[25] V . P. Amin, J. Zemen, and M. D. Stiles, Interface -\nGenera ted Spin Currents. Phys. Rev. Lett. 121, 136805 \n(2018).  \n[26] L. Wang, R. J. H. Wesselink, Y . Liu, Z. Yuan, K. Xia, \nand P. J. Kelly, Giant Room Temperature Interface Spin \nHall and Inverse Spin Hall Effects. Phys. Rev. Lett. 116, \n196602 (2016).  \n[27] J. Borge  and I. V . Tokatly, Boundary conditions for \nspin and charge diffusion in the presence of interfacial \nspin-orbit coupling. Phys. Rev. B 99, 241401 (2019).  \n[28] S.-h. C. Baek, V . P. Amin, Y . -W. Oh, G. Go, S. -J. Lee, \nG.-H. Lee, K. -J. Kim, M. D. Stiles, B. -G. Park, and K. -J. \nLee, Spin currents and spin –orbit torques in ferromagnetic \ntrilayers. Nat. Mater. 17, 509 (2018).  \n[29] S. Ding, A. Ross, D. Go, L. Baldrati, Z. Ren, F. \nFreimuth, S. Becker, F. Kammerbauer, J. Yang, G. Jakob, \nY . Mokrousov, and M. Kläui, Harn essing Orbital -to-Spin \nConversion of Interfacial Orbital Currents for Efficient \nSpin-Orbit Torques. Phys. Rev. Lett. 125, 177201 (2020).  \n[30] T. Taniguchi, J. Grollier, and M. D. Stiles, Spin -\nTransfer Torques Generated by the Anomalous Hall Effect \nand Anis otropic Magnetoresistance. Phys. Rev. Appl. 3, \n044001 (2015).  \n[31] A. Bose, D. D. Lam, S. Bhuktare, S. Dutta, H. Singh, \nY . Jibiki, M. Goto, S. Miwa, and A. A. Tulapurkar, \nObservation of Anomalous Spin Torque Generated by a \nFerromagnet. Phys. Rev. Appl. 9, 064026 (2018).  \n[32] C. Safranski, E. A. Montoya, and I. N. Krivorotov, \nSpin–orbit torque driven by a planar Hall current. Nat. \nNanotechnol. 14, 27 (2019).  \n[33] C. Safranski, J. Z. Sun, J. -W. Xu, and A. D. Kent, \nPlanar Hall Driven Torque in a \nFerromagnet/No nmagnet/Ferromagnet System. Phys. Rev. \nLett. 124, 197204 (2020).  \n[34] M. Kimata, H. Chen, K. Kondou, S. Sugimoto, P. K. \nMuduli, M. Ikhlas, Y . Omori, T. Tomita, A. H. Macdonald, \nS. Nakatsuji, and Y . Otani, Magnetic and magnetic  inverse \nspin Hall effects in a non -collinear antiferromagnet. Nature \n565, 627 (2019).  \n[35] X. R. Wang, Anomalous spin Hall and inverse spin \nHall effects in magnetic systems. Commun. Phys. 4, 55 \n(2021).  \n[36] A. Mook, R. R. Neumann, A. Johansson, J. Henk, and \nI. Mertig, Origin of the ma gnetic spin Hall effect: Spin \ncurrent vorticity in the Fermi sea. Physical Review \nResearch 2, 023065 (2020).  \n[37] M. Jiang, H. Asahara, S. Sato, T. Kanaki, H. Yamasaki, \nS. Ohya, and M. Tanaka, Efficient full spin –orbit torque \nswitching in a single layer of  a perpendicularly magnetized \nsingle -crystalline ferromagnet. Nat. Commun. 10, 2590 \n(2019).  \n[38] T. Nan, T. J. Anderson, J. Gibbons, K. Hwang, N. \nCampbell, H. Zhou, Y . Q. Dong, G. Y . Kim, D. F. Shao, T. R. Paudel, N. Reynolds, X. J. Wang, N. X. Sun, E. Y . \nTsymbal, S. Y . Choi, M. S. Rzchowski, Y . B. Kim, D. C. \nRalph, and C. B. Eom, Anisotropic spin -orbit torque \ngeneration in epitaxial SrIrO 3 by symmetry design. \nProceedings of the National Academy of Sciences 116, \n16186 (2019).  \n[39] A. Bose, J. N. Nelson, X. S. Zhang, P. Jadaun, R. Jain, \nD. G. Schlom, D. C. Ralph, D. A. Muller, K. M. Shen, and \nR. A. Buhrman, Effects of Anisotropic Strain on Spin –\nOrbit Torque Produced by the Dirac Nodal Line Semimetal \nIrO 2. ACS Appl. Mater. Interfaces 12, 55411 (2020).  \n[40] S. V . Aradhya, G. E. Rowlands, J. Oh, D. C. Ralph, \nand R. A. Buhrman, Nanosecond -Timescale Low Energy \nSwitching of In -Plane Magnetic Tunnel Junctions through \nDynamic Oersted -Field -Assisted Spin Hall Effect. Nano \nLett. 16, 5987 (2016).  \n[41] L. Zhu, L. Zhu, M. Sui, D. C. Ralph, and R. A. \nBuhrman, Variation of the giant intrinsic spin Hall \nconductivity of Pt with carrier lifetime. Sci. Adv. 5, \neaav8025 (2019).  \n[42] E. Sagasta, Y . Omori, M. Isasa, M. Gradhand, L. E. \nHueso, Y . Niimi, Y . Otani, and F. Casano va, Tuning the \nspin Hall effect of Pt from the moderately dirty to the \nsuperclean regime. Phys. Rev. B 94, 060412 (2016).  \n[43] Z. Chi, Y . -C. Lau, X. Xu, T. Ohkubo, K. Hono, and \nM. Hayashi, The spin Hall effect of Bi -Sb alloys driven by \nthermally excited Di rac-like electrons. Sci. Adv. 6, \neaay2324 (2020).  \n[44] T.-Y . Chen, C. -W. Peng, T. -Y . Tsai, W. -B. Liao, C. -T. \nWu, H. -W. Yen, and C. -F. Pai, Efficient Spin –Orbit Torque \nSwitching with Nonepitaxial Chalcogenide \nHeterostructures. ACS Appl. Mater. Interfaces 12 , 7788 \n(2020).  \n[45] L. Zhu, X. S. Zhang, D. A. Muller, D. C. Ralph, and \nR. A. Buhrman, Observation of Strong Bulk Damping -Like \nSpin-Orbit Torque in Chemically Disordered \nFerromagnetic Single Layers. Adv. Funct. Mater. 30, \n2005201 (2020).  \n[46] L. Zhu, D. C.  Ralph, and R. A. Buhrman, Unveiling \nthe Mechanism of Bulk Spin -Orbit Torques within \nChemically Disordered Fe xPt1-x Single Layers. Adv. Funct. \nMater. 31, 2103898 (2021).  \n[47] Q. Liu, L. Zhu, X. S. Zhang, D. A. Muller, and D. C. \nRalph, Giant bulk spin –orbit  torque and efficient electrical \nswitching in single ferrimagnetic FeTb layers with strong \nperpendicular magnetic anisotropy. Appl. Phys. Rev. 9, \n021402 (2022).  \n[48] Y . Hibino, T. Taniguchi, K. Yakushiji, A. Fukushima, \nH. Kubota, and S. Yuasa, Large Spin -Orbit-Torque \nEfficiency Generated by Spin Hall Effect in Paramagnetic \nCo-Ni-Pt Alloys. Phys. Rev. Appl. 14, 064056 (2020).  \n[49] C.-F. Pai, L. Liu, Y . Li, H. W. Tseng, D. C. Ralph, and \nR. A. Buhrman, Spin transfer torque devices utilizing the \ngiant spin Hall  effect of tungsten. Appl. Phys. Lett. 101, \n122404 (2012).  \n[50] D. Macneill, G. M. Stiehl, M. H. D. Guimaraes, R. A. \nBuhrman, J. Park, and D. C. Ralph, Control of spin –orbit \ntorques through crystal symmetry in WTe 2/ferromagnet \nbilayers. Nat. Phys. 13, 300 (2017).  \n[51] G. M. Stiehl, R. Li, V . Gupta, I. E. Baggari, S. Jiang, \nH. Xie, L. F. Kourkoutis, K. F. Mak, J. Shan, R. A. \nBuhrman, and D. C. Ralph, Layer -dependent spin -orbit 14 \n torques generated by the centrosymmetric transition metal \ndichalcogenide MoTe. Phy s. Rev. B 100, 184402 (2019).  \n[52] Q. Xie, W. Lin, S. Sarkar, X. Shu, S. Chen, L. Liu, T. \nZhao, C. Zhou, H. Wang, J. Zhou, S. Gradečak, and J. Chen, \nField -free magnetization switching induced by the \nunconventional spin –orbit torque from WTe 2. APL Mater. \n9, 051114 (2021).  \n[53] I. H. Kao, R. Muzzio, H. Zhang, M. Zhu, J. Gobbo, S. \nYuan, D. Weber, R. Rao, J. Li, J. H. Edgar, J. E. Goldberger, \nJ. Yan, D. G. Mandrus, J. Hwang, R. Cheng, J. Katoch, and \nS. Singh, Deterministic switching of a perpendicularly \npolariz ed magnet using unconventional spin –orbit torques \nin WTe 2. Nat. Mater. 21, 1029 (2022).  \n[54] C. K. Safeer, N. Ontoso, J. Ingla -Aynés, F. Herling, V . \nT. Pham, A. Kurzmann, K. Ensslin, A. Chuvilin, I. Robredo, \nM. G. Vergniory, F. de Juan, L. E. Hueso, M. R. Calvo, and \nF. Casanova, Large Multidirectional Spin -to-Charge \nConversion in Low -Symmetry Semimetal MoTe 2 at Room \nTemperature. Nano Lett. 19, 8758 (2019).  \n[55] P. Song, C. -H. Hsu, G. Vignale, M. Zhao, J. Liu, Y . \nDeng, W. Fu, Y . Liu, Y . Zhang, H. Lin, V . M. Pereira, and \nK. P. Loh, Coexistence of large conventional and planar \nspin Hall effect with long spin diffusion length in a low -\nsymmetry semimetal at room temperature. Nat. Mater. 19, \n292 (2020).  \n[56] L. Liu, C. Zhou, X. Shu, C. Li, T. Zhao, W. Lin, J. \nDeng , Q. Xie, S. Chen, J. Zhou, R. Guo, H. Wang, J. Yu, S. \nShi, P. Yang, S. Pennycook, A. Manchon, and J. Chen, \nSymmetry -dependent field -free switching of perpendicular \nmagnetization. Nat. Nanotechnol. 16, 277 (2021).  \n[57] Y . Liu, Y . Liu, M. Chen, S. Srivastav a, P. He, K. L. \nTeo, T. Phung, S. -H. Yang, and H. Yang, Current -induced \nOut-of-plane Spin Accumulation on the (001) Surface of \nthe IrMn 3 Antiferromagnet. Phys. Rev. Appl. 12, 064046 \n(2019).  \n[58] T. Nan, C. X. Quintela, J. Irwin, G. Gurung, D. F. \nShao, J. G ibbons, N. Campbell, K. Song, S. Y . Choi, L. \nGuo, R. D. Johnson, P. Manuel, R. V . Chopdekar, I. \nHallsteinsen, T. Tybell, P. J. Ryan, J. W. Kim, Y . Choi, P. G. \nRadaelli, D. C. Ralph, E. Y . Tsymbal, M. S. Rzchowski, \nand C. B. Eom, Controlling spin current po larization \nthrough non -collinear antiferromagnetism. Nat. Commun. \n11, 4671 (2020).  \n[59] Y . You, H. Bai, X. Feng, X. Fan, L. Han, X. Zhou, Y . \nZhou, R. Zhang, T. Chen, F. Pan, and C. Song, Cluster \nmagnetic octupole induced out -of-plane spin polarization \nin antiperovskite antiferromagnet. Nat. Commun. 12, 6524 \n(2021).  \n[60] X. Chen, S. Shi, G. Shi, X. Fan, C. Song, X. Zhou, H. \nBai, L. Liao, Y . Zhou, H. Zhang, A. Li, Y . Chen, X. Han, S. \nJiang, Z. Zhu, H. Wu, X. Wang, D. Xue, H. Yang, and F. \nPan, Observation of t he antiferromagnetic spin Hall effect. \nNat. Mater. 20, 800 (2021).  \n[61] R. González -Hernández, L. Šmejkal, K. Výborný, Y . \nYahagi, J. Sinova, T. Jungwirth, and J. Železný, Efficient \nElectrical Spin Splitter Based on Nonrelativistic Collinear \nAntiferromagnet ism. Phys. Rev. Lett. 126, 127701 (2021).  \n[62] A. Bose, N. J. Schreiber, R. Jain, D. -F. Shao, H. P. \nNair, J. Sun, X. S. Zhang, D. A. Muller, E. Y . Tsymbal, D. \nG. Schlom, and D. C. Ralph, Tilted spin current generated \nby the collinear antiferromagnet ruthen ium dioxide. Nat. \nElectron. 5, 267 (2022).  [63] S. Karube, T. Tanaka, D. Sugawara, N. Kadoguchi, M. \nKohda, and J. Nitta, Observation of Spin -Splitter Torque in \nCollinear Antiferromagnetic RuO 2. Phys. Rev. Lett. 129, \n137201 (2022).  \n[64] H. Bai, L. Han, X. Y . Feng, Y . J. Zhou, R. X. Su, Q. \nWang, L. Y . Liao, W. X. Zhu, X. Z. Chen, F. Pan, X. L. Fan, \nand C. Song, Observation of Spin Splitting Torque in a \nCollinear Antiferromagnet RuO 2. Phys. Rev. Lett. 128, \n197202 (2022).  \n[65] A. M. Humphries, T. Wang, E. R. J.  Edwards, S. R. \nAllen, J. M. Shaw, H. T. Nembach, J. Q. Xiao, T. J. Silva, \nand X. Fan, Observation of spin -orbit effects with spin \nrotation symmetry. Nat. Commun. 8, 911 (2017).  \n[66] H. Kurebayashi, J. Sinova, D. Fang, A. C. Irvine, T. \nD. Skinner, J. Wunde rlich, V . Novák, R. P. Campion, B. L. \nGallagher, E. K. Vehstedt, L. P. Zârbo, K. Výborný, A. J. \nFerguson, and T. Jungwirth, An antidamping spin –orbit \ntorque originating from the Berry curvature. Nat. \nNanotechnol. 9, 211 (2014).  \n[67] C. Ciccarelli, L. Ander son, V . Tshitoyan, A. J. \nFerguson, F. Gerhard, C. Gould, L. W. Molenkamp, J. \nGayles, J. Železný, L. Šmejkal, Z. Yuan, J. Sinova, F. \nFreimuth, and T. Jungwirth, Room -temperature spin –orbit \ntorque in NiMnSb. Nat. Phys. 12, 855 (2016).  \n[68] L. Chen, M. Decker , M. Kronseder, R. Islinger, M. \nGmitra, D. Schuh, D. Bougeard, J. Fabian, D. Weiss, and \nC. H. Back, Robust spin -orbit torque and spin -galvanic \neffect at the Fe/GaAs (001) interface at room temperature. \nNat. Commun. 7, 13802 (2016).  \n[69] T. Wang, S. Lendinez, M. B. Jungfleisch, J. Kolodzey, \nJ. Q. Xiao, and X. Fan, Detection of spin -orbit torque with \nspin rotation symmetry. Appl. Phys. Lett. 116, 012404 \n(2020).  \n[70] Y . Hibino, T. Taniguchi, K. Yakushiji, A. Fukushima, \nH. Kubota, and S. Yuasa, Giant cha rge-to-spin conversion \nin ferromagnet via spin -orbit coupling. Nat. Commun. 12, \n6254 (2021).  \n[71] L. Zhu, D. C. Ralph, and R. A. Buhrman, Maximizing \nspin-orbit torque generated by the spin Hall effect of Pt. \nAppl. Phys. Rev. 8, 031308 (2021).  \n[72] K. S. No voselov, A. K. Geim, S. V . Morozov, D. Jiang, \nY . Zhang, S. V . Dubonos, I. V . Grigorieva, and A. A. Firsov, \nElectric Field Effect in Atomically Thin Carbon Films. \nScience 306, 666 (2004).  \n[73] Z. Y . Zhu, Y . C. Cheng, and U. Schwingenschlögl, \nGiant spin -orbit-induced spin splitting in two -dimensional \ntransition -metal dichalcogenide semiconductors. Phys. \nRev. B 84, 153402 (2011).  \n[74] J. Jiang, F. Tang, X. C. Pan, H. M. Liu, X. H. Niu, Y . \nX. Wang, D. F. Xu, H. F. Yang, B. P. Xie, F. Q. Song, P. \nDudin, T. K. Ki m, M. Hoesch, P. K. Das, I. V obornik, X. G. \nWan, and D. L. Feng, Signature of Strong Spin -Orbital \nCoupling in the Large Nonsaturating Magnetoresistance \nMaterial WTe 2. Phys. Rev. Lett. 115, 166601 (2015).  \n[75] Y . Wang, P. Deorani, K. Banerjee, N. Koirala, M . \nBrahlek, S. Oh, and H. Yang, Topological Surface States \nOriginated Spin -Orbit Torques in Bi 2Se3. Phys. Rev. Lett. \n114, 257202 (2015).  \n[76] J. Han, A. Richardella, S. A. Siddiqui, J. Finley, N. \nSamarth, and L. Liu, Room -Temperature Spin -Orbit \nTorque Switc hing Induced by a Topological Insulator. Phys. \nRev. Lett. 119, 077702 (2017).  15 \n [77] H. Wu, Y . Xu, P. Deng, Q. Pan, S. A. Razavi, K. Wong, \nL. Huang, B. Dai, Q. Shao, G. Yu, X. Han, J. -C. Rojas -\nSánchez, S. Mangin, and K. L. Wang, Spin -Orbit Torque \nSwitching o f a Nearly Compensated Ferrimagnet by \nTopological Surface States. Adv. Mater. 31, 1901681 \n(2019).  \n[78] Y . Deng, Y . Yu, Y . Song, J. Zhang, N. Z. Wang, Z. Sun, \nY . Yi, Y . Z. Wu, S. Wu, J. Zhu, J. Wang, X. H. Chen, and Y . \nZhang, Gate -tunable room -temperature f erromagnetism in \ntwo-dimensional Fe 3GeTe 2. Nature 563, 94 (2018).  \n[79] G. Zheng, W. -Q. Xie, S. Albarakati, M. Algarni, C. \nTan, Y . Wang, J. Peng, J. Partridge, L. Farrar, J. Yi, Y . \nXiong, M. Tian, Y . -J. Zhao, and L. Wang, Gate -Tuned \nInterlayer Coupling in v an der Waals Ferromagnet \nFe3GeTe 2 Nanoflakes. Phys. Rev. Lett. 125, 047202 (2020).  \n[80] Y . Wang, C. Wang, S. -J. Liang, Z. Ma, K. Xu, X. Liu, \nL. Zhang, A. S. Admasu, S. -W. Cheong, L. Wang, M. Chen, \nZ. Liu, B. Cheng, W. Ji, and F. Miao, Strain -Sensitive \nMagn etization Reversal of a van der Waals Magnet. Adv. \nMater. 32, 2004533 (2020).  \n[81] Q. Li, M. Yang, C. Gong, R. V . Chopdekar, A. T. \nN’Diaye, J. Turner, G. Chen, A. Scholl, P. Shafer, E. \nArenholz, A. K. Schmid, S. Wang, K. Liu, N. Gao, A. S. \nAdmasu, S. -W. Ch eong, C. Hwang, J. Li, F. Wang, X. \nZhang, and Z. Qiu, Patterning -Induced Ferromagnetism of \nFe3GeTe 2 van der Waals Materials beyond Room \nTemperature. Nano Lett. 18, 5974 (2018).  \n[82] C. Gong, L. Li, Z. Li, H. Ji, A. Stern, Y . Xia, T. Cao, \nW. Bao, C. Wang, Y . Wang, Z. Q. Qiu, R. J. Cava, S. G. \nLouie, J. Xia, and X. Zhang, Discovery of intrinsic \nferromagnetism in two -dimensional van der Waals crystals. \nNature 546, 265 (2017).  \n[83] B. Huang, G. Clark, E. Navarro -Moratalla, D. R. \nKlein, R. Cheng, K. L. Seyler, D . Zhong, E. Schmidgall, \nM. A. McGuire, D. H. Cobden, W. Yao, D. Xiao, P. Jarillo -\nHerrero, and X. Xu, Layer -dependent ferromagnetism in a \nvan der Waals crystal down to the monolayer limit. Nature \n546, 270 (2017).  \n[84] W. Zhang, J. Sklenar, B. Hsu, W. Jiang,  M. B. \nJungfleisch, J. Xiao, F. Y . Fradin, Y . Liu, J. E. Pearson, J. \nB. Ketterson, Z. Yang, and A. Hoffmann, Research Update: \nSpin transfer torques in permalloy on monolayer MoS 2. \nAPL Mater. 4, 032302 (2016).  \n[85] Q. Shao, G. Yu, Y . -W. Lan, Y . Shi, M. -Y . Li, C. Zheng, \nX. Zhu, L. -J. Li, P. K. Amiri, and K. L. Wang, Strong \nRashba -Edelstein Effect -Induced Spin –Orbit Torques in \nMonolayer Transition Metal Dichalcogenide/Ferromagnet \nBilayers. Nano Lett. 16, 7514 (2016).  \n[86] W. Lv, Z. Jia, B. Wang, Y . Lu, X. Luo,  B. Zhang, Z. \nZeng, and Z. Liu, Electric -Field Control of Spin –Orbit \nTorques in WS 2/Permalloy Bilayers. ACS Appl. Mater. \nInterfaces 10, 2843 (2018).  \n[87] L. A. Benítez, W. Savero Torres, J. F. Sierra, M. \nTimmermans, J. H. Garcia, S. Roche, M. V . Costache, and \nS. O. Valenzuela, Tunable room -temperature spin galvanic \nand spin Hall effects in van der Waals heterostructures. Nat. \nMater. 19, 170 (2020).  \n[88] S. Husain, X. Chen, R. Gupta, N. Behera, P. Kumar, \nT. Edvinsson, F. García -Sánchez, R. Brucas, S. Chaudha ry, \nB. Sanyal, P. Svedlindh, and A. Kumar, Large Damping -\nLike Spin –Orbit Torque in a 2D Conductive 1T -TaS 2 \nMonolayer. Nano Lett. 20, 6372 (2020).  [89] M. H. D. Guimarães, G. M. Stiehl, D. Macneill, N. D. \nReynolds, and D. C. Ralph, Spin –Orbit Torques in \nNbSe2/Permalloy Bilayers. Nano Lett. 18, 1311 (2018).  \n[90] P. Li, W. Wu, Y . Wen, C. Zhang, J. Zhang, S. Zhang, \nZ. Yu, S. A. Yang, A. Manchon, and X. -X. Zhang, Spin -\nmomentum locking and spin -orbit torques in magnetic \nnano -heterojunctions composed of Weyl semim etal WTe 2. \nNat. Commun. 9, 3990 (2018).  \n[91] S. Shi, S. Liang, Z. Zhu, K. Cai, S. D. Pollard, Y . \nWang, J. Wang, Q. Wang, P. He, J. Yu, G. Eda, G. Liang, \nand H. Yang, All -electric magnetization switching and \nDzyaloshinskii –Moriya interaction in WTe 2/ferroma gnet \nheterostructures. Nat. Nanotechnol. 14, 945 (2019).  \n[92] H. Xu, J. Wei, H. Zhou, J. Feng, T. Xu, H. Du, C. He, \nY . Huang, J. Zhang, Y . Liu, H. -C. Wu, C. Guo, X. Wang, Y . \nGuang, H. Wei, Y . Peng, W. Jiang, G. Yu, and X. Han, High \nSpin Hall Conductivity i n Large -Area Type -II Dirac \nSemimetal PtTe 2. Adv. Mater. 32, 2000513 (2020).  \n[93] S. Liang, S. Shi, C. -H. Hsu, K. Cai, Y . Wang, P. He, \nY . Wu, V . M. Pereira, and H. Yang, Spin -Orbit Torque \nMagnetization Switching in MoTe 2/Permalloy \nHeterostructures. Adv. Mat er. 32, 2002799 (2020).  \n[94] Y . Ou, W. Yanez, R. Xiao, M. Stanley, S. Ghosh, B. \nZheng, W. Jiang, Y . -S. Huang, T. Pillsbury, A. Richardella, \nC. Liu, T. Low, V . H. Crespi, K. A. Mkhoyan, and N. \nSamarth, ZrTe 2/CrTe 2: an epitaxial van der Waals platform \nfor sp intronics. Nat. Commun. 13, 2972 (2022).  \n[95] Y.-Y . Lv, X. Li, B. -B. Zhang, W. Y . Deng, S. -H. Yao, \nY . B. Chen, J. Zhou, S. -T. Zhang, M. -H. Lu, L. Zhang, M. \nTian, L. Sheng, and Y . -F. Chen, Experimental Observation \nof Anisotropic Adler -Bell-Jackiw Anomaly in  Type -II \nWeyl Semimetal WTe 1.98 Crystals at the Quasiclassical \nRegime. Phys. Rev. Lett. 118, 096603 (2017).  \n[96] D. MacNeill, G. M. Stiehl, M. H. D. Guimarães, N. D. \nReynolds, R. A. Buhrman, and D. C. Ralph, Thickness \ndependence of spin -orbit torques gener ated by WTe 2. Phys. \nRev. B 96, 054450 (2017).  \n[97] C.-W. Peng, W. -B. Liao, T. -Y . Chen, and C. -F. Pai, \nEfficient Spin -Orbit Torque Generation in Semiconducting \nWTe 2 with Hopping Transport. ACS Appl. Mater. \nInterfaces 13, 15950 (2021).  \n[98] X. Li, P. Li, V . D. H. Hou, M. Dc, C. -H. Nien, F. Xue, \nD. Yi, C. Bi, C. -M. Lee, S. -J. Lin, W. Tsai, Y . Suzuki, and \nS. X. Wang, Large and robust charge -to-spin conversion in \nsputter ed conductive WTe x with disorder. Matter 4, 1639 \n(2021).  \n[99] H. Xie, A. Talapatra, X. Chen, Z. Luo, and Y . Wu, \nLarge damping -like spin –orbit torque and perpendicular \nmagnetization switching in sputtered WTe x films. Appl. \nPhys. Lett. 118, 042401 (2021).  \n[100] Q. Liu and L. Zhu, Current -induced perpendicular \neffective magnetic field in magnetic heterostructures. Appl. \nPhys. Rev. 9, 041401 (2022).  \n[101]  P. Tsipas, D. Tsoutsou, S. Fragkos, R. Sant, C. \nAlvarez, H. Okuno, G. Renaud, R. Alcotte, T. Baron, and \nA. Dimoulas, Massless Dirac Fermions in ZrTe 2 Semimetal \nGrown on InAs(111) by van der Waals Epitaxy. ACS Nano \n12, 1696 (2018).  \n[102]  Y . Zhang, K. He, C. -Z. Chang, C. -L. Song, L. -L. \nWang, X. Chen, J. -F. Jia, Z. Fang, X. Dai, W. -Y . Shan, S. -\nQ. Shen, Q. Niu, X. -L. Qi, S. -C. Zhang, X. -C. Ma, and Q. -\nK. Xue, Crossover of the three -dimensional topological 16 \n insulator Bi 2Se3 to the two -dimensional limit. Nat. Phys. 6, \n584 (2010).  \n[103]  J. Han and L. Liu, Topological insulators for efficient \nspin–orbit torques. APL Mater . 9, 060901 (2021).  \n[104]  Q. L. He, T. L. Hughes, N. P. Armitage, Y . Tokura, \nand K. L. Wang, Topological spintronics and \nmagnetoelectronics. Nat. Mater. 21, 15 (2022).  \n[105]  H. Wu, P. Zhang, P. Deng, Q. Lan, Q. Pan, S. A. \nRazavi, X. Che, L. Huang, B. Dai, K. Wong, X. Han, and \nK. L. Wang, Room -Temperature Spin -Orbit Torque from \nTopological Surface States. Phys. Rev. Lett. 123, 207205 \n(2019).  \n[106]  N. H. D. Khang, S. Nakano, T. Shirokura, Y . \nMiyamoto, and P. N. Hai, Ultralow power spin –orbit \ntorque magne tization switching induced by a non -epitaxial \ntopological insulator on Si substrates. Sci. Rep. 10, 12185 \n(2020).  \n[107]  J. C. Y . Teo, L. Fu, and C. L. Kane, Surface states and \ntopological invariants in three -dimensional topological \ninsulators: Application to Bi 1-xSbx. Phys. Rev. B 78, \n045426 (2008).  \n[108]  X. Wang, J. Tang, X. Xia, C. He, J. Zhang, Y . Liu, C. \nWan, C. Fang, C. Guo, W. Yang, Y . Guang, X. Zhang, H. \nXu, J. Wei, M. Liao, X. Lu, J. Feng, X. Li, Y . Peng, H. Wei, \nR. Yang, D. Shi, X. Zhang, Z. Han, Z . Zhang, G. Zhang, G. \nYu, and X. Han, Current -driven magnetization switching in \na van der Waals ferromagnet Fe 3GeTe 2. Sci. Adv. 5, \neaaw8904 (2019).  \n[109]  M. Alghamdi, M. Lohmann, J. Li, P. R. Jothi, Q. Shao, \nM. Aldosary, T. Su, B. P. T. Fokwa, and J. Shi, Highly \nEfficient Spin –Orbit Torque and Switching of Layered \nFerromagnet Fe 3GeTe 2. Nano Lett. 19, 4400 (2019).  \n[110]  W. Zhu, C. Song, L. Han, H. Bai, Q. Wang, S. Yin, L. \nHuang, T. Chen, and F. Pan, Interface -Enhanced \nFerromagnetism with Long -Distance Effect  in van der \nWaals Semiconductor. Adv. Funct. Mater. 32, 2108953 \n(2022).  \n[111]  V . Gupta, T. M. Cham, G. M. Stiehl, A. Bose, J. A. \nMittelstaedt, K. Kang, S. Jiang, K. F. Mak, J. Shan, R. A. \nBuhrman, and D. C. Ralph, Manipulation of the van der \nWaals Magnet C r2Ge2Te6 by Spin –Orbit Torques. Nano \nLett. 20, 7482 (2020).  \n[112]  X. Liu, D. Wu, L. Liao, P. Chen, Y . Zhang, F. Xue, Q. \nYao, C. Song, K. L. Wang, and X. Kou, Temperature \ndependence of spin —orbit torque -driven magnetization \nswitching in in situ grown Bi 2Te3/MnTe heterostructures. \nAppl. Phys. Lett. 118, 112406 (2021).  \n[113]  M. Dc, R. Grassi, J. -Y . Chen, M. Jamali, D. \nReifsnyder Hickey, D. Zhang, Z. Zhao, H. Li, P. \nQuarterman, Y . Lv, M. Li, A. Manchon, K. A. Mkhoyan, T. \nLow, and J. -P. Wang, Room -temperature hi gh spin –orbit \ntorque due to quantum confinement in sputtered Bi xSe(1–x) \nfilms. Nat. Mater. 17, 800 (2018).  \n[114]  K. Kondou, R. Yoshimi, A. Tsukazaki, Y . Fukuma, J. \nMatsuno, K. S. Takahashi, M. Kawasaki, Y . Tokura, and Y . \nOtani, Fermi -level -dependent charge -to-spin current \nconversion by Dirac surface states of topological  insulators. \nNat. Phys. 12, 1027 (2016).  \n[115]  N. H. D. Khang, Y . Ueda, and P. N. Hai, A conductive \ntopological insulator with large spin Hall effect for \nultralow power spin –orbit torque swi tching. Nat. Mater. 17, \n808 (2018).  [116]  Z. Zheng, Y . Zhang, D. Zhu, K. Zhang, X. Feng, Y . \nHe, L. Chen, Z. Zhang, D. Liu, Y . Zhang, P. K. Amiri, and \nW. Zhao, Perpendicular magnetization switching by large \nspin–orbit torques from sputtered Bi 2Te3. Chin. Ph ys. B 29, \n078505 (2020).  \n[117]  N. D. Mermin and H. Wagner, Absence of \nFerromagnetism or Antiferromagnetism in One - or Two -\nDimensional Isotropic Heisenberg Models. Phys. Rev. Lett. \n17, 1133 (1966).  \n[118]  Y . Zhang, H. Xu, C. Yi, X. Wang, Y . Huang, J. Tang, \nJ. Jiang, C. He, M. Zhao, T. Ma, J. Dong, C. Guo, J. Feng, \nC. Wan, H. Wei, H. Du, Y . Shi, G. Yu, G. Zhang, and X. \nHan, Exchange bias and spin –orbit torque in the Fe 3GeTe 2-\nbased heterostructures prepared by vacuum exfoliation \napproach. Appl. Phys. Lett. 118,  262406 (2021).  \n[119]  R. Zhu, W. Zhang, W. Shen, P. K. J. Wong, Q. Wang, \nQ. Liang, Z. Tian, Y . Zhai, C. -w. Qiu, and A. T. S. Wee, \nExchange Bias in van der Waals CrCl 3/Fe 3GeTe 2 \nHeterostructures. Nano Lett. 20, 5030 (2020).  \n[120]  L. Zhang, X. Huang, H. Dai, M. Wang, H. Cheng, L. \nTong, Z. Li, X. Han, X. Wang, L. Ye, and J. Han, \nProximity -Coupling -Induced Significant Enhancement of \nCoercive Field and Curie Temperature in 2D van der Waals \nHeterostructures. Adv. Mater. 32, 2002032 (2020).  \n[121]  C. Kittel, Theory of Antiferromagnetic Resonance. \nPhys. Rev. 82, 565 (1951).  \n[122]  C. D. Stanciu, A. V . Kimel, F. Hansteen, A. \nTsukamoto, A. Itoh, A. Kirilyuk, and T. Rasing, Ultrafast \nspin dynamics across compensation points in ferrimagnetic \nGdFeCo: The role of angular mom entum compensation. \nPhys. Rev. B 73, 220402 (2006).  \n[123]  Y . Xu, M. Deb, G. Malinowski, M. Hehn, W. Zhao, \nand S. Mangin, Ultrafast Magnetization Manipulation \nUsing Single Femtosecond Light and Hot -Electron Pulses. \nAdv. Mater. 29, 1703474 (2017).  \n[124]  V . Baltz, A. Manchon, M. Tsoi, T. Moriyama, T. Ono, \nand Y . Tserkovnyak, Antiferromagnetic spintronics. Rev. \nMod. Phys. 90, 015005 (2018).  \n[125]  P. Qin, H. Yan, X. Wang, H. Chen, Z. Meng, J. Dong, \nM. Zhu, J. Cai, Z. Feng, X. Zhou, L. Liu, T. Zhang, Z. Zeng, \nJ. Zhang, C. Jiang, and Z. Liu, Room -temperature \nmagnetoresistance in an all -antiferromagnetic tunnel \njunction. Nature 613, 485 (2023).  \n[126]  X. Chen, T. Higo, K. Tanaka, T. Nomoto, H. Tsai, H. \nIdzuchi, M. Shiga, S. Sakamoto, R. Ando, H. Kosaki, T. \nMatsuo, D. Nishio -Hamane, R. Arita, S. Miwa, and S. \nNakatsuji, Octupole -driven magnetoresistance in an \nantiferromagnetic tunnel junction. Nature 613, 490 (2023).  \n[127]  Z. Zhang, J. Shang, C. Jiang, A. Rasmita, W. Gao, \nand T. Yu, Direct Photoluminescence Pro bing of \nFerromagnetism in Monolayer Two -Dimensional CrBr 3. \nNano Lett. 19, 3138 (2019).  \n[128]  W. Chen, Z. Sun, Z. Wang, L. Gu, X. Xu, S. Wu, and \nC. Gao, Direct observation of van der Waals stacking –\ndependent interlayer magnetism. Science 366, 983 (2019).  \n[129]  C. Chen, X. Chen, C. Wu, X. Wang, Y . Ping, X. Wei, \nX. Zhou, J. Lu, L. Zhu, J. Zhou, T. Zhai, J. Han, and H. Xu, \nAir-Stable 2D Cr 5Te8 Nanosheets with Thickness -Tunable \nFerromagnetism. Adv. Mater. 34, 2107512 (2022).  \n[130]  Z. Lin, B. Huang, K. Hwangbo, Q. Jiang, Q. Zhang, \nZ. Liu, Z. Fei, H. Lv, A. Millis, M. McGuire, D. Xiao, J. -\nH. Chu, and X. Xu, Magnetism and Its Structural Coupling 17 \n Effects in 2D Ising Ferromagnetic Insulator VI 3. Nano Lett. \n21, 9180 (2021).  \n[131]  L. Kang, C. Ye, X. Zhao, X. Z hou, J. Hu, Q. Li, D. \nLiu, C. M. Das, J. Yang, D. Hu, J. Chen, X. Cao, Y . Zhang, \nM. Xu, J. Di, D. Tian, P. Song, G. Kutty, Q. Zeng, Q. Fu, Y . \nDeng, J. Zhou, A. Ariando, F. Miao, G. Hong, Y . Huang, S. \nJ. Pennycook, K. -T. Yong, W. Ji, X. Renshaw Wang, and Z.  \nLiu, Phase -controllable growth of ultrathin 2D magnetic \nFeTe crystals. Nat. Commun. 11, 3729 (2020).  \n[132]  J. Seo, D. Y . Kim, E. S. An, K. Kim, G. -Y . Kim, S. -\nY . Hwang, D. W. Kim, B. G. Jang, H. Kim, G. Eom, S. Y . \nSeo, R. Stania, M. Muntwiler, J. Lee, K. W atanabe, T. \nTaniguchi, Y . J. Jo, J. Lee, B. I. Min, M. H. Jo, H. W. Yeom, \nS.-Y . Choi, J. H. Shim, and J. S. Kim, Nearly room \ntemperature ferromagnetism in a magnetic metal -rich van \nder Waals metal. Sci. Adv. 6, eaay8912 (2019).  \n[133]  A. F. May, D. Ovchinni kov, Q. Zheng, R. Hermann, \nS. Calder, B. Huang, Z. Fei, Y . Liu, X. Xu, and M. A. \nMcGuire, Ferromagnetism Near Room Temperature in the \nCleavable van der Waals Crystal Fe 5GeTe 2. ACS Nano 13, \n4436 (2019).  \n[134]  H. Wu, W. Zhang, L. Yang, J. Wang, J. Li, L. Li,  Y . \nGao, L. Zhang, J. Du, H. Shu, and H. Chang, Strong \nintrinsic room -temperature ferromagnetism in freestanding \nnon-van der Waals ultrathin 2D crystals. Nat. Commun. 12, \n5688 (2021).  \n[135]  D. C. Freitas, R. Weht, A. Sulpice, G. Remenyi, P. \nStrobel, F. Gay , J. Marcus, and M. Núñez -Regueiro, \nFerromagnetism in layered metastable 1T -CrTe 2. J. Phys. \nCondens. Matter 27, 176002 (2015).  \n[136]  X. Sun, W. Li, X. Wang, Q. Sui, T. Zhang, Z. Wang, \nL. Liu, D. Li, S. Feng, S. Zhong, H. Wang, V . Bouchiat, M. \nNunez Regueir o, N. Rougemaille, J. Coraux, A. Purbawati, \nA. Hadj -Azzem, Z. Wang, B. Dong, X. Wu, T. Yang, G. Yu, \nB. Wang, Z. Han, X. Han, and Z. Zhang, Room temperature \nferromagnetism in ultra -thin van der Waals crystals of 1T -\nCrTe 2. Nano Res. 13, 3358 (2020).  \n[137]  X. Zhang, Q. Lu, W. Liu, W. Niu, J. Sun, J. Cook, M. \nVaninger, P. F. Miceli, D. J. Singh, S. -W. Lian, T. -R. Chang, \nX. He, J. Du, L. He, R. Zhang, G. Bian, and Y . Xu, Room -\ntemperature intrinsic ferromagnetism in epitaxial CrTe 2 \nultrathin films. Nat. Commun. 1 2, 2492 (2021).  \n[138]  A. L. Coughlin, D. Xie, X. Zhan, Y . Yao, L. Deng, H. \nHewa -Walpitage, T. Bontke, C. -W. Chu, Y . Li, J. Wang, H. \nA. Fertig, and S. Zhang, Van der Waals Superstructure and \nTwisting in Self -Intercalated Magnet with Near Room -\nTemperature Pe rpendicular Ferromagnetism. Nano Lett. 21, \n9517 (2021).  \n[139]  Y . Wen, Z. Liu, Y . Zhang, C. Xia, B. Zhai, X. Zhang, \nG. Zhai, C. Shen, P. He, R. Cheng, L. Yin, Y . Yao, M. \nGetaye Sendeku, Z. Wang, X. Ye, C. Liu, C. Jiang, C. Shan, \nY . Long, and J. He, Tunable Room -Temperature \nFerromagnetism in Two -Dimensional Cr 2Te3. Nano Lett. \n20, 3130 (2020).  \n[140]  R. Chua, J. Zhou, X. Yu, W. Yu, J. Gou, R. Zhu, L. \nZhang, M. Liu, M. B. H. Breese, W. Chen, K. P. Loh, Y . P. \nFeng, M. Yang, Y . L. Huang, and A. T. S. Wee, Room \nTemperature Ferromagnetism of Monolayer Chromium \nTelluride with Perpendicular Magnetic Anisotropy. Adv. \nMater. 33, 2103360 (2021).  \n[141]  Y . Zhang, J. Chu, L. Yin, T. A. Shifa, Z. Cheng, R. \nCheng, F. Wang, Y . Wen, X. Zhan, Z. Wang, and J. He, Ultrathin Magneti c 2D Single -Crystal CrSe. Adv. Mater. \n31, 1900056 (2019).  \n[142]  G. Zhang, F. Guo, H. Wu, X. Wen, L. Yang, W. Jin, \nW. Zhang, and H. Chang, Above -room -temperature strong \nintrinsic ferromagnetism in 2D van der Waals Fe 3GaTe 2 \nwith large perpendicular magnetic anisotropy. Nat. \nCommun. 13, 5067 (2022).  \n[143]  V . Ostwal, T. Shen, and J. Appenzeller, Efficient \nSpin-Orbit Torque Switching of the Semiconducting Van \nDer Waals Ferromagnet Cr 2Ge2Te6. Adv. Mater. 32, \n1906021 (2020).  \n[144]  M. Mogi, K. Yasuda, R. Fujimura, R. Yoshimi, N. \nOgawa, A. Tsukazaki, M. Kawamura, K. S. Takahashi, M. \nKawasaki, and Y . Tokura, Current -induced switching of \nproximity -induced ferromagnetic surface states in a \ntopological insulator. Nat. Commun. 12, 1404 (2021).  \n[145]  R. Fujimura, R. Yoshim i, M. Mogi, A. Tsukazaki, \nM. Kawamura, K. S. Takahashi, M. Kawasaki, and Y . \nTokura, Current -induced magnetization switching at \ncharge -transferred interface between topological insulator \n(Bi,Sb) 2Te3 and van der Waals ferromagnet Fe 3GeTe 2. Appl. \nPhys. Lett. 119, 032402 (2021).  \n[146]  I. Shin, W. J. Cho, E. -S. An, S. Park, H. -W. Jeong, S. \nJang, W. J. Baek, S. Y . Park, D. -H. Yang, J. H. Seo, G. -Y . \nKim, M. N. Ali, S. -Y . Choi, H. -W. Lee, J. S. Kim, S. D. \nKim, and G. -H. Lee, Spin –Orbit Torque Switching in an \nAll-Van der Waals Heterostructure. Adv. Mater. 34, \n2101730 (2022).  \n[147]  D. R. Klein, D. MacNeill, Q. Song, D. T. Larson, S. \nFang, M. Xu, R. A. Ribeiro, P. C. Canfield, E. Kaxiras, R. \nComin, and P. Jarillo -Herrero, Enhancement of interlayer \nexchange in an ultrat hin two -dimensional magnet. Nat. \nPhys. 15, 1255 (2019).  \n[148]  D. J. O’Hara, T. Zhu, A. H. Trout, A. S. Ahmed, Y . K. \nLuo, C. H. Lee, M. R. Brenner, S. Rajan, J. A. Gupta, D. \nW. McComb, and R. K. Kawakami, Room Temperature \nIntrinsic Ferromagnetism in Epitaxi al Manganese Selenide \nFilms in the Monolayer Limit. Nano Lett. 18, 3125 (2018).  \n[149]  K.-S. Lee, S. -W. Lee, B. -C. Min, and K. -J. Lee, \nThreshold current for switching of a perpendicular \nmagnetic layer induced by spin Hall effect. Appl. Phys. \nLett. 102, 1124 10 (2013).  \n[150]  K.-S. Lee, S. -W. Lee, B. -C. Min, and K. -J. Lee, \nThermally activated switching of perpendicular magnet by \nspin-orbit spin torque. Appl. Phys. Lett. 104, 072413 \n(2014).  \n[151]  A. Thiaville, S. Rohart, É. Jué, V . Cros, and A. Fert, \nDynamics of  Dzyaloshinskii domain walls in ultrathin \nmagnetic films. Europhysics Letters 100, 57002 (2012).  \n[152]  D. Bhowmik, M. E. Nowakowski, L. You, O. Lee, D. \nKeating, M. Wong, J. Bokor, and S. Salahuddin, \nDeterministic Domain Wall Motion Orthogonal To Current \nFlow Due To Spin Orbit Torque. Sci. Rep. 5, 11823 (2015).  \n[153]  T.-Y . Chen, C. -T. Wu, H. -W. Yen, and C. -F. Pai, \nTunable spin -orbit torque in Cu -Ta binary alloy \nheterostructures. Phys. Rev. B 96, 104434 (2017).  \n[154]  L. Zhu, D. C. Ralph, and R. A. Buhrman, Lack of \nSimple Correlation between Switching Current Density \nand Spin -Orbit -Torque Efficiency of Perpendicularly \nMagnetized Spin -Current -Generator -Ferromagnet \nHeterostructures. Phys. Rev. Appl. 15, 024059 (2021).  \n "    },    {        "title": "1606.08334v2.Nonreciprocal_Transverse_Photonic_Spin_and_Magnetization_Induced_Electromagnetic_Spin_Orbit_Coupling.pdf",        "content": "1 \n NONRECIPROCAL TRANSVERSE PHOTONIC SPIN AND MAGNETIZATION -INDUCED \nELECTROMAGNETIC SPIN -ORBIT  COUPLING  \n  \nMiguel Levy and  Dolendra Karki  \nPhysics Department, Michigan Technological University  \nHenes Center for Quantum Phenomena  \n \nABSTRACT  \nA study of nonreciprocal transverse -spin a ngular -momentum -density shift s for evanescent waves \nin magneto -optic waveguide media is presented. Their functional relation to electromagnetic spin- \nand orbital -momenta is presented and analyzed. It is shown that the magneto -optic gyrotropy can \nbe re -interpreted as the nonreciprocal electromagnetic spin-density shift per unit energy flux , thus \nproviding an interesting alternative physical picture for the magneto -optic gyrotropy . The \ntransverse spin -density shift is fou nd to be thickness -dependent in slab optical waveguides. This \ndependence is traceable t o the admixture of  minority helicity component s in the transverse spin \nangular momentum.  It is also shown that the transverse spin is magnetically tunable . A formulation \nof electromagnetic spin-orbit coupling in magneto -optic media is presented,  and an alternative \nsource of spin -orbit coupling  to non -paraxial optics vortices is proposed. It is shown that  \nmagnetization -induced electromagnetic spin -orbit coupli ng is possible, and that it  leads to spin to \norbit al angular momentum conversion in magneto -optic media evanescent waves . \n \n \nINTRODUCTION  \nIn 1939, F. J. Beli nfante introduced a spin momentum density expression for vector fields to \nexplain the spin of quantum part icles and symmetrize the energy -momentum tensor [ 1]. For \nmonochromatic electromagnetic waves in free -space , the corresponding time-averaged  spin \nmomentum density reads  \n \n1\n2BBps \n ,         (Eq. 1)  \nand the  spin angular momentum density  is \n* 1Im ( )2Bos E E \n.       (Eq. 2 ) \n\nis the optical frequency and\n0 the permittivity of free -space  [2].  \nThis spin angular momentum , in its transverse electromagnetic form , has merited much \nattention in recent years, as  it can be studied in evanescent waves [ 3-7]. There are fundamental \nand practical reasons for this.  \nUntil recently, the quantum field theory of the electromagnetic field has lacked a description of \nseparate local conservation laws for the spin and orbital angular momentum -generating  currents \n[7]. Whether such spin -generating momenta , as opposed to the actual spin angular momenta they \ninduce,  are indeed observable or merely ‘virtual’ is of fundamental interest . Moreover, if the \nelectromagnetic spin and orbital momenta are separable,  the question arises as to  whether there are \nany photonic spin -orbit interaction  effects . Bliokh and co -workers give a positive answer for non -\nparaxial fields.  [7] Using the conservation laws proposed by these authors, we show that it is \npossible to magnetically induce electromagnetic spin -orbit coupling in magneto -optic media.  2 \n We know  that the transverse optical spin  is a physically meaningful  quantity that can be \ntransferred  to material particles  [3-8]. This has  potential ly appealing consequences for  optical -\ntweezer particle manipulation , or to locate  and track nanoparticles with  a high degree of te mporal \nand spatial resolution [9 ]. Thus,  developing means of control for  the transverse optical spin is of \npractical interest.  \nIn this paper, w e address the latter question for both  spin momenta  \nBp\n and angular momenta\nBs\n. We show that the ir magnitude s and sense of circulation can be accessed and controlled in a \nsingle structure , and propose a specific configuration to this end .  Explicit expressions for these \nphysical quantities and for the spin -orbit coupling are presented. Moreover, we develop our  \ntreatment for nonreciprocal slab optical waveguides , resulting in a different response upon time \nreversal s.  \nWe consider the behavior of evanescent waves in transverse -magnetic (TM) modes in magnetic \ngarnet claddings on silicon -on-insulator guides.  This allows us to obtain explicit expressions for \nthe transverse Beli nfante spin momenta and angular momenta and to propose a means for \nmagnetically controlling  these objects, with potential applicabilit y to nanoparticle manipulation.  \nMAGNETIC -GYROTROPY -DEPENDENT EVANESCENT WAVES  \nConsider  a silicon -on-insulator slab waveguide with iron garnet top cladding , as in Fig. 1 . The \noff-diagonal component s of the garnet’s dielectric permittivity tensor contr ol the structure’s \nmagneto -optic response.  Infrared 1550nm  wavelength  light propagates in the slab, in the presence \nof a magnetic field transverse to the dir ection of propagation . \nThe electromagnetic field-expressions in the top cladding  for transverse magnetization  (y-\ndirection) and monochromatic light propagating in the z -direction  are, \n()\n()( , )e\n( , )i z t\no\ni z t\noE E x y\nH H x y e\n\n\n\n        (Eq. 3)  \nMaxwell -Ampere’s and Faraday’s laws in ferrimagnetic media are  \n00\nˆ 0 0 0 0\n00cc\no o c o c\nccig ig\nEEH i Ettig ig\n      \n                  \n   (Eq. 4)  \nooHE i Ht    \n        (Eq. 5)  \nThe off -diagonal component  of the dielectric permittivity tensor\nˆ  is the gyrotropy parameter , \nparameterized by g.  \nWe examine  transverse -magnetic (TM)  propagation in the slab. Vertical and transverse -\nhorizontal directions are x, and y, respectively,  is the propagation constant, and the wave equation \nin the iron garnet is given by  3 \n \n22\n22\n20y o c y\ncgH k Hx      , with\n02k , for wavelength\n  [10, 11 ]. (Eq. 6)  \nDefining  \n2\neff c\ncg\n  as an effective permittivity in the cover  layer, we get:  \n,0effx\nycH H e x\n                                                 (Top cladding )  (Eq. 7)  \ncos( ), x 0y f x cH H k x d      \n                             (Core )   (Eq. 8)  \nexp( ( )),y s sH H x d x d   \n,                                (Substrate)   (Eq. 9)  \nwhere \n22\neff o eff k   \n ,         (Eq. 10)  \n22\nx o fkk\n,         (Eq. 11)  \n22\ns o s k   \n         (Eq. 12)  \nf\n, and\ns are the silicon -slab and substrate dielectric -permittivity  constants, respectively, and d is \nthe slab thickness.  \nSolving for the electric -field components in the top cladding layer, we get,  \n22\n0\n22\n0()\n()c eff\nzy\nc\nc eff\nxy\ncgE i Hg\ngEHg  \n \n \n \n\n        (Eq. 13 ) \nNotice that these two electric field components are \n/2  out of phase, hence the  polarization  is \nelliptical  in the cover laye r, with  optical spin transverse to the propagation direction.  In addition, \nthe polarization evinces opposite  helicities  for counter -propagating beam s, as\n/zxEE  changes sign \nupon  propagation direction reversal.  \nThis result already contains  an important difference with reciprocal  non-gyrotropic \nformulations, where\n//zxE E i  , and\n the decay constant in the top cladding. E quation 13  \ndepends on the gyrotropy parameter g, both explicitly and implicitly through\n , and is therefore \nmagnetically tunable, as we shall see below.  \nWe emphasize that the magnitude and sign of the propagation constant\n change  upon \npropagation direction reversal, and , separately,  upon magnetization direction reversal. The \ndifference between forward and backward propagation constants is also gyrotropy dependent. This \nnonreciprocal quality of magneto -optic waveguides is central to the proper functioning of certain \non-chip devices, such as Mach -Zehnder -based optical isolat ors [10, 11 ].  \nAs pointed out before, Eq. 2  applies to  free-space Maxwell electromagnetism. In a dielectric \nmedium, the momentum density expression must account for the electronic response to the optical 4 \n wave. Minkowski’s  and Abraham’s formulations describe  the canonical and the kinetic \nelectromagnetic momenta , respectively  [12]. Here we w ill focus on Minkowski’s version ,\np D B\n, as it is intimately linked  to the generation of translations in the host medium, and hence \nto optical phase shifts , of interest in nonreciprocal phenomena.  \nD\n is the dis placement vector, and\nB\nthe magnetic flux density.   \nDual -symmetric versions of electromagnetic field theory in free space have been considered  by \nvarious authors [ 2, 7, 8, 13]. However, t he interaction of light and matter at the local level often  \nhas an electric character. Dielectric probe particles will generally sense the electric part  of the \nelectromagnetic momentum and spin densities  [2, 7, 8, 13]. Hence, we treat the standard (electric -\nbiased) formulation of the electromagnetic spin and orbital angular momenta. In the presence of \ndielectric media , such as  iron garnets  in the near -infrared range , the expression  for spin angular \nmomentum become s \n*\n, Im ( )2o\nBMs E E\n\n.        (Eq. 14)   \nThe orbital momentum is  \n*Im ( ( ) )2Oop E E  \n, where        (Eq. 15)  \n()x x y y z z X Y X Y X Y X Y       \n, and\n is the relative dielectric permittivity of the medium [4, \na, D, E] .  \nIn magneto -optic media, the dielectric permittivity\n is\ncg , depending on the helicity of the \npropagating transverse circular polarization.  This is usually a small correction to\nc , as g is two -, \nor three -, orders of magnitude smaller in iron garnets , in the near infrared range . For elliptical \nspins, where one he licity component dominates, we account for the admixture level of the minority \ncomponent  in\nthrough a  weighted average . \nNONRECIPROCAL ELECTROMAGNETIC TRANSVERSE SPIN ANGULAR MOMENTUM  \nAND SPIN -ORBIT COUPLING  \n \n1. Transverse Spin Momentum and Angular M omentum Densities in Non -Reciprocal Media  \nIn thi s section we present a  formulation for the transverse -spin momentum and angular \nmomentum densities , as well as  the orbi tal angular momentum density, induced by  evanescent \nfields in nonreciprocal magneto -optic media . The magnitude and  tuning range  of these objects in \nterms of waveguide geometry  and optical gyrotropy are expounded  and discussed.  We detail t heir \nunequal response to given optical energy fluxe s in opposite propagation directions  and to changes \nin applied magnetic fields.  And we apply the recently proposed Bliokh -Dressel -Nori \nelectromagnetic  spin-orbit correction term to calculate the spin -orbit interaction for evanescent \nwaves in gyrotropic media [7].  \nEquation (13), together with Eq. (14) and Eq. (15) yield the following expressions for the \ntransv erse Belinfante -Minkowski spin angular momentum, spin momentum  and the orbital \nmomentum densities in evanescent nonreciprocal electromagnetic waves,  5 \n \n2\n, 3 2 2 2 2ˆc eff c eff\nB M y\no c cggs H y\ngg    \n         \n     (Eq. 16)  \n2\n, 3 2 2 2 2ˆeff c eff c eff\nB M y\no c cggp H z\ngg     \n         \n     (Eq. 17)  \n22\n2\n3 2 2 2 2ˆ\n2c eff c eff\nOy\no c cggp H z\ngg    \n                \n     (Eq.18 ) \nAnd the ratio  \n, 2c eff c eff O\nB M c eff c effgg p\ns g g    \n    \n      (Eq. 19)   \nThese  expression s depend  on the magneto -optic gyrotropy parameter g and the dielectric \npermittivity  of the waveguide core channel and of its cover layer under  transverse magnetization. \nThey yield different values under magnetic field tuning, magnetization and beam propagation \ndirection reversals, and as a function of waveguide core thickness as discussed below . The \npropagation constant\n is gyrotropy -, propagation -direction -, and waveguide -core-thickness -\ndependent, and this behavior strongly impacts the electromag netic spin and orbital momenta.  \nThe time -averaged electromagnetic energy flux (Poynting’s vector) in the iron garnet layer is  \n2* 1\n2 22\n01ˆ Re( )2 ()c eff\ny\ncgS E H H z\ng \n   \n\n.     (Eq. 20)  \nRe-expressing the transverse Belinfante -Minkowski spin angular momentum and spin \nmomentum densities in terms of the energy flow\nS\n , \n, 2 2 22ˆc eff\nBM\ncgs S y\ng  \n\n        (Eq. 21)  \n, 2 2 22eff c eff\nBM\ncgpS\ng   \n\n       (Eq. 22)  \nFigure 1 plots the nonreciprocal Belinfante -Minkowski transverse spin-angular -momentum -\ndensity shift per unit energy flux, as a function of silicon slab thickness  in an SOI  slab waveguide \nwith Ce 1Y2Fe5O12 garnet top cladding . Calculations are performed for the same electromagnetic \nenergy flux in opposite propagation directions, at a wavel ength of 1550 nm, \n0.0086 g . The \nnonreciprocal shift is  normalized to the average spin angular momentum,  as follows,   6 \n \n\n,,2f c eff b c efffb\nB M S\nf c eff b c efffbgg\ns\ngg       \n         \n  \n.    (Eq. 23)  \nSubscripts f and b stand for forward, and backward  propagation , respectively. This expression  \nevinces a relatively stable value , close to  0.7% above 0 .3m thickness. What  is the explanation for \nthis? It has to do with the ellipticity of the transverse polarization in the x -z plane.  Above 0.3m, \nthe ellipticity ranges from 31.4° to 36.9°, where  45° corresponds to circular polarization. In other \nwords, the ellipticity  stays fairly constants, with a moderately small  admixture of the minority \ncircularly polarized component , ranging  from  25% to 14%. Below 0.3m, the minority component \nadmixture increases precipitously, reaching 87% at 0.13m. \n Magnetization reversals produce the same effect. Consider the nonreciprocal Belinfante -\nMinkowski transverse spin -angular -momentum -density shif t, as a function of silicon slab \nthickness. Figure 2 plots the normalized shift in  Eq. 16 pre -factor,  \n \n2 2 2 2 2 2 2 2\n,\n2 2 2 2 2 21\n2c eff c eff c eff c eff\nc c c cgg\nBM\nc eff c eff c eff c eff\nc c c cgg g g g\ng g g g\ns\ng g g g\ng g g         \n   \n         \n                                     \n                  22\ngg              (Eq. 24)  \nWe observe the same qualitative thickness dependence as in Fig. 1 , corresponding to the \nmoderate, and relatively stable, admixture of minority circularly -polarized component above \n0.3m thickness.  7 \n  \nFig. 1.  Normalized nonreciprocal Belinfante -Minkowski transverse spin -angular -momentum -\ndensity shift per unit energy flux as a function of silicon slab thickness for\n0.0086 g , \ncorresponding to Ce 1Y2Fe5O12 garnet top cladding on SOI  at\n1.55 m wavelength . The inset \nshows the slab waveguide structure. M stands for the magnetization in the garnet.  \nThe magneto -optic gyrotropy of an iron  garnet can be controlled through an app lied magnetic field. \nThese ferri magnetic materials evince a hysteretic response, suc h as the one displayed in Fig. 3 \n(inset)  for 532nm wavelength in a sputter -deposited  film. The target composition is \nBi1.5Y1.5Fe5.0O12. Shown here are actual experimental data extracted from Faraday rotation \nmeasurements. Below saturation, the magneto -optic response exhibits an effective gyrotropy value \nthat can be tuned through th e applic ation of a  magnetic field. These measurements correspond to \na 0.5m-thick film on a (100) -oriented terbium gallium garnet (Tb GG) substrate. The optical beam \nis incident normal to the surface, and the hysteresis loop probes the degree of magnetization norma l \nto the surface as a function of applied magnetic field. These data show that the electromagnetic \nspin angular momentum can b e tuned below saturation and between opposite magnetization \ndirections.   \nFigure 3 also revea ls an interesting feature about  the magneto -optic gyrotropy.  The normalized \nnonreciprocal Belinfante -Minkowski transverse spin -angular -momentum -density shift per unit \nenergy flux, \n,,B M Ss\n , linearly tracks the gyrotropy, and  is of the same order of magnitude as  g, \nalthough  thickness -dependent . Yet, as pointed out before, this thickness dependence  reflects  the \nadmixture of the minor helicity component in the spin ellipticity.  At 0.4m, for example,\n,, 0.0072B M Ss\n when\n0.0086 g . However, the major polarization helicity component \ncontribution to \n,,B M Ss\n  is 84.4% at this thickness, translating into 0.00853 at 100%. At 0.25m,\n8 \n \n,, 0.00655B M Ss\n, and the major polarization helicity component contribution is 76.2%, \ntranslating into 0.0086 at 100%. We, thus,  re-interpret the  magneto -optic al gyrotropy as the \nnormalized Belinfante -Minkowski spin -angular -momentum density shift per unit energy flux. \n \nFig. 2 . Normalized nonreciprocal Belinfante -Minkowski transverse spin -angular -momentum -\ndensity pre-factor shift as a function of silicon slab thickness for\n0.0086 g , corresponding to \nCe1Y2Fe5O12 garnet top cladding on SOI at\n1.55 m wavelength.  \n \n9 \n Fig. 3. Normalized nonreciprocal Belinfante -Minkowski transverse spin -angular -momentum -\ndensity shift per unit energy flux as a function of magneto -optical gyrotropy. Data correspond to  \n0.25m silicon -slab thickness  with Ce1Y2Fe5O12 garnet top cladding , \n1.55 m wavelength.  The \ninset shows the gyrotropy  versus  magnetic field  hysteresis loop of  a magnetic garnet film at\n532nm\n, sputter -deposited using a Bi1.5Y1.5Fe5.0O12 target . \n2. Magnetization -Induced Electro magnetic Spin -Orbit Coupling  \nBliokh and co -authors have studied  the electromagnetic spin -orbit coupling in non -paraxial optical \nvortex beams  [7, 13]. They find that there is a spin dependent term in the orbital angular \nmomentum expression that leads to spin -to-orbit angular momentum conversion. This \nphenomenon occurs under tight focusing or the scattering of light [ 7, 13]. Here we consider an \nalternativ e source of electromagnetic spin -orbit coupling, magnetization -induced coupling in  \nevanescent waves.  \nThe time-averaged spin- and orbital -angular momenta co nservation laws put forth  in [7] are \n* * * * 1Im ( ) Im22o\nt i j ij i j j i E E H E H E H E      \n, and    (Eq. 25)  \n22* * * 11Im ( ) Im ( )2 2 4o\nt j jkl l i k j i ijk oiE r E H r E H E H E                 \n \n           (Eq. 26)  \nLatin indices i, j,… take on values x, y, z  and \nijk  is the Levi -Civita symbol.  Summation over \nrepeated indices is assumed.  \nThe interesting term in these equations, responsible for spin -orbit coupling, is\n*Im2o\njiHE\n . Notice \nthat it appears with opposite signs in the above equations , signaling a transfer of angular \nmomentum from spin to orbital motion.  As it stands, so far in our treatment , this term equals zero, \nsince the spin points in the y -direction and the electric -field components of the TM wave point in \nthe x -, and z -directions. A way to overcome this null coupling, and enable the angular momentum \ntransfer,  is to partially rotate the  applied magnetic field about the x -axis away from the y -direction , \nas in Fig. 4 . This action induces a Faraday rotation about the z -axis, generating a spin -orbit \ncoupling term in the angula r momentum  conservation laws.  A slight rotation or directional gradient \nin the magnetization  M will induce electromagnetic spin-orbit interaction in the magneto -optic \nmedium .  \nMaxwell -Ampere’s law acquires off -diagonal components \nig  in the dielectric permittivity \ntensor  upon rotation of the magnetic moment in the iron garnet film away from the y -axis, as shown \nin Eq. 27  [14]. Hence, non -zero electromagnetic field components \nyE  and\nxH , and spin -orbit \ncoupling, are induced in the propagating wave. The spatial, non -intrinsic, component, \ncharacteristic of orbital motion, emerges in the form of a z -dependence  in the angular momentum, 10 \n embodied in the partial or total evanescence  of the major circularly -polarized component as the \nwave propagates along th e guide.  \n \nˆ 0\n0c\no o c\nci g ig\nEH i g i Etig\n     \n   \n     (Eq. 27)  \nIn what sense is there an  angular momentum transfer from spin to orbital, in this case? As the \npolarization rotates in the x -y plane due to the Faraday Effect, there will be a spatially -dependent \nreduction in the circulating electric field component of the electromagnetic wave alo ng the \npropagation -direction.  This can be seen as a negative increase in circular polarization with z, i.e. , \nan orbital angular momentum in the opposite direction to the electromagn etic spin.  \n \nFig. 4. Rotated magnetization M generates TM to TE waveguide mode coupling and \nelectromagnetic spin -orbit coupling.  \n \nFinally, we derive an explicit expression for the spin -orbit coupling term. The relevant term  \nappearing in the orbital angular momentum flux in the z -direction is  \n* 0Im2z z yHE\n \n         Eq. (28)  \nWe assume t hat Faraday rotation induces  the \n,yzEH terms via TM to transverse -electric (TE) \nmode conversion, where  \n0zyiHEx\n,         Eq.  (29)  \nand  \n ,0 sinTE TEx i z\ny y FE E e e z\n.       Eq.  (30)  \n,0yE\nis the electric field amplitude corresponding to full TM to TE conversion, \nF is the specific \nFaraday rotation angle, \nTE  and \nTE are the cover -layer decay constant  and the propagation \nconstant for the TE mode, respectively. For simplicity, we assume no linear birefringence in the \nwavegu ide, so\nTE TM . Hence, the spin to orbital angular momentum coupling term is  \n 22 * 0\n,0 2Im sin 22 2TEx TE\nz z y y F FH E E e z     \n    Eq. (31)  \n11 \n  \n \nAcknowledgment  \nThe authors thank Ramy El -Ganainy for suggesting this problem  and for useful discussion s.  12 \n References  \n1. F. J. Beli nfante, “On the current and the density of electric charge, the energy, the linear \nmomentum and the angular momentum of arbitrary fields,” Physica 7, 449 -474 (1940).  \n2. K. Y. Bliokh, A. Y. Bekshaev, and F. Nori, “Extraordinary momentum and spin in \nevanescent waves,” Nature Communications 5, 4300  (2014).  \n3. M. Neuberger an co -workers, “Experimental demonstration  of the geometric so  in Hall \neffect of light in highly focused vector beams,” Conference on Lasers and Electro -Optics,” \nQW1E.A (OSA 2012).  \n4. P. Banzer a co -workers, “The photonic wheel demonstration of a state of light with purely \ntransverse momentum,” J. Eur. Opt. Soc. Rap. Publ. 8, 13032 (2013).  \n5. K. Y. Bliokh and F. Nori, “Transverse spin of a surface polariton,” Phys. Rev. A 85, 061801 \n(2012) . \n6. K. –Y. Kim, I. –M. Lee, J. Kim, J. Jung, and B. Lee, “Time reversal and the spin angular \nmomentum  of tra nsverse electric and transverse -magnetic surface modes,” Phys. Rev. A \n86, 063805 (2012).   \n7. K. Y. Bliokh, J. Dressel and F. Nori, “Conservation of the s pin and orbital angular momenta \nin electromagnetism,” New Journal of Physics 16, 093037 (2014).  \n8. A. Aiello, P. Banzer, M. Neugebauer and G. Leuchs, “From transverse angular momentum \nto photonic wheels,” Nature Photonics 9, 789 -795 (2015).  \n9. M.E.J. Friese, J. Enger, H. Rubinsztein -Dunlop, and N. R. Heckenberg, “Optical angular -\nmomentum transfer to trapped absorbing particles, Phys. Rev. A 54 1593 -1596 (1996).  \n10. J. Fujita, M. Levy, R. M. Osgood, Jr. , L. Wilkens, and H. Dötsch, “Waveguide Optical \nIsolator Based on Mach -Zehnder Interferometer,” Appl. Phys. Lett. 76, 2158 (2000).  \n11. B. J. Stadler and T. Mizumoto, “Integrated magneto -optical materials and isolators: a \nreview,” IEEE Photonics Journal 6, 0600215 (2014).  \n12. S. M. Barnett, “ Resolution of the Ab raham -Minkowski Dilemma ,” Phys. Rev. Lett. 104, \n070401 (2010).  \n13. Y. Bliokh, A. Y. Bekshaev, and F. Nori, “Dual electromagnetism: helicity, spin, \nmomentum and angular momentum,” New Journal of Physics 15, 033026 (2013).  \n14. D. C. Hutchings, B. M. Holmes, C. Zhang, P. Dulal, A. D. Block, S. -Y. Sung, N. C. A. \nSeaton, B. J. H. Stadler, “Quasi -phase -matched Faraday rotation in semiconductor \nwaveguides with a magneto -optic cledding for monolithically integrated optical isolators,”  \nIEEE Photonics Journal 5, 6602512 (2013).  \n "    },    {        "title": "1201.4842v2.Strong_Enhancement_of_Rashba_spin_orbit_coupling_with_increasing_anisotropy_in_the_Fock_Darwin_states_of_a_quantum_dot.pdf",        "content": "arXiv:1201.4842v2  [cond-mat.mes-hall]  24 Jan 2012Strong Enhancement of Rashba spin-orbit coupling with incr easing anisotropy in the\nFock-Darwin states of a quantum dot\nSiranush Avetisyan,1Pekka Pietil¨ ainen,2and Tapash Chakraborty‡1\n1Department of Physics and Astronomy, University of Manitob a, Winnipeg, Canada R3T 2N2\n2Department of Physics/Theoretical Physics, University of Oulu, Oulu FIN-90014, Finland\nWe have investigated the electronic properties of elliptic al quantum dots in a perpendicular ex-\nternal magnetic field, and in the presence of the Rashba spin- orbit interaction. Our work indicates\nthat the Fock-Darwin spectra display strong signature of Ra shba spin-orbit coupling even for a low\nmagnetic field, as the anisotropy of the quantumdot is increa sed. An explanation of this pronounced\neffect with respect to the anisotropy is presented. The stron g spin-orbit coupling effect manifests\nitself prominently in the corresponding dipole-allowed op tical transitions, and hence is susceptible\nto direct experimental observation.\nIn recent years our interest in understanding the\nunique effects of the spin-orbit interaction (SOI) in semi-\nconductor nanostructures [2] has peaked, largely due to\nthe prospect of the possible realization of coherent spin\nmanipulation in spintronic devices [3], where the SOI is\ndestined to play a crucial role [4]. As the SOI couples\nthe orbital motion of the charge carriers with their spin\nstate, an all-electrical control of spin states in nanoscale\nsemiconductor devices could thus be a reality. In this\ncontext the Rashba SOI [5] has received particular at-\ntention, largelybecauseinatwo-dimensionalelectrongas\nthestrengthoftheRashbaSOIhasalreadybeenshownto\nbe tuned by the application of an electric field [6]. While\nthe earlier studies were primarily in a two-dimensional\nelectron gas, the attention has now been focused on the\nrole of SOI in a single InAS quantum dot [7]. The quan-\ntum dot (QD) [8], a system of few electrons confined in\nthe nanometer region has the main advantage that the\nshape and size of the confinement can be externally con-\ntrolled, which provides an unique opportunity to study\nthe atomic-likepropertiesofthesesystems[8,9]. SOcou-\nplinginquantumdotsgeneratesanisotropicspinsplitting\n[10] which provides important information about the SO\ncoupling strength.\nExtensivetheoreticalstudiesofthe influenceofRashba\nSOI in circularly symmetric parabolic confinement have\nalready been reported earlier [11], where the SO cou-\npling wasfound to manifest itself mainly in multiple level\ncrossings and level repulsions. They were attributed to\nan interplay between the Zeeman and the SOI present\nin the system Hamiltonian. Those effects, in particular,\nthe level repulsions were however weak and as a result,\nwouldrequireextraordinaryefforts todetect the strength\nof SO coupling [12] in those systems. Here we show that,\nby introducing anisotropy in the QD, i.e., by breaking\nthe circular symmetry of the dot, we can generate a ma-\njor enhancement of the Rashba SO coupling effects in a\nquantum dot. As shown below, this can be observed di-\nrectly in the Fock-Darwin states of a QD, and therefore\nshould be experimentally observable [8, 9]. We show be-\nlow that the Rashba SO coupling effects are manifestlystrongin an elliptical QD [13], which should providea di-\nrect route to unambigiously determine (and control) the\nSO coupling strength. It has been proposed recently that\nthe anisotropy of a quantum dot can also be tuned by an\nin-plane magnetic field [14].\nThe Fock-Darwin energy levels in elliptical QDs sub-\njected to a magnetic field was first reported almost two\ndecadesago[13], whereit wasfound that the majoreffect\nof anisotropy was to lift the degeneracies of the single-\nparticle spectrum [15]. The starting point of our present\nstudy is the stationary Hamiltonian\nHS=1\n2m∗/parenleftBig\np−e\ncAS/parenrightBig2\n+Vconf(x,y)+HSO+Hz\n=H0+HSO+Hz\nwhere the confinement potential is chosen to be of the\nform\nVconf=1\n2m∗/parenleftbig\nω2\nxx2+ω2\nyy2/parenrightbig\n,\nHSO=α\n/planckover2pi1/bracketleftbig\nσ×/parenleftbig\np−e\ncAS/parenrightbig/bracketrightbig\nzis the Rashba SOI, and Hz\nisthe Zeemancontribution. Here m∗is the effective mass\nof the electron, σare the Pauli matrices, and we choose\nthe symmetric gauge vector potential AS=1\n2(−y,x,0).\nAs in Ref. [13], we introduce the rotated coordinates and\nmomenta\nx=q1cosχ−χ2p2sinχ,\ny=q2cosχ−χ2p1sinχ,\npx=p1cosχ+χ1q2sinχ,\npy=p2cosχ+χ1q1sinχ,\nwhere\nχ1=−/bracketleftbig1\n2/parenleftbig\nΩ2\n1+Ω2\n2/parenrightbig/bracketrightbig1\n2, χ2=χ−1\n1,\ntan2χ=m∗ωc/bracketleftbig\n2/parenleftbig\nΩ2\n1+Ω2\n2/parenrightbig/bracketrightbig1\n2//parenleftbig\nΩ2\n1−Ω2\n2/parenrightbig\n,\nΩ2\n1,2=m∗2/parenleftbig\nω2\nx,y+1\n4ω2\nc/parenrightbig\n, ωc=eB/m∗c.\nIn terms of the rotated operators introduced above, the\nHamiltonian H0is diagonal [13]\nH0=1\n2m∗/summationdisplay\nν=1,2/bracketleftbig\nβ2\nνp2\nν+γ2\nνq2\nν/bracketrightbig\n,2\n/s48/s52/s56/s49/s50/s49/s54\n/s32\n/s32/s32/s69 /s32/s40/s109 /s101/s86 /s41\n/s40/s97/s41\n/s52/s56/s49/s50/s49/s54/s50/s48\n/s32/s32\n/s32/s40/s99/s41\n/s48/s52/s56/s49/s50/s49/s54\n/s32/s32/s69 /s32/s40/s109 /s101/s86 /s41\n/s40/s98/s41\n/s48 /s49 /s50 /s51 /s52/s52/s56/s49/s50/s49/s54/s50/s48\n/s66/s32/s40/s84/s41\n/s32/s32\n/s66/s32/s40/s84/s41/s49 /s50/s51 /s52 /s48/s40/s100/s41\nFIG. 1: Magnetic field dependence of the low-lying Fock-\nDarwin energy levels of an elliptical dot without the Rashba\nSO interaction ( α= 0). The results are for (a) ωx= 4 meV\nandωy= 4.1 meV, (b) ωx= 4 meV and ωy= 6 meV, (c)\nωx= 4 meV and ωy= 8 meV, and (d) ωx= 4 meV and\nωy= 10 meV.\nwhere\nβ2\n1=Ω2\n1+3Ω2\n2+Ω2\n3\n2(Ω2\n1+Ω2\n2), γ2\n1=1\n4/parenleftbig\n3Ω2\n1+Ω2\n2+Ω2\n3/parenrightbig\n,\nβ2\n2=3Ω2\n1+Ω2\n2−Ω2\n3\n2(Ω2\n1+Ω2\n2), γ2\n2=1\n4/parenleftbig\nΩ2\n1+3Ω2\n2−Ω2\n3/parenrightbig\n,\nΩ2\n3=/bracketleftBig/parenleftbig\nΩ2\n1−Ω2\n2/parenrightbig2+2m∗2ω2\nc/parenleftbig\nΩ2\n1+Ω2\n2/parenrightbig/bracketrightBig1\n2.\nSince the operator H0is obviously equivalent to the\nHamiltonianoftwoindependentharmonicoscillators,the\nstates of the electron can be described by the state vec-\ntors|n1,n2;sz/angbracketright. Here the oscillator quantum numbers\nni= 0,1,2,...correspond to the orbital motion and\nsz=±1\n2to the spin orientation of the electron.\nThe Rashba Hamiltonian, in terms of the rotated op-\nerators is now written as,\n/planckover2pi1\nαHSO=σx(sinχχ1−cosχω0)q1\n−σy(sinχχ1+cosχω0)q2\n−σy(cosχ−sinχω0χ2)p1\n+σx(cosχ+sinχω0χ2)p2,\nwhereω0=eB/2c. The effect of the SO coupling is\nreadily handled by resortingto the standard ladder oper-\nator formalism of harmonic oscillators and by diagonal-\nizingHSOin the complete basis formed by the vectors\n|n1,n2;sz/angbracketright.\nThe Fock-Darwin states in the absence of the Rashba\nSOI (α= 0) are shown in Fig. 1, for ωx= 4 meV and/s48/s52/s56/s49/s50/s49/s54/s32\n/s69 /s32/s40/s109 /s101/s86 /s41\n/s32\n/s32/s32\n/s40/s97/s41\n/s52/s56/s49/s50/s49/s54/s50/s48\n/s32/s32/s32\n/s50/s40/s100/s41\n/s48/s52/s56/s49/s50/s49/s54\n/s32/s32/s69 /s32/s40/s109 /s101/s86 /s41\n/s40/s98/s41/s52/s56/s49/s50/s49/s54/s50/s48\n/s66/s32/s40/s84/s41\n/s66/s32/s40/s84/s41\n/s32/s32\n/s40/s99/s41\n/s48 /s49 /s50 /s51 /s52/s48 /s49 /s51 /s52\nFIG. 2: Same as in Fig. 1, but for α= 20.\nωy= 4.1,6,8,10 meV in (a)-(d) respectively. We have\nconsidered the parameters of an InAs QD [11] through-\nout, because in such a narrow-gap semiconductor sys-\ntem, the dominant source of the SO interaction is the\nstructural inversion asymmetry [16], which leads to the\nRashbaSO interaction. As expected, breakingof circular\nsymmetry in the dot results in lifting of degeneracies at\nB= 0, that is otherwise present in a circular dot [13, 15].\nIn Fig. 1 (a), the QD is very close to being circularly\nsymmetric, and as a consequence, the splittings of the\nzero-field levels are vanishingly small. As the anisotropy\nof the QD is increased [(b) – (d)], splitting of the levels\nbecomes more appreciable.\nAs the SO term is linear in the position and momen-\ntum operators it is also linear in the raising and lowering\nladder operators. It is also off-diagonal in the quantum\nnumber sz. As a consequence, the SOI can mix only\nstates which differ in the spin orientation, and differ by 1\neither in the quntum number n1or inn2but not in both.\nIn the case of rotationally symmetric confinements these\nselection rules translate to the conservation of the total\nangular momentum j=m+szin the planar motion of\nthe electron.\nAt the field B= 0 the ground states |0,0;±1\n2/angbracketrightare two-\nfold degenerate. Due to the selection rules, this degener-\nacy cannot be lifted either by the eccentricity of the dot\nor by the Rashba coupling. Many of the excited states,\nsuch as |n1,n2;±1\n2/angbracketrightretain their degeneracy no matter\nhow strong the SO coupling is or how eccentric the dot\nis, as we can see in the Figs. 1-3. At the same time, many\nother degeneracies are removed by squeezing or streching\nthe dot. At non-zero magnetic fields some of the cross-\nings of the energy spectra are turned to anti-crossings by\nthe Rashba term in the Hamiltonian. For example, the3\n/s48/s52/s56/s49/s50/s49/s54/s69 /s32/s40/s109 /s101/s86 /s41\n/s32/s32\n/s32/s69 /s32/s40/s109 /s101/s86 /s41\n/s40/s97/s41\n/s48 /s49 /s50 /s51 /s52/s48/s52/s56/s49/s50/s49/s54\n/s32/s32\n/s40/s98/s41\n/s48 /s49 /s50 /s51 /s52/s52/s56/s49/s50/s49/s54/s50/s48\n/s32/s32\n/s40/s100/s41/s52/s56/s49/s50/s49/s54/s50/s48\n/s66/s32/s40/s84/s41/s66/s32/s40/s84/s41\n/s32/s32\n/s40/s99/s41\nFIG. 3: Same as in Fig. 1, but for α= 40.\nsecond and third excited states in Fig. 2 (a) – Fig. 2 (d)\nare composed mainly of the states |0,0;1\n2/angbracketrightand|1,0;−1\n2/angbracketright\nwhich are mixed by the HSOaroundB= 3T causing a\nlevel repulsion. We can also see that the squeezing of the\ndot enhances the SO coupling. This can be thought of as\na consequence of pushing some states out of the way, just\nas in our example of the state |1,1;1\n2/angbracketright. SOI mixes it with\nthe state |1,0;−1\n2/angbracketrightcausing the latter state to shift down-\nward in energy thereby reducing the anti-crossing gap.\nSqueezing the dot, however moves the state energetically\nfarther away from |1,0;−1\n2/angbracketrightand so weakens this gap re-\nduction effect. It is abundantly clear from the features\nrevealed in the energy spectra that for a combination of\nstrong anisotropy of the dot and higher values of the SO\ncoupling strength, large anti-crossing gaps would appear\neven for relatively low magnetic fields.\nThe effects of anisotropy and spin-orbit interaction on\nthe energy spectra above are also reflected in the optical\nabsorption spectra. Let us turn our attention on the\nabsorption spectra for transitions from the ground state\nto the excited states. For that purpose we subject the\ndot to the radiation field\nAR=A0ˆǫ/parenleftBig\nei(ω/c)ˆn·r−iωt+e−i(ω/c)ˆn·r+iωt/parenrightBig\n,\nwhereˆǫ,ωandˆnare the polarization, frequency and\nthe direction of propagation of the incident light, respec-\ntively. We let the radiation enter the dot along the direc-\ntion perpendicular to the motion of the electron, that is\nparallel to the z-axis. Due to the transversalitycondition\nthe polarization vector will then lie in the xy-plane.\nAs usual, we shall make two approximations. First we\nassume the intensity of the field be so weak that only the\nterms linear in ARhas to be taken into account. Then\nthe effect of the radiative magnetic field on the spin can/s48/s51/s54/s57/s69/s32/s40/s109/s101/s86/s41 /s69/s32/s40/s109/s101/s86/s41\n/s32/s32/s69/s32/s40/s109/s101/s86/s41\n/s40/s97/s41\n/s48/s51/s54/s57/s49/s50\n/s32/s32/s32\n/s40/s98/s41\n/s48 /s49 /s50/s48/s51/s54/s57/s49/s50\n/s40/s99/s41/s48/s51/s54/s57\n/s32/s32\n/s40/s100/s41\n/s48/s51/s54/s57/s49/s50\n/s32/s32\n/s40/s101/s41\n/s48 /s49 /s50/s48/s51/s54/s57/s49/s50\n/s66/s40/s84/s41\n/s32/s32\n/s66/s40/s84/s41/s40/s102/s41\nFIG.4: Opticalabsorption(dipoleallowed) spectaofellip tical\nQDs for various choice of parameters: (a) i α= 0,ωx= 4\nmeV,ωy= 6, (b) α= 20,ωx= 4 meV, ωy= 8 meV, and\n(c)α= 40,ωx= 4,ωy= 6. The polarization of the incident\nradiation is along the x-axis. The parameters for (d)-(f) are\nthe same, except that the incident radiation is polarized al ong\nthey-axis. The areas of the filled circles are proportional to\nthe calculated absorption cross-section.\nbe neglected as well. So we can simply replace in the\nstationary Hamiltonian HSthe vector potential ASwith\nthe field A=AS+AR. Discarding terms higher than\nlinear order in ARleads to the total Hamiltonian\nH=HS+HR,\nwhere the radiative part HRis given by\nHR=−e\nmecAR·/parenleftBig\np−e\ncAS/parenrightBig\n−αe\n/planckover2pi1c[σ×AR]z.\nThe radiative Hamiltonian, even in the presence of the\nRashba SO coupling can be expressed in the well-known\nform\nHR= ie\nc/planckover2pi1AR·[x,HS],\nxbeing the position operator in the xy-plane.\nOur second approximation is the familiar dipole ap-\nproximation. We assume that the amplitude of radiation\ncan be taken as constantwithin the quantum dot, so that\nwe are allowed to write the field as\nAR≈A0ˆǫ/parenleftbig\ne−iωt+eiωt/parenrightbig\n.4\nSince the transition energies expressed in terms of radia-\ntion frequences are of the order of THz, the correspond-\ning wavelengths are much larger than the typical size of\na dot, thus justifying our approximation. Applying now\nthe Fermi Golden Rule leads to the dipole approximation\nform\nσabs(ω) = 4π2αfωni|/angbracketleftn|ˆǫ·x|i/angbracketright|2δ(ωni−ω)\nof the absorption cross section for transitions from the\ninital state |i/angbracketrightto the final state |n/angbracketright. Hereαfis the fine\nstructure constant and ωniis the frequency correspond-\ning to the transition energy /planckover2pi1ω.\nThe familiar dipole selection rules for oscillator states\ndictate largely the features seen in Fig. 4. In the absence\nof the SOI, these rules – the spin state is preserved and\neithern1orn2is changed by unity – completely deter-\nmine the allowed two transitions/vextendsingle/vextendsingle0,0;−1\n2/angbracketrightbig\n→/vextendsingle/vextendsingle1,0;−1\n2/angbracketrightbig\nand/vextendsingle/vextendsingle0,0;−1\n2/angbracketrightbig\n→/vextendsingle/vextendsingle0,1;−1\n2/angbracketrightbig\n. In contrast to the case of\ncircular dots the absorption in the elliptical dot depends\nstrongly on the polarization. This is explained by noting\nthat the oscillator strengths\nfni=2m∗ωni\n/planckover2pi1|/angbracketleftn|ˆǫ·x|i/angbracketright|2.\nactually probe the occupations of quantum states related\nto oscillations in the direction of the polarization ˆǫ. In a\ncircular dot all oscillation directions are equally probable\nat all energies implying that the oscillator strengths are\nindependent of the polarization and depend only slightly\non the transitionenergyvia ωni, and the final state quan-\ntum numbers n1,2. When the dot is squeezed in the y-\ndirection,say,theoscillatorstatesrelatedtothe y-motion\nare pushed up in energy. This means that the polariza-\ntion being along x-axis most of the oscillator strength\ncomes from transitions to allowed states with lowest en-\nergies. Similarly, when the incident radiation is polarized\nalong the y-axis most of the contribution is due to the\ntransitions to the oscillator states pushed up in the en-\nergy. Inellipticaldotstheoscillatorstatesarenotpure x-\nandy-oscillators but their superpositions. Therefore in\naddition to the main absorption lines, other allowed final\nstates have also non-vanishing oscillator strength. Fur-\nthermore, as one can see by looking at the phase space\nrotation formulas the external magnetic field tends to ro-\ntate directions of the oscillator motion causing a shift of\nthe oscillator strength from an allowed transition to an-\nother. This is exactly what we see in Fig. 4 (a) and Fig. 4\n(d).\nEven in the presence of the SOI the two allowed final\noscillator states provide major contributions to the cor-\nresponding corrected states. Hence we still see two dom-\ninant absorption lines. However, now many forbidden\ntransitions have become allowed. The lowest absorption\nlinecorrespondingtothetransitionbetweenZeemansplit\nstates with main components/vextendsingle/vextendsingle0,0;−1\n2/angbracketrightbig\nand/vextendsingle/vextendsingle0,0;1\n2/angbracketrightbig\npro-\nvides a typical example. The transition involves a spinflip and is therefore strongly forbidden without the SOI.\nBecause the SOI mixes the state/vextendsingle/vextendsingle1,0;1\n2/angbracketrightbig\ninto the for-\nmer one and the/vextendsingle/vextendsingle0,1;−1\n2/angbracketrightbig\ninto the latter one, the tran-\nsition becomes possible. The appearance of other new\nlines can be explained by analogous arguments. There\nare also additional features involving discontinuities and\nanti-crossings in Fig. 4. A comparision with the energy\nspectra indicates that these are the consequences of the\nanti-crossings present in the energy spectra.\nIt is also readily verified that the oscillator strengths\nsatisfy the Thomas-Reiche-Kuhn sum rule [17]\n/summationdisplay\nnfni= 1.\nIn terms of the cross section this translates to the condi-\ntion\n/integraldisplay∞\n−∞σabs(ω)dω=2π2/planckover2pi1αf\nm∗.\nThe absorptions visible in Fig. 4 practically saturate the\nsum rule, the saturation being, of course complete in the\nabsence of the SOI in panels (a) and (d). The largest\nfraction (of the order of 1/10) of the cross section either\nfalling outside of the displayed energy scale or having too\nlow intensity to be discernible in our pictures is found at\nthe strongest Rashba coupling in the panels (c) and (f)\nfor large magnetic fields, as expected.\nThe results presented here clearly indicate that, the\nanisotropyofaQDalonecausesliftingofthedegeneracies\noftheFock-DarwinlevelsatB=0,asreportedearlier[13].\nHowever, for large SO coupling strengths α, the effects\nof the Rashba SOI, mainly the level repulsions at finite\nmagnetic fields, are maginified rather significantly as one\nintroduces anisotropy in the QD. This is reflected also\nin the corresponding dipole-allowed optical transitions\nwhere the distinct anti-crossingbehavioris observedthat\nis a direct manifestation of the anti-crossings in the en-\nergy spectra. This prominent effect of the Rashba SOI\npredicted here could be confirmed experimentally in op-\ntical spectroscopy and the Fock-darwin spectra of few-\nelectron QDs [9, 18, 19]. It would also provide a very\nuseful step to control the SO coupling in nanostructures,\nen route to semiconductor spintronics [3].\nThe work was supported by the Canada Research\nChairs Program of the Government of Canada.\n[‡] Electronic address: tapash@physics.umanitoba.ca\n[2] Y. Oreg, P.W. Brouwer, X.Waintal, andB.I. Halperin, in\nNano-Physics & Bio-Electronics: A New Odyssey , edited\nby, T. Chakraborty, F. Peeters, and U. Sivan (Elsevier,\nAmsterdam, 2002).\n[3] For recent comprehensive reviews, see, T. Dietl, D.D.\nAwschalom, M. Kaminska, and H. Ono, (Eds.) Spintron-\nics(Elsevier, Amsterdam, 2008); I. Zutic, J. Fabian, and5\nS.DasSarma, Rev.Mod.Phys. 76, 323(2004); J. Fabian,\nA. Matos-Abiague, C. Ertler, P. Stano, and I. Zutic, Acta\nPhysica Slovaca 57, 565 (2007); M.W. Wu, J.H. Jiang,\nand M.Q. Weng, Phys. Rep. 493, 61 (2010).\n[4] H.-A. Engel, B.I. Halperin, and E.I. Rashba, Phys. Rev.\nLett.95, 166605 (2005).\n[5] Y.A. Bychkov and E.I. Rashba, J. Phys. C 17, 6039\n(1984).\n[6] J. Nitta, T. Akazaki, H. Takayanagi, and T. Enoki, Phys.\nRev. Lett. 78, 1335 (1997); M. Studer, G. Salis, K. En-\nsslin, D.C. Driscoll, and A.C. Gossard, ibid.103, 027201\n(2009); D. Grundler, ibid.84, 6074 (2000).\n[7] H. Sanada, T. Sogawa, H. Gotoh, K. Onomitshu, M. Ko-\nhda, J. Nitta, and P.V. Santos, Phys. Rev. Lett. 106,\n216602 (2011); S. Takahashi, R.S. Deacon, K. Yoshida,\nA. Oiwa, K. Shibata, K. Hirakawa, Y. Tokura, and S.\nTarucha, ibid.104, 246801 (2010).\n[8] T. Chakraborty, Quantum Dots (North-Holland, Am-\nsterdam, 1999); T. Chakraborty, Comments Condens.\nMatter Phys. 16, 35 (1992); P.A. Maksym and T.\nChakraborty, Phys. Rev. Lett. 65, 108 (1990).\n[9] D. Heitmann (Ed.), Quantum Materials (Springer, Hei-\ndelberg, 2010).\n[10] J. K¨ onemann, R.J. Haug, D.K. Maude, V. Fal’ko, and\nB.L. Altshuler, Phys. Rev. Lett. 94, 226404 (2005).\n[11] T. Chakraborty and P. Pietil¨ ainen, Phys. Rev. Lett.\n95, 136603 (2005); P. Pietil¨ ainen and T. Chakraborty,\nPhys. Rev. B 73, 155315 (2006); T. Chakraborty and P.\nPietil¨ ainen, ibid.71, 113305 (2005); A. Manaselyan andT. Chakraborty, Europhys. Lett. 88, 17003 (2009); and\nthe references therein.\n[12] H.-Y. Chen, V. Apalkov, andT. Chakraborty, Phys. Rev.\nB75, 193303 (2007).\n[13] A.V.MadhavandT. Chakraborty, Phys.Rev.B 49, 8163\n(1994); See also, P.A. Maksym, Physica B 249-251 , 233\n(1998).\n[14] M.P. Nowak, B. Szafran, F.M. Peeters, B. Partoens, and\nW.J. Pasek, Phys. Rev. B 83, 245324 (2011).\n[15] A. Singha, V. Pellegrini, S. Kalliakos, B. Karmakar, A.\nPinczuk, L.N. Pfeiffer, and K.W. West, Appl. Phys. Lett.\n94, 073114 (2009); D.G. Austing, S. Sasaki, S. Tarucha,\nS.M. Reimann, M. Koskinen, M. Manninen, Phys. Rev.\nB60, 11514 (1999).\n[16] W. Zawadzki and P. Pfeffer, Semicond. Sci. Technol. 19,\nR1 (2004).\n[17] J.J. Sakurai and J. Napolitano, Modern Quantum Me-\nchanics, second edition, (Addison-Wesley, New York,\n1994), p. 368; W. Thomas, Naturwissenschaften 13, 627\n(1925): W. Kuhn, Z. Phys. 33, 408 (1925); F. Reiche and\nW. Thomas, Z. Phys. 34, 510 (1925).\n[18] L.P. Kouwenhoven, D.G Austing, and S. Tarucha, Rep.\nProg. Phys. 64, 701 (2001); A. Babinski, M. Potemski, S.\nRaymond, J. Lapointe, and Z.R. Wasilewski, phys. stat.\nsol. (c)3, 3748 (2006).\n[19] V. Pellegrini, and A. Pinczuk, phys. stat. sol. (b) 243,\n3617 (2006)."    },    {        "title": "1505.04301v1.Dynamics_of_a_macroscopic_spin_qubit_in_spin_orbit_coupled_Bose_Einstein_condensates.pdf",        "content": "arXiv:1505.04301v1  [quant-ph]  16 May 2015Dynamics of a macroscopic spin qubit in spin-orbit\ncoupled Bose-Einstein condensates\nSh Mardonov1,2,3, M Modugno4,5and E Ya Sherman1,4\n1Department of Physical Chemistry, The University of the Basque C ountry, 48080\nBilbao, Spain\n2The Samarkand Agriculture Institute, 140103 Samarkand, Uzbek istan\n3The Samarkand State University, 140104 Samarkand, Uzbekistan\n4IKERBASQUE Basque Foundation for Science, Bilbao, Spain\n5Department of Theoretical Physics and History of Science, Univer sity of the Basque\nCountry UPV/EHU, 48080 Bilbao, Spain\nE-mail:evgeny.sherman@ehu.eus\nAbstract. We consider a macroscopic spin qubit based on spin-orbit coupled Bos e-\nEinstein condensates, where, in addition to the spin-orbit coupling, spin dynamics\nstrongly depends on the interaction between particles. The evolut ion of the spin\nfor freely expanding, trapped, and externally driven condensate s is investigated. For\ncondensates oscillating at the frequency corresponding to the Ze eman splitting in the\nsynthetic magnetic field, the spin Rabi frequency does not depend on the interaction\nbetween the atoms since it produces only internal forces and does not change the\ntotal momentum. However, interactions and spin-orbit coupling br ing the system\ninto a mixed spin state, where the total spin is inside rather than on t he Bloch\nsphere. This greatly extends the available spin space making it three -dimensional, but\nimposes limitations on the reliable spin manipulation ofsuch a macroscop icqubit. The\nspin dynamics can be modified by introducing suitable spin-dependent initial phases,\ndetermined by the spin-orbit coupling, in the spinor wave function.\nPACS numbers: 03.75.Mn, 67.85.-d\nKeywords : Two-component Bose-Einstein condensate, spin-orbit coupling, spin\ndynamics\nSubmitted to: J. Phys. B: At. Mol. Opt. Phys.Dynamics of a macroscopic spin qubit in spin-orbit coupled B EC 2\n1. Introduction\nThe experimental realization of synthetic magnetic fields and spin-o rbit coupling (SOC)\n[1, 2] in Bose-Einstein condensates (BECs) of pseudospin-1/2 par ticles has provided\nnovel opportunities for visualizing unconventional phenomena in qu antum condensed\nmatter [3, 4]. More recently, also ultracold Fermi gases with synthe tic SOC have been\nproduced and studied [5, 6]. These achievements have motivated an d intense activity,\nand a rich variety of new phases and phenomena induced by the SOC h as been discussed\nboth theoretically and experimentally [7, 8, 9, 10, 11, 12, 13, 14, 15 , 16, 17, 18, 19, 20].\nRecently, it has also been experimentally demonstrated [3, 21] the a bility of a reliable\nmeasurement of coupled spin-coordinate motion.\nOne of the prospective applications of spin-orbit coupled Bose-Eins tein condensates\nconsists in the realization of macroscopic spin qubits [8]. A more detaile d analysis of\nquantum computation based on a two-component BEC was propose d in [22]. The\ngates for performing these operations can be produced by means of the SOC and of an\nexternal synthetic magnetic field. Due to the SOC, a periodic mecha nical motion of\nthe condensate drives the spin dynamics and can cause spin-flip tra nsitions at the Rabi\nfrequency depending on the SOC strength. This technique, known in semiconductor\nphysics as the electric dipole spin resonance, is well suitable for the m anipulation of\nqubits based on the spin of a single electron [23, 24, 25]. For the macr oscopic spin qubits\nbased onBose-Einstein condensate, the physics is different in at lea st two aspects. First,\na relative effect of the SOC compared to the kinetic energy can be mu ch stronger here\nthan in semiconductors. Second, the interaction between the bos ons can have a strong\neffect on the entire spin dynamics.\nHere we study how the spin evolution of a quasi one-dimensional Bos e-Einstein\ncondensate depends on the repulsive interaction between the par ticles and on the SOC\nstrength. The paper is organized as follows. In Section 2 we remind t he reader the\nground state properties of a quasi-one dimensional condensate a nd consider simple spin-\ndipole oscillations. In Section 3 we analyze, by means of the Gross-Pit aevskii approach,\nthedynamics of free, harmonically trapped, andmechanically driven macroscopic qubits\nbased on such a condensate. We assume that the periodic mechanic al driving resonates\nwith the Zeeman transition in the synthetic magnetic field and find diffe rent regimes of\nthe spin qubit operation in terms of the interaction between the ato ms, the driving\nfrequency and amplitude. We show that some control of the spin qu bit state can\nbe achieved by introducing phase factors, dependent on the SOC, in the spinor wave\nfunction. Conclusions will be given in Section 4.\n2. Ground state and spin-dipole oscillations\n2.1. Ground state energy and wave function\nBefore analyzing the spin qubit dynamics, we remind the reader how t o obtain the\nground state of an interacting BEC. In particular, we consider a ha rmonically trappedDynamics of a macroscopic spin qubit in spin-orbit coupled B EC 3\nquasi one-dimensional condensate, tightly bounded in the transv erse directions. The\nsystem can be described by the following effective Hamiltonian, where the interactions\nbetween the atoms are taken into account in the Gross-Pitaevskii form:\n/hatwideH0=/hatwidep2\n2M+Mω2\n0\n2x2+g1|ψ(x)|2. (1)\nHereψ(x) is the condensate wave function, Mis the particle mass, ω0is the frequency\nof the trap (with the corresponding oscillator length aho=/radicalbig\n/planckover2pi1/Mω0), andg1= 2as/planckover2pi1ω⊥\nis the effective one-dimensional interaction constant, with asbeing the scattering length\nof interacting particles, and ω⊥≫ω0being the transverse confinement frequency. For\nfurther calculations we put /planckover2pi1≡M≡1, and measure energy in units of ω0and length\nin units of aho, respectively. All the effects of the interaction are determined by the\ndimensionless parameter /tildewideg1N, where/tildewideg1= 2/tildewideas/tildewideω⊥, where/tildewideasis the scattering length in\nthe units of aho,/tildewideω⊥is the transverse confinement frequency in the units of ω0, andN\nis the number of particles. In physical units, for a condensate of87Rb and an axial\ntrapping frequency ω0= 2π×10 Hz, the unit of time corresponds to 0 .016 s, the unit\nof lengthahocorresponds to 3 .4µm, and the unit of speed ahoω0becomes 0.021 cm/s,\nrespectively. In addition, considering that as= 100aB,aBbeing the Bohr radius, in the\npresence of a transverse confinement with frequency ω⊥= 2π×100 Hz the dimensionless\ncoupling constant /tildewideg1turns out to be of the order of 10−3.\nIn order to find the BEC ground state we minimize the total energy in a properly\ntruncated harmonic oscillator basis. To design the wave function we take the real sum\nof even-order eigenfunctions\nψ0(x) =N1/2nmax/summationdisplay\nn=0C2nϕ2n(x). (2)\nHere\nϕ2n(x) =1/radicalbig\nπ1/2(2n)!22nH2n(x)exp/bracketleftbigg\n−x2\n2/bracketrightbigg\n, (3)\nwhereH2n(x) are the Hermite polynomials, and the normalization is fixed by requirin g\nthat\nnmax/summationdisplay\nn=0C2\n2n= 1. (4)\nThe coefficients C2nare determined by minimizing the total energy Etot, such that\nEmin= min\nC2n{Etot}, (5)\nwhere\nEtot=1\n2/integraldisplay/bracketleftBig\n(ψ′(x))2+x2ψ2(x)+/tildewideg1ψ4(x)/bracketrightBig\ndx, (6)\nand|C2nmax| ≪1.\nFormulas (5) and (6) yield the ground state energy, while the width o f the\ncondensate is defined as:\nwgs=/bracketleftbigg2\nN/integraldisplay\nx2|ψ0(x)|2dx/bracketrightbigg1/2\n. (7)Dynamics of a macroscopic spin qubit in spin-orbit coupled B EC 4\nFigure 1. (Color online) Ground-state probability density of the condensate obtained\nfrom (2)-(6) (blue solid line), compared with the Thomas-Fermi app roximation in (8)\n(red dashed line) for /tildewideg1N= 40.\nIn the non interacting limit, /tildewideg1= 0,ψ0(x) is the ground state of the harmonic\noscillator (nmax= 0), that is a Gaussian function with wgs= 1. In the opposite, strong\ncoupling limit /tildewideg1N≫1, the exact wave function (2) is well reproduced (see Figure 1)\nby the Thomas-Fermi expression\nψTF(x) =√\n3\n2√\nN\nw3/2\nTF/parenleftbig\nw2\nTF−x2/parenrightbig1/2;|x| ≤wTF, (8)\nwherewTF= (3/tildewideg1N/2)1/3.\nIn general, for a qualitative description of the ground state one ca n use instead of\nthe exact wave function (2), the Gaussian ansatz\nψG(x) =/parenleftbiggN\nπ1/2w/parenrightbigg1/2\nexp/bracketleftbigg\n−x2\n2w2/bracketrightbigg\n, (9)\nwhere the width wis single variational parameter for the energy minimization. Then\nthe total energy (6) becomes:\nEtot=N/bracketleftbigg1\n4/parenleftbigg\nw2+1\nw2/parenrightbigg\n+/tildewideg1N\n2(2π)1/2w/bracketrightbigg\n. (10)\nThe latter is minimized with respect to wby solving the equation\ndEtot\ndw=N/bracketleftbigg1\n2/parenleftbigg\nw−1\nw3/parenrightbigg\n−/tildewideg1N\n2(2π)1/2w2/bracketrightbigg\n= 0. (11)\nInthestrongcouplingregime, /tildewideg1N≫1, wehavew≫1sothat-toafirstapproximation\n- the kinetic term ∝1/w3in (11) can be neglected, yielding the following value for the\nwidth of the ground state:\n/tildewidewG=/parenleftbigg/tildewideg1N√\n2π/parenrightbigg1/3\n. (12)\nThe first order correction can be obtained by writing w=/tildewidew+ǫ(ǫ≪1), so that from\n(11) it follows:\nw=/parenleftbigg/tildewideg1N√\n2π/parenrightbigg1/3\n+√\n2π\n3/tildewideg1N. (13)Dynamics of a macroscopic spin qubit in spin-orbit coupled B EC 5\n0 10 20 30 40012345\ng/OverTilde\nl1NEmin,wgs\nFigure 2. (Color online) Ground state energy (black solid line) and condensate width\n(red dashed line) vs. the interaction parameter /tildewideg1N.\nBy substituting (13) in (10) we obtain that the leading term in the gro und state energy\nfor/tildewideg1N≫1 is:\nE[G]\nmin=3\n4N/parenleftbigg/tildewideg1N√\n2π/parenrightbigg2/3\n. (14)\nIn figure 2 we plot the ground state energy and the condensate wid th as a function\nof the interaction, as obtained numerically from (5) and (7), respe ctively. As expected,\nin the strong coupling regime /tildewideg1N≫1 both quantities nicely follow the behavior (not\nshown in the Figure) predicted both by the Gaussian approximation a nd by the exact\nsolution, namely Emin∝(/tildewideg1N)2/3andwgs∝(/tildewideg1N)1/3.\n2.2. Simple spin-dipole oscillations\nLet us now turn to the case of a condensate of pseudospin 1/2 ato ms. Here the system\nis described by a two-component spinor wave function Ψ = [ ψ↑(x,t),ψ↓(x,t)]T, still\nnormalized to the total number of particles N.The interaction energy (third term in\nthe functional (6)) now acquires the form (see, e.g. [9])\nEint=1\n2/tildewideg1/integraldisplay/bracketleftbig\n|ψ↑(x,t)|2+|ψ↓(x,t)|2/bracketrightbig2dx, (15)\nwhere, for simplicity and qualitative analysis, we neglect the depende nce of interatomic\ninteraction on the spin component ↑or↓and characterize all interactions by a single\nconstant/tildewideg1.\nHere we consider spin dipole oscillations, induced by a given small initial s ymmetric\ndisplacement of the two spin components ±ξ. For a qualitative understanding, we\nassume a negligible spin-orbit coupling and a Gaussian form of the wave function\npresented as\nΨG(x) =1√\n2/bracketleftBigg\nψG(x−ξ)\nψG(x+ξ)/bracketrightBigg\n, (16)\nwhereψGis given by (9), and ξ≪w. The corresponding energy is given by:\nE=E[G]\nmin+Esh, (17)Dynamics of a macroscopic spin qubit in spin-orbit coupled B EC 6\nwhereE[G]\nminis defined by (14) and Eshis the shift-dependent contribution:\nEsh=N\n2ξ2/parenleftbigg\n1−/tildewideg1N√\n2πw3/parenrightbigg\n. (18)\nThen, it follows that the corresponding oscillation frequency is\nωsh=/radicalBigg\n1−/tildewideg1N√\n2πw3. (19)\nFor strong interaction ( /tildewideg1N≫1) by substituting (13) in (19) we obtain:\nωsh≈/parenleftBigg√\n2π\n/tildewideg1N/parenrightBigg2/3\n. (20)\nTherefore, for strong interaction the frequency of the spin dipo le oscillations falls as\n(/tildewideg1N)−2/3, and this result is common for the Gaussian ansatz and for the exac t solution;\nit will be useful in the following section.\n3. Spin evolution and particles interaction\n3.1. Hamiltonian, spin density matrix, and purity\nTo consider the evolution of the driven quasi one dimensional pseud ospin-1/2 SOC\ncondensate we begin with the effective Hamiltonian\n/hatwideH=α/hatwideσz/hatwidep+/hatwidep2\n2+∆\n2/hatwideσx+1\n2(x−d(t))2+/tildewideg1|Ψ|2. (21)\nHereαis the SOC constant (see [11] and [12] for comprehensive review on t he SOC in\ncoldatomicgases), /hatwideσxand/hatwideσzarethePauli matrices, ∆isthesynthetic Zeemansplitting,\nandd(t) is the driven displacement of the harmonic trap center as can be ob tained by\na slow motion of the intersection region of laser beams trapping the c ondensate.\nThetwo-componentspinorwavefunctionΨisobtainedasasolutiono fthenonlinear\nSchr¨ odinger equation\ni∂Ψ\n∂t=/hatwideHΨ. (22)\nTo describe spin evolution we introduce the reduced density matrix\nρ(t)≡ |Ψ∝an}bracketri}ht∝an}bracketle{tΨ|=/bracketleftBigg\nρ11(t)ρ12(t)\nρ21(t)ρ22(t)/bracketrightBigg\n, (23)\nwhere we trace out the x−dependence by calculating integrals\nρ11(t) =/integraldisplay\n|ψ↑(x,t)|2dx, ρ22(t) =/integraldisplay\n|ψ↓(x,t)|2dx,\nρ12(t) =/integraldisplay\nψ∗\n↑(x,t)ψ↓(x,t)dx, ρ21(t) =ρ∗\n12(t), (24)\nand, as a result,\ntr(ρ(t))≡ρ11(t)+ρ22(t) =N. (25)Dynamics of a macroscopic spin qubit in spin-orbit coupled B EC 7\nFigure 3. (Color online) (a) Separation of a freely expanding condensate in tw o spin-\nup and spin-down components with opposite anomalous velocities. (b ) Oscillation of\nthe spin-up and spin-down components in the harmonic trap.\nAlthough the |Ψ∝an}bracketri}htstate is pure, integration in (24) produces ρ(t) formally describing a\nmixed state in the spin subspace. One can characterize the resultin g spin state purity\nby a parameter Pdefined as\nP=2\nN2/parenleftbigg\ntr/parenleftbig\nρ2/parenrightbig\n−N2\n2/parenrightbigg\n, (26)\nwhere 0≤P≤1,\ntr/parenleftbig\nρ2/parenrightbig\n=N2+2(|ρ12|2−ρ11ρ22), (27)\nand we omitted the explicit t−dependence for brevity. The system is in the pure state\nwhenP= 1,that is tr(ρ2) =N2withρ11ρ22=|ρ12|2. In the fully mixed state, where\ntr(ρ2) =N2/2, we have P= 0 with\nρ11=ρ22=N\n2, ρ 12= 0. (28)\nThe spin components defined by ∝an}bracketle{t/hatwideσi∝an}bracketri}ht ≡tr(/hatwideσiρ)/Nbecome\n∝an}bracketle{t/hatwideσx∝an}bracketri}ht=2\nNRe(ρ12),∝an}bracketle{t/hatwideσy∝an}bracketri}ht=−2\nNIm(ρ12),\n∝an}bracketle{t/hatwideσz∝an}bracketri}ht=2\nNρ11−1, (29)\nand the purity P=/summationtext3\ni=1∝an}bracketle{t/hatwideσi∝an}bracketri}ht2, which allows one to match the value of Pand the length\nof the spin vector inside the Bloch sphere. For a pure state/summationtext3\ni=1∝an}bracketle{t/hatwideσi∝an}bracketri}ht2= 1, and the\ntotal spin is on the Bloch sphere. Instead, for a fully mixed state/summationtext3\ni=1∝an}bracketle{t/hatwideσi∝an}bracketri}ht2= 0, and\nthe spin null.\n3.2. A simple condensate motion\nLet us suppose that a condensate of interacting spin-orbit couple d particles is located\nin a harmonic trap and characterized by an initial wave function\nΨ0(x,0) =1√\n2ψin(x)/bracketleftBigg\n1\n1/bracketrightBigg\n, (30)Dynamics of a macroscopic spin qubit in spin-orbit coupled B EC 8\n(a)\n024680.00.20.40.60.81.0\ntP/LParen1t/RParen1(b)\n0246810120.00.20.40.60.81.0\nt/LAngleBracket1Σ/Hat\nx/LParen1t/RParen1/RAngleBracket1\nFigure 4. (Color online) ( a) Purity and ( b) spin component as a function of time for\na condensate released from the trap, for α= 0.2. The lines correspond to /tildewideg1N= 0\n(black solid line; for the purity cf. (32)), /tildewideg1N= 10 (red dashed line), and /tildewideg1N= 20\n(blue dot-dashed line).\nwith the spin parallel to the x−axis.\nThe spin-orbit coupling modifies the commutator corresponding to t he velocity\noperator by introducing the spin-dependent contribution as:\n/hatwidev≡i/bracketleftbigg/hatwidep2\n2+α/hatwideσz/hatwidep,/hatwidex/bracketrightbigg\n=/hatwidep+α/hatwideσz. (31)\nThe effect of the spin-dependent anomalous velocity term on the co ndensate motion\nwas clearly observed experimentally in [3] as the spin-induced dipole os cillations and in\n[21] as the Zitterbewegung . Since the initial spin in (30) is parallel to the x-axis, the\nexpectation value of the velocity vanishes, ∝an}bracketle{t/hatwidev∝an}bracketri}ht= 0.\nFree and oscillating motion of the BEC is shown in figure 3(a) and figure 3(b),\nrespectively. When one switches off the trap, the condensate is se t free, and the two\nspin components start to move apart and the condensate splits up , see figure 3(a).\nEach spin-projected component broadens due to the Heisenberg momentum-coordinate\nuncertainty and interaction. The former effect is characterized b y a rate proportional\nto 1/wgs.At large/tildewideg1N,the width wgs∼(/tildewideg1N)1/3,and, as a result, the quantum\nmechanical broadening rate decreases as ( /tildewideg1N)−1/3.At the same time, the repulsion\nbetween the spin-polarized components accelerates the peak sep aration [26] and leads\nto the asymptotic separation velocity ∼(/tildewideg1N)1/2. This acceleration by repulsion leads\nto opposite time-dependent phase factors in ψ↑(x,t) andψ↓(x,t) in (24) and, therefore,\nresults in decreasing in |ρ12(t)|and in the purity. Thus, with the increase in the\ninteraction, the purity and the x−spin component asymptotically tend to zero faster,\nas demonstrated in figure 4. For a noninteracting condensate with the initial Gaussian\nwave function ψin∼exp(−x2/2w2) the purity can be written analytically as\nP0(t) = exp/bracketleftBigg\n−2/parenleftbiggαt\nw/parenrightbigg2/bracketrightBigg\n. (32)\nIn the presence of the trap (figure 3(b)), the anomalous velocity in (31) causes spin\ncomponents (spin-dipole) oscillations with a characteristic frequen cy of the oder of ωshDynamics of a macroscopic spin qubit in spin-orbit coupled B EC 9\n(a)\n0102030405060700.00.20.40.60.81.0\ntP/LParen1t/RParen1\n(b)\n0102030405060700.00.20.40.60.81.0\nt/LAngleBracket1Σ/Hat\nx/LParen1t/RParen1/RAngleBracket1\n(c)\n010203040506070/MinuΣ3/MinuΣ2/MinuΣ10123\nt/LAngleBracket1xΣ/Hat\nz/RAngleBracket1\nFigure 5. (Color online) (a) Purity, (b) spin component, and (c) spin dipole mom ent\nas a function of time for the system in the harmonic trap with α= 0.2,∆ = 0, d0= 0.\nThe different lines correspond to /tildewideg1N= 0 (black solid line), /tildewideg1N= 10 (red dashed\nline),/tildewideg1N= 20 (blue dot-dashed line), and /tildewideg1N= 60 (green dotted line).\nin (20). With the increase in the interatomic interaction, the freque ncyωshdecreases\nand, therefore, the amplitude of the oscillations arising due to the a nomalous velocity\n(∼α/ωsh) increases. As a result, the acceleration and separation of the sp in-projected\ncomponents increase, the off-diagonal components of the densit y matrix in (24) became\nsmaller, and the minimum in P(t) rapidly decreases to P(t)≪1 as shown by the exact\nnumerical results presented in figures 5(a) and (b) [27]. In figure 5 (c) we show the\ncorresponding evolution of spin density dipole moment\n∝an}bracketle{tx/hatwideσz∝an}bracketri}ht=1\nN/integraldisplay\nΨ†x/hatwideσzΨdx. (33)\nHere the oscillation frequency is a factor of two larger than that of the spin density\noscillation.\n3.3. Spin-qubit dynamics and the Rabi frequency\nTo manipulate the macroscopic spin qubit, the center of the trap is d riven harmonically\nat the frequency corresponding to the Zeeman splitting ∆ as\nd(t) =d0sin(t∆), (34)\nwhered0is an arbitrary amplitude and the corresponding spin rotation Rabi f requency\nΩRis defined as αd0∆.At ∆≪1,as will be considered here, for a noninteractingDynamics of a macroscopic spin qubit in spin-orbit coupled B EC 10\n(a)\n0 50 100 150 2000.00.20.40.60.81.0\ntP/LParen1t/RParen1(b)\n050100 150 200/MinuΣ1.0/MinuΣ0.50.00.51.0\nt/LAngleBracket1Σ/Hat\nx/LParen1t/RParen1/RAngleBracket1\nFigure 6. (Color online) ( a) Purity and ( b) spin component as a function of time for\na driven condensate with α= 0.1,∆ = 0.1, d0= 2. The lines correspond to /tildewideg1N= 0\n(black solid line), /tildewideg1N= 10 (red dashed line), and /tildewideg1N= 20 (blue dot-dashed line).\ncondensate and a very weak spin-orbit coupling, the spin componen t∝an}bracketle{t/hatwideσx(t)∝an}bracketri}htis expected\nto oscillate approximately as\n∝an}bracketle{t/hatwideσx(t)∝an}bracketri}ht= cos(Ω Rt). (35)\nThe corresponding spin-flip time Tsfis\nTsf=π\nΩR. (36)\nFigure 6 shows the time dependence of the purity and the spin of the condensate for\ngivenα,d0, and ∆ at different interatomic interactions. In figure 6(a) one can see that\nthe increase of /tildewideg1Nenhances the variation of the purity (cf. Fig 5(a)). This variation\nprevents a precise manipulation of the spin-qubit state in the conde nsate [28]. It follows\nfrom figure 6(b) that although increasing the interaction strongly modifies the spin\ndynamics, it roughly conserves the spin-flip time Tsf= 50π, see (36). To demonstrate\nthe role of the SOC coupling strength αand interatomic interaction at nominally the\nsame Rabi frequency Ω R,we calculated the spin dynamics presented in Figure 7. By\ncomparing Figures 6 and 7(a),(b) we conclude that the increase in th e SOC, at the\nsame Rabi frequency, causes an increase in the variation of the pu rity and of the spin\ncomponent. These results show that to achieve a required Rabi fr equency and a reliable\ncontrol of the spin, it is better to increase the driving amplitude d0rather than the spin-\norbit couping α.The increase in the SOC strength can result in losing the spin state\npurity and decreasing the spin length. Figure 7(c) shows the irregu lar spin evolution of\nthe condensate inside the Bloch sphere. In figure 7(a), for α= 0.2 and/tildewideg1N= 20,the\npurity decreases almost to zero, placing the spin close to the cente r of the Bloch sphere,\nas can be seen in figure 7(c). It follows from Figures 6(b) and 7(b) t hat in order to\nprotect pure macroscopic spin-qubit states, the Rabi frequenc y should be small. Then,\ntaking into account that the displacement of the spin-projected w ave packet is of the\norder ofα(/tildewideg1N)2/3and the packet width is of the order of ( /tildewideg1N)1/3, we conclude that\nforα/greaterorsimilar(/tildewideg1N)−1/3, the purity of the driven state tends to zero. As a result, the Rab i\nfrequency for the pure state evolution is strongly limited by the inte raction between the\nparticles and cannot greatly exceed d0∆/(/tildewideg1N)1/3.Dynamics of a macroscopic spin qubit in spin-orbit coupled B EC 11\n(a)\n0 50 100 150 2000.00.20.40.60.81.0\ntP/LParen1t/RParen1(b)\n050100 150 200/MinuΣ1.0/MinuΣ0.50.00.51.0\nt/LAngleBracket1Σ/Hat\nx/LParen1t/RParen1/RAngleBracket1\nFigure 7. (Color online) ( a) Purity, ( b) spin component, and ( c) spatial trajectory of\nthe spin inside the Bloch sphere for the driven BEC with α= 0.2,∆ = 0.1, d0= 1\nresulting in the same Rabi frequency as in Figure (6). In Figures ( a) and (b) the lines\ncorrespond to: /tildewideg1N= 0 (black solid line), /tildewideg1N= 10 (red dashed line), and /tildewideg1N= 20\n(blue dotted line). At /tildewideg1N= 0,the time dependence of ∝an}bracketle{tσx∝an}bracketri}htis rather accurately\ndescribed by cos(Ω Rt) formula, corresponding to a relatively small variation in the\npurity, 1 −P(t)≪1.With the increase in /tildewideg1N,the purity variation increases and the\nbehavior of ∝an}bracketle{tσx∝an}bracketri}htdeviates stronger from the conventional cos(Ω Rt) dependence. ( c)\nHere the interaction is fixed to /tildewideg1N= 20. The green and red vectors correspond to the\ninitial and final states of the spin, respectively. Here the final time is fixed to tfin=Tsf,\nsee (36).\nIn addition, it is interesting to note that for /tildewideg1N≫1,where the spin dipole\noscillates at the frequency of the order of ( /tildewideg1N)−2/3(as given by (20)), the perturbation\ndue to the trap motion is in the high-frequency limit already at ∆ ≥(/tildewideg1N)−2/3, having\na qualitative influence on the spin dynamics [29, 30, 31].\n3.4. Phase factors due to spin-orbit coupling\nThe above results show that the spin-dependent velocity in (31), a long with the\ninteratomic repulsion, results in decreasing the spin state purity an d produces irregular\nspin motion inside the Bloch sphere. To reduce the effect of these an omalous velocities\nand to prevent the resulting fast separation (with the relative velo city of 2α) of the spin\ncomponents, we compensate them by introducing coordinate-dep endent phases (similar\nto the Bragg factors) in the wave function [32]. To demonstrate th e effect of theseDynamics of a macroscopic spin qubit in spin-orbit coupled B EC 12\n(a)\n0 50 100 150 2000.00.20.40.60.81.0\ntP/LParen1t/RParen1(b)\n050100 150 200/MinuΣ1.0/MinuΣ0.50.00.51.0\nt/LAngleBracket1Σ/Hat\nx/LParen1t/RParen1/RAngleBracket1\nFigure 8. (Color online) ( a) Purity, ( b) spin component, and ( c) trajectory of the\nspin inside the Bloch sphere for a driven BEC with initial phases as in (37 ) and\nα= 0.2,∆ = 0.1, d0= 1.In Figures ( a) and (b) the black solid line is for /tildewideg1N= 0,\nthe red dashed line is for /tildewideg1N= 10, and the blue dotted line is for /tildewideg1N= 20. In Figure\n(c) the interaction is /tildewideg1N= 20. The green and red vectors correspond to the initial\nand final states of the spin, respectively ( tfin=Tsf). The initial spin state (a solid-line\ncircle with white filling) is inside the Bloch sphere since ∝an}bracketle{tσx(t= 0)∝an}bracketri}ht=/radicalbig\nP(0), and\nP(0)<1 due to the mixed character in the spin subspace of the state in (37 ).\nphase factors, we construct the initial spinor Ψ α(x,0) by a coordinate-dependent SU(2)\nrotation [33] of the state with ∝an}bracketle{tσx∝an}bracketri}ht= 1 in (30) as\nΨα(x,0) =e−iαx/hatwideσzΨ0(x,0) =ψin(x)√\n2/bracketleftBigg\ne−iαx\neiαx/bracketrightBigg\n. (37)\nThe expectation value of the velocity (31) at each component ψin(x)exp(±iαx) is zero,\nand, as a result, the α-induced separation of spin components vanishes, making, as can\nbe easily seen [33], the spinor (37) the stationary state of the spin- orbit coupled BEC\nin the Gross-Pitaevskii approximation.\nIn terms of the spin density matrix (24), the state (37) is mixed. Fo r a Gaussian\ncondensate with the width w, we get the following expression for the purity at t= 0\nP[G]\nα(0) = exp/bracketleftbig\n−2(αw)2/bracketrightbig\n. (38)Dynamics of a macroscopic spin qubit in spin-orbit coupled B EC 13\nInstead, in the case of a Thomas-Fermi wave function as in (8), in t he limitαwTF≫1\nthe initial purity behaves as\nP[TF]\nα(0)∼cos2(2αwTF)\n(αwTF)4. (39)\nBoth cases are characterized by a rapid decrease as αwTFis increased [25].\nIn the absence of external driving, the spin components and purit y of (37) state\nremain constant. With the driving, spinor components evolve with tim e and the\nobservables show evolution quantitatively different from that pres ented in Figure 7. In\nfigure 8 we show the analog of figure 7 for the initial state in (37), wit hψin(x) =ψ0(x)\ngiven by (2)-(5). By comparing these Figures one can see that the inclusion of the\nspin-dependent phases in (37) strongly reduces the oscillations in t hex−component of\nthe spin, making the spin trajectory more regular and decreasing t he variations in the\npurityP(t) compared to the initial state without these phase factors.\nA general effect of the interatomic interaction can be seen in both fi gures 7 and 8.\nNamely, for smaller interactions /tildewideg1N, the destructive role of the interatomic repulsion\non the spin state purity is reduced and the spin dynamics becomes mo re regular. As a\nresult, at smaller /tildewideg1Nthe spin trajectory is located closer to the Bloch sphere.\n4. Conclusions\nWe have considered the dynamics of freely expanding and harmonica lly driven\nmacroscopic spin qubits based on quasi one-dimensional spin-orbit coupled Bose-\nEinstein condensates in a synthetic Zeeman field. The resulting evolu tion strongly\ndepends in a nontrivial way on the spin-orbit coupling and interaction between the\nbosons. On one hand, spin-orbit coupling leads to the driven spin qub it dynamics. On\nthe other hand, it leads to a spin-dependent anomalous velocity cau sing spin splitting\nof the initial wave packet and reducing the purity of the spin state b y decreasing the\noff-diagonal components of the spin density matrix. This destruct ive influence of spin-\norbit coupling is enhanced by interatomic repulsion. The effects of th e repulsion can be\ninterpreted in terms of the increase in the spatial width of the cond ensate and the\ncorresponding decrease in the spin dipole oscillation frequency with t he interaction\nstrength. The joint influence of the repulsion and spin-orbit couplin g can spatially\nseparate and modify the spin components stronger than just the spin-orbit coupling\nand result in stronger irregularities in the spin dynamics. The spin-flip Rabi frequency\nremains, however, almost unchangedinthepresence oftheintera tomicinteractions since\nthey lead to only internal forces and do not change the condensat e momentum. As a\nresult, to preserve the evolution within a high-purity spin-qubit sta te, with the spin\nbeing always close to the Bloch sphere, the spin-orbit coupling should be weak and, due\nto this weakness, the spin-rotation Rabi frequency should be sma ll and spin rotation\nshould take a long time. The destructive effect of both the spin-orb it coupling and\ninteratomicrepulsiononthepurityofthespinstatecanbecontrolla blyandconsiderablyDynamics of a macroscopic spin qubit in spin-orbit coupled B EC 14\nreduced, although not completely removed, by introducing spin-de pendent Bragg-like\nphase factors in the initial spinor wave function.\nAcknowledgments\nThis work was supported by the University of Basque Country UPV/ EHU under\nprogram UFI 11/55, Spanish MEC (FIS2012-36673-C03-01 and FI S2012-36673-C03-\n03), and Grupos Consolidados UPV/EHU del Gobierno Vasco (IT-47 2-10). S.M.\nacknowledges EU-funded Erasmus Mundus Action 2 eASTANA, “evr oAsian Starter for\nthe Technical Academic Programme” (Agreement No. 2001-2571/ 001-001-EMA2).\nReferences\n[1] Lin Y-J, Compton R L, K Jim´ enez-Garcia, Porto J V and Spielman I B 2009Nature462628\n[2] Lin Y-J, Jim´ enez-Garc´ ıa K and Spielman I B 2011 Nature47183\n[3] Zhang J-Y, Ji S-C, Z Chen, Zhang L, Z-D Du, Yan B, G-S Pan, Zha o B, Deng Y-J, Zhai H, Chen\nS and Pan J-W 2012 Phys. Rev. Lett. 109115301\n[4] Ji S-C, Zhang J-Y, L Zhang, Du Z-D, Zheng W, Deng Y-J, Zhai H, Chen Sh and Pan J-W 2014\nNature Physics 10314\n[5] Wang P, Yu Z-Q, Fu Z, Miao J, Huang L, Chai Sh, Zhai H and Zhang J 2012Phys. Rev. Lett. 109\n095301\n[6] Cheuk L W, Sommer A T, Hadzibabic Z, Yefsah T, Bakr W S and Zwierle in M W 2012 Phys.\nRev. Lett. 109095302\n[7] Liu X-J, Borunda M F, Liu X and Sinova J 2009 Phys. Rev. Lett. 102046402\n[8] Stanescu T D, Anderson B and Galitski V 2008 Phys. Rev. A78023616\n[9] Li Y, Martone G I and Stringari S 2012 EPL9956008\nMartone G I, Li Y, Pitaevskii L P and Stringari S 2012 Phys. Rev. A86063621\n[10] Zhang Y, Mao L and Zhang Ch 2012 Phys. Rev. Lett. 108035302\n[11] Zhai H 2012 Int. J. Mod. Phys. B261230001\n[12] Galitski V and Spielman I B 2013 Nature49449\n[13] Zhang Y, Chen G and Zhang Ch 2013 Scientific Reports 31937\n[14] Achilleos V, Frantzeskakis D J, Kevrekidis P G and Pelinovsky D E 20 13Phys. Rev. Lett. 110\n264101\n[15] Ozawa T, Pitaevskii L P and Stringari S 2013 Phys. Rev. A87063610\n[16] Wilson R M, Anderson B M and Clark Ch W 2013 Phys. Rev. Lett. 111185303\n[17] L¨ u Q-Q and Sheehy D E 2013 Phys. Rev. A88043645\n[18] Xianlong G 2013 Phys. Rev. A87023628\n[19] Anderson B M, Spielman I B and Juzeliu˜ nas G 2013 Phys. Rev. Lett. 111125301\n[20] Dong L, Zhou L, Wu B, Ramachandhran B and Pu H 2014 Phys. Rev. A89011602\n[21] Qu Ch, Hamner Ch, Gong M, Zhang Ch and Engels P 2013 Phys. Rev. A88021604\n[22] Byrnes T, Wen K and Yamamoto Y 2012 Phys. Rev. A85040306\n[23] Nowack K C, Koppens F H L, Nazarov Yu V and Vandersypen L M K 2 007Science3181430\n[24] Rashba E I and Efros Al L 2003 Phys. Rev. Lett. 91126405\n[25] It is interesting to mention that an increase in spin-orbit coupling strength after a certain value\nleads to a less efficient spin driving as shown with perturbation theory approach by Li R, You\nJ Q, Sun C P and Nori F 2013 Phys. Rev. Lett. 111086805\n[26] This procedure models the von Neumann quantum spin measurem ent, see Sherman E Ya and\nSokolovski D 2014 New J. Phys. 16015013\n[27] An analysis of the spin-dipole oscillations based on the sum rules wa s presented in [9]Dynamics of a macroscopic spin qubit in spin-orbit coupled B EC 15\n[28] This low precision of the spin control can be seen as a general fe ature of the systems where\nexternal perturbation strongly drives the orbital motion. See, e .g. Khomitsky D V, Gulyaev L\nV and Sherman E Ya 2012 Phys. Rev. B85125312\n[29] A technique to study and engineer the high-frequency behavio r has been introduced in Bukov M,\nD’Alessio L and Polkovnikov A 2014 arXiv:1407.4803. Its application to t he spin dynamics of\nthe spin-orbit coupled BEC goes far beyond the scope of this paper , however.\n[30] Optimal control theory proposed by Budagosky J A and Castr o A 2015 The European Physical\nJournal B8815 can potentially be applied to the engineering of a d(t)−dependence more\ncomplicated than a harmonic oscillation.\n[31] Xiong B, Zheng J-H and Wang D-W 2014 arXiv:1410.8444 analyzed t he condensate driving in\nterms of the multichannel quantum interference.\n[32] These phase-related Josephson effect in the spin-orbit couple d BEC in a double-well potential\nhas been recently analyzed in Garcia-March M A, Mazzarella G, Dell’Ann a L, Juli´ a-D´ ıaz B,\nSalasnich L and Polls A 2014 Phys. Rev. A89063607\nCitro R and Naddeo A 2014 arXiv:1405.5356\n[33] Tokatly I V and Sherman E Ya 2010 Phys. Rev. B82161305"    },    {        "title": "1203.2795v1.Impact_of_Dresselhaus_vs__Rashba_spin_orbit_coupling_on_the_Holstein_polaron.pdf",        "content": "arXiv:1203.2795v1  [cond-mat.str-el]  13 Mar 2012Impact of Dresselhaus vs. Rashba spin-orbit coupling on the Holstein polaron\nZhou Li1, L. Covaci2, and F. Marsiglio1\n1Department of Physics, University of Alberta, Edmonton, Al berta, Canada, T6G 2J1\n2Departement Fysica, Universiteit Antwerpen, Groenenborg erlaan 171, B-2020 Antwerpen, Belgium\n(Dated: November 6, 2018)\nWe utilize an exact variational numerical procedure to calc ulate the ground state properties of a\npolaron in the presence of Rashba and linear Dresselhaus spi n-orbit coupling. We find that when\nthe linear Dresselhaus spin-orbit coupling approaches the Rashba spin-orbit coupling, the Van-Hove\nsingularity in the density of states will be shifted away fro m the bottom of the band and finally\ndisappear when the two spin-orbit couplings are tuned to be e qual. The effective mass will be\nsuppressed; the trend will become more significant for low ph onon frequency. The presence of two\ndominant spin-orbit couplings will make it possible to tune the effective mass with more varied\nobservables.\nI. INTRODUCTION\nOne of the end goals in condensed matter physics is to\nachieve a sufficient understanding of materials fabrica-\ntion and design so as to ‘tailor-engineer’ specific desired\nproperties into a material. Arguably pn-junctions long\nago represented some of the first steps in this direction;\nnowadays, heterostructures1and mesoscopic geometries2\nrepresent further progress towards this goal.\nInthefieldof spintronics , wherethespindegreeoffree-\ndom is specifically exploited for potential applications,3,4\nspin-orbit coupling5plays a critical role because con-\ntrol of spin will require coupling to the orbital mo-\ntion. Spin orbit coupling, as described by Rashba6and\nDresselhaus,7is expected to be prominent in two dimen-\nsional systems that lack inversion symmetry, including\nsurface states. These different kinds of coupling are in\nprinciple independently controlled.8,9\nThe coexistence of Rashba and Dresselhaus spin-orbit\ncoupling has now been realized in both semiconductor\nquantum wells4,9and more recently in a neutral atomic\nBose-Einstein condensate.10When the Rashba and (lin-\near) Dresselhaus spin-orbit coupling strengths are tuned\nto be equal, SU(2) symmetry is predicted to be recov-\nered and the persistent spin helix state will emerge.4,10,11\nThis symmetry is expected to be robust against spin-\nindependent scattering but is broken by the cubic Dres-\nselhaus spin-orbit coupling and other spin-dependent\nscattering which may be tuned to be negligible.4\nWhile we focus on the spin-orbit interaction, other in-\nteractionsarepresent. In particular, the electron-phonon\ninteraction will be present and may be strong in semi-\nconductor heterostructures. Moreover, optical lattices12\nwith cold polar molecules may be able to realize a tune-\nable Holstein model.13The primary purpose of this work\nis to investigate the impact of electron-phonon coupling\n(as modelled by the Holstein model14) on the proper-\nties of the spin-orbit coupled system. We will utilize a\ntight-binding framework; previously it was noted that in\nthe presence of Rashba spin-orbit coupling the vicinity\nof a van Hove singularity near the bottom of the elec-\ntron band15–17had a significant impact on the polaronic\npropertiesofanelectron; withadditional(linear)Dressel-haus spin-orbit coupling the van Hove singularity shifts\nwell away from the band bottom, as the two spin-orbit\ncouplings acquire equal strength. As we will illustrate\nbelow, the presence of two separately tunable spin-orbit\ncouplings will result in significant controllability of the\nelectron effective mass.\nII. MODEL AND METHODOLOGIES\nWe use a tight-binding model with dimensionless Hol-\nstein electron-phonon coupling of strength g, and with\nlinear Rashba ( VR) and Dresselhaus ( VD) spin-orbit cou-\npling:\nH=−t/summationdisplay\n<i,j>,s =↑↓(c†\ni,scj,s+c†\nj,sci,s)\n+i/summationdisplay\nj,α,β(c†\nj,αˆV1cj+ˆy,β−c†\nj,αˆV2cj+ˆx,β−h.c.)\n−gωE/summationdisplay\ni,s=↑↓c†\ni,sci,s(ai+a†\ni)+ωE/summationdisplay\nia†\niai(1)\nwherec†\ni,s(ci,s) creates (annihilates) an electron at site\niwith spin index s, anda†\ni(ai) creates (annihilates) a\nphononatsite i. The operators ˆVj,j= 1,2arewritten in\nterms of the spin-orbit coupling strengths and the Pauli\nmatrices as ˆV1=VRˆσx−VDˆσy, andˆV2=VRˆσy−VDˆσx,\nThe sum over iis over all sites in the lattice, whereas\n< i,j > signifies that only nearest neighbour hopping\nis included. Other parameters in the problem are the\nphonon frequency, ωE, and the hopping parameter t,\nwhich hereafter is set equal to unity.\nWithout the electron-phononinteractionthe electronic\nstructure is readily obtained by diagonalizing the Hamil-\ntonian in momentum space. With the definitions\nS1≡VRsin(ky)+VDsin(kx),\nS2≡VRsin(kx)+VDsin(ky), (2)\nwe obtain the eigenvalues\nεk,±=−2t[cos(kx)+cos(ky)]±2/radicalBig\nS2\n1+S2\n2(3)2\n-5-4-3-2-1 0 1 2 3 4\n-3-2-1 0 1 2 3-3-2-1 0 1 2 3VR/t = 0.5 VD/t = 0.5\n-5-4-3-2-1 0 1 2 3 4\n-3-2-1 0 1 2 3-3-2-1 0 1 2 3VR/t = 0.8 VD/t = 0.2\n-5-4-3-2-1 0 1 2 3 4\n-3-2-1 0 1 2 3-3-2-1 0 1 2 3VR/t = 0.9 VD/t = 0.1\n-5-4-3-2-1 0 1 2 3 4\n-3-2-1 0 1 2 3-3-2-1 0 1 2 3VR/t = 0.99 VD/t = 0.01\nFIG. 1. Contour plots for the bare energy bands with Rashba-\nDresselhaus spin-orbit coupling, for different values of VRand\nVDwhile the sum is kept constant: VR+VD=tfor these\ncases. (a) VR=VD= 0.5t, (b)VR= 0.8t,VD= 0.2t, (c)\nVR= 0.9t,VD= 0.1t, and (d) VR= 0.99t,VD= 0.01t. Note\nthe clear progression from a two-fold degenerate ground sta te\nto a four-fold degenerate one.and eigenvectors\nΨk±=1√\n2/bracketleftBigg\nc†\nk↑±S1−iS2/radicalbig\nS2\n1+S2\n2c†\nk↓/bracketrightBigg\n|0/angbracketright.(4)\nThe ground state energy is\nE0=−4t/radicalbig\n1+(VR+VD)2/(2t2). (5)\nWithout loss of generality we can consider only VR≥0\nandVD≥0. Either Rashba and Dresselhaus spin-orbit\ncoupling independently behave in the same manner, and\ngive rise to a four-fold degenerate ground state with\nwave vectors, ( kx,ky) = (±arctan(VR√\n2t),±arctan(VR√\n2t),\n(VD= 0), and similarly for VD/negationslash= 0 and VR= 0. With\nboth couplings non-zero, however, the degeneracy be-\ncomes two-fold, with the ground state wave vectors,\n(kx0,ky0) =±(k0,k0); where k0= tan−1(VR+VD√\n2t).\n(6)\nIt is clear that the sum of the coupling strengths replaces\nthe strength of either in these expressions, so that hence-\nforth in most plots we will vary one of the spin-orbit\ninteraction strengths while maintaining their sum to be\nfixed. Similarly, the effective mass, taken along the diag-\nonal, is\nmSO\nm0=1/radicalbig\n1+(VR+VD)2/(2t2), (7)\nwherem0≡1/(2t) (lattice spacing, a≡1, and/planckover2pi1≡1)\nis the bare mass in the absence of spin-orbit interaction,\nandmSOis the effective mass due solely to the spin-orbit\ninteraction. As detailed in the Appendix, the effective\nmass becomes isotropic when the Rashba and Dressel-\nhaus spin-orbit coupling strengths are equal.\nThe non-interacting electron density of states (DOS)\nis defined for each band, as\nDs(ǫ) =/summationdisplay\nkδ(ǫ−ǫks) (8)\nwiths=±1.\nIn Fig.2(a) we show the low energy DOS for various\nvalues of the spin-orbit coupling strengths, VRandVD,\nwhile keeping their sum constant; the low energy van\nHove singularity disappears for VR=VD. Note that only\nD−(ǫ) is shown, as the upper band, with DOS D+(ǫ),\nexists only at higher energies. Furthermore, informa-\ntion concerning the upper band can always be obtained\nthrough the symmetry\nD+(ǫ) =D−(−ǫ). (9)\nIn Fig.2(b) we show the value of the density of states\nat the bottom of the band vs. VD; as derived in the\nAppendix, theDOSvalueattheminimumenergyisgiven\nby\nD−(E0) =1\n2πt1/radicalBig\n1+(VR+VD)2\n2t2−(VR−VD)2\n(VR+VD)2.(10)3\n/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48 /s49/s46/s50/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52/s48/s46/s53/s48/s46/s54\n/s32/s86\n/s82/s61/s48/s46/s57/s57/s44/s32/s86\n/s68/s61/s48/s46/s48/s49\n/s32/s86\n/s82/s61/s48/s46/s57/s44/s32/s32/s32/s86\n/s68/s61/s48/s46/s49\n/s32/s86\n/s82/s61/s48/s46/s56/s44/s32/s32/s32/s86\n/s68/s61/s48/s46/s50\n/s32/s86\n/s82/s61/s48/s46/s53/s44/s32/s32/s32/s86\n/s68/s61/s48/s46/s53/s68/s95/s40/s69/s41\n/s69/s40/s97/s41\n/s32\n/s74/s117/s109/s112/s32/s111/s102/s32/s100/s101/s110/s115/s105/s116/s121/s32/s111/s102/s32/s115/s116/s97/s116/s101/s115\n/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52/s48/s46/s53/s48/s46/s54\n/s32/s86\n/s82/s61/s48/s44/s32/s86\n/s68/s61/s49/s32/s111/s114/s32/s32/s86\n/s68/s61/s48/s44/s32/s86\n/s82/s61/s49\n/s32/s86\n/s82/s43/s86\n/s68/s61/s48/s46/s52\n/s32/s86\n/s82/s43/s86\n/s68/s61/s49/s68/s95/s40/s69\n/s48/s41\n/s86\n/s82/s47/s40/s86\n/s82/s43/s86\n/s68/s41\n/s32/s40/s98/s41\nFIG. 2. (a)The non-interacting density of states D−(E)\nnear the bottom of the band for four values of the spin-\norbit coupling strengths: ( VR,VD)/t= (0.5,0.5) (dot-dashed\ncurve), (0 .8,0.2) (dotted curve), (0 .9,0.1) (dashed curve),\nand (0.99,0.01) (solid curve). Note that for equal coupling\nstrengths there is no van Hove singularity at low energies.\n(b) The value of the density of states at the bottom of the\nband (ground state) as a function of VD(while the total cou-\npling strength, VR+VD, is held constant. The value of the\ndensity of states achieves a minimum value when VR=VD.\nForVR= 0 orVD= 0 there is a discontinuity, caused by\nthe transition from a doubly degenerate ground state to a\nfour-fold degenerate ground state.Note that when the coupling strengths are equal, the\ndensity of states has a minimum. Also note that when\none kind of spin-orbit coupling vanishes, e.g. VR= 0,\norVD= 0, there will be a discontinuity for the density\nof states (the density of states jumps to twice its value).\nThis is caused by a transition from a doubly degenerate\ngroundstate to a four-fold degenerateground state. This\ndiscontinuity will also appear for VD≃0 orVR≃0 near\nthe bottom of the band as can be seen from Fig.2(a) for\nVR= 0.99,VD= 0.01.\nIII. RESULTS WITH THE\nELECTRON-PHONON INTERACTION\nAs the electron phonon interaction is turned on, the\nground state energy (effective mass) will decrease (in-\ncrease) due to polaron effects. To study the polaron\nproblem numerically, we adopt the variational method\noutlined by Trugman and coworkers,18,19which is a con-\ntrolled numerical technique to determine polaron prop-\nerties in the thermodynamic limit exactly. This method\nwas recently further developed20,21to study the polaron\nproblem near the adiabatic limit with Rashba spin-orbit\ncoupling.17This case was also studied in Ref. [16] using\nthe Momentum Average Approximation.22\nIn Fig. 3, we show the ground state energy and\nthe effective mass correction as a function of the elec-\ntron phonon coupling λ≡2g2ωE/(4πt),20for various\nspin-orbit coupling strengths, but with the sum fixed:\nVR+VD=t. These are compared with the results\nfrom the Rashba-Holstein model with VD= 0. Here the\nphonon frequency is set to be ωE/t= 1.0, which is the\ntypical value used in Ref.[16], and for each value of VR,\nthe ground state energy is compared to the correspond-\ning result for λ= 0. The numerical results are compared\nwith results from the MA method and from Lang-Firsov\nstrong coupling theory23,24(see Appendix). In Fig. 3(a),\nthe ground state energycrossesoversmoothly (at around\nλ≈0.8)fromthe delocalizedelectronregimetothe small\npolaron regime. In the whole regime, the ground state\nenergy is shifted up slightly as the Dresselhaus spin-orbit\ncoupling, VD, is increased in lieu of the Rashba spin-\norbit coupling. We show results for VD≤VR, as the\ncomplementary regime is completely symmetric. The\nMA results agree very well with the exact results and\nthe Lang-Firsov strong coupling results agree well in the\nλ≥1 regime. Similarly, weak coupling perturbation\ntheory17agrees with the exact results for λ≤1 (not\nshown). Fig. 3(b) shows the effective mass as a func-\ntion of coupling strength; it decreases slightly, for a given\nvalue ofλ, by increasing VDin lieu of VR.\nAll these results are plotted as a function of the elec-\ntron phonon coupling strength, λ, as defined above; this\ndefinition requires the value of the electron density of\nstates at the bottom of the band, and we have elected to\nuse, for any value of spin-orbit coupling, the value 1 /(4πt\nappropriate to nospin-orbit coupling. If the actual DOS4\n/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48 /s49/s46/s50/s45/s53/s45/s52/s45/s51/s45/s50/s45/s49/s48\n/s69/s47/s116/s61/s49/s46/s48\n/s32/s76/s70/s44/s32 /s86\n/s82/s47/s116/s61/s48/s46/s53\n/s32/s76/s70/s44/s32 /s86\n/s82/s47/s116/s61/s48/s46/s56\n/s32/s76/s70/s44/s32 /s86\n/s82/s47/s116/s61/s49/s46/s48/s86\n/s82/s47/s116/s43/s86\n/s68/s47/s116/s61/s49/s46/s48/s32/s69/s120/s97/s99/s116/s44/s32/s86\n/s82/s47/s116/s61/s48/s46/s53\n/s32/s69/s120/s97/s99/s116/s44/s32/s86\n/s82/s47/s116/s61/s48/s46/s56\n/s32/s69/s120/s97/s99/s116/s44/s32/s86\n/s82/s47/s116/s61/s49/s46/s48\n/s32/s77/s46/s65/s46/s32/s86\n/s82/s47/s116/s61/s48/s46/s53\n/s32/s77/s46/s65/s46/s32/s86\n/s82/s47/s116/s61/s48/s46/s56\n/s32/s77/s46/s65/s46/s32/s86\n/s82/s47/s116/s61/s49/s46/s48/s40/s69\n/s71/s83/s45/s69\n/s48/s41/s47/s116\n/s32/s40/s97/s41\n/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s49/s46/s48/s49/s46/s53/s50/s46/s48/s50/s46/s53/s51/s46/s48/s51/s46/s53/s52/s46/s48/s52/s46/s53/s53/s46/s48\n/s86\n/s82/s47/s116/s43/s86\n/s68/s47/s116/s61/s49/s46/s48/s32/s77/s46/s65/s46/s32/s86\n/s82/s47/s116/s61/s48/s46/s53\n/s32/s77/s46/s65/s46/s32/s86\n/s82/s47/s116/s61/s48/s46/s56\n/s32/s77/s46/s65/s46/s32/s86\n/s82/s47/s116/s61/s49/s46/s48\n/s32/s32/s69/s120/s97/s99/s116/s44/s32/s86\n/s82/s47/s116/s61/s48/s46/s53\n/s32/s32/s69/s120/s97/s99/s116/s44/s32/s86\n/s82/s47/s116/s61/s48/s46/s56\n/s32/s32/s69/s120/s97/s99/s116/s44/s32/s86\n/s82/s47/s116/s61/s49/s46/s48/s109/s42/s47/s109\n/s83/s79\n/s32/s69/s47/s116/s61/s49/s46/s48/s40/s98/s41\nFIG. 3. (a) Ground state energy difference EGS−E0vs.λfor\nVR/t= 0.5,0.8,1.0 andωE/t= 1.0 while the total coupling\nstrength is kept fixed: VR+VD=t. Exact numerical results\nare compared with those from the Momentum Average (MA)\nmethod. Agreement is excellent. Strong coupling results ar e\nalso plotted (in red) by utilizing the Lang-Firsov (LF) stro ng\ncoupling approximation. Agreement in the strong coupling\nregime ( λ≥1) is excellent. (b) Effective mass m∗/mSOvs.\nλ. MA results are plotted (symbols) with the exact numerical\nresults, and again, agreement is excellent. In both (a) and ( b)\nthe polaronic effects are minimized for VR=VD./s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s45/s48/s46/s53/s45/s48/s46/s52/s45/s48/s46/s51/s45/s48/s46/s50/s45/s48/s46/s49\n/s86\n/s82/s47/s116/s43/s86\n/s68/s47/s116/s61/s49/s46/s48/s32/s77/s46/s65/s46\n/s69/s47/s116/s61/s48/s46/s49\n/s32/s77/s46/s65/s46\n/s69/s47/s116/s61/s48/s46/s50\n/s32/s77/s46/s65/s46\n/s69/s47/s116/s61/s49/s46/s48\n/s32/s69/s120/s97/s99/s116\n/s69/s47/s116/s61/s48/s46/s49\n/s32/s69/s120/s97/s99/s116\n/s69/s47/s116/s61/s48/s46/s50\n/s32/s69/s120/s97/s99/s116\n/s69/s47/s116/s61/s49/s46/s48/s40/s69\n/s71/s83/s45/s69\n/s48/s41/s47/s116\n/s86\n/s68/s47/s116\n/s32/s61/s48/s46/s51/s50/s40/s97/s41\n/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s49/s46/s49/s48/s49/s46/s49/s53/s49/s46/s50/s48/s49/s46/s50/s53/s49/s46/s51/s48/s49/s46/s51/s53/s49/s46/s52/s48/s49/s46/s52/s53/s49/s46/s53/s48/s49/s46/s53/s53\n/s32/s77/s46/s65/s46\n/s69/s47/s116/s61/s48/s46/s49\n/s32/s77/s46/s65/s46\n/s69/s47/s116/s61/s48/s46/s50\n/s32/s77/s46/s65/s46\n/s69/s47/s116/s61/s49/s46/s48\n/s32/s69/s120/s97/s99/s116\n/s69/s47/s116/s61/s48/s46/s49\n/s32/s69/s120/s97/s99/s116\n/s69/s47/s116/s61/s48/s46/s50\n/s32/s69/s120/s97/s99/s116/s44\n/s69/s47/s116/s61/s49/s46/s48/s86\n/s82/s47/s116/s43/s86\n/s68/s47/s116/s61/s49/s46/s48/s109/s42/s47/s109\n/s83/s79\n/s86\n/s68/s47/s116\n/s32/s61/s48/s46/s51/s50/s40/s98/s41\nFIG.4. (a)Groundstateenergy EGS−E0asafunctionofspin\norbit coupling VD/tforωE/t= 0.1,0.2,1.0 with weakelectron\nphonon coupling, λ= 0.32, and moderate spin-orbit coupling,\nVR+VD=t. (b) Effective mass m∗/mSOas a function of\nspin orbit coupling VD/tfor the same parameters. MA results\nare again compared with the exact numerical results, and are\nreasonably accurate for these parameters.5\nappropriate to the value of spin-orbit coupling were used\nin the definition of λ, then the effective mass, for ex-\nample, would vary even more with varying VDvs.VR\n(see Fig. 2(b)). Moreover, this variation would be more\npronounced for lower values of ωE.\nIn Fig. 4, we show results for the ground state energy\nand effective mass for different values of the Einstein\nphonon frequency, ωE; MA results are also shown for\ncomparison. In these plots the electron phonon coupling\nstrength is kept fixed and VDis varied while maintaining\nthe total spin-orbit coupling constant. The ground state\nenergy has a maximum when the two spin-orbit coupling\nstrengths, VDandVR, are tuned to be equal; similarly,\ntheeffectivemasshasaminimumwhenthetwoareequal.\nAs the phonon frequency is reduced the minimum in the\neffective mass becomes more pronounced. The MA re-\nsults track the exact results, and, as found previously,17\nare slightly less accurate as the phonon frequency be-\ncomes much lower than the hopping matrix element, t.\nIV. SUMMARY\nLinear spin-orbit coupling can arise in two varieties;\ntaken on their own, they are essentially equivalent, and\ntheir impact on a single electron, even in the presence of\nelectron phonon interactions, will be identical. However,with the ability to tune either coupling constant, in both\nsolid state and cold atom experiments, one can probe\nthe degree of Dresselhaus vs. Rashba spin-orbit coupling\nthroughtheimpactonpolaronicproperties. Theprimary\neffect of this variation is the electron density of states,\nwhere the van Hove singularity can be moved as a func-\ntion of chemical potential (i.e. doping) through tuning of\nthe spin-orbit parameters. These conclusions are based\non exact methods (the so-called Trugman method), and\nare not subject to approximations. These results have\nbeen further corroborated and understood through the\nMomentum Average approximation, and through weak\nand strong coupling perturbation theory. The effect is\nexpected to be experimentally relevant since in typi-\ncal materials with large spin-orbit couplings the phonon\nfrequency is small when compared to the bandwidth,\nωE/t≪1.\nACKNOWLEDGMENTS\nThis work was supported in part by the Natural\nSciences and Engineering Research Council of Canada\n(NSERC), by ICORE (Alberta), by the Flemish Science\nFoundation (FWO-Vl) and by the Canadian Institute for\nAdvanced Research (CIfAR).\n1See, for example, Quantum Dot Heterostructures , byDieter\nBimberg, Marius Grundmann, and Nikolai N. Ledentsov\n(John Wiley and Sons, Toronto, 1999).\n2For example, Mesoscopic Systems , by Y. Murayama\n(Wiley-VCH, Toronto, 2001).\n3S. A. Wolf, D. D. Awschalom, R. A. Buhrman, J. M.\nDaughton, S. von Molnar, M. L. Roukes, A. Y. Chtchelka-\nnova and D. M. Treger, Science 294, 1488, (2001).\n4J. D. Koralek, C. P. Weber, J. Orenstein, B. A. Bernevig,\nShou-Cheng Zhang, S. Mack and D. D. Awschalom, Na-\nture,458, 610-613(2009).\n5R. Winkler, Spin-Orbit Coupling Effects in Two-\nDimensional Electron and Hole Systems (Springer, Berlin,\n2003).\n6E.I. Rashba, Sov. Phys. Solid State 2, 1109 (1960).\n7Dresselhaus, G. Phys. Rev. 100, 580–586 (1955).\n8S.K. Maiti, S. Sil, and A. Chakrabarti, arXiv:1109.5842v2.\n9L. Meier, G. Salis, I. Shorubalko, E. Gini, S. Sch¨ on, and\nK. Ensslin, Nature Physics 3, 650 (2007).\n10Y. J. Lin, K. Jimenez-Garcia and I. B. Spielman, Nature,\n471, 83 (2011). See alsoT. Ozawa and G. Baym, Phys.\nRev. A85, 013612 (2012).\n11B.A. Bernevig, J. Orenstein, and S.-C. Zhang, Phys. Rev.\nLett.97, 236601 (2006).\n12I. Bloch, J. Dalibard, and W. Zwerger, Rev. Mod. Phys.\n80, 885 (2008).\n13F. Herrera and R.V. Krems, Phys. Rev. A 84, 051401(R),\n(2011).\n14T. Holstein, Ann. Phys. (New York) 8, 325 (1959).15E. Cappelluti, C. Grimaldi and F. Marsiglio, Phys. Rev.\nLett98, 167002 (2007); Phys. Rev. B 76, 085334 (2007).\nSee also, C. Grimaldi, E. Cappelluti, and F. Marsiglio,\nPhys. Rev. Lett. 97, 066601 (2006); Phys. Rev. B 73,\n081303(R) (2006).\n16L. Covaci and M. Berciu, Phys. Rev. Lett 102, 186403\n(2009).\n17Zhou Li, L.Covaci, M. Berciu, D. Baillie and F. Marsiglio,\nPhys. Rev. B 83, 195104, (2011).\n18S.A. Trugman, in Applications of Statistical and Field The-\nory Methods to Condensed Matter , edited by D. Baeriswyl,\nA.R. Bishop, and J. Carmelo (Plenum Press, New York,\n1990).\n19J. Bonˇ ca, S.A. Trugman, and I. Batist´ ıc, Phys. Rev.\nB60,1633 (1999).\n20Zhou Li, D. Baillie, C. Blois, and F. Marsiglio, Phys. Rev.\nB81, 115114, (2010).\n21A. Alvermann, H. Fehske, and S.A. Trugman, Phys. Rev.\nB81, 165113 (2010).\n22M. Berciu, Phys. Rev. Lett 97, 036402 (2006).\n23I.G. Lang and Yu. A. Firsov, Sov. Phys. JETP16, 1301\n(1963); Sov. Phys. Solid State 52049 (1964).\n24F. Marsiglio, Physica C 24421, (1995).\nAppendix A: Density of States and effective mass\nExpanding εk,−around the minimum energy E0,\nby defining k′\nx=kx±arctan(VR+VD√\n2t),k′\ny=ky±6\narctan(VR+VD√\n2t),we have\nεk,−=E0+˜t1/braceleftbig\nk′2\nx+k′2\ny/bracerightbig\n±˜t2k′\nxk′\ny,(A1)\nwhere\n˜t1=t/braceleftBigg\n1+(VR+VD)2\n2t2−(VR−VD)2\n2(VR+VD)2/radicalbig\n1+(VR+VD)2/(2t2)/bracerightBigg\n,(A2)\nand\n˜t2=t/braceleftBigg(VR−VD)2\n(VR+VD)2/radicalbig\n1+(VR+VD)2/(2t2)/bracerightBigg\n,(A3)\nNote that, with generic spin-orbit coupling, the effective\nmass is in general anisotropic, but when VD=VR, it\nbecomes isotropic.\nTo calculate the density of states at the bottom of the\nband, from the definition, we have\nD−(E0+δE) =1\n4π2/integraldisplayπ\n−πdkx/integraldisplayπ\n−πdkyδ(E0+δE−εk,−),\n(A4)\nwhereδEis a small amount of energy above the bottom\nof the band, E0. Around the two energy minimum points\nthere are two small regions which will contribute to this\nintegral. We choose one of them (and then multiply our\nresultbyafactorof2),thenusethedefinitionsof k′above\ninstead of k, and introduce a small cutoff kc, which is the\nradius of a small circle around kmin. Thus the integral\nbecomes\nD−(E0+δE) = 2×1\n4π2/integraldisplaykc\n0k′dk′/integraldisplayπ\n−πdθ\nδ/bracketleftbigg\nδE−/braceleftbig˜t1+1\n2˜t2sin2θ/bracerightbig\nk′2/bracketrightbigg\n=1\n2πt1/radicalBig\n1+(VR+VD)2\n2t2−(VR−VD)2\n(VR+VD)2(A5)\nIn the weak electron-phonon coupling regime, perturba-\ntion theory can be applied to evaluate the effective mass;\nthe self energy to first order in λis given by\nΣweak(ω+iδ) =πλtωE/summationdisplay\nk,s=±1\nω+iδ−ωE−εk,s.(A6)Theeffectivemasscanbeobtainedthroughthederivative\nof the self energy\nm∗\nweak\nmSO= 1−∂\n∂ωΣweak(ω+iδ)|ω=E0.(A7)\nBy inserting the expansion of εk,−around the minimum\nenergyE0into Eqn.[A6] and Eqn.[A7], we obtain the\neffective mass near the adiabatic limit as\nm∗\nweak\nmSO= 1+λ\n21/radicalBig\n1+(VR+VD)2\n2t2−(VR−VD)2\n(VR+VD)2.(A8)\nThe effective mass has a minimum for VR=VDwhile\nVR+VDis a constant.\nAppendix B: Strong coupling theory\nTo investigate the strong coupling regime of the\nRashba-Dresselhaus-Holstein model for a single polaron,\nwe use the Lang-Firsov2324unitary transformation H=\neSHe−S, whereS=g/summationtext\ni,σni,σ(ai−a†\ni). Following pro-\ncedures similar to those in Ref. (17), we obtain the first\norder perturbation correction to the energy as\nE(1)\nk±=e−g2εk±−g2ωE, (B1)\nwheregis the band narrowing factor, as used in the Hol-\nstein model. To find the second order correction to the\nground state energy, we proceed as in Ref. (17), and find\nE(2)\nk−=−4e−2g2t2+(VR)2+(VD)2\nωE\n×/bracketleftbig\nf(2g2)−f(g2)/bracketrightbig\n−e−2g2f(g2)ǫ2\nk−\nωE,(B2)\nwheref(x)≡∞/summationtext\nn=11\nnxn\nn!≈ex/x/bracketleftbig\n1 + 1/x+ 2/x2+.../bracketrightbig\n.\nThus the ground state energy, excluding exponentially\nsuppressed corrections, is\nEGS=−2πtλ/parenleftbig\n1+2t2+(VR)2+(VD)2\n(2πtλ)2/parenrightbig\n,(B3)\nand there is a correction of order 1 /λ2compared to the\nzeroth order result. Corrections in the dispersion enter\nin strong coupling only with an exponential suppression.\nThe ground state energy predicted by strong coupling\ntheory has a maximum for VR=VDwhileVR+VDis a\nconstant."    },    {        "title": "1606.05758v1.Spin_Transport_at_Interfaces_with_Spin_Orbit_Coupling__Formalism.pdf",        "content": "Spin Transport at Interfaces with Spin-Orbit Coupling: Formalism\nV. P. Amin1, 2,\u0003and M. D. Stiles2\n1Maryland NanoCenter, University of Maryland, College Park, MD 20742\n2Center for Nanoscale Science and Technology, National Institute\nof Standards and Technology, Gaithersburg, Maryland 20899, USA\n(Dated: June 21, 2016)\nWe generalize magnetoelectronic circuit theory to account for spin transfer to and from the atomic\nlattice via interfacial spin-orbit coupling. This enables a proper treatment of spin transport at inter-\nfaces between a ferromagnet and a heavy-metal non-magnet. This generalized approach describes\nspin transport in terms of drops in spin and charge accumulations across the interface (as in the\nstandard approach), but additionally includes the responses from in-plane electric \felds and o\u000bsets\nin spin accumulations. A key \fnding is that in-plane electric \felds give rise to spin accumulations\nand spin currents that can be polarized in any direction, generalizing the Rashba-Edelstein and\nspin Hall e\u000bects. The spin accumulations exert torques on the magnetization at the interface when\nthey are misaligned from the magnetization. The additional out-of-plane spin currents exert torques\nvia the spin-transfer mechanism on the ferromagnetic layer. To account for these phenomena we\nalso describe spin torques within the generalized circuit theory. The additional e\u000bects included in\nthis generalized circuit theory suggest modi\fcations in the interpretations of experiments involving\nspin orbit torques, spin pumping, spin memory loss, the Rashba-Edelstein e\u000bect, and the spin Hall\nmagnetoresistance.\nI. INTRODUCTION\nThe spin-orbit interaction couples the spin and mo-\nmentum of carriers, leading to a variety of important\ne\u000bects in spintronic devices. It enables the conversion\nbetween charge and spin currents [1, 2], allows for the\ntransfer of angular momentum between populations of\nspins [3{9], couples charge transport and thermal trans-\nport with magnetization orientation [10{17], and results\nin magnetocrystalline anisotropy [18{20]. Many of these\ne\u000bects already facilitate technological applications. The\ndevelopment of such applications can be assisted by both\npredictive (yet complicated) \frst-principles calculations\nand clear phenomenological models, which would aid the\ninterpretation of experiments and help to predict device\nbehavior.\nIn multilayer systems, bulk spin-orbit coupling plays a\ncrucial role in spin transport but the role of interfacial\nspin-orbit coupling remains largely unknown. This un-\ncertainty derives from the uncharacterized transfer of an-\ngular momentum between carriers and the atomic lattice\nwhile scattering from interfaces with spin-orbit coupling.\nThis transfer of angular momentum occurs because a car-\nrier's spin is coupled via spin-orbit coupling to its orbital\nmoment, which is coupled via the Coulomb interaction to\nthe crystal lattice. Such interfaces behave as either a sink\nor a source of spin polarization for carriers in a way that\ndoes not yet have an accurate phenomenological descrip-\ntion. In this paper we develop a formal generalization of\nmagnetoelectronic circuit theory to treat interfaces with\nspin-orbit coupling. In a companion paper, we extract\nthe most important consequences of this generalization\n\u0003vivek.amin@nist.govand show that they capture the dominant e\u000bects found\nin more complicated Boltzmann equation calculations.\nTo understand the impact of interfacial spin-orbit cou-\npling we consider a heavy metal/ferromagnet bilayer,\nwhere in-plane currents generate torques on the magne-\ntization through various mechanisms that involve spin-\norbit coupling [6, 7, 21{24]. For example, bulk spin-orbit\ncoupling converts charge currents in the heavy metal\ninto orthogonally-\rowing spin currents, through a pro-\ncess known as the spin Hall e\u000bect [25{31]. Upon entering\nthe ferromagnetic layer these spin currents transfer angu-\nlar momentum to the magnetization through spin trans-\nfer torques [32{36]. Both the spin Hall e\u000bect and spin-\ntransfer torques have been extensively studied, but addi-\ntional sources contribute to the total spin torque. These\nremaining contributions arise from interfacial spin-orbit\ncoupling, which enables carriers of the in-plane charge\ncurrent to develop a net spin polarization at the inter-\nface [37{41]. In systems with broken inversion symmetry\n(such as interfaces) the generation of such spin polariza-\ntion is known as the Rashba-Edelstein e\u000bect. This spin\npolarization can exert a torque on any magnetization at\nthe interface via the exchange interaction [7, 42]. A re-\ncent experiment suggests that this mechanism can induce\nmagnetization switching alone, without relying on the\nbulk spin Hall e\u000bect [24].\nThe spin torque driven by the Rashba-Edelstein e\u000bect\nis typically studied by con\fning transport to the two-\ndimensional interface. Semiclassical models can capture\nthe direct and inverse Rashba-Edelstein e\u000bects [43{46] in\nthis scenario. However, such models are not realistic de-\nscriptions of bilayers, in which carriers scatter both along\nand across the interface. Since spin transport across the\ninterface is a\u000bected by the transfer of angular momentum\nto the atomic lattice, the resulting spin torques are modi-\n\fed in ways that two-dimensional models cannot capture.arXiv:1606.05758v1  [cond-mat.mes-hall]  18 Jun 20162\nFIG. 1. (Color online) (a) A heavy metal/ferromagnet bilayer subject to an in-plane electric \feld. The axes directly below the\nbilayer is used to describe electron \row, where the z-axis points normal to the interface plane. The other axes is used to describe\nspin orientation, where the direction `points along the magnetization while the directions dandfspan the plane transverse to\n`. (b) Depiction of the physics described by the spin mixing conductance. Spins incident from the heavy metal brie\ry precess\naround the magnetization when re\recting o\u000b of the interface. The imaginary part of the spin mixing conductance describes the\nextent of this precession. Interfacial spin-orbit coupling changes the e\u000bective magnetic \feld seen by carriers during this process\nin a momentum-dependent way; this alters the precession axis for each carrier and thus modi\fes the spin mixing conductance.\n(c) Depiction of the loss of spin polarization that carriers experience while crossing interfaces with spin-orbit coupling. Without\ninterfacial spin-orbit coupling, carriers retain the portion of their spin polarization aligned with the magnetization, but lose the\nportion polarized transversely to the magnetization due to dephasing processes just within the ferromagnet. With interfacial\nspin-orbit coupling, carriers trade angular momentum with the atomic lattice; this leads to changes in all components of the\nspin polarization. This phenomenon, known as spin memory loss, a\u000bects each component di\u000berently. The panel illustrates only\nthe loss in spin polarization aligned with magnetization. (d) Depiction of interfacial spin-orbit scattering in the presence of\nan in-plane electric \feld. Interfacial spin-orbit coupling allows for spins aligned with the magnetization to become misaligned\nupon re\rection and transmission. For the scattering potential discussed in Sec. IV, the spin of a single re\rected carrier cancels\nthe spin of a single carrier transmitted from the other side of the interface. However, a net cancellation of spin is prevented if\nthe total number of incoming carriers di\u000bers between sides, as can happen in the presence of in-plane current \row. This occurs\nbecause an in-plane electric \feld drives two di\u000berent charge currents within each layer; this forces the number of carriers with\na given in-plane momentum to di\u000ber on each side of the interface. The scattered carriers then carry a net spin polarization\nand a net spin current.\nThe various contributions to spin torques in bilayers re-\nmain di\u000ecult to distinguish experimentally [22, 23] in\npart because of the lack of models that accurately cap-\nture interfacial spin-orbit coupling [42].\nInterfacial spin-orbit coupling may play an impor-\ntant role in other phenomena. Spin pumping is one\nexample; it describes the process in which a precess-\ning magnetization generates a spin current [8]. In\nheavy metal/ferromagnet bilayers, the pumped spin cur-\nrent \rows from the ferromagnet into the heavy metal,\nwhere the inverse spin Hall e\u000bect generates an orthog-\nonal charge current [47{51]. However, because inter-\nfacial spin-orbit coupling transfers spin polarization to\nthe atomic lattice, it modi\fes the pumped spin current\nas it \rows across the interface. This transfer of spin\npolarization remains uncharacterized in many systems,\nthus contributing to inconsistencies in the quantitative\ninterpretation of experiments [9, 52{54]. Another exam-\nple, known as the spin Hall magnetoresistance, describes\nthe magnetization-dependent in-plane resistance of heavy\nmetal/ferromagnet bilayers [55{61]. Currently this e\u000bect\nis attributed to magnetization-dependent scattering at\nthe interface, but may also contain a contribution frominterfacial spin-orbit scattering. The impact of interfa-\ncial spin-orbit coupling on these e\u000bects remains unclear\ndue to the absence of appropriate models with which to\nanalyze the data.\nMagnetoelectronic circuit theory is the most frequently\nused approach to model spin currents at the interface\nbetween a non-magnet and a ferromagnet. It describes\nspin transport in terms of four conductance parameters,\nwhere drops in spin-dependent electrochemical poten-\ntials across the interface play the role of traditional volt-\nages. However, the theory cannot describe interfaces with\nspin-orbit coupling because it does not consider spin-\n\rip processes due to spin-orbit coupling at the interface.\nFig. 1(a) depicts a typical scattering process described by\none of these conductance parameters. Given its success\nin describing spin transport in normal metal/ferromagnet\nbilayers, generalizing magnetoelectronic circuit theory to\ninclude interfacial spin-orbit coupling would make it a\nvaluable tool for describing heavy metal/ferromagnet bi-\nlayers.\nTo generalize magnetoelectronic circuit theory one\nmust consider all the ways that interfacial spin-orbit cou-\npling potentially a\u000bects spin transport. One such e\u000bect,3\nknown as spin memory loss, describes a loss of spin cur-\nrent across interfaces due to spin-orbit coupling. We il-\nlustrate a process that contributes to spin memory loss\nin Fig. 1(b). This loss occurs when the atomic lattice\nat the interface behaves as a sink of angular momen-\ntum. Recent work [62] incorporates this behavior into a\ntheory for spin pumping, but descriptions of this e\u000bect\ndate back to over a decade ago [63{66]. Thus generaliz-\ning magnetoelectronic circuit theory for interfaces with\nspin-orbit coupling requires accounting for spin memory\nloss. By incorporating spin-\rip processes at the inter-\nface into magnetoelectronic circuit theory, one can treat\nthis aspect of the phenomenology of interfacial spin-orbit\ncoupling.\nAnother important consequence of interfacial spin-\norbit coupling is that in-plane electric \felds can create\nspin currents that \row away from the interface. First\nprinciples calculations of Pt/Py bilayers suggest that a\ngreatly enhanced spin Hall e\u000bect occurs at the inter-\nface (as compared to the bulk) that could generate such\nspin currents [67]. This suggests that in-plane electric\n\felds (and not just drops in spin and charge accumula-\ntions across the interface) must play a role in generaliza-\ntions of magnetoelectronic circuit theory. It also suggests\nthat one cannot con\fne transport to the two-dimensional\ninterface when describing the e\u000bect of in-plane electric\n\felds. Instead, one must consider transport both along\nand across the interface. Some of the consequences of this\nthree-dimensional picture have been investigated in mul-\ntilayer systems containing an insulator [68, 69]. The only\nsemiclassical calculations of three-dimensional metallic\nbilayers are based on the Boltzmann equation [42]. Like\nspin memory loss, these spin currents must be included\nin generalizations of magnetoelectronic circuit theory to\nfully capture the e\u000bect of interfacial spin-orbit coupling.\nIn the following we give a semiclassical picture of how\nsuch spin currents arise, and how they exert magnetic\ntorques that are typically not considered in bilayers.\nFig. 1(c) depicts how spins aligned with the magneti-\nzation scatter from an interface with spin-orbit coupling.\nFor the scattering potential discussed in Sec. IV, single\nre\rected and transmitted spins cancel on each side of the\ninterface. However, the netcancellation of spin is avoided\nif the number of incoming carriers di\u000bers between sides.\nIn the simplest scenario, this occurs if the in-plane elec-\ntric \feld drives di\u000berent currents within each layer, so\nthat the occupancy of carriers di\u000bers on either side for\na given in-plane momentum. We \fnd that through this\nmechanism, carriers subject to interfacial spin-orbit scat-\ntering can carry a net spin current in addition to exhibit-\ning a net spin polarization. If the net spin polarization is\nmisaligned with the magnetization, it can exert a torque\non the magnetization at the interface. This describes the\ncontribution to the spin torque normally associated with\nthe Rashba-Edelstein e\u000bect (discussed earlier). However,\nthe spin currents created by interfacial spin-orbit scatter-\ning can \row away from the interface, and those that \row\ninto the ferromagnet exert additional torques. Althoughthese spin currents generate torques via the spin-transfer\nmechanism, they arise from interfacial spin-orbit scat-\ntering instead of the spin Hall e\u000bect. This mechanism,\nwhich cannot be captured by con\fning transport to the\ntwo-dimensional interface, is not usually considered when\nanalyzing spin torques in bilayers. However, it can con-\ntribute to the total spin torque in important ways. For\ninstance, it allows for spin torques generated by inter-\nfacial spin-orbit coupling to point in directions typically\nassociated with the spin Hall e\u000bect. The spin polariza-\ntion and \row directions of these spin currents are not\nrequired to be orthogonal to each other or the electric\n\feld, unlike the spin currents generated by the spin Hall\ne\u000bect in in\fnite bulk systems. More work is needed to\ndetermine how this semiclassical description of interfacial\nspin current generation compares with the \frst principles\ndescription of an enhanced interfacial spin Hall e\u000bect [67].\nIn this paper, we generalize magnetoelectronic circuit\ntheory to include interfacial spin-orbit coupling. Not only\ndoes interfacial spin-orbit coupling modify the conduc-\ntance parameters introduced by magnetoelectronic cir-\ncuit theory, it requires additional conductivity parame-\nters to capture the spin currents that arise from in-plane\nelectric \felds and spin-orbit scattering. Furthermore, the\ntransfer of angular momentum between carriers and the\natomic lattice at the interface alters the spin torque that\ncarriers can exert on the magnetization; this introduces\nadditional parameters that are needed to distinguish spin\ntorques from spin currents. However, we \fnd that many\nof the parameters in this generalized circuit theory may\nbe neglected when modeling spin-orbit torques in bilayer\nsystems, and that including the conductivity and spin\ntorque parameters is more important than modifying the\nconductance parameters. As with magnetoelectronic cir-\ncuit theory, we provide microscopic expressions for most\nparameters.\nIn a companion paper, to highlight the utility of the\nproposed theory, we produce an analytical model describ-\ning spin-orbit torques caused by the spin-Hall and inter-\nfacial Rashba-Edelstein e\u000bects. We achieve this by solv-\ning the drift-di\u000busion equations with this generalization\nof magnetoelectronic circuit theory. In that paper, we\nfocus on only the parameters that describe the response\nof in-plane electric \felds, and neglect all other changes\nto magnetoelectronic circuit theory. We show that this\nsimpli\fed approach captures the most important e\u000bects\nfound in Boltzmann equation calculations of a model sys-\ntem. In this paper, we discuss the complete generaliza-\ntion of magnetoelectronic circuit theory in the presence\nof interfacial spin-orbit coupling.\nIn Sec. II of this paper we describe spin transport at in-\nterfaces with and without interfacial spin-orbit coupling.\nIn Sec. III we motivate the derivation of all parame-\nters, leaving some details for appendices A and B. In\nSec. IV we perform a numerical analysis of each bound-\nary parameter for a scattering potential relevant to heavy\nmetal/ferromagnet bilayers. This analysis allows us to\ndetermine which parameters matter the most in these4\nsystems. Finally, in Sec. V we discuss implications of our\ntheory on experiments involving spin orbit torque, spin\npumping, the Rashba-Edelstein e\u000bect, and the spin Hall\nmagnetoresistance.\nII. SPIN AND CHARGE TRANSPORT AT\nINTERFACES\nIn the following we discuss the general phenomenology\nof spin transport at interfaces with and without spin or-\nbit coupling. We \frst describe some conventional spin\ntransport models to build up to the proposed model, and\nrefrain from presenting explicit expressions of any param-\neters until later sections.\nA. Collinear spin transport\nIn the absence of spin-\rip processes one often assigns\nseparate current densities for majority ( j\") and minority\n(j#) carriers, i.e.\nj\"=G\"\u0001\u0016\"j#=G#\u0001\u0016#: (1)\nHereG\"=#denotes the spin-dependent interfacial conduc-\ntance, while \u0001 \u0016\"=#refers to the drop in quasichemical\npotential for each carrier population across the interface.\nWe may then de\fne charge ( c) and spin ( s) components\nfor the drop in quasichemical potential\n\u0001\u0016c= \u0001\u0016\"+ \u0001\u0016# (2)\n\u0001\u0016s= \u0001\u0016\"\u0000\u0001\u0016#; (3)\nand for the current densities\njc=j\"+j# (4)\njs=j\"\u0000j#: (5)\nacross the interface. Using the following modi\fed con-\nductance parameters\nG\u0006=1\n2\u0000\nG\"\u0006G#\u0001\n; (6)\nwe may rewrite Eq. (1) as\n \njs\njc!\n= \nG+G\u0000\nG\u0000G+! \n\u0001\u0016s\n\u0001\u0016c!\n(7)\ninstead. In this case both spin and charge currents are\ncontinuous across the interface.\nB. Magnetoelectronic Circuit Theory\nWhen describing spin orientation in bulk ferromagnetic\nsystems, the magnetization direction provides a naturalspin quantization axes. However, at the interface be-\ntween a non-magnet and a ferromagnet, the net spin po-\nlarizations of each region need not align. To account\nfor this, one must consider spins in the non-magnet that\npoint in any direction. In the ferromagnet, spins are mis-\naligned with the magnetization near the interface but be-\ncome aligned in the bulk. This occurs because spins pre-\ncess incoherently around the magnetization; eventually\nthe net spin polarization transverse to the magnetization\nvanishes. In transition metal ferromagnets and their al-\nloys, this dephasing happens over distances smaller than\nthe spin di\u000busion length.\nTo describe electron \row and spin orientation in non-\nmagnet/ferromagnet bilayers, we use two separate coor-\ndinate systems. For electron \row, we choose the x=y\nplane to lie along the interface and the z-axis to point\nperpendicular to it. The interface is located at the z-axis\norigin, and z= 0\u0000andz= 0+describe the regions just\nwithin the non-magnet and ferromagnet respectively. To\ndescribe spin orientation, we choose the direction `to\nbe along the magnetization ( ^`=^m) and the directions\ndandfto be perpendicular to ^`. The damping-like\n(d) and \feld-like ( f) directions point along the vectors\n^d/^m\u0002[^m\u0002(\u0000E\u0002^z)] and ^f/^m\u0002(\u0000E\u0002^z) re-\nspectively. This provides a convenient coordinate system\nfor describing spin-orbit torques, because torques with a\ndamping-like component push the magnetization towards\nthe\u0000E\u0002^zdirection, while those with a \feld-like com-\nponent force the magnetization to precess about \u0000E\u0002^z.\nWe \frst de\fne the spin and charge accumulations at\nthe interface ( \u0016\u000b), where the index \u000b2[d;f;`;c;`\u0003;c\u0003]\ndescribes the type of accumulation. The \frst four indices\ndenote the spin ( d,f,l) and charge ( c) accumulations in\nthe non-magnet at z= 0\u0000. The last two indices describe\nthe spin (`\u0003) and charge ( c\u0003) accumulations in the fer-\nromagnet at z= 0+. In the ferromagnet we omit spin\naccumulations aligned transversely to the magnetization,\ndue to the dephasing processes discussed above. Note\nthat the charge and spin components of \u0016\u000bhave units\nof voltage. We then de\fne the spin and charge current\ndensities \rowing out-of-plane ( jz\u000b) in an identical fash-\nion. The charge and spin components of jz\u000bhave the\nunits of number current density [70]. We refer to \u000bas\nthe spin/charge index.\nOne may rede\fne any tensor that contains spin/charge\nindices in another basis when useful. For instance, we\nmay write the spin accumulations and spin current den-\nsities with longitudinal spin polarization in terms of av-\nerages and di\u000berences across the interface:\n\u0001\u0016`=1\n2\u0000\n\u0016`\u0000\u0016`\u0003\u0001\n; \u0016\u0016`=1\n2\u0000\n\u0016`+\u0016`\u0003\u0001\n; (8)\n\u0001jz`=1\n2\u0000\njz`\u0000jz`\u0003\u0001\n; \u0016jz`=1\n2\u0000\njz`+jz`\u0003\u0001\n:(9)\nWe may de\fne similar expressions for the charge accumu-\nlations and charge current densities. As we shall see, this\nbasis (\u000b2[d;f;\u0001`;\u0001c;\u0016`;\u0016c]) provides a more physically\ntransparent representation of all quantities.5\nIn the absence of interfacial spin-orbit coupling, the\nspin current polarized along the magnetization direction\nremains conserved. However, the spin current with po-\nlarization transverse to the magnetization dissipates en-\ntirely upon leaving the normal metal. The interface ab-\nsorbs part of this spin current, while the remaining por-\ntion quickly dissipates within the ferromagnet due to a\nprecession-induced dephasing of spins. The total loss\nof spin current then results in a spin transfer torque.\nFigure 2 depicts this process by use of solutions to the\ndrift-di\u000busion equations. In this situation, one may show\n[71, 72] that the spin and charge current densities at\nz= 0\u0006become\njz\u000b=GMCT\n\u000b\f\u0016\f (10)\nfor a conductance tensor GMCT\n\u000b\f given by\nGMCT=0\nBBBBBBBB@d f \u0001l\u0001c\u0016l\u0016c\nd Re[G\"#]\u0000Im[G\"#] 0 0 0 0\nf Im[G\"#] Re[G\"#] 0 0 0 0\n\u0016l 0 0 G+G\u00000 0\n\u0016c 0 0 G\u0000G+0 0\n\u0001l 0 0 0 0 0 0\n\u0001c 0 0 0 0 0 01\nCCCCCCCCA:\n(11)\nThis formalism|known as magnetoelectronic circuit\ntheory|disregards spin currents and accumulations in\nthe ferromagnet with polarization transverse to the mag-\nnetization (due to the precession-induced dephasing de-\nscribed above). This amounts to assuming that the pro-\ncesses occurring in the shaded regions of Fig. 2(b) hap-\npen entirely at the interface instead. While this restric-\ntion helps to reduce the number of required parameters,\nit need not apply to non-ferromagnetic systems or ex-\ntremely thin ferromagnetic layers. Note that the rows\ncorresponding to average and discontinuous quantities\nare switched from the columns corresponding to those\nquantities. This is done to emphasize that drops in ac-\ncumulations cause average currents in magnetoelectronic\ncircuit theory.\nEquation (11) implies that spin populations polarized\ntransverse to the magnetization decouple from those po-\nlarized longitudinal to it. The charge and longitudinal\nspin current densities still obey Eq. (7), whereas the\ntransverse (non-collinear) spin current densities experi-\nences a \fnite rotation in polarization about the magneti-\nzation axis. Note that the spin mixing conductance G\"#\ngoverns the latter phenomenon. In general, one obtains\nall parameters via integrals of the transmission and/or\nre\rection amplitudes over the relevant Fermi surfaces.\nFIG. 2. (Color online) Spin current densities plotted ver-\nsus distance from the interface, calculated using the drift-\ndi\u000busion equations. Panel (a) treats the case without in-\nterfacial spin-orbit coupling using magnetoelectronic circuit\ntheory as boundary conditions, whereas panel (b) treats the\ncase with interfacial spin-orbit coupling by using Eq. (12) as\nboundary conditions instead. Due to precession-induced de-\nphasing,jzdandjzfdissipate entirely within the ferromag-\nnet some distance from the interface (denoted by the purple\ndashed line). With no interfacial spin-orbit coupling, the spin\ncurrent density polarized along the magnetization ( jzl) is con-\nserved, while the spin current densities polarized transversely\n(jzdandjzf) exhibit discontinuities at the interface. With\ninterfacial spin-orbit coupling, all spin currents are discon-\ntinuous at the interface. Furthermore, interfacial spin-orbit\ncoupling introduces additional sources of spin current via the\nconductivity \u001bi\u000band torkivity \rFM\n\u001btensors (when an in-plane\nelectric \feld is present). These sources may oppose the spin\ncurrents that develop in the bulk. For example, the inclusion\nof interfacial spin-orbit coupling leads jzfto switch signs near\nto the interface, as seen by comparing panels (a) and (b).\nC. Spin transport with interfacial spin orbit\ncoupling\nTo generalize magnetoelectronic circuit theory, i.e.\nEq. (10), to account for interfacial spin orbit coupling\nand in-plane electric \felds, we introduce the following\nexpression for the spin and charge current densities at\nthe interface:\nji\u000b=Gi\u000b\f\u0016\f+\u001bi\u000b~E: (12)\nHere we use a scaled electric \feld de\fned by ~E\u0011\u0000E=e\nso that the elements of the tensor \u001bi\u000bhave units of con-\nductivity. Without loss of generality, we assume that the\nelectric \feld points along the xaxis.\nThe explosion of new parameters (relative to magneto-6\nelectronic circuit theory) is an unfortunate consequence\nof spin-\rip scattering at the interface. Like magnetoelec-\ntronic circuit theory, one may express each parameter\nas an integral of scattering amplitudes over the relevant\nFermi surfaces; to discover which parameters may be ne-\nglected we numerically study these integrals in Sec. IV.\nHere, we discuss the overarching implications of this\nmodel. In particular, three new concepts emerge from\nthe above expression:\nFirst of all, the current density ji\u000bnow includes an\nindex describing its direction of \row ( i2[x;y;z ]), which\nwas previously assumed to be out-of-plane. In this gen-\neralization, a buildup of spin and charge accumulation\nat interfaces may lead to spin and charge currents that\n\row both in-plane and out-of-plane. The treatment of\nin-plane currents close to the interface requires not only\nthe evaluation of Eq. (12), but also an extension of the\ndrift-di\u000busion equations themselves.\nSecondly, Eq. (12) depends on values of the spin and\ncharge accumulations from each side of the interface,\nrather than di\u000berences in those values across the inter-\nface. This suggests that currents result from both drops\nin accumulations andnon-zero averages of spin accumu-\nlation at the interface [73].\nFinally, interfacial spin-orbit scattering results in a\nconductivity tensor ( \u001bi\u000b) that drives spin currents in the\npresence of an in-plane electric \feld. This feature rep-\nresents the greatest conceptual departure from previous\ntheories describing spin transport and is motivated by\nresults from the Boltzmann equation. Figure 2 describes\nhow some of these properties alter solutions of the drift-\ndi\u000busion equations, as compared with magnetoelectronic\ncircuit theory.\nWithout interfacial spin-orbit coupling the in-plane\nconductance tensors ( Gx\u000b\fandGy\u000b\f) vanish, implying\nthat accumulations do not create in-plane currents in this\nscenario. The conductivity tensor vanishes as well. Spin\ntransport transverse to the magnetization still decouples\nfrom that longitudinal to it, and magnetoelectronic cir-\ncuit theory is recovered. In the presence of interfacial\nspin-orbit coupling, none of the tensors elements intro-\nduced in Eq. (12) necessarily vanish, and spin transport\nin all polarization directions becomes coupled. However,\nfor the interfacial scattering potential studied in Sec. IV,\nmany parameters di\u000ber by orders of magnitude; thus cer-\ntain parameters may be neglected on a situational basis.\nD. Spin-orbit torques\nWithout interfacial spin-orbit coupling, spin and\ncharge accumulations at an interface create both a spin\npolarization and spin currents. The spin polarization de-\nvelops atz= 0 and exerts a torque on any magnetization\nat the interface via the exchange interaction. The spin\ncurrent that develops at z= 0+exerts an additional\ntorque by transferring angular momentum to the ferro-\nmagnetic region via dephasing processes. For simplicity,we assume that this spin current transfers all of its angu-\nlar momentum to the magnetization rather than the bulk\natomic lattice. We do so under the assumption that the\ndephasing processes within the ferromagnet diminish spin\ncurrents faster than the spin di\u000busive processes caused by\nbulk spin-orbit coupling. All of the incident transverse\nspin current is then lost at the interface ( z= 0) or in\nthe bulk of the ferromagnet ( z > 0), and carriers can\nonly exchange angular momentum with the magnetiza-\ntion. Thus the spin current at z= 0\u0000, which represents\nthe incident \rux of angular momentum on the magne-\ntized part of the bilayer, equals the total spin torque on\nthe system. Furthermore, the spin torques at z= 0 and\nz>0 add up to equal the spin current at z= 0\u0000.\nHowever, at interfaces with spin-orbit coupling, the\natomic lattice behaves as a reservoir that carriers may\ntransfer angular momentum to. In this scenario, carri-\ners exert spin torques on both the magnetization and the\nlattice. We cannot compute spin torques solely from the\nspin currents described by Eq. (12) if we are to account\nfor the losses to this additional reservoir of angular mo-\nmentum. Thus, we introduce a separate expression for\nthe total spin torque on the bilayer:\n\u001c\u001b= \u0000\u001b\f\u0016\f+\r\u001b~E; (13)\nNote that the index \u001b2[d;f] describes the directions\ntransverse to the magnetization, since spin torques only\npoint in those directions. The tensor \u0000, known as the\ntorkance, describes contributions to the spin torque from\nthe buildup of spin and charge accumulation at an in-\nterface. The tensor \r, which we call the torkivity , cap-\ntures the corresponding contributions from an external,\nin-plane electric \feld. The torkivity tensor originates\nfrom interfacial spin-orbit scattering, much like the con-\nductivity tensor introduced earlier.\nWe may separate the total spin torque into two contri-\nbutions:\n\u0000\u001b\f= \u0000mag\n\u001b\f+ \u0000FM\n\u001b\f (14)\n\r\u001b=\rmag\n\u001b+\rFM\n\u001b: (15)\nThe \frst tensors on the right hand side of Eqs. (14) and\n(15) describe torques exerted by the spin polarization\natz= 0. The second tensors describe the spin torque\nexerted in the bulk of the ferromagnet ( z > 0). Both\ntorques are exerted on the magnetization rather than on\nthe atomic lattice. Here we assume that the torque at\nz > 0 equals the transverse spin current at z= 0+as\nbefore. Thus, the spin torques exerted at z= 0 and\nz > 0 are both included in the torkance and torkivity\ntensors.\nWithout interfacial spin-orbit coupling, the torkivity\ntensor vanishes and the torkance tensor \u0000 \u001b\fbecomes\nidentical to Gz\u001b\f. This indicates that the transverse spin\ncurrent at z= 0\u0000equals the total spin torque, as ex-\npected. In the presence of interfacial spin-orbit coupling,\nthe lattice also receives angular momentum from carriers;7\nin this case \u0000 \u001b\f6=Gz\u001b\fand\r\u001b6= 0. Thus, by comput-\ning the tensors introduced in Eq. (13), one may calculate\nspin-orbit torques such that the lattice torques are ac-\ncounted for. Furthermore, Eqs. (14) and (15) allow one\nto separate the total spin torque into its interfacial and\nbulk ferromagnet contributions.\nIII. DERIVATION OF BOUNDARY\nPARAMETERS\nInterfacial spin-orbit coupling causes both momen-\ntum and spin-dependent scattering at interfaces. If\nthe incident distribution of carriers depends on mo-\nmentum and/or spin, outgoing carriers may become\nspin-polarized via interfacial spin-orbit scattering. This\ngives rise to non-vanishing accumulations, currents, and\ntorques, which are related by Eqs. (12) and (13). We now\nmotivate these relationships, which can be expressed in\nterms of scattering amplitudes. We do so by approx-\nimating the non-equilibrium distribution function near\nthe interface.\nWe \frst consider the total distribution function f \u000b(k),\nwhich gives the momentum-dependent occupancy of car-\nriers described by the spin/charge index \u000b. In equilib-\nrium, this distribution function equals the Fermi-Dirac\ndistribution feq\n\u000b(\"\u000bk). Just out of equilibrium, f \u000b(k) is\nperturbed as follows\nf\u000b(k) = feq\n\u000b(\"\u000bk) +@feq\n\u000b\n@\"\u000bkg\u000b(k); (16)\nwhereg\u000b(k) denotes the non-equilibrium distribution\nfunction. The equilibrium distribution functions vanish\nfor\u000b2[d;f;` ] since the non-magnet exhibits no equi-\nlibrium spin polarization. However, the non-equilibrium\ndistribution functions for all spin/charge indices are gen-\nerally non-zero.\nTo obtain the tensors introduced in Eqs. (12) and (13),\nwe must evaluate g\u000b(k) near the interface. One could\nevaluateg\u000b(k) by solving the spin-dependent Boltzmann\nequation for the bilayer system. In this approach one cap-\ntures spin transport both in the bulk and at the interface.\nA more practical approach is to assume some generic\nform forg\u000bnear the interface that is physically plau-\nsible. This approach yields boundary conditions suit-\nable for simpler bulk models of spin transport such as\nthe drift-di\u000busion equations. In the companion paper,\nwe show that solving the drift-di\u000busion equations using\nthese boundary conditions produces quantitatively simi-\nlar results to solving the Boltzmann equation.\nFor simplicity, we assume that spherical Fermi surfaces\ndescribe carriers in both layers. Later we generalize this\nformalism to describe non-trivial electronic structures.\nIn the non-magnet, all carriers belong to the same Fermi\nsurface. In the ferromagnet, majority ( \") and minor-\nity (#) carriers belong to di\u000berent Fermi surfaces. Thus\nwe use the spin/charge basis \u000b2[d;f;`;c;\";#], since\nin this model carriers belonging to those populationshave well-de\fned Fermi surfaces and velocities. The ten-\nsors derived in this section may be expressed in other\nspin/charge bases by straightforward linear transforma-\ntions.\nTo approximate g\u000bat the interface we use the following\nexpression:\ngin\n\u000b(kjj) =\u0000e\u0010\nq\u000b+~EfE\n\u000b(kjj)\u0011\n; (17)\nEquation (17) represents the portion of g\u000bincident on\nthe interface, where kjjdenotes the in-plane momentum\nvector and eequals the elementary charge. The right\nhand side of Eq. (17) describes two pieces of the incom-\ning distribution function; Fig. 3 depicts both pieces over\nk-space for each side of the interface. The \frst term\ncaptures spin/charge currents incident on the interface.\nThey may arise, for example, from the bulk spin Hall\ne\u000bect or ferromagnetic leads. The quantities q\u000bdenote\nthe isotropic spin/charge polarization of those currents.\nThe second term represents the anisotropic contribution\nto the distribution function caused by an external electric\n\feld. We remind the reader that the scaled electric \feld\n~Epoints along the xaxis. The simplest approximation\nforfE\n\u000b(kjj) is to use the particular solution of the Boltz-\nmann equation in the relaxation time approximation:\nfE\n\u000b(kjj) =\u0000evx\u000b(kjj)\u00028\n>>>>><\n>>>>>:0\u000b2[d;f;` ]\n\u001c \u000b =c\n\u001c\"\u000b=\"\n\u001c#\u000b=#(18)\nThis term describes the in-plane charge current caused\nby the external electric \feld, but also describes an in-\nplane spin current polarized opposite to the magnetiza-\ntion in the ferromagnet. The momentum relaxation times\nin the ferromagnet di\u000ber between majority ( \u001c\") and mi-\nnority (\u001c#) carriers. In the non-magnet, the momentum-\nrelaxation time ( \u001c) is renormalized by bulk spin-\rip pro-\ncesses (see appendix A).\nThe outgoing distribution function\ngout\n\u000b(kjj) =S\u000b\f(kjj)gin\n\f(kjj); (19)\nis speci\fed by the incoming distribution function and the\nunitary scattering coe\u000ecients S\u000b\f, given by\nS\u000b\f\u0011jvz\u000b(kjj)j\njvz\f(kjj)jS0\n\u000b\f(kjj); (20)\nwhere\nS0\n\u000b\f=8\n>>>>>>><\n>>>>>>>:1\n2tr\u0002\nry\u001b\u000br\u001b\f\u0003\n\u000b;\f2[d;f;`;c ]\n1\n2tr\u0002\nty\u001b\u000bt\u001b\f\u0003\n\u000b2[d;f;`;c ]; \f2[\";#]\n1\n2tr\u0002\n(t\u0003)y\u001b\u000bt\u0003\u001b\f\u0003\n\u000b2[\";#]; \f2[d;f;`;c ]\n1\n2tr\u0002\n(r\u0003)y\u001b\u000br\u0003\u001b\f\u0003\n\u000b;\f2[\";#]\n(21)8\nFIG. 3. (Color online) Non-equilibrium distribution functions g\u000b(k) in the presence of interfacial spin-orbit scattering, resulting\nfrom an (a) incident spin and charge accumulation and an (b) in-plane external electric \feld. The images depict g\u000b(k) on each\nside of the interface plotted over k-space. The gray spheres represent the equilibrium Fermi surface. The colored surfaces\nrepresent the non-equilibrium perturbation to the Fermi surface, given by the charge distribution gc(k) (not to scale). The\narrows denote the spin distribution g\u001b(k). The blue and red regions represent the wavevectors pointing incident and away\nfrom the interface respectively. (a) Scenario in which the incident carriers exhibit a net spin and charge accumulation. The\nspin-polarization of the outgoing carriers di\u000bers from the incident carriers due to interfacial spin-orbit scattering. The total\nspin/charge current density ( ji\u000b) and the resulting spin torques ( \u001c\u001b) are related to the total spin/charge accumulation ( \u0016\u000b) by\nthe tensors Gi\u000b\fand \u0000\u001b\frespectively. (b) Scenario in which the incident carriers are subject to an in-plane electric \feld. The\nin-plane electric \feld drives two di\u000berent charge currents on each side of the interface, since each layer possesses a di\u000berent bulk\nconductivity. This shifts the occupancy of carriers (i.e. the charge distribution) di\u000berently on each side of the interface. When\nspin- unpolarized carriers scatter o\u000b of an interface with spin-orbit coupling they become spin-polarized. Because the occupancy\nof incident carriers was asymmetrically perturbed at the interface, a net cancellation of spin is avoided in even the simplest\nscattering model. The resulting spin/charge currents and spin torques are captured by the tensors \u001bi\u000band\r\u001brespectively.\nNote that for a ferromagnetic layer, in-plane electric \felds also create incident in-plane spin currents as well (suppressed for\nclarity in this \fgure).\nHere we de\fne the Pauli vector \u001b\u000bsuch that\u001bd=\u001bx,\n\u001bf=\u001by,\u001b`=\u001bz, and\n\u001bc= \n1 0\n0 1!\n; \u001b\"= \n1 0\n0 0!\n; \u001b#= \n0 0\n0 1!\n:(22)\nThe coe\u000ecient S0\n\u000b\fgives the strength of scattering\nfor carriers with spin/charge index \finto those with\nspin/charge index \u000b. The scattering coe\u000ecients depend\non the 2\u00022 re\rection and transmission matrices for spins\npointing along the magnetization axis. In particular, the\nmatricesr\u0003andt\u0003describe re\rection and transmission\nrespectively into the ferromagnet. The matrices randt\ndescribe re\rection and transmission into the non-magnet.\nNote that the density of states and Fermi surface area\nelement di\u000ber between incoming and outgoing carriers.\nThus to conserve particle number one must include the\nratio of velocities within the scattering coe\u000ecients, as\ndone in Eq. (20).\nWe obtain all non-equilibrium quantities near the in-\nterface by integrating g\u000bover the relevant Fermi surfaces.\nWe note that the outgoing part of g\u000bincludes the con-\nsequences of interfacial scattering, since it depends on\nthe scattering coe\u000ecients. For example, the interfacial\nexchange interaction leads to spin-dependent scattering,\nwhich is captured by the di\u000berence in the diagonal ele-ments of the 2\u00022 re\rection and transmission matrices.\nOn the other hand, the interfacial spin-orbit interaction\nintroduces spin-\rip scattering, which is captured by the\no\u000b-diagonal elements within these matrices. Thus, to\ndescribe the consequences of interfacial spin-orbit scat-\ntering we must not limit the form of the re\rection and\ntransmission matrices as was often done in the past.\nWe write the current density ji\u000bfor carriers with\nspin/charge index \u000b\rowing in direction i2[x;y;z ] as\nfollows:\nji\u000b=1\n~(2\u0019)31\nvF\u000bZ\nFS\u000bd2kvi\u000b(k)g\u000b(k) (23)\nNote that all integrals run over the Fermi surface cor-\nresponding to the population with spin/charge index \u000b.\nThe quantity vF\u000bdenotes the Fermi velocity for that\npopulation. To de\fne the accumulations \u0016\u000bwe follow\nthe example of magnetoelectronic circuit theory [71, 72]\nand assume that the incoming currents behave as if they\noriginate from spin-dependent reservoirs. This implies\nthat the incoming polarization q\u000bapproximately equals\nthe accumulation \u0016\u000bat the interface.\nWe have now discussed the requirements for deriv-\ning the conductance and conductivity tensors found in\nEq. (12). We obtain these tensors by plugging Eqs. (17)\nand (19) into Eq. (23) and noting that q\u000b\u0019\u0016\u000b. In doing9\nso we write the currents ji\u000bin terms of the accumulations\n\u0016\u000band the in-plane electric \feld ~E. From the resulting\nexpressions one then obtains formulas for the conduc-\ntance and conductivity tensors in terms of the interfacial\nscattering coe\u000ecients. We outline this remaining process\nin appendix A. In appendix C we generalize those ex-\npressions for the case of non-trivial electronic structures,\nwhich allows one to compute the conductance and con-\nductivity tensors for realistic systems.\nHaving discussed the currents that arise from interfa-\ncial spin-orbit scattering, we now discuss the spin torques\ncaused by the same phenomenon. The transverse spin po-\nlarization at z= 0 exerts a torque on any magnetization\nat the interface via the exchange interaction. The trans-\nverse spin current at z= 0+exerts a torque by transfer-\nring angular momentum to the ferromagnet. The total\nspin torque then equals the sum of these two torques. To\ndescribe the spin torque at z= 0, we must compute the\nspin polarization at the interface. To accomplish this we\nde\fne the following matrix\nT\u001b\f=8\n<\n:1\n2tr\u0002\n(t\u0003)y\u001b\u001bt\u0003\u001b\f\u0003\n\f2[d;f;`;c ]\n1\n2tr\u0002\nty\u001b\u001bt\u001b\f\u0003\n\f2[\";#](24)\nwhich describes phase-coherent transmission from all\npopulations into transverse spin states at the interface.\nWe may then compute the ensemble average of spin den-\nsityhs\u001biatz= 0 as follows:\nhs\u001bi=1\n~(2\u0019)3X\n\f1\nvF\fZ\nFS\f2ind2kT\u001b\f(kjj)gin\n\f(kjj):\n(25)\nThe torque at z= 0 is then given by\n\u001cmag\n\u001b=\u00000+Z\n0\u0000dzJex\n~\u0002\nhsi\u0002^m\u0003\n\u001b; (26)\nwhereJexequals the exchange energy at the interface.\nWe evaluate this integral over the region that describes\nthe interface, where the exchange interaction and strong\nspin-orbit coupling overlap. Note that the cross prod-\nuct\u0002\nhsi\u0002^m\u0003\n\u001b=\u000f\u001b\u001b0hs\u001b0iis evaluated by computing\nEq. (25).\nTo describe the spin torque at z= 0+, we introduce\nan additional scattering matrix:\n\u0016S\u001b\f=8\n<\n:1\n2tr\u0002\n(t\u0003)y\u001b\u001bt\u0003\u001b\f\u0003\n\f2[d;f;`;c ]\n1\n2tr\u0002\n(r\u0003)y\u001b\u001br\u0003\u001b\f\u0003\n\f2[\";#](27)\nThis scattering matrix is used to calculate the transverse\nspin current at z= 0+. Since this spin current rapidly de-\nphases, it contributes entirely to the spin torque exerted\non the ferromagnet. Note that the currents discussedpreviously corresponded to carriers with well-de\fned ve-\nlocities. However, transverse spin states in the ferromag-\nnet consist of linear combinations of majority and minor-\nity spin states. Since these spin states possess di\u000berent\nphase velocities, the velocities of transverse spin states\noscillate over position. These states also posses di\u000berent\ngroup velocities, and wave packets with transverse spin\ntravel with the average group velocity. The transverse\nspin current at z= 0+then equals\n\u001cFM\n\u001b=1\n~(2\u0019)3X\n\fZ\n2DBZdkjj\u0016vz(kjj)\nvz\f(kjj)\u0016S\u001b\f(kjj)gin\n\f(kjj);\n(28)\nwhere\n\u0016vz(kjj)\u00111\n2\u0010\nvz\"(kjj) +vz#(kjj)\u0011\n(29)\ngives the average group velocity of carriers in the ferro-\nmagnet. Note that we write this integral over the max-\nimal two-dimensional Brillouin zone common to all car-\nriers (see appendices A and B). The total torque then\nequals the sum of torques at the interface and in the bulk\nferromagnet:\n\u001c\u001b=\u001cmag\n\u001b+\u001cFM\n\u001b: (30)\nAs before we assume that the incoming polarizations\napproximately equal the accumulations at the interface.\nThus we obtain \u001cmag\n\u001b and\u001cFM\n\u001bin terms of \u0016\u000band ~E\nby plugging Eqs. (17) and (19) into Eqs. (25), (26), and\n(28). From the resulting expressions we may de\fne the\ntorkance and torkivity tensors introduced in Eq. (13). In\nappendix B we discuss this process, and in appendix C we\npresent generalized expressions for non-trivial electronic\nstructures.\nWe note that the conductance and conductivity tensors\ndescribe the charge current and longitudinal spin current\nin the ferromagnet, but not the transverse spin currents.\nIn the ferromagnet, the transverse spin currents dissipate\nnot far from the interface, while the charge current and\nlongitudinal spin current can propagate across the entire\nlayer. Thus the transverse spin currents in the ferromag-\nnet are best described as spin torques given by \u001cFM\n\u001b; this\nexplains why we include them in the torkance and torkiv-\nity tensors instead of the conductance and conductivity\ntensors. If we derive a similar formalism to describe a\nnon-magnetic bilayer, spin currents polarized in all di-\nrections should be included in the conductance and con-\nductivity tensors. With no magnetism, no spin torques\nare exerted at or near the interface and the torkance and\ntorkivity tensors are not meaningful.\nIV. NUMERICAL ANALYSIS OF BOUNDARY\nPARAMETERS\nIn the following we numerically analyze the boundary\nparameters introduced in Eqs. (12) and (13) in the pres-\nence of an interfacial exchange interaction and spin-orbit10\nFIG. 4. (Color online) Contour plots of various boundary parameters versus the interfacial exchange ( uex) and Rashba ( uR)\nstrengths. The magnetization points away from the electric \feld 45oin-plane and 22 :5oout-of-plane. Note that the parameters\nplotted in panels (a)-(c) describe the scattering processes illustrated in Figs. 1(a)-(c). (a) Plot of Gzdf, which generalizes\nIm[G\"#] in the presence of interfacial spin-orbit coupling. It describes a rotation of spin currents polarized transversely to the\nmagnetization. (b) Plot of Gz\u0016l\u0001l, which contributes to spin memory loss longitudinal to the magnetization. It varies mostly\nwithuR, since interfacial spin-orbit coupling provides a sink for angular momentum. (c) Plot of \rFM\nd, which describes the\nout-of-plane, damping-like spin current created by an in-plane electric \feld and spin-orbit scattering. It exceeds its \feld-like\ncounterpart ( \rFM\nf); thus, the resulting spin current exerts a (mostly) damping-like spin torque upon entering the ferromagnet.\n(d) An array of contour plots, with each plot shown over an identical range as those in (a)-(c). The plot in row \u000band column\n\fcorresponds to the parameter Gz\u000b\f. From this one may visualize the coupling between spin/charge indices for this tensor,\nshown across the parameter space of the scattering potential given by Eq. (31). The overall structure of Gz\u000b\fresembles that\nof magnetoelectronic circuit theory, given by Eq. (11). The corresponding \fgures for (e) \u001bz\u000b, (f)\rFM\n\u001b, and (g)\rmag\n\u001bare also\nshown.\nscattering. We do so to provide intuition as to the rel-\native strengths of each boundary parameter. We use a\nscattering potential localized at the interface [42] that is\nbased on the Rashba model of spin orbit coupling\nV(r) =~2kF\nm\u000e(z)\u0000\nu0+uex\u001b\u0001^m+uR\u001b\u0001(^k\u0002^z)\u0001\n(31)\nwhereu0represents a spin-independent barrier, uexgov-\nerns the interfacial exchange interaction, and uRdenotesthe Rashba interaction strength. Plane waves comprise\nthe scattering wavefunctions in both regions.\nIn Fig. 4 we plot various boundary parameters versus\nthe exchange interaction strength ( uex) and the Rashba\ninteraction strength ( uR). Figures 4(a)-(c) display indi-\nvidual boundary parameters, while Figs. 4(d)-(g) display\nmultiple boundary parameters for a given tensor. The\nplots in Figs. 4(d)-(g) are arranged as arrays to help visu-11\nalize the coupling between spin/charge components. The\nspin-orbit interaction misaligns the preferred direction of\nspins from the magnetization axis. Thus, no two tensor\nelements are identical, though many remain similar. As\nexpected, the coupling between the transverse spin com-\nponents and the charge and longitudinal spin components\ndoes not vanish.\nThe conductance tensor Gz\u000b\fgeneralizes GMCT\n\u000b\f in the\npresence of interfacial spin-orbit coupling. Comparison\nto Eq. (11) suggests that the parameters GzddandGzdf\nrepresent the real and imaginary parts of a generalized\nmixing conductance ( ~G\"#). Each element of the conduc-\ntance tensor experiences a similar perturbation due to\nspin-orbit coupling. However, the tensor elements from\nthe 2\u00022 o\u000b-diagonal blocks in Fig. 4(d) either vanish or\nremain two orders of magnitude smaller than those from\nthe diagonal blocks. This remains true even for values\nofuRapproaching the spin-independent barrier strength\nu0. While these blocks are small for the simple model\ntreated here, they may become important for particular\nrealistic electronic structures. The fact that the elements\nGz\u0001c\u000bandGz\u000b\u0016cvanish for all \u000bensures the conserva-\ntion of charge current and guarantees no dependence on\nan o\u000bset to the charge accumulations. Note that four\nadditional parameters vanish in the conductance tensor\nshown in Fig. 4(d); this occurs because identical scatter-\ning wavefunctions were used for both sides of the inter-\nface when computing the scattering coe\u000ecients. These\nparameters do not vanish in general.\nThe results shown in Fig. 4 were computed for a mag-\nnetization with out-of-plane components. In magneto-\nelectronic circuit theory, the parameters are independent\nof the magnetization direction. With interfacial spin-\norbit coupling, this is no longer the case. In general all\nof the parameters in Eqs. (12) and (13) depend on the\nmagnetization direction. However, we \fnd that this de-\npendence is weak for the model we consider here. For in-\nplane magnetizations (not shown) the 2 \u00022 o\u000b-diagonal\nblocks vanish, but spin-orbit coupling still modi\fes the\ndiagonal blocks in the manner described above.\nIn the presence of interfacial spin-orbit coupling the\nlattice also receives angular momentum from carriers.\nThis results in a loss of spin current across the inter-\nface, or spin memory loss, which the elements Gz\u0001l\u000b\npartly characterize. The computation of these parame-\nters for realistic electronic structures should help predict\nspin memory loss in experimentally-relevant bilayers. In\nparticular, spin memory loss might play a crucial role\nwhen measuring the spin Hall angle of heavy metals via\nspin-pumping from an adjacent ferromagnet [9]. Here\nGz\u0001l\u0016lprovides the strongest contribution to spin mem-\nory loss that is caused by accumulations, and approaches\nthe imaginary part of the generalized mixing conductance\nin magnitude.\nUntil now, we have discussed the tensors that describe\nhow accumulations a\u000bect transport. However, in-plane\nelectric \felds and spin-orbit scattering create additional\ncurrents that form near the interface. In particular, theconductivity parameters \u001bi\u000bdescribe the currents that\ncan propagate into either layer without signi\fcant de-\nphasing. For instance, the element \u001bz\u0016ldescribes an out-\nof-plane longitudinal spin current driven by an in-plane\nelectric \feld. The element \u001bz\u0001lthen gives the disconti-\nnuity in this spin current across the interface. This dis-\ncontinuity arises because of coupling to the lattice, and\nthus contributes to spin memory loss.\nLikewise, the torkivity tensors describe contributions\nto the total spin torque that arise from in-plane electric\n\felds and spin-orbit scattering. This includes the torques\nexerted by the spin polarization at z= 0 and by the\ntransverse spin currents at z= 0+. The tensors \rmag\n\u001b\nand\rFM\n\u001bdescribe these torques respectively. Since the\ntransverse spin currents at z= 0+quickly dephase in the\nferromagnet, we treat them as spin torques and do not\ninclude them in the conductivity tensor.\nTo understand how the boundary parameters con-\ntribute to spin-orbit torques, we note that \rmag\nf> \rmag\nd\nover the swept parameter space. This implies that the\ntorque exerted at z= 0 is primarily \feld-like, which\nagrees with previous studies of interfacial Rashba spin or-\nbit torques [42]. However, we also \fnd that \rFM\nd>\rFM\nf\nfor stronguR; in this case the resulting spin current exerts\na damping-like torque by \rowing into the ferromagnet.\nBoth spin torque contributions result from the interfacial\nRashba interaction. This implies that interfacial spin-\norbit scattering provides a crucial mechanism to the cre-\nation of damping-like Rashba spin torques. In the com-\npanion paper we support this claim by comparing spin-\norbit torques computed using both the drift-di\u000busion and\nBoltzmann equations.\nV. OUTLOOK\nIn the previous section we demonstrated that only cer-\ntain boundary parameters remain important when mod-\neling spin orbit torques. The interfacial conductivity and\ntorkivity parameters capture physics due to in-plane ex-\nternal electric \felds. They depend on the di\u000berence in\nbulk conductivities, which are typically easier to estimate\nthan interfacial spin/charge accumulations. For this rea-\nson, calculating the conductivity and torkivity tensors\nfor a realistically-modeled system should provide direct\ninsight into its spin transport behavior. In particular,\nwe showed that conductivity and torkivity parameters\nstrongly indicate the potential to produce damping-like\nand \feld-like torques. Further studies may yield signi\f-\ncant insight into the underlying causes of these and other\nphenomena for speci\fc material systems. Even so, treat-\ning the elements of these tensors as phenomenological\nparameters should bene\ft the analysis of a variety of ex-\nperiments, which we discuss now.\n(1) Spin pumping/memory loss | Spin pumping ex-\nperiments in Pt-based multilayers suggest that the mea-\nsured interfacial spin current di\u000bers from the actual spin\ncurrent in Pt, leading to inconsistent predictions of the12\nspin Hall angle [9, 53]. Rojas-Sanchez et al. [9] ex-\nplain this discrepancy in terms of spin memory loss while\nZhang et al. [53] attribute it to interface transparency.\nThe latter characterizes the actual spin current generated\nat an interface when backscattering is accounted for; it\ndepends on G\"#and does not require interfacial spin-\norbit coupling. Though further experimental evidence is\nneeded to resolve these claims, the elements of Gz\u000b\fchar-\nacterize both spin memory loss and transparency. Fig-\nure 4 implies that transparency depends on interfacial\nspin-orbit coupling, while spin memory loss also depends\non the interfacial exchange interaction. Thus, the gen-\neralized boundary conditions introduced here unify these\ntwo interpretations and allow for further investigation us-\ning a single theory.\n(2) Rashba-Edelstein e\u000bect | Sanchez et al. [41] mea-\nsure the inverse Rashba-Edelstein e\u000bect in an Ag/Bi in-\nterface, in which interfacial spin orbit coupling converts\na pumped spin current into a charge current. The the-\noretical methods that describe this phenomena to date\n[37, 44{46, 68] assume orthogonality between the direc-\ntional and spin components of the spin current tensor.\nHowever, the conductivity tensor introduced here is ro-\nbust in general; this implies that interfacial spin-orbit\nscattering converts charge currents into spin currents\nwith polarization and \row directions not orthogonal to\nthe charge current. Onsager reciprocity implies that spin\ncurrents should give rise to charge currents at the inter-\nface that \row in all directions as well. Thus, the con-\nductivity tensor describes a generalization of the direct\nand inverse Rashba-Edelstein e\u000bects as they pertain to\ninterfaces with spin-orbit coupling.\n(3) Spin Hall magnetoresistance | The conductivity\ntensor also leads to in-plane charge currents. These cur-\nrents depend on magnetization direction via the scat-\ntering amplitudes, and thus suggest a new contribution\nto the spin Hall magnetoresistance based on the Rashba\ne\u000bect in addition to that from the spin Hall e\u000bect. Pre-\nliminary calculations of this mechanism suggest a mag-\nnetoresistance in Pt/Co of a few percent, which is com-\nparable or greater than experimentally measured values\nin various systems [58{61].\nWe expect that the most useful approach for inter-\npreting experiments as above is to treat the new trans-\nport parameters as \ftting parameters. In the future, this\napproach can be checked by calculating the parameters\nfrom \frst principles [74, 75] as has been done for magne-\ntoelectronic circuit theory. This requires computing the\nboundary parameters for realistic systems using the ex-\npressions given in appendix C. Such calculations would\nprovide a useful bridge between direct \frst-principles cal-\nculations of spin torques [76{79] and drift-di\u000busion cal-\nculations done to analyze experiments.\nTo conclude, we present a theory of spin transport at\ninterfaces with spin-orbit coupling. The theory describes\nspin/charge transport in terms of resistive elements,\nwhich ultimately describe measurable consequences of\ninterfacial spin-orbit scattering. In particular, the pro-posed conductivity and torkivity tensors model the phe-\nnomenology of in-plane electric \felds in the presence of\ninterfacial spin-orbit coupling, which was previously inac-\ncessible to the drift-di\u000busion equations. We calculate all\nparameters in a simple model, but also provide general\nexpressions in the case of realistic electronic structure.\nWe found that elements of the conductivity and torkiv-\nity tensors are more important than the modi\fcations of\nother transport parameters (such as the mixing conduc-\ntance) in many experimentally-relevant phenomena, such\nas spin orbit torque, spin pumping, the Rashba-Edelstein\ne\u000bect, and the spin Hall magnetoresistance.\nACKNOWLEDGMENTS\nThe authors thank Kyoung-Whan Kim, Paul Haney,\nGuru Khalsa, Kyung-Jin Lee, and Hyun-Woo Lee for\nuseful conversations and Robert McMichael and Thomas\nSilva for critical readings of the manuscript. VA ac-\nknowledges support under the Cooperative Research\nAgreement between the University of Maryland and the\nNational Institute of Standards and Technology, Cen-\nter for Nanoscale Science and Technology, Grant No.\n70NANB10H193, through the University of Maryland.\nAppendix A: Derivation of the conductance and\nconductivity tensors\nTo derive the conductance and conductivity tensors\nwe must approximate the distribution function f \u000b(k)\nat the interface. The distribution function gives the\nmomentum-dependent occupancy of carriers described\nby the spin/charge index \u000b. Just out of equilibrium, it is\nperturbed by the linearized non-equilibrium distribution\nfunctiong\u000b(k), as seen in Eq. (16). In the following we\ncomplete the derivation begun in Sec. III.\nWe write the portion of g\u000b(kjj) incident on the inter-\nface as done in Eq. (17). The \frst term on the right\nhand side of Eq. (17) captures the spin and charge cur-\nrents incident on the interface, while the second term\ngives an anisotropic contribution caused by an external\nelectric \feld. As discussed in Sec. III, the simplest ap-\nproximation for g\u000b(kjj) is to use the particular solution\nof the Boltzmann equation in the relaxation time approx-\nimation, given by Eq. (18). The momentum relaxation\ntimes that we use account for di\u000bering majority ( \u001c\") and\nminority (\u001c#) relaxation times in the ferromagnet, and\nare renormalized by bulk spin-\rip scattering in the non-\nmagnet:\n(\u001c)\u00001= (\u001cmf)\u00001+ (\u001csf)\u00001: (A1)\nWe may better approximate Eq. (18) by forcing the dis-\ntribution function to obey outer boundary conditions as\nwell. In the companion paper we present a more sophisti-\ncated approximation for Eq. (18) that accomplishes this13\nby using solutions to the homogeneous Boltzmann equa-\ntion.\nThe outgoing distribution, given by Eq. (19), is speci-\n\fed by the incoming distribution and the scattering co-\ne\u000ecientsS\u000b\f. The scattering coe\u000ecients are given by\nEq. (20) and Eq. (21). Here we compute non-equilibrium\naccumulations analogously to the currents de\fned by\nEq. (23),\n\u0016\u000b=\u00001\ne1\nAFS\u000bZ\nFS\u000bd2kg\u000b(k); (A2)\nwhere\u0016\u000bdenotes the accumulation. Furthermore, AFS\u000b\ngives the Fermi surface area while vF\u000bgives the Fermi\nvelocity. The quantities just de\fned apply to the popula-\ntion with spin/charge index \u000b. Likewise, all integrals are\nevaluated over the Fermi surface that corresponds to the\nspin/charge index \u000b. Note that we express the accumula-\ntions in units of voltage and the current densities in units\nof number current density. Using Eqs. (19) and (20) we\nmay rewrite these expressions as integrals over the max-\nimal two-dimensional Brillouin zone common (2DBZ) to\nall carriers\n\u0016\u000b=\u0000c\u0016\neX\n\fZ\n2DBZdkjj1\nvz\u000b(kjj)\u0000\n\u000e\u000b\f+S\u000b\f\u0001\ngin\n\f(kjj)\n(A3)\nji\u000b=\u0000cj\neX\n\fZ\n2DBZdkjjvi\u000b(kjj)\nvz\u000b(kjj)\u0000\n\u000e\u000b\f\u0012iz+S\u000b\f\u0001\ngin\n\f(kjj);\n(A4)\nwhere\nc\u0016\u0011vF\u000b\nAFS\u000b; cj\u0011\u0000e\n~(2\u0019)3: (A5)\nNote that the velocities correspond to outgoing carriers.\nThe factor \u0012iz\u0011(1\u00002\u000eiz) accounts for the fact that\nincoming and outgoing currents have the opposite sign\nfori=zbut the same sign for i2[x;y]. By integrat-\ning over the maximal two-dimensional Brillouin zone we\nencounter evanescent states, since kjjvectors not corre-\nsponding to real Fermi surfaces have imaginary kzvalues.\nHere we neglect the contributions to the currents and ac-\ncumulations due to evanescent states. Such contributions\nvanish very close to the interface.\nWe must now express the accumulations and currents\nin terms of the incoming polarizations and the in-plane\nelectric \feld. Plugging Eqs. (17) and (19) into Eqs. (A3)\nand (A4), we obtain the following\n\u0016\u000b=A\u000b\fq\f+a\u000b~E (A6)\nji\u000b=Bi\u000b\fq\f+bi\u000b~E (A7)\nwhere the tensors that contract with the incident\nspin/charge polarization are given by\nA\u000b\f=c\u0016Z\n2DBZdkjj1\nvz\u000b\u0000\n\u000e\u000b\f+S\u000b\f\u0001\n(A8)\nBi\u000b\f=cjZ\n2DBZdkjjvi\u000b\nvz\u000b\u0000\n\u000e\u000b\f\u0012iz+S\u000b\f\u0001\n(A9)while the tensors that multiply the in-plane electric \feld\nbecome\na\u000b=c\u0016X\n\fZ\n2DBZdkjj1\nvz\u000b\u0000\n\u000e\u000b\f+S\u000b\f\u0001\nfE\n\f (A10)\nbi\u000b=cjX\n\fZ\n2DBZdkjjvi\u000b\nvz\u000b\u0000\n\u000e\u000b\f\u0012iz+S\u000b\f\u0001\nfE\n\f (A11)\nIn the same spirit as magnetoelectronic circuit theory,\nthese tensors represent moments of the scattering coe\u000e-\ncients weighted by velocities.\nTo determine exactly how the currents depend on the\naccumulations, we solve for ji\u000bin terms of \u0016\u000b. Doing so\nyields the following conductance and conductivity tensors\nGi\u000b\f=Bi\u000b\r[A\u00001]\r\f\n\u001bi\u000b=bi\u000b\u0000Gi\u000b\fa\f:\nTo further simplify these expressions, we follow the ex-\nample of magnetoelectronic circuit theory [71, 72] and\nassume that the incoming spin-currents behave as if they\noriginate from spin-dependent reservoirs. This implies\nthat the incoming spin polarization q\u000bequals the qua-\nsichemical potential \u0016\u000bat the interface. For this to be\ntrue, we must \fnd that A\u000b\f/\u000e\u000b\fanda\u000b= 0 by in-\nspection of Eq. (A6). These relationships hold if one\nevaluates Eqs. (A8) and (A10) over the incoming portion\nof the Fermi surface only. We \fnd that the contributions\nfrom the outgoing portion of the Fermi surface cancel to\na good approximation, which suggests that:\nGi\u000b\f=Bi\u000b\f (A12)\n\u001bi\u000b=bi\u000b: (A13)\nThe above equations give simpler expressions for the con-\nductance and conductivity tensors in terms of interfacial\nscattering coe\u000ecients.\nAppendix B: Derivation of the torkance and\ntorkivity tensors\nTo describe the spin torque at z= 0, we must compute\nthe ensemble average of spin density hs\u001biusing Eq. (25).\nThe resulting torque is given by Eq. (26). To describe the\nspin torque at z= 0+, we must calculate the transverse\nspin current in the ferromagnet using Eqs. (28) and (29).\nWe then express the spin torque in terms of the incoming\npolarizations and the in-plane electric \feld by plugging\nEqs. (17) and (19) into Eqs. (25), (26), and (28). In doing\nso we obtain\n\u001c\u001b=C\u001b\fq\f+c\u001b~E; (B1)\nwhere\nC\u001b\f=CFM\n\u001b\f+Cmag\n\u001b\f(B2)\nc\u001b=cFM\n\u001b+cmag\n\u001b (B3)14\ndescribes the separation of the spin torque into its bulk\nferromagnet and interface contributions. The tensors\nthat contract with the incident spin/charge polarization\nare given by\nCFM\n\u001b\f=cjZ\n2DBZdkjj\u0016vz\nvz\f\u0016S\u001b\f; (B4)\nCmag\n\u001b\f=\u0000Jex\n~cjX\n\u001b0Z\n2DBZdkjj1\nvz\f\u000f\u001b\u001b0T\u001b0\f; (B5)\nwhile the tensors that multiply the in-plane electric \feld\nbecome\ncFM\n\u001b=cjX\n\fZ\n2DBZdkjj\u0016vz\nvz\f\u0016S\u001b\ffE\n\f: (B6)\ncmag\n\u001b=\u0000Jex\n~cjX\n\fZ\n2DBZdkjj1\nvz\f\u000f\u001b\u001b0T\u001b0\ffE\n\f (B7)\nwhere the velocity \u0016 vz(kjj) corresponds to the outgoing\nportion of the Fermi surface in the ferromagnet.\nAs we did for the currents, we solve for \u001c\u001bin terms of\n\u0016\u000b. Doing so yields the following torkance and torkivity\ntensors\n\u0000\u001b\f=C\u001b\r[A\u00001]\r\f\n\r\u001b=c\u001b\u0000\u0000\u001b\fa\f:\nThe torkance tensor describes the contribution to the to-\ntal spin torque that arises from the accumulations at the\ninterface. The torkivity tensor describes the subsequent\ncontribution from interfacial spin-orbit scattering when\ndriven by an in-plane electric \feld. Following the argu-\nments made for Eq. (A12):\n\u0000\u001b\f=C\u001b\f (B8)\n\r\u001b=c\u001b: (B9)\nAs seen in the companion paper, this approximation pro-\nduces good agreement with the interfacial charge cur-\nrents, spin currents, and spin torques computed via the\nBoltzmann equation.\nAppendix C: Boundary Parameters for Realistic\nInterfaces\nTo generalize the expressions from the previous section\nto include electronic structure, we must consider the non-\nequilibrium distribution function for all bands relevant to\ntransport:\nfm\u000b(k) = feq\nm\u000b(\"m\u000bk) +@feq\nm\u000b\n@\"m\u000bkgm\u000b(k): (C1)\nHeremdescribes the spin-independent band number and\n\u000bdenotes the spin/charge index. If the case of a non-\nmagnet, for each spin-independent band there are two\ndegenerate states. Linear combinations of these statescan produce phase coherent spin states that point in any\ndirection. Thus, for the non-magnet, the spin/charge in-\ndex should span \u000b2[d;f;`;c ], where the `direction is\naligned with the magnetization in the neighboring ferro-\nmagnet for convenience. In the ferromagnet all bands are\nnon-degenerate, so each state possesses a di\u000berent phase\nvelocity. As a result, linear combinations of these states\nhave spin expectation values that oscillate over position,\ncomplicating the description presented above. There is\nno natural pairing of non-degenerate spin states. How-\never, if states are quantized along a particular axis, the\nspin accumulations and spin currents with polarization\nalong that axis are well-de\fned regardless of the choice\nof pairing. Thus for each spin-independent band in the\nferromagnet, the spin/charge index spans the states de-\nscribing majority and minority carriers, i.e. \u000b2[\";#].\nWe generalize the approximate distribution function\nfE\nm\u000b(kjj) caused by an external electric \feld to allow for\na band dependence. We do so because the velocities now\ndepend on band number and the scattering times may as\nwell. However, we assume that the incoming polarization\nq\u000bdoes not depend on band number; thus we treat in-\ncident currents as if they originate from spin-dependent\n(but not band-dependent) reservoirs. The momentum\nrelaxation times for each spin-independent band in the\nnon-magnet are renormalized using Eq. (A1).\nTo account for coherence between bands, we begin with\na more general expression for the ensemble average of the\noutgoing current:\nhhjout\ni\u000bii=1\n~X\nmnn0\fZ\n2DBZdkjjgin\nm\f\njvmz\fj\n\u0002tr\u0002\n(sn0m)yJout\nn0n;i\u000bsnm\u001b\f\u0003\n:\n(C2)\nHeresstands for re\rection or transmission, depending on\nwhat region(s) incoming and outgoing carriers are from.\nThe indices mand\fcorrespond to incoming carriers,\nwhilen,n0, and\u000bdescribe the outgoing carriers. The\ncurrent operator Jout\nn0n;i\u000bis given by\nJout\nn0n;i\u000b=i~\n2mZ\n2DPCdrjj(\tn0k)y( \u0000@i\u001b\u000b\u0000\u001b\u000b\u0000 !@i)\tnk\n(C3)\nwhere the integral runs over a two-dimensional slice of\nthe primitive cell (aligned parallel to the interface). The\n2\u00022 matrix \t nkis de\fned for outgoing modes in the\nferromagnet as\n\tnk=eikjj\u0001rjj \nu\"\nnk(r)eik\"\nnzz0\n0u#\nnk(r)eik#\nnzz!\n;(C4)\nwhereu\"=#\nnk(r) andk\"=#\nnzdenote the Bloch wavefunction\nand out-of-plane wavevector for majority/minority car-\nriers. Both are de\fned at kjjon the Fermi surface cor-15\nresponding to band n. For outgoing modes in the non-\nmagnet, \t nksimpli\fes to:\n\tnk=eikjj\u0001rjjeiknzzunk(r)I2\u00022: (C5)\nThe incoming current is de\fned as follows\nhhjin\ni\u000bii=1\n~X\nm\fZ\n2DBZdkjjgin\nm\f\njvmz\fjtr\u0002\nJin\nm;i\u000b\u001b\f\u0003\n(C6)\nwhere\nJin\nm;i\u000b=\u0012izi~\n2mZ\n2DPCdrjj(\tmk)y( \u0000@i\u001b\u000b\u0000\u001b\u000b\u0000 !@i)\tmk\n(C7)\ngives the current operator for the incoming current. The\ntotal current is then\nhhji\u000bii=hhjin\ni\u000bii+hhjout\ni\u000bii\n=1\n~X\nmnn0\fZ\n2DBZdkjjgin\nm\f\njvmz\fj\n\u0002trh\u0000\n\u0012izJin\nm;i\u000b+ (sn0m)yJout\nn0n;i\u000bsnm\u0001\n\u001b\fi\n(C8)\nwhere the choice of scattering matrix depends on the in-\ncoming spin/charge index \fand outgoing spin/charge\nindex\u000bas follows:\nsnm=8\n>>>>><\n>>>>>:rnm\u000b;\f2[d;f;`;c ]\ntnm\u000b2[d;f;`;c ]; \f2[\";#]\nt\u0003\nnm\u000b2[\";#]; \f2[d;f;`;c ]\nr\u0003\nnm\u000b;\f2[\";#]\n(C9)\nBy plugging in the generalizations of Eqs. (17) and (18)\ninto Eq. (C8), we \fnd that Eq. (A9) generalizes to the\nfollowing\nBi\u000b\f=\u0000e\n~X\nmnn0Z\n2DBZdkjj1\njvmz\fj\n\u0002trh\u0000\n\u0012izJin\nm;i\u000b+ (sn0m)yJout\nn0n;i\u000bsnm\u0001\n\u001b\fi\n;\n(C10)\nwhile Eq. (A11) now becomes:\nbi\u000b=\u0000e\n~X\nmnn0\fZ\n2DBZdkjjfE\nm\f\njvmz\fj\n\u0002trh\u0000\n\u0012izJin\nm;i\u000b+ (sn0m)yJout\nn0n;i\u000bsnm\u0001\n\u001b\fi\n:\n(C11)Assuming as before that the incoming spin-currents be-\nhave as if they originate from spin-dependent reservoirs\n(\u0016\u000b\u0019q\u000b), we have:\nGi\u000b\f=Bi\u000b\f (C12)\n\u001bi\u000b=bi\u000b: (C13)\nThus, Eqs. (C10) and (C11) generalize the conductance\nand conductivity tensors respectively to include non-\ntrivial electronic structure.\nThe transverse spin current that develops in the ferro-\nmagnet atz= 0+may be obtained by using similar ex-\npressions. The tensor CFM\n\u001b\f, originally given by Eq. (B4),\nnow becomes\nCFM\n\u001b\f=\u0000e\n~X\nmnn0Z\n2DBZdkjj1\njvmz\fj\n\u00028\n<\n:tr\u0002\n(t\u0003\nn0m)yJout\nn0n;z\u001bt\u0003\nnm\u001b\f\u0003\n\f2[d;f;`;c ]\ntr\u0002\n(r\u0003\nn0m)yJout\nn0n;z\u001br\u0003\nnm\u001b\f\u0003\n\f2[\";#]:\n(C14)\nLikewise, the tensor cFM\n\u001b, \frst described by Eq. (B6),\ngeneralizes to the following:\ncFM\n\u001b=\u0000e\n~X\nmnn0\fZ\n2DBZdkjjfE\nm\f\njvmz\fj\n\u00028\n<\n:tr\u0002\n(t\u0003\nn0m)yJout\nn0n;z\u001bt\u0003\nnm\u001b\f\u0003\n\f2[d;f;`;c ]\ntr\u0002\n(r\u0003\nn0m)yJout\nn0n;z\u001br\u0003\nnm\u001b\f\u0003\n\f2[\";#]:\n(C15)\nEvaluating the trace in Eq. (C15) gives the ensemble av-\nerage of velocity for the transverse spin states in the fer-\nromagnet. Here we do not assume that the velocity of\nthese states equals the average velocity of majority and\nminority carriers. However, for the simple model dis-\ncussed in the previous section, one can show that the\ncurrent operator Jout\nn0n;z\u001bsimpli\fes to the following:\nJout\nn0n;z\u001b!Jout\nz\u001b/1\n2\u0000\nvz\"+vz#\u0001\n\u001b\u001b (C16)\nIn this scenario, Eqs. (C14) and (C15) reduce to\nEqs. (B4) and (B6) as expected. This justi\fes the use\nof the average velocity to describe transverse spin states\nin the simple model. For \u0016\u000b\u0019q\u000bwe have:\n\u0000FM\n\u001b\f=CFM\n\u001b\f (C17)\n\rFM\n\u001b=cFM\n\u001b: (C18)\nThus we have generalized the torkance and torkivity\ntensors that describe bulk ferromagnet torques for non-\ntrivial electronic structures.\nFor realistic systems, the interface should be modeled\nover a few atomic layers so that an exchange potential16\nParameter Value\nE\u000bective mixing conductance\nRe[~G\"#]GzddorGzff\nIm[~G\"#]GzdforGzfd\nSpin current due to interfacial spin-orbit scattering\njE\nd(0\u0000)\u001bzd~E\njE\nf(0\u0000)\u001bzf~E\nSpin torque on the lattice at the interface\n\u001clatt\nd\u0000\n\u001bzd\u0000\rd\u0001~E\n\u001clatt\nf\u0000\n\u001bzf\u0000\rf\u0001~E\nTABLE I. Table of phenomenological parameters relevant to\nthe drift-di\u000busion model of spin-orbit torque developed in the\ncompanion paper, chosen by the numerical study performed\nin Sec.IV. All other boundary parameters are discarded in\nthat model. As can be seen in section IIIA of the companion\npaper, the \frst four parameters govern the total spin torque\nthickness dependence, while the last two parameters describe\nthe spin torque's zero-thickness intercept. Note that here all\nboundary parameters obey the sign convention that positive\ncurrents \row towards from the ferromagnet.\nand spin-orbit coupling may simultaneously exist. If\nthese atomic layers make up the scattering region used\nto obtain the scattering coe\u000ecients, then the expressions\npresented here describe the currents on either side of the\ninterface as intended. However, in order to describe the\ninterfacial torque, the tensors Cmag\n\u001b\fandcmag\n\u001bmust be\nwritten as sums of the layer-resolved torques within the\ninterfacial scattering region. We save the generalization\nof Eqs. (B5) and (B7) for future work, since in this paper\nwe treat the interface as a plane rather than a region of\n\fnite thickness.\nAppendix D: Boundary parameters relevant to\nbilayer spin-orbit torques\nIn Sec. IV, we numerically analyze each boundary pa-\nrameter for an interfacial scattering potential that in-\ncludes the exchange interaction and spin-orbit coupling.\nWe \fnd that many parameters di\u000ber by several orders\nof magnitude. In the companion paper, we use this in-\nformation to derive an analytical drift-di\u000busion model of\nspin-orbit torques in heavy metal/ferromagnet bilayers.\nIn the following we discuss the minimal set of parameterscrucial to that solution.\nTable I includes six parameters important to the inter-\nface of heavy metal/ferromagnet bilayers. Along with the\nspin di\u000busion length ( lsf), the bulk conductivity ( \u001bNM\nbulk),\nand the spin Hall current density ( jsH\nd) in the non-\nmagnet, they describe all of the phenomenological pa-\nrameters used by the analytical drift-di\u000busion model in\nthe companion paper. The \frst two parameters are the\nreal and imaginary parts of the spin mixing conductance.\nThe generalized version of these parameters may be ex-\ntracted from the conductance tensor Gz\u000b\f. Numerical\nstudies show that these parameters depend weakly on\nmagnetization direction. In the companion paper, the\nungeneralized spin mixing conductance is used. The pa-\nrametersjE\nd(0\u0000) andjE\nf(0\u0000) denote the interfacial spin\ncurrents just within the non-magnet that arise due to\nin-plane electric \felds and spin-orbit scattering. In anal-\nogy to the bulk spin Hall current, these parameters act\nas sources of spin current for the drift-di\u000busion equa-\ntions. Thus, in the absence of jE\nd(0\u0000),jE\nf(0\u0000), andjsH\nd,\nall bulk currents and accumulations vanish. In addition\nto the spin mixing conductance, these parameters deter-\nmine the non-magnet thickness-dependence of spin-orbit\ntorques. The \fnal two parameters give the approximate\nloss of angular momentum to the interface. They equal\nthe damping-like and \feld-like spin-orbit torques in the\nlimit of vanishing non-magnet thickness. They are de-\nrived by subtracting the interfacial torque from the loss\nin out-of-plane spin current density across the interface.\nOur numerical analysis suggests that spin and charge\naccumulations cause negligible di\u000berences in these two\nquantities. Thus, we assume that \u001clatt\ndand\u001clatt\nfstem\nprimarily from spin-orbit scattering at the interface. The\ntreatment of the lattice torque presented in the compan-\nion paper begins from this assumption.\nThe model introduced in the companion paper general-\nizes the drift-di\u000busion model used in Ref. [42] to include\ninterfacial spin-orbit e\u000bects. Only two additional phe-\nnomenological parameters ( jE\nd(0\u0000) andjE\nf(0\u0000)) are re-\nquired to capture the non-magnet thickness dependence,\nwhile an additional two parameters ( \u001clatt\ndand\u001clatt\nf) de-\nscribe the corresponding zero-thickness intercept. Table I\nprovides formulas for these phenomenological parameters\nin terms of the boundary parameters contained within\nEqs. (12) and (13). We note that in magnetoelectronic\ncircuit theory, the conductance parameters are given by\nsums of interfacial scattering coe\u000ecients over the avail-\nable scattering states. All of the boundary parameters\nintroduced here possess a similar form, as discussed in\nappendices A, B, and C.\n[1] T. Jungwirth, J. Wunderlich, and K. Olejnk, Nature\nMaterials 11, 382 (2012).\n[2] E. M. Hankiewicz and G. Vignale, Journal of Physics:\nCondensed Matter 21, 253202 (2009).\n[3] A. Manchon and S. Zhang, Phys. Rev. B 78, 212405(2008).\n[4] A. Manchon and S. Zhang, Phys. Rev. B 79, 094422\n(2009).\n[5] A. Matos-Abiague and R. L. Rodr\u0013inguez-Su\u0013 arez, Phys.\nRev. B 80, 094424 (2009).17\n[6] K. Ando, S. Takahashi, K. Harii, K. Sasage, J. Ieda,\nS. Maekawa, and E. Saitoh, Phys. Rev. Lett. 101, 036601\n(2008).\n[7] I. M. Miron, K. Garello, G. Gaudin, P.-J. Zermatten,\nM. V. Costache, S. Au\u000bret, S. Bandiera, B. Rodmacq,\nA. Schuhl, and P. Gambardella, Nature 476, 189 (2011).\n[8] Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys.\nRev. Lett. 88, 117601 (2002).\n[9] J.-C. Rojas-S\u0013 anchez, N. Reyren, P. Laczkowski,\nW. Savero, J.-P. Attan\u0013 e, C. Deranlot, M. Jamet, J.-M.\nGeorge, L. Vila, and H. Ja\u000br\u0012 es, Phys. Rev. Lett. 112,\n106602 (2014).\n[10] W. Thomson, Proceedings of the Royal Society of London\n8, 546 (1856).\n[11] T. McGuire and R. Potter, IEEE Transactions on Mag-\nnetics 11, 1018 (1975).\n[12] C. Gould, C. Ruster, T. Jungwirth, E. Girgis, G. M.\nSchott, R. Giraud, K. Brunner, G. Schmidt, and L. W.\nMolenkamp, Phys. Rev. Lett. 93, 117203 (2004).\n[13] A. N. Chantis, K. D. Belashchenko, E. Y. Tsymbal, and\nM. van Schilfgaarde, Phys. Rev. Lett. 98, 046601 (2007).\n[14] N. Nagaosa, J. Sinova, S. Onoda, A. H. MacDonald, and\nN. P. Ong, Rev. Mod. Phys. 82, 1539 (2010).\n[15] G. E. W. Bauer, E. Saitoh, and B. J. van Wees, Nature\nMaterials 11, 391 (2012).\n[16] V. P. Amin, J. Zemen, J. \u0014Zelezn\u0013 y, T. Jungwirth, and\nJ. Sinova, Phys. Rev. B 90, 140406 (2014).\n[17] C. L\u0013 opez-Mon\u0013ins, A. Matos-Abiague, and J. Fabian,\nPhys. Rev. B 90, 174426 (2014).\n[18] L. N\u0013 eel, J. Physique Rad. 15, 225 (1954).\n[19] G. H. O. Daalderop, P. J. Kelly, and M. F. H. Schuur-\nmans, Phys. Rev. B 41, 11919 (1990).\n[20] M. T. Johnson, P. J. H. Bloemen, F. J. A. den Broeder,\nand J. J. de Vries, Reports on Progress in Physics 59,\n1409 (1996).\n[21] K. Garello, I. M. Miron, C. O. Avci, F. Freimuth,\nY. Mokrousov, S. Blugel, S. Au\u000bret, O. Boulle,\nG. Gaudin, and P. Gambardella, Nature Nanotechnology\n8, 587593 (2013).\n[22] X. Fan, H. Celik, J. Wu, C. Ni, K.-J. Lee, V. O. Lorenz,\nand J. Q. Xiao, Nature Communications 5, (2014).\n[23] G. Allen, S. Manipatruni, D. E. Nikonov, M. Doczy, and\nI. A. Young, Phys. Rev. B 91, 144412 (2015).\n[24] S. Emori, T. Nan, A. M. Belkessam, X. Wang, A. D.\nMatyushov, C. J. Babroski, Y. Gao, H. Lin, and N. X.\nSun, Phys. Rev. B 93, 180402 (2016).\n[25] M. I. D'yakonov and V. I. Perel, 13, 467 (1971).\n[26] J. E. Hirsch, Phys. Rev. Lett. 83, 1834 (1999).\n[27] S. Zhang, Phys. Rev. Lett. 85, 393 (2000).\n[28] S. Murakami, N. Nagaosa, and S.-C. Zhang, Science 301,\n1348 (2003).\n[29] J. Sinova, D. Culcer, Q. Niu, N. A. Sinitsyn, T. Jung-\nwirth, and A. H. MacDonald, Phys. Rev. Lett. 92,\n126603 (2004).\n[30] Y. K. Kato, R. C. Myers, A. C. Gossard, and D. D.\nAwschalom, Science 306, 1910 (2004).\n[31] J. Wunderlich, B. Kaestner, J. Sinova, and T. Jungwirth,\nPhys. Rev. Lett. 94, 047204 (2005).\n[32] J. C. Slonczewski, Phys. Rev. B 39, 6995 (1989).\n[33] J. Slonczewski, Journal of Magnetism and Magnetic Ma-\nterials 159, L1 (1996).\n[34] L. Berger, Phys. Rev. B 54, 9353 (1996).\n[35] D. Ralph and M. Stiles, Journal of Magnetism and Mag-\nnetic Materials 320, 1190 (2008).[36] M. D. Stiles and A. Zangwill, Phys. Rev. B 66, 014407\n(2002).\n[37] V. Edelstein, Solid State Communications 73, 233\n(1990).\n[38] S. D. Ganichev, E. L. Ivchenko, V. V. Bel'kov, S. A.\nTarasenko, M. Sollinger, D. Weiss, W. Wegscheider, and\nW. Prettl, Nature 417, 153156 (2002).\n[39] Y. K. Kato, R. C. Myers, A. C. Gossard, and D. D.\nAwschalom, Phys. Rev. Lett. 93, 176601 (2004).\n[40] A. Y. Silov, P. A. Blajnov, J. H. Wolter, R. Hey, K. H.\nPloog, and N. S. Averkiev, Applied Physics Letters 85,\n5929 (2004).\n[41] J. C. R. S\u0013 anchez, L. Vila, G. Desfonds, S. Gambarelli,\nJ. P. Attan\u0013 e, J. M. D. Teresa, C. Mag\u0013 en, and A. Fert,\nNature Communications 4, (2013).\n[42] P. M. Haney, H.-W. Lee, K.-J. Lee, A. Manchon, and\nM. D. Stiles, Phys. Rev. B 87, 174411 (2013).\n[43] C. Gorini, P. Schwab, R. Raimondi, and A. L. Shelankov,\nPhys. Rev. B 82, 195316 (2010).\n[44] C. Gorini, R. Raimondi, and P. Schwab, Phys. Rev. Lett.\n109, 246604 (2012).\n[45] R. Raimondi, P. Schwab, C. Gorini, and G. Vignale,\nAnnalen der Physik 524, n/a (2012).\n[46] K. Shen, G. Vignale, and R. Raimondi, Phys. Rev. Lett.\n112, 096601 (2014).\n[47] O. Mosendz, J. E. Pearson, F. Y. Fradin, G. E. W. Bauer,\nS. D. Bader, and A. Ho\u000bmann, Phys. Rev. Lett. 104,\n046601 (2010).\n[48] H. Nakayama, K. Ando, K. Harii, T. Yoshino, R. Taka-\nhashi, Y. Kajiwara, K. Uchida, Y. Fujikawa, and\nE. Saitoh, Phys. Rev. B 85, 144408 (2012).\n[49] Z. Feng, J. Hu, L. Sun, B. You, D. Wu, J. Du, W. Zhang,\nA. Hu, Y. Yang, D. M. Tang, B. S. Zhang, and H. F.\nDing, Phys. Rev. B 85, 214423 (2012).\n[50] V. Vlaminck, J. E. Pearson, S. D. Bader, and A. Ho\u000b-\nmann, Phys. Rev. B 88, 064414 (2013).\n[51] W. Zhang, V. Vlaminck, J. E. Pearson, R. Divan, S. D.\nBader, and A. Ho\u000bmann, Applied Physics Letters 103,\n242414 (2013).\n[52] M. Obstbaum, M. H artinger, H. G. Bauer, T. Meier,\nF. Swientek, C. H. Back, and G. Woltersdorf, Phys. Rev.\nB89, 060407 (2014).\n[53] W. Zhang, W. Han, X. Jiang, S.-H. Yang, and S. S. P.\nParkin, Nature Physics 11, 496 (2015).\n[54] Y. Liu, Z. Yuan, R. J. H. Wesselink, A. A. Starikov, and\nP. J. Kelly, Phys. Rev. Lett. 113, 207202 (2014).\n[55] Y.-T. Chen, S. Takahashi, H. Nakayama, M. Althammer,\nS. T. B. Goennenwein, E. Saitoh, and G. E. W. Bauer,\nPhys. Rev. B 87, 144411 (2013).\n[56] M. Weiler, M. Althammer, F. D. Czeschka, H. Huebl,\nM. S. Wagner, M. Opel, I.-M. Imort, G. Reiss,\nA. Thomas, R. Gross, and S. T. B. Goennenwein, Phys.\nRev. Lett. 108, 106602 (2012).\n[57] S. Y. Huang, X. Fan, D. Qu, Y. P. Chen, W. G. Wang,\nJ. Wu, T. Y. Chen, J. Q. Xiao, and C. L. Chien, Phys.\nRev. Lett. 109, 107204 (2012).\n[58] H. Nakayama, M. Althammer, Y.-T. Chen, K. Uchida,\nY. Kajiwara, D. Kikuchi, T. Ohtani, S. Gepr ags,\nM. Opel, S. Takahashi, R. Gross, G. E. W. Bauer, S. T. B.\nGoennenwein, and E. Saitoh, Phys. Rev. Lett. 110,\n206601 (2013).\n[59] C. Hahn, G. de Loubens, O. Klein, M. Viret, V. V. Nale-\ntov, and J. Ben Youssef, Phys. Rev. B 87, 174417 (2013).\n[60] N. Vlietstra, J. Shan, V. Castel, B. J. van Wees, and18\nJ. Ben Youssef, Phys. Rev. B 87, 184421 (2013).\n[61] M. Althammer, S. Meyer, H. Nakayama, M. Schreier,\nS. Altmannshofer, M. Weiler, H. Huebl, S. Gepr ags,\nM. Opel, R. Gross, D. Meier, C. Klewe, T. Kuschel, J.-M.\nSchmalhorst, G. Reiss, L. Shen, A. Gupta, Y.-T. Chen,\nG. E. W. Bauer, E. Saitoh, and S. T. B. Goennenwein,\nPhys. Rev. B 87, 224401 (2013).\n[62] K. Chen and S. Zhang, Phys. Rev. Lett. 114, 126602\n(2015).\n[63] K. Eid, D. Portner, J. A. Borchers, R. Loloee, M. Al-\nHaj Darwish, M. Tsoi, R. D. Slater, K. V. O'Donovan,\nH. Kurt, W. P. Pratt, and J. Bass, Phys. Rev. B 65,\n054424 (2002).\n[64] H. Kurt, R. Loloee, K. Eid, W. P. Pratt, and J. Bass,\nApplied Physics Letters 81, 4787 (2002).\n[65] H. Nguyen, W. P. Jr., and J. Bass, Journal of Magnetism\nand Magnetic Materials 361, 30 (2014).\n[66] J. Bass and W. P. P. Jr, Journal of Physics: Condensed\nMatter 19, 183201 (2007).\n[67] L. Wang, R. J. H. Wesselink, Y. Liu, Z. Yuan, K. Xia,\nand P. J. Kelly, Phys. Rev. Lett. 116, 196602 (2016).\n[68] J. Borge, C. Gorini, G. Vignale, and R. Raimondi, Phys.\nRev. B 89, 245443 (2014).\n[69] S. S.-L. Zhang, G. Vignale, and S. Zhang, Phys. Rev. B\n92, 024412 (2015).[70] To convert these quantities back into their traditional\nunits, one must multiply the components describing\ncharge current densities by \u0000eand those describing spin\ncurrent densities by ~=2.\n[71] A. Brataas, Y. V. Nazarov, and G. E. W. Bauer, Phys.\nRev. Lett. 84, 2481 (2000).\n[72] A. Brataas, Y. Nazarov, and G. Bauer, The European\nPhysical Journal B - Condensed Matter and Complex\nSystems 22, 99 (2001).\n[73] Note, however, that an average charge accumulation\namounts to an o\u000bset in electric potential; thus it can-\nnot a\u000bect the interfacial spin currents.\n[74] M. D. Stiles and D. R. Penn, Phys. Rev. B 61, 3200\n(2000).\n[75] K. Xia, P. Kelly, G. Bauer, A. Brataas, and I. Turek,\nPhysical Review B: Condensed matter and materials\nphysics 65, 220401 (2002).\n[76] P. M. Haney, H.-W. Lee, K.-J. Lee, A. Manchon, and\nM. D. Stiles, Phys. Rev. B 88, 214417 (2013).\n[77] F. Freimuth, S. Bl ugel, and Y. Mokrousov, Phys. Rev.\nB90, 174423 (2014).\n[78] F. Freimuth, S. Bl ugel, and Y. Mokrousov, Phys. Rev.\nB92, 064415 (2015).\n[79] G. G\u0013 eranton, F. Freimuth, S. Bl ugel, and Y. Mokrousov,\nPhys. Rev. B 91, 014417 (2015)."    },    {        "title": "1002.0441v2.Spin_resolved_scattering_through_spin_orbit_nanostructures_in_graphene.pdf",        "content": "arXiv:1002.0441v2  [cond-mat.mes-hall]  26 Apr 2010Spin-resolved scattering through spin-orbit nanostructu res in graphene\nD. Bercioux1,2,∗and A. De Martino3,†\n1Freiburg Institute for Advanced Studies, Albert-Ludwigs- Universit¨ at, D-79104 Freiburg, Germany\n2Physikalisches Institut, Albert-Ludwigs-Universit¨ at, D-79104 Freiburg, Germany\n3Institut f¨ ur Theoretische Physik, Universit¨ at zu K¨ oln, Z¨ ulpicher Straße 77, D-50937 K¨ oln, Germany\n(Dated: November 10, 2018)\nWe address the problem of spin-resolved scattering through spin-orbit nanostructures in graphene,\ni.e., regions of inhomogeneous spin-orbit coupling on the nanom eter scale. We discuss the phe-\nnomenon of spin-double refraction and its consequences on t he spin polarization. Specifically, we\nstudy the transmission properties of a single and a double in terface between a normal region and\na region with finite spin-orbit coupling, and analyze the pol arization properties of these systems.\nMoreover, for the case of a single interface, we determine th e spectrum of edge states localized at\nthe boundary between the two regions and study their propert ies.\nPACS numbers: 72.80.Vp, 73.23.Ad, 72.25.-b, 72.25.Mk, 71. 70.Ej\nI. INTRODUCTION\nGraphene1,2— a singlelayerofcarbonatoms arranged\nin a honeycomb lattice — has attracted huge atten-\ntion in the physics community because of many unusual\nelectronic, thermal and nanomechanical properties.3,4In\ngraphene the Fermi surface, at the charge neutrality\npoint, reduces to two isolated points, the two inequiv-\nalent corners KandK′of the hexagonal Brillouin zone\nof the honeycomb lattice. In their vicinity the charge\ncarriers form a gas of chiral massless quasiparticles with\na characteristic conical spectrum. The low-energy dy-\nnamics is governed by the Dirac-Weyl (DW) equation5,6\nin which the role of speed of light is played by the elec-\ntron Fermi velocity. The chiral nature of the quasipar-\nticles and their linear spectrum lead to remarkable con-\nsequences for a variety of electronic properties as weak\nlocalization, shot noise, Andreev reflection, and many\nothers. Also the behavior in a perpendicular magnetic\nfield discloses new physics. Graphene exhibits a zero-\nenergy Landau level, whose existence gives rise to an un-\nconventional half-integer quantum Hall effect, one of the\npeculiar hallmarks of the DW physics.\nDriven by the prospects of using this material in spin-\ntronic applications,7,8the study of spin transport is one\nof the most active field in graphene research.9–14Sev-\neral experiments have recently demonstrated spin injec-\ntion, spin-valve effect, and spin-coherent transport in\ngraphene, with spin relaxation length of the order of\nfew micrometers.10,14In this context a crucial role is\nplayedbythe spin-orbitinteraction. Ingraphenesymme-\ntries allow for two kinds of spin-orbit coupling (SOC).15\nTheintrinsic SOC originates from carbon intra-atomic\nSOC. It opens a gap in the energy spectrum and con-\nverts graphene into a topological insulator with a quan-\ntized spin-Hall effect.15This term has been estimated\nto be rather weak in clean flat graphene.16–19Theex-\ntrinsicRashba-like SOC originates instead from inter-\nactions with the substrate, presence of a perpendic-\nular external electric field, or curvature of graphene\nmembrane.16–18,20This term is believed to be responsiblefor spin polarization21and spin relaxation22,23physics in\ngraphene. Optical-conductivitymeasurementscouldpro-\nvide a way to determine the respective strength of both\nSOCs.24\nIn this article we address the problem of ballistic spin-\ndependent scattering in the presence of inhomogeneous\nspin-orbit couplings. Our main motivation stems from\na recent experiment that reported a large enhancement\nof Rashba SOC splitting in single-layer graphene grown\non Ni(111) intercalated with a Au monolayer.25Further\nexperimental results show that the intercalation of Au\natoms between graphene and the Ni substrate is essen-\ntial in order to observe sizable Rashba effect.26,27The\npreparation technique of Ref. 25seems to provide a sys-\ntem with properties very close to those of freestanding\ngraphene in spite of the fact that graphene is grown on\na solid substrate. The presence of the substrate does not\nseem to fundamentally alter the electronic properties ob-\nserved in suspended systems, i.e., the existence of Dirac\npoints at the Fermi energy and the gapless conical dis-\npersion in their vicinity.\nThese results suggest that a certain degree of control\non the SOC can be achieved by appropriate substrate\nengineering, with variations of the SOC strength on sub-\nmicrometer scales, without spoiling the relativistic gap-\nless nature of quasiparticles. This could pave the way for\nthe realization of spin-optics devices for spin filtration\nand spin control for DW fermions in graphene. An opti-\nmal design would require a detailed understanding of the\nspin-resolved ballistic scattering through such spin-orbit\nnanostructures , which is the aim of this paper.\nThe problem of spin transport through nanostructures\nwith inhomogeneous SOC has already been thoroughly\nstudied in the case of two-dimensional electron gas in\nsemiconductor heterostructures with Rashba SOC.28–30\nHere the Rashba SOC31— arising from the inversion\nasymmetry of the confinement potential — couples the\nelectron momentum to the spin degree of freedom and\nthereby lifts the spin degeneracy. In this case, a region\nwith finite SOC between two normal regions has prop-\nerties similar to biaxial crystals: an electron wave inci-2\nN region SO region \nky\nkxky\nkxφE k+k-\nξ+ξ-\nFigure 1: (Color online) Illustration of the kinematics of t he\nscattering at a N-SO interface in graphene. The circles rep-\nresent constant energy contours.\ndent from the normal region splits at the interface and\nthe two resulting waves propagate in the SO region with\ndifferent Fermi velocities and momenta.28This effect —\nanalogousto the opticaldouble-refraction— producesan\ninterference pattern when the electron waves emerge in\nthe second normal region. Moreover, electrons that are\ninjected in an spin unpolarized state emerge from the SO\nregion in a partially polarized state.\nHerewe shallfocus on the twosimplest examplesofSO\nnanostructuresingraphene: (i)asingleinterfacebetween\ntwo regions with different strengths of SOC; (ii) a SOC\nbarrier, or double interface, i.e., a region of finite SOC\nin between two regions with vanishing SOC.\nOur analysis shows — in analogy to the case of 2DEG\n— that the ballistic propagation of carriers is governed\nbythespin-doublerefraction. Wefind thatthescattering\nproperties of the structure strongly depend on the injec-\ntion angle. As a consequence, an initially unpolarized\nDW quasiparticle emerges from the SOC barrier with a\nfinite spin polarization. In analogy to the edge states in\nthe quantum spin-Hall effect,15we also consider the pos-\nsibility of edge states localized at the interface between\nregions with and without SOC.\nThis paper is organized as follows. In Sec. IIwe intro-\nduce the model and the transfer matrix formalism used\nin the rest of the paper. In Sec. IIIwe discuss the scat-\ntering problem at a single interface and the spectrum of\nedge states. In Sec. IVwe address the case of a dou-\nble interface — a SOC barrier — and the final Sec. Vis\ndevoted to the discussion of results and conclusions.\nII. MODEL AND FORMALISM\nWe consider a clean graphene sheet in the xy-\nplane with SOCs15,16,21,32,33inhomogeneous along the\nx-direction. We shall restrict ourselves to a single-\nparticle picture and neglect electron-electron interaction\neffects. The length scale over which the SOCs vary is as-\nsumed to be much largerthan graphene’slattice constant\n(a= 0.246 nm) but much smaller than the typical Fermiwavelength of quasiparticles λF. Since close to the Dirac\npointsλF∼1/|E|, at low energy Ethis approximation\nis justified. This assumption ensures that we can use the\ncontinuum DW description, in which the two valleys are\nnot coupled. Yet close to a Dirac point we can approxi-\nmate the variation of SOCs as a sharp change. Focusing\non a single valley, the single-particle Hamiltonian reads\nH=vFσ·p+HSO, (1)\nHSO=λ(x)\n2(σ×s)z+∆(x)σzsz, (2)\nwherevF≈106m/s is the Fermi velocity in graphene.\nIn the following we set /planckover2pi1=vF= 1. The vector of Pauli\nmatrices σ= (σx,σy) [resp.s= (sx,sy)] acts in sublat-\ntice space [resp. spin space]. The term HSOcontains the\nextrinsic or Rashba SOC of strength λand the intrinsic\nSOC of strength ∆. While experimentally the Rashba\nSOC can be enhanced by appropriate optimization of\nthe substrate up to values of the order of 14 meV,25the\nintrinsic SOC seems at least two orders of magnitude\nsmaller. Yet, the limit of large intrinsic SOC is of con-\nsiderable interest since in this regime graphene becomes\na topological insulator.15Thus in this paper we shall\nnot restrict ourself to the experimentally relevant regime\nλ≫∆ but consider also the complementary regime.\nThe wave function Ψ is expressed as\nΨT= (ΨA↑,ΨB↑,ΨA↓,ΨB↓),\nwhere the superscriptTdenotes transposition. Spectrum\nand eigenspinors of the Hamiltonian ( 1) with uniform\nSOCs are briefly reviewed in Appendix A. The spec-\ntrum consists of four branches Eα,ǫ(k) labelled by the\ntwo quantum numbers ǫ=±1 andα=±1. Here, the\nfirst distinguishes particle and hole branches, the second\ngives the sign of the expectation value of the spin pro-\njection along the in-plane direction perpendicular to the\npropagation direction k. The spectrum strongly depends\non the ratio\nη=∆\nλ. (3)\nForη >1/2 a gap separates particle and hole branches.\nThe gap closes at η= 1/2 and forη <1/2 one particle\nbranch and one hole branch are degenerate at k= 0 (see\nFig.8in App.A).\nWe now briefly summarize the transfer matrix ap-\nproach employed in this paper to solve the DW scat-\ntering problem.35–38We assume translational invariance\nin they-direction, thus the scattering problem for the\nHamiltonian( 1)reducestoaneffectivelyone-dimensional\n(1D) one. The wave function factorizes as Ψ( x,y) =\neikyyχ(x), wherekyis the conserved y-component of the\nmomentum, which parameterizes the eigenfunctions of\nthe Hamiltonian of given energy E.\nFor simplicity we consider piecewise constant profiles\nof SOCs, and solve the DW equation in each region of\nconstant couplings. Then we introduce the x-dependent3\n0  !/4  !/2 0 !/4  !/2 \nIncident angle !Refraction angles \"#\"-\"+\n0  !/4  !/2 !-!+(a) (b)\nFigure 2: (Color online) Refraction angles as function of th e\nincidence angle for fixed energy and fixed SOCs. Panel (a):\nE= 5,λ= 0.5, ∆ = 2; panel (b): E= 5,λ= 2, ∆ = 0 .5.\n4×4 matrix Ω( x), whose columns are given by the com-\nponents of the independent, normalized eigenspinor of\nthe 1D DW Hamiltonian at fixed energy.39Due to the\ncontinuity of the wave function at each interface between\nregions of different SOC, the wave function on the left\nof the interface can be expressed in terms of the wave\nfunction on the right via the transfer matrix\nM=/bracketleftbig\nΩ(x−\n0)/bracketrightbig−1Ω(x+\n0), (4)\nwherex0is the position of the interface and x±\n0=x0±δ\nwith infinitesimal positive δ. The condition det M= 1\nguarantees conservation of the probability current across\ntheinterface. Thegeneralizationtothecaseofasequence\nofNinterfaces at positions xi,i= 1,...,N, is straight-\nforward since the transfer matrices relative to individual\ninterfaces combine via matrix multiplication:\nM=N/productdisplay\ni=1/bracketleftbig\nΩ(x−\ni)/bracketrightbig−1Ω(x+\ni). (5)\nFrom the transfer matrix it is straightforward to deter-\nmine transmission and reflection matrices, which encode\nall the relevant information on the scattering properties.\nIII. THE N-SO INTERFACE\nFirst we concentrate on the elastic scattering problem\nat the interface separating a normal region N ( x <0),\nwhere SOCs vanish, and a SO region ( x >0), where\nSOCs are finite and uniform.\nWe consider a quasiparticle of energy E, withEas-\nsumed positive for definiteness and outside the gap pos-\nsibly opened by SOCs. This quasiparticle incident from\nthe normal region is characterized by the y-component\nof the momentum, or equivalently, the incidence angle φ\nmeasured with respect to the normal at the interface, seeFig.1. Conservation of kyimplies that\nkN\ny=Esinφ=Eαsinξα=kSO\ny (6a)\nkN\nx=Ecosφ (6b)\nkSO\nxα=Eαcosξα (6c)\nwhereα=±1 andEα=/radicalbig\n(E−∆)(E+∆−αλ). The\nrefraction angles ξαare fixed by momentum conservation\nalong the interface ( 6a) and read\nξα= arcsin/parenleftbiggE\nEαsinφ/parenrightbigg\n. (7)\nFigure1illustrates the refraction process at the N-\nSO interface. The incident wave function, assumed to\nhave fixed spin projection in the z-direction, in the SO\nregion splits in a superposition of eigenstates of the\nSOCs Hamiltonian corresponding to states in the differ-\nent branches of the spectrum. These eigenstates prop-\nagate along two distinct directions characterized by the\nanglesξα, whose difference depends on SOC and is an in-\ncreasing function of the incidence angle, see Fig. 2. The\nanglesξαcoincide only for normal incidence or for λ= 0.\nEquation ( 7) implies that there exists a critical angle\nfor each of the two modes given by\n˜φα= arcsin/parenleftbiggEα\nE/parenrightbigg\n. (8)\nForφlarger than both critical angles ˜φα, the quasipar-\nticle is fully reflected, since there are no available trans-\nmission channels in the SO region. For φin between the\ntwo critical angles the quasiparticletransmits only in one\nchannel.40\nAfter this qualitative discussion of the kinematics, we\nnow present the exact solution of the scattering problem.\nIn the N region x <0 a normalized scattering state of\nenergyE >0, incident from the left on the interface with\nincidence angle φand spin projection sis given by\nχN(x) =[δ↑,s|↑/angb∇acket∇ight+δ↓,s|↓/angb∇acket∇ight]/parenleftbigg1\neiφ/parenrightbiggeikxx\n/radicalbig2vx\nF\n+[r↑s|↑/angb∇acket∇ight+r↓s|↓/angb∇acket∇ight]/parenleftbigg1\n−e−iφ/parenrightbigge−ikxx\n/radicalbig2vx\nF,(9)\nwherekx≡kN\nx(cf. Eq. 6b). Here the index s=↑,↓\nspecifies the spin projection of the incoming quasipar-\nticle with |↑/angb∇acket∇ightand|↓/angb∇acket∇ighteigenstates of szandδi,jis the\nKronecker delta. The velocity vx\nF= cosφis included to\nensure proper normalization of the scattering state. The\ncomplex coefficients rs′sare reflection probability ampli-\ntudes for a quasiparticle with spin sto be reflected with\nspins′. The associated matrix Ω N(x) reads\nΩN(x) =1/radicalbig2vx\nF\neikxxe−ikxx0 0\nei(kxx+φ)−e−i(kxx+φ)0 0\n0 0 eikxxe−ikxx\n0 0 ei(kxx+φ)−e−i(kxx+φ)\n.4\n00.1 0.2 0.3 0.4 0.5 \n-90 -75 -60 -45 -30 -15 015 30 45 60 75 90 (a)\n00.1 0.2 0.3 0.4 0.5 \n-90 -75 -60 -45 -30 -15 015 30 45 60 75 90 (b)\n00.1 0.2 0.3 0.4 0.5 0.6 \n-90 -75 -60 -45 -30 -15 015 30 45 60 75 90 (c)\nFigure 3: (Color online) Angular dependence of the transmis -\nsion probabilities T+↑(blue dashed line) and T−↑(red solid\nline) at energy E= 2.5. The SOC are fixed as follows: (a)\nλ= 0.1 and ∆ = 0, (b) λ= 0 and ∆ = 0 .1, and (c) λ= 0.5\nand ∆ = 0 .1.\nSimilarly the wave function in the SO region ( x>0) can\nbe expressed in general form as\nχSO(x) =1/radicalbigvx\n++/bracketleftbig\nt+ψ++(x)+r+¯ψ++(x)/bracketrightbig\n+1/radicalbigvx\n−+/bracketleftbig\nt−ψ−+(x)+r−¯ψ−+(x)/bracketrightbig\n(10)\nwheret±(resp.r±) are complex amplitudes for right-\nmoving (resp. left-moving) states. The coefficient tα\nrepresents the transmission amplitude into mode α. The\nwave functions ψα+and the Fermi velocities vx\nα+in the\nSO region are obtained from the expressions given in\nApp.Awith the replacement kx→kSO\nxα, where for no-\ntational simplicity the label SO will be understood. The\nwave functions ¯ψα+are in turn obtained from ψα+by\nreplacingkxα→ −kxα. The matrix Ω SO(x) then reads\nΩSO(x) = (11)\n\ne−iξ+−θ+\n2−eiξ+−θ+\n2e−iξ−−θ−\n2−eiξ−−θ−\n2\neθ+\n2 eθ+\n2 eθ−\n2 eθ−\n2\nieθ+\n2 ieθ+\n2 −ieθ−\n2−ieθ−\n2\nieiξ+−θ+\n2−ie−iξ+−θ+\n2−ieiξ−−θ−\n2ie−iξ−−θ−\n2\n\n\nN+eikx++x0 0 0\n0N+e−ikx+x0 0\n0 0 N−eikx−x0\n0 0 0 N−e−ikx−x\n\nwhere in the second matrix the normalization factors are\ndefined as Nα= 1/(2/radicalbig\nvα+sinhθα).\nAccording to Eq. ( 4) the transfer matrix for the sin-\ngle interface problem is given by the matrix product\nM= [ΩN(0−)]−1ΩSO(0+). FromMwe obtain the trans-\nmission and the reflection probabilities for a spin-up orspin-down incident quasiparticle:\nT+s=/vextendsingle/vextendsingle/vextendsingle/vextendsingleM33δs,↑+M13δs,↓\nM13M31−M11M33/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\nΥ+(φ), (12)\nT−s=/vextendsingle/vextendsingle/vextendsingle/vextendsingleM31δs,↑+M11δs,↓\nM13M31−M11M33/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\nΥ−(φ), (13)\nR↑s=/vextendsingle/vextendsingle/vextendsingle/vextendsingleM31M23−M33M21\nM13M31−M11M33/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\nδs,↑\n+/vextendsingle/vextendsingle/vextendsingle/vextendsingleM13M21−M11M23\nM13M31−M11M33/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\nδs,↓,(14)\nR↓s=/vextendsingle/vextendsingle/vextendsingle/vextendsingleM31M43−M33M41\nM13M31−M11M33/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\nδs,↑\n+/vextendsingle/vextendsingle/vextendsingle/vextendsingleM13M41−M11M43\nM13M31−M11M33/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\nδs,↓,(15)\nwhere Υ α(φ) =θ(˜φα−φ)θ(˜φα+φ) withθ(x) the Heav-\niside step function. Here, Tαsis the probability for an\nincident quasiparticle with spin projection sto be trans-\nmittedinmode αintheSOregion. Ofcourse,probability\ncurrent conservation enforces T+s+T−s+R↑s+R↓s= 1.\nFigures3(a)–(c) show the angular dependence of the\ntransmission probabilities for an incident spin-up quasi-\nparticle into the (+) and ( −) modes of the SO region for\ndifferent values of the SOCs. Panel (a) refers to the case\nof vanishing intrinsic SOC (∆ = 0). Here the (+) and\nthe (−) energy bands are separated by a SOC-induced\nsplitting ∆Eext=λ. Therefore at fixed energy the two\npropagating modes in the SO region have two different\nmomenta, which gives rise to the two different critical\nangles (cf. Eq. ( 8) with ∆ = 0). Panel (b) refers to the\ncaseλ= 0, where the SOC opens a gap ∆ Eint= 2∆\nbetween the particle- and the hole-branches, however the\n(+)/(−)-modes remain degenerate. Therefore at fixed\nenergy these modes have the same momentum and, as a\nconsequence, the same critical angles (cf. Eq. ( 8) for\nλ= 0 and ∆ /negationslash= 0). When both SOCs are finite —\nthe situation illustrated in panel (c) — the transmission\nprobabilitiesexhibit morestructure. Forincidenceangles\nsmaller than ˜φ+no particular differences with the cases\nof panels (a) and (b) are visible. When the (+) mode\nis closed, an increase (resp. decrease) of the ( −) mode\ntransmission is observed for positive (resp. negative) an-\ngles, before the transmission drops to zero for incidence\nangles approaching ˜φ−. The asymmetry between posi-\ntive and negative angles is reversed if the spin state of\nthe incident quasiparticle is reversed.\nThese symmetryproperties can be rationalizedby con-\nsidering the operator of mirror symmetry through the\nx-axes.41This consists of the transformation y→ −y\nand at the same time the inversion of the spin and the\npseudo-spin states. It reads\nSy= (σx⊗sy)Ry, (16)\nwhereRytransforms y→ −y. The operator Sycom-\nmutes with the total Hamiltonian of the system [ Sy,H0+5\nHSO] = 0, therefore allows for a common basis of eigen-\nstates. For the scattering states in the SO region ( 10)\nwe haveSyχ+(ξ+) =χ+(ξ+) andSyχ−(ξ−) =−χ−(ξ−).\nInstead, it induces the following transformation on the\nscattering states ( 9) in the normal region: Syχs(φ) =\niχ−s(−φ). By comparing the original scattering matrix\nwith the Sy-transformed one we find that\nTα,s(φ) =Tα,−s(−φ) (17)\nwithα=±ands=↑,↓, which is indeed the symmetry\nobservedintheplots. Theasymmetryofthetransmission\ncoefficients occurs only when both SOCs are finite.\nA. Edge states at the interface\nIn addition to scatteringsolutions ofthe DW equation,\nit is interesting to study the possibility that edge states\nexist at the N-SO interface, which propagate alongthe\ninterface but decay exponentially away from it. The in-\nterestinthesetypesofsolutionsisconnectedtothestudy\nof topological insulators. It has been shown — first by\nKane and Mele15— that a zig-zag graphene nanoribbon\nwith intrinsic SOC supports dissipationless edge states\nwithin the SOC gap. In fact, similar states are always\nexpected to exist at the interface between a topologically\ntrivial and a topologically non-trivial insulator. In our\ncase, the latter is represented by graphene with intrinsic\nSOC. Of course SOC-free graphene is not an insulator,\nhoweverit is topologicallytrivial and edgestate solutions\ndo arise for |ky|>|E|. WhenEis within the gap in the\nSO region the corresponding mode is evanescent along\nthexdirection on both sides of the interface. Note that\n012345012345\nkyE(ky)\nFigure 4: (Color online). Energy dispersion of the edge stat e\nat the N-SO interface as a function of the momentum along\nthe interface kyfor different values of SOCs. Solution of the\ntranscendental equation is allowed only for |ky|>|E|(white\narea). In all three cases shown η >1/2: ∆ = 1 and λ= 0.4\n(lower-red line), ∆ = 1 .5 andλ= 0.7 (middle-blue line), and\n∆ = 2 and λ= 0.9 (upper-green line).the edge state we find is different from the one discussed\nin Refs.15,32where zig-zag or hard-wall boundary con-\nditions at the edge of the SOC region were imposed.\nThe wave function on the N side then reads\nχN(x) =/parenleftbigg1\n−iq+iky\nE/parenrightbigg\n(A|↓/angb∇acket∇ight+B|↑/angb∇acket∇ight)eqx(18)\nwithq=/radicalbig\n|ky|2−E2. In the SO region the wave func-\ntion can be written as\nχSO(x) =C\n(−q++ky)\ni(E−∆)\nE−∆\ni(q++ky)\ne−q+x+D\n(q−−ky)\n−i(E−∆)\nE−∆\n(q−+iky)\ne−q−x\nwithqα=/radicalig\nk2y−(E−∆)(E+∆−αλ). The continu-\nity of the wave function at the N-SO interface leads to\na linear system of equations for the amplitudes AtoD.\nThe matrix of coefficients must have a vanishing deter-\nminant for a non-trivial solution to exist. This condition\nprovides a transcendental equation for the energy of pos-\nsible edge states, whose solutions are illustrated in Fig. 4\nfor different values of the intrinsic and extrinsic SOCs.\nThe condition |ky|>|E|implies that solutions only exist\noutside the shadowed area. In addition, they are allowed\nonlyin the caseSOCsopena gapinthe energyspectrum,\nwhich occurs when η>1/2 (see App. Aand Eq. ( 3)). As\ncan be seen in Fig. 4the result is quite insensitive to the\nprecise value of the extrinsic SOC.\nEdge states exist only for values of the momentum\nalong the interface larger than the intrinsic SOC, i.e.,\nky>kmin\ny= ∆. The apparent breaking of time-reversal\ninvariance (the dispersion is not even in ky) is due to the\nfact that we are considering a single-valley theory. The\nfull two-valley SOC Hamiltonian is invariant under time-\nreversal symmetry, that interchanges the valley quantum\nnumber. This invariance implies that solutions for neg-\native values of kycan be obtained by considering the\nDirac-Weyl Hamiltonian relative to the other valley. The\ntwocounter-propagatingedge states live then at opposite\nvalleysand haveoppositespinstateandrealizeapeculiar\n1D electronic system.\nAs mentioned in the Introduction, the intrinsic SOC\nis estimated to be much smaller than the extrinsic one,\ntherefore in a realistic situation one would not expect the\nopening of a significant energy gap and the presence of\nedge states. It would be interesting to explore the pos-\nsibility to artificially enhance the intrinsic SOC, thereby\nrealizing the condition for the occurrence of edge states.\nIV. THE N-SO-N INTERFACE\nThe analysis of the scattering problem on a N-SO in-\nterface of the previous section can be straightforwardly\ngeneralized to the case of a double N-SO-N interface (SO\nbarrier). Here the transmission matrix Dis given by6\n00.2 0.4 0.6 0.8 1\n-90 -75 -60 -45 -30 -15 015 30 45 60 75 90 (a)\n00.2 0.4 0.6 0.8 1\n-90 -75 -60 -45 -30 -15 015 30 45 60 75 90 (b)\n00.2 0.4 0.6 0.8 1\n-90 -75 -60 -45 -30 -15 015 30 45 60 75 90 (c)\nFigure 5: (Color online). Panel (a): Angular plots for T↑↑as\na function of the injection angle for E= 2, ∆ = 1 and λ= 0.\nThe three lines correspond to different distance between the\ninterfaces: d=π/2 (dashed black), d=π(dotted red), and\nd= 2π(solid blue). The spin-precession length is ℓSO=π.\nWhenλ= 0 the transmission probability in the spin state\nopposed to the injected spin is always zero. Panel (b) and\n(c): angular plots of T↑↑(solid-blue) and T↓↑(dashed red) as\na function of the injection angle for E= 2,λ= 1 and ∆ = 0.\nThe distance between the two interfaces is d=πin panel\n(a) and d= 2πin panel (b). The spin-precession length is\nℓSO= 2π.\nEq. (5) withN= 2. The transmission and the reflection\nprobabilities in the case of a spin-up or -down incident\nquasiparticle read\nT↑s=/vextendsingle/vextendsingle/vextendsingle/vextendsingleD33δs,↑+D13δs,↓\nD13D31−D11D33/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n, (19)\nT↓s=/vextendsingle/vextendsingle/vextendsingle/vextendsingleD31δs,↑+D11δs,↓\nD13D31−D11D33/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n, (20)\nR↑s=/vextendsingle/vextendsingle/vextendsingle/vextendsingleD31D23−D33D21\nD13D31−D11D33/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\nδs,↑\n+/vextendsingle/vextendsingle/vextendsingle/vextendsingleD13D21−D11D23\nD13D31−D11D33/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\nδs,↓,(21)\nR↓s=/vextendsingle/vextendsingle/vextendsingle/vextendsingleD31D43−D33D41\nD13D31−D11D33/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\nδs,↑\n+/vextendsingle/vextendsingle/vextendsingle/vextendsingleD13D41−D11D43\nD13D31−D11D33/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\nδs,↓.(22)\nInthiscasethereisanadditionalparameterwhichcon-\ntrols the scattering properties of the structure, namely\nthe widthdof the SO region. In order to compare this\nlength to a characteristic length scale of the system, we\nintroduce the spin-precession length defined as\nℓSO= 2π/planckover2pi1vF\nλ+2∆. (23)\nThe intrinsic SOC alone cannot induce a spin preces-\nsion on the carriers injected into the SO barrier — an\ninjected spin state, say up, is obviously never converted\ninto a spin-down state. Figure 5(a) shows the angular00.2 0.4 0.6 0.8 1\n-90 -75 -60 -45 -30 -15 015 30 45 60 75 90 (a)\n00.2 0.4 0.6 0.8 1\n-90 -75 -60 -45 -30 -15 015 30 45 60 75 90 (b)\n00.2 0.4 0.6 0.8 1\n-90 -75 -60 -45 -30 -15 015 30 45 60 75 90 (c)\nFigure 6: (Color online). Angular plot of T↑↑(solid-blue)\nandT↓↑(dashed-red) as a function of the injection angle for\nE= 2,λ= 1 and (a) ∆ = λ/4, (b) ∆ = λ/2, and ∆ = λ. The\ndistance between the two interfaces is kept fixed to d=ℓSO.\ndependence of the transmission in the case of injection of\nspin-up. The behavior of the transmission as a function\nof the injection angle depends sensitively on the width d\ncompared to the spin-precession length. For small width\nd < ℓSO(dashed line) the transmission is a smooth de-\ncreasing function of φand stays finite also for φlarger\nthan the critical angle. In the case d≥ℓSO(dotted- and\nsolid-lines) instead the transmission probability exhibits\na resonant behavior and drops to zero as soon as the\ninjection angle equals the critical angle.\nWhen only the extrinsic SOC is finite, the transmis-\nsion behavior changes drastically. Two different critical\nangles appear — the biggest coincides usually with π/2.\nThe extrinsic SOC induces spin precession because of the\ncoupling between the pseudo- and the real-spin. This is\nillustrated in Fig. 5(b)-(c). In Panel (b) we consider the\ncase of spin-up injection with d=ℓSO/2. At normal inci-\ndence the transmission is entirely in the spin-down chan-\nnel (dashed line). Moving away from normal incidence,\nthe transmission in the spin-up channel (solid line) in-\ncreases from zero and, after the first critical angle, the\ntransmissions in spin-up and spin-down channels tend to\ncoincide. In panel (c) the width of the barrier is set to\nd=ℓSO. Here, the width of the SO region permits to an\ninjected carrier at normal incidence to perform a com-\nplete precession of its spin state — the transmission is in\nthe spin-up channel. For finite injection angles the spin-\ndown transmission (dashed line) also becomes finite. For\nφ/lessorsimilar˜φ+the transmission in the spin-up channel is almost\nfully suppressed while that in the spin-down channel is\nlarge. Finally, for φ >˜φ+the two transmission coeffi-\ncients do not show appreciable difference.\nIn the case where both extrinsic and intrinsic SOC\nare finite, the transmission probability exhibits a richer\nstructure. We focus again on the case of injection of\nspin-up quasiparticles. Moreover we fix the width of the\nSO regionsothat it is alwaysequalto the spin-precession\nlengthd=ℓSO. Fig.6illustrates the transmission proba-\nbilitiesTs↑forthree values ofthe ratio∆ /λ= 1/4,1/2,1.\nNotice that from panel (a) to (c) the width of SO region7\n00.2 0.4 0.6 0.8 1-90 -75 -60 -45 -30 -15 015 \n30 \n45 \n60 \n75 \n90 (a)\n-90 -45 0 45 90 -0.4 -0.2 00.2 0.4 \nϕ  Injection angle Pz Polarization (b) \nFigure 7: (Color online). Panel (a): total transmission T\nas a function of the injection angle for E= 2,d= 2πand\nseveral values of SOCs: λ= 1 and ∆ = 0 (blue-solid line),\nλ= 0 and ∆ = 0 .5 (red-dotted line), λ= 1 and ∆ = λ/4\n(yellow-dashed line), ∆ = λ/2 (orange-dashed-dotted line),\nandλ= ∆ (black-dotted-dotted-dashed line). Panel (b): z-\ncomponent of the spin polarization Pzas a function on the\ninjection angle for E= 2 and d= 2πand the following values\nof the SOCs: λ= 1, ∆ = 0 and λ= 0, ∆ = 1 (same black-\ndashed line), λ= 1 and ∆ = λ/4 (red-dotted), ∆ = λ/2\n(blue-dotted-dashed line), and ∆ = λ(green-solid line).\ndecreases.\nThe symmetry properties of the transmission func-\ntion can be rationalized by using the symmetry opera-\ntion (16). Proceeding in a similar manner as in the case\nof the single interface, for the SO barrier we find the\nfollowing symmetry relations\nTs,s(φ) =Ts,s(−φ), (24a)\nTs,−s(φ) =T−s,s(−φ), (24b)\nwhich are confirmed by the explicit calculations.\nSo far we have considered the injection of a pure spin\nstate — the injected carrier was either in a spin-up state\nor a spin-down state. Following Ref. 30we now address\nthe transmission of an unpolarized statistical mixture of\nspin-up and spin-down carriers. This will characterize\nthe spin-filtering properties of the SO barrier. In the\ninjection N region, an unpolarized statistical mixture of\nspins is defined by the density matrix\nρin=1\n2|χ↑/angb∇acket∇ight/angb∇acketleftχ↑|+1\n2|χ↓/angb∇acket∇ight/angb∇acketleftχ↓|, (25)\nwhere|χs/angb∇acket∇ight ≡ |s/angb∇acket∇ight ⊗ |σ/angb∇acket∇ightwith|σ/angb∇acket∇ight= (1/√\n2)(1,eiφ) cor-\nresponds to a pure spin state. When traveling throughthe SO region, the injected spin-unpolarized state is sub-\njected to spin-precession. The density matrix in the out-\nput N region can be expressed in terms of the transmis-\nsion functions ( 19) as\nρout=1\n2T↑|ζ↑/angb∇acket∇ight/angb∇acketleftζ↑|+1\n2T↓|ζ↓/angb∇acket∇ight/angb∇acketleftζ↓|,(26)\nwhere the coefficients Ts=T↑s+T↓sare the total trans-\nmissions for fixed injection state. The spinor part is de-\nfined as\n|ζs/angb∇acket∇ight=1√Ts/parenleftbigg\nt↑s\nt↓s/parenrightbigg\n⊗|σ/angb∇acket∇ight, (27)\nwherets′,sare the transmission amplitudes for incoming\n(resp. outgoing) spin s(resp.s′). The output density\nmatrix is used to define the total transmission\nT=T↑+T↓\n2(28)\nandthe expectation valueofthe zcomponent ofthe spin-\npolarization\nPz=1\n2(T↑↑+T↑↓−T↓↑−T↓↓).(29)\nIn Fig.7we report the total transmission (panel (a))\nand thez-component of the spin-polarization (panel (b))\nas a function of the injection angle for fixed energy and\nwidth of the SO region. We observe that for an un-\npolarized injected state the transmission probability is\nan even function of the injection angle T(φ) =T(−φ).\nMoreover, for injection angles larger than the first criti-\ncal angleφ >˜φ+, the transmission has an upper bound\natT= 1/2. On the contrary Pzis an odd function of\nthe injection angle Pz(φ) =−Pz(−φ). It is zero when\nat least one SOC is zero. When both SOC parameters\nare finite Pzis finite and reaches the largest values for\nφ>˜φ+. The maxima in this case increase as a function\nof the intrinsic SOC.\nTo experimentally observe this polarization effect the\nmeasurement should not involve an average over the an-\ngleφ, which, otherwise — due to the antisymmetry of Pz\n— would wash out the effect. To achieve this, one could\nuse,e.g., magnetic barriers,37,42which are known to act\nas wave vector filters.\nV. CONCLUSIONS\nIn this paper we have studied the spin-resolved trans-\nmissionthroughSOnanostructuresin graphene, i.e., sys-\ntems where the strength of SOCs — both intrinsic and\nextrinsic — is spatially modulated. We have considered\nthe case ofan interface separatinga normal regionfrom a\nSO region, and a barrier geometry with a region of finite\nSOC sandwiched between two normal regions. We have\nshown that — because of the lift of spin degeneracy due\nto the SOCs — the scatteringat the single interface gives8\nrise to spin-double refraction: a carrier injected from the\nnormal region propagates into the SO region along two\ndifferent directions as a superposition of the two avail-\nable channels. The transmission into each of the two\nchannels depends sensitively on the injection angle and\non the values of SOC parameters. In the case of a SO\nbarrier, this result can be used to select preferential di-\nrections along which the spin polarization of an initially\nunpolarized carrier is strongly enhanced.\nWe have also analyzed the edge states occurring in the\nsingle interface problem in an appropriate range of pa-\nrameters. These states exist when the SOCs open a gap\nin the energy spectrum and correspond to the gapless\nedge states supported by the boundary of topological in-\nsulators.\nA natural follow-up to this work would be the detailed\nanalysis of transport properties of such SO nanostruc-\ntures. From our results for the transmission probabil-\nities, spin-resolved conductance and noise could easily\nbe calculated by means of the Landauer-B¨ uttiker formal-\nism. Moreoverweplantostudyothergeometries,as, e.g.,\nnanostructureswith aperiodic modulation ofSOCs. The\neffects of various types of impurities on the properties\ndiscussed here is yet another interesting issue to address.\nWe hope that our work will stimulate further theo-\nretical and experimental investigations on spin transport\nproperties in graphene nanostructures.\nAcknowledgments\nWe gratefully acknowledge helpful discussions with\nL. Dell’Anna, R. Egger, H. Grabert, M. Grifoni, W.\nH¨ ausler, V. M. Ramaglia, P. Recher and D. F. Urban.\nThe work of DB is supported by the Excellence Initia-\ntive of the German Federal and State Governments. The\nwork of ADM is supported by the SFB/TR 12 of the\nDFG.\nAppendix A: Graphene with uniform spin-orbit\ninteractions.\nIn this appendix we briefly review the basic proper-\nties of DW fermions in graphene with homogeneous SO\ninteractions.21The energy eigenstates are plane waves\nψ∼Φ(k)eik·rwith Φ a four-componentspinorand eigen-\nvalues given by ( vF=/planckover2pi1= 1)\nEα,ǫ(k) =αλ\n2+ǫ/radicaligg\nk2x+k2y+/parenleftbigg\n∆−αλ\n2/parenrightbigg2\n,(A1)\nwhereα=±andǫ=±. The energy dispersion as a\nfunction of kxat fixedky= 0 is illustrated in Fig. 8for\nseveral values of ∆ and λ. The index ǫ=±specifies the\nparticle/holebranchesofthe spectrum. The eigenspinorsΦα,ǫ(k) read\nΦT\nα,ǫ(k) =1\n2√coshθα× (A2)\n(e−iφ−ǫθα/2,ǫeǫθα/2,iαǫeǫθα/2,iαeiφ−ǫθα/2),\nwhereTdenotes transposition and\nsinhθα=αλ/2−∆\nk, (A3)\neiφ=kx+iky\nk, (A4)\nwithk=/radicalig\nk2x+k2y. The spin operator components are\nexpressed as Sj=1\n2sj⊗σ0. Their expectation values in\nthe eigenstate Φ α,ǫread\n/angb∇acketleftSx/angb∇acket∇ight=−ǫαsinφ\n2coshθα, (A5a)\n/angb∇acketleftSy/angb∇acket∇ight=ǫαcosφ\n2coshθα, (A5b)\n/angb∇acketleftSz/angb∇acket∇ight= 0, (A5c)\nwhich shows that the product ǫαcoincides with the sign\nof the expectation value of the spin projection along the\ninplanedirectionperpendiculartothedirectionofpropa-\ngation. For vanishing extrinsic SOC, the eigenstates Φ α,ǫ\nreduce to linear combinations of eigenstates of Sz.\nSimilarly, the expectation value of the pseudo-spin op-\n(b)\n-4 -2 024\nMomentum (d)-4 -2 024Energy (a)\n-4 -2 024\nMomentum -4 -2 024Energy (c)\nFigure 8: Spectrum of the DW Hamiltonian with intrinsic\nand Rashba SOC as a function of kxforky= 0 . Panel (a):\ndashed lines refer to ∆ = 0 .5 andλ= 0; solid and dotted lines\nrefer to ∆ = 0 and λ= 1. Panel (b): ∆ = 0 .4 andλ= 1.\nPanel (c): ∆ = 0 .5 andλ= 1. Panel (d): ∆ = 0 .8 andλ= 1.9\neratorσis given by\n/angb∇acketleftσx/angb∇acket∇ight=ǫcosφ\ncoshθα, (A6a)\n/angb∇acketleftσy/angb∇acket∇ight=ǫsinφ\ncoshθα. (A6b)\nSince the SOCs in graphene do not depend on momen-\ntum, thevelocityoperatorstillcoincideswiththepseudo-\nspin operator: v=˙r= i[H,r] =σ. Thus the velocityexpectationvalueinthestateΦ α,ǫisgivenbyEqs. ( A6a–\nA6b). Alternatively, it can be obtained from the energy\ndispersion as\nvα,ǫ=∇kEα,ǫ=ǫk/radicalig\nk2+/parenleftbig\n∆−αλ\n2/parenrightbig2.(A7)\nThe groupvelocity is then independent of the modulus of\nthe wave vector if either the SOCs vanish or ∆ = αλ/2.\n∗Electronic address: dario.bercioux@frias.uni-freiburg.de\n†Electronic address: ademarti@thp.uni-koeln.de\n1K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y.\nZhang, S. V. Dubonos, I. V. Griegorieva, and A. A. Firsov,\nScience306, 666(2004); Nature(London) 438, 197(2005).\n2Y. Zhang, Y. W. Tan, H. L. Stormer, and P. Kim, Nature\n(London) 438, 201 (2005).\n3For recent reviews, see A. K. Geim and K. S. Novoselov,\nNature Mat. 6, 183 (2007); A.H. Castro Neto, F. Guinea,\nN. M. R. Peres, K. S. Novoselov, and A. K. Geim, Rev.\nMod. Phys. 81, 109 (2009); A. K. Geim, Science 324, 1530\n(2009).\n4C. W. J. Beenakker, Rev. Mod. Phys. 80, 1337 (2008).\n5G. W. Semenoff, Phys. Rev. Lett. 53, 2449 (1984).\n6D. P. Di Vincenzo and E.J. Mele, Phys. Rev. B 29, 1685\n(1984).\n7I.ˇZuti´ c, J. Fabian, S. Das Sarma, Rev. Mod. Phys. 76,\n323 (2004).\n8B. Trauzettel, D.V. Bulaev, D. Loss, and G. Burkard, Na-\nture Phys. 3, 192 (2007).\n9E. W. Hill, A.K.Geim, K. S. Novoselov, F. Schedin, and P.\nBlake, IEEE Trans. Magn. 42(10),2694 (2006).\n10N. Tombros, C. Jozsa, M. Popinciuc, H. T. Jonkman, and\nB. J. van Wees, Nature 448, 571 (2007).\n11S. Cho, Y.F. Chen, and M. S. Fuhrer, Appl. Phys. Lett.\n91123105 (2007).\n12M. Nishioka and A. M. Goldman, Appl. Phys. Lett. 90\n252505 (2007).\n13C. J´ ozsa, M. Popinciuc, N. Tombros, H. T. Jonkman, and\nB. J. van Wees, Phys. Rev. Lett. 100, 236603 (2008)\n14N. Tombros, S. Tanabe, A. Veligura, C. Jozsa, M. Popin-\nciuc, H. T. Jonkman, and B. J. van Wees, Phys. Rev. Lett.\n101, 046601 (2008).\n15C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 226801\n(2005).\n16D. Huertas-Hernando, F. Guinea, and A. Brataas, Phys.\nRev B74155426 (2006).\n17Hongki Min, J.E. Hill, N. A. Sinitsyn, B. R. Sahu, L.\nKleinman, and A.H. MacDonald, Phys. Rev. B 74, 165310\n(2006).\n18Y. Yao, F. Ye, X. L. Qi, S. C. Zhang, and Z. Fang, Phys.\nRev. B75, 041401(R) (2007).\n19J. C. Boettger and S. B. Trickey, Phys. Rev. B 75,\n121402(R) (2007); Phys. Rev. B 75, 199903(E) (2007).\n20M. Zarea and N. Sandler, Phys. Rev. B 79, 165442 (2009).\n21E. I. Rashba, Phys. Rev. B 79, 161409(R) (2009).\n22D. Huertas-Hernando, F. Guinea, and A. Brataas, Phys.\nRev. Lett. 103, 146801 (2009).\n23C. Ertler, S. Konschuh, M. Gmitra, and J. Fabian, Phys.Rev. B80, 041405(R) (2009).\n24P. Ingenhoven, J. Z. Bern´ ad, U. Z¨ ulicke, and R. Egger\nPhys. Rev. B 81, 035421 (2010) .\n25A. Varykhalov, J. S´ anchez-Barriga, A. M. Shikin, C.\nBiswas, E. Vescovo, A. Rybkin, D. Marchenko, and O.\nRader, Phys. Rev. Lett. 101, 157601 (2008).\n26O. Rader, A. Varykhalov, J. S´ anchez-Barriga, D.\nMarchenko, A. Rybkin, and A. M. Shikin, Phys. Rev. Lett.\n102, 057602 (2009).\n27A. Varykhalov and O. Rader, Phys. Rev. B 80, 035437\n(2009).\n28V. M. Ramaglia, D. Bercioux, V. Cataudella, G. De Filip-\npis, C. A. Perroni, and F. Ventriglia, Eur. Phys. J. B 36,\n365 (2003).\n29M. Khodas, A. Shekhter, and A. M. Finkel’stein, Phys.\nRev. Lett. 92, 086602 (2004).\n30V. M. Ramaglia, D. Bercioux, V. Cataudella, G. De Fil-\nippis, and C. A. Perroni, J. Phys.: Condens. Matter 16,\n9143 (2004).\n31E. I. Rashba, Fiz. Tverd. Tela (Leningrad) 2, 1224 (1960)\n[Sov. Phys. Solid State 2, 1109 (1960)].\n32T. Stauber and J. Schliemann, New J. Phys. 11, 115003\n(2009).\n33A. Yamakage, K.-I. Imura, J. Cayssol, and Y. Kuramoto,\nEPL87, 47005 (2009).\n34F.Guinea, M.I.Katsnelson, andA.G.Geim, NaturePhys.\n6, 30 (2009).\n35B. H. J. McKellar and G. J. Stephenson, Jr., Phys. Rev.\nC35, 2262 (1987).\n36M. Barbier, F. M. Peeters, P. Vasilopoulos, and J. M.\nPereira , Phys. Rev. B 77, 115446.\n37L. Dell’Anna and A. De Martino, Phys. Rev. B 79, 045420\n(2009); Phys. Rev. B 80, 089901(E) (2009).\n38L. Dell’Anna and A. De Martino, Phys. Rev. B 80, 155416\n(2009).\n39Eigenspinors are normalized in order to ensure probability\nflux conservation across the interface.\n40For incidence angle larger than one of the two critical an-\ngles, the respective refraction angle becomes complex:\nξα=π\n2−iξ′\nα,\nwhere the correct determination of the imaginary part is\nobtained for ξ′>0 by the relations sin ξα= coshξ′\nαand\ncosξα= isinhξ′\nα.\n41F. Zhai and H. Q. Xu, Phys. Rev. Lett. 94, 246601 (2005).\n42A. De Martino, L. Dell’Anna, and R. Egger, Phys. Rev.\nLett.98, 066802 (2007); Sol. St. Comm. 144, 547 (2007)."    },    {        "title": "1509.06118v1.Nature_of_Valance_Band_Splitting_on_Multilayer_MoS2.pdf",        "content": " 1 Nature of Valance Band Splitting on M ultilayer  MoS 2 \n \nXiaofeng  Fana, *, W.T. Zhenga, and David J. Singha,b, † \na. College of Materials Science and Engineering, Jilin University, Changchun 1300 12, China  \nb. Department of Physics and Astronomy, University of Missouri, Columbia, Missouri \n65211 -7010, USA  \n*E-mail: xffan@jlu.edu.cn ; † E-mail:  singhdj@ missouri.edu  \n \n \nAbstract  \n \nUnderstanding the origin  of splitting of valance band is important  since it govern s the \nunique spin and valley ph ysics in few -layer MoS 2. With first principle methods, we explore  \nthe effects of spin -orbit coupling and layer ’s coupling on few -layer  MoS 2. It is found that \nintra-layer spin -orbit coupling has a major contribution to t he splitting  of valance band at K. \nIn double -layer MoS 2, the layer ’s coupling result s in the widen ing of energy gap  of splitted \nstates  induced by intra -layer spin -orbit coupling. The valance band splitting of bulk MoS 2 \nin K can follow this model. We also f ind the effect of inter -layer spin -orbit coupling in \ntriple -layer MoS 2. In addition , the inter -layer spin -orbit coupling is found to become to be \nstronger under the pressure and results  in the decrease  of main energy gap in the splitting \nvalance bands at K.  .  \n \nIntroduction  \nA new class of 2D materials, the single -layer and/or few -layer of hexagonal transition  \nmetal dichalcogenides  (h-TMDs) , have attracted broad attentions due to the extraordinary \nphysic al properties  and promising applications in electric and optoelectronic devices1-5. As \nthe prototypical 2D materials, single -layer h -TMDs are direct band gap semiconductors \nwith spin -splitting at valance  band maximum  which is much different from graphene6-9. \nThis promises  a chance to manipulate  the spin degree of freedom and valley \npolarization10-12. In addition , with extreme dimensional confinement , tightly -bound \nexcitons and strong electron -electron interactions due to weak screening, h -TMDs have \nbeen ideal low -dimensional  compounds to explore many interesting quantum phenomena11, \n13-16, such as  spin- and valley - Hall effects and superconductivity17-19. There are also a lot \nof fascinating  optical properties in single -layer h -TMDs, such as the strong band gap \nphotoluminescence at edge5, surface sensitive luminescence20, 21 and strain -controlled \noptical band gap22-25, and so on.  \n    Among the se h-TMDs, MoS 2 is a representative . Bulk MoS 2 is a layered compound  \nstacked with the weak van der Waals interaction26. Due to the highly anisotropic  \nmechanic al property , it is used in dry lubrication . It has also made the interest due to the \nspecial catalytic  activity from its edge27. In each layer of MoS 2, there are three atomic \nlayers  with a center layer of Mo around S layers in both sides. The state s near band gap are  2 well-known to be mainly from the d -orbitals of Mo28. There  is a priori  proposal that the \nlayer ’s coupling is possible to have a very weak effect to the states near band gap. Bulk \nMoS 2 is an indirect -gap semiconductor with a band gap of 1.29  eV. However, following the \nreduction of layers to sinlge -layer, there is a transition between indirect band  gap and direct \ngap3. The single -layer MoS 2 is found to have a direct band gap of about 1.8 eV5, 29. \nTherefore, the layer’s coupling has a strong effect to the states near band gap  with recent \nreports30. Especially, the states  of valance band top a t  point  (VB-) and cond uction band \nbottom along Λ (CB-Λ) is much sensitive  to the layer ’s coupling  (LC) . Compared with the \nstates of VB - and CB - Λ, the LC effects on the states of valance band top and conduction  \nband bottom  at K point (VB -K, CB -K) are very  weak. Therefore, there remains an open \nquestion  about the origin of the splitting at valance band of K point which govern s the \nunique spin and valley ph ysics. In the single -layer limit, the splitting can be attributed  \nentirely to spin -orbit coupling (SOC). In bulk limit, it is considered to be a result of \ncombination of SOC and LC. However, there is disagreement about the relative  strength  of \nboth mechanisms31-37.    \n    In the work, we explore  the effect of SOC on few -layer  MoS 2 with th e rule of LC by \nfirst principle methods in details. We analyze the splitting  of states at VB -, VB -K CB -Λ, \nand CB -K and explore the change of splitting by following the increase of distance of both \nlayers for double -layer MoS 2. It is found that intra-layer  SOC (intra -SOC) has a major \ncontribution to the splitting  at VB -K, while LC can open effectively the degeneracy  of \nstates at VB -K. With the analysis of charge distribution in real space, the double - \ndegeneracy  of states at the valance band maximum  of K po int, which isn't broken due to \nthe inter-layer inverse  symmetry for both layers which result s in the forbidding  of \ninter-layer SOC, are mainly from the spin -up state of first -layer and spin -down state of \nsecond layer. For triple -layer  MoS 2, the LC with int er-layer SOC due to the absence of \ninter-layer inverse  symmetry in t hree-layer system makes the splitting complicated. The \nintra-layer SOC results in two main bands splitting, while in each main band, the \ntriple -degeneracy  is broken mainly due to the inter -layer SOC. With the pressure, it is \nfound in double -layer MoS 2 that the  double -degeneracy  of states in each main band isn ’t \nbroken when the splitting of both main bands is increased due to the strengthening  of LC. \nFor triple -layer Mo S2 under large pressure, the splitting  of triple -degeneracy in each main \nband is very obvious.  \n \nComputational Method  \nThe present calculations are performed within density functional theory using \naccurate frozen -core full -potential projector augmented -wave (P AW) pseudopotentials , as \nimplemented in the VASP code38-40. The generalized gradient approximation (GGA) with \nthe parametrization of Perdew -Burke -Ernzerhof (PBE)  and with added van der Waals \ncorrections  is used41. The k-space integrals and the plane -wave basis sets  are chosen to \nensure that the total energy is converged at the 1 meV/atom level. A kinetic energy  cutoff of \n500 eV for the plane  wave expansion  is found to be sufficient . The effect  of dispersion  \ninteraction  is included  by the empirical  correction  scheme  of Grimme  (DFT +D/PBE)42. \nThis approach has  been  successful  in describing  layered structures43, 44.  \nThe lattice constants a and c of bulk MoS 2 are 3.191  Å and 12.374  Å which is similar  3 to that from the experiments (3.160  Å and 12.295 Å). For the different layered MoS 2, the \nsupercells are constructed with a vacuum space of 20 Å along z direction. The Brillouin \nzones are sampled with the Γ -centered k -point grid of 18 181. With the state -of-the-art \nmethod of adding the stress to stress tensor in V ASP code39, 40, the structure of bulk MoS 2 \nis optimized under a specified hydrostatic pressure  of 15GPa . With these structural \nparameters from bulk MoS 2, the double - and triple -layer MoS 2 structures under the \npressures are constructed. The electronic properties can analyzed with and/or  without \nspin-orbit coupling  to explore the band splitting  near band gap. The calculated band gap of \nsingle -layer MoS 2 without  the consideration of spin -orbit interaction is 1.66 eV and less \nthan the experimental report of about 1.8 eV . Obviously , the band gap from PBE is \nunderestimated  as in common in usual density functional calculations . Though the band \ngap is underestimated b y PBE, the band structure near Fermi level doesn ’t have obvious \ndifference from that from other many body method s. \n \nResults and discussion  \n    The structure of single -layer MoS 2 has the hexagonal symmetry with space group \nP-6m2. The six sulfur atoms near each Mo atom form a trigonal prismatic structure with \nthe mirror symmetry in c direction. Obviously, the reversal symmetry is absent, and the \nintra-SOC in the band structure becomes to be free. An obvious band splitting on the \nvalance band maximum around K (K’) point  has been observed  and is contributed to the \nSOC. In addition , the SOC also results in the band splitting on conduction  band minimum  \naround Λ point, while the splitting at VB - and CB -K is not opened.  The states of VB- \nand CB -K are  contribut ed mostly from d z2 orbital of Mo and the effect of the spin-orbit \neffect is very  weak . At the same time, the states of VB-K and CB -Λ are mainly  from d x2\n-y2 \nand d xy orbitals of Mo and the spin -orbit effect on Mo can be revealed  in the case of the \nabsence of reversal symmetry . The band splitting at VB -K (149 meV) is larger than that at \nCB-Λ (about 79 meV) . It may be that the distribution  of change (or wave function) at \nCB-Λ around Mo atoms in xy plane is more localized  than that at VB -K30. We also notice \nthat the charge distribution of spin -up state is much different from that of spin -down state \nat VB -K. The spin -down state around Mo is more localized than the spin -up state.  \n    For double -layer MoS 2, the interaction between two layers become s important  to the \nstates near Fermi level. One of much evident  effect s is the direct band gap  (K-K) of \nsingle -layer becomes to be indirect  band gap (-K) due to the uplift of state at VB -. It can \nbe ascribe d to large band splitting (0.618 eV) at VB -. It is also observed that the band \nsplitting  at the conduction band bottom around  is about 0.352 eV . However, without the \nconsideration  of SOC effect , the band splitting  at VB -K is just 73.8 meV . The difference  of \nLC’s strength of different  states at VB-, VB -K and CB - is ascribed to the charge \ndistribution near sulfur atoms. The large contribution  of charge on sulfur atoms for the \nstates at VB - makes the LC to become easier30. The weak LC at  VB-K may make the \nSOC important. In order to explore the rules of SOC and LC in double -layer MoS 2, we \ncalculat ed the change of band structures  by following the change of distance between  both \nlayers with and without the consideration of spin -orbit effect.  As shown in Fig. 1, the band \nsplitting including that at VB -, VB -K and CB - approach es zero quickly following the \nincrease of distance, especially that of VB -K, if the spin -orbit effect is not considered. With  4 SOC, the band splitting at VB-K and CB - converge towards some constants (about 149 \nmeV and 79 meV), while the band splitting at VB - approaches zero.  Obviously, there is \nno spin -orbit effect at  VB-. With the large distance between both layers, the effect of LC \ncan be ignored  and the splitting is from SOC.  \n    It has been well-known that the spin -up and spin -down states at VB -K’ are reversed  \nby compared  with that at VB -K in single -layer MoS 2. For double -layer MoS 2, both \nsplitting bands at VB -K are two -degeneracy . As shown in Fig. 2c, the upper band of both \nbands is composed by the spin -up state of first layer and spin -down state of second layer \n(~|1, ~|2). The low er band is with the spin -down state of first layer and spin-up state  of \nsecond layer  (~|1, ~|2). Obviously, the energies of spin-up and spin -down of second \nlayer at VB -K are reversed by compared with that of first layer. Since there is reversal \nsymmetry for double -layer system, the energ ies of states |1  and |2 with same energy \ncannot be split due to the absence  of inter-layer SOC (inter -SOC). Therefore, we can \nunderstand the splitting at VB -K based on the intra -SOC and LC with the theoretical  model \nshown in Fig. 2a. Because of the splitting of intra -SOC, the energies of  |1 and |2 are \nvery different . This reduces largely the coupling  of both states due to layer ’s interaction. \nFrom the band splitting value (166 meV) of double layer at VB -K, the increased splitting \nfrom LC effect is about 17 meV and is much less than that (73.8 meV) from LC without \nthe consideration of spin -orbit effect. For the spin -down channel , the mechanism  of band \nsplitting is same to that of spin -up channel. Therefore , the contribution of intra -SOC (149 \nmeV) to the band splitting at VB -K is much larger than that of LC. The same  mechanism  \nabout the splitting at VB -K (shown in Fig. 2a) can be used for bulk MoS 2. The contribution  \nof LC is increased to about 59 meV since the band splitting at VB -K is about 0.208 eV . If \nno considering the spin -orbit effect, the band splitting due to LC is abou t 145.7 meV which \nmuch similar to the value from SOC(149 meV) in single -layer.  This may be the reason that \nthere is disagreement about the relative strength of both effects in bulk limit. Based on the \nmodel mentioned above and analysis, the intra -SOC effec t is the main mechanism for the \nsplitting at VB -K in bulk limit .  \n    For triple -layer MoS 2, the band splitting  near band gap is complicated , since there are \nthree states from three layers which are coupling with each other and  hybridized  with \npossible int er-SOC . For the states at VB -, there is no SOC effect and the three degenerate  \nstates will be splitting due to LC. It is found that the two splitting values ( 1 and 2 in \nFig. 3a) which control  the relative energy difference  of three states after the hybridization  \nare 0.293 eV and 0.502 eV , respectively . The much different value of both splitting implies \nthat there is strong coupling between  first layer and third layer since both splitting value s \nshould be equivalent  if the nearest -neighbor interaction  is just considered for the three \ndegenerate  states . Without the consideration of spin -orbit effect, the splitting values at \nCB- (1 and 2) are 0.241 eV and 0.225 eV and that at VB -K (K1 and K2) are 49 \nmeV and 55 meV , respectively . Based on the nearest -neighbor LC strength (73.8/2 meV) at \nVB-K from double layer, the LC strength between  first-layer and third layer  of triple layer \nis about 2 meV  at VB -K and may be ignored. Therefore, we propose a coupling model \nbased on the in tra-SOC a nd nearest -neighbor LC, as shown in Fig. 3c. With this model, the \nspin-up and spin -down bands of each layer are splitted by the intra -SOC. Then the LC will \nperturb these states  for each spin channel . For example , the spin -up states are composed  5 with two degenerate  upper states (|1 , |3) and one lower state (|2 ) in Fig. 3c and LC \nwill result in the splitting of two degenerate  upper states with the increase of energy gap \nbetween |2 and |3. With the spin -up and spin -down channels together after LC, there \nshould two main bands and each main band is composed with two degenerate states and \none single states. The LC doesn't  change the energy gap between  the main bands. It is \nfound that the energy gap ’SOC is 148.7 meV and similar  to the splitting from i ntra-SOC \n(SOC =149 meV). However, it is interesting  that the two degenerate states in each main \nband, such as the upper states ~|1  and ~|2 and the lower states ~|2  and ~|3, are \nsplitted, as shown in the inset of Fig. 3c. In addition , it is found that the splitting values are \nso large  (such as, 11.3 meV between  ~|1 and ~|2) that the contribution of LC between  \nfirst layer and third layer is not enough. We propose that the splitting of degenerate states \nin each main band is from the  inter-SOC.  \n    While the small pressure doesn ’t induce the obvious splitting from intra -SOC in \nsingle -layer MoS 2, it is possible there is strong effect to the splitting from inter -SOC in \ntriple -layer MoS 2. For double -layer MoS 2, the inter -SOC is forbidden and the de generate \nstate in each main band isn ’t opened and the energy gap between both main bands is \nincreased with the strengthening of LC under the pressure in Fig. 4a. In triple -layer, the \nstrengthening  of LC under pressure should have the obvious  effect, such as the increase of \nenergy gap between ~|1  and ~|3 in Fig. 4c. Besides  the enhanced  LC effect, an \napparent  observation  is the energy gap between main bands ’SOC has been decreased to \n135.8 meV under 15 GPa in Fig. 4b. This should be the typical  evidence  for the inter -SOC.   \n \nConclusions   \nWe study the band splitting at valance band maximum of multi -layer MoS 2 by first \nprinciple methods in details. We propose a model based on the int ra-layer spin -orbit \ncoupling to solve t he valance  band splitting  at K point  of multi -layer MoS 2 and bulk MoS 2 \nwith the perturbation  of layer ’s coupling and inter -layer spin -orbit coupling. It is also found \nthat the direct interaction between second near -neighbor  layers is weak  at VB -K. While the \ninter-layer spin -orbit coupling is forbidden  in double -layer MoS 2, this effect appear s in \ntriple -layer MoS 2. Especially, under the pressure, the inter -layer spin -orbit coupling is \nraised  with the decrease of energy gap between main bands from intr a-layer spin -orbital \ncoupling.   \n \nReferences  \n \n1. Wang, Q. H.; Kalantar -Zadeh, K.; Kis, A.; Coleman, J. N.; Strano, M. S., Electronics and \nOptoelectronics of Two -Dimensional Transition Metal Dichalcogenides. Nat. Nanotech. 2012,  7, 699 -712. \n2. Radisavljevic, B.; Radenovic, A.; Brivio, J.; Giacometti, V .; Kis, A., Single -Layer MoS 2 Transistors. Nat \nNanotech. 2011,  6, 147 -150. \n3. Mak, K. F.; Lee, C.; Hone, J.; Shan, J.; Heinz, T. F., Atomically Thin MoS 2: A New Direct -Gap \nSemiconductor. Phys. Rev. Lett. 2010,  105, 136805.  \n4. Lee, C.; Yan, H.; Brus, L. E.; Heinz, T. F.; Hone, J.; Ryu, S., Anomalous Lattice Vibrations of Single - \nand Few -Layer MoS 2. ACS Nano 2010,  4, 2695 -2700.   6 5. Splendiani, A.; Sun, L.; Zhang, Y .; Li, T.; Kim, J.; Chim, C.-Y .; Galli, G.; Wang, F., Emerging \nPhotoluminescence in Monolayer MoS 2. Nano Lett. 2010,  10, 1271 -1275.  \n6. Zhang, Y .; Chang, T. -R.; Zhou, B.; Cui, Y . -T.; Yan, H.; Liu, Z.; Schmitt, F.; Lee, J.; Moore, R.; Chen, Y .; \nLin, H.; Jeng, H. -T.; Mo, S. -K.; Hussai n, Z.; Bansil, A.; Shen, Z. -X., Direct observation of the transition from \nindirect to direct bandgap in atomically thin epitaxial MoSe2. Nat Nanotech 2014,  9, 111 -115. \n7. Yeh, P. -C.; Jin, W.; Zaki, N.; Zhang, D.; Liou, J. T.; Sadowski, J. T.; Al -Mahboob, A .; Dadap, J. I.; \nHerman, I. P.; Sutter, P.; Osgood, R. M., Layer -dependent electronic structure of an atomically heavy \ntwo-dimensional dichalcogenide. Phys. Rev. B 2015,  91, 041407.  \n8. Zhu, Z. Y .; Cheng, Y . C.; Schwingenschl 枚gl, U., Giant spin -orbit -induce d spin splitting in \ntwo-dimensional transition -metal dichalcogeni de semiconductors. Phys. Rev. B 2011,  84, 153402.  \n9. Zeng, H.; Dai, J.; Yao, W.; Xiao, D.; Cui, X., Valley polarization in MoS 2 monolayers by optical \npumping. Nat. Nanotech. 2012,  7, 490 -493. \n10. Cao, T.; Wang, G.; Han, W.; Ye, H.; Zhu, C.; Shi, J.; Niu, Q.; Tan, P.; Wang, E.; Liu, B.; Feng, J., \nValley -selective circular dichroism of monolayer molybdenum disulphide. Nat Commun 2012,  3, 887.  \n11. Xiao, D.; Liu, G. -B.; Feng, W.; Xu, X.; Yao, W., Coupled Spin and Valley Physics in Monolayers of \nMoS2 and Other Group -VI Dichalcogenides. Phys. Rev. Lett. 2012,  108, 196802.  \n12. Yao, W.; Xiao, D.; Niu, Q., Valley -Dependent Optoelectronics from Inversion Symmetry Breaking. \nPhys. Rev. B 2008,  77, 235406.  \n13. Klots, A. R.; Newaz, A. K. M.; Wang, B.; Prasai, D.; Krzyzanowska, H.; Lin, J.; Caudel, D.; Ghimire, N. \nJ.; Yan, J.; Ivanov, B. L.; Velizhanin, K. A.; Burger, A.; Mandrus, D. G.; Tolk, N. H.; Pantelides, S. T.; \nBolotin, K. I., Probing excitonic states in suspended two -dimensional semiconductors by photocurrent \nspectroscopy. Scientific Reports 2014,  4, 6608.  \n14. Li, X.; Zhang, F.; Niu, Q., Unconventional Quantum Hall Effect and Tunable Spin Hall Effect in Dirac \nMaterials: Application to an Isolated MoS 2 Trilayer. Phys. Rev. Lett. 2013,  110, 066803.  \n15. Ross, J. S.; Wu, S.; Yu, H.; Ghimire, N. J.; Jones, A. M.; Aivazian, G.; Yan, J.; Mandrus, D. G.; Xiao, D.; \nYao, W.; Xu, X., Electrical control of neutral and charged excitons in a monolayer semiconductor. Nat \nCommun 2013,  4, 1474.  \n16. Mak, K. F.; He, K.; Lee, C.; Lee, G. H.; Hone, J.; Heinz, T. F.; Shan, J., Tightly bound trions in \nmonolayer MoS 2. Nat Mater 2013,  12, 207 -211. \n17. Mak, K. F.; McGill, K. L.; Park, J.; McEuen, P. L., The valley Hall effect in MoS2 transistors. Science \n2014,  344, 1489 -1492.  \n18. Roldá n, R.; Cappelluti, E.; Guinea, F., Interactions and superconductivity in heavily doped \nMoS${}_{2}$. Phys. Rev. B 2013,  88, 054515.  \n19. Geim, A. K.; Grigorieva, I. V ., Van der Waals heterostructures. Nature 2013,  499, 419 -425. \n20. Mouri, S.; Miyauchi, Y .; Matsuda, K., Tunable Photoluminescence of Monolayer MoS 2 via Chemical \nDoping. Nano Letters 2013 , (12), 5944 -5948.  \n21. Tongay, S.; Suh, J.; Ataca, C.; Fan, W.; Luce, A.; Kang, J. S.; Liu, J.; Ko, C.; R aghunathanan, R.; Zhou, \nJ.; Ogletree, F.; Li, J.; Grossman, J. C.; Wu, J., Defects activated photoluminescence in two -dimensional \nsemiconductors: interplay between bound, charged, and free excitons. Scientific Reports 2013,  3, 2657.  \n22. Feng, J.; Qian, X.;  Huang, C. -W.; Li, J., Strain -Engineered Artificial Atom as a Broad -Spectrum Solar \nEnergy Funnel. Nat. Photon. 2012,  6, 866 -872. \n23. Fan, X.; Zheng, W.; Kuo, J. -L.; Singh, D. J., Structural Stability of Single -layer MoS 2 under Large \nStrain. J. Phys.: Conde ns. Matter 2015,  27, 105401.  \n24. Scalise, E.; Houssa, M.; Pourtois, G.; Afanasev, V .; Stesmans, A., Strain -Induced Semiconductor to  7 Metal Transition in the Two -Dimensional Honeycomb Structure of MoS 2. Nano Research 2012,  5, 43 -48. \n25. Conley, H. J.; Wang, B.; Ziegler, J. I.; Haglund, R. F.; Pantelides, S. T.; Bolotin, K. I., Bandgap \nEngineering of Strained Monolayer and Bilayer MoS2. Nano Lett. 2013,  13, 3626 -3630.  \n26. Mattheiss, L. F., Band Structures of Transition -Metal -Dichalcogenide Layer Compounds. Phys. Rev. B \n1973,  8, 3719 -3740.  \n27. Kibsgaard, J.; Chen, Z.; Reinecke, B. N.; Jaramillo, T. F., Engineering the Surface Structure of MoS 2 to \nPreferentially Expose Active Edge Sites for Electrocatalysis. Nat. Mater. 2012,  11, 963 -969. \n28. Chang, C. -H.; Fan , X.; Lin, S. -H.; Kuo, J. -L., Orbital analysis of electronic structure and phonon \ndispersion in MoS 2, MoSe 2, WS 2, and WSe 2 monolayers under strain. Phys. Rev. B 2013,  88, 195420.  \n29. Kuc, A.; Zibouche, N.; Heine, T., Influence of Quantum Confinement on the  Electronic Structure of the \nTransition Metal Sulfide TS 2. Phys. Rev. B 2011,  83, 245213.  \n30. Fan, X.; Chang, C. H.; Zheng, W. T.; Kuo, J. -L.; Singh, D. J., The Electronic Properties of Single -Layer \nand Multilayer MoS2 under High Pressure. J. Phys. Chem. C  2015,  119, 10189 -10196.  \n31. Latzke, D. W.; Zhang, W.; Suslu, A.; Chang, T. -R.; Lin, H.; Jeng, H. -T.; Tongay, S.; Wu, J.; Bansil, A.; \nLanzara, A., Electronic structure, spin -orbit coupling, and interlayer interaction in bulk MoS 2 and WS 2. Phys. \nRev. B 2015 , 91, 235202.  \n32. Klein, A.; Tiefenbacher, S.; Eyert, V .; Pettenkofer, C.; Jaegermann, W., Electronic band structure of \nsingle -crystal and single -layer WS 2: Influence of interlayer van der Waals interactions. Phys. Rev. B 2001,  64, \n205416.  \n33. Molina -Sanchez, A.; Sangalli, D.; Hummer, K.; Marini, A.; Wirtz, L., Effect of Spin -Orbit Interaction \non the Optical Spectra of Single -Layer, Double -Layer, and Bulk MoS 2. Phys. Rev. B 2013,  88, 045412.  \n34. Alidoust, N.; Bian, G.; Xu, S. -Y .; Sankar, R.; Neu pane, M.; Liu, C.; Belopolski, I.; Qu, D. -X.; Denlinger, \nJ. D.; Chou, F. -C.; Hasan, M. Z., Observation of monolayer valence band spin -orbit effect and induced \nquantum well states in MoX 2. Nat Commun 2014,  5, 1312.  \n35. Eknapakul, T.; King, P. D. C.; Asakawa , M.; Buaphet, P.; He, R. H.; Mo, S. K.; Takagi, H.; Shen, K. M.; \nBaumberger, F.; Sasagawa, T.; Jungthawan, S.; Meevasana, W., Electronic Structure of a Quasi -Freestanding \nMoS 2 Monolayer. Nano Letters 2014,  14, 1312 -1316.  \n36. Jin, W.; Yeh, P. -C.; Zaki, N.;  Zhang, D.; Sadowski, J. T.; Al -Mahboob, A.; van der Zande, A. M.; Chenet, \nD. A.; Dadap, J. I.; Herman, I. P.; Sutter, P.; Hone, J.; Osgood, R. M., Direct Measurement of the \nThickness -Dependent Electronic Band Structure of MoS 2 Using Angle -Resolved Photoem ission \nSpectroscopy. Phys. Rev. Lett. 2013,  111, 106801.  \n37. Suzuki, R.; Sakano, M.; Zhang, Y . J.; Akashi, R.; Morikawa, D.; Harasawa, A.; Yaji, K.; Kuroda, K.; \nMiyamoto, K.; Okuda, T.; Ishizaka, K.; Arita, R.; Iwasa, Y ., Valley -dependent spin polarization  in bulk MoS 2 \nwith broken inversion symmetry. Nat Nano 2014,  9, 611 -617. \n38. Hohenberg, P.; Kohn, W., Inhomogeneous Electron Gas. Phys. Rev. 1964,  136, B864.  \n39. Kresse, G.; Furthmü ller, J., Efficient Iterative Schemes for Ab Initio Total -Energy Calculatio ns Using a \nPlane -wave Basis Set. Phys. Rev. B 1996,  54, 11169 –11186  \n40. Kresse, G.; Furthmü ller, J., Efficiency of Ab -Initio Total Energy Calculations for Metals and \nSemiconductors Using a Plane -wave Basis Set Computat. Mater. Sci. 1996,  6, 15 -50. \n41. Perdew, J. P.; Burke, K.; Ernzerhof, M., Generalized Gradient Approximation Made Simple. Phys. Rev. \nLett. 1996,  77, 3865 -3868.  \n42. Grimme, S., Semiempirical GGA -Type Density Functional Constructed with a Long -Range Dispersion \nCorrection. J. Comput. Chem. 2006,  27, 1787.  \n43. Fan, X. F.; Zheng, W. T.; Chihaia, V .; Shen, Z. X.; Kuo, J. -L., Interaction Between Graphene and the  8 Surface of SiO 2. J. Phys.: Condens. Matter 2012,  24, 305004.  \n44. Mercurio, G.; McNellis, E. R.; Martin, I.; Hagen, S.; Leyssner, F.; So ubatch, S.; Meyer, J.; Wolf, M.; \nTegeder, P.; Tautz, F. S.; Reuter, K., Structure and Energetics of Azobenzene on Ag(111): Benchmarking \nSemiempirical Dispersion Correction Approaches. Phys. Rev. Lett. 2010,  104, 036102.  \n \n \n \n \n \n \n \n \n \n  9 Fig. 1.  \n \n \n \nFig.1 Band structure  of double -layer MoS 2 calculated  without spin -orbit coupling (a) and with spin -orbit \ncoupling, and the changes of conduction band splitting ( ) at  point  and valance band splitting at  \npoint () and K point (K) following the distance between two layers of double -layer MoS 2, calculated \nwithout spin -orbit coupling (c) and with spin -orbit coupling (d). Note the red circle s in Fig. 1c and d \nrepresent  the data from the equilibrium  (or stable) state (ES) and the dot and dash dot lines presen t the \nconduction band splitting ( ) at  point and valance band splitting ( K) at K point of single -layer \nMoS 2, respectively.  \n \n \n \n 10 Fig.2  \n \n \n \nFig. 2 Schematic  of valance band splitting of valance band maximum  at K point due to the spin -orbit \ncoupling in the each layer (intra -SOC) and layer ’s coupling (LC) in band structure of double -layer  MoS 2 \n(a), schematic structure of double -layer  MoS 2 (b), the isosurface of band -decomposed charge density of \nfour states at valance band maximum  of K point including the states ~|1 , ~|2, ~|1  and ~|2 \nshown in Fig. 2a  after considering the effects of intra -SOC and LC.  \n \n \n \n \n 11 Fig. 3  \n \n \n \nFig. 3 Band structure  of tri ple-layer MoS 2 calculated  without spin -orbit coupling (a) and with spin -orbit \ncoupling (b), S chematic  of valance band splitting of valance band maximum  at K point due to the \nspin-orbit (SOC) and layer ’s coupling (LC) in band structure of triple -layer MoS 2 (c). Note that in the \ninset of Fig. 3c, the band structure is plotted with two directions K  and KM and the lengths for \nK and KM are the 1/10 of total lengths in the two directions,  respectively.  \n \n \n 12 Fig. 4  \n \n \n \nFig. 4 Band structure s of double -layer MoS 2 (a) and triple -layer MoS 2 (b) under the pressure of 15 GPa \ncalculated  with spin -orbit coupling, and valance  bands of triple -layer MoS 2 under 15 GPa near K Point \nand the related s chematic  about band splitting due to the spin -orbit (SOC) and layer ’s couplin g (LC).  \n \n \n \n \n"    },    {        "title": "0805.1028v1.Frustration_and_entanglement_in_the__t__2g___spin__orbital_model_on_a_triangular_lattice__valence__bond_and_generalized_liquid_states.pdf",        "content": "arXiv:0805.1028v1  [cond-mat.str-el]  7 May 2008Frustration and entanglement in the t2gspin–orbital model on a triangular lattice:\nvalence–bond and generalized liquid states\nBruce Normand\nD´ epartement de Physique, Universit´ e de Fribourg, CH–170 0 Fribourg, Switzerland\nTheoretische Physik, ETH–H¨ onggerberg, CH–8093 Z¨ urich, Switzerland\nAndrzej M. Ole´ s\nMarian Smoluchowski Institute of Physics, Jagellonian Uni versity, Reymonta 4, PL–30059 Krak´ ow, Poland\nMax-Planck-Institut f¨ ur Festk¨ orperforschung, Heisenb ergstrasse 1, D-70569 Stuttgart, Germany\n(Dated: November 1, 2018)\nWe consider the spin–orbital model for a magnetic system wit h singly occupied but triply degen-\neratet2gorbitals coupled into a planar, triangular lattice, as woul d be exemplified by NaTiO 2. We\ninvestigate the ground states of the model for interactions which interpolate between the limits of\npure superexchange and purely direct exchange interaction s. By considering ordered and dimerized\nstates at the mean–field level, and by interpreting the resul ts from exact diagonalization calculations\non selected finite systems, we demonstrate that orbital inte ractions are always frustrated, and that\norbital correlations are dictated by the spin state, manife sting an intrinsic entanglement of these\ndegrees of freedom. In the absence of Hund coupling, the grou nd state changes from a highly res-\nonating, dimer–based, symmetry–restored spin and orbital liquid phase, to one based on completely\nstatic, spin–singlet valence bonds. The generic propertie s of frustration and entanglement survive\neven when spins and orbitals are nominally decoupled in the f erromagnetic phases stabilized by a\nstrong Hund coupling. By considering the same model on other lattices, we discuss the extent to\nwhich frustration is attributable separately to geometry a nd to interaction effects.\nPACS numbers: 71.10.Fd, 74.25.Ha, 74.72.-h, 75.30.Et\nI. INTRODUCTION\nFrustration in magnetic systems may be of geometri-\ncal origin, or may arise due to competing exchange in-\nteractions, or indeed both.1For quantum spins, frustra-\ntion acts to enhance the effects of quantum fluctuations,\nleading to a number of different types of magnetically\ndisordered state, among which some of the more familiar\nare static and resonating valence–bond (VB) phases. A\nfurther form of solution in systems with frustrated spin\ninteractionsis the emergenceof novelorderedstates from\nahighlydegeneratemanifoldofdisorderedstates,andthe\nmechanism for their stabilizationhas become known sim-\nply as “order–by–disorder”.1,2Many materials are now\nknown whose physical properties could be understood\nonly by employing microscopic models with frustrated\nspin interactions in which some of these theoretical con-\ncepts operate.\nA different and still richer situation occurs in the class\nof transition–metal oxides or fluorides with partly filled\n3dorbitals and near–degeneracy of active orbital degrees\nof freedom. In undoped systems, large Coulomb interac-\ntions on the transition–metal ions localize the electrons,\nand the low–energy physics is that of a Mott (or charge–\ntransfer3) insulator. Their magnetic properties are de-\nscribed by superexchange spin–orbital models, derived\ndirectly from the real electronic structure and contain-\ning linearly independent but strongly coupled spin and\norbital operators.4Such models emerge from the charge\nexcitations which involve various multiplet states,5,6in\nwhich ferromagnetic (FM) and antiferromagnetic (AF)interactions, as well the tendencies towards ferro–orbital\n(FO) and alternating orbital (AO) order, compete with\neachother. This leadsto a profound, intrinsic frustration\nof spin–orbital exchange interactions, which occurs even\nin case of only nearest–neighbor interactions for lattices\nwith unfrustrated geometry, such as the square and cu-\nbic lattices.7The underlying physics is formulated in the\nGoodenough–Kanamorirules,8which imply that the two\ntypes of order are complementary in typical situations:\nAO order favorsa FM state while FO order coexists with\nAF spin order. Only recently have exceptions to these\nrules been noticed,9and the search for such exceptions,\nand thus for more complex types of spin–orbital order or\ndisorder, have become the topic of much active research.\nA case study for frustration in coupled spin–orbital\nsystems is provided by the one–dimensional (1D) SU(4)\nmodel.10One expects a priori no frustration in one di-\nmension and with only nearest–neighbor interactions.\nHowever, spin and orbital interactions, the latter for-\nmulated in terms of pseudospin operators, appear on a\ncompletelysymmetricalfooting foreverybond, and favor\nrespectively AF and AO ordering tendencies, which com-\npete with each other. In fact a low–energy but magnet-\nically disordered spin state also frustrates the analogous\npseudospin–disordered state, and conversely. This com-\npetition results in strong, combined spin–orbital quan-\ntum fluctuations which make it impossible to separate\nthe two subsystems, and it is necessary to treat explicitly\nentangled spin–pseudospin states.9,11While in one sense\nthis may be considered as a textbook example of frustra-\ntion and entanglement, the symmetry of the entangled2\nsectorsissohighthatjointspin–pseudospinoperatorsare\nasfundamentalasthe separatespinandpseudospinoper-\nators, forming parts of a larger group of elementary (and\ndisentangled) generators. The fact that the 1D SU(4)\nmodelisexactlysolvablealsoresultsinfundamentalsym-\nmetriesbetween the intersitecorrelationfunctions forthe\nspin and orbital (and spin–orbital) sectors.12We return\nbelow to a more detailed discussion of entanglement and\nits consequences. Although indicative of the rich under-\nlying physics (indeed, unconventional behavior has been\nidentified for the SU(4) Hamiltonian on the triangular\nlattice,10,13)theimplicationsofthismodelareratherlim-\nited because it does not correspond to the structure of\nsuperexchange interactions in real correlated materials.\nRealistic superexchange models for perovskite\ntransition–metal oxides with orbital degrees of freedom\nhave been known for more than three decades,5,6but\nthe intrinsic frustrating effects of spin–orbital interac-\ntions have been investigated only in recent years.7,14\nA primary reason for this delay was the complexity\nof the models and the related quantum phenomena,\nwhich require advanced theoretical methods beyond a\nstraightforward mean–field theory. The structure of\nspin–orbital superexchangeinvolves interactionsbetween\nSU(2)–symmetric spins {/vectorSi,/vectorSj}on two nearest–neighbor\ntransition–metal ions {i,j}, each coupled to orbital op-\nerators{/vectorTi,/vectorTj}which obey only much lower symmetry\n(at most cubic for a cubic lattice), and its general form\nis4\nHJ=J/summationdisplay\n/angbracketleftij/angbracketright/bardblγ/braceleftBig\nˆJ(γ)\nij/parenleftBig\n/vectorSi·/vectorSj/parenrightBig\n+ˆK(γ)\nij/bracerightBig\n.(1.1)\nThe energy scale Jis determined (Sec. II) by the in-\nteraction terms and effective hopping matrix elements\nbetween pairs of directional egorbitals [(ddσ) element]\nort2gorbitals [(ddπ) element] The orbital operators ˆJ(γ)\nij\nandˆK(γ)\nijspecifytheorbitalsoneachbond /an}bracketle{tij/an}bracketri}ht /bardblγ,which\nparticipate in dn\nidn\nj⇀↽dn+1\nidn−1\njvirtual excitations, and\nthus have the symmetry of the lattice. The form of the\norbital operators depends on the valence n, on the type\n(egort2g) of the orbitals and, crucially, on the bond di-\nrectionin realspace.15It is clearfrom Eq.(1.1) that indi-\nvidualtermsintheHamiltonian HJcanbeminimized for\nparticularly chosen spin and orbital configurations,4but\nin general the structure of the orbital operators ensures\na competition between the different bonds.\nThis directional nature is the microscopic origin of the\nintrinsic frustration mentioned above, which is present\neven in the absence of geometrical frustration. Both\nspin andorbitalinteractionsarefrustrated, makinglong–\nrange order more difficult to realize in either sector, and\nenhancing the effects of quantum fluctuations. Quite\ngenerally, because insufficient (potential) energy is avail-\nable from spin or orbital order, instead the system is\ndriven to gain (kinetic) energy from resonance processes,\npromoting phases with short–range dynamical correla-\ntions and leading naturally to spin and/or orbital dis-order. Disordering tendencies are particularly strong in\nhighlysymmetricsystems, whichforcrystallinematerials\nmeans cubic and hexagonal structures. Among possible\nmagnetically disordered phases for spin systems, tenden-\ncies towards dimer formation are common in the regime\nof predominantly AF spin interactions, and new phases\nwith VB correlations occur. This type of physics was\ndiscussed first for egorbitals on the cubic lattice,7and,\nin the context of BaVS 3, for one version of the problem\noft2gorbitals on a triangular lattice.16The same generic\nbehaviorhassincebeenfoundfor t2gorbitalsonthecubic\nlattice,17eg–orbitalsystemsonthetriangularlattice,18,19\nand fort2gorbitals in the pyrochlore geometry.20,21By\nanalogy with spin liquids, the orbital–liquid phase1has\nbeen introduced for systems with both eg7,22andt2g14,23\norbital degrees of freedom. The orbital liquid is a phase\nin which strongorbital fluctuations restorethe symmetry\nof the orbital sector, in the sense that the instantaneous\norbital state of any site is pure, but the time average\nis a uniform occupation of all available orbital states.\nWe note that in the discussion of orbital liquids in t2g\nsystems,14,23it was argued that the spin sector would be\nordered. To date little is known concerning the behavior\noforbitalcorrelationsin an orbitalliquid, the possible in-\nstabilities of the orbital liquid towards dimerized or VB\nphases, or its interplay with lattice degrees of freedom.\nOnepossiblemechanismforthe formationofanorbital\nliquid state is the positional resonanceof VBs. There has\nbeen considerable recent discussion of spin–orbital mod-\nels in the continuing search for a realistic system real-\nizing such a resonating VB (RVB) state,19including in\na number of the references cited in the previous para-\ngraph. While the RVB state was first proposed for the\nS=1\n2Heisenberg model on a triangular lattice,24exten-\nsive analysis of spin–only models has not yet revealed\na convincing candidate system, although the nearest–\nneighbor dimer basis has been shown to deliver a very\ngood description of the low–energy sector for the S=1\n2\nHeisenberg model on a kagome lattice.25To date, the\nonly rigorous proof for RVB states has been obtained in\nrather idealized quantum dimer models (QDMs),26most\nnotably on the triangular lattice.27The insight gained\nfrom this type of study can, however, be used19to for-\nmulate some qualitative criteria for the emergence of an\nRVB ground state. These combine energetic and topo-\nlogical requirements, both of which are essential: the\nenergetics of the system must establish a proclivity for\ndimer formation, a high quasi–degeneracy of basis states\nin the candidate ground manifold, and additional energy\ngains from dimer resonance; exact degeneracy between\ntopological sectors (determined by a non–local order pa-\nrameter related to winding of wave functions around the\nsystem) is a prerequisite to remove the competing possi-\nbility of a “solid” phase with dimer, plaquette or other\n“crystalline” order.28\nWe comment here that the “problem” of frustration,\nand the resulting highly degenerate manifolds of states\nwhichmaypromoteresonancephenomena,isoftensolved3\nNa-ionO-ion\nTi-ionYZ\nXYZX\n(a) (b)\nFIG. 1: (Color online) Structure of the transition–metal ox ide\nwith edge–sharing octahedra realized for NaTiO 2: (a) frag-\nment of crystal structure, with Ti and Na ions shown respec-\ntively by black and green (grey) circles separated by O ions\n(open circles); (b) titanium /an}bracketle{t111/an}bracketri}htplane with adjacent oxy-\ngen layers, showing each Ti3+ion coordinated by six oxygen\natoms (open circles). The directions of the Ti–Ti bonds are\nlabeled as XY,YZ, andZX, corresponding to the plane\nspanned by the connecting Ti–O bonds. This figure is repro-\nduced from Ref. 33, where it served to explain the structure\nof LiNiO 2.\nby interactions with the lattice. Lattice deformations act\nto lift degeneracies and to stabilize particular patterns\nof spin and orbital order, the most familiar situation\nbeing that in colossal–magnetoresistance manganites.29\nThe samephysicsis alsodominantin anumberofspinels,\nwhere electron–lattice interactions are responsible both\nfor the Verwey transition in magnetite30and fort2gor-\nbital order below it, as well as for inducing the Peierls\nstate in CuIr 2S4and MgTi 2O4.31Similar phenomena are\nalso expected31to play a role in NaTiO 2. Here, however,\nwe will not introduce a coupling to phonon degrees of\nfreedom, and focus only on purely electronic interactions\nwhose frustration is not quenched by the lattice.\nThe spin–orbital interactions on a triangular lattice\nare particularly intriguing. This lattice occurs for edge–\nsharing MO 6octahedra in structures such as NaNiO 2or\nLiNiO 2, where the consecutive /an}bracketle{t111/an}bracketri}htplanes of Ni3+ions\nare well separated. These two eg–electron systems be-\nhavequitedifferently: while NaNiO 2undergoesacooper-\nativeJahn–Tellerstructuraltransitionfollowedbyamag-\nnetic transition at low temperatures ( TN= 20 K), both\ntransitions are absent in LiNiO 2.32Possible reasons for\nthis remarkable contrast were discussed in Ref. 33, where\nthe authors noted in particular that realistic spin–orbital\nsuperexchange neither has an SU(2) ⊗SU(2) structure,18\nnor can it ever be reduced only to the consideration of\nFM spin terms.34These studies showed in addition that\nLiNiO 2is not a spin–orbital liquid, and that the rea-\nsons for the observed disordered state are subtle, as spins\nand orbitals are thought likely to order in a strictly two–\ndimensional (2D) spin–orbital model.33\nThepossibilitiesofferedforexoticphasesinthistypeof\nmodel and geometry motivate the investigation of a real-\nistic spin–orbital model with active t2gorbitals, focusing\nfirst on 3d1electronic configurations. The threefold de-generacyofthe orbitalsismaintained, although, asnoted\nabove, this condition may be hard to maintain in real\nmaterials at low temperatures. A material which should\nexemplify this system is NaTiO 2(Fig. 1), which is com-\nposed of Ti3+ions int1\n2gconfiguration, but has to date\nhad rather limited experimental35,36and theoretical37\nattention. Considerably more familiar is the set of tri-\nangular cobaltates best known for superconductivity in\nNaxCoO2: here the Co4+ions havet5\n2gconfiguration and\nare expected to be analogous to the d1case by particle–\nhole symmetry. The effects of doping have recently been\nremoved by the synthesis of the insulating end–member\nCoO2.38Another system for which the same spin–orbital\nmodel could be applied is Sr 2VO4, where the V4+ions\noccupy the sites of a square lattice.39\nThe model with hopping processes of pure superex-\nchangetypewasconsideredinthecontextofdopedcobal-\ntates by Koshibae and Maekawa.40These authors noted\nthat, like the cubic system, two t2gorbitals are active\nfor each bond direction in the triangular lattice, but that\nthesuperexchangeinteractionsareverydifferentfromthe\ncubic case because the effective hopping interchanges the\nactiveorbitals. Here we focus onlyon insulating systems,\nwhose entire low–energy physics is described by a spin–\norbital model. In addition to superexchange processes\nmediated by the oxygen ions, on the triangular lattice\nit is possible to have direct–exchange interactions, which\nresult from charge excitations due to direct d−dhop-\nping between those t2gorbitals which do not participate\nin the superexchange. The ratio of these two types of\ninteraction ( α, defined in Sec. II) is a key parameter of\nthe model. Further, in transition–metal ions4the coef-\nficients of the different microscopic processes depend on\nthe Hund exchange JHarising from the multiplet struc-\nture of the excited intermediate d2state,41and we intro-\nduce\nη=JH\nU, (1.2)\nas the second parameter of the model. The aim of this\ninvestigation is to establish the general properties of the\nphase diagram in the ( α,η) plane.\nWe conclude our introductory remarks by returning to\nthe question of entanglement. In the analysis to follow\nwewillshowthat thepresenceofconflictingorderingten-\ndencies driven by different components of the frustrated\nintersite interactions can be related to the entanglement\nof spin and orbital interactions. By “entanglement” we\nmean that the correlations in the ground state involve\nsimultaneous fluctuations of the spin and orbital com-\nponents of the wave function which cannot be factor-\nized. We will introduce an intersite spin–orbital corre-\nlation function to identify and quantify this type of en-\ntanglement in different regimes of the phase diagram. It\nhas been shown9that such spin–orbital entanglement is\npresent in cubic titanates or vanadates for small values\nof the Hund exchange η. Here we will find entanglement\nto be a generic feature of the model for all exchange in-4\nteractions, even in the absence of dimer resonance, and\nthat only the FM regime at sufficiently high η, which\nis fully factorizable, provides a counterpoint where the\nentanglement vanishes.\nThe paper is organized as follows. In Sec. II we derive\nthe spin–orbitalmodel for magneticions with the d1elec-\ntronicconfiguration(Ti3+orV4+)onatriangularlattice.\nThe derivation proceeds from the degenerate Hubbard\nmodel, and the resulting Hamiltonian contains both su-\nperexchange and direct exchange interactions. We begin\nour analysis of the model, which covers the full range\nof physical parameters, in Sec. III by considering pat-\nterns of long–ranged spin and orbital order representa-\ntive of all competitive possibilities. These states compete\nwith magnetically or orbitally disordered phases domi-\nnated by VB correlations on the bonds, which are in-\nvestigated in Sec. IV. The analysis suggests strongly\nthat all long–range order is indeed destabilized by quan-\ntum fluctuations, leading over much of the phase dia-\ngram to liquid phases based on fluctuating dimers, with\nspin correlations of only the shortest range. In Sec. V\nwe present the results of exact diagonalization calcula-\ntions performed for small clusters with three, four, and\nsix bonds, which reinforce these conclusions and provide\ndetailed information about the local physical processes\nleading to the dominance of resonating dimer phases. In\neach of Secs. III, IV, and V, we conclude with a short\nsummary of the primary results, and the reader who is\nmore interested in an overview, rather than in detailed\nenergetic comparisons and actual correlation functions\nfor the different phases, may wish to read only these.\nSome insight into the competition and collaboration be-\ntween frustration effects of different origin can be ob-\ntainedbyvaryingthegeometryofthesystem,andSec.VI\ndiscusses the properties of the model on related lattices.\nA discussion and concluding summary are presented in\nSec. VII.\nII. SPIN–ORBITAL MODEL\nA. Hubbard model for t2gelectrons\nWe consider the spin–orbital model on the triangular\nlattice which follows from the degenerate Hubbard–like\nmodel fort2gelectrons. It contains the electron kinetic\nenergy and electronic interactions for transition–metal\nions arranged on the /an}bracketle{t111/an}bracketri}htplanes of a compound with\nlocalcubic symmetryandwith the d1ionicconfiguration,\nand as such is applicable to Ti3+or V4+[Fig. 1(a)]. The\nkinetic energy is given by\nHt=−/summationdisplay\n/angbracketleftij/angbracketright/bardblγ,µν,σt(γ)\nµν/parenleftBig\nd†\niµσdjνσ+d†\njνσdiµσ/parenrightBig\n,(2.1)\nwhered†\niµσare creation operators for an electron with\nspinσ=↑,↓andorbital“color” µat sitei, and the sum is\n/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0\n/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1\n/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0\n/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0\n/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1\n/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0\n/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1\n/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0\n/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1\n/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0\n/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1\n/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0\n/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0\n/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0\n/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1\n(a)\nc2\n(c)3\n23\n1 1(bc) (ca)\n(ab)ba\n(b)\nFIG. 2: (Color online) (a) Schematic representation of the\nhopping processes in Eq. (2.1) which contribute to magnetic\ninteractions on a representative bond /an}bracketle{tij/an}bracketri}htalong the c–axis\nin the triangular lattice. The t2gorbitals are represented by\ndifferent colors (greyscale intensities). Superexchange p ro-\ncesses involve O 2 pzorbitals (violet), and couple pairs of a\nandborbitals (red, green) with effective hopping elements t,\ninterchanging their orbital color. Direct exchange couple sc\norbitals (blue) with hopping strength t′. (b) Pairs of t2gor-\nbitals active in superexchange and (c) single orbitals acti ve in\ndirect exchange; horizontal bonds correspond to the situat ion\ndepicted in panel (a).\nmade over all the bonds /an}bracketle{tij/an}bracketri}ht/bardblγspanning the three direc-\ntions,γ=a,b,c, of the triangular lattice. This notation\nisadoptedfromthesituationencounteredinacubicarray\nof magnetic ions, where only two of the three t2gorbitals\nareactiveonanyonebond /an}bracketle{tij/an}bracketri}ht, andcontribute t(γ)\nµνtothe\nkineticenergy, whilethe third liesinthe planeperpendic-\nular to the γaxis and thus hopping processes involving\nthe 2pπoxygen orbitals is forbidden by symmetry.42,43\nWe introduce the labels a≡yz,b≡xz, andc≡xyalso\nfor the three orbital colors, and in the figures to follow\ntheir respective spectral colors will be red, green, and\nblue.\nFor the triangular lattice formed by the ions on the\n/an}bracketle{t111/an}bracketri}htplanes of transition–metal oxides (Fig. 1) it is also\nthecasethatonlytwo t2gorbitalsparticipatein(superex-\nchange)hoppingprocessesviatheoxygensites. However,\nunlike the cubic lattice, where the orbital color is con-\nserved, here any one active orbital color is exchanged for\nthe other one [Fig. 2(a)]. Using the same convention,\nthat each direction in the triangular lattice is labeled by\nits inactive orbital color44γ=a,b,c, the hopping ele-5\nments for a bond oriented (for example) along the c–axis\nin Eq. (2.1) are t(c)\nab=t(c)\nba=t, whilet(c)\naa=t(c)\nbb= 0. In\naddition, and also in contrastto the cubic system, for the\ntriangular geometry a direct hopping from one corbital\ntothe other, i.e.withoutinvolvingtheoxygenorbitals, is\nalso permitted on this bond (Fig. 2), and this element is\ndenoted by t′=t(c)\ncc. We will also refer to these hopping\nprocesses as off–diagonal and diagonal. We stress that\nwhile the lattice structure of magnetic ions is triangular,\nthe system under consideration retains local cubic sym-\nmetry of the metal–oxygen octahedra, which is crucial\nto ensure that the degeneracy of the three t2gorbitals is\npreserved.\nThe electron–electroninteractionsaredescribedby the\non–site terms45\nHint=U/summationdisplay\niµniµ↑niµ↓+/parenleftbigg\nU−5\n2JH/parenrightbigg/summationdisplay\ni,µ<ν,σσ′niµσniνσ′\n−2JH/summationdisplay\ni,µ<ν/vectorSiµ·/vectorSiν+JH/summationdisplay\ni,µ/negationslash=νd†\niµ↑d†\niµ↓diν↓diν↑,(2.2)\nwhereUandJHrepresent respectively the intraorbital\nCoulomb and on–site Hund exchange interactions. Each\npair of orbitals {µ,ν}is included only once in the inter-\naction terms. The Hamiltonian (2.2) describes rigorously\nthe multiplet structureof d2ions within the t2gsubspace,\nand is rotationally invariant in the orbital space.45\nWhen the Coulomb interaction is large compared with\nthe hopping elements ( U≫t,t′), the system is a Mott\ninsulator with one delectron per site in the t2gorbitals,\nwhence the local constraint in the strongly correlated\nregime is\nnia+nib+nic= 1, (2.3)\nwhereniγ=niγ↑+niγ↓. The operators act in the re-\nstricted space niγ= 0,1. The low–energy Hamiltonian\nmay be obtained by second–order perturbation theory,\nandconsistsofasuperpositionoftermswhichfollowfrom\nvirtuald1\nid1\nj⇀↽d2\nid0\njexcitations. Because each hopping\nprocess may be of either off–diagonal ( t) [Fig. 2(b)] or\ndiagonal (t′) type [Fig. 2(c)], the Hamiltonian consists\nof several contributions which are proportional to three\ncoupling constants,\nJs=4t2\nU, Jd=4t′2\nU, Jm=4tt′\nU.(2.4)\nThese represent in turn the superexchange term, the di-\nrect exchange term, and mixed interactions which arise\nfrom one diagonal and one off–diagonal hopping process.\nWe chooseto parameterizethe Hamiltonian by the sin-\ngle variable\nα= sin2θ, (2.5)\nwith\ntanθ=t′\nt, (2.6)which gives Js=Jcos2θ,Jm=Jsinθcosθ, andJd=\nJsin2θ;Jis the energy unit, which specifies respectively\nthe superexchange ( J=Js) and direct–exchange ( J=\nJd) constants in the two limits α= 0 andα= 1. The\nHamiltonian\nH=J/braceleftBig\n(1−α)Hs+/radicalbig\n(1−α)αHm+αHd/bracerightBig\n(2.7)\nconsists of three terms which follow from the processes\ndescribed by the exchange elements in Eqs. (2.4), each of\nwhich contains contributions from both high– and low–\nspin excitations.\nB. Superexchange\nSuperexchange contributions to Hcan be expressed in\nthe form\nHs=1\n2/summationdisplay\n/angbracketleftij/angbracketright/bardblγ/braceleftBig\nr1/parenleftBig\n/vectorSi·/vectorSj+3\n4/parenrightBig/bracketleftBig\nA(γ)\nij+1\n2(niγ+njγ)−1/bracketrightBig\n+r2/parenleftBig\n/vectorSi·/vectorSj−1\n4/parenrightBig/bracketleftBig\nA(γ)\nij−1\n2(niγ+njγ)+1/bracketrightBig\n−2\n3(r2−r3)/parenleftBig\n/vectorSi·/vectorSj−1\n4/parenrightBig\nB(γ)\nij/bracerightBig\n, (2.8)\nwhere one recognizes a structure similar to that for su-\nperexchange in cubic vanadates,4,14with separation into\naspinprojectionoperatoronthetripletstate,( /vectorSi·/vectorSj+3\n4),\nand an operator ( /vectorSi·/vectorSj−1\n4) which is finite only for\nlow–spin excitations. These operators are accompanied\nby coefficients ( r1,r2,r3) which depend on the Hund ex-\nchange parameter (1.2), and are given from the multiplet\nstructure of d2ions41by\nr1=1\n1−3η, r2=1\n1−η, r3=1\n1+2η.(2.9)\nTheCoulombandHundexchangeelementsdeducedfrom\nthe spectroscopic data of Zaanen and Sawatzky46are\nU= 4.35 eV and JH= 0.59 eV, giving a realistic value\nofη≃0.136 for Ti2+ions. For V2+one finds46U= 4.98\neV andJH= 0.64 eV, whence η≃0.13, and the values\nfor V3+ions are expected to be very similar. Finally,\nfor Co3+ions,47U= 6.4 eV andJH= 0.84 eV, giving\nagainη≃0.13. The value η= 0.13 therefore appears to\nbe quite representative for transition–metal oxides with\npartly filled t2gorbitals, whereas somewhat larger values\nhave been found for systems with active egorbitals due\nto a stronger Hund exchange.4\nThe orbital operators AijandBijin Eq. (2.8) depend\non the bond direction γand involve two active orbital\ncolors,\nA(γ)\nij=/parenleftBig\nT+\niγT+\njγ+T−\niγT−\njγ/parenrightBig\n−2Tz\niγTz\njγ+1\n2n(γ)\nin(γ)\nj,(2.10)\nB(γ)\nij=/parenleftBig\nT+\niγT−\njγ+T−\niγT+\njγ/parenrightBig\n−2Tz\niγTz\njγ+1\n2n(γ)\nin(γ)\nj.(2.11)6\nFor illustration, in the case γ=c(/an}bracketle{tij/an}bracketri}ht /bardblc), the orbitals\naandbat siteiare interchanged (off–diagonal hopping)\nat sitej, and the electron number operator is n(γ)\ni=\nnia+nib. The quantity niγin Eq. (2.8) is the number\noperator for electrons on the site in orbitals inactive for\nhopping on bond γ,niγ= 1−n(γ)\ni, ornicin this example.\nFor a single bond, the orbital operators in Eq. (2.10)\nmay be written in a very suggestive form by performing\na local transformation in which the active orbitals are\nexchanged on one bond site, specifically |a/an}bracketri}ht → |b/an}bracketri}htand\n|b/an}bracketri}ht → |a/an}bracketri}hton bondγ=c.40Then\nA(γ)\nij= 2/parenleftBig\n/vectorTiγ·/vectorTjγ+1\n4n(γ)\nin(γ)\nj/parenrightBig\n, (2.12)\nB(γ)\nij= 2/parenleftBig\n/vectorTiγ×/vectorTjγ+1\n4n(γ)\nin(γ)\nj/parenrightBig\n,(2.13)\nwhere the scalar product in Aijis the conventional\nexpression for pseudospin–1/2 variables, and the cross\nproduct in Bijis defined as\n/vectorTiγ×/vectorTjγ=1\n2(T+\niγT+\njγ+T−\niγT−\njγ)+Tz\niγTz\njγ.(2.14)\nEquations (2.8) and (2.12) make it clear that for a sin-\ngle superexchange bond, the minimal energy is obtained\neither by forming an orbital singlet, in which case the op-\ntimal spin state is a triplet, or by forming a spin singlet,\nin which case the preferred orbital state is a triplet; we\nrefer to these bond wavefunctions respectively as (os/st)\nand (ss/ot). The two states are degenerate for η= 0,\nwhile for finite Hund exchange\nE(os/st)=−Jr1, (2.15)\nE(ss/ot)=−1\n3J(2r2+r3), (2.16)\nand the (os/st) state is favored. This propensity for sin-\nglet formation in the α= 0 limit will drive much of the\nphysics to be analyzed in what follows.\nBecauseoftheoff–diagonalnatureofthehoppingterm,\nin the original electronic basis (before the local transfor-\nmation) the orbital singlet is the state\n|ψos/an}bracketri}ht=1√\n2(|aa/an}bracketri}ht−|bb/an}bracketri}ht), (2.17)\nwhile the orbital triplet states are\n|ψot+/an}bracketri}ht=|ab/an}bracketri}ht, (2.18)\n|ψot0/an}bracketri}ht=1√\n2(|aa/an}bracketri}ht+|bb/an}bracketri}ht), (2.19)\n|ψot−/an}bracketri}ht=|ba/an}bracketri}ht. (2.20)\nThe locally transformed basis then gives a clear analogy\nwhich can be used for single bonds and dimer phases in\ncombination with all of the understanding gained for the\nHeisenberg model. However, we stress here that the local\ntransformationfailsforsystemswithmorethan1bondinthe absence of static dimer formation. This arises due to\nfrustration, and can be shown explicitly in numerical cal-\nculations, but we will not enter into this point in more\ndetail here. However, we take the liberty of retaining\nthe notation of the local transformation, particularly in\nSec. IV when considering dimers. Because the transfor-\nmation interchanges the definitions of FO and AO con-\nfigurations, we will state clearly in each section the basis\nin which the notation is chosen.\nC. Direct Exchange\nThe direct exchange part is obtained by considering\nvirtual excitations of active γorbitals on a bond /an}bracketle{tij/an}bracketri}ht /bardblγ,\nwhich yield\nHd=1\n4/summationdisplay\n/angbracketleftij/angbracketright/bardblγ/braceleftBig/bracketleftBig\n−r1/parenleftBig\n/vectorSi·/vectorSj+3\n4/parenrightBig\n+r2/parenleftBig\n/vectorSi·/vectorSj−1\n4/parenrightBig/bracketrightBig\n×/bracketleftBig\nniγ(1−njγ)+(1−niγ)njγ/bracketrightBig\n+1\n3(2r2+r3)/parenleftBig\n/vectorSi·/vectorSj−1\n4/parenrightBig\n4niγnjγ/bracerightBig\n.(2.21)\nHere there are no orbital operators, but only number\noperators which select electrons of color γon bonds ori-\nentedalongthe γ–axis. Whenonlyonlyoneactiveorbital\nis occupied [ niγ(1−njγ)], this electron can gain energy\n−1\n4Jfrom virtual hopping at η= 0, a number which\nhas only a weak dependence on the bond spin state at\nη>0. When both active orbitals are occupied ( niγnjγ),\nplacing the two electrons in a spin singlet yields the far\nlower bond energy −J, and thus again one may expect\nmuchofthe discussionto followtocenterondimer–based\nstates of the extended system. Again the triplet d2spin\nexcitation corresponds to the lowest energy, ( U−3JH),\nand only the lower two excitations involve spin singlets\nwhich could minimize the bond energy. The structure\nof these terms is the same as in the 1D egspin–orbital\nmodel,48or the case of the spinel MgTi 2O4.20A simpli-\nfied model for the triangular–lattice model in this limit,\nusing a lowest–order expansion in ηfor the spin but not\nfor orbital interactions, was introduced in Ref. 49.\nD. Mixed Exchange\nFinally, the twodifferent types ofhopping channelmay\nalso contribute to two–step, virtual d1\nid1\nj⇀↽d2\nid0\njexci-\ntations with one off–diagonal ( t) and one diagonal ( t′)\nprocess. The occupied orbitals are changed at both sites\n(Fig. 2), and as for the superexchange term the result-\ning effective interaction may be expressed in terms of or-\nbital fluctuation operators. To avoid a more general but\ncomplicated notation, we write this term only for c–axis\nbonds,\nH(c)\nm=−1\n4/summationdisplay\n/angbracketleftij/angbracketright/bardblc/bracketleftBig\nr1/parenleftBig\n/vectorSi·/vectorSj+3\n4/parenrightBig\n−r2/parenleftBig\n/vectorSi·/vectorSj−1\n4/parenrightBig/bracketrightBig7\n×/parenleftBig\nT+\niaT+\njb+T−\nibT−\nja+T+\nibT+\nja+T−\niaT−\njb/parenrightBig\n,(2.22)\nwhere the orbital operators are\nT+\nia=b†\nici, T+\nib=c†\niai,\nT−\nia=c†\nibi, T−\nib=a†\nici. (2.23)\nThese definitions are selected to correspond to the ↑–\npseudospin components of both operators being |bi/an}bracketri}htfor\nTz\niaand|ci/an}bracketri}htforTz\nib. The form of the H(a)\nmandH(b)\nmterms\nis obtained from Eq. (2.22) by a cyclic permutation of\nthe orbital indices. By inspection, this type of term is\nfinite only for bonds whose sites are occupied by linearsuperpositions of different orbital colors, and creates no\nstrong preference for the spin configuration at small η.\nE. Limit of vanishing Hund exchange\nIn the subsequent sections we will give extensive con-\nsideration to the model of Eq. (2.7) at η= 0. In this\nspecial case the multiplet structure collapses (spin sin-\nglet and triplet excitations are degenerate), one finds a\nsinglechargeexcitationofenergy U, andtheHamiltonian\nreduces to the form\nH(η= 0) =J/summationdisplay\n/angbracketleftij/angbracketright/bardblγ/braceleftBig\n(1−α)/bracketleftBig\n2/parenleftBig\n/vectorSi·/vectorSj+1\n4/parenrightBig/parenleftBig\n/vectorTiγ·/vectorTjγ+1\n4n(γ)\nin(γ)\nj/parenrightBig\n+1\n2(niγ+njγ)−1/bracketrightBig\n+α/bracketleftBig/parenleftBig\n/vectorSi·/vectorSj−1\n4/parenrightBig\nniγnjγ−1\n4/parenleftBig\nniγ(1−njγ)+(1−niγ)njγ/parenrightBig/bracketrightBig\n−1\n4/radicalbig\nα(1−α)/parenleftBig\nT+\ni¯γT+\nj˜γ+T−\ni˜γT−\nj¯γ+T+\ni˜γT+\nj¯γ+T−\ni¯γT−\nj˜γ/parenrightBig/bracerightBig\n, (2.24)\nwhich depends only on the ratio of superexchange to di-\nrect exchange (0 ≤α≤1). The first line of Eq. (2.24)\nmakes explicit the fact that the spin and orbital sectors\nare completely equivalent and symmetrical at α= 0, at\nleast at the level of a single bond. However, we will\nshowthatthisequivalenceisbrokenwhenmorebondsare\nconsidered, and no higher symmetry emerges because of\nthe color changes involved for different bond directions,\nwhich change the SU(2) orbital subsector. The second\nline of Eq. (2.24) emphasizes the importance of bond oc-\ncupation and singlet formation at α= 1 (Sec. IIC).\nIn the third line of Eq. (2.24), the labels ¯ γ/ne}ationslash= ˜γre-\nfer to the two mixed orbital operators on each bond\n[Eq. (2.23)]. Orbital fluctuations are the only processes\ncontributing to the mixed terms in this limit, where the\nspin state of the bond has no effect. We draw the at-\ntention of the reader to the fact that for the parameter\nchoiceα= 0.5, anelectronofanycoloratanysitehasthe\nsame matrix element to hop in any direction. However,\nbecause of the different color changes involved in these\nprocesses, again the spin–orbital Hamiltonian does not\nexhibit ahigher symmetry at this point, a result reflected\nin the different operator structures of superexchange and\ndirect exchange components.\nIII. LONG–RANGE–ORDERED STATES\nIn this Section we study possible ordered or partially\nordered states for the Hamiltonian of Eq. (2.7). As ex-\nplained in Sec. II, the parameters of the problem arethe ratio of the direct and superexchange interactions,\nα(2.5), and the strength of the Hund exchange interac-\ntion,η(1.2). Regarding the latter, we will discuss briefly\nthe transition to ferromagnetic (FM) spin order for in-\ncreasingηin this framework.\nThe first necessary step in any analysis of such an\ninteracting system is to establish the energies of differ-\nent (magnetically and orbitally) ordered states. The\nhigh connectivity of the triangular–lattice system sug-\ngests that ordered states will dominate, and claims of\nmore exotic ground states are justifiable only when these\nare shown to be uncompetitive. The calculations in this\nSection will be performed for static orbital and spin con-\nfigurations, with the virtualprocessesresponsiblefor(su-\nper)exchange as the only fluctuations. In the language\nof the discussion in Sec. I, fully ordered states gain only\npotential energy at the cost of sacrificing the kinetic (res-\nonance) energy from fluctuation processes, which we will\nshow in Secs. IV and V is of crucial importance here.\nA. Possible orbital configurations\nThe results to follow will be obtained by first fixing\nthe orbital configuration, either on every site or on par-\nticular bonds, and then computing the spin interaction\nand optimizing the spin state accordingly. While this is\nequivalent to the converse, the procedure is more trans-\nparent and offers more insight into the candidate phases.\nWe limit the number of states to ordered phases with\nsmall unit cells, and the orbital states to be considered8\n(a) (b)\n(c) (d)\n(e) (f)\nFIG. 3: (Color online) Schematic representation of possibl e\norbital states with a single color on each site of the triangu -\nlar lattice: (a) one–color state; (b) and (c) two inequivale nt\ntwo–color states; (d) three–sublattice three–color state ; (e)\nand (f) two inequivalent three–color states. The latter two\nconfigurations are degenerate with similar states where the\nlines of occupied aandborbitals repeat rather than being\nstaggered along the direction perpendicular to the lines of\noccupied corbitals. The three–sublattice state (3d) is nonde-\ngenerate ( d= 1), states (3a), (3b), and (3e) have degeneracy\nd= 3, and states (3c) and (3f) have degeneracy d= 6.\nare enumerated in this subsection. For clarity we adopt\nthe conventionofFig. 2(c) that horizontal( c) bonds have\ndiagonal (direct exchange) hopping of corbitals, which\nareshowninblue, andoff–diagonal(superexchange)hop-\npingprocessesfor aandborbitals[Fig.2(b)], respectively\nred and green; up–slanting ( a) bonds have diagonal hop-\nping foraorbitals and off–diagonal hopping between b\nandcorbitals; down–slanting ( b) bonds have diagonal\nhopping for borbitals and off–diagonal hopping between\naandcorbitals. All Hamiltonians and energies are func-\ntions ofαandη, as given by Eqs. (2.7), (2.8), (2.21),\nand (2.22). To minimize additional notation, they will\nbe quoted in this and in the next section as functions of\nthe single argument α, with implicit η–dependence con-\ntained in the parameters ( r1,r2,r3). The orbital bond\nindexγwill also be suppressed here and in Sec. IV.\nWe continue to refer to the orbital type as a “color”,\nand begin by listing symmetry–inequivalent states where(a)\n(b) (c)\n(d) (e)\nFIG. 4: (Color online) Schematic representation of possibl e\norbital configurations with superpositions of (a) two orbit als\nin a two–color state, (b) three orbitals, (c) two orbitals wi th\nequal net weight, and (d) and (e) two orbitals with differing\nnet weights of all three orbitals. State (a) has degeneracy\nd= 3, states (b) and (c) have d= 1, and the degeneracies of\nstates (d) and (e) are d= 6 and d= 3.\neach site has a unique color. If the same orbital is occu-\npied at every site [Fig. 3(a)], the three states with a,b,\norcorbitals occupied are physically equivalent (degener-\nacy isd= 3). When lines of the same occupied orbitals\nalternate along the perpendicular direction there are two\nbasicpossibilities, whichareshowninFigs.3(b) and3(c).\nThese two–color states differ in their numbers of active\nsuperexchange or direct–exchange bonds, which depend\non how the monocolored lines are oriented relative to the\nactive hopping direction(s) of the orbital color. There\nis only one three–color configuration with equal occupa-\ntions, which is shown in Fig. 3(d).\nTurning to orbital states with unequal occupations,\nmotivated by the tendency of Hto favor dimer forma-\ntion in certain limits we extend our considerations to the\npossibility of a four–site unit cell [Figs. 3(e) and 3(f)].\nMore elaborate three–color unit cells are not considered.\nIn this case the same state is obtained when the fourth\nsite is occupied by electrons whose orbital color is any of\nthe other three. Again this state, which breaksrotational\nsymmetry, differs depending on its orientation relative to\nthe active hopping axes.9\nStates involving a superposition of either two or three\norbitals at each site can be expected to allow a signif-\nicantly greater variety of hopping processes. When ei-\nther two or three orbital states are partially occupied\nat each site (we stress that the condition of Eq. (2.3)\nis always obeyed rigorously), one finds the two uniform\nstates represented in Figs. 4(a) and 4(b). These denote\nthe symmetric wavefunctions |ψ2/an}bracketri}ht= (|φa/an}bracketri}ht+|φb/an}bracketri}ht)/√\n2\nand|ψ3/an}bracketri}ht= (|φa/an}bracketri}ht+|φb/an}bracketri}ht+|φc/an}bracketri}ht)/√\n3 at every site, where\n|φγ/an}bracketri}ht=γ†|0/an}bracketri}ht. The remaining states shown in Fig. 4 in-\nvolveonlytwoorbitalspersite, but with allthreeorbitals\npartly occupied in the lattice. The average electron den-\nsitypersiteandperorbitalis1 /3in thestateofFig.4(c),\nwhile in Figs. 4(d) and 4(e) it is nc=1\n2,na=nb=1\n4.\nThe latter two states are neither unique nor (for general\ninteractions) equivalent to each other, and represent two\nclasses of states with respective degeneracies 3 and 6.\nB. Ordered–state energies: superexchange\nBefore analyzing the different possible ordered states\nfor any of the model parameters, we stress that the spin\ninteractions on a given bond depend strongly on the or-\nbital occupation of that bond. We begin with the pure\nsuperexchange model Hs(2.8), meaning α= 0, for which\nthe question of spin and orbital singlets was addressed\nin Sec. IIB. Here the spin and orbital scalar products\n/an}bracketle{t/vectorSi·/vectorSj/an}bracketri}htand/an}bracketle{t/vectorTi·/vectorTj/an}bracketri}htmay take only values consistent\nwith long–range order throughout the system and thus\nvary between −1/4 and +1/4.\nFor a bond on which both electrons occupy active or-\nbitals, one has the possibility of either FO or AO states.\nFor the FO state, /an}bracketle{t/vectorTi·/vectorTj/an}bracketri}ht= 1/4 =/an}bracketle{t/vectorTi×/vectorTj/an}bracketri}htand\n/an}bracketle{tAij/an}bracketri}ht=/an}bracketle{tBij/an}bracketri}ht= 1, whence the terms of Hscan be sepa-\nrated into the physically transparent form\nH(FO)\n1(0) =1\n2Jr1/parenleftbigg1\n2/an}bracketle{tniγ+njγ/an}bracketri}ht/parenrightbigg/parenleftbigg\n/vectorSi·/vectorSj+3\n4/parenrightbigg\n= 0,\nH(FO)\n2(0) =1\n2Jr2/parenleftbigg\n2−1\n2/an}bracketle{tniγ+njγ/an}bracketri}ht/parenrightbigg/parenleftbigg\n/vectorSi·/vectorSj−1\n4/parenrightbigg\n=Jr2/parenleftbigg\n/vectorSi·/vectorSj−1\n4/parenrightbigg\n, (3.1)\nH(FO)\n3=1\n3J(r3−r2)/parenleftbigg\n/vectorSi·/vectorSj−1\n4/parenrightbigg\n,\nspecifying a net spin interaction which, because niγ= 0,\nmust be AF if any hopping processes are to occur. In\nthe AO case, /an}bracketle{t/vectorTi·/vectorTj/an}bracketri}ht=−1/4 =/an}bracketle{t/vectorTi×/vectorTj/an}bracketri}htand/an}bracketle{tAij/an}bracketri}ht=\n/an}bracketle{tBij/an}bracketri}ht= 0, giving\nH(AO)\n1(0) =−1\n2Jr1/parenleftbigg\n/vectorSi·/vectorSj+3\n4/parenrightbigg\n,\nH(AO)\n2(0) =1\n2Jr2/parenleftbigg\n/vectorSi·/vectorSj−1\n4/parenrightbigg\n,(3.2)\nH(AO)\n3(0) = 0,and the spin interaction is constant at η= 0, with only a\nweak FM preference emerging at finite η. We remind the\nreader here that the designations FO and AO continue\nto be based on the conventional notation22obtained by a\nlocal transformation on one bond site, and in the basis of\nthe original orbitals correspond respectively to opposite\nactive orbitals and to equal active orbitals. Cases where\nonly one orbital is active on a bond are by definition AO,\nbut do contribute a finite spin interaction\nH1\n1(0) =−1\n4Jr1/parenleftbigg\n/vectorSi·/vectorSj+3\n4/parenrightbigg\n,\nH1\n2(0) =1\n4Jr2/parenleftbigg\n/vectorSi·/vectorSj−1\n4/parenrightbigg\n, (3.3)\nH1\n3(0) = 0,\nwhich again has only a weak FM tendency at η >0.\nClearly, when neither electron may hop, the bond does\nnot contribute a finite energy.\nWe begin with the uniform, one–color orbital state of\nFig. 3(a), meaning that all bonds are AO by the defini-\ntion of the previous paragraph. In two directions both\nelectrons are active, while in the third none are. The\nenergy per bond is\nE(3a)\nFM(0) =−1\n3Jr1. (3.4)\nand the spin configuration is FM. However, an antifer-\nromagnetic (AF) spin configuration on the square lattice\ndefined by the active hopping directions has energy\nE(3a)\nAF(0) =−1\n6J(r1+r2), (3.5)\nfrom which one observes that all spin states are degen-\nerate atη= 0. The ordered spin state spin is then FM\nfor any finite η. We note in passing that the energy per\nbond for a square lattice would have the significantly\nlower value −1\n2Jfor the same Hsconvention, by which\nis meant the presence of the constants +3\n4and−1\n4in\nEq. (2.8). This result is a direct reflection of the geo-\nmetrical frustration of the triangular lattice, an issue to\nwhich we return in Sec. VI.\nThe state of Fig. 3(b) involves one set of (alternating)\nAO lines with two active orbitals and two sets of (AO)\nlines each with one active orbital. All sets of lines favor\nFM order at finite η, with\nE(3b)\nFM(0) =−1\n3Jr1. (3.6)\nHere the square–lattice state which becomes degenerate\natη= 0, with\nE(3b)\nAF(0) =−1\n6J(r1+r2), (3.7)\nis more accurately described as one with two lines of AF\nspins andone ofFM spins [Fig. 5(a)], and will be denoted\nhenceforth as AFF.10\n(a) (b)\nFIG. 5: (Color online) Spin configurations minimizing the\ntotal energy of the superexchange Hamiltonian Hs(α= 0)\nfor given fixed patterns of orbital order: (a) AFF state for\nthe orbital ordering pattern of Fig. 3(c), showing how the FM\nlineisselected bythedirection(here b)givingzerofrustration;\n(b) 60–120◦ordered spin configuration minimizing the total\nenergy for the orbital ordering pattern of Fig. 3(d).\nThe state of Fig. 3(c) involves one set of FO lines with\ntwoactiveorbitals,onesetoflineswithoneactiveorbital,\none half set of AO lines with two active orbitals and one\nhalf set of inactive lines. The two–active FO lines will\nfavor AF order, while the AO and the one–active lines\nwill favor FM order only at η>0, giving\nE(3c)\nAFF(0) =−1\n72J(9r1+11r2+4r3) (3.8)\nfrom the AFF configuration, but with 2 equivalent di-\nrections for the FM line. At η= 0 the energy is again\n−1\n3J. BothE(3b)\nAF(0) andE(3c)\nAFF(0) can be regarded as the\nenergy of an unfrustrated system, in the sense that the\nspin order enforced in any one direction by the orbital\nconfiguration at no time denies the system the ability to\nadopt the energy–minimizing configuration in other di-\nrections. However, at finite ηthe configurations shown\nin Figs. 3(b) and 3(c) will be penalized relative to the\nuniform (AO) order of Fig. 3(a) due to the presence of\nAF bonds.\nWe insert here an important observation: the orbital\nstate of Fig. 3(c) also admits the formation of 1D AF\nHeisenberg spin chains on the FO ( b–axis) lines. The\nenergy per bond of such a state includes constant inter-\nchain contributions which are independent of the spin\nstate (/an}bracketle{t/vectorSi·/vectorSj/an}bracketri}ht= 0) on these bonds. Of these interchain\nbonds, 1/4 are FO with two active orbitals and 1/2 have\none active orbital. One finds\nE(3c)\n1D(0) =−1\n9Jln2 (2r2+r3)−1\n24J(3r1+r2),(3.9)\nwhich gives E(3c)\n1D(0) =−0.3977Jatη= 0. This energy\nis significantly lower than that of an ordered magnetic\nstate, a result showing that the kinetic energy gained\nfrom resonance processes on the chains is far more signif-\nicantthan minimalpotential energygainobtainablefrom\nan ordering of the magnetic moments on the active inter-\nchain bonds which are active, and thus provides strong\nevidence in favorof the hypothesis that any orderedstatewill “melt” to a quantum disordered one in this system.\nWe will return to this issue below.\nFor the two–color superposition [Fig. 4(a)], one set\nof bonds always has two active orbitals, but with equal\nprobability of being FO or AO, while the other two sets\nof bonds have a 1/4 probability of having two active or-\nbitals, which are FO, or a 1/2 probability of having one\nactive orbital (and a 1/4 probability of having none).\nUnder these circumstances, the net system Hamiltonian\ncan be expressedby summing overall the possible orbital\nstates, although this is not necessarily a useful exercise\nwhen the spin state may not be isotropic. By insert-\ning the three most obvious ordered spin states, FM, AF\n(meaning here the AF state of the triangular lattice with\n120◦bond angles and /an}bracketle{t/vectorSi·/vectorSj/an}bracketri}ht=−1\n8) and AFF, the can-\ndidate energies are\nE(4a)\nFM(0) =−1\n6Jr1, (3.10)\nE(4a)\nAF(0) =E(4a)\nAFF(0) =−1\n48J(5r1+7r2+2r3).\nThe coincidence for the results for the AF and AFF or-\ndered states in this case is an accidental degeneracy. The\nfinal energy E(4a)\nAF(F)=−7\n24Jatη= 0 shows that both\nstates are compromises, and it is not possible to put all\nbonds in their optimal spin state simultaneously. This\narises because of the presence of two–active FO compo-\nnents in all three lattice directions, and will emerge as a\nquite generic feature of superposition states, albeit not\none without exceptions.\nIn general there is no compelling reason (given by H\nfor any value of α) to expect that two–color superposi-\ntions of this type may be favorable. While the 120◦state\nofatriangular–latticeantiferromagnetisonecompromise\nwithinaspaceofSU(2)operators,thistypeofsymmetry–\nbreaking is not relevant within the orbital sector, where\nthere are three colors and the two–color subsector of ac-\ntive orbitals in the α= 0 limit changes as a function of\nthe bond orientation.\nIn the equally weighted three–color state [Fig. 3(d)],\nall bonds are FO and it is easy to show that 1/3 of them\n(arranged as isolated triangles) have two active orbitals\nwhile the other 2/3 have one active orbital. The two–\nactive bonds favor AF order while the one–active bonds\nhave only a weak preference for FM order at finite η.\nIn this case the problem becomes frustrated and is best\nresolved by a kind of AF state on the triangular lattice\nwhere the strong triangles have 120◦angles and alternat-\ning triangles have spins either all pointing in or all point-\ning out [Fig. 5(b)]; then 2/3 of the intertriangle bonds\nhave 60◦angles while the other 1/3 have 120◦angles.\nThe energy of this state is\nE(3d)(0) =−1\n144J(19r1+17r2+6r3),(3.11)\nandE(3d)(0) =−7\n24Jatη= 0, a value again inferior to\nthe optimal energy due to the manifest spin frustration.11\nIn the state of Fig. 3(e), the only AO bonds (1/6 of the\ntotal) contain inactive orbitals. Of the remaining bonds,\n3/6 have two active FO orbitals (in all three directions)\nand 2/6 have one active orbital. Once again the system\nis composed of strongly coupled triangles, but this time\nin a square array and with strong coupling in their basal\ndirection by one set of two–active FO bonds. Possible\ncompetitive spin–ordered states would be AF or AFF,\nwith energies\nE(3e)\nAF(0) =−1\n96J(5r1+15r2+6r3),(3.12)\nE(3e)\nAFF(0) =−1\n288J(15r1+35r2+16r3).\nThe lowest energy is obtained for 120◦AF order, with\nthe frustrated value E(3e)\nAF(0) =−13\n48Jforη= 0.\nFor the state in Fig. 3(f) the FO bonds (1/6) and only\n1/6 of the AO bonds have two active orbitals, while the\nother 2/3 of the bonds have one active orbital. In this\ncase\nE(3f)\nFM(0) =−1\n4Jr1,\nE(3f)\nAF(0) =−1\n96J(15r1+13r2+2r3),(3.13)\nE(3f)\nAFF(0) =−1\n192J(36r1+17r2+5r3),\nleading again to an AF spin state. At η= 0 one has\nE(3f)\nAF(0) =−5\n16J,i.e.relatively weaker frustration.\nTurning now to three–color superpositions, the “uni-\nform” orbital state [Fig. 4(b)] is one in which on every\nbond there is a probability 2/9 of having two active FO\norbitals, 2/9 for two active AO orbitals, 4/9 of one active\norbital and 1/9 of no active orbitals. The appropriately\nweighted bond interaction strengths may be summed to\ngive the net interaction, which for the three spin states\nconsidered results in the energies\nE(4b)\nFM(0) =−2\n9Jr1,\nE(4b)\nAF(0) =−1\n36J(5r1+5r2+r3),(3.14)\nE(4b)\nAFF(0) =−1\n81J(12r1+10r2+2r3),\nandthusthe AFstateislowest, with thevalue E(4b)\nAF(0) =\n−11\n36Jatη= 0. While this orbital configuration does not\nattain the minimal energy of −1\n3J, it is a close competi-\ntor: although it involves every bond, the fractional prob-\nabilities of each being in a two–active state mean that\nit cannot maximize individual bond contributions. How-\never, we will see in Sec. IIID that state (4b) lies lowest\nover much of the phase diagram (0 <α<1) as a result\nof the contributions from mixed terms.\nFor states with unequal site occupations, in Fig. 4(c)\none has a situation where on 1/3 of the bonds (arranged\nin separate triangles) there is a 1/4 probability of twoactive FO orbitals and a 1/2 probability of one active\norbital, while on the remaining 2/3 of the bonds there is\na 1/4 probability of two active AO orbitals, 1/4 of two\nactive FO orbitals and 1/2 of having one active orbital.\nOn computing the net energies for the three standard\nspin configurations, one obtains\nE(4c)\nFM(0) =−5\n24Jr1,\nE(4c)\nAF(0) =−1\n192J(25r1+25r2+8r3),(3.15)\nE(4c)\nAFF(0) =−1\n216J(30r1+25r2+8r3),\nwherethe AF statewith E(4c)\nAF(0) =−29\n96Jisthe lowestat\nη= 0. However, this state is also manifestly frustrated.\nIn the unequally weighted state of Fig. 4(d), the prob-\nlemisbestconsideredonceagainaslinesofdifferentbond\ntypes. Here 1/6 of the lines have two active orbitals (1/2\nFO and 1/2 AO), 1/6 of the lines have probability 1/4\nof two active orbitals (AO) and 1/2 of one active orbital,\n1/3 of the lines have probability 1/4 of two active FO or-\nbitals, 1/4oftwoactiveAOorbitalsand1/2ofoneactive\norbital, and the remaining 1/3 of the lines have probabil-\nity 1/4 of two active orbitals (FO) and 1/2 of one active\norbital. The ordered spin states yield the energies\nE(4d)\nFM(0) =−5\n24Jr1,\nE(4d)\nAF(0) =−1\n192J(25r1+27r2+6r3),(3.16)\nE(4d)\nAFF(0) =−1\n144J(21r1+17r2+4r3),\nwhenceitisagaintheAFstate, withasmalldegreeofun-\nrelieved frustration in its energy E(4d)\nAF(0) =−29\n96J, which\nlies lowest at η= 0.\nFinally, the state of Fig. 4(e) has the orbital pattern of\nFig. 4(d) rotated in such a way that the number of active\norbitals in different bond directions is changed. Now 1/3\nof the bonds have probabilities 1/4 of two active orbitals\n(AO) and 1/2 of one active orbital, while the remaining\n2/3 have probabilities 1/4 of two active orbitals (FO),\n1/4 of two active orbitals (AO) and 1/2 of one active\norbital. The ordered–state energies are\nE(4e)\nFM(0) =−1\n4Jr1,\nE(4e)\nAF(0) =−1\n96J(15r1+13r2+2r3),(3.17)\nE(4e)\nAFF(0) =−1\n36J(6r1+5r2+r3),\nof which the AFF states lies lowest at η= 0, achieving\nthe unfrustrated value E(4e)\nAFF(0) =−1\n3J. That it is pos-\nsible to obtain this energy in an orbital superposition is\nbecause of the absence of FO bond contributions in one\ndirection, which can then be chosen to be FM.\nThe results of this section and the conclusions one may\ndraw from them are summarized in Subsec. IIIE below.12\nC. Ordered–state energies: direct exchange\nIn the limit of only direct exchange, the analysis is\nsomewhat simpler. The Hamiltonian is Hdof Eq. (2.21),\nand in this case a particle on any site is active in only\none direction, which leads to the immediate observation\nthatin astaticorbitalconfigurationit isneverpossibleto\nhave, on average,activeexchangeprocessesonmorethan\n2/3 of the bonds. For simplicity we repeat the Hamilto-\nnianforthe twocasesofAO orderbetweensites, inwhich\ncase by definition at most one of the orbitals is active,\nand FO order between sites, which is restricted to the\ncase where neighboring sites have the same orbital color\nand the correct bond orientation. We stress that in this\nsubsection the definitions FO and AO are entirely con-\nventional, as the local transformation of Sec. IIB is not\nrelevant at α= 1, and thus the designation FO implies\norbitals of the same color, and AO orbitals of different\ncolors. One obtains the expressions\nH(AO)(1) =1\n4J/bracketleftbigg\n−r1/parenleftbigg\n/vectorSi·/vectorSj+3\n4/parenrightbigg\n+r2/parenleftbigg\n/vectorSi·/vectorSj−1\n4/parenrightbigg/bracketrightbigg\n,(3.18)\nH(FO)(1) =1\n3J(2r2+r3)/parenleftbigg\n/vectorSi·/vectorSj−1\n4/parenrightbigg\n,(3.19)\nwhich in the η= 0 limit reduce to the forms\nH(AO)(1) =−1\n4J, (3.20)\nH(FO)(1) =J/parenleftbigg\n/vectorSi·/vectorSj−1\n4/parenrightbigg\n.(3.21)\nIt is clear (Sec. II) that for a single bond, the most fa-\nvorablestate is aspin singlet, which would contribute en-\nergy−J, but at the possible expense of placing all of the\nneighboring bonds in suboptimal states. The very strong\npreference for such singlet bonds means that any mean–\nfield study of the minimal energy is incomplete without\nthe consideration of dimerized (or valence–bond) states\n(Sec. IV). The analysis of this section can be considered\nas elucidating the optimal energies to be gained from\nlong–ranged magnetic and orbital order on these bonds,\nwhere the optimal energy of any one is −1\n2J. Also as\nnoted in Sec. II, any active AO bond gains an exchange\nenergy (−1\n4J) simply because it does not prevent one of\nthe two particles from performing virtual hopping pro-\ncesses, and this we term “avoided blocking”. In the limit\nof zero Hund exchange, these will give a highly degener-\nate manifold of all possible spin states, from which FM\nstates are selected at finite η.\nWe beginagainwith one–colorstateofFig. 3(a), which\nwe denote henceforth as (3a). Only one set of lattice\nbonds has finite interactions, all FO, and therefore the\nsystem behaves as a set of AF Heisenberg spin chains\nwith energy per bond\nE(3a)\nAF1D(1) =−1\n9Jln2 (2r2+r3),(3.22)whenceE(3a)\nAF1D(1) =−0.2310Jatη= 0.\nIn state (3b), the FO lines do not correspond to active\nhopping directions. The remaining two directions then\nform an AO square lattice with\nE(3b)\nFM(1) =−1\n6Jr1. (3.23)\nThis can be called a “pure avoided–blocking” energy.\nThe spins are unpolarized at η= 0, where all bond spin\nstates are equivalent, but any finite ηwill select FM or-\nder (hence the notation). We will see in the remainder of\nthis section that E=−1\n6Jis the optimal energy obtain-\nable by a 2D ordered state in the direct–exchange limit\n(α= 1), where the net energy is generically higher than\natα= 0quitesimplybecausetherearehalfasmanyhop-\nping channels. Thus the “melting” of such ordered states\ninto quasi–1D states becomes clear from the outset, and\ncan be understood due to the very low connectivity of\nthe active hopping network on the triangular lattice.\nIn state (3c), one of the FO lines is active, and forms\nAF Heisenberg spin chains. Electrons in the other FO\nline areactiveonly in across–chaindirection, wheretheir\nbonds areAO,and gainavoided–blockingenergy, whence\nE(3c)\nAF(1) =−1\n12J(2ln2+1) = −0.1988J(3.24)\natη= 0. As in the preceding subsection, the coherent\nstate of each Heisenberg chain is not altered by the pres-\nence of additional electrons from other chains executing\nvirtual hopping processes onto empty orbitals of individ-\nual sites. The spin chains remain uncorrelated and only\nquasi–long–range–ordereduntil a finite value of η, where\nFM spin polarization and a long–range–orderedstate are\nfavored.\nIn the two–color superposition (4a), 1/3 of the bonds\nare inactive, while on the other 2/3 one has probabil-\nity 1/4 of two active electrons (FO), 1/2 of one active\n(AO) and 1/4 of two inactive electrons. In this case, one\nobtains an effective square lattice on which an AF spin\nconfiguration is favored by the FO processes, with\nE(4a)\nAF(1) =−1\n72J(3r1+7r2+2r3),(3.25)\nso againE(4a)\nAF(1) =−1\n6Jatη= 0.\nThe uniform three–color state (3d) maximizes AO\nbonds, but 1/3 of the bonds on the lattice remain in-\nactive. Thus\nE(3d)\nFM(1) =−1\n6Jr1, (3.26)\nand Hund exchange will select the FM spin state.\nThe three–color state (3e) has FO lines oriented in\ntheir active direction and will, as in state (3c), form\nHeisenberg chains linked by bonds with AO order. While\nthe geometry of the interchain coupling can differ de-\npending on the orbital alignment in the inactive chains,\nit does not create a frustrated spin configuration and13\nthe net energy is E(3e)\nAF(1) =E(3c)\nAF(1). The state (3f)\nhas only inactive FO lines and so gains only avoided–\nblocking energy, from 2/3 of the bonds in the system,\nwhenceE(3f)\nFM(1) =E(3d)\nFM(1).\nIn the uniform three–color superposition (4b), every\nbond has probability 1/9 of containing two active elec-\ntrons (FO), 4/9 of one active electron and 4/9 of remain-\ning inactive. For the three different ordered spin config-\nurations considered in Subsec. IIIB the energies are\nE(4b)\nFM(1) =−1\n9Jr1,\nE(4b)\nAF(1) =−1\n72J(5r1+5r2+r3),(3.27)\nE(4b)\nAFF(1) =−1\n81J(6r1+5r2+r3),\nand one finds the energy E(4b)\nAF(1) =−11\n72Jfor the 1200\nAF state at η= 0.\nThe three–color state (4c) is one in which 1/3 of the\nbonds (arranged on isolated triangles) have probability\n1/4 of being in a state with two active electrons and 1/2\nof containing one active electron, while on the other 2/3\nof the bonds there is simply a 1/2 probability of one\nactive orbital. The respective energies are\nE(4c)\nFM(1) =−1\n8Jr1,\nE(4c)\nAF(1) =−1\n192J(15r1+13r2+2r3),(3.28)\nE(4c)\nAFF(1) =−1\n216J(18r1+13r2+2r3).\nAtη= 0, the energy E(4c)\nFM(1) =−5\n32Jis minimized\nby a 120◦state on the triangles, which are also isolated\nmagnetically in this limit. Finite values of ηresult in\nFM interactions between the triangles, and a frustrated\nproblem in the spin sectorwhich by inspection is resolved\nin favor of a net FM configuration only at large η(η >\n0.23).\nFinally, the three–color states (4d) and (4e) yield two\npossibilities in the α= 1 limit, namely where one of the\nminority colors is aligned with its active direction and\nwhere neither is. In the former case,\nE(4d)\nFM(1) =−5\n48Jr1,\nE(4d)\nAF(1) =−1\n384J(25r1+27r2+6r3),(3.29)\nE(4d)\nAFF(1) =−1\n96J(7r1+7r2+2r3),\nand the lowest energy E(4d)\nAFF(1) =−1\n6Jatη= 0 is given\nby the directionally anisotropic AFF spin configuration.\nThis is because 1/2 of the lines, in two of the three di-\nrections, have some AF preference from their 1/4 prob-\nability of containing two active orbitals, while the third\ndirection has no preference at η= 0, and in any case\nfavors FM spins at η >0. In the latter case, the onlyAF tendencies arise along lines in a single direction, but\navoided–blocking energy is sufficient to exclude the pos-\nsibility of a Heisenberg chain state. Here\nE(4e)\nFM(1) =−1\n8Jr1,\nE(4e)\nAF(1) =−1\n192J(15r1+13r2+2r3),(3.30)\nE(4e)\nAFF(1) =−1\n72J(6r1+5r2+r3),\nwhenceE(4e)\nAFF(1) =−1\n6Jatη= 0, in fact with two de-\ngenerate possibilities for the orientation of the FM line.\nD. Ordered–state energies: α= 0.5\nToillustratethepropertiesofthemodelinthepresence\nof finite direct and superexchange contributions, i.e.at\nintermediate values of α, we consider the point α= 0.5.\nAs shown in Sec. II, there is no special symmetry at this\npoint, because the contributions from diagonal and off–\ndiagonal hopping remain intrinsically different. States\nwith long–ranged orbital (and spin) order at α= 0.5\nare mostly very easy to characterize, because all virtual\nprocesses, of both types, allowed by the given configura-\ntion are able to contribute in full to the net energy. For\nthe many of the states considered in this section, the en-\nergetic calculation for α= 0.5 is merely an exercise in\nadding the α= 0 andα= 1 results with equal weight.\nExceptions occur for superposition states gaining energy\nfrom processes contained in Hm[Eq. (2.22)], and are in\nfact decisive here. Because these terms involve explicitly\na finite density of orbitals of all three colors on the bond\nin question, with the active diagonalcolor representedon\nboth sites, only for states (4b), (4c), and (4d), but not\n(4e) [Figs. 4(b–e)], will it be necessary to consider this\ncontribution.\nFor state (3a), in two directions both electrons are\nactive by off–diagonal hopping, while in the third both\nmay hop diagonally. Diagonal hopping favorsan AF spin\nconfiguration, while the off–diagonalhopping bonds have\nonly a weak preference (by Hund exchange) for FM or-\nder. The ordered–state spin solution is then a doubly\ndegenerate AFF state with energy per bond\nE(3a)(0.5) =−1\n72J(9r1+7r2+2r3),(3.31)\ngivingE(3a)(0.5) =−1\n4Jatη= 0. We remind the\nreader that the prefactor of the superexchange and di-\nrect exchange contributions is only half as large as in\nSubsecs. IIIB and IIIC [Eq. (2.7)], so the overall effect\nof additional hopping processes in this state is in fact\nan unfrustrated energy summation. We also comment\nthat, exactly at η= 0, there is no obvious preference\nfor any magnetic order between the diagonal–hopping\nchains. Only at unrealistically large values of ηwould14\nthe system sacrifice this diagonal–hopping energy to es-\ntablish a square–lattice FM state. At finite η, the one–\ncolororbitalstate representsa compromisebetween com-\npeting spin states preferred by the two types of hopping\ncontribution.\nState (3b) has no diagonal–hopping chains, and these\nprocesses therefore enforce only a weak preference for a\nFMsquarelattice. Becausetheoff–diagonalhoppingpro-\ncesses also favor FM order at finite η(Subsec. IIIB), the\ntwo types of contribution cooperate and one obtains\nE(3b)(0.5) =−1\n4Jr1. (3.32)\nState (3c) contains one half set of diagonal–hopping\nchains, which fall along one of the directions which in\nthe spin state favored by the off–diagonal hopping pro-\ncesses could be FM or AF; this degeneracy will therefore\nbe broken. The other half set of chains will gain only\navoided–blocking energy from diagonal processes, which\nwill take place in the FM direction and thus cause no\nfrustration even at finite η. One obtains\nE(3c)(0.5) =−1\n144J(3r1+7r2+2r3),(3.33)\nand thusE(3c)(0.5) =−1\n4Jatη= 0 from this AFF\nconfiguration. The additive contributions from superex-\nchange and direct exchange remove the possibility that\nHeisenberg–chain states in either of the directions fa-\nvored separately by off–diagonal (Sec. IIIB) or diagonal\n(Sec. IIIC) hopping could result in an overall lowering of\nenergy.\nAs in Subsec. IIIC, in the two–colorsuperposition (4a)\nthe diagonal hopping processes are optimized by an AFF\nspin configuration. Although this is one of the degener-\nate states minimizing the off–diagonal Hamiltonian, the\ndirections of the FM lines do not match. Insertion of the\nfour possible spin states yields\nE(4a)\nFM(0.5) =−1\n8Jr1,\nE(4a)\nAF(0.5) =−1\n96J(8r1+10r2+3r3),\nE(4a)\nAFF(0)(0.5) =−1\n72J(6r1+7r2+r3),(3.34)\nE(4a)\nAFF(1)(0.5) =−1\n144J(12r1+14r2+3r3),\nwhence the lowest final energy is E(4a)\nAF(0.5) =−7\n32Jat\nη= 0. As noted in the previous sections for this spin\nconfiguration, the optimal energy for all bonds is not\nattainable within the off–diagonal hopping sector, and\nthe addition of the (small) diagonal–hopping contribu-\ntion causes little overall change.\nThe equally weighted three–color state (3d) has no\nlines of diagonal–hopping bonds, and in fact these con-\ntribute only avoided–blocking energy on the bonds be-\ntween the strong triangles defined by the off–diagonalproblem, adding to the weak propensity for FM intertri-\nangle bonds arising only from the Hund exchange. The\ndiagonal processes can be taken only to alter this energy,\nand not to promote any tendency towards an alteration\nof the spin state, whose energy is then\nE(3d)(0.5) =−1\n144J(19r1+11r2+3r3),(3.35)\nwithE(3d)(0.5) =−11\n48Jatη= 0.\nState (3e) is already frustrated in the off–diagonal sec-\ntor, and diagonal–hoppingprocessescontributeprimarily\non otherwise inactive bonds without changing the frus-\ntration conditions. For the two candidate spin configu-\nrations\nE(3e)\nAF(0.5) =−1\n96J(5r1+12r2+4r3),\nE(3e)\nAFF(0.5) =−1\n192J(11r1+19r2+8r3),(3.36)\na competition won by the 120◦AF–ordered state with\nE(3e)\nAF(0.5) =−7\n32Jatη= 0.\nState (3f) lacks active lines of diagonal–hopping pro-\ncesses, and thus the avoided–blocking energy may be\nadded simply to the results for the off–diagonal sector,\ngiving\nE(3f)\nFM(0.5) =−5\n24Jr1,\nE(3f)\nAF(0.5) =−1\n192J(25r1+19r2+2r3),(3.37)\nE(3f)\nAFF(0.5) =−5\n384J(12r1+5r2+r3),\nor a minimum of E(3f)\nAF(0.5) =−23\n96Jatη= 0.\nIn the uniform three–color superposition (4b), on ev-\nery bond there is a probability 4/9 of having only off–\ndiagonal hopping processes, 2/9 for 2 active FO orbitals\nand 2/9 for two active AO orbitals, a probability 1/9 of\nhaving only diagonal hopping processes, and a probabil-\nity 4/9 of other processes. These last include the contri-\nbutions fromoneactivediagonaloroff–diagonalelectron,\nand mixed processes contained in the Hamiltonian Hm\n(2.22); none of these three possibilities favors any given\nbond spin configuration other than a FM orientation at\nfiniteη. The net energy contributions are\nE(4b)\nFM(0.5) =−2\n9Jr1,\nE(4b)\nAF(0.5) =−1\n144J(20r1+18r2+3r3),(3.38)\nE(4b)\nAFF(0.5) =−1\n54J(8r1+6r2+r3),\nand thus the AF state is lowest, with E(4b)\nAF(0.5) =−41\n144J\natη= 0. While this energy differs from that for the AFF\nspin configuration by only1\n144J, its crucial property is\nthat it lies below the value −1\n4Jobtained by direct sum-\nmation of the superexchangeand direct–exchangecontri-\nbutions.15\nFor this orbital configuration, all three spin states gain\na net energy of −1\n18Jatη= 0 from mixed processes,\nand these are sufficient, as we shall see, to reduce the\notherwisepartiallyfrustratedordered–stateenergytothe\nglobal minimum for this value of α. By a small extension\nof the calculation, the energy of the 1200AF spin state\nmay be deduced at η= 0 for all values of α, and is given\nby\nE(4b)\nAF(α) =−1\n72J/parenleftbig\n22−11α+8/radicalbig\nα(1−α)/parenrightbig\n.(3.39)\nComparison with the value obtained by direct summa-\ntion,E=−1\n6(2−α), revealsthat state(4b) isthe lowest–\nlying fully spin and orbitally ordered configuration in the\nregion0.063<α<0.983. Thatthis statedominatesover\nthe majorityofthe phasediagramis a directconsequence\nof its ability to gain energy from mixed processes.\nThe non–uniform three–colorstate (4c) also presents a\ndelicate competition between spin configurations of very\nsimilar energies. From the preceding subsections, it is\nclearthat inthis casediagonalandoff–diagonalprocesses\nfavor different ground states, while there will also be a\nmixed contribution from 1/3 of the bonds. The energies\nof the three standard spin configurations are\nE(4c)\nFM(0.5) =−1\n12Jr1,\nE(4c)\nAF(0.5) =−1\n384J(45r1+41r2+10r3),(3.40)\nE(4c)\nAFF(0.5) =−1\n432J(54r1+41r2+10r3),\nwhere the AF state, obtaining E(4c)\nAF(0.5) =−1\n4Jis the\nlowest atη= 0.\nFinally, in the three–color states (4d) and (4e), which\nare composed of lines of two–colorsites, this delicate bal-\nance between different spin configurations persists. For\nconfiguration (4d), an AFF state with the same orien-\ntation of the FM line (along the b–axis) is both favored\nby diagonal hopping processes and competitive for off–\ndiagonal processes. With inclusion of a small contribu-\ntion due to mixed processes, the three orderedspin states\nhave energies\nE(4d)\nFM(0.5) =−51\n288Jr1,\nE(4d)\nAF(0.5) =−1\n768J(85r1+84r2+18r3),(3.41)\nE(4d)\nAFF(0.5) =−1\n576J(71r1+59r2+14r3),\nfrom which the AFF state minimizes the energy at η= 0\nwithE(4d)\nAFF(0.5) =−1\n4J.\nFor state (4e), which has no mixed contribution, the\norientationsofthe FM lines in the optimal AFF states do\nnot match, and it is necessary, as above, to consider both\npossibilities when performing a full comparison. These\nfour ordered spin states yield the energies\nE(4e)\nFM(0.5) =−3\n16Jr1,E(4e)\nAF(0.5) =−1\n128J(15r1+13r2+2r3),\nE(4e)\nAFF(0)(0.5) =−1\n144J(18r1+13r2+2r3),\nE(4e)\nAFF(1)(0.5) =−1\n48J(6r1+4r2+r3),(3.42)\namong which the AF state in fact lies lowest at η= 0,\nachievingthe weaklyfrustratedvalue E(4e)\nAF(0.5) =−15\n64J.\nE. Summary\nHere we summarize the results of this section in a con-\ncise form. For the superexchange model ( α= 0), a con-\nsiderable number of 2D ordered orbital and spin states\nexist which return the energy −1\n3Jatη= 0. This de-\ngeneracy is lifted at any finite Hund exchange in favor\nof orbital states [(3a), (3b)] permitting a fully FM spin\nalignment. Most other orbital configurations introduce a\nfrustration in the spin sector at small η, while some offer\nthe possibility of a change of ground–state spin configu-\nration at finite η, wherer1exceeds the r2andr3contri-\nbutions and begins to favor states with more FM bonds.\nHowever, the value E=−1\n3Jper bond remains a\nrather poor minimum for a system as highly connected\nas the triangular lattice, even if, as in the superexchange\nlimit, active hopping channels exist only in two of the\nthree lattice directions for each orbital color. Indeed,\nthe limitations of the available ordering (potential) en-\nergy are clearly visible from the fact that a significantly\nlower overall energy is attained in systems which aban-\ndon spin order in favor of the resonance (kinetic) en-\nergy gains available in one lattice direction. The result\nE(3c)\n1D(0) =−0.3977Jis the single most important ob-\ntained in this section, and in a sense obviates all of the\nconsiderations made here for fully ordered states, man-\ndating the full consideration of 2D magnetically and or-\nbitally disordered phases.\nIn the study of ordered states, it becomes clear that\nthe Hund exchange acts to favor FM spin alignments at\nhighη. Because the “low–spin” states of minimal energy\nare in fact stabilized by quantum corrections due to AF\nspin fluctuations, the lowest energies at η= 0 are never\nobtained for FM states, and therefore increasing ηdrives\na phase transition between states of differing spin and\norbital order. We show in Fig. 6 the transitions from\nquasi–1D AF–correlated states at low η, for bothα= 0\nandα= 1, to FM states of fixed orbital and spin order\n(3b). The transitions occur at the values ηc(0) = 0.085\nandηc(1) = 0.097, indicating that FM ordered states\nmay well compete in the physical parameter regime. We\nnote again that the energies in the superexchange limit\nare lower by approximately a factor of two compared to\nthe direct–exchange limit simply because of the number\nof available hopping channels.\nWenotealsothatthereisneverasituationinwhichthe\nspin Hamiltonian becomes that of a Heisenberg model on16\n0 0.05 0.1 0.15 0.2\nη−0.8−0.6−0.4−0.2EAF/J, EFM/J\nFIG. 6: (Color online) Minimum energies per bond obtained\nfor orbitally ordered phases, showing a transition as a func -\ntion of Hund exchange ηfrom quasi–1D, AF–correlated to\nFM ordered spin states. For the superexchange Hamiltonian\nH∫of Sec. II ( α= 0), the transition is from the quasi–1D\nspin state on orbital configuration (3c) [black, dashed line\nfrom Eq. (3.9)] to the one–color orbital state (3a) [red, sol id\nline from Eq. (3.4)]. For the direct–exchange Hamiltonian\nHd(α= 1), the transition is from the purely 1d spin state\non the one–color orbital state (3a) [green, dot–dashed line\nfrom Eq. (3.22)] to the two–colour, avoided–blocking state\n(3b) [blue, dotted line from Eq. (3.23)]. The transitions to\nFM order as obtained from the mean–field considerations of\nthis section are marked by arrows.\na triangular lattice. This demonstrates again the inher-\nentfrustrationintroducedbytheorbitalsector. However,\nthe fact that the ordered–state energy can never be low-\nered to the value EHAF=−3\n8J, which might be expected\nfor a two–active FO situation on every bond, far less the\nvalue−1\n2Jwhich could be achieved if it were possible to\noptimize every bond in some ordered configuration, can\nbe taken as a qualitative reflection of the fact that on\nthe triangular lattice the orbital degeneracy “enhances”\nrather than relieves the (geometrical) frustration of su-\nperexchange interactions (Sec. VI).\nThe limit of direct exchange ( α= 1) is found to be\nquite different: the very strong tendency to favor spin\nsinglet states, and the inherent one–dimensionalityof the\nmodel in this limit (one active hopping direction per or-\nbital color), combine to yield no competitive states with\nlong–ranged magnetic order. Their optimal energy is\nvery poor because of the restricted number of hopping\nchannels, and coincides with the (“avoided–blocking”)\nvalue for the model with only AO bonds, E=−1\n6J.\nThus these states form part of a manifold with very\nhigh degeneracy. However, even at this level it is clear\nthat more energy, meaning kinetic (from resonance pro-\ncesses) rather than potential, may be gained by forming\nquasi–1D Heisenberg–chain states with little or no inter-\nchain coupling and only quasi–long–ranged magnetic or-\nder. Studies of orbital configurations permitting dimer-\nized states are clearly required (Sec. IV). Finite Hund\nexchange acts to favor ordered FM configurations, whichwill take over from chain–like states at sufficiently high\nvalues ofη(Fig. 6).\nFinally, orderedstatesofthemixedmodelshowanum-\nber of compromises. At α= 0.5, where the coefficients of\nsuperexchange and direct–exchange are equal, some con-\nfigurations are able to return the unfrustrated sum of the\noptimal states in each sector when considered separately,\nnamely−1\n4J. However, superposition states, which are\nnot optimal in either limit, can redeem enough energy\nfrom mixed processes to surpass this value, and in fact\nthe maximally superposed configuration (4b) is found\nto minimize the energy over the bulk of the phase di-\nagram. Still, the net energy of such states remains small\ncompared to expectations for a highly connected state\nwith three available hopping channels per orbital color.\nBecause of the directional mismatch between the diago-\nnal and off–diagonal hopping sectors, no quasi–1D states\nwith only chain–like correlations are able to lower the\nordered–state energy in the intermediate regime.\nIV. DIMER STATES\nAs shown in Sec. II, the spin–orbital model on a sin-\ngle bond favors spin or orbital dimer formation in the\nsuperexchange limit, and spin dimer formation in the\ndirect–exchange limit. The physical mechanism respon-\nsible for this behavior is, as always, the fluctuation en-\nergy gain from the highly symmetric singlet state. On\nthe basis of this result, combined with our failure to find\nany stable, energetically competitive states with long–\nranged spin and orbital order in either limit of the model\n(Sec. III), we proceed to examine states based on dimers.\nGiven the high connectivity of the triangular lattice,\ndimer–based states are not expected a priorito be capa-\nble of attaining lower energies than ordered ones, and\nif found to be true it would be a consequence of the\nhigh frustration, which as noted in Sec. I has its ori-\ngin in both the interactions and the geometry. Here we\nconsider static dimer coverings of the lattice, and com-\npute the energies they gain due to inter–singlet corre-\nlations. The tendency towards the formation of singlet\ndimerstates will be supportedby the numericalresultsin\nSec. V, which will also address the question of resonant\ndimer states.\nA. Superexchange model\nMotivated by the fact that the spin and orbital sectors\ninHs(2.8) are not symmetrical, we proceed with a sim-\nple decoupling of spin and orbital operators. Extensive\nresearch on spin–orbital models has shown that this pro-\ncedure is unlikely to capture the majority of the physical\nprocesses contributing to the final energy, particularly\nin the vicinity of highly symmetric points of the general\nHamiltonian. The results to follow are therefore to be\ntreated as a preliminary guide, and a basis from which17\nto consider a more accurate calculation of the missing\nenergetic contributions. We remind the reader that the\nnotation FO and AO used in this subsection is again that\nobtained by performing a local transformation on one\nsite of every dimer. As noted in Sec. IIB, this procedure\nis valid for the discussion of states based on individual\ndimerized bonds, where it represents merely a notational\nconvenience. For FO configurations, which in the origi-\nnal basis have different orbital colors, one might in prin-\nciple expect that, because of the color degeneracy, there\nshould be more ways to realize these without frustration\nthan thereareto realizeAF spinconfigurations; however,\nbecause of the directional dependence of the hopping, we\nwill find that this is not necessarily the case (below).\nThebasicpremiseofthespin–orbitaldecouplingisthat\nif the spin (orbital) degrees of freedom on a dimer bond\nform a singlet state, their expectation value /an}bracketle{t/vectorSi·/vectorSj/an}bracketri}ht\n(/an}bracketle{t/vectorTi·/vectorTj/an}bracketri}ht) on the neighboring interdimer bonds will be\nprecisely zero. The optimal orbital (spin) state of the\ninterdimer bond may then be deduced from the effec-\ntive bond Hamiltonian obtained by decoupling. Because\nHsdepends on the number of electrons on the sites of\na given bond which are in active orbitals, and this num-\nber is well defined only for the dimer bonds, the effective\nHamiltonian will be obtained by averaging over all occu-\npation probabilities. In contrast to the pure Heisenberg\nspin Hamiltonian, here the interdimer bonds contribute\nwith finite energies, and the dimer distribution must be\noptimized. A systematic optimization will not be per-\nformed in this section, where we consider only represen-\ntative dimer coverings giving the semi–quantitative level\nofinsightrequiredasapreludetoaddingdimerresonance\nprocesses (Sec. V).\nOn the triangular lattice there are three essentially\ndifferent types of interdimer bond, which are shown in\nFig. 7). For a “linear” configuration [Fig. 7(a)], the num-\nber of electrons in active orbitals on the interdimer bond\nis two; for the 8 possible configurations where one dimer\nbond is aligned with the interdimer bond under consider-\nation [Fig. 7(b)], the number is one on the corresponding\nsite and one or zero with equal probability on the other;\nfor the 14 remaining configurations where neither dimer\nbond is aligned with the interdimer bond [Fig. 7(c)], the\nnumber is one or zero for both sites. The number of elec-\ntrons in active orbitals is then two for type (7a), two or\none, each with probability 1/2, for type (7b), and two,\none or zero with probabilities 1/4, 1/2, and 1/4 for type\n(7c).\nThe effective interdimer interactions for each type of\nbond can be deduced in a manner similar to the treat-\nment of the previous section. Considering first the situa-\ntion for a bond of type (7a) with (os/st) dimers, setting\n/an}bracketle{t/vectorTi·/vectorTj/an}bracketri}ht= 0 yields one high–spin and two low–spin terms\nwhich contribute\nH(os,7a)\n1(0) =−1\n4Jr1/parenleftBig\n/vectorSi·/vectorSj+3\n4/parenrightBig\n,\nH(os,7a)\n2(0) =3\n4Jr2/parenleftBig\n/vectorSi·/vectorSj−1\n4/parenrightBig\n, (4.1)(a) (b)\n(c)\nFIG. 7: (Color online) Types of interdimer bond differing in\neffective interaction due to dimer coordination: (a) “linea r”,\n(b) “semi–linear”, (c) “non–linear”.\nH(os,7a)\n3(0) =1\n6J(r3−r2)/parenleftBig\n/vectorSi·/vectorSj−1\n4/parenrightBig\n.\nClearlyH(os,7a)\n1favors FM (high–spin) interdimer spin\nconfigurations with coefficient1\n4, whileH(os,7a)\n2and\nH(os,7a)\n3favor AF (low–spin) configurations with coeffi-\ncient3\n8(both atη= 0). Because r1exceedsr2andr3\nwhen Hund exchangeis finite, one expects a criticalvalue\nofηwhere FM configurations will be favored. Simple al-\ngebraic manipulations using all three terms suggest that\nthis value, which should be relevant for a linear chain of\n(os/st)dimers, is ηc=1\n8. In the limit η→0, the effective\nbond Hamiltonian simplifies to\nH(os,7a)\neff(0) =1\n2J/parenleftbigg\n/vectorSi·/vectorSj−3\n4/parenrightbigg\n.(4.2)\nFor a bond of type (7a) with (ss/ot) dimers, setting\n/an}bracketle{t/vectorSi·/vectorSj/an}bracketri}ht= 0 on the interdimer bond yields\nH(ss,7a)\n1(0) =3\n4Jr1/parenleftbigg\n/vectorTi·/vectorTj−1\n4/parenrightbigg\n,\nH(ss,7a)\n2(0) =−1\n4Jr2/parenleftbigg\n/vectorTi·/vectorTj+3\n4/parenrightbigg\n, (4.3)\nH(ss,7a)\n3(0) =−1\n6J(r3−r2)/parenleftbigg\n/vectorTi×/vectorTj+1\n4/parenrightbigg\n.\nHereH(ss,7a)\n1favorsAO configurations with coefficient3\n8,\nwhileH(ss,7a)\n2andH(ss,7a)\n3both favor FO configurations\nwith coefficient1\n4(atη= 0). Over the relevant range of\nHund exchangecoupling, 0 <η<1/3, there is no change\nin sign and AO configurations are always favored. The\neffective bond Hamiltonian for η→0 is\nH(ss,7a)\neff(0) =1\n2J/parenleftbigg\n/vectorTi·/vectorTj−3\n4/parenrightbigg\n.(4.4)\nFor bonds of type (7b), when only one electron occu-\npies an active orbital the corresponding decoupled inter-\ndimer bond Hamiltonians are, for (os/st) dimers,\nH(os,1)\n1(0) =−1\n4Jr1/parenleftBig\n/vectorSi·/vectorSj+3\n4/parenrightBig\n,18\nH(os,1)\n2(0) =1\n4Jr2/parenleftBig\n/vectorSi·/vectorSj−1\n4/parenrightBig\n,(4.5)\nH(os,1)\n3(0) = 0.\nThe final interdimer interaction is obtained by averag-\ning over these expressions and those (4.1) for two active\norbitalsper bond, and takes the rathercumbersome form\nH(os,7b)\neff(0) =1\n12J(r3+5r2−3r1)/vectorSi·/vectorSj\n−1\n48J(9r1+5r2+r3),(4.6)\nwhich reduces in the limit η→0 to\nH(os,7b)\neff(0) =1\n4J/parenleftbigg\n/vectorSi·/vectorSj−5\n4/parenrightbigg\n.(4.7)\nFor (ss/ot) dimers, the situation cannot be formulated\nanalogously, because if only one electron on the bond\nis active, the orbital state of the other electron has no\ninfluence on the hopping process, i.e./vectorTi·/vectorTjis not a\nmeaningful quantity. The resulting expressions lead then\nto\nH(ss,7b)\neff(0) =1\n8J(3r1−r2)/vectorTi·/vectorTj−1\n12J(r3−r2)/vectorTi×/vectorTj\n−1\n48J(9r1+5r2+r3), (4.8)\nwhich has the η→0 limit\nH(ss,7b)\neff(0) =1\n4J/parenleftbigg\n/vectorTi·/vectorTj−5\n4/parenrightbigg\n.(4.9)\nFinally, forabondoftype(7c), thereisnocontribution\nfrom interdimer bond states with no electrons in active\norbitals, so the above results [(4.1, 4.5) and (4.3, 4.8)]\nare already sufficient to perform the necessary averaging.\nWith (os/st) dimers\nH(os,7c)\neff(0) =1\n48J(2r3+13r2−9r1)/vectorSi·/vectorSj\n−1\n192J(27r1+13r2+2r3),(4.10)\nwhich reduces in the limit η→0 to\nH(os,7c)\neff(0) =1\n8J/parenleftbigg\n/vectorSi·/vectorSj−7\n4/parenrightbigg\n,(4.11)\nwhile for (ss/ot) dimers,\nH(ss,7c)\neff(0) =1\n16J(3r1−r2)/vectorTi·/vectorTj−1\n24J(r3−r2)/vectorTi×/vectorTj\n−1\n192J(27r1+13r2+2r3),(4.12)\nwhich in the η→0 limit gives\nH(ss,7c)\neff(0) =1\n8J/parenleftbigg\n/vectorTi·/vectorTj−7\n4/parenrightbigg\n.(4.13)These results have clear implications for the nearest–\nneighbor correlations in an extended system. By inspec-\ntion, systems composed of either type of dimer would\nfavor AF (spin) and AO interdimer bonds, to the extent\nallowed by frustration, and “linear” [type (7a)] bonds\nover “semi–linear” [type (7b)] bonds over “non–linear”\n[type (7c)] bond types in Fig. 7, to the extent allowed\nby geometry. Discussion of this type of state requires in\nprincipletheconsiderationofallpossibledimercoverings,\nbut will be restricted here to a small number of periodic\narrays which illustrate much of the essential physics of\nextended dimer systems within this model.\nWe begin by considering the periodic covering of\nFig. 8(a), a fully linear conformation (of ground–state\ndegeneracy 12) whose interdimer bond types (Table I)\nmaximize the possible number of bonds of type (7a).\nThe counterpoint shown in Fig. 8(b) consists of pairs\nof dimer bonds with alternating orientations in two of\nthe three lattice directions, and constitutes the simplest\nconfiguration minimizing (to zero) the number of type–\n(7a) interdimer bonds. The coverings in Figs. 8(c) and\n(d) have the same property. These configurations exem-\nplify a quite general result, that any dimer covering in\nwhich there are no linear configurations [type (7a)] of\nany pair of dimers will have 1/3 type–(7b) bonds, and\nthus the remaining 1/2 of the bonds must be of type\n(7c). The coverings shown in Figs. 8(a) and (b, c, d)\nrepresent the limiting cases on numbers of each type of\nbond, in that any random dimer covering will have val-\nues between these. Indeed, it is straightforward to argue\nthat, in changes of position of any set of dimers within\na covering, the creation of any two bonds of type (7b)\nwill destroy one of type (7a) and one of type (7c), and\nconversely.\nHavingestablishedthiseffectivesumrule, weturnnext\nto the energies of the dimer configurations. First, for\nboth types of dimer [(os/st) and (ss/ot)], all states with\nequal numbers of each bond type are degenerate, sub-\nject to equal solutions of the frustration problem. Next,\nif frustration is neglected, it is clear from Eqs. (4.2,4.4),\n(4.7,4.9), and (4.11,4.13), that the AF and AO energy\nvalues for the three bond types (obtained by substitut-\ning−1\n4for/vectorSi·/vectorSjand/vectorTi·/vectorTj) are respectively −1\n2J,−3\n8J\nand−1\n4J, which, when taken together with the sum rule,\nsuggest a very large degeneracy of dimer covering ener-\ngies.\nReturning to the question of frustration, a covering of\nminimal energy is one which both minimizes the number\nTABLE I: Occurrence probabilities for bonds of each type for\nfour simple periodic dimer coverings of the triangular latt ice.\nconfiguration dimer bond (7a) bond (7b) bond (7c)\nFig. 8(a)1\n61\n602\n3\nFig. 8(b)1\n601\n31\n2\nFig. 8(c)1\n601\n31\n2\nFig. 8(d)1\n601\n31\n219\n(a) (b)\n(c) (d)\nFIG. 8: (Color online) Periodic dimer coverings on the trian -\ngular lattice, each representative of a class of coverings: (a)\nlinear; (b) plaquette; (c) 12–site unit cell; (d) “zig–zag” .\nof FM or FO bonds, and ensures that they fall on bonds\nof type (7c); both criteria are equally important. For\nthe dimer covering (8a), with maximal aligned bonds,\nit is possible by using the spin (for (os/st) dimers) or\norbital (for (ss/ot) dimers) configuration represented by\nthe arrows in Fig. 9(a) to make the number of frustrated\n(FM/FO) interdimer bonds equal to 1/6 of the total.\nBearing in mind that the 1/6 of bonds covered by dimers\nare also FM/FO, and that at least 1/3 of bonds on the\ntriangular lattice must be frustrated for collinear spins,\nthis number is an absolute minimum. [Here we do not\nconsiderthe possibilityof non–collinearorderof the non–\nsinglet degree of freedom.] Further, for this configuration\none observes that all of the FM/FO bonds already fall on\nbondsoftype(7c), providinganoptimalcasewith energy\nEdim(0) =−J/parenleftbigg1\n6+1\n6·1\n2+1\n2·1\n4+1\n6·3\n16/parenrightbigg\n=−13\n32J (4.14)\natη= 0. This value constitutes a basic bound which\ndemonstrates that a simple, static dimer covering has\nlowerenergythan any long–range–orderedspin ororbital\nstate discussed in Sec. III in this limit ( α= 0) of the\nmodel.\nIt remains to establish the degeneracy of the ground–\nstate manifold of such coverings, and we provide only\na qualitative discussion using further examples. If al-\nternate four–site (dimer pair) clusters in Fig. 8(a) are\nrotated to give the covering of Fig. 8(b), the minimal\nfrustration is spoiled: by analogy with Fig. 9, it is easy\nto show that, if only 1/6 of the bonds are to be frus-(a) (b)\nFIG. 9: (Color online) Spin or orbital configurations (black\narrows) within (a) linear and (b) zig–zag orbital– or spin–\nsinglet dimer coverings of the triangular lattice. The numb er\nof frustrated interdimer bonds is reduced to 1/6 of the total ,\nand all are of type (7c). This figure emphasizes that for the\nspin–orbital model, dimer singlet formation does not exhau st\nthe available degrees of freedom.\ntrated, then they are of type (7b), and otherwise 1/3\nof the bonds are frustrated if all are to be of type (7c).\nOn the periodic 12–site cluster [Fig. 8(c)], one may place\nthree four–site clusters in each of the possible orienta-\ntions, which as above removes all bonds of type (7a) and\nmaximizes those of type (7b). Within this cluster it is\npossible to have only four frustrated interdimer bonds\nout of 18, while between the clusters there is again an\narrangement of the spin or orbital arrows ( cf.Fig. 9)\nwith only six FM or FO bonds out of 24, for a net total\nof 1/6 frustrated interdimer bonds, of which half are of\ntype (7b). The covering of Fig. 8(d) represents an exten-\nsion ofthe procedureof enlargingunit cells and removing\nfour–site plaquettes, which demonstrates that it remains\npossiblein the limit ofno type–(7a) bonds to reduce frus-\ntration to 1/6 of the bonds, and to bonds of type (7c)\n[Fig. 9(b)], whence the energy of the covering is again\nEdim(0) =−13\n32(4.14). Thus it is safe to conclude that,\nfor the static–dimer problem, the ground–state manifold\nforα= 0 consists of a significant number of degener-\nate coverings. We do not pursue these considerations\nfurther because of degeneracy lifting by dimer resonance\nprocesses, and because the energetic differences between\nstaticdimerconfigurationsarelikelytobedwarfedbythe\ncontributions from dimer resonance, the topic to which\nwe turn in Sec. V.\nB. Direct exchange model\nThe very strong preference for bond spin singlets (the\nfactor of 4 in Eq. (2.21)] suggests that dimer states will\nalso be competitive in this limit, even though only 1/6 of\nthe bonds may redeem an energy of −J. Following the\nconsiderations and terminology of the previous subsec-\ntion, we note (i) that /an}bracketle{t/vectorSi·/vectorSj/an}bracketri}ht= 0 on interdimer bonds\nand (ii) that in this case, interdimer bonds have energy\n−1\n4Jatη= 0 for types (7a) and (7b), and 0 for type\n(7c). Because any state with a maximal number (1/6) of20\ntype–(7a) bonds must have only bonds of type (7c) for\nthe other 2/3 [states (8a)], such a state is manifestly less\nfavorable at α= 1 than those of type (8b)–(8d), where\nthere are no aligned pairs of dimers. In this latter case,\nthe full calculation gives\nEdim(1) =−1\n144J(9r1+19r2+8r3),(4.15)\nandEdim(1) =−1\n4Jforη= 0. This energy does now ex-\nceed that availablefrom the formationof Heisenbergspin\nchains in one of the three lattice directions (Sec. IIIC),\nwhich gave the value E1DAF(1) =−0.231J.\nAt the level of these calculations, the manifold of de-\ngenerate states with this energy is very large, and its\ncounting is a problem which will not be undertaken here.\nWe will show in Sec. V that, precisely in this limit, no\ndimerresonanceprocessesoccurandthestaticdimercov-\nerings do already constitute a basis for the description of\nthe ground state. The question of fluctuations leading to\nthe selection of a particular linear combination of these\nstates which is of lowest energy, i.e.of a type of order–\nby–disorder mechanism, is addressed in Ref. 49.\nAt finite values of the Hund exchange, this type of\nstatewillcomeintocompetitionwiththesimpleavoided–\nblocking states which gain, with a FM spin state, an\nenergy\nEFM(1) =−1\n6Jr1, (4.16)\nas 2/3 of the bonds contribute with an energy of −1\n4Jr1.\nThe critical value of ηrequired to drive the transition\nfrom the low–spin dimerized state to the FM state is\nfound to be\nηc= 0.1589. (4.17)\nC. Mixed model\nBecause both of the endpoints, α= 0 andα= 1, favor\ndimerized states over states of long–ranged order, it is\nnatural to expect that a dimer state will provide a lower\nenergy also at α= 0.5. However, we remind the reader\nthat there are no intermediate dimer bases, and caution\nthere is no strong reason to expect one or other of the\nlimiting dimer states to be favored close to α= 0.5. By\ninspection, the energy of an α>0 state can be obtained\nby direct addition of the diagonal interdimer bond con-\ntributions in an (ss/ot) or (os/st) dimer state, which is\nestablished by pure off–diagonalhopping, because no site\noccupancies arise which allow mixed processes. For the\nsame reason, no interdimer terms impede a calculation\nof the energy of an α <1 state by summing the off–\ndiagonal interdimer bond contributions in a spin–singlet\ndimer state stabilized by purely diagonal processes. We\nwill not analyze the static dimer solutions for the inter-\nmediate regime in great detail, and provide only a crudeestimate of the α= 0.5 energy by averagingoverboth re-\nsults at the limits of their applicability. We will make no\nattempt here to exclude other forms of disordered state\natα= 0.5, and return to this question in Sec. V.\nFor each type of bond it is straightforward to compute\nthe energy gained from interdimer hopping processes of\nthe type not constituting the dimer state, and the re-\nsults are shown in Table II. The first four lines give the\nenergies per bond from diagonal hopping processes oc-\ncurring on the bonds of the different α= 0 dimer states,\nand conversely for the final two lines. It is clear that the\noccupations of type (7a) bonds preclude any hopping of\nthe opposite type. For α= 0 dimer configurations, the\ninterdimer diagonal hopping on (7b) bonds is always of\navoided–blocking type, while on (7c) bonds a blocking\ncan occur, and like the other terms is evaluated using\n/an}bracketle{t/vectorSi·/vectorSj/an}bracketri}ht. Forα= 1, off–diagonal hopping on the inter-\ndimer bonds is evaluated with /an}bracketle{t/vectorSi·/vectorSj/an}bracketri}ht= 0 between the\nspinsinglets: allprocesseson(7b)bondsarethoseforone\nactive orbital; complications arise only for (7c) bonds,\nwhere an interdimer bond between parallel dimers has\ntwo active AO orbitals, while one between dimers which\nare not parallel has two active FO orbitals.\nAtα= 0.5, the energy of an (os/st) or (ss/ot) dimer\nstate augmented by diagonal hopping processes is mini-\nmized by states (8a) and (8d): the interdimer bond con-\ntributions of all coverings in Fig. 8 are equal, despite the\ndifferenttypecounts, soonlythe α= 0energyisdecisive.\nAtη= 0,\nE(8a)\no(0.5) =−1\n2/parenleftbigg13\n32+2\n3·1\n4/parenrightbigg\nJ=−55\n192J,(4.18)\nE(8d)\no(0.5) =−1\n2/parenleftbigg13\n32+1\n3·1\n8+1\n2·1\n4/parenrightbigg\nJ\n=−55\n192J. (4.19)\nThe energy of a spin–singlet dimer state augmented by\noff–diagonal hopping is minimal in states (8b) and, cu-\nriously, (8a): although the latter has explicitly a worse\nground–state energy than the other states shown, the\neffect of the additional hopping is strong, not least be-\ncause all interdimer type–(8c) bonds arebetween parallel\nTABLE II: Additional interdimer bond energies at α= 0.5\ndue respectively to (i) diagonal hopping occurring in a stat e\n(designated by α= 0) stabilized by off–diagonal processes\nand (ii) off–diagonal hopping in a state ( α= 1) stabilized by\ndiagonal processes.\nbond (7a) (7b) (7c)\nα= 0, (os/st), AF 0 −1\n16(r1+r2)−1\n8(r1+r2)\nα= 0, (os/st), FM 0 −1\n8r1 −1\n4r1\nα= 0, (ss/ot), AO 0 −1\n32(3r1+r2)−1\n16(3r1+r2)\nα= 0, (ss/ot), FO 0 −1\n32(3r1+r2)−1\n16(3r1+r2)\nα= 1,/bardbldimers 0 −1\n16(3r1+r2)−1\n8(3r1+r2)\nα= 1, non– /bardbldimers 0 −1\n16(3r1+r2)−1\n12(2r2+r3)21\ndimers. Thus at η= 0,\nE(8a)\nd(0.5) =−1\n2/parenleftbigg5\n24+1\n2·2\n3/parenrightbigg\nJ=−13\n48J,\nE(8b)\nd(0.5) =−1\n2/parenleftbigg1\n4+1\n3·1\n4+1\n2·2\n3·1\n2+1\n2·1\n3·1\n4/parenrightbigg\nJ\n=−13\n48J. (4.20)\nDespite the fact that these are two completely differ-\nent expansions, it is worth noting that the two sets of\nnumbers are rather similar, which occurs because the\nsignificantly inferior energy of the α= 1 ground state\nis compensated by the significantly greater interdimer\nbond energies available from off–diagonal hopping pro-\ncesses. However, this result also implies that no special\ncombinations of diagonal and off–diagonal dimers can be\nexpected to yield additional interdimer energies beyond\nthis value.\nTakingthe covering(8a)asrepresentativeofthelowest\navailable energy, but bearing in mind that many other\nstates lie very close to this value, an average over the two\napproaches yields\nE(8a)\ndim(0.5) =−107\n384J (4.21)\natη= 0. This number is no longer lower than the value\nobtained in Sec. IIID for fully ordered states gaining en-\nergy from mixed processes, raising the possibility that\nnon–dimer–based phases may be competitive in the in-\ntermediate regime, where neither of the limiting types\nof dimer state alone is expected to be particularly suit-\nable. However, we will not investigate this question more\nsystematicallyhere, and cautionthat the approximations\nmade both in Sec. IIID and here make it difficult to draw\na definitive conclusion.\nD. Summary\nThe results of this section make it clear that static\ndimer states, while showing the same energetic trend,\nare considerably more favorable than any long–range–\nordered states (Sec. III) over most of the phase diagram.\nAs a function of α, the dimer energy increases mono-\ntonically from −13\n32Jto−1\n4J, and both end–point values\nalso lie below the results obtained for quasi–1D spin–\ndisordered states in Sec. III. We stress that the results of\nthis section are provisional in the sense that we have not\nperformed a systematic exploration of all possible dimer\ncoverings, but rather have focused on a small number\nof examples illustrative of the limiting cases in terms of\ninterdimer bond types. More importantly, we have con-\nsidered only static dimer coverings with effective inter-\ndimer interactions: the kinetic energy contributions due\nto dimer resonance processes for all values of α <1 are\nmissing in this type of calculation. For this reason, wehave also refrained from investigating higher–order pro-\ncesses, which may select particular dimer states from a\nmanifold of static coverings degenerate at the level of the\ncurrent considerations. Gaining some insight into the\nmagnitude and effects of resonance contributions is the\nsubject of the following section.\nV. EXACT DIAGONALIZATION\nA. Clusters and correlation functions\nIn this Section we present results obtained for small\nsystems by full exact diagonalization (ED). Because each\nsite has two spin and three orbital states, the dimension\nof the Hilbert space increases with cluster size as 6N,\nwhereNis the number of sites. As a consequence, we\nfocus here only on systems with N= 2, 3, and 4 sites:\nall three clusters can be considered as two–, three– or\nfour–site segments of an extended triangular lattice, con-\nnected with periodic boundary conditions. For the single\nbond and triangle this only alters the bond energies by\na factor of two, a rescaling not performed here, but for\nthe four–site system it is easy to see that the intercluster\nbonds ensure that the system connectivity is tetrahedral.\nWe will also compare some of the single–bond and tetra-\nhedron results with those for a four–site chain. Other ac-\ncessible cluster sizes ( N= 5 and 6) yield awkwardshapes\nwhich disguise the intrinsic system properties. Indeed we\nwill emphasize throughout this Section those features of\nour very small clusters which can be taken to be generic,\nand those which are shape–specific.\nGiven the clear tendency to dimerization illustrated in\nSecs. III and IV, it is to be expected that spin correla-\ntion lengths in all regimes of αare very small. To the\nextent that the behavior of the model for any parameter\nsetis drivenby localphysics, the clusterresultsshouldbe\nhighly instructivefor such trends asdimer formation, rel-\native roles of diagonal and off–diagonal hopping, dimer\nresonance processes, lifting of degeneracies both in the\norbital sector and between states of (os/st) and (ss/ot)\ndimers, and the importance of joint spin–orbital corre-\nlations. However, generic features of extended systems\nwhich cannot be accessed in small clusters are those con-\ncerning questions of high system degeneracy and subtle\nselection effects favoring specific states.\nWe will compute and discuss the cluster energies, de-\ngeneracies, site occupations, bond hopping probabili-\nties in diagonal and off–diagonal channels (discussed in\nSec. VC), and the spin, orbital, and spin–orbital (four–\noperator) correlation functions. All of these quantities\nwill be calculated for representative values of αandη\ncoveringthefullphasediagram,andeachcontainsimpor-\ntant information of direct relevance to the local physics\nproperties listed in the previous paragraph. Although\nthe systems we study are perforce rather small, we will\nshowthat onemayrecognizein them anumber ofgeneral\ntrends valid also in the thermodynamic limit.22\nWe introduce here the three correlation functions,\nwhich for a bond /an}bracketle{tij/an}bracketri}htoriented along axis γare given\nrespectively by\nSij≡1\nd/summationdisplay\nn/angbracketleftbig\nn/vextendsingle/vextendsingle/vectorSi·/vectorSj/vextendsingle/vextendsinglen/angbracketrightbig\n, (5.1)\nTij≡1\nd/summationdisplay\nn/angbracketleftbig\nn/vextendsingle/vextendsingle/vectorTiγ·/vectorTjγ/vextendsingle/vextendsinglen/angbracketrightbig\n, (5.2)\nCij≡1\nd/summationdisplay\nn/angbracketleftbig\nn/vextendsingle/vextendsingle(/vectorSi·/vectorSj)(/vectorTiγ·/vectorTjγ)/vextendsingle/vextendsinglen/angbracketrightbig\n−1\nd2/summationdisplay\nn/angbracketleftbig\nn/vextendsingle/vextendsingle/vectorSi·/vectorSj/vextendsingle/vextendsinglen/angbracketrightbig/summationdisplay\nm/angbracketleftbig\nm/vextendsingle/vextendsingle/vectorTiγ·/vectorTjγ/vextendsingle/vextendsinglem/angbracketrightbig\n,(5.3)\nwheredis the degeneracy of the ground state. The\ndefinitions of the spin ( Sij) and orbital ( Tij) correla-\ntion functions are standard, and we have included ex-\nplicitly all of the quantum states {|n/an}bracketri}ht}which belong to\nthe ground–state manifold. The correlation function Cij\n(5.3) contains information about spin–orbital entangle-\nment, as defined in Sec. I: it represents the difference\nbetween the average over the complete spin–orbital op-\nerators and the product of the averagesover the spin and\norbital parts taken separately. It is formulated in such\na way that Cij= 0 means the mean–field decoupling of\nspin and orbital operators on every bonds is exact, and\nbothsubsystemsmaybetreatedindependentlyfromeach\nother. Such exact factorizability is found9in the high–\nspin states at large η; its breaking, and hence the need\nto handle coupled spin and orbital correlations in a sig-\nnificantly more sophisticated manner, is what is meant\nby “entanglement” in this context.\nB. Single bond\nWe consider first a single bond oriented along the c–\naxis (Fig. 10). In the superexchange limit the active or-\nbitals areaandb, while for direct exchange only the cor-\nbitalscontributeinEq.(2.7). AsdiscussedinSec.IVA, a\nsingle bond gives energy −Jin the superexchange model\n(α= 0) [Fig. 10(a)], where the ground state has degener-\nacyd= 6 atη= 0, from the two triply degenerate wave\nfunctions (ss/ot) and (os/st). At finite η, the latter is\nfavored as it permits a greater energy gain from excita-\ntions to the lowest triplet state in the d2configuration\n[Eqs. (2.15) and (2.16)].\nAlthough orbital fluctuations which appear in the\nmixed exchange terms in Eq. (2.22) may in principle\ncontribute at α >0, one finds that the wave function\nremains precisely that for α= 0,i.e.(ss/ot) degenerate\nwith (os/st), all the way to α= 0.5. Thus for the param-\neter choice specified in Sec. II, the ground–state energy\nincreases to a maximum of E0=−0.5Jhere [Fig. 10(a)].\nThe degeneracy d= 6 is retained throughout the regime\nα<0.5, and only at α= 0.5 do several additional states\njoin the manifold, causing the degeneracy to increase to\nd= 15. For the entire regime α∈(0.5,1], the ground0 0.2 0.4 0.6 0.8 1\nα−0.8−0.6−0.4−0.20.0Sij , Tij , Cij0 0.2 0.4 0.6 0.8 1−1.0−0.8−0.6−0.4−0.20.0En/J\n{a,b} orbitals c orbitals(ss/ot)\n(st/os)(ss/cc)(a)\n(b)\nresonating static13\n18\n86\n6\nFIG. 10: (Color online) Evolution of the properties of a sing le\nbondγ≡cas a function of αatη= 0: (a) energy spectrum\n(solid lines) with degeneracies as shown; (b) spin ( Sij, filled\ncircles), orbital ( Tij, empty circles), and spin–orbital ( Cij,×)\ncorrelations: Sij=Tij=Cij=−0.25 forα <0.5, while\nTij=Cij= 0 for α >0.5. The ground–state energy E0is\n−Jfor both the superexchange ( α= 0) and direct–exchange\n(α= 1) limits, and its increase between these is a result of the\nscaling convention. The transition between the two regimes\noccurs by a level crossing at α= 0.5. For α <0.5, the\ntwo types of dimer wave function [(ss/ot) and (os/st)] are\ndegenerate ( d= 6) for resonating orbital configurations {ab},\nwhile at α >0.5, the nondegenerate spin singlet is supported\nby occupation of corbitals at both sites [(ss/cc)].\nstate is a static orbitalconfigurationwith corbitals occu-\npied at both sites to support the spin singlet, and d= 1.\nThe evolution of the spectrum with αdemonstrates not\nonly that superexchange and direct exchange are physi-\ncally distinct, unable to contribute at the same time, but\nthat the two limiting wave functions are extremely ro-\nbust, their stability quenching all mixed fluctuations for\na single bond. In this situation it is not the ground–state\nenergy but the higher first excitation energy which re-\nveals the additional quantum mechanical degrees of free-\ndom active at α= 0 compared to α= 1 [Fig. 10(a)].\nThe spin, orbital, and composite spin–orbital corre-\nlation functions defined in Eqs. (5.1)–(5.3) give more\ninsight into the nature of the single–bond correlations.\nThe degeneracy of wavefunctions (ss/ot) and (os/st) for\n0≤α <0.5 leads to equal spin and orbital correlation\nfunctions, as shown in Fig. 10(b), and averaging over the\ndifferent states gives Sij=Tij=−1\n4. As a singlet for\nonequantityismatchedbyatripletfortheother, the two23\nsectors are strongly correlated, and indeed Cij=−1\n4, in-\ndicating an entangled ground state. However, a consider-\nablymoredetailedanalysisispossible. Eachofthe sixin-\ndividual states {|n/an}bracketri}ht}within the ground manifold has the\nexpectation value/angbracketleftbig\nn/vextendsingle/vextendsingle(/vectorSi·/vectorSj)(/vectorTiγ·/vectorTjγ)/vextendsingle/vextendsinglen/angbracketrightbig\n=−3\n16, which\nwe assert is the minimum possible when the spin and\npseudospin are the quantum numbers of only two elec-\ntrons. It is clear that if the operator in Cijis evaluated\nfor any one of these states alone, the result is zero. En-\ntanglement arisesmathematically because of the product\nofaveragesinthesecondtermofEq.(5.3), andphysically\nbecause the ground state is a resonant superposition of\na number of degenerate states. We emphasize that the\nresulting value, Cij=−1\n4, is the minimum obtainable\nin this type of model, reflecting the maximum possible\nentanglement. We will show in Sec. VE that this value\nis also reproduced for the Hamiltonian of Eq. (2.7) on\na linear four–site cluster, whose geometry ensures that\nthe system is at the SU(4) point of the 1D SU(2) ⊗SU(2)\nmodel.9\nBy contrast, for α >0.5 those states favored by su-\nperexchangebecomeexcited, andthespin–singletground\nstate hasSij=−3\n4. The orbital configuration is charac-\nterized by /an}bracketle{tnicnjc/an}bracketri}ht= 1, a rigid order which quenches all\norbital fluctuations (indeed, the orbital pseudospin vari-\nables/vectorTiγare zero). Thus the spin and orbital parts are\ntrivially decoupled, giving Cij= 0. Finally, at the tran-\nsition point α= 0.5, averaging over all 15 degenerate\nstates yields Sij=−0.15,Tij=−0.10, andCij=−0.09.\nIn summary, the very strong tendency to dimer forma-\ntion in the two limits α= 0 andα= 1 precludes any\ncontribution from mixed terms on a single bond, lead-\ning to a very simple interpretation of the ground–state\nproperties for all parameters.\nC. Triangular cluster\nWe turn next to the triangle, which has one bond in\neach of the lattice directions a,b, andc. Unlike the case\nof the single bond, here the spin–orbital interactions are\nstrongly frustrated, in a manner deeper than and quali-\ntatively different from the Heisenberg spin Hamiltonian.\nNot only can interactions on all three bonds not be sat-\nisfied at the same time, but also the actual form of these\ninteractions changes as a function of the occupied or-\nbitals. The triangle is sufficient to prove (numerically\nand analytically) the inequivalence in general of the orig-\ninal model and the model after local transformation, for\nfrustration reasons discussed in Sec. IIB.\nWebeginwiththeobservationthattheresultstofollow\nare interpreted most directly in terms of resonant dimer\nstates on the triangle. This fact is potentially surprising,\ngiven that the number of sites is odd and dimer forma-\ntion must always exclude one of them, but emphasizes\nthe strong tendencies to dimer formation in all param-\neter regimes of the model. For their interpretation we\nuse a VB ansatz where it is assumed that one bond isoccupied by an optimal dimer state, minimizing its en-\nergy, and the final state of the system is determined by\nthe contributions of the other two bonds. This ansatz is\nperforce only static, and breaks the symmetry at a crude\nlevel, but enables one to understand clearly the effects of\nthe resonance processes captured by the numerical stud-\nies in restoring symmetries and lowering the total energy.\nConsidering first the VB ansatz for the superexchange\nmodel, the energy −Jmay be gained only on a single\nbond, in one of two ways. For the bond spin state to\nbe a singlet ( S= 0, (ss/ot) wave function), two differ-\nent active orbitals are occupied at both sites in one of\nthe orbital triplet states. The other two bonds lower the\ntotal energy when the third site has an electron of the\nthird orbital color, each gaining an energy of −0.25Jdue\nto the orbital interactions in Eq. (2.8). The energy of\nthe triangle is then EVB(0) =−0.5Jper bond, and the\ncluster has a low–spin ( S=1\n2) ground state with degen-\neracyd= 6 from the combination of the orbital triplet\nand the spin state of the third electron. We stress that\nthe location ( a,b, orcbond) of the spin singlet does not\ncontribute to the degeneracybecause the three VB states\nare mixed within the ground state by the contributing\noff–dimer hopping processes. The same considerations\napplied to an (os/st) dimer on one of the bonds of the\ntriangle shows that there is no color and spin state of\nthe third electron which allows both non–dimer bonds\nto gain the energy −0.25Jsimultaneously, so the cluster\nhas a higher energy of −5\n12Jper bond. Thus the VB\nansatz illustrates a lifting of the degeneracy between the\ntwo types singlet state, the physical origin of which lies\nin the permitted off–dimerfluctuation processes,and this\nwill be borne out in the calculations below. However, the\nnet spin state of the cluster has little effect on the esti-\nmated energy of the (os/st) case, and its high–spin ver-\nsion (S=3\n2) will be a strong candidate for the ground\nstate at higher values of η. In the direct–exchange limit\n(α= 1), the VB ansatz for spin singlets again returns an\nenergyEVB(1) =−5\n12J, alsobecauseonlyonenon–dimer\nbond cancontribute. Herethe off–dimerprocessesarere-\nstricted to the third electron, which has arbitrary color\nand spin, and cannot mix the three VB states, whence\nthe degeneracy is d= 12.\nWith this framework in mind, we turn to a description\nof the numerical calculations at all values of α, beginning\nwiththemostimportantresults: at α= 0thedegeneracy\nisd= 6, and hence VB resonance is confirmed, yielding\nan energy very much lower than the static estimate, at\nE0=−0.75Jper bond [Fig. 11(a)]. Thus strong orbital\ndynamics and positional resonance effects operate in the\nground–state manifold. These break the (ss/ot)/(os/st)\nsymmetry, but act to restore other symmetries broken in\nthe VB ansatz. At α= 1, the energy and degeneracy\nfrom the VB ansatz are exact, showing that the orbital\nsector is classical andintroduces no resonance effects.\nFigure 11(a) shows the complete spectrum of the tri-\nangular cluster for all ratios of superexchange to di-\nrect exchange, and in the absence of Hund coupling.24\n0.0 0.2 0.4 0.6 0.8 1.0−0.80−0.60−0.40−0.200.000.20En/J\n0.0 0.2 0.4 0.6 0.8 1.0−0.3−0.2−0.10.00.1Sij , Tij , Cij\n0.0 0.2 0.4 0.6 0.8 1.0α0.00.20.40.6n1γ\n(ss/ot) (ss/cc) orbitalsfluctuating{a,b} orbitals c orbitals(b)(a)\nresonating static488\n(c)\nFIG. 11: (Color online) (a) Energy spectrum per bond for\na triangular cluster as a function of αforη= 0. Ground–\nstate degeneracies are as indicated, with d= 6 at α= 0\nandd= 12 atα= 1. The arrows mark two transitions in the\nnature of the (low–spin) ground state, which are further cha r-\nacterized in panels (b) and (c). (b) Spin ( Sij, filled circles),\norbital (Tij, empty circles), and spin–orbital ( Cij,×) correla-\ntion functions on the cbond. (c) Average electron densities in\nthet2gorbitals at site 1 [Figs. 2(b,c)], showing n1b(solid line)\nandn1a=n1c(dashed). The orbital labels are shown for a c\nbond. All three panels show clearly a superexchange regime\nforα <0.32, a direct–exchange regime for α >0.69, and an\nintermediate regime (0 .32< α <0.69). A full description is\npresented in the text.\nFrustration of spin–orbital interactions is manifest in\nrather dense energy spectra away from the symmetric\npoints, and in a ground–state energy per bond signifi-\ncantly higher than the minimal value −J. Atα= 0\nthe spectrum is rather broad, with a significant number\nof states of relatively low degeneracy due to the strong\nfluctuations and consequent mixing of VB states in this\nregime. However, even in this case the ground state is\nwell separated from the first excited state. As empha-\nsized above, the ground–state energy, E0(0) =−0.75J,\nis quite remarkable, demonstrating a very strong energy\ngain from dimer resonance processes. By contrast, thevalueE0(1) =−5\n12Jper bond found at α= 1 is exactly\nequal to that deduced from the VB ansatz, demonstrat-\ning that this wave function is exact. Here the excited\nstates have high degeneracies, mostly of orbital origin,\nand thus the spectrum shows wide gaps between these\nmanifolds of states; this effect is more clearly visible in\nFig. 12(c). The degeneracies shown in Fig. 11(a) are dis-\ncussed below. In the intermediate regime, many of the\ndegeneracies at the end–points are lifted, leading to a\nvery dense spectrum. The two transitions at α= 0.32\nandα= 0.69 appear as clear level–crossings: the inter-\nmediate ground state is a highly excited state in both\nof the limits ( α= 0,1), reinforcing the physical picture\nof a very different type of wave function dominated by\norbital fluctuations and, as we discuss next, with little\novert dimer character.\nThecorrelationfunctionsforanyonebondofthetrian-\ngle are shown in Fig. 11(a). That Sijis constant for all α\ncan be understood in the dimer ansatz by averaging over\nthe three configurations with one (ss/ot) or (ss) bond\nand one ’decoupled’ spin on the third site, which gives\nSij=−1\n4everywhere. The orbital and spin–orbital cor-\nrelation functions show a continuous evolution accompa-\nnied by discontinuous changes at two transitions, where\nthe nature of the ground state is altered. The orbital\ncorrelation function Tij=−1\n12atα= 0 may be under-\nstood as an average over the orbital triplet (+1\n4) and the\ntwo non–dimer bonds (each −1\n4). Whenαincreases, this\nvalue is weakened by orbital fluctuations, and undergoes\natransitionat α= 0.32toaregimewhereorbitalfluctua-\ntions dominate, and Tijis close to zero. Above α= 0.69,\nTijbecomes positive, and approaches +1\n12asα→1,\nindicating that the wave function changes to the static–\ndimer limit. While /vectorTicvanishes on the cbond here, the\ncluster average has a finite value due to the contribution\nTij=1\n4from the active non–singlet bond.\nThe spin–orbital correlation function Cijalso marks\nclearly the three different regimes of α. Whenα<0.32,\nCijhasasignificantnegativevalue[Fig.11(b)]whosepri-\nmary contributions are given by the four–operator com-\nponent/an}bracketle{t(/vectorSi·/vectorSj)(/vectorTiγ·/vectorTjγ)/an}bracketri}ht. Bycontrast, Cijis closetozero\nin the intermediate regime, increasing again to positive\nvalues forα>0.69. For all α>0.32,Cijcan be shown\nto be dominated by the term −SijTijin Eq. (5.3), while\nthe four–operator contribution is small, and vanishes as\nα→1. Thus entanglement, defined as the lack of fac-\ntorizability of the spin and orbital sectors, can be finite\neven for vanishing joint spin–orbital dynamics.\nFurther valuable information is contained in the or-\nbital occupancies at individual sites [Fig. 11(b)], which\nshow clearly the three different regimes. Although there\nis always on average one electron of each orbital color\non the cluster, these are not equally distributed, as each\nsite participates only in two bonds and the symmetry is\nbroken. A representative site, labelled 1 in Figs. 2(b,c)]\nhas onlyaandcbonds, and hence the electron density\nin theborbital is expected to differ from the other two.\nThe values nb= 2/3 andna=nc= 1/6 found in the25\nregimeα<0.32 is understood readily as following from\na 1/3 average occupation of ( ab) and (bc) orbital triplet\nstates on the candabonds, respectively, and of an ( ac)\norbitaltriplet state on the bbond, which ensuresthat the\nelectron at site 1 is in orbital b[Fig. 2(b)]. By contrast,\nin the regime α >0.69, only the two static orbital con-\nfigurations ( cc) and (bb) on thecandbbonds contribute,\nandna=nc=1\n2, whilenb= 0; when the system is in the\nthird possible spin–singlet state, with a ( bb) orbital state\non thebbond, the third electron is either aorc. Between\nthese two regimes(0 .32<α<0.69) is an extended phase\nwith equal averageoccupancy of all three orbitals at each\nsite, a potentially surprising result given the broken site\nsymmetry of the cluster. While this may be interpreted\nas a restoration of the symmetry of the orbital sector by\nstrong orbital fluctuations, including those due to terms\ninHm(2.22), it does not imply a higher symmetry of the\nstrongly frustrated interactions at α= 0.5.\nThespectraasafunctionofHundcoupling ηareshown\nin Fig. 12 for the α= 0 andα= 1 limits, and at α= 0.5\nto represent the intermediate regime. The lifting of de-\ngeneracies as a function of ηis a generic feature. States\nof higher spin are identifiable by their stronger depen-\ndence onη, and in all three panels a transition is vis-\nible from a low–spin to a high–spin state. At α= 0\n[Fig. 12(a)], the large low– ηgap to the next excited state\nresultsinthetransitionoccurringattheratherhighvalue\nofηc= 0.158. This can be taken as a further indication\nof the exceptional stability of the resonance–stabilized\nground state in the low–spin sector. The degeneracy\nd= 12 of the high–spin state is discussed below.\nThe transition to the high–spin state at α= 1 also\noccurs at a high critical value, ηc= 0.169 [Fig. 12(a)],\ndue in this case quite simply to the lack of competition\nfor the strong singlet states on individual bonds. Only\nin the intermediate regime, 0 .32< α <0.69, where we\nhave shown already that the orbital state is quite differ-\nent from that in either limit [Fig. 11], is the transition\nto the high–spin state much more sensitive to η. The or-\nbitalfluctuations inthis phaseoccurboth in the low–spin\nand the high–spin channel, making these very similar in\nenergy, and the transition occurs for α= 0.5 at only\nηc= 0.033 [Fig. 12(b)]. As expected from the α= 0\nlimit, where fluctuations are also strong, the characteris-\ntic features of this energy spectrum are low degeneracy\nand a semicontinuous nature. The location of the high–\nspin transition as a function of αmay be used to draw a\nphase diagram for the triangular cluster, which has the\nrather symmetric form shown in Fig. 13.\nYet more information complementary to that in the\nenergy spectra and correlation functions can be obtained\nby considering the average “occupation correlations” for\na bond/an}bracketle{tij/an}bracketri}ht /bardblγ,\nP=/an}bracketle{tniγnjγ/an}bracketri}ht, (5.4)\nQ=/an}bracketle{tniγ(1−njγ)/an}bracketri}ht+/an}bracketle{t(1−niγ)njγ/an}bracketri}ht,(5.5)\nR=/an}bracketle{t(1−niγ)(1−njγ)/an}bracketri}ht. (5.6)\nThese probabilities ( P+Q+R= 1) reflect directly the0 0.05 0.1 0.15 0.2−1.0−0.6−0.20.2En/J0 0.05 0.1 0.15 0.2−1.2−0.8−0.40.0En/J\n0 0.05 0.1 0.15 0.2\nη−0.6−0.4−0.20.0En/J12\n6\n128(a)\n(b)\n(c)44\nFIG.12: (Color online) Energyspectrafor atriangular clus ter\nas a function of Hund exchange η. Energies are quoted per\nbond, and shown for: (a) α= 0, (b) α= 0.5, and (c) α= 1.\nThe arrows indicate transitions at ηcfrom the low–spin ( S=\n1/2) to the high–spin ( S= 3/2) ground state. The numbers\nin all panels give degeneracies for the two lowest states for\nη < ηcandη > ηc, respectively.\nnature of the resonance processes contributing to the en-\nergy of the cluster states, in that they show the relative\nimportance of diagonal and off–diagonal hopping in the\nground states, and the evolution of these contributions\nwithαandη. We do not present these quantities in de-\ntail here, but only summarize the overall picture of the\nground state whose understanding they help elucidate.\nFor this summary we return to the VB framework,\nwhich accounts for many of the basic properties illus-\ntrated in the numerical results presented above. Con-\nsidering first the low–spin states ( η= 0), atα= 0 the\nground state is given by one (ss/ot) dimer resonating\naround the three bonds of the cluster; the third site has\nthe third color, its hopping gives a largevalue of Q= 1/3\n(R= 2/3 from the pure superexchange channel) and its26\n0 0.2 0.4 0.6 0.8 1\nα0.000.040.080.120.160.20η\nS=1/2S=3/2\nFIG. 13: (Color online) Phase diagram of the triangular clus -\nter in the plane ( α,η). The spin states below and above the\ntransition line ηc(α) are respectively spin doublet ( S= 1/2)\nand spin quartet ( S= 3/2).\nspin an addition twofold degeneracy ( d= 3×2 = 6);\nthe orbital occupation of the (ss/ot) dimer is responsible\nfor the net 1/6:1/6:2/3 occupation distribution. When\nα >0 the state remains essentially one with a resonat-\ning spin singlet, large Qand dominant R, but the orbital\ntriplet degeneracy is lifted to 2+1 and the ground–state\ndegeneracy to d= 2×2. All quantities, including P,Q,\nandR, undergo discontinuous changes at α≃0.32, and\nin this regime there is no longer strong evidence for an\ninterpretation in terms of resonating spin singlets: large\nQ≃2/3 and the equal site occupations suggest the dom-\ninance of mixed hopping processes which are not con-\nsistent with either mechanism of singlet formation. The\nretention of fourfold degeneracy across this transition is\nlargely accidental, and stems from twofold spin and or-\nbital contributions. Only for α >0.69 is a spin–singlet\ndescription once again valid: here Pbecomes significant,\nastheresonatingsingletisstabilizedbydiagonalhopping\nwhere the orbital has the bond color. The third site now\nhas one of two possible colors, its hopping keeps Qlarge,\nand its spin yields another twofold degeneracy, as do the\norbital states, whence the net degeneracy is d= 2×2×2.\nOnly atα= 1 does the spin singlet become static, while\nthe third site still has either of the other colors, yield-\ning the symmetric result P= 1/9,Q=R= 4/9, and\ndegeneracy d= 12.\nA similar description is possible in the high–spin states\natη >ηc. Atα= 0 the (os/st) dimer is rendered static\nby the fact that hopping to the third site is now excluded\nif it has the third color, and so instead this site takes one\nof the singlet colors, a twofold degree of freedom which,\nhowever, does not allow singlet motion; as a consequence\nthe orbital occupation is uniform (1/3:1/3:1/3), the hop-\npingprocessesincludecontributionsinthediagonalchan-\nnel (P= 1/6,Q= 1/3,R= 1/2) and the degeneracy\nisd= 3×4 = 12. For α >0 the orbital singlet mayagain resonate, but the third site retains one of the sin-\nglet colors, orbital degeneracy is broken and d= 4. Once\nagain strong mixed processes dominate the intermedi-\nate regime, in which the spin state is not an important\ndetermining factor. Above α= 0.69 the critical value\nηcrequired to overcome spin singlet formation becomes\nlargeagain,andthehigh–spinstateisonewhereavoided–\nblocking processes (large Q) dominate, while broken or-\nbital degeneracy keeps d= 4. Finally, at α= 1 one\nobtains a pure avoided–blocking state with orbital con-\nfigurations acborcbafor the sites (1 ,2,3) of Fig. 2(c),\nand consequent degeneracy d= 4×2 = 8. Thus it is\nclear that the high– ηregion is also one yielding interest-\ning orbital models with nontrivial ground states, some\nincluding orbital singlet states.\nD. Tetrahedral cluster\nAsinthecaseofthetriangularlattice,interpretationof\nthenumericalresultsforthetetrahedralcluster(four–site\nplaquette of the triangular lattice) is aided by considera-\ntion of the VB ansatz in the two limits of superexchange\nand direct–exchange interactions. The tetrahedral clus-\nter can accommodate exactly two dimers, with all inter-\ndimer bonds of type (7c), and may thus be expected to\nfavor dimer–based states by simple geometry. However,\nbecause the considerations and comparisons of this sub-\nsectionaregivenonlyforthis singleclustertype, anybias\nof this sort would not invalidate the results and trends\ndiscussed here.\nBecause of the different forms and symmetries of the\nspin and orbital sectors, there is no possibility of elemen-\ntary spin–orbital operators, or of a ground–state wave\nfunction which is a net singlet of a higher symmetry\ngroup. The state with two orbital singlets on one pair\nof bonds, two spin singlets on a second pair and pure\ninterdimer bonds on the third pair does exist, but is not\ncompetitive: the energy cost forremovingthe orbital sin-\nglets from the spin state maximizing their energy is by\nno means compensated by the energy gain from having\ntwo spin singlet bonds in an orbital state which also does\nnot maximize their energy. This result may be taken as\na further indication for the stability of dimers only in\nthe forms (os/st) or (ss/ot) in this model, and states of\nsharedorbitalandspinsingletsarenotconsideredfurther\nhere. We return to this point in the following subsection,\nin the context of the four–site chain.\nWe discuss only the energies of the VB wave functions\natη= 0. The minimal values obtainable for /an}bracketle{t/vectorSi·/vectorSj/an}bracketri}htand\n/an}bracketle{t/vectorTiγ·/vectorTjγ/an}bracketri}htonthe interdimerbondsis −1/4,corresponding\nto the AF/AO order. Thus at α= 0 the energy per bond\nis\nEos/st(0) =Ess/ot(0) =−1\n2J, (5.7)\nwith the degeneracy of the (ss/ot) and (os/st) wave func-\ntionsrestoredasforthe singlebond. In the limit ofdirect27\nexchange, the VB wave function consists of spin singlets\nwith twoactiveorbitalsofthe bond. Thegeometryofthe\ncluster precludes these orbitals from being active on any\nof the interdimer bonds, as a result of which the energy\nper bond at η= 0 is\nE(1) =−1\n3J, (5.8)\nand the ground state has degeneracy d= 3.\nThe most important results for the tetrahedron, which\nwe discuss in detail in the remainder of the subsection,\nare the following. At α= 0, the exact ground state\nenergy isE0=−0.5833J: while not as large as in the\ncase of the triangle (Sec. VC), the resonance energy\ncontribution is very significant also for an even number\nof cluster sites. The degeneracy of the numerical ground\nstate,d= 6, has its origin in only one of the (ss/ot) or\n(os/st) wavefunctions (below), demonstratingagainthat\nthereisnosensein whichthe quantumfluctuationsinthe\nspinandorbitalsectorsaresymmetrical,andthattheVB\nansatz is capturing the essence of the local physics only\nat a very crude level. At α= 1, as also for the triangular\ncluster, the numerical results confirm not only the energy\ngiven by the VB ansatz but every detail (degeneracies,\noccupations, correlations) of this state.\nWe begin the systematic presentation of results by dis-\ncussing the energy spectra at η= 0 [Fig. 14(a)]. As soon\nas the degeneracies of the superexchange limit ( α= 0)\nare broken, the spectrum becomes very dense, and re-\nmains so across almost the complete phase diagram until\na level–crossing at αc= 0.92. The ground–state energy\nfor all intermediate values of αinterpolates smoothly to-\nwards the transition, showing an initial decrease not ob-\nserved in the triangle: for the tetrahedron, mixed hop-\nping terms make a significant contribution, leading to an\noverall energy minimum around α= 0.15. The domi-\nnance of these terms is indicated by both the extremely\nhigh value of αcand the steepness of the low– αcurve\nwhere the transition to the static VB phase is finally\nreached.\nThe bond correlation functions shown in Fig. 14(b)\nillustrate the effects of corrections to the VB ansatz.\nThe spin correlations always have the constant value\nSij=−1\n4, which is the most important indication of the\nbreakingofsymmetrybetween(ss/ot)and(os/st)sectors\nat lowα: this value is an average over the spin–singlet\nresult−3/4 (on two bonds) and four bonds with value 0,\nand thus it is clear that (ss/ot) dimers afford more res-\nonance energy. However, the proximity of (os/st) states\nsuggests that a low value of ηc, the critical Hund cou-\npling for the transition to the high–spin state, is to be\nexpected (below).\nThe orbital correlations average to zero at α= 0, a\nnon–trivial result whose origin lies in the breaking of\nnine–fold degeneracy within the orbital sector, and re-\nmain close to this value until the transition at αc. It is\nworth noting here that Tij= 0 implies a higher frustra-\ntion in the orbital sector than would be obtained in the0.0 0.2 0.4 0.6 0.8 1.0\nα−0.3−0.2−0.10.00.1Sij , Tij , Cij0.0 0.2 0.4 0.6 0.8 1.0−0.6−0.5−0.4−0.3−0.2En/J\n(a)\n(b)2\n1\nFIG. 14: (Color online) (a) Energy spectrum per bond for\na tetrahedral cluster as a function of αforη= 0. Ground–\nstate degeneracies are as indicated, with d= 6 at α= 0\nandd= 150 at α= 1. The arrow marks a transition in the\nnature of the (low–spin) ground state. (b) Spin ( Sij, filled\ncircles), orbital ( Tij, empty circles), and spin–orbital ( Cij,×)\ncorrelation functions on the cbond of the tetrahedral cluster\nas functions of αforη= 0.\nspin sector for an (os/st) state ( Sij=−1\n12), which is due\nto the complex direction–dependence of the orbital de-\ngrees of freedom. This phase is maintained across much\nof the phase diagram, with only small changes to the cor-\nrelation functions, the negative value of Tijreflecting an\neasing of orbital frustration. The lack of a phase transi-\ntion throughout the region in which mixed processes are\nalso important suggests that a dimer–based schematic\npicture of the ground state remains appropriate for the\nfour–site system, with only quantitative evolution as a\nfunction of αuntilαc= 0.92. Atα= 1, the result\nTij=−1\n6is the consequence of c–orbital operators on\nthe interdimer aandbbonds.\nSignificant spin–orbital correlations, Cij≃ −0.1 at\nα= 0 [Fig. 14(b)], are found to be due exclusively to\nthe four–operator term at low α. While these negative\ncontributions drop steadily through most of the regime\nα < α c, signifying a gradual decoupling of orbitals and\nspins as the static limit ( α= 1) is approached, near αc\nthe negative value of Cijis again enhanced by the con-\ntribution −SijTijdue to the interdimer bonds. Thus,\nas for the triangle (Sec. VC), the entanglement is finite,28\n0 0.05 0.1 0.15 0.2−1.0−0.8−0.6−0.4−0.20.0En/J\n0 0.05 0.1 0.15 0.2\nη−0.4−0.3−0.2−0.10.0En/J0 0.05 0.1 0.15 0.2−1.0−0.8−0.6−0.4−0.20.00.2En/J(a)\n(b)\n(c)\nFIG. 15: (Color online) Energy spectra for a tetrahedral clu s-\nter as a function of Hundexchange η. Energies are quoted per\nbond, and shown for: (a) α= 0, (b) α= 0.5, and (c) α= 1.\nThe arrows indicate transitions from the low–spin ( S= 0) to\nthe high–spin ( S= 2) ground state.\ncomplete factorization is not possible, and a finite value\nCij=−1\n24is found even at α= 1. We note here that on\nthe tetrahedron there is little information in the orbital\noccupations, which are constant ( nγ=1\n3) over the entire\nphase diagram, demonstrating only the symmetry of this\ncluster geometry, and are therefore not shown.\nThespectraasafunctionofHundcoupling ηareshown\nfor the three parameter choices α= 0, 0.5, and 1 in\nFig. 15. Once again, the spectra become very dense away\nfromη= 0. Atα= 0 [Fig. 15(a)] high–spin states are\nfound also in the low–energy sector, as a consequence of\nthe near–degeneracy of (ss/ot) and (os/st) states, and\nthe high–spin transition occurs at a very low value of ηc\n[Fig. 15(a)]. The direct–exchange limit is both qualita-\ntively and quantitatively different, because the quantum\nfluctuations and the corresponding energy gains are lim-\nited to the spin sector, making the low–spin states con-0 0.2 0.4 0.6 0.8 1\nα0.000.040.080.120.160.20η\nS=0S=2\nFIG.16: (Color online)Phase diagram of thetetrahedral clu s-\nter in the plane ( α,η). As for the triangular cluster, the spin\nstates below and above the line ηc(α) are respectively singlet\n(S= 0) and quintet ( S= 2), with no intermediate triplet\nphase.\nsiderably more stable and giving ηc= 0.175 [Fig. 15(c)].\nThe spin excitation gap decreasesgraduallywith increas-\ningη, but until just below ηc, for all values of α, the spin\nexcitation is to S= 1 states. However, these triplet\nstates are never the ground state in the entire regime\nofη, a single transition always occurring directly into\nanS= 2 state. In the intermediate regime represented\nbyα= 0.5, the energy spectrum is so dense that indi-\nvidual states are difficult to follow (a more systematic\nanalysis of the spectra in different subspaces of Szis not\npresented here). The high–spin transition occurs at the\nrelatively high value ηc= 0.136, due mainly to the large\nenergygains in the low–spinsector from mixed exchange.\nFurther evidence for the importance of the orbital exci-\ntations in Hm(2.22) can be found in the broadening of\nthe spectrum which leads to the occurrence of quantum\nstates with weakly positive energies: for both superex-\nchange and direct–exchange processes, the Hamiltonians\nare constructed as products of projection operators with\nnegative coefficients, so positive energies are excluded.\nThe low– to high–spin transition points at all values of\nαcan be collected to give the full phase diagram of the\ntetrahedron shown in Fig. 16. As shown above, in the\nsuperexchange limit the high–spin state lies very close\nto the low–spin ground state, and the transition to an\nS= 2 spin quintet occurs at ηc= 0.017. We comment\nhere that this high–spin state is in no sense classical or\ntrivial, being based on orbital singlets which are stabi-\nlized by strong orbital fluctuations, and emphasize again\nthat the high–spin sector also contains a manifold of rich\nproblems in orbital physics, which we will not consider\nfurther here. The near degeneracy of (ss/ot) and (os/st)\nstates is further lifted in the presence of the mixed terms\ninHm, raisingηcto values on the order of 0.12 acrossthe\nbulk of the phase diagram. For no choice of parameters29\nis a spin triplet state found at intermediate values of η.\nThe reentrant behavior close to α= 0.5 is an indication\noftheimportanceofmixedtermsinstabilizingalow–spin\nstate, the tetrahedral geometry providing one of the few\nexamples we have found of anything other than a direct\ncompetition, andhenceaninterpolation,betweenthetwo\nlimiting cases. The rapid upturn in the limit of α→1\nreflects the anomalous stability of the static VB states\nin the direct–exchange limit. The very strong asymme-\ntry of the transition line in Fig. 16 contrasts sharply with\nthe near–symmetryabout α= 0.5 observed for the trian-\ngle (Fig. 13), and shows directly the differences between\nthose features of the phase diagram which are universal\nand those which are effects of even or odd cluster sizes in\na dimer–based system.\nWe close our discussion of the tetrahedral cluster with\na brief discussion of degeneracies and summary of the\npicture provided by the VB ansatz with additional res-\nonance. For the orbital occupation correlations and de-\ngeneracies, we begin with the low–spin sector ( η= 0).\nAtα= 0 one has two (ss/ot) VBs resonating around the\n6 bonds of the cluster, a state characterized by P= 1/6,\nQ= 1/3, andR= 1/2; however, a mixing of the orbital\ntriplet states lowers the degeneracy from 9 to d= 6.\nForα >0 the state is the same, with slow evolution of\nP <1/6,Q >1/2, andR >1/3, but now mixed hop-\nping terms break all orbital degeneracies, giving d= 1.\nOnly when α>0.92 is the ground state more accurately\ncharacterized as one based on spin singlets of the bond\ncolor, with significant values of Pand the restoration of\nanorbitaldegeneracy d= 2. Asα→1, thediagonalhop-\nping component is strengthened ( P→1/3) as the pair of\nbond–colored spin singlets resonates, until at α= 1 they\nbecome static and the degeneracy is d= 3.\nFor the high–spin states in the regime η>ηc, atα= 0\none has two resonating (os/st) VBs, with the hopping\nchannels unchanged and only the spin degeneracy d= 5.\nThis state is not altered qualitatively for any α <0.92,\na transition value independent of η. For 0.92< α <\n1, orbital correlations are strongly suppressed and the\nstate is characterized by hopping processes largely of the\navoided–blocking type (one active orbital, Qdominant),\nstill withd= 5. Finally, α= 1 represents the limit of a\npure avoided–blocking state ( P= 0,Q= 2/3,R= 1/3),\nwhere the degeneracy jumps to 150, a number which can\nbe understood as 5 (spin degeneracy) ×[6 (number of\ntwo–colorstates with no bonds requiring spin singlets) +\n24 (number of three–color states with no bonds requiring\nspin singlets)].\nE. Four–site chain\nAs a fourth and final case, we present results from\na linear four–site cluster. While not directly relevant\nto the study of the triangular lattice, this system offers\nfurther valuable insight into the intrinsic physics of the\nspin–orbital model. The cluster is oriented along the c–0 0.2 0.4 0.6 0.8 1\nα−0.6−0.4−0.20.0Sij , Tij , Cij0 0.2 0.4 0.6 0.8 1−1.0−0.8−0.6−0.4−0.20.0En/J\n{a,b} orbitals c orbitals(ss/os) (ss/cc)(a)\n(b)\nresonating static1 1h\nFIG. 17: (Color online) Evolution of the properties of the\nfour–site chain as a function of αatη= 0: (a) energy spec-\ntrum and (b) spin ( Sij, filled circles), orbital ( Tij, empty cir-\ncles), and spin–orbital ( Cij,×) correlation functions. Both\npanels show a transition occurring at a level crossing at\nα= 4/7. In panel (a), the labels show a nondegenerate\nground state ( d= 1) in both regimes, which has predimi-\nnantly spin singlet character at α >0.571, but both spin\nand orbital singlet components at α <0.571. In panel (b),\nSij=Tij=Cij=−1\n4forα <0.571 due to a resonating ( ab)\norbital configuration, while Tij=Cij= 0 forα >0.571 as a\nconsequence of the static corbital configuration.\naxis with periodic boundary conditions. As for the single\nbond (Sec. VB), only the aandborbitals contribute at\nα= 0, where indeed one finds average electron densities\nper sitenia=nib=1\n2, andnic= 0. Likewise, at α= 1\nonly thecorbitals are occupied, with nic= 1, a result\ndictated by the spin singlet correlations, which are fully\ndeveloped only for complete orbital occupation.\nThe energy per bond for the four–site chain in the\nsuperexchange limit is again −J, as for a single bond\n[Fig. 17(a)]: somewhat surprisingly, the bonds do not\n”disturb” each other, and joint spin–orbital fluctuations\nextend over the entire chain. However, in contrast to\na single bond, this behavior is due to only one quan-\ntum state, the SU(4) singlet. In this geometry, only\none SU(2) orbital subsector is selected, and the result-\ningSU(2) ⊗SU(2) systemislocatedpreciselyattheSU(4)\npoint ofthe Hamiltonian.50Thus, exactly asin the SU(4)\nchain, all spin, orbital and spin–orbital correlation func-\ntions are equal, Sij=Tij=Cij=−0.25, as shown\nin Fig. 17(b). For SijandTij, this result may be under-30\nstoodasan averageoverequalprobabilitiesofsinglet and\ntriplet states on each bond. In more detail, the condition\nset on the correlation functions by SU(4) symmetry12is\n4\n3/an}bracketle{t(/vectorSi·/vectorSj)(/vectorTic·/vectorTjc)/an}bracketri}ht=Sij=Tij, an equality also obeyed\nby the single bond (Sec. VB). The product of Sijand\nTijin its definition ensures the identity for Cij. The\nunique ground state is nevertheless a linear superposi-\ntion of states expressed in the spin and orbital bases,\nand has not only finite but maximal entanglement. This\nstate persists, with a perfectly linear α–dependence, all\nthe way to α= 1, but ceases to be the ground state\natα=4\n7[Fig. 17(a)], where there is a level–crossing\nwith theα= 1 ground state (also perfectly linear). This\nlatter state has a completely different, fluctuation–free\norbital configuration, with pure c–orbital occupation at\nevery site, and gains energy solely in the direct–exchange\nchannel. The spins and orbitals are decoupled, Tijand\nCijvanish, and the spin state has Sij=−0.50: this re-\nsult can be understood as an equal average over bond\nstates with /vectorSi·/vectorSj=−3\n4and−1\n4, and matches that ob-\ntained for the four–site AF Heisenberg model with a res-\nonating VB (RVB) ground state.2The energy at α= 1,\nE0=−0.75J[Fig. 17(a)], is given directly by including\nthe constant term, −1\n4Jper bond, in the definition of the\nHamiltonian (2.21).\nThe results for the linear four–site cluster demonstrate\nagain the competition between superexchange and direct\nexchange. The orbital fluctuations arising due to the\nmixed exchange term, Hm(2.22), are responsible for re-\nmoving the high degeneracies of the eigenenergies in the\nlimitsα= 0 andα= 1 [Fig. 17(a)]. In fact the spectrum\nof the excited states is quasi–continuous in the regime\naroundα= 0.5, but has a finite spin and orbital gap ev-\nerywhereother than the quantum critical point at α=4\n7.\nThese chain results raise a further possibility for the\nspontaneous formation at α= 0 of a 1D state not dis-\ncussed in Sec. III. A set of (for example) c–axis chains,\nwith onlyaandborbitalsoccupied in the pseudospinsec-\ntor, would createexactly the 1D SU(4) model, and would\ntherefore redeem an energy E=−3\n4Jper bond from the\nformation of linear, four–site spin–orbital singlets. The\nenergy of the triangular lattice would receive a further,\nconstant contribution from the cross–chain bonds, which\nwascalculatedin Eq.(3.9) forgeneral η, andhence would\nbe given at η= 0 by\nESU(4)\n1D(0) =−1\n3·3\n4J−1\n6J=−5\n12J. (5.9)\nThis energy represents a new minimum compared with\nall of the results in Sec. III. That it was obtained from\na melting of both spin and orbital order confirms the\nconclusion that ordered phases are inherently unstable\nin this class of model, being unable to provide sufficient\nenergy to compete with the kinetic energy gains avail-\nable through resonance processes. That its value is now\nlower than that obtained for a static, 2D dimer covering\n(Sec. IV) is not of any quantitative significance, given\nthe results of Sec. V confirming the importance of the0.0 0.2 0.4 0.6 0.8 1.0\nα−0.8−0.6−0.4−0.20.0E0/J\nFIG. 18: (Color online) Ground–state energy per bond as a\nfunctionof α, obtainedwith η= 0for atriangular clusterwith\n3 bonds (blue, dashed line), and a tetrahedral cluster with 6\nbonds (red, solid line). For comparison, the energies obtai ned\nfrom the VB ansatz in the limiting cases α= 0 and α=\n1 are shown for the triangular cluster (blue, diamonds) and\ntetrahedral cluster (red, yellow–filled, open circles); at α= 0\nboth VB energies are the same, while at α= 1 they match\nthe exact solutions. Green, upward–pointing triangles sho w\nthe static–dimer results of Sec. IV for the extended system,\nand the black, dot–dashed line the lowest energy per bond\nobtained for fully spin and orbitally ordered phases in Sec. III.\nThe violet, downward–pointing triangle shows the energy of\nthe orbitally ordered but spin–disordered Heisenberg–cha in\nstate at α= 0 [Eq. (3.9)] and the open, yellow–filled square\nthat of the analogous state at α= 1 [Eq. (3.22)], while the\ncross shows the energy of the spin– and orbitally disordered ,\nSU(4)–chain state [Eq. (5.9)].\npositional resonance of dimers.\nF. Summary\nTo summarize, we have shown in this section the re-\nsults of exact numerical diagonalization calculations per-\nformed on small clusters. Detailed analysis of ground–\nstate energies, degeneracies, site occupancies and a num-\nber of correlation functions can be used to extract valu-\nable information about the local physics of the model\nacross the full regime of parameters. Essentially all of\nthe quantities considered show strong local correlations\nand the dominance of quantum fluctuations of the short-\nest range, with ready explanations in terms of resonating\ndimer states.\nWe draw particular attention to the extremely low\nground–state energy of the triangular cluster, which\nshows large gains from dimer resonance. The tetrahedral\ncluster also has a very significant resonance contribution,\nalthough more of its ground–state energy is captured at31\nthe level of a static dimer model. Such a VB ansatz\nprovides the essential framework for the understanding\nof all the results obtained, even for systems with odd\nsite numbers. The energies and their evolution with α\ncontain some quantitative contrasts between even– and\nodd–site systems, allowing further insight concerning the\nrange over which the qualitative features of the cluster\nresults extend.\nFocusing in detail upon these energies, Fig. 18 sum-\nmarizes the exact diagonalization results at zero Hund\ncoupling, and provides a comparison not only with the\nVB ansatz, but with all of the other results obtained\nin Secs. III–V. From bottom to top are shown: the ex-\nact cluster energies including all physical processes; the\nclusterVB ansatz, showingthe importance ofdimer reso-\nnance energy; the static VB ansatz for extended systems,\nsuggestingbycomparisonwithclusterstheeffectsofreso-\nnance; the energies of “melted” states with 1D spin (and\norbital) correlations; the optimal energy of states with\nfull, long–ranged spin and orbital order.\nReturning to the cluster results, their degeneracies can\nbe understood precisely, and demonstrate the restora-\ntion of various symmetries due to resonance processes.\nWe provide a complete explanation for all the correla-\ntion functions computed, and use these to quantify the\nentanglement as a function of α,ηand the system size.\nThere is a high–spin transition as a function of ηfor all\nvalues ofα, which sets the basic phase diagram and es-\ntablishes a new set of disentangled orbital models at high\nη.\nThe extrapolation of the cluster results to states of\nextended systems, some approximations for which are\nshown in Fig. 18, is not straightforward, and cannot be\nexpected to include any information relevant to subtle\nselection effects within highly degenerate manifolds of\nstates. However, with the exception of the static–dimer\nregime around α= 1, our calculations suggest that noth-\ning subtle is happening in this model over the bulk of\nthe phase diagram, where the physics is driven by large\nenergetic contributions from strong, local resonance pro-\ncesses.\nVI. RHOMBIC, HONEYCOMB, AND KAGOME\nLATTICES\nIn Sec. I we alluded to the question of different sources\noffrustrationincomplexsystemssuchasthe spin–orbital\nmodel of Eq. (2.7). More specifically, this refers to the\nrelative effects of pure geometrical frustration, as under-\nstood for AF spin interactions, and ofinteraction frustra-\ntion of the type which can arise in spin–orbital models\neven on bipartite lattices.7Because the interaction frus-\ntration depends in a complex manner on system geom-\netry, no simple separation of these contributions exists.\nIn this section we alter the lattice geometry to obtain\nsomequalitativeresultswithabearingonthisseparation,\nby considering the same spin–orbital model on the three(a) (b)\n(c)\nFIG. 19: (Color online) (a) Rhombic lattice, showing a two–\ncolor orbitally ordered state. (b) Honeycomb lattice, show ing\na one–color orbitally ordered state. (c) Kagome lattice, sh ow-\ning a three–color orbitally ordered state.\nsimple lattice geometries which can be obtained from the\ntriangular lattice by the removal of active bonds or sites.\nThe geometries we discuss are rhombic, obtained by\nremoving all bonds in one of the three triangular lat-\ntice directions [Fig. 19(a)], honeycomb, or hexagonal, ob-\ntained by removing every third lattice site [Fig. 19(b)],\nand kagome, obtained by removing every fourth lattice\nsite in a 2 ×2 pattern [Fig. 19(c)]. Simple geometrical\nfrustration is removed in the rhombic and honeycomb\ncases, but for Heisenberg spin interactions the kagome\ngeometry is generally recognized (from the ground–state\ndegeneracy of both classical and quantum problems) to\nbe even more frustrated than the triangular lattice. We\nconsider only the α= 0 andα= 1 limits of the model,\nandη= 0. We discuss the results for long–range–ordered\nstates (Sec. III) and for static dimer states (Sec. IV) for\nall three lattice geometries. Here we do not enter into nu-\nmerical calculations on small clusters, and comment only\non those systems for which exact diagonalization may be\nexpected to yield valuable information not accessible by\nanalytical considerations.\nA. Rhombic lattice\nWhile the connectivity of this geometry is precisely\nthat of the square lattice, we refer to it here as rhombic\nto emphasize the importance of the bond angles of the32\n(a) (b)\nFIG. 20: (Color online) Rhombic lattice with (a) columnar\nand (b) plaquette dimer coverings.\nchemical structure in maintaining the degeneracy of the\nt2gorbitalsandindeterminingthenatureoftheexchange\ninteractions. It is worth noting that the spin–orbital\nmodel (2.7) on this lattice may be realized in Sr 2VO4\n(below). In the absence of geometrical frustration, the\nspin problem created by imposing any fixed orbital con-\nfiguration selected from Sec. III (Figs. 3 and 4) is gen-\nerally rather easy to solve. Further, at η= 0 both FM\nand AF, and by extension AFF, spin states have equal\nenergies, leading to a high spin degeneracy.\nFollowing Sec. III, the α= 0 energies for the majority\nof the orbitally ordered states of Fig. 3 are\nErh\nlro(0) =−1\n2J (6.1)\nper bond at η= 0 for a number of possible spin con-\nfigurations, whose degeneracy is lifted (in favor of FM\nlines or planes) at finite η. Indeed, the only exceptions\nto this rule occur for the three–color state [Fig. 3(d)]\nand for orientations of the other states which preclude\nhopping in one of the two lattice directions, whose tri-\nangular symmetry properties are broken by the missing\nbond. As noted in Sec. III, for superpositions is it the\nexception rather than the rule for all hopping processes\nto be maximized, but on the rhombic lattice this is pos-\nsible for the states in Fig. 4(a) and some orientations of\nthose in Figs. 4(d) and 4(e).\nForα= 1, the energy limit even on the triangular lat-\ntice wasset ratherby the number ofactive bonds than by\nthe problem of minimizing their frustration. Similar to\ntheα= 0 case, all states where the active hopping direc-\ntion is one of the two lattice directions, plus in this case\nstate (3d), can redeem the maximum energy available,\nErh\nlro(1) =−1\n4J (6.2)\natη= 0, which is simply the avoided–blocking energy,\nfor a large number of possible spin configurations. Finite\nHund exchange favors FM spin states.\nTurning to dimerized states, the calculation of the en-\nergy of any given dimer covering proceeds as in Sec. IV,\nnamely by counting for each the respective numbers of\nbonds of types (7a), (7b), and (7c) [Fig. 7]. For the\nrhombic lattice, lack of geometrical frustration meansthat all interdimer bonds can be chosen to be AF/AO.\nThe two most regular dimer coveringsof the rhombic lat-\ntice with small unit cells may be designated as “colum-\nnar” [Fig. 20(a)] and “plaquette” [Fig. 20(b)]. In both\ncases, 1/4 of the bonds are the dimers, and by inspection\n1/4 of the interdimer bonds in the columnar state are of\ntype (7a), while the remainder are (7c); by contrast, the\nplaquette state has no type–(7a) bonds, 1 /2 type–(7b)\nbonds, and the remainder are of type (7c). For α= 0,\nthe energies are\nErh\ndc(0) =−1\n4J−1\n4·1\n2J−1\n2·1\n4J=−1\n2J,\nErh\ndp(0) =−1\n4J−1\n2·3\n8J−1\n4·1\n4J=−1\n2J(6.3)\natη= 0, both for (ss/ot) and for (os/st) dimers. The de-\ngeneracyofthese two limiting cases, in the sense of maxi-\nmalandminimalnumbersoftypes–(7a)and–(7b)bonds,\nsuggests a degeneracy of all dimer coverings at this level\nof analytical sophistication. Further, all of these dimer\ncoverings are degenerate with all of the unfrustrated or-\ndered states at η= 0. The selection of a true ground\nstate from this large manifold of static states (order–\nby–disorder) would hinge on higher–order processes, but\nthese considerations are likely to be rendered irrelevant\nby dimer resonance (Sec. V).\nFor the spin–singlet dimer states at α= 1 one finds\nErh\ndc(1) =−1\n4J−1\n4·1\n4J−1\n2·0J=−5\n16J,\nErh\ndp(1) =−1\n4J−1\n2·1\n4J−1\n4·0J=−3\n8J,(6.4)\natη= 0, and thus that, as for the triangular lattice, the\nenergy is minimized by dimer configurations excluding\nlinear interdimer bonds. This remains a large manifold\nofdimer coverings,whose energyis manifestly lowerthan\nany of the possible orbitally ordered states in this limit\nof the model, and within which order–by–disorder is ex-\npected to operate (Sec. V).49\nThe considerations of this subsection, extended to fi-\nnite values of η, may be relevant in the understanding of\nexperimentalresultsforSr 2VO4. ThesesuggestweakFM\norder,51accompanied by an AO order52which could be\ninterpreted as arising from the formation of dimer pairs.\nWhen the oxygen octahedra distort, the threefold degen-\neracy of the t2gorbitals is lifted, to give a model con-\ntaining only two degenerate orbitals, dyzanddxz. This\nleads to a situation with Ising–like superexchange inter-\nactionsand quasi–1Dholepropagationin aneffective t–J\nmodel.53\nB. Honeycomb lattice\nThe situation for the honeycomb lattice is very simi-\nlar to that for the rhombic case. Again the absence of\ngeometrical frustration makes it possible to obtain the33\n(a)\n(b)\nFIG. 21: (Color online) Honeycomb lattice with (a) columnar\nand (b) three–way dimer coverings.\nminimal energy for a number of orbital orderings, with a\nhigh spin degeneracy at η= 0. For pure superexchange\ninteractions, once again\nEh\nlro(0) =−1\n2J (6.5)\nper bond, while in the direct–exchange limit\nEh\nlro(1) =−1\n4J, (6.6)\nboth atη= 0, for the same physical reasons as above.\nFor dimer states, on the honeycomb lattice all inter-\ndimer bonds are by definition of type (7c), and again can\nbe made AF/AO because frustration is absent, so the\nenergies of all dimer coverings are de facto identical. By\nway of demonstration, the two simplest regular configu-\nrations, which we label “columnar”and “three–way”,are\nshown in Fig. 21, and, from the fact that now 1 /3 of the\nbonds contain dimers, their energies are\nEh\ndc(0) =−1\n3J−2\n3·1\n4J=−1\n2J,\nEh\nd3(0) =−1\n3J−2\n3·1\n4J=−1\n2J,(6.7)\nper bond at α= 0 =η. Thus static dimer states are\nagain degenerate with unfrustrated ordered states in thesuperexchangelimit, anddetailedconsiderationofkinetic\nprocesses would be required to deduce the lowest total\nenergy. In this context, the dimer coverings shown in\nFig. 21 exemplify two limits about which little kinetic\nenergy can be gained from resonance (Fig. 21(a), where\nlarge numbers of dimers must be involved in any given\nprocess)and in which kinetic energy gainsfrom processes\ninvolving short loops [the three dimers around 2/3 of the\nhexagons, Fig. 21(b)] are maximized.\nAtα= 1, only the dimer energy is redeemed, and this\non 1/3 of the bonds, so\nEh\nd(1) =−1\n3J (6.8)\natη= 0 for a large manifold of coverings. This energy\nis once again significantly better than any of the possi-\nble ordered states, a result which can be ascribed to the\nlow connectivity. That the ground state of the extended\nsystem in this limit for both the rhombic and honeycomb\nlattices involves a selection from a large number of nearly\ndegeneratestates suggeststhat numerical calculationson\nsmall clusters would not be helpful in resolving detailed\nquestions about its nature. The same model for the hon-\neycomb geometry in the α= 1 limit has been discussed\nfor theS= 1 compound Li 2RuO3,54where the authors\ninvoked the lattice coupling, in the form of a structural\ndimerization driven by the formation of spin singlets, to\nselect the true ground state.\nC. Kagome lattice\nThe kagome lattice occupies something of a special\nplace among frustrated spin systems1as one of the most\nhighly degenerate and intractable problems in existence,\nfor both classical and quantum spins, and even with\nonly nearest–neighbor Heisenberg interactions. Inter-\nest in this geometry has been maintained by the dis-\ncovery of a number of kagome spin systems, and has\nrisen sharply with the recent synthesis of a true S= 1/2\nkagome material, ZnCu 3(OH)6Cl2.55Preliminary local–\nprobe experiments56,57show a state of no magnetic or-\nder and no apparent spin gap, whose low–energy spin\nexcitations have been interpreted58as evidence for an\nexotic spin–liquid phase. Both experimentally and the-\noretically, kagome systems of higher spins ( S= 3/2 and\n5/2) are found to have flat bands of magnetic excita-\ntions, reflecting the very high degeneracy of the spin\nsector.59While no kagome materials are yet known with\nboth spin and orbital degrees of freedom, Maekawa and\ncoworkers40,44have considered the itinerant electron sys-\ntem on the triangular lattice for α= 0 (actually for the\nmotion of holes in Na xCoO2), demonstrating that the\ncombination of orbital, hopping selection, and geometry\nleads to any one hole being excluded from every fourth\nsite, and thus moving on a system of four interpenetrat-\ning kagome lattices.34\n(a) (b)\nFIG. 22: (Color online) Kagome lattice with unequally\nweighted two–color states oriented (a) with and (b) against\nthe lattice direction corresponding to the majority orbita l\ncolor.\nConsidering first the energies per bond for states of\nlong–ranged spin and orbital order, in a number of cases\nthe values for the kagome lattice are identical to those of\nthe triangular lattice. This is easy to show by inspection\nfor the one–color state (3a), and for the superposition\nstates (4a), (4b), and (4c), where bonds of all types are\nremoved in equal number. However, for the less symmet-\nrical orbital color configurations a more detailed analysis\nof the type performed in Sec. III is required, and yields\nprovocative results. The two simple possibilities for or-\ndered two–color states with a single color per site are\nshown in Fig. 22, and differ only in the orientation of\nthe continuous lines (the majority color) relative to the\nactive orbitals. These can be considered as the kagome–\nlattice analogs of states (3b) and (3c), as well as of (3e)\nand (3f).\nWhen the lines of c–orbitalsarealignedwith the c–axis\n[Fig. 22(a)], this direction is inactive at α= 0, and only\nthe other two directions contribute, one with two active\nFO orbitals, mandating an AF spin state to give energy\n−1\n2Jper bond, and the other with energy −1\n4Jand no\nstrong spin preference, whence\nE(k3b)(0) =−1\n4J (6.9)\natη= 0 for sets of unfrustrated AF chains. By contrast,\nwhen the lines of c–orbitals fall along the b–direction\n[Fig. 22(b)], the α= 0 problem contains one FO and\none AO line each with two active orbitals, and one line\nwith one active orbital. Only the first requires AF spin\nalignment, while the other two lines are not frustrating,\nwith the result that an energy\nE(k3c)(0) =−5\n12J (6.10)\ncan be obtained. This value is lower than that on the\ntriangular lattice, showing that for the class of models\nunder consideration, where not all hopping channels are\nactivein alldirections, asystem oflowerconnectivitycan\nlead to frustration relief even when its geometry remains\npurely that of connected triangles.(a) (b)\nFIG. 23: (Color online) Kagome lattice with two different,\nequally weighted three–color states: (a) two–color lines o ri-\nented such that only one superexchange channel, plus the di-\nrect exchange channel, is active on every bond. (b) two–colo r\nlines oriented such that all superexchange channels are act ive,\nbut no direct exchange channels.\nWith this result in mind, we consider again the possi-\nbilities offered by different three–color states, specifically\nthose shown in Fig. 23. With reference to the superex-\nchange problem, the state in Fig. 19(c), which by anal-\nogy with (3d) we denote as (k3d), contains only a small\nnumberofremnanttrianglesand isolatedbonds still with\ntwo active orbitals. However, the state (k3d1), shown in\nFig. 23(a) is that which ensures that no such bonds re-\nmain, and every single bond of the lattice has one active\nsuperexchange channel. The state (k3d2) in Fig. 23(b)\nis that in which every single bond of the lattice has two\nactive (FO) superexchange channels: this possibility can\nbe realized for the kagome geometry, at the cost of creat-\ning a frustrated magnetic problem requiring a 120◦spin\nstate to minimize the energy,\nE(k3d)(0) =−5\n16J, (6.11)\nE(k3d1)(0) =−1\n4J, (6.12)\nE(k3d2)(0) =−3\n8J. (6.13)\nThus one finds that lower energies than the value −1\n3J\nper bond, which was the lower bound for fully (or-\nbitally and spin–)ordered states on the triangular lattice,\nare again possible for three–color ordered states. How-\never, the residual spin frustration means that the lowest\nordered–stateenergyonthekagomelatticeisgivenbythe\nunfrustrated, two–color AFF state, E(k3c)(0) =−5\n12J.\nWe present briefly the energies of the same states at\nα= 1,whereonlyamaximumofonehoppingchannelper\nbond can be active, and as noted abovethis is generallya\nstricter energetic limit than any frustration constraints.\nThe results at η= 0 are\nE(k3b)(1) =−1\n4J (6.14)\nfor an AFF state gaining most of its energy from the35\nc–axis chains, and\nE(k3c)(1) =−1\n12J (6.15)\ndue to the dearth of active orbitals in this orientation.\nSimilarly, by counting active orbitals in the three–color\nstates,\nE(k3d)(1) =−1\n6J, (6.16)\nE(k3d1)(1) =−1\n4J, (6.17)\nE(k3d2)(1) = 0, (6.18)\nand it is the state of Fig. 23(a) which achieves the un-\nfrustrated value −1\n4Jby permitting one active hopping\nchannel on every bond of the kagome lattice.\nWe will not discuss the orbital superposition states\nwhich are the analogs of (4d) and (4e), noting only\nthat these present again two different possibilities on the\nkagome lattice, depending on the orientation of the ma-\njority lines. Even with the frustration relief offered by\nthis geometry for the type of model under consideration,\nsuperposition states contain too many hopping channels\nfor all to be satisfied simultaneously, and it is not possi-\nble to equal the energy values found respectively for the\nconfigurations in Figs. 23(a) and (b) at α= 1 andα= 0.\nIt remains to consider dimer states on the kagome lat-\ntice, as these have been of equal or lower energy for ev-\nery case analyzed so far. The set of nearest–neighbor\ndimer coverings of the kagome lattice is large, and for\ntheS= 1/2 Heisenberg model in this geometry the spin\nsinglet manifold has been proposed as the basis for an\nRVB description.25Two dimer coverings degenerate at\nthe level of the current treatment are shown in Fig. 24.\nDimer coverings of the kagome lattice have the prop-\nerty that 3 /4 of the triangles contain one dimer. In\nthis case, the other bonds of the triangle are interdimer\nbonds, one of which is of type (7b) while the other is of\ntype (7c). The other 1/4 of the triangles, known60as\n“defect triangles”, have no dimers, and their three bonds\nare either all of type (7b), with probability 1/4, or one\neach of types (7a), (7b), and (7c), with probability 3/4.\nThe frustration of the system is contained in the problem\nof minimizing the number of FM/FO interdimer bonds;\nthis exercise is complex and no solution is known, so only\nan upper bound will be estimated here.\nThe bonds of a defect triangle connect three differ-\nent dimers, and so one (or all three) must be FM/FO. A\nhexagonofthekagomelatticewithnodimersonitsbonds\nissurroundedbysixnon–defectivetriangles,onewithone\ndimer by one defective neighbor, with two dimers two,\nand a hexagon with three dimers shares its non–dimer\nbonds with three defect triangles. Hexagons with odd\ndimer numbers must create a FM/FO bond between at\nleast one pair of dimers, and it is reasonable to place this\nbond on the defect triangle(s) where an energy cost is al-\nready incurred. We note immediately that the cost of re-\nversing the type–(7a) bond,1\n4J(Sec. IVA), exceeds that(a)\n(b)\nFIG. 24: (Color online) Kagome lattice with two different\ndimer coverings, (a) and (b). In both examples, only two of\nthe twelve triangles shown explicitly on the cluster are “de fec-\ntive”(contain nodimer), butthereader maynotice thatmany\nof the next twelve triangles adjoining the boundary must als o\nbe so.\nof reversingboth interdimer bonds of a non–defective tri-\nangle, which is1\n8J+1\n16J. As a consequence, we take this\ncost, which is equal to that of reversing both a non–\ndefective triangle and the weakest bond of the defect tri-\nangle, to be an upper bound on the effect of frustration.\nThe net energy of a dimer state for α= 0 =ηis then\nestimated to be\nEkd(0) =−3\n4·1\n3J−3\n4·1\n3J/parenleftbigg3\n8+1\n4/parenrightbigg\n−1\n4J/bracketleftbigg1\n4/parenleftbigg2\n3·3\n8+1\n3·1\n4/parenrightbigg\n+3\n4·1\n3/parenleftbigg1\n4+3\n8+1\n4/parenrightbigg/bracketrightbigg\n=−209\n384J≃ −13\n24J. (6.19)\nThis is a very large number for the kagome lattice, ex-\nceeding even the value −1\n2Jper bond (which, however, is\nof no special significance here). Thus we find that dimer\nstates in this type of model are strongly favored, gaining\na very much higher energy than even the best ordered\nstates. Qualitatively, the dimer energy shares with the\nordered–state energy the feature that it is considerably36\nbetterthananythingobtainableforthetriangularlattice.\nThis implies that the reduced connectivity of the lattice\ngeometry for a model where the orbital degeneracy pro-\nvides a number of mutually exclusive hopping channels\nmakesit easier to find states where every remainingbond\ncan support a favorable hopping process without strong\nfrustration.\nApplying all of the above geometrical considerations\nto the direct–exchange model ( α= 1), where there is no\nfrustration problem between the spin singlets, one finds\nEkd(1) =−3\n4·1\n3J−3\n4·1\n3J/bracketleftbigg1\n4+0/bracketrightbigg\n−1\n4·1\n3J/bracketleftbigg1\n4·3\n4+1\n4·1\n2/bracketrightbigg\n=−21\n64J (6.20)\natη= 0. Once again this energy is significantly lower\nthan the value Edim(1) =−1\n4Jobtained for the triangu-\nlar lattice in Eq. (4.15), demonstrating that the multi-\nchannel spin–orbital model of the type considered here is\nless frustrated in the kagome geometry.\nWecommentinclosingthatthe dimerenergieswehave\nestimatedareonlythoseofstaticVBconfigurations,and,\naway from α= 1, the possibility remains of a signifi-\ncant resonance energy gain from quantum fluctuations\nbetween these states ( cf.Sec. V). Numerical calculations\non small clusters of sufficient size (here at least 6 sites for\na unit cell) would be helpful in this frustrated case.\nTo summarize this section, the spin–orbital model on\nbipartite lattices appears to present competing ordered\nand dimerized states with the prospect of high degenera-\ncies. Among “frustrated” systems (in the sense of being\nnon–bipartite), the kagome lattice provides an example\nwhere geometrical and orbital frustration effects cancel\npartially, affording favorable dimerized solutions. Thus,\nwhile it is possible to ascribe some of the frustration ef-\nfects we have studied in the triangular lattice to a purely\ngeometrical origin, for more complex models it is in gen-\neralnecessaryto extend the concept of “geometricalfrus-\ntration” beyond that applicable to pure spin systems.\nVII. DISCUSSION AND SUMMARY\nWe have considered a spin–orbital model representa-\ntive of a strongly interacting 3 d1electron system with\nthe cubic structural symmetry of edge–sharing metal–\noxygen octahedra, conditions which lead to a triangular\nlattice of magnetic interactions between sites with un-\nbroken, threefold orbital degeneracy. We have elucidated\nthe qualitativephasediagram,whichturns outto be very\nrich, in the physicalparameterspace presented by the ra-\ntio (α) of superexchange to direct–exchange interactions\nand the Hund exchange ( η).\nDespite the strong changes in the fundamental nature\nof the model Hamiltonian as a function of αandη, anumber of generic features persist throughout the phase\ndiagram. With the exception ofthe ferromagneticphases\nat highη, which effectively suppressesquantum spin fluc-\ntuations (below), there is no long–rangedmagnetic or or-\nbital order anywhere within the entire parameter regime.\nThis shows a profound degree of frustration whose origin\nlies both in the geometry and in the properties of the\nspin–orbital coupling; a qualitative evaluation of these\nrespective contributions is discussed below.\nAll of the phases ofthe model show a strong preference\nfor the formation of dimers. This can be demonstrated\nin a simple, static valence–bond (VB) ansatz, and is re-\ninforced by the results of numerical calculations. The\nstaticansatzis alreadyanexactdescriptionofthe direct–\nexchange limit, α= 1, and gives the best analytic frame-\nwork for understanding the properties of much of the re-\nmainder of the phase diagram. The most striking single\nnumerical result is the prevalence of VB states even on a\ntriangular cluster, and the underlying feature reinforced\nby all of the calculations is the very large additional “ki-\nnetic” contribution to the ground–state energy arising\nfrom the resonance of VBs due to quantum fluctuations.\nIt is this resonance which drives symmetry restoration in\nsome or all of the spin, orbital, and translational sectors\nover large regions of the parameter space. The sole ex-\nception to dimerization is found at high ηand around\nα= 1, where the only mechanism for virtual hopping is\nthe adoption of orbital configurations which permit one\norbital to be active (“avoided blocking”).\nThe “most exotic” region of the phase diagram is that\nat smallαandη, and this we have assigned tentatively\nas an orbital liquid. In this regime, quantum fluctua-\ntions are at their strongest and most symmetrical, and\nevery indication obtained from energetic considerations\nof extended systems, and from microscopic calculations\nof a range of local quantities on small clusters, suggests\na highly resonant, symmetry–restored phase. While this\norbital liquid is in all probability (again from the same\nindicators) based on resonating dimers, an issue we dis-\ncuss in full below, we cannot exclude fully the possibility\nof a type of one–dimensional physics: short, fluctuating\nsegments of frustration–decoupled spin or orbital chains,\nwhose character persists despite the high site coordina-\ntion. It should be stressed here that the point ( α,η) =\n(0,0)isnotinanysenseaparentphaseforexoticstatesin\nthe restofthe phasediagram: mixed and direct exchange\nprocesses are qualitatively different elements, which in-\ntroduce different classes of frustrated model at finite α.\nWhile the matter is somewhat semantic, we comment\nonly that one cannot argue for the point α= 0.5 being\n“more exotic” than α= 0 despite having the maximal\nnumber of equally weighted hopping channels, because it\ndoes not possess any additional symmetries which man-\ndate qualitative changes to the general picture. In this\nsense, the limit α= 1 serves as a valuable fixed point\nwhich is understood completely, and yet is still domi-\nnated by the purely quantum mechanical concept of sin-\nglet formation.37\nOne indicatorwhich can be employed to quantify “how\nexotic” a phase may be is the entanglement of spin and\norbital degrees of freedom. We define entanglement as\nthe deviation of the spin and orbital sectors from the\nfactorized limit in which their fluctuations can be treated\nseparately. We compute a spin–orbital correlation func-\ntion and use it to measure entanglement, finding that\nthis is significant over the whole phase diagram. Qual-\nitatively, entanglement is maximal around the superex-\nchange limit, which is dominated by dimers where sin-\nglet formation forces the other sector to adopt a local\ntriplet state. However, for particular clusters and dimer\nconfigurations, the high symmetry may allow less entan-\ngled possibilities to intervene exactly at α= 0. The\ndirect–exchange limit, α= 1, provides additional in-\nsight into the entanglement definition: the four–operator\nspin–orbital correlation function vanishes, reflecting the\nclear decoupling of the two sets of degrees of freedom at\nthis point, but the finite product of separate spin and\norbital correlation functions violates the factorizability\ncondition.\nThis preponderance of evidence for quantum states\nbased on robust, strongly resonating dimers implies fur-\nther that the (spin and orbital) liquid phase is gapped.\nSuch a state would have only short–ranged correlation\nfunctions. However, these gapped states are part of a\nlow–energy manifold, and for the extended system we\nhave shown that this consists quite generally of large\nnumbers of (nearly) degenerate states. The availability\nof arbitrary dimer rearrangements at no energy cost has\nbeen suggested to be sufficient for the deconfinement of\nelementary S= 1/2 (and by analogy T= 1/2) excita-\ntions with fractional statistics.61However, the spinons\n(orbitons) are massive in such a model, in contrast to\nthe properties of algebraic liquid phases.62\nAlow–spintohigh–spintransition,occurringasafunc-\ntion ofη, is present for all values of α. The quantitative\nestimation of ηcin the extended system remains a prob-\nlem for a more sophisticated analysis. At the qualitative\nlevel, largeηcan be considered to suppressquantum spin\nfluctuations by promoting parallel–spin (ferromagnetic)\nintermediate states on the magnetic ions. However, even\nwhen this sector is quenched, the orbital degrees of free-\ndom remain frustrated, and contain non–trivialproblems\nin orbital dynamics. In the superexchange (low– α, high–\nη) region, frustration is resolved by the formation of or-\nbital singlet (spin triplet) dimers, whose resonance min-\nimizes the ground–state energy. The frustration in the\ndirect–exchange (high– α, high–η) region is resolved by\navoided–blocking orbital configurations, and order–by–\ndisorder effects are responsible for the selection of the\ntrue ground state from a degenerate manifold of possi-\nbilities; this is the only part of the phase diagram not\ndisplaying dimer physics. Thus the ferromagnetic orbital\nmodels in both limits exhibit a behavior quite different\nfrom that of systems with only S= 1/2 spin degrees of\nfreedom on the triangular lattice.\nWe have commented on both geometry and spin–orbital interactions as the origin of frustration in the\nmodels under consideration. However, a statement such\nas “on the triangular lattice, geometrical frustration en-\nhances interaction frustration for spin–orbital models”\nmust be qualified carefully. We have obtained anecdotal\nevidence concerning such an assertion in Sec. VI by con-\nsideringotherlatticegeometries,andfind thatindeed the\nsame model on an unfrustrated geometry appears capa-\nble of supporting ordered states; however, the interplay\nof the two effects is far from direct, as the kagome lattice\npresents a case where dimer formation acts to reduce the\nnet frustation. Quite generally, spin–orbital models con-\ntain in principle more channels which can be used for re-\nlievingfrustration,buttheexactnatureofthecouplingof\nspin and orbital sectors may result in the opposite effect.\nSpecificdatacharacterizingmutualfrustrationcanbeob-\ntained from the spin and orbital correlations computed\non small clusters: as shown in Sec. V, for the triangular\nlattice there are indeed regimes where, for example, the\neffective orbital interactions enforced by the spin sector\nmake the orbital sector more frustrated (higher Tij) than\nwould be the analogous pure spin problem (measured by\nSij), and conversely.\nWe comment briefly on other approaches which might\nbe employed to obtain more insight into the states of the\nextended system, with a view to establishing more defini-\ntively the nature and properties of the candidate orbital\nliquid phase. More advanced numerical techniques could\nbe used to analyze larger unit cells, but while Lanczos\ndiagonalization, contractor renormalization63or other\ntruncation schemes might afford access to systems two,\nor even four, times larger, it seems unlikely that these\nclusters could provide the qualitatively different type of\ndata required to resolve the questions left outstanding\nin Sec. V. An alternative, but still non–perturbative and\npredominantly unbiased, approach would be the use of\nvariational wave functions, either formulated generally\nor in the more specific projected wave function technique\nwhich leads to different types of flux phase.64,65Adapt-\ning this type of treatment to the coupled spin and orbital\nsectorswithout undue approximationremainsa technical\nchallenge.\nWithin the realm of effective models which could be\nobtained by simplification of the ground–state manifold,\nwe cite only the possibility motivated by the current re-\nsults of constructing dimer models based on (ss/ot) and\n(os/st) dimers. Dimer models26are in general highly\nsimplified, and there is no systematic procedure for their\nderivation from a realistic Hamiltonian, but they are\nthought to capture the essential physics of certain classes\nof dimerized systems. Because QDM Hamiltonians pro-\nvide exact solutions, and in some cases genuine exam-\nples of exotica long sought in spin systems, including the\nRVB phase and deconfined spinon excitations, they rep-\nresent a valuable intermediate step in understanding how\nsuch phenomena may emerge in real systems. Here we\nhave found (i) a very strong tendency to dimer forma-\ntion, (ii) a large semi–classical degeneracy of basis states38\nformed from these dimers, and (iii) that resonance pro-\ncesses even at the four–site plaquette scale providea very\nsignificant energetic contribution. From the final obser-\nvation alone, a minimal QDM, meaning only exchange of\nparallel dimers of all three directions and on all possible\nplaquette units, would already be expected to contain\nthe most significant corrections to the VB energy. At\nthis point we emphasize that, because of the change of\nSU(2) orbitalsectorwith lattice direction, our2Dmodels\nare not close to the SU(4) point where four–site plaque-\ntte formation, and hence very probably a crystallization,\nwould be expected.13From the results of Secs. IV and\nV, a rather more likely phase of the QDM would be one\nwith complete plaquette resonance through all three col-\nors, and without breaking of translational symmetry.\nRigorous proof of a liquid phase, such as that repre-\nsented by an RVB state, is more complex, and as noted\nin Sec. I it requires satisfying both energetic and topo-\nlogical criteria. Following the prescription in Ref. 19,\nthree conditions must be obeyed: (i) a propensity for\ndimer formation, (ii) a highly degenerate manifold of ba-\nsis states from which the RVB ground state may be con-\nstructed, and (iii) a mapping of the system to a liquid\nphase of a QDM. Criteria (i) and (ii) match closely the\nlabels in the previous paragraph, and both dimer forma-\ntion and high degeneracy have been demonstrated ex-\ntensively here. The energetic part of criterion (iii) also\nappears to be obeyed here: static dimers have an en-\nergy (V), and allowing their location and orientation\nto change gains more ( t). The regime V/t <1 of the\ntriangular–lattice QDM is the RVB phase demonstrated\nin Ref. 27, whose properties include short–range corre-\nlation functions and gapped, deconfined spinons. This\nmapping also contains the criterion of togological degen-\neracy, andcouldinprinciplebe partiallycircumventedby\na direct demonstration. However, no suitable numericalstudies are available of non–simply connected systems,\nand so here we can present only plausibility arguments\nbased on the high degeneracy and spatial topology of the\ndimer systems analyzed in Secs. IV and V. It is safe to\nconcludethatthe threefold–degenerate t2gorbitalsystem\non the triangular lattice is one of best candidates yet for\na true spin–orbital RVB phase.\nIn closing, spin–orbital models have become a frontier\nof intense current interest for both experimental and the-\noretical studies of novel magnetic and electronic states\nemerging as a consequence of intrinsic frustration. Our\nmodel has close parallels to, and yet crucial differences\nfrom, similar studies of manganites (cubic systems of eg\norbitals), LiNiO 2(triangular, eg), YTiO 3and CaVO 3\n(cubic,t2g), and many other transition–metal oxides, ap-\npearing in some respects to be the most frustrated yet\ndiscussed. One of its key properties, arising from the ex-\ntreme(geometricalandinteraction–driven)frustration,is\nthat ordered states become entirely uncompetitive com-\npared to the resonance energy gained by maximizing\nquantum (spin and orbital) fluctuations. In the orbital\nsector, the restoration of symmetry by orbital fluctua-\ntions makes the model a strong candidate to display an\norbital liquid phase. Because this liquid is based on ro-\nbustdimerstates, the mechanismforitsformationisvery\nlikely to be spin–orbital RVB physics.\nAcknowledgments\nWe thank G. Khaliullin and K. Penc for helpful\ndiscussions, and J. Chaloupka for technical assistance.\nA. M. Ole´ s acknowledges support by the Foundation for\nPolish Science (FNP) and by the Polish Ministry of Sci-\nenceandEducationunderProjectNo.N20206832/1481.\n1Frustrated Spin Systems , edited by H. T. Diep (World Sci-\nentific, Singapore, 2004).\n2P. Fazekas, Lectures on Electron Correlation and Mag-\nnetism(World Scientific, Singapore, 1999).\n3J. Zaanen, G. A. Sawatzky, and J. W. Allen, Phys. Rev.\nLett.55, 418 (1985).\n4A. M. Ole´ s, P. Horsch, G. Khaliullin, and L. F. Feiner,\nPhys. Rev. B 72, 214431 (2005).\n5K. I. Kugel and D. I. Khomskii, Usp. Fiz. Nauk 136, 621\n(1982) [Sov. Phys. Usp. 25, 231 (1982)].\n6C. Castellani, C. R. Natoli, and J. Ranninger, Phys. Rev.\nB18, 4945 (1985); Phys. Rev. B 18, 4967 (1985); Phys.\nRev. B18, 5001 (1985).\n7L. F. Feiner, A. M. Ole´ s, and J. Zaanen, Phys. Rev. Lett.\n78, 2799 (1997).\n8J. B. Goodenough, Magnetism and the Chemical Bond (In-\nterscience, New York, 1963); J. Kanamori, J. Phys. Chem.\nSolids10, 87 (1959).\n9A. M. Ole´ s, P. Horsch, L. F. Feiner, and G. Khaliullin,\nPhys. Rev. Lett. 96, 147205 (2006).10Y. Q. Li, M. Ma, D. N. Shi, and F. C. Zhang, Phys. Rev.\nLett.81, 3527 (1998).\n11A. M. Ole´ s, P. Horsch, and G. Khaliullin, Phys.Stat. Solidi\nB244, 3478 (2007).\n12B. Frischmuth, F. Mila, and M. Troyer, Phys. Rev. Lett.\n82, 835 (1999); F. Mila, B. Frischmuth, A. Deppeler, and\nM. Troyer, ibid.82, 3697 (1999).\n13K.Penc, M. Mambrini, P.Fazekas, andF. Mila, Phys. Rev.\nB68, 012408 (2003).\n14G. Khaliullin and S. Maekawa, Phys. Rev. Lett. 85, 3950\n(2000); G. Khaliullin, Phys. Rev. B 64, 212405 (2001).\n15J. van den Brink, New J. Phys. 6, 201 (2004).\n16G. Mihaly, I. Kezsmarki, F. Zamborszky, M. Miljak,\nK. Penc, P. Fazekas, H. Berger, and L. Forro, Phys. Rev.\nB61, R7831 (2000); P. Fazekas, K. Penc, K. Radnoczi,\nN. Barisic, H. Berger, L. Forro, S. Mitrovic, A. Gauzzi,\nL. Demko, I. Kezsmarki, and G. Mihaly, J. Magn. Magn.\nMater.310, 928 (2007).\n17P. Horsch, G. Khaliullin, and A. M. Ole´ s, Phys. Rev. Lett.\n91, 257203 (2003).39\n18F. Vernay, K. Penc, P. Fazekas, and F. Mila, Phys. Rev.\nB70, 014428 (2004); F. Vernay, A. Ralko, F. Becca, and\nF. Mila, ibid.74, 054402 (2006).\n19F. Mila, F. Vernay, A. Ralko, F. Becca, P. Fazekas, and\nK. Penc, J. Phys.: Condens. Matter 19, 145201 (2007).\n20S. Di Matteo, G. Jackeli, C. Lacroix, and N. B. Perkins,\nPhys. Rev. Lett. 93, 077208 (2004); S. Di Matteo, G. Jack-\neli, and N. B. Perkins, Phys. Rev. B 72, 024431 (2005).\n21S. Di Matteo, G. Jackeli, and N. B. Perkins, Phys. Rev. B\n72, 020408(R) (2005).\n22L. F. Feiner and A. M. Ole´ s, Phys. Rev. B 71, 144422\n(2005).\n23G. Khaliullin, Prog. Theor. Phys. Suppl. 160, 155 (2005).\n24P. Fazekas and P. W. Anderson, Philos. Mag. 30, 423\n(1974).\n25F. Mila, Phys. Rev. Lett. 81, 2356, (1988); M. Mambrini\nand F. Mila, Eur. Phys. J. B 17, 651 (2000).\n26D. S. Rokhsar and S. A. Kivelson, Phys. Rev. Lett. 61,\n2376 (1988).\n27R. Moessner and S. L. Sondhi, Phys. Rev. Lett. 86, 1881\n(2001).\n28A. Ralko, M. Ferrero, F. Becca, D. Ivanov, and F. Mila\nPhys. Rev. B 71, 224109 (2005).\n29A. Weisse and H. Fehske, New J. Phys. 6, 158 (2004);\nE. Dagotto, New J. Phys. 7, 67 (2005); K. Ro´ sciszewski\nand A. M. Ole´ s, J. Phys.: Cond. Matter 19, 186223 (2007).\n30P. Piekarz, K. Parlinski, and A. M. Ole´ s, Phys. Rev. Lett.\n97, 156402 (2006); Phys. Rev. B 76, 165124 (2007).\n31D. I. Khomskii and T. Mizokawa, Phys. Rev. Lett. 94,\n156402 (2005).\n32M. Holzapfel, S. de Brion, C. Darie, P. Bordet, E. Chappel,\nG. Chouteau, P. Strobel, A. Sulpice, and M. D. N´ u˜ nez–\nRegueiro, Phys. Rev. B 70, 132410 (2004).\n33A. Reitsma, L. F. Feiner, and A. M. Ole´ s, New J. Phys. 7,\n121 (2005).\n34M. V. Mostovoy and D. I. Khomskii, Phys. Rev. Lett. 89,\n227203 (2002).\n35K. Hirakawa, H.Kadowaki, andK. Ubukoshi, J. Phys. Soc.\nJpn.54, 3526 (1985).\n36K. Takeda, K. Miyake, K. Takeda, and K. Hirakawa,\nJ. Phys. Soc. Jpn. 61, 2156 (1992).\n37H. F. Pen, J. van den Brink, D. I. Khomskii, and G. A.\nSawatzky, Phys. Rev. Lett. 78, 1323 (1997).\n38T. Motohashi, Y.Katsumata, T. Ono, R.Kanno, M. Karp-\npinen, and H. Yamauchi, J. Appl. Phys. 103, 07C902\n(2008).\n39M. Itoh, M. Shikano, H. Kawaji, and T. Nakamura, Solid\nState Commun. 80, 545 (1991); Y. Imai, I. Solovyev, and\nM. Imada, Phys. Rev. Lett. 95, 176405 (2005).\n40W. Koshibae and S. Maekawa, Phys. Rev. Lett. 91, 257003\n(2003).\n41J. S. Griffith, The Theory of Transition Metal Ions (Cam-\nbridge University Press, Cambridge, 1971).\n42G. Khaliullin, P. Horsch, and A. M. Ole´ s, Phys. Rev. Lett.\n86, 3879 (2001); A. M. Ole´ s, P. Horsch, and G. Khaliullin,\nPhys. Rev. B 75, 184434 (2007).43A. B. Harris, T. Yildirim, A. Aharony, O. Entin-Wohlman,\nand I. Ya. Korenblit, Phys. Rev. Lett. 91, 087206 (2003).\n44This convenient notation for the triangular lattice was in-\ntroduced by G. Khaliullin, W. Koshibae, and S. Maekawa,\nPhys. Rev. Lett. 93, 176401 (2004).\n45A. M. Ole´ s, Phys. Rev. B 28, 327 (1983).\n46J. Zaanen and G. A. Sawatzky, J. Solid State Chem. 88, 8\n(1990).\n47T. Mizokawa and A. Fujimori, Phys. Rev. B 54, 5368\n(1996).\n48M. Daghofer, A. M. Ole´ s, and W. von der Linden, Phys.\nRev. B70, 184430 (2004); Phys. Stat. Solidi B 242, 311\n(2005).\n49G. Jackeli and D. A. Ivanov, Phys. Rev. B 76, 132407\n(2007).\n50C. Itoi, S. Qin, and I. Affleck, Phys. Rev. B 61, 6747\n(2000).\n51A. Nozaki, H. Yoshikawa, T. Wada, H. Yamauchi, and\nS. Tanaka, Phys. Rev. B 43, 181 (1991).\n52J. Matsuno, Y. Okimoto, M. Kawasaki, and Y. Tokura,\nPhys. Rev. Lett. 95, 176404 (2005).\n53M. Daghofer, K. Wohlfeld, A. M. Ole´ s, E. Arrigoni, and\nP. Horsch, Phys. Rev. Lett. 100, 066403 (2008).\n54G. Jackeli and D. I. Khomskii, Phys. Rev. Lett. 100,\n147203 (2008).\n55M. P. Shores, E. A. Nytko, B. M. Barlett, and D. G. No-\ncera, J. Am. Chem. Soc. 127, 13462 (2005).\n56T. Imai, E. A. Nytko, B. M. Bartlett, M. P. Shores, and\nD. G. Nocera, Phys. Rev. Lett. 100, 077203 (2008).\n57P. Mendels, F. Bert, M. A. de Vries, A. Olariu, A. Har-\nrison, F. Duc, J. C. Trombe, J. S. Lord, A. Amato, and\nC. Baines, Phys. Rev. Lett. 98, 077204 (2007); F. Bert, S.\nNakamae, F. Ladieu, D. L’Hˆ ote, P. Bonville, F. Duc, J.–C.\nTrombe, and P. Mendels, Phys. Rev. B 76, 132411 (2007);\nA. Olariu, P. Mendels, F. Bert, F. Duc, J.–C. Trombe, M.\nA. de Vries, and A. Harrison, Phys. Rev. Lett. 100, 087202\n(2008).\n58J. S. Helton, K. Matan, M. P. Shores, E. A. Nytko, B. M.\nBarlett, Y. Yoshida, Y. Takano, A. Suslov, Y. Qiu, J.–H.\nChung, D. G. Nocera, and Y. S. Lee, Phys. Rev. Lett. 98,\n107204 (2007).\n59K. Matan, D. Grohol, D. G. Nocera, T. Yildirim, A. B.\nHarris, S. H. Lee, S. Nagler, and Y. S. Lee, Phys. Rev.\nLett.96, 247201 (2006).\n60V. Elser, Phys. Rev. Lett. 62, 2405 (1989).\n61S. Dommange, M. Mambrini, B. Normand, and F. Mila,\nPhys. Rev. B 68, 224416 (2003).\n62M. Hermele, T. Senthil, and M. P. A. Fisher, Phys. Rev.\nB72, 104404 (2005).\n63S. Capponi, A. L¨ auchli, and M. Mambrini, Phys. Rev. B\n70, 104424 (2004).\n64M. B. Hastings, Phys. Rev. B 63, 014413 (2000).\n65Y. Ran, M. Hermele, P. A. Lee, and X.–G. Wen, Phys.\nRev. Lett. 99, 117205 (2007)."    },    {        "title": "1604.04623v3.Determining_the_spin_orbit_coupling_via_spin_polarized_spectroscopy_of_magnetic_impurities.pdf",        "content": "arXiv:1604.04623v3  [cond-mat.mes-hall]  15 Oct 2016Determining the spin-orbit coupling via spin-polarized sp ectroscopy of magnetic\nimpurities\nV. Kaladzhyan,1,2,∗P. Simon,2and C. Bena1,2\n1Institut de Physique Th´ eorique, CEA/Saclay, Orme des Meri siers, 91190 Gif-sur-Yvette Cedex, France\n2Laboratoire de Physique des Solides, CNRS, Univ. Paris-Sud ,\nUniversit´ e Paris-Saclay, 91405 Orsay Cedex, France\n(Dated: October 1, 2018)\nWe study the spin-resolved spectral properties of the impur ity states associated to the presence of\nmagnetic impurities in two-dimensional, as well as one-dim ensional systems with Rashba spin-orbit\ncoupling. We focus on Shiba bound states in superconducting materials, as well as on impurity\nstates in metallic systems. Using a combination of a numeric al T-matrix approximation and a\ndirect analytical calculation of the bound state wave funct ion, we compute the local density of\nstates (LDOS) together with its Fourier transform (FT). We fi nd that the FT of the spin-polarized\nLDOS, a quantity accessible via spin-polarized STM, allows to accurately extract the strength of the\nspin-orbit coupling. Also we confirm that the presence of mag netic impurities is strictly necessary\nfor such measurement, and that non-spin-polarized experim ents cannot have access to the value of\nthe spin-orbit coupling.\nI. INTRODUCTION\nThe electronic bands of materials that lack an inver-\nsion center are split by the spin-orbit (SO) coupling. A\nstrong SO coupling implies that the spin of the electron\nis tied to to the direction of its momentum. Materials\nwith strong SO coupling have been receiving a consider-\nable attention in the past decade partly because SO is\nplaying an important role for the discovery of new topo-\nlogical classes of materials.1,2Two-dimensional topolog-\nical insulators, first predicted in graphene,3have been\ndiscovered in HgTe/CdTe heterostructures4following a\ntheoretical prediction by Bernevig et al..5They are char-\nacterized by one-dimensional helical edge states where\nthe spin is locked to the direction of propagation due\nto the strong SO coupling. Similar features occur for\nthe surface states of 3D topological insulators which also\nhaveastrongbulk SO coupling.1The spin-to-momentum\nlocking was directly observed by angle-resolved photoe-\nmission spectroscopy (ARPES) experiments.6,7\nTopological superconductors share many properties\nwith topological insulators. They possess exotic edge\nstatescalled Majoranafermions, particleswhich aretheir\nown antiparticles.1Topological superconductivity can be\neither induced by the proximity with a standard s-wave\nsuperconductor or be intrinsic. In the former case, Ma-\njorana states have thus been proposed to form in one-\ndimensional8,9and two-dimensional semiconductors10,11\nwithstrongSOcouplingwhenproximitizedwithas-wave\nsuperconductor, and in the presence of a Zeeman field.\nFollowing this strategy, many experiments have reported\nsignatures of Majorana fermions through transport spec-\ntroscopy in one dimensional topological wires.12–16How-\never, there are presently only a few material candidates\nsuch as strontium ruthenate,17certain heavy fermion\nsuperconductors,18or some doped topological insulators\nsuch as Cu xBi2Se3,19that may host intrinsic topological\nsuperconductivity.Although SO coupling has been playing an essential\nrole in the discovery of new topological materials, it is\nalso of crucial importance in the physics of spin Hall\neffect,20in spintronics21and quantum (spin) computa-\ntion since it allows to electrically detect and manipulate\nspin currents in confined nanostructures (see Ref. 22 for\na recent review).\nBased on the prominent role played by SO in the\npast decades, it is thus of great interest to be able\nto evaluate the SO coupling value in a given mate-\nrial accurately, though in general this is a very dif-\nficult task. Inferences can be made from ARPES\nmeasurements;23–25in particular spin-polarized ARPES\nmeasurements have been used to evaluate the SO cou-\npling in variousmaterials.26–32Other possibilities involve\nmagneto-transport measurements in confined nanostruc-\ntures: this technique has been used to measure the\nSO coupling in clean carbon nanotubes33or in InAs\nnanowires.34\nHere we propose a method to measure the magnitude\nof the SO coupling directly using spin-polarized scan-\nning tunnelling microscopy (STM),35and the Fourier\ntransform (FT) of the local density of states (LDOS)\nnear magnetic impurities (FT-STM). The FT-STM tech-\nnique has been used in the past in metals, where it\nhelped in mapping the band structure and the shape\nand the properties of the Fermi surface,36–43as well\nas in extracting information about the spin properties\nof the quasiparticles.44More spectacularly, it was used\nsuccessfully in high-temperature SCs to map with high\nresolution the particular d-wave structure of the Fermi\nsurface, as well as to investigate the properties of the\npseudogap.45–47\nIn this paper, on one hand, we calculate the Fourier\ntransform of the spin-polarized local density of states\n(SP LDOS) of the so-called Shiba bound state48–51as-\nsociated with a magnetic impurity in a superconductor.\nShiba bound states have been measured experimentally\nby STM52–54and it has actually been shown that the ex-2\ntentoftheShibawavefunction canreachtensofnanome-\nters in 2D superconductors, which allows one to mea-\nsure the spatial dependence of the LDOS of such states\nwithhighresolution.55Weconsiderbothone-dimensional\nand two-dimensional superconductors with SO coupling.\nWhile two-dimensional systems such as e.g. Sr 2RuO4,17\nor NbSe 255,56become superconducting when brought at\nlowtemperature, one-dimensionalwiressuchasInAs and\nInSb are not superconducting at low temperature. In or-\nder to see the formation of Shiba states one would need\nto proximitize them by a SC substrate. The formation of\nShiba states in such systems,57,58as well as in p-wave\nsuperconductors,59,60has been recently touched upon,\nbut the effect of the SO coupling on the FT of the SP\nLDOS in the presence of magnetic impurities has not\npreviously been analyzed.\nOn the other hand we focus on the effects of the spin-\norbit coupling on the impurity states of a classical mag-\nnetic impurity in one-dimensional and two-dimensional\nmetallic systems such as Pb61and Bi, as well as InAs\nand InSb semiconducting wires that can be also mod-\neled as metals in the energy range that we consider. We\nshould note that for these systems no bound state forms\nat a specific energy, but the impurity is affecting equally\nthe entire energy spectrum.\nBy studying the two classesofsystems described above\nwe show that the SO coupling can directly be read-off\nfrom the FT features of the SP LDOS in the vicinity\nof the magnetic impurity. We note that such a signa-\nture appearsonly formagneticimpurities, and onlywhen\nthesystemisinvestigatedusingspin-polarizedSTMmea-\nsurements, the non-spin-polarized measurements do not\nprovide information on the SO, as it has also been previ-\nously noted.62The main difference between the SC and\nmetallic systems, beyond the existence of a bound state\nin the former case, is that the spin-polarized Friedel os-\ncillations around the impurity haveadditional features in\ntheSCphase, themostimportantonebeingtheexistence\nof oscillations with a wavelength exactly equal to the SO\ncoupling length scale; such oscillations are not present in\nthe metallic phase. Another difference is the broadening\nof the FT features in the superconducting phase com-\npared to the non-SC phase in which the sole broadening\nis due to the quasiparticle lifetime.\nWe focus on Rashba SO coupling as assumed to be\nthe most relevant for the systems considered, but we\nhave checked that our conclusion holds for other types\nof SO. To obtain the SP LDOS we use a T-matrix\napproximation,43,63,64and we present both numerical\nand analytical results which allow us to obtain a full un-\nderstandingofthe observedfeatures, ofthe splittings due\nto the SO, as well as of the spin-polarization of the im-\npurity states and of the symmetry of the FT features.\nIn Sec. II, we present the general model for two-\ndimensional and one-dimensional cases and the basics of\nthe T-matrix technique. In Sec. III we show our results\nfor the SP LDOS, calculated both numerically and an-\nalytically, for 2D systems, both in the SC and metallicphase. Sec. IV is devoted to SP LDOS of impurity in\none-dimensional systems. Our Conclusions are presented\nin section V. Details of the analytical calculations are\ngiven in the Appendices.\nII. MODEL\nWe consider an s-wave superconductor with a SC\nparing ∆ s, and Rashba SO coupling λ, for which\nthe Hamiltonian, written in the Nambu basis Ψ p=\n(ψ↑p,ψ↓p,ψ†\n↓−p,−ψ†\n↑−p)T, is given by:\nH0=/parenleftbigg\nξpσ0∆sσ0\n∆sσ0−ξpσ0./parenrightbigg\n+HSO. (1)\nThe energy spectrum is ξp≡p2\n2m−εF, whereεFis the\nFermi energy. The operator ψ†\nσpcreates a particle of spin\nσ=↑,↓of momentum p≡(px,py) in 2D and p≡pxin\n1D. Below we set /planckover2pi1to unity. The system is considered to\nlay in the (x,y) plane in 2D case, whereas in 1D case we\nsetpyto zero in the expressions above, and we consider\na system lying along the x-axis. The metallic limit is\nrecovered by setting ∆ s= 0. The Rashba Hamiltonian\ncan be written as\nHSO=λ(pyσx−pxσy)⊗τz, (2)\nin 2D and simply as HSO=λpxσy⊗τzin 1D. We\nhave introduced σandτ, the Pauli matrices acting re-\nspectively in the spin and the particle-hole subspaces.\nThe unperturbed retarded Green’s function can be ob-\ntained from the above Hamiltonian via G0(E,p) =\n[(E+iδ)I4−H0(p)]−1, whereδis the inverse quasipar-\nticle lifetime.\nIn what follows we study what happens when a sin-\ngle localized impurity is introduced in this system. We\nconsider magnetic impurities of spin J= (Jx,Jy,Jz) de-\nscribed by the following Hamiltonian:\nHimp=J·σ⊗τ0·δ(r)≡V·δ(r),(3)\nwhereJis the magnetic strength. We only consider here\nclassical impurities oriented either along the z-axis,J=\n(0,0,Jz), or along the x-axis,J= (Jx,0,0). This is\njustifiedprovidedtheKondotemperatureismuchsmaller\nthan the superconducting gap.51\nTo find the impurity states in the model described\nabove we use the T-matrix approximation described in\n[51, 63, and 64] and [43]. We also neglect the renormal-\nization of the superconducting gap because it is mainly\nlocal51,65and therefore only introduces minor effects for\nour purposes. Since the impurity is localized, the T-\nmatrix is given by:\nT(E) =/bracketleftbigg\n1−V/integraldisplayd2p\n(2π)2G0(E,p)/bracketrightbigg−1\nV.(4)3\nThe real-space dependence of the non-polarized,\nδρ(r,E), and SP LDOS, Sˆn(r,E), with ˆn=x,y,z, can\nbe found as\nSx(r,E) =−1\nπℑ[∆G12+∆G21],\nSy(r,E) =−1\nπℜ[∆G12−∆G21],\nSz(r,E) =−1\nπℑ[∆G11−∆G22],\nδρ(r,E) =−1\nπℑ[∆G11+∆G22],\nwith\n∆G(E,r)≡G0(E,−r)T(E)G0(E,r),\nwhere ∆Gijdenotes the ij-th component of the matrix\n∆G, andG0(E,r) is the unperturbed retarded Green’s\nfunction in real space, given by the Fourier transform\nG0(E,r) =/integraldisplaydp\n(2π)2G0(E,p)eipr. (5)\nThe FT of the SP LDOS components in momentum\nspace,Sˆn(p,E) =/integraltextdrSˆn(r,E)e−ipr, with ˆn=x,y,z,\nas well as the FT of the non-polarized LDOS, δρ(p,E) =/integraltext\ndrδρ(r,E)e−iprare given by\nSx(p,E) =i\n2π/integraldisplaydq\n(2π)2[˜g12(E,q,p)+ ˜g21(E,q,p)],(6)\nSy(p,E) =1\n2π/integraldisplaydq\n(2π)2[g21(E,q,p)−g12(E,q,p)],(7)\nSz(p,E) =i\n2π/integraldisplaydq\n(2π)2[˜g11(E,q,p)−˜g22(E,q,p)],(8)\nδρ(p,E) =i\n2π/integraldisplaydq\n(2π)2[˜g11(E,q,p)+ ˜g22(E,q,p)],(9)\nwheredq≡dqxdqy,\ng(E,q,p) =G0(E,q)T(E)G0(E,p+q)\n+G∗\n0(E,p+q)T∗(E)G∗\n0(E,q),\n˜g(E,q,p) =G0(E,q)T(E)G0(E,p+q)\n−G∗\n0(E,p+q)T∗(E)G∗\n0(E,q),\nandgij, ˜gijdenote the corresponding components of the\nmatricesgand ˜g. Note that while the non-polarized and\nthe SP LDOS are of course real functions when evalu-\nated in position space, their Fourier transforms need not\nbe. Sometimes we get either or both real and imaginary\ncomponents for the FT, depending on their correspond-\ning symmetries. In the figureswe shallindicate eachtime\nif we plot the real or the imaginarycomponent of the FT.\nTo obtain the FT of the non-polarized and the SP\nLDOS, we first evaluate the momentum integrals in\nEqs. (4-9) numerically. For this we use a square lattice\nversion of the Hamiltonians (1) and (2), where we take\nthe tight-binding spectrum Ξ p≡µ−2t(cospx+cospy)\nwith chemical potential µand hopping parameter t. We\nset the lattice constant to unity. It is also worth noting\nthatallthe numericalintegrationsareperformedoverthe\nfirst Brillouin zone and that we use dimensionless units\nby settingt= 1.Alternatively, as detailed in the appendices, we find\nthe exact form for the non-polarized and SP LDOS in\nthe continuum limit by performing the integrals in the\nFTofthe Green’sfunctions analytically. Moreover,when\nconsidering the SC systems, the energies Eof the Shiba\nstatestogetherwith the correspondingeigenstatesforthe\nShibawavefunctionsΦattheorigincanbeobtainedfrom\nthe corresponding eigenvalue equation66\n[I4−VG0(E,r=0)]Φ(0) = 0. (10)\nThe spatial dependence of the Shiba state wave function\nis determined using\nΦ(r) =G0(E,r)VΦ(0). (11)\nThe real-space Green’s function is obtained simply by a\nFourier transformof the unperturbed Green’s function in\nmomentum space, G0(E,p). The non-polarized and the\nSP LDOS are given by\nρ(E,r) = Φ†(r)/parenleftbigg\n0 0\n0σ0/parenrightbigg\nΦ(r), (12)\nand\nS(E,r) = Φ†(r)/parenleftbigg\n0 0\n0σ/parenrightbigg\nΦ(r), (13)\nwhere we take into account only the hole components of\nthe wave function, and not the electron ones. This is\nbecause the physical observables are related to only one\nof the two components, for example in a STM measure-\nment one injects an electron at a given energy and thus\nhave access to the allowed number of electronic states,\nnot to both the electronic and hole states simultaneously.\nThe Bogoliubov-de Gennes Hamiltonian contains the so-\ncalled particle-hole redundancy, and the electron and the\nholecomponentscanbesimplyrecoveredfromeachother\nby overall changes of sign, and/or changing the sign of\nthe energy. Belowwecompute only the hole components,\nbut there would have been no qualitative differenced had\nwe computed the electron component.\nIII. RESULTS FOR TWO DIMENSIONAL\nSYSTEMS\nA. Real and momentum space dependence of the\n2D Shiba bound states\nFor a 2D superconductorwith SO coupling in the pres-\nence of a magnetic impurity one expects the formation\nof a single pair of Shiba states.57,58The energies of the\nparticle-hole symmetric Shiba states67are given by (in-\ndependent of the direction of the impurity):\nE1,¯1=±1−α2\n1+α2∆s,\nwhereα=πνJandν=m\n2π. (See Appendix A for details\nof how the energies of the Shiba states are calculated.)4\nUp to the critical value αc= 1 these energies are ordered\nthe following way: E1>E¯1. As soon as α>α c, energy\nlevelsE1andE¯1exchange places, making the order the\nfollowing:E¯1> E1. This corresponds to a change of\nthe ground state parity.51,68,69Forα≫1 the subgap\nstates approach the gap edge and eventually merge with\nthe continuum. For the type of impurities considered\nhere, there is no dependence of these energies on the SO\ncouplingin thelow-energyapproximation,thoughaweak\ndependence is introduced when one takes into account\nthe non-linear form of the spectrum. The dependence\nof energy of the Shiba states on the impurity strength\nJis depicted in Fig. 1 where we plot the total spin of\nthe impurity state S(p= 0) as a function of energy and\nimpurity strength. Note that the two opposite-energy\nShiba states have opposite spins.\nFIG. 1. (Color) The averaged SP LDOS induced by an impu-\nrity as a function of the impurity strength for an in-plane\nmagnetic impurity. The dashed line shows the supercon-\nducting gap. A similar result is obtained when the impu-\nrity spin is perpendicular to the plane. Note that the two\nShiba states with opposite energies have opposite spin. We\nsett= 1,µ= 3,δ= 0.01,λ= 0.5,∆s= 0.2.\nWe are interested in studying the spatial structure of\nthe Shiba states in the presence of magnetic impurities\noriented both perpendicular to the plane, and in plane.\nThiscanbedonebothinrealspaceandmomentumspace\nbycalculatingthe Fouriertransformofthe spin-polarized\nLDOSusingtheT-matrixtechniquedetailedintheprevi-\nous section. We focus on the positive-energy Shiba state,\nnoting that its negative energy counterpart exhibits a\nqualitativelysimilarbehavior. InFig. 2weshowthereal-\nspace dependence of the non-polarized and SP LDOS.\nEach of the panels corresponds to the interference pat-\nterns originating from different types of scattering. Note\nthat the spin-orbit value cannot be accurately extracted\nfrom these type of measures, since the system contains\noscillationswith manydifferentsuperposingwavevectors.\nToovercomethisproblemwefocusontheFTofthesefea-\ntures, as it is oftentimes done in spatially resolved STM\nexperiments, which allow for a more accurate separation\nof the different wavevectors.36–43Thus in Fig. 3 we focus\non the FT of the SP LDOS for two types of impuritieswith spin oriented along zandxaxes respectively.\nz-impurity x-impurity\nJz(Jx=Jy= 0) Jx(Jy=Jz= 0)\nFIG. 2. (Color) The real-space dependence of the non-\npolarized as well as of the SP LDOS components for the\npositive energy Shiba state, for a magnetic impurity with\nJz= 2 (left column), and Jx= 2 (right column). We take\nt= 1,µ= 3,δ= 0.01,λ= 0.5,∆s= 0.2.\nNote that the SO introducesnon-zerospin components\nin the directions different from that of the impurity spin.\nThese components exhibit either two-fold or four-fold\nsymmetric patterns. Also the SO is affecting strongly\nthe spin component parallelto the impurity, in particular\nwhen the impurity is in-plane, in which case the struc-\nture of the SP LDOS around the impurity is no longer\nradially symmetric. However, as can be seen in the bot-\ntom panel of Fig. 3, the non-spin-polarized LDOS is5\nnot affected by the presence of SO, preserving a radially\nsymmetric shape quasi-identical to that obtained in the\nabsence of SO. Thus the SO coupling can be measured\nonlyviathe spin-polarizedcomponentsofthe LDOS, and\nnot the non-polarized LDOS.\nThese results, which are obtained using a numerical\nintegration of the T-matrix equations, are also supported\nby analytical calculations which help to understand the\nfine structure of the FT of the SP LDOS (see Appendices\nfor details). These calculations yield for the SP LDOS\ngenerated by a magnetic impurity perpendicular to the\nplane\nSx(r) = +J2\nz/parenleftbigg\n1+1\nα2/parenrightbigge−2psr\nrcosφr×\n×/braceleftigg/summationdisplay\nσσν2\nσ\npσ\nFcos(2pσ\nFr−θ)+2ν2v2\nF\nv2pFsinpλr/bracerightigg\n,\nSy(r) = +J2\nz/parenleftbigg\n1+1\nα2/parenrightbigge−2psr\nrsinφr×\n×/braceleftigg/summationdisplay\nσσν2\nσ\npσ\nFcos(2pσ\nFr−θ)+2ν2v2\nF\nv2pFsinpλr/bracerightigg\n,\nSz(r) =−J2\nz/parenleftbigg\n1+1\nα2/parenrightbigge−2psr\nr×\n×/braceleftigg/summationdisplay\nσν2\nσ\npσ\nFsin(2pσ\nFr−θ)−2ν2v2\nF\nv2pFcospλr/bracerightigg\n,\nρ(r) = +J2\nz/parenleftbigg\n1+1\nα2/parenrightbigge−2psr\nr×\n×/braceleftbigg\n2ν2\nmv+2ν2v2\nF\nv2pFsin(2mvr−θ)/bracerightbigg\n,(14)\nwith\ntanθ=/braceleftigg\n2α\n1−α2,ifα/ne}ationslash= 1\n+∞,ifα= 1. (15)\nWe have introduced eiφr=x+iy\nr, and\npσ\nF=−σmλ+mv, (16)\npλ= 2mλ, (17)\nps=/radicalig\n∆2s−E2\n1/v, (18)\nwithv=/radicalbig\nv2\nF+λ2, andvF=/radicalbig\n2εF/m. Herepσ\nF, pλ\nandpsare the different momenta which can be read off\nfrom the SP LDOS. For an in-plane magnetic impuritywe have\nSs\nx(r) =−J2\nx/parenleftbigg\n1+1\nα2/parenrightbigg/braceleftbigg/summationdisplay\nσν2\nσ\npσ\nF[1+sin(2pσ\nFr−2β)],\n+2ν2v2\nF\nv2pF[cospλr+sin(2mvr−2β)]/bracerightbigge−2psr\nr\nSa\nx(r) = +J2\nx/parenleftbigg\n1+1\nα2/parenrightbigg/braceleftbigg/summationdisplay\nσν2\nσ\npσ\nF[1−sin(2pσ\nFr−2β)]\n−2ν2v2\nF\nv2pF[cospλr+sin(2mvr−2β)]/bracerightbigge−2psr\nrcos2φr,\nSy(r) = +J2\nx/parenleftbigg\n1+1\nα2/parenrightbigg/braceleftbigg/summationdisplay\nσν2\nσ\npσ\nF[1−sin(2pσ\nFr−2β)]\n−2ν2v2\nF\nv2pF[cospλr−sin(2mvr−θ)]/bracerightbigge−2psr\nrsin2φr,\nSz(r) =−J2\nx/parenleftbigg\n1+1\nα2/parenrightbigg/braceleftbigg\n2/summationdisplay\nσσν2\nσ\npσ\nFcos(2pσ\nFr−θ)\n+4ν2v2\nF\nv2pFsinpλr/bracerightbigge−2psr\nrcosφr,\nρ(r) = +J2\nx/parenleftbigg\n1+1\nα2/parenrightbigg/braceleftbigg\n4ν2\nmv+4ν2v2\nF\nv2pF×\n×sin(2mvr−θ)/bracerightbigge−2psr\nr, (19)\nwith tanβ=α.\nTheSxcomponent is the sum of symmetric part a Ss\nx\nand an asymmetric part Sa\nx. Note that the features ob-\nserved in the FT of the SP LDOS plots are well captured\nby the analytical calculations. In particular we note that\nthe oscillations in the SP LDOS are dominated by the\nfollowing four wavevectors:\n2p±\nF, p+\nF+p−\nF= 2mv,andp−\nF−p+\nF=pλ≡2mλ,\nwhich should give rise in the FT to high-intensity fea-\ntures at these wavevectors (the red arrows in Fig. 3).\nIndeed, we note in the numerical results for the FT of\nthe SP LDOS the existence of four rings, correspond-\ning to 2p±\nF,p+\nF+p−\nF= 2mvandp−\nF−p+\nF=pλ, hav-\ning the proper two-fold or fold-fold symmetries, consis-\ntent with the cos /sinφrand cos/sin2φrdependence of\nthe SP LDOS obtained analytically. For example, in the\nxcomponent of the SP LDOS induced by an ximpu-\nrity, the 2p+\nF, 2p−\nFandpλrings have a maximum along x\nand a minimum along y, while the 2 mvring has a sym-\nmetry corresponding to a rotation by 90 degrees. The\nycomponent of the FT of the SP LDOS has a four-\nfold symmetry in which we can again identify the same\nwavevectors,while the Szcomponent has a two-foldsym-\nmetry, and the 2 mvvector is absent. Similarly, for the\nSxand theSycomponents of the SP LDOS induced by\nazimpurity (these components should be zero in the\nabsence of the SO coupling) only the 2 p±\nFandpλwave\nvectors are present, with similar symmetries, while the\nSzcomponent is symmetric. Note also the central peak6\nz-impurity x-impurity\nJz(Jx=Jy= 0) Jx(Jy=Jz= 0)\nFIG. 3. (Color) The FT of the non-polarized as well as of the\nSP LDOS components for the positive energy Shiba state as a\nfunction of momentum, for a magnetic impurity with Jz= 2\n(left column), and Jx= 2 (right column). We take t= 1,µ=\n3,δ= 0.01,λ= 0.5,∆s= 0.2. For a z-impurity we depict\nthe real part of the FT for δρand forSz, and the imaginary\npart for and SxandSy, whereas for an x-impurity we take\nthe imaginary part only for the Szcomponent. Black two-\nheaded arrows correspond to the value of 2 pλ≡4mλ(see the\nanalytical results) and thus allow to extract the SO couplin g\nconstant directly from these strong features in momentum\nspace. The other arrows correspond to the other important\nwavevectors that can be observed in these FTs, as identified\nwith the help of the analytical results.\natpx=py= 0 which is due to the terms independent of\nFIG. 4. (Color) The FT of various SP LDOS component for a\nShiba state as a function of the SO coupling λand ofpy(for\npx= 0 - vertical cut). We take t= 1,µ= 3,δ= 0.01,∆s=\n0.2,Jz= 2.\nposition in the SP LDOS.\nThe most important observation is that all the compo-\nnents of the FT of the SP LDOS exhibit a strong feature\nat wave vector pλ. Thus an experimental observation of\nthis feature via spin-polarized STM would allow one to\nread-off directly the value of the SO coupling. The spin\norbit can also be read-off from the distance between the\n2p+and 2p−peaks, though the intensity of these fea-\ntures is not as strong. This appears clearly in Fig. 4, in\nwhich we plot a horizontal cut though two of the FT –\nSP LDOS above as a function of the SO coupling λ.\nNote that the only wave vector present in the non-\npolarized LDOS is 2 mv, which has only a very weak de-\npendence on λfor not too large values of the SO with re-\nspect to the Fermi velocity, thus it is quasi-impossible to\ndetermine the SO coupling from a measurement without\nspin resolution. Note also the typical two-dimensional\n1/rdecay of the Friedel oscillations is overlapping in this\ncase with an exponential decay with wave vector ps.\nB. Comparison to the metallic phase\nAsimilaranalysiscanbe performedforimpurity states\nforming in the vicinity of a magnetic impurity in a metal-\nlic system. Here the classical magnetic impurity does not\nlead to any localized bound states at a specific energy,\nand the intensity of the impurity contribution is roughly\nindependent of energy.\nThus in Fig. 5 we plot the FT of the impurity con-\ntribution to the LDOS and SP LDOS at a fixed energy\nE= 0.1. We note that we have similar features to those\nobserved in the SC regime, with the main differences be-\ning that the long-wavelengthcentral features are now ab-\nsent, andthat the FT peaksaremuch sharperthan in the\nSC regime. This behavior can be explained from the an-\nalytical expressions of the non-polarized and SP LDOS,\nwhose derivation is presented in Appendix B. The results7\nz-impurity x-impurity\nJz(Jx=Jy= 0) Jx(Jy=Jz= 0)\nFIG. 5. (Color) The FT of the impurity contributions to the\nnon-polarized and SP LDOS for an energy E= 0.1 and for\na magnetic impurity with Jz= 2 (left column), and Jx= 2\n(right column). We take the inverse quasiparticle lifetime\nδ= 0.03 and we set t= 1,µ= 3,λ= 0.5,∆s= 0. For\naz-impurity we depict the real part of the FT for δρand\nforSz, and the imaginary part for SxandSy, whereas for\nanx-impurity we take the imaginary part only for the Sz\ncomponent. UnlikeintheSCcase, thestrongpeaksappearing\nin the center and at pλare absent here. The arrows denote\nthe wavevectors of the observed features as identified from t he\nanalytical calculations.are presented below for an out-of-plane spin impurity:\nSx(r)∼J\n1+α2cosφr\nr/summationdisplay\nσσν2\nσ\npσsin2pσr,\nSy(r)∼J\n1+α2sinφr\nr/summationdisplay\nσσν2\nσ\npσsin2pσr,\nSz(r)∼ −J\n1+α22\nr/summationdisplay\nσν2\nσ\npσcos2pσr,\nρ(r)∼ −J\n1+α24αν2v2\nF\nv21/radicalbig\np2\nF+2mE+E2/v2×\n×sinpεr\nr, (20)\nwhile for an xdirected impurity (in-plane):\nSx(r)∼ −J\n1+α2/braceleftbigg\n2ν2v2\nF\nv21−cos2φr\nrcospεr/radicalbig\np2\nF+2mE+E2/v2\n+/summationdisplay\nσν2\nσ\npσ1+cos2φr\nrcos2pσr,\nSy(r)∼ −J\n1+α2sin2φr\nr/bracketleftbigg\n−2ν2v2\nF\nv2cospεr/radicalbig\np2\nF+2mE+E2/v2\n+/summationdisplay\nσν2\nσ\npσcos2pσr/bracketrightbigg\n,\nSz(r)∼ −J\n1+α2cosφr\nr/summationdisplay\nσσν2\nσ\npσsin2pσr,\nρ(r)∼ −J\n1+α2·α\nr·4ν2v2\nF\nv2sinpεr/radicalbig\np2\nF+2mE+E2/v2,(21)\nwithpF=mvF,pσ=pσ\nF+E/v/ne}ationslash= 0,pε≡p++p−=\n2(mv+E/v) andνσ=ν/bracketleftbig\n1−σλ\nv/bracketrightbig\n.\nNote that these expressions are very similar to those\nobtained in the SC regime, except that the wave vec-\ntors of the oscillations now do not include pλ. However,\nthis could still be read-off experimentally from the differ-\nence between p−andp+. Another important difference\nbetween the SC and non-SC regimes is the presence of\nthe exponentially decaying term in the expressions de-\nscribing the LDOS dependence for the Shiba states in\nthe SC regime. The Shiba states have an exponential\ndecay for distances larger than the superconducting co-\nherence length, while the impurity states in the non-SC\nregime only decay algebraically as 1 /r. In the Fourier\nspace this is translated into a much larger broadening of\nthe features corresponding to the Shiba states in the SC\nregime with respect to that of the features corresponding\nto the impurity contributions in metals. The width of\nthe peaks in the latter is solely controlled by the inverse\nquasiparticle lifetime δand is generally quite small.\nNote also that in both regimes one needs to use the\nspin-polarized LDOS and magnetic impurities to be able\nto extract the value of the SO coupling, while the non-\npolarized LDOS is not sensitive to this wavevector. Last\nbut notleast, asdescribedin Appendix B,theLDOS per-\nturbations induced by a non-magnetic impurity do not8\nshow any direct signature of the SO coupling (the only\ncontributing wavevector is 2 mvin the metallic regime,\nwhile in the SC regime no Shiba state form for a non-\nmagnetic impurity), thus the only manner to have access\nto the SO coupling is via spin-polarized STM in the pres-\nence of magnetic impurities.\nIV. ONE-DIMENSIONAL SYSTEMS\nWhile in one-dimensional systems superconductivity is\nnot intrinsic, a superconducting gap can be opened via\nproximitizing them with a superconducting substrate.\nFor such systems it is thus particularly interesting to\nstudy the FT of the SP LDOS for both the supercon-\nducting and non-superconducting regimes, as both these\nregimes can be achieved experimentally at low tempera-\nture for the same materials.\nWeconsidertheHamiltoniangivenbyEqs.(1-3), where\nwe setpy→0, and we perform a T-matrix analysis sim-\nilar to that described in the previous section for both\nthe SC and non-SC phases, for different directions of the\nmagnetic impurity. The wire is considered to be oriented\nalong thexdirection, and the SO coupling is oriented\nalongy.8,9We thus expect a similar and more exotic be-\nhavior for impurities directed along xandz, and a more\nclassical behavior for impurities with the spin parallel to\nthe direction of the SO, thus oriented along y.\nThe energies and wave functions of the Shiba states\ncan be found using the same procedure as for the two-\ndimensional systems (see Appendix C). This yields for\nthe energies of the states:\nE1,¯1=±1−α2\n1+α2∆s,whereα=J/v.\nThe FT of the positive energy state as a function of\nmomentum and the SO coupling is presented in Fig. 6\nfor a SC (left column) and non-SC state (right column),\nfor an impurity directed along z. For this situation the\nspin of the Shiba state has two non-zero components,\none parallel to the wire, and one parallel to the impurity\nspin, and these two components are depicted in Fig. 6.\nNote that, similar to the two-dimensional case, there is\na split of the FT features increasing linearly with the SO\ncoupling strength. Also note that in the non-SC phase\nthe central feature, whose wave vector is given by pλ,\nis absent, and that the FT features are broadened in\nthe SC regime with respect to the non-SC one. Also,\nsame as in the two-dimensional case, the SO affects the\nspin-polarized components but almost do not change the\nnon-polarized LDOS, as it can be seen in Fig. 6 where\nit appears that the non-polarized LDOS FT features do\nnot evolve with the SO coupling.\nThese results are confirmed by analytical calculations.\nBelow we give the spin components and the LDOS in\nthe SC state for an impurity directed along zobtained\nanalytically (see Appendix C), for the positive energy\nShiba state:SC case Non-SC case\nFIG. 6. (Color) The FT of various SP LDOS component\nfor a Shiba state (left column), and for an impurity state at\nE= 0.1 (right column), as a function of the SO coupling λ\nand of momentum p, for an impurity perpendicular to the\nwire and directed along z. We set t= 1,µ= 1. We take\n∆s= 0.2,Jz= 4,δ= 0.01 in the SC case and ∆ s= 0,Jz=\n2,δ= 0.05 in the non-SC case.\nSx(x) =1+α2\n4[2sinpλx+sin(2mv|x|+pλx−2θ)\n−sin(2mv|x|−pλx−2θ)]·e−2ω|x|/v\nSy(x) = 0\nSz(x) =−1+α2\n4[2cospλx+cos(2mv|x|+pλx−2θ)\n+cos(2mv|x|−pλx−2θ)]·e−2ω|x|/v\nρ(x) =1+α2\n2[1+cos(2mv|x|−2θ)]·e−2ω|x|/v(22)\nwhere tanθ=α. We also present the FT of the SP\nLDOSforthenon-SCphasefortheimpuritycontribution9\ncorresponding to the energy E(see Appendix D):\nSx(x) = +α\n1+α2·1\nπv[cos(pε|x|−pλx)−cos(pε|x|+pλx)]\nSy(x) = 0\nSz(x) = +α\n1+α2·1\nπv[sin(pε|x|−pλx)+sin(pε|x|+pλx)]\nρ(x) =−2α2\n1+α2·1\nπvcospεx\nAs before, in the expressions above pε= 2(mv+\nE/v),pλ= 2mλ.\nIndeed these calculations confirm our observations, in\nthe SC state the dominantwavevectorsare2 p±\nF= 2mv±\npλ, 2mvandpλ, while in the non-SC phase only pǫ±pλ,\nand 2mv.\nSimilar results are obtained if the impurity is oriented\nalongx, with the only difference that the xandzcompo-\nnents will be interchanged, up to on overall sign change\n(see Appendices C and D). For impurities parallel to y,\nand thus to the SO vector, we expect the SP LDOS to\nbe less exotic, and indeed in this case the only non-zero\ncomponent of the impurity SP LDOS is Sy. In the SC\nregime we thus find\nSx(x) = 0\nSy(x) =−(1+α2)[1+cos(2mv|x|−2θ)]·e−2ω|x|/v\nSz(x) = 0\nρ(x) = +(1+α2)[1+cos(2mv|x|−2θ)]·e−2ω|x|/v\nwhile in the non-SC regime we have\nSx(x) = 0\nSy(x) = +2α\n1+α2·1\nπvsinpε|x|\nSz(x) = 0\nρ(x) =−2α2\n1+α2·1\nπvcospεx\nWe see that Syexhibits features only at the 2 mvand\ncorrespondingly at the pǫwave vectors, same as the non-\npolarized LDOS, thus not allowing for the detection of\nthe SO coupling.\nFor intermediate directions of the impurity spin, all\nthree components will be present, with the xandzex-\nhibiting all the wave vectors, while the ycomponent\nsolely the 2 mv, and with relative intensities given by the\nrelative components of the impurity spin.\nThus, we conclude that, same as in the 2D case, the\nSO can be measured using spin-polarized STM and mag-\nnetic impurities; moreover, in the 1D case one needs to\nconsider impurities that have a non-zero component per-\npendicular to the direction of the SO.\nV. CONCLUSIONS\nWe have analyzed the formation of Shiba states and\nimpuritystatesin1Dand2Dsuperconductingandmetal-lic systems with Rashba SO coupling. In particular we\nhave studied the Fourier transform of the local density of\nstates of Shiba states in SCs and of the impurity states\nin metals, both non-polarized and spin-polarized. We\nhave shown that the spin-polarized density of states con-\ntains information that allows one to extract experimen-\ntally the strength of the SO coupling. In particular the\nfeatures observed in the FT of the SP LDOS split with\na magnitude proportional to the SO coupling strength.\nMoreover, the Friedel oscillations in the SP LDOS in the\nSC regime show a combination of wavelengths, out of\nwhich the SO length can be read off directly and non-\nambiguously. We note that these signatures are only vis-\nible in the spin-polarized quantities and in the presence\nof magnetic impurities. For non-spin-polarized measure-\nments, no such splitting is present and the wave vectors\nobserved in the FT of the SP LDOS basically do not\ndepend on the SO coupling. When comparing the re-\nsults for the SC Shiba states to the impurity contribu-\ntion in the metallic state and we find a few interesting\ndifferences, such as a broadening of the FT features cor-\nresponding to a spatial exponential decay of the Shiba\nstates compared to the non-SC case. Moreover, the FT\nof the SP LDOS in the SC regime exhibits extra fea-\ntures with a wavelengthequal to the SO length which are\nnot present in the non-SC phase. It would be interest-\ning to generalize our results to more realistic calculations\nwhich may include some specific lattice characteristics,\nmore realistic material-dependent tight-binding parame-\nters for the band structure and the SO coupling values.\nHowever, we should note that our results have a fully\ngeneral characteristic, independent of the band structure\nor other material characteristics, and that the features\nin the FT of the non-polarized LDOS will correspond to\nsplit features in the spin-polarized LDOS, and thus the\nspin-orbit can be measured unequivocally from the split\nobtained from the comparison between the non-polarized\nand spin-polarized measurements. We have checked that\nup to a rotation in the spin space our results hold also\nfor other types of SO coupling such as Dresselhaus.\nAccording to our knowledge, the FT-STM is a well-\nestablished experimental technique which does not deal\nwith large systematic errors.36–43The experimental data\npresented e.g. in Ref. 43 shows that the resolution in\nthe Fourier space (momentum space) reaches 0 .05˚A−1,\nwhereas a typical value of spin-orbit coupling wave vec-\ntorpλ∼0.15˚A−1(see e.g. Ref. 22), and thus it is suf-\nficient to resolve the features originating from the spin-\norbit coupling. Moreover, we would like to point that\nthe exponent e−2psrdefines in the real space how far the\nimpurity-induced states areextended, and it manifests in\nthe momentum space as the widening of the ring-like fea-\ntures appearing at particular momenta. The condition of\nresolving the spin-orbit is thus 2 ps< pλ, otherwise the\nwidening is large enough to blur the spin-orbit feature.\nThis condition can be rewritten in a more explicit way,10\nnamely\n1/radicalbig\n1+(λ/vF)2·α\n1+α2·∆s\nεF<λ\nvF\nFor any realistic parameters the first two factors on the\nleft side are of the order of unity, and ∆ s/εF∼10−3\nfor superconductors. However, for realistic values of the\nspin-orbit coupling λ, this inequality holds and therefore\nthere should not be any technical problem with resolving\nthose features.\nOur results can be tested using for example materials\nsuch as Pb, Bi, NbSe 2or InAs and InSb wires, which\nare known to have a strong SO coupling, using spin-\npolarized STM which is nowadays becoming more andmore available.35\nWhile finalizing this manuscript we became aware of\na recent work70focusing on issues similar to some of the\nsubjects (in particular the real space Friedel oscillations\nin the metallic regime) addressed in our work.\nACKNOWLEDGMENTS\nThis work is supported by the ERC Starting Indepen-\ndent Researcher Grant NANOGRAPHENE 256965. PS\nwould like to acknowledge interesting discussions with T.\nCren and financial support from the French Agence Na-\ntionale de la Recherche through the contract Mistral.\n∗vardan.kaladzhyan@cea.fr\n1M. Z. Hasan and C. L. Kane,\nRev. Mod. Phys. 82, 3045 (2010).\n2X.-L. Qi and S.-C. Zhang,\nRev. Mod. Phys. 83, 1057 (2011).\n3C. L. Kane and E. J. Mele,\nPhys. Rev. Lett. 95, 226801 (2005).\n4M. Konig and et al., Science 318, 766 (2007).\n5B. A. Bernevig, T. L. Hughes, and S.-C. Zhang,\nScience314, 1757 (2006).\n6D. Hsieh, Y. Xia, D. Qian, L. Wray, F. Meier, J. H. Dil,\nJ. Osterwalder, L. Patthey, A. V. Fedorov, H. Lin, A. Ban-\nsil, D. Grauer, Y. S. Hor, R. J. Cava, and M. Z. Hasan,\nPhys. Rev. Lett. 103, 146401 (2009).\n7D. Hsieh, , Y. Xia, D. Qian, L. Wray, J. H. Dil, F. Meier,\nJ. Osterwalder, L. Patthey, J. G. Checkelsky, N. P. Ong,\nA. V. Fedorov, H. Lin, A. Bansil, D. Grauer, Y. S. Hor,\nR. J. Cava, and M. Z. Hasan, Nature 460, 1101 (2009).\n8Y. Oreg, G. Refael, and F. von Oppen,\nPhys. Rev. Lett. 105, 177002 (2010).\n9R. M. Lutchyn, J. D. Sau, and S. Das Sarma,\nPhys. Rev. Lett. 105, 077001 (2010).\n10J. D. Sau, R. M. Lutchyn, S. Tewari, and S. Das Sarma,\nPhys. Rev. Lett. 104, 040502 (2010).\n11J. Alicea, Phys. Rev. B 81, 125318 (2010).\n12V. Mourik, K. Zuo, S. M. Frolov, P. S. M., E. P. A. M.\nBakkers, and L. P. Kouwenhoven, Science 336, 1003\n(2012).\n13M. T. Deng, C. L. Yu, G. Y. Huang, M. Larsson, P. Caroff,\nand H. Q. Xu, Nano lett. 12, 64146419 (2012).\n14A. Das, Y. Ronen, Y. Most, Y. Oreg, M. Heiblum, and\nH. Shtrikman, Nature Physics 8, 887 (2012).\n15H. O. H. Churchill, V. Fatemi, K. Grove-Rasmussen,\nM. T. Deng, P. Caroff, H. Q. Xu, and C. M. Marcus,\nPhys. Rev. B 87, 241401 (2013).\n16S.M. Albrecht, A.P.Higginbotham, M.Madsen, F.Kuem-\nmeth, T. S. Jespersen, J. Nygard, P. Krogstrup, and C. M.\nMarcus, Nature 531, 206 (2016).\n17A. P. Mackenzie and Y. Maeno,\nRev. Mod. Phys. 75, 657 (2003).\n18R.Joynt andL.Taillefer, Rev. Mod. Phys. 74, 235 (2002).\n19S. Sasaki, M. Kriener, K. Segawa, K. Yada,\nY. Tanaka, M. Sato, and Y. Ando,Phys. Rev. Lett. 107, 217001 (2011).\n20J. Sinova, S. O. Valenzuela, J. Wunderlich, C. H. Back,\nand T. Jungwirth, Rev. Mod. Phys. 87, 1213 (2015).\n21I.ˇZuti´ c, J. Fabian, and S. Das Sarma,\nRev. Mod. Phys. 76, 323 (2004).\n22A. Manchon, A. H. C. Koo, J. Nitta, S. M. Frolov, and\nR. A. Duine, Nature Materials 14, 871 (2015).\n23S. LaShell, B. A. McDougall, and E. Jensen,\nPhys. Rev. Lett. 77, 3419 (1996).\n24C. R. Ast, J. Henk, A. Ernst, L. Moreschini, M. C.\nFalub, D. Pacil´ e, P. Bruno, K. Kern, and M. Grioni,\nPhys. Rev. Lett. 98(2007), 10.1103/physrevlett.98.186807.\n25A. Varykhalov, D. Marchenko, M. R.\nScholz, E. D. L. Rienks, T. K. Kim,\nG. Bihlmayer, J. S´ anchez-Barriga, and O. Rader,\nPhys. Rev. Lett. 108(2012), 10.1103/physrevlett.108.066804.\n26M. Hoesch, T. Greber, V. Petrov, M. Muntwiler,\nM. Hengsberger, W. Auw¨ arter, and J. Osterwalder,\nJournal of Electron Spectroscopy and Related Phenomena 124, 263 (2002).\n27M. Hochstrasser, J. G. Tobin,\nE. Rotenberg, and S. D. Kevan,\nPhys. Rev. Lett. 89(2002), 10.1103/physrevlett.89.216802.\n28M. Hoesch, M. Muntwiler, V. N. Petrov,\nM. Hengsberger, L. Patthey, M. Shi,\nM. Falub, T. Greber, and J. Osterwalder,\nPhys. Rev. B 69(2004), 10.1103/physrevb.69.241401.\n29T. Hirahara, K. Miyamoto, I. Matsuda,\nT. Kadono, A. Kimura, T. Nagao, G. Bihlmayer,\nE. V. Chulkov, S. Qiao, K. Shimada, H. Na-\nmatame, M. Taniguchi, and S. Hasegawa,\nPhys. Rev. B 76(2007), 10.1103/physrevb.76.153305.\n30D.Hsieh, Y.Xia, L.Wray, D.Qian, A.Pal, J. H.Dil, J. Os-\nterwalder, F. Meier, G. Bihlmayer, C. L. Kane, Y. S. Hor,\nR. J. Cava, and M. Z. Hasan, Science 323, 919 (2009).\n31K. Yaji, Y.Ohtsubo, S.Hatta, H.Okuyama, K. Miyamoto,\nT. Okuda, A. Kimura, H. Namatame, M. Taniguchi, and\nT. Aruga, Nature Communications 1, 1 (2010).\n32C. Jozwiak, Y. L. Chen, A. V. Fedorov, J. G. An-\nalytis, C. R. Rotundu, A. K. Schmid, J. D. Den-\nlinger, Y.-D. Chuang, D.-H. Lee, I. R. Fisher, R. J.\nBirgeneau, Z.-X. Shen, Z. Hussain, and A. Lanzara,\nPhys. Rev. B 84(2011), 10.1103/physrevb.84.165113.11\n33F. Kuemmeth, S. Ilani, D. C. Ralph, and P. L. McEuen,\nNature , 448 (2004).\n34C. Fasth, A. Fuhrer, L. Samuelson, V. N. Golovach, and\nD. Loss, Phys. Rev. Lett. 98, 266801 (2007).\n35R.Wiesendanger,Reviews of Modern Physics 81, 1495 (2009).\n36Y. Hasegawa and P. Avouris,\nPhys. Rev. Lett. 71, 1071 (1993).\n37M. F. Crommie, C. P. Lutz, and D. M. Eigler,\nNature363, 524 (1993).\n38P. T. Sprunger, L. Petersen, E. W. Plummer, E. Lægs-\ngaard, and F. Besenbacher, Science 275, 1764 (1997).\n39L. Petersen, P. Hofmann, E. Plum-\nmer, and F. Besenbacher,\nJournal of Electron Spectroscopy and Related Phenomena 109, 97 (2000).\n40P. Hofmann, B. G. Briner, M. Doering, H.-P.\nRust, E. W. Plummer, and A. M. Bradshaw,\nPhys. Rev. Lett. 79, 265 (1997).\n41F. Vonau, D. Aubel, G. Gewinner, C. Pirri,\nJ. C. Peruchetti, D. Bolmont, and L. Simon,\nPhys. Rev. B 69, 081305 (2004).\n42F. Vonau, D. Aubel, G. Gewinner, S. Zabrocki,\nJ. C. Peruchetti, D. Bolmont, and L. Simon,\nPhys. Rev. Lett. 95, 176803 (2005).\n43L. Simon, C. Bena, F. Vonau, M. Cranney, and D. Aubel,\nJournal of Physics D: Applied Physics 44, 464010 (2011).\n44J. I. Pascual, G. Bihlmayer, Y. M. Koroteev, H.-P.\nRust, G. Ceballos, M. Hansmann, K. Horn, E. V.\nChulkov, S. Bl¨ ugel, P. M. Echenique, and P. Hofmann,\nPhys. Rev. Lett. 93, 196802 (2004).\n45J. E. Hoffman, K. McElroy, D.-H. Lee, K. M.\nLang, H. Eisaki, S. Uchida, and J. C. Davis,\nScience297, 1148 (2002).\n46K. McElroy, R. W. Simmonds, J. E. Hoffman, D.-H. Lee,\nJ. Orenstein, H. Eisaki, S. Uchida, and J. C. Davis,\nNature422, 592 (2003).\n47M. Vershinin, S. Misra, S. Ono, Y. Abe, Y. Ando, and\nA. Yazdani, Science 303, 1995 (2004).\n48L. Yu, Acta Physica Sinica 21, 75 (1965).\n49H. Shiba, Progress of Theoretical Physics 40, 435 (1968).\n50A. I. Rusinov, Soviet Journal of Experimental and Theo-\nretical Physics Letters 9, 85 (1969).\n51A. V. Balatsky, I. Vekhter, and J.-X. Zhu,\nRev. Mod. Phys. 78, 373 (2006).\n52A. Yazdani, B. A. Jones, C. P. Lutz, M. F. Crommie, and\nD. M. Eigler, Science 275, 1767 (1997).53S.-H. Ji, T. Zhang, Y.-S. Fu, X. Chen, X.-C. Ma,\nJ. Li, W.-H. Duan, J.-F. Jia, and Q.-K. Xue,\nPhys. Rev. Lett. 100, 226801 (2008).\n54N. Hatter, B. W. Heinrich, M. Ruby, J. I. Pascual, and\nK. J. Franke, Nat. Comm. 6, 8988 (2015).\n55G. M´ enard and et al., Nature Physics 11, 1013 (2015).\n56M. M. Ugeda, A. J. Bradley, Y. Zhang, S. Onishi, Y. Chen,\nW. Ruan, C. Ojeda-Aristizabal, H. Ryu, M. T. Edmonds,\nH.-Z. Tsai, A. Riss, S.-K. Mo, D. Lee, A. Zettl, Z. Hussain,\nZ.-X. Shen, and M. F. Crommie, Nat Phys 12, 92 (2015).\n57P. M. R. Brydon, S. Das Sarma, H.-Y. Hui, and J. D. Sau,\nPhys. Rev. B 91, 064505 (2015).\n58Y. Kim, J. Zhang, E. Rossi, and R. M. Lutchyn,\nPhys. Rev. Lett. 114, 236804 (2015).\n59V. Kaladzhyan, C. Bena, and P. Simon,\nPhys. Rev. B 93, 214514 (2016).\n60V. Kaladzhyan, C. Bena, and P. Simon, arXiv:1606.06338\n(2016).\n61C. Brun and et al., Nature Physics 10, 444 (2014).\n62L. Petersen, L. Brgi, H. Brune, F. Besenbacher, and\nK. Kern, Surface Science 443, 154 (1999).\n63G. Mahan, Many-Particle Physics , Physics of Solids and\nLiquids (Springer, 2000).\n64H. Bruus and K. Flensberg,\nMany-Body Quantum Theory in Condensed Matter Physics: An In troduction ,\nOxford Graduate Texts (OUP Oxford, 2004).\n65T. Meng, J. Klinovaja, S. Hoffman, P. Simon, and D. Loss,\nPhys. Rev. B 92, 064503 (2015).\n66F. Pientka, L. I. Glazman, and F. von Oppen,\nPhys. Rev. B 88, 155420 (2013).\n67In this paper, we use the plural when refering to Shiba\nstates in order to facilitate the discussion. However, it\nshould be kept in mind that for a given localized mag-\nnetic impurity, there is a single Shiba state with particle\nand hole components whose real space wave function can\nactually behave differently.\n68A.Sakurai,Progress of Theoretical Physics 44, 1472 (1970).\n69M. I. Salkola, A. V. Balatsky, and J. R. Schrieffer,\nPhys. Rev. B 55, 12648 (1997).\n70N. Khotkevych, Y. Kolesnichenko, and J. van Ruitenbeek,\narXiv:1601.03154v1 (2016).\nAppendix A: Analytical calculation of the Shiba states wave functions for a 2D system\nWecancalculateanalyticallythenon-polarizedandtheSPLDOSforth eShibastatesexploitingthemodeldescribed\nby the Hamiltonians in Eqs. (1-3). All the integrations below are perf ormed using a linearization around the Fermi\nenergy. The energies of the Shiba states can be found by solving th e corresponding eigenvalue equation66\n[I4−VG0(E,r=0)]Φ(0) = 0 (A1)\nwhereG0(E,r) is the retarded Green’s function in real space obtained by a Fourie r transform from the retarded\nGreen’s function in momentum space G0(E,p) = [(E+iδ)I4−H0(p)]−1, whereδis the inverse quasiparticle lifetime.\nIn all the calculations below we take the limit of δ→+0, and we specify + i0 only in the cases when it affects the\nresults. The wave functions of the Shiba states at r= 0 are given by the eigenfunctions obtained from the equation\nabove. Their spatial dependence is determined using\nΦ(r) =G0(E,r)VΦ(0) (A2)12\nConsequently, the non-polarized and the SP LDOS are given by\nρ(E,r) = Φ†(r)/parenleftbigg\n0 0\n0σ0/parenrightbigg\nΦ(r), (A3)\nS(E,r) = Φ†(r)/parenleftbigg\n0 0\n0σ/parenrightbigg\nΦ(r). (A4)\nThus, in order to find the energies and the wave functions corresp onding to the Shiba states we need to find the\nreal-space Green’s function. This is obtained simply by a Fourier tran sform of the unperturbed Green’s function in\nmomentum space, G0(E,p). We start by writing down the unperturbed Green’s function in mom entum space, which\nis given by G0(E,p) =1\n2/summationtext\nσ=±Gσ\n0(E,p), where\nGσ\n0(E,p) =−1\nξ2σ+ω2/parenleftbigg\n1iσe−iφp\n−iσeiφp1/parenrightbigg\n⊗/parenleftbigg\nE+ξσ∆s\n∆sE−ξσ/parenrightbigg\n, (A5)\nwhereω=/radicalbig\n∆2s−E2, ξσ=ξp+σλp. Toobtainitsreal-spacedependence oneneedstoperformthe Fo uriertransform:\nGσ\n0(E,r) =/integraldisplaydp\n(2π)2Gσ\n0(E,p)eipr\nWe will have four types of integrals:\nXσ\n0(r) =−/integraldisplaydp\n(2π)2eipr\nξ2σ+ω2(A6)\nXσ\n1(r) =−/integraldisplaydp\n(2π)2ξσeipr\nξ2σ+ω2(A7)\nXσ\n2(s,r) =−/integraldisplaydp\n(2π)2−isσeisφpeipr\nξ2σ+ω2(A8)\nXσ\n3(s,r) =−/integraldisplaydp\n(2π)2−isσeisφpξσeipr\nξ2σ+ω2(A9)\nSince the spectrum is split by SO coupling, there will be two Fermi mome nta which can be found the following way:\np2\n2m+σλp−εF= 0, pσ\nF=−σλ+/radicalbig\nλ2+2εF/m\n1/m\nForp>0 we linearize the spectrum around the Fermi momenta, thus:\nξσ≈/parenleftbiggpσ\nF\nm+σλ/parenrightbigg\n(p−pσ\nF) =/radicalbig\nλ2+2εF/m(p−pσ\nF)≡v(p−pσ\nF),\nthereforep=pF+ξσ/v, wherev=/radicalbig\nv2\nF+λ2. We rewrite:\ndp\n(2π)2=m\n2π/bracketleftbigg\n1−σλ\nv/bracketrightbigg\ndξσdφ\n2π=νσdξσdφ\n2π,\nwhereνσ=ν/bracketleftbig\n1−σλ\nv/bracketrightbig\n, withν=m/2π. Due to the symmetry all the integrals are zero at r=0except for the first\none, namely,\nXσ\n0(0) =−νσπ\nω. (A10)\nAll the coordinate dependences can be calculated using the formalis m introduced in Ref. 60. Finally we get:\nXσ\n0(r) =−2νσ·1\nω·ℑK0[−i(1+iΩσ)pσ\nFr] (A11)\nXσ\n1(r) =−2νσ·ℜK0[−i(1+iΩσ)pσ\nFr] (A12)\nXσ\n2(s,r) = 2sσνσ·1\nω·eisφr·ℜK1[−i(1+iΩσ)pσ\nFr] (A13)\nXσ\n3(s,r) =−2sσνσ·eisφr·ℑK1[−i(1+iΩσ)pσ\nFr], (A14)13\nwhere Ω σ=ω/pσ\nFvdefines the inverse superconducting decay length, and pS=ω/v. Therefore, the Green’s function\ncan be written as\nGσ\n0(E,r) =\nEXσ\n0(r)+Xσ\n1(r)EXσ\n2(−,r)+Xσ\n3(−,r) ∆ sXσ\n0(r) ∆ sXσ\n2(−,r)\nEXσ\n2(+,r)+Xσ\n3(+,r)EXσ\n0(r)+Xσ\n1(r) ∆ sXσ\n2(+,r) ∆ sXσ\n0(r)\n∆sXσ\n0(r) ∆ sXσ\n2(−,r)EXσ\n0(r)−Xσ\n1(r)EXσ\n2(−,r)−Xσ\n3(−,r)\n∆sXσ\n2(+,r) ∆ sXσ\n0(r)EXσ\n2(+,r)−Xσ\n3(+,r)EXσ\n0(r)−Xσ\n1(r)\n.\n(A15)\nThus we have:\nG0(E,r=0) =−πν/radicalbig\n∆2s−E2/parenleftbigg\nEσ0∆sσ0\n∆sσ0Eσ0/parenrightbigg\n. (A16)\n1. z-impurity\nThe coordinate dependence of the eigenfunctions is given by\nΦ¯1(r) = +Jz\n2/summationdisplay\nσ=±\n(E¯1−∆s)Xσ\n0(r)+Xσ\n1(r)\n(E¯1−∆s)Xσ\n2(+,r)+Xσ\n3(+,r)\n−(E¯1−∆s)Xσ\n0(r)+Xσ\n1(r)\n−(E¯1−∆s)Xσ\n2(+,r)+Xσ\n3(+,r)\n,Φ1(r) =−Jz\n2/summationdisplay\nσ=±\n(E1+∆s)Xσ\n2(−,r)+Xσ\n3(−,r)\n(E1+∆s)Xσ\n0(r)+Xσ\n1(r)\n(E1+∆s)Xσ\n2(−,r)−Xσ\n3(−,r)\n(E1+∆s)Xσ\n0(r)−Xσ\n1(r)\n.\n(A17)\nUsing these expressions we can compute the asymptotic behavior o f the non-polarized and SP LDOS in coordinate\nspace for the state with positive energy (thus we omit index 1 below) :\nSx(r) = +J2\nz/parenleftbigg\n1+1\nα2/parenrightbigg/braceleftigg/summationdisplay\nσσν2\nσcos(2pσ\nFr−θ)\npσ\nF+2ν2v2\nF\nv2·sinpλr\npF/bracerightigg\n·e−2psr\nrcosφr (A18)\nSy(r) = +J2\nz/parenleftbigg\n1+1\nα2/parenrightbigg/braceleftigg/summationdisplay\nσσν2\nσcos(2pσ\nFr−θ)\npσ\nF+2ν2v2\nF\nv2·sinpλr\npF/bracerightigg\n·e−2psr\nrsinφr (A19)\nSz(r) =−J2\nz/parenleftbigg\n1+1\nα2/parenrightbigg/braceleftigg/summationdisplay\nσν2\nσsin(2pσ\nFr−θ)\npσ\nF−2ν2v2\nF\nv2·cospλr\npF/bracerightigg\n·e−2psr\nr(A20)\nρ(r) = +J2\nz/parenleftbigg\n1+1\nα2/parenrightbigg/braceleftbigg\n2ν2\nmv+2ν2v2\nF\nv2·sin(2mvr−θ)\npF/bracerightbigg\n·e−2psr\nr(A21)\nwith tanθ=/braceleftigg\n2α\n1−α2,ifα/ne}ationslash= 1\n+∞,ifα= 1,andpλ= 2mλ. Performing the Fourier transforms of these expressions we can\nobtain information about the main features and symmetries that we observe in momentum space:\nSx(p) = +2πiJ2\nz/parenleftbigg\n1+1\nα2/parenrightbigg\ncosφp+∞/integraldisplay\n0drJ1(pr)/braceleftigg/summationdisplay\nσσν2\nσcos(2pσ\nFr−θ)\npσ\nF+2ν2v2\nF\nv2·sinpλr\npF/bracerightigg\n·e−2psr(A22)\nSy(p) = +2πiJ2\nz/parenleftbigg\n1+1\nα2/parenrightbigg\nsinφp+∞/integraldisplay\n0drJ1(pr)/braceleftigg/summationdisplay\nσσν2\nσcos(2pσ\nFr−θ)\npσ\nF+2ν2v2\nF\nv2·sinpλr\npF/bracerightigg\n·e−2psr(A23)\nSz(p) =−2πJ2\nz/parenleftbigg\n1+1\nα2/parenrightbigg+∞/integraldisplay\n0drJ0(pr)/braceleftigg/summationdisplay\nσν2\nσsin(2pσ\nFr−θ)\npσ\nF−2ν2v2\nF\nv2·cospλr\npF/bracerightigg\n·e−2psr(A24)\nρ(p) = +2πJ2\nz/parenleftbigg\n1+1\nα2/parenrightbigg+∞/integraldisplay\n0drJ0(pr)/braceleftbigg\n2ν2\nmv+2ν2v2\nF\nv2·sin(2mvr−θ)\npF/bracerightbigg\n·e−2psr(A25)14\n2. x-impurity\nThe coordinate dependence of the eigenfunctions is given by\nΦ¯1(r) = +Jx\n2/summationdisplay\nσ=±\n+(E¯1−∆s)[Xσ\n0(r)+Xσ\n2(−,r)]+Xσ\n1(r)+Xσ\n3(−,r)\n+(E¯1−∆s)[Xσ\n0(r)+Xσ\n2(+,r)]+Xσ\n1(r)+Xσ\n3(+,r)\n−(E¯1−∆s)[Xσ\n0(r)+Xσ\n2(−,r)]+Xσ\n1(r)+Xσ\n3(−,r)\n−(E¯1−∆s)[Xσ\n0(r)+Xσ\n2(+,r)]+Xσ\n1(r)+Xσ\n3(+,r)\n, (A26)\nΦ1(r) =−Jx\n2/summationdisplay\nσ=±\n+(E1+∆s)[Xσ\n0(r)−Xσ\n2(−,r)]+Xσ\n1(r)−Xσ\n3(−,r)\n−(E1+∆s)[Xσ\n0(r)−Xσ\n2(+,r)]−Xσ\n1(r)+Xσ\n3(+,r)\n+(E1+∆s)[Xσ\n0(r)−Xσ\n2(−,r)]−Xσ\n1(r)+Xσ\n3(−,r)\n−(E1+∆s)[Xσ\n0(r)−Xσ\n2(+,r)]+Xσ\n1(r)−Xσ\n3(+,r)\n. (A27)\nFor the positive energy state we compute the asymptotic behavior of the non-polarized and SP LDOS in coordinate\nspace. We write Sx(r) =Ss\nx(r)+Sa\nx(r):\nSs\nx(r) =−J2\nx/parenleftbigg\n1+1\nα2/parenrightbigg/braceleftigg/summationdisplay\nσν2\nσ1+sin(2pσ\nFr−2β)\npσ\nF+2ν2v2\nF\nv2·cospλr+sin(2mvr−2β)\npF/bracerightigg\n·e−2psr\nr(A28)\nSa\nx(r) = +J2\nx/parenleftbigg\n1+1\nα2/parenrightbigg/braceleftigg/summationdisplay\nσν2\nσ1−sin(2pσ\nFr−2β)\npσ\nF−2ν2v2\nF\nv2·cospλr+sin(2mvr−2β)\npF/bracerightigg\n·e−2psr\nrcos2φr(A29)\nSy(r) = +J2\nx/parenleftbigg\n1+1\nα2/parenrightbigg/braceleftigg/summationdisplay\nσν2\nσ1−sin(2pσ\nFr−2β)\npσ\nF−2ν2v2\nF\nv2·cospλr−sin(2mvr−θ)\npF/bracerightigg\n·e−2psr\nrsin2φr(A30)\nSz(r) =−J2\nx/parenleftbigg\n1+1\nα2/parenrightbigg/braceleftigg\n2/summationdisplay\nσσν2\nσcos(2pσ\nFr−θ)\npσ\nF+4ν2v2\nF\nv2·sinpλr\npF/bracerightigg\n·e−2psr\nrcosφr (A31)\nρ(r) = +J2\nx/parenleftbigg\n1+1\nα2/parenrightbigg/braceleftbigg\n4ν2\nmv+4ν2v2\nF\nv2·sin(2mvr−θ)\npF/bracerightbigg\n·e−2psr\nr(A32)\nwith tanβ=α. Sameasbefore, performingthe Fouriertransformsoftheseex pressionsallowsus to obtaininformation\nabout the most important features and symmetries we observe in m omentum space:\nSs\nx(p) =−2πJ2\nx/parenleftbigg\n1+1\nα2/parenrightbigg+∞/integraldisplay\n0drJ0(pr)/braceleftigg/summationdisplay\nσν2\nσ1+sin(2pσ\nFr−2β)\npσ\nF+ (A33)\n+2ν2v2\nF\nv2·cospλr+sin(2mvr−2β)\npF/bracerightbigg\n·e−2psr(A34)\nSa\nx(p) =−2πJ2\nx/parenleftbigg\n1+1\nα2/parenrightbigg\ncos2φp+∞/integraldisplay\n0drJ2(pr)/braceleftigg/summationdisplay\nσν2\nσ1−sin(2pσ\nFr−2β)\npσ\nF− (A35)\n−2ν2v2\nF\nv2·cospλr+sin(2mvr−2β)\npF/bracerightbigg\n·e−2psr(A36)\nSy(p) =−2πJ2\nx/parenleftbigg\n1+1\nα2/parenrightbigg\nsin2φp+∞/integraldisplay\n0drJ2(pr)/braceleftigg/summationdisplay\nσν2\nσ1−sin(2pσ\nFr−2β)\npσ\nF− (A37)\n−2ν2v2\nF\nv2·cospλr−sin(2mvr−θ)\npF/bracerightbigg\n·e−2psr(A38)\nSz(p) =−2πiJ2\nx/parenleftbigg\n1+1\nα2/parenrightbigg\ncosφp+∞/integraldisplay\n0drJ1(pr)/braceleftigg\n2/summationdisplay\nσσν2\nσcos(2pσ\nFr−θ)\npσ\nF+4ν2v2\nF\nv2·sinpλr\npF/bracerightigg\n·e−2psr(A39)\nρ(p) = +2πJ2\nx/parenleftbigg\n1+1\nα2/parenrightbigg+∞/integraldisplay\n0drJ0(pr)/braceleftbigg\n4ν2\nmv+4ν2v2\nF\nv2·sin(2mvr−θ)\npF/bracerightbigg\n·e−2psr(A40)15\nAppendix B: The SPDOS for a 2D metallic system in the presence of a magnetic impurity\nThe low-energy Hamiltonian can be written as\nH0=ξpσ0+λ(pyσx−pxσy) =/parenleftbigg\nξpiλp−\n−iλp+ξp/parenrightbigg\n, (B1)\nwhereξp=p2\n2m−εF. The corresponding spectrum is given by E=ξp±λp. The retarded Green’s function reads\nG0(E,p) =1\n(E−ξp+i0)2−λ2p2/parenleftbigg\nE−ξp+i0iλp−\n−iλp+E−ξp+i0/parenrightbigg\n(B2)\nTo compute the eigenvalues for a single localized impurity we calculate\nG0(E,r=0) =/integraldisplaydp\n(2π)2E−ξp+i0\n(E−ξp+i0)2−λ2p2/parenleftbigg\n1 0\n0 1/parenrightbigg\n=1\n2/summationdisplay\nσ/integraldisplaydp\n(2π)21\nE−ξσ+i0/parenleftbigg\n1 0\n0 1/parenrightbigg\n,\nwhereξσ=ξp+σλp. Forp>0 we linearize the spectrum around Fermi momenta, thus:\nξσ≈/parenleftbiggpσ\nF\nm+σλ/parenrightbigg\n(p−pσ\nF) =/radicalbig\nλ2+2εF/m(p−pσ\nF)≡v(p−pσ\nF),\nwithpσ\nF=m[−σλ+v], and thus we rewrite:\ndp\n(2π)2=m\n2π/bracketleftbigg\n1−σλ\nv/bracketrightbigg\ndξσdφ\n2π=νσdξσdφ\n2π,\nwhereνσ=ν/bracketleftbig\n1−σλ\nv/bracketrightbig\n, withν=m/2π. Thus we get:\n/integraldisplaydp\n(2π)21\nE−ξσ+i0=νσ/integraldisplay\ndξσ1\nE−ξσ+i0=−iπνσ,\nand therefore:\nG0(E,r=0) =1\n2/summationdisplay\nσ(−iπνσ)/parenleftbigg\n1 0\n0 1/parenrightbigg\n=−iπν/parenleftbigg\n1 0\n0 1/parenrightbigg\n(B3)\nSince there is no energy dependence, there will be no impurity-induc ed states. To find the coordinate dependence of\nthe Green’s function we calculate:\nXσ\n0(r) =/integraldisplaydp\n(2π)2eipr\nE−ξσ+i0(B4)\nXσ\n1(s,r) =/integraldisplaydp\n(2π)2−iseisφpeipr\nE−ξσ+i0(B5)\nBelow we use the Sokhotsky formula:\n1\nx+i0=P1\nx−iπδ(x)\nXσ\n0(r) =/integraldisplaydp\n(2π)2eipr\nE−ξσ+i0=νσ/integraldisplay\ndξσ/integraldisplaydφp\n2πeiprcos(φp−φr)\nE−ξσ+i0=νσ/integraldisplay\ndξσJ0[(pσ\nF+ξσ/v)r]\nE−ξσ+i0=\n=νσ/braceleftbigg\nP/integraldisplay\ndξσJ0[(pσ\nF+ξσ/v)r]\nE−ξσ−iπ/integraldisplay\ndξσδ(E−ξσ)J0[(pσ\nF+ξσ/v)r]/bracerightbigg\n=♠\nWe calculate separately the first integral:\nP/integraldisplay\ndξσJ0[(pσ\nF+ξσ/v)r]\nE−ξσ=2\nπ+∞/integraldisplay\n1du√\nu2−1P/integraldisplay\ndξσsin[(pσ\nF+ξσ/v)r]\nE−ξσ=\n=2\nπℑ+∞/integraldisplay\n1du√\nu2−1P/integraldisplay\ndξσei(pσ\nF+ξσ/v)r\nE−ξσ=2\nπℑ+∞/integraldisplay\n1du√\nu2−1eipσru·P/integraldisplay\ndxe−ir\nvx\nx=♣16\nP/integraldisplay\ndxe−ir\nvx\nx=P/integraldisplaycosr\nvx\nxdx−iP/integraldisplaysinr\nvx\nxdx= 0−iπ=−iπ\nTherefore:\n♣=−2ℑ+∞/integraldisplay\n1ieipσru\n√\nu2−1du=−2+∞/integraldisplay\n1cospσru√\nu2−1du=πY0(pσr), pσ/ne}ationslash= 0\n♠=πνσ[Y0(pσr)−iJ0(pσr)].\nThe second integral is\nXσ\n1(s,r) =/integraldisplaydp\n(2π)2−iseisφpeipr\nE−ξσ+i0=νσ/integraldisplay\ndξσ/integraldisplaydφp\n2π−iseisφpeiprcos(φp−φr)\nE−ξσ+i0=seisφrνσ/integraldisplay\ndξσJ1[(pσ\nF+ξσ/v)r]\nE−ξσ+i0=\n=seisφr·νσ/braceleftbigg\nP/integraldisplay\ndξσJ1[(pσ\nF+ξσ/v)r]\nE−ξσ−iπ/integraldisplay\ndξσδ(E−ξσ)J1[(pσ\nF+ξσ/v)r]/bracerightbigg\n=♥\nWe calculate separately the first integral:\nP/integraldisplay\ndξσJ1[(pσ\nF+ξσ/v)r]\nE−ξσ=P/integraldisplay\ndxJ1[(pσ−x/v)r]\nx=−∂\n∂rP/integraldisplay\ndxJ0[(pσ−x/v)r]\nx(pσ−x/v)=\n=−∂\n∂rP/integraldisplay\ndyJ0[(pσ−y)r]\ny(pσ−y)=−∂\n∂(pσr)/bracketleftbigg\nP/integraldisplay\ndyJ0[(pσ−y)r]\ny+P/integraldisplay\ndyJ0[(pσ−y)r]\npσ−y/bracketrightbigg\n=\n=−∂\n∂(pσr)2\nπℑ+∞/integraldisplay\n1du√\nu2−1/bracketleftbigg\nP/integraldisplayei(pσ−y)ru\nydy+P/integraldisplayei(pσ−y)ru\npσ−ydy/bracketrightbigg\n=−2∂\n∂(pσr)ℑ+∞/integraldisplay\n1idu√\nu2−1/bracketleftbig\n1−eipσru/bracketrightbig\n=\n=−2+∞/integraldisplay\n1usinpσru√\nu2−1du= 2∂\n∂(pσr)+∞/integraldisplay\n1cospσru√\nu2−1du=−π∂\n∂(pσr)Y0(pσr) =πY1(pσr), pσ/ne}ationslash= 0\nTherefore:\n♥=πνσ[Y1(pσr)−iJ1(pσr)].\nFinally:\nXσ\n0(r) =πνσ[Y0(pσr)−iJ0(pσr)] (B6)\nXσ\n1(s,r) =seisφr/braceleftig\nπνσ[Y1(pσr)−iJ1(pσr)]/bracerightig\n≡seisφr˜Xσ\n1(r), (B7)\nwherepσ=pσ\nF+E/v/ne}ationslash= 0. Thus the Green’s function for r/ne}ationslash=0can be written as:\nG0(E,r) =1\n2/summationdisplay\nσ/parenleftbigg\nXσ\n0(r)−σe−iφr˜Xσ\n1(r)\nσeiφr˜Xσ\n1(r)Xσ\n0(r)/parenrightbigg\n(B8)\nBelow we compute the T-matrix for different types of impurities. Imp urity potentials take the following forms:\nVsc=U/parenleftbigg\n1 0\n0 1/parenrightbigg\n, Vz=Jz/parenleftbigg\n1 0\n0−1/parenrightbigg\n, Vx=Jx/parenleftbigg\n0 1\n1 0/parenrightbigg\n(B9)\nThe corresponding T-matrices are\nTsc=U\n1+iπνU/parenleftbigg\n1 0\n0 1/parenrightbigg\n, Tz=/parenleftbiggJ\n1+iπνJ0\n0−J\n1−iπνJ/parenrightbigg\n, T x=J\n1+π2ν2J2/parenleftbigg\n−iπνJ 1\n1−iπνJ/parenrightbigg\n(B10)\nFor each type of impurity we can compute the SP and non-polarized L DOS using\n∆G(E,r) =G0(E,−r)T(E)G0(E,r) (B11)17\nSx(E,r) =−1\nπ[ℑ∆G12+ℑ∆G21] (B12)\nSy(E,r) =−1\nπ[ℜ∆G12−ℜ∆G21] (B13)\nSz(E,r) =−1\nπ[ℑ∆G11−ℑ∆G22] (B14)\n∆ρ(E,r) =−1\nπ[ℑ∆G11+ℑ∆G22] (B15)\nAsymptotic expansions of Bessel functions\nSince the integrals are expressed in terms of Neumann function and Bessel function of the first kind, we give their\nasymptotic behavior for x→+∞:\nJ0(x)∼+/radicalbigg\n2\nπxcos/parenleftig\nx−π\n4/parenrightig\n, J 1(x)∼ −/radicalbigg\n2\nπxcos/parenleftig\nx+π\n4/parenrightig\nY0(x)∼ −/radicalbigg\n2\nπxcos/parenleftig\nx+π\n4/parenrightig\n, Y 1(x)∼ −/radicalbigg\n2\nπxcos/parenleftig\nx−π\n4/parenrightig\nFourier transforms in 2D\nF[f(r)] = 2π+∞/integraldisplay\n0rJ0(pr)f(r)dr (B16)\nF[cosφrf(r)] = 2πicosφp·+∞/integraldisplay\n0rJ1(pr)f(r)dr,F[sinφrf(r)] = 2πisinφp·+∞/integraldisplay\n0rJ1(pr)f(r)dr(B17)\nF[cos2φrf(r)] =−2πcos2φp+∞/integraldisplay\n0rJ2(pr)f(r)dr,F[sin2φrf(r)] =−2πsin2φp+∞/integraldisplay\n0rJ2(pr)f(r)dr(B18)\n1. z-impurity\nWe denote α=πνJand write the asymptotic expansions of the non-polarized and SP LD OS components in\ncoordinate space:\nSx(r)∼J\n1+α2cosφr\nr/summationdisplay\nσσν2\nσ\npσsin2pσr (B19)\nSy(r)∼J\n1+α2sinφr\nr/summationdisplay\nσσν2\nσ\npσsin2pσr (B20)\nSz(r)∼ −J\n1+α22\nr/summationdisplay\nσν2\nσ\npσcos2pσr (B21)\nρ(r)∼ −J\n1+α24αν2v2\nF\nv21/radicalbig\np2\nF+2mE+E2/v2·sinpεr\nr, (B22)18\nwherepε= 2(mv+E/v). and we get for pσ>0:\nSx(p)∼+J\n1+α2·2πicosφp+∞/integraldisplay\n0drJ1(pr)/summationdisplay\nσσν2\nσ\npσsin2pσr (B23)\nSy(p)∼+J\n1+α2·2πisinφp+∞/integraldisplay\n0drJ1(pr)/summationdisplay\nσσν2\nσ\npσsin2pσr (B24)\nSz(p)∼ −J\n1+α2·4π+∞/integraldisplay\n0drJ0(pr)/summationdisplay\nσν2\nσ\npσcos2pσr (B25)\nρ(p)∼ −J\n1+α2·8παν2v2\nF\nv21/radicalbig\np2\nF+2mE+E2/v2+∞/integraldisplay\n0drJ0(pr)sinpεr (B26)\n2. x-impurity\nSx(r)∼ −J\n1+α21\nr/braceleftigg\n2ν2v2\nF\nv2cospεr/radicalbig\np2\nF+2mE+E2/v2+/summationdisplay\nσν2\nσ\npσcos2pσr+ (B27)\n+cos2φr/bracketleftigg\n−2ν2v2\nF\nv2cospεr/radicalbig\np2\nF+2mE+E2/v2+/summationdisplay\nσν2\nσ\npσcos2pσr/bracketrightigg/bracerightigg\n(B28)\nSy(r)∼ −J\n1+α2sin2φr\nr/bracketleftigg\n−2ν2v2\nF\nv2cospεr/radicalbig\np2\nF+2mE+E2/v2+/summationdisplay\nσν2\nσ\npσcos2pσr/bracketrightigg\n(B29)\nSz(r)∼ −J\n1+α2cosφr\nr/summationdisplay\nσσν2\nσ\npσsin2pσr (B30)\nρ(r)∼ −J\n1+α2·α\nr·4ν2v2\nF\nv2sinpεr/radicalbig\np2\nF+2mE+E2/v2(B31)\nWith the corresponding Fourier transforms:\nSx(p) =Ssym\nx(p)+Sasym\nx(p) =−J\n1+α2·2π+∞/integraldisplay\n0drJ0(pr)/bracketleftigg\n2ν2v2\nF\nv2cospεr/radicalbig\np2\nF+2mE+E2/v2+/summationdisplay\nσν2\nσ\npσcos2pσr/bracketrightigg\n−(B32)\n−J\n1+α2·2πcos2φp+∞/integraldisplay\n0drJ2(pr)/bracketleftigg\n2ν2v2\nF\nv2cospεr/radicalbig\np2\nF+2mE+E2/v2−/summationdisplay\nσν2\nσ\npσcos2pσr/bracketrightigg\n(B33)\nSy(p) =−J\n1+α2·2πsin2φp+∞/integraldisplay\n0drJ2(pr)/bracketleftigg\n2ν2v2\nF\nv2cospεr/radicalbig\np2\nF+2mE+E2/v2−/summationdisplay\nσν2\nσ\npσcos2pσr/bracketrightigg\n(B34)\nSz(p)∼ −J\n1+α2·2πicosφp+∞/integraldisplay\n0drJ1(pr)/summationdisplay\nσσν2\nσ\npσsin2pσr (B35)\nρ(p)∼ −J\n1+α2·8παν2v2\nF\nv21/radicalbig\np2\nF+2mE+E2/v2+∞/integraldisplay\n0drJ0(pr)sinpεr (B36)19\nAppendix C: Analytical calculation of the Shiba states wave functions for a 1D system\nThe unperturbed Green’s function in momentum space is G0(E,p) =1\n2/summationtext\nσ=±Gσ\n0(E,p), where\nGσ\n0(E,p) =−1\nξ2σ+∆2s−E2/parenleftbigg\n1iσ\n−iσ1/parenrightbigg\n⊗/parenleftbigg\nE+ξσ∆s\n∆sE−ξσ/parenrightbigg\n, (C1)\nwhereξσ=ξp+σλp. To get the coordinate value one needs to perform the Fourier tra nsform:\nGσ\n0(E,x) =/integraldisplaydp\n2πGσ\n0(E,p)eipx\nWe will have two types of integrals:\nXσ\n0(x) =−/integraldisplaydp\n2πeipx\nξ2σ+ω2, (C2)\nXσ\n1(x) =−/integraldisplaydp\n2πξσeipx\nξ2σ+ω2, (C3)\nwhereω2= ∆2\ns−E2. Since the spectrum is split by SO coupling, there will be two Fermi mom enta which can be\nfound the following way:\np2\n2m+σλp−εF= 0, pσ\nF=−σλ+/radicalbig\nλ2+2εF/m\n1/m≡m[−σλ+v]\nForp>0 we linearize the spectrum around Fermi momenta, thus:\nξσ≈/parenleftbiggpσ\nF\nm+σλ/parenrightbigg\n(p−pσ\nF) =/radicalbig\nλ2+2εF/m(p−pσ\nF)≡v(p−pσ\nF),\nthereforep=pσ\nF+ξσ/vand we get:\nXσ\n0(x) =−/integraldisplaydp\n2πeipx\nξ2σ+ω2=−\n+∞/integraldisplay\n0dp\n2πeipx\nξ2σ+ω2++∞/integraldisplay\n0dp\n2πe−ipx\nξ2\n−σ+ω2\n=♣\n+∞/integraldisplay\n0dp\n2πeipx\nξ2σ+ω2≈1\n2πveipσ\nFx/integraldisplay\ndξσeiξσx/v\nξ2σ+ω2=1\n2vωeipσ\nFxe−ω|x|/v\n+∞/integraldisplay\n0dp\n2πe−ipx\nξ2\n−σ+ω2≈1\n2πve−ip−σ\nFx/integraldisplay\ndξ−σe−iξ−σx/v\nξ2\n−σ+ω2=1\n2vωe−ip−σ\nFxe−ω|x|/v\n♣=−1\n2vω/bracketleftig\neim[−σλ+v]x+e−im[σλ+v]x/bracketrightig\ne−ω|x|/v=−1\nv·1\nωcosmvxe−iσmλxe−ω|x|/v\nXσ\n1(x) =−/integraldisplaydp\n2πξσeipx\nξ2σ+ω2=−\n+∞/integraldisplay\n0dp\n2πξσeipx\nξ2σ+ω2++∞/integraldisplay\n0dp\n2πξ−σe−ipx\nξ2\n−σ+ω2\n=♠\n+∞/integraldisplay\n0dp\n2πξσeipx\nξ2σ+ω2≈1\n2πveipσ\nFx/integraldisplay\ndξσξσeiξσx/v\nξ2σ+ω2=i\n2vsgnxeipσ\nFxe−ω|x|/v\n+∞/integraldisplay\n0dp\n2πξ−σe−ipx\nξ2\n−σ+ω2≈1\n2πve−ip−σ\nFx/integraldisplay\ndξ−σξ−σe−iξ−σx/v\nξ2\n−σ+ω2=−i\n2vsgnxe−ip−σ\nFxe−ω|x|/v20\n♠=−i\n2vsgnx/bracketleftig\neim[−σλ+v]x−e−im[σλ+v]x/bracketrightig\ne−ω|x|/v=1\nv·sinmv|x|e−iσmλxe−ω|x|/v\nFinally:\nXσ\n0(x) =−1\nv·1\nωcosmvxe−iσmλxe−ω|x|/v(C4)\nXσ\n1(x) = +1\nv·sinmv|x|e−iσmλxe−ω|x|/v(C5)\nand\nG0(E,x) =1\n2/summationdisplay\nσ=±/parenleftbigg\n1iσ\n−iσ1/parenrightbigg\n⊗/parenleftbigg\nEXσ\n0(x)+Xσ\n1(x) ∆ sXσ\n0(x)\n∆sXσ\n0(x)EXσ\n0(x)−Xσ\n1(x)/parenrightbigg\n(C6)\nG0(ǫ,x= 0) =−1\nv1√\n1−ǫ2/parenleftbigg\nǫσ0σ0\nσ0ǫσ0/parenrightbigg\n,whereǫ=E\n∆s(C7)\nThe eigenvalues and eigenfunctions at r=0can be obtained using Eq. (10) The energy levels are\nE1,¯1=±1−α2\n1+α2∆s,whereα=J/v. (C8)\nIn case of an impurity along the z-axis the corresponding eigenvectors are\nΦ¯1(0) =/parenleftbig1 0−1 0/parenrightbigT,Φ1(0) =/parenleftbig0 1 0 1/parenrightbigT(C9)\nand in case of an impurity along the x-axis:\nΦ¯1(0) =/parenleftbig1 1−1−1/parenrightbigT,Φ1(0) =/parenleftbig1−1 1−1/parenrightbigT. (C10)\n1. z-impurity\nΦ¯1(x) = +Jz\n2/summationdisplay\nσ\n+(E¯1−∆s)Xσ\n0(x)+Xσ\n1(x)\n−iσ[(E¯1−∆s)Xσ\n0(x)+Xσ\n1(x)]\n−(E¯1−∆s)Xσ\n0(x)+Xσ\n1(x)\n+iσ[(E¯1−∆s)Xσ\n0(x)−Xσ\n1(x)]\n,Φ1(x) =−Jz\n2/summationdisplay\nσ\n+iσ[(E1+∆s)Xσ\n0(x)+Xσ\n1(x)]\n(E1+∆s)Xσ\n0(x)+Xσ\n1(x)\n+iσ[(E1+∆s)Xσ\n0(x)−Xσ\n1(x)]\n(E1+∆s)Xσ\n0(x)−Xσ\n1(x)\n.\n(C11)\nUsing these expressions we can compute the non-polarized and SP L DOS in both coordinate and momentum space\nfor the positive energy state (omitting the index 1):\nSx(x) =1+α2\n4[2sinpλx+sin(2mv|x|+pλx−2θ)−sin(2mv|x|−pλx−2θ)]·e−2ω|x|/v(C12)\nSy(x) = 0 (C13)\nSz(x) =−1+α2\n4[2cospλx+cos(2mv|x|+pλx−2θ)+cos(2mv|x|−pλx−2θ)]·e−2ω|x|/v(C14)\nρ(x) =1+α2\n2[1+cos(2mv|x|−2θ)]·e−2ω|x|/v(C15)21\nwhere tanθ=α. We perform the Fourier transform to get the momentum space be havior, exploiting the following\n’standard’ integrals:\n/integraldisplay\ne−2ω|x|/ve−ipxdx= 22ω/v\np2+(2ω/v)2(C16)\n/integraldisplay\ncospλx·e−2ω|x|/ve−ipxdx=2ω\nv/bracketleftbigg1\n(p+pλ)2+(2ω/v)2+1\n(p−pλ)2+(2ω/v)2/bracketrightbigg\n(C17)\n/integraldisplay\nsinpλx·e−2ω|x|/ve−ipxdx=i2ω\nv/bracketleftbigg1\n(p+pλ)2+(2ω/v)2−1\n(p−pλ)2+(2ω/v)2/bracketrightbigg\n(C18)\n/integraldisplay\nsin2mv|x|·e−2ω|x|/ve−ipxdx=p+2mv\n(p+2mv)2+(2ω/v)2−p−2mv\n(p−2mv)2+(2ω/v)2(C19)\nWe rewrite these expressions using p±\nF, thus we get:\n/integraldisplay\ncospλx·e−2ω|x|/ve−ipxdx=2ω\nv/braceleftigg\n1\n/bracketleftbig\np+(p−\nF−p+\nF)/bracketrightbig2+(2ω/v)2+1\n(/bracketleftbig\np−(p−\nF−p+\nF)/bracketrightbig2+(2ω/v)2/bracerightigg\n(C20)\n/integraldisplay\nsinpλx·e−2ω|x|/ve−ipxdx=i2ω\nv/braceleftigg\n1\n/bracketleftbig\np+(p−\nF−p+\nF)/bracketrightbig2+(2ω/v)2−1\n/bracketleftbig\np−(p−\nF−p+\nF)/bracketrightbig2+(2ω/v)2/bracerightigg\n(C21)\n/integraldisplay\nsin2mv|x|·e−2ω|x|/ve−ipxdx=p+(p−\nF+p+\nF)\n/bracketleftbig\np+(p−\nF+p+\nF)/bracketrightbig2+(2ω/v)2−p−(p−\nF+p+\nF)\n/bracketleftbig\np−(p−\nF+p+\nF)/bracketrightbig2+(2ω/v)2(C22)\nFor the last two integrals we introduce symbols/summationtext\np′and/tildewider/summationtext\np′(wide tilde signify that we take the difference, not sum),\nwherep′∈ {p−pλ,p+pλ}. Thus we have\n/integraldisplay\ncos(2mv|x|−2θ)cospλx·e−2ω|x|/ve−ipxdx= (C23)\n=1\n2/summationdisplay\np′/braceleftbigg1−α2\n1+α2·2ω\nv/bracketleftbigg1\n(p′+2mv)2+(2ω/v)2+1\n(p′−2mv)2+(2ω/v)2/bracketrightbigg\n+ (C24)\n+2α\n1+α2·/bracketleftbiggp′+2mv\n(p′+2mv)2+(2ω/v)2+p′−2mv\n(p′−2mv)2+(2ω/v)2/bracketrightbigg/bracerightbigg\n(C25)\n/integraldisplay\ncos(2mv|x|−2θ)sinpλx·e−2ω|x|/ve−ipxdx= (C26)\n=1\n2i/tildewidest/summationdisplay\np′/braceleftbigg1−α2\n1+α2·2ω\nv/bracketleftbigg1\n(p′+2mv)2+(2ω/v)2+1\n(p′−2mv)2+(2ω/v)2/bracketrightbigg\n+ (C27)\n+2α\n1+α2·/bracketleftbiggp′+2mv\n(p′+2mv)2+(2ω/v)2+p′−2mv\n(p′−2mv)2+(2ω/v)2/bracketrightbigg/bracerightbigg\n(C28)\nWe rewrite these expressions using p±\nF, thus we get:\n/integraldisplay\ncos(2mv|x|−2θ)cospλx·e−2ω|x|/ve−ipxdx= (C29)\n=1−α2\n1+α2·ω\nv/bracketleftbigg1\n(p+2p+\nF)2+(2ω/v)2+1\n(p−2p−\nF)2+(2ω/v)2/bracketrightbigg\n+\n+α\n1+α2·/bracketleftbiggp+2p+\nF\n(p+2p+\nF)2+(2ω/v)2+p−2p−\nF\n(p−2p−\nF)2+(2ω/v)2/bracketrightbigg\n+\n+1−α2\n1+α2·ω\nv/bracketleftbigg1\n(p+2p−\nF)2+(2ω/v)2+1\n(p−2p+\nF)2+(2ω/v)2/bracketrightbigg\n+\n+α\n1+α2·/bracketleftbiggp+2p−\nF\n(p+2p−\nF)2+(2ω/v)2+p−2p+\nF\n(p−2p+\nF)2+(2ω/v)2/bracketrightbigg22\n/integraldisplay\ncos(2mv|x|−2θ)sinpλx·e−2ω|x|/ve−ipxdx= (C30)\n=1\ni/braceleftbigg1−α2\n1+α2·ω\nv/bracketleftbigg1\n(p+2p+\nF)2+(2ω/v)2+1\n(p−2p−\nF)2+(2ω/v)2/bracketrightbigg\n+\n+α\n1+α2·/bracketleftbiggp+2p+\nF\n(p+2p+\nF)2+(2ω/v)2+p−2p−\nF\n(p−2p−\nF)2+(2ω/v)2/bracketrightbigg/bracerightbigg\n−\n−1\ni/braceleftbigg1−α2\n1+α2·ω\nv/bracketleftbigg1\n(p+2p−\nF)2+(2ω/v)2+1\n(p−2p+\nF)2+(2ω/v)2/bracketrightbigg\n+\n+α\n1+α2·/bracketleftbiggp+2p−\nF\n(p+2p−\nF)2+(2ω/v)2+p−2p+\nF\n(p−2p+\nF)2+(2ω/v)2/bracketrightbigg/bracerightbigg\nUsing the formula cos2γ= (1+cos2 γ)/2 we can write the momentum space expressions for the non-polariz ed and\nSP LDOS components:\nSx(p) =i(1+α2)ω\nv/braceleftigg\n1\n/bracketleftbig\np+(p−\nF−p+\nF)/bracketrightbig2+(2ω/v)2−1\n/bracketleftbig\np−(p−\nF−p+\nF)/bracketrightbig2+(2ω/v)2/bracerightigg\n+ (C31)\n+1\ni/braceleftbigg1−α2\n2·ω\nv/bracketleftbigg1\n(p+2p+\nF)2+(2ω/v)2+1\n(p−2p−\nF)2+(2ω/v)2/bracketrightbigg\n+\n+α\n2·/bracketleftbiggp+2p+\nF\n(p+2p+\nF)2+(2ω/v)2+p−2p−\nF\n(p−2p−\nF)2+(2ω/v)2/bracketrightbigg/bracerightbigg\n−\n−1\ni/braceleftbigg1−α2\n2·ω\nv/bracketleftbigg1\n(p+2p−\nF)2+(2ω/v)2+1\n(p−2p+\nF)2+(2ω/v)2/bracketrightbigg\n+\n+α\n2·/bracketleftbiggp+2p−\nF\n(p+2p−\nF)2+(2ω/v)2+p−2p+\nF\n(p−2p+\nF)2+(2ω/v)2/bracketrightbigg/bracerightbigg\nSz(p) =−(1+α2)ω\nv/braceleftigg\n1\n/bracketleftbig\np+(p−\nF−p+\nF)/bracketrightbig2+(2ω/v)2+1\n/bracketleftbig\np−(p−\nF−p+\nF)/bracketrightbig2+(2ω/v)2/bracerightigg\n− (C32)\n−1−α2\n2·ω\nv/bracketleftbigg1\n(p+2p+\nF)2+(2ω/v)2+1\n(p−2p−\nF)2+(2ω/v)2/bracketrightbigg\n−\n−α\n2·/bracketleftbiggp+2p+\nF\n(p+2p+\nF)2+(2ω/v)2−p−2p−\nF\n(p−2p−\nF)2+(2ω/v)2/bracketrightbigg\n−\n−1−α2\n2·ω\nv/bracketleftbigg1\n(p+2p−\nF)2+(2ω/v)2+1\n(p−2p+\nF)2+(2ω/v)2/bracketrightbigg\n−\n−α\n2·/bracketleftbiggp+2p−\nF\n(p+2p−\nF)2+(2ω/v)2+p−2p+\nF\n(p−2p+\nF)2+(2ω/v)2/bracketrightbigg\nρ(p) = (1+α2)/braceleftigg\n2ω/v\np2+(2ω/v)2+/bracketleftigg\nω/v\n/bracketleftbig\np+(p−\nF+p+\nF)/bracketrightbig2+(2ω/v)2+ω/v\n(/bracketleftbig\np−(p−\nF+p+\nF)/bracketrightbig2+(2ω/v)2/bracketrightigg/bracerightigg\n+ (C33)\n+α/braceleftigg\np+(p−\nF+p+\nF)\n/bracketleftbig\np+(p−\nF+p+\nF)/bracketrightbig2+(2ω/v)2−p−(p−\nF+p+\nF)\n/bracketleftbig\np−(p−\nF+p+\nF)/bracketrightbig2+(2ω/v)2/bracerightigg23\n2. x-impurity\nΦ¯1(x) = +Jx\n2/summationdisplay\nσ\n(1+iσ)[(E¯1−∆s)Xσ\n0(x)+Xσ\n1(x)]\n(1−iσ)[(E¯1−∆s)Xσ\n0(x)+Xσ\n1(x)]\n−(1+iσ)[(E¯1−∆s)Xσ\n0(x)−Xσ\n1(x)]\n−(1−iσ)[(E¯1−∆s)Xσ\n0(x)−Xσ\n1(x)]\n,Φ1(x) = +Jx\n2/summationdisplay\nσ\n−(1−iσ)[(E1+∆s)Xσ\n0(x)+Xσ\n1(x)]\n(1+iσ)[(E1+∆s)Xσ\n0(x)+Xσ\n1(x)]\n−(1−iσ)[(E1+∆s)Xσ\n0(x)−Xσ\n1(x)]\n(1+iσ)[(E1+∆s)Xσ\n0(x)−Xσ\n1(x)]\n.\n(C34)\nUsing these expressions we can compute the non-polarized and SP L DOS in both coordinate and momentum space.\nWe perform the calculation for the positive-energy state, and we fi nd, omitting index 1:\nSx(x) =−1+α2\n2[2cospλx+cos(2mv|x|+pλx−2θ)+cos(2mv|x|−pλx−2θ)]·e−2ω|x|/v(C35)\nSy(x) = 0 (C36)\nSz(x) =−1+α2\n2[2sinpλx+sin(2mv|x|+pλx−2θ)−sin(2mv|x|−pλx−2θ)]·e−2ω|x|/v(C37)\nρ(x) = (1+α2)[1+cos(2mv|x|−2θ)]·e−2ω|x|/v(C38)\nwhere tanθ=α. Momentum space dependence can be derived from the z-impurity expressions since everything\ncoincides up to coefficients.\n3. y-impurity\nΦ¯1(x) = +Jy\n2/summationdisplay\nσ\n(1−σ)[(E¯1−∆s)Xσ\n0(x)+Xσ\n1(x)]\ni(1−σ)[(E¯1−∆s)Xσ\n0(x)+Xσ\n1(x)]\n−(1−σ)[(E¯1−∆s)Xσ\n0(x)−Xσ\n1(x)]\n−i(1−σ)[(E¯1−∆s)Xσ\n0(x)−Xσ\n1(x)]\n,Φ1(x) = +Jy\n2/summationdisplay\nσ\n−(1+σ)[(E1+∆s)Xσ\n0(x)+Xσ\n1(x)]\ni(1+σ)[(E1+∆s)Xσ\n0(x)+Xσ\n1(x)]\n−(1+σ)[(E1+∆s)Xσ\n0(x)−Xσ\n1(x)]\ni(1+σ)[(E1+∆s)Xσ\n0(x)−Xσ\n1(x)]\n,\n(C39)\nafter summation over σ:\nΦ¯1(x) = +Jy\n+/bracketleftbig\n(E¯1−∆s)X−\n0(x)+X−\n1(x)/bracketrightbig\ni/bracketleftbig\n(E¯1−∆s)X−\n0(x)+X−\n1(x)/bracketrightbig\n−/bracketleftbig\n(E¯1−∆s)X−\n0(x)−X−\n1(x)/bracketrightbig\n−i/bracketleftbig\n(E¯1−∆s)X−\n0(x)−X−\n1(x)/bracketrightbig\n,Φ1(x) = +Jy\n−/bracketleftbig\n(E1+∆s)X+\n0(x)+X+\n1(x)/bracketrightbig\ni/bracketleftbig\n(E1+∆s)X+\n0(x)+X+\n1(x)/bracketrightbig\n−/bracketleftbig\n(E1+∆s)X+\n0(x)−X+\n1(x)/bracketrightbig\ni/bracketleftbig\n(E1+∆s)X+\n0(x)−X+\n1(x)/bracketrightbig\n.(C40)\nUsing these expressions we can compute the non-polarized and SP L DOS in coordinate space\nSx(x) = 0, (C41)\nSy(x) =−(1+α2)[1+cos(2mv|x|−2θ)]·e−2ω|x|/v, (C42)\nSz(x) = 0, (C43)\nρ(x) = +(1+α2)[1+cos(2mv|x|−2θ)]·e−2ω|x|/v. (C44)\nAppendix D: The SPDOS for a non-superconducting one-dimens ional system in the presence of a magnetic\nimpurity\nThe low-energy Hamiltonian in the non-SC regime can be written as\nH0=ξpσ0+λ(pyσx−pxσy) =/parenleftbigg\nξpiλp\n−iλp ξ p/parenrightbigg\n(D1)24\nwhereξp=p2\n2m−εF. The corresponding spectrum is given by E=ξp±λpand the retarded Green’s function reads\nG0(E,p) =1\n(E−ξp+i0)2−λ2p2/parenleftbigg\nE−ξp+i0iλp\n−iλp E −ξp+i0/parenrightbigg\n. (D2)\nTo compute the eigenvalues for a single localized impurity we calculate\nG0(E,x= 0) =/integraldisplaydp\n2πE−ξp+i0\n(E−ξp+i0)2−λ2p2/parenleftbigg\n1 0\n0 1/parenrightbigg\n=1\n2/summationdisplay\nσ/integraldisplaydp\n2π1\nE−ξσ+i0/parenleftbigg\n1 0\n0 1/parenrightbigg\n, (D3)\nwhereξσ=ξp+σλp. Forp>0 we linearize the spectrum around the Fermi momenta, thus:\nξσ≈/parenleftbiggpσ\nF\nm+σλ/parenrightbigg\n(p−pσ\nF) =/radicalbig\nλ2+2εF/m(p−pσ\nF)≡v(p−pσ\nF),\nwherepσ\nF=m[−σλ+v], and thus we get:\n/integraldisplaydp\n2π1\nE−ξσ+i0≈1\n2πv/bracketleftbigg/integraldisplaydξσ\nE−ξσ+i0+/integraldisplaydξ−σ\nE−ξ−σ+i0/bracketrightbigg\n=−i\nv\nThis leads to:\nG0(E,x= 0) =1\n2/summationdisplay\nσ/parenleftbigg\n−i\nv/parenrightbigg/parenleftbigg\n1 0\n0 1/parenrightbigg\n=−i\nv/parenleftbigg\n1 0\n0 1/parenrightbigg\n(D4)\nSince there is no energy dependence, there will be no impurity-induc ed states. The Green’s function coordinate\ndependence is given by the following expression:\nG0(E,x) =1\n2/summationdisplay\nσ/integraldisplaydp\n2πeipx\nE−ξσ+i0/parenleftbigg\n1iσ\n−iσ1/parenrightbigg\n(D5)\nTo find the coordinate dependence of the Green’s function we calcu late:\nXσ\n0(x) =/integraldisplaydp\n2πeipx\nE−ξσ+i0(D6)\nIntegral calculation\nBelow we use the Sokhotsky formula1\nx+i0=P1\nx−iπδ(x):\nXσ\n0(x) =/integraldisplaydp\n2πeipx\nE−ξσ+i0=1\n2πv/bracketleftbigg\neipσ\nFx/integraldisplay\ndξσeiξσx/v\nE−ξσ+i0+e−ip−σ\nFx/integraldisplay\ndξ−σe−iξ−σx/v\nE−ξ−σ+i0/bracketrightbigg\nWe compute explicitly only one of the integrals in the brackets since th e other one can be computed in the similar\nfashion:\n/integraldisplay\ndξσeiξσx/v\nE−ξσ+i0=P/integraldisplay\ndξσeiξσx/v\nE−ξσ−iπ/integraldisplay\ndξσδ(E−ξσ)eiξσx/v=−iπ(1+sgnx)eiEx/v\nFinally we have:\nXσ\n0(x) =−i\nvexp/bracketleftbigg\ni/parenleftbigg\nmv+E\nv/parenrightbigg\n|x|/bracketrightbigg\ne−iσmλx, (D7)\nand the Green’s function can be written as:\nG0(E,x) =1\n2/summationdisplay\nσ/parenleftbigg\n1iσ\n−iσ1/parenrightbigg\nXσ\n0(x). (D8)25\nBelow we compute the T-matrix for different types of impurities. Imp urity potentials take the following forms:\nVsc=U/parenleftbigg\n1 0\n0 1/parenrightbigg\n, Vz=Jz/parenleftbigg\n1 0\n0−1/parenrightbigg\n, Vx=Jx/parenleftbigg\n0 1\n1 0/parenrightbigg\n(D9)\nThe corresponding T-matrices are\nTsc=U\n1+iU/v/parenleftbigg\n1 0\n0 1/parenrightbigg\n, Tz=/parenleftiggJ\n1+iJ/v0\n0−J\n1−iJ/v/parenrightigg\n, T x=J\n1+J2/v2/parenleftbigg\n−iJ/v1\n1−iJ/v/parenrightbigg\n(D10)\nFor each type of impurity we can compute the non-polarized and SP L DOS using Eq. (B11) and Eqs. (B15) where\nwe replace rbyx. By taking the Fourier transforms of the expressions above we ge t the the momentum space\ndependence. Below we denote α=J/v.\n1. z-impurity\nSx(x) = +α\n1+α2·1\nπv[cos(pε|x|−pλx)−cos(pε|x|+pλx)] (D11)\nSy(x) = 0 (D12)\nSz(x) = +α\n1+α2·1\nπv[sin(pε|x|−pλx)+sin(pε|x|+pλx)] (D13)\nρ(x) =−2α2\n1+α2·1\nπvcospεx (D14)\nwhere we denote pε= 2(mv+E/v),pλ= 2mλ. After taking the Fourier transform we get:\nSx(p) = +α\n1+α2·i\nπv/bracketleftbigg1\np+pε+pλ−1\np+pε−pλ−1\np−pε+pλ+1\np−pε−pλ/bracketrightbigg\n(D15)\nSy(p) = 0 (D16)\nSz(p) = +α\n1+α2·1\nπv/bracketleftbigg1\np+pε+pλ+1\np+pε−pλ−1\np−pε+pλ−1\np−pε−pλ/bracketrightbigg\n(D17)\nρ(p) =−2α2\n1+α2·1\nv[δ(p−pε)+δ(p+pε)] (D18)\n2. x-impurity\nSx(x) = +α\n1+α2·1\nπv[sin(pε|x|−pλx)+sin(pε|x|+pλx)] (D19)\nSy(x) = 0 (D20)\nSz(x) =−α\n1+α2·1\nπv[cos(pε|x|−pλx)−cos(pε|x|+pλx)] (D21)\nρ(x) =−2α2\n1+α2·1\nπvcospεx (D22)\nWe do not give the Fourier transform for these expressions since t hey coincide with the ones for a z-impurity if we\nexchangeSzandSxand change the overall sign.26\n3. y-impurity\nSx(x) =Sz(x) = 0 (D23)\nSy(x) = +2α\n1+α2·1\nπvsinpε|x| (D24)\nρ(x) =−2α2\n1+α2·1\nπvcospεx (D25)\nThe corresponding Fourier transform is:\nSy(p) =2α\n1+α2·1\nπv/bracketleftbigg1\np+pε−1\np−pε/bracketrightbigg\n(D26)"    },    {        "title": "1403.4728v1.Spin_orbit_coupling_effects_on_spin_dependent_inelastic_electronic_lifetimes_in_ferromagnets.pdf",        "content": "arXiv:1403.4728v1  [cond-mat.mtrl-sci]  19 Mar 2014Spin-Orbit Coupling Effects on Spin Dependent Inelastic Ele ctronic Lifetimes in\nFerromagnets\nSteffen Kaltenborn and Hans Christian Schneider∗\nPhysics Department and Research Center OPTIMAS,\nUniversity of Kaiserslautern, 67663 Kaiserslautern, Germ any\n(Dated: July 4, 2021)\nFor the 3d ferromagnets iron, cobalt and nickel we compute th e spin-dependentinelastic electronic\nlifetimes due to carrier-carrier Coulomb interaction incl uding spin-orbit coupling. We find that\nthe spin-dependent density-of-states at the Fermi energy d oes not, in general, determine the spin\ndependence of the lifetimes because of the effective spin-fli p transitions allowed by the spin mixing.\nThe majority and minority electron lifetimes computed incl uding spin-orbit coupling for these three\n3-d ferromagnets do not differ by more than a factor of 2, and ag ree with experimental results.\nPACS numbers: 71.70.Ej,75.76.+j,75.78.-n,85.75.-d\nI. INTRODUCTION\nThe theoretical and experimental characterization of\nspin dynamics in ferromagnetic materials due to the in-\nteraction with short optical pulses has become an impor-\ntant part of research in magnetism.1–6In this connec-\ntion, spin-dependent hot-electron transport processes in\nmetallic heterostructures have received enormous inter-\nest in the past few years.7In particular, superdiffusive-\ntransport theory has played an increasingly important\nrole in the quantitative interpretationof experimental re-\nsults.4,6,8Superdiffusive transport-theory, which was in-\ntroduced and comprehensively described in Refs. 9and\n10, uses spin- and energy-dependent electron lifetimes\nas input,10and its quantitative results for hot-electron\ntransportonultrashorttimescalesinferromagneticmate-\nrials rely heavily, to the best of our knowledge, on the re-\nlation between majority and minority electrons for these\nmaterials.\nThe spin-dependent lifetimes that are used for hot-\nelectrontransport,bothinferromagnetsandnormalmet-\nals, are the so-called “inelastic lifetimes.” These state\n(orenergy)dependentlifetimesresultfromout-scattering\nprocesses due to the Coulomb interaction between an ex-\ncited electron and the inhomogeneous electron gas in the\nsystem. These lifetimes can be measured by tracking op-\ntically excited electrons using spin- and time-resolved 2-\nphoton photoemission (2PPE)11,12and can be calculated\nasthebroadeningoftheelectronicspectralfunctionusing\nmany-body Green function techniques.13,14The problem\noftheaccuratedeterminationoftheselifetimes hasfueled\nmethod development on the experimental and theoreti-\ncal side,15but has always suffered from the presence of\ninteractions (electron-phonon, surface effects) that can-\nnot be clearly identified in experiment and are difficult\nto include in calculations. Qualitative agreement was\nreached for the spin-integrated lifetimes in simple met-\nals and iron,16but even advanced quasiparticle calcula-\ntions including many-body T-matrix contributions, have\nyielded a ratio between majority and minority lifetimes,\nwhich is in qualitative disagreement with experiment forsome ferromagnets. A particularly important material\nin recent studies has been nickel,4,6,10for which the the-\noretical ratio comes out between 6 and 8,16while the\nexperimental result17is 2. Recent experimental results\npoint toward a similar disagreement for cobalt.12\nIn light normal metals and ferromagnets spin-orbit\ncoupling generally leads to very small corrections to\nthe single-particle energies, i.e., the band structure , but\nit changes the single-particle states qualitatively by in-\ntroducing a state-dependent spin mixing. With spin-\norbit coupling, the average spin of an electron can be\nchanged in transitions due to any spin-diagonal interac-\ntion, in particular by electron-phonon momentum scat-\ntering.18–21This isalsotrue forthetwo-particleCoulomb\ninteraction,22,23as long as one monitors only the average\nspin of one of the scattering particles, as is done in life-\ntime measurements by 2-photon photoemission experi-\nments. While this spin mixing due to spin-orbit cou-\npling has recently been included in lifetime calculations\nfor lead,24it was not included in DFT codes used for ex-\nisting lifetime calculations for 3d-ferromagnets and alu-\nminum,16,25,26whose results are nowadays widely used.\nThis paper presents results for electron lifetimes in\nmetals and spin-dependent lifetimes in ferromagnets in-\ncluding spin-orbit coupling . We show that spin-orbit cou-\npling can be important for electron lifetimes in metals\nin general. Moreover, the ratio between the calculated\nmajority and minority lifetimes is, for the first time, in\nagreement with experiment.11,12,17We believe that our\ncalculated electronic lifetimes should be used as an accu-\nrate input for calculationsof spin-dependent hot-electron\ndynamics in ferromagnets.\nII. SPIN-DEPENDENT ELECTRON AND HOLE\nLIFETIMES IN CO AND NI\nWe first discuss briefly our theoretical approach to cal-\nculate the lifetimes. We start from the dynamical and\nwave-vector dependent dielectric function ε(/vector q,ω) in the\nrandom phase approximation (RPA).13,14,25,26Our ap-\nproach, cf. Ref. 27, evaluates the wave-vector summa-2\ntions inε(/vector q,ω) without introducing an additional broad-\nening of the energy-conserving δfunction. This proce-\ndure removes a parameter whose influence on the calcu-\nlation for small qis not easy to control and which would\notherwise need to be separately tested over the whole\nenergy range.\nThe/vectork- and band-resolved electronic scattering rates,\ni.e., the inverse lifetimes, γν\n/vectork= (τν\n/vectork)−1, are calculated\nusing the expression25,26\nγν\n/vectork=2\n¯h/summationdisplay\nµ/vector q∆q3\n(2π)3Vq/vextendsingle/vextendsingleBµν\n/vectork/vector q/vextendsingle/vextendsingle2fµ\n/vectork+/vector qℑε(/vector q,∆E)\n|ε(/vector q,∆E)|2.(1)\nHere, the band indices aredenoted by µandν, and/vectorkand\n/vector qdenote wave-vectors in the first Brillouin zone (1. BZ).\nThe energies ǫµ\n/vectork, occupation numbers fµ\n/vectorkand overlap ma-\ntrix elements Bµν\n/vectork/vector q=/angbracketleftψµ\n/vectork+/vector q|ei/vector q·/vector r|ψν\n/vectork/angbracketrightare extracted from\nthe ELK DFT (density functional theory) code,28which\nemploys a full-potential linearized augmented plane wave\n(FP-LAPW) basis. Last, Vq=e2/(ε0q2) denotes the\nFouriertransformedCoulombpotentialand∆ E=ǫµ\n/vectork+/vector q−\nǫν\n/vectorkis the energy difference between initial and final state.\nFor negative ∆ E, the distribution function has to be re-\nplaced by −(1−fµ\n/vectork+/vector q). By using the overlap matrix el-\nements as defined above we neglect corrections due to\nlocal field effects. In the language of many-body Green\nfunctions, this corresponds to an on-shell G0W0calcula-\ntion,16,26where the screened Coulomb interaction ( W0)\nis obtained from the full RPA dielectric function. The\n/vectork- and band-dependent wave-functions that result from\nthe DFT calculations including spin-orbit coupling are of\nthe form |ψµ\n/vectork/angbracketright=aµ\n/vectork|↑/angbracketright+bµ\n/vectork|↓/angbracketright,18where|σ/angbracketrightare spinors\nidentified by the spin projection σ=↑,↓along the mag-\nnetization direction. According to whether |aµ\n/vectork|2or|bµ\n/vectork|2\nis larger, we relabel each eigenstate by its dominant spin\ncontribution σ, so that we obtain spin-dependent life-\ntimes,τσ\n/vectork. Our choice of quantization axis is such that\nσ=↑denotes majority carriers states and σ=↓minor-\nity carrier states. Due to the existence of several bands\n(partially with different symmetries) in the energy range\nof interest and the anisotropy of the DFT bands ǫ(ν)(/vectork),\nseveral lifetimes τν\n/vectorkcan be associated with the same spin\nand energy. When we plot these spin and energy depen-\ndent lifetimes τσ(E) in the following, in particularFigs. 1\nand2, this leads to a scatter of τσ(E) values.\nFigures1and2displaythecalculatedenergy-andspin-\nresolved carrier lifetimes τσ(E) around the Fermi energy\nfor cobalt and nickel. The spread of lifetimes at the same\nenergy, which was mentioned above, can serve as an indi-\ncation for the possible range of results for measurements\nof energy resolved lifetimes. These “raw data” are im-\nportant for the interpretation of the theoretical results\nbecause they already show two important points. First,\nwe checked that there is no good Fermi-liquid type fit to\nthese lifetimes. Second, even if one fits the lifetimes in\na restricted energy range by a smooth τ(E) curve, this−2−1 012051015202530τCo(fs)\n  \nE−EF(eV)\nFIG. 1. (Color online) Energy-resolved majority ( τ↑, blue +)\nand minority ( τ↓, red◦) carrier lifetimes for cobalt. There are\nin general several different lifetime points at the same ener gy\n(see text). We used 173/vectork-points in the full BZ.\n−2−1 012020406080\nE−EF(eV)τNi(fs)\n  \nFIG. 2. (Color online) Same as Fig. 1for nickel.\nignoresthe spread of lifetimes, which can be quite sizable\nas shown in Figs. 1and2. We believe that such a spread\nof electronic lifetimes, in particular in the range around\n1eV above the Fermi energy should be important for the\ninterpretation of photoemission experiments in this en-\nergy range, and when these results are used as input in\nhot-electron transport calculations.\nFigure1shows the energy- and spin-resolved lifetimes\nin cobalt. In addition to the longer lifetimes close to the\nFermi energy, hole lifetimes in excess of 5fs occur at the\ntop of some d-bands around −1.5,−1.2, and−1eV. For\nelectronic states with energies above 0.5eV longer life-\ntimesoccuratsome /vectork-points. Therearealso kstateswith\napronouncedspin-asymmetryinthelifetimes(seediscus-\nsion below). Another important property of cobalt is the\nexistence of two different conduction bands, which inter-\nsect the Fermi surface with different slope. This leads to3\ntworather well-defined lifetime curves, both for electrons\nand holes. This can be best seen between −0.6 and 0eV,\nwhere the two curves are shifted by about 0.2eV.\nThe calculated lifetimes in nickel, see Fig. 2, do\nnot show a pronounced influence of d-bands and/or\nanisotropy below the Fermi energy as in cobalt, which is\ndue to the smaller number of bands in the vicinity of the\nFermi energy. However, there is a clear spin-dependence\nof electronic lifetimes, which is most pronounced around\n0.4eV, but persists almost up to 2eV.\nIII. SPIN ASYMMETRY OF ELECTRON\nLIFETIMES IN FE, CO, AND NI\nIn the following, we will mainly be concerned with life-\ntimes above 0.3eV above the Fermi energy, which is the\ninteresting energy range for the interpretation of photoe-\nmission experiments and hot-electron transport calcula-\ntions, because close to the Fermi energy the influence\nof phonons is expected to become more pronounced and\nleadtosignificantlyshorterlifetimes thanthosepredicted\nby a calculation that includes only the Coulomb interac-\ntion. To facilitate comparison with experiment we aver-\nage the lifetimes in each spin channel in bins of 100meV\nand denote the result by ¯ τ(E). The standard deviation\nof the averaging process then yields “error bars” on the\n¯τ(E) values. Note that this procedure does not corre-\nspond to a “random k” approximation.\nFigure3displays the averagedelectron lifetimes deter-\nmined from the data shown in Figs. 1and2. As insets\nwe have included the ratio of majority and minority life-\ntimes,τ↑/τ↓, together with experimental data11,12,17for\niron, cobalt and nickel. Figure 3(a) shows that there is\nonly a veryweakspin dependence for iron, and the agree-\nment of the ratio τ↑/τ↓with experiment17and recent in-\nvestigations15,16,29is quite good, but there is a slight dis-\nagreement with earlier, semiempirical studies.30,31How-\never, even an increase of the ratio around 0 .5eV in the\nexperiment17is well reproduced in our results.\nThe averaged lifetimes of cobalt, which are shown in\nFig.3(b), agree quite well with the experimental life-\ntimes,11,12but the large error bars extend to a much\nwider energy range than in iron. This can be traced\nback to the scatterof lifetimes in Fig. 1. The correspond-\ning figure for iron (not shown) exhibits a much smaller\nscatter. The ratio of majority and minority electron life-\ntimes, see inset in Fig. 3(b), is around 1 below 0.5eV\nand increases to τ↑/τ↓≃2 for larger energies, a trend\nthat agrees extremely well with measurements.11,12,17To\nput this result into perspective we note that the experi-\nmental data in Ref. 17were compared with a theoretical\nmodel based on the random kapproximation.32If the\nrandom-kinteraction matrix elements are taken to be\nspin and energy independent, the majority and minor-\nity relaxation times are determined by double convolu-\ntions over the spin-dependent density-of-states (DOS).16\nIt was found that the experimental results were not in050100150200¯τFe(fs)\n  \n01020¯τCo(fs)\n  \n00.511.522.530204060\nE−EF(eV)¯τNi(fs)\n  0.30.50.70.91.10.511.52\nE−EF(eV)τ↑\nFe/τ↓\nFe\n0.30.50.70.91.1123\nE−EF(eV)τ↑\nCo/τ↓\nCo\n0.30.50.70.91.1123\nE−EF(eV)τ↑\nNi/τ↓\nNi(c)(a)\n(b)\nFIG. 3. (Color online) Energetically averaged majority (bl ue\nup triangles) and minority (red down triangles) lifetimes f or\n(a) Fe, (b) Co and (c) Ni. The error bars denote the standard\ndeviation obtained from the scatter of the lifetimes as show n\nin Figs.1and2. The insets show the calculated ratio of ma-\njority and minority electrons (“ ◦”),τ↑/τ↓, in comparison to\nexperimental data, where the “ •” (“×”) correspond to values\nextracted from Ref. 11and17(12).\nagreement with the ratio of the DOS at the Fermi en-\nergy, which led the authors of Ref. 17to speculate that\nthe matrix elements for parallel and antiparallel spins\nshould be different due to the Pauli exclusion principle.\nIn our calculations, the effective spin-dependence of the\nmatrix elements is caused exclusively by the spin-mixing\ndue to spin-orbit coupling, but the effect is the same: It\nmakes the ratio of the lifetimes different from the spin-\ndependent DOS at the Fermi energy.\nIn Fig.3(c) we turn to nickel. Here, as in the case\nof iron, the average lifetimes are slightly larger than\nthe measured ones17(not shown), but due to the small\nanisotropy in the band structure, the lifetimes in nickel\nshow the smallest error bars and thus an extremely well-\ndefined spin dependence. Only our calculated majority\nelectron lifetimes are similar to earlier ab-initio evalua-\ntions,15,16,29but there is an important discrepancy in the\nratioτ↑/τ↓: The inset of Fig. 3(c) shows a ratio of about\nτ↑/τ↓≃2, which is independent of energy above 0.4eV.4\n012345020406080100\nE−EF(eV)τAl(fs)\n  \nFIG. 4. (Color online) Calculated energy-resolved electro nic\nlifetimes for aluminum (blue “ ◦”) in comparison to earlier in-\nvestigations without spin-orbit interaction. The black sq uares\n(stars) correspond to some data extracted from Ref. 25(33).\nThere are in general several different lifetime points at the\nsame energy (see text). We used 173/vectork-points in the full BZ.\nThis results compares extremely well with experiment,\nand should be contrasted with the calculated result of\nRef.16forτ↑/τ↓≃8 around 0.5eV. These GW calcu-\nlations (even with a T-matrix approach) gave very simi-\nlar results to those of the random kapproximation16in\nthe energy range above 0.5eV. This indicates that the\nresulting spin asymmetry τ↑/τ↓≃6–8 is solely deter-\nmined by the spin-dependent DOS Dσ(E). Indeed, one\nhasD↑(EF)/D↓(EF)≃8. With the inclusion of spin-\norbit coupling, which gives rise to effective spin-flip tran-\nsitions, the spin asymmetry is no longer determined by\nthe spin-dependent DOS alone. This interpretation is\nagain supported by Ref. 17where a strongly enhanced\nspin-flip matrix element had to be introduced by hand to\nimprove the agreement between a random- kcalculation\nand experiment.\nTo conclude the discussion of the ferromagnets, we\ncomment on the spin-integrated lifetimes which can be\nobtained from the spin-dependent lifetimes, but are not\nshown here. Compared with experimental lifetimes of\nRef.17we generally find an agreement for energies above\n0.5eV that is on par with earlier calculations.15–17,29For\nenergies below 0.5eV where the error bars on the av-\neraged lifetimes are largest, the calculated lifetimes are\nlarger than the measured ones, but in this energy range\na good agreement with experiments cannot be expected\nbecause of scattering processes, which appear as elastic\ndue to the energy resolution of the photoemission exper-\niments.IV. INFLUENCE OF SPIN-ORBIT COUPLING\nON ELECTRON LIFETIMES IN AL\nTo underscore the importance of spin-orbit coupling\nfor lifetime calculations, we also briefly discuss our calcu-\nlated results for electronic lifetimes in aluminum in com-\nparison with earlier investigations25,33without spin-orbit\neffects. Fig. 4shows that smaller electronic lifetimes re-\nsult for aluminum when spin-orbit coupling is included.\nIn particular in the energy range between 1 and 3eV\nthe lifetimes differ by almost a factor of two. Thus the\ninclusion of spin-orbit coupling improves the agreement\nwithexperiment(see,forinstance, Ref. 34), whichwasal-\nready quite good for the existing calculations.33It is con-\nceivable that the use of more sophisticated many-body\ntechniques, such as the inclusion of vertex corrections\nor using a T-matrix approach,33might lead to further\nimprovements. As in the case of the ferromagnets, the\nelectronic band structure is practically unchanged by the\ninclusion of spin-orbit coupling, but the rather large ef-\nfect of the spin-orbit coupling on spin relaxation in alu-\nminum through spin hot-spots has already been demon-\nstrated.18Another argument for the importance of the\nspin-orbit coupling is that the spin-mixing allows transi-\ntion between the Kramersdegeneratebands. These tran-\nsitionsbetweenKramersdegeneratebandsmayhaveare-\nmarkable influence even on electron-gas properties that\nare usually assumed to be spin-independent, such as the\nintraband plasma frequency.27\nV. CONCLUSION\nIn conclusion, we presented ab-initio results for spin-\ndependent electronic lifetimes in ferromagnets and alu-\nminum including spin-orbit coupling. We found that\nthe electronic lifetimes in iron exhibit no visible spin de-\npendence in the range of −2 up to 3eV in agreement\nwith earlier results, whereas the ratio τ↑/τ↓between\nmajority and minority lifetimes does not exceed 2 for\ncobalt and nickel. Our results agree well with experi-\nmental data, but differ from earlier calculations, which\nfound thatτ↑/τ↓was essentially determined by the spin-\ndependent density-of-states. We showed that, by allow-\ning for effectively spin-changing transitions as contribu-\ntions to the lifetime, spin-orbit coupling is the essential\ningredient that can make the spin asymmetry of the elec-\ntroniclifetimes much smallerthan the spin-asymmetryof\nthe density-of-states. Inclusion of our calculated spin de-\npendentlifetimesintransportcalculationsshouldmakeit\npossible to more accurately characterize the influence of\nspin-dependent hot-electron transport on magnetization\ndynamics.\nACKNOWLEDGMENTS\nWe are grateful for a CPU-time grant from the J¨ ulich\nSupercomputer Centre (JSC). We acknowledge helpful\ndiscussions with M. Aeschlimann, M. Cinchetti, and S.\nMathias.5\n∗hcsch@physik.uni-kl.de\n1B. Koopmans, G. Malinowski, F. Dalla Longa, D. Steiauf,\nM. F¨ ahnle, T. Roth, M. Cinchetti, and M. Aeschlimann,\nNature Materials 9, 259 (2010).\n2G. M. M¨ uller, J. Walowski, Marija Djordjevic1, G.-X.\nMiao, A. Gupta, A. V. Ramos, K. Gehrke, V. Moshnyaga,\nK. Samwer, J. Schmalhorst, A. Thomas, A. H¨ utten, G.\nReiss, J. S. Moodera, and M. M¨ unzenberg, Nature Mate-\nrials8, 56 (2009).\n3B. Pfau, S. Schaffert, L. M¨ uller, C. Gutt, A. Al-Shemmary,\nF. B¨ uttner, R. Delaunay, S. D¨ usterer, S. Flewett, R.\nFr¨ omter, J. Geilhufe, E. Guehrs, C.M. G¨ unther, R. Hawal-\ndar, M. Hille, N. Jaouen, A. Kobs, K. Li, J. Mohanty, H.\nRedlin, W.F. Schlotter, D. Stickler, R. Treusch, B. Vo-\ndungbo, M. Kl¨ aui, H.P. Oepen, J. L¨ uning, G. Gr¨ ubel, and\nS. Eisebitt, Nature Comm. 3, 1100 (2012).\n4D. Rudolf, C. La-O-Vorakiat, M. Battiato, R. Adam, J.\nM. Shaw, E. Turgut, P. Maldonado, S. Mathias, P. Grych-\ntol, H. T. Nembach, T. J. Silva, M. Aeschlimann, H. C.\nKapteyn, M. M. Murnane, C. M. Schneider, and P. M.\nOppeneer, Nature Comm. 3, 1037 (2012).\n5G. P. Zhang, W. H¨ ubner, G. Lefkidis, Y. Bai and T. F.\nGeorge, Nature Physics 5, 499 (2009).\n6A. Eschenlohr, M. Battiato, P. Maldonado, N. Pontius,\nT. Kachel, K. Holldack, R. Mitzner, A. F¨ ohlisch, P. M.\nOppeneer, and C. Stamm, Nature Mat. 12, 332 (2013).\n7A. Melnikov, I. Razdolski, T. O. Wehling, E. T. Papaioan-\nnou, V. Roddatis, P. Fumagalli, O. Aktsipetrov, A. I.\nLichtenstein, and U. Bovensiepen, Phys. Rev. Lett. 107,\n076601 (2011).\n8T. Kampfrath, M. Battiato, P. Maldonado, G. Eilers,\nJ. N¨ otzold, S. M¨ ahrlein, V. Zbarsky, F. Freimuth, Y.\nMokrousov, S. Bl¨ ugel, M. Wolf, I. Radu, P.M. Oppeneer,\nand M. M¨ unzenberg, Nature Nano. 8, 256 (2013).\n9M. Battiato, K. Carva, and P. M. Oppeneer, Phys. Rev.\nLett.105, 027203 (2010).\n10M. Battiato, K. Carva, and P. M. Oppeneer, Phys. Rev.\nB86, 024404 (2012).\n11M. Aeschlimann, M. Bauer, S. Pawlik, W. Weber, R.Burg-\nermeister, D. Oberli, and H. C. Siegmann, Phys. Rev.\nLett.79, 5158 (1997).\n12A. Goris, K. M. D¨ obrich, I. Panzer, A. B. Schmidt, M. Do-\nnath, and M. Weinelt, Phys. Rev. Lett. 107, 026601\n(2011).\n13G. Giuliani and G. Vignale, Quantum Theory of the Elec-\ntron Liquid , Cambridge University Press (2005).14G. Onida, L. Reining, and A. Rubio, Rev. Mod. Phys. 74\n601 (2002).\n15V. P. Zhukov, E. V. Chulkov, and P. M. Echenique, Phys.\nRev. Lett. 93, 096401 (2004).\n16V. P. Zhukov, E. V. Chulkov, and P. M. Echenique, Phys.\nRev. B73, 125105 (2006).\n17R. Knorren, K. H. Bennemann, R. Burgermeister, and\nM. Aeschlimann, Phys. Rev. B 61, 9427 (2000).\n18J. Fabian and S. Das Sarma, Phys. Rev. Lett. 81, 5624\n(1998).\n19S. Essert and H. C. Schneider, Phys. Rev. B 84, 224405\n(2011).\n20K. Carva, M. Battiato, and P. M. Oppeneer, Phys. Rev.\nLett.107, 207201 (2011).\n21D. Steiauf and M. F¨ ahnle, Phys. Rev. B 79, 140401(R)\n(2009).\n22B. Y. M¨ uller, A. Baral, S. Vollmar, M. Cinchetti, M.\nAeschlimann, H. C. Schneider, and B. Rethfeld Phys. Rev.\nLett.111, 167204 (2013).\n23M. Krauß, T. Roth, S. Alebrand, D. Steil., M. Cinchetti,\nM. Aeschlimann, and H. C. Schneider, Phys. Rev. B 80,\n180407(R) (2009).\n24X. Zubizarreta, V. M. Silkin, and E. V. Chulkov, Phys.\nRev. B84, 115144 (2011).\n25F. Ladst¨ adter, U. Hohenester, P. Puschnig, and\nC. Ambrosch-Draxl, Phys. Rev. B 70, 235125 (2004).\n26V. P. Zhukov, F. Aryasetiawan, E. V. Chulkov,\nI. G. de Gurtubay, and P. M. Echenique, Phys. Rev. B 64,\n195122 (2001).\n27S. Kaltenborn and H. C. Schneider, Phys. Rev. B 88,\n045124 (2013).\n28The Elk FP-LAPW Code, http://elk.sourceforge.net, ver-\nsion 1.4.18.\n29V. P. Zhukov and E. V. Chulkov, Physics Uspekhi 52, 105-\n136 (2009).\n30D. R. Penn, S. P. Apell, and S. M. Girvin, Phys. Rev.\nLett.55, 518 (1985).\n31D. R.Penn, S.P. Apell, and S.M. Girvin, Phys.Rev.B 32,\n7753 (1985).\n32W. F. Krolikowski and W. E. Spicer, Phys. Rev. 185, 882\n(1969).\n33V. P. Zhukov, E. V. Chulkov, and P. M. Echenique, Phys.\nRev. B72, 155109 (2005).\n34M. Bauer, S. Pawlik, and M. Aeschlimann, Proc.\nSPIE3272, 201 (1998)."    },    {        "title": "0710.2866v1.Intersubband_spin_orbit_coupling_and_spin_splitting_in_symmetric_quantum_wells.pdf",        "content": "arXiv:0710.2866v1  [cond-mat.mtrl-sci]  15 Oct 2007Intersubband spin-orbit coupling and spin splitting in sym metric quantum wells\nF. V. Kyrychenko and C. A. Ullrich\nDepartment of Physics and Astronomy, University of Missour i, Columbia, Missouri, 65211, USA\nI. D’Amico\nDepartment of Physics, University of York, York YO10 5DD, Un ited Kingdom\nIn semiconductors with inversion asymmetry, spin-orbit co upling gives rise to the well-known\nDresselhaus and Rashba effects. If one considers quantum wel ls with two or more conduction sub-\nbands, an additional, intersubband-induced spin-orbit te rm appears whose strength is comparable\nto the Rashba coupling, and which remains finite for symmetri c structures. We show that the con-\nduction band spin splitting due to this intersubband spin-o rbit coupling term is negligible for typical\nIII-V quantum wells.\nPACS numbers: 73.50.-h, 73.40.-c, 73.20.Mf, 73.21.-b\nKeywords: spintronics, spin Coulomb drag, spin-orbit coup ling, quantum wells\nI. INTRODUCTION\nResearch in nanoscience is crucial for its technologi-\ncal implications and for the fundamental exploration of\nthe quantum properties of nanostructures such as quan-\ntum wells, wires and dots. Of particular interest is the\nstudy of spin dynamics, which hopes to revolutionizetra-\nditional electronics using the spin properties of the carri-\ners (spintronics) [1]. In this context, the theoretical pre-\ndiction [2] and experimental confirmation [3] of the spin-\nCoulomb drag (SCD) effect was of great importance, as\nthis effect results in the natural decay ofspin current and\nintrinsic dissipation in AC-spintronic circuits [4]. Due\nto Coulomb interactions between spin-up and spin-down\nelectrons, theupanddowncomponentsofthetotallinear\nmomentum are not separately conserved. This momen-\ntum exchange between the two populations represents\nan intrinsic source of friction for spin currents, known as\nspin-transresistivity [5].\nIn [4] we demonstrated that the SCD produces an in-\ntrinsic linewidth in spin-dependent optical excitations,\nwhich can be as big as a fraction of a meV for intersub-\nband spin plasmons in parabolic semiconductor quan-\ntum wells (QWs). This intrinsic linewidth would be\nideal to experimentally verify the behavior of the spin-\ntransresistivity in the frequency domain.\nIn our proposed experiment, we suggested to use sym-\nmetricparabolic QWs to avoid an undesired splitting of\nthe spin plasmons due to Rashba spin-orbit (SO) cou-\npling. We based our discussion on earlier work [6], in\nwhich collective intersubband spin excitations in QWs\nweredescribedinthepresenceofDresselhausandRashba\nSO interaction terms [7, 8] for strictly two-dimensional\n(2D) systems [9]. In symmetric QWs, the Rashba term\nvanishes and only bulk inversion asymmetry (Dressel-\nhaus) interaction is present.\nHowever, as shown recently by Bernardes et al. [10],\nthe Rashba SO coupling gives finite contributions even\nfor symmetric structures, if treated in higher order per-\nturbation theory. As a consequence, for QWs with morethan one subband, there appears an additional intersub-\nband SO interaction, whose magnitude can become com-\nparable to that of 2D Dresselhaus and Rashba interac-\ntions. This interaction gives rise to a nonzero spin-Hall\nconductivity and renormalizes the bulk mass by ∼5% in\nInSb double QWs [10]. This raises the question whether\nthis effect must be accounted for when extracting the\nSCD from intersubband spin plasmon linewidths [4].\nIn this paper we are going to show that while intersub-\nband SO interaction may manifest itself in some special\ncases, as for example in the double well analyzed in Ref.\n[10], it has little to no effect on spin splitting and spin\nmixing in QWs once the 2D Dresselhaus and/or Rashba\nterms are taken into account.\nIn Sec. II we present the general formalism of calcu-\nlating conduction band states in quantum structures in-\ncluding both 2D and intersubband SO interaction. In\nSec. III we consider the specific case of symmetric single-\nwell quantum structures, and in Sec. IV we present re-\nsults for a parabolic model QW. Sec. V gives a brief\nsummary.\nII. GENERAL FORMALISM\nWe consider conduction electrons in a QW described\nby the Hamiltonian\nˆH=ˆH0+ˆHso, (1)\nwhereˆH0is spin independent and ˆHsois the SO inter-\naction projected on the conduction band. For simplic-\nity we will consider only spin off-diagonal (spin-mixing)\nterms in ˆHso. The eigenfunctions associated with ˆH0\nalone can be obtained by solving a single-particle equa-\ntion of the Schr¨ odinger-Poisson or Kohn-Sham type, re-\nsulting in spin-independent subband envelope functions\nψi(z,k/bardbl) and energy eigenvalues εi, whereiis the sub-\nband index and zis the direction of quantum confine-\nment.2\nLet us now consider the two lowest conduction sub-\nbands of the QW. In the basis of the first two subband\nspinors|ψ1↑/angbracketright,|ψ1↓/angbracketright,|ψ2↑/angbracketright,|ψ2↓/angbracketright, the Schr¨ odinger\nequation with the full Hamiltonian (1) has the form\n\nε1α10β\nα∗\n1ε1β′0\n0β′∗ε2α2\nβ∗0α∗\n2ε2\nA=εA, (2)\nwhere\nα1=/angbracketleftψ1↑ |ˆHso|ψ1↓/angbracketright\nα2=/angbracketleftψ2↑ |ˆHso|ψ2↓/angbracketright,\nβ=/angbracketleftψ1↑ |ˆHso|ψ2↓/angbracketright,\nβ′=/angbracketleftψ1↓ |ˆHso|ψ2↑/angbracketright. (3)\nTo remove the off-diagonal terms mixing the ↑,↓states\nwithin the same subband, we apply the unitary transfor-\nmationB=U·Awith\nU=1√\n2\n1−α1\n|α1|0 0\n1α1\n|α1|0 0\n0 0 1 −α2\n|α2|\n0 0 1α2\n|α2|\n. (4)\nEquation (2) then transforms into\n\nε1−|α1|0 −γ1γ2\n0ε1+|α1| −γ2γ1\n−γ∗\n1−γ∗\n2ε2−|α2|0\nγ∗\n2γ∗\n1 0ε2+|α2|\nB=εB,\n(5)\nwhere the off-diagonal matrix elements\nγ1,2=1\n2/bracketleftig\nβα∗\n2\n|α2|±β′α1\n|α1|/bracketrightig\n(6)\nconnect the first and second subbands. We treat these\ncontributions to the conduction band Hamiltonian per-\nturbatively to second order, and obtain the following so-\nlutions of Eq. (5):\nε±\n1=ε1±|α1|\n+|γ1|2\nε1±|α1|−ε2∓|α2|+|γ2|2\nε1±|α1|−ε2±|α2|,\nε±\n2=ε2±|α2|\n+|γ1|2\nε2±|α2|−ε1∓|α1|+|γ2|2\nε2±|α2|−ε1±|α1|\nand\nB−\n1=\n1\n0\n−γ∗\n1\nε1−|α1|−ε2+|α2|\nγ∗\n2\nε1−|α1|−ε2−|α2|\n, (7)B+\n1=\n0\n1\n−γ∗\n2\nε1+|α1|−ε2+|α2|\nγ∗\n1\nε1+|α1|−ε2−|α2|\n, (8)\nB−\n2=\n−γ1\nε2−|α2|−ε1+|α1|\n−γ2\nε2−|α2|−ε1−|α1|\n1\n0\n, (9)\nB+\n2=\nγ2\nε2+|α2|−ε1+|α1|γ1\nε2+|α2|−ε1−|α1|\n0\n1\n.(10)\nThe eigenvectors B±\niare normalized up to first order in\nthe off-diagonal perturbation.\nIn the absence of intrasubband (2D) terms, α1=α2=\n0, theintersubband SO interactiongives rise to spin mix-\ning without lifting the spin degeneracy( ε+\ni=ε−\ni); it only\ncauses a spin-independent shift of the subband energies.\nBy contrast, if an intrasubband interaction is present (or\nifspindegeneracyislifted byothermeans, e.g.,byamag-\nnetic field), the spin splitting is affected. For the lowest\nsubband it is given by ε+\n1−ε−\n1= ∆ε1, where\n∆ε1= 2|α1|+2|γ1|2 |α2|−|α1|\n(ε2−ε1)2−(|α2|−|α1|)2\n−2|γ2|2 |α2|+|α1|\n(ε2−ε1)2−(|α2|+|α1|)2.(11)\nTo proceed further we need the explicit form of the SO\nHamiltonian ˆHso.\nIII. RASHBA AND DRESSELHAUS SO\nINTERACTION IN SYMMETRIC QWS\nBy folding down the 14 ×14k·pHamiltonian for a\nQW grown in [001] direction in a zinc-blende crystal to\na 2×2 conduction band problem [11], one obtains an\neffective SO Hamiltonian in the conduction band:\nˆHso≈/parenleftbigg\n0hso\nh∗\nso0/parenrightbigg\n, (12)\nwhere\nhso=R(z)k−−iλk+∂2\n∂z2−iλ\n4(k2\n−−k2\n+)k−,(13)\nwith\nλ= 4√\n2\n3PQP′/parenleftbigg1\n(E∆−ε)(E′v−ε)−1\n(Ev−ε)(E′\n∆−ε)/parenrightbigg3\nand\nR(z) =√\n2\n3P2/bracketleftbigg∂\n∂z/parenleftbigg1\nEv−ε−1\nE∆−ε/parenrightbigg/bracketrightbigg\n+√\n2\n3P′2/bracketleftbigg∂\n∂z/parenleftbigg1\nE′v−ε−1\nE′\n∆−ε/parenrightbigg/bracketrightbigg\n.(14)\nHere,k±=1√\n2(kx±iky),εis the electron energy, Ev(z)\nandE∆(z) are the position-dependent Γ 8and Γ7valence\nbandedges, and P=−i/planckover2pi1\nm/angbracketleftS|ˆpx|X/angbracketright=/radicalig\nEp/planckover2pi12\n2misthe mo-\nmentum matrix element. Primed quantities correspond\nto the higher lying Γ 8−Γ7conduction band and Qis\nthe momentum matrix element between the valenceband\nand the higher conduction band. Along with the Rashba\nand linear Dresselhaus terms in Eq. (12) we keep the cu-\nbic Dresselhaus term as well. During the derivation we\nassumed that the variation of the band edges is small\ncompared with the energy gaps in the material.\nIn symmetric structures, due to parity conservation\ntheintrasubband SO interaction contains only the Dres-\nselhaus contribution,\nα1=−λ\n4√\n2k3sin(2ϕ)e−iϕ+D11√\n2kei(ϕ+π\n2),(15)\nα2=−λ\n4√\n2k3sin(2ϕ)e−iϕ+D22√\n2kei(ϕ+π\n2),(16)\nand theintersubband SO interaction(between the lowest\ntwo subbands) involves only the Rashba term\nβ=β′∗=R12√\n2ke−iϕ, (17)\nwhereϕis the polar angle of the in-plane vector k/bardblmea-\nsured from the [100] direction, and k=|k/bardbl|. Further-\nmore,\nDii=−λ/angbracketleftbigg\nψi(z)/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂2\n∂z2/vextendsingle/vextendsingle/vextendsingle/vextendsingleψi(z)/angbracketrightbigg\n(18)\nand\nR12=/angbracketleftψ1(z)|R(z)|ψ2(z)/angbracketright. (19)\nThe quantity R12corresponds to the coupling parameter\nηderived in Ref. [10] using an 8-band k·pmodel.\nFor smallkthe linear term in Eqs. (15)-(16) dominates\nand we can approximate\nα1\n|α1|≈α2\n|α2|≈ei(ϕ+π\n2). (20)\nThen,\nγ1=1√\n2R12kcos/parenleftig\n2ϕ+π\n2/parenrightig\n(21)\nγ2=−i√\n2R12ksin/parenleftig\n2ϕ+π\n2/parenrightig\n, (22)and the ground state spin splitting follows from Eq. (11)\nas\n∆ε1≈2|α1|−R2\n12D11√\n2(ε2−ε1)2k3−R2\n12D22√\n2(ε2−ε1)2k3cos(4ϕ).\n(23)\nTheintersubband interaction results thus in an addi-\ntional spin splitting proportional to k3.\nNext, we expand the spin splitting that is induced by\ntheintrasubband SO interaction. Up to order k3we ob-\ntain\n|α1| ≈D11√\n2k+λ\n8√\n2k3−λ\n8√\n2k3cos(4ϕ),(24)\nwhich givesthe final expressionforthe subband splitting:\n∆ε1=√\n2D11k+/parenleftbiggλ\n4−R2\n12D11\n(ε2−ε1)2/parenrightbiggk3\n√\n2\n−/parenleftbiggλ\n4+R2\n12D22\n(ε2−ε1)2/parenrightbiggk3\n√\n2cos(4ϕ).(25)\nOne finds that the intersubband SO interaction produces\nan additional spin splitting of the same symmetry as the\nintrasubband cubic Dresselhaus term. We will now es-\ntimate the magnitude of this additional contribution for\nGaAs parabolic QWs.\nIV. SUBBAND SPIN SPLITTING IN\nPARABOLIC WELLS\nLet us consider a parabolic QW with conduction band\nconfining potential\nV(z) =1\n2Kz2, (26)\nresulting in the noninteracting energy spectrum\nεj=/radicalbigg\n/planckover2pi12K\nm∗/parenleftbigg\nj−1\n2/parenrightbigg\n, j = 1,2,...(27)\nThe first and second subband envelope functions are\nψ1(z) =4/radicalbigg\n2ξ\nπe−ξz2, (28)\nψ2(z) =4/radicalbigg\n32ξ3\nπze−ξz2, (29)\nwhereξ=/radicalbig\nm∗K/4/planckover2pi12. Straightforward calculations\ngive\n/angbracketleftbigg\nψ1/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂2\n∂z2/vextendsingle/vextendsingle/vextendsingle/vextendsingleψ1/angbracketrightbigg\n=−ξ, (30)\n/angbracketleftbigg\nψ2/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂2\n∂z2/vextendsingle/vextendsingle/vextendsingle/vextendsingleψ2/angbracketrightbigg\n=−3ξ, (31)\n/angbracketleftψ1|z|ψ2/angbracketright=−1\n2√ξ. (32)4\nFor our parabolic well, the positional dependence of the\nvalence band edge (the valence band potential) is\nEv=−1\n4Kz2,\ncorresponding to a valence band offset VBO=0.33. For\nGaAs parameters ( Eg= 1.42 eV, ∆ = 0 .34 eV,Ep=\n22 eV) Eq. (14) gives R(z)≈ −/parenleftbig∂Ev\n∂z/parenrightbig\n7˚A2. Using Eqs.\n(18), (19) and (30)–(32) we then get\nR12=−(7˚A2)K\n4√ξ, D 11=λξ, D 22= 3λξ,\nand\nR2\n12D22\n(ε2−ε1)2=\n∆ε\n/planckover2pi12\n2m∗˚A2\n2\n147\n64λ∼10−6λ,\nform∗= 0.065m0and ∆ε=ε2−ε1= 40 meV. The con-\ntribution of the intersubband SO interaction to the spin\nsplitting of the lowest conduction subband is six orders\nof magnitude weaker than that of the cubic Dresselhaus\nintrasubband termand thus canbe completely neglected.\nThe spin mixing induced by the intersubband SO in-\nteraction can be estimated from Eq. (7):\n/vextendsingle/vextendsingle/vextendsingle/vextendsingleγ2\nε2−ε1/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n≈R2\n12k2\n2(∆ε)2=49\n32\n∆ε\n/planckover2pi12\n2m∗˚A2\nk2˚A2∼10−7,\nfork= 0.01˚A−1. This is seven orders of magnitude\nweaker than the spin mixing induced by intrasubband\nSO interaction and also can be completely neglected.\nSimilarresultswereobtainedforGaAssymmetricrect-\nangular QWs.V. CONCLUSIONS\nIn this paper, we have considered the effects of SO\ncoupling on the conduction subband states in symmetric\nQWs. Our work was motivated by Ref. [10], which dis-\ncussed a SO coupling effect specific to QWs with more\nthan one subband and showed that it can affect the elec-\ntronic and spin transport properties in some systems.\nWe found that although the magnitude of this in-\ntersubband SO interaction can be comparable to that\nof the 2D Dresselhaus and Rashba terms, its effect on\nthe spin splitting and spin mixing of conduction band\nstates is several orders of magnitude weaker since it con-\nnects states with different energies. This is due to the\nfact that the spin splitting and spin mixing of conduc-\ntion band states are renormalized by the intersubband\nenergy difference.\nTherefore, ifone considerssystem with non-degenerate\nsubbands, one can completely neglect the intersubband\nSO interaction compared to the usual 2D Dresselhaus\nand Rashba terms. These findings provide an a posteri-\norijustification for the approach used to calculate sub-\nband splittings and spin plasmon dispersions carried out\nin Ref. [6]. This opens the way for a comprehensive the-\nory of collective intersubband excitations in QWs in the\npresence of SCD and SO coupling.\nAcknowledgments\nThis work was supported by DOE Grant No.\nDE-FG02-05ER46213, the Nuffield Foundation Grant\nNAL/01070/G, and by the Research Fund 10024601 of\nthe Department of Physics of the University of York.\n[1]Semiconductor spintronics and quantum computation ,\nedited by D. D. Awschalom, N. Samarth, and D. Loss\n(Springer, Berlin, 2002)\n[2] I.D’AmicoandG.Vignale, Phys.Rev.B 62, 4853(2000).\n[3] C. P. Weber, N. Gedik, J. E. Moore, J. Orenstein, J.\nStephens, and D. D. Awschalom, Nature (London) 437,\n1330 (2005).\n[4] I. D’Amico and C. A. Ullrich, Phys. Rev. B 74,\n121303(R) (2006); I. D’Amico and C. A. Ullrich, Physica\nStatus Solidi (b) 243, 2285 (2006); I. D’Amico and C. A.\nUllrich, Journal of Magnetism and Magnetic Materials\n316, 484 (2007)\n[5] I. D’Amico I and G. Vignale, Europhys. Lett. 55, 566\n(2001); I. D’Amico and G. Vignale, Phys. Rev. B 65,\n085109 (2002); I. D’Amico and G. Vignale, Phys. Rev. B\n68, 045307 (2003); K. Flensberg, T. S. Jensen, and N. A.Mortensen, Phys. Rev. B 64, 245308 (2001)\n[6] C. A. Ullrich and M. E. Flatte, Phys. Rev. B 66, 205305\n(2002); C. A. Ullrich and M. E. Flatte, Phys. Rev. B 68,\n235310 (2003)\n[7] G. Dresselhaus, Phys. Rev. 100, 580 (1955)\n[8] Yu. L. Bychkov and E. I. Rashba, J. Phys. C 17, 6039\n(1984)\n[9] R. Winkler, Spin-orbit coupling effects in two-\ndimensional electron and hole systems (Springer,\nBerlin, 2003)\n[10] E. Bernandes, J. Schliemann, M. Lee, J.C. Egues, and D.\nLoss, Phys. Rev. Lett 99, 076603 (2007)\n[11] P. Pfeffer and W. Zawadzki, Phys. Rev. B 41, 1561\n(1990); P. Pfeffer, Phys. Rev. B 59, 15902 (1999); P. Pf-\neffer and W. Zawadzki, Phys. Rev. B 15, R14332 (1995)"    },    {        "title": "1805.00047v1.Superconducting_tunneling_spectroscopy_of_spin_orbit_coupling_and_orbital_depairing_in_Nb_SrTiO__3_.pdf",        "content": "Superconducting tunneling spectroscopy of spin-orbit coupling\nand orbital depairing in Nb:SrTiO 3\nAdrian G. Swartz,1, 2, 3,\u0003Alfred K. C. Cheung,4Hyeok Yoon,1, 2, 3Zhuoyu\nChen,1, 2, 3Yasuyuki Hikita,2Srinivas Raghu,2, 4and Harold Y. Hwang1, 2, 3\n1Geballe Laboratory for Advanced Materials, Stanford University, Stanford, California 94305, USA\n2Stanford Institute for Materials and Energy Sciences,\nSLAC National Accelerator Laboratory, Menlo Park, California 94025, USA\n3Department of Applied Physics, Stanford University, Stanford, California 94305, USA\n4Department of Physics, Stanford University, Stanford, California 94305, USA\n(Dated: May 2, 2018)\nWe have examined the intrinsic spin-orbit coupling (SOC) and orbital depairing in thin \flms of\nNb-doped SrTiO 3by superconducting tunneling spectroscopy. The orbital depairing is geometrically\nsuppressed in the two-dimensional limit, enabling a quantitative evaluation of the Fermi level spin-\norbit scattering using Maki's theory. The response of the superconducting gap under in-plane\nmagnetic \felds demonstrates short spin-orbit scattering times \u001cso\u00141:1 ps. Analysis of the orbital\ndepairing indicates that the heavy electron band contributes signi\fcantly to pairing. These results\nsuggest that the intrinsic spin-orbit scattering time in SrTiO 3is comparable to those associated\nwith Rashba e\u000bects in SrTiO 3interfacial conducting layers and can be considered signi\fcant in all\nforms of superconductivity in SrTiO 3.\nThe relativistic spin-orbit interaction is fundamental\nin the solid state, connecting the conduction electron\nspin to the atomic, electronic, orbital, and structural\nsymmetry properties of the material [1]. SrTiO 3is an\noxide semiconductor with highly mobile t2gconduction\nelectrons and exhibits superconductivity at the lowest\nknown carrier density of any material [2{4]. The rele-\nvance of the intrinsic spin-orbit coupling (SOC) for su-\nperconductivity in the bulk material remains an open\nquestion: the atomic SOC produces a relatively small\nsplitting (29 meV [2]) of the t2gbands, butmight be an\nimportant energy scale considering the small supercon-\nducting gap in SrTiO 3. Moreover, SrTiO 3is the host\nmaterial for unconventional two-dimensional (2D) super-\nconductors such as FeSe/SrTiO 3[5],\u000e-doped SrTiO 3[3],\nand LaAlO 3/SrTiO 3[6]. Spin-orbit coupling in SrTiO 3\ninterfacial accumulation layers has been extensively stud-\nied both experimentally and theoretically [7{13]. In these\nsystems, Rashba SOC has been suggested to give rise to\nmany of the unusual normal- and superconducting-state\nproperties due to the broken inversion symmetry and the\nhighly asymmetric con\fnement potential. Understand-\ning the competition between the intrinsic and Rashba\ncoupling scales is critical to understanding the spin-orbit\ntextures and superconducting phases in both bulk and\n2D systems.\nThe spin-orbit coupling strength can be quantitatively\nextracted from superconducting tunneling spectra of thin\n\flms in large parallel magnetic \felds [14{16]. In a con-\nventionals-wave superconductor, a magnetic \feld acts\nin two ways on the conduction electrons: by inducing\ncyclotron orbits and via the electron magnetic moment\n(spin). Both of these e\u000bects lead to the breaking of\nCooper pairs once their energy scale competes with the\ncondensation energy. For thin \flms in the 2D limit, theorbital depairing can be geometrically suppressed, lead-\ning to highly anisotropic upper-critical \felds with large\nin-planeHc2;k. In the absence of spin-orbit coupling, spin\nis a good quantum number and Hc2;kis determined by\nthe Pauli paramagnetic limit ( HP= \u0001 0=p\n2\u0016B, where\n\u00010is the superconducting gap at T= 0 and\u0016Bis the\nBohr magneton) [14, 17, 18]. The application of an in-\nplane magnetic \feld splits the spin-up and spin-down\nsuperconducting quasiparticle density of states (DOS)\nthrough the Zeeman e\u000bect (Fig. 1 left panel) [14]. In-\ncreasing the spin-orbit coupling leads to a mixing of the\nspin-up and spin-down states and lifts Hc2;kabove the\nPauli limit [14, 19, 20]. If the spin-orbit scattering rate is\nvery fast (\u0016h=\u001cso>\u00010, where\u001csois the normal-state spin-\norbit scattering time), then the superconducting DOS\ndoes not exhibit measurable Zeeman splitting (Fig. 1\nright panel). Fitting the tunneling spectra using Maki's\ntheory [21{23] enables a quantitative extraction of both\nthe orbital depairing parameter ( \u000bo) and\u001csofrom the\ntunneling spectra. This approach, pioneered by Tedrow\nand Meservey, has been used extensively to explore de-\npairing mechanisms of conventional elemental supercon-\nductors [14{16, 22, 23].\nHere we examine spin-orbit coupling and orbital de-\npairing in thin \flms of Nb-doped SrTiO 3(NSTO) using\ntunneling spectroscopy. Recently, we have developed an\napproach for realizing high-quality tunneling junctions\nfor bulk NSTO with \u0016eV resolution of the superconduct-\ning gap [24, 25]. By carefully engineering the band align-\nments using polar tunneling barriers, the interfacial car-\nrier density probed by tunneling corresponds to the nom-\ninal density of dopants. We study the tunneling conduc-\ntance (di=dv ) of NSTO \flms in the 2D limit ( d < \u0018 GL,\nwheredis the \flm thickness and \u0018GLis the Ginzburg-\nLandau coherence length). We \fnd a single supercon-arXiv:1805.00047v1  [cond-mat.supr-con]  30 Apr 20182\n-4-2024E/Δ2.52.01.51.00.50.0ρ↑,↓ /ρ0 -4-2024E/Δ\n!∥!#SrTiO3(001)Nb:SrTiO3LaAlO3AgdΩΩa)\nb)\nSrTiLaAlO(SrO)0(TiO2)0(LaO)+(AlO2)-(AlO2)-(LaO)+\n%=0(=0.1+,-∆/=0.6%=6(=0.1+,-∆/=0.61↑+1↓\nFIG. 1. (Color online) a) Schematic of the tunneling junc-\ntion device structure and atomic stacking of the oxide het-\nerostructure. b) Expected e\u000bect of Zeeman splitting on the\nspin-dependent DOS for two cases: zero spin-orbit coupling\n(b= 0) (left panel) and large spin-orbit coupling ( b= 6) (right\npanel). The dimensionless SOC parameter b= \u0016h=(3\u001cso\u00010) re-\n\rects the strength of the SOC relative to the gap energy scale.\nDashed blue (dashed grey) and solid red (solid grey) curves\nrepresent the spin-up and spin-down DOS, respectively, while\nthe solid black curve gives the total DOS from \u001a\"+\u001a#(shifted\nupwards by 1 for clarity). The spectra were calculated using\nMaki's theory (Eq. (2)) at T= 0 K, lifetime broadening\nparameter\u0010= 0:1, and magnetic \feld \u0016BH=\u00010= 0:6.\nducting gap which closes at the superconducting transi-\ntion temperature ( Tc). We extract Hc2;kfrom the tun-\nneling spectra and \fnd that it greatly exceeds the Pauli\nlimit. Under in-plane applied \felds, Zeeman splitting is\nnot observed and an apparent single gap persists at all\n\felds until closing completely near 1.6 T, indicating that\nthe spin-orbit coupling scale (\u0016 h=\u001cso) is larger than \u0001 0.\nWe analyze the data using Maki's theory [21{23] and ex-\namine the relative contributions from orbital depairing\nand spin-orbit scattering. Due to the heavy mixing of\nthe spin states, Maki's theory provides an upper-bound\nfor the spin-orbit scattering time of \u001cso\u00141:1 ps and spin\ndi\u000busion length \u0015s\u001432 nm.\nWe fabricated tunneling junctions consisting of super-\nconducting NSTO thin \flms of thickness d= 18 nm, with\na 2 unit cell (u.c.) epitaxial LaAlO 3tunneling barrier,\nand Ag counter electrodes as described elsewhere [24, 25].\nNSTO with 1 at.% Nb-doping was homoepitaxially de-\nposited on undoped SrTiO 3(001) by pulsed-laser deposi-\ntion [26]. Films grown by this technique exhibit full car-\n6050403020100 Δ (µeV)0.60.40.20.0T (K)1.00.80.60.40.20.0R/RN0.60.50.40.30.20.10.0di/dv (mS)-400-2000200400V (µV)!\"#= 0 T!\"#= 0 Ta)b)FIG. 2. (Color online) Tunneling spectroscopy and resis-\ntivity in zero \feld. a) Tunneling conductance ( di=dv ) of 18\nnm thick Nb-doped SrTiO 3thin \flm measured at the base\ntemperature of the dilution refrigerator. b) Superconducting\ngap amplitude (\u0001) (open circles, left axis) compared to the\nnormalized resistance (solid blue (grey) line, right axis). The\nsuperconducting gap closes at T= 315\u00065 mK, which is very\nclose to the resistive transition temperature Tc= 330 mK\nde\fned as 50% of the normal state resistivity at T= 0:6 K.\nrier activation and bulk-like electron mobility. The polar\nLaAlO 3tunnel barrier plays a crucial role in enabling\naccess to the electronic structure of NSTO in the 2D su-\nperconducting limit. The LaAlO 3layer provides an in-\nterfacial electric dipole which shifts the band alignments\nbetween the Ag electrode and semiconducting SrTiO 3by\n\u00190:5 eV/u.c. [27{29]. Aligning the Fermi-level between\nthe two electrodes signi\fcantly reduces the Schottky bar-\nrier and eliminates the long depletion length which pro-\nhibits direct tunneling.\nFirst, we report the zero-\feld superconducting behav-\nior of the sample. Figure 2a shows di=dv measured at\nbase temperature ( T= 20 mK) and \u00160H= 0 T exhibit-\ning a single superconducting gap (\u0001). Although we ob-\nserve high-energy coupling to longitudinal-optic phonon\nmodes (not shown) as reported recently [24], we do not\n\fnd other strong-coupling renormalizations (i.e. McMil-\nlan and Rowell [30]) in the tunneling spectra. The su-\nperconducting gap is well \ft by the Bardeen-Cooper-\nSchrie\u000ber (BCS) equation for the density of states with\n\u00010= 47\u00061\u0016eV. Due to the \fnite resolution of the\nmeasurement and thermal broadening, the minimum of\nthe superconducting gap is \fnite. Here, the gap mini-\nmum is two-orders of magnitude smaller than the normal\nstate conductance, demonstrating the dominance of elas-\ntic tunneling and the high quality of the junction, even\nin the 2D limit. The superconducting gap closes near\nTc= 330 mK as measured by four-point resistivity (Fig.\n2b). Importantly, we do not observe a pseudogap as was\nrecently observed in LaAlO 3/SrTiO 3[31], indicating the\npseudogap is speci\fc to the LAO/STO interface and not\na generic feature in the 2D limit.\nWe now turn to the magnetic-\feld response of the su-\nperconducting gap. Figure 3a shows the superconducting\ngap at several characteristic values of applied magnetic3\n\feld (left panel: H?, right panel: Hk). Figure 3b dis-\nplays the zero-bias conductance (gap minimum) normal-\nized to the normal-state zero-bias conductance for both\n\feld orientations. We \fnd a large anisotropy between\nHc2;?andHc2;kwith a ratio Hc2;?/Hc2;k= 0.052. We\nextract the Ginzburg-Landau superconducting coherence\nlength\u0018GL=p\n\b0=(2\u0019Hc2;?) = 62 nm > d, con\frming\nthe superconducting state is in the 2D regime. SrTiO 3\nis a type-II superconductor with large London penetra-\ntion depth compared to \u0018GLandd, and the quenching\nof superconductivity due to an out-of-plane \feld can be\nattributed to the formation of vortices. For \felds applied\nin-plane, the large size of a vortex core is energetically un-\nfavorable to form in the 2D limit and the orbital depair-\ning is dramatically suppressed leading to enhanced Hc2;k.\nWe \fnd that the superconducting gap exhibits large Hc2;k\nfar in excess of the Pauli limit ( HP= \u0001 0=p\n2\u0016B= 0.574\nT), and in agreement with a study of upper-critical \felds\nfrom resistivity measurements in \u000e-doped SrTiO 3quan-\ntum wells [20]. Here, we can examine the spin-dependent\nresponse of the superconducting gap spectra to extract\nthe relevant contributions to orbital and spin depairing\nmechanisms.\nThe superconducting DOS has been given by Maki's\ntheory, which takes into account orbital depairing, Zee-\nman splitting of the spin states, and SOC [15, 21]. The\nspin-dependent DOS is given by,\n\u001a\";#=\u001a0\n2sgn(E)Re0\n@u\u0006q\nu2\n\u0006\u000011\nA; (1)\nwhere\u001a0is the normal-state DOS and u\u0006are de\fned by,\nu\u0006=E\u0007\u0016BH\n\u00010+\u0010u\u0006q\n1\u0000u2\n\u0006+b0\n@u\u0007\u0000u\u0006q\n1\u0000u2\n\u00071\nA;(2)\nfor whichEis the energy relative to the Fermi level ( EF),\nb= \u0016h=(3\u001cso\u00010) is a dimensionless quantity representing\nthe strength of the spin-orbit scattering relative to \u0001 0,\nand\u0010represents spin-independent lifetime corrections.\nMaki's equation (Eq. (2)) reduces to the BCS DOS in\nthe limit of vanishing \u0010andb. The quantity \u0016BHrepre-\nsents the Zeeman splitting of the spin-dependent states\nand observation of this splitting in the experimental data\ndepends on the strength of b(see Fig. 1). The parame-\nter\u0010=\u000bi+\u000boH2\nkincludes \feld-independent broadening\n(\u000bi) and\u000bo=De2d2=(6\u0016h\u00010) is the standard orbital de-\npairing for a thin \flm in a parallel magnetic \feld ( Dis\nthe di\u000busion coe\u000ecient) [14, 15, 21]. We follow the nu-\nmerical approach of Worledge and Geballe in applying\nEq. (2) to the tunneling data [21{23].\nWe now focus on the spectra shown in Fig. 3a (right\npanel) for in-plane applied \felds. The magnetic \felds ex-\nplored here ( \u0016BHk=\u00010<2) are large enough to observe\nZeeman splitting in the weak spin-orbit limit ( b<1) [32].\n1.21.00.80.60.40.20.0(σ / σN)|V=0 \n2.01.51.00.50.0µ0H (T)0.60.50.40.30.20.10.0di/dv (mS)-400-2000200400V (µV) 0.11 T 0.09 T 0.08 T  0.07 T 0.06 T 0.05 T 0.04 T 0.03 T 0.02 T 0.01 T 0 T-400-2000200400V (µV) 2 T 1.6 T 1.5 T 1.4 T   1.2 T 1.0 T 0.8 T 0.5 T 0.3 T 0 T!\"#,%!\"#,∥'(!∥='(!%=\n!)a)\nb)FIG. 3. (Color online) Tunneling spectroscopy of the super-\nconducting gap under applied magnetic \feld. a) Raw di=dv\ndata for several values of magnetic \felds applied out-of plane\n(\u00160H?, left panel) and in-plane ( \u00160Hk, right panel). b) Zero-\nbias conductivity ( \u001b=di=dv ) of the gap minimum normalized\nto the normal-state conductance ( \u001bN) for both \feld orienta-\ntions. The out-of plane ( Hc2;?) and in-plane ( Hc2;k) upper\ncritical \felds are indicated. The vertical dashed blue line in-\ndicates the Pauli paramagnetic limiting \feld ( HP).\nHowever, for all measured magnetic \felds, the data does\nnot exhibit a clear signature of Zeeman splitting indicat-\ning strong spin scattering relative to the superconducting\ngap (compare Fig. 1 with Fig. 3a right panel) and con-\nsistent with the violation of the Pauli-limit. While the\nspin-orbit parameter bis \feld-independent, the e\u000bect of\nZeeman splitting in combination with rapid spin mixing\nis to produce an e\u000bective broadening of the total DOS\n(\u001a\"+\u001a#, see Fig. 1) following an H2dependence [15].\nTherefore since both orbital depairing and the large SOC\nproduce quasiparticle broadening under an applied \feld,\nit is a useful exercise to \frst consider a reduced version of\nMaki's theory which ignores the spin-degree of freedom\nin the problem, such that,\nu\u0006!u=E\n\u00010+\u00100up\n1\u0000u2; (3)\nwhich in zero-\feld is equivalent to the Dynes formulation\nwere the phenomenological Dynes quasiparticle broaden-\ning parameter is given by \u0000 = \u00100\u00010[33]. We \frst \ft the\ndata of Fig. 3a right panel using Eq. (3) where \u00100is4\nthe only free parameter. The results for \u00100are shown in\nFig. 4a as a function of H2\nkand are well described by\n\u00100=\u000bi+\u0011H2with\u000bi= 0.056 and \u0011= 0.4 T\u00002. The\nsmall intrinsic quasiparticle broadening ( \u000bi) gives \u0000 =\n2\u0016eV and identical to our previous report in the bulk\nlimit [24]. The extracted \u0011value re\rects the total contri-\nbution to \feld-induced broadening from both spin-orbit\ncoupling and orbital depairing.\nTo quantify the spin-orbit and orbital depairing con-\ntributions, we apply Maki's full theory (Eq. (2)) to\nthe set of tunneling data between 300 and 700 mT\n(0:3< \u0016 BH=\u00010<0:86) including the spin-dependent\ndensity of states, spin-orbit parameter b, and depairing\nparameter\u0010=\u000bi+\u000boH2. The only free parameters are b\nand\u000bowhich must both be singly valued at all \felds. We\n\fnd that the best \fts are statistically equivalent for b>4\n(with varying \u000b0) [32], indicating short spin-orbit scat-\ntering times \u001cso<1:1 ps. In this regime ( b>4),\u000b0and\nbare correlated. This can be understood as a competi-\ntion between the spin-orbit induced e\u000bective broadening\nand orbital depairing. For instance, in the limit b!1 ,\nthe broadening from SOC vanishes and orbital depairing\nmust asymptotically approach \u0011to account for the exper-\nimentally observed broadening. Fig. 4b shows a charac-\nteristic best \ft for \u00160Hk= 0.5 T (\u0016BH=\u00010= 0.61) with\nb= 6 and\u000b0= 0:11. Additionally, an upper-bound on\nthe spin di\u000busion length is given by \u0015s=q\n3\n4Dtr\u001cso<32\nnm [34], where Dtr=v2\nF\u001ctr=3\u00190:0012 m2/s is the trans-\nport di\u000busion coe\u000ecient. Here we have estimated the\nFermi velocity vFin a single-band approximation with\ne\u000bective mass m\u0003= 1.24m0[3] and Fermi level EF=\n61 meV [2, 24]. We have used the Drude scattering time\n\u001ctr=m\u0003\u0016e=ewhere\u0016e= 300 cm2/Vs is the experimen-\ntally measured electron mobility.\nThe contribution from orbital depairing in the tunnel-\ning data provides additional information on the super-\nconducting phase. The best \fts from Maki's theory in\nthe rangeb >4 correspond to 0.016 T\u00002< \u000bo\u0014\u0011, for\nwhich\u000boincreases commensurately with b. Thus, even\nthough spin-orbit and orbital depairing cannot be quan-\nti\fed independently, there are clear experimental limits\non\u000bo. We can compare the experimental \u000bowith the ex-\npected orbital contribution from normal-state transport\nparameters with \u000bo=Dtre2d2=(6\u0016h\u00010)\u00192 T\u00002, which is\nfar in excess of the measured total broadening of \u0011= 0:4\nT\u00002. This apparent discrepancy can be resolved by con-\nsidering the multi-band nature of bulk SrTiO 3with three\noccupiedt2gorbitals comprised of two light- and one\nheavy-electron bands [2, 10]. Normal-state transport co-\ne\u000ecients are dominated by the highly mobile light elec-\ntrons, but these carriers only make-up a fraction of the\ntotal DOS, whereas the lowest lying heavy band com-\nprises the majority of the electrons at EF[2, 3, 10, 35]. In\nother words, the experimental data cannot be explained\nby solely considering highly mobile, light electrons in\n1.61.20.80.40.0σ / σN-400-2000200400V (µV)0.80.60.40.20.0ζ´1.61.20.80.40.0(µ0H||)2 (T)a)b)\n!\"#∥= 0.5 T%=6(\"=0.11FIG. 4. (Color online) Maki analysis of the superconducting\ngap spectra under in-plane magnetic \felds. a) Total quasi-\nparticle broadening \u00100(black dots) determined by \ftting the\ntunneling data of Fig. 3a right panel using Eq. (3). The\ntotal broadening exhibits a dependence on the square of the\napplied magnetic \feld and the solid line represents a \ft to\n\u00100=\u000bi+\u0011H2. b) Normalized di=dv data (solid black line)\nmeasured at \u00160Hk= 0.5 T and theoretical \ft (dashed red\n(grey) line) using Maki's full theory as expressed in Eq. (2)\nwithb= 6,\u000bi= 0.056,\u000bo= 0.11, and \u0001 0= 47\u0016eV.\nforming the superconducting phase. We can transpose\nthe orbital depairing extracted from the superconduct-\ning tunneling data to DSCrepresenting the di\u000busion co-\ne\u000ecient for electrons which contribute to pairing. We\n\fnd 0:1\u000210\u00004m2/s< D SC<2:3\u000210\u00004m2/s, which\nagrees very well with a simplistic estimate of the di\u000bu-\nsion constant for the heavy electron band with m\u0003\u00196m0\n[36] and momentum scattering time \u001che\u0019100 fs, giving\nDhe\u00191\u000210\u00004m2/s. Therefore, the robustness of super-\nconductivity at high magnetic \felds is consistent with the\nestablished bulk band structure for which the heavy elec-\ntron band dominates the total DOS and results in weak\norbital depairing. We note that the importance of the\nheavy bands for superconductivity has been suggested in\nLaAlO 3/SrTiO 3[37, 38]\nThe spin-orbit scattering times observed here are com-\nparable to the momentum scattering time ( \u001ctr=\u001cso\u00180:1)\nand signi\fcantly shorter than those suggested theoreti-\ncally in a single band limit [39]. Additionally, we can\nexpect that Rashba and Dresselhaus \felds are minimal\nin the current sample structure under investigation [32].\nTherefore, the rapid spin mixing near the Fermi level\ncan be understood in the context of the multiband elec-\ntronic structure of bulk SrTiO 3with hybridized orbital\ncharacter arising from the tetragonal crystal \feld split-\nting and the intrinsic atomic spin-orbit interaction [2].\nThis picture is analogous to p-type Si where short spin\nrelaxation times are characteristic despite the modest\nSOC [40, 41]. The spin-orbit scattering explored here\nre\rects the electrons with the largest contribution to\nthe density of states and the superconducting conden-\nsate, which in bulk SrTiO 3is the heavy electron band.\nThis is in contrast to transport experiments exploring5\nspin-orbit coupling in the normal state (i.e. weak (anti-\n)localization, Subnikov de Haas oscillations) which are\nmost sensitive to the highly mobile subset of carriers\n[3, 35]. Therefore, careful analysis of the sub-band struc-\nture and orbital character in con\fned SrTiO 3-based het-\nerostructures (e.g. LaAlO 3/SrTiO 3) is critical to un-\nderstanding the spin-orbit properties of the normal and\nsuperconducting phases. Regardless, it is interesting to\nnote that the scattering times found here ( \u001cso\u00181 ps),\nare in the ballpark of the vast majority of experimen-\ntal \fndings in LaAlO 3/SrTiO 3[7, 8], suggesting that the\nspin-orbit scattering at the Fermi level arising from the\nintrinsic atomic spin-orbit interaction contributes at least\non equal footing with Rashba e\u000bects.\nIn conclusion, we have performed tunneling experi-\nments on the dilute superconductor SrTiO 3doped with\n1 at.% Nb in the 2D superconducting limit. These re-\nsults were enabled by precisely designing the tunneling\njunction with epitaxial dipole tunnel barriers, which shift\nband alignments and facilitates high-resolution tunneling\nspectroscopy. The data indicates a single superconduct-\ning gap which closes at Tc. By geometrically suppressing\nthe orbital depairing, we show that the large intrinsic\nSOC can be observed directly in the tunneling spectra\nby the violation of the Pauli-limit and the absence of\nZeeman splitting. Surprisingly short spin-orbit scatter-\ning times of order 1 ps were obtained. Examination of\nthe orbital depairing parameter indicates that the heavy\nelectron band, which is di\u000ecult to explore in transport\nexperiments, plays an important role in the formation of\nthe superconducting phase.\nWe thank M. E. Flatt\u0013 e for useful discussions. This\nwork was supported by the Department of Energy,\nO\u000ece of Basic Energy Sciences, Division of Mate-\nrials Sciences and Engineering, under Contract No.\nDE-AC02-76SF00515; and the Gordon and Betty\nMoore Foundation's EPiQS Initiative through Grant\nGBMF4415 (dilution fridge measurements).\n\u0003aswartz@stanford.edu\n[1] R. Winkler, Spin-Orbit Coupling E\u000bects in Two-\nDimensional Electron and Hole Systems .\n[2] D. van der Marel, J. L. M. van Mechelen, and I. I. Mazin,\nPhys. Rev. B 84, 205111 (2011).\n[3] M. Kim, C. Bell, Y. Kozuka, M. Kurita, Y. Hikita, and\nH. Y. Hwang, Phys. Rev. Lett. 107, 106801 (2011).\n[4] X. Lin, Z. Zhu, B. Fauqu\u0013 e, and K. Behnia, Phys. Rev.\nX3, 021002 (2013).\n[5] W. Qing-Yan, L. Zhi, Z. Wen-Hao, Z. Zuo-Cheng, Z. Jin-\nSong, L. Wei, D. Hao, O. Yun-Bo, D. Peng, C. Kai, et al. ,\nChin. Phys. Lett. 29, 037402 (2012).\n[6] A. D. Caviglia, S. Gariglio, N. Reyren, D. Jac-\ncard, T. Schneider, M. Gabay, S. Thiel, G. Hammerl,\nJ. Mannhart, and J.-M. Triscone, Nature 456, 624(2008).\n[7] M. Ben Shalom, M. Sachs, D. Rakhmilevitch,\nA. Palevski, and Y. Dagan, Phys. Rev. Lett. 104, 126802\n(2010).\n[8] A. D. Caviglia, M. Gabay, S. Gariglio, N. Reyren, C. Can-\ncellieri, and J.-M. Triscone, Phys. Rev. Lett. 104, 126803\n(2010).\n[9] H. Nakamura, T. Koga, and T. Kimura, Phys. Rev. Lett.\n108, 206601 (2012).\n[10] Z. Zhong, A. T\u0013 oth, and K. Held, Phys. Rev. B 87, 161102\n(2013).\n[11] G. Khalsa, B. Lee, and A. H. MacDonald, Phys. Rev. B\n86, 125121 (2012).\n[12] P. King, S. M. Walker, A. Tamai, A. De La Torre,\nT. Eknapakul, P. Buaphet, S. Mo, W. Meevasana,\nM. Bahramy, and F. Baumberger, Nature Commun. 5,\n3414 (2014).\n[13] P. K. Rout, E. Maniv, and Y. Dagan, Phys. Rev. Lett.\n119, 237002 (2017).\n[14] R. Meservey and P. Tedrow, Phys. Rep. 238, 173 (1994).\n[15] P. Fulde, Adv. Phys. 22, 667 (1973).\n[16] R. Meservey, P. Tedrow, and R. C. Bruno, Phys. Rev.\nB17, 2915 (1978).\n[17] A. M. Clogston, Phys. Rev. Lett. 9, 266 (1962).\n[18] B. Chandrasekhar, Appl. Phys. Lett. 1, 7 (1962).\n[19] X. Wu, P. Adams, Y. Yang, and R. McCarley, Phys.\nRev. Lett. 96, 127002 (2006).\n[20] M. Kim, Y. Kozuka, C. Bell, Y. Hikita, and H. Hwang,\nPhys. Rev. B 86, 085121 (2012).\n[21] K. Maki, Pog. Theor. Phys. 32, 29 (1964).\n[22] J. Alexander, T. Orlando, D. Rainer, and P. Tedrow,\nPhys. Rev. B 31, 5811 (1985).\n[23] D. Worledge and T. Geballe, Phys. Rev. B 62, 447 (2000).\n[24] A. G. Swartz, H. Inoue, T. A. Merz, Y. Hikita, S. Raghu,\nT. P. Devereaux, S. Johnston, and H. Y. Hwang, Proc.\nNatl. Acad. Sci. 115, 1475 (2018).\n[25] H. Inoue, A. G. Swartz, N. J. Harmon, T. Tachikawa,\nY. Hikita, M. E. Flatt\u0013 e, and H. Y. Hwang, Phys. Rev.\nX5, 041023 (2015).\n[26] Y. Kozuka, Y. Hikita, C. Bell, and H. Y. Hwang, Appl.\nPhys. Lett. 97, 012107 (2010).\n[27] T. Yajima, M. Minohara, C. Bell, H. Kumigashira,\nM. Oshima, H. Y. Hwang, and Y. Hikita, Nano Lett.\n15, 1622 (2015).\n[28] T. Tachikawa, M. Minohara, Y. Hikita, C. Bell, and\nH. Y. Hwang, Adv. Mater. 27, 7458 (2015).\n[29] Y. Hikita, K. Nishio, L. C. Seitz, P. Chakthranont,\nT. Tachikawa, T. F. Jaramillo, and H. Y. Hwang, Adv.\nEnergy Mater. 6, 1502154 (2016).\n[30] W. L. McMillan and J. M. Rowell, Phys. Rev. Lett. 14,\n108 (1965).\n[31] C. Richter, H. Boschker, W. Dietsche, E. Fillis-Tsirakis,\nR. Jany, F. Loder, L. Kourkoutis, D. Muller, J. Kirtley,\nC. Schneider, et al. , Nature 502, 528 (2013).\n[32] See Supplementary Information.\n[33] R. Dynes, V. Narayanamurti, and J. P. Garno, Phys.\nRev. Lett. 41, 1509 (1978).\n[34] I. \u0014Zuti\u0013 c, J. Fabian, and S. D. Sarma, Rev. Mod. Phys.\n76, 323 (2004).\n[35] A. D. Caviglia, S. Gariglio, C. Cancellieri, B. Sac\u0013 ep\u0013 e,\nA. Fete, N. Reyren, M. Gabay, A. F. Morpurgo, and\nJ.-M. Triscone, Phys. Rev. Lett. 105, 236802 (2010).\n[36] Y. J. Chang, A. Bostwick, Y. S. Kim, K. Horn, and\nE. Rotenberg, Phys. Rev. B 81, 235109 (2010).6\n[37] A. Joshua, S. Pecker, J. Ruhman, E. Altman, and\nS. Ilani, Nature Commun. 3, 1129 (2012).\n[38] Y. Nakamura and Y. Yanase, J. Phys. Soc. Jpn. 82,\n083705 (2013).[39] C. S \u0018ahin, G. Vignale, and M. E. Flatt\u0013 e, Phys. Rev. B\n89, 155402 (2014).\n[40] K. Ando and E. Saitoh, Nature Commun. 3, 629 (2012).\n[41] R. Jansen, S. P. Dash, S. Sharma, and B. C. Min, Semi-\ncond. Sci. Technol. 27, 083001 (2012)."    },    {        "title": "1803.03549v1.Spin_vorticity_coupling_in_viscous_electron_fluids.pdf",        "content": "arXiv:1803.03549v1  [cond-mat.mes-hall]  9 Mar 2018Spin-vorticity coupling in viscous electron fluids\nRuben J. Doornenbal,1Marco Polini,2and Rembert A. Duine1,3\n1Institute for Theoretical Physics, Utrecht University,\nLeuvenlaan 4, 3584 CE Utrecht, The Netherlands\n2Istituto Italiano di Tecnologia, Graphene Labs, Via Morego 30, I-16163 Genova, Italy\n3Department of Applied Physics, Eindhoven University of Tec hnology,\nP.O. Box 513, 5600 MB Eindhoven, The Netherlands\n(Dated: November 7, 2021)\nWe consider spin-vorticity coupling—the generation of spi n polarization by vorticity—in viscous\ntwo-dimensional electron systems with spin-orbit couplin g. We first derive hydrodynamic equations\nfor spin and momentum densities in which their mutual coupli ng is determined by the rotational\nviscosity. We then calculate the rotational viscosity micr oscopically in the limits of weak and strong\nspin-orbit coupling. We provide estimates that show that th e spin-orbit coupling achieved in recent\nexperiments is strong enough for the spin-vorticity coupli ng to be observed. On the one hand, this\ncoupling provides a way to image viscous electron flows by ima ging spin densities. On the other\nhand, we show that the spin polarization generated by spin-v orticity coupling in the hydrodynamic\nregime can, in principle, be much larger than that generated , e.g. by the spin Hall effect, in the\ndiffusive regime.\nPACS numbers: 85.75.-d, 75.30.Ds, 04.70.Dy\nIntroduction. —The field of spintronics is concerned\nwith electric control of spin currents [1]. For the de-\nscription of experimentally relevant systems it has, until\nvery recently, been sufficient to consider their coupled\nspin-charge dynamics in the diffusive regime where the\ntime scale for electron momentum scattering is fast com-\npared to other time scales. The celebrated Valet-Fert\ntheory for electron spin transport in magnetic multilay-\ners [2] and the Dyakonov-Perel drift-diffusion theory for\nspin generation by the spin Hall effect [3], for example,\nfall within this paradigm.\nVery recent experimental developments have brought\nabout solid-state systems, such as ultra-clean encapsu-\nlated graphene, in which the momentum scattering time\ncan be much longer than the time scale for electron-\nelectron interactions [4–7]. In this so-called hydrody-\nnamic regime, the electron momentum needs to be in-\ncluded as a hydrodynamic variable and the viscosity of\nthe electron system cannot be neglected [8–17]. The\nfinite electron viscosity leads to several physical conse-\nquences, such as a negative nonlocal resistance [4] and\nsuper-ballistic transport through point contacts [7, 18].\nThese developments have spurred on a great deal of re-\nsearch, including proposals for measuring the Hall vis-\ncosity [19–21] and connections to strong-coupling predic-\ntions from string theory [22].\nIn a seemingly unrelated development, spin-\nhydrodynamic generation, i.e. the generation of voltages\nfrom vorticity, was recently experimentally observed in\nliquid Hg [23]. Spin-hydrodynamic generation is believed\nto be a consequence of spin-vorticity coupling. Phe-\nnomenological theories of spin-vorticity coupling were\ndeveloped early on [24] and have been applied to fluids\nconsisting of particles with internal angular momentum\nsuch as ferrofluids [25], molecular nanofluids [26], andnematic liquid crystals [27]. In these phenomenological\ntheories, the coupling between orbital angular momen-\ntum, i.e. vorticity of the fluid, and internal angular\nis governed by a dissipative coefficient, the so-called\n“rotational viscosity”. This type of viscosity has been\nestimated microscopically for classical systems (see e.g.\n[27]) and Hg [23], but not for viscous electrons in a\ncrystal.\nMotivated by the recent realization of solid-state sys-\ntems hosting viscous electron fluids, we develop in this\nLetter the theory for spin-vorticity coupling in such sys-\ntems. Wederivethephenomenologicalequationsdescrib-\ning coupled spin and momentum diffusion, and compute\nthe rotational viscosity microscopically. We apply our\ntheory to viscous electron flow through a point contact\nand show that the spin densities generated hydrodynam-\nically can be much larger than the ones that are gen-\nerated by the spin Hall effect in the diffusive transport\nregime. Our results may therefore stimulate experimen-\ntal research towards novel ways of spin detection and\ngeneration.\nPhenomenology. —We consider two-dimensional (2D)\nelectron systems with approximate translation invari-\nance and approximate rotation invariance around the\naxis perpendicular to the plane (chosen to be the ˆz-\ndirection). The conserved quantities of this system are\nenergy, charge, linear momentum in the plane and an-\ngular momentum in the ˆz-direction. For brevity, we do\nnot consider energy conservation explicitly and focus on\nmomentum and angular momentum conservation. In the\nfollowing, we follow the discussion of Ref. [24] and gen-\neralize it to include spin diffusion and lack of Galilean\ninvariance. The momentum density is denoted by p(r,t)\nand is a 2D vector p= (px,py) in the ˆx-ˆy-plane with\nr= (x,y) = (rx,ry). The total angular momentum den-2\nsity in the ˆz-direction is the sum of orbital angular mo-\nmentum density ǫαβrαpβand spin density s(r,t) (in the\nˆz-direction). Here, ǫαβis the 2D Levi-Civita tensor and\nsummation over repeated indices α,β,γ,δ ∈ {x,y}is im-\nplied. We denote with vthe conjugate variable to the\nmomentum density, i.e., the velocity, whereas the spin\nchemical potential, commonly referred to as spin accu-\nmulation,µsis the conjugate variable to the spin density.\nConservation of linear momentum yields\n∂pα(r,t)\n∂t=−∂Παβ(r,t)\n∂rβ, (1)\nwith Π αβ(r,t) the stress tensor. Conservation of angular\nmomentum in the z-direction is expressed as\n∂[ǫαβrαpβ(r,t)+s(r,t)]\n∂t=−∂jJ\nα(r,t)\n∂rα,(2)\nwithjJ\nα(r,t) theα-th component of the angular momen-\ntum current and in the above equations the summation\nis overboth αandβ. The equation for the spin density is\nfound by subtracting the cross-product of rwith Eq. (1)\nfrom Eq. (2) and yields\n∂s(r,t)\n∂t=−∂js\nα(r,t)\n∂rα−2Πa(r,t),(3)\nwith Πa(r,t) =ǫαβΠβα(r,t)/2 the antisymmetricpart of\nthe stress tensor and js\nα(r,t) =jJ\nα(r,t)−ǫβγrβΠγα(r,t)\nthe spin current.\nAnonzerovelocityandspindensityincreasetheenergy\nof the system. By symmetry, a nonzero velocity leads to\na contribution ρkinv2/2 to the energy density. This ex-\npression defines the kinetic mass density ρkin, such that\np(r,t) =ρkinv(r,t)[28]. Forthecasethatisofinterestto\nus, i.e., 2D electrons with spin-orbit coupling, the kinetic\nmass density is not equal to the average mass density ρ\nbecause spin-orbit coupling breaks Galilean invariance.\nLikewise, a nonzero spin density contributes χsµ2\ns/2 to\nthe energy density, where χsis the static spin suscepti-\nbility, so that s(r,t) =/planckover2pi1χsµs(r,t). These terms in the\nenergy density lead to contributions to the entropy pro-\nduction from which relations between the fluxes (the spin\ncurrent and antisymmetric part of the pressure tensor)\nand the forces (spin accumulation and velocity) are de-\nrived phenomenologically. In terms of µs(r,t) andv(r,t)\nwe havefor the antisymmetric part of the pressuretensor\nthat [24]\nΠa(r,t) =−ηr[ω(r,t)−2µs(r,t)//planckover2pi1],(4)\nwithω(r,t) =ǫαβ∂vβ(r,t)/∂rαthe vorticity and ηrthe\nrotational viscosity . The aboveexpression showsthat an-\ngular momentum is transferred, by spin-orbit coupling,\nbetween orbital and spin degrees of freedom until the an-\ntisymmetric part of the pressure tensor is zero. For the\nspin current we have that js\nα(r,t) =−σs∂µs(r,t)/∂rα=−Ds∂s(r,t)/∂rαwhich defines the spin diffusion con-\nstantDsand spin conductivity σs, which obey the Ein-\nstein relation σs=/planckover2pi1Dsχs. Note that we are omitting an\nadvective contribution ∼vαsto the spin current as we\nrestrict ourselves to the linear-responseregime. Inserting\nthese results for the fluxes into Eq. (3) and using Eq. (1)\nleads to\n∂s(r,t)\n∂t=Ds∇2s(r,t)\n+2ηr/bracketleftbigg\nω(r,t)−2s(r,t)\n/planckover2pi12χs/bracketrightbigg\n−s(r,t)\nτsr;\nρkin∂vα(r,t)\n∂t=−eρEα\nm+νρkin∇2vα(r,t)\n+ηrǫαβ∂\n∂rβ/bracketleftbigg\nω(r,t)−2s(r,t)\n/planckover2pi12χs/bracketrightbigg\n−ρkinvα(r,t)\nτmr.(5)\nIn the above we have assumed the linear-response regime\nand introduced the kinematic viscosity νusing that the\nsymmetric part of the stress tensor is given by Π αβ=\nνρkin∂vα/∂rβ. Furthermore, we have added spin and\nmomentum relaxation terms, parameterized by the phe-\nnomenological time scales τsrandτmr, respectively. We\nhave also included an electric field E(the electron has\ncharge−e).\nEqs. (5) are the main phenomenological equations for\nspin density and velocity. The term proportional to ηrin\nthe first equation describes generation of spin accumula-\ntion in response to vorticity, e.g., spin-vorticity coupling.\nIn the steady state the hydrodynamic equations are\ncharacterized by three length scales. The first is a length\nscalethatresultsfromthespin-vorticitycouplingequalto\nℓsv=/radicalbig\nDs/planckover2pi12χs/(2ηr), which is the characteristic length\nover which the orbital and spin angular momentum equi-\nlibrate. Furthermore, we have the spin diffusion length\nℓsr=√Dsτsrthat determines the length scales for relax-\nation of spin due to impurities, and the momentum dif-\nfusion length ℓmr=√ντmr. The most interesting regime,\nwhichoccursinthe limitofstrongspin-orbitcouplingrel-\native to momentum and spin relaxation, is the one where\nℓsvis the shortest length scale. In this case the spin den-\nsity locally follows the vorticity, which is determined by\nthe electron flow.\nApplication. —We consider electron flow through a\npoint contact (PC) [7, 18] driven by a voltage V. Tak-\ningτmr,τsr→ ∞we have from Ref. [18] for the velocity\ndistribution at the PC that\nvy(x) =−πρeV\n4mνρkin/radicalbigg/parenleftBigw\n2/parenrightBig2\n−x2, (6)\nwheretheflowisinthe y-directionand wisthePCwidth.\nFrom Eq. (5), in the limit ℓsv≪wthe steady-state spin\ndensity generated at the PC by spin-vorticity coupling in\nthe hydrodynamic regime is then\ns(x)\n/planckover2pi12χsjc=−m\nπewρ4x/radicalbig\n(w/2)2−x2, (7)3\nwherejc=−eρ/integraltext\ndxvy(x)/(mw) is the average current\ndensity.\nLet us compare Eq. (7) with the spin density gener-\nated by the spin Hall effect in the diffusive limit. In\nthe latter case, the spin accumulation is determined\nby∂2µs/∂x2=µs/ℓ2\nsr, which follows from Eqs. (5) in\nthe limitℓsr≪ℓsv, together with the expression js\ny=\n−σs∂µx/∂x+θSH/planckover2pi1jc\ny/(2e) for the spin current. Here\njc\ny=σeEyis the diffusive charge current through the\nPC, withσe=e2ρ2τmr/(m2ρkin) the electrical conduc-\ntivity andθSHthe spin Hall angle. Using the boundary\nconditionsjs(−w/2) =js(w/2) = 0, we find for the spin\ndensity in the diffusive limit that\nsdiff(x)\n/planckover2pi12χsjcy=θSHℓsr\n2eσssech/parenleftbiggw\n2ℓsr/parenrightbigg\nsinh/parenleftbiggx\nℓsr/parenrightbigg\n.(8)\nA crucial difference is thus that for diffusive spin trans-\nport and when w≫ℓsr, the spin density is only nonzero\nwithin a distance ∼ℓsraway from the edges of the PC,\nwhile when w≫ℓsvand in the hydrodynamic limit, the\nspin density [see Eq. (7)] is nonzero everywhere (except\natx= 0 where it vanishes by symmetry).\nIn both hydrodynamic and diffusive limits, the max-\nimum spin density occurs at the edges. In the hy-\ndrodynamic limit the spin density formally diverges as\n|x| →w/2, since the vorticity that results from the ve-\nlocity in Eq. (6) diverges in the same limit. This diver-\ngence is, however, unphysical, as there will be a micro-\nscopic length scale ℓedgeover which the velocity goes to\nzero near the edge of the sample, resulting in a maxi-\nmum spin density of |s(±w/2)|/(/planckover2pi12χsjc)∼m/(eρℓedge)\nnear the edges of the sample. We expect the lat-\nter to be much larger than the maximum spin density\n|sdiff(±w/2)/(/planckover2pi12χsjc)| ∼m2θSHℓsr/(e/planckover2pi1ρτmr) generated\nby the spin Hall effect in the diffusive regime (where we\nestimatedσs∼/planckover2pi1ρτmr/m2), because /planckover2pi1τmr/(mθSHℓsr)∼\nℓmr/(θSHkFℓsr) is expected to be much larger than the\nmicroscopic length scale ℓedge. Here,kFis the Fermi\nwave number.\nMicroscopic theory. —We proceed by calculating the\nrotational viscosity microscopically. This is most easily\nachieved [29] by noting that even when spin relaxation\ndue to impurities is absent ( τsr→ ∞), the spin-vorticity\ncoupling opens a channel for spin relaxation, with rate\n4ηr//planckover2pi12χs, which microscopically stems from the com-\nbined effect of spin-orbit coupling and electron-electron\ninteractions. Hence, ηrcan be extracted from the re-\ntarded spin-spin response function (for spin in the ˆz-\ndirection) at zero wave vector, denoted by χ(+)\ns(ω), when\nthis response function is computed for a clean system\nwith spin-orbit coupling and interactions. From Eqs. (5)\nwe find that for v=0this response function has the\nform\nχ(+)\ns(ω) =χs\n1−iω/planckover2pi12χs/(4ηr). (9)Hence, we have that\n1\nηr=−/parenleftbigg2\n/planckover2pi1χs/parenrightbigg2\nlim\nω→0Im[χ(+)\ns(ω)]\nω.(10)\nAs a representative example, we compute the rota-\ntionalviscosityusingstandardlinear-responsetechniques\nfor a 2D electron gas with Rashba spin-orbit coupling,\nwhich has the following Hamiltonian [30]:\nˆH=/integraldisplay\ndr/summationdisplay\nσ∈{↑,↓}ˆψ†\nσ(r)/bracketleftbigg\n−/planckover2pi12∇2\n2m+λ/planckover2pi1ˆz·/parenleftbigg∇\ni×τ/parenrightbigg/bracketrightbigg\nˆψσ(r),\n(11)\nwhereˆψσ(r) [ˆψ†\nσ(r)] is an electron annihilation [creation]\noperator and τis a vector of Pauli matrices. The unit\nvector in the ˆz-direction is denoted by ˆz. The con-\nstantλparametrizes the strength of spin-orbit interac-\ntions. The spin density operator in imaginary time τis\nˆs(r,τ) =/planckover2pi1[ˆψ†\n↑(r,τ)ˆψ↑(r,τ)−ˆψ†\n↓(r,τ)ˆψ↓(r,τ)]/2, where\nthe dependence on τof the electron creation and annihi-\nlationoperatorsindicates theircorrespondingHeisenberg\nevolution in imaginary time. We have for the imaginary-\ntime spin-spin response function\nχs(iωn) =1\n/planckover2pi1/integraldisplay\ndr/integraldisplay/planckover2pi1β\n0dτ∝an}b∇acketle{tˆs(r,τ)ˆs(r,0)∝an}b∇acket∇i}ht0eiωnτ,(12)\nwhereiωn= 2πn/(/planckover2pi1β) is a bosonic Matsubara frequency\nwithβ= 1/(kBT) the inverse thermal energy, and the\nexpectation value ∝an}b∇acketle{t···∝an}b∇acket∇i}ht0is taken at equilibrium. Neglect-\ning vertex corrections due to interactions, this is worked\nout to yield\nχs(iωn) =−1\n4/planckover2pi1V/summationdisplay\nk/summationdisplay\nδ/ne}ationslash=δ′/integraldisplay\nd/planckover2pi1ωd/planckover2pi1ω′Aδ(k,ω)Aδ′(k,ω′)\n×/bracketleftbiggN(/planckover2pi1ω)−N(/planckover2pi1ω′)\nω−ω′+iωn/bracketrightbigg\n, (13)\nwithN(/planckover2pi1ω) =/bracketleftbig\neβ(/planckover2pi1ω−µ)+1/bracketrightbig−1the Fermi-Dirac distri-\nbution function at chemical potential µ. The spectral\nfunctionsAδ(k,ω) are labeled by the Rashba spin-orbit-\nsplit band index δ=±. We incorporateelectron-electron\ninteractions into the spectral function by taking them\nequal to Lorentzians broadened by the electron collision\ntimeτee[this corresponds to dressing bare propagator\nlines in the spin bubble in Eq. (12) by self-energy inser-\ntions], i.e.,\nAδ(k,ω) =/planckover2pi1\n2πτee1\n[/planckover2pi1ω−/planckover2pi1ωδ(k)]2+/parenleftBig\n/planckover2pi1\n2τee/parenrightBig2,(14)\nwhere/planckover2pi1ωδ(k) =/planckover2pi12k2/2m+δ/planckover2pi1λkis the Rashba band dis-\npersion. Inserting Eq. (14) into Eq. (13) and performing\na Wick rotation iωn→ω+i0+yields\nηr=4π2/planckover2pi14χ2\ns\nmτee/bracketleftBigg\n2π+8/parenleftbigµτee\n/planckover2pi1/parenrightbig\n1+4/parenleftbigµτee\n/planckover2pi1/parenrightbig2+4tan−1/parenleftbigg2µτee\n/planckover2pi1/parenrightbigg/bracketrightBigg\n,\n(15)4\nwhere we took λ→0. In the limit µτee//planckover2pi1≫1, we have\nηr=π/planckover2pi14χ2\ns/(mτee).\nSince we have neglected vertex corrections, the result\nin Eq. (15) does not vanish in the λ→0 limit and is\nstrictly speaking only valid when spin-orbit coupling is\nso strong that the spin-vorticity coupling is limited by\nelectron-electron interactions, i.e., when λkFτee≫1. In\nthe opposite limit, where the bottleneck for spin relax-\nation is the spin-orbit coupling, we perform a Fermi’s\nGolden Rule calculation to determine the decay rate of\na spin polarization to second order in the strength of the\nspin-orbit interactions. This gives at low temperatures\nthat\nηr=−π/planckover2pi1\n8/integraldisplaydk\n(2π)2A2(k,µ)(λ/planckover2pi1k)2,(16)\nwhereA(k,µ) is the spectral function obtained from\nEq. (14) by replacing /planckover2pi1ωδ(k)→/planckover2pi12k2/2m. Carrying out\nthe remaining integral gives\nηr=mλ2\n2/planckover2pi1/bracketleftbigg\n1+π/parenleftBigµτee\n/planckover2pi1/parenrightBig\n+2/parenleftBigµτee\n/planckover2pi1/parenrightBig\ntan−1/parenleftbigg2µτee\n/planckover2pi1/parenrightbigg/bracketrightbigg\n,\n(17)\nwhich indeed vanishes as λ→0. Whenµτee//planckover2pi1≫1, we\nhave that /planckover2pi1ηr∼(λkF)(λkFτee), showing the dependence\non the small parameter λkFτee≪1 explicitly. Inter-\nestingly, since the kinematic viscosity ν∝τee, we have\nthat the rotational viscosity ηr∝1/νin the limit of\nstrongspin-orbit coupling and ηr∝νin the limit ofweak\nspin-orbit coupling, with a maximum rotational viscosity\nwhenλkFτee∼1.\nEstimates. —Next, we estimate the spin-vorticity cou-\npling for graphene with proximity-induced spin-orbit\ncoupling. We take λ/planckover2pi1kFto be on the order of 1 meV\n[32]. Furthermore, we take τee∼100 fs [4]. We thus\nhave thatλ/planckover2pi1kFis about one order of magnitude smaller\nthan/planckover2pi1/τeeand use the weak spin-orbit coupling expres-\nsion in Eq. (16). Evaluating Eq. (16) for a linear disper-\nsion/planckover2pi1vFk, wherevF∼106m/s is the graphene Fermi\nvelocity, we find that\nηr∼(λ/planckover2pi1kF)2\n/planckover2pi1v2\nF/parenleftBigµτee\n/planckover2pi1/parenrightBig\n, (18)\nusingµτee≫/planckover2pi1. We estimate the corresponding inverse\ntime scale as\nηr\n/planckover2pi12χs∼(λ/planckover2pi1kF)2\n/planckover2pi13χsv2\nF/parenleftBigµτee\n/planckover2pi1/parenrightBig\n∼100 GHz,(19)\nwhere we took µτee//planckover2pi1∼10, and estimated the spin sus-\nceptibility as χs∼D(µ), with the density of states at the\nFermi level D(µ)∼√ne/(/planckover2pi1vF), and the electron number\ndensityne∼1012cm−2[4].\nTo estimate the corresponding length scale ℓsv, we as-\nsume that spin diffusion is in the hydrodynamic regime\ndetermined by electron-electron interactions that lead tospin drag [33]. We then have for the spin diffusion con-\nstant thatDs∼/planckover2pi1ρτee/(m2χs). The spin-vorticity length\nscale is then ℓsv∼vF/planckover2pi1/radicalbig\nτeeχs/ηr∼1µm. This is\nthe same order of magnitude as the momentum relax-\nation length scale ℓmr[4], so that the rotational viscosity\nappears to be high enough to lead to observable spin-\nvorticity coupling. Moreover, the limit where ℓsv< ℓmr\nseems to be within experimental reach. Note that in the\nregime of weak spin-orbit coupling we have for the spin\nrelaxation the Dyakonov-Perel result that 1 /τsr∝τmr\n[36], which yields that in the hydrodynamic regime we\nhaveℓsr∼ℓsv/radicalbig\nτee/τmr≫ℓsv.\nA simple interpretation of the spin-vorticity coupling\nis that the electron spins are polarized by an effective\nmagnetic field /planckover2pi1ω(r,t)/µB, withµBthe Bohr magneton,\nin the frame that rotates with the electron flow vorticity.\nWe estimate the vorticity ω∼v/ℓmrusingℓmr∼0.1-\n1µm, and a drift velocity of v∼100 m/s [4], which\nyields a substantial effective magnetic field of 1-10 mT.\nDiscussion and conclusions. —We have developed the\ntheory for spin-vorticity coupling in viscous electron flu-\nids, both phenomenologically and microscopically, and\nwe have estimated that the proximity-induced spin-orbit\ncoupling in graphene is large enough for observable ef-\nfects. As an example, we predict alargespin polarization\ninduced by spin-hydrodynamic generation in a PC. This\nlarge spin density may e.g. be observed optically [37] or\nvia nitrogen-vacancy centre magnetometry [34, 35]. The\nimaged spin density would provide a fingerprint of the\nvorticity of the electron flow.\nAn interesting direction for future research is general-\nization of the phenomenological and microscopic deriva-\ntion to other spin-orbit couplings, including, in particu-\nlar, also the effects of violation of translational and ro-\ntational invariance beyond the phenomenological relax-\nation terms that we included here. One example would\nbe that of Weyl semi-metals that naturally have size-\nable spin-orbit coupling and have also been reported to\nbe able to reach the hydrodynamic regime [38]. Other\ncandidates are bismuthene [39] and stanene [40] that\ncombine strong spin-orbit coupling with high mobility.\nFurther interesting directions of research include incor-\nporating effects of a magnetic field and computation of\nthe rotational viscosity in the regime where spin-orbit\ninteractions and electron-electron interactions are com-\nparable in magnitude. In this regime, the crossover from\nweak-to-strong spin-orbit coupling takes place, whereas\ninclusion of momentum-relaxing scattering would lead to\na crossover from the spin-vorticity coupling to the spin\nHall effect.\nAcknowledgements. —We thank Denis Bandurin, Eu-\ngene Chudnovsky, and Harold Zandvliet for useful com-\nments. R.D. is member of the D-ITP consortium, a\nprogram of the Netherlands Organisation for Scientific\nResearch (NWO) that is funded by the Dutch Min-\nistry of Education, Culture and Science (OCW). This5\nwork is in part funded by the Stichting voor Funda-\nmenteel Onderzoek der Materie (FOM) and the Euro-\npean Research Council (ERC). M.P. is supported by\nthe European Union’s Horizon 2020 research and inno-\nvation programme under grant agreement No. 696656\n“GrapheneCore1”.\n[1] S. A. Wolf, D. D. Awschalom, R. A. Buhrman, J.\nM. Daughton, S. von Moln´ ar, M. L. Roukes, A. Y.\nChtchelkanova, and D. M. Treger, Science 294, 1488\n(2001).\n[2] T. Valet and A. Fert, Phys. Rev. B 48, 7099 (1993).\n[3] M. I. Dyakonov and V. I. Perel, Sov. Phys. JETP Lett.\n13, 467 (1971).\n[4] D. A. Bandurin, I. Torre, R. K. Kumar, M. Ben Shalom,\nA. Tomadin, A. Principi, G. H. Auton, E. Khestanova,\nK. S. Novoselov, I. V. Grigorieva, L. A. Ponomarenko,\nA. K. Geim, and M. Polini, Science 351, 1055 (2016).\n[5] J. Crossno, J. K. Shi, K. Wang, X. Liu, A. Harzheim, A.\nLucas, S. Sachdev, P. Kim, T. Taniguchi, K. Watanabe,\nT. A. Ohki, and K. C. Fong, Science 351, 1058 (2016).\n[6] P. J. W. Moll, P. Kushwaha, N. Nandi, B. Schmidt, and\nA. P. Mackenzie, Science 351, 1061 (2016).\n[7] R. K. Kumar, D. A. Bandurin, F. M. D. Pellegrino,\nY. Cao, A. Principi, H. Guo, G. H. Auton, M. Ben\nShalom, L. A. Ponomarenko, G. Falkovich, K. Watan-\nabe, T. Taniguchi, I. V. Grigorieva, L. S. Levitov, M.\nPolini, and A. K. Geim, Nature Phys. 13, 1182 (2017).\n[8] R. Gurzhi, Sov. Phys. JETP 44, 771 (1963).\n[9] J. E. Black, Phys. Rev. B 21, 3279 (1980).\n[10] Z. Z. Yu, M. Haerle, J. W. Zwart, J. Bass, W. P. Pratt,\nand P. A. Schroeder, Phys. Rev. Lett. 52, 368 (1984).\n[11] M. J. M. de Jong and L. W. Molenkamp, Phys. Rev. B\n51, 13389 (1995).\n[12] M. Dyakonov and M. Shur, Phys. Rev. Lett. 71, 2465\n(1993).\n[13] E. Chow, H. P. Wei, S. M. Girvin, and M. Shayegan,\nPhys. Rev. Lett. 77, 1143 (1996).\n[14] A. V. Andreev, S. A. Kivelson, and B. Spivak, Phys. Rev.\nLett.106, 256804 (2011).\n[15] I. Torre, A. Tomadin, A. K. Geim, and M. Polini, Phys.\nRev. B92, 165433 (2015).\n[16] A. Lucas, New J. Phys. 17, 113007 (2015).\n[17] L. Levitov and G. Falkovich, Nature Phys. 12, 672\n(2016).\n[18] H. Guo, E. Ilseven, G. Falkovich, and L. S. Levitov, Proc .\nNatl. Acad. Sci. (USA) 114, 3068 (2017).\n[19] T. Scaffidi, N. Nandi, B. Schmidt, A. P. Mackenzie, and\nJ. E. Moore, Phys. Rev. Lett. 118, 226601 (2017).\n[20] F. M. D. Pellegrino, I. Torre, and M. Polini, Phys. Rev.\nB96, 195401 (2017).\n[21] L. V. Delacr´ etaz and A. Gromov, Phys. Rev. Lett. 119,\n226602 (2017).\n[22] J. M. Link, B. N. Narozhny, and J. Schmalian,arXiv:1708.02759.\n[23] R. Takahashi, M. Matsuo, M. Ono, K. Harii, H. Chudo,\nS. Okayasu, J. Ieda, S. Takahashi, S. Maekawa and E.\nSaitoh, Nature Phys. 12, 52 (2016).\n[24] S. R. de Groot and P. Mazur, Non-equilibrium thermo-\ndynamics (North-Holland, Amsterdam, 1962).\n[25] B. U. Felderhof, Phys. Fluids 23, 042001 (2011).\n[26] J. S. Hansen, P. J. Daivis, and B. D. Todd, Microfluids\nNanofluidics 6, 785 (2009).\n[27] D.L. Cheung, S.J. Clark, M.R. Wilson, Chemical Physics\nLetters356, 140 (2002).\n[28] J. de Boer, J. Hartong, N. A. Obers, W. Sybesma, and\nS. Vandoren, arXiv:1710.04708.\n[29] Alternatively, the rotational viscosity may be evalua ted\nusing the linear-response relation in Eq. (4). From this\nrelation we find that ηrcan be extracted from the corre-\nlation function K(r−r′;τ−τ′) =/angbracketleftˆΠa(r,τ)ˆs(r′,τ′)/angbracketright0\nat zero wave vector so that ηr=K(k→0,iωn)/2,\nwhere the factor 1 /2 compensates for the factor 2 in\nEq. (4). Using Eq. (3) at zero wave vector we have that\nK(k→0;iωn) =−i/planckover2pi12ωnχs(iωn)/2, so that insertion of\nEq. (9) indeed yields that K(k→0,iωn) = 2ηrin the\n“fast”ωn≫ηr/(/planckover2pi12χs) limit [31]. For completeness, we\nalso compute the kinetic mass density for the Rashba\nmodel in the noninteracting limit. We then have that the\nmomentum density is given by\np=/summationdisplay\nδ/integraldisplaydk\n(2π)2/planckover2pi1kAδ(k,ω)N(/planckover2pi1ω−µ−/planckover2pi1k·v),\nwhich yields p=ρkinvforv→0, with the kinetic mass\ndensityρkin=ρ[1 +λ2m/2µ+O(λ4)]. The correction,\ni.e. the second term between the brackets, is typically\nsmall since one almost always has that λ2m/2µ≪1.\n[30] A. Manchon, H. C. Koo, J. Nitta, S. M. Frolov and R.\nA. Duine, Nature Mater. 14, 871 (2015).\n[31] J.M. Luttinger, Phys. Rev. 135, A1505 (1964).\n[32] Z. Wang et al., Nat. Commun. 6, 8339 (2015).\n[33] I. D’??Amico and G. Vignale, Phys. Rev. B 62, 4853\n(2000).\n[34] J. R. Maze, P. L. Stanwix, J. S. Hodges, S. Hong, J.\nM. Taylor, P. Cappellaro, L. Jiang, M. V. G. Dutt, E.\nTogan, A. S. Zibrov, A. Yacoby, R. L. Walsworth, and\nM. D. Lukin, Nature 455, 644 (2008).\n[35] L. Rondin, J.-P. Tetienne, T. Hingant, J.-F. Roch, P.\nMaletinsky, and V. Jacques, Rep. Prog. Phys. 77, 56503\n(2014).\n[36] M. I. Dyakonov and V. I. Perel, Sov. Phys. Solid State\n13, 3023 (1972)\n[37] V. Sih, R. C. Myers, Y. K. Kato, W. H. Lau, A. C. Gos-\nsard and D. D. Awschalom, Nature Phys. 1, 31 (2005).\n[38] J. Gooth, F. Menges, C. Shekhar, V. S¨ uss, N. Kumar, Y.\nSun, U. Drechsler, R. Zierold, C. Felser, B. Gotsmann,\narXiv:1706.05925.\n[39] F. Reis et al., Science 10.1126/science.aai8142 (2017).\n[40] F. Zhu et al., Nature Material 14. 1020 (2015)."    },    {        "title": "1402.5817v2.Low_Energy_Effective_Hamiltonian_for_Giant_Gap_Quantum_Spin_Hall_Insulators_in_Honeycomb_X_Hydride_Halide__X_N_Bi__Monolayers.pdf",        "content": "Low-Energy Effective Hamiltonian for Giant-Gap Quantum Spin Hall Insulators in\nHoneycomb X-Hydride/Halide ( X= N-Bi) Monolayers\nCheng-Cheng Liu,1Shan Guan,1Zhigang Song,2Shengyuan A. Yang,3Jinbo Yang,2,4and Yugui Yao1,\u0003\n1School of Physics, Beijing Institute of Technology, Beijing 100081, China\n2State Key Laboratory for Mesoscopic Physics, and School of Physics, Peking University, Beijing 100871, China\n3Engineering Product Development, Singapore University of Technology and Design, Singapore 138682, Singapore\n4Collaborative Innovation Center of Quantum Matter, Beijing, China\nUsingthetight-bindingmethodincombinationwithfirst-principlescalculations,wesystematically\nderive a low-energy effective Hilbert subspace and Hamiltonian with spin-orbit coupling for two-\ndimensional hydrogenated and halogenated group-V monolayers. These materials are proposed to\nbe giant-gap quantum spin Hall insulators with record huge bulk band gaps opened by the spin-orbit\ncoupling at the Dirac points, e.g., from 0.74 to 1.08 eV in Bi X(X= H, F, Cl, and Br) monolayers.\nWe find that the low-energy Hilbert subspace mainly consists of pxandpyorbitals from the group-V\nelements, and the giant first-order effective intrinsic spin-orbit coupling is from the on-site spin-orbit\ninteraction. These features are quite distinct from those of group-IV monolayers such as graphene\nand silicene. There, the relevant orbital is pzand the effective intrinsic spin-orbit coupling is from\nthe next-nearest-neighbor spin-orbit interaction processes. These systems represent the first real 2D\nhoneycomb lattice materials in which the low-energy physics is associated with pxandpyorbitals.\nA spinful lattice Hamiltonian with an on-site spin-orbit coupling term is also derived, which could\nfacilitate further investigations of these intriguing topological materials.\nPACS numbers: 73.43.-f, 73.22.-f, 71.70.Ej, 85.75.-d\nI. INTRODUCTION\nRecent years have witnessed great interest in two-\ndimensional (2D) layered materials with honeycomb lat-\ntice structures. Especially, the 2D group-IV honey-\ncomb lattice materials, such as successively fabricated\ngraphene,1,2and silicene,3,4have attracted considerable\nattention both theoretically and experimentally due to\ntheir low-energy Dirac fermion behavior and promising\napplications in electronics. Recently, we have discovered\nstable 2D hydrogenated and halogenated group-V hon-\neycomb lattices via first-principles (FP) calculations.5\nTheir structures are similar to that of a hydrogenated\nsilicene (silicane), as shown in Fig. 1(a). In the absence\nof spin-orbit coupling (SOC), the band structures show\nlinear energy crossing at the Fermi level around Kand\nK0points of the hexagonal Brillouin zone. It is quite\nunusual that the low-energy bands of these materials are\nofpxandpyorbital character. Previous studies in the\ncontext of cold atoms systems have shown that pxand\npyorbital character could lead to various charge and or-\nbital ordered states as well as topological effects.6,7Our\nproposed materials, being the first real condensed mat-\nter systems in which the low-energy physics is associated\nwithpxandpyorbitals, are therefore expected to exhibit\nrich and interesting physical phenomena.\nThe quantum spin Hall (QSH) insulator state has gen-\nerated great interest in condensed matter physics and\nmaterialscienceduetoitsscientificimportanceasanovel\nquantum state and its potential technological applica-\ntions ranging from spintronics to topological quantum\ncomputation.8–10This novel electronic state is gaped in\nthe bulk and conducts charge and spin in gapless edge\nstates without dissipation protected by time-reversal\n(b)\nΓΚ\nΜ\n(a)FIG. 1. (Color online). (a) The lattice geometry for 2D\nX-hydride/halide ( X= N-Bi) monolayer from the side view\n(top) and top view (bottom). Note that two sets of sublat-\ntice in the honeycomb group V element Xare not coplanar\n(a buckled structure). The monolayer is alternatively hydro-\ngenated or halogenated from both sides. (b) The first Bril-\nlouin zone of 2D X-hydride/halide monolayer and the points\nof high symmetry.\nsymmetry. The concept of QSH effect was first proposed\nby Kane and Mele in graphene in which SOC opens a\nnontrivial band gap at the Dirac points.11,12Subsequent\nworks, however, showed that the SOC for graphene is\ntiny, hence the effect is difficult to be detected experi-\nmentally.13–15So far, QSH effect has only been demon-\nstrated in HgTe/CdTe quantum wells,16,17and experi-\nmental evidence for helical edge modes has been pre-\nsented for inverted InAs-GaSb quantum wells.18–20Nev-\nertheless, these existing systems more or less have serious\nlimitationsliketoxicity,difficultyinprocessing,andsmall\nbulk gap opened by SOC. Therefore, an easy and envi-arXiv:1402.5817v2  [cond-mat.mtrl-sci]  22 Sep 2014ronmental friendly realization of a QSH insulator is much\ndesired. Extensive effort has been devoted to the search\nfor new QSH insulators with large SOC gap.21–28For\ninstance, new layered honeycomb lattice type materials\nsuch as silicene, germanene24or stanene25, and chemi-\ncally modified stanene27have been proposed. Ultrathin\nBi(111) films have drawn attention as a candidate QSH\ninsulator, whose 2D topological properties have been re-\nported.29An approach to design a large-gap QSH state\non a semiconductor surface by a substrate orbital filter-\ning process was also proposed.30However, desirable QSH\ninsulators preferably with huge bulk gaps are still rare.\nA sizable bulk band gap in QSH insulators is essential for\nrealizing many exotic phenomena and for fabricating new\nquantum devices that can operate at room temperature.\nUsing FP method, we have recently demonstrated that\nthe QSH effect can be realized in the 2D hydrogenated\nand halogenated group-V honeycomb monolayers family,\nwith a huge gap opened at the Dirac points due to SOC.5\nAlthough the low-energy spectrum of these materials is\nsimilar to the 2D group-IV honeycomb monolayers such\nas graphene and silicene, the low-energy Hilbert space\nchanges from the pzorbital to orbitals mainly consisting\nofpxandpyfrom the group-V atoms (N-Bi). More-\nover, the nature of the effective SOC differs between\nthe two systems. Motivated by the fundamental inter-\nest associated with the QSH effect and huge SOC gaps\nin these novel 2D materials, we develop a low-energy ef-\nfective model Hamiltonian that captures their essential\nphysics. In addition, we propose a minimal four-band\nlatticeHamiltonianwiththeon-siteSOCtermusingonly\nthepxandpyorbitals.\nFrom the symmetry analysis, the next-nearest-\nneighbor (NNN) intrinsic Rashba SOC should exist in\nthese systems due to the low-buckled structure, similar\nto the case of silicene.25However, as we shall see, the\ndominant effect is from the much larger first-order SOC\nof on-site origin. Therefore, in the following discussion,\nwe shall focus on the first-order on-site SOC and neglect\nthe higher-order effects. This point will be further dis-\ncussed later in this paper.\nThe paper is organized as follows. In Sec. II, we de-\nrivestepbystepthelow-energyeffectiveHilbertsubspace\nand Hamiltonian for honeycomb X-hydride ( X= N-Bi)\nmonolayers, and also investigate in detail the effective\nSOC.SectionIIIpresentsthederivationofthelow-energy\neffectivemodelfor X-halide( X=N-Bi, halide=F-I)hon-\neycomb monolayers. In Sec. IV, a simple spinful lattice\nHamiltonian for the honeycomb X-hydride/halide mono-\nlayers family is constructed. We conclude in Sec. V with\na brief discussion of the effective SOC and present a sum-\nmary of our results.II. LOW-ENERGY EFFECTIVE HAMILTONIAN\nFOR HONEYCOMB XH(X= N-BI)\nMONOLAYERS\nA. Low-energy Hilbert subspace and effective\nHamiltonian without SOC\nAs is shown in Fig. 1(a), there are two distinct sites\nA and B in the unit cell of X-hydride ( X= N-Bi)\nhoneycomb lattice with full hydrogenation from both\nsides of the 2D Xhoneycomb sheet. The primitive\nlattice vectors are chosen as ~ a1=a(1=2;p\n3=2)and\n~ a2=a(\u00001=2;p\n3=2), whereais the lattice constant.\nWe consider the outer shell orbitals of textitX ( X=\nN-Bi), namely s,px,py,pz, and also the sorbital of\nH in the modeling. Therefore, in the representation\nfjpA\nyi;jpA\nxi;jpA\nzi;jsA\nHi;jsAi;jpB\nyi;jpB\nxi;jpB\nzi;jsB\nHi;jsBig\n(for simplicity, the Dirac ket symbol is omitted in the\nfollowing), the Hamiltonian (without SOC) at Kpoint\nwith the nearest-neighbor hopping considered in the\nSlater-Koster formalism31reads\nH0=\u0012HAA\n0HAB\n0\nHABy\n0HBB\n0\u0013\n; (1)\nwith\nHAA\n0=2\n666640 0 0 0 0\n0 0 0 0 0\n0 0 0\u0000VH\nsp\u001b 0\n0 0\u0000VH\nsp\u001b \u0001HVH\nss\u001b\n0 0 0 VH\nss\u001b \u00013\n77775;(2)\nHAB\n0=2\n66664\u0000V0\n1\u0000iV0\n10 0V0\n2\n\u0000iV0\n1V0\n10 0\u0000iV0\n2\n0 0 0 0 0\n0 0 0 0 0\n\u0000V0\n2iV0\n20 0 03\n77775;(3)\nHBB\n0=2\n666640 0 0 0 0\n0 0 0 0 0\n0 0 0VH\nsp\u001b 0\n0 0VH\nsp\u001b \u0001HVH\nss\u001b\n0 0 0VH\nss\u001b \u00013\n77775; (4)\nwhereVH\nsp\u001b(VH\nss\u001b) is the hopping between the pz(s)\norbital from Xatom and the sorbital from H, and\nV0\n1\u0011(3=4) (Vpp\u0019\u0000Vpp\u001b)andV0\n2\u0011(3=2)Vsp\u001bwithVpp\u0019,\nVsp\u001b, andVpp\u001bbeing the standard Slater-Koster hopping\nparameters. \u0001and\u0001Hare on-site energies for sorbitals\nof atom Xand of atom H, respectively. The on-site en-\nergies forporbitals are taken to be zero.\nTo diagonalize the Hamiltonian, we first perform the\n2(c)(a) (b)\n(d)FIG. 2. (Color online). (a)(b) The partial band structure projection for NH and NF without SOC, respectively. Symbol size\nis proportional to the population in the corresponding states. The Fermi level is indicated by the dotted line. (c)(d) Band\nstructures for BiH and BiF without (black dash lines) and with (red solid lines) SOC. The four band structures are obtained\nfrom the first-principles methods implemented in the VASP package32using projector augmented wave pseudo-potential, and\nthe exchange-correlation is treated by PAW-GGA. The Fermi level is indicated by the solid line.\nfollowing unitary transformation:\n'A\n1=\u00001p\n2\u0000\npA\nx+ipA\ny\u0001\n=jpA\n+i;\n'B\n2=1p\n2\u0000\npB\nx\u0000ipB\ny\u0001\n=jpB\n\u0000i;\n'3=1p\n2\u0014\n\u00001p\n2\u0000\npA\nx\u0000ipA\ny\u0001\n\u00001p\n2\u0000\npB\nx+ipB\ny\u0001\u0015\n;\n'4=1p\n2\u00141p\n2\u0000\npA\nx\u0000ipA\ny\u0001\n\u00001p\n2\u0000\npB\nx+ipB\ny\u0001\u0015\n:(5)\nIn the basisf'A\n1;sB;sB\nH;pB\nz;'B\n2;sA;sA\nH;pA\nz;'3;'4g,\nthe Hamiltonian can be written as a block-diagonal form\nwith three decoupled blocks H\u000b,H\f, andH\r:\nH0\u0000!H1=Uy\n1H0U1; (6)U1=2\n666666666666664\u0000ip\n20i\n2\u0000i\n20 0 0 0 0 0\n\u00001p\n20\u00001\n21\n20 0 0 0 0 0\n0 0 0 0 1 0 0 0 0 0\n0 0 0 0 0 1 0 0 0 0\n0 0 0 0 0 0 1 0 0 0\n0\u0000ip\n2\u0000i\n2\u0000i\n20 0 0 0 0 0\n01p\n2\u00001\n2\u00001\n20 0 0 0 0 0\n0 0 0 0 0 0 0 1 0 0\n0 0 0 0 0 0 0 0 1 0\n0 0 0 0 0 0 0 0 0 13\n777777777777775;(7)\nH1=H\u000b\bH\f\bH\r; (8)\nwith\nH\u000b=2\n6640iV20 0\n\u0000iV2\u0001VH\nss\u001b 0\n0VH\nss\u001b\u0001HVH\nsp\u001b\n0 0VH\nsp\u001b 03\n775; (9)\n3H\f=2\n6640\u0000iV2 0 0\niV2\u0001VH\nss\u001b 0\n0VH\nss\u001b \u0001H\u0000VH\nsp\u001b\n0 0\u0000VH\nsp\u001b 03\n775;(10)\nH\r=diagfV1;\u0000V1g; (11)\nwhereV1= 2V0\n1andV2=p\n2V0\n2.\nThe eigenvectors for the first diagonal block H\u000bcan\nbe easily obtained as\nj\"ii=1\nNi2\n66641\n\u0000i\"i\nV2\n\u0000i\"2\ni\u0000\u0001\"i\u0000V2\n2\nV2VHss\u001b\n\u0000iVH\nsp\u001b\n\"i\"2\ni\u0000\u0001\"i\u0000V2\n2\nV2VHss\u001b3\n7775;(12)\nwhere\"iandNi(i= 1;2;3;4)are the correspond-\ning eigenvalues and normalization factors, respectively.\nTherefore, upon performing the unitary transforma-\ntionf\u001e1;\u001e2;\u001e3;\u001e4g=f'A\n1;sB;sB\nH;pB\nzgU\u000bwithU\u000b=\nfj\"iigi=1;2;3;4\u0011fu\u000b\njig, the above upper-left 4\u00024block\nH\u000bis diagonalized.\nFor the second diagonal block H\f, its eigenvalues are\ndenoted as\"4+i(i= 1;2;3;4), and it can be easily shown\nthat\"4+i=\"i, where\"iare eigenvalues of H\u000b. This is\nconsistent with FP results, i.e., there are four two-fold\ndegeneracy points at Kpoint as shown in Fig. 2(a). The\neigenvectors of H\fare given by\nj\"ii=1\nNi2\n66641\ni\"i\nV2\ni\"2\ni\u0000\u0001\"i\u0000V2\n2\nV2VH\nss\u001b\n\u0000iVH\nsp\u001b\n\"i\"2\ni\u0000\u0001\"i\u0000V2\n2\nV2VHss\u001b3\n7775;(13)\nwhere\"iandNi(i= 5;6;7;8)are the corresponding\neigenvalues and normalization factors. Similar to the\ncase ofH\u000b, upon performing the unitary transforma-\ntionf\u001e5;\u001e6;\u001e7;\u001e8g=f'B\n2;sA;sA\nH;pA\nzgU\fwithU\f=\nfj\"i+4igi=1;2;3;4\u0011fu\f\njig, the block H\fis diagonalized.\nThe third block H\ris already diagonal\nwith eigenvalues fV1;\u0000V1gand eigenvectors\nf'3;'4g \u0011 f\u001e9;\u001e10g. Therefore, in the new\nbasisf\u001e1;\u001e2;\u001e3;\u001e4;\u001e5;\u001e6;\u001e7;\u001e8;\u001e9;\u001e10g \u0011\b\n'A\n1;sB;sB\nH;pB\nz;'B\n2;sA;sA\nH;pA\nz;'3;'4\t\nU2, where\nU2\u0011u\u000b\bu\f\bI2\u00022, the total Hamiltonian (1) takes\na fully diagonlized form. The whole diagonalization\nprocess can be summarized as follows:\nf\u001e1;\u001e2;\u001e3;\u001e4;\u001e5;\u001e6;\u001e7;\u001e8;\u001e9;\u001e10g\n=\b\npA\ny;pA\nx,pA\nz;sA\nH;sA;pB\ny;pB\nx;pB\nz;sB\nH;sB\t\nU;(14)\nwhere\nU=U1U2; (15)\nH0\u0000!H0\n0=UyH0U; (16)H0\n0=diagf\"1;\"2;\"3;\"4;\"5;\"6;\"7;\"8;V1;\u0000V1g:(17)\nFrom the band components projection as shown in\nFig. 2(a), in the vicinity of the Dirac points (around\nFermilevel), themaincomponentsofthebandcomefrom\nthepxandpyorbitals of group-V element textitX mixed\nwith a small amount of sorbital of textitX. Compared\nwith the expressions of the eigenstates obtained above,\nwe find that the orbital features agree with that of j\"1i\nandj\"5iifwetaketheireigenenergiesastheFermienergy.\nTherefore the corresponding states \u001e1and\u001e5constitute\nthelow-energyHilbertsubspace. Inthefollowing, wewill\ngive the explicit forms of the low-energy states \u001e1and\u001e5\nas well as their eigenvalues.\nNote that, in the above 4\u00024H\u000b, the scale of the 2\u00022\nnon-diagonal block H\u000b12is smaller than the difference of\nthe typical eigenvalues between the upper 2\u00022diagonal\nblockH\u000b11and the lower 2\u00022diagonal block H\u000b22.\nHence, through the downfolding procedure33, we could\nobtain the low-energy effective Hamiltonian as\nHeff\n\u000b11=H\u000b11+H\u000b12(\"\u0000H\u000b22)\u00001H\u000b21:(18)\nUp to the second order, one obtains\n\"1=1\n2\u0012\n\u00010+q\n\u000102+ 4V2\n2\u0013\n; (19)\nwith\n\u00010= \u0001 +\"VH\nss\u001b2\n\"2\u0000\u0001H\"\u0000VHsp\u001b2;\n\"=1\n2\u0012\n\u0001 +q\n\u00012+ 4V2\n2\u0013\n:(20)\nConsequently, we can obtain the explicit expressions of\nj\"1i\u0011\b\nu\u000b\nj1\t\nj=1;4and\u001e1. In a similar way, the explicit\nexpressions ofj\"5i\u0011fu\f\nj1gj=1;4and\u001e5can also be ob-\ntained. So far, we have obtained the eigenvalues \"1=\"5\n[Eqs. (19) and (20)] and the corresponding low-energy\nHilbert subspace consisting of \u001e1and\u001e5,\n\u001e1=u\u000b\n11'A\n1+u\u000b\n21sB+u\u000b\n31sB\nH+u\u000b\n41pB\nz;\n\u001e5=u\u000b\n11'B\n2\u0000u\u000b\n21sA\u0000u\u000b\n31sA\nH+u\u000b\n41pA\nz:(21)\nThe above coefficients fu\u000b\nj1gj=1;4are given in Eq. (12).\nFurther simplification could be made in order to cap-\nture the main physics. We can omit the second-order\ncorrection for the eigenvalues and the first-order correc-\ntion for the eigenvectors, i.e., the terms (u\u000b\n31sB\nH+u\u000b\n41pB\nz)\nfor\u001e1and(\u0000u\u000b\n31sA\nH+u\u000b\n41pA\nz)for\u001e5, and only keep the\nzeroth-order eigenvectors and eigenvalues,\n\u001e1=u\u000b\n11'A\n1+u\u000b\n21sB;\n\u001e5=u\u000b\n11'B\n2\u0000u\u000b\n21sA;\n\"1=\"=1\n2\u0012\n\u0001 +q\n\u00012+ 4V2\n2\u0013\n:(22)\n4This approximation is justified by our FP calculations,\nnamely in the vicinity of the Fermi level, px,py, and\nsorbitals overwhelmingly dominate over the sHandpz\norbitals in the band components.\nIn the Hamiltonian (17), one can take the Fermi en-\nergyEF=\"1=\"5as energy zero point. Hence, states\n\u001e1and\u001e5, which constitute the low-energy Hilbert sub-\nspace, take the following explicit forms:\n\u001e1=u\u000b\n11\u0014\n\u00001p\n2\u0000\npA\nx+ipA\ny\u0001\u0015\n+u\u000b\n21sB;\n\u001e5=u\u000b\n11\u00141p\n2\u0000\npB\nx\u0000ipB\ny\u0001\u0015\n\u0000u\u000b\n21sA;(23)\nwith\nu\u000b\n11=\u0010\n\u0000\u0001 +q\n\u00012+ 18V2sp\u001b\u0011\nr\n2\u00012+ 36V2sp\u001b\u00002\u0001q\n\u00012+ 18V2sp\u001b;\nu\u000b\n21=\u00003p\n2iVsp\u001br\n2\u00012+ 36V2sp\u001b\u00002\u0001q\n\u00012+ 18V2sp\u001b:\nSince we are interested in the low-energy physics near\ntheDiracpoint, weperformthesmall ~kexpansionaround\nKby~k!~k+Kand keep the terms that are first order\nin~k. We find that\nHK=\u0012\n0vFk\u0000\nvFk+ 0\u0013\n; (24)\nwithvFbeing the Fermi velocity\nvF=p\n3a\n2\u00141\n2ju\u000b\n11j2(Vpp\u001b\u0000Vpp\u0019) +ju\u000b\n21j2Vss\u001b\u0015\n;\n(25)\nand\nk\u0006=kx\u0006iky:\nEither following similar procedures, or using the in-\nversion symmetry (or time-reversal symmetry ) of the\nsystem, we can easily obtain the low-energy Hilbert sub-\nspace and the low-energy effective Hamiltonian around\ntheK0point. Finally, we can summarize the basis for\nthe low-energy Hilbert subspace as\n\u001e1=u\u000b\n11\u0014\n\u00001p\n2\u0000\npA\nx+i\u001czpA\ny\u0001\u0015\n+u\u000b\n21\u001czsB;\n\u001e5=u\u000b\n11\u00141p\n2\u0000\npB\nx\u0000i\u001czpB\ny\u0001\u0015\n\u0000u\u000b\n21\u001czsA;(26)\nand the low-energy effective Hamiltonian without SOC\nreads\nH\u001c=vF(kx\u001bx+\u001czky\u001by); (27)\nwhere Pauli matrices \u001bdenote the orbital basis degree\nof freedom, and \u001cz=\u00061labels the two valleys Kand\nK0. Note that under the space inversion operation P=\n\u001bx\u001cxand the time-reversal operation T=\u001cx^K(^Kis\nthe complex conjugation operator), the above low-energy\neffective Hamiltonian [Eq. (27)] is invariant.B. Low-energy effective Hamiltonian involving\nSOC\nThe SOC can be written as\nHso=\u00180^L\u0001^s=\u00180\n2\u0012L+s\u0000+L\u0000s+\n2+Lzsz\u0013\n;(28)\nwheres\u0006=sx\u0006isyandL\u0006=Lx\u0006iLydenote the ladder\noperators for the spin and orbital angular momenta, re-\nspectively. Here ^s= (~=2)~ s, and in the following we shall\ntake ~= 1.\u00180is the magnitude of atomic SOC. Because\nof the presence of pxandpyorbital component in the\nlow-energy Hilbert subspace [Eq. (26)] f\u001e1;\u001e5g\nf\";#g,\nan on-site effective SOC is generated with\nHso=\u0015so\u001cz\u001bzsz; (29)\nwhere\n\u0015so=1\n2ju\u000b\n11j2\u00180\n=1\n22\n41\u00009V2\nsp\u001b\n\u00012\u0000\u0001q\n\u00012+ 18V2sp\u001b+ 18V2sp\u001b3\n5\u00180:\n(30)\nAgain we stress that in the honeycomb textitX-hydride\nmonolayers the dominant intrinsic effective SOC is on-\nsite rather than from the NNN hopping processes as in\nthe original Kane-Mele model.\nConsequently, from the above Hamiltonian (27) and\n(29), we obtain the generic low-energy effective Hamil-\ntonian around the Dirac points acting on the low-energy\nHilbert subspace:\nHeff=H\u001c+Hso=vF(kx\u001bx+\u001czky\u001by) +\u0015so\u001cz\u001bzsz;\n(31)\nwhere the analytical expressions for Fermi velocity vF\nand magnitude of intrinsic effective SOC \u0015soare given in\nEqs. (25) and (30), whose explicit values are presented\nin Table I via FP calculations. Again we note that the\nabove spinful low-energy effective Hamiltonian is invari-\nant under both the space-inversion symmetry operation\nand time-reversal symmetry operation with T=isy\u001cx^K.\nThe two model parameters vFand\u0015socan be obtained\nby fitting the band dispersions of the FP results. Their\nvalues are listed in Table I.\nIII. LOW-ENERGY EFFECTIVE\nHAMILTONIAN FOR HONEYCOMB\nTEXTITX-HALIDE ( X= N-BI) MONOLAYERS\nA. Low-energy Hilbert subspace and effective\nHamiltonian without SOC\nFor the textitX-halide ( X= N-Bi) systems, the outer\nshellorbitalsofXlabeledas Xs,Xpx,Xpy,Xpz, andthe\n5TABLE I. Values of Fermi velocity vFand magnitude of in-\ntrinsic SOC \u0015sofor textitX-hydride honeycomb monolayers\nobtained from FP calculations. Note that \u0015so=Eg=2, with\nEgthe gap opened by SOC at the Dirac point.\nsystem vF\u0000\n105m=s\u0001\n\u0015so(eV)\nNH 6.8 6:7\u000210\u00003\nPH 8.3 18\u000210\u00003\nAsH 8.7 97\u000210\u00003\nSbH 8.6 0.21\nBiH 8.9 0.62\nouter shell orbitals of halogen labeled as Hs,Hpx,Hpy,\nHpzwith(H=F-I)aretakenintoaccountinthefollowing\nderivation. As is shown in Fig. 1(a), there are also two\ndistinct sites A and B in the honeycomb lattice unit cell\nof textitX-halide with full halogenation from both sides\nofthe2DtextitXhoneycombsheet. Intherepresentation\nfXpA\ny,XpA\nx,XpA\nz,HpA\nz,HpA\ny,HpA\nx,HsA,XsA,XpB\ny,\nXpB\nx,XpB\nz,HpB\nz,HpB\ny,HpB\nx,HsB,XsBgand at theK\npoint, the total Hamiltonian with the nearest-neighbor\nhopping considered in the Slater-Koster formalism reads\nHha\n0= \nhAA\n0hAB\n0\nhAB\n0yhBB\n0!\n; (32)\nwith\nhAA\n0=\n2\n66666666666640 0 0 0 Vha\npp\u0019 0 0 0\n0 0 0 0 0 Vha\npp\u0019 0 0\n0 0 0 Vha\npp\u001b 0 0\u0000Vha\nsp\u001b 0\n0 0Vha\npp\u001b \u0001ha\np 0 0 0 Vha\nsp\u001b\nVha\npp\u0019 0 0 0 \u0001ha\np 0 0 0\n0Vha\npp\u0019 0 0 0 \u0001ha\np 0 0\n0 0\u0000Vha\nsp\u001b 0 0 0 \u0001ha\nsVha\nss\u001b\n0 0 0 Vha\nsp\u001b 0 0Vha\nss\u001b \u00013\n7777777777775;\n(33)\nhAB\n0=2\n6666666666664\u0000V0\n1\u0000iV0\n10 0 0 0 0 V0\n2\n\u0000iV0\n1V0\n10 0 0 0 0\u0000iV0\n2\n0 0 0 0 0 0 0 0\n0 0 0 0 0 0 0 0\n0 0 0 0 0 0 0 0\n0 0 0 0 0 0 0 0\n0 0 0 0 0 0 0 0\n\u0000V0\n2iV0\n20 0 0 0 0 03\n7777777777775;(34)hBB\n0=\n2\n66666666666640 0 0 0 Vha\npp\u0019 0 0 0\n0 0 0 0 0 Vha\npp\u0019 0 0\n0 0 0 Vha\npp\u001b 0 0Vha\nsp\u001b 0\n0 0Vha\npp\u001b \u0001ha\np 0 0 0\u0000Vha\nsp\u001b\nVha\npp\u0019 0 0 0 \u0001ha\np 0 0 0\n0Vha\npp\u0019 0 0 0 \u0001ha\np 0 0\n0 0Vha\nsp\u001b 0 0 0 \u0001ha\nsVha\nss\u001b\n0 0 0\u0000Vha\nsp\u001b 0 0Vha\nss\u001b \u00013\n7777777777775;\n(35)\nwhere \u0001ha\npis the on site energy for the porbitals of the\nhalogen atom, \u0001(\u0001ha\ns) is the on site energy for the sor-\nbital of textitX (halogen) atom, the on site energies for p\norbitals of textitX atoms are taken to be zero. Vha\npp\u0019(Vha\npp\u001b\n)isthehoppingbetweenthe pzorbitalfromtextitXatom\nand thepzorbital from halogen atom in the \"shoulder by\nshoulder\" (\"head to tail\") type. Vha\nsp\u001bis the hopping be-\ntween thepz(s) orbital from textitX atom and the s(pz)\norbital from halogen atom. Vha\nss\u001bis the hopping between\nthesorbital from textitX atom and the sorbital from\nhalogen atom. The parameters V0\n1andV0\n2take the same\nexpressions as in Sec.II A.\nFirstly, we perform the unitary transformation as in\nEq. (5), as well as the following unitary transformation\nH'A\n1=\u00001p\n2\u0000\nHpA\nx+iHpA\ny\u0001\nH'B\n2=1p\n2\u0000\nHpB\nx\u0000iHpB\ny\u0001\nH'A\n3=\u00001p\n2\u0000\nHpA\nx\u0000iHpA\ny\u0001\nH'B\n4=\u00001p\n2\u0000\nHpB\nx+iHpB\ny\u0001: (36)\nIn the new basis fX'A\n1,XsB,H'A\n1,HsB,XpB\nz,HpB\nz,\nX'B\n2,XsA,H'B\n2,HsA,XpA\nz,HpA\nz,X'3,X'4,H'A\n3,\nH'B\n4g=fXpA\ny,XpA\nx,XpA\nz,HpA\nz,HpA\ny,HpA\nx,HsA,\nXsA,XpB\ny,XpB\nx,XpB\nz,HpB\nz,HpB\ny,HpB\nx,HsB,XsBg\nUha\n1, we could rewrite the Hamiltonian in the following\nblock-diagonalformwiththreedecoupleddiagonalblocks\nHha\n1=Hha\n1;\u000b\bHha\n1;\f\bHha\n1;\r; (37)\nHha\n1;\u000b=2\n6666666640iV2Vha\npp\u0019 0 0 0\n\u0000iV2\u0001 0 Vha\nss\u001b 0\u0000Vha\nsp\u001b\nVha\npp\u0019 0 \u0001ha\np 0 0 0\n0Vha\nss\u001b 0 \u0001ha\nsVha\nsp\u001b 0\n0 0 0 Vha\nsp\u001b 0Vha\npp\u001b\n0\u0000Vha\nsp\u001b 0 0Vha\npp\u001b \u0001ha\np3\n777777775;\n(38)\n6Hha\n1;\f=2\n6666666640\u0000iV2Vha\npp\u0019 0 0 0\niV2\u0001 0 Vha\nss\u001b 0Vha\nsp\u001b\nVha\npp\u0019 0 \u0001ha\np 0 0 0\n0Vha\nss\u001b 0 \u0001ha\ns\u0000Vha\nsp\u001b 0\n0 0 0\u0000Vha\nsp\u001b 0Vha\npp\u001b\n0Vha\nsp\u001b 0 0 Vha\npp\u001b \u0001ha\np3\n777777775;\n(39)\nHha\n1;\r=2\n6666664V1 0Vha\npp\u0019p\n2Vha\npp\u0019p\n2\n0\u0000V1\u0000Vha\npp\u0019p\n2Vha\npp\u0019p\n2\nVha\npp\u0019p\n2\u0000Vha\npp\u0019p\n2\u0001ha\np 0\nVha\npp\u0019p\n2Vha\npp\u0019p\n20 \u0001ha\np3\n7777775:(40)\nFor the first diagonal block Hha\n1;\u000b, in the presentation\nfX'A\n1;XsB;H'A\n1;HsB;XpB\nz;HpB\nzgits eigenvectors can\nbe written as\nj\"ha\nii=1\nNha\ni\u0002\n2\n666666666641\ni\nC\nVha\npp\u0019\n\"ha\ni\u0000\u0001hap\ni[Vha\npp\u001b(Vha2\nsp\u001b+Vha\npp\u001bVha\nss\u001b)\u0000\"ha\niVha\nss\u001b(\"ha\ni\u0000\u0001ha\np)]\nDC\n\u0000iVha\nsp\u001b[\u0001ha\nsVha\npp\u001b\u0000\u0001ha\npVha\nss\u001b\u0000\"ha\ni(Vha\npp\u001b\u0000Vha\nss\u001b)]\nDC\n\u0000iVha\nsp\u001b[Vha2\nsp\u001b+Vha\nss\u001bVha\npp\u001b\u0000\"ha\ni(\"ha\ni\u0000\u0001ha\ns)]\nDC3\n77777777775;(41)\nwith\nD\u0000\n\"ha\ni\u0001\n\u0011\u0000\n\"ha\ni\u0000\u0001ha\ns\u0001\u0002\nVha2\npp\u001b\u0000\"ha\ni\u0000\n\"ha\ni\u0000\u0001ha\np\u0001\u0003\n+\n\u0000\n\"ha\ni\u0000\u0001ha\np\u0001\nVha2\nsp\u001b;\n(42)\nand\nC\u0011V2\u0000\n\"ha\ni\u0000\u0001ha\np\u0001\nVha2pp\u0019\u0000\"ha\ni\u0000\n\"ha\ni\u0000\u0001hap\u0001: (43)\nHere,\"ha\niandNha\ni(i= 1;2;\u0001\u0001\u0001;6)are the corresponding\neigenvalues and the normalization factors, respectively.\nTherefore, by the unitary transformation\n\b\n\u001eha\n1;\u001eha\n2;\u001eha\n3;\u001eha\n4;\u001eha\n5;\u001eha\n6\t\n=\b\nX'A\n1;XsB;H'A\n1;HsB;XpB\nz;HpB\nz\t\nU\u000b;(44)\nwithU\u000b=fj\"ha\niigi=1;2;\u0001\u0001\u0001;6\u0011fu\u000b\njig, the above 6\u00026block\nHha\n1;\u000bis diagonalized.\nFrom our FP calculations [Fig. 2(b)], the main compo-\nnents of the band around the Dirac points and the Fermi\nlevel come from the XpxandXpyorbitals, mixed with\na small amount of the HpxandHpyorbitals as well as\nXsorbital. The orbital features are identical with the\neigenvectors of \"ha\n1. When we take its eigenvalue as theFermi energy EF. Following similar procedures as in the\nprevious section, we can obtain the eigenvalues up to the\nsecond-order correction and the eigenvectors up to the\nfirst-order correction with\n\"ha\n1=1\n20\n@\u00010+vuut\u000102+ 4V2\n2\u00002\u00010Vha2pp\u0019\n\"\u0000\u0001hap+Vha4pp\u0019\u0000\n\"\u0000\u0001hap\u000121\nA;\n(45)\nwhere\n\u00010= \u0001\u0000\n\"ha02\n1\u0000\nVha2\nss\u001b+Vha2\nsp\u001b\u0001\n\u0000\"ha0\n1\u0000\n\u0001ha\npVha2\nss\u001b+ \u0001ha\nsVha2\nsp\u001b\u0001\nD\u0000\n\"ha0\ni\u0001 +\n+\u0000\nVha2\nsp\u001b+Vha\nss\u001bVha\npp\u001b\u0001\nD\u0000\n\"ha0\ni\u0001;\n(46)\n\"ha0\n1=1\n20\n@\u0001 +vuut\u00012+ 4V2\n2\u00002\u0001Vha2pp\u0019\n\"\u0000\u0001hap+Vha4pp\u0019\u0000\n\"\u0000\u0001hap\u000121\nA;\n(47)\nand\n\"=1\n2\u0012\n\u0001 +q\n\u00012+ 4V2\n2\u0013\n: (48)\nUptothispoint, wehavefoundthelow-energyeigenvalue\n\"ha\n1and the corresponding basis \u001eha\n1. Again, in order to\ncapture the essential physics, we simply the above ex-\npressions by taking only the zeroth-order terms. So in\nthe following, we take \"ha\n1=\"ha0\n1and omit the correc-\ntion withfHsB;XpB\nz;HpB\nzgfor the eigenvector fj\"ha\n1ig.\nConsequently, the eigenvector has the following form in\nthe basisfX'A\n1;XsB;H'A\n1g\nj\"ha\n1i=1\nnha\n12\n6641\n\u0000iV2\n\"ha0\n1\u0000\u0001\nVha\npp\u0019\n\"ha0\n1\u0000\u0001hap3\n775\u00112\n64uha\n11\nuha\n21\nuha\n313\n75;(49)\nwithnha\n1being a normalization constant, and the eigen-\nvalue\"ha0\n1is given in Eqs. (47) and (48).\nThe eigenvalues of the second diagonal block Hha\n1;\fare\ndenoted as \"ha\n6+i(i= 1;2;\u0001\u0001\u0001;6), and one finds that\n\"ha\n6+i=\"ha\ni(i= 1;2;\u0001\u0001\u0001;6), where\"ha\niare eigenvalues of\nHha\n1;\u000b. Through similar procedures, the low-energy eigen-\nvectorfj\"ha\n7ighas the following simple form in the basis\nfX'B\n2;XsA;H'B\n2g:\nj\"ha\n7i=1\nnha\n12\n6641\niV2\n\"ha0\n1\u0000\u0001\nVha\npp\u0019\n\"ha0\n1\u0000\u0001hap3\n775=2\n64uha\n11\n\u0000uha\n21\nuha\n313\n75:(50)\n7The third diagonal block Hha\n1;\rare of high energy hence\nis not of interest here.\nFromtheaboveanalysis, thelow-energystates \u001eha\n1and\n\u001eha\n7constitute the low-energy Hilbert subspace. They\nhave the following explicit forms:\n\u001eha\n1=uha\n11\u0014\n\u00001p\n2\u0000\nXpA\nx+iXpA\ny\u0001\u0015\n+uha\n21XsB\n+uha\n31\u0014\n\u00001p\n2\u0000\nHpA\nx+iHpA\ny\u0001\u0015\n;\n\u001eha\n7=uha\n11\u00141p\n2\u0000\nXpB\nx\u0000iXpB\ny\u0001\u0015\n\u0000uha\n21XsA\n+uha\n31\u00141p\n2\u0000\nHpB\nx\u0000iHpB\ny\u0001\u0015\n:(51)\nAgainweperformthesmall ~kexpansionintheabovelow-\nenergy Hilbert subspace around Kpoint by~k!~k+K\nand keep the first-order terms in ~k,\nHK= \n0vFk\u0000\nvFk+ 0!\n; (52)\nwithvFthe Fermi velocity\nvF=p\n3a\n2\u00141\n2juha\n11j2(Vpp\u001b\u0000Vpp\u0019) +juha\n21j2Vss\u001b\u0015\n:\n(53)\nNote that for the textitX-halide systems, juha\n11j2is\nmuch larger than juha\n21j2andjuha\n31j2. Either follow-\ning similar procedures, or via the inversion symmetry\n(or time-reversal symmetry ), one can obtain the low-\nenergy Hilbert subspace and and the low-energy effective\nHamiltonian around the K0point. Finally the basis for\nlow-energy Hilbert subspace can be summarized as\n\u001eha\n1=uha\n11\u0014\n\u00001p\n2\u0000\nXpA\nx+i\u001czXpA\ny\u0001\u0015\n+uha\n21\u001czXsB\n+uha\n31\u0014\n\u00001p\n2\u0000\nHpA\nx+i\u001czHpA\ny\u0001\u0015\n;\n\u001eha\n7=uha\n11\u00141p\n2\u0000\nXpB\nx\u0000i\u001czXpB\ny\u0001\u0015\n\u0000uha\n21\u001czXsA\n+uha\n31\u00141p\n2\u0000\nHpB\nx\u0000i\u001czHpB\ny\u0001\u0015\n:(54)\nand the low-energy effective Hamiltonian without SOC\nreads\nH\u001c=vF(kx\u001bx+\u001czky\u001by); (55)\nwhere Pauli matrices \u001bdenote the orbital basis degree of\nfreedom, and \u001czlabels the two valleys KandK0. Note\nthatunderthespacereversaloperation P=\u001bx\u001cxandthe\ntime-reversal operation T=\u001cx^K, the above low-energy\neffective Hamiltonian Eq. (55) is also invariant.B. Low-energy effective Hamiltonian involving\nSOC\nIn a similar way as in Sec. II B, we obtain an on-\nsite SOC in the spinful low-energy Hilbert subspace\nf\u001e1;\u001e7g\nf\";#g,\nHso=\u0015so\u001cz\u001bzsz; (56)\n\u0015so=1\n2juha\n11j2\u0018X\n0+1\n2juha\n31j2\u0018ha\n0;(57)\nwhereuha\n11anduha\n31are given in Eq. (49), and \u0018X\n0(\u0018ha\n0)\nis the magnitude of atomic SOC of pnictogen (halogen).\nIt should be noted that due to the presence of major px\nandpyorbital components, the first-order on-site effec-\ntive SOC also dominates in the textitX-halide systems.\nEquation (49) explains the tendency that the \u0015soin-\ncreases with the atomic number of halogen for the same\npnictogen element, as shown in Table II.\nFrom Eqs. (55) and (56), we obtain the generic low-\nenergyeffectiveHamiltonianaroundtheDiracpointsact-\ning on the low-energy Hilbert subspace f\u001e1;\u001e7g\nf\";#g\nHeff=H\u001c+Hso=vF(kx\u001bx+\u001czky\u001by) +\u0015so\u001cz\u001bzsz;\n(58)\nwhere Fermi velocity vFand magnitude of intrinsic effec-\ntive SOC\u0015soare given in Eqs. (53) and (57), and their\nvalues are listed in Table II. One notes that this Hamil-\ntonian is also invariant under both the space-inversion\nsymmetry and time-reversal symmetry with T=isy\u001cx^K.\nThe two model parameters vFand\u0015sofor halides ob-\ntained by fitting the band dispersions of the FP results\nare listed in Table II.\nIV. A SIMPLE SPINFUL LATTICE\nHAMILTONIAN FOR THE HONEYCOMB\nTEXTITX-HYDRIDE/HALIDE ( X= N-BI)\nMONOLAYERS FAMILY\nFor the purpose of studying the topological proper-\nties of the honeycomb textitX-hydride/halide ( X= N-\nBi) monolayers family, as well as their edge states, it is\nconvenient to work with a lattice Hamiltonian via lat-\ntice regularization of the low-energy continuum models\n(Eq. (31) and Eq. (58)). Taking into account the main\nphysics involving pxandpyorbitals, we construct the fol-\nlowing spinful lattice Hamiltonian for the 2D honeycomb\ntextitX-hydride/halide ( X= N-Bi) monolayers\nH=X\nhi;ji;\u000b;\f=px;pyt\u000b\f\nijcy\ni\u000bcj\f\n+X\ni;\u000b;\f=px;py;\u001b;\u001b0=\";#\u0015\u000b\f\n\u001b;\u001b0cy\ni\u000b\u001bci\f\u001b0sz\n\u001b;\u001b0;(59)\nwherehi;jimeansiandjsites are nearest neighbors,\n\u000band\fare the orbital indices. The first term is the\nhoppingtermandthesecondoneistheon-siteSOCterm.\n8TABLE II. Values of two model parameters vFand\u0015sofor honeycomb textitX-halide ( X= N-Bi) monolayers obtained from\nFP calculations. Note that \u0015so=Eg=2, withEgthe gap opened by SOC at the Dirac point.\nsystem vF\u0000\n105m=s\u0001\n\u0015so(eV) system vF\u0000\n105m=s\u0001\n\u0015so(eV)\nNF 5.5 8:5\u000210\u00003NBr 4.2 19\u000210\u00003\nPF 7.2 13\u000210\u00003PBr 8.0 17\u000210\u00003\nAsF 7.3 80\u000210\u00003AsBr 8.2 98\u000210\u00003\nSbF 6.6 0.16 SbBr 7.7 0.20\nBiF 7.2 0.55 BiBr 7.3 0.65\nNCl 4.3 9:7\u000210\u00003NI 3.8 28\u000210\u00003\nPCl 7.8 17\u000210\u00003PI 8.1 19\u000210\u00003\nAsCl 8.0 95\u000210\u00003AsI 9.1 0:10\nSbCl 7.3 0.19 SbI 7.7 0.21\nBiCl 6.9 0.56 BiI 7.7 0.65\nAfter Fourier transformation of the above lattice\nHamiltonian, its energy spectrum over the entire Bril-\nlouin zone can be obtained. Since here spin is good quan-\ntum number, we can divide the model Hamiltonian into\ntwo sectors for spin up and spin down separately. For\neach sector, the corresponding model Hamiltonian reads\nH\"(k) =2\n66640\u0000i\u00180\n2hAB\nxx(k)hAB\nxy(k)\n0hAB\nxy(k)hAB\nyy(k)\n0\u0000i\u00180\n2y03\n7775;(60)\nH#(k) =2\n66640i\u00180\n2hAB\nxx(k)hAB\nxy(k)\n0hAB\nxy(k)hAB\nyy(k)\n0i\u00180\n2y03\n7775;(61)\nwhere\nhAB\nxx(k)\u00111\n2(3Vpp\u001b+Vpp\u0019) cos\u0012kx\n2\u0013\nexp\u0012\niky\n2p\n3\u0013\n+Vpp\u0019exp\u0012\n\u0000ikyp\n3\u0013\n;\nhAB\nxy(k)\u0011ip\n3\n2(Vpp\u001b\u0000Vpp\u0019) sin\u0012kx\n2\u0013\nexp\u0012\niky\n2p\n3\u0013\n;\nand\nhAB\nyy(k)\u00111\n2(Vpp\u001b+ 3Vpp\u0019) cos\u0012kx\n2\u0013\nexp\u0012\niky\n2p\n3\u0013\n+Vpp\u001bexp\u0012\n\u0000ikyp\n3\u0013\n:\nFor simplicity, we choose the lattice constant a= 1. The\non-site energies for porbitals are taken to be zero. Near\ntheKandK0points, the above model Hamiltonian re-\nduces to the low-energy effective Hamiltonian [Eq. (31)\nand (58)] with vF=p\n3a\n4(Vpp\u001b\u0000Vpp\u0019)and\u0015so=\u00180=2.\n-4-3-2-101234 \n Energy(eV)Κ\nΜΓ Γ FIG. 3. (Color online). A comparison of the band structures\nfor monolayer SbH calculated using FP and TB methods with\nSOC . The dashed green curve is the FP result. The solid red\nand blue curves are the TB model results. The red curve is\nwith the NN hopping only, while the blue curve also includes\nthe NNN hopping terms. For the NN case, the parameters\nare taken as Vpp\u001b= 1:68eV,Vpp\u0019=\u00000:60eV. For the NNN\ncase, the parameters are taken as Vpp\u001b= 1:69eV,Vpp\u0019=\n\u00000:62eV,VNNN\npp\u001b = 0eV,VNNN\npp\u0019 =\u00000:23eV. For both cases,\n\u0015so= 0:21eV. The superscript NNN means the next-nearest-\nneighbor hoping. The Fermi level is set to zero.\nTaking SbH as an example, we compare the results\nfrom FP calculations and from the lattice models. As\nshown in Fig. 3, there is a good agreement between the\ntwo results around the Kpoint. The fitting away from\nKpoint can be improved by including hopping terms\nbetween far neighbors. In Fig. 3, we also show the result\nwith NNN hopping, for which a fairly good agreement\nwith the FP low-energy bands over the whole Brillouin\nzone can be achieved.\n9V. DISCUSSION AND SUMMARY\nWehaveobtainedthelow-energyeffectiveHamiltonian\nfor the textitX-hydride and textitX-halide ( X= N-Bi)\nfamily of materials, which is analogous to the Kane-Mele\nmodel proposed for the QSH effect in graphene.11The\nimportant difference is that in Kane-Mele model the ef-\nfective SOC is of second-order NNN type, which is much\nweaker than the on-site SOC in our systems. The SOC\nterm in our Hamiltonian opens a large nontrivial gap\nat the Dirac points. From KtoK0the mass term\nchanges sign for each spin species and the band is in-\nverted. As a result, the QSH effect can be realized in the\ntextitX-hydride and textitX-halide ( X= N-Bi) mono-\nlayers. Some of these materials, such as BiH/BiF, have\nrecord huge SOC gap with magnitude around 1 eV, far\nhigher than the room-temperature energy scale, hence\nmaking their detection much easier.\nOn the experimental side, the buckling honeycomb\nBi(111) monolayer and film have been manufactured via\nmolecular beam epitaxy (MBE).23,29,34On the other\nhand, chemical functionalization of 2D materials is a\npowerful tool to create new materials with desirable\nfeatures, such as modifying graphene into graphane,\ngraphone, and fluorinated graphene via H and F, re-\nspectively.35Therefore, it is very promising that Bi-\nHydride/Halidemonolayer, thehugegapQSHinsulators,\nmay be synthesized by chemical reaction in the solvents\nor by the exposure of the Bi (111) monolayer and film\nto the atomic or molecular gases. It is noted that even\nthough one side (full passivation) instead of both sides\n(alternatingpassivation)ofBi(111)bilayersispassivated,\nthe band structure is almost unchanged and the topol-\nogy properties remain nontrivial. This will provide more\nfreedom to realize these kinds of materials.\nIt is known that the low-energy Hilbert space for\ngraphene consists of the pzorbital from carbon atoms. In\nthat system, the SOC term from NNN second-order pro-\ncesses is vanishingly small, and the on-site SOC as well\nas the nearest neighbor SOC are forbidden by symme-\ntry constraint. In contrast, for the honeycomb textitX-\nhydride/halide monolayers, pxandpyorbitals from the\ngroup V elements constitute the low-energy Hilbert sub-\nspace. In fact, this represents the first class of materials\nfor which the Dirac fermion physics is associated with\npxandpyorbitals. Because of this, the effective on-site\nSOC can has nonzero matrix elements and results in the\nhuge SOC gap at the Dirac points.\nThe leading-order effective SOC processes in the\ntextitX-hydride and textitX-halide systems, silicene, and\ngraphene are schematically shown in Fig. 4. As shown\nin Figs. 44(a) and 4(b), the representative leading-order\neffective SOCprocesses aroundthe Kpoint inthe honey-\ncomb textitX-hydride and textitX-halide monolayers are\njpA\n+\"i\u0015so\u0000!jpA\n+\"i;jpA\n+#i\u0000\u0015so\u0000!jpA\n+#i;\njpB\n\u0000\"i\u0000\u0015so\u0000!jpB\n\u0000\"i;jpB\n\u0000#i\u0015so\u0000!jpB\n\u0000#i;(62)where\u0015sorepresents the atomic spin-orbit interaction\nstrength, which is given in Eq. (30) for textitX-hydride\nsystems and Eq. (57) for textitX-halide systems. In a\nHilbert subspace consisting of pxandpyorbitals, such\neffective SOC arises in the first-order on-site processes,\nwhich leads to its huge magnitude.\nAs for silicene, which has a low-buckled structure, the\ntypical leading-order SOC is from the (first-order) NNN\nprocesses,25as shown in Fig. 4(c),\njpA\nz\"iV\u0000!jpB\n\u0000\"i\u0000\u00180\n2\u0000!jpB\n\u0000\"iV\u0000!jpA\nz\"i;\njpA\nz#iV\u0000!jpB\n\u0000#i\u00180\n2\u0000!jpB\n\u0000#iV\u0000!jpA\nz#i;\njpB\nz\"iV\u0000!jpA\n+\"i\u00180\n2\u0000!jpA\n+\"iV\u0000!jpB\nz\"i;\njpB\nz#iV\u0000!jpA\n+#i\u0000\u00180\n2\u0000!jpA\n+#iV\u0000!jpB\nz#i;(63)\nwhereVis the nearest-neighbor direct hopping ampli-\ntudeand\u00180representstheatomicintrinsicSOCstrength.\nThe whole process can be divided into three steps. For\nexample, we consider the pA\nzorbital. Firstly, due to the\nlow-buckled structure, pA\nzcouples topB\n\u0000. Carriers in pA\nz\norbital then hop to the nearest neighbor pB\n\u0000orbital. Sec-\nondly, the atomic intrinsic SOC shifts the energy of the\nspin up and spin down carriers by \u0007\u00180\n2. In the third step,\ncarriersin the pB\n\u0000orbitalhop toanothernearest-neighbor\npA\nzorbital, making the resulting effective SOC an NNN\nprocess and of first order in \u00180.\nAs for graphene, around Dirac point, the leading-order\neffective SOC is from (second-order) NNN effective SOC\nprocess, as shown in Fig. 4(d):\njpA\nz\"i\u00180=p\n2\u0000! jpA\n+#iV\u0000!jsB\n#iV\u0000!jpA\n+#i\u00180=p\n2\u0000! jpA\nz\"i;\njpB\nz#i\u00180=p\n2\u0000! jpB\n\u0000\"iV\u0000!jsA\n\"iV\u0000!jpB\n\u0000\"i\u00180=p\n2\u0000! jpB\nz#i:(64)\nDuring the whole NNN hopping process, the atomic SOC\nappears twice, making the effective SOC second order in\n\u00180and hence much weaker.\nIn summary, using the TB method and the FP calcu-\nlation, we have derived the low-energy effective Hilbert\nsubspace and Hamiltonian for the honeycomb textitX-\nhydride/halide monolayers materials. These 2D group-V\nhoneycomb lattice materials have the same low-energy\neffective Hamiltonian due to their same D3dpoint group\nsymmetry and the same D3small group at the Kand\nK0points. The low-energy model contains two key pa-\nrametersvFand\u0015so. We have obtained their analytic\nexpressions and also their numerical values by fitting the\nFP calculations. Moreover, we have found that the low-\nenergy Hilbert subspace consists of pxandpyorbitals\nfrom the group-V elements, which is a key reason for the\nhuge SOC gap. This feature is distinct from the group-\nIV honeycomb lattice monolayers such as silicene and\ngraphene. Finally, we construct a spinful lattice Hamil-\ntonian for these materials. Our results will be useful for\nfurther investigations of this intriguing class of materials.\n10ππ\nπσ\nσsoc\nππ\nπσσsocsocσ\nσsoc\nσσσ\nσ\nσsocσ\nAB\nAB\nBB\nA A(a) (b)\n(c) (d)FIG. 4. (Color online). The leading-order effective SOC pro-\ncesses in textitX-hydride or textitX-halide ( X= N-Bi), sil-\nicene and graphene. (a) and (b) Sketches of the huge effective\non-site SOC in textitX-hydride systems and textitX-halide\nsystems. (c) Sketch of the effective SOC from NNN hopping\nprocesses caused by the buckling in silicene. (d) Sketch of the\nsecond-order effective SOC from NNN hopping processes in\ngraphene.ACKNOWLEDGMENTS\nThis work was supported by the MOST Project\nof China (Nos. 2014CB920903, 2010CB833104, and\n2011CBA00100), the National Natural Science Foun-\ndation of China (Grant Nos. 11225418, 51171001,\nand 11174337), SUTD-SRG-EPD2013062, and the Spe-\ncialized Research Fund for the Doctoral Program of\nHigherEducationofChina(GrantNo. 20121101110046).\nCheng-Cheng Liu was supported Excellent young schol-\nars Research Fund of Beijing Institute of Technology\n(Grant No. 2014CX04028).\nNote added Recently, we notice another relevant\nwork36discussing effective models of a honeycomb lat-\ntice withpxandpyorbitals.\n\u0003ygyao@bit.edu.cn\n1A. K. Gelm and K. S. Novoselov, Nat. Mater. 6, 183-191\n(2007).\n2A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S.\nNovoselov, and A. K. Geim, Rev. Mod. Phys. 81, 109\n(2009).\n3P. Vogt, P. De Padova, C. Quaresima, J. Avila, E.\nFrantzeskakis, M. C. Asensio, A. Resta, B. Ealet, and G.\nLe Lay, Phys. Rev. Lett. 108, 155501 (2012).\n4L. Chen, C.-C. Liu, B. Feng, X. He, P. Cheng, Z. Ding, S.\nMeng, Y. Yao, and K. Wu, Phys. Rev. Lett. 109, 056804\n(2012).\n5Z. Song, C.-C. Liu, J. Yang, J. Han, B. Fu, Y. Yang, Q.\nNiu, J. Lu, and Y.G. Yao, arXiv:cond-mat/1402.2399.\n6C. Wu, D. Bergman, L. Balents, and S. Das Sarma, Phys.\nRev. Lett. 99, 070401 (2007).\n7C. Wu, Phys. Rev. Lett. 100, 200406 (2008).\n8M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045\n(2010).\n9X. Qi and S. Zhang, Rev. Mod. Phys. 83, 1057 (2011).\n10B. Yan and S.-C. Zhang, Rep. Prog. Phys. 75, 096501\n(2012).\n11C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 226801\n(2005).\n12C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 146802\n(2005).\n13Y. G. Yao, F. Ye, X. L. Qi, S. C. Zhang, and Z. Fang,\nPhys. Rev. B 75, 041401(R) (2007).\n14H.Min, J.E.Hill, N.A.Sinitsyn, B.R.Sahu, L.Kleinman,\nand A. H. MacDonald, Phys. Rev. B 74, 165310 (2006).\n15S. Konschuh, M. Gmitra, and J. Fabian, Phys. Rev. B 82,\n245412 (2010).16B. A. Bernevig, T. L. Hughes, and S. C. Zhang, Science\n314, 1757-1761 (2006).\n17M. K onig, S. Wiedmann, C. Br une, A. Roth, H. Buhmann,\nL. W. Molenkamp, X. L. Qi, and S.C. Zhang, Science 318,\n766-770 (2007).\n18C. X. Liu, T. L. Hughes, X. L. Qi, K. Wang, and S. C.\nZhang, Phys. Rev. Lett. 100, 236601 (2008).\n19I. Knez, R.R. Du, and G. Sullivan, Phys. Rev. Lett. 107,\n136603 (2011).\n20I. Knez, R.R. Du, and G. Sullivan, Phys. Rev. Lett. 109,\n186603 (2012).\n21S. Murakami, Phys. Rev. Lett. 97, 236805 (2006).\n22Z. Liu, C. X. Liu, Y.S. Wu, W.H. Duan, F. Liu, and Jian\nWu, Phys. Rev. Lett. 107, 136805 (2011).\n23T. Hirahara, G. Bihlmayer, Y. Sakamoto, M. Yamada, H.\nMiyazaki, S.I. Kimura, S. Bl ugel, and S. Hasegawa, Phys.\nRev. Lett. 107, 166801 (2011).\n24C.-C. Liu, W. Feng, and Y. Yao, Phys. Rev. Lett. 107,\n076802 (2011).\n25C.-C. Liu, H. Jiang, and Y. Yao, Phys. Rev. B 84, 195430\n(2011).\n26C. Weeks, J. Hu, J. Alicea, M. Franz, and R. Wu, Phys.\nRev. X 1, 021001 (2011).\n27Y. Xu, B. Yan, H.-J. Zhang, J. Wang, G. Xu, P. Tang,\nW. Duan, and S.-C. Zhang, Phys. Rev. Lett. 111, 136804\n(2013).\n28Z.F. Wang, Z. Liu, and F. Liu, Nat. Commun. 4, 1471\n(2013).\n29F. Yang,L. Miao, Z. F. Wang, M.-Y. Yao, F. Zhu, Y. R.\nSong, M.-X. Wang, J.-P. Xu, A. V. Fedorov, Z. Sun, G. B.\nZhang, C. Liu, F. Liu, D. Qian, C. L. Gao, and J.-F. Jia,\nPhys. Rev. Lett. 109, 016801 (2012).\n1130M. Zhou, W. Ming, Z. Liu, Z. Wang, Y.G. Yao, and F.\nLiu, arXiv:cond-mat/1401.3392.\n31J. C. Slater and G. F. Koster, Phys. Rev. 94, 1498 (1954).\n32G. Kresse and J. Furthm uller, Phys. Rev. B 54, 11169-\n11186 (1996).\n33R. Winkler, Spin-Orbit Coupling Effects in Two-\nDimensional Electron and Hole Systems , 1st ed. (Springer,\nBerlin, 2003).\n34C. Sabater, D. Gos \u0013albez-Mart \u0013inez, J. Fern \u0013andez-Rossier,\nJ. G. Rodrigo, C. Untiedt, and J. J. Palacios, Phys. Rev.\nLett.110, 176802 (2013).\n35J. O. Sofo, A. S. Chaudhari, and G. D. Barber, Phys. Rev.B75, 153401 (2007); D. C. Elias, R. R. Nair, T. M. G.\nMohiuddin, S. V. Morozov, P. Blake, M. P. Halsall, A.\nC. Ferrari, D. W. Boukhvalov, M. I. Katsnelson, A. K.\nGeim, K. S. Novoselov, Science 323, 610 (2009); J. Zhou,\nQ. Wang, Q. Sun, X. S. Chen, Y. Kawazoe, and P. Jena,\nNano Lett. 9, 3867 (2009); J. T. Robinson, J. S. Burgess,\nC. E. Junkermeier, S. C. Badescu, T. L. Reinecke, F. K.\nPerkins, M. K. Zalalutdniov, J. W. Baldwin, J. C. Cul-\nbertson, P. E. Sheehan, and E. S. Snow, ibid. 10, 3001\n(2010).\n36G.-F. Zhang, Y. Li, and C. Wu, arXiv:cond-\nmat/1403.0563.\n12"    },    {        "title": "2206.05041v1.Quantum_heat_engine_based_on_a_spin_orbit_and_Zeeman_coupled_Bose_Einstein_condensate.pdf",        "content": "Quantum heat engine based on a spin-orbit and Zeeman-coupled Bose-Einstein condensate\nJing Li,1,\u0003E. Ya Sherman,2, 3, 4and Andreas Ruschhaupt1\n1Department of Physics, University College Cork, T12 H6T1 Cork, Ireland\n2Departamento de Qu´ ımica-F´ ısica, UPV /EHU, Apartado 644, 48080 Bilbao, Spain\n3IKERBASQUE, Basque Foundation for Science, 48011 Bilbao, Spain\n4EHU Quantum Center, University of the Basque Country UPV /EHU\nWe explore the potential of a spin-orbit coupled Bose-Einstein condensate for thermodynamic cycles. For\nthis purpose we propose a quantum heat engine based on a condensate with spin-orbit and Zeeman coupling\nas a working medium. The cooling and heating are simulated by contacts of the condensate with an external\nmagnetized media and demagnetized media. We examine the condensate ground state energy and its dependence\non the strength of the synthetic spin-orbit and Zeeman couplings and interatomic interaction. Then we study\nthe e\u000eciency of the proposed engine. The cycle has a critical value of spin-orbit coupling related to the engine\nmaximum e \u000eciency.\nIntroduction Quantum cycles are of much importance both\nfor fundamental research and for applications in quantum-\nbased technologies[1, 2]. Quantum heat engines have been\ndemonstrated in recent on several quantum platforms, such as\ntrapped ions [3, 4], quantum dots [5] and optomechanical os-\ncillators [6–9]. Well-developed techniques for experimental\ncontrol make Bose-Einstein condensates (BECs) [10] a suit-\nable system for a quantum working medium of a thermal ma-\nchine [11–13].\nRecently, a quantum Otto cycle was experimentally realized\nusing a large quasi-spin system with individual cesium (Cs)\natoms immersed in a quantum heat bath made of ultracold ru-\nbidium (Rb) atoms [14, 15]. Several spin heat engines have\nbeen theoretically and experimentally implemented using a\nsingle-spin qubit [16], ultracold atoms [17], single molecule\n[18], a nuclear magnetic resonance setup [19] and a single-\nelectron spin coupled to a harmonic oscillator flywheel [20].\nThese examples have motivated our exploration of the spin-\norbit coupled BEC considered in this paper.\nSpin-orbit coupling (SOC) links a particle’s spin to its\nmotion, and artificially introduces charge-like physics into\nbosonic neutral atoms [21]. The experimental generation\n[22–25] of SOC is usually accompanied by a Zeeman field,\nwhich breaks various symmetries of the underlying system\nand induces interesting quantum phenomena, e.g. topological\ntransport[26]. In addition, in the spin-orbit coupled BEC sys-\ntem, more studies on moving solitons [27–29], vortices [30],\nstripe phase [31] and dipole oscillations [32] have been re-\nported.\nIn this paper, we propose a BEC with SOC as a working\nmedium in a quantum Stirling cycle. The classic Stirling cy-\ncle is made of two isothermal branches, connected by two\nisochore branches. The BEC is characterized by SOC, Zee-\nman splitting, a self-interaction, and is located in a quasi-\none-dimensional vessel with a moving piston that changes the\nlength of the vessel. The external ”cooling” and ”heating”\nreservoirs are modelled by the interaction of the spin-1 /2 BEC\nwith an external magnetized and demagnetized medias. The\nexpansion and compression works depend on the SOC and\n\u0003Corresponding author: jli@ucc.ieZeeman coupling. A main goal is to examine the condensate\nground state energy and its dependence on the strength of the\nsynthetic spin-orbit, Zeeman couplings, interatomic interac-\ntion and length of the vessel. For the semiquantitative analy-\nsis, perturbation theory is applied to understand the e \u000bects of\nSOC and Zeeman splitting. We further analyze several impor-\ntant parameters and investigate how they a \u000bect the e \u000eciency\nof the cycle, e.g. the critical SOC strength for di \u000berent self-\ninteractions.\nModel of the heat engine: Working medium We consider a\nquasi-one dimensional BEC, extended along the x\u0000axis and\ntightly confined in the orthogonal directions. The mean-field\nenergy functional of the system is then given by E=R+1\n\u00001\"dx\nwith spin-independent self-interaction of the Manakov’s sym-\nmetry [33]:\n\"=\tyH0\t +g\n2(j \"j2+j #j2)2; (1)\nwhere \t\u0011( \";  #)T(here T stands for transposition) and\nthe wavefunctions  \"and #are related to the two pseudo-\nspin components. The parameter grepresents the strength of\nthe atomic interaction which can be tuned by atomic s\u0000wave\nscattering length using Feshbach resonance [34, 35] with g>\n0,g<0, and g=0 giving the repulsive, attractive, and no\natomic interaction, respectively. The Hamiltonian H0in Eq.\n(1) of the spin-1 /2 BEC, trapped in an external potential V(x),\nis given by\nH0=ˆp2\n2mˆ\u001b0+\u000b\n~ˆpˆ\u001bx+~\n2\u0001ˆ\u001bz+V(x); (2)\nwith ˆ p=\u0000i~@xbeing the momentum operator in the longi-\ntudinal direction, ˆ \u001bx;zbeing the Pauli matrices, and ˆ \u001b0being\nthe identity matrix. Here \u000bis the SOC constant and \u0001is the\nZeeman field. We choose a convenient length unit \u0018, an energy\nunit~2=(m\u00182) and a time unit m\u00182=~and express the following\nequations in the corresponding dimensionless variables. The\ncoupled Gross-Pitaevskii equations are now given by\ni@\n@t \"= \n\u00001\n2@2\n@x2+\u0001\n2+g n(x)+V(x)!\n \"\u0000i\u000b@\n@x #;(3)\ni@\n@t #= \n\u00001\n2@2\n@x2\u0000\u0001\n2+g n(x)+V(x)!\n #\u0000i\u000b@\n@x \";(4)arXiv:2206.05041v1  [cond-mat.quant-gas]  10 Jun 20222\na a aP\n=0\n>0A\nB\nCD\nmagnetization \nsource\nmagnetization source\nB     C demagnetization source\nD     Ademagnetization \nsource\n1 2(a)\n(b)\nFIG. 1. (a) The schematic diagram of the quantum Stirling cycle\nbased on the Zeeman and SOC. (b) Visualization of the demagne-\ntization (left) and magnetization (right) processes with the external\nsources; the blue dots represent the BEC atoms and the orange dots\nrepresent the external source.\nwhere the density is given by n(x)=j \"j2+j #j2. We fix the\nnorm N=R1\n\u00001n(x)dx=1.\nWe consider a hard-wall potential V(x) of half width a:\nV(x)=0;(jxj\u0014a); V(x)=1(jxj>a):(5)\nThis potential is analogous to a piston in a thermodynamic\ncycle and it allows one to define the work of the quantum cy-\ncle. The ground state \tof the BEC then depends on the half\nwidth a, the detuning \u0001, the interactions gand the SOC \u000b, i.e.\n\t\u000b;g(a;\u0001), and the corresponding total ground state energy of\nthe BEC is then denoted as E\u000b;g(a;\u0001). We define also the pres-\nsure P\u000b;g(a;\u0001) as a measure of the energy E\u000b;g(a;\u0001) stored per\ntotal length 2 a:\nP\u000b;g(a;\u0001)\u0011\u0000@E\u000b;g(a;\u0001)\n2@a: (6)\nIn the special case of \u0001 = 0 and for the spin-independent\nself-interaction proportional to n(x), the energy [36, 37]\nis given by E\u000b;g(a;0)=E0;g(a;0)\u0000\u000b2=2 resulting in\n\u000b\u0000independent pressure P\u000b;g(a;0). Notice that at both\nnonzero\u000band\u0001, the system is characterized by a magne-\ntostriction in the form M\u000b;g(a;\u0001)=@P\u000b;g(a;\u0001)=@\u0001:\nModel of the heat engine: Quantum Stirling cycle We con-\nsider a quantum Stirling cycle keeping the interaction gand\nthe SOC\u000bfixed during the whole process. The key idea is that\nthe external ”cooling” and ”heating” reservoirs are modelled\nby the interaction of the spin 1 /2 BEC with an external mag-\nnetized media (see Fig. 1(b), right) resp. demagnetized media\n(see Fig. 1(b), left). This external, (de)magnetized source\nleads to a random magnetic field in the condensate and be-\ncause of the Zeeman-e \u000bect this corresponds to a detuning ofthe condensate to \u0001with some probability density distribution\np(\u0001). We assume that this external source brings the system to\na stationary state with the condensate described by a density\noperator\nˆ\u001a=Z\np(\u0001)j\t\u000b;g(a;\u0001)ih\t\u000b;g(a;\u0001)jd\u0001: (7)\nThe probability density distribution of the demagnetizing\nsource pdm(\u0001) is centered around h\u0001idm\u0011R\n\u0001pdm(\u0001)d\u0001 =0\nwhile the one of the magnetizing source pm(\u0001) is centered\naround a positive value h\u0001im>0. As an increase in \u0001de-\ncreases the BEC energy [10] by an \u000b\u0000dependent amount, the\ndemagnetization source plays the role of a “hot thermal bath”\nhere and the magnetization source plays the role of a “cold\nthermal bath”. In general there could exist a stationary exter-\nnal magnetic field leading to an additional detuning during the\ncycle. We neglect this possibility in the following in order to\nsimplify the notation.\nThe realization of the Stirling cycle is described by a four-\nstroke protocol, illustrated in Fig. 1(a). We start at point\nAwith the BEC being in contact with the demagnetiza-\ntion source, leading to an e \u000bective detuning centered around\nh\u0001idm=0. The potential is of half width a1. The BEC state is\ngiven by Eq. 7 with p(\u0001)=pA(\u0001)\u0011pdm(\u0001).\nQuantum “isothermal” expansion stroke A !B:Dur-\ning this stroke, the working medium stays in contact with\nthe external demagnetization source while the potential ex-\npands adiabatically from a1toa2without excitation in the\nBEC. The probability density distribution p(\u0001) stays con-\nstant during this ”isothermal” stroke, i.e. we have pA(\u0001)=\npB(\u0001)=pdm(\u0001) (e\u000bective detuning centered around h\u0001idm=\n0). The average work done during this “isothermal” ex-\npansion stroke can be then calculated as [38] hWei=R\npdm(\u0001)\u0010\nE\u000b;g(a1;\u0001)\u0000E\u000b;g(a2;\u0001)\u0011\nd\u0001.\nQuantum isochore cooling stroke B !C:The contact\nwith the demagnetization source is switched o \u000band the work-\ning medium is brought into contact with the magnetization\nsource while keeping a2constant. The probability distribu-\ntion p(\u0001) is changed to pC(\u0001)\u0011pm(\u0001), this corresponds\nto a ”cooling” (as the total energy of the BEC is lowered).\nThe average heat exchange in this stroke can be calculated as\nhQci=R\n(pm(\u0001)\u0000pdm(\u0001))E\u000b;g(\u0001;a2)d\u0001.\nQuantum “isothermal” compression stroke C ! D:\nDuring this stroke, the working medium stays in contact\nwith the external magnetization source while the BEC com-\npresses adiabatically from potential half width a2toa1with-\nout excitation in the BEC. The probability density distribu-\ntion p(\u0001) remains constant during this ”isothermal” stroke,\ni.e. we have pD(\u0001)=pC(\u0001)=pm(\u0001) leading to an ef-\nfective detuning centered around h\u0001im>0. The average\nwork done during this “isothermal” compression is hWci=R\npm(\u0001)\u0010\nE\u000b;g(a2;\u0001)\u0000E\u000b;g(a1;\u0001)\u0011\nd\u0001.\nQuantum isochore heating stroke D !A:The contact\nwith the magntetization source is switched o \u000band the work-\ning medium is brought again into contact with the demagne-\ntization source while keeping a1constant. The probability\ndistribution p(\u0001) is changed back to pA(\u0001)=pdm(\u0001), this\ncorresponds to a ”heating” (as the total energy of the BEC is3\nincreased). The average heat exchange in this stroke can be\ncalculated ashQhi=R\n(pdm(\u0001)\u0000pm(\u0001))E(\u0001;a1)d\u0001.\nTo study this quantum cycle, it is important to examine and\nunderstand the dependence of the BEC ground-state energy on\nthe di \u000berent parameters. This will be done in the following.\nPerturbation theory for the ground state energy The com-\nplex BEC system used in the thermodynamic cycle does not\nhave an exact analytical solution. However, we can obtain\nanalytical insight by considering perturbation theory of the\nground state energy E\u000b;0(a;\u0001) of the non-selfinteracting BEC\n(i.e. g=0) at small\u000b(and nonzero \u0001), as well as at small \u0001\n(and nonzero \u000b).\nIn the case of small \u000bthen\u000b\u001c1=a, the Hamiltonian in\nEq. (1) can be written as H0=H0;0+H0\n0whereH0=ˆp2=2+\n\u0001ˆ\u001bz=2+V(x) and the perturbation term being H0\n0=\u000bˆpˆ\u001bx.\nThe eigenstate basis of H0;0is given by  (0)\nn;#(x)=\u00020; n(x)\u0003T,\n (0)\nn;\"(x)=\u0002 n(x);0\u0003T, where n(x) are the eigenstates of the\npotential in Eq.(5). The first-order correction to the energy\nvanishes and the second-order correction becomes:\n\u000f(0)\n2=\u0000X\nn>1jh (0)\nn;\"(x)jH0\n0j (0)\n0;#(x)ij2\n(n2\u00001)\u00192=(8a2)+ \u0001: (8)\nThus, the total ground state energy E\u000b;0(a;\u0001) of the system up\nto second order in \u000bis given by\nE\u000b;0(a;\u0001)\u0019\u00192\n8a2\u0000\u0001\n2\u0000\u00192\u000b2\n4\u0001a2+\u00192\u000b2\n8a4\u00012\u001f(a;\u0001) cot\u001f(a;\u0001)\n2;(9)\nwhere\u001f(a;\u0001)\u0011p\n\u00192\u00008a2\u0001:We can simplify Eq. (9) by\napproximating the expression up to first order in \u0001:\nE\u000b;0(a;\u0001)\u0019\u00192\n8a2\u0000\u0001\n2\u0000\u000b2\n2+\u00192\u00006\n3\u00192 a\n`sr!2\n\u0001: (10)\nThe first three terms on the right-hand side of Eq. (10) corre-\nspond to kinetic energy, Zeeman energy (at \u000b=0) and SOC\nenergy (at \u0001 =0). Here we introduced the spin rotation length\n`sr\u00111=\u000bwith a=`sr\u001c1.\nAlternatively, in the case of large \u000bthen\u000b>1=aand small\ndetuning \u0001, the Hamiltonian can be written as H0=H0;1+H0\n1\nwhereH0=ˆp2=2+\u000bˆpˆ\u001bx+V(x);and the perturbation term\nH0\n1= \u0001 ˆ\u001bz=2:The unperturbedH0;1has pairs of degenerate\neigenstates  (0)\naand (0)\nbwith the energy E\u000b;0(a;0):\n (0)\na(x)= n(x)e\u0000i\u000bx \n1\n1!\n;  (0)\nb(x)= n(x)ei\u000bx \n1\n\u00001!\n:(11)\nBased on the perturbation theory for degenerate states and tak-\ning into account that the diagonal matrix elements of the per-\nturbation, \u0001h (0)\nij\u001bzj (0)\nii=2=0, we obtain at a=`sr\u001d1 the\nground state energy in the form:\nE\u000b;0(a;\u0001)\u0019\u00192\n8a2\u0000\u000b2\n2\u0000\u00192\n4`sr=a\f\f\f(2a=`sr)2\u0000\u00192\f\f\f\f\f\f\f\f\fsin 2a\n`sr!\f\f\f\f\f\f\u0001:(12)\nWhen we look at the corresponding pressure following from\nEq. (12), we can calculate approximately the pressure di \u000ber-\nence\u000ePbetween the points BandCin the cycle (at a2, see Fig.\n1.56 1.58 1.60 1.62 1.64-0.04-0.020.000.02\n1.0 1.2 1.4 1.6 1.8 2.00.00.20.40.60.81.01.2FIG. 2. Pressure P\u000b;0(a2;\u0001) versus potential half width afor the cases\nof\u0001 = 0;\u000b=1:6 (solid black, essentially, \u000b-independent), \u0001 =\n1;\u000b=1 (dashed blue) and \u0001 = 1;\u000b=1:6 (dot-dashed red). (Inset)\nThe pressure di \u000berence between points CandB,\u000eP=P\u000b;0(a2;1)\u0000\nP\u000b;0(a2;0).\n0.5 1 1.5 21.41.61.82.2.2\nFIG. 3. Critical \u000bc(g;\u0001) versus detuning \u0001for di \u000berent nonlineari-\nties: attractive g=\u00001 (black solid line), non-interaction g=0 (blue\ndashed line), and repulsive g=1 (dot-dashed red line).\n1). The di \u000berence\u000ePjumps from negative to positive at cer-\ntain widths where 2 a2=`sr\u0019(n+1)\u0019or\u000b\u0019(n+1)\u0019=(2a2) with\nn=1;2;3;4;:::. In addition, there is always an \u000bbetween two\nconsecutive “jump points” where \u000ePbecomes zero. We will\ndenote the first corresponding value of \u000b;where the change of\n\u000ePfor negative to positive occurs, as the critical \u000bc(g;\u0001).\nEnergy and pressure We examine now the exact numerical\nvalues of energy and pressure where we fix a1=1 and a2=2.\nThe corresponding pressure is illustrated in Fig. 2 for a non-\ninteracting BEC ( g=0). The shown pressure P\u000b;0(a2;0) for\n\u0001 =0 does not depend on the strength of SOC \u000bas discussed\nabove. We can also see that the pressure P\u000b;0(a2;1) is approx-\nimately equal to the pressure P\u000b;0(a2;0) at\u000ba2\u0019\u0019;providing\ncrossing of the red and dotted lines; this corresponds then to a\ncritical\u000bc(0;1)\u00191:6. The corresponding di \u000berence in pres-\nsure\u000ePis shown in detail in the inset; it can be seen that \u000eP\nchanges from negative to positive at \u000bc(0;1) as one expects it\nfrom the perturbation theory above.\nIn Fig. 3, the relations between the critical \u000bc(g;\u0001) and de-\ntuning \u0001for di \u000berent nonlinearities gare plotted. From the\nperturbation theory for g=0 and for small \u0001, one expects a\nvalue of\u000bc(g;\u0001)\u0019\u0019=a2\u00191:57. The figure shows that the ex-4\nact\u000bcis increasing with increasing \u0001for all cases of g. There\nis a competition between SOC and Zeeman field, therefore, a\nlarger detuning \u0001requires automatically a larger \u000b(and there-\nfore a larger \u000bc) to have an e \u000bect. We also see that \u000bcis larger\n(smaller) for attractive g=\u00001 (repulsive g=1) for all \u0001. The\nheuristic reason is that there is a (kind of) compression (ex-\npansion) of the wavefunction for g<0 (g>0) and, therefore,\na weaker (stronger) e \u000bect of SOC. This requires heuristically\na larger (smaller) \u000b(and therefore \u000bc) to show an e \u000bect.\nWork, heat and e \u000eciency of the engine Here we are mainly\ninterested in the properties of the cycle originating from the\nBEC and not in the details of the (de)magnetization source.\nTherefore, we assume that the probability density distribu-\ntions pdmresp. pmare strongly peaked around h\u0001idm=0\nresp.h\u0001im= \u0001 0>0 such that we approximate pdm(\u0001)=\u000e(\u0001)\nand pm(\u0001)=\u000e(\u0001\u0000\u00010) (where\u000eis the Dirac distribution).\nIn this case, the black-solid line and the blue-dashed line\nin Fig. 2 present an example of the expansion and com-\npression strokes of the cycle shown in the schematic Fig.\n1. The work done during the “isothermal” expansion pro-\ncess in Fig. 1,hWei, is then given by the energy di \u000ber-\nences:hWei=E\u000b;g(a1;0)\u0000E\u000b;g(a2;0). The cooling heat\nexchange from BtoChQcithrough contact with the mag-\nnetization source, becomes hQci=E\u000b;g(a2;\u00010)\u0000E\u000b;g(a2;0):\nThe workhWcidone during the compression stroke is then\nhWci=E\u000b;g(a2;\u00010)\u0000E\u000b;g(a1;\u00010). The heat in the last stroke\ncan be calculated by hQhi=E\u000b;g(a1;0)\u0000E\u000b;g(a1;\u00010). The\ntotal work then becomes:\nA=hWci+hWei=I\nABCDP\u000b;g(a;\u00010)da: (13)\nFor small \u00010,\nA=\u0000\u00010Z2a2\n2a1M\u000b;g(a;\u00010!0)da: (14)\nAs defined above, at \u000b=\u000bc(g;\u00010), the pressures at a2for\n\u0001 = 0 and \u00010>0 approximately coincide. If \u000b > \u000b c(g;\u00010);\nthe pressure-dependencies on afor\u0001 = 0 and \u00010>0 cross\nat a certain half width eawith a1<ea<a2. In that case, the\nwork done at the interval ( ea;a2) provides a negative contri-\nbution while the contribution of the interval ( a1;ea) can still\nincrease. In the following, we restrict our analysis to the case\n\u000b\u0014\u000bc(g;\u00010) while we expect a maximum of the total work\nclose to\u000bc(g;\u00010).\nThe e \u000eciency of each quantum cycle is now defined as\n\u0011=A\nhQhi: (15)\nAt small\u000b\u001c1=(2a2) we may approximate the e \u000eciency of\nthe quantum cycle in terms of \u00010as\n\u0011\u0019266664\u00192\n2\u00012\n00BBBB@1\na2\n1\u00001\na2\n21CCCCA+\u00192\n4\u00013\n0\u0010377775\u000b2; (16)\nwhere the coe \u000ecient\u0010is\n\u0010=\u001f(a1;\u00010)\na4\n1cot\u001f(a1;\u00010)\n2\u0000\u001f(a2;\u00010)\na4\n2cot\u001f(a2;\u00010)\n2:(17)\n0.0 0.5 1.0 1.50.00.20.40.60.81.0\n0.0 0.5 1.0 1.5 2.00.00.20.40.60.81.0(a)\n(b)\nFIG. 4. E \u000eciency\u0011versus\u000bwith \u00010=0:5 (solid black), \u00010=\n1:0 (solid blue) and \u00010=2:0 (solid red); the dotted vertical lines\ndenote the critical SOC strength \u000bc(g;\u00010). (a) g=0; results based\non perturbation theory in Eq. (16) (blue, red and black dashed lines);\nthe dashed pink line is given by Eq. (18) for \u00010!0. (b) g=\u00001.\nIn the limit of \u00010!0, the e \u000eciency\u0011simplifies to\n\u0011=2\u0010\n\u00192\u00006\u0011\n3\u00192\u0010\na2\n2\u0000a2\n1\u0011\n\u000b2: (18)\nIt is worth noticing that Eq. (18) has two limits with respect to\nthe value of a2:Let us define \u0011cas the e \u000eciency at the critical\n\u000bc. First, Eq. (18) is applicable only at \u000ba2<\u0019; thus, limiting\nthe critical\u0011cto the values of the order of 0.1. Secondly, for\ng<0 the value of a2is limited to 2 =jgj[39], thus\u0011cis limited\ncorrespondingly. (Note that Eq. (18) is not directly applicable\ntog,0 BEC).\nFigure 4 shows that the e \u000eciency\u0011grows as\u000bincreases.\nThe approximate e \u000eciency in Eq. (16) is a quadratic function\nof\u000b, and this is in good agreement with the numerical results\nin Fig. 4(a) for the case g=0. In the limit of \u00010!0,\nthe e\u000eciency\u0011\u0018\u000b2, see Eq. (18). This limit case is also\nshown by the dashed pink line in Fig. 4(a). As one expects\na maximum of the total work close to \u000bc(g;\u00010), one expects\nalso that the e \u000eciency reaches the maximum at \u000bclose to\n\u000bc(g;\u00010). The e \u000eciency\u0011cat a critical \u000bcwith respect to \u00010\nis shown in Fig. 5. The e \u000eciency decreases with increasing\n\u0001. This corresponds to Eq. (16) when \u000b=\u000bc\u0019\u0019=a2for all\nthree cases of g(see Fig. (3)).\nDiscussion and conclusions Here we return to the physical\nunits and discuss the possibility of experimental realization of\nthe present Stirling cycle. In the one-dimensional realization5\n0.5 1 1.5 20.50.60.70.80.9\nFIG. 5. E \u000eciency\u0011cat\u000bc(g;\u00010) versus \u00010. Nonlinearities: attrac-\ntiveg=\u00001 (black solid), non-interaction g=0 (blue dashed), and\nrepulsive g=1 (dot-dashed red). Values of \u000bc(g;\u00010) are the same as\nthose in Fig. 3.\nconsidered above, with the physical unit of length \u0018, the re-\nsulting dimensionless coupling constant gcan be estimated as\n\u00182Naat\u0018=sp;where spis the condensate cross-section, physi-\ncally corresponding to the piston cross-section. Here aatis the\ninteratomic scattering length (typically of the order of 10 aB,\nwhere aBis the Bohr radius) dependent on the Feshbach res-\nonance realization, and N\u0018 103is the total number of atoms\nin the condensate. A reasonable \u0018for optical setups is of the\norder of 10 \u0016m. Thus, the choice of a1;a2of the order of 10\n\u0016m allows one to achieve dimensionless \u000band\u00010of the order\nof unity [25], and thus explore the operational regimes of the\nStirling cycle up to the critical values.\nIn summary, we have explored the potential of a spin-orbit coupled Bose-Einstein condensate in a thermodynamic\nStirling-like cycle. It takes advantage of both the non-\ncommuting synthetic spin-orbit and Zeeman-like contribu-\ntions. The ”cooling” and ”heating” is assumed to originate\nby interaction with external magnetization and demagnetiza-\ntion media. We have examined the ground-state energy of\nthe condensate and how the corresponding pressure depends\non the di \u000berent parameters of the system. We have studied\nthe e\u000eciency of the corresponding engine in the dependence\non the strength of these spin-related couplings. The cycle is\ncharacterized by a critical spin-orbit coupling, corresponding,\nessentially, to the maximum e \u000eciency. The dependence of\nthe e\u000eciency on the spin-dependent coupling and nonlinear\nself-interaction paves the way to applications of these cycles.\nWhile we have concentrated here on e \u000bects originating from\nthe BEC, it will be interesting to study the details of the e \u000bects\nof the external magnetization and demagnetization sources in\nthe future.\nACKNOWLEDGMENTS\nWe are grateful to C. Whitty and D. Rea for commenting\non the manuscript. J.L. and A.R. acknowledge support from\nthe Science Foundation Ireland Frontiers for the Future Re-\nsearch Grant “Shortcut-Enhanced Quantum Thermodynam-\nics” No.19 /FFP/6951. The work of E.S. is financially sup-\nported through the Grant PGC2018-101355-B-I00 funded by\nMCIN /AEI/10.13039 /501100011033 and by ERDF “A way\nof making Europe”, and by the Basque Government through\nGrant No. IT986-16.\n[1] J. Goold, M. Huber, A. Riera, L. del Rio, and P. Skrzypczyk,\nJournal of Physics A: Mathematical and Theoretical 49, 143001\n(2016).\n[2] S. De \u000bner and S. Campbell, Quantum Thermodynamics , 2053-\n2571 (Morgan & Claypool Publishers, 2019).\n[3] O. Abah, J. Roßnagel, G. Jacob, S. De \u000bner, F. Schmidt-Kaler,\nK. Singer, and E. Lutz, Phys. Rev. Lett. 109, 203006 (2012).\n[4] J. Roßnagel, S. T. Dawkins, K. N. Tolazzi, O. Abah, E. Lutz,\nF. Schmidt-Kaler, and K. Singer, Science 352, 325 (2016).\n[5] B. Sothmann, R. S ´anchez, and A. N. Jordan, Nanotechnology\n26, 032001 (2014).\n[6] K. Zhang, F. Bariani, and P. Meystre, Phys. Rev. Lett. 112,\n150602 (2014).\n[7] C. Bergenfeldt, P. Samuelsson, B. Sothmann, C. Flindt, and\nM. B ¨uttiker, Phys. Rev. Lett. 112, 076803 (2014).\n[8] C. Elouard, M. Richard, and A. Au \u000b`eves, New Journal of\nPhysics 17, 055018 (2015).\n[9] T. Hugel, N. B. Holland, A. Cattani, L. Moroder, M. Seitz, and\nH. E. Gaub, Science 296, 1103 (2002).\n[10] C. J. Pethick and H. Smith, Bose–Einstein Condensation in\nDilute Gases , 2nd ed. (Cambridge University Press, 2008) p.\n316–347.\n[11] C. Charalambous, M. A. Garcia-March, M. Mehboudi, and\nM. Lewenstein, New Journal of Physics 21, 083037 (2019).\n[12] N. M. Myers, F. J. Pe ˜na, O. Negrete, P. Vargas, G. D. Chiara,and S. De \u000bner, New Journal of Physics 24, 025001 (2022).\n[13] J. Li, T. Fogarty, S. Campbell, X. Chen, and T. Busch, New\nJournal of Physics 20, 015005 (2018).\n[14] F. Schmidt, D. Mayer, Q. Bouton, D. Adam, T. Lausch, J. Net-\ntersheim, E. Tiemann, and A. Widera, Phys. Rev. Lett. 122,\n013401 (2019).\n[15] Q. Bouton, S. Nettersheim, Jensand Burgardt, D. Adam,\nE. Lutz, and A. Widera, Nature Communications 12, 2063\n(2021).\n[16] K. Ono, S. N. Shevchenko, T. Mori, S. Moriyama, and F. Nori,\nPhys. Rev. Lett. 125, 166802 (2020).\n[17] J.-P. Brantut, C. Grenier, J. Meineke, D. Stadler, S. Krinner,\nC. Kollath, T. Esslinger, and A. Georges, Science 342, 713\n(2013).\n[18] W. H ¨ubner, G. Lefkidis, C. D. Dong, D. Chaudhuri, L. Chotorl-\nishvili, and J. Berakdar, Phys. Rev. B 90, 024401 (2014).\n[19] J. P. S. Peterson, T. B. Batalh ˜ao, M. Herrera, A. M. Souza, R. S.\nSarthour, I. S. Oliveira, and R. M. Serra, Phys. Rev. Lett. 123,\n240601 (2019).\n[20] D. von Lindenfels, O. Gr ¨ab, C. T. Schmiegelow, V . Kaushal,\nJ. Schulz, M. T. Mitchison, J. Goold, F. Schmidt-Kaler, and\nU. G. Poschinger, Phys. Rev. Lett. 123, 080602 (2019).\n[21] Y . Zhang, M. E. Mossman, T. Busch, P. Engels, and C. Zhang,\nFrontiers of Physics 11, 118103 (2016).\n[22] Y .-J. Lin, K. Jim ´enez-Garc ´ıa, and I. B. Spielman, Nature 471,6\n83 (2011).\n[23] C. Wang, C. Gao, C.-M. Jian, and H. Zhai, Phys. Rev. Lett. 105,\n160403 (2010).\n[24] T.-L. Ho and S. Zhang, Phys. Rev. Lett. 107, 150403 (2011).\n[25] D. L. Campbell, G. Juzeli ¯unas, and I. B. Spielman, Phys. Rev.\nA84, 025602 (2011).\n[26] D. Xiao, M.-C. Chang, and Q. Niu, Rev. Mod. Phys. 82, 1959\n(2010).\n[27] V . Achilleos, D. J. Frantzeskakis, P. G. Kevrekidis, and D. E.\nPelinovsky, Phys. Rev. Lett. 110, 264101 (2013).\n[28] Y . Xu, Y . Zhang, and B. Wu, Phys. Rev. A 87, 013614 (2013).\n[29] Y . V . Kartashov and V . V . Konotop, Phys. Rev. Lett. 118,\n190401 (2017).\n[30] J. Radi ´c, T. A. Sedrakyan, I. B. Spielman, and V . Galitski, Phys.\nRev. A 84, 063604 (2011).\n[31] S. Sinha, R. Nath, and L. Santos, Phys. Rev. Lett. 107, 270401(2011).\n[32] J.-Y . Zhang, S.-C. Ji, Z. Chen, L. Zhang, Z.-D. Du, B. Yan, G.-\nS. Pan, B. Zhao, Y .-J. Deng, H. Zhai, S. Chen, and J.-W. Pan,\nPhys. Rev. Lett. 109, 115301 (2012).\n[33] S. V . Manakov, Sov. Phys. JETP 65, 248 (1974).\n[34] E. Tiesinga, A. J. Moerdijk, B. J. Verhaar, and H. T. C. Stoof,\nPhys. Rev. A 46, R1167 (1992).\n[35] P. Courteille, R. S. Freeland, D. J. Heinzen, F. A. van Abeelen,\nand B. J. Verhaar, Phys. Rev. Lett. 81, 69 (1998).\n[36] D. S ´anchez and L. Serra, Phys. Rev. B 74, 153313 (2006).\n[37] I. V . Tokatly and E. Y . Sherman, Annals of Physics 325, 1104\n(2010).\n[38] P. Talkner, E. Lutz, and P. H ¨anggi, Phys. Rev. E 75, 050102\n(2007).\n[39] C. Sulem and P.-L. Sulem, in The Nonlinear Schr¨ odinger Equa-\ntion: Self-Focusing and Wave Collapse (Applied Mathematical\nSciences Book 139) (Springer, 1999)."    },    {        "title": "0810.5405v1.Berry_Phase_in_a_Single_Quantum_Dot_with_Spin_Orbit_Interaction.pdf",        "content": "arXiv:0810.5405v1  [cond-mat.mes-hall]  30 Oct 2008APS/123-QED\nBerry Phase in a Single Quantum Dot with Spin-Orbit Interact ion\nHuan Wang∗, Ka-Di Zhu†\nDepartment of Physics, Shanghai Jiao Tong University, Shan ghai 200240, People’s Republic of China\n(Dated: November 3, 2018)\nBerry phase in a single quantum dot with Rashba spin-orbit co upling is investigated theoretically.\nBerry phases as functions of magnetic field strength, dot siz e, spin-orbit coupling and photon-spin\ncoupling constants are evaluated. It is shown that the Berry phase will alter dramatically from 0\nto 2πas the magnetic field strength increases. The threshold of ma gnetic field depends on the dot\nsize and the spin-orbit coupling constant.\nPACS numbers: 71.70.Ej, 03.65.Vf\nI. INTRODUCTION\nDue to its important role in encoding information, the\nphase of wavefunction attracts a lot of interest in infor-\nmation science. Those properties can also be used in fu-\nture quantum information and quantum computer. Thus\nselectingone kindofphaseswhich canbe manipulated by\nquantumeffectisveryimportant. Berryphaseisbelieved\nto be a promising candidate. As a quantum mechanical\nsystem evolvescyclically in time such that it return to its\ninitial physical state, its wavefunction can acquire a ge-\nometric phase factor in addition to the familiar dynamic\nphase[1, 2]. If the cyclic change of the system is adia-\nbatic, this additional factor is known as Berry’s phase[3],\nand is, in contrast to dynamic phase, independent of en-\nergy and time. Fuentes-Guridi et al.[4] calculated the\nBerry phase of a particle in a magnetic field in consid-\neration of the quantum nature of the light field. Yi et\nal.[5] studied the Berry phase in a composite system and\nshowed how the Berry phases depend on the coupling be-\ntween the two subsystems. In a recent paper, San-Jose\net al.[6] have described the effect of geometric phases in-\nduced by either classical or quantum electric fields acting\non single electron spins in quantum dots. Wang and Zhu\n[7] have investigated the voltage-controlled Berry phases\nin two vertically coupled InGaAs/GaAs quantum dots.\nMost recently, observations of Berry phases in solid state\nmaterials are reported[8, 9, 10]. Leek et al.[10] demon-\nstrated the controlled Berry phase in a superconducting\nqubit which manipulates the qubit geometrically using\nmicrowave radiation and observes the phase in an inter-\nference experiment.\nSpin-relatedeffectshavepotentialapplicationsinsemi-\nconductor devices and in quantum computation. Rashba\net al.[11] have described the orbital mechanisms of\nelectron-spin manipulation by an electric field. Sonin[12]\nhas demonstrated that an equilibrium spin current in a\n2D electron gas with Rashba spin-orbit interaction can\nresult in a mechanical torque on a substrate near an edge\nof the medium. Serebrennikov[13] considered that the\n∗Email: wanghuan2626@sjtu.edu.cn\n†Email: zhukadi@sjtu.edu.cncoherent transport properties of a charge carrier. The\ntransportation will cause a spin precession in zero mag-\nnetic fields and can be described in purely geometric\nterms as a consequence of the corresponding holonomy.\nThe spin-orbit interaction in semiconductor heter-\nstructures is increasingly coming to be seen as a tool\nwhichcan manipulate electronicspin states[14, 15]. Two\nbasic mechanisms of the spin-orbit coupling of 2D elec-\ntrons are directly related to the symmetry properties of\nQDs. They stem from the structure inversion asymme-\ntry mechanism described by the Rashba term[11, 16] and\nthe bulk inversion asymmetry mechanism described by\nthe Dresselhaus term[17]. Recently, Debald and Emary\n[18] have investigated a spin-orbit driven Rabi oscilla-\ntion in a single quantum dot with Rashba spin-orbit cou-\npling. However, the influence of spin-orbit interaction\non Berry phase in a single quantum dot is still lacking.\nIn the present paper we will give a detail study on the\nBerry phase evolution of a single quantum dot with spin-\norbit interaction in a time-dependent quantized electro-\nmagnetic environment. We will borrow quantum optics\nmethod to investigate the impact of the spin-orbit inter-\naction and spin-photon interaction on Berry phase.\nThe paper is organized as follows. In Sec.II, we give\nthe model Hamiltonian including both spin-orbit inter-\naction and spin-photon interaction and calculated Berry\nphases as functions of magnetic field strength, dot size,\nspin-orbit coupling and photon-spin coupling constants.\nIn Sec.III, we draw the figures of the Berry phase as a\nfunction of magnetic field strength and some discussions\nare given. The final conclusion is presented in Sec.IV.\nII. THEORY\nWe consider a simple two-dimensional quantum dot\nwith parabolic lateral confinement potential in a perpen-\ndicular magnetic field Bwhich points along zdirection.\nThen the electron system can be described by the Hamil-\ntonian [18],\nHs=(p+e\ncA)2\n2m∗+m∗\n2ω2\n0(x2+y2)+1\n2gµBBσz,(1)\nwherepis the linear momentum operatorof the electron,\nA(r) =B\n2(−y,x,0)isthevectorpotentialinthesymmet-2\nric gauge,ω0is the characteristic confinement frequency,\nandσ= (σx,σy,σz) is the vector Pauli matrices. m∗\nis the effective mass of the electron and gits gyromag-\nnetic factor. µBis the Bohr magneton. In the second\nquantized notation, Eq.(1) becomes\nHs= (a+\nxax+a+\nyay+1)/planckover2pi1/tildewideω+/planckover2pi1ωc\n2i(a+\nxay−axa+\ny)\n+1\n2gµBBσz,(2)\nwhereωc=eB\nm∗cand/tildewideω2=ω2\n0+ω2\nc\n4. If we set\na+=1√\n2(ax−iay),a−=1√\n2(ax+iay),(3)\nThen, the Hamiltonian (2) can be written as\nHs=n+/planckover2pi1ω++n−/planckover2pi1ω−+1\n2gµBBσz, (4)\nwhereω±=/tildewideω±ωc/2,n+=a+\n+a+andn−=a+\n−a−. In\nwhat follows we include the spin-orbit interaction which\nis described as Rashba Hamiltonian in this system [11]\nHso=−α\n/planckover2pi1[(p+e\ncA)×σ]z, (5)\nwhereαis the spin-orbit coupling constant which can be\ncontrolledbygatevoltageinexperiment. Onsubstituting\nEq.(3) into Eq.(5) and then\nHso=α\n/tildewidel[γ+(σ+a++σ−a+\n+)−γ−(σ−a−+σ+a+\n−)],(6)\nwhereγ±= 1±1\n2(/tildewidel/lB)2,/tildewidel= (/planckover2pi1/m∗/tildewideω)1\n2andlB=\n(/planckover2pi1/m∗ωc)1\n2.\nThe Hamiltonians of photons and the coupling to the\nelectron spin can be written as follows:\nHp=/planckover2pi1ωpb+b, (7)\nHp−s=gc(σ++σ−)(b†+b), (8)\nwhereb+(b) andωpare the creation (annihilation) op-\nerator and energy of the photons, respectively. gcis the\nspin-photon coupling constant. Hence we obtain the to-\ntal Hamiltonian of the electron and photons:\nH=Hs+Hso+Hp+Hp−s\n=/planckover2pi1ω+a+\n+a++/planckover2pi1ω−a+\n−a−+1\n2gµBBσz\n+α\n/tildewidel[γ+(σ+a++σ−a+\n+)−γ−(σ−a−+σ+a+\n−)]\n+/planckover2pi1ωpb+b+gc(σ++σ−)(b†+b).(9)\nPerforming a unitary rotation of the spin such that σz→\n−σzandσ±→ −σ∓, we arrive at the Hamiltonian\nH=/planckover2pi1ω+a+\n+a++/planckover2pi1ω−a+\n−a−−1\n2gµBBσz\n+α\n/tildewidel[γ−(σ+a−+σ−a+\n−)−γ+(σ−a++σ+a+\n+)]\n+/planckover2pi1ωpb+b−gc(σ++σ−)(b†+b).(10)We now derive an approximation form of this Hamilto-\nnian by borrowing the observation from quantum optics\nthat the terms preceded by γ+in Eq.(10) are counterro-\ntating, and thus negligible under the rotating-wave ap-\nproximation when the spin-orbit coupling is small com-\npared to the confinement [18]. The last term in Eq.(10)\ntreats in the conventional rotaing-waveapproximation of\nquantum optics.\nH=/planckover2pi1ω+a+\n+a++/planckover2pi1ω−a+\n−a−+1\n2|g|µBBσz\n+λ(σ+a−+σ−a+\n−)+/planckover2pi1ωpb+b−gc(σ+b+σ−b+),(11)\nwhereλ=αγ−//tildewidel. Sincegis negative in InGaAs, we\nchoose the absolute value |g|ofg. It is obvious that the\nω+mode is decoupled from the rest of the system, giving\nH=/planckover2pi1ω+n++HJCwhere\nHJC=/planckover2pi1ω−a+\n−a−+1\n2|g|µBBσz+/planckover2pi1ωpb+b\n+λ(σ+a−+σ−a+\n−)−gc(σ+b+σ−b+).(12)\nThis is the well-known two mode Jaynes-Cummings\nmodel of quantum optics. In general this Hamiltonian\ncan not be solved exactly except ωp=ω−. In what fol-\nlows, for the sake of analytical simplicity, we consider\nωp=ω−which we can use a frequency-controllable laser\nand a special circuit to satisfy this condition in real ex-\nperiments.\nIn order to solve the above Hamiltonian, we define the\nnormal-mode operators:\nA=e1a−+e2b, (13)\nK=e2a−−e1b, (14)\nwhere\ne1=λ/radicalbig\nλ2+g2c,e2=−gc/radicalbig\nλ2+g2c,(15)\nwithe2\n1+e2\n2= 1. The new operators satisfy the commu-\ntation relations[19]\n[A,A†] = 1,[NA,A] =−A,[NA,A†] =A†,\n[K,K†] = 1,[NK,K] =−K,[NK,K†] =K†,\n[A,K] = 0,[A,K†] = 0,[NA,NK] = 0,(16)\nwhereNA=A†A(NK=K†K) is the number opera-\ntor related to the normal-mode operator A(K). Intro-\nducing the number-sum operator S=NA+NKand\nthe number-difference D=NA−NK, we can verify\nthat the Hamiltonian (12) transforms into the follow-\ning Hamiltonian:(i) S=na+nbis a conserved quan-\ntity (na=a†\n−a−andnb=b†b); (ii) the operator D can\nbe written in terms of the generators {Q+,Q−,Q}of the\nSU(2) Lie algebra,\nD= 2(e2\n1−e2\n2)Q0+2e1e2(Q++Q−),(17)3\nwhereQ−=a−b†,Q+=a†\n−b, andQ0=1\n2(a†\n−a−−b†b),\nwith [Q−,Q+] =−2Q0and [Q0,Q±] =±Q±;(iii) the\ncommutation relation between the operators S and D is\nnull, i.e., [S,D] = 0; and consequently, (iv) the Hamilto-\nnianHJCsimplifies to HJC=H0+V, where\nH0=/planckover2pi1ωp(S+1\n2σz),\nV=1\n2δσz+λA(σ−A++σ+A),(18)\nwith [H0,V] = 0.λA=/radicalbig\nλ2+g2cis an effective coupling\nconstantand δ=ωp−|g|µBB//planckover2pi1. The aboveHamiltonian\ncan be solved exactly. The eigenstates of this Hamilto-\nnian are given by\n|Ψ(n,±)/an}bracketri}ht=cosθ(n,±)|n,↑/an}bracketri}ht+sinθ(n,±)|n+1,↓/an}bracketri}ht,(19)\ntanθ(n,±)= (δ±∆n)/2λA/radicalbig\n(n+1),(20)\nwhere ∆ n=/radicalbig\nδ2+4λ2\nA(n+1) and | ↑>(| ↓>) is the\nspin-up (down) state.\nAccording to Ref.[4], since only the quasi-mode Ais\ncoupled with the spin of the electron, so the phase shift\noperatorU(ϕ) =e−iϕA†Ais introduced. Applied adi-\nabatically to the Hamiltonian (18), the phase shift op-\nerator alters the state of the field and gives rise to the\nfollowing eigenstates:\n|ψ(n,±)>=e−inϕcosθ(n,±)|n,↑>+\ne−i(n+1)ϕsinθ(n,±)|n+1,↓>.(21)\nChangingϕslowly from 0 to 2 π, the Berry phase is cal-\nculated as Γ l=i/integraltext2π\n0l/an}bracketle{tψ|∂\n∂ϕ|ψ/an}bracketri}htldϕwhich is given by\nΓl= 2π[sinθ(n,l)]2. (22)\nThis Berry phase is composed of two parts. One is in-\nduced by spin-orbit interaction, the other is induced by\nquantized light. Therefore if we can measure the total\nBerry phase and either part of two Berry phase, we will\nmeasure the other part of Berry phase.\nIII. NUMERICAL RESULTS\nFor the illustration of the numerical results, we choose\nthe typical parametersof the InGaAs: g=−4,m∗/me=\n0.05 (meis the mass of free electron). The dot size is\ndefined by l0=/radicalbig\n/planckover2pi1/m∗ω0. Figure 1 depicts the Berry\nphases Γ +as a function of the magnetic field strength\nfor three spin-orbit couplings. In Figure 1, we can find\nthat all the Berry phases change almost from 0 to 2 πas\nthe magnetic field strength varies from 20 mTto 50mT.\nWhen other parametersare fixed, the spin-orbit coupling\nconstantchangesas α= 0.4×10−12eVm, 0.8×10−12eVm\nand 1.2×10−12eVm, the Berry phases Γ +will have a\nFIG. 1: The Berry phase Γ +as a function of magnetic field\nstrength Bwith three spin-orbit coupling constants ( α=\n0.4×10−12eVm, 0.8×10−12eVmand 1.2×10−12eVm). The\nother parameters used are g=−4,m∗/me= 0.05,gc=\n0.01meV,l0= 80nm, andn= 0.\nFIG. 2: The Berry phases of Γ +and Γ −as a function of\nmagnetic field strength B. The parameters used are α=\n0.4×10−12eVm,g=−4,m∗/me= 0.05,gc= 0.01meV,\nl0= 80nm, andn= 0.\nslight movement in the figure. When B <20mTand\nB >50mT, the Berry phase changes gradually, while\nwhen 20mT <B < 50mT, the Berry phase changes dra-\nmatically. As the coupling constant increases, the Berry\nphase changes from sharply to slowly. The Sh¨ ordinger\nequationhastwodifferenteigenenergieswhen n= 0. The\ntwo eigenenergies will give two different Berry phases.\nFigure 2 illustrates these two Berry phases. In Figure\n2, when the others parameter are fixed, one of the Berry\nphasechangesfrom0to2 π, whiletheotherchangesfrom\n2πto 0 as the magnetic field strength varies from 20 mT\nto 50mT. Two Berry phases have an intersecting point\nat approximatively B= 33mT, which is corresponding\nto the resonant point.4\nFIG. 3: The Berry phases Γ +as a function of Bwith three\nthree different dot sizes ( l0= 70nm,80nm,90nm). The pa-\nrameters used are α= 0.4×10−12eVm,g=−4,m/me=\n0.05,gc= 0.01meV, andn= 0.\nFIG. 4: The Berry phases Γ +as a function of B\nwith three different light coupling constants ( gc=\n0.01meV,0.02meV,0.03meV). The parameters used are α=\n0.4×10−12eVm,g=−4,m/me= 0.05,l0= 80nm, and\nn= 0.\nFigure 3 shows the effect of the dot size on the Berry\nphase. When dot size becomes large from 70nm to 90nm,\nalthough all three Berry phases change from 0 to 2 π, the\nthreshold points of the magnetic field have a large move-\nment. When the dot size is 70nm, the Berry phase will\nchange dramatically at approximately 40mT, while the\ndot sizes are 80nm and 90nm, the turning points are ap-\nproximately at 30mT and 20mT, respectively. This im-\nplies that the bigger the dot, the smaller the threshold of\nthe magnetic field strength. Figure 4 illustrates the in-\nfluence of spin-photon coupling constant on Berry phase.\nAs the coupling constant becomes large, the Berry phase\nbecomes less drastic as shown in Figure 4.\nIn a recent paper, Giuliano et al.[20] have designed an\nexperimental arrangement, which is capacitively coupledthe dot to one arm of a double-path electron interferome-\nter. The phase carried by the transported electrons may\nbe influenced by the dot. The dot’s phase gives raise to\nan interference term in the total conductance across the\nring. More recently, Leek et al.[10] have measured Berry\nphase in a Ramsey fringe interference experiment. Our\nexperimentalsetupproposedhereisanalogouswith these\ntwo arrangements as shown in Figure 5. A beam light is\nsplit into two beams, one of the beams passes through\nthe dot, and interferes with the other one. Accurate con-\ntrol of the light field for dot is achieved through phase\nand amplitude modulation of laser radiation coupled to\nthe dot. We choose a special designed electric circuit\nto ensure the magnetic and laser vary synchronistically.\nThrough detecting the interfered light, we can measure\nthe Berry phase.\nFIG. 5: A sketch of a possible experimental setup to detect\nthe Berry phase. a, b and c are three mirror, d is a beam\nsplitter.\nIV. CONCLUSIONS\nIn conclusion, we have theoretically investigated the\nBerry phase in a single quantum dot in the presence of\nRashba spin-orbit interaction. Berry phases as functions\nof magnetic field strength, dot size, spin-orbit coupling\nand photon-spin coupling constants are evaluated. It\nis shown that for a given quantum dot, the spin-orbit\ncoupling constant and photon-spin coupling constant the\nBerry phase will alter dramatically from 0 to 2 πas the\nmagnetic field strength increases. The threshold of mag-\nneticfieldisdependent onthe Rashbaspin-orbitcoupling\nconstant, spin-photon coupling constantand the dot size.\nWealsoproposeapracticablemethod todetectthe Berry\nphase in such a quantum dot system. Finally, we hope\nthat our predictions in the present work can be testified\nby experiments in the near future.5\nAcknowledgments\nThis work has been supported in part by National\nNatural Science Foundation of China (No.10774101) andthe National MinistryofEducationProgramforTraining\nPhD.\n[1] A.Shapere and F.Wilczek, Geometric Phases in Physics,\n(World Scientific, Singapore, 1989)\n[2] J.Anandan, Nature (London)360 , 307(1992).\n[3] M.V.Berry, Proc.R.Soc.London Ser. A 392 , 45(1984).\n[4] I.Fuentes-Guridi, A.Carollo, S.Bose and V.Vedral,\nPhys.Rev.Lett.89 , 220404(2002).\n[5] X.X.Yi, L.C.Wang, and T.Y.Zheng, Phys.Rev.Lett.92 ,\n150406(2004).\n[6] P.San-Jose, B.Scharfenberger, G.Shon, A.Shnirman, an d\nG.Zarand, arXiv:cond-mat/0710.3931(2007).\n[7] H.Wang and K.D.Zhu, Euro.Phys.Lett82 , 60006(2008).\n[8] Y.Zhang, Y.W.Tan, H.L.Stormer and P.Kim, Nature\n(London)438 ,201(2005).\n[9] M.M¨ ott¨ onen, J.J.Vartiainen, and J.P.Pekola, Phys. Rev.\nLett.100, 177201(2008).\n[10] P.J.Leek, J.M.Fink, A.Blais, Science318 ,1889(2007).[11] E.I.Rashba, Sov.Phys.Solid State2 , 1109(1960)\n[12] E.B.Sonin, Phys. Rev. Lett. 99 ,266602(2007).\n[13] Y.A.Serebrennikov, Phys.Rev.B73 ,195317(2006).\n[14] J.C.Egues, G.Burkard, and D.Loss, Phys.Rev.Lett.89 ,\n176401(2002).\n[15] E.I.Rashba, and Al.L.Efros, Phys. Rev. Lett.91 , 126405\n(2003).\n[16] Y.A.Bychkov and E.I.Rashba, JEPT Lett.39 , 78(1984).\n[17] G.Dresselhaus Phys.Rev.B 100 , 580(1955).\n[18] S.Debald and C.Emary Phys. Rev. Lett.94 ,226803(2005)\n[19] M.A.Marchiolli, R.J.Missori and J.A.Roversi arXiv:\nquant-ph/0404.008v1 (2004).\n[20] D.Giuliano, P.Sodano, and A.Tagliacozzo Phys.Rev.B67 ,\n155317(2003)."    },    {        "title": "2307.12177v2.Spin_orbit_coupling_induced_phase_separation_in_trapped_Bose_gases.pdf",        "content": "Spin-orbit-coupling-induced phase separation in trapped Bose gases\nZhiqian Gui,1Zhenming Zhang,2Jin Su,3Hao Lyu,4and Yongping Zhang1,∗\n1Department of Physics, Shanghai University, Shanghai 200444, China\n2CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei 230026, China\n3Department of Basic Medicine, Changzhi Medical College, Changzhi 046000, China\n4Quantum Systems Unit, Okinawa Institute of Science and Technology Graduate University, Onna, Okinawa 904-0495, Japan\nIn a trapped spin-1 /2 Bose-Einstein condensate with miscible interactions, a two-dimensional spin-\norbit coupling can introduce an unconventional spatial separation between the two components. We\nreveal the physical mechanism of such a spin-orbit-coupling-induced phase separation. Detailed\nfeatures of the phase separation are identified in a trapped Bose-Einstein condensate. We further\nanalyze differences of phase separation in Rashba and anisotropic spin-orbit-coupled Bose gases. An\nadiabatic splitting dynamics is proposed as an application of the phase separation.\nI. INTRODUCTION\nPhase separation is a generic phenomenon from clas-\nsical physics to quantum physics, for example, the\noil-water separation and spin Hall effect [1]. Two-\ncomponent atomic Bose-Einstein condensates (BECs)\nprovide a tunable platform for the investigations of phase\nseparation [2–8]. The two components can be real-\nized by using different atomic species or same species\nwith different electric hyperfine states. Such a sys-\ntem features intra- and inter-component interactions.\nWhen the inter-component interactions dominate over\nthe intra-component interactions, two components pre-\nfer to be phase-separated in order to minimize the inter-\ncomponent interactions [9, 10]. The interactions for\nphase separation are called immiscible. The immiscibility\nof two-component BECs are completely tunable in exper-\niments. Phase separation effect induces rich physics in\nquantum gases, such as the formation of vector solitons\nand vortex-soliton structures, coherent spin dynamics,\nand pattern formations [11–19].\nIn a two-component BEC, an artificial spin-orbit cou-\npling can be synthesized between different hyperfine\nstates via Raman lasers [20–23]. Such a Raman-induced\nspin-orbit coupling is one dimensional. Rashba spin-\norbit coupling, which is two dimensional, has also been\nexperimentally realized in BECs [24, 25]. The imple-\nmentation of spin-orbit coupling in BECs gives rise to\nexotic quantum phases and rich superfluid properties,\nwhich opens an avenue for simulating topological mat-\nters and exploring superfluid dynamics [25–32]. In a\nRaman-type spin-orbit-coupled BEC, a stripe phase [33]\ncan exist in miscible interactions [26, 28]. In contrast,\nfor a Rashba spin-orbit-coupled BEC, the stripe phase\nmay appear in the immiscible regime [34]. Very inter-\nestingly, Refs. [35, 36] have numerically found a spatially\nphase-separated ground state in a Rashba-coupled and\nharmonically trapped BEC with miscible interactions.\nSuch a ground state in the miscible regime is unexpected\n∗yongping11@t.shu.edu.cnfor a usual two-component BEC without spin-orbit cou-\npling. Reference [35] identifies that the exotic phase sep-\naration satisfies a combined symmetry of parity and a\nspin flip. The existence of this state is attributed by\nRefs. [36, 37] to a spin-dependent force. The force is in-\ntrinsic in the presence of Rashba spin-orbit coupling and\ndrives the two components moving in opposite directions.\nThe force concept provides an intuitive picture for the un-\nexpected phase separation. However, its weakness is ob-\nvious. The force is proportional to the square of Rashba\nspin-orbit coupling strength. Therefore, a large strength\nis expected to generate a larger spatial separation. In\ncontrast, numerical results show that the separation de-\ncreases with an increasing strength [36, 37]. So far, the\nphysical origin of the unconventional phase separation in\nthe miscible regime is yet to be addressed. It calls for an\nunambiguous interpretation, since the phase separation\nhas already found a broad application in other excited\nstates. In Refs. [38, 39], a spin-orbit-coupled bright soli-\nton is found to be spatially separated in center-of-mass\nbetween the two components. Dynamics of the separa-\ntion in bright solitons is analyzed by varying spin-orbit\ncoupling strength [40]. A spin-orbit-coupled single-vortex\nstate, in which each component carries a singly quan-\ntized vortex, shows spatial separation between two com-\nponents, and the separation is inversely proportional to\nspin-orbit coupling strength [41]. Recently, dynamics of\nthe separation is triggered by a sudden quench of spin-\norbit coupling strength in a trapped BEC [42]. All men-\ntioned separations occur in the miscible regime, causing\na counterintuitive expectation.\nIn this paper, we provide the physical mechanism for\nthe unconventionally spin-orbit-coupling-induced phase\nseparation. Eigenstates of a two-dimensional spin-orbit\ncoupling have a momentum-dependent relative phase\nφ(⃗k) between the two components. Closely around a\nfixed momentum ⃗k0, the relative phase may present a\nlinear dependence φ(⃗k)∝(⃗k−⃗k0)·⃗ r0with a constant\n⃗ r0. The linear dependence is a momentum kick to move\ntwo components relatively. The superposition of these\neigenstates distributing around ⃗k0constitutes a spatially\nseparated wave packet. The separation, whose amplitude\ncan be calculated, is a completely single-particle effect ofarXiv:2307.12177v2  [cond-mat.quant-gas]  12 Oct 20232\nspin-orbit coupling. A weakly trapped BEC with two-\ndimensional spin-orbit coupling is a perfect platform to\nsimulate the phase separation. The miscible interactions\nforce atoms to condense at a certain spin-orbit-coupled\nmomentum state with a momentum-dependent relative\nphase. Meanwhile, weak traps broaden the condensed\nmomentum so that the condensation occupies momen-\ntum states in a narrow regime, which give rise to the\nlinear dependence of the relative phase. We numerically\nidentify detailed features of spin-orbit-coupling-induced\nphase separation in a trapped BEC with miscible inter-\nactions by analyzing its ground states. The phase sepa-\nration matches with the single-particle prediction when\nspin-orbit coupling strength dominates. We also com-\npare the separation differences between Rashba and an\nanisotropic spin-orbit coupling. Finally, as an applica-\ntion of the phase separation, we propose an adiabatic\nsplitting dynamics.\nThe paper is organized as follows. In Sec. II, the phys-\nical mechanism of spin-orbit-coupling-induced phase sep-\naration is unveiled. The separation amplitudes are pre-\ndicted. From the mechanism, we know that the sepa-\nration is a single-particle effect. In Sec. III, we identify\nseparated features of ground states in a trapped BEC\nwith Rashba spin-orbit coupling by the imaginary-time\nevolution method and the variational method. In Sec. IV,\nwe reveal the effect of the anisotropy of spin-orbit cou-\npling on the phase separation. In Sec. V, we propose an\nadiabatic dynamics to dynamically split two components\nbasing on the phase separation. For the completeness\nof our discussion, immiscible-interaction-induced phase\nseparation is shown in Sec. VI. The conclusion follows in\nSec. VII.\nII. SPIN-ORBIT-COUPLING-INDUCED PHASE\nSEPARATION\nRashba spin-orbit-coupling-induced phase separation\nis a completely single-particle effect. We reveal the phys-\nical origin of such phase separation. The Rashba spin-\norbit-coupled Hamiltonian is\nHSOC=p2\nx+p2\ny\n2+λ(pxσy−pyσx), (1)\nwhere pxandpyare the momenta along the xandydirec-\ntions respectively, λis the spin-orbit coupling strength,\nandσx,yare spin-1/2 Pauli matrices. The eigenenergy of\nthe Hamiltonian has two bands. The lower band is\nE=k2\nx+k2\ny\n2−λq\nk2x+k2y, (2)\nwith associated eigenstates being\nΦ =1√\n2eikxx+ikyy\u0012\neiφ\n2\ne−iφ\n2\u0013\n. (3)\nSince the Hamiltonian possesses continuously transla-\ntional symmetry, the eigenstates are plane waves withkx,ybeing the quasimomenta along the xand ydi-\nrections, respectively. The outstanding feature is that\nRashba spin-orbit coupling generates a relative phase φ\nbetween the two components, which satisfies\ntan(φ) =kx\nky. (4)\nIt is noted that ( kx, ky) = (0 ,0) is a singularity, closely\naround which the relative phase cannot be defined.\nTherefore, the eigenstate in Eq. (3) works beyond the\nregime around the singularity.\nWe construct a wave packet by superposing these\neigenstates,\nΨ =Z∞\n−∞dkxdkyG(k−¯k)Φ, (5)\nwith the superposition coefficient Gbeing a momentum-\ndependent localized function centering around ¯k. For a\nstraightforward illustration, we take a Gaussian distribu-\ntion as an example,\nG(k−¯k) =1\n2πp\n∆x∆ye−(kx−¯kx)2\n2∆x−(ky−¯ky)2\n2∆y.(6)\nThe Gaussian distributed superposition coefficient is cen-\ntered at ¯k= (¯kx,¯ky) with the packet widthsp\n∆x,yalong\nxandydirections. If the widths are narrow, the superpo-\nsition mainly happens around ¯k. Therefore, we analyze\nthe eigenstates around ¯k, and the relative phase becomes\nφ(k)≈φ(¯k) + (kx−¯kx)∂φ\n∂kx\f\f\f\f¯k+ (ky−¯ky)∂φ\n∂ky\f\f\f\f¯k,(7)\nwhich is linearly dependent on the momenta kx,y. This\nis true since expanding any continuous function around\na certain parameter point leads to dominant linear-\ndependence. Such linear dependence in Eq. (7) induced a\nmomentum kick, generating the relative motion between\nthe two components. After substituting the Gaussian\ndistribution in Eq. (6) and φin Eq. (7) into Eq. (5) and\nperforming integration, we get the wave packet,\nΨ =1√\n2ei¯kxx+i¯kyy\n×\ne−∆x\n2h\nx+1\n2∂φ(¯k)\n∂kxi2−∆y\n2h\ny+1\n2∂φ(¯k)\n∂kyi2+iφ(¯k)\n2\ne−∆x\n2h\nx−1\n2∂φ(¯k)\n∂kxi2−∆y\n2h\ny−1\n2∂φ(¯k)\n∂kyi2−iφ(¯k)\n2\n.(8)\nThe outstanding feature of the resultant wave packet is\nthat the two components have a relative position dis-\nplacement. The displacements along the xandydirec-\ntions are\n∂φ(k)\n∂kx\f\f\f\f¯k=¯ky\n¯k2x+¯k2y,∂φ(k)\n∂ky\f\f\f\f¯k=−¯kx\n¯k2x+¯k2y. (9)\nThe nonzero displacements give rise to a phase separation\nbetween two components. From the construction of the3\nphase-separated wave packets, we can see that the origin\nof the phase separation is the existence of the momentum-\ndependent relative phase in eigenstates and the occupa-\ntion of these eigenstates confined in a narrow momentum\nregime.\nRashba spin-orbit coupled BEC is an ideal platform to\ngenerate such a phase-separated state. The lower band\nin Eq. (2) has infinite energy minima which locate at\nthe quasimomenta satisfying k2\nx+k2\ny=λ2; therefore,\nkx=λcos(θ) and ky=λsin(θ) with θbeing an angle.\nThe interacting atoms spontaneously choose one of en-\nergy minima to condense and form a BEC [33, 35, 36].\nThis means that θis spontaneously chosen to be a value\n¯θ. In real atomic BEC experiments, traps are inevitable.\nA weak harmonic trap naturally broadens the BEC mo-\nmentum giving rise to a Gaussian distribution centered\nat (¯kx,¯ky) =λ(cos( ¯θ),sin(¯θ)). Furthermore, the broad-\nening is narrow so that Eq. (7) is satisfied. Conse-\nquently, the Rashba-coupled BEC presents as a phase\nseparated state in Eq. (8) with ∂φ(k)/∂kx|¯k= sin( ¯θ)/λ\nand∂φ(k)|/∂ky|¯k=−cos(¯θ)/λ. The position displace-\nment is inversely proportional to the spin-orbit coupling\nstrength λ, which clearly indicates Rashba spin-orbit-\ncoupling-induced phase separation. When the strength\ngoes to zero ( λ≈0), the momentum ( ¯kx,¯ky)≈(0,0)\nbecomes a singularity so that the eigenstate in Eq. (3)\nis not physical. Without the spin-orbit coupling, the\nBEC becomes the conventional one, and there is no phase\nseparation between two components. If the strength\nis enhanced gradually from zero, the position displace-\nments should continuously increase from zero to catch up\nwith the predicted value (sin( ¯θ)/λ,−cos(¯θ)/λ). When\nthe strength λis large enough, the position displace-\nments decrease towards zero again since they are in-\nversely proportional to λ. In this case, the plane-wave\nphase ( ¯kxx+¯kyy) dominates, while the relative phase\nin the eigenstates [Eq. (3)] is independent of λ, i.e.,\ntan(φ) = ¯kx/¯ky= cot( ¯θ). Consequently, the effect of\nthe relative phase is obliterated by the plane-wave phase,\nand phase separation disappears.\nAccording to the above mechanism of the phase sepa-\nration, if there is no weak trap to broaden the condensed\nmomentum, the spin-orbit-coupled BEC can not present\nthe position displacement. This is why a spatially ho-\nmogeneous BEC with spin-orbit coupling does not show\nphase separation as studied in most literature. Never-\ntheless, in order to broaden the condensed momentum\nwithout traps, we may consider spatially localized exci-\ntation states, such as bright solitons and vortices. These\nspatially self-trapped states naturally broadens the con-\ndensed momentum. Therefore, the resultant phase sep-\naration between two components in Rashba spin-orbit-\ncoupled bright solitons and quantum vortices, which have\nbeen numerically revealed in Refs. [38, 39, 41], can be\nunderstood by a generalization of our mechanism. Inter-\nestingly, the position displacement of quantum vortex is\ninversely proportional to the spin-orbit coupling strength\nas uncovered numerically in [41], can be explained unam-biguously.\nWe emphasize that the spin-orbit-coupling-induced\nphase separation only works for a two-dimensional spin-\norbit coupling. For an one-dimensional spin-orbit cou-\npling, i.e., the Raman-induced one, the single-particle\nHamiltonian is H′=p2\nx/2+λpxσz+Ωσxwith Ω being the\nRabi frequency due to Raman lasers [20, 23]. The lower\nenergy band of this system is E=k2\nx/2−p\nλ2k2x+ Ω2\nwith eigenstates being Φ = eikxx(−sin(Θ) ,cos(Θ))T.\nHere, tan(Θ) = Ω /(λkx), and Tis the transpose oper-\nator. It is noted that there is no momentum-dependent\nrelative phase in the eigenstates. Therefore, the Raman-\ninduced spin-orbit coupling, in principle, can not gener-\nate the phase separation.\nIn above, we have revealed the physical mecha-\nnism of Rashba spin-orbit-coupling-induced phase sep-\naration. We demonstrate that a weakly trapped spin-\norbit-coupled BEC satisfies requirements of the mech-\nanism. Quantum phase in trapped spin-orbit-coupled\nBECs may be phase separated states. The spin-orbit-\ncoupling-induced phase separation is a single-particle ef-\nfect. The role of nonlinearity in the BEC is to sponta-\nneously choose one energy minimum for condensation. In\nthe following, we study ground states of a trapped spin-\norbit-coupled BEC and identify features of spin-orbit-\ncoupling-induced phase separation.\nIII. RASHBA\nSPIN-ORBIT-COUPLING-INDUCED PHASE\nSEPARATION IN TRAPPED BECS\nWe consider a quasi-two-dimensional spin-1/2 BEC\nwith Rashba spin-orbit coupling. The trap frequency ωz\nalong the zdirection is assumed to be very large so that\nthe dynamics is completely frozen into the ground state of\nthez-directional harmonic trap. Such the strong trap can\nbe implemented by an optical lattice in the zdirection\nin experiments. After integrating the atomic state along\nzdirection, we are left with a quasi-two-dimensional sys-\ntem. Rashba spin-orbit coupling can be artificially im-\nplemented by an optical Raman lattice [24], generating\nthe Hamiltonian HSOC shown in Eq. (1). The spin-\norbit-coupled BEC is described by the following Gross-\nPitaevskii (GP) equation,\ni∂Ψ\n∂t= (HSOC+V+Hint) Ψ. (10)\nwith Ψ = (Ψ 1,Ψ2)Tbeing the two-component wave\nfunction. The harmonic trap in the x−yplane is\nV=1\n2ω2(x2+y2) with ωthe dimensionless trap fre-\nquency. Hintdenotes nonlinear interactions,\nHint=\u0012\ng|Ψ1|2+g12|Ψ2|20\n0 g12|Ψ1|2+g|Ψ2|2\u0013\n,(11)\nThe GP equation is dimensionless, and the units of\nlength, time, momentum and energy are chosen as lz=4\n- 808y / lz( a )\n- 202lzky/s47 /s1115( c )\n- 8 0 8- 808\nx / lz( b )\n- 2 0 2- 202\nlzkx/s47 /s1115( d )\nFIG. 1. Ground state of a trapped Rashba spin-orbit-coupled\nBEC with miscible interactions. (a) and (b) Density distri-\nbutions |Ψ1|2and|Ψ2|2in the coordinate space. (c) and (d)\nDensity distributions |Ψ1|2and|Ψ2|2in the momentum space.\nThe parameters are ω/ωz= 0.1,lzλ/ℏ= 0.2,g= 12 and\ng12= 8.\np\nℏ/(mωz), 1/ωz,ℏ/lzandℏωz, respectively. With the\nunits, the inter-and intra-component interaction coeffi-\ncients become g=Na√\n8π/lzandg12=Na12√\n8π/lz.\nHere, Nis the atom number, and aand a12are\ncorresponding s-wave scattering lengths, respectively.\nThe wave functions satisfy the normalization condition,R\ndxdy(|Ψ1|2+|Ψ2|2) = 1. In numerical calculations, ex-\nperimentally accessible parameters are used. The typical\ntrap frequency is ωz= 2π×200 Hz, leading to the unites\nof length and time lz= 0.76µmand 1 /ωz= 0.8msre-\nspectively. a∼100a0with a0being the Bohr radius,\nandN∼300, lead to g∼10. The spin-orbit coupling\nstrength can be changed by tuning parameters of Raman\nlasers in experiments [24].\nWhen g > g 12, the interactions are miscible. We\nfirst study ground states of the system in this regime\nby performing the imaginary-time evolution of the GP\nequation. The evolution is numerically implemented\nby the split-step Fourier method. The window of two-\ndimensional space is chosen as ( x, y)∈[−6π,6π] and is\ndiscretized into a 256 ×256 grid.\nA typical result is shown in Fig. 1. As expected from\nthe prediction in the previous section, the ground state\nis phase-separated. The two components are spatially\nseparated along the xdirection, as shown by Figs. 1(a)\nand 1(b). The ground state spontaneously chooses ¯θ=\n−π/2 so that the atoms condense at ( ¯kx,¯ky) = (0 ,−λ),\nwhich can be seen from the momentum-space density dis-\ntributions in Figs. 1(c) and 1(d). In this case, according\nto Eq. (9), the position displacement occurs along the x\ndirection, and the first component shifts by 1 /(2λ) on the\nright side and the second component shifts oppositely by\n1/(2λ) on the left side.\nIn the presence of interactions, it is impossible to con-\nstruct analytical wave function of ground state from the\nprocedure demonstrated in the previous section. Never-\ntheless, the single-particle wave functions in Eq. (8) and\n-101-\n2-100\n1 2 0.0580.0950\n1 2 0.0560.063x/lz(a) \nlz k'y//s295(b)Δ'x/lzl\nz /s108/ /s295(c)Component 1C\nomponent 2Δ\n'y/lzl\nz /s108/ /s295(d)FIG. 2. Rashba spin-orbit-coupling-induced phase sepa-\nration in a trapped BEC. The parameters are ω/ωz= 0.1,\ng= 12 and g12= 8. (a) The center of mass for the two com-\nponents along the xdirection as a function of the spin-orbit\ncoupling strength λ. The solid lines are from the variational\nmethod, and the circles are obtained by the imaginary-time\nevolution of the GP equation. The dashed lines are ±1/(2λ)\npredicted from the single-particle model. The red (blue) color\nrepresents the first (second) component. (b) The condensed\nmomentum ¯k′\nyas a function of λ. The red solid line and blue\ncircles are obtained by the imaginary-time evolution and the\nvariational method, respectively. The variational parameters\n∆′\nx,yare shown in (c) and (d).\nphase-separated results shown in Fig. 1 stimulate us to\nuse a trial wave function to study the phase separation\nby the variational method [37]. The trial wave function\nis assumed to be\nΨ(x, y) =(∆′\nx∆′\ny)1\n4\n√\n2πei¯k′\nyy\ne−∆′\nx\n2(x−δx)2−∆′\ny\n2y2\ne−∆′\nx\n2(x+δx)2−∆′\ny\n2y2\n.(12)\nHere, we have assumed that the atoms spontaneously\ncondenses at (0 ,¯k′\ny) in momentum space and therefore\nthe phase separation only happens along the xdirection\nwith the relative position displacement 2 δx. 1/p∆′x,y\ncharacterize the widths of the wave packet along the x, y\ndirections. The unknown parameters ¯k′\ny, δx,∆′\nx,yare to\nbe determined by minimizing the energy functional,\nE=Z\ndxdyΨ∗(HSOC+V)Ψ\n+Z\ndxdyhg\n2(|Ψ1|4+|Ψ2|4) +g12|Ψ1|2|Ψ2|2i\n,\n(13)\nSubstituting the trial wave function into the energy func-5\n4\n6\n8\n1\n0\n-\n1\n.\n0\n-\n0\n.\n8\n0\n.\n8\n1\n.\n0\n0\n.\n1\n0\n.\n2\n0\n.\n3\n-\n0\n.\n7\n0\n.\n0\n0\n.\n7\nx\n/\nl\nz\ng\n1\n2\n(\na\n)\nx\n/\nl\nz\n/s119\n \n/\n/s119\nz\n(\nb\n)\nFIG. 3. The center of mass of two components for a non-\ndominant lzλ/ℏ= 0.5. The solid lines are obtained by the\nvariational method and the circles are from the imaginary-\ntime evolution of the GP equation. The red and blue colors\nrepresent the first and second components, respectively. (a)\nThe center-of-mass as a function of the inter-component in-\nteraction coefficient g12.ω/ωz= 0.1 and g= 12. (b) The\ncenter-of-mass as a function of the trap frequency ω.g= 12\nandg12= 8.\ntional Eleads to\nE=¯k′2\ny\n2+∆′\nx+ ∆′\ny\n4\u0012\n1 +ω2\n∆′x∆′y\u0013\n+1\n2ω2δ2\nx\n+λ(∆′\nxδx−¯k′\ny)e−∆′\nxδ2\nx+p∆′x∆′y\n8π\u0010\ng+g12e−2∆′\nxδ2\nx\u0011\n.\n(14)\nBy minimizing the energy functional with respect to the\nunknown parameters, ∂E/∂X = 0 ( X=¯k′\ny, δx,∆′\nx,y), we\nobtain all information of the trial wave function. The\nphase separation can be characterized by the center of\nmass of each component,\n¯r1,2=Z\nr|Ψ1,2(r)|2dr, (15)\nwithr= (x, y). In Fig. 2(a), the solid lines show\n¯x1,2=±δxcalculated from the variational method, while\nthe results obtained by the imaginary-time evolution of\nthe GP equation are demonstrated by the circles. We\nfind that the results from the two calculation methods\nagree very well. Without spin-orbit coupling ( λ= 0),\nthe conventional BEC has ¯ x1,2= 0 and condensates at\n¯k′\ny= 0, as shown in Fig. 2(b). With the growth of λ,\n¯k′\nyalways increases linearly [see Fig. 2(b)]. The displace-\nment ¯ xfirst increases drastically to a maximum value and\nthen declines to the predicted ±1/(2λ) obtained by the\nsingle-particle model [see the dashed lines in Fig. 2(a)].\nThe dependence of the displacement on λexactly follows\nthe expectation in the previous section. In the dramatic\nincrease regime for ¯ x, the variational parameters ∆′\nx,y\nalso change dramatically [see Figs. 2(c) and 2(d)].\nRashba spin-orbit coupling introduces an intrinsic\nforce,\nF=dp\ndt=−\u0002\n[r, HSOC], HSOC\u0003\n= 2λ2(p×ez)σz, (16)withezbeing the unit vector along the zdirection and\npthe atom momentum. The force originates from spin-\norbit-coupling-induced anomalous velocity [43–45]. Con-\nsidering the ground states shown in Fig. 2, the force op-\nerator in momentum space becomes Fx= 2λ2¯k′\nyσzand\nFy= 0. The two components feel opposite force Fxalong\nthexdirection. Ground states must compensate the in-\ntrinsic force to reach equilibrium. It can be implemented\nby displacing two component opposite to the force. Since\n¯k′\ny<0 in the case shown in Fig. 2, the first component is\ndisplaced towards to the right side and the second to the\nleft side. The force concept has been used in Refs. [36, 37]\nto explain the phase separation. Since the force is pro-\nportional to λ2, it seems that a large displacement would\nbe induced for a large λ. However, as shown in Fig. 2(a),\nthe dependence of the displacement on λdoes not fol-\nlow the force. We can see that the intrinsic force cannot\nexplain the phase separation in the large λregime.\nFigure 2(a) shows that the separation follows the\nsingle-particle prediction ±1/(2λ) when λdominates.\nWhen λis weak, the displacement also depends on other\nparameters, such as nonlinear coefficients and the har-\nmonic trap. In Fig. 3(a), we plot the displacement ¯ xas\na function of the inter-component interaction coefficient\ng12for a non-dominant λ. The displacement slightly\nrises with an increasing g12, and it reaches the maxi-\nmum when g12=g. If g12> g, the interactions be-\ncome immiscible, leading to ground states different from\nthe trial wave function in Eq. (12). The dependence\nof the displacement on the trap frequency is shown in\nFig. 3(b). We find that the displacement decreases as\nthe trap frequency increases. This is because the dis-\nplacement requires more kinetic energy in a tight trap.\nIt is notice that there is a slight mismatching between\nthe results from the variational method (the solid line)\nand the imaginary-time evolution (the circles) in Fig. 3.\nThe origin of such the mismatching is that the Gaus-\nsian profile in the trial wave function in Eq. (12) cannot\nexactly describe the imaginary-time-evolution-generated\nwave function as shown in Fig. 1.\nIV. THE ANISOTROPIC\nSPIN-ORBIT-COUPLING-INDUCED PHASE\nSEPARATION IN TRAPPED BECS\nRashba-spin-orbit-coupling-induced phase separation\nhas been analyzed in the previous section. In the two-\ndimensional spin-orbit-coupled BEC experiment [24], the\nspin-orbit coupling strengths are tunable, which leads\nto an anisotropic coupling. It has been revealed that\nthe anisotropic spin-orbit coupling has a great impact\non ground states of a spatially homogeneous BEC [46].\nIn this section, we study anisotropic-spin-orbit-coupling-\ninduced phase separation. The single-particle Hamilto-\nnian of the anisotropic spin-orbit coupling is\nH′\nSOC=p2\nx+p2\ny\n2+λ1pxσy−λ2pyσx, (17)6\nFIG. 4. The anisotropic spin-orbit-coupling-induced phase separation in a trapped BEC. The parameters are ω/ωz= 0.1,\ng= 12, and g12= 8. (a1) The lower band of H′\nSOCin Eq. (17) with lzλ1/ℏ= 0.3 and lzλ2/ℏ= 0.6. (a2) and (a4) show\ncorresponding ground-state density distributions of the first component |Ψ1|2in coordinate and momentum spaces, respectively.\n(a3) and (a5) are for the second component |Ψ2|2. (b1)-(b5) are same as (a1)-(a5) but with lzλ1/ℏ= 0.6 and lzλ2/ℏ= 0.3. In\n(a2), (a3), (b2), and (b3), white stars represent the center of wave packets predicted by the single-particle model.\nwith the anisotropic strengths λ1̸=λ2. The lower band\nofH′\nSOCis\nE=k2\nx+k2\ny\n2−q\nλ2\n1k2x+λ2\n2k2y. (18)\nwith the associated eigenstates being the same as Eq. (3)\nbut having the different relative phase which can be writ-\nten as\ntan(φ) =λ1kx\nλ2ky. (19)\nAccording to the mechanism of the spin-orbit-coupling-\ninduced phase separation, the anisotropic coupling can\ngenerate position displacements related to the derivatives\nof the relative phase. The displacements along the xand\nydirections are\n∂φ(k)\n∂kx\f\f\f\f¯k=λ1λ2¯ky\nλ2\n1¯k2x+λ2\n2¯k2y,\n∂φ(k)\n∂ky\f\f\f\f¯k=−λ1λ2¯kx\nλ2\n1¯k2x+λ2\n2¯k2y. (20)Here, ¯k= (¯kx,¯ky) is the momentum at which the atoms\ncondense. The lowest energy minima of the lower band\ndepend on the anisotropy. When λ1< λ 2, the two\nminima locate at ( ¯kx,¯ky) = (0 ,±λ2) [see Fig. 4(a1)].\nThey locate at ( ¯kx,¯ky) = (±λ1,0) when λ1> λ 2[see\nFig. 4(b1)]. With the miscible interactions, the BEC\nspontaneously chooses one of these two minima to con-\ndense. The ground state that spontaneously condenses\nat (¯kx,¯ky) = (0 ,−λ2) for λ1< λ 2is demonstrated\nin Figs. 4(a2)-(a5). We obtain ground states by the\nimaginary-time evolution of the GP equation with H′\nSOC.\nFrom the single-particle prediction in Eq. (20), the phase\nseparation of this ground state happens only along the\nxdirection, and the center-of-mass of the first compo-\nnent is λ1/(2λ2\n2) and that of the second component is\n−λ1/(2λ2\n2) [see the white stars in Figs. 4(a2) and 4(a3)].\nDensity distributions shown in Figs. 4(a2) and 4(a3)\nclearly indicate the phase separation following the pre-\ndictions. The ground state that spontaneously condenses\nat (¯kx,¯ky) = ( −λ1,0) for λ1> λ 2is demonstrated\nin Figs. 4(b2)-(b5). The single-particle mechanism in7\n0.00 .51 .01 .52 .0-0.40.00.4l\nz /s1082//s295x/lz-\n0.40.00.4y\n/lz\nFIG. 5. Anisotropic-spin-orbit-coupling-induced phase sepa-\nration as a function of λ2with a fixed lzλ1/ℏ= 1. Circles are\nfor the first component and crosses are for the second compo-\nnent. The blue (red) color represents separation along the x\n(y) direction. Other parameters are ω/ωz= 0.1,g= 12 and\ng12= 8.\nEq. (20) predicts that for this ground state the separa-\ntion happens along the ydirection and the center-of-mass\nare∓λ2/(2λ2\n1) for two components [see the white stars in\nFigs. 4(b2) and 4(b3)]. The results from the imaginary-\ntime evolution shown in Figs. 4(b2) and 4(b3) match with\nthe single-particle predictions.\nThese analyses have shown that the center of mass\nof each component strongly depends on the ratio of\nthe spin-orbit coupling strengths. To reveal the depen-\ndence of phase separation on λ2/λ1, we calculate ground\nstates with a fixed λ1and a changeable λ2by using the\nimaginary-time evolution. The results are summarized in\nFig. 5, where the circles (crosses) represent the center of\nmass for the first (second) component. For λ2< λ1= 1,\nthe phase separation occurs along the ydirection and\n|¯y|increases with the increase of λ2[see red circles and\ncrosses in Fig. 5], while ¯ xis zero [see blue circles and\ncrosses in Fig. 5]. When λ2= 0, the spin-orbit coupling\nbecomes one-dimensional, there is no phase separation\ndue to the absence of the relative phase. The results\nchange for λ2> λ 1= 1 and the phase separation along\nthexdirection is observed. In this case, the separation\ndecreases with λ2increasing. For a very large λ2, the\nseparation disappears since the spin-orbit coupling effec-\ntively turns to be one-dimensional. The results in Fig. 5\ndemonstrate that the maximum separation happens for\nλ1=λ2which is Rashba spin-orbit coupling. This is also\nexpected from the single-particle prediction in Eq. (20).\nV. ADIABATIC SPLITTING DYNAMICS\nWe have shown that ground states of a trapped BEC\nwith two-dimensional spin-orbit coupling and miscible in-teractions are phase-separated. As an important appli-\ncation, we study adiabatic dynamics of the phase separa-\ntion. As pointed out by previous works, a linear coupling\nbetween two component favors miscibility regardless of\ninteractions [47, 48]. Therefore, a miscible-to-immiscible\ntransition may occur by decreasing the coupling. The\nadiabatic dynamics is stimulated by slowly switching off\nthe linear coupling. Theoretically, the process is de-\nscribed by the time-dependent GP equation,\ni∂Ψ\n∂t= [HSOC+ Ω(t)σx+V+Hint] Ψ. (21)\nHere, Ω( t)σxrepresents the linear coupling between the\ntwo components, and can be experimentally achieved by\nusing a radio-frequency coupling [6]. The time-dependent\nRabi frequency is\nΩ(t) = Ω 0(1−t/τq), (22)\nwith Ω 0being the initial value of the linear coupling and\nτqis the quench duration. At t= 0, the presence of Ω 0\ngreatly suppresses the ground-state phase separation. We\nobtain ground state by the imaginary-time evolution of\nEq. (21) with Ω( t) = Ω 0. A typical ground state is shown\nin insets (a) and (b) of Fig. 6, and the separation be-\ntween two components is not obvious. Using this ground\nstate as initial state, we evolve the time-dependent GP\nequation. The center-of-mass ¯ xfor two components is\nrecorded during the time evolution in Fig. 6. By de-\ncreasing the linear coupling adiabatically, the separation\nbetween two component gradually increases. When it\nis completely switched off, i.e., t=τq, the separation\n04 08 01 20-0.6-0.30.00.30.6x/lz/s119\nz t\n-808-808x\n/lzy/lz\n-808-808x\n/lzy/lz\n-808-808x\n/lzy/lz\n-808-808x\n/lzy/lz(\na\n)\n(\nb\n)\n(\nc\n)\n(\nd\n)\nFIG. 6. Adiabatic splitting dynamics of a trapped BEC with\nRashba spin-orbit coupling by slowly switching off the linear\ncoupling. The parameters are ω/ωz= 0.1,g= 12, g12= 8,\nΩ0/ωz= 3, and ωzτq= 150. The red (blue) dots represent\nthe center-of-mass of the first (second) component. Insets\n(a,b) [(c,d)] are density distributions of the first and second\ncomponents at t= 0 [t=τq], respectively.8\nFIG. 7. Immiscible-interaction-induced phase-separated ground states in a trapped BEC with Rashba spin-orbit coupling.\nThe parameters are g= 4, g12= 8 and ω/ωz= 0.1. (a1)-(a4) When lzλ/ℏ= 0.5 the ground state is a half-quantum vortex\nstate. (a1) [a(3)] and (a2) [(a4)] are the coordinate (momentum) space density distributions of the first and second components\nrespectively. (b1)-(b4) When lzλ/ℏ= 1.5 the ground state is a stripe state. (b1) [b(3)] and (b2) [(b4)] are the coordinate\n[momentum] space density distributions of the first and second components respectively.\nis maximized [see corresponding density distributions in\ninsets (c) and (d)]. The two components can realize a\ndynamically spatial splitting, which move along opposite\ndirections. Such adiabatic splitting dynamics are remi-\nniscent of a kind of “atomic spin Hall effect” [49].\nVI. IMMISCIBLE INTERACTIONS INDUCED\nPHASE SEPARATION\nIn all above, the interactions are miscible ( g > g 12),\nwhich support atoms to condense at a particular momen-\ntum state. On the other hand, immiscible interactions\n(g < g 12) prefer a spatial separation between two com-\nponents in order to minimize the inter-component inter-\nactions proportional to g12. In the presence of spin-orbit\ncoupling, the immiscible-interaction-induced phase sepa-\nration presents interesting features [35, 36, 50]. In Fig. 7,\nwe show two different kinds of immiscible-interaction-\ninduced phase separated ground states with different val-\nues of spin-orbit coupling strength λ. For λ= 0.5, the\nground state obtained by the imaginary-time evolution\nis a half-quantum vortex state which has been first re-\nvealed in Refs. [35, 50]. The first component distribution\nhas a Gaussian shape [see Figs. 7(a1) and 7(a3)], and\nthe second component is a vortex with a winding num-\nberw= 1 [see Figs. 7(a2) and 7(a4)]. The first compo-\nnent is filled in the density dip of the second one, form-\ning a spatial separation along the radial direction. For\nλ= 1.5, the ground state becomes a stripe state which\nhas been first revealed in Refs. [33, 34]. The ground state\ncondenses simultaneously at two different momenta [see\nFigs. 7(b3) and 7(b4)]. Such momentum occupation gen-\nerates spatially periodic modulations in density distribu-\ntions [Figs. 7(b1) and 7(b2)]. Meanwhile, stripes of the\ntwo components are spatially separated.We emphasize that phase separations induced by spin-\norbit coupling and immiscible interaction have different\nphysical origins. The spin-orbit-coupling-induced phase\nseparation only works for a two-dimensional spin-orbit\ncoupling. However, phase separations have been also\nstudied for a BEC with a one-dimensional spin-orbit cou-\npling, the mechanism of which is different. In the pi-\noneered spin-orbit-coupled experiment, experimentalists\nobserved a spatial separation between two dressed states\nwith a Raman-induced spin-orbit coupling [20]. The\nspin-orbit coupling generates two energy minima, whose\noccupations can be considered as two dressed states. In\nthe dressed state space, atomic interactions turn to be\nimmiscible between two dressed states in the presence\nof the Rabi frequency. The phase separation happens\nin the dressed state space due to immiscibility. In ad-\ndition, Ref. [47] reveals the existence of phase separa-\ntion in a spin-1 BEC with the Raman-induced spin-orbit\ncoupling. The single-particle Hamiltonian of the system\nisH= (px+λ′Fz)2/2 + Ω′Fx+ϵF2\nz. Here, Fx,y,z are\nthe spin-1 Pauli matrices, λ′is the spin-orbit coupling\nstrength, Ω′is the Rabi frequency, and ϵis the quadratic\nZeeman shift. The spinor interactions include density-\ndensity part with the coefficient c0and spin-spin part\nwith the coefficient c2. In particular, a very negative\nquadratic Zeeman shift ϵ=−λ2/2 was considered. With\nsuch a large negative ϵ, the occupation in the second\ncomponent can be eliminated. The spinor only occu-\npies the first and third components. Interestingly, the\nspinor interactions between the first and third compo-\nnents are immiscible for a negative spin-spin interaction\n(c2<0). Different phase-separated states between the\nfirst and third component are due to immiscible interac-\ntions [47] .9\nVII. CONCLUSION\nIn summary, we have revealed the physical mechanism\nof spin-orbit-coupling-induced phase separation. The\nmechanism, which is very different from the conventional\nimmiscible-interaction-induced separation, is a complete\nsingle-particle effect of spin-orbit coupling. We have ana-\nlyzed separation features in a trapped BEC with Rashba\nspin-orbit coupling and miscible interactions and studied\nthe effects of the anisotropy of spin-orbit coupling on theseparation. All features can be explained by the single-\nparticle mechanism. As an interesting application of the\nphase separation, we propose an adiabatic dynamics that\ncan dynamically split two components spatially.\nACKNOWLEDGMENTS\nThis work was supported by National Natural Sci-\nence Foundation of China with Grants No.12374247 and\n11974235. H.L. acknowledges support from Okinawa In-\nstitute of Science and Technology Graduate University.\n[1] J. Sinova, S. O. Valenzuela, J. Wunderlich, C. H. Back,\nand T. Jungwirth, Spin Hall effects, Rev. Mod. Phys. 87,\n1213 (2015).\n[2] Y. Shin, M. W. Zwierlein, C. H. Schunck, A. Schirotzek,\nand W. Ketterle, Observation of Phase Separation in a\nStrongly Interacting Imbalanced Fermi Gas, Phys. Rev.\nLett.97, 030401 (2006).\n[3] S. B. Papp, J. M. Pino, and C. E. Wieman, Tunable\nMiscibility in a Dual-Species Bose-Einstein Condensate,\nPhys. Rev. Lett. 101, 040402 (2008).\n[4] G. Thalhammer, G. Barontini, L. De Sarlo, J. Catani,\nF. Minardi, and M. Inguscio, Double Species Bose-\nEinstein Condensate with Tunable Interspecies Interac-\ntions, Phys. Rev. Lett. 100, 210402 (2008).\n[5] S. Tojo, Y. Taguchi, Y. Masuyama, T. Hayashi, H. Saito,\nand T. Hirano, Controlling phase separation of binary\nBose-Einstein condensates via mixed-spin-channel Fesh-\nbach resonance, Phys. Rev. A 82, 033609 (2010).\n[6] E. Nicklas, H. Strobel, T. Zibold, C. Gross, B. A. Mal-\nomed, P. G. Kevrekidis, and M. K. Oberthaler, Rabi\nFlopping Induces Spatial Demixing Dynamics, Phys.\nRev. Lett. 107, 193001 (2011).\n[7] K. Jim´ enez-Garc´ ıa, A. Invernizzi, B. Evrard, C. Frapolli,\nJ. Dalibard, and F. Gerbier, Spontaneous formation and\nrelaxation of spin domains in antiferromagnetic spin-1\ncondensates, Nat. Commun. 10, 1422 (2019).\n[8] L. He, P. Gao, and Z.-Q. Yu, Normal-Superfluid Phase\nSeparation in Spin-Half Bosons at Finite Temperature,\nPhys. Rev. Lett. 125, 055301 (2020).\n[9] M. Trippenbach, K. G´ oral, K. Rzazewski, B. Malomed,\nand Y. B. Band, Structure of binary Bose-Einstein con-\ndensates, J. Phys. B 33, 4017 (2000).\n[10] D. S. Petrov, Quantum Mechanical Stabilization of a Col-\nlapsing Bose-Bose Mixture, Phys. Rev. Lett. 115, 155302\n(2015).\n[11] U. Shrestha, J. Javanainen, and J. Ruostekoski, Pulsat-\ning and Persistent Vector Solitons in a Bose-Einstein\nCondensate in a Lattice upon Phase Separation Insta-\nbility, Phys. Rev. Lett. 103, 190401 (2009).\n[12] K. J. H. Law, P. G. Kevrekidis, and L. S. Tuckerman, Sta-\nble Vortex–Bright–Soliton structures in Two-Component\nBose-Einstein Condensates, Phys. Rev. Lett. 105, 160405\n(2010).\n[13] K. L. Lee, N. B. Jørgensen, I.-K. Liu, L. Wacker, J. J.\nArlt, and N. P. Proukakis, Phase separation and dynam-\nics of two-component Bose-Einstein condensates, Phys.Rev. A 94, 013602 (2016).\n[14] A. Roy, M. Ota, A. Recati, and F. Dalfovo, Finite-\ntemperature spin dynamics of a two-dimensional Bose-\nBose atomic mixture, Phys. Rev. Res. 3, 013161 (2021).\n[15] Y. Eto, M. Takahashi, M. Kunimi, H. Saito, and\nT. Hirano, Nonequilibrium dynamics induced by misci-\nble–immiscible transition in binary Bose–Einstein con-\ndensates, New J. Phys. 18, 073029 (2016).\n[16] N. R. Bernier, E. G. Dalla Torre, and E. Demler, Un-\nstable Avoided Crossing in Coupled Spinor Condensates,\nPhys. Rev. Lett. 113, 065303 (2014).\n[17] W. E. Shirley, B. M. Anderson, C. W. Clark, and R. M.\nWilson, Half-Quantum Vortex Molecules in a Binary\nDipolar Bose Gas, Phys. Rev. Lett. 113, 165301 (2014).\n[18] S. I. Mistakidis, G. C. Katsimiga, P. G. Kevrekidis, and\nP. Schmelcher, Correlation effects in the quench-induced\nphase separation dynamics of a two species ultracold\nquantum gas, New J. Phys. 20, 043052 (2018).\n[19] R. S. Lous, I. Fritsche, M. Jag, F. Lehmann, E. Kir-\nilov, B. Huang, and R. Grimm, Probing the Interface of\na Phase-Separated State in a Repulsive Bose-Fermi Mix-\nture, Phys. Rev. Lett. 120, 243403 (2018).\n[20] Y.-J. Lin, K. Jim´ enez-Garc´ ıa, and I. B. Spielman, Spin–\norbit-coupled Bose–Einstein condensates, Nature (Lon-\ndon)471, 83 (2011).\n[21] N. Goldman, G. Juzeli¯ unas, P. ¨Ohberg, and I. B. Spiel-\nman, Light-induced gauge fields for ultracold atoms, Rep.\nProg. Phys. 77, 126401 (2014).\n[22] H. Zhai, Degenerate quantum gases with spin–orbit cou-\npling: a review, Rep. Prog. Phys. 78, 026001 (2015).\n[23] Y. Zhang, M. E. Mossman, T. Busch, P. Engels, and\nC. Zhang, Properties of spin–orbit-coupled Bose–Einstein\ncondensates, Front. Phys. 11, 118103 (2016).\n[24] Z. Wu, L. Zhang, W. Sun, X.-T. Xu, B.-Z. Wang, S.-C. Ji,\nY. Deng, S. Chen, X.-J. Liu, and J.-W. Pan, Realization\nof two-dimensional spin-orbit coupling for Bose-Einstein\ncondensates, Science 354, 83 (2016).\n[25] A. Vald´ es-Curiel, D. Trypogeorgos, Q.-Y. Liang, R. P.\nAnderson, and I. B. Spielman, Topological features with-\nout a lattice in Rashba spin-orbit coupled atoms, Nat.\nCommun. 12, 593 (2021).\n[26] Y. Li, L. P. Pitaevskii, and S. Stringari, Quantum Tri-\ncriticality and Phase Transitions in Spin-Orbit Coupled\nBose-Einstein Condensates, Phys. Rev. Lett. 108, 225301\n(2012).10\n[27] A. Manchon, H. C. Koo, J. Nitta, S. M. Frolov, and R. A.\nDuine, New perspectives for Rashba spin–orbit coupling,\nNat. Mater. 14, 871 (2015).\n[28] J.-R. Li, J. Lee, W. Huang, S. Burchesky, B. Shteynas,\nF. C ¸. Top, A. O. Jamison, and W. Ketterle, A stripe\nphase with supersolid properties in spin–orbit-coupled\nBose–Einstein condensates, Nature (London) 543, 91\n(2017).\n[29] M. A. Khamehchi, K. Hossain, M. E. Mossman,\nY. Zhang, T. Busch, M. M. Forbes, and P. Engels,\nNegative-Mass Hydrodynamics in a Spin-Orbit–Coupled\nBose-Einstein Condensate, Phys. Rev. Lett. 118, 155301\n(2017).\n[30] Y. V. Kartashov and D. A. Zezyulin, Stable Multiring\nand Rotating Solitons in Two-Dimensional Spin-Orbit-\nCoupled Bose-Einstein Condensates with a Radially Pe-\nriodic Potential, Phys. Rev. Lett. 122, 123201 (2019).\n[31] M. Hasan, C. S. Madasu, K. D. Rathod, C. C. Kwong,\nC. Miniatura, F. Chevy, and D. Wilkowski, Wave Packet\nDynamics in Synthetic Non–Abelian Gauge Fields, Phys.\nRev. Lett. 129, 130402 (2022).\n[32] A. Fr¨ olian, C. S. Chisholm, E. Neri, C. R. Cabrera,\nR. Ramos, A. Celi, and L. Tarruell, Realizing a 1D topo-\nlogical gauge theory in an optically dressed BEC, Nature\n(London) 608, 293 (2022).\n[33] C. Wang, C. Gao, C.-M. Jian, and H. Zhai, Spin-Orbit\nCoupled Spinor Bose-Einstein Condensates, Phys. Rev.\nLett.105, 160403 (2010).\n[34] T.-L. Ho and S. Zhang, Bose-Einstein Condensates with\nSpin-Orbit interaction, Phys. Rev. Lett. 107, 150403\n(2011).\n[35] H. Hu, B. Ramachandhran, H. Pu, and X.-J. Liu, Spin-\nOrbit Coupled Weakly Interacting Bose-Einstein Con-\ndensates in Harmonic Traps, Phys. Rev. Lett. 108,\n010402 (2012).\n[36] Y. Zhang, L. Mao, and C. Zhang, Mean-Field Dynamics\nof Spin-Orbit Coupled Bose-Einstein Condensates, Phys.\nRev. Lett. 108, 035302 (2012).\n[37] S.-W. Song, Y.-C. Zhang, L. Wen, and H. Wang,\nSpin–orbit coupling induced displacement and hidden\nspin textures in spin-1 Bose–Einstein condensates, J.\nPhys. B 46, 145304 (2013).\n[38] H. Sakaguchi, B. Li, and B. A. Malomed, Creation of\ntwo-dimensional composite solitons in spin-orbit-coupled\nself-attractive Bose-Einstein condensates in free space,Phys. Rev. E 89, 032920 (2014).\n[39] Y. Xu, Y. Zhang, and C. Zhang, Bright solitons\nin a two-dimensional spin-orbit-coupled dipolar Bose-\nEinstein condensate, Phys. Rev. A 92, 013633 (2015).\n[40] Y.-J. Wang, L. Wen, G.-P. Chen, S.-G. Zhang, and X.-\nF. Zhang, Formation, stability, and dynamics of vec-\ntor bright solitons in a trapless Bose–Einstein conden-\nsate with spin–orbit coupling, New J. Phys. 22, 033006\n(2020).\n[41] M. Kato, X.-F. Zhang, and H. Saito, Vortex pairs in a\nspin-orbit-coupled Bose-Einstein condensate, Phys. Rev.\nA95, 043605 (2017).\n[42] R. Ravisankar, H. Fabrelli, A. Gammal, P. Muruganan-\ndam, and P. K. Mishra, Effect of Rashba spin-orbit and\nRabi couplings on the excitation spectrum of binary\nBose-Einstein condensates, Phys. Rev. A 104, 053315\n(2021).\n[43] S. Mardonov, M. Modugno, and E. Y. Sherman, Dynam-\nics of spin-orbit coupled bose-einstein condensates in a\nrandom potential, Phys. Rev. Lett. 115, 180402 (2015).\n[44] S. Mardonov, E. Y. Sherman, J. G. Muga, H.-W. Wang,\nY. Ban, and X. Chen, Collapse of spin-orbit-coupled\nbose-einstein condensates, Phys. Rev. A 91, 043604\n(2015).\n[45] S. Mardonov, V. V. Konotop, B. A. Malomed, M. Mod-\nugno, and E. Y. Sherman, Spin-orbit-coupled soliton in\na random potential, Phys. Rev. A 98, 023604 (2018).\n[46] T. Ozawa and G. Baym, Ground-state phases of ultra-\ncold bosons with Rashba-Dresselhaus spin-orbit coupling,\nPhys. Rev. A 85, 013612 (2012).\n[47] S. Gautam and S. K. Adhikari, Phase separation in a\nspin-orbit-coupled Bose-Einstein condensate, Phys. Rev.\nA90, 043619 (2014).\n[48] I. M. Merhasin, B. A. Malomed, and R. Driben, Transi-\ntion to miscibility in a binary Bose–Einstein condensate\ninduced by linear coupling, J. Phys. B 38, 877 (2005).\n[49] M. C. Beeler, R. A. Williams, K. Jim´ enez-Garc´ ıa, L. J.\nLeBlanc, A. R. Perry, and I. B. Spielman, The spin\nHall effect in a quantum gas, Nature (London) 498, 201\n(2013).\n[50] B. Ramachandhran, B. Opanchuk, X.-J. Liu, H. Pu, P. D.\nDrummond, and H. Hu, Half-quantum vortex state in a\nspin-orbit-coupled Bose-Einstein condensate, Phys. Rev.\nA85, 023606 (2012)."    },    {        "title": "1601.06935v2.Double_Quantum_Spin_Vortices_in_SU_3__Spin_Orbit_Coupled_Bose_Gases.pdf",        "content": "arXiv:1601.06935v2  [cond-mat.quant-gas]  24 Sep 2016Double-quantum spin vortices in SU(3) spin-orbit coupled B ose gases\nWei Han,1,2Xiao-Fei Zhang,1Shu-Wei Song,3Hiroki Saito,4Wei Zhang∗,5Wu-Ming Liu†,2and Shou-Gang Zhang‡1\n1Key Laboratory of Time and Frequency Primary Standards,\nNational Time Service Center, Chinese Academy of Sciences, Xi’an 710600, China\n2Beijing National Laboratory for Condensed Matter Physics,\nInstitute of Physics, Chinese Academy of Sciences, Beijing 100190, China\n3State Key Laboratory Breeding Base of Dielectrics Engineer ing,\nHarbin University of Science and Technology, Harbin 150080 , China\n4Department of Engineering Science, University of Electro- Communications, Tokyo 182-8585, Japan\n5Department of Physics, Renmin University of China, Beijing 100872, China\nWe show that double-quantum spin vortices, which are charac terized by doubly quantized circu-\nlating spin currents and unmagnetized filled cores, can exis t in the ground states of SU(3) spin-orbit\ncoupled Bose gases. It is found that the SU(3) spin-orbit cou pling and spin-exchange interaction\nplay important roles in determining the ground-state phase diagram. In the case of effective fer-\nromagnetic spin interaction, the SU(3) spin-orbit couplin g induces a three-fold degeneracy to the\nmagnetized ground state, while in the antiferromagnetic sp in interaction case, the SU(3) spin-orbit\ncoupling breaks the ordinary phase rule of spinor Bose gases , and allows the spontaneous emergence\nof double-quantum spin vortices. This exotic topological d efect is in stark contrast to the singly\nquantized spin vortices observed in existing experiments, and can be readily observed by the current\nmagnetization-sensitive phase-contrast imaging techniq ue.\nPACS numbers: 03.75.Lm, 03.75.Mn, 67.85.Bc, 67.85.Fg\nI. INTRODUCTION\nThe recent experimental realization of synthetic spin-\norbit (SO) coupling in ultracold quantum gases [1–10] is\nconsidered as an important breakthrough, as it provides\nnew possibilities for ultracold quantum gases to be used\nas quantum simulation platforms, and paves a new route\ntowards exploring novel states of matter and quantum\nphenomena [11–19]. It has been found that the SO cou-\npling can not only stabilize various topological defects,\nsuch as half-quantum vortex, skyrmion, composite soli-\nton and chiral domain wall, contributing to the design\nand exploration of new functional materials [20–23], but\nalsoleadtoentirelynew quantum phases, suchasmagne-\ntized phase and stripe phase [24–26], providing support\nfor the study of novel quantum dynamical phase transi-\ntions [27, 28] and exotic supersolid phases [29–31].\nAll the intriguing features mentioned above are based\non the characteristics that the SO coupling (either of the\nNIST[1], Rashba[24]orWeyl[32]types)makestheinter-\nnal states coupled to their momenta via the SU(2) Pauli\nmatrices. However, if the (pseudo)spin degree of freedom\ninvolves more than two states, the SU(2) spin matrices\ncannot describe completely all the couplings among the\ninternal states. For example, a direct transition between\nthe states |1∝angbracketrightand|−1∝angbracketrightis missing in a three-component\nsystem [24, 33]. From this sense, an SU(3) SO coupling\nwith the spin operator spanned by the Gell-Mann matri-\n∗wzhangl@ruc.edu.cn\n†wliu@iphy.ac.cn\n‡szhang@ntsc.ac.cnces is more effective in describing the internal couplings\namong three-component atoms [33, 34]. The SU(3) SO\ncoupled system has no analogue in ordinary condensed\nmatter systems, hence may lead to new quantum phases\nand topological defects.\nIn this article, we show that a new type of topological\ndefects, double-quantum spin vortices, can exist in the\nground states of SU(3) SO coupled Bose-Einstein con-\ndensates (BECs). It is found that the SU(3) SO coupling\nleads to two distinct ground-state phases, a magnetized\nphase or a lattice phase, depending on the spin-exchange\ninteraction being ferromagnetic or antiferromagnetic. In\nthe magnetized phase, the SU(3) SO coupling leads to a\ngroundstatewiththree-folddegeneracy,in starkcontrast\nto the SU(2) case where the degeneracy is two, thus may\noffer new insights into quantum dynamical phase transi-\ntions [27]. In the lattice phase, the SU(3) SO coupling\nbreaks the ordinary phase requirement 2w 0= w1+w−1\nfor ordinary spinor BECs, where w iis the winding num-\nber of thei-th spin component [35–37], and induces three\ntypes of exotic vortices with cores filled by different mag-\nnetizations. The interlaced arrangementof these vortices\nleads to the spontaneous formation of multiply quantized\nspin vortices with winding number 2. This new type of\ntopological defects can be observed in experiments us-\ning magnetization-sensitive phase-contrast imaging tech-\nnique.\nII. SU(3) SPIN-ORBIT COUPLING\nWe consider the F= 1 spinor BECs with SU(3)\nSO coupling. Using the mean-field approximation, the\nHamiltonian can be written in the Gross-Pitaevskii form2\nas\nH=/integraldisplay\ndr/bracketleftbigg\nΨ†/parenleftbigg\n−/planckover2pi12∇2\n2m+Vso/parenrightbigg\nΨ+c0\n2n2+c2\n2|F|2/bracketrightbigg\n,(1)\nwhere the order parameter Ψ= [Ψ1(r),Ψ0(r),Ψ−1(r)]⊤\nis normalized with the total particle num-\nberN=/integraltextdrΨ†Ψ. The particle density is\nn=/summationtext\nm=1,0,−1Ψ∗\nm(r)Ψm(r), and the spin density\nvectorF= (Fx,Fy,Fz) is defined by Fν(r) =Ψ†fνΨ\nwithf= (fx,fy,fz) being the vector of the spin-1 ma-\ntrices given in the irreducible representation [35, 38–40].\nTheSOcouplingtermischosenas Vso=κλ·p, whereκis\nthe spin-orbit coupling strength, p= (px,py) represents\n2D momentum, and λ= (λx,λy) is expressed in terms\nofλx=λ(1)+λ(4)+λ(6)andλy=λ(2)−λ(5)+λ(7),\nwithλ(i)(i= 1,...8) being the Gell-Mann matrices, i.e.,\nthe generators of the SU(3) group [41]. Note that the\nSU(3) SO coupling term in the Hamiltonian involves all\nthe pairwise couplings between the three states. This is\ndistinct from the previously discussed SU(2) SO coupling\nin spinor BECs, where the states Ψ 1(r) and Ψ −1(r)\nare coupled indirectly [24, 42, 43]. The parameters c0\nandc2describe the strengths of density-density and\nspin-exchange interactions, respectively.\nThe Hamiltonian with SU(3) SO coupling can be re-\nalized using a similar method of Raman dressing as in\nthe SU(2) case [1, 9, 44]. As shown in Fig. 1(a), three\nlaser beams with different polarizations and frequencies,\nintersecting at an angle of 2 π/3, are used for the Ra-\nman coupling. Each of the three Raman lasers dresses\none hyperfine spin state from the F= 1 manifold ( |F=\n1,mF= 1∝angbracketright,|F= 1,mF= 0∝angbracketrightand|F= 1,mF=−1∝angbracketright) to\nthe excited state |e∝angbracketright[See Fig. 1(b)]. When the standard\nrotatingwaveapproximationisusedandtheexcitedstate\nis adiabatically eliminated due to far detuning, one can\nobtain the effective Hamiltonian in Eq. (1), as discussed\nin Appendix A.\nBy diagonalizing the kinetic energy and SO coupling\nterms, we can obtain the single-particleenergyspectrum,\nwhich can provide useful information about the ground\nstateofBosecondensates. FortheSU(2)case,itisknown\nthat the single-particle spectrum with the NIST type SO\ncoupling acquires either a single or two minima, depend-\ning on the strength of the Raman coupling [1], while for\nthe case of Rashba type there exist an infinite number\nof minima locating on a continuous ring in momentum\nspace [45]. For the SU(3) SO coupling discussed here, we\nfind that there are in general three discrete minima re-\nsiding on the vertices of an equilateral triangle [See Figs.\n1(c)-1(d)]. This unique property of the energy band im-\nplies the possibilityofa three-folddegeneratemany-body\nmagnetized state [27] or a topologically nontrivial lattice\nstate, depending on the choices among the three minima\nmade by the many-body interactions.\nFIG. 1: (Color online) Scheme for creating SU(3) spin-orbit\ncoupling in spinor BECs. (a) Laser geometry. Three laser\nbeams with different frequencies and polarizations, inters ect-\ning at an angle of 2 π/3, illuminate the cloud of atoms. (b)\nLeveldiagram. EachofthethreeRamanlasers dresses onehy-\nperfine Zeeman level from |F= 1,mF= 1/angbracketright,|F= 1,mF= 0/angbracketright\nand|F= 1,mF=−1/angbracketrightof the87Rb 5S1/2,F= 1 ground state.\nδ1,δ2andδ3correspond to the detuning in the Raman tran-\nsitions. (c) Triple-well dispersion relation. The SU(3) sp in-\norbit coupling induces three discrete minima of the single-\nparticle energy band on the vertices of an equilateral trian gle\nin thekx-kyplane. (d) Projection of the first energy band on\na 2D plane. Units with /planckover2pi1=m= 1 are used for simplicity.\nIII. PHASE DIAGRAM\nNext, we discuss the phase diagram of the many-body\nground states. For the case of SU(2) SO coupling, it is\nshown that two many-body ground states, magnetized\nstate and stripe state, can be stabilized in a homoge-\nneous system [7, 24, 26]. Although the Rashba SO cou-\npling provides infinite degenerate minima in the single-\nparticle spectrum, a many-body ground state condensed\nin one or two points in momentum space is always ener-\ngetically favorable due to the presence of spin-exchange\ninteraction [24]. As a result, a lattice state with the con-\ndensates occupying three or more momentum points for\nSU(2) SO coupling is unstable, unless a strong harmonic\ntrap is introduced [21, 42, 46].\nFor the present case of SU(3) SO coupling, we first\nanalytically calculate the possible ground states using\na variational approach with a trial wave function Ψ=\nα1Ψ1+α2Ψ2+α3Ψ3, where3\nΨ1=1√\n3\n1\n1\n1\ne−i2κx, (2a)\nΨ2=1√\n3\ne−iπ\n3\neiπ\n3\neiπ\neiκ(x−√\n3y),(2b)\nΨ3=1√\n3\neiπ\n3\ne−iπ\n3\neiπ\neiκ(x+√\n3y), (2c)\ncorrespond to the many-body states with all particles\ncondensing on one of the three minima of the single-\nparticlespectrum, and αi=1,2,3areexpansioncoefficients.\nSubstituting Eqs. (2a)-(2c) into the interaction energy\nfunctional\nE=/integraldisplay\ndr/parenleftigc0\n2n2+c2\n2|F|2/parenrightig\n, (3)\none obtains\nE\nN=/parenleftbiggc0\n2+4c2\n9/parenrightbigg\n¯n−7c2\n9¯n/summationdisplay\ni/negationslash=j|αi|2|αj|2,(4)\nwhere ¯n=|α1|2+|α2|2+|α3|2is the mean particle den-\nsity. By minimizing the interaction energy with respect\nto the variation of |αi|2, one finds that the spin-exchange\ninteraction plays an important role in determining the\nphase diagram.\nWhenc2>0, it favors |α1|2=|α2|2=|α3|2= ¯n/3,\nindicating that the ground state is a triangular lattice\nphasewithanequallyweightedsuperpositionofthe three\nsingle-particle minima. On the other hand, as c2<0,\nthe system prefers a state with either |α1|2=¯n,|α2|2=\n|α3|2= 0, or |α2|2= ¯n,|α1|2=|α3|2= 0, or |α3|2=\n¯n,|α1|2=|α2|2= 0, indicating that the ground state\noccupies one single minimum in the momentum space,\nand corresponds to a three-fold degenerate magnetized\nphase.\nNote that the variational wave function Eqs. (2a)-(2c)\nis a good starting point as the SO coupling is strong\nenough to dominate the chemical potential. For the case\nwith weak SO coupling, one must rely on numerical sim-\nulations to determine the many-body ground state. In\nsuch a situation, we find a stripe phase with two minima\nin momentum space occupied for c2≫κ2, which will be\ndiscussed latter.\nThe many-body ground states can be numerically ob-\ntained by minimizing the energy functional associated\nwith the Hamiltonian Eq. (1) via the imaginarytime evo-\nlution method. It is found that the numerical results are\nconsistent with the analytical analysis discussed above\nfor rather weak interaction with c2/lessorsimilarκ2. Figure 2 il-\nlustrates the two possible ground states of spinor BECs\nwith SU(3) SO coupling. When c2>0, the three compo-\nnents are immiscible and arranged as an interlaced tri-\nangular lattice with the spatial translational symmetry\nFIG. 2: (Color online) Two distinct phases present in SU(3)\nspin-orbit coupled BECs. (a)-(d) The topologically nontri vial\nlattice phase for antiferromagnetic spin interaction ( c2>0)\nwith (a) the density and phase of Ψ 1represented by heights\nand colors, (b) the phase within one unit cell showing the\npositions of vortices (white circles) and antivortices (bl ack\ncircles), (c) the corresponding momentum distributions, a nd\n(d) the structural schematic drawing of the phase separatio n.\n(e)-(f) The three-fold degenerate magnetized phase for fer ro-\nmagnetic spin interaction ( c2<0) with (e) the density and\nphase distributions of Ψ 1and (f) the corresponding momen-\ntum distributions.\nspontaneously broken [See Figs. 2(a)-2(d)]. This lattice\nis topologically nontrivial and embedded by vortices and\nantivortices as shown in Fig. 2(b). From this result,\nwe conclude that a lattice phase can be stabilized in a\nuniform SU(3) SO coupled BEC, which is in clear con-\ntrast to the SU(2) case where a strong harmonic trap is\nrequired[21,42,46]. Moredetailsonthestructureofvor-\ntices aswell astheir unique spin configurationswill be in-\nvestigated later. On the other hand, as c2<0, the three\ncomponents are miscible, and the system forms a magne-\ntized phase with the spatial transitional symmetry pre-\nserved but the time-reversal symmetry broken [See Figs.\n2(e)-2(f)]. This magnetized phase occupies one of the\nthree minima of the single-particle spectrum by sponta-\nneous symmetry breaking, hence is three-fold degenerate\ninstead of doubly degenerate in the SU(2) case [26, 27].\nFor strong antiferromagnetic spin interaction with\nc2≫κ2, however, a stripe phase is identified with two\nof three minima occupied in the momentum space. We\ntake the states with two or three minima occupied in\nthe momentum space as trial wave functions, and per-\nform imaginary time evolution to find their respective\noptimized ground state energy. A typical set of results\nare summarized in Figure 3(a), showing the energy com-\nparison with different values of interatomic interactions.\nObviously, one finds that the stripe phase will has lower\nenergy than the lattice phase when the interatomic in-\nteraction exceeds a critical value. Due to the finite mo-\nmentum in vertical direction of the stripe [See Fig. 3(d)],\nboth the spatial translational and time-reversal symme-\ntries are broken [See Figs. 3(b)-3(c)]. This is distinct4\nFIG. 3: (color online) (a) Energy comparison between the\nlattice and stripe phases. The energy difference ∆ Ebetween\nthe numerical simulation and the variational calculation a re\nshown by solid (lattice state) and dashed (stripe state) lin es.\n(b)-(d) The ground-state density, phase and momentum dis-\ntributions of the stripe phase with the parameters c2= 20κ2\nandc0= 10c2.\nfrom the stripe phase induced by SU(2) SO coupling,\nwhere the time-reversal symmetry is preserved [24].\nIV. PHASE REQUIREMENT\nThe vortex configuration of spinor BECs depends on\nthe phase relation between the three components. We\nnext discuss the influence of SO coupling on the phase\nrequirement of the vortex configuration. We first assume\nthat the spinor order parameter of a vortex in the polar\ncoordinate ( r,θ) can be described as\nψj(r,θ) =φjeiwjθ+αj, (5)\nwherej= 0,±1 andφj≥0.\nA. Without spin-orbit coupling\nIn the absence of SO coupling, the phase-dependent\nterms in the Hamiltonian are\nHphase=Ephase\nkin+Ephase\nint\n=−1\n2/integraldisplay\nΨ∗1\nr2∂2\n∂θ2Ψdr+2c2/integraldisplay\nℜ(ψ∗\n1ψ∗\n−1ψ2\n0)dr,(6)\nwhere the first term results from the kinetic energy and\nthe second from the spin-exchange interaction. Substi-tuting Eq. (5) into (6), one obtains\nEphase\nkin=/summationdisplay\nj=1,0,−1w2\nj/integraldisplayπφ2\nj\nrdr, (7)\nEphase\nint= 2c2/integraldisplay\nφ1φ−1φ2\n0rdr\n/integraldisplay\ncos[(w 1−2w0+w−1)θ+(α1−2α0+α−1)]dθ.(8)\nIt is easy to read from Eq. (7) that the system favors\nsmall winding numbers energetically. Moreover, from\nEq. (8) the energy minimization requires the winding\nnumber and phase satisfy the following relations\nw1−2w0+w−1= 0, (9a)\nα1−2α0+α−1=nπ, (9b)\nwherenis odd forc2>0 and even for c2<0. The phase\nrequirementofEq. (9a)indicatesthatthefollowingtypes\nof winding combination, such as ∝angbracketleft±1,×,0∝angbracketright,∝angbracketleft0,×,±1∝angbracketright,\n∝angbracketleft±1,0,∓1∝angbracketright,∝angbracketleft±1,±1,±1∝angbracketright,∝angbracketleft±2,±1,0∝angbracketrightand∝angbracketleft0,±1,±2∝angbracketrightare\nallowed in a spinor BEC, where the symbol “ ×” denotes\nthe absence of the Ψ 0component.\nB. With SU(2) spin-orbit coupling\nFor the case of SU(2) SO coupling, we take the Rashba\ntype as an example, and write the Hamiltonian as\nEsoc=/integraldisplay\nκψ†\n0−i∂x−∂y0\n−i∂x+∂y0−i∂x−∂y\n0−i∂x+∂y0\nψdr,(10)\nwhereψ= [ψ1,ψ0,ψ−1]⊤. Substituting Eq. (5)into(10),\none can obtain\nEsoc=/integraldisplay\ndrdθ/bracketleftig\n(φ0r∂rφ1−w1φ0φ1)ei[(w1−w0+1)θ+(α1−α0−π\n2)]\n−(φ1r∂rφ0+w0φ1φ0)e−i[(w1−w0+1)θ+(α1−α0−π\n2)]\n+(φ0r∂rφ−1+w−1φ0φ−1)ei[(w−1−w0−1)θ+(α−1−α0−π\n2)]\n−(φ−1r∂rφ0−w0φ−1φ0)e−i[(w−1−w0−1)θ+(α−1−α0−π\n2)]/bracketrightig\n.\n(11)\nIn order to minimize the SO coupling energy, it is pre-\nferred that\nw1−w0+1 = 0, (12a)\nw−1−w0−1 = 0, (12b)\nα1−α0−π\n2=mπ, (12c)\nα−1−α0−π\n2=nπ. (12d)\nThen the SO coupling energy is rewritten as\nEsoc= 2π/integraldisplay\n[φ0r∂rφ1−φ1r∂rφ0−(w1+w0)φ0φ1]drcosmπ\n+2π/integraldisplay\n[φ0r∂rφ−1−φ−1r∂rφ0+(w−1+w0)φ0φ−1]drcosnπ,\n(13)5\nwheremandnare odd or even, which can be deter-\nmined by minimizing the energy expressed in Eq. (13).\nIt is found that the SU(2) SO coupling does not vio-\nlate the ordinary requirement on the winding combina-\ntion in Eq. (9a), but introduces further requirements in\nEqs.(12a)-(12b). Asaresult, while ∝angbracketleft−1,0,1∝angbracketright,∝angbracketleft−2,−1,0∝angbracketright\nand∝angbracketleft0,1,2∝angbracketrightare still allowed, some winding combinations\nsuch as ∝angbracketleft±1,±1,±1∝angbracketright,∝angbracketleft±1,×,0∝angbracketright,∝angbracketleft0,×,±1∝angbracketright,∝angbracketleft1,0,−1∝angbracketright,\n∝angbracketleft2,1,0∝angbracketrightand∝angbracketleft0,−1,−2∝angbracketrightare forbidden. Obviously, one\ncan see that the SO coupling break the chiral symmetry,\nthus may lead to chiral spin textures.\nC. With SU(3) spin-orbit coupling\nFor the case of SU(3) SO coupling, the effective Hamil-\ntonian can be written as\nEsoc=/integraldisplay\nκψ†\n0−i∂x−∂y−i∂x+∂y\n−i∂x+∂y0−i∂x−∂y\n−i∂x−∂y−i∂x+∂y0\nψdr.(14)\nSubstituting Eq. (5) into (14), we get\nEsoc=/integraldisplay\ndrdθ/bracketleftig\n(φ0r∂rφ1−w1φ0φ1)ei[(w1−w0+1)θ+(α1−α0−π\n2)]\n−(φ1r∂rφ0+w0φ1φ0)e−i[(w1−w0+1)θ+(α1−α0−π\n2)]\n+(φ0r∂rφ−1+w−1φ0φ−1)ei[(w−1−w0−1)θ+(α−1−α0−π\n2)]\n−(φ−1r∂rφ0−w0φ−1φ0)e−i[(w−1−w0−1)θ+(α−1−α0−π\n2)]\n+(φ−1r∂rφ1+w1φ−1φ1)ei[(w1−w−1−1)θ+(α1−α−1−π\n2)]\n−(φ1r∂rφ−1−w−1φ1φ−1)e−i[(w1−w−1−1)θ+(α1−α−1−π\n2)]/bracketrightig\n.\n(15)\nBy minimizing the SO coupling energy, one obtains the\nfollowing relations\nw1−w0+1 = 0, (16a)\nw−1−w0−1 = 0, (16b)\nw1−w−1−1 = 0, (16c)\nα1−α0−π\n2=mπ, (16d)\nα−1−α0−π\n2=nπ, (16e)\nα1−α−1−π\n2=lπ. (16f)\nThen the SO coupling energy can be rewritten as\nEsoc=2π/integraldisplay\n[φ0r∂rφ1−φ1r∂rφ0−(w1+w0)φ0φ1]drcosmπ\n+2π/integraldisplay\n[φ0r∂rφ−1−φ−1r∂rφ0+(w−1+w0)φ0φ−1]drcosnπ\n+2π/integraldisplay\n[φ−1r∂rφ1−φ1r∂rφ−1+(w1+w−1)φ−1φ1]drcoslπ,\n(17)wherem,nandlare odd or even, which can be\ndetermined from Eq. (17). However, the three winding\nrequirements Eqs. (16a)-(16c) can not be satisfied simul-\ntaneously. Thus the SU(3) SO coupling may choose two\nout of the three winding requirements for the following\nthree cases:\nCase I:\nw1−w0+1 = 0, (18a)\nw−1−w0−1 = 0, (18b)\nα1−α0−π\n2=mπ, (18c)\nα−1−α0−π\n2=nπ. (18d)\nCase II:\nw1−w0+1 = 0, (19a)\nw1−w−1−1 = 0, (19b)\nα1−α0−π\n2=mπ, (19c)\nα1−α−1−π\n2=lπ. (19d)\nCase III:\nw−1−w0−1 = 0, (20a)\nw1−w−1−1 = 0, (20b)\nα−1−α0−π\n2=nπ, (20c)\nα1−α−1−π\n2=lπ. (20d)\nFor case I, the winding combination ∝angbracketleft−1,0,1∝angbracketrightis allowed,\nwhile∝angbracketleft1,0,−1∝angbracketrightis not allowed, indicating the chiral sym-\nmetry is broken. For case II and case III, one can find\nthat the SU(3) SO coupling breaks the ordinary require-\nment on the winding combination in Eq. (9a), thus new\nwinding combinations, such as ∝angbracketleft0,1,−1∝angbracketrightand∝angbracketleft1,−1,0∝angbracketright,\nare possible.\nV. VORTEX CONFIGURATIONS\nThe vortex configurations of spinor BECs can be clas-\nsified according to the combination of winding numbers\nand the magnetization of vortex core [35–37]. For ex-\nample, a Mermin-Ho vortex has winding combination\n∝angbracketleft±2,±1,0∝angbracketrightwith a ferromagnetic core, where the plus\nand minus signs represent different chirality of the vor-\ntices [47], and the expression of ∝angbracketleftw1,w0,w−1∝angbracketrightindicates\nthat the components of Ψ 1, Ψ0and Ψ −1in the wave\nfunction acquire winding numbers of w 1, w0and w −1,\nrespectively. Using this notation, a polar-core vortex has\nwinding combination ∝angbracketleft±1,0,∓1∝angbracketrightwith an antiferromag-\nnetic core, and a half-quantum vortex has winding com-\nbination ∝angbracketleft±1,×,0∝angbracketrightwith a ferromagnetic core, where the\nsymbol “ ×” denotes the absence of the Ψ 0component.6\nFIG. 4: (Color online) Vortex configurations in antiferro-\nmagnetic spinor BECs with SU(3) spin-orbit coupling. (a)\nVortex arrangement among the three components of the con-\ndensates. One can identify three types of vortices, includ-\ning a polar-core vortex with winding combination /angbracketleft−1,0,1/angbracketright\n(blue line) and two ferromagnetic-core vortices with wind-\ning combinations /angbracketleft1,−1,0/angbracketright(green line) and /angbracketleft0,1,−1/angbracketright(red\nline). (b)-(d) Spherical-harmonic representation of the t hree\ntypes of vortices. The surface plots of |Φ(θ,φ)|2for (b) the\npolar-core vortex /angbracketleft−1,0,1/angbracketright, (c) the ferromagnetic-core vor-\ntex/angbracketleft1,−1,0/angbracketright, and (d) the ferromagnetic-core vortex /angbracketleft0,1,−1/angbracketright\nare shown with the colors representing the phase of Φ(θ,φ).\nHere,Φ(θ,φ) =/summationtext1\nm=−1Y1m(θ,φ)ΨmandY1mis the rank-1\nspherical-harmonic function.\nIn the lattice phase induced by the SU(3) SO cou-\npling with antiferromagnetic spin interaction, there ex-\nists three types of vortices: one is a polar-core vortex\nwith winding combination ∝angbracketleft−1,0,1∝angbracketright, and the other two\nare ferromagnetic-core vortices with winding combina-\ntions∝angbracketleft1,−1,0∝angbracketrightand∝angbracketleft0,1,−1∝angbracketright[See Fig. 4(a)]. However,\nthe vortex configurations with opposite chirality of each\ntype, such as ∝angbracketleft1,0,−1∝angbracketright,∝angbracketleft−1,1,0∝angbracketrightand∝angbracketleft0,−1,1∝angbracketright, are not\nallowed, because the chiral symmetry is intrincically bro-\nkenin SU(3) SO coupledsystems, asdiscussed inSec. IV.\nSurprisingly, one finds that the two types of\nferromagnetic-core vortices ∝angbracketleft1,−1,0∝angbracketrightand∝angbracketleft0,1,−1∝angbracketrightvio-\nlate the conventional phase requirement 2w 0= w1+w−1\nfor ordinary spinor BECs [35–37]. This can be under-\nstood by noting that the relative phase among different\nwave function components are no longer uniquely deter-\nmined by the spin-exchange interaction but also affected\nby the SU(3) SO coupling, as qualitatively explained in\nSec. IV. Thus, the interlaced arrangement of the three\ntypes of vortices forms a new class of vortexlattice which\nhas no analogue in systems without SO coupling.\nThe configurations of the three types of vortices in-\nduced by the SU(3) SO coupling with antiferromagnetic\ninteraction are essentially different from those usually\nobserved in ferromagnetic spinor BECs, as can be il-\nlustrated by the spherical-harmonic representation [35].\nFrom Figs. 4(b)-4(d) one can find that for the polar-corevortex,theantiferromagneticorderparametervariescon-\ntinuously everywhere, while for the ferromagnetic-core\nvortex,themagneticorderparameteracquiresasingular-\nity at the vortex core. In contrast, in the ordinary ferro-\nmagnetic spinor BECs, the ferromagnetic order parame-\nter varies continuously everywhere for the ferromagnetic-\ncorevortex,buthasasingularityatthecoreforthepolar-\ncore vortex [35].\nVI. DOUBLE-QUANTUM SPIN VORTICES\nSpin vortex is a complex topological defect resulting\nfrom symmetry breaking, and is characterized by zero\nnet mass current and quantized spin current around an\nunmagnetized core [35, 38, 48–51]. It is not only different\nfrom the magnetic vortex found in magnetic thin films\n[52–54], but alsofromthe 2Dskyrmion[55,56] dueto the\nexistence of singularity in the spin textures [57]. Single-\nquantum spin vortex with the spin current showing one\nquantum ofcirculation has been experimentally observed\nin ferromagnetic spinor BECs [58]. Multi-quantum spin\nvortices with l(l≥2) quanta circulating spin current,\nhowever, are considered to be topologically unstable and\nhave not been discovered yet [35].\nA particularly important finding of our present work\nis that the polar-core vortex in the lattice phase has a\nspin current with two quanta of circulation around the\nunmagnetized core, hence can be identified as a double-\nquantum spin vortex. Figure 5 presents the transverse\nmagnetization F+=Fx+iFy, longitudinal magnetiza-\ntionFz, and amplitude of the total magnetization |F|\nin the lattice phase, which are experimentally observ-\nable by magnetization-sensitive phase-contrast imaging\ntechnique [59]. From these results, one can find two dis-\ntinct types of topological defects, double-quantum spin\nvortex (DSV) and half skyrmion (HS) [60, 61], which\ncorrespond to the polar-core vortex with winding com-\nbinations ∝angbracketleft−1,0,1∝angbracketrightand the ferromagnetic-core vortex\nwith winding combinations ∝angbracketleft1,−1,0∝angbracketrightor∝angbracketleft0,1,−1∝angbracketright, re-\nspectively. In particular, for the double-quantum spin\nvortex, the core is unmagnetized and the orientation of\nthe magnetization along a closed path surrounding the\ncore acquires a rotation of 4 π. This finding indicates\nthat a regular lattice of multi-quantum spin vortices can\nemerge spontaneously in antiferromagnetic spinor BECs\nwith SU(3) SO coupling. By exploring the effect of a\nsmall but finite temperature, we confirm that the double-\nquantum spin vortices are robust against thermal fluctu-\nations and hence are observable in experiments, as dis-\ncussed in Appendix B.\nThe emergence of spin current with two quanta of cir-\nculation can be analytically understood by expanding\nthe wave function obtained by the variational methods\naround the center of a double-quantum spin vortex. We\nsuppose that the wave function of the lattice phase is7\nFIG. 5: (Color online) Double-quantum spin vortex in an-\ntiferromagnetic spinor BECs with SU(3) spin-orbit couplin g.\n(a) Spatial maps of the transverse magnetization with color s\nindicating the magnetization orientation. (b) Longitudin al\nmagnetization. (c) Amplitude of the total magnetization |F|.\nTwo kinds of topological defects, double-quantum spin vor-\ntex (DSV) and half skyrmion (HS) are marked by big and\nsmall circles, respectively. The transverse magnetizatio n ori-\nentation arg F+along a closed path (indicated by big circles)\nsurrounding the unmagnetized core shows a net winding of\n4π, revealing the presence of a double-quantum spin vortex.\nwritten as\nψ=1\n3\n1\n1\n1\ne−i2κx+1\n3\ne−iπ\n3\neiπ\n3\neiπ\neiκ(x−√\n3y)+1\n3\neiπ\n3\ne−iπ\n3\neiπ\neiκ(x+√\n3y).\n(21)\nThen one can expand ψaround the center of a vortex\nwith winding number ∝angbracketleft−1,0,1∝angbracketright, e.g., at the location of\n(x,y) = (0,π/(3√\n3κ)). Substituting x=ǫcosθandy=\nπ/(3√\n3κ)+ǫsinθintoψand expanding with respect to\nthe infinitesimal ǫ, we obtain\nψ=\n−iκe−iθǫ−1\n2κ2ei2θǫ2\n1−κ2ǫ2\n−iκeiθǫ−1\n2κ2e−i2θǫ2\n+O/parenleftbig\nǫ3/parenrightbig\n.(22)\nNotice that the second-order terms with e±i2θhave\nno essential influence on the phases, thus the wind-\ning number for each component can still be represented\nas∝angbracketleft−1,0,1∝angbracketright[See Figs. 6(a)-6(c)]. However, since the\nfirst-order terms are canceled out when calculating the\ntransverse magnetization F+=√\n2[ψ∗\n1ψ0+ψ∗\n0ψ−1], the\nsecond-order terms play a dominant role, leading to the\nemergence of spin current with two quanta of circulation\naround an unmagnetized core\nF+∝ǫ2e−i2θ, (23)\nas illustrated in Fig. 6(d).\nFIG. 6: (color online) (a)-(c) Phases of the polar-core vort ex\ndescribed by the wave function in Eq. (22), displaying the\nwinding combination /angbracketleft−1,0,1/angbracketright. (d) Direction of the trans-\nverse magnetization, indicating the emergence of spin curr ent\nwith two quanta of circulation.\nVII. CONCLUSION\nTo summarize, we have mapped out the ground-state\nphase diagram of SU(3) spin-orbit coupled Bose-Einstein\ncondensates. Several novel phases are discovered includ-\ning a three-fold degenerate magnetized phase, a vortex\nlattice phase, as well as a stripe phase with time-reversal\nsymmetry broken. We also investigate the influence of\nSU(3) spin-orbit coupling on the phase requirement of\nthevortexconfiguration,anddemonstratethattheSU(3)\nspin-orbit coupling breaks the ordinary phase rule of\nspinor Bose-Einstein condensates, and allows the sponta-\nneous emergence of stable double-quantum spin vortices.\nAs a new member in the family of topological defects,\ndouble quantum spin vortex has never been discovered\nin any other systems. Our work deepen the understand-\ning of spin-orbit phenomena, and will attract extensive\ninterest of scientists in the cold atom community.\nACKNOWLEDGMENTS\nThis work was supported by NKRDP under grants\nNos. 2016YFA0301500, 2012CB821305; NSFC under\ngrants Nos. 61227902, 61378017, 11274009, 11434011,\n11434015, 11447178, 11522436, 11547126; NKBRSFC\nunder grant No. 2012CB821305; SKLQOQOD un-\nder grant No. KF201403; SPRPCAS under grant\nNos. XDB01020300, XDB21030300; JSPS KAKENHI8\ngrant No. 26400414 and MEXT KAKENHI grant No.\n25103007. W. H. and X.-F. Z. contributed equally to\nthis work.\nAPPENDIX A: DERIVING THE EFFECTIVE\nHAMILTONIAN\nWe consider spinor Bose-Einstein condensates (BECs)\nilluminated by three Raman laser beams, which couple\ntwo of the three hyperfine spin components respectively,\nas illustrated in Figs. 1(a)-1(b) of the main text. The\ninternal dynamics of a single particle under this scheme\ncan be described by the Hamiltonian\nH=3/summationdisplay\nj=1/parenleftbigg/planckover2pi12k2\n2m+εj/parenrightbigg\n|j∝angbracketright∝angbracketleftj|+n/summationdisplay\nl=1El|l∝angbracketright∝angbracketleftl|\n+3/summationdisplay\nj=1n/summationdisplay\nl=1/bracketleftig\nΩjei(Kj·r+ωjt)Mlj|l∝angbracketright∝angbracketleftj∝angbracketright+h.c./bracketrightig\n,(A1)\nwhere/planckover2pi1kis the momentum of the particles, and εjand\nElare the energies of the ground and excited states, re-\nspectively. In the atom-light coupling term, Kjandωj\nare the wave vectors and frequencies of the three Raman\nlasers with Ω jthe corresponding Rabi frequencies, and\nMljis the matrix element of the dipole transition. One\ncan see that this Hamiltonian is similar to that used in\nthe scheme for creating 2D spin-orbit (SO) coupling in\nultracold Fermi gases [9], thus can be readily realized in\nBose gases. Taking the standard rotating wave approxi-\nmation to get rid of the time dependence of the Hamilto-\nnian, and adiabatically eliminating the excited states for\nfar detuning, the Hamiltonian can be rewritten as\nH=\n/planckover2pi12(k+K1)2\n2m+δ1Ω12 Ω13\nΩ21/planckover2pi12(k+K2)2\n2m+δ2Ω23\nΩ31 Ω32/planckover2pi12(k+K3)2\n2m+δ3\n,(A2)\nwhereδ1,δ2andδ3arethetwo-photondetunings, andthe\nrealparametersΩ jj′= Ωj′jdescribe the Raman coupling\nstrength between hyperfine ground states |j∝angbracketrightand|j′∝angbracketright,\nwhich can be expressed as [9, 62]\nΩjj′=−/radicalbigIjIj′\n/planckover2pi12cǫ0/summationdisplay\nm′∝angbracketleftj′|erq|m′∝angbracketright∝angbracketleftm′|erq|j∝angbracketright\n∆.(A3)\nHere,Ijis the intensity of each Raman laser, and ∆\ndenotes the one-photon detuning. Other parameters c,\nǫ0andein Eq. (A3) are the speed of light, permittiv-\nity of vacuum and elementary charge, respectively. In\nEq. (A3),q=x,y,zis an index labeling the components\nofrin the spherical basis, and |m′∝angbracketrightdescribes the middle\nexcited hyperfine spin state in the Raman process. For\nsimplicity, we assume Ω = Ω 12= Ω13= Ω23, which can\nalways be satisfied by adjusting the system parameters,\nsuch as the laser intensity.\nIntroducing a unitary transformation\nU=1√\n3\n1 1 1\n−e−iπ\n3−eiπ\n31\n−eiπ\n3−e−iπ\n31\n (A4)and a time-dependent unitary transformation U(t) =\nei/parenleftbigg\n/planckover2pi12K2\n0\n2m+δ2−Ω/parenrightbigg\nt, the effective Hamiltonian becomes\nH=\n/planckover2pi12k2\n2m+δ1−δ20 0\n0/planckover2pi12k2\n2m0\n0 0/planckover2pi12k2\n2m+δ3−δ2+3Ω\n+Vso,(A5)\nwhere the laser vectors K1=−K0ˆ ey,K2=√\n3K0\n2ˆ ex+\nK0\n2ˆ eyandK3=−√\n3K0\n2ˆ ex+K0\n2ˆ eyare defined with K0=\n2mκ//planckover2pi1. The spin-dependent uniform potential induced\nby the Raman detuning δiand Raman coupling strength\nΩ can be eliminated by applying a Zeeman field, leading\nto\nH=\n/planckover2pi12k2\n2m+ǫ10 0\n0/planckover2pi12k2\n2m0\n0 0/planckover2pi12k2\n2m+ǫ2\n+Vso,(A6)\nwhereǫ1=δ1−δ2+∆1+∆2andǫ2=δ3−δ2−∆1+∆2+\n3Ω with ∆ 1and ∆ 2denoting the linear and quadratic\nZeeman energy respectively. By tuning the detuning, the\nZeeman energy and the Raman coupling strength, one\ncan reach the regime ∆ 1=δ3−δ1+3Ω\n2and ∆ 2=δ2−\nδ1+δ3+3Ω\n2which satisfying ǫ1=ǫ2= 0. Then we have\nH=/planckover2pi12k2\n2m+Vso, (A7)\nwhich is the single-particle Hamiltonian with SU(3) SO\ncoupling considered in the main text.\nAPPENDIX B: STABILITY OF THE\nDOUBLE-QUANTUM SPIN VORTEX STATES\nIn order to verify the stability of the phases discovered\nin this manuscript, we have explored the effects of a\nsmall but finite temperature, and concluded that the\ndouble-quantum spin vortex states are robust against\nthe thermal fluctuations. In particular, we considered\na random fluctuation ∆ φin the real-time evolution\nof the Gross-Pitaevskii equation, which causes an\nenergy fluctuation about ∆ E= 0.03EgwithEgthe\nground-state energy. An estimation shows that this\nlevel of fluctuation corresponds to the energy scale\nkBTwithT∼300 nK, which is higher enough for a\nusual system of Bose-Einstein condensates in realistic\nexperiments. According to numerical simulations, we\nfind that the structure of the double-quantum spin\nvortex state is stable under this level of fluctuation in\ntens of millisenconds [See Fig. 7], suggesting that this\nphase is indeed observable in experiments.9\nFIG. 7: (color online) Stable double-quantum\nspin vortex under a random fluctuation. The\nimages are taken at t= 20msin the real-time\nevolution, with thermal fluctuation in the en-\nergy scale of kBTwithT∼300 nK. (a) Spa-\ntial maps of the transverse magnetization with\ncolors indicating the magnetization orientation.\n(b) Longitudinal magnetization. (c) Amplitude\nof the total magnetization |F|. It is shown that\nthe double-quantum spin vortices are topologi-\ncally stable under external fluctuations with a\nfairly long lifetime of tens of ms.\n[1] Y. J. Lin, K. Jim´ enez-Garc´ ıa, and I. B. Spielman, Natur e\n471, 83 (2011).\n[2] P. Wang, Z. Q. Yu, Z. Fu, J. Miao, L. Huang, S. Chai, H.\nZhai, and J. Zhang, Phys. Rev. Lett. 109, 095301 (2012).\n[3] L. W. Cheuk, A. T. Sommer, Z. Hadzibabic, T. Yefsah,\nW. S. Bakr, and M. W. Zwierlein, Phys. Rev. Lett. 109,\n095302 (2012).\n[4] J. Y. Zhang, S. C. Ji, Z. Chen, L. Zhang, Z. D. Du, B.\nYan, G. S. Pan, B. Zhao, Y. J. Deng, H. Zhai, S. Chen,\nand J. W. Pan, Phys. Rev. Lett. 109, 115301 (2012).\n[5] C. Qu, C. Hamner, M. Gong, C. Zhang, and P. Engels,\nPhys. Rev. A 88, 021604(R) (2013).\n[6] Z. Fu, L. Huang, Z. Meng, P. Wang, L. Zhang, S. Zhang,\nH. Zhai, P. Zhang, and J. Zhang, Nat. Phys. 10, 110\n(2014).\n[7] S. C. Ji, J. Y. Zhang, L. Zhang, Z. D. Du, W. Zheng, Y.\nJ. Deng, H. Zhai, S. Chen, and J. W. Pan, Nat. Phys.\n10, 314 (2014).\n[8] A. J. Olson, S.-J. Wang, R. J. Niffenegger, C.-H. Li, C.\nH. Greene, and Y. P. Chen, Phys. Rev. A 90, 013616\n(2014).\n[9] L. Huang, Z. Meng, P. Wang, P. Peng, S.-L. Zhang, L.\nChen, D. Li, Q. Zhou, and J. Zhang, Nat. Phys. 12, 540\n(2016).\n[10] Z. Wu, L. Zhang, W. Sun, X.-T. Xu, B.-Z. Wang, S.-\nC. Ji, Y. Deng, S. Chen, X.-J. Liu, and J.-W. Pan,\narXiv:1511.08170 (2015).\n[11] A. M. Dudarev, R. B. Diener, I. Carusotto, and Q. Niu,\nPhys. Rev. Lett. 92, 153005 (2004).\n[12] V. Galitski and I. B. Spielman, Nature 494, 49 (2013).\n[13] J. Dalibard, F. Gerbier, G. Juzeli¯ unas, and P. ¨Ohberg,\nRev. Mod. Phys. 83, 1523 (2011).\n[14] N. Goldman, G. Juzeli¯ unas, P. ¨Ohberg, and I. B. Spiel-\nman, Rep. Prog. Phys. 77, 126401 (2014).\n[15] H. Zhai, Int. J. Mod. Phys. B 26, 1230001 (2012).\n[16] H. Zhai, Rep. Prog. Phys. 78, 026001 (2015).\n[17] W. Yi, W. Zhang, and X. Cui, Sci. China Phys. Mech.\nAstron. 58,014201 (2015).\n[18] J. Zhang, H. Hu, X. J. Liu, and H. Pu, Annu. Rev. Cold\nAt. Mol. 2, 81 (2014).[19] C. Wu, Mod. Phys. Lett. B 23, 1 (2009).\n[20] C. Wu, I. Mondragon-Shem, and X. F. Zhou, Chin. Phys.\nLett. 28, 097102 (2011).\n[21] H. Hu, B. Ramachandhran, H. Pu, and X. J. Liu, Phys.\nRev. Lett. 108, 010402 (2012).\n[22] H. Sakaguchi, B. Li, and B. A. Malomed, Phys. Rev. E\n89, 032920 (2014).\n[23] W. Han, G. Juzeli¯ unas, W. Zhang, and W. M. Liu, Phys.\nRev. A91, 013607 (2015).\n[24] C. Wang, C. Gao, C. M. Jian, and H. Zhai, Phys. Rev.\nLett.105, 160403 (2010).\n[25] T. L. Ho and S. Zhang, Phys. Rev. Lett. 107, 150403\n(2011).\n[26] Y. Li, L. P. Pitaevskii, and S. Stringari, Phys. Rev. Let t.\n108, 225301 (2012).\n[27] T. F. J. Poon and X. J. Liu, Phys. Rev. A 93, 063420\n(2016).\n[28] A. Polkovnikov, K. Sengupta, A. Silva, and M. Vengalat-\ntore, Rev. Mod. Phys. 83, 863 (2011).\n[29] Y. Li, G. I. Martone, L. P. Pitaevskii, and S. Stringari,\nPhys. Rev. Lett. 110, 235302 (2013).\n[30] G. I. Martone, Y. Li, L. P. Pitaevskii, and S. Stringari,\nPhys. Rev. A 86, 063621 (2012).\n[31] K. Sun, C. Qu, Y. Xu, Y. Zhang, and C. Zhang, Phys.\nRev. A93, 023615 (2016).\n[32] B. M. Anderson, G. Juzeli¯ unas, V. M. Galitski,and I. B.\nSpielman, Phys. Rev. Lett. 108, 235301 (2012).\n[33] T. Graß, R. W. Chhajlany, C. A. Muschik, and M.\nLewenstein, Phys. Rev. B 90, 195127 (2014).\n[34] R. Barnett, G. R. Boyd, andV.Galitski, Phys.Rev.Lett.\n109, 235308 (2012).\n[35] Y. Kawaguchi and M. Ueda, Phys. Rep. 520, 253 (2012).\n[36] T. Isoshima, K. Machida, and T. Ohmi, J. Phys. Soc.\nJpn.70, 1604 (2001).\n[37] T. Mizushima, N. Kobayashi, and K. Machida, Phys.\nRev. A70, 043613 (2004).\n[38] D. M. Stamper-Kurn and M. Ueda, Rev. Mod. Phys. 85,\n1191 (2013).\n[39] D. S. Wang, Y. R. Shi, K. W. Chow, Z. X. Yu, and X.\nG. Li, Eur. Phys. J. D 67, 242 (2013).10\n[40] D. S. Wang, Y. Q. Ma, and X. G. Li, Commun. Nonlinear\nSci. Numer. Simulat. 19, 3556 (2014).\n[41] G. B. Arfken, H. J. Weber, and F. E. Harris, Mathemat-\nical Methods for Physicists (7th ed.) (2000).\n[42] Z. F. Xu, Y. Kawaguchi, L. You, and M. Ueda, Phys.\nRev. A86, 033628 (2012).\n[43] Z. Lan and P. Ohberg, Phys. Rev. A 89, 023630 (2014).\n[44] Y. Zhang, L. Mao, and C. Zhang, Phys. Rev. Lett. 108,\n035302 (2012).\n[45] T. D. Stanescu, B. Anderson, andV. Galitski, Phys. Rev.\nA78, 023616 (2008).\n[46] S. Sinha, R. Nath, and L. Santos, Phys. Rev. Lett. 107,\n270401 (2011).\n[47] H. Saito, Y. Kawaguchi, and M. Ueda, Phys. Rev. Lett.\n96, 065302 (2006).\n[48] H. Saito, Y. Kawaguchi, and M. Ueda, Phys. Rev. A 75,\n013621 (2007).\n[49] Y. Kawaguchi, H. Saito, K. Kudo, and M. Ueda, Phys.\nRev. A82, 043627 (2010).\n[50] J. Lovegrove, M. O. Borgh, and J. Ruostekoski, Phys.\nRev. A86, 013613 (2012).\n[51] S.-W. Su, I.-K. Liu, Y.-C. Tsai, W. M. Liu, and S.-C.\nGou, Phys. Rev. A 86, 023601 (2012).\n[52] A. Hubert and R. Schafer, Magnetic Domains (Springer,Berlin, 1998).\n[53] T. Shinjo, T. Okuno, R. Hassdorf, K. Shigeto, and T.\nOno, Science 289, 930 (2000).\n[54] A. Wachowiak, J. Wiebe, M. Bode, O. Pietzsch, M. Mor-\ngenstern, and R. Wiesendanger, Science 298, 577 (2002).\n[55] L. S. Leslie, A. Hansen, K. C. Wright, B. M. Deutsch,\nand N. P. Bigelow, Phys. Rev. Lett. 103, 250401 (2009).\n[56] Jae-yoon Choi, Woo Jin Kwon, and Yong-il Shin, Phys.\nRev. Lett. 108, 035301 (2012).\n[57] S. Yi and H. Pu, Phys. Rev. Lett. 97, 020401 (2006).\n[58] L. E. Sadler, J. M. Higbie, S. R. Leslie, M. Vengalattore ,\nand D. M. Stamper-Kurn, Nature 443, 312 (2006).\n[59] Other experimental techniques, such as selective abso rp-\ntion and time-of-flight measurement, can also be used to\nobserve the double-quantum spin vortex states. We sug-\ngest the magnetization-sensitive phase-contrast imaging\nmethod just because it has advantage in observing the\nphase information of the spin-vortex structure.\n[60] G. E. Brown and M. Rho, The Multifaceted Skyrmion\n(World Scientific, Singapore, 2010).\n[61] M. DeMarco and H. Pu, Phys. Rev. A 91, 033630 (2015).\n[62] T. A.Savard, S. R.Granade, K. M. O’Hara, M. E. Gehm,\nand J. E. Thomas, Phys. Rev. A 60, 4788 (1999)."    },    {        "title": "1208.2923v2.Chaos_driven_dynamics_in_spin_orbit_coupled_atomic_gases.pdf",        "content": "Chaos-driven dynamics in spin-orbit coupled atomic gases\nJonas Larson,1, 2,\u0003Brandon M. Anderson,3and Alexander Altland2\n1Department of Physics, Stockholm University, Se-106 91 Stockholm, Sweden\n2Institut f ur Theoretische Physik, Universit at zu K oln, K oln, De-50937, Germany\n3Joint Quantum Institute, national Institute of Standards and technology\nand the University of Maryland, Gaithersburg, Maryland 20899-8410, USA\n(Dated: May 24, 2022)\nThe dynamics, appearing after a quantum quench, of a trapped, spin-orbit coupled, dilute atomic\ngas is studied. The characteristics of the evolution is greatly in\ruenced by the symmetries of the\nsystem, and we especially compare evolution for an isotropic Rashba coupling and for an anisotropic\nspin-orbit coupling. As we make the spin-orbit coupling anisotropic, we break the rotational sym-\nmetry and the underlying classical model becomes chaotic; the quantum dynamics is a\u000bected ac-\ncordingly. Within experimentally relevant time-scales and parameters, the system thermalizes in a\nquantum sense. The corresponding equilibration time is found to agree with the Ehrenfest time,\ni.e. we numerically verify a \u0018log(\u0016h\u00001) scaling. Upon thermalization, we \fnd the equilibrated\ndistributions show examples of quantum scars distinguished by accumulation of atomic density for\ncertain energies. At shorter time-scales we discuss non-adiabatic e\u000bects deriving from the spin-orbit\ncoupled induced Dirac point. In the vicinity of the Dirac point, spin \ructuations are large and, even\nat short times, a semi-classical analysis fails.\nPACS numbers: 03.75.Kk, 03.75.Mn\nI. INTRODUCTION\nThe physics of ultracold atomic gases has greatly ad-\nvanced in recent years [1]. The high control of system pa-\nrameters, together with the isolation of the system from\nits environment, have made it possible to use such se-\ntups to simulate various theoretical models of condensed\nmatter physics [1, 2]. Of signi\fcance in many condensed\nmatter models is the response to external magnetic \felds.\nSince atoms are neutral, there is no direct way to imple-\nment a Lorentz force in these systems. Early experiments\ncreated a synthetic magnetic \feld via rotation [3]. While\nsimple theoretically, these methods are impractical for\ncertain setups, and they are limited to weak, uniform\n\felds. The \frst experimental demonstration of laser-\ninduced synthetic magnetic \felds for neutral atoms [4],\non the other hand, paves the way for an avenue of new\nsituations to be studied in a versatile manner [5{7]. Ow-\ning to numerous fundamental applications in the con-\ndensed matter community [8, 9], maybe the most im-\nportant direction appears when the laser \felds induce\na synthetic spin-orbit (SO) coupling. Indeed, a certain\nkind of SO-coupling for neutral atoms has already been\ndemonstrated [10], and it is expected that more general\nSO-couplings will be attainable within the very near fu-\nture [11, 12].\nWhile SO-couplings can in principle bear identical\nforms in condensed matter and cold atom models, there is\nan inevitable di\u000berence, often overlooked, between these\ntwo systems. The presence of a con\fning potential for the\natomic gas can qualitatively change the physics [1, 3],\n\u0003jolarson@fysik.su.seand has only recently been addressed [13{17]. Fur-\nthermore, most of these studies are concerned with\nground/stationary state properties of the system [13{\n15], while few works discuss dynamics or non-equilibrium\nphysics. Notwithstanding, the experimental isolation of\nthese systems suggests that they are well suited for stud-\nies of closed quantum dynamics [18].\nHistorically, some of the \fnest experiments regard-\ning dynamics of closed quantum systems have been per-\nformed in quantum optics [19, 20]. An early example\nproved quantization of the electromagnetic \feld by mak-\ning explicit use of quantum revivals [21]. Such quantum\nrecurrences, in general connected to integrability or small\nsystem sizes, are now well understood. The situation be-\ncomes more complex for non-integrable systems [18] or\nsystems with a large number of degrees-of-freedom [22].\nOne particularly interesting question is whether any ini-\ntial state relaxes to an asymptotic state, and if so, what\nare then the properties of this \\equilibrated\" state and\nthe mechanism behind the equilibration. Both these\nquestions have inspired numerous publications during\nthe last decade, both theoretical [23, 24] as well as ex-\nperimental [25{27]. A rule of thumb is that in order\nfor a closed quantum system to thermalize , i.e. all ex-\npectation values can be obtained from a microcanonical\nstate, its underlying classical Hamiltonian should be non-\nintegrable [18]. While true in most cases studied so far,\nexceptions to this hypothesis has been found [28]. More-\nover, the behavior near the transition from regular to\nchaotic dynamics, classically explained by Kolmogorov-\nArnold-Moser theory [29], is not well understood for a\nquantum system [30]. It is therefore desirable to study\na system where these two regimes can be explored by\ntuning an external parameter, and for which the exper-\nimental methods in terms of preparation and detectionarXiv:1208.2923v2  [quant-ph]  28 Jan 20132\nare already well developed.\nMotivated by the above arguments, in this paper we\nconsider dynamics of a trapped SO-coupled cold dilute\natomic gas. The SO-coupling is assumed tunable from\nisotropic (Rashba-like) to anisotropic, and hence the sys-\ntem can be tuned between regular and chaotic. Note\nthat even though this crossover is generated by a change\nin the form of the SO-coupling, the con\fning trap causes\nthe system to become non-integrable. We distinguish\nbetween short and long time evolution, where by \\long\ntime\" we mean times similar to the Ehrenfest time. In\nfact, the corresponding time-scale for the thermalization\nis found to agree with the Ehrenfest time, and thereby\nscale as log(\u0016 h\u00001)=\u0015where\u0015is the maximum Lyaponov\nexponent. This scaling for the thermalization has been\nconjectured in Ref. [31], but was not numerically ver-\ni\fed in these works. At shorter times when the wave\npacket remains localized, we especially study the rapid\nchanges in the spin as the wave packet evolves in the\nvicinity of the Dirac point (DP). For energies below the\nDP (E < 0), we utilize an adiabatic model derived in the\nBorn-Oppenheimer approximation (BOA) [32]. Aside\nfrom some special initial states, we encounter thermal-\nization in all cases. These exceptions correspond to\nstates evolving within a regular \\island\" in the otherwise\nchaotic sea. Among the thermalized states, the equi-\nlibrated distributions are found to show quantum scars\noriginating from periodic orbits of the underlying classi-\ncal model. The experimental relevance of all our theoret-\nical predictions are discussed and put in a state-of-the-art\nexperimental perspective.\nThe paper is outlined as follows. The following sec-\ntion introduces the system Hamiltonian and discusses\nits symmetries. Section II B derives the adiabatic model\nby imposing the BOA. A semi-classical analysis, demon-\nstrating classical chaos for anisotropic SO-couplings, is\npresented in Sec. III. The following section considers the\nfull quantum model at short times, Sec. IV A, and long\ntimes, Sec. IV B. Section IV C contains a discussion re-\ngarding experimental relevance of our results. Finally,\nSec. V gives some concluding remarks.\nII. SPIN-ORBIT COUPLED COLD ATOMS\nA. Model spin-orbit Hamiltonian\nSeveral proposals exist for implementing spin-orbit\ncouplings in cold atoms [33{35]. In general, these syn-\nthetic spin-orbit \felds are generated through the appli-\ncation of optical and Zeeman \felds to produce a set of\ndressed states that are well separated energetically from\nthe remaining dressed states [5]. We denote these states\nas pseudo-spin, but emphasize that there is no connection\nto real space rotations. Spatial variation of the dressed\nstates will couple the pseudo-spin to the orbital motion\nof the atom. An atom prepared in a pseudo-spin state\nwill therefore see an e\u000bective Hamiltonian, provided theatom is su\u000eciently cold.\nFor a speci\fc con\fguration of optical \felds, one can\ninduce the e\u000bective Hamiltonian [35]\n^HSO=^p2\n2m+1\n2m!2r2+vx^px^\u001bx+vy^py^\u001by;(1)\nwhere ^p= (^px;^py) is the momentum operator, ^r= (^x;^y)\nis the position operator, mis the mass of the atom, and\n!the frequency of a harmonic trap. The operator ^ \u001bi\nis thei-th Pauli matrix in pseudo-spin space, and the\nvelocitiesvicouple pseudo-spin to an e\u000bective momen-\ntum dependent Zeeman \feld, B(p) = (vxpx;vypy). This\nmomentum-dependent Zeeman \feld can simulate any\ncombination of the Rashba [38] and Dresselhaus [39] SO-\ncouplings experienced in semiconductor quantum wells\nand systems alike.\nIn the absence of a trap, != 0, the spectrum of (1) is\nE\u0016(px;py) =1\n2m\u0000\np2\nx+p2\ny\u0001\n+\u0016q\n(vxpx)2+ (vypy)2(2)\nwith the corresponding eigenfunctions\nj \u0016;pi=eim(vxx+vyy)j'\u0016i (3)\nwhere\nj'\u0016i=1p\n2\u0010\ne\u0000i'=2j\"i\u0000\u0016ei'=2j#i\u0011\n; (4)\nis a spinor with helicity \u0016=\u00061 and'=\narctan(vypy=vxpx). These states have well de\fned mo-\nmentum, but have no velocity since h_ri=hrpHi= 0,\nprovided the optical \felds are maintained. Note further\nthat the eigenstates are parametrically dependent on px\nandpy.\nWe remark that for an isotropic SO-coupling, vx=vy,\nthe Hamiltonian (1) is equivalent to the dual E\u0002\"Jahn-\nTeller model, frequently appearing in chemical/molecular\nphysics and condensed matter theories [37]. With a\nsimple unitary rotation of the Pauli matrices, the SO-\ncoupling attains the more familiar Rashba form [38] (or\nequivalently Dresselhaus form [39]). For vx6=vy, i.e.\nwhen the SO-coupling is anisotropic, the model becomes\nthe dualE\u0002(\fx+\fy) Jahn-Teller model [37]. In par-\nticular, the ^ z-projection of total angular momentum,\n^Jz=^Lz+^\u001bz\n2, is a constant-of-motion for the isotropic\nbut not for the anisotropic model. More precisely, break-\ning of the SO isotropy implies a reduction in symmetry\nfromU(1) toZ2.\nThroughout we will use dimensionless parameters\nwhere the oscillator energy Eo= \u0016h!sets the energy-\nscale,l=p\n\u0016h=m! the length-scale, and the characteristic\ntime is\u001c=!\u00001. We note that for typical experimental\nsetups,!\u001810\u0000100 Hz and m(v2\nx+v2\ny)=\u0016h\u00181\u000010 kHz.\nMoreover, in what follows we will refer to pseudo-spin\nsimply as spin. When necessary, we introduce a param-\neterhserving as a dimensionless Planck's constant, i.e.\nh\u0016h. In this way, hcontrols the strength of Planck's con-\nstant and by varying it we can explore how the dynamics\ndepends on \u0016 h.3\nB. Adiabatic model\nThe large ratio of the SO energy to trapping energy,\ntypicallymv2=\u0016h!\u001810\u00001000, suggests that a BOA [32]\nwill be valid for experimental implementations. The sep-\naration of timescales of the spin and orbital degrees of\nfreedom implies that in some regimes we can factorize\nthe wavefunction as the product of spin and orbital wave-\nfunctions. A spin initially aligned with the adiabatic\nmomentum-dependent magnetic \feld B(p) will remain\nlocked to that \feld at future times, provided the center\nof mass motion avoids the DP. We then solve for the spin\nwavefunction at an instantaneous orbital con\fguration\nand use this answer to \fnd an adiabatic potential for the\norbital motion. This is in analogy with the traditional\nBOA, where the electronic and nuclear wavefunctions are\napproximated as a product, and the electron degrees of\nfreedom instantaneously adjust to the adiabatic potential\ngiven by the nuclear degrees of freedom.\nIn our BOA, we have chosen the adiabatic states [32]\nfor the orbital motion to be the spin-helicity states, given\nby (4). If we project the Hamiltonian into the basis j'\u0016i,\nwe arrive at the adiabatic potential\n^H(\u0016)\nad=^x2\n2+^y2\n2+^p2\nx\n2+^p2\ny\n2+\u0016q\nv2x^p2x+v2y^p2y:(5)\nThe trap thus takes the role of kinetic energy and (5) can\nbe pictured as a particle in a (dual) adiabatic potential\nV\u0016(^px;^py) =^p2\nx\n2+^p2\ny\n2+\u0016q\nv2x^p2x+v2y^p2y: (6)\nshown in Fig. 1 for both the isotropic (a) and anisotropic\n(b) cases. We have neglected non-adiabatic corrections\narising from the vector potential and the Born-Huang\nterm [40]. For example, an additional scalar potential\nVnad(px;py)\u0018(vxvy)2(p2\nx+p2\ny)\n\u0000\nv2xp2x+v2yp2y\u00012: (7)\nwill emerge from the action of the SO-coupling on the\nspinorj'\u0016i. This term is order Vnad\u0018h'jr2\npj'i\u00181=p2.\nThere will also be an additional vector potential term\nA\u00181=p. The non-adiabatic corrections diverge near the\nDP, but then fall o\u000b rapidly at \fnite p. The adiabatic\napproximation, i.e. BOA, will be valid if the particle\navoidsp= 0. We will show later that this condition is\nmet if the particle is in the lower band, \u0016=\u00001, and has\nenergyE < 0.\nImposing the BOA, any state propagating on the lower\nadiabatic potential will be denoted \b( px;py;t), and it is\nunderstood that\n\b(px;py;t) =\u001e(px;py;t)j'\u0000i: (8)\nThe real space wave function \t( x;y;t ) is given as\nusual from the Fourier transform of \u001e(px;py;t).\nThe time-evolution follows from \u001e(px;py;t) =exp\u0010\n\u0000i^H(\u0000)\nadt\u0011\n\u001e(px;py;0). It is also clear that the\nstate \b(px;py;t) determines the spin orientation which\nis inherent in the ket-vector j'ii. More explicitly, the\ntime-evolved Bloch vector\nR(t) = (Rx(t);Ry(t);Rz(t))\u0011(h^\u001bxi;h^\u001byi;h^\u001bzi) (9)\ntakes the form\nRx(t) =Z\ndpxdpyj\u001e(px;py;t)j2cos(');\nRy(t) =Z\ndpxdpyj\u001e(px;py;t)j2sin('); (10)\nRz(t) = 0\nin the BOA, and it is remembered that the parameter '\ndepends on pxandpy. Note that the Bloch vector pre-\ncesses in the equatorial spin xy-plane. If the wave packet\n\b(px;py;t) is sharply localized, a crude approximation\nfor the Bloch vector is given by\n\u0016Rx(t) =vx\u0016px(t)q\n(vx\u0016px(t))2+ (vy\u0016py(t))2; (11)\n\u0016Ry(t) =vy\u0016py(t)q\n(vx\u0016px(t))2+ (vy\u0016py(t))2; (12)\n\u0016Rz(t) = 0; (13)\nwhere \u0016p\u000b(t) =R\ndpxdpyj\b(px;py;t)j2p\u000bwith\u000b=x; y.\nIII. CLASSICAL DYNAMICS\nQuantum chaos is often de\fned by having an under-\nlying chaotic classical model. For the full model (1),\nthe spin degrees-of-freedom cannot be eliminated in a\nstraightforward manner in the vicinity of the Dirac point\nand as a consequence it is not a priori clear what the\nunderlying classical model would be in this regime. On\nthe other hand, in the BOA, the adiabatic Hamiltonian\n^H(\u0000)\nadcan serve as our classical model Hamiltonian. Still,\nit should be noted that we assume h^H(\u0000)\nadi\u001c0, such that\nthe spectrum contains a su\u000eciently large number of en-\nergies below E= 0. Furthermore, we point out that jus-\nti\fcation of the BOA does not necessarily imply approval\nof a semi-classical approximation which depends on the\nsystem energy and the actual shape of the dual poten-\ntialV\u0000(px;py). Nevertheless, as we will demonstrate in\nthe following, for the chosen parameters, the agreement\nis indeed very good.\nThe classical equations-of-motion of the Hamiltonian4\nV ( )p+V ( )p-V ( )p+\nV ( )p-py\npypxpx(a)\n(b)\nFIG. 1. Adiabatic potentials of the isotropic (a) and\nanisotropic (b) SO-coupled models. In both \fgures, the E= 0\nplane is the one including the DP at px=py= 0. A nec-\nessary, but not su\u000ecient, condition for the validity of the\nBOA is that E < 0. In (a), the lower adiabatic potential\nV\u0000(px;py) has the characteristic sombrero shape. By con-\nsidering an anisotropic SO-coupling, the rotational symme-\ntry is broken and V\u0000(px;py) possesses two global minima at\n(px;py) = (0;\u0006vy).\n−20 0 20−20−1001020\nx−20 0 20−50050\nxpx(a) (b)\nFIG. 2. Two examples of classical trajectories (( x(t);Px(t))\nfor regular (a) and chaotic (b) dynamics. In (a), typical for\nregular motion the trajectories evolve upon a tori. Contrary,\nin (b) the trajectory is much more irregular which is char-\nacteristic for the chaotic evolution. The regular motion is\ncalculated for the SO-coupling strengths vx=vy= 30, and\nthe chaotic motion with vx= 20 andvy= 30. In both cases,\nthe energy is E=\u0000192.\nFIG. 3. Poincar\u0013 e sections of the Rashba SO-coupled adiabatic\nmodel (5) for the intersections y= 0 (a) and py= 0 (b).\nThe initial energy is E=\u0000192, the SO-coupling strengths\nvx=vy= 30, and the number of simulated semi-classical\ntrajectories 18.\n^H(\u0000)\nadare\n_x=px\u0000v2\nxpxq\nv2xp2x+v2yp2y; (14)\n_px=\u0000x; (15)\n_y=py\u0000v2\nypyq\nv2xp2x+v2yp2y; (16)\n_py=\u0000y: (17)\nFor the Rashba SO-coupling, vx=vy=v, there is one\nunstable \fx point ( px;py) = (0;0) and a seam of sta-\nble \fx points p2\nx+p2\ny=v2, see Fig. 1 (a). For the\nanisotropic case, vy> vx, there are three unstable \fx\npoints, (px;py) = (0;0) and (px;py) = (\u0006vx;0), while\nthere are two stable \fx points ( px;py) = (0;\u0006vy), see\nFig. 1 (b).\nThe classical energy E(x;px;y;py) =p2\nx=2 +p2\ny=2 +\nx2=2 +y2=2\u0000q\nv2xp2x+v2yp2ydetermines a hypersurface\nin phase space for any given energy E(x;px;y;py) =E0.\nThe semi-classical trajectories ( x(t);px(t);y(t);py(t)) live\non this surface. For the integrable case, vx=vy,\nthese surfaces form di\u000berent tori characteristic for quasi-\nperiodic motion. As the rotational symmetry is slightly\nbroken,vx6=vy, the tori deforms and the motion loses\nits quasi-periodic structure [29]. This is the generic\ncrossover from regular to chaotic classical dynamics. As\nan example of this generic behavior, we show in Fig. 2\ntwo randomly sampled trajectories in the xpx-plane for\nregular (a) and chaotic (b) evolution. For all results of\nthis section, we solve the set of coupled di\u000berential equa-\ntions (14) using the Runge-Kutta (4,5) algorithm mod-\ni\fed by Gear's method , suitable for sti\u000b equations. We\nhave also numerically veri\fed our results employing dif-\nferent algorithms [41]. As will be discussed further below,\neven in the chaotic regime, periodic orbits may persist\nand will greatly a\u000bect the dynamics, both at a classical\nand a quantum level [42]. Such orbits are not, however,\nvisible from Fig. 2.\nThe semi-classical behavior of classical dynamical sys-\ntems is favorable visualized using Poincar\u0013 e sections [43].5\nCorresponding sections for the system (14)-(17) are de-\npicted in Figs. 3 and 4. In the \frst \fgure we display the\nPoincar\u0013 e sections in the xpxplane for the intersections\ndetermined by y= 0 (a) or py= 0 (b) of the isotropic\nmodel with the SO-coupling amplitudes vx=vy= 30.\nThe initial energy is taken as E=\u0000192, well below\nthe DP, consistent with the BOA. In (b), the section\nde\fned bypy= 0, the evolution results in ellipses in the\nPoincar\u0013 e section, characteristic of quasi periodic motion.\nThe structure of the Poincar\u0013 e section for y= 0 (a) is\nsomewhat more complex. This can be understood from\nthe sombrero shape of the adiabatic potential V\u0000(px;py);\nfor givenx=x0,px=p0\nx,y= 0, and energy E0, there\nare four possible values of py, and this multiplicity of\npossiblepy's allow the \\curves\" in Fig. 3 (a) to cross.\nIt should be noted that any single curve does not cross\nitself. Furthermore, by adding the pyvalues to Fig. 3\nwe have veri\fed that neither of the corresponding three\ndimensional curves cross.\nFigure 4 presents two examples for anisotropic SO-\ncouplings, both with vx= 20 and vy= 30. The\nquasi-periodic evolution is lost and the dynamics become\nmixed, with regions of both chaos and regular dynamics.\nThe same conclusions were found in Ref. [44] where a re-\nlated Jahn-Teller model was studied. The two lower plots\nconsider the same energies as in Fig. 3, i.e. E=\u0000192,\nwhile for (a) and (b) E=\u000088. Expectedly, the higher\nenergy increases the accessible volume of phase space.\nFor both energies we \fnd islands free from chaotic tra-jectories. As will be demonstrated in the next section,\nwithin these islands the evolution is regular and the sys-\ntem does not thermalize. The plots also demonstrate\nclear structures also appearing in the chaotic regimes of\nthe Poincar\u0013 e sections in which the density of solutions\nchanges.\nIV. QUANTUM DYNAMICS\nThe idea of this section is to analyze how the corre-\nsponding quantum evolution is a\u000bected by whether the\nclassical dynamics is regular or chaotic. Of particular im-\nportance is the long time evolution in which the system\nstate may or may not equilibrate. However, we study\nalso the short time dynamics arising for a localized wave\npacket traversing the Dirac point. In this regime, clearly\nthe classical results of the previous section does not hold.\nTo study the system beyond the classical approxima-\ntion, we solve the time-dependent Schr odinger equation,\nrepresented by the Hamiltonians (1) or (5), to obtain\nthe corresponding wave function \t( x;y;t ) at time t.\nNote that for the full model (1), the wave function con-\ntains the spin degree-of-freedom \t( x;y;t ) = \"(x;y;t )j\"\ni+ #(x;y;t )j#i. The non-equilibrium initial state ap-\npears after a quench in the center of the trap. We prepare\nthe system in a quasi-ground state for a shifted trap, and\natt= 0 suddenly move the trap center to xs=ys= 0,\nV(x;y) =(x\u0000xs)2\n2+(y\u0000ys)2\n2;\u001a\nxs6= 0 and=orys6= 0;t<0;\nxs=ys= 0; t\u00150:(18)\nBy \\quasi-ground state\" in an anisotropic SO-coupled\nsystem, we consider an initial state predominantly pop-\nulated in one of the two minima of the adiabatic po-\ntentialV\u0000(px;py). This seems experimentally reasonable\nwhere small \ructuations will favor one of the two min-\nima. For the isotropic case, the phase of \b( px;py;t= 0)\nis taken randomly in agreement with symmetry breaking.\nGiven the evolved states \t( x;y;t ), we are interested in\nthe Bloch vector (10) or its components, and the distri-\nbutionsj\b(px;py;t)j2andj\t(x;y;t )j2.\nThe numerical calculation is performed employing the\nsplit-operator method [45] which relies on factorizing, for\nshort times \u000et, the time-evolution operator into a spatial\nand a momentum part. For small SO-couplings vxand\nvy, the method is relatively fast, while as vxand/orvyare\nincreased the time-steps \u000etmust be considerably reduced\nand the necessary computational power rises rapidly. In\naddition, for large vxandvy, the grid sizes of position\nand momentum space must be increased, which also in-\ncreases the computation time. Thus, we will limit the\nanalysis to SO-couplings vx; vy\u001430. Furthermore, we\nhave found by convergence tests that the full model (1)requires much smaller time-steps \u000etthan the adiabatic\none (5), and most of our simulations will therefore be re-\nstricted to energies E < 0 for which the BOA is justi\fed.\nThe full quantum simulations are complemented by the\nsemi-classical truncated Wigner approximation (TWA),\nwhich has turned out very e\u000ecient in order to re-\nproduce quantum dynamics [46]. The TWA considers\na set ofNdi\u000berent initial values ( xi;yi;pxi;pyi) ran-\ndomly drawn from the distributions j\t(x;y;0)j2and\nj\b(px;py;0)j2. These are then propagated according to\nthe classical equations-of-motion (14). The propagated\nset (xi(t);yi(t);pxi(t);pyi(t)) gives the semi-classical dis-\ntributions, from which expectation values can be evalu-\nated.\nA. Short time dynamics\nBefore investigating the prospects of thermalization,\nwe \frst consider short time dynamics, by which we mean\ntime-scales where the wave packet remains localized. In6\nFIG. 4. Poincar\u0013 e sections of the anisotropic SO-coupled adia-\nbatic model (5) for y= 0 (a) and (c), and for py= 0 (b)\nand (d). The initial energies are E=\u000088 (a) and (b),\nandE=\u0000192 (c) and (d), and the SO-coupling strengths\nvx= 20 andvy= 30 for both cases. The corresponding maxi-\nmum Lyaponov exponents have been derived to \u0015\u00190:12 and\n\u0015= 0:090 respectively. The number of semi-classical trajec-\ntories is the same as for Fig. 3, namely 18.\nthis respect, it is tempting to think of the dynamics as\nsemi-classical. However, in the vicinity of the the DP any\nclassical description would fail. Equivalently, the spin\ndegrees-of-freedom will show large \ructuations which are\ndi\u000ecult to capture classically. The short time dynamics\nis consequently most interesting for situations with ener-\ngiesE > 0 where both the semi-classical approximation\nand the BOA break down, implying that the simulation\n0 5 10 15 20 25 30−101\ntRα−101Rα  (a)\n(b)FIG. 5. Bloch vector components Rx(dashed lines) and Ry\n(solid lines). For the upper plot (a), the trap has been dis-\nplaced in th y-direction,xs= 0 andys= 28, while in the lower\nplot (b) the displace direction is the perpendicular, xs= 28\nandys= 0. In both \fgures, vx= 10 andvy= 15, and the\naverage energy \u0016E\u0019280.\nis performed using the full model Hamiltonian (1). For\nthese energies, the wave packet can traverse the DP and\npopulation transfer between the two adiabatic potentials\nV\u0016(^px;^py) typically occurs. It is known that such non-\nadiabatic transitions can play important roles for the dy-\nnamics, and that the actual transition probabilities be-\ntween the two potentials may be extremely sensitive to\nsmall \ructuations in the state [31, 48]. In this subsection\nwe especially address such non-adiabatic e\u000bects.\nThere are indeed several relevant time-scales in the dy-\nnamics: (i) The spin precession time Tspgives the typical\ntime for spin evolution and is proportional to the e\u000bective\nmagnetic \feldjB(p)j, (ii) the classical oscillation period\nTcl= 2\u0019, and (iii) the thermalization time Tth, which\nestimates the time it takes for the system to thermalize,\ni.e. when expectation values become approximately time\nindependent. Normally, the magnitudes of these times\nfollow the list above (in growing order), except in the\nvicinity of the DP where Tsp\u0018Tclor evenTsp\u001cTcl\nvery close to the DP. While the \frst two are well de\fned,\nde\fning the last one is non-trivial. We can say that ( i)\nand (ii) characterizes short time-time scales, and ( iii)\nlong time-scales. As will be numerically demonstrated,\nthe thermalization time turns out to scale as log( h\u00001)=\u0015,\nwherehis the e\u000bective dimensionless Planck's constant\nand\u0015the maximum Lyaponov exponent. This suggests\nthat the thermalization time agrees with the Ehrenfest\ntime\nTE= log(V=h)=\u0015; (19)\nwithVthe e\u000bective occupied phase space volume. TE\nis also the typical time-scale where semi-classical (TWA)\nexpectation values no longer agree with quantum expec-\ntation values, which can be seen as a breakdown of Ehren-\nfest's theorem [49].\nFrom the form of the non-adiabatic coupling (7), it fol-\nlows that transitions between the adiabatic states (4) are\nrestricted to the vicinity of the DP. These non-adiabatic\ntransitions are manifested as rapid changes in the Bloch7\nvector (10). In Fig. 6 we present two examples of the\nBloch vector evolution (in both examples Rz(t)\u00190). In\nFig. 6 (a), the trap has been shifted in the y-direction.\nFor short times, the shift of the trap induces a build-up of\nmomentum in the opposite y-direction as a consequence\nof the Ehrenfest theorem. This adds with the non-zero\ny-component of momentum before the quench. The av-\nerage momentum in the x-direction remains zero and as\na consequence Rx(t)\u00190, see Eq. (11).\nThese dynamics change qualitatively if the trap is\nshifted in the x-direction instead of the y-direction. For\nsu\u000eciently large shifts of xs, the wave packet will set o\u000b\nalong the adiabatic potentials and encircle the DP. The\nspin dynamics should therefore not display the same type\nof \\jumps\" that appear when the wave packet traverses\nthe DP. Moreover, since the average momentum in the\nx-direction is in general non-zero, Rx(t) will also be non-\nzero. The results are demonstrated in Fig. 6 (b). Com-\npared to the \frst example in (a), the wave packet does\nnot spend much time near the DP so the wave packet\ndelocalization occurs more slowly. To a large extent the\nevolution is driven by harmonicity, in contrast to the ex-\nample of Fig. 6 (a) where the anharmonicity of the Born-\nHuang term, and the non-adiabatic transitions near the\nDP, push the system away from semi-classical evolution.\nThe \fgure demonstrates how the dynamics can depend\non the initial conditions, in both (a) and (b), \u0016E\u0019280 but\nthe wave packet broadening starts earlier in (a) than in\n(b). This type of state-dependence has been discussed in\nRef. [47]; generically there is a period tswhere the width\nof the wave packet stays nearly constant, followed by a\nrapid broadening. The time-scale tsdepends strongly\non the initial conditions, while the proceeding evolution\naftertsseems pretty generic for chaotic systems.\nB. Long time dynamics; thermalization\nWhenever we consider an anisotropic SO-coupling,\nvx6=vy, from the Figs. 3 and 4 it is clear how the\nadiabatic classical model becomes chaotic. Beyond the\nadiabatic model, it has been shown [50] that the full\nanisotropic model, i.e., E\u0002(\fx+\fy) Jahn-Teller model,\nis chaotic in the sense of level repulsion [51] of eigenen-\nergies. For the isotropic E\u0002\"Jahn-Teller model, on\nthe other hand, the level repulsion e\u000bect is not as evi-\ndent, however a weak repulsion also in this model signals\nemergence of quantum chaos [52].\nThe goal of this subsection is to study the long time\ndynamics of the system; speci\fcally if equilibration oc-\ncurs, and if so, does the equilibrated state mimic a ther-\nmal state. A distinguishing property of thermal states is,\nfor example ergodicity, i.e., the distributions j\t(x;y;t )j2\nandj\b(px;py;t)j2spread out over their accessible energy\nshells. Moreover, for a thermally equilibrated state, the\ndistributions show seemingly irregular interference struc-\ntures on scales of the order of the Planck cells, which\nnormally become even \fner in the Wigner quasi distri-\nFIG. 6. (Color online) Distributions j\t(x;y;t f)j2((a) and\n(c)) andj\b(px;py;tf)j2((b) and (d)) at tf= 400 for the\nRashba SO-coupled model. At time t= 0, the trap is sud-\ndenly displaced from x0=y0= 16 tox0=y0= 0. The\ninitial ground state is then quenched into a localized excited\nstate. The upper two plots (a) and (b) display the results\nfrom full quantum simulations of the adiabatic model (5),\nwhile the lower plots (c) and (d) show the corresponding\nsemi-classical TWA distributions. The average semi-classical\nenergy \u0016E\u0019\u0000192 with a standard deviation \u000e\u0016E\u001922. The\ndimensionless SO-coupling strengths vx=vy= 30.\nbution [53{55]. Non-thermalized states, on the contrary,\ntypically leave much more regular traces of quantum in-\nterference in their distributions. While such often sym-\nmetrical structures are absent for thermalized states, we\nwill demonstrate that thermalized distributions may still\nshow clear density \ructuations on scales larger than the\nPlanck cells. These are examples of quantum scars and\nthey are remnants of classical periodic orbits [42].\nWe begin by considering the adiabatic isotropic model\nwithvx=vy= 30, and trap shifts xs=ys= 16. Af-\nter a quench of the trap position, the initial energy is\n\u0016E=h^H(\u0000)\nadi\u0019\u0000 192. This energy corresponds to the\nenergy of the Poincar\u0013 e section presented in Fig. 3. The\nresulting distributions are shown in Fig. 7 (a) and (b)\nafter a propagation time tf= 400 . The \fnal time tfap-\nproximates 60 classical oscillations. Both the real space\ndensityj\t(x;y;t )j2and momentum density j\b(px;py;t)j2\nreveal clear interference patterns as anticipated. The DP\nat the origin ( px;py) = (0;0) repels the wave function\nforming a \\hole.\" The lack of zero momentum states in-\nduces a mass \row in real space and a similar \\hole\" in\nits distribution. The classically energetically accessible8\nFIG. 7. (Color online) Same as Fig. 6 but for the anisotropic\nSO-coupled model with vx= 20 andvy= 30. The largely\npopulated regions are so called quantum scars and derive from\nproperties of the underlying classical model, i.e. they are not\noutcomes of some coherent quantum mechanism.\nregions are given by\nx2+y2\u00142Emax+v2\ny;\np2\nx+p2\ny\u00002q\nv2xp2x+v2yp2y\u00142Emax;(20)\nwhereEmaxis the maximum energy component notice-\nably populated by the state.\nThe quantum results are compared with the TWA dis-\ntributions displayed in the lower plots (c) and (d) of the\nsame Fig. 7. The same kind of ring-shape is obtained,\nand the concentration in density appears at the same lo-\ncations for both the quantum and classical simulations.\nExpectedly, the quantum interference taking place within\nthe wave packet is not captured by the TWA. This fol-\nlows since single semi-classical trajectories are treated\nindependently, i.e. added incoherently, while a quantum\nwave packet must be considered as one entity. For a TWA\napproach of the full isotropic E\u0002\"Jahn-Teller model (1)\nwe refer to Ref. [56].\nThe situation is drastically changed when we break\nthe rotational U(1) symmetry by assuming vx6=vy. The\nresult for low initial energy is depicted in Fig. 7 (a) and\n(b). The energy is comparable to the potential barrier\nseparating the two minima in the adiabatic potentials,\nand as a consequence, the wave packet is predominantly\nlocalized in the left minima. The density modulations\nseems now much more irregular in comparison to Fig. 6.\nIn the seemingly random density distribution, some clear\ndensity maxima emerge, both in momentum as well as\nin real space. These density accumulations derive from\nperiodic orbitals of the underlying classical model and\nFIG. 8. (Color online) Same as Fig. 7 but for an initial energy\nE > 0. The dimensionless SO-couplings vx= 14 andvy= 21,\nwhile the shifts xs=ys= 16 giving an average energy \u0016E=\nh^HSOi\u001936:5.\n0100200300∆x(t)\n0 10 20 30 40 50 60 70 800100200300\nt∆x(t+δ)(a)\n(b)h\nFIG. 9. Examples of the phase space area \u0001 x(t) for di\u000berent\nh-values (h= 1;2;3; :::; 10). The upper plot (a) gives \u0001 x(t)\nwithout shifting the time, while for the lower one (b) time has\nbeen shifted by \u000e= log(h)=\u0015. The arrow indicates increasing\nh-values. It is clear how the spread in \u0001 x(t) between di\u000berent\nhvalues is suppressed when we shift the time. The trap shifts\nxs=ys= 19 resulting in an energy \u0016E\u0019\u000088. The maximum\nLyaponov exponent \u0015= 0:18.\nare termed quantum scars [42, 57, 58]. The appearance\nof scars is an example of the classically chaotic model\nleaving a trace in its quantum counterpart. The scars\nare also captured in the semi-classical TWA, shown in\nFig. 7 (c) and (d), supporting their classical origin.\nWhen we shift the trap for larger values on xsand\nys, the energy is increased and at some point the BOA\nbreaks down. An example, obtained from integrating the\nfull model (1), is presented in Fig. 8. For these higher\nenergies there are no signs of quantum scars. As for the\nsituation of Fig. 7, the spread of the wave packet and the\nirregular interference patterns indicates thermalization.\nThis far we have demonstrated thermalization for the\nanisotropic SO coupled model, but not discussed corre-\nsponding time-scales. One related question is how the\nevolution of various expectation values scale with h(di-\nmensionless Planck constant). It has been argued that\nthe Ehrenfest time, Eq. (19), can be a measure of the\nthermalization time [31]. We will now explore how the\nphase space area \u0001 \u000b(t) = \u0001\u000b\u0001p\u000b(\u000b=x; y), where9\n−7 −6.5 −6 −5.5 −5 −4.5 −4−0.02−0.0100.010.020.030.04\nx|ψ(x,0)|\nFIG. 10. (Color online) Sections of j (x;y= 0)jfor di\u000berent\nvalues on the dimensionless Planck's constant h:h= 1 (black\nsolid line), h= 2 (blue dotted line), and h= 3 (red dashed\nline). The \fnal time tf= 80,xs=ys= 16, and vx= 14\nandvy= 20. As a comparison between classical and quan-\ntum results, we also include the TWA results as a green solid\nline, calculated for h= 1. The green line has been shifted\ndownward with 0.02 for clarity.\n\u0001\u000band \u0001p\u000bare the variances of ^ \u000band ^p\u000brespectively,\nevolves for di\u000berent values of h. Since \u0001 x(t) and \u0001y(t)\nbehave similarly we focus only on \u0001 x(t). For thermaliza-\ntion, \u0001x(t)\u0001y(t) is an e\u000bective measure of the covered\nphase space volume, and for large times tit should more\nor less approach the accessible phase space volume as the\ndistribution spreads over the whole energy shell. We have\nchosen to study \u0001 x(t) since it \ructuates relatively little\nbefore reaching its asymptotic value. In Fig. 9 (a) we dis-\nplay \u0001x(t) for 10 di\u000berent values on hranging from h= 1\ntoh= 10. The arrow in the plot shows the direction of\nincreasingh's. As is seen, by increasing hthe wave packet\nbroadening starts earlier and the state equilibrates faster.\nIf the Ehrenfest time TEsets the typical time scale in the\nprocess, by shifting the time with \u000e= log(h)=\u0015we should\nrecover a \\clustering\" of the curves. This is indeed ver-\ni\fed in Fig. 9 (b) where the curves have been shifted in\ntime by\u000e. The corresponding Lyaponov exponent \u0015has\nbeen optimized in order to minimize the spread in the\ncurves. The obtained value \u0015= 0:18 is somewhat larger\nthan the numerically calculated one \u0015= 0:12 but still of\nthe same order. The picture also makes clear that the\nwave packet broadening kicks in after some time tsas\nanticipated above.\nThe route to thermalization can typically be di-\nvided into; ( i) a classical drift, and ( ii) quantum dif-\nfusion [31]. The role of the quantum di\u000busion for ther-\nmalization was analyzed in Ref. [31], where it was found\nto \\smoothen\" the phase space distributions preventing\nsub-Planck structures. For the classical drift there is no\nlower bound on the \fneness of density structures that can\nform, and characteristic for classical chaotic dynamics is\nthat ever \fner formations build-up as a result of the typ-\nical \\stretching-and-folding\" mechanism. However, in aquantum chaotic system, when the structures reach the\nPlanck cell regime, the quantum pressure becomes too\nstrong and the quantum di\u000busion then prevents any fur-\nther structures to form. Thus, Planck's constant sets\na lower bound for the \ructuations in the distributions.\nThis quantum smoothening is demonstrated in Fig. 10,\nwhere we plot a section of j (x;y= 0)jfor di\u000berent values\non the scaled dimensionless Planck's constant h(= 1;2;3\nfor black, blue, and red lines respectively). The e\u000bect is\nclearly seen in the \fgure. A similar pattern is found\n(not shown) also for the momentum distributions. For\nthe classical system, corresponding to h= 0, there is no\nlower limit on how \fne the structures can be. We indicate\nthis by also plotting the TWA results in the same \fgure\nas a green line (note that the green line has been shifted\ndownward in order to separate it from the quantum re-\nsults). The number of trajectories used for the \fgure is\n250 000, and if we would like to produce \fner structures\n(by propagating the system for longer times) we would\nneed many more trajectories and the simulation would\nrapidly become very time consuming.\nRelated to the above discussion a note on quantum\nphase space distributions is in order. It is well known\nthat sub-Planck structures are common in the Wigner\ndistribution [53]. This is not contradicting any quan-\ntum uncertainty relation. After all, the Wigner distribu-\ntion is not a proper probability distribution, despite the\nfact that its marginal distributions reproduce the cor-\nrect real and momentum space probability distributions.\nThe Husimi Q-function, while not possessing the proper\nmarginal distributions, is positive de\fnite and lacking\nsingularities, and it is indeed found that the Q-function\ndoes not support sub-Planck structures [60].\nWe \fnish this subsection by analyzing the dynamics in\nthe islands of the Poincar\u0013 e sections of Fig. 4 where the\nclassical theory predicts regular evolution. From Fig. 4\n(c) we have that for px\u001920 andx\u0019y\u00190 the evolu-\ntion should be regular. We can achieve such a situation\nby using the quench-shifts xs= 20 andys= 0. As for\nthe examples above, we propagate the state for a time\ntf= 400, and the resulting distributions are given in\nFig. 11. The striking di\u000berence with Figs. 7 and 8 is evi-\ndent; no irregular structure is apparent, but clear regular\ninterference patterns are. We have veri\fed that the in-\nterference structure prevails also after doubling the time,\ntf= 800.\nC. Proposed experimental realization\nMuch of the above dynamics can be observed in a sys-\ntem of cold atoms with synthetic SO-coupling, for exam-\nple, a system of87Rb with a synthetic \feld induced by\nthe 4-level scheme [33]. In this system, the recoil energy\nEr=mv2\u0018\u0016h\u000250 kHz. The synthetic \feld limits the\nlifetime of the experiment to tl\u00181s[4, 10]. To push\nthe experiment into the long time regime, we will use\na trapping frequency of !=2\u0019= 30 Hz. These parame-10\nFIG. 11. (Color online) Same as Fig. 7 but for the shifts\nxs= 20 andys= 0. For the given dimensionless parameters,\nthe initial state is such that its dynamics should be regular\naccording to the corresponding Poincar\u0013 e section, Fig. 4. The\nenergy \u0016E\u0019\u0000250.\nters will give a dimensionless value of vy=q\nEr\n\u0016h!\u001811,\nwithvxtunable between 0 and 11. The large trapping\nfrequency will provide a su\u000ecient number of oscillations\nfor thermalization to occur. We could consider values of\nvy\u001830 by decreasing the trapping frequency to 10Hz,\nbut then the lifetime of the system may be at the boarder\nfor thermalization.\nThe condensate can be adiabatically loaded to one of\nthe two states at the bottom of the momentum-space\npotential, de\fned by p=\u0006mvy^y. The quench can then\nbe preformed by shifting the minimum of the real-space\ntrapping potential. We then let the system evolve until\nwe reach either the thermalization time, or the lifetime\nof the experiment. The momentum distribution can be\nmeasured with a destructive time-of-\right (TOF) mea-\nsurement [4, 10], which should reveal thermalization as\nwell as signatures of quantum scars. Repeated experi-\nmental measurements allow for time-resolved calculation\nof expectation values. Similarly, the quantum spin jumps\nnear the DP, as discussed in Sec. IV A, can be observed\nusing a spin-resolved TOF measurement.\nAs a \fnal remark, for a weakly interacting gas we\nwork near a Feschbach resonance [61]. However, for re-\nalistic parameters [62], we estimate a scattering length\nas\u00183\u000210\u00009m,N\u00185\u0002105atoms, and a transverse\nharmonic trapping frequency !z\u0018100 Hz. For these\nparameters, the characteristic scale of the non-linearity\nis\u0016\u0018h\u00021kHz, which is smaller than the recoil energy\nabove, suggesting the non-linear term will play only a\nminor role. We have numerically veri\fed that the results\ndo not change qualitatively by solving the corresponding\nnon-linear Gross-Pitaevskii equation. Indeed, we \fnd the\ndeviations with a non-linearity are not large enough to\nbe seen by eye.\nV. CONCLUSIONS\nIn this paper we studied dynamics, deriving from a\nquantum quench, in anisotropic SO-coupled cold gases,\nfocusing primary on aspects arising from the fact thatthe underlying classical model is chaotic. The evolution\nof the initially localized wave packet on its way to equili-\nbration has been analyzed, and we have shown how a clas-\nsical period of limited spreading is followed by a collapse\nregime dominated by rapid spreading. After the collapse\nperiod, the wave packet is maximally delocalized, but\nstill possesses quantum interference structures. At the\nEhrenfest time, the state has approximately equilibrated\nas is seen in the decay of expectation values, as well as\nseemingly irregular density \ructuations both in real and\nmomentum space. We showed that the \fne structure of\nthese \ructuations are limited by the quantum di\u000busion,\nand thereby the size of the Planck's constant h. For the\nisotropic model, after the collapse no thermalization is\nfound, as is expected from the integrability of the under-\nlying classical model.\nFor smaller energies, when the wave packet predom-\ninantly populates one of the dual potential wells, ther-\nmalization is again seen. Here, however, an additional\nphenomenon appears in terms of quantum scars. These\ndensity enhancements emerge along classically periodic\norbits. They are classical in nature and long lived. Quan-\ntum scars have also been studied in di\u000berent cold atom\nsettings; atoms in an optical lattice and con\fned in an\nanisotropic harmonic trap [58]. The results on thermal-\nization presented in this work is most likely also applica-\nble to the set-up of Ref. [58]. We also demonstrated that\nfor certain \fne tuned initial states, the dynamics stays\nregular even in the anisotropic model. In the classical\npicture, these solutions correspond to the ones belong-\ning to regular islands in the otherwise chaotic Poincar\u0013 e\nsections.\nWe argue that the present system is ideal for studies\nof quantum chaos and quantum thermalization for nu-\nmerous reasons. The system parameters can be tuned\nexternally by adjusting the wavelength of the lasers in-\nducing the SO-coupling, and as we discussed in Sec. IV C\nthe SO dominated regime is reachable in current exper-\niments. Moreover, both state preparation and detection\nare relatively easily performed in these setups. Equally\nimportant, the system is well isolated from any environ-\nment and coherent dynamics can be established up to\nhundreds of oscillations which is well beyond the themral-\nization time. The energy of the state is simply controlled\nby the trap displacement, and it should for example be\npossible to give the system small energies such that the\natoms reside mainly in one potential well where quantum\nscars develop.\nWe \fnish by pointing out that the present model is also\ndi\u000berent from most earlier studies on quantum thermal-\nization [18, 24] in the sense that the dynamics is essen-\ntially \\single-particle\" and not arising from many-body\nphysics. Related to this, we have numerically veri\fed\nthat adding a non-linear term gj\t(x;y;t )j2to the Hamil-\ntonian does not change our results qualitatively for mod-\nerate realistic interaction strengths g. In order to enter\ninto the regime where interaction starts to a\u000bect the re-\nsults, one would need a condensate with a large number11\nof atoms (\u0018millions of atoms) or alternatively externally\ntune the scattering length via the method of Feshback\nresonances.\nACKNOWLEDGMENTS\nThe authors thank Ian Spielman for helpful comments.\nSFB/TR 12 is acknowledged for \fnancial support. JLacknowledges Vetenskapsr\u0017 adet (VR), DAAD (Deutscher\nAkademischer Austausch Dienst), and the Royal Re-\nsearch Council Sweden (KVA) for \fnancial help. BA\nacknowledges the sponsorship of the US Department of\nCommerce, National Institute of Standards and Technol-\nogy, and was supported by the National Science Founda-\ntion under Physics Frontiers Center Grant PHY-0822671\nand by the ARO under the DARPA OLE program.\n[1] M. Lewenstein, A. Sanpera, V. Ahu\fnger, B. Damski,\nA. Sen(De), and U. Sen, Adv. Phys. 56, 243 (2007); I.\nBloch, J. Dalibard, and W. Zwerger, Rev. Mod. Phys.\n80, 885 (2008).\n[2] M. Greiner, O. Mandel,T. Esslinger, T. W. H ansch, and\nI. Bloch, Nature 415, 39 (2002).\n[3] C. J. Pethick and H. Smith, Bose-Einstein Condensa-\ntion in Dilute Gases , (Cambridge University Press, Cam-\nbridge, 2008).\n[4] Y. J. Lin, R. L. Compton, K. Jimenez-Garcia, J. V.\nPorto, and I. B. Spielman, Nature 462, 628 (2009); Y. J.\nLin, R. L. Compton, A. R. Perry, W. D. Phillips, J. V.\nPorto, and I. B. Spielman, Phys. Rev. Lett. 102, 130401\n(2009).\n[5] J. Dalibard, F. Gerber, G. Juzelunas, and P. Ohberg,\nRev. Med. Phys. 83, 1523 (2011).\n[6] G. Juzeli\u0016 unas, J. Ruseckas, P. Ohberg, and M. Fleis-\nchhauer, Phys. Rev. A 73, 025602 (2006).\n[7] N. R. Cooper and J. Dalibard, Euro. Phys. Lett. 95,\n66004 (2011).\n[8] M. Sigrist and K. Ueda, Rev. Mod. Phys. 63, 239 (1991);\nI. Zutic, J. Fabian, S. Das Sarma, Rev. Mod. Phys. 76,\n323 (2004)\n[9] M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045\n(2010).\n[10] Y. J. Lin, K. Kimenez-Garcia, and I. Spielman, Nature\n471, 83 (2011).\n[11] G. Juzeli\u0016 unas, J. Ruseckas, and J. Dalibard, Phys. Rev.\nA81, 053403 (2010).\n[12] B. M. Anderson, G. Juzeli\u0016 unas, I. B. Spielman, and V. M.\nGalitski, Phys. Rev. Lett. 108, 235301 (2012).\n[13] H. Hu, B. Ramachandhran, H. Pu, and X.-J. Liu, Phys.\nRev. Lett. 108, 010402 (2012).\n[14] S. Sinha, R. Nath, and L. Santos, Phys. Rev. Lett. 107,\n270401 (2011).\n[15] J. Radic, T. A. Sedrakyan, I. B. Spielman, and V. Galit-\nski, Phys. Rev. A 84, 063604 (2011); H. Hu, H. Pu, and\nX.-J. Liu, Phys. Rev. Lett. 108, 010402 (2012); H. Hu\nand X.-J. Liu, Phys. Rev. A 85, 013619 (2012).\n[16] J. Larson and E. Sj oqvist, Phys. Rev. A 79, 043627\n(2009); Y. Zhang, L. Mao, and C. Zhang, Phys. Rev.\nLett. 108, 035302 (2012).\n[17] S. K. Ghosh, J. P. Vyasanakere and V. B. Shenoy, Phys.\nRev. A 84, 053629 (2011)\n[18] A. Polkovnikov, K. Sengupta, A. Silva, and M. Vengalat-\ntore, Rev. Mod. Phys. 83, 863, (2011); V. I. Yukalov,\nLaser Phys. Lett. 8, 485 (2011).\n[19] D. Leibfried, R. Blatt, C. Monroe, and D. Wineland, Rev.\nMod. Phys. 75, 281 (2003); S. Haroche and J.-M. Rai-mond, Exploring the Quantum , (Oxford University Press,\nOxford, 2006); R. Islam, E. E. Edwards, K. Kim, S. Ko-\nrenblit, C. Noh, H. J. Carmichael, G. D. Lin, L. M. Duan,\nC. C. J. Wang, J. K. Freericks, and C. Monroe, Nature\nCommun. 2, 377 (2011).\n[20] J. M. Fink, M. G oppl, M. Baur, R. Bianchetti, P. J. Leek,\nA. Blais, and A. Wallra\u000b, Nature 454, 315 (2008).\n[21] H. Walther, B. T. H. Varcoe, T. H. Benjamin B. G. En-\nglert, and T. Becker, Rep. Prog. Phys. 69, 1325 (2006).\n[22] Q. Y. He, M. D. Reid, B. Opanchuk, R. Polk-\ninghorne, L. E. C. Rosales-Zarate, and P. D. Drummond,\narXiv:1112.0380.\n[23] J. M. Deutsch, Phys. Rev. A 43, 2046 (1991); M. Sred-\nnicki, Phys. Rev. E 50, 888 (1994).\n[24] M. Rigol, V. Dunjko, and M. Olshanii, Nature 452, 854\n(2008).\n[25] T. Kinoshita, T. Wenger, and S. D. Weiss, Nature 440,\n900 (2006).\n[26] V. Milner, J. L. Hanssen, W. C. Campbell, and M. G.\nRaizen, Phys. Rev. Lett. 86, 1514 (2001); M. F. Ander-\nsen, A. Kaplan, T. Gr unzweig, and N. Davidson, Phys.\nRev. Lett. 97, 104102 (2006); M. Gring, M. Kuhnert, T.\nLangen, T. Kitagawa, B. Rauer, M. Schreitl, I. Mazets,\nD. A. Smith, E. Demler, and J. Schmiedmayer, Science\n337, 1318 (2012).\n[27] S. Chaudhury, A. Smith, B. E. Anderson, S. Ghose, and\nP. S. Jessen, Nature 461, 768 (2009).\n[28] C. Gogolin, M. P. M uller, and J. Eisert, Phys. Rev. Lett.\n106, 230502 (2011).\n[29] R. C. Hilborn, Chaos and Nonlinear Dynamics , 2nd ed.\n(Oxford University Press, Oxford, 1980).\n[30] V. A. Yurovsky and M. Oslhanii, Phys. Rev. Lett. 106,\n025303 (2011); M. Olshanii, K. Jacobs, M. Rigol, V. Dun-\njko, H. Kennard, and V. A. Yurovsky, Nature Commun.\n3, 641 (2012).\n[31] A. Altland and F. Haake, Phys. Rev. Lett. 108, 073601\n(2012); A. Altland and F. Haake, New J. Phys. 14,\n073011 (2012).\n[32] A. Bohm, A. Mostafazadeh, H. Koizumi, Q. Niu, and J.\nZwanziger, The Geomtric Phases in Quantum Systems ,\n(Springer Verlag, Berlin, 2003); M. Baer, Beyond Born-\nOppenheimer , (John Wiley & Sons, New Jersey, 2006).\n[33] D. L. Campbell, G. Juzeli\u0016 unas, and I. B. Spielman, Phys.\nRev. A 84025602 (2011).\n[34] Z. F. Xu, L. You, Phys. Rev. A 85, 043605 (2012)\n[35] J. Ruseckas, G. Juzeli\u0016 unas, P. Ohberg, and M. Fleis-\nchhauer, Phys. Rev. Lett. 95, 010404 (2005); T. D.\nStanescu, C. Zhang, and V. Galitski, Phys. Rev. Lett.\n99, 110403 (2007); T. D. Stanescu, B. Anderson, and12\nV. Galitski, Phys. Rev. A 78, 023616 (2008).\n[36] H. C. Longuet-Higgens, U. Opik, M. H. L. Pryce, and R.\nA. Sack, Proc. R. Soc. London Ser. A 244, 1 (1958); J.\nLarson, Phys. Rev. A 78, 033833 (2008).\n[37] R. Englman, The Jahn-Teller E\u000bect in Molecules and\nCrystals , (Wiley, New York, 1972); G. Grosso and G. P.\nParravicini, Solid State Physics , (Academic Press, 2003).\n[38] Y. A. Bychkov and E. I. Rashba, J. Phys. C 17, 6039\n(1984).\n[39] G. Dresselhaus, Phys. Rev. 100, 580 (1955).\n[40] M. V. Berry and R. Lim, J. Phys. A: Math. Gen. 23,\nL655 (1990).\n[41] Regular Runge-Kutta (4,5), and the Adams-Bashforth-\nMoulton PECE solver.\n[42] E. J. Heller, Phys. Rev. Lett. 53, 1515 (1984); M. V.\nBerry, Proc. Roy. Soc. London Ser. A 423, 219 (1989);\nL. Kaplan and E. J. Heller, Annals of Phys. 264, 171\n(1998); L. Kaplan, Nonlinearity 12, 1 (1998).\n[43] S. H. Strogatz, Nonlinear Dynamics and Chaos , (West-\nview Press, Cambridge, 2000).\n[44] R. S. Markiewicz, Phys. Rev. E 64, 026216 (2001).\n[45] M. D. Fleit, J. A. Fleck, and A. Steiger, J. Comput. Phys.\n47, 412 (1982).\n[46] A. Polkovnikov, Annals of Phys. 325, 1790 (2010).\n[47] W.-G. Wang and B. Li, Phys. Rev. E 66, 056208 (2002).\n[48] D. Wang, T. Hansson, \u0017A. Larson, H. O. Karlsson, and J.\nLarson, Phys. Rev. A 77, 053808 (2008).\n[49] S. Habib, K. Shizume, and W. H. Zurek, Phys. Rev. Lett.\n80, 4361 (1998).\n[50] E. Majernikova and S. Shpyrko, Phys. Rev. E 73, 057202\n(2006).\n[51] F. Haake, Quantum Signatures of Chaos , (Springer-\nVerlag, Berlin, 2010).\n[52] E. Majernikova and S. Shpyrko, Phys. Rev. E 73, 066215(2006).\n[53] W. H. Zurek, Nature 412, 712 (2001).\n[54] Ph. Jacquod, I. Adagideli, and C. W. Beenakker, Phys.\nRev. Lett. 89, 154103 (2002).\n[55] Normally, the sub-Planck cell structures are studied in\nthe phase space distributions. However, there exist sev-\neral di\u000berent phase psace distributions having di\u000berent\nproperties. For example, the sub-Planck cell structures\ntypically appear in the Wigner distributions, while in the\nHusimiQ-function, being strictly positive, small \ructua-\ntions have been \\averaged out\" and it is normally much\nsmoother than the Wigner distribution.\n[56] J. Larson, E. N. Ghassemi, and \u0017A. Larson,\narXiv:1111.4647.\n[57] P. B. Wilkinson, T. M. Fromhold, L. Eaves, F. W.\nSheard, N. Miura, and T. Takamasu, Nature 380, 608\n(1996); A. M. Burke, R. Akis, T. E. Day, G. Speyer,\nD. K. Ferry, and B. R. Bennett, Phys. Rev. Lett. 104,\n176801 (2010).\n[58] T. M. Fromhold, C. R. Tench, S. Bujkiewicz, P. B.\nWilkinson, and F. W. Sheard, J. Opt. B: Quant. Semi-\nclass. Opt. 2, 628 (2999); R. G. Scott, S. Bujkiewicz, T.\nM. Fromhold, P. B. Wilkinson, and F. W. Sheard, Phys.\nRev. A 66, 023407 (2002).\n[59] J. C. Sprott, Chaos and Time-Series Analysis , (Oxford\nUniversity Press, Oxford, 2003).\n[60] D. J. O'Dell and J. Larson, To be submitted.\n[61] G. Roati, M. Zaccanti, C. D'Errico, J. Catani, M. Mod-\nugno, A. Simoni, M. Inguscio, and G. Modugno, Phys.\nRev. Lett. 99, 010403 (2007).\n[62] R. A. Williams, L. J. LeBlanc, K. Jimenez-Garcia, M. C.\nBeeler, A. R. Perry, W. D. Phillips, and I. B. Spielman,\nScience 335, 314 (2012)."    },    {        "title": "2207.06347v1.Giant_orbital_Hall_effect_and_orbital_to_spin_conversion_in_3d__5d__and_4f_metallic_heterostructures.pdf",        "content": "PHYSICAL REVIEW RESEARCH 4, 033037 (2022)\nGiant orbital Hall effect and orbital-to-spin conversion in 3 d,5d,a n d4 fmetallic heterostructures\nGiacomo Sala*and Pietro Gambardella†\nDepartment of Materials, ETH Zurich, 8093 Zurich, Switzerland\n(Received 4 April 2022; revised 21 June 2022; accepted 23 June 2022; published 13 July 2022)\nThe orbital Hall effect provides an alternative means to the spin Hall effect to convert a charge current into\na flow of angular momentum. Recently, compelling signatures of orbital Hall effects have been identified in 3 d\ntransition metals. Here, we report a systematic study of the generation, transmission, and conversion of orbitalcurrents in heterostructures comprising 3 d,5d,a n d4 fmetals. We show that the orbital Hall conductivity of\nCr reaches giant values of the order of 5 ×10\n5[¯h\n2e]/Omega1−1m−1and that Pt presents a strong orbital Hall effect in\naddition to the spin Hall effect. Measurements performed as a function of thickness of nonmagnetic Cr, Mn, andPt layers and ferromagnetic Co and Ni layers reveal how the orbital and spin currents compete or assist eachother in determining the spin-orbit torques acting on the magnetic layer. We further show how this interplaycan be drastically modulated by introducing 4 fspacers between the nonmagnetic and magnetic layers. Gd and\nTb act as very efficient orbital-to-spin current converters, boosting the spin-orbit torques generated by Cr by afactor of 4 and reversing the sign of the torques generated by Pt. To interpret our results, we present a generalizeddrift-diffusion model that includes both spin and orbital Hall effects and describes their interconversion mediatedby spin-orbit coupling.\nDOI: 10.1103/PhysRevResearch.4.033037\nI. INTRODUCTION\nThe interconversion of charge and spin currents underpins\na variety of phenomena and applications in spintronics, in-cluding spin-orbit torques, spin pumping, the excitation ofmagnons, and the tuning of magnetic damping [ 1,2]. The spin\nHall effect (SHE) mediates this interconversion through thecombination of intrinsic and extrinsic scattering processes,all of which require sizable spin-orbit coupling [ 3]. Recent\ntheoretical work has shown that the intrinsic SHE is accompa-nied by a complementary process involving the orbital angularmomentum, the so-called orbital Hall effect (OHE), whichconsists in the flow of orbital momentum perpendicular tothe charge current [ 4–10]. According to theoretical calcula-\ntions, the OHE is more common and fundamental than theSHE because it does not require spin-orbit coupling and canthus occur in a wider range of materials. The intrinsic SHEthen emerges as a by-product of the OHE resulting from theorbital-to-spin conversion in materials with nonzero spin-orbitcoupling. In this case, the spin Hall conductivity has the samesign as the product between the orbital conductivity and theexpectation value of spin-orbit coupling: σ\nS∼σL/angbracketleftL·S/angbracketright.T h e\nOHE was first predicted in 4 dand 5 dtransition elements\n[11,12] and recently in light metals [ 4] and their interfaces\n*giacomo.sala@mat.ethz.ch\n†pietro.gambardella@mat.ethz.ch\nPublished by the American Physical Society under the terms of the\nCreative Commons Attribution 4.0 International license. Further\ndistribution of this work must maintain attribution to the author(s)\nand the published article’s title, journal citation, and DOI.[13] as well as in two-dimensional (2D) materials [ 14,15].\nThe theoretical orbital Hall conductivity of light elements iscomparable to or even larger than the spin Hall conductivityof Ta, W, and Pt, which provide a strong SHE [ 4]. The OHE\nis thus intrinsically more efficient than the SHE, and orbitalcurrents are expected to contribute to magnetotransport effectssuch as the anisotropic, spin Hall, and unidirectional mag-netoresistance as well as spin-orbit torques [ 6,16–20]. The\nubiquity and strength of the OHE, besides making it funda-mentally interesting, broaden the material palette available forspintronic applications and provide an additional handle tooptimize the efficiency of spin-orbit torques. Yet, differentlyfrom spins, nonequilibrium orbital currents do not coupledirectly to the magnetization of magnetic materials and cantorque magnetic moments only indirectly through spin-orbitcoupling [ 19–21]. Optimizing the orbital-to-spin conversion\nis thus a prerequisite for taking advantage of large orbitalcurrents.\nThe prediction of the OHE in light elements has triggered\nintense research on current-induced orbital effects. Recentexperiments have identified signatures of the OHE in ma-terials with low [ 18,19] and high [ 20] spin-orbit coupling\nand revealed its contribution to spin-orbit torques [ 16,18],\nwhose strength can be tuned by improving the orbital-to-spinconversion ratio [ 19,22]. However, experimental values of the\norbital Hall conductivity are smaller than theoretical estimates[16,18,19,23], and a systematic investigation of orbital effects\nas a function of the type and thickness of nonmagnetic, ferro-magnetic, and spacer layers is still missing.\nHere, we present a comprehensive study of the interplay of\nthe OHE and SHE in structures combining different light andheavy nonmagnetic metals (NM =Cr, Mn, Pt), ferromagnets\n(FM=Co, Ni), and rare-earth spacers ( X=G d ,T b ) .W e\n2643-1564/2022/4(3)/033037(14) 033037-1 Published by the American Physical SocietyGIACOMO SALA AND PIETRO GAMBARDELLA PHYSICAL REVIEW RESEARCH 4, 033037 (2022)\n(a) (b)\n(a)LS\nMTSTLSJCxz\nL SE\nMTSTLSLSJC E\nNM\nFML\nS\nFIG. 1. (a) The spin Hall effect and orbital Hall effect induced\nby an electric field Ein a nonmagnet (NM) produce spin ( TS)a n d\norbital ( TLS) torques on the magnetization Mof an adjacent ferro-\nmagnet (FM). The strength of the torques depends on the intensityof the spin and orbital currents and on the spin-orbit coupling of the\nferromagnet. The schematic shows the direction of the induced spin\n(S, blue dots and crosses) and orbital ( L, red circling arrows) angular\nmomenta when the spin and orbital Hall conductivities σ\nS,L>0.\n(b) The insertion of a spacer layer may increase the orbital torque\nrelative to the spin torque by converting the orbital current (red) intoa spin current (blue) prior to their injection into the ferromagnet.\nprovide evidence of the OHE in Pt and Mn and report giant\nvalues of the orbital Hall conductivity in Cr, which extrapolateto the theoretical limit of 10\n6[¯h\n2e](/Omega1m)−1in Cr films thicker\nthan the orbital diffusion length [ 4], which we estimate to\nbe/greaterorsimilar20 nm. Because of the simultaneous presence of strong\nOHE and SHE in Pt and Cr, we argue that experimental resultsare best described by a combined spin-orbital conductivityrather than by separating the two effects. We show that theinterplay between orbital and spin currents can be tailoredby varying the thickness of the ferromagnetic layer as wellas by inserting a Gd or Tb conversion layer between thenonmagnet and the ferromagnet. Rare-earth spacers do notgenerate significant spin-orbit torques by themselves, but theyenhance the torque efficiency up to four times when Cr isthe source of spin and orbital currents and reverse the signof the torques generated by Pt. The latter effect is attributedto the OHE overcoming the SHE in Pt. Finally, we present aphenomenological extension of the spin drift-diffusion modelthat includes orbital effects and the conversion between spinand orbital moments, which accounts for both the thicknessdependence and sign change of the spin-orbit torques gen-erated by the interplay of OHE and SHE in NM/FM andNM/ X/FM heterostructures.\nII. BACKGROUND\nAccording to the theory of the OHE, an electric field ap-\nplied along the xdirection in a material with orbital texture\ninkspace induces interband mixing that results in electron\nstates with finite orbital angular momentum [ 5–7]. Electrons\noccupying these nonequilibrium states carry the angular mo-mentum as they travel in real space. Therefore, although thetotal orbital momentum vanishes, a nonzero orbital current isproduced along the z(y) direction with orbital polarization\nparallel to ±y(±z), similar to the SHE [see Fig. 1(a)]. Thelatter occurs concomitantly with the OHE when the nonmag-\nnet has nonzero spin-orbit coupling /angbracketleftL·S/angbracketright\nNM. The primary\nspin current injected into the adjacent ferromagnet exerts adirect torque on the local magnetization (spin torque). Or-bitals, instead, act indirectly through the spin-orbit couplingof the ferromagnet that converts the orbital current into asecondary spin current. We refer to the torque generated bythis secondary spin current as orbital torque. The (indepen-dence) dependence of the (spin) orbital torque on /angbracketleftL·S/angbracketright\nFM\nis the key difference between SHE and OHE. In the SHE\nscenario, the angular momentum is entirely generated in thenonmagnet, and the ferromagnet behaves almost as a passivelayer since it only contributes to the properties of the NM/FMinterface. In contrast, the OHE in a NM/FM bilayer dependson both the interfacial and bulk properties of the ferromagnet,which is directly involved in the torque generation. Sincethe orbital conductivity is typically large [ ≈10\n5(/Omega1m)−1][4]\nbut the spin-orbit coupling of 3 dferromagnets is relatively\nweak [ 24], the orbital torque efficiency in NM/FM bilayers\nis finite but small. Alternatively, the orbital torque may beenhanced by realizing most of the orbital-to-spin conversionin a spacer layer sandwiched between the nonmagnet and theferromagnet [Fig. 1(b)]. The effectiveness of this approach\ndepends on the conversion efficiency of the spacer, its spinand orbital diffusion lengths, and the quality of the additionalinterfaces, as discussed later.\nHere, we summarize fundamental theoretical predictions\nand experimental confirmations of the OHE. We list ap-proaches to distinguish orbital and spin effects by meansof torque measurements in heterostructures with differentelements, thickness, and stacking order. Furthermore, we es-tablish a parallel between known spin-transport effects andpossible orbital counterparts that have not been observed yetbut could contribute to answering open questions about orbitaltransport.\n(i) Large orbital Hall conductivities have been predicted in\nseveral 3 d,4d, and 5 dtransition elements [ 4,11,12] and 2D\nmaterials [ 9,10]. Experimental evidence is so far limited to Cr\n[19,25], Cu [ 20,21,26], Zr [ 18], and Ta [ 20]. Recent experi-\nments on V [ 23,27] can also be reinterpreted in light of the\nOHE. The coexistence of the OHE and the SHE, especially inheavy metals, makes it difficult to distinguish the two effects.\n(ii) The spin and orbital torques are expected to add\nconstructively (destructively) when /angbracketleftL·S/angbracketright\nNM·/angbracketleftL·S/angbracketrightFM>0\n(<0). This competition can be tailored by properly choosing\nthe ferromagnet, as recently observed in Refs. [ 19,20].\n(iii) In a NM/FM bilayer, the orbital Hall efficiency should\ndepend on the thickness of both the nonmagnet ( tNM) and\nthe ferromagnet ( tFM). In contrast, the spin Hall efficiency is\nnominally independent of the latter and results in an inversedependence of the spin torque on t\nFM[1]. The dependence\nof the orbital Hall efficiency on tNMhas been addressed in\nRef. [ 19], but the role of tFMis still unknown.\n(iv) The spin diffusion in transition metals with strong\nSHE is typically limited to a few nanometers [ 28]. Al-\nthough recent measurements suggest longer orbital dif-fusion lengths [ 19,29], the length scale of the orbital\ndiffusion and its conversion into spins remain to be es-tablished. These quantities and the nature of the mech-anisms underlying the orbital scattering may be ad-\n033037-2GIANT ORBITAL HALL EFFECT AND ORBITAL-TO-SPIN … PHYSICAL REVIEW RESEARCH 4, 033037 (2022)\ndressed by torque measurements in thick nonmagnetic\nfilms and by nonlocal transport measurements, which couldalso verify the existence of the inverse OHE.\n(v) Spacer layers between the nonmagnet and the ferro-\nmagnet can alter spin torques in several ways, namely, byintroducing an additional interface with different spin scat-tering properties, by suppressing the spin backflow, and bymodifying the spin memory loss [ 1,30–33]. Such effects are\nexpected to influence also the orbital torque. In addition,spacers can either increase or decrease the orbital torquedepending on the sign of their orbital and spin Hall conduc-tivities, and spin-orbit coupling, which converts orbitals intospins and vice versa. Pt spacers have been shown to increasethe orbital torques in light metal systems [ 19,21]; however, Pt\nis also a well-known SHE material. A systematic investigationof the enhancement or suppression of spin and orbital currentsin materials with different combinations of orbital and spinconductivities is required.\n(vi) The spin diffusion in multilayer structures is usu-\nally modeled by semiclassical drift-diffusion equations thataccount for, e.g., spin backflow at interfaces, the spin-orbittorque dependence on the thickness of the nonmagnet, and thespin Hall and unidirectional magnetoresistance [ 34–39]. The\nmodel has not been extended yet to the OHE, which requiresthe inclusion of the spin-orbital interconversion mediated byspin-orbit coupling.\n(vii) The orbital transmission at the NM/FM interface is\nmore sensitive to the interface quality than spins and, hence,to growth conditions and stacking order [ 7,18]. It is an open\nquestion whether the transmission can be described by a singleparameter equivalent to the spin-mixing conductance, whichwe dub orbital mixing conductance.\n(viii) The SHE generates dampinglike and fieldlike spin-\norbit torques of comparable strength [ 1,40]. So far, no\ntheoretical or experimental work has determined with cer-tainty the relative magnitude of the two components ofthe orbital torque. Assessing their strength may help us tounderstand the mechanism of accumulation, transfer, and con-version of orbitals.\n(ix) The OHE has been attributed to an intrinsic scattering\nmechanism in elements with orbital texture. The analogy withthe SHE [ 41,42] suggests that also extrinsic processes may\ncontribute to the generation of orbital currents. Measuring theorbital Hall efficiency as a function of the element resistivitymay reveal extrinsic orbital effects.\n(x) The transmission and absorption of spins and orbitals\nat the interface with an insulating ferromagnet [ 43], e.g.,\nyttrium iron garnet (YIG), may be fundamentally differentsince the latter do not interact with the magnetization. Earlyexperiments reported spin pumping effects in YIG/light metalbilayers, but they were interpreted in terms of the inverse SHE[44].\n(xi) The generation and accumulation of orbitals at the\nNM/FM interface can modulate the longitudinal resistance bythe combination of direct and inverse OHE, as recently foundin Ref. [ 22]. Compared with the spin Hall magnetoresistance\n[35], such orbital Hall magnetoresistance may have a different\ndependence on the type of ferromagnet, its thickness, and thethickness of the nonmagnet.(xii) Orbital accumulation might also give rise to a unidi-\nrectional magnetoresistance, in analogy to the unidirectionalspin Hall magnetoresistance [ 45]. The underlying mechanism,\nhowever, would be intrinsically different since orbitals wouldnot directly alter the magnon population, whereas orbital-dependent scattering might contribute to the conductivityin addition to spin-dependent scattering [ 46]. On the other\nhand, the injection into the ferromagnet of electrons withfinite orbital momentum may induce an additional source oflongitudinal magnetoresistance analogous to the anisotropicmagnetoresistance [ 17].\nOrbital effects are thus rich and intertwined with spin\ntransport, allowing for additional means to tune the spin-orbittorque efficiency as well as to understand the transport ofangular momentum in thin-film heterostructures. In the fol-lowing, we address points (i)–(vii) listed above. We providecomprehensive evidence for the occurrence of giant OHEsin 3dand 5 dtransition metals, reveal the interplay of the\nOHE and SHE in ferromagnets of variable thickness withand without spacer layers, and establish a phenomenologicalframework to analyze and efficiently exploit the interplay ofspin and orbital currents in metallic heterostructures.\nIII. EXPERIMENTS\nWe studied NM/FM and NM/ X/FM multilayers where NM\n=Cr, Mn, or Pt, FM =Co or Ni, and X=Gd or Tb.\nThe samples were grown by magnetron sputtering on a SiNsubstrate, capped with either Ti(2) or Ru(3.5) (thicknessesin nanometers), and patterned in Hall-bar devices by opticallithography and lift-off. All samples have in-plane magneti-zation. Current-induced spin-orbit torques were quantified bythe harmonic Hall voltage method [ 40] using angle-scan mea-\nsurements [ 51]. We detected the first- and second-harmonic\nHall voltage while applying an alternate current with 10 Hzfrequency and rotating a constant magnetic field in the easyplane of the magnetization [ xyplane; see Fig. 2(a)]. The\nharmonic signals were measured as a function of currentamplitude and field strength [Figs. 2(b) and2(c)]. The second-\nharmonic resistance depends on the field angle φasR\n2ω\nxy=\n/Theta1Tcosφ+/Phi1T(2 cos3φ−cosφ). Here, /Theta1Tis the sum of the\ndampinglike spin-orbit field BDLand contribution from the\nthermal gradient along z, and /Phi1Tdepends on the fieldlike\nspin-orbit field BFLand the Oersted field BOe. Thus the anal-\nysis of R2ω\nxymeasured at different magnetic fields allows for\nthe separation of torques and thermal effects, yielding themagnitude of the spin-orbit fields for a given electric field[51]. These spin-orbit fields exert spin and orbital torques\non the magnetization T\nDL=MsBDLm×(p×m) and TFL=\nMsBFLm×p, where pis the net spin polarization direction,\nmis the magnetization vector, and Msis the saturation mag-\nnetization. In the following, we consider uniquely BDLsince,\napart from Ni/Cr and Co/Pt samples, BFLwas too small to\ndistinguish from the Oersted field. The difficult detection ofB\nFLin our samples originates from the very small planar Hall\ncoefficient (of the order of 1 m /Omega1) to which /Phi1Tis propor-\ntional. To compare samples with different elements, thickness,and stacking order, we converted B\nDLinto a spin-orbital\n033037-3GIACOMO SALA AND PIETRO GAMBARDELLA PHYSICAL REVIEW RESEARCH 4, 033037 (2022)\n(b) (c)JCzmϕ Vxy(a)B\nxy\n0.0 0.5 1.0 1.5 2.0-2.2-2.02.02.22.4 Co/Cr(12)\n Ni/Cr(12)ΘT/RAHE-3 (10 )\n1/Beff (1/T)0 60 120 180 240 300 3600.00.10.2 180 mT\n 1350 mTRxy2ω (mΩ)\nϕ (° )\nFIG. 2. (a) Schematic of the harmonic Hall effect measurements.\nAn alternating current Jcflows along the Hall bar and generates\ntransverse first- and second-harmonic Hall signals that depend on\nthe angle φrelative to xof a magnetic field Bof constant ampli-\ntude. (b) Representative second-harmonic Hall resistance measuredin Co(2) /Cr(12) during the rotation of Bin the xyplane. The solid\nlines are fits to the function R\n2ω\nxy=/Theta1Tcosφ+/Phi1T(2 cos3φ−cosφ)\n(see text). (c) Dependence of /Theta1T(dampinglike field BDL+ther-\nmal signal) normalized to the anomalous Hall resistance ( RAHE)\non the effective field given by the sum of the applied magnetic\nfield and demagnetizing field. Data are shown for Co(2) /Cr(12) and\nNi(4)/Cr(12). The slope of the linear fit (solid lines) is proportional\ntoBDL, while its intercept with the vertical axis corresponds to the\nthermal contribution, which is field independent and can be easilydistinguished.\nconductivity according to the formula\nξLS=2e\n¯hMstFMBDL\nE, (1)\nwhere eis the electron charge, ¯ his Planck’s constant, tFMis\nthe thickness of the ferromagnet, and E=ρJcis the applied\nelectric field ( ρis the longitudinal resistivity, and Jcis the\ncurrent density) [ 1,52]. The normalization to the applied elec-\ntric field avoids the ambiguities intrinsic to the calculation ofthe current density in a heterostructure. Since in our samplesthe ferromagnet lies below the nonmagnet, we invert the signof the measured B\nDLto follow the convention that Pt has\npositive spin Hall conductivity. In the literature, Eq. ( 1)i s\nusually referred to as the spin Hall conductivity orspin-orbit\ntorque efficiency , which is related to the effective spin Hall\nangle of the NM layer by θLS=ρξLS(Appendix A). Here,\nwe point out that, when the SHE and OHE are consideredtogether, both spin and orbital currents influence ξ\nLSand their\nindividual quantitative contributions cannot be disentangledbecause the spin-orbit torques depend on the total nonequi-librium spin angular momentum in the ferromagnet (primaryspins+converted spins) but not on the orbital component.\nThis reasoning implies the impossibility of determining sep-arately the spin and orbital Hall conductivities of a materialby measuring nonequilibrium effects on an adjacent ferro-magnet, even for transparent interfaces. Thus we call ξ\nLSthe\nspin-orbital conductivity and θLSthe spin-orbital Hall angle.\nHowever, spin and orbital effects can still be distinguished(b)σ > 0S σ > 0LL S\nσ < 0S σ > 0LL S0.10.2 Ni/Cr\n Ni/Mn\n02468 1 0 1 2 1 4 1 6-2-10\n Co/Cr\n Co/MnξLS (105 Ω-1m-1)\nt (nm)NMNM02468 1 0 1 201234\n Co/Pt\n Ni/PtξLS (105 Ω-1m-1)\nt (nm)(a)\nFIG. 3. (a) Spin-orbital conductivity as a function of the thick-\nness of the Pt layer in Co(2) /Pt(tNM) and in Ni(4)/Pt(5). The solid\nline is a fit to the drift-diffusion equation [Eq. ( 2)]. The sign of\nthe spin and orbital Hall conductivities in the nonmagnet is indi-\ncated and color-coded in the schematic representing the generation,transmission, and conversion of orbital (red) and positive (blue) or\nnegative (white) spin currents. (b) The same as (a) in FM /Cr(t\nNM)\nand FM /Mn(tNM), where FM =Co(2) or Ni(4).\nat a qualitative level, as discussed in the following. We also\nnote that a finite OHE could explain, at least in part, the largevariability of the spin-orbit torque efficiency found in sampleswith different ferromagnets, thicknesses, stacking order, andpreparation conditions [ 1].\nIV . OHE in Cr, Mn, and Pt\nA. Dependence of ξLSon the thickness of the NM layer\nFigure 3compares ξLSmeasured in FM/NM bilayers,\nwhere FM is an in-plane magnetized Co(2) or Ni(4) layer andNM is a Cr, Mn, or Pt layer of variable thickness t\nNM.W e\nfind that the two 3 dlight metals generate sizable spin-orbit\ntorques, similar to previous measurements in materials withweak spin-orbit coupling such as V , Cr, and Zr [ 16,23,44,53].\nThe torques are remarkably strong in Cr-based samples, forwhich ξ\nLSreaches values similar to those for Co/Pt. To the best\n033037-4GIANT ORBITAL HALL EFFECT AND ORBITAL-TO-SPIN … PHYSICAL REVIEW RESEARCH 4, 033037 (2022)\nTABLE I. Sign of the spin-orbit coupling /angbracketleftL·S/angbracketrightand orbital\nand spin Hall conductivity σL,Sin selected transition metals (see\nRefs. [ 4,11,12,47–49]). A positive orbital (spin) Hall conductivity\nmeans that a charge current along +xinduces an orbital current (spin\ncurrent) along +zwith orbital (spin) angular momentum along −y\n[50].\nCr Mn Co Ni Gd Tb Pt\n/angbracketleftL·S/angbracketright –– ++ –– +\nσL ++ + +\nσS –– ++ –– +\nof our knowledge, this is the highest torque efficiency reported\nin the literature for a FM/NM bilayer made of light elements.However, the dependence of ξ\nLSon the type of ferromagnet\nand on tNMis very different in Cr and Mn with respect to Pt.\nCo/Pt(tNM) and Ni/Pt(5) have torque efficiencies of com-\nparable magnitude and identical sign. In contrast, when Cror Mn is used, ξ\nLSchanges sign when Co is replaced with\nNi [see Fig. 3(b)]. A comparison between Fig. 3and Table I\nindicates that Co/Cr and Co/Mn behave as expected withinthe framework of the SHE, namely, the sign of the torques isopposite to Co/Pt because the spin Hall conductivity σ\nShas\nopposite sign in Cr and Mn relative to Pt. The same argu-ment, however, cannot explain the positive sign of ξ\nLSin the\nNi-based samples since the direction of the spin polarizationinduced by the SHE is fixed and determined by /angbracketleftL·S/angbracketright\nNM.\nThe sign change can be accounted for only by consideringthe OHE and the opposite sign of the spin and orbital Hallconductivities of Cr and Mn. In this case, the negative ξ\nLS\nmeasured in Co/Cr and Co/Mn indicates that in these samples\nthe spin torque overwhelms the orbital torque. The positivespin-orbital conductivity found with Ni shows instead that theorbital-to-spin conversion in this ferromagnet is so efficientas to make the orbital torque stronger than the spin torque[19]./angbracketleftL·S/angbracketright\nFMis indeed predicted to be larger in Ni than in\nCo and positive [ 7,20,54]; thus a larger amount of the orbital\ncurrent can be converted into a spin current of opposite signto the primary spin current generated by Cr or Mn. Thereforethe torques exerted on Co are mostly generated outside theferromagnet thanks to the orbital-to-spin conversion occurringin the nonmagnet. In contrast, the torques on Ni result fromthe orbital-to-spin conversion inside the ferromagnet.\nThe variation of ξ\nLSwith the thickness tNMof Cr and\nMn is also different from the thickness dependence of thetorque efficiency in heavy elements [ 1,52]. In Co/Pt( t\nNM),ξLS\nsaturates at about 9 nm [Fig. 3(a)]. The fit to the drift-diffusion\nequation\nξLS(t)=σLS/bracketleftbigg\n1−sech/parenleftbiggt\nλ/parenrightbigg/bracketrightbigg\n(2)\nyields a diffusion length λ=2.2 nm and an intrinsic spin-\norbital Hall conductivity σLS=3.5×105[¯h\n2e](/Omega1m)−1.T h i s\nvalue, which agrees with previous works [ 1,3,41,52,55], as-\nsumes a transparent interface and is thus an underestimationof the intrinsic spin-orbital Hall conductivity of Pt. In Cr andMn,ξ\nLSincreases with tNMand does not saturate, even at\ntNM=15 nm. The intrinsic spin-orbital Hall conductivity ofCr is thus significantly larger than ξLSreported in Fig. 3(b).\nIndeed, fitting ξLSin Co/Cr(tNM) with λfixed in the range 15–\n25 nm yields 5 ×105<|σLS|<12×105(/Omega1m)−1, in good\nagreement with the predicted giant orbital Hall conductivityof Cr [ 4].\nThe trend of ξ\nLS(tNM) hints at two alternatives. The first\npossibility is that the spin ( λS) and orbital ( λL) diffusion\nlengths of Cr and Mn are larger than the typical spin diffusionlength of heavy elements. For example, λ\nSis found to be\nabout 13 and 11 nm in Cr and Mn, respectively, in Ref. [ 44],\nwhereas λS=1.8 nm and λL=6.1 nm in Cr according to\nRef. [ 19]. Alternatively, we argue that it suffices to have a\nlarge orbital diffusion length and a nonzero /angbracketleftL·S/angbracketrightNMfor spins\nto accumulate over long distances, even if the spin diffusionlength in the nonmagnet is short (see Sec. VI). Spin torque\nmeasurements cannot distinguish between the two possibili-ties. Nonetheless, the trends in Fig. 3suggest the possibility\nto increase the spin-orbital conductivity in FM/Cr sampleswith large t\nNMup to and beyond the maximal efficiency of\nCo/Pt. This possibility has gone unnoticed so far because thinnonmagnetic films ( t\nNM≈5 nm) are typically considered in\ntorque measurements.\nA very long orbital diffusion length in Mn may also ex-\nplain why ξLSis smaller in Mn than in Cr at any thickness\nand independently of the ferromagnet. This result contrastswith theoretical calculations that predict large and comparableorbital conductivities in Cr and Mn [ 4] but agrees with the spin\npumping measurements of Ref. [ 44]. We notice that Ref. [ 4]\nconsidered the bcc structure to calculate the orbital conduc-tivity of Mn, but different crystalline phases can compete andcoexist in Mn thin films [ 56]. This difference may account for\nthe small experimental value of ξ\nLS. Alternatively, the small\nspin-orbital conductivity may be determined by a differentquality of the FM/Cr and FM/Mn interfaces, to which theorbital current is very sensitive [ 7,18], possibly because of\nCo and Mn intermixing [ 57]. Owing to the larger resistivity\nof Mn compared with Cr, however, we note that the effectivespin-orbital Hall angle of Co(2)/Mn(9) is θ\nLS=−0.03, which\nis comparable to θLS=−0.05 of Co(2)/Cr(9) (Appendix A).\nWe also notice that interfacial effects (interfacial torques,\nspin memory loss, and spin transparency) can influence thestrength of the torques and hence the spin-orbital conductivity,as shown by the different ξ\nLSmeasured in Co/Pt and Ni/Pt\nsamples [ 58]. However, interfacial effects cannot explain our\nresults, namely, the sign change of ξLSwith the ferromagnet\nand its monotonic increase with tNM, because they should be\nindependent of the thickness of the nonmagnet and becomenegligible in thick films.\nOverall, these measurements provide strong evidence of\nthe OHE and orbital torques in Cr and Mn, in agreement withtheoretical predictions and previous studies of Cr-based sam-ples [ 4,19,25]. Additionally, they show that the spin-orbital\ndiffusion length is much longer in light elements than in Pt,a difference that could be exploited to boost the effectivespin-orbital conductivity beyond the limit of FM/Pt samples.\nB. Dependence of ξLSon the thickness of the FM layer\nTheoretical calculations of the spin and orbital transfer at\nthe FM/NM interface predict a different dependence of the\n033037-5GIACOMO SALA AND PIETRO GAMBARDELLA PHYSICAL REVIEW RESEARCH 4, 033037 (2022)\nspin and orbital torque on the thickness tFMof the ferromagnet\n[59]. The former is dominant when tFMis small, whereas the\norbital torque can be comparable to or larger than the spintorque in thick ferromagnets. As a consequence, the totaltorque may change sign when t\nFMincreases if /angbracketleftL·S/angbracketrightNM·\n/angbracketleftL·S/angbracketrightFM<0, as, for instance, in the case of Co/Cr. To test\nthis possibility, we measured the torque on the magnetizationof Co( t\nFM)/Cr(9) and Co( tFM)/Pt(5) as a function of tFM.\nFigures 4(a) and4(b) show the torque per unit electric field\ncalculated as T=MsBDL\nE. The sign of the torque is opposite\nin the two sets of samples and does not change in the exploredthickness range. This result might suggest that in Co( t\nFM)/Cr\nthe orbital torque is always negligible compared with thespin torque. However, a careful analysis indicates a differentscenario. After taking into account the dead magnetic layer(0.5 and 0.3 nm in the samples with Cr and Pt, respectively;see Appendix B), we tentatively fit the dependence of Ton the\nferromagnet thickness to ∼1/t\nFM. This scaling should reflect\nthe inverse proportionality of the torque amplitude to the mag-netic volume when the current-induced angular momentumis generated outside the ferromagnet. In this case, the spinHall conductivity is constant and solely determined by thecharge-to-spin conversion efficiency of the nonmagnet [ 1],\nand Eq. ( 1) yields\nT=¯h\n2eξLS\ntFM. (3)\nEquation ( 3) captures well the variation of the torque only\nfortFM<1–2 nm, in both Co( tFM)/Cr and Co( tFM)/Pt. The\ndiscrepancy at large thicknesses suggests the presence of atorque mechanism additional to the spin current injection fromthe nonmagnetic layer. This possibility is corroborated by thethickness dependence of the spin-orbital conductivity, whichis different in the two series of samples. In Co( t\nFM)/Pt,|ξLS|\nis approximately constant up to 3 nm and increases at larger\ntFMby about 20% [see Fig. 4(c)]. In Co( tFM)/Cr, instead,\n|ξLS|initially increases as the ferromagnet becomes thicker,\npossibly due to the formation of a continuous Co/Cr interface;then it decreases starting from t\nFM=1 nm and drops by more\nthan 50% at tFM=3 nm relative to the maximum. Beyond this\nthickness, it remains approximately unchanged. The distinctthickness dependence in Co( t\nFM)/Cr and Co( tFM)/Pt cannot be\nascribed to strain [ 60] since Co is grown on an amorphous\nsubstrate. In addition, strain-induced effects should be similarin the two sample series. Moreover, it cannot be attributed to avariation of the interface quality. Since the latter is expected toimprove as Co becomes thicker, the spin-orbital conductivityshould increase or remain approximately constant for t\nFM>\n1 nm. Furthermore, we exclude that the measured trend de-pends on uncertainties in the saturation magnetization due toproximity effects since ξ\nLSdepends on the areal magnetization\n[see Eq. ( 1)], which is free from ambiguity (see Appendix B).\nFinally, we rule out self-torques due to the SHE inside the Colayer [ 49] since control measurements in Co(7)/Ti(3) do not\ngive evidence of torques within the experimental resolution.\nAlternatively, we propose that the decrease of the spin-\norbital conductivity with t\nFMin Co/Cr results from the\ncompetition between spin and orbital torques in the ferromag-net. As sketched in Figs. 4(d) and4(e), the spin and orbital\ncurrents J\nSandJLdecay inside the ferromagnet on a lengthzJL\nJSJLS\nzJL\nJSJLS\ntFM tFM|ξ |LS(a) (b)\n(c)\n(d) (e)\n|ξ |LS02468 1 0 1 2-1.6-0.4-0.200.2\n Co/Cr Co/PtT (ΩTm-2)\ntFM (nm)\n0123456789 1 0 1 1 1 201234|ξLS| (105 Ω-1m-1)\nt (nm)FM02468 1 0 1 2-0.04-0.0200.020.04T (ΩTm-2)\ntFM (nm)\nFIG. 4. (a) Dependence of the spin-orbit torque normalized to the\napplied electric field on tFMin Co( tFM)/Cr(9) and Co( tFM)/Pt(5). The\nsolid lines are fits to1\ntFM. (b) Enlarged view of (a). (c) Dependence of\n|ξLS|ontFMin the two sample series. (d) and (e) Schematics showing\nqualitatively the interplay of the spin JSand orbital JLcurrents, which\nare injected into the ferromagnet from the interface with the nonmag-\nnetic metal and decay with the distance z. Part of the orbital moments\nis converted into spin moments and generates a spin current JLSwith\nthe same (opposite) polarization as the primary spin current in Pt/Co\n(Cr/Co). JSyields the spin torque, and JLSyields the orbital torque.\nThe spin-orbital conductivity ξLSis constant when the orbital-to-spin\nconversion is negligible (dashed line). It increases with tFMwhen JS\nandJLSadd up and decreases when JSandJLScompete (solid line).\nscale determined by the respective dephasing lengths. In the\nabsence of orbital-to-spin conversion, the spin-orbital conduc-tivity, which depends on the absorption of the injected spincurrent ξ\nLS∼JS(0)−JS(tFM), increases rapidly with tFMand\n033037-6GIANT ORBITAL HALL EFFECT AND ORBITAL-TO-SPIN … PHYSICAL REVIEW RESEARCH 4, 033037 (2022)\nremains constant afterwards because spin dephasing occurs\nwithin a few atomic layers from the interface [ 61,62]. On the\nother hand, if we assume that the orbital current is transmittedover a distance longer than its spin counterpart [ 59] and that\npart of it is also converted into a spin current J\nLS, then ξLS∼\nJS(0)−JS(tFM)±JLS(tFM) can increase or decrease with tFM\ndepending on the relative sign of JSandJLS, i.e., on the prod-\nuct/angbracketleftL·S/angbracketrightNM·/angbracketleftL·S/angbracketrightFM. Since the latter is positive (negative)\nin Co( tFM)/Pt [Co( tFM)/Cr], our qualitative model envisages\nan increase (decrease) of the spin-orbital conductivity with\ntFM, in agreement with our measurements and the thickness\ndependence predicted in Ref. [ 59].\nThe dependence of both T andξLSontFMshows that\nCo, rather than being a passive layer subject to an externallygenerated spin current, participates in the overall generationof spin-orbit torques. The active role of the ferromagnet inval-idates the assumption on which Eq. ( 3) rests and explains the\ndeviation of the torque measured at large thicknesses from the1/t\nFMdependence. Interestingly, these measurements point to\na non-negligible OHE in Pt, in accordance with the measure-ments discussed next.\nV . ORBITAL-TO-SPIN CONVERSION IN A SPACER LAYER\nThe results presented in Sec. IVshow that the spin-orbital\nconductivity of a light metal can be maximized by a properchoice of the ferromagnet and its thickness. There is, however,a limitation from both a practical and theoretical point of view.According to Hund’s third rule, light metals have oppositespin-orbit coupling relative to ferromagnetic Fe, Co, and Ni;thusξ\nLScannot be maximized in such bilayers. As proposed\nin Sec. II, this optimization may be possible, instead, if the\norbital current is converted into the spin current prior to the in-jection into the ferromagnetic layer [Fig. 1(b)]. This approach\nrequires materials with high spin-orbit coupling between thelight metal and the ferromagnet [ 21]. Although the additional\nlayer can itself be a source of spin current, we show in thefollowing how thickness-dependent measurements reveal theunderlying orbital-to-spin conversion and indicate the optimalconversion conditions.\nFigure 5shows the spin-orbital conductivity measured in\nCo(2)/X(t\nX)/Cr(9) and Co(2) /X(tX)/Pt(5) as a function of\nthe rare-earth thickness, where Xis either Gd or Tb. We find\na drastic change of the magnitude and sign of the torquesupon increasing t\nX. As the rare-earth layer becomes thicker\nin Co/X(tX)/Cr,|ξLS|first increases, reaching its maximum\nmagnitude at about t=3 nm, and then decreases [notice the\nnegative sign of ξLSin Fig. 5(a)]. At this thickness, |ξLS|\nof Co/ X(3)/Cr is three to four times larger than in Co/Cr\nand is thus comparable to or larger than the highest spin-orbital conductivity of Co/Pt [cf. Figs. 3(a) and 5(a)]. In\nCo/X(t\nX)/Pt, instead, ξLSdecreases rapidly with tX, changes\nsign at 2 nm, and saturates thereafter. This variation, which issimilar in samples containing Gd and Tb, is in direct contrastwith the widespread assumption that the positive spin Hallconductivity of Pt determines the sign and magnitude of thedampinglike spin-orbit torque in Pt heterostructures.\nIndeed, our findings cannot be attributed to the sole SHE\nin the nonmagnetic layer, nor can they be attributed to thespin-orbit torques generated by the rare-earth layer, which,(b)σ < 0S σ > 0LL S\nL S\nσ > 0S σ > 0L(a)\n0123456-5-4-3-2-10\n Co/Gd/Cr\n Co/Tb/CrξLS (105 Ω-1m-1)\nt (nm)X\n02468 1 0-3-2-10123 Co/Gd/Pt\n Co/Tb/PtξLS (105 Ω-1m-1)\ntX (nm)\nFIG. 5. (a) Dependence of the spin-orbital conductivity on\nthe thickness of the rare-earth spacer in Co(2) /Gd(tX)/Cr(9) and\nCo(2)/Tb(tX)/Cr(9). The schematic depicts the conversion of the\norbital current into a spin current. Since the spin and orbital Hallconductivities of Cr are opposite and the spin-orbit coupling of\nGd and Tb is negative, the primary and converted spin currents\nhave the same sign. (b) The same as (a) in Co(2) /Gd(t\nX)/Pt(5) and\nCo(2)/Tb(tX)/Pt(5). In this case, the primary (blue) and converted\n(white) spin currents have opposite sign because Pt has positive\nspin and orbital Hall conductivities and Gd and Tb have negativespin-orbit coupling.\nalthough present, are too small to explain the sizable change\nofξLSin the trilayers with respect to the Co/Cr and Co/Pt\nbilayers (see control measurements of Co/Tb and Co/Gd in theConclusions). Moreover, samples with inverted position of Gdand Tb with respect to the Co layer present spin-orbital Hallconductivities similar to the samples without the spacer, whichindicates that the rare-earth layer is not the dominant source ofspin-orbit torques (see the Conclusions). Instead, the results inFig. 5can be rationalized by considering the combination of\nOHE, SHE, and orbital-to-spin conversion in the spacer. Thenet spin current transferred from Cr or Pt to Co depends onthe transmission at the interface, the spin and orbital diffusionin the rare-earth layer, and its orbital-to-spin conversion effi-ciency. Whereas the first two effects always diminish the spincurrent reaching the ferromagnet, the orbital-to-spin conver-sion enhances it when /angbracketleftL·S/angbracketright\nNM·/angbracketleftL·S/angbracketrightX>0 and weakens\nit when /angbracketleftL·S/angbracketrightNM·/angbracketleftL·S/angbracketrightX<0. This is the case for samples\ncontaining Cr and Pt, respectively (see Table I). The length\nscale over which the effect takes place is determined by thecombination of the spin and orbital diffusion lengths of Gdand Tb. When the spacer is thin relative to these two lengths,the orbital-to-spin conversion supplies the spin current withmore spins than those lost by scattering. On the other hand,\n033037-7GIACOMO SALA AND PIETRO GAMBARDELLA PHYSICAL REVIEW RESEARCH 4, 033037 (2022)\nspin-flip events become dominant at large thicknesses and\ndecrease the transmitted spin current. The spin-orbital con-ductivity saturates then to a finite value determined by theSHE of the rare-earth layer, as indicated by the similar ξ\nLS\nmeasured in samples with either Cr or Pt and thick spacers\n(tX/greaterorequalslant6n m ) .\nThese findings highlight the importance of achieving ef-\nficient orbital-to-spin conversion. This can be pursued bysandwiching a rare-earth spacer of optimal thickness betweenthe ferromagnet and the nonmagnet because rare-earth met-als are effective enhancers of the conversion but not strongsources of spin-orbit toques [ 63]. Remarkably, our results also\nprovide evidence of a strong OHE in Pt.\nVI. GENERALIZED DRIFT-DIFFUSION MODEL\nOF ORBITAL AND SPIN CURRENTS\nTo shed light on the interplay between spin and orbital\ncurrents, we developed a 1D model that takes into accountthe generation and diffusion of both spin and orbital angu-lar momenta as well as their interconversion mediated byspin-orbit coupling. We consider first a single nonmagneticlayer where an electric field Eapplied along xinduces the\nSHE and OHE. Let μ=μ\nS,Lbe the spin or orbital chemical\npotential and Jμ=JS,Lbe the corresponding spin or orbital\ncurrent along zwith spin and orbital polarization along y.\nThe generation, drift, and diffusion of spins and orbitals aregoverned by [ 34–39]\nd\n2μ\ndz2=μ\nλ2μ, (4)\nJμ=−σ\n2edμ\ndz+σHE, (5)\nwhere λμis the diffusion length, σis the longitudinal electri-\ncal conductivity, and σHis the spin or orbital conductivity, i.e.,\nthe off-diagonal element of the conductivity tensor. Solvingthese equations yields μ=Ae\nz/λμ+Be−z/λμ, with the coeffi-\ncients AandBobtained by imposing the boundary condition\nthatJμvanishes at the edges of the nonmagnet. In this form,\nhowever, the equations of the spin and orbital components areindependent and cannot account for the orbital-to-spin andspin-to-orbital conversion mediated by spin-orbit coupling.To capture this process, we add a phenomenological termto Eq. ( 4) for the spin (orbital) chemical potential that is\nproportional to its orbital (spin) counterpart, i.e.,\nd\n2μS\ndz2=μS\nλ2\nS±μL\nλ2\nLS, (6)\nd2μL\ndz2=μL\nλ2\nL±μS\nλ2\nLS, (7)\nJS=−σ\n2edμS\ndz+σSE, (8)\nJL=−σ\n2edμL\ndz+σLE, (9)\nwhere the +(−) sign corresponds to negative (positive) spin-\norbit coupling. Physically, this additional term represents theconversion between spins and orbitals at a rate proportional tothe respective chemical potential. Thus, even when the SHE\nis negligible, a finite spin imbalance is produced in responseto the orbital accumulation. The parameter controlling thisprocess is the coupling length λ\nLS, which is a measure of both\nthe efficiency and length scale over which the conversion takesplace.\nWe remark that Eqs. ( 6)–(9) are phenomenological and\nbased on the hypothesis that spin and orbital transport canbe described on an equal footing. They assume implicitly thepossibility of defining spin and orbital potentials and currentseven if the spin and orbital angular momenta are not conservedin the presence of spin-orbit coupling and the crystal field[64]. In this regard, we notice that the spin diffusion model\nhas found widespread use in the quantitative analysis of spin-orbit torques [ 1,52,65], spin Hall magnetoresistance [ 35], and\nsurface spin accumulation [ 55] despite the nonconservation\nof spin angular momentum. Moreover, there is a fundamentaldifference between spin and orbital transport that makes theapproximations underlying the orbital drift-diffusion modelless critical. Contrary to intuition, the crystal field does notquench the nonequilibrium orbital moment as efficiently as it\nsuppresses the equilibrium orbital moment. This is because\nthe orbital moment is carried by a relatively narrow subsetof conduction electron states, namely, its transport is medi-ated by “hot spots” in kspace. Since the orbital degeneracy\nof the hot spots is in general protected against the crystalfield splitting, the orbital momentum can be transported overlonger distances than its spin counterpart [ 5,59]. This orbital\ntransport mechanism has no spin equivalent and is supportedby the experimental evidence that orbital diffusion lengths innonmagnets and dephasing lengths in ferromagnets are signif-icantly longer than the corresponding spin lengths, as shownin this paper and in Refs. [ 19,29]. Further theoretical work\nis required to ascertain the limits of our spin-orbital modeland determine how to capture analytically the spin-orbitalinterconversion. However, our model is consistent with theBoltzmann approach proposed in Ref. [ 66] and also repro-\nduces the experimental results, as explained in the following.\nTo solve the coupled equations ( 6) and ( 7), we substitute\nthe former into the latter and obtain\nd\n4μS\ndz4−/parenleftbigg1\nλ2\nS+1\nλ2\nL/parenrightbiggd2μS\ndz2+/parenleftbigg1\nλ2\nSλ2L−1\nλ4\nLS/parenrightbigg\nμS=0.(10)\nThe solution to Eq. ( 10) reads\nμS(z)=Aez/λ1+Be−z/λ1+Cez/λ2+De−z/λ2, (11)\nwhere\n1\nλ2\n1,2=1\n2⎡\n⎣1\nλ2\nS+1\nλ2\nL±/radicalBigg/parenleftbigg1\nλ2\nS−1\nλ2\nL/parenrightbigg2\n+4\nλ4\nLS⎤\n⎦ (12)\nare the combined spin-orbital diffusion lengths that result\nfrom the coupling of the spin and orbital degrees of freedomintroduced by λ\nLS. Equation ( 11) is the generalization of the\nstandard diffusion of spins valid in the absence of spin-orbitalinterconversion. Two additional exponentials appear becauseof the coupling between LandS. For the same reason, the\nspin-orbital diffusion lengths λ\n1,2are a combination of the\nspin, orbital, and coupling lengths. The same formal solutionas Eqs. ( 11) and ( 12) holds for the orbital chemical potential\n033037-8GIANT ORBITAL HALL EFFECT AND ORBITAL-TO-SPIN … PHYSICAL REVIEW RESEARCH 4, 033037 (2022)\nbecause our model treats μSandμLon an equal footing.\nHowever, the eight unknown coefficients in Eq. ( 11) (four\nforμSand four for μL) are in general different between μS\nandμL. They are found by imposing that the spin and orbital\ncurrents vanish at the edges of the nonmagnet and that the pairof solutions for μ\nSandμL[Eq. ( 11)] satisfies Eqs. ( 6) and ( 7)\nat any z. Then, we find that the spin chemical potential at the\nsurface of the nonmagnet increases with the thickness tNMas\nμS(tNM)=2eλ1/parenleftBiggσS∓σL\nλ2\nLSγ2\n1−γ2\nγ1/parenrightBigg\nE\nσtanh/parenleftbiggtNM\n2λ1/parenrightbigg\n+2eλ2/parenleftBiggσS∓σL\nλ2\nLSγ1\n1−γ1\nγ2/parenrightBigg\nE\nσtanh/parenleftbiggtNM\n2λ2/parenrightbigg\n, (13)\nwhere γi=1\nλ2\ni−1\nλ2S. Equation ( 13) captures the interplay be-\ntween the SHE and OHE, which reinforce or weaken each\nother depending on the sign of the spin-orbit coupling and onλ\nLS. In comparison, in the absence of coupling between Sand\nL,E q .( 13) would read\nμS(tNM)=2eλSσS\nσEtanh/parenleftbiggtNM\n2λS/parenrightbigg\n, (14)\nconsistent with the standard spin drift-diffusion model. We\nnote that Eqs. ( 11) and ( 13) are valid under the condition\nλLS>√λSλLbecause for smaller values of λLSthe solution\nto Eq. ( 10) is a linear combination of complex exponential\nfunctions, i.e., μSandμLhave an oscillatory dependence on z.\nSimilar oscillations have been predicted in Ref. [ 9]. However,\nwe argue that complex solutions to Eq. ( 12) are incompati-\nble with experimental results since an oscillatory dependenceof spin-orbit torques or spin Hall magnetoresistance on thethickness of the nonmagnetic layer has never been observed.The condition λ\nLS>√λSλLalso implies that the conversion\nbetween spins and orbitals cannot occur on a length scaleshorter than the shortest distance over which either spins ororbitals diffuse. At the same time, it shows that the conver-sion is always less efficient than the intrinsic spin and orbitalrelaxation.\nWe apply our model to study the interplay of nonequi-\nlibrium spins and orbitals induced by the SHE and OHEin two exemplary situations. First, we take a single non-magnetic layer with negative spin-orbit coupling, e.g., Cr.Figure 6shows the spin and orbital chemical potentials in\nthree different conditions. In Fig. 6(a), the OHE is turned\noff (σ\nL=0), and the SHE is active [ σS=−105(/Omega1m)−1]. In\nFigs. 6(b)and6(c), the situation is opposite, namely, σL=105\n(/Omega1m)−1andσS=0. In all cases, orbitals (spins) accumulate\nat the interfaces even if the OHE (SHE) is set to zero. Since/angbracketleftL·S/angbracketright\nNM<0, the two chemical potentials are of opposite\nsign. When λLSdecreases, the spin accumulation resulting\nfrom the orbital conversion increases approximately as λ−2\nLS\n[Fig. 6(b) and Eq. ( 13)]. Interestingly, we find that even if\nλSis small, spins accumulate on a long distance because the\nspin-orbital diffusion lengths λ1,2are dominated by λL[cf.\nFigs. 6(b) and6(c)].\nAsλLSdecreases and the spin-orbit conversion becomes\nmore efficient, both the spin accumulation and the orbitalaccumulation increase at the sample edges. This effect might(a) (b)\nσ= 0L \nσ= 0S (c)\n-30030\n0 5 10 15 20-0.80.00.8L (μeV ) S (μeV)\nz (nm) λ =  5  n mL λ  = λ  = 2 nmLSσ= 0S \n λ  = λ  = 2 nmLS\n λ  = 2 nmS-0.20.00.2\n05 1 0 1 5 2 0-10010L (μeV) S (μeV)\nz (nm)-10010\n05 1 0 1 5 2 0-202 λLS= 10 nm\n λLS= 3 nmL (μeV)\n λLS= 10 nm\n λLS= 3 nmS (μeV)\nz (nm)\nFIG. 6. (a) Orbital and spin chemical potentials in a single non-\nmagnetic layer with tNM=20 nm, σS=−105(/Omega1m)−1,σL=0,\nλS=λL=2n m ,a n d λLS=10 nm. (b) The same as (a) with σS=0,\nσL=105(/Omega1m)−1,λS=λL=2n m ,a n d λLS=3 or 10 nm. (c) The\nsame as (a) with σS=0,σL=105(/Omega1m)−1,λS=2n m , λL=5n m ,\nandλLS=10 nm. In all cases, the resistivity of the NM layer was\nset to ρ=56×10−8/Omega1m as measured for Cr, and an electric field\nE=5×104V/m was considered.\nseem counterintuitive, because spin-orbit coupling usually\ninduces dissipation of angular momentum. In our model, how-ever, the dissipation of SandLis included in the parameters\nλ\nSandλL, respectively, whereas λLSdescribes the nondissipa-\ntive exchange of angular momentum between the orbital andspin reservoirs. Thus λ\nLSeffectively increases the spatial ex-\ntent of orbital and spin accumulation. Formally, this happensbecause one of the two spin-orbital diffusion lengths λ\n1,2in-\ncreases while the other changes weakly when λLSis reduced.\nAs a consequence, more spins and orbitals can accumulate atthe sample edges. This result is similar to the model withoutspin-orbit coupling, which predicts an increase in μ\nSwith the\nspin diffusion length: μS(tNM/greatermuchλS)∼λS[see Eq. ( 14)].\nNext, we consider a trilayer structure representative of\nthe samples Co(2) /X(tX)/Cr(9) and Co(2) /X(tX)/Pt(5). We\nmodel the spatial variations of μSandμLby four equations of\nthe same type as Eq. ( 11), two for the rare-earth layer and\ntwo for Cr (or Pt). We assume that μS,μL,JS, and JLare\ncontinuous at the X/NM interface ( z=tX) and impose the\nconstraint that JSandJLrelate to μSandμL, respectively,\n033037-9GIACOMO SALA AND PIETRO GAMBARDELLA PHYSICAL REVIEW RESEARCH 4, 033037 (2022)\n(a) (b)\n-1.5-1.0-0.50.0\n01234560510JS (109 A/m2) JL (109 A/m2)\ntX (nm)012\n02468 1 0048JS (109 A/m2) JL (109 A/m2)\ntX (nm)σ > 0S σ > 0L\nPt\nX\nCotXCr\nX\nCotXσ < 0S σ > 0L\nFIG. 7. (a) Calculated spin and orbital currents at the FM/ X\ninterface as a function of tXin Co(2) /X(tX)/Pt(5). (b) The same as\n(a) for Co(2) /X(tX)/Cr(9). The orbital and spin Hall conductivities\nof Pt and Cr are indicated above the graphs. The parameters usedto calculate the orbital and spin currents can be found in Table II\n(Appendix C).\nthrough the mixing conductance GS,L:\nJX\nS(tX)=GX\nS\neμS(tX), (15)\nJX\nL(tX)=GX\nL\neμL(tX). (16)\nIn doing so, we introduce the orbital equivalent of the spin\nmixing conductance, which is expected to depend on thespin-orbit coupling of the ferromagnet and to influence thestrength of the orbital torque. Thus, in our model, G\nLtakes\ninto account the additional orbital-to-spin conversion occur-ring in the ferromagnet or at the interface. Furthermore, weonly consider the real part of G\nS,Lsince the fieldlike torque\nis small in our samples. Finally, we assume a finite SHE inboth the nonmagnetic and rare-earth layers, but smaller inthe latter, whereas the OHE is present only in the nonmagnet(see Appendix Cfor a list of the parameters). We set σ\nS>0\nin Pt, σS<0 in Cr and in the spacer, and σL>0 in both\nCr and Pt. The spin-orbit coupling is assumed positive in Ptand negative in Cr and in the rare-earth layer. With thesereasonable assumptions, we can reproduce qualitatively theresults of Fig. 5, namely, the enhancement of the spin-orbital\nconductivity upon insertion of a rare-earth spacer betweenCo and Cr and the sign change of the torques when thesame layer is sandwiched between Co and Pt. Figure 7shows\nthe calculated spin and orbital currents, to which spin-orbittorques are proportional, that reach the FM/ Xinterface as a\nfunction of the rare-earth thickness t\nX. In both the case of\nCr and the case of Pt the orbital current decreases monoton-ically as t\nXincreases because of the orbital diffusion away\nfrom the X/NM interface. In contrast, the spin current variesdifferently with tXdepending on whether Cr or Pt is chosen\nbecause the primary spin current and the current obtainedupon orbital-to-spin conversion in the rare-earth element havethe same sign with Cr and have opposite sign with Pt. Thuscalculations based on a generalized drift-diffusion model con-firm the interpretation of the data in Fig. 5, which cannot\nbe explained without the inclusion of the OHE. We believethat a better quantitative agreement with the measurementscould be obtained by including additional effects that we havedisregarded, namely, the interfacial resistance ( μ\nSandμLnot\ncontinuous), the interfacial spin and orbital scattering ( JSand\nJLnot continuous), and the thickness dependence of the spacer\nresistivity and, possibly, of the diffusion lengths. The modelcould be extended to account for the orbital conversion in theferromagnet, which is hidden here behind the orbital mixingconductance. Finally, it may be employed to investigate othertransport effects such as the spin Hall magnetoresistance andits orbital counterpart.\nVII. CONCLUSIONS\nOur measurements of spin-orbit torques in FM/NM and\nFM/X/NM multilayers with light and heavy metals provide\ncomprehensive evidence for strong OHE effects in 3 dand 5 d\nmetals and establish a systematic framework to analyze andefficiently exploit the interplay of spin and orbital currents.Owing to the entanglement of the orbital and spin degreesof freedom in materials with finite spin-orbit coupling, thisinterplay is best described by combined spin-orbital conduc-tivity ( ξ\nLS) and diffusion length ( λLS) parameters rather than\nby considering the OHE and SHE as two separate effects.The experimental values of ξ\nLSfor the different systems and\ncontrol samples are summarized in Fig. 8. Corresponding\nvalues of the spin-orbital Hall angle θLSare reported in Ap-\npendix A. We found strong spin-orbit torques produced by the\nlight elements Cr and Mn, whose sign depends on the adja-cent ferromagnet, in contrast with torques generated by theSHE. The spin-orbital conductivity increases with the thick-ness of the light metal layer without indications of saturation.This trend is compatible with spin-orbital diffusion lengthsλ\nLS/greaterorsimilar20 nm in these elements and extrapolates to a giant\nintrinsic spin-orbital conductivity as predicted by theory [ 4].\nBecause of the competition between spin and orbital torques,the spin-orbital conductivity varies with the thickness of theferromagnet in a monotonic or nonmonotonic way dependingon the relative sign of /angbracketleftL·S/angbracketright\nNMand/angbracketleftL·S/angbracketrightFM. Furthermore,\nwe show that the interplay between spin and orbital torquescan be drastically enhanced by inserting a 4 fspacer layer\nbetween the nonmagnet and the ferromagnet. As summarizedin Fig. 8, the inclusion of a Tb (Gd) spacer results in a fourfold\n(threefold) increase of the torques generated by Cr and Mnthat cannot be attributed to spin currents generated by therare-earth element. Instead, the enhancement results from theconversion of the orbital current into a secondary spin currentof the same sign as the primary spin current. The orbital-to-spin conversion has a striking effect in Pt, when the primaryspin current generated by the SHE and the secondary spincurrents generated by the OHE interfere destructively. Thiseffect results in the reversal of the spin-orbit torque generatedby Pt when the orbital-to-spin conversion rate is stronger than\n033037-10GIANT ORBITAL HALL EFFECT AND ORBITAL-TO-SPIN … PHYSICAL REVIEW RESEARCH 4, 033037 (2022)\nFM/Cr(9)\nCo(2)/Gd(3)/Cr(9)\nCo(2)/Tb(3)/Cr(9)\nCo(2)/Gd(4)\nCo(2)/Tb(4) \nGd(3)/Co(2)/Cr(9)-4-3-2-101(a) (b)\n(c)23ξLS (105 Ω-1m-1)\nCoNi\nCr\nFM/Mn(9)\nCo(2)/Gd(3)/Mn(9)\nGd(3)/Co(2)/Mn(9)CoNi\nMn\nFM/Pt(5)\nCo(2)/Gd(3)/Pt(5)\nCo(2)/Tb(3)/Pt(5)\nGd(4)/Co(2)/Pt(5)\nTb(4)/Co(2)/Pt(5)NiCo\nPt\nFIG. 8. Comparison of the effective spin-orbital conductivity ξLSmeasured in NM/FM and NM/ X/FM layers where (a) NM =Cr,\n(b) NM =Mn, and (c) NM =Pt; FM =Co,Ni,X=Gd,Tb. The thickness of each layer is indicated in nanometers in parentheses. The\nresults of control experiments on X/FM and NM/FM/ Xlayers are also shown.\nthe primary spin current. These findings indicate the presence\nof a strong OHE and SHE in Cr, Mn, and Pt and highlightthe importance of orbital-to-spin conversion phenomena indifferent types of heterostructures. The largest ξ\nLS=−4.3×\n105(/Omega1m)−1andθLS≈0.25 are found in Co(2)/Gd(3) /Cr(9)\nand Co(2)/Tb(3) /Cr(9) layers. Both of these parameters are\nlarger compared with Co/Pt and previous measurements, in-dicating that optimization of the thickness of 3 dmetal layers\nand the insertion of 4 fspacers lead to giant spin-orbital Hall\neffects and ensuing spin-orbit torques. The fits of ξ\nLSas a\nfunction of thickness indicate that the spin-orbital conductiv-ity of Cr saturates at values of the order of 10\n6(/Omega1m)−1,i n\nagreement with theoretical estimates [ 4]. Finally, we propose\nan extended drift-diffusion model that treats the orbital andspin moment on an equal footing and includes the orbital-to-\nFM/Cr(9)\nCo(2)/Gd(3)/Cr(9)\nCo(2)/Tb(3)/Cr(9)Co(2)/Gd(4)\nCo(2)/Tb(4) \nGd(3)/Co(2)/Cr(9)-0.3-0.2-0.10.00.10.2θH\nCoNi\nCr\nFM/Mn(9)\nCo(2)/Gd(3)/Mn(9)\nGd(3)/Co(2)/Mn(9)MnCoNi\nFM/Pt(5)\nCo(2)/Gd(3)/Pt(5)\nCo(2)/Tb(3)/Pt(5)\nGd(4)/Co(2)/Pt(5)\nTb(4)/Co(2)/Pt(5)Co\nNi\nPt075150225300ρ (μΩ cm) Co\n Ni\nFIG. 9. Resistivity (top) and effective spin-orbital Hall angle\n(bottom) of the samples in Fig. 8in the main text.spin conversion mediated by spin-orbit coupling. The model\nexplains both the monotonic and nonmonotonic behavior ofξ\nLSobserved in the FM/NM and FM/ X/NM multilayers as\na function of thickness and spin-orbit coupling of the con-stituent layers. It also shows how the spatial profiles of theorbital and spin accumulation are determined by the combinedspin-orbital diffusion lengths and spin and orbital mixing con-ductances. Overall, our results provide a useful framework tomaximize the orbital-to-spin conversion efficiency, interpretexperimental results, and address open fundamental questionsabout orbital transport.\n(a) (b)\n01234560.00.51.01.52.02.5\n Co/Gd( tX)/Cr\n Co/Tb( tX)/CrMs*t (mA )\ntX (nm)02468 1 00.00.40.81.21.62.02.4\n Co/Gd( tX)/Pt\n Co/Tb( tX)/PtMs*t (mA)\ntX (nm)\n02468 1 0 1 20481216\n Co(tFM)/Cr\n Co(tFM)/PtMs*t (mA)\ntFM (nm)(c)\nFIG. 10. (a) Dependence of the areal magnetization on the\nthickness of the rare-earth layer in Co(2) /X(tX)/Cr(9) samples.\n(b) The same as (a) in Co(2) /X(tX)/Pt(5) samples. (c) Dependence\nof the areal magnetization on the thickness of the ferromagnet in\nCo(tFM)/Cr(9) and Co( tFM)/Pt(5) samples.\n033037-11GIACOMO SALA AND PIETRO GAMBARDELLA PHYSICAL REVIEW RESEARCH 4, 033037 (2022)\nTABLE II. Parameters used in the drift-diffusion model to calculate the spin and orbital currents in a FM/ X/NM trilayer, where NM is\neither Cr or Pt. λS,Lis the spin or orbital diffusion length, λLSis the spin-orbital conversion length, σS,Lis the spin or orbital Hall conductivity,\nα=±1 is the sign of the spin-orbit coupling, GS,Lis the spin or orbital mixing conductance, and ρis the electrical resistivity. The thickness\nof the Cr (Pt) layer was 9 (5) nm. An electric field E=5×104V/m was considered in both cases.\nλNM\nLλXLλNMSλXSλNMLSλXLS σNM\nL σX\nL σNM\nS σX\nS GL GS ρNM ρX\n(nm) (nm) (nm) (nm) (nm) (nm) [( /Omega1m)−1][ (/Omega1m)−1][ (/Omega1m)−1][ (/Omega1m)−1]αNMαX[(/Omega1m2)−1][ (/Omega1m2)−1](/Omega1m) ( /Omega1m)\nC r 8262 2 0 2 . 5 8 .2×1050 −0.7×105−0.15×105−1−13×1014101456×10−8115×10−8\nP t 12222 2 . 5 8 .8×10503 .5×105−0.15×105+1−13×1014101433×10−8115×10−8\nACKNOWLEDGMENT\nWe acknowledge the support of the Swiss National Science\nFoundation (Grant No. 200020_200465).\nAPPENDIX A: EFFECTIVE SPIN-ORBITAL HALL ANGLE\nFigure 9shows the effective spin-orbital Hall angle of\nthe samples presented in Fig. 8in the main text. The Hall\nangle was calculated according to θLS=ξLSρ, where ρis the\nresistivity of the entire stack. However, we refrain from esti-mating the resistivity of the individual layers and comparingquantitatively θ\nLSof the NM layers alone in this way, because\nthe resistivity of the heterostructures depends strongly on in-terfaces and the thickness of all layers. Similarly to ξ\nLS,w e\ninterpret θLSas a parameter that describes the simultaneous\noccurrence of the OHE, the SHE, and orbital-to-spin conver-sion. The values reported in Figs. 8and9are measured in\nsamples with the thickness specified in the axis labels.\nAPPENDIX B: SATURATION MAGNETIZATION\nFigure 10shows the surface saturation magnetization\nof samples belonging to the series Co(2) /X(tX)/Cr(9),\nCo(2)/X(tX)/Pt(5), Co( tFM)/Cr(9), and Co( tFM)/Pt(5) as a\nfunction of the corresponding thickness. The magnetiza-tion was measured by superconducting quantum interferencedevice (SQUID) magnetometry on blanket films grown si-multaneously to the measured devices. The measurementyields the magnetic moment of the sample, which, after nor-malization to the sample area, defines the areal saturationmagnetization M\nstFM. This parameter is to be preferred over\nthe volume saturation magnetization, since the latter dependson the thickness of the ferromagnetically active material. Thisis in turn difficult to define with certainty in the studied sam-ples because of interdiffusion at interfaces, proximity effects,and possible ferrimagnetic coupling. Such a complexity, how-ever, does not impinge on the calculation of the spin-orbitalconductivity because the quantity appearing in Eq. ( 1)i st h e\nareal saturation magnetization M\nstFM, not the volume magne-\ntization.\nFigures 10(a) and 10(b) show that the areal magnetiza-\ntion decreases upon increasing the thickness of either Gdor Tb in both Cr- and Pt-based samples. We attribute thisreduction to the antiferromagnetic interaction between Co and\nthe rare-earth layer. We note that the ferrimagnetic couplingcannot explain our results, namely, the trends presented inFig. 5. First, the torque efficiency rises by a factor of 3–4\nwhen the thickness of the rare earth t\nXincreases from 0 to\n3 nm, while the areal magnetization decreases only by 20%.Second, the areal magnetization decreases monotonically with\nt\nX, while the trends in Fig. 5are not monotonic with respect\nto the thickness. For example, the spin-orbital conductivity ofCo(2)/X(t\nX)/Pt(5) saturates in the limit of large tX, whereas\nthe magnetization does not.\nThe areal magnetization of Co( tX)/Cr(9) and Co( tX)/Pt(5)\nsamples increases linearly with tX, as expected. The linear fits\nyield dead layers of about 0.5 and 0.3 nm in the two series, re-spectively. The dead layer is likely located at the substrate/Cointerface and is probably thinner in the Co( t\nFM)/Pt(5) series\nbecause of proximity effects with Pt. These values have beentaken into account in the torque calculation in Fig. 4.\nFinally, the saturation magnetization of Co(2) /NM( t\nNM)\nand Ni(4) /NM( tNM) was found to be independent of tNM,\nexcept for Co(2) /Pt(tNM), where the areal magnetization in-\ncreases by 7% from tPt=1n mt o tPt=12 nm (not shown).\nAPPENDIX C: PARAMETERS OF THE\nDRIFT-DIFFUSION MODEL\nTable IIlists all the parameters used for the calculation of\nthe orbital and spin currents in the Co(2) /X(tX)/Cr(9) and\nCo(2)/X(tX)/Pt(5) samples (Fig. 7). Some of them have been\nmeasured (spin diffusion length of Pt; spin Hall conductivityof Cr and Pt from Co/Cr and Co/Pt samples, respectively;and resistivity). Others have been chosen in accordance withthe literature (spin mixing conductance, sign of the spin-orbitcoupling, orbital conductivity). The remaining parameters,mostly involving orbitals and the spin-orbital interconver-sion, are not available in the literature and have been chosensuch that the calculations agree qualitatively with the mea-surements. From this perspective, the extended drift-diffusionmodel can be used to estimate the order of magnitude ofthe unknown parameters. For instance, the spin and orbitaldiffusion length and the spin-orbital coupling length of therare-earth layer must be of the order of a few nanometers atmost for the model to reproduce the measurements in Fig. 5.\n[1] A. Manchon, J. Železný, I. M. Miron, T. Jungwirth, J. Sinova,\nA. Thiaville, K. Garello, and P. Gambardella, Current-inducedspin-orbit torques in ferromagnetic and antiferromagnetic sys-\ntems, Rev. Mod. Phys. 91, 035004 (2019) .\n033037-12GIANT ORBITAL HALL EFFECT AND ORBITAL-TO-SPIN … PHYSICAL REVIEW RESEARCH 4, 033037 (2022)\n[2] V . E. Demidov, S. Urazhdin, G. de Loubens, O. Klein, V . Cros,\nA. Anane, and S. O. Demokritov, Magnetization oscillationsand waves driven by pure spin currents, Phys. Rep. 673,1\n(2017) .\n[3] J. Sinova, S. O. Valenzuela, J. Wunderlich, C. H. Back, and T.\nJungwirth, Spin Hall effects, Rev. Mod. Phys. 87, 1213 (2015) .\n[4] D. Jo, D. Go, and H. W. Lee, Gigantic intrinsic orbital Hall\neffects in weakly spin-orbit coupled metals, P h y s .R e v .B 98,\n214405 (2018) .\n[5] D. Go, D. Jo, C. Kim, and H. W. Lee, Intrinsic Spin and Orbital\nHall Effects from Orbital Texture, Phys. Rev. Lett. 121, 086602\n(2018) .\n[6] D. Go and H.-W. Lee, Orbital torque: Torque generation by\norbital current injection, Phys. Rev. Research 2, 013177 (2020) .\n[7] D. Go, F. Freimuth, J.-P. Hanke, F. Xue, O. Gomonay, K.-J. Lee,\nS. Blügel, P. M. Haney, H.-W. Lee, and Y . Mokrousov, Theoryof current-induced angular momentum transfer dynamics inspin-orbit coupled systems, P h y s .R e v .R e s . 2, 033401 (2020) .\n[8] D. Go, D. Jo, H.-W. Lee, M. Kläui, and Y . Mokrousov, Orbi-\ntronics: Orbital currents in solids, Europhys. Lett. 135, 37001\n(2021) .\n[9] L. Salemi, M. Berritta, and P. M. Oppeneer, Quantitative com-\nparison of electrically induced spin and orbital polarizations inheavy-metal/3 d-metal bilayers, P h y s .R e v .M a t e r i a l s 5, 074407\n(2021) .\n[10] P. Sahu, S. Bhowal, and S. Satpathy, Effect of the inversion\nsymmetry breaking on the orbital Hall effect: A model study,Phys. Rev. B 103, 085113 (2021) .\n[11] T. Tanaka, H. Kontani, M. Naito, T. Naito, D. S. Hirashima, K.\nYamada, and J. Inoue, Intrinsic spin Hall effect and orbital Halleffect in 4 dand 5 dtransition metals, Phys. Rev. B 77, 165117\n(2008) .\n[12] H. Kontani, T. Tanaka, D. S. Hirashima, K. Yamada, and J.\nInoue, Giant Orbital Hall Effect in Transition Metals: Origin ofLarge Spin and Anomalous Hall Effects, Phys. Rev. Lett. 102,\n016601 (2009) .\n[13] D. Go, D. Jo, T. Gao, K. Ando, S. Blügel, H.-w. Lee, and\nY . Mokrousov, Orbital Rashba effect in a surface-oxidized Cufilm, P h y s .R e v .B 103, L121113 (2021) .\n[14] S. Bhowal and S. Satpathy, Intrinsic orbital moment and predic-\ntion of a large orbital Hall effect in two-dimensional transitionmetal dichalcogenides, Phys. Rev. B 101, 121112(R) (2020) .\n[15] L. M. Canonico, T. P. Cysne, A. Molina-Sanchez, R. B. Muniz,\nand T. G. Rappoport, Orbital Hall insulating phase in transitionmetal dichalcogenide monolayers, Phys. Rev. B 101, 161409(R)\n(2020) .\n[16] Z. C. Zheng, Q. X. Guo, D. Jo, D. Go, L. H. Wang, H. C. Chen,\nW. Yin, X. M. Wang, G. H. Yu, W. He, H.-W. Lee, J. Teng,and T. Zhu, Magnetization switching driven by current-inducedtorque from weakly spin-orbit coupled Zr, Phys. Rev. Res. 2\n,\n013127 (2020) .\n[17] H. W. Ko, H. J. Park, G. Go, J. H. Oh, K. W. Kim, and K. J. Lee,\nRole of orbital hybridization in anisotropic magnetoresistance,Phys. Rev. B 101, 184413 (2020) .\n[18] J. Kim, D. Go, H. Tsai, D. Jo, K. Kondou, H. W. Lee, and Y . C\nOtani, Nontrivial torque generation by orbital angular momen-tum injection in ferromagnetic-metal/Cu/Al\n2O3trilayers, Phys.\nRev. B 103, L020407 (2021) .\n[19] S. Lee, M.-G. Kang, D. Go, D. Kim, J.-H. Kang, T. Lee, G.-H.\nLee, J. Kang, N. J. Lee, Y . Mokrousov, S. Kim, K.-J. Kim, K.-J.Lee, and B.-G. Park, Efficient conversion of orbital Hall current\nto spin current for spin-orbit torque switching, Commun. Phys.\n4, 234 (2021) .\n[20] D. Lee, D. Go, H.-J. Park, W. Jeong, H.-W. Ko, D. Yun,\nD. Jo, S. Lee, G. Go, J. H. Oh, K.-J. Kim, B.-G. Park,B.-C. Min, H. C. Koo, H.-W. Lee, O. Lee, and K.-J. Lee,Orbital torque in magnetic bilayers, Nat. Commun. 12, 6710\n(2021) .\n[21] S. Ding, A. Ross, D. Go, L. Baldrati, Z. Ren, F. Freimuth, S.\nBecker, F. Kammerbauer, J. Yang, G. Jakob, Y . Mokrousov, andM. Kläui, Harnessing Orbital-to-Spin Conversion of InterfacialOrbital Currents for Efficient Spin-Orbit Torques, Phys. Rev.\nLett.125, 177201 (2020) .\n[22] S. Ding, Z. Liang, D. Go, C. Yun, M. Xue, Z. Liu, S. Becker,\nW. Yang, H. Du, C. Wang, Y . Yang, G. Jakob, M. Kläui, Y .Mokrousov, and J. Yang, Observation of the Orbital Rashba-Edelstein Magnetoresistance, Phys. Rev. Lett. 128, 067201\n(2022) .\n[23] Q. Guo, Z. Ren, H. Bai, X. Wang, G. Yu, W. He, J. Teng, and T.\nZhu, Current-induced magnetization switching in perpendicu-larly magnetized V/CoFeB/MgO multilayers, Phys. Rev. B 104,\n224429 (2021) .\n[24] K. V . Shanavas, Z. S. Popovi ´c, and S. Satpathy, Theoretical\nmodel for Rashba spin-orbit interaction in delectrons, Phys.\nRev. B 90, 165108 (2014) .\n[25] C.-Y . Hu, Y .-F. Chiu, C.-C. Tsai, C.-C. Huang, K.-H. Chen, C.-\nW. Peng, C.-M. Lee, M.-Y . Song, Y .-L. Huang, S.-J. Lin, andC.-F. Pai, Toward 100% spin-orbit torque efficiency with highspin-orbital Hall conductivity Pt-Cr alloys, ACS Appl. Electron.\nMater. 4, 1099 (2022) .\n[26] H. An, Y . Kageyama, Y . Kanno, N. Enishi, and K. Ando, Spin-\ntorque generator engineered by natural oxidation of Cu, Nat.\nCommun. 7, 13069 (2016) .\n[27] T. Wang, W. Wang, Y . Xie, M. A. Warsi, J. Wu, Y . Chen,\nV . O. Lorenz, X. Fan, and J. Q. Xiao, Large spin Hall anglein vanadium film, Sci. Rep. 7, 1306 (2017) .\n[28] J. Bass and W. P. Pratt Jr, Spin-diffusion lengths in metals and\nalloys, and spin-flipping at metal/metal interfaces: An experi-mentalist’s critical review, J. Phys.: Condens. Matter 19, 183201\n(2007) .\n[29] Y .-G. Choi, D. Jo, K.-h. Ko, D. Go, and H.-w. Lee, Observation\nof the orbital Hall effect in a light metal Ti, arXiv:2109.14847 .\n[30] J.-C. Rojas-Sánchez, N. Reyren, P. Laczkowski, W. Savero, J.-P.\nAttané, C. Deranlot, M. Jamet, J.-M. George, L. Vila, and H.Jaffrès, Spin Pumping and Inverse Spin Hall Effect in Platinum:The Essential Role of Spin-Memory Loss at Metallic Interfaces,Phys. Rev. Lett. 112, 106602 (2014) .\n[31] X. Tao, Q. Liu, B. Miao, R. Yu, Z. Feng, L. Sun, B. You, J.\nDu, K. Chen, S. Zhang, L. Zhang, Z. Yuan, D. Wu, and H.Ding, Self-consistent determination of spin Hall angle and spindiffusion length in Pt and Pd: The role of the interface spin loss,Sci. Adv. 4, eaat1670 (2018) .\n[32] L. Zhu, D. C. Ralph, and R. A. Buhrman, Spin-Orbit Torques in\nHeavy-Metal–Ferromagnet Bilayers with Varying Strengths ofInterfacial Spin-Orbit Coupling, Phys. Rev. Lett. 122, 077201\n(2019) .\n[33] C. O. Avci, G. S. D. Beach, and P. Gambardella, Effects of\ntransition metal spacers on spin-orbit torques, spin Hall magne-toresistance, and magnetic anisotropy of Pt/Co bilayers, Phys.\nRev. B 100, 235454 (2019) .\n033037-13GIACOMO SALA AND PIETRO GAMBARDELLA PHYSICAL REVIEW RESEARCH 4, 033037 (2022)\n[34] P. C. van Son, H. van Kempen, and P. Wyder, Boundary Resis-\ntance of the Ferromagnetic-Nonferromagnetic Metal Interface,Phys. Rev. Lett. 58, 2271 (1987) .\n[35] Y .-T. Chen, S. Takahashi, H. Nakayama, M. Althammer, S. T. B.\nGoennenwein, E. Saitoh, and G. E. W. Bauer, Theory of spinHall magnetoresistance, P h y s .R e v .B 87, 144411 (2013) .\n[36] P. M. Haney, H. W. Lee, K. J. Lee, A. Manchon, and\nM. D. Stiles, Current induced torques and interfacial spin-orbitcoupling: Semiclassical modeling, P h y s .R e v .B 87, 174411\n(2013) .\n[37] V . P. Amin and M. D. Stiles, Spin transport at interfaces\nwith spin-orbit coupling: Formalism, Phys. Rev. B 94, 104419\n(2016) .\n[38] V . P. Amin and M. D. Stiles, Spin transport at interfaces with\nspin-orbit coupling: Phenomenology, P h y s .R e v .B 94, 104420\n(2016) .\n[39] K.-W. Kim, Spin transparency for the interface of an ultrathin\nmagnet within the spin dephasing length, Phys. Rev. B 99,\n224415 (2019) .\n[40] K. Garello, I. M. Miron, C. O. Avci, F. Freimuth, Y . Mokrousov,\nS. Blügel, S. Auffret, O. Boulle, G. Gaudin, and P. Gambardella,Symmetry and magnitude of spin-orbit torques in ferromagneticheterostructures, Nat. Nanotechnol. 8, 587 (2013) .\n[41] E. Sagasta, Y . Omori, M. Isasa, M. Gradhand, L. E. Hueso, Y .\nNiimi, Y . Otani, and F. Casanova, Tuning the spin Hall effectof Pt from the moderately dirty to the superclean regime, Phys.\nRev. B 94, 060412(R) (2016) .\n[42] H. Moriya, A. Musha, S. Haku, and K. Ando, Observation of\nthe crossover between metallic and insulating regimes of thespin Hall effect, Commun. Phys. 5, 12 (2022) .\n[43] H. Nakayama, M. Althammer, Y .-T. Chen, K. Uchida, Y .\nKajiwara, D. Kikuchi, T. Ohtani, S. Geprägs, M. Opel, S.Takahashi, R. Gross, G. E. W. Bauer, S. T. B. Goennenwein,and E. Saitoh, Spin Hall Magnetoresistance Induced by aNonequilibrium Proximity Effect, Phys. Rev. Lett. 110, 206601\n(2013) .\n[44] C. Du, H. Wang, F. Yang, and P. C. Hammel, Systematic\nvariation of spin-orbit coupling with d-orbital filling: Large\ninverse spin Hall effect in 3 dtransition metals, Phys. Rev. B\n90, 140407(R) (2014) .\n[45] C. O. Avci, K. Garello, A. Ghosh, M. Gabureac, S. F. Alvarado,\nand P. Gambardella, Unidirectional spin Hall magnetoresis-tance in ferromagnet-normal metal bilayers, Nat. Phys. 11, 570\n(2015) .\n[46] C. O. Avci, J. Mendil, G. S. D. Beach, and P. Gambardella,\nOrigins of the Unidirectional Spin Hall Magnetoresistance inMetallic Bilayers, P h y s .R e v .L e t t . 121, 087207 (2018) .\n[47] K. Ueda, C.-f. Pai, A. J. Tan, M. Mann, and G. S. D. Beach,\nEffect of rare earth metal on the spin-orbit torque in magneticheterostructures, Appl. Phys. Lett. 108, 232405 (2016) .\n[48] Q. Y . Wong, C. Murapaka, W. C. Law, W. L. Gan, G. J. Lim,\nand W. S. Lew, Enhanced Spin-Orbit Torques in Rare-EarthPt/[Co/Ni]\n2/Co/Tb Systems, Phys. Rev. Applied 11, 024057\n(2019) .\n[49] W. Wang, T. Wang, V . P. Amin, Y . Wang, A. Radhakrishnan, A.\nDavidson, S. R. Allen, T. J. Silva, H. Ohldag, D. Balzar, B. L.Zink, P. M. Haney, J. Q. Xiao, D. G. Cahill, V . O. Lorenz, andX. Fan, Anomalous spin-orbit torques in magnetic single-layerfilms, Nat. Nanotechnol. 14, 819 (2019) .[50] M. Schreier, G. E. W. Bauer, V . I. Vasyuchka, J. Flipse, K.-\ni. Uchida, J. Lotze, V . Lauer, A. V . Chumak, A. A. Serga, S.Daimon, T. Kikkawa, E. Saitoh, B. J. van Wees, B. Hillebrands,R. Gross, and S. T. B. Goennenwein, Sign of inverse spin Hallvoltages generated by ferromagnetic resonance and temperaturegradients in yttrium iron garnet platinum bilayers, J. Phys. D:\nAppl. Phys. 48, 025001 (2015) .\n[51] C. O. Avci, K. Garello, M. Gabureac, A. Ghosh, A. Fuhrer,\nS. F. Alvarado, and P. Gambardella, Interplay of spin-orbittorque and thermoelectric effects in ferromagnet-normal-metalbilayers, Phys. Rev. B 90, 224427 (2014) .\n[52] M. H. Nguyen, D. C. Ralph, and R. A. Buhrman, Spin Torque\nStudy of the Spin Hall Conductivity and Spin Diffusion Lengthin Platinum Thin Films with Varying Resistivity, Phys. Rev.\nLett.116, 126601 (2016) .\n[53] T. C. Chuang, C. F. Pai, and S. Y . Huang, Cr-induced Perpen-\ndicular Magnetic Anisotropy and Field-Free Spin-Orbit-TorqueSwitching, Phys. Rev. Applied 11, 061005 (2019) .\n[54] V . P. Amin, J. Li, M. D. Stiles, and P. M. Haney, Intrinsic spin\ncurrents in ferromagnets, P h y s .R e v .B 99, 220405(R) (2019) .\n[55] C. Stamm, C. Murer, M. Berritta, J. Feng, M. Gabureac, P. M.\nOppeneer, and P. Gambardella, Magneto-Optical Detection ofthe Spin Hall Effect in Pt and W Thin Films, P h y s .R e v .L e t t .\n119, 087203 (2017) .\n[56] K. Ounadjela, P. Vennegues, Y . Henry, A. Michel, V . Pierron-\nBohnes, and J. Arabski, Structural changes in metastableepitaxial Co/Mn superlattices, Phys. Rev. B 49, 8561 (1994) .\n[57] G. M. Luo, H. W. Jiang, C. X. Liu, Z. H. Mai, W. Y . Lai,\nJ. Wang, and Y . F. Ding, Chemical intermixing at FeMn/Cointerfaces, J. Appl. Phys. (Melville, NY) 91, 150 (2002) .\n[58] W. Zhang, W. Han, X. Jiang, S.-H. Yang, and S. S. P. Parkin,\nRole of transparency of platinum-ferromagnet interfaces in de-termining the intrinsic magnitude of the spin Hall effect, Nat.\nPhys. 11, 496 (2015) .\n[59] D. Go, D. Jo, K.-W. Kim, S. Lee, M.-G. Kang, B.-G. Park,\nS. Blügel, H.-W. Lee, and Y . Mokrousov, Long-range orbitaltransport in ferromagnets, arXiv:2106.07928 .\n[60] P. Chowdhury, P. D. Kulkarni, M. Krishnan, H. C. Barshilia, A.\nSagdeo, S. K. Rai, G. S. Lodha, and D. V . Sridhara Rao, Ef-fect of coherent to incoherent structural transition on magneticanisotropy in Co/Pt multilayers, J. Appl. Phys. (Melville, NY)\n112, 023912 (2012) .\n[61] M. D. Stiles and A. Zangwill, Anatomy of spin-transfer torque,\nPhys. Rev. B 66, 014407 (2002) .\n[62] A. Ghosh, S. Auffret, U. Ebels, and W. E. Bailey, Penetration\nDepth of Transverse Spin Current in Ultrathin Ferromagnets,Phys. Rev. Lett. 109, 127202 (2012) .\n[63] N. Reynolds, P. Jadaun, J. T. Heron, C. L. Jermain, J. Gibbons,\nR. Collette, R. A. Buhrman, D. G. Schlom, and D. C. Ralph,Spin Hall torques generated by rare-earth thin films, Phys. Rev.\nB95, 064412 (2017) .\n[64] J. Shi, P. Zhang, D. Xiao, and Q. Niu, Proper Definition of Spin\nCurrent in Spin-Orbit Coupled Systems, P h y s .R e v .L e t t . 96,\n076604 (2006) .\n[65] L. Liu, T. Moriyama, D. C. Ralph, and R. A. Buhrman, Spin-\nTorque Ferromagnetic Resonance Induced by the Spin HallEffect, Phys. Rev. Lett. 106, 036601 (2011) .\n[66] S. Han, H.-W. Lee, and K.-W. Kim, Orbital Dynamics in Cen-\ntrosymmetric Systems, P h y s .R e v .L e t t . 128, 176601 (2022) .\n033037-14"    },    {        "title": "1211.2055v1.Metal_insulator_transition_in_three_band_Hubbard_model_with_strong_spin_orbit_interaction.pdf",        "content": "arXiv:1211.2055v1  [cond-mat.str-el]  9 Nov 2012Metal-insulator transition in three-band Hubbard model wi th strong spin-orbit\ninteraction\nLiang Du and Xi Dai\nBeijing National Laboratory for Condensed Matter Physics,\nand Institute of Physics, Chinese Academy of Sciences, Beij ing 100190, China\nLi Huang\nBeijing National Laboratory for Condensed Matter Physics, and Institute of Physics,\nChinese Academy of Sciences, Beijing 100190, China and\nScience and Technology on Surface Physics and Chemistry Lab oratory,\nP.O. Box 718-35, Mianyang 621907, Sichuan, China\n(Dated: June 19, 2018)\nRecent investigations suggest that both spin-orbit coupli ng and electron correlation play very\ncrucial roles in the 5 dtransition metal oxides. By using the generalized Gutzwill er variational\nmethodanddynamicalmean-fieldtheorywith thehybridizati onexpansioncontinuoustime quantum\nMonte Carlo as impurity solver, the three-band Hubbard mode l with full Hund’s rule coupling and\nspin-orbit interaction terms, which contains the essentia l physics of partially filled t2gsub-shell of\n5dmaterials, is studied systematically. The calculated phas e diagram of this model exhibits three\ndistinct phase regions, including metal, band insulator an d Mott insulator respectively. We find that\nthe spin-orbit coupling term intends to greatly enhance the tendency of the Mott insulator phase.\nFurthermore, the influence of the electron-electron intera ction on the effective strength of spin-orbit\ncoupling in the metallic phase is studied in detail. We concl ude that the electron correlation effect\non the effective spin-orbit coupling is far beyond the mean-fi eld treatment even in the intermediate\ncoupling region.\nI. INTRODUCTION\nThe Mott metal-insulator transition (MIT) induced by\nelectron-electron correlation has attracted intensive re-\nsearchactivities in the past several decades1–4. Although\nthe main features of Mott transition have already been\ncaptured by single-band Hubbard model, most of Mott\ntransition in realistic materials have multi-orbital nature\nand should be described by multi-band Hubbard model.\nUnlike the situation in single-band case, where the Mott\ntransition is completely driven by the local Coulomb in-\nteraction U, the Mott transition in multi-band case is\naffected by not only Coulomb interaction but also crys-\ntal field splitting and Hund’s rule coupling among dif-\nferent orbitals5–7. The interplay between Hund’s rule\ncoupling and crystal field splitting generates lots of inter-\nesting phenomena in the multi-band Hubbard model, for\nexamples, orbital selective Mott transition, high-spin to\nlow-spin transition and orbital ordering. Therefore, most\nofthe intriguingphysicsin 3 dor4dtransitionmetalcom-\npounds can be well described bythe multi-band Hubbard\nmodel with both Hund’s rule coupling and crystal field\nsplitting.\nIn the present paper, we would like to concentrate our\nattention on the Mott physics in another group of inter-\nesting compounds, the 5 dtransition metal compounds,\nwhere spin-orbit coupling (SOC), the new physical in-\ngredient in Mott physics, plays an important role. Com-\npared to 3 dorbitals, the 5 dorbitals are much more ex-\ntended and the correlation effects are not expected to be\nimportant here. While as firstly indicated in reference8,\nthe correlation effects can be greatly enhanced by SOC,which is commonly strong in 5 dmaterials. The first well\nstudied 5 dMott insulator with strong SOC is Sr 2IrO4,\nwheretheSOCsplitsthe t2gbandsinto(upper) jeff= 1/2\ndoublet and (lower) jeff= 3/2 quartet bands and greatly\nsuppresses their bandwidths8–12. Since there are totally\nfive electrons in its 5 dorbitals, the jeff= 1/2 bands are\nhalf filled and the jeff= 3/2 bands are fully occupied,\nwhich makes the system being effectively a jeff= 1/2\nsingle-band Hubbard model with reduced bandwidth.\nTherefore the checkerboard anti-ferromagnetic ground\nstate of Sr 2IrO4can be well described by the single-band\nHubbard model with half filling.\nHere, we will focus on the 5 dmaterials with four elec-\ntrons in the t2gsub-shell. These materials include the\nnewly discovered BaOsO 3, CaOsO 3and NaIrO 3etc13.\nAll these materials share one important common feature:\nin low temperature, these materials are insulators with-\nout magnetic long-range order. The origin of the insu-\nlator behavior can be due to two possible reasons, the\nstrong enough Coulomb interaction and SOC. We will\nhave Mott insulator in the former and band insulator in\nthe latter case respectively. Therefore it is interesting\nto study the features of metal-insulator transition in a\ngenerict2gsystem occupied by four electrons with both\nCoulomb interaction and SOC.\nIn the present paper, we study the t2gHubbard model\nwithSOCandfourelectronsfillingbyusingrotationalin-\nvariantGutzwillerapproximation(RIGA)anddynamical\nmean-field theory combined with the hybridization ex-\npansion continuous time quantum Monte Carlo (DMFT\n+ CTQMC) respectively. The paramagnetic U−ζphase\ndiagram is derived carefully. Further, the interplay be-2\ntween SOC ζand Coulomb interaction Uis analyzed in\ndetail. We will mainly focus on the following two key is-\nsues: (i) How does the SOC affect the boundary of Mott\ntransitions in this three-band model? (ii) How does the\nCoulomb interaction modify the effective SOC strength?\nThispaperisorganizedasfollows. InSec. II,thethree-\nband model is specified, and the generalized multi-band\nGutzwiller variational wave function is introduced. In\nSec. IIIA, the calculated results, including U−ζphase\ndiagram, quasi-particle weight and charge distribution,\nfor the three-band model are presented. The effect of\nCoulomb interaction on SOC is analyzed in Sec. IIIB.\nFinally we make conclusions in section IV.\nII. MODEL AND METHOD\nThe three-band Hubbard model with full Hund’s rule\ncoupling and SOC terms is defined by the Hamiltonian:\nH=−/summationdisplay\nij,aσtijd†\ni,aσdj,aσ+/summationdisplay\niHi\nloc=Hkin+Hloc,(1)\nwhereσdenotes electronic spin, and arepresents the\nthreet2gorbitals with a= 1,2,3 corresponding to\ndyz,dzx,dxyorbitals respectively. The first term de-\nscribes the hopping process of electrons between spin-\norbital state “ aσ” on different lattice sites iandj. Local\nHamiltonian terms Hi\nloc=Hi\nu+Hi\nsoccontain Coulomb\ninteraction Hi\nuand SOC Hi\nsoc(In the following, the site\nindex is suppressed for sake of simplicity).\nHu=U/summationdisplay\nana↑na↓+U′/summationdisplay\na<b,σσ′naσnbσ′−Jz/summationdisplay\na<b,σnaσnbσ\n−Jxy/summationdisplay\na<b/parenleftig\nd†\na↑da↓d†\nb↓db↑+d†\na↑d†\na↓db↑db↓+h.c./parenrightig\n,(2)\nHsoc=/summationdisplay\naσ/summationdisplay\nbσ′ζ/angbracketleftaσ|lxsx+lysy+lzsz|bσ′/angbracketrightd†\naσdbσ′,(3)\nwhereU(U′) denotes the intra-orbital (inter-orbital)\nCoulomb interaction, Jzterm describes the longitudinal\npart of the Hund’s coupling. While the other two Jxy\nterms describe the spin-flip and pair-hopping process re-\nspectively. ζis SOC strength, l(s) is orbital (spin) an-\ngular momentum operator. Here we assume the studied\nsystem experiences approximately cubic symmetry ( Oh\nsymmetry), in which two parameters U′andJxyfollow\nthe constraints U′=U−2JandJxy=Jz=J. Here we\nchooseJ/U= 0.25 for the systems studied in this paper\nunless otherwise noted. This lattice model is solved in\nthe framework of RIGA14–17and DMFT(CTQMC)18,19\nmethods respectively, which are both exact in the limit\nof infinite spacial dimensions20,21. In this work, a semi-\nelliptic bare density of states ρ(ǫ) = (2/πD)/radicalbig\n1−(ǫ/D)2\nis adopted, which corresponds to Bethe lattice with infi-\nnite connectivity. In the present study, the energy unitis set to be half bandwidth D= 1 and all orbitals are\nassumed to have equal bandwidth.\nNext, we will briefly introduce the recently developed\nRIGA method16. The generalized Gutzwiller trial wave\nfunction |ΨG/angbracketrightcan be constructed by acting a many-\nparticle projection operator Pon the uncorrelated wave\nfunction |Ψ0/angbracketright22–24,\n|ΨG/angbracketright=P|Ψ0/angbracketright, (4)\nwith\nP=/productdisplay\niPi=/productdisplay\ni/summationdisplay\nΓΓ′λΓΓ′|Γ/angbracketrightii/angbracketleftΓ′|. (5)\n|Ψ0/angbracketrightis normalized uncorrelated wave function in which\nWick’s theorem holds. |Γ/angbracketrightiare atomic eigenstates on\nsiteiandλΓΓ′are Gutzwiller variational parameters. In\nour work, |Γ/angbracketrightiare eigenstates of atomic Hamiltonian Hu,\neach|Γ/angbracketrightiis labeled by good quantum number N,J,Jz,\nwhereNis total number of electrons, Jis total angular\nmomentum, Jzis projection of total angular momen-\ntum along zdirection. The non-diagonal elements of the\npreviously defined variational parameter matrix λΓΓ′are\nassumed to be finite only for state |Γ/angbracketright,|Γ′/angbracketrightbelonging to\nthe sameatomic multiplet, i.e, with the same three quan-\ntum labels15. In the following, we assume the local Fock\nterms are absent in |Ψ0/angbracketright,\n/angbracketleftΨ0|c†\niαciβ|Ψ0/angbracketright=δαβ/angbracketleftΨ0|c†\niαciα|Ψ0/angbracketright=δαβn0\niα.(6)\nFor general case, a local unitary transformation matrix\nAis needed to transform the original diα-basis into the\nso-called natural cim-basis16, i.e,dα=/summationtext\nmAαmcm. In\nthe original single particle basis ( dyz↑,dyz↓,dzx↑,dzx↓,\ndxy↑,dxy↓), SOC term is expressed as :\nHsoc=−ζ\n2\n0 0−i0 0 1\n0 0 0 i−1 0\ni0 0 0 0 −i\n0−i0 0−i0\n0−1 0i0 0\n1 0i0 0 0\n.(7)\nThen the transformation matrix Ais as follows:\nA=1√\n6\n−√\n3 0 1 0 0 −√\n2\n0−1 0√\n3−√\n2 0\n−i√\n3 0−i0 0 i√\n2\n0−i0−i√\n3−i√\n2 0\n0 2 0 0 −√\n2 0\n0 0 2 0 0√\n2\n,(8)\nwhere the t2gorbitals have been treated as a system with\nleff= 1.\nIn the natural single particle basis, SOC matrix is\ntransformed into:\nHsoc=\n−ζ/2 0 0 0 0 0\n0−ζ/2 0 0 0 0\n0 0 −ζ/2 0 0 0\n0 0 0 −ζ/2 0 0\n0 0 0 0 ζ0\n0 0 0 0 0 ζ\n.(9)3\nMeanwhile the Coulomb interaction term is transformed\nas:\n/summationdisplay\nαβδγUαβδγd†\nαd†\nβdδdγ=/summationdisplay\nmnkl˜Umnklc†\nmc†\nnckcl,(10)\nwith\n˜Umnkl=/summationdisplay\nαβδγUαβδγA†\nmαA†\nnβAδkAγl.(11)\nFixn0in each orbital\ninitial guess of renormalization factor R\nconstruct Gutzwiller effective one-particle Hamiltonian,\nˆHeff\n0=/summationtext\ni/negationslash=j/summationtext\nαβ/summationtext\nγδtαβ\nijR†\nαγˆc†\niγˆcjδRδβ+/summationtext\niαηαˆc†\niαˆciα\nand solve the Hamiltonian to get ∂E/∂R.\n∂E\n∂λΓΓ′=/summationdisplay\nδβ/parenleftigg\n∂Ekin\n∂Rδβ∂Rδβ\n∂λΓΓ′+∂Ekin\n∂R†\nβδ∂R†\nβδ\n∂λΓΓ′/parenrightigg\n+∂Eloc\n∂λΓΓ′+/summationdisplay\nαηα∂n′\nα\n∂λΓΓ′= 0\nsolve the equations to get renormalization factor R\ncheck ifRis\nself-consistent.\ncalculate quantities:\ntotal energy, occupation ...YesNo\nFIG. 1. Flowchart of the RIGA self-consistent loop to mini-\nmize total energy E(n0) with respect to |Ψ0/angbracketrightandλΓΓ′.\nIn this paper, we define expectation value with uncor-\nrelated wave function:\nO0=/angbracketleftΨ0|ˆO|Ψ0/angbracketright, (12)\nwhile expectation value with Gutzwiller wave function is\ndefined as:\nO=OG=/angbracketleftΨG|ˆO|ΨG/angbracketright. (13)\nDuring the minimization process, two following con-\nstraints are forced,\n/angbracketleftΨ0|P†P|Ψ0/angbracketright= 1, (14)\nand\n/angbracketleftΨ0|P†Pniα|Ψ0/angbracketright=/angbracketleftΨ0|niα|Ψ0/angbracketright. (15)\nIn the present paper, the second constraint is satisfied\nin the following way. We first calculate the total energy\nof the trial wave function with both the left-hand andright-hand side of the above equation equaling to some\ndesired occupation number n0\nα. Then we minimize the\nenergy respect to n0\nαat the last step.\nThe remaining task is to minimize the variational\nground energy E=Ekin+Elocwith respect to λΓΓ′\nand|Ψ0/angbracketright, and fulfill the previous two constraints. Here,\nEkin=/angbracketleftΨG|Hkin|ΨG/angbracketright=/summationdisplay\nij/summationdisplay\nγδ˜tγδ\nij/angbracketleftΨ0|c†\niγcjδ|Ψ0/angbracketright,\n(16)\nand\nEloc=/angbracketleftΨG|Hloc|ΨG/angbracketright= Tr(φ†Hlocφ),(17)\nwith˜t,Randφdefined as:\n˜tγδ\nij=/summationdisplay\nαβtαβ\nijR†\nαγRδβ (18)\nR†\nαγ=Tr/parenleftbig\nφ†c†\nαφcγ/parenrightbig\n/radicalig\nn0γ(1−n0γ), (19)\nφII′=/angbracketleftI|P|I′/angbracketright/radicalbig\n/angbracketleftΨ0|I′/angbracketright/angbracketleftI′|Ψ0/angbracketright, (20)\nwhere|I/angbracketright(|I′/angbracketright) stands for a many-body Fock state and\nn0\nγ=/angbracketleftΨ0|nγ|Ψ0/angbracketright.\nThe flowchart of RIGA method is shown in Fig.1. For\nfixedn0\nαin each orbital, minimizing Ewith respect to\n|Ψ0/angbracketrightandλΓΓ′can be divided into two steps in each iter-\native process. Firstly, fix Gutzwiller variational parame-\ntersλΓΓ′and find optimal uncorrelated wave function by\nsolving effective single particle Hamiltonian,\nHeff\n0=/summationdisplay\ni/negationslash=j/summationdisplay\nγδ˜tγδ\nijc†\niγcjδ+/summationdisplay\niαηαc†\niαciα,(21)\nwhere Lagrange parameters ηαare used to minimize the\nvariational energy fulfilling Gutzwiller constraints. Sec-\nondly, we fix the uncorrelated wave function, and opti-\nmize the variational energy with respect to Gutzwiller\nvariational parameters λΓΓ′,\n∂E\n∂λΓΓ′=/summationdisplay\nδβ/parenleftigg\n∂Ekin\n∂Rδβ∂Rδβ\n∂λΓΓ′+∂Ekin\n∂R†\nβδ∂R†\nβδ\n∂λΓΓ′/parenrightigg\n+∂Eloc\n∂λΓΓ′+/summationdisplay\nαηα∂n′\nα\n∂λΓΓ′= 0, (22)\nwheren′\nα=/angbracketleftΨ0|P†Pnα|Ψ0/angbracketright. In this way, λΓΓ′and|Ψ0/angbracketright\nare self-consistently determined.\nFor the fix n0\nαalgorithm, we need to scan the n0\nαto\nget the global ground state of the studied system. In\nthis paper, because SOC will split the t2gorbitals into\ntwo fold jeff= 1/2 and four fold jeff= 3/2 states, we can\nintroduce an alternative variable δn0to determine n0\nαfor4\neach orbital. The occupation polarization δn0is defined\nas:\nδn0=n0\n3/2−n0\n1/2, (23)\nin which n0\n3/2andn0\n1/2stand for the average occupa-\ntion number of lower ( jeff= 3/2) and upper ( jeff= 1/2)\norbitals respectively. Since total electron number of\nthe system is fixed to be 4 n0\n3/2+ 2n0\n1/2= 4, we have\n0≤δn0≤1.δn0(n0) corresponding to ground state is\ndenoted by δn0\ng(n0\ng).\nIn the present paper, we also use DMFT+CTQMC\nmethod to crosscheck our results derived by RIGA. For\nDMFT+CTQMC method, the system temperature is set\nto beT= 0.025 (corresponding to inverse temperature\nβ= 40). In each DMFT iteration, typically 4 ×108\nQMC samplings have been performed to reach sufficient\nnumerical accuracy25.\nIII. RESULTS AND DISCUSSION\nA.U-ζphase diagram\nIn this subsection, we mainly focus on phase diagram\nfor the three-band model proposed in Eq.(1). The ob-\ntainedU−ζphase diagrams with J/U= 0.25 are shown\nin Fig.2. The upper panel shows the phase diagram cal-\nculated by zero temperature RIGA method, while the\ncalculated results by DMFT+CTQMC method at finite\ntemperature is shown in the lower panel. The results ob-\ntained by two different methods are consistent with each\nother quite well except that DMFT+CTQMC can not\ndistinguish between band insulator and Mott insulator,\nwhich will be explained later. Apparently, there exists\nthree different phasesin U−ζplane: metallic state in the\nlower left corner, band insulator in the lower right region\nand Mott insulator in the upper right region. The gen-\neral shape of the phase diagram can be easily understood\nby considering two limiting cases: (i) For ζ= 0, one has\na degenerate three-band Hubbard model populated by 4\nelectrons per site. The model will undergo an interaction\ndriven Mott transition at critical Uc/D∼11.0 with each\nband filled by 4 /3 electrons. (ii) For non-interacting case\n(U= 0.0), the model is exactly soluble. The three bands\nare degenerate and filled by 4 /3 electrons at ζ=0.0. Fi-\nniteζwill split the three degenerate bands into a (lower)\njeff= 3/2quadrupletand(upper) jeff= 1/2doubletwith\nenergy separation being 1 .5ζ. Increasing ζwill pump\nelectrons from upper to lower orbitals until the upper\nbands are completely empty and the system undergoes a\nmetal to band insulator transition, which is expected at\nζ/D= 1.33.\nIn order to clarify the way we determine the metal,\nband insulator, and Mott insulator phases by RIGA\nmethod, in Fig.3 we plot the total energy and quasipar-\nticle weight as a function of δn0defined in the previous\nsection , where the SOC strength is fixed at ζ/D= 0.7,\nU/D\nζ/D 0 2 4 6 8 10 12\n 0  0.2  0.4  0.6  0.8  1  1.2  1.4MetalBand InsulatorMott InsulatorU/D\nζ/D 0 2 4 6 8 10 12\n 0  0.2  0.4  0.6  0.8  1  1.2  1.4MetalInsulator\nFIG. 2. (Color online) Phase diagram of three-band Hub-\nbard model with full Hund’s coupling terms in the plane of\nCoulomb interaction U(J/U= 0.25) and spin-orbit coupling\nζ. Upper panel: The phase diagram is calculated by RIGA at\nzero temperature. Lower panel: The phase diagram is calcu-\nlated by DMFT+CTQMC with finite temperature T= 0.025.\nand from top to bottom the Coulomb interaction is\nU/D= 1.0,3.0,6.0. The ground state of the system\nis the state with the lowest energy respect to δn0. The\ntypical solution for the metal phase is shown in Fig.3a,\nwhere the energy minimum occurs at 0 < δn0\ng<1.0 cor-\nresponding to the case that all orbitals being partially\noccupied. While for a band insulator, as shown in Fig.3b\nas a typical situation, the energy minimum happens at\nδn0\ng= 1.0 corresponding to the case that the jeff= 3/2\norbitals are fully occupied and jeff= 1/2 orbitals are\nempty, and more over the quasiparticle weight Zkeeps\nfinite when δn0approching unit. And finally the situ-\nation of a Mott insulator is illustrated in Fig.3c, where\nthe quasiparticle weight Zvanishes at some critical δn0,\nabove which the system is in Mott insulator phase and\ncan no longer be described by the Gutzwiller variational\nmethod.\nWhile in the DMFT+CTQMC calculations, the phase\nboundary between metal and insulator is identified by\nmeasuring the imaginary-time Green function at τ=\nβ/226,27. SinceG(β/2) can be viewed as a representa-5\n1.601.802.00\n(a) 0.91.0\n(d)\n7.407.607.80E\n(b) 0.50.60.7\nZ\n(e)\n16.116.216.3\n0.0 0.2 0.4 0.6 0.8\nδn0(c)\n0.0 0.2 0.4 0.6 0.8 1.00.00.10.20.3\nδn0(f)U=1 U=1,j=3/2\nU=1,j=1/2\nU=3 U=3,j=3/2\nU=3,j=1/2\nU=6 U=6,j=3/2\nU=6,j=1/2\nFIG. 3. (Color online) Total energy E(δn0) and quasipar-\nticle weight Z(δn0) as a function of occupation polarization\nδn0=n0\n3/2−n0\n1/2for different values of interaction strength\nU/D= 1,3,6 (J/U= 0.25) at fixed ζ/D= 0.7 and zero\ntemperature, where n0\n3/2is average occupation number of\njeff= 3/2 quadruplet, n0\n1/2is average occupation number of\njeff= 1/2 doublet.\ntion of the integrated spectral weight within a few kBT\nofEF, so it can be used as an important criterion to\njudge whether metal-insulator transition occurs. The\ncorresponding results for SOC strength ζ/D= 0.5 are\nshown in Fig.4. Clearly seen in this figure, the critical Uc\nis about 3.5, and both the jeff= 3/2 andjeff= 1/2\norbitals undergo metal-insulator transitions simultane-\nously. Since there is chemical potential ambiguity in\nthe insulator phase, it is difficult to further distinguish\nMott insulator from band insulator by DMFT+CTQMC\nmethod. Therefore we only calculatethe phase boundary\nbetween the metal and insulator phase, which is in good\nagreement with the results obtained by RIGA.\nNow, we come back to discussed the phase diagram\nobtained by RIGA and DMFT. When both the Coulomb\ninteraction Uand SOC ζare finite, the phase diagram\nlooks a bit complicated. By considering different values\nofSOC, we divide the phase diagramverticallyinto three\nregions.\nFirstly, for 0 .00< ζ/D < 0.24, with increasing\nCoulomb interaction U, our calculation by RIGA pre-\ndicts a transition from metal to Mott insulator. The\ntransition is characterizedby the vanishing of quasiparti-\ncle weight as discussed previously. The critical Coulomb\ninteraction Ucdecreasesdrasticallywith the increment of\nSOC, the effect of SOC tends to enhance the Mott MIT\ngreatly. The DMFT results show very similar behavior\nin this region as shown in lower panel of Fig.2, except\nthat the Ucobtained by DMFT has weaker dependence\non the strength of SOC compared to that of RIGA. From\nthe view point of DMFT, the suppression of the metallic\nphase by SOC can be explained quite clearly. For the\neffective impurity model in DMFT, the metallic phase\ncorresponds to a solution when the local moments on the0.20.40.60.81.0\n 1  2  3  4  5  6  7  8  9  10|G(β/2)|\nU/Dζ=0.5j=3/2\nj=1/2\nFIG. 4. (Color online) The imaginary-time Green function\natτ=β/2 as a function of Coulomb interaction strength U.\nThe SOC strength ζis chosen to be 0.5 as a illustration. The\ncalculation is done by DMFT+CTQMC method at β= 40.\nIn this figure the normalized quantities by G(β/2) atU/D=\n1.0 are shown and the arrows correspond to phase transition\npoints.\nimpurity site are fully screened by the electrons in heat-\nbath through the Kondo like effect. With the SOC, there\nis an additional channel to screen the local spin moment\notherthanthe Kondoeffect, whichleadstothe formation\nof spin-orbital singlets. This additional screening chan-\nnel, which is completely local, will thus compete with the\nKondo effect and suppress the metallic solution. There\nis no net local moment left in this type of Mott phase,\nand the ground state is simply a product state of local\nspin-orbital singlets on each site.\nFor 0.24< ζ/D < 1.33, two successive phase transi-\ntions are observed with the increment of U. The transi-\ntion from metallic to band insulator phases occurs firstly,\nand followed by another transition to the Mott insula-\ntor phase. In the intermediate Uregion, the effective\nband width of system is reduced by the correlation ef-\nfects, whichdrivesthe systemintoabandinsulatorphase\nwith relative small band splitting induced by SOC. Fur-\nther increasing interaction strength Uwill push the sys-\ntem to the Mott limit. Although this process is believed\nto be a crossover rather than a sharp phase transition,\nour RIGA calculation provides a mean field description\nfor these two different insulators, where in the band in-\nsulator phase the interaction effects only renormalize the\neffective band structure and do not suppress the coher-\nent motion of the electrons entirely. Similar behavior can\nalso be obtained by DMFT method, where the quasipar-\nticle weight determined by DMFT selfenergy keeps finite\nfor the band insulator phase and vanishes for the Mott\ninsulator phase.\nAt last, for ζ/D >1.33 region, the orbitals are fully\npolarized with electrons fully occupied jeff= 3/2 bands\nand fully empty jeff= 1/2 bands at U= 0.0, indicating\nthat the system is in the band insulator state already6\nin the non-interacting case. Similar band insulator to\nMott insulator transition will be induced with further\nincrement of Uin the RIGA description, as discussed\nbefore.\n0.00.20.40.60.8Zζ=0.1, j=3/2\nζ=0.5, j=3/2\nζ=1.5, j=3/2\n0.00.20.40.60.8\n0 1 2 3 4 5 6 7Z\nU/Dζ=0.1, j=1/2\nζ=0.5, j=1/2\nζ=1.5, j=1/2\nFIG. 5. (Color online) Quasiparticle renormalization fact ors\nZof the lower orbitals ( jeff= 3/2 quadruplet) and upper\norbitals ( jeff= 1/2 doublet) as function of Coulomb inter-\nactionU(J/U= 0.25) for different values of SOC ( ζ/D=\n0.1,0.5,1.5). The dashed lines label the critical Ufor transi-\ntion to Mott state. The results are obtained by zero temper-\nature RIGA method.\n0.650.700.750.800.850.900.951.00\n0 1 2 3 4n(jeff= 3/2)\nU/Dζ=0.1, RIGA\nζ=0.1, DMFT\nζ=0.5, RIGA\nζ=0.5, DMFT\nζ=1.5, RIGA\nζ=1.5, DMFT\nFIG. 6. (Color online) Occupation number of the lower\norbitals ( jeff= 3/2 quadruplet) with increasing Coulomb\nU(J/U= 0.25) for selected SOC ( ζ/D= 0.1,0.5,1.5). Both\nthecalculated results byRIGAandDMFT(CTQMC) are pre-\nsented.\nFor several typical SOC parameters ( ζ/D=\n0.1,0.5,1.5) in the three regions defined above, we study\nthe evolutions of quasiparticle weight and band specific\noccupancy with Coulomb interaction. The quasiparti-\ncle weight for selected SOC with increasing Uis plotted\nin Fig.5. The upper (lower) panel shows the quasiparti-\ncle weight for jeff= 3/2 (1/2) orbitals. Note the quasi-\nparticle weight in RIGA is defined as the eigenvalues of\nthe Hermite matrix R†R. For both ζ/D= 0.1 and 1.5,1.71.92.1/angbracketleftL2/angbracketright(a)\n1.01.41.8/angbracketleftS2/angbracketright(b)\n0.00.51.01.5\n0 0 .5 1 1 .5 2 2 .5 3 3 .5 4/angbracketleftJ2/angbracketright\nU/D(c)DMFT\nRIGA\nDMFT\nRIGA\nDMFT\nRIGA\nFIG. 7. (Color online) Expectation value of orbital angular\nmomentum /angbracketleftL2/angbracketright, spin angular momentum /angbracketleftS2/angbracketright, and total an-\ngular momentum /angbracketleftJ2/angbracketrightas function of Coulomb interaction U\nwith fixed spin-orbit coupling strength ζ/D= 0.7. It is de-\nrived by RIGA at zero temperature and DMFT+CTQMC at\nβ= 40 respectively.\nthe quasiparticle weights decrease from 1 to 0 monoton-\nically with the increasing interaction strength UandJ\nuntil the transition to Mott insulator phase. While for\nζ/D= 0.5, there exists a kink at U/D= 2.7 in the lower\npanel (jeff= 1/2), which corresponds to the transition\nfrom metal to band insulating state. For transition to\nMott insulating state, quasiparticle weights for all the\norbitals reach zero simultaneously, with Uc/D= 6.7 for\nζ/D= 0.1,Uc/D= 4.5 for ζ/D= 0.5, and Uc/D= 4.0\nforζ/D= 1.5.\nThe occupation number of the (lower) jeff= 3/2 or-\nbitals as a function of on-site Coulomb interaction U\nis ploted in Fig.6 for three typical SOC strength. For\nζ/D= 0.1, to some extent, the occupation behavior is\nsimilar to ζ= 0 case, in which the occupation number\nis only slightly changed by the interaction. The situa-\ntion is quite different for ζ/D= 0.5, where the occupa-\ntion of the jeff= 3/2 orbital increases with interaction\nat the beginning and decreases slightly after the tran-\nsition to the band insulator phase. The non-monotonic\nbehavior here is mainly due to the competition between\nthe repulsive interaction Uand Hund’s rule coupling J.\nThe effect of Uwill always enhance the splitting of the\nlocal orbitals to reduce the repulsive interaction among\nthese orbitals. While the Hund’s rule coupling intents\nto distribute the electrons more evenly among different\norbitals. For ζ/D= 1.5 case, occupation number in the\ntwo subsets is fully polarized at U= 0 and the effect of\nHund’s coupling term will reduce the occupation of the\njeff= 3/2 orbital monotonically.\nAt last, the expectation value of the total orbital angu-\nlar momentum /angbracketleftL2/angbracketright, spin angular momentum /angbracketleftS2/angbracketrightand\ntotal angular momentum /angbracketleftJ2/angbracketrightas a function of Coulomb\ninteraction Uare plotted in Fig.7, where ζ/Dis fixed\nto 0.70. In the non-interacting case, all the three ex-7\npectation values can be calculated exactly and they will\napproach the atomic limit with the increment of inter-\nactionUandJ. In the atomic limit the SOC strength\nis much weaker than the Hund’s coupling J, the ground\nstate can be well described by the LScoupling scheme,\nwhere the four electrons will first form a state with total\norbital angular momentum L= 1 and total spin momen-\ntumS= 1,andthenformaspin-orbitalsingletstatewith\ntotal angularmomentum J= 0. From Fig.7, we can find\nthat the system approaches the spin-orbital singlet quite\nrapidly after the transition to the band insulator phase.\n0.81.01.21.41.61.82.02.22.42.62.8\n0 0.5 1 1.5 2 2.5ζeff/ζ\nU/Dζ=0.1, RIGA\nζ=0.1, HFA  \nζ=0.5, RIGA\nζ=0.5, HFA  \nFIG. 8. (Color online) Evolution of effective SOC strength\nwith increasing Coulomb U(J/U= 0.25) for selected SOC\n(ζ/D= 0.1,0.5), A comparison of results derived by RIGA\nand HFA are presented.\nB. Effective spin-orbit coupling\nIn the multi-orbital system, the interaction effects will\nmainly cause two consequences for the metallic phases:\n(1) It will introduce renormalizationfactor for the energy\nbands; (2) It will modify the local energy level for each\norbital which splits the bands. For the present model,\nthe second effect will modify the effective SOC, which\nis another very important problem for the spin orbital\ncoupled correlation system. Within the Gutzwiller vari-\national scheme used in the present paper, the effective\nSOC can be defined as:\nζeff=−1\n2∂Eint(δn0)\n∂δn0−1\n2∂Esoc(δn0)\n∂δn0,(24)\nwhereEintandEsocare the ground state expectation\nvalues of interaction and SOC terms in the Hamilto-\nnian respectively. Note the second term is different from\nthe bare SOC ζ0unlessnαis a good quantum num-\nber. If the interaction energy is treated by Hartree Fock\nmean field approximation (HFA), the above equation\ngivesζeff=−∂EHF\nint(δn0)/(2∂δn0) +ζ0, which will al-\nways greatly enhance the spin-orbital splitting with the\nincreasing Uas found by some works based on LDA+Umethod28,29. In this section, we compare the results ob-\ntained by RIGA and HFA. As shown in Fig.8, the effec-\ntive SOC obtained by HFA increases quite rapidly with\nthe interaction UandJ. While the results obtained\nby RIGA show very different behavior. For weak SOC\nstrength, i.e. ζ/D= 0.1, the effective SOC obtained\nby RIGA increases first then decrease. This interesting\nnon-monotonicbehaviorreflectsthecompetitionbetween\nthe repulsive interaction U, which intends to increase the\noccupation difference for jeff= 1/2 andjeff= 3/2 or-\nbitals, and the Hund’s rule coupling J, which intents to\ndecrease the occupation difference. While for relatively\nstrong SOC strength, the effective SOC increases with\ninteraction U(andJ) monotonically all the way to the\nphase boundary indicating the repulsive interaction U\nplays a dominate role here. But compared to HFA, the\nenhancement of effective SOC induced by the interaction\nis much weaker even for the latter case. This is mainly\ndue to the reduction of the high energy local configura-\ntions in the Gutzwiller variational wave function com-\npared to the Hatree Fock wave function, which greatly\nreduces the interaction energy and its derivative to the\norbital occupation.\nIV. CONCLUDING REMARKS\nThe Mott MIT in three-band Hubbard model with full\nHund’s rule coupling and SOC is studied in detail using\nRIGA and DMFT+CTQMC methods. First, we propose\nthe phase diagram with the strength of electron-electron\ninteraction and SOC. Three different phases have been\nfound in the U−ζplane, which are metal, band insula-\ntor and Mott insulator phases. For 0 .00< ζ/D < 0.24,\nincreasing Coulomb interaction will induce a MIT tran-\nsition from metal to Mott insulator. For 0 .24< ζ/D <\n1.33,effect of Uwill causetwosuccessivetransitions, first\nfrommetaltobandinsulator, thentoMottinsulator. For\nζ/D >1.33, a transition from band insulator to Mott in-\nsulatorisobserved. Fromthephasediagram,wefindthat\nthe critical interaction strength Ucis strongly reduced by\nthe presence of SOC, which leads to the conclusion that\nthe SOC will greatly enhance the strong correlation ef-\nfects in these systems. Secondly, we have studied the ef-\nfect of electron-electron interaction on the effective SOC.\nOur conclusion is that the enhancement of effective SOC\nfound in HFA is strongly suppressed once we go beyond\nthe mean field approximationand include the fluctuation\neffects by RIGA or DMFT methods.\nACKNOWLEDGMENT\nWe acknowledge valuable discussions with profes-\nsor Y.B. Kim and professor K. Yamaura, and finan-\ncial support from the National Science Foundation of\nChina and that from the 973 program under Contract\nNo.2007CB925000 and No.2011CBA00108. The DMFT8\n+ CTQMC calculations have been performed on the SHENTENG7000 at Supercomputing Center of Chinese\nAcademy of Sciences (SCCAS).\n1M. Imada, A. Fujimori, and Y. Tokura,\nRev. Mod. Phys. 70, 1039 (1998).\n2A. Georges, G. Kotliar, W. Krauth, and M. J. Rozenberg,\nRev. Mod. Phys. 68, 13 (1996).\n3G. Kotliar, S. Y. Savrasov, K. Haule, V. S.\nOudovenko, O. Parcollet, and C. A. Marianetti,\nRev. Mod. Phys. 78, 865 (2006).\n4L. de’ Medici, J. Mravlje, and A. Georges,\nPhys. Rev. Lett. 107, 256401 (2011).\n5L. de’ Medici, S. R. Hassan, M. Capone, and X. Dai,\nPhys. Rev. Lett. 102, 126401 (2009).\n6P. Werner and A. J. Millis,\nPhys. Rev. Lett. 99, 126405 (2007).\n7T. Kita, T. Ohashi, and N. Kawakami,\nPhys. Rev. B 84, 195130 (2011).\n8B. J. Kim, H. Jin, S. J. Moon, J. Y. Kim, B. G. Park,\nC. S. Leem, J. Yu, T. W. Noh, C. Kim, S. J. Oh,\nJ. H. Park, V. Durairaj, G. Cao, and E. Rotenberg,\nPhys. Rev. Lett. 101, 076402 (2008).\n9D. Pesin and L. Balents, Nature Physics 6, 376 (2010).\n10B. J. Kim, H. Ohsumi, T. Komesu, S. Sakai, T. Morita,\nH. Takagi, and T. Arima, Science 323, 1329 (2009).\n11H. Watanabe, T. Shirakawa, and S. Yunoki,\nPhys. Rev. Lett. 105, 216410 (2010).\n12G. Jackeli and G. Khaliullin,\nPhys. Rev. Lett. 102, 017205 (2009).\n13M. Bremholm, S. Dutton, P. Stephens, and R. Cava,\nJournal of Solid State Chemistry 184, 601 (2011).14J. B¨ unemann, W. Weber, and F. Gebhard,\nPhys. Rev. B 57, 6896 (1998).\n15J. B¨ unemann, F. Gebhard, T. Ohm, S. Weiser, and\nW. Weber, Phys. Rev. Lett. 101, 236404 (2008).\n16N. Lanat` a, H. U. R. Strand, X. Dai, and B. Hellsing,\nPhys. Rev. B 85, 035133 (2012).\n17X. Deng, L. Wang, X. Dai, and Z. Fang,\nPhys. Rev. B 79, 075114 (2009).\n18P. Werner and A. Millis, Phys. Rev. B 74, 1 (2006).\n19E. Gull, A. J. Millis, A. I. Lichtenstein,\nA. N. Rubtsov, M. Troyer, and P. Werner,\nRev. Mod. Phys. 83, 349 (2011).\n20W. Metzner and D. Vollhardt,\nPhys. Rev. Lett. 59, 121 (1987).\n21W. Metzner and D. Vollhardt,\nPhys. Rev. B 37, 7382 (1988).\n22M. C. Gutzwiller, Phys. Rev. Lett. 10, 159 (1963).\n23M. C. Gutzwiller, Phys. Rev. 134, A923 (1964).\n24M. C. Gutzwiller, Phys. Rev. 137, A1726 (1965).\n25L. Huang, L. Du, and X. Dai,\nPhys. Rev. B 86, 035150 (2012).\n26A. Liebsch, Phys. Rev. Lett. 91, 226401 (2003).\n27L. Huang, Y. Wang, and X. Dai,\nPhys. Rev. B 85, 245110 (2012).\n28G.-Q. Liu, V. N. Antonov, O. Jepsen, and O. K. Ander-\nsen., Phys. Rev. Lett. 101, 026408 (2008).\n29A. Subedi, Phys. Rev. B 85, 020408 (2012)."    },    {        "title": "1110.6798v1.Spin_Orbit_Coupled_Quantum_Gases.pdf",        "content": "November 1, 2011 0:15 WSPC/INSTRUCTION FILE Review_SOC\nSpin-Orbit Coupled Quantum Gases\nHui Zhai\nInstitute for Advanced Study, Tsinghua University, Beijing, 100084, China\nIn this review we will discuss the experimental and theoretical progresses in studying\nspin-orbit coupled degenerate atomic gases during the last two years. We shall \frst review\na series of pioneering experiments in generating synthetic gauge potentials and spin-orbit\ncoupling in atomic gases by engineering atom-light interaction. Realization of spin-orbit\ncoupled quantum gases opens a new avenue in cold atom physics, and also brings out a\nlot of new physical problems. In particular, the interplay between spin-orbit coupling and\ninter-atomic interaction leads to many intriguing phenomena. Here, by reviewing recent\ntheoretical studies of both interacting bosons and fermions with isotropic Rashba spin-\norbit coupling, the key message delivered here is that spin-orbit coupling can enhance\nthe interaction e\u000bects, and make the interaction e\u000bects much more dramatic even in the\nweakly interacting regime.\nKeywords : Cold Atoms, Synthetic Gauge Potential, Spin-Orbit Coupling, Super\ruidity,\nFeshbach Resonance\nMany interesting phenomena in condensed matter physics occur when electrons\nare placed in an electric or magnetic \feld, or possess strong spin-orbit (SO) cou-\npling. However, in the cold atom systems, neutral atoms neither possess gauge\ncoupling to electromagnetic \felds nor have SO coupling. Recently, by controlling\natom-light interaction, one can generate an arti\fcial external abelian or non-abelian\ngauge potential coupled to neutral atoms. The basic principle is based on the Berry\nphase e\u000bect1and its non-abelian generalization2. An important application of this\nscheme is creating an e\u000bective SO coupling in degenerate atomic gases. Since 2009,\nSpielman's group in NIST has successfully implemented this principle and gener-\nated both synthetic uniform gauge \feld6, magnetic \feld7, electric \feld8and SO\ncoupling9. We shall discuss the experimental progresses along this line in Sec. 1.\nThe e\u000bects of SO coupling in electronic systems have been extensively discussed\nin condensed matter physics before and are also important topics nowadays. One of\nthe most famous example is recently discovered topological insulators3;4;5. In this\nreview, we try to convey the message that SO coupling in degenerate atomic gases\nwill bring out new physics which have not been considered before, mainly due to\nthe interplay between SO coupling and the unique properties of atomic gases. For\nbosonic atoms, SO coupled interacting bosons is a system never explored in physics\nbefore. For fermionic atoms, since a lot of intriguing physics have been revealed\nduring the last ten years by utilizing Feshbach resonance technique to achieve in-\nteraction as strong as Fermi energy, the interplay between resonance physics and\nSO coupling is de\fnitely a subject of great interests.\n1arXiv:1110.6798v1  [cond-mat.quant-gas]  31 Oct 2011November 1, 2011 0:15 WSPC/INSTRUCTION FILE Review_SOC\n2Hui Zhai\nA key point we want to emphasize in this review is that a nearly isotropic\nSO coupling will dramatically enhance the e\u000bects of inter-particle interactions, so\nthat the interaction e\u000bects are not weak even in the regime where the interaction\nstrength itself is small . This is because an isotropic Rashba SO coupling or a nearly\nisotropic SO coupling signi\fcantly changes the low-energy states of single particle\nHamiltonian, as we shall discuss in Sec 2. In Sec. 3, we discuss many-body system\nof bosons. Because the single particle ground state has large degeneracy, it is the\ninter-particle interaction that selects out a unique many-body ground state and\ndetermines its low-energy \ructuations. In Sec. 4, we discuss many-body system of\nfermions. Because the low-energy density-of-state (DOS) is largely enhanced, the\ninteraction e\u000bects become much more profound, in particular for weak attractions.\nA brief summary and future perspective are given in Sec. 5.\n1. Synthetic Gauge Potentials and Spin-Orbit Coupling\nIn 2009, Spielman's group in NIST \frst realized a uniform vector potential in Bose-\nEinstein condensate (BEC) of87Rb6. In this experiment, two counter propagating\nRaman laser beams couple jF;m Fi=j1;\u00001ilevel of87Rb toj1;0ilevel, and couple\nj1;0ilevel toj1;1ilevel, as shown in Fig. 1(a) and (b), which can be described by\nthe Hamiltonian\nH=0\nB@k2\nx\n2m+\u000f1\n2ei2k0x0\n\n2e\u0000i2k0xk2\nx\n2m\n2ei2k0x\n0\n2e\u0000i2k0xk2\nx\n2m\u0000\u000f21\nCA (1)\nwherek0= 2\u0019=\u0015,\u0015is the wave length of two lasers. 2 k0is therefore the momentum\ntransfer during the two-photon processes. \u000f1= \u0001 1+\u000e!+\u00012, and\u000f2= \u0001 1+\u000e!\u0000\u00012,\nwhere \u0001 1denotes the linear Zeeman energy, \u000e!denotes the frequency di\u000berence of\ntwo Raman lasers, and \u0001 2is the quadratic Zeeman energy.\nApplying a unitary transformation to wave function \b = U\t, where\nU=0\n@e\u0000i2k0x0 0\n0 1 0\n0 0ei2k0x1\nA (2)\nand the e\u000bective Hamiltonian becomes\nHe\u000b=UHUy=0\nB@(kx+2k0)2\n2m+\u000f1\n20\n\n2k2\nx\n2m\n2\n0\n2(kx\u00002k0)2\n2m\u0000\u000f21\nCA: (3)\nWhen both \u000f1and\u000f2are large, the single particle energy dispersion of He\u000bis shown\nas Fig. 1(c), which displays a single energy minimum at \fnite kx. In this regime the\nlow energy physics is dominated by a single dressed state described by1\n2m(kx\u0000Ax)2,\nwhereAxis a constant. This leads to a uniform vector gauge \feld.\nIn a follow up experiment, Spielman's group applied a Zeeman \feld gradient\nalong ^ydirection to this system7. In this case, Axbecomes a function of yinsteadNovember 1, 2011 0:15 WSPC/INSTRUCTION FILE Review_SOC\nSpin-Orbit Coupled Quantum Gases 3\n/Minus3/Minus2/Minus1123k/Slash1k00.51.01.5E-101(b)BEC\n(a)(c)(d)/Minus3/Minus2/Minus1123k/Slash1k0/Minus224E\nFig. 1. (a) A schematic of NIST experiment, in which two counter propagating Raman beams are\napplied to87Rb BEC. (b) A schematic of how three F= 1 levels are coupled by Raman beams. (c)\nDispersion in the regime of uniform vector potential. (d) Dispersion in the regime of non-abelian\ngauge \feld.\nof a constant. It gives rise to a non-zero synthetic magnetic \feld Bz=\u0000@yAx6= 0.\nThey observed a critical magnetic \feld above which many vortices are generated in\nthe BEC7. In another experiment8, they made Axtime dependent which gives rise\nto a non-zero electric \fled Ex=\u0000@tAx6= 0. They observed collective oscillation of\nBEC after a pulse of electric \feld8.\nBy tuning the Zeeman energy and the laser frequency, one can also reach the\nregime where \u0001 1+\u000e!\u0019\u00012, and thus\u000f2\u00190, while\u000f1\u00192\u00012is still large. In 2011,\nSpielman's group \frst reached this regime and showed that a SO coupling can be\ngenerated9. As shown in Fig. 1(d), in this regime the low-energy physics contains\ntwo energy minima which are dominated by j1;\u00001iandj1;0istates, respectively.\nTherefore we can deduce the low-energy e\u000bective Hamiltonian by keeping both\nj1;\u00001iandj1;0i, and rewrite the Hamiltonian as\nH= k2\nx\n2m+h\n2\n2ei2k0x\n\n2e\u0000i2k0xk2\nx\n2m\u0000h\n2!\n(4)\nwhereh=\u000f2. Similarly, by applying a unitary transformation to the wave function\nwith\nU=\u0012e\u0000ik0x0\n0eik0x\u0013\n(5)\none reaches an e\u000bective Hamiltonian that describes SO coupling\nHSO=UHUy= \n(kx+k0)2\n2m+h\n2\n2\n\n2(kx\u0000k0)2\n2m\u0000h\n2!\n=1\n2m(kx+k0\u001bz)2+\n2\u001bx+h\n2\u001bz:\n(6)November 1, 2011 0:15 WSPC/INSTRUCTION FILE Review_SOC\n4Hui Zhai\nIn fact, upon a pseudo-spin rotation \u001bx!\u0000\u001bzand\u001bz!\u001bx, the above Hamiltonian\nis equivalent to\nHSO=1\n2m(kx+k0\u001bx)2\u0000\n2\u001bz+h\n2\u001bx; (7)\nwhere the \frst term can be viewed as an equal weight of Rashba ( kx\u001bx+ky\u001by)\nand Dresselhaus ( kx\u001bx\u0000ky\u001by) SO couplinga. The Hamiltonian of Eq. 7 can also\nbe viewed as a Hamiltonian with synthetic non-abelian gauge \feld, since the vector\npotentialAx=\u0000k0\u001bzdoes not commute with the scale potential \b =\n2\u001bx+h\n2\u001bz.\nLater on, Campbell et al. discussed how to generalize the NIST scheme to create\nSO coupling in both ^ xand ^ydirection, and nearly isotropic Rashba SO coupling10.\nXu and You recently introduce a dynamics generalization of the NIST scheme11.\nSau et al. discussed an explicit method to create Rashba SO coupling in fermionic\n40K system at \fnite magnetic \feld, where a magnetic Feshbach resonance is avail-\nable12. Beside the NIST scheme, there are also other theoretical proposals that use\n\u0003-type or tripod system to generate synthetic magnetic \feld13;14;15;16, non-abelian\ngauge \feld with a monopole17and SO coupling18;19;20;21. However, for those pro-\nposals using dark states13;14;15;16;20;21, collisional stability is a concern since there\nis always one eigen-state whose energy is below the dark state manifold, and multi-\nparticle collision can lead to decay out of the dark state manifold. A recent review\npaper by Dalibard et al. has discussed di\u000berent proposals in detail22.\n2. Single Particle Properties with Rashba Spin-Orbit Coupling\nIn the rest part of this review we will consider SO coupling in both ^ xand ^ydirections,\nwhose Hamiltonian is given by\nH0=1\n2m\u0002\n(kx\u0000\u0014x\u001bx)2+ (ky\u0000\u0014y\u001by)2+k2\nz\u0003\n(8)\nand in particular, we will consider the most symmetric Rashba case \u0014x=\u0014y=\n\u0014>0, where the physics is most interesting. In this case, the Hamiltonian can be\nrewritten as\nH0=1\n2m\u0000\nk2\n?\u00002\u0014k?\u0001\u001b+\u00142+k2\nz\u0001\n(9)\nObviously, spin is no longer a good quantum number. However \\helicity\" is a good\nquantum number. \\Helicity\" \u0006means that the spin direction is either parallel or\nanti-parallel to the in-plane momentum direction. For these two helicity branches,\ntheir dispersion are given by\n\u000fk\u0006=1\n2m(k2\n?\u00072\u0014k?+\u00142+k2\nz) (10)\naIn many literature, Rashba SO coupling denotes kx\u001by\u0000ky\u001bx, while Dresselhaus SO coupling\ndenoteskx\u001by+ky\u001bx. They are equivalent to the notations used in this paper by a pseudo-spin\nrotation\u001bx!\u0000\u001byand\u001by!\u001bx.November 1, 2011 0:15 WSPC/INSTRUCTION FILE Review_SOC\nSpin-Orbit Coupled Quantum Gases 5\n(a)(b)/Minus2/Minus10120.00.51.01.52.0\nk/UpTee/Slash1ΚE/Slash1ER012340.000.050.100.150.20E/Slash1ERDOS/Slash1/LParen1Κm/RParen1\nFig. 2. Energy dispersion with kz= 0 (a) and density-of-state (b) for Rashba spin-orbit coupled\nsingle particle Hamiltonian. In (a), \\helicity\" is \\ + \" for red solid line and is \\ \u0000\" for blue dashed\nline.ER=\u00142=(2m) is introduced as energy unit.\nwhere k?= (kx;ky) andk?=q\nk2x+k2y. This Hamiltonian displays a symmetry\nof simultaneously rotation of spin and momentum along ^ zdirection.\nAs shown in Fig. 2(a), helicity plus branch has lower energy. The single par-\nticle energy minimum has \fnite in-plane momentum k?=\u0014, and all the single\nparticle states with same k?=\u0014andkz= 0 but di\u000berent azimuthal angle are\ndegenerate ground states. The single particle DOS also has non-trival feature. In a\nthree-dimensional system without SO coupling, DOS vanishes asp\u000fat low ener-\ngies. However, in this system, as shown in Fig. 2(b), the low-energy DOS becomes\na constant when \u000f<E R, similar to a conventional two-dimensional system.\nWithout SO coupling, the single particle Hamiltonian has a unique ground state\natk= 0. At zero-temperature, bosons are all condensed at k= 0 state. If the inter-\naction is weak, interaction e\u000bect is perturbative which only creates \fnite quantum\ndepletion to the zero-momentum condensate. Also, because the vanishing DOS at\nlow energies, two-body bound state appears only when the attractive interaction is\nabove a threshold. Far below the threshold, when the attractive interaction is weak,\nthe strength of fermion pairing and the fermion super\ruid transition temperature\nare both exponentially small. That is to say, without SO coupling, not surprisingly,\nthe interaction e\u000bect is weak in the regime where the interaction strength is small.\nWith SO coupling, the e\u000bects of interaction are signi\fcantly enhanced even when\nits strength is still weak, because SO coupling signi\fcantly changes the single par-\nticle behaviors as discussed above. First, because the single particle ground state\nis not unique now, the ground state of a boson condensate is also not unique if\nthere is no interactions. That is to say, it is the interaction that selects out a unique\nground state among many possibilities. In this sense, the e\u000bect of interaction is\nnon-perturbative even for very weak interactions. Secondly, because the DOS is a\nconstant at low energies, a two-body bound state appears for any weak attractions.\nTherefore, for weak attractive interactions, both pairing gap and super\ruid transi-\ntion temperature are largely enhanced by SO coupling. In particular, the super\ruid\ntransition temperature can reach the same order as the Fermi temperature even forNovember 1, 2011 0:15 WSPC/INSTRUCTION FILE Review_SOC\n6Hui Zhai\nweak attractions. These points will be discussed in the next two sections in more\ndetail.\nOn the other hand, we shall always keep in mind that in real experiment, one can\nnever achieve a perfect isotropic Rashba SO coupling. Therefore, in our following\ndiscussions, we always \frst obtain interesting results in the isotropic Rashba limit,\nand then discuss how robust the results are if there is a little anisotropy, namely, a\nslight di\u000berence between \u0014xand\u0014y.\n3. Spin-Orbit Coupled Bose Gases\nWe will \frst consider (pseudo-)spin-1 =2 bosons. We shall \frst consider the mean-\n\feld ground stateband then discuss the e\u000bects of quantum and thermal \ructuations\non top of mean-\feld saddle points. Let us \frst consider the most simpli\fed form of\ninteractions\n^Hint=Z\nd3r\u0000\ngn2\n1(r) +gn2\n2(r) + 2g12n1(r)n2(r)\u0001\n(11)\nBy minimizing Gross-Pitaevskii energy functional, Wang et al. found that the mean-\n\feld ground state has two di\u000berent phases depending on the sign of g12\u0000g23. When\ng12<g, the system is in the \\plane wave phase\", where all bosons are condensed into\na single plane wave state, and the direction of plane wave is spontaneously chosen\nin thexyplane. For instance, if the plane wave momentum is along ^ xdirection, the\ncondensate wave function is given by\n =r\u001a\n2ei\u0014x\u00121\n1\u0013\n: (12)\nThe density of each component is uniform, but their phases modulate from zero\nto 2\u0019periodically, as found from numerical solution of Gross-Pitaevskii equation\nand shown in Fig. 3(a). This state spontaneously breaks time-reversal, rotational\nsymmetry and the U(1) symmetry of super\ruid phase. When g12>g, all bosons are\ncondensed into a superposition of two plane wave states with opposite momentums,\nwhose condensate wave function is given by\n =p\u001a\n2\u0014\nei\u0014x\u00121\n1\u0013\n+e\u0000i\u0014x\u00121\n\u00001\u0013\u0015\n=p\u001a\u0012cos(\u0014x)\nisin(\u0014x)\u0013\n(13)\nIn this phase, the spin density n1\u0000n2=\u001acos(2\u0014x) which has a periodic modulation\nin space, as shown in Fig. 3(b), and therefore is named as \\stripe super\ruid\". The\ndirection of the stripe is also spontaneously chosen in the xyplane. Here, without\nloss of generality, we choose it along ^ xdirection. In this state, the high density regime\nof one component coincides with the low-density regime of the other component, so\nthat the inter-component repulsive interaction is maximumly avoided. That is the\nbThere are also proposals of non-mean-\feld fragmented state, like N00N state, as ground state of\nSO coupled bosons24, however, the conventional wisdom is that such a state is very fragile when\nexternal perturbations are present.November 1, 2011 0:15 WSPC/INSTRUCTION FILE Review_SOC\nSpin-Orbit Coupled Quantum Gases 7\n(b)\n(a)\nFig. 3. (a) Phase of condensate wave function in the \\plane wave phase\"; (b) Spin density in the\n\\stripe super\ruid\" phase.\nreason why the spin stripe state is favored when g12is larger than g. In addition to\ntheU(1) super\ruid phase, this state also breaks the rotational symmetry (but keeps\nthe re\rection symmetry), and the translational symmetry along the stripe direction.\nHence, symmetry wise, it can also been called \\smectic super\ruid\". In contrast to\nthe \\plane wave phase\", this state does not break time-reversal symmetry.\nIn principle, the condensate wave function can be any superposition of single\nparticle ground states as\n =X\n'c'ei\u0014(cos'x+sin'y)\u00121\nei'\u0013\n(14)\nOne may wonder why only a single state or a superposition of a pair of states\nare favored by interactions. In fact, if there are more than a pair of single particle\nstates in the superposition, the condensate wave function will exhibit interesting\nstructure, such as various types of skyrmion lattices. And if all the degenerate\nstates enter the condensate wave function with equal weight and speci\fed relative\nphases, condensate will exhibit interesting structure of half vortices, as \frst proposed\nby Stanescu, Anderson, Galitski24and Wu, Mondragon-Shem25. However, such a\nstate is not energetically favorable in spin-1 =2 case for a uniform (or nearly uniform)\nsystem. This is because that the interaction part can be rewritten as\n^Hint=Z\nd3r\u0012g+g12\n2n2(r) +g\u0000g12\n2S2\nz(r)\u0013\n; (15)\nwheren(r) =n1(r) +n2(r) andSz=n1(r)\u0000n2(r). TheS2\nz-term can be satis\fed\nby either choosing the \\plane wave phase\" or the \\stripe phase\", while the n2-\nterm with positive coe\u000ecient always favors a uniform density. One can easily show\nthat, if there are more than a pair of states in the superposition, the condensate\ndensity will always have non-uniform modulation, which causes energy of n2-term.\nSimilar situation has also been found for spin-1 Hamiltonian26;23. However, there\nare several situations where skyrmion lattices or half vortices are found as ground\nstate. When a strong harmonic con\fnement potential V(r) =m!r2\n?=2 is applied\nto the system, condensate density is no longer uniform because of the trapping\npotential and the requirement from n2-term becomes less restrictive. In addition,November 1, 2011 0:15 WSPC/INSTRUCTION FILE Review_SOC\n8Hui Zhai\none notes in the limit of zero interaction, the ground state in a harmonic trap can\nbe solved exactly and it is a half-vortex state31;32;54. Recently, three groups have\nfound that if one largely increases the con\fnement potential so that a=p\n~=m! is\ncomparable to 1 =\u0014, or reduces the interaction energy to be comparable to ~!, the\nground state will evolve continuously to skyrmion lattice phases, and \fnally to half\nvortex phases31;32;54. Besides, Xu et al. and Kawakami et al. found in spin-2 case,\nbecause of an addition interaction term (which favors cyclic phase in absence of SO\ncoupling29;30), there exists a parameter regime where various types of skyrmion\nlattice phases are ground state even for a uniform system.\nBack to the discussion of a nearly uniform spin-1 =2 systemc, the \\plane wave\nphase\" and the \\stripe phase\" are in fact two very robust mean-\feld states. In\nreal situation, the interactions between two pseudo-spin states have much more\ncomplicated form than the simpli\fed form of Eq. (11). Both Yip and Zhang et al.\nconsidered a speci\fc type of complicated interaction form using a concrete real-\nization of Rashba SO coupling, and they found the ground state is still either the\n\\plane wave phase\" or the \\stripe phase\"34;80. And also, the SO coupling is always\nnot perfectly isotropic, say, \u0014x>\u0014y, then the Hamiltonian itself does not have rota-\ntional symmetry anymore. The single particle energy has two minima at ( \u0006\u0014;0;0)\ninstead of a continuous circle of degenerate states. At mean-\feld level, the e\u000bect of\nanisotropic SO coupling is to pin the direction of plane wave or stripe into certain\ndirection (^xdirection for this case). The NIST experimental situation discussed in\nSec. 1 corresponds to the case \u0014y= 0 and also with a Zeeman \feld. As shown by\nHo and Zhang36, and also in the experimental paper9, the phase diagram of this\nsystem also only contains such two phases.\nTo go beyond the mean-\feld description, three di\u000berent approaches have been\ntried so far. The \frst is the e\u000bective \feld theory approach37, which treats Gaussian\n\ructuations of low-energy modes. The second is Bogoliubov approach41;42;43, which\nmore focuses on the gapless phonon excitations. And the third is the renormalization\napproach38;39;40, which discusses how scattering vertices are renormalized by high\norder processes. These di\u000berent approaches address beyond-mean-\feld e\u000bects from\ndi\u000berent perspectives, and the results are consistent with each other where they\noverlap.\nTaking e\u000bective \feld theory approach as an example, for the \\stripe\" phase, the\nsuper\ruid phase \u0012and the relative phase ubetween two momentum components\nare two low-lying modes:\n'ST=p\u001a\n2ei\u0012\u0014\nei(\u0014x+u)\u00121\n1\u0013\n+e\u0000i(\u0014x+u)\u00121\n\u00001\u0013\u0015\n: (16)\nIn fact, the relative phase udescribes the phonon mode of stripe order. The dy-\ncFor typical experimental parameters, the BEC is in the nearly uniform regime rather than strong\nharmonic con\fnement regime.November 1, 2011 0:15 WSPC/INSTRUCTION FILE Review_SOC\nSpin-Orbit Coupled Quantum Gases 9\nnamics of\u0012andu\felds are governed by an e\u000bective energy function37\nHST\ne\u000b=\u001a\n2m\"\n(@x\u0012)2+(@y\u0012)2\n\u000b2+ (@xu)2+\u0000\n@2\nyu\u00012\n4\u00142#\n; (17)\nwhere\u000b>1 is a constant. For the \\plane wave\" phase, the superfuid phase is the\nonly low energy mode\n'PW=p\u001aei(\u0014x+\u0012)\u00121\n1\u0013\n(18)\nand its e\u000bective energy is derived as37\nHPW\ne\u000b=\u001a\n2m\u0014\n(@x\u0012)2+1\n4\u00142(@2\ny\u0012)2\u0015\n: (19)\nOne notes that in the \\stripe\" phase, the quadratic term ( @yu)2is absent in Eq.\n(17), and in the \\plane wave\" phase, the quadratic term ( @y\u0012)2is absent in Eq.\n(19). This is in fact a manifestation of rotational symmetry in this system. Similar\ne\u000bective theory has also been found for \\FFLO\" state in fermion super\ruid44;46.\nSuch an energy function is also the classical energy of smectic liquid crystal.\nIn two dimensions, the \fnite temperature phase transition is driven by prolif-\neration of topological defects. For usual XYmodel, both the energy of topological\nvortex and its entropy logarithmically depend on system size. Thus, only above\na critical temperature, entropy wins energy and the topological defects proliferate\nwhich drives system into normal phase. However, in the \\stripe\" phase, because the\nabsence of ( @yu)2term, the energy for a topological defect of u, i.e. a dislocation in\nthe stripe, no longer logarithmically depends on system size. Hence it will lose to en-\ntropy at any \fnite temperature, and the proliferation of these dislocations will melt\nthe stripe order. This restores translational symmetry and drives the system into a\nboson paired super\ruid phase37. Such a boson pairing state can also be predicted\nby looking at pairing instability of normal state with renormalized interactions38.\nBy considering the renormalization e\u000bects of scattering vertex from high energy\nstates, Gopalakrishnan et al.38and Ozawa, Baym39;40found that the scattering\namplitudes between two states with opposite momentum become smaller and even\nvanishing, which makes the \\stripe phase\" more favorable at the low-density limit,\nand meanwhile leads to more signi\fcant the \ructuation of stripe order38. For same\nreason, in the \\plane wave\" phase, the super\ruid phase \u0012will immediately disorder\nat \fnite temperature and the system becomes normal.\nIn addition, the e\u000bective theory Eq.(17) and Eq. (19) also imply that there is a\nGoldstone mode which has linear dispersion along ^ xdirection (direction of stripe\nor plane wave momentum), and quadratic dispersion along ^ ydirection (direction\nperpendicular to the direction of stripe or plane wave momentum). Same results\nhave been reached by Bogoliubov calculation41;42;43. Similar behaviors of Goldstone\nmodes also exist in \\FFLO\" phase of fermion super\ruid44;45;46.November 1, 2011 0:15 WSPC/INSTRUCTION FILE Review_SOC\n10Hui Zhai\nWhen the SO coupling is anisotropic ( \u0014x6=\u0014y), there is no rotational symmetry\nin the Hamiltonian. Therefore, one will have ( @yu)2in the stripe phase, and ( @y\u0012)2\nin the plane wave phase. However, the coe\u000ecients of those terms are propositional\nto 1\u0000(\u0014y=\u0014x)2. In the regime \u0014y=\u0014xis very close to unity, the energy of a disloca-\ntion is much smaller than the energy of a vortex or a half vortex. Hence, the system\nwill undergo two Kosterlitz-Thouless phase transitions. At a lower critical tempera-\nture, dislocations proliferate and the system becomes paired super\ruid. Then, at a\nhigher critical temperature, vortices proliferate and the system becomes normal. A\ncompleted phase diagram in term of interaction parameter, SO coupling anisotropy\nand temperature is given by Jian and Zhai37.\nFor SO coupled bosons, many questions remain open. Recently several works\nbegin to address the questions about the super\ruid critical velocity42, vortices in\npresence of external rotation47;48;49, the e\u000bects of dipolar interactions50, the collec-\ntive modes51, super\ruid to Mott insulator transition in a lattice52, the interplay\nbetween magnetic \feld and SO coupling53, and the dynamical e\u000bects nearby Dirac\npoint due to SO coupling54;55;56. The research along this direction will de\fnitely\nreveal more interesting physics and stimulate more interesting experiments.\n4. Spin-Orbit Coupled Fermi Gases across a Feshbach Resonance\nEven for a non-interacting system, the thermodynamic behavior of a Fermi gas is\ndramatically changed by a strong SO coupling because of the change of the low-\nenergy DOS57. In this section, we focus on Fermi gas with attractive interaction,\nand in particular, across a Feshbach resonance. The interaction part is modeled as\nHint=gZ\nd3r y\n\"(r) y\n#(r) #(r) \"(r) (20)\nwhere\n1\ng=m\n4\u0019~2as\u0000X\nk1\n~2k2=m: (21)\nHere we relate the bare interaction gtos-wave scattering length via Eq. (21), which\nis the same as the scheme widely used in free space without SO coupling. This is\nbased on the assumption that the interaction Hamiltonian is not changed by SO\ncoupling, and the ashere should be understood as scattering length in free space.d\nUsing this interaction Hamiltonian, Vyasanakere and Shenoy \frst studied the\ntwo-body problem across a Feshbach resonance59. Because the low-energy DOS is\nnow a constant, an arbitrary weak attractive interaction will give rise to a bound\nstate. Similar situations are two-body problem in two-dimension and Cooper prob-\nlem in three-dimension in absence of SO coupling, where DOS are also constants.\ndThis is equivalent to assume that the s-wave pseudo-potential is still a valid approximation\nfor a short-range realistic potential in presence of SO coupling. In fact, the validity of such an\napproximation is not quite obvious, and it has only been examined recently by Cui58.November 1, 2011 0:15 WSPC/INSTRUCTION FILE Review_SOC\nSpin-Orbit Coupled Quantum Gases 11\nFig. 4. (a) Gap size \u0001 as a function of \u0014=kFfor di\u000berent values of 1 =(kFas). (b) Size of Cooper\npair inxyplanel?and along ^zdirectionlzas a function of \u0014=kFat resonance with as=1; (c)\nSuper\ruid transition temperature Tc=TFas a function of 1 =(kFas) for di\u000berent values of \u0014=kF.\nReprinted from arXiv: 1105.2250 (Phys. Rev. Lett. to be published).\nThe binding energy can be easily calculated by looking at poles of T-matrix59;61\nor by reducing the two-body Schr odinger equation to a self-consistent equation62.\nThe two-body properties at the BCS side and at resonance regime are signi\fcantly\nchanged by SO coupling. At weakly interacting BCS side with small negative \u0014as,\nthe binding energy behaves as\nEb\u0019\u0000~2\u00142\n2m4\ne2e\u00002\n\u0014jasj: (22)\nwhere a large binding energy can always been reached by increasing the strength\nof SO coupling. At resonance with as=1, 1=\u0014is the only length scale in the\ntwo-body problem and therefore one has a universal result\nEb=\u00000:88~2\u00142\n2m: (23)\nWhile for the BEC side with small positive \u0014as, at leading order Ebis still given by\n~2=(2ma2\ns) and is not a\u000bected by SO coupling. Moreover, because of SO coupling,\nthe two-body wave function has both singlet and triplet components. For a two-\nbody bound state with zero center-of-mass momentum, the wave function behaves\nas59\n = s(r)j\"#\u0000#\"i + \u0003\na(r)j\"\"i + a(r)j##i (24)\nwhere s(r) and a(r) are symmetric and anti-symmetric functions, respectively.\nFurthermore, by looking at binding energy with \fnite center-of-mass momentum,\none can determine the e\u000bective mass of molecules (two-body bound state). Hu et al.\n61and Yu and Zhai62found that at the BCS limit, the e\u000bective mass of molecule\n\fnally saturates to 4 m, and at resonance, the e\u000bective mass is a universal number\nof 2:40m. In the BEC limit, the e\u000bective mass saturates to 2 mas conventional case.\nGeneralizing conventional BEC-BCS crossover mean-\feld theory to the case\nwith SO coupling and equal population hn\"i=hn#i, one can show that the system\nremains gapped for all kFas, although there are triplet p-wave components60;61;62;63,November 1, 2011 0:15 WSPC/INSTRUCTION FILE Review_SOC\n12Hui Zhai\nand the pair wave function obtained from the mean-\feld theory \fnally approaches\nthe wave function of two-body bound state (i.e. molecule wave function)60;62. Hence,\nit is still a crossover as 1 =(kFas) changes from negative to positive. However, on the\nother hand, the change of low-energy DOS and the presence of two-body bound\nstate at the BCS side and resonance regime will signi\fcantly change the properties\nof crossover60;61;62;64. For example, as shown in Fig. 4(a), the pairing gap at the\nBCS side is dramatically enhanced when \u0014=kFis comparable or larger than unitye.\nFor such a strong SO coupling, the DOS at Fermi energy becomes a constant, and\nis much larger than the DOS in absence of SO coupling. That is the reason why\nthe pairing e\u000bects become much dramatic even for same interaction strength. For\nanother example, in absence of SO coupling, Fermi energy is the only energy scale at\nresonance, and therefore the size of Cooper pair is a universal constant times 1 =kF.\nSO coupling introduces another scale at resonance, which is 1 =\u0014. For large \u0014, as\nthe pairing gap approaches two-body binding energy, the size of Cooper pairs also\napproaches 1 =\u0014. Fig. 4(b) shows that the behavior of lundergoes a crossover from\n1=kFto 1=\u0014as\u0014=kFincreases. This plot also shows that the size of Cooper pair in\nthexyplane is di\u000berent from the size along ^ zdirection, namely, the Cooper pairs\nare anisotropic.61;62. In addition, one can also show that the super\ruid transition\ntemperature at the BCS side can also be enhanced a lot by SO coupling. For large\nenough SO coupling, it eventually approaches the BEC temperature of molecules\nwith mass 4 m62;64, which is a sizable fraction of Fermi temperature. The critical\ntemperature across resonance is \frst estimated by Yu and Zhai62as shown in Fig.\n4(c).\nIf SO coupling is slightly anisotropic, DOS at very low-energy will \fnally vanish.\nHowever, there is still a large energy window .~2\u00142=(2m) where the DOS is greatly\nenhanced by SO coupling. Hence, pairing gap will still be enhanced as long as the\ndensity of fermions is not extremely low. Moreover, although it is no longer true\nthat arbitrary small attraction can cause a bound state, the critical value for the\nappearance of a bound state will move from unitary point as=1to the BCS side\nwith negative as59. Once the bound state is present, it will in\ruence the universal\nbehavior of pair size at resonance and super\ruid critical temperature as discussed\nabove.\nFor the imbalanced case, the phase diagram becomes more richer. Several groups\nhave studied the phase diagram in presence of a Zeeman \feld65;66;67;68;69and\nin various other circumstance70;71;72;73;74;75;76;77;78;79;80 f. They have shown that,\neIn cold atom system, SO coupling is generated by atom-light coupling, and therefore \u0014is on the\norder of the inverse of the laser wave length. And since the laser wave length and the inter-particle\ndistance are comparable (between \u00180:1\u0016mand\u00181\u0016m) in atomic gases, the strength of SO\ncoupling in cold atom systems can naturally reach the regime \u0014=kF\u00181.\nfIskin and Subasi studied SO coupled Fermi gas with mass imbalance72. They use mixture of\ndi\u000berent species as motivation of this study. We caution that the current way of generating SO\ncoupling is based on light coupling of di\u000berent internal state of atoms, which can not generate SO\ncoupling if di\u000berent internal states are di\u000berent atomic speciesNovember 1, 2011 0:15 WSPC/INSTRUCTION FILE Review_SOC\nSpin-Orbit Coupled Quantum Gases 13\ninstead of a crossover, there are phase transitions between topological and non-\ntopological phases65;66;67;68. They have also discussed how SO coupling in\ruences\nthe competition between a uniform super\ruid and a phase separation66;67.\n5. Summary and Future Developments\nSO coupled quantum gases with interactions are new systems in cold atom physics.\nMoreover, SO coupled bosonic system has never been thought in physics before.\nCurrently our understanding of this system is still very limited, and many questions\nremain open. However, even from our limited experience with this new system, one\ncan already get a feeling that this system has many unusual behaviors. This gives\na lot of opportunities for theorists and experimentalists.\nAcknowledgements\nI would like to thank Chao-Ming Jian and Zeng-Qing Yu for collaboration on this\nsubject, and thank Xiaoling Cui for helpful discussions. This work is supported by\nTsinghua University Initiative Scienti\fc Research Program, NSFC under Grant No.\n11004118 and No. 11174176, NKBRSFC under Grant No. 2011CB921500.\nReferences\n1. M. V. Berry, Proc. Roy. Soc. London A 392, 45 (1984)\n2. F. Wilczek and A. Zee, Phys. Rev. Lett. 52, 2111 (1984)\n3. X. L. Qi and S. C. Zhang, Physics Today, 63, 33 (2010);\n4. M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045 (2010)\n5. X. L. Qi, S. C. Zhang, arXiv: 1008.2026, to be published in Rev. Mod. Phys\n6. Y. J. Lin, R. L. Compton, A. R. Perry, W. D. Phillips, J. V. Porto, and I. B. Spielman,\nPhys. Rev. Lett. 102, 130401 (2009)\n7. Y. J. Lin, R. L. Compton, K. Jimnez-Garca,, J. V. Porto, I. B. Spielman, Nature 462,\n628 (2009)\n8. Y. J Lin, R. L. Compton, K. Jimnez-Garca, W. D. Phillips, J. V. Porto, I. B. Spielman,\nNature Physics, 7, 531, (2011)\n9. Y.-J. Lin, K. Jimnez-Garca, I. B. Spielman, Nature, 471, 83 (2011)\n10. D. L. Campbell, G. Juzeli unas, I. B. Spielman, arXiv: 1102.3945\n11. Z. F. Xu and L. You, arXiv:1110.5705\n12. J. D. Sau, R. Sensarma, S. Powell, I. B. Spielman, S. Das Sarma, Phys. Rev. B 83,\n140510(R) (2011)\n13. R. Dum and M. Olshanii, Phys. Rev. Lett. 76, 1788 (1996).\n14. G. Juzeli\u0016 unas and P. Ohberg, Phys. Rev. Lett. 93, 033602 (2004);\n15. G. Juzeli\u0016 unas, J. Ruseckas, P. Ohberg, and M. Fleischhauer, Phys. Rev. A 73, 025602\n(2006).\n16. K. J. G unter, M. Cheneau, T. Yefsah, S. P. Rath, and J. Dalibard, Phys. Rev. A 79,\n011604(R) (2009)\n17. J. Ruseckas, G. Juzeli\u0016 unas, P. Ohberg, and M. Fleischhauer, Phys. Rev. Lett. 95,\n010404 (2005);\n18. S. L. Zhu, H. Fu, C. J. Wu, S. C. Zhang, and L. M. Duan, Phys. Rev. Lett. 97, 240401\n(2006);November 1, 2011 0:15 WSPC/INSTRUCTION FILE Review_SOC\n14Hui Zhai\n19. X. J. Liu, X. Liu, L. C. Kwek, and C. H. Oh, Phys. Rev. Lett. 98, 026602 (2007);\n20. T. D. Stanescu, C. Zhang, and V. Galitski, Phys. Rev. Lett. 99, 110403 (2007).\n21. G. Juzeliunas, J. Ruseckas, J. Dalibard, Phys. Rev. A 81, 053403 (2010)\n22. J. Dalibard, F. Gerbier, G. Juzeli unas, P. Ohberg, arXiv: 1008.5378, to be published\nin Rev. Mod. Phys.\n23. C. Wang, C. Gao, C. M. Jian, and H. Zhai, Phys. Rev. Lett. 105, 160403 (2010)\n24. T. D. Stanescu, B. Anderson, and V. Galitski, Phys. Rev. A 78, 023616 (2008)\n25. C. J. Wu, I. Mondragon-Shem, arXiv:0809.3532v1\n26. T. L. Ho, Phys. Rev. Lett. 81, 742 (1998)\n27. Z. F. Xu, R. L u, and L. You, Phys. Rev. A 83, 053602 (2011)\n28. T. Kawakami, T. Mizushima, and K. Machida, Phys. Rev. A 84, 011607 (2011)\n29. C. V. Ciobanu, S. K. Yip, and T. L. Ho, Phys. Rev. A 61, 033607 (2000)\n30. M. Koashi and M. Ueda, Phys. Rev. Lett. 84, 1066 (2000)\n31. C. J. Wu, I. Mondragon-Shem, X. F. Zhou, Chin. Phys. Lett. 28, 097102 (2011)\n32. H. Hu, H. Pu, X. J. Liu, arXiv:1108.4233\n33. S. Sinha, R. Nath, L. Santos, arXiv: 1109.2045\n34. S. K. Yip, Phys. Rev. A 83, 043616 (2011)\n35. Y. Zhang, L. Mao, C. Zhang, arXiv: 1102.4045\n36. T. L. Ho, S. Zhang, Phys. Rev. Lett. 107, 150403 (2011)\n37. C. M. Jian and H. Zhai, Phys. Rev. B 84, 060508 (2011)\n38. S. Gopalakrishnan, A. Lamacraft, P. M. Goldbart, arXiv: 1106.2552\n39. T. Ozawa, G. Baym, arXiv:1107.3162\n40. T. Ozawa, G. Baym, arXiv:1109.4954\n41. Q. Zhou and X. L. Cui, to be published\n42. Q. Zhu, C. Zhang and B. Wu, arXiv: 1109.5811\n43. R. Barnett, S. Powell, T. Gra \f, M. Lewenstein and S. Das Sarma, arXiv:1109.4945\n44. L. Radzihovsky, A. Vishwanath, Phys. Rev. Lett. 103, 010404 (2009),\n45. L. Radzihovsky and S. Choi, Phys. Rev. Lett. 103, 095302 (2009)\n46. Radzihovsky, arXiv: 1102.4903\n47. X. Q. Xu, J. H. Han, arXiv: 1107.4845\n48. X. F. Zhou, J. Zhou, C. J. Wu, arXiv: 1108.1238\n49. J. Radic, T. Sedrakyan, I. Spielman, V. Galitski, arXiv: 1108.4212\n50. Y. Deng, J. Cheng, H. Jing, C.-P. Sun, S. Yi, arXiv: 1110.0558\n51. E. van der Bijl, R.A. Duine, arXiv: 1107.0578\n52. T. Grass, K. Saha, K. Sengupta, M. Lewenstein, arXiv: 1108.2672\n53. M. Burrello and A. Trombettoni, Phys. Rev. Lett. 105, 125304 (2010)\n54. M. Merkl, A. Jacob, F. E. Zimmer, P. Ohberg, and L. Santos, Phys. Rev. Lett. 104,\n073603 (2010)\n55. M. J. Edmonds, J. Otterbach, R. G. Unanyan, M. Fleischhauer, M. Titov, P. Ohberg,\narXiv: 1106.5925\n56. D. W. Zhang, Z. Y. Xue, H. Yan, Z. D. Wang, S. L. Zhu, arXiv: 1104.0444\n57. L. He and Z. Q. Yu, Phys. Rev. A 84, 025601 (2011)\n58. X. L. Cui, to be published\n59. J. P. Vyasanakere, V. B. Shenoy, Phys. Rev. B 83094515 (2011)\n60. J. P. Vyasanakere, S. Zhang, V. B. Shenoy, Phys. Rev. B 84, 014512 (2011)\n61. H. Hu, L. Jiang, X. J. Liu, H. Pu, arXiv: 1105.2488, to be published in Phys. Rev.\nLett.\n62. Z. Q. Yu and H. Zhai, arXiv: 1105.2250, to be published in Phys. Rev. Lett.\n63. L. Han, C. A. R. S\u0013 a de Melo, arXiv: 1106.3613;\n64. J. P. Vyasanakere, V. B. Shenoy, arXiv: 1108.4872November 1, 2011 0:15 WSPC/INSTRUCTION FILE Review_SOC\nSpin-Orbit Coupled Quantum Gases 15\n65. M. Gong, S. Tewari, C. Zhang, arXiv: 1105.1796, to be published in Phys. Rev. Lett.\n66. M. Iskin and A. L. Subas, Phys. Rev. Lett. 107, 050402 (2011);\n67. W. Yi, G. C. Guo, arXiv: 1106.5667;\n68. K. Seo, L. Han, C. A. R. S\u0013 a de Melo, arXiv:1108.4068;\n69. R. Liao, Y. X. Yu and W. M. Liu, arXiv: 1110.5818\n70. S. L. Zhu, L. B. Shao, Z. D. Wang, L. M. Duan, Phys. Rev. Lett. 106, 100404 (2011)\n71. L. Jiang, T. Kitagawa, J. Alicea, A. R. Akhmerov, D. Pekker, G. Refael, J. Ignacio\nCirac, E. Demler, M. D. Lukin, P. Zoller, Phys. Rev. Lett. 106, 220402 (2011)\n72. M. Iskin, A. L. Subasi, arXiv:1107.2376\n73. M. Iskin and A. L. Subas, arXiv: 1108.4263\n74. G. Chen, M. Gong, C. Zhang, arXiv: 1107.2627;\n75. L. Dell'Anna, G. Mazzarella and L. Salasnich, arXiv: 1108.1132\n76. L. He and X. G. Huang, arXiv: 1109.5777\n77. S. K. Ghosh, J. P. Vyasanakere and V. B. Shenoy, arXiv: 1109.5279\n78. J. Zhou, W. Zhang and W. Yi, arXiv:1110.2285\n79. J. N. Zhang. Y. H. Chan and L. M. Duan, arXiv: 1110.2241\n80. K. Zhou and Z, Zhang, arXiv: 1110.3565"    },    {        "title": "2010.01970v1.Detection_of_the_Orbital_Hall_Effect_by_the_Orbital_Spin_Conversion.pdf",        "content": "Detection of the Orbital Hall E\u000bect by the Orbital-Spin Conversion\nJiewen Xiao,1Yizhou Liu,1and Binghai Yan1,\u0003\n1Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot 7610001, Israel\n(Dated: October 6, 2020)\nThe intrinsic orbital Hall e\u000bect (OHE), the orbital counterpart of the spin Hall e\u000bect, was pre-\ndicted and studied theoretically for more than one decade, yet to be observed in experiments. Here\nwe propose a strategy to convert the orbital current in OHE to the spin current via the spin-orbit\ncoupling from the contact. Furthermore, we \fnd that OHE can induce large nonreciprocal magne-\ntoresistance when employing the magnetic contact. Both the generated spin current and the orbital\nHall magnetoresistance can be applied to probe the OHE in experiments and design orbitronic\ndevices.\nI. INTRODUCTION\nThe intrinsic orbital Hall e\u000bect (OHE), where an elec-\ntric \feld induces a transverse orbital current, was pro-\nposed by the Zhang group [1] soon after the prediction\nof the intrinsic spin Hall e\u000bect (SHE) [2, 3]. The SHE\nwas soon observed [4, 5], later applied for the spintronic\ndevices[6, and references therein], and also led to the sem-\ninal discovery of the quantum SHE, i.e., the 2D topolog-\nical insulator [7, 8]. Di\u000berent from the SHE, the OHE\ndoes not rely on the spin-orbit coupling (SOC), and thus,\nit was predicted to exist in many materials [1, 9{18] with\neither weak or strong SOC, for example, in metals Al,\nCu, Au, and Pt.\nIn an OHE device, the transverse orbital current leads\nto the orbital accumulation at transverse edges, similar\nto the spin accumulation in a SHE device. Zhang et al\n[1] proposed to measure the edge orbital accumulation by\nthe Kerr e\u000bect. Recently, Ref. 19 predicted the orbital\ntorque generated by the orbital current. However, the\nOHE is yet to be detected in experiments until today.\nThe detection of the orbital is rather challenging, because\nthe orbital is highly non-conserved compared to the spin,\nespecially at the device boundary.\nA very recent work by us proposed [20] that the lon-\ngitudinal current through DNA-type chiral materials is\norbital-polarized, and contacting DNA to a large-SOC\nmaterial can transform the orbital current into the spin\ncurrent. Thus, we are inspired to conceive a similar way\nto detect the transverse OHE by converting the orbital\nto the spin by the SOC proximity.\nIn this article, we propose two ways to probe the OHE,\nwhere the strong SOC from the contact transforms the\norbitronic problem to the spintronic measurement. One\nway is to generate spin current or spin polarization from\nthe transverse orbital current by connecting the edge to a\nthird lead with the strong interfacial SOC. Then the edge\nspin polarization and spin current is promising to be mea-\nsured by the Kerr e\u000bect [4] and the inverse SHE [21{23],\nrespectively. The other way is to introduce a third mag-\nnetic lead and measure the magnetoresistance. We call\n\u0003binghai.yan@weizmann.ac.il\nFIG. 1. Illustration of the orbital-spin conversion and\nthe orbital Hall magnetoresistance (OHME). (a) The\norbital Hall e\u000bect and the spin polarization/current genera-\ntion. Opposite orbitals (red and blue circular arrows) from\nthe left lead de\rect into opposite boundaries. The red and\nblue backgrounds represent the orbital accumulation at two\nsides. Because of the SOC region (yellow) at one side, the\norbital current is converted into the spin current (indicated\nby black arrows). (b) The two-terminal (2T) OHMR. The\nthird lead is magnetized but open. (c) The three-terminal\n(3T) OHMR. The third lead is magnetized and conducts cur-\nrent. The 2T/3T conductance between di\u000berent leads relies\non the magnetization sensitively. The thickness of grey curves\nrepresent the relative magnitude of the conductance.\nit the orbital Hall magnetoresistance (OHMR), similar\nto the spin Hall magnetoresistance [24, 25]. In our pro-\nposal, the OHE refers to orbitals that resemble atomic-\nlike orbitals, which naturally couple to the spin via the\natomic SOC. We \frst demonstrate detection principles in\na lattice model by transport calculations. Then we incor-\nporate these principles into the metal copper, which has\nnegligible SOC and avoids the co-existence of the SHE,\nas a typical example of realistic materials. In the copper\nbased device, we demonstrate the resultant spin polariza-\ntion/current and very large OHMR (0 :3\u00181:3 %), which\nare measurable by present experiment techniques.\nII. RESULTS AND DISCUSSIONS\nA. Methods and General Scenario\nTo detect the OHE, we introduce an extra contact with\nthe strong SOC on the boundary of the OHE material,arXiv:2010.01970v1  [cond-mat.mtrl-sci]  5 Oct 20202\nas shown in Figure 1. This device can act for both two-\nterminal (2T) and three-terminal (3T) measurements (or\nmore terminals). In theoretical calculations, we com-\npletely exclude SOC from all leads so that we can well\nde\fne the spin current. We also remove SOC in the OHE\nmaterial, the device regime in the center, to avoid the ex-\nistence of SHE. Only \fnite atomic SOC is placed in the\ninterfacial region (highlighted by yellow in Figure 1) be-\ntween the OHE and the third lead.\nWe \frst prove the principle by a simple square-lattice\nmodel that hosts OHE. As shown in the inset of Figure\n2(a), a tight-binding spinless model is constructed, withthree orbitals s,pxandpyassigned to each site. Under\nthe above basis, the atomic orbital angular momentum\noperator ^Lzis written as\n^Lz= \u0016h2\n40 0 0\n0 0\u0000i\n0i03\n5 (1)\nAnd three eigenstates p\u0006\u0011(px\u0006ipy)=p\n2;scorrespond\nto eigenvalues Lz=\u00061;0, respectively. After consider-\ning the nearest neighboring hopping, the Hamiltonian is\nwritten as\nH(kx;ky) =0\n@Es+ 2tscoskxa+ 2tscoskya\u00002itspsinkxa \u00002itspsinkya\n2itspsinkxa E px+ 2tp\u001bcoskxa+ 2tp\u0019coskya 0\n2itspsinkya 0 Epy+ 2tp\u0019coskxa+ 2tp\u001bcoskya1\nA\n(2)\nwhereEs,EpxandEpyare onsite energies of s,pxandpy\norbitals.ts,tp\u001b,tp\u0019,tspare electron hopping integrals\nbetweensorbitals,\u001btype oriented porbitals,\u0019type\norientedporbitals, and sandporbitals, respectively. In\nthe following calculations, their values are speci\fed as\nEs= 1:3,Epx=Epy=\u00001:9,ts=\u00000:3,tp\u001b= 0:6,\ntp\u0019= 0:3, andtsp= 0:5, in the unit of eV. To realize\nthe OHE, it requires the inter-orbital hopping to induce\nthe transverse Lzcurrent. Since pxandpyorbital are\northogonal under the square lattice geometry, the inter-\norbital hopping tspbecomes the critical parameter that\ncontrols the existence of the OHE. Then we introduce the\natomic SOC on the boundary to demonstrate the OHE\ndetection by \u0015soc^Sz\u0001^Lz, where ^Szis the spin operator.\nWe estimate the OHE conductivity ( \u001bOH) with the\norbital Berry curvature in the Kubo formula [26, 27],\n\u001bOH=e\n\u0016hX\nnZd3k\n(2\u0019)3fnk\nLz\nn(k) (3)\n\nLz\nn(k) = 2\u0016h2X\nm6=nIm[hunkjjLzyjumkihumkj^vxjunki\n(Enk\u0000Emk)2]\n(4)\nwhere \nLzn(k) is the \\orbital\" Berry curvature for the\nnthband with Bloch state junkiand energy eigenvalue\nEnk.fnkis the Fermi-Dirac distribution function. vx\nis thexcomponent of the band velocity operator while\njLzyis the orbital current operator in the ydirection, de-\n\fned asjLzy= (^Lz^vy+ ^vy^Lz)=2. Therefore, the above\nformula indicates that the interband perturbation in-\nduces the orbital Berry curvature, further reiterating\nthe importance of inter-orbital hopping. We also note\nthat, the orbital Berry curvature is even under the time-\nreversal symmetry or the spatial inversion symmetry,\n\nLzn(k) = \nLzn(\u0000k).\nFor the device schematically presented in Figure 1, wecalculated the conductance by the Landauer-B uttiker for-\nmula [28] with the scattering matrix from lead ito lead\nj,\nGi!j=e2\nhX\nn2j;m2ijSnmj2; (5)\nwhereSmnis the scattering matrix element from the m-\nth eigenstate in lead ito thentheigenstate in lead j.\nIn all three leads ( i;j= 1;2;3), spin (Sz=\"#) is a con-\nserved quantity because of the lack of SOC. We turn\no\u000b the inter-orbital hopping in leads so that Lzis also\nconserved, i.e., Lzcommutes with the Hamiltonian (See\nSupplementary Materials). Therefore, with the spin and\norbital conserved leads, we can specify the conductance\nin eachSzandLzchannel, and de\fne the orbital- and\nspin-polarized conductance as:\nGij\nSz=Gi!j\"\u0000Gj!j#(6)\nGij\nLz=Gi!j+\u0000Gi!j\u0000; (7)\nwhereGij\nSz(Lz)is the conductance from lead ito the\nSz(Lz) channel of lead j.Gi!j0is omitted here since\nLz= 0 contributes no polarization. We performed the\nconductance calculations with the quantum transport\npackage Kwant [29].\nAs illustrated in Figure 1(a), electrons with the op-\nposite orbital angular momentum de\rect into transverse\ndirections in the OHE region, resulting in the transverse\norbital current. Therefore, orbital accumulates at two\nsides, and the orbital polarization emerges. To detect the\norbital polarization, atomic SOC is added at one side, as\nhighlighted by yellow in Figure 1(a). After electron de-\n\recting into the SOC region, the right-handed orbital\n(red circular arrows) is converted to the up spin polar-\nization. If a third lead is further attached, the SOC re-\ngion converts the orbital current into the spin current.3\nIf the third lead exhibits magnetization along z(Mz)\n(Figure 1(b) and 1(c)), inversely, the OHE induces the\nOHMR, relying on whether Mzis parallel or anti-parallel\nto the generated spin polarization. In the 2T measure-\nment (Figure 1(b)), the conductance from lead 1 to lead 2\n(G1!2) changes when the Mzdirection is reversed. And\nthe changing direction of G1!2depends sensitively on\nthe size of the device, due to the complex orbital accu-\nmulation and re\rection with an open lead. While for its\nspin counterpart, the SHE-induced magnetoresistance is\ncommonly measured in a 2T setup [24, 25]. In the 3T\ndevice (Figure 1(c)), the situation is simpler since the\ntransverse orbital current can \row into the third lead.\nIfMzand spin polarization is parallel (anti-parallel), the\ntransverse orbital current matches (mismatches) the lead\nmagnetization, resulting in the high (low) G1!3and low\n(high)G1!2accordingly. We point out that the 3T mea-\nsurement is usually more favorable than 2T, since the 3T\ndevice avoids the 2T reciprocity constrain [30] and the\nconductance change [\u0001 G=G(Mz)\u0000G(\u0000Mz)] is also\nrelatively larger in the third lead, as discussed in the fol-\nlowing.\nB. Spin Polarization and Spin Current Generated\nby the OHE\nThe band structure weighted by the orbital Berry cur-\nvature for the square lattice is plotted in Figure 2(a).\nThe highest band corresponds to the sorbital dispersion,\nwhile two lower bands are dominated by porbitals. The\norbital Berry Curvature concentrates near the \u0000 point,\nMpoint and \u0000- Mline in the Brillouin zone, where band\nhybridization is strong. After integrating \n Lzin the Bril-\nlouin zone, the orbital Hall conductivity is derived and\npresented in Figure 2(a). It shows that, due to the inter-\norbital hopping tsp, states below ( porbitals) and above ( s\norbital) Fermi level both exhibits signi\fcant \u001bOH. How-\never, iftspis turned o\u000b so that Lzis conserved, both \n Lz\nand\u001bOHvanishes.\nBased on the square lattice with \fnite tsp, the 2T de-\nvice is constructed, as shown in Figure 2(b). Without\nSOC at two sides, the orbital density distribution is plot-\nted in Figure 2(c), which shows that opposite orbitals\naccumulate and polarize at two boundaries. With SOC\nturned on, spin density appears and largely concentrates\non the local SOC atoms, which is promising to be de-\ntected by the Kerr e\u000bect [4]. Since SOC couples the p+\n(p\u0000) orbital to the\"(#) spin and forms the jjm=3\n2i\n(jjm=\u00003\n2i) state, the spin density near the SOC region\nlargely follows the orbital density pattern: positive at\nthe upper side and negative at the lower side. To ver-\nify that the spin polarization is directly induced by the\nOHE rather than SOC, we turned o\u000b the OHE by setting\ntsp= 0 eV and preserve the SOC at the interface. The\nsupplementary Figure S2 shows that both the orbital and\nspin polarization disappear.\nOn the basis of 2T device, a third lead is attached to\nFIG. 2. Orbital-spin conversion in the two terminal\n(2T) and three terminal (3T) device. (a) Band struc-\nture of the square lattice with tsp= 0:5 eV (left) and the\norbital Hall conductivity with tsp= 0:5 eV andtsp= 0:0 eV\n(right). In the inset, the tight binding model of the square\nlattice is presented. (b) 2T and 3T detection devices, where\nlarger spheres at two sides represent SOC regions. The yel-\nlow spheres at left, right and upper sides represent leads. (c)\nOrbital and spin density distribution in the 2T setup, at the\nenergy level of 0.2 eV. (d) Total, orbital and spin conductance\nfrom lead 1 to lead 3 with (left) and without (right) SOC.\nthe SOC side to form a 3T device, as shown in Figure\n2(b). Therefore, rather than the orbital accumulation,\nthe orbital current will \row into the third lead and gen-\nerate the spin current. Figure 2(d) shows that the orbital\ncurrent from lead 1 to lead 3 ( G13\nLz) exists with and with-\nout SOC at the interface. For instance, for states above\nFermi level, Lz= +1 states are more easily transported\ninto lead 3 than Lz=\u00001 states, and thus polarizes the\nlead, being consistent with the positive orbital polariza-\ntion at the upper side in Figure 2(c). On the other hand,\nfor the spin conductance, it only appears when turning on\nSOC, and the energy dependence of G13\nSzlargely follows\nthe orbital conductance, further demonstrating the spin\ngeneration process from the orbital. If we increase the\nSOC strength, G13\nSzincreases accordingly, because of the\nhigher orbital-spin conversion e\u000eciency (see Figure S3).\nWe also test orbital non-conserved leads with nonzero\ntsp, whose spin conductance remains the similar feature\n(see Figure S4).\nC. Orbital Hall Magnetoresistance\nAs discussed above, the current injected into lead 3 is\nspin-polarized. When lead 3 is magnetized along the z\naxis, we expect the existence of magnetization-dependent\nconductance, i.e. G13(Mz)6=G13(\u0000Mz). From the cur-4\nFIG. 3. Orbital Magnetoresistance with magnetic leads. (a) 2T and 3T detection devices with the exchange \feld \u0006Mz\nin the open and conducting lead 3. Larger spheres represent the interfacial SOC region. (b) Total conductance from lead 1 to\nlead 2 in \u0006Mz\feld in the 2T device, with the dephasing term \u0011set to 0.001. In the inset, the dephasing dependent \u0001 G12for\nthe peak inside the circle is presented. (c) Total conductance from lead 1 to lead 2 in \u0006Mz\feld in the 3T device. (d) Total,\n(e) Spin and (f) Orbital conductance from lead 1 to lead 3 in \u0006Mz\feld in the 3T device. In all these calculations, \u0015SOC and\nMzis set to 0.2 eV and 0.4 eV, respectively.\nrent conservation [30], we deduce the relation,\n\u0001G13=\u0000\u0001G12(8)\nwhere \u0001Gij\u0011Gij(Mz)\u0000Gij(\u0000Mz). To demonstrate\nthis,Mzis introduced to lead 3 as an exchange \feld to\nthe spin, as shown in the 3T setup in Figure 3(a). Results\nin Figure 3(c) and 3(d) indicate that \u0001 G12and \u0001G13\ncan reach several percentage of the total conductance at\nsome energies. We also con\frm that \u0001 G12and \u0001G13are\nproportional to the exchange \feld strength (see Figure\nS5).\nTo understand the orbital induced magnetoresistance,\nthe spin and orbital conductance from lead 1 to lead 3 are\ncalculated. As shown in Figure 3(e), G13\nSzalmost changes\nits sign when \ripping Mzin lead 3, as expected. And G13\nSz\nnow inversely a\u000bects G13\nLzbecause of the interfacial SOC.\nWhen further comparing Figure 3(d) and 3(f), we found\nthat the change of the magnitude of G13\nLzis proportional\nto the change of total conductance \u0001 G13. Therefore,\nit veri\fes the scenario in Figure 1(c): when the orbital\nmatches the spin in magnetic leads, G13\nLzand thusG13is\nhigher while G12is accordingly lower. Thus, it indicates\nthe essential role of the orbital in connecting charge and\nspin in the transport.However, the 2T results exhibit qualitatively di\u000berent\nfeatures from the 3T results. According to the reciprocity\nrelation [30], the 2T conductance obeys G12(Mz) =\nG12(\u0000Mz). Only when the current conservation is bro-\nken, one may obtain the 2T magentoresistance. There-\nfore, we introduce a dephasing term i\u0011to leak electrons\ninto virtual leads [31] to release the above constrain. As\nshown in the inset of Figure 3(b), the \u0001 G12is zero at\n\u0011= 0, \frst increases quickly and soon decreases as fur-\nther increasing \u0011. In the large \u0011limit, the system is to-\ntally out of coherence and thus, the conductance cannot\nremember the spin and orbital information. We note that\nthe dephasing exists ubiquitously in experiments due to\nthe dissipative scattering for example by electron-phonon\ninteraction and impurities.\nFor the same Mz, the 2T \u0001G12(Figure (3b)) roughly\nexhibits the opposite sign compared to the 3T \u0001 G12(Fig-\nure (3c)) in the energy window investigated. Unlike that\n\u0001G13follows the change of G13\nLz(Figure (3f)), the change\ndirection of G12depends on the geometry of the 2T de-\nvice (see Figure S6). The magnitude of the 2T \u0001 G12\nalso depends sensitively on the value of \u0011. Its peak value\n(Figure (3b)), with \u0011around 0.001, is comparable with\nthe 3T value in the same parameter regime. However,5\nthe 3T conductance avoids the strict constrain of the 2T\nreciprocity, and the existence of 3T OHMR does not rely\non the dephasing. Furthermore, in the 3T setup, the\nmagnetoresistance ratio \u0001 G13=G13in the third lead is\nalso larger than \u0001 G12=G12, because of the lower total\nconductance of G13. Therefore, we propose that the 3T\nsetup may be more advantageous to detect the OHE.\nD. Realistic Material Cu\nBased on the simple square lattice model, we demon-\nstrate two main phenomena, the OHE-induced spin po-\nlarization / spin current current assisted by the atomic\nSOC on the boundary and the existence of OHMR. We\nfurther examine them in a realistic material Cu. This\nlight noble metal is predicted to exhibit the strong OHE.\nAs shown in Figure 4(a), the 2T (without lead 3) and\n3T devices are composed of Cu (without SOC) in both\nthe scattering region and leads, and the heavy metal Au\n(with SOC) at two boundaries. We adopted the tight-\nbinding method to describe the Cu and leads, where 9\natomic orbitals ( s,px,py,pz,dz2,dx2\u0000y2,dxy,dyz,dzx)\nare assigned to each site. The nearest-neighboring and\nthe second-nearest-neighboring hoppings are considered\nwith the Slater-Koster type parameters from Ref. 32. For\nthe heavy metal Au at two sides, the SOC strength is\nset to 0.37 eV as suggested by Ref. 32. With the tight-\nbinding approach, the \frst-principles band structure of\nCu is reproduced (see Figure S7).\nAs shown in Figure 4(b), the orbital Berry curvature\nconcentrates on the dorbital region (\u00004 eV\u0018\u00002 eV)\ndue to the orbital hybridization, consistent with previ-\nous works [12, 15]. After integrating \n Lz, Figure 4(b)\nshows that the orbital Hall conductivity is around 6000\n(\u0016h=e)(\ncm)\u00001in thedorbital region, even larger than\nthe spin Hall conductivity of Pt. Near the Fermi level, the\norbital Hall conductivity is determined by the s-orbital\nderived bands and reduces to around 1000 (\u0016 h=e)(\ncm)\u00001.\nFor the 2T device, the orbital and spin density at Fermi\nlevel are plotted in Figure 4(c). The orbital polarization\nexists at two sides as a consequence of the OHE. With the\nheavy metal Au attached, spin polarization is generated,\nwhich concentrates on Au atoms and follows the orbital\ndensity pattern. To con\frm that the spin polarization\nis induced by the OHE, we arti\fcially turn o\u000b the inter-\norbital hopping in Cu to eliminate the OHE, but still\nkeep the SOC in the Au region. Result show that both\nthe orbital and spin polarization disappear (see Figure\nS8), in accordance with our prediction.\nFor the 3T device, we add a third Cu lead to one SOC\nside and calculate the spin conductance from lead 1 to\nlead 3 (G13\nSz). As shown in Figure 4(d), the generatedspin conductance displays an energy-dependence similar\nto the bulk \u001bOH. Near the Fermi level, the spin polar-\nization rate can reach 4 %, and it is even around 20%\nin thedorbital region. Therefore, a sizable spin current\ncan also be generated from the OHE by adding an inter-\nfacial SOC layer. Similarly, when arti\fcially switching\no\u000b the OHE of Cu but keeping the Au part, the spin cur-\nrent disappears, eliminating the contribution of the SHE\nbrought by the thin Au layer (see Figure S9).\nWe also studied the OHMR by applying an exchange\n\feldMzin the lead 3. We choose Mz= 0:95 eV ac-\ncording to the approximate spin splitting in the tran-\nsition metal Co (see Figure S10). As shown in Figure\n4(e) and 4(f), the 3T OHMR is rather large, where we\n\fnd \u0001G12=G12\u00190:3% and \u0001G13=G13\u00191:3% at the\nFermi level. In experiment, the SHE magentoresistance is\naround 0:05\u00180:5% (see Ref. 33 for example). Therefore,\nthe sizable OHMR in copper can be fairly measurable by\npresent experimental techniques. We should point out\nthat similar e\u000bects can be generalized to other OHE ma-\nterials like Li and Al [15].\nIII. SUMMARY\nIn summary, we have proposed the OHE detec-\ntion strategies by converting the orbital to spin by\nthe interfacial SOC, and inducing the strong spin cur-\nrent/polarization. Inversely, the OHE can also generate\nthe large nonreciprocal magnetoresistance when employ-\ning the magnetic contact. We point out that, compared\nto the two-terminal one, the three-terminal OHMR does\nnot require the dephasing term , and may be more ad-\nvantageous to detect the OHE. Using the device setup\nbased on the metal Cu, we demonstrate that the gener-\nated spin polarization and OHMR are strong enough to\nbe measured in the present experimental condition. Our\nwork will pave a way to realize the OHE in experiment,\nand further design orbitronic or even orbitothermal de-\nvices for future applications.\nIV. ACKNOWLEDGEMENT\nWe honor the memory of Prof. Shoucheng Zhang. This\narticle follows his earlier works on the intrinsic orbital\nHall e\u000bect and spin Hall e\u000bect. B.Y. acknowledges the\n\fnancial support by the Willner Family Leadership Insti-\ntute for the Weizmann Institute of Science, the Benoziyo\nEndowment Fund for the Advancement of Science, Ruth\nand Herman Albert Scholars Program for New Scientists,\nand the European Research Council (ERC) under the\nEuropean Union's Horizon 2020 research and innovation\nprogramme (Grant No. 815869, NonlinearTopo).\n[1] B. A. Bernevig, T. L. Hughes, and S.-C. Zhang, Physical\nReview Letters 95, 066601 (2005).[2] S. Murakami, N. Nagaosa, and S.-C. Zhang, Science 301,\n1348 (2003).6\nFIG. 4. The orbital-spin conversion and orbital magnetoresistance in Cu. (a) 2T/3T device based on real materials,\nwhere the scattering region and leads are treated as Cu and the SOC region is treated as Au. (b) The band structure of Cu,\nweighted by the orbital Berry curvature \n Lz, and the corresponding orbital Hall conductivity. (c) Orbital and spin density\ndistribution at the Fermi level in the 2T setup. (d) Spin conductance from lead 1 to lead 3 in nonmagnetic leads in the 3T setup.\n(e) Total conductance from lead 1 to lead 2 in \u0006Mzexchange \feld in the 3T setup. (f) Magnetoresistance ratio \u0001 G12=G12and\n\u0001G13=G13in the 3T setup.\n[3] J. Sinova, D. Culcer, Q. Niu, N. Sinitsyn, T. Jungwirth,\nand A. H. MacDonald, Physical review letters 92, 126603\n(2004).\n[4] Y. K. Kato, R. C. Myers, A. C. Gossard, and D. D.\nAwschalom, science 306, 1910 (2004).\n[5] J. Wunderlich, B. Kaestner, J. Sinova, and T. Jungwirth,\nPhysical review letters 94, 047204 (2005).\n[6] T. Jungwirth, J. Wunderlich, and K. Olejn\u0013 \u0010k, Nature\nmaterials 11, 382 (2012).\n[7] C. L. Kane and E. J. Mele, Physical review letters 95,\n226801 (2005).\n[8] B. A. Bernevig, T. L. Hughes, and S.-C. Zhang, science\n314, 1757 (2006).\n[9] G. Y. Guo, Y. Yao, and Q. Niu, Phys. Rev. Lett. 94,\n226601 (2005).\n[10] H. Kontani, T. Tanaka, D. S. Hirashima, K. Yamada,\nand J. Inoue, Physical Review Letters 100, 096601\n(2008), cond-mat/0702447.\n[11] H. Kontani, T. Tanaka, D. S. Hirashima, K. Yamada,\nand J. Inoue, Physical Review Letters 102, 016601\n(2008), 0806.0210.\n[12] T. Tanaka, H. Kontani, M. Naito, T. Naito, D. S. Hi-\nrashima, K. Yamada, and J. Inoue, Physical Review B\n77(2008), 10.1103/physrevb.77.165117.\n[13] I. V. Tokatly, Physical Review B 82, 161404 (2010),\n1004.0624.\n[14] D. Go, D. Jo, C. Kim, and H.-W. Lee, Physical Review\nLetters 121, 086602 (2018).[15] D. Jo, D. Go, and H.-W. Lee, Phys. Rev. B 98, 214405\n(2018).\n[16] V. T. Phong, Z. Addison, S. Ahn, H. Min, R. Agarwal,\nand E. J. Mele, Phys. Rev. Lett. 123, 236403 (2019).\n[17] L. M. Canonico, T. P. Cysne, A. Molina-Sanchez,\nR. Muniz, and T. G. Rappoport, arXiv preprint\narXiv:2001.03592 (2020).\n[18] S. Bhowal and S. Satpathy, Phys. Rev. B 101, 121112\n(2020).\n[19] D. Go and H.-W. Lee, Physical Review Research 2,\n013177 (2020).\n[20] Y. Liu, J. Xiao, J. Koo, and B. Yan, arXiv:2008.08881\n(2020).\n[21] E. Saitoh, M. Ueda, H. Miyajima, and G. Tatara, Ap-\nplied physics letters 88, 182509 (2006).\n[22] S. O. Valenzuela and M. Tinkham, Nature 442, 176\n(2006).\n[23] H. Zhao, E. J. Loren, H. M. van Driel, and A. L. Smirl,\nPhys. Rev. Lett. 96, 246601 (2006).\n[24] S. Y. Huang, X. Fan, D. Qu, Y. P. Chen, W. G. Wang,\nJ. Wu, T. Y. Chen, J. Q. Xiao, and C. L. Chien, Phys.\nRev. Lett. 109, 107204 (2012).\n[25] M. Weiler, M. Althammer, F. D. Czeschka, H. Huebl,\nM. S. Wagner, M. Opel, I.-M. Imort, G. Reiss,\nA. Thomas, R. Gross, and S. T. B. Goennenwein, Phys.\nRev. Lett. 108, 106602 (2012).\n[26] D. Xiao, M.-C. Chang, and Q. Niu, Rev. Mod. Phys. 82,\n1959 (2010).7\n[27] N. Nagaosa, J. Sinova, S. Onoda, A. H. MacDonald, and\nN. P. Ong, Reviews of Modern Physics 82, 1539 (2010).\n[28] M. B uttiker, Phys. Rev. Lett. 57, 1761 (1986).\n[29] C. W. Groth, M. Wimmer, A. R. Akhmerov, and\nX. Waintal, New Journal of Physics 16, 063065 (2014).[30] M. B uttiker, IBM Journal of Research and Development\n32, 317 (1988).\n[31] M. B uttiker, Physical Review B 33, 3020 (1986).\n[32] D. A. Papaconstantopoulos et al. ,Handbook of the band\nstructure of elemental solids (Springer, 1986).\n[33] C. O. Avci, K. Garello, A. Ghosh, M. Gabureac, S. F.\nAlvarado, and P. Gambardella, Nature Physics 11, 570\n(2015), 1502.06898."    },    {        "title": "2206.00784v2.Substrate_Effects_on_Spin_Relaxation_in_Two_Dimensional_Dirac_Materials_with_Strong_Spin_Orbit_Coupling.pdf",        "content": "Substrate E\u000bects on Spin Relaxation in Two-Dimensional Dirac Materials with Strong\nSpin-Orbit Coupling\nJunqing Xu1,\u0003and Yuan Ping1,y\n1Department of Chemistry and Biochemistry, University of California, Santa Cruz, CA 95064, USA\n(Dated: December 6, 2022)\nUnderstanding substrate e\u000bects on spin dynamics and relaxation in two-dimensional (2D) mate-\nrials is of key importance for spintronics and quantum information applications. However, the key\nfactors that determine the substrate e\u000bect on spin relaxation, in particular for materials with strong\nspin-orbit coupling, have not been well understood. Here we performed \frst-principles real-time\ndensity-matrix dynamics simulations with spin-orbit coupling (SOC) and quantum descriptions of\nelectron-phonon and electron-impurity scattering for the spin lifetimes of supported/free-standing\ngermanene, a prototypical strong SOC 2D Dirac material. We show that the e\u000bects of di\u000ber-\nent substrates on spin lifetime ( \u001cs) can surprisingly di\u000ber by two orders of magnitude. We \fnd\nthat substrate e\u000bects on \u001csare closely related to substrate-induced modi\fcations of the SOC-\feld\nanisotropy, which changes the spin-\rip scattering matrix elements. We propose a new electronic\nquantity, named spin-\rip angle \u0012\"#, to characterize spin relaxation caused by intervalley spin-\rip\nscattering. We \fnd that the spin relaxation rate is approximately proportional to the averaged value\nof sin2\u0000\n\u0012\"#=2\u0001\n, which can be used as a guiding parameter of controlling spin relaxation.\nINTRODUCTION\nSince the long spin di\u000busion length ( ls) in large-area\ngraphene was \frst reported by Tombros et al.[1], sig-\nni\fcant advances have been made in the \feld of spin-\ntronics, which has the potential to realize low-power\nelectronics by utilizing spin as the information car-\nrier. Various 2D materials have shown promising spin-\ntronic properties[2], e.g., long lsat room temperatures\nin graphene[3] and ultrathin black phosphorus[4], spin-\nvalley locking (SVL) and ultralong spin lifetime \u001csat\nlow temperatures in transition metal dichalcogenides\n(TMDs)[5] and germanene[6], and persistent spin helix\nin 2D hybrid perovskites[7].\nUnderstanding spin relaxation and transport mecha-\nnism in materials is of key importance for spintronics\nand spin-based quantum information technologies. One\ncritical metric for ideal materials in such applications is\nspin lifetime ( \u001cs), often required to be su\u000eciently long\nfor stable detection and manipulation of spin. In 2D-\nmaterial-based spintronic devices, the materials are usu-\nally supported on a substrate. Therefore, for the design\nof those devices, it is crucial to understand substrate ef-\nfects on spin relaxation. In past work, the substrate ef-\nfects were mostly studied for weak SOC Dirac materials\nlike graphene[8{12]. How substrates a\u000bect strong SOC\nDirac materials like germanene is unknown. In partic-\nular, the spin relaxation mechanism between weak and\nstrong SOC Dirac materials was shown to be drastically\ndi\u000berent. [6] Therefore, careful investigations are required\nto unveil the distinct substrate e\u000bects on these two types\nof materials.\nHere we focus on the dangling-bond-free insulating\n\u0003jxu153@ucsc.edu\nyyuanping@ucsc.edusubstrates, which interact weakly with the material thus\npreserve its main physical properties. Insulating sub-\nstrates can a\u000bect spin dynamics and relaxation in sev-\neral aspects: (i) They may induce strong SOC \felds, so\ncalled internal magnetic \felds Binby breaking inversion\nsymmetry[9] or through proximity e\u000bects[10]. For ex-\nample, the hexagonal boron nitride substrate can induce\nRashba-like \felds on graphene and dramatically accel-\nerate its spin relaxation and enhance the anisotropy of\n\u001csbetween in-plane and out-of-plane directions[8]. (ii)\nSubstrates may introduce additional impurities [11, 12]\nor reduce impurities/defects in material layers, e.g.,\nby encapsulation[13]. In consequence, substrates may\nchange the electron-impurity (e-i) scattering strength,\nwhich a\u000bects spin relaxation through SOC. (iii) Ther-\nmal vibrations of substrate atoms can introduce addi-\ntional spin-phonon scattering by interacting with spins\nof materials[9].\nPreviously most theoretical studies of substrate e\u000bects\non spin relaxation were done based on model Hamil-\ntonian and simpli\fed spin relaxation models[9, 11, 12].\nWhile those models provide rich mechanistic insights,\nthey are lack of predictive power and quantitative ac-\ncuracy, compared to \frst-principles theory. On the other\nhand, most \frst-principles studies only simulated the\nband structures and spin polarizations/textures of the\nheterostructures[14{16], which are not adequate for un-\nderstanding spin relaxation. Recently, with our newly-\ndeveloped \frst-principles density-matrix (FPDM) dy-\nnamics approach, we studied the hBN substrate e\u000bect on\nspin relaxation of graphene, a weak SOC Dirac material.\nWe found a dominant D'yakonov-Perel' (DP) mechanism\nand nontrivial modi\fcation of SOC \felds and electron-\nphonon coupling by substrates[8]. However, strong SOC\nDirac materials can have a di\u000berent spin relaxation mech-\nanism - Elliott-Yafet (EY) mechanism[17], with only\nspin-\rip transition and no spin precession, unlike the DParXiv:2206.00784v2  [cond-mat.mes-hall]  4 Dec 20222\nmechanism. How substrates a\u000bect spin relaxation of ma-\nterials dominated by EY mechanism is the key question\nhere. Furthermore, how such e\u000bects vary among di\u000ber-\nent substrates is another outstanding question for guiding\nexperimental design of interfaces.\nIn our recent study, we have predicted that mono-\nlayer germanene (ML-Ge) is a promising material for\nspin-valleytronic applications, due to its excellent prop-\nerties including spin-valley locking, long \u001csandls, and\nhighly tunable spin properties by varying gates and ex-\nternal \felds[6]. As discussed in Ref. 6, ML-Ge has strong\nintrinsic SOC unlike graphene and silicene. Under an\nout-of-plane electric \feld (in consequence broken inver-\nsion symmetry), a strong out-of-plane internal magnetic\n\feld forms, which may lead to mostly EY spin relax-\nation [6]. Therefore, predicting \u001csof supported ML-Ge\nis important for future applications and our understand-\ning of substrate e\u000bects on strong SOC materials. Here,\nwe examine the substrate e\u000bects on spin relaxation in\nML-Ge through FPDM simulations, with self-consistent\nSOC and quantum descriptions of e-ph and e-i scatter-\ning processes[6, 8, 18{20]. We study free-standing ML-\nGe and ML-Ge supported by four di\u000berent insulating\nsubstrates - germanane (GeH), silicane (SiH), GaTe and\nInSe. The choice of substrates is based on similar lat-\ntice constants to ML-Ge, preservation of Dirac Cones,\nand experimental synthesis accessibility[21, 22]. We will\n\frst show how electronic structures and \u001csof ML-Ge\nare changed by di\u000berent substrates - while \u001csof ML-\nGe on GeH and SiH are similar to free-standing ML-\nGe, the GaTe and InSe substrates strongly reduce \u001csof\nML-Ge due to stronger interlayer interactions. We then\ndiscuss what quantities are responsible for the disparate\nsubstrate e\u000bects on spin relaxation, which eventually an-\nswered the outstanding questions we raised earlier.\nRESULTS AND DISCUSSIONS\nSubstrate e\u000bects on electronic structure and spin\ntexture\nWe begin with comparing band structures and spin\ntextures of free-standing and supported ML-Ge in Fig. 1,\nwhich are essential for understanding spin relaxation\nmechanisms. Since one of the most important e\u000bects of\na substrate is to induce an out-of-plane electric \feld Ez\non the material layer, we also study ML-Ge under a con-\nstantEzas a reference. The choice of the Ezis based on\nreproducing a similar band splitting to the one in ML-Ge\nwith substrates. The band structure of ML-Ge is similar\nto graphene with two Dirac cones at KandK0\u0011\u0000K,\nbut a larger band gap of 23 meV. At Ez= 0, due to\ntime-reversal and inversion symmetries of ML-Ge, every\ntwo bands form a Kramers degenerate pair[17]. A \fnite\nEzor a substrate breaks the inversion symmetry and in-\nduces a strong out-of-plane internal B \feld Bin(Eq. 21),\nwhich splits the Kramers pairs into spin-up and spin-down bands[6]. Interestingly, we \fnd that band struc-\ntures of ML-Ge-SiH (Fig. 1c) and ML-Ge-GeH (Fig. S4)\nare quite similar to free-standing ML-Ge under Ez=-7\nV/nm (ML-Ge@-7V/nm, Fig. 1b), which indicates that\nthe impact of the SiH/GeH substrate on band structure\nandBinmay be similar to a \fnite Ez(see Fig. S4). This\nsimilarity is frequently assumed in model Hamiltonian\nstudies[9, 11]. On the other hand, the band structures of\nML-Ge-InSe (Fig. 1d) and ML-Ge-GaTe (Fig. S4) have\nmore di\u000berences from the free-standing one under Ez,\nwith larger band gaps, smaller band curvatures at Dirac\nCones, and larger electron-hole asymmetry of band split-\ntings. This implies that the impact of the InSe/GaTe\nsubstrates can not be approximated by applying an Ez\nto the free-standing ML-Ge, unlike SiH/GeH substrates.\nWe further examine the spin expectation value vectors\nSexpof substrate-supported ML-Ge. Sexpis parallel to\nBinby de\fnition (Eq. 21). Sexp\u0011\u0000\nSexp\nx;Sexp\ny;Sexp\nz\u0001\nwithSexp\nibeing spin expectation value along direction i\nand is the diagonal element of spin matrix siin Bloch\nbasis. Importantly, from Fig. 1e and 1f, although Sexpof\nML-Ge on substrates are highly polarized along z(out-of-\nplane) direction, the in-plane components of Sexpof ML-\nGe-InSe (and ML-Ge-GaTe) are much more pronounced\nthan ML-Ge-SiH (and ML-Ge-GeH). Such di\u000berences are\ncrucial to the out-of-plane spin relaxation as discussed in\na later subsection.\nSpin lifetimes of germanene on substrates and spin\nrelaxation mechanism\nWe then perform our \frst-principles density-matrix\ncalculation [6, 18{20] at proposed interfaces, and examine\nthe role of electron-phonon coupling in spin relaxation of\nML-Ge at di\u000berent substrates. Throughout this paper,\nwe focus on out-of-plane \u001csof ML-Ge systems, since their\nin-plane\u001csis too short and less interesting. We com-\npare out-of-plane \u001csdue to e-ph scattering between the\nfree-standing ML-Ge (with/without an electric \feld) and\nML-Ge on di\u000berent substrates in Fig. 2a. Here we show\nelectron\u001csfor most ML-Ge/substrate systems as intrin-\nsic semiconductors, except hole \u001csfor the ML-Ge-InSe\ninterface. This choice is because electron \u001csare mostly\nlonger than hole \u001csat lowTexcept for the one at the\nML-Ge-InSe interface; longer lifetime is often more ad-\nvantageous for spintronics applications. From Fig. 2, we\n\fnd that\u001csof ML-Ge under Ez= 0 and -7 V/nm are\nat the same order of magnitude for a wide range of tem-\nperatures. The di\u000berences are only considerable at low\nT, e.g, by 3-4 times at 20 K. On the other hand, \u001csof\nsupported ML-Ge are very sensitive to the speci\fc sub-\nstrates. While \u001csof ML-Ge-GeH and ML-Ge-SiH have\nthe same order of magnitude as the free-standing ML-\nGe, in particular very close between ML-Ge-GeH and\nML-Ge@-7 V/nm, \u001csof ML-Ge-GaTe and ML-Ge-InSe\nare shorter by at least 1-2 orders of magnitude in the\nwhole temperature range. This separates the substrates3\nFIG. 1. Band structures and spin textures around the Dirac cones of ML-Ge systems with and without substrates. (a)-(d) show\nband structures of ML-Ge under Ez= 0 and under -7 V/nm and ML-Ge on silicane (SiH) and on InSe substrates respectively.\n(e) and (f) show spin textures in the kx-kyplane and 3D plots of the spin vectors Sexp\nk1on the circlej\u0000 !kj= 0:005 bohr\u00001of\nthe band at the band edge around Kof ML-Ge on SiH and InSe substrates respectively. Sexp\u0011\u0000\nSexp\nx;Sexp\ny;Sexp\nz\u0001\nwithSexp\ni\nbeing spin expectation value along direction iand is the diagonal element of spin matrix siin Bloch basis. The red and blue\nbands correspond to spin-up and spin-down states. Due to time-reversal symmetry, band structures around another Dirac cone\natK0=\u0000Kare the same except that the spin-up and spin-down bands are reversed. The grey, white, blue, pink and green\nballs correspond to Ge, H, Si, In and Se atoms, respectively. Band structures of ML-Ge on germanane (GeH) and GaTe are\nshown in Fig. S4 in the Supporting Information, and are similar to those of ML-Ge on SiH and InSe substrates, respectively.\nIn subplots (e) and (f), the color scales Sexp\nzand the arrow length scales the vector length of in-plane spin expectation value.\ninto two categories, i.e. with a weak e\u000bect (ML-Ge-GeH\nand ML-Ge-SiH) and a strong e\u000bect (ML-Ge-GaTe and\nML-Ge-InSe).\nWe further investigate the role of electron-impurity (e-\ni) scattering in spin relaxation under di\u000berent substrates,\nby introducing defects in the material layer. We consider\na common type of impurity - single neutral Ge vacancy,\nwhose formation energy was found relatively low in previ-\nous theoretical studies[23, 24]. From Fig. 2b, we can see\nthat\u001csof all \fve systems decrease with impurity density\nni. Since carrier scattering rates \u001c\u00001\np(carrier lifetime \u001cp)\nincreases (decrease) with ni, we then obtain \u001csdecreases\nwith\u001cp's decrease, an evidence of EY spin relaxation\nmechanism. Moreover, we \fnd that \u001csis sensitive to the\ntype of the substrate with all values of ni, and for each of\nfour substrates, \u001csis reduced by a similar amount with\ndi\u000berentni, from low density limit (109cm\u00002, where e-\nph scattering dominates) to relatively high density (1012\ncm\u00002, where e-i scattering becomes more important).\nSince the bands near the Fermi energy are composed\nof the Dirac cone electrons around KandK0valleys in\nML-Ge, spin relaxation process arises from intervalleyand intravalley e-ph scatterings. We then examine rel-\native intervalley spin relaxation contribution \u0011(see its\nde\fnition in the Fig. 2 caption) in Fig. 2c. \u0011being close\nto 1 or 0 corresponds to intervalley or intravalley scatter-\ning being dominant in spin relaxation. \u0011becomes close\nto 1 below 70 K for electrons of ML-Ge-SiH, and below\n120 K for holes of ML-Ge-InSe. This indicates that at\nlowTonly intervalley scattering processes are relevant\nto spin relaxation in ML-Ge on substrates. This is a re-\nsult of spin-valley locking (SVL), i.e. large SOC-induced\nband splittings lock up or down spin with a particular K\nor K' valley [6]. According to Fig. 1 and 2c, the SVL\ntransition temperature ( TSVL; below which the propor-\ntion of intervalley spin relaxation rate \u0011is close to 1)\nseems approximately proportional to SOC splitting en-\nergy \u0001SOC, e.g. for electrons (CBM) of ML-Ge-GaTe and\nML-Ge-SiH, and for holes (VBM) of ML-Ge-InSe, \u0001SOC\nare\u001815,\u001824 and 40 meV respectively, while TSVLare\n50, 70 and 120 K respectively. As \u0001SOCcan be tuned by\nEzand the substrate, TSVLcan be tuned simultaneously.\nUnder SVL condition, spin or valley lifetime tends to be\nexceptionally long, which is ideal for spin-/valley-tronic4\nFIG. 2. The out-of-plane spin lifetime \u001csof intrinsic free-standing and substrate-supported ML-Ge. (a) \u001csof ML-Ge under\nEz= 0, -7 V/nm and substrate-supported ML-Ge as a function of temperature without impurities. Here we show electron \u001cs\nfor intrinsic ML-Ge systems except that hole \u001csis shown for ML-Ge-InSe, since electron \u001csare longer than hole \u001csat lowT\nexcept ML-Ge-InSe. (b) \u001csas a function of impurity density niat 50 K. The impurities are neutral ML-Ge vacancy with 50% at\nhigher positions and 50% at lower ones of a Ge layer. The dashed vertical line corresponds to the impurity density where e-ph\nand e-i scatterings contribute equally to spin relaxation ( ni;s). And e-ph (e-i) scattering is more dominant if ni<(>)ni;s. (c)\nThe proportion of intervalley spin relaxation contribution \u0011of (electrons of) ML-Ge-SiH and (holes of) ML-Ge-InSe without\nimpurities. \u0011is de\fned as \u0011=(\u001cinter\ns;z)\u00001\n(\u001cinters;z)\u00001+w(\u001cintras;z)\u00001, where\u001cinter\ns;z and\u001cintra\ns;z are intervalley and intravalley spin lifetimes,\ncorresponding to scattering processes between KandK0valleys and within a single KorK0valley, respectively. \u0011being close\nto 1 or 0 corresponds to dominant intervalley or intravalley spin relaxation, respectively. wis a weight factor related to what\npercentage of total Szcan be relaxed out by intravalley scattering itself. wbeing close to 0 and 1 correspond to the cases that\nintravalley scattering can only relax a small part (0) and most of excess spin (1) respectively. In Supporting Information Sec.\nSII, we give more details about de\fnition of w. (d) Electron and hole \u001csat 20 K of ML-Ge without impurities on hydrogen-\nterminated multilayer Si, labeled as Si nH withnbeing number of Si layers. Si nH is silicane if n= 1, and hydrogen-terminated\nSilicon (111) surface if n=1.\napplications.\nAdditionally, the studied substrates here are mono-\nlayer, while practically multilayers or bulk are more com-\nmon, thus it is necessary to understand how \u001cschanges\nwith the number of substrate layers. In Fig. 2d, we\nshow\u001csat 20 K of ML-Ge on hydrogen-terminated mul-tilayer Si, ML-Ge-Si nH, withnbeing number of Si layer.\nSinH becomes hydrogen-terminated Silicon (111) surface\nifn=1. We \fnd that \u001csare changed by only 30%-40%\nby increasing nfrom 1 to 3 and kept unchanged after\nn\u00153. For generality of our conclusion, we also test\nthe layer dependence of a di\u000berent substrate. We found5\nthe\u001csof ML-Ge on bilayer InSe ( n= 2) is changed by\n\u00188% compared to monolayer InSe at 20 K, even smaller\nchange than the one at Si nH substrates. Given the dis-\nparate properties of these two substrates, we conclude\nusing a monolayer is a reasonable choice for simulating\nthe substrate e\u000bects on \u001csin this work.\nThe correlation of electronic structure and phonon\nproperties to spin relaxation at di\u000berent substrates\nWe next analyze in detail the relevant physical quan-\ntities, and determine the key factors responsible for sub-\nstrate e\u000bects on spin relaxation. We focus on results\nunder lowTas spin relaxation properties are superior at\nlowerT(the realization of SVL and longer \u001cs).\nFirst, to have a qualitative understanding of the\nmaterial-substrate interaction strength, we show charge\ndensity distribution at the cross-section of interfaces in\nFig. 3a-d. It seems that four substrates can be catego-\nrized into two groups: group A contains GeH and SiH\nwith lower charge density distribution in the bonding re-\ngions (pointed by the arrows); group B contains GaTe\nand InSe with higher charge density distribution in the\nbonding regions. In Fig. S5, we investigate the charge\ndensity change \u0001 \u001ae(de\fned by the charge density dif-\nference between interfaces and individual components).\nConsistent with Fig. 3, we \fnd that \u0001 \u001aefor GaTe and\nInSe substrates overall has larger magnitude than the\none for GeH and SiH substrates. Therefore the material-\nsubstrate interactions of group B seem stronger than\nthose of group A. Intuitively, we may expect that the\nstronger the interaction, the stronger the substrate e\u000bect\nis. The FPDM simulations in Fig. 2a-b indeed show that\nthe substrate e\u000bects of group B being stronger than those\nof group A on \u001cs, consistent with the above intuition.\nNext we examine electronic quantities closely related\nto spin-\rip scattering responsible to EY spin relaxation.\nQualitatively, for a state k1, its spin-\rip scattering rate\n\u001c\u00001\ns(k1) is proportional to the number of its pair states\nk2allowing spin-\rip transitions between them. The num-\nber of pair states is approximately proportional to den-\nsity of states (DOS) around the energy of k1. Moreover,\nfor EY mechanism, it is commonly assumed that spin\nrelaxation rate is proportional to the degree of mixture\nof spin-up and spin-down states (along the zdirection\nhere), so called \\spin-mixing\" parameter[17] b2\nz(see its\nde\fnition in Sec. SII), i.e., \u001c\u00001\ns/\nb2\nz\u000b\n, where\nb2\nz\u000b\nis the\nstatistically averaged spin mixing parameter as de\fned in\nRef. 6. Therefore, we show DOS, energy-resolved spin-\nmixingb2\nz(\") and\nb2\nz\u000b\nas a function of temperature in\nFig. 3e-g.\nWe \fnd that in Fig. 3e DOS of ML-Ge-GeH and ML-\nGe-SiH are quite close to that of ML-Ge@-7V/nm, while\nDOS of ML-Ge-GaTe and ML-Ge-InSe are 50%-100%\nhigher around the band edge. Such DOS di\u000berences\nare qualitatively explained by the staggered potentials of\nML-Ge-GaTe and ML-Ge-InSe being greater than thoseof ML-Ge-GeH and ML-Ge-SiH according to the model\nHamiltonian proposed in Ref. 25. In Fig. 3f-g, b2\nzof ML-\nGe-GeH and ML-Ge-SiH are found similar to ML-Ge@-\n7 V/nm, and not sensitive to energy and temperature.\nOn the contrast, for ML-Ge-GaTe and ML-Ge-InSe, their\nb2\nz(\") and\nb2\nz\u000b\nincrease rapidly with energy and temper-\nature. Speci\fcally, we can see at 300 K,\nb2\nz\u000b\nof ML-Ge-\nGaTe and ML-Ge-InSe are about 4-20 times of the one\nof ML-Ge-GeH and ML-Ge-SiH in Fig. 3g. Thus the one\norder of magnitude di\u000berence of \u001csbetween group A (ML-\nGe-GeH and ML-Ge-SiH) and group B (ML-Ge-GaTe\nand ML-Ge-InSe) substrates at 300 K can be largely ex-\nplained by the substrate-induced changes of DOS and\nb2\nz\u000b\n. On the other hand, at low T, e.g., at 50 K,\nb2\nz\u000b\nof ML-Ge-GaTe and ML-Ge-InSe are only about 1.5 and\n2.5 times of the ones of ML-Ge-GeH and ML-Ge-SiH,\nand DOS are only tens of percent higher. However, there\nis still 1-2 order of magnitude di\u000berence of \u001csbetween\ndi\u000berent substrates. Therefore, the substrate e\u000bects on\n\u001cscan not be fully explained by the changes of\nb2\nz\u000b\nand\nDOS, in particular at relatively low temperature.\nWe then examine if substrate-induced modi\fcations of\nphonon can explain the changes of spin relaxation at dif-\nferent substrates, especially at low T. We emphasize that\nat lowT, since spin relaxation is fully determined by\nintervalley processes (Fig. 2c), the related phonons are\nmostly close to wavevector K. From Fig. 4, we \fnd that\nthe most important phonon mode for spin relaxation at\nlowThas several similar features: (i) It contributes to\nmore than 60% of spin relaxation (see Fig 4a). (ii) Its\nenergy is around 7 meV in the table of Fig. 4a. (iii)\nIts vibration is \rexural-like, i.e., atoms mostly vibrate\nalong the out-of-plane direction as shown in Fig. 4b-\nd. Moreover, for this mode, the substrate atoms have\nnegligible thermal vibration amplitude compared to the\none of the materials atoms. This is also con\frmed in\nthe layer-projected phonon dispersion of ML-Ge-InSe in\nFig. 4e. The purple box highlights the critical phonon\nmode around K, with most contribution from the mate-\nrial layer. (iv) The critical phonon mode does not couple\nwith the substrate strongly, since its vibration frequency\ndoes not change much when substrate atoms are \fxed\n(by comparing Fig. 4e with f). We thus conclude that\nthe substrate-induced modi\fcations of phonons and ther-\nmal vibrations of substrate atoms seem not important for\nspin relaxation at low T(e.g. below 20 K).\nTherefore, neither the simple electronic quantities\nb2\u000b\nand DOS nor the phonon properties can explain the sub-\nstrate e\u000bects on spin relaxation at low T.\nThe determining factors of spin relaxation derived\nfrom spin-\rip matrix elements\nOn the other hand, with a simpli\fed picture of spin-\n\rip transition by the Fermi's Golden Rule, the scattering\nrate is proportional to the modulus square of the scat-\ntering matrix elements. For a further mechanistic un-6\nFIG. 3. Charge density, density of states (DOS), and spin mixing parameters of free-standing and substrate-supported ML-Ge.\nCross-section views of charge density at interfaces of ML-Ge on (a) GeH, (b) SiH, (c) GaTe, and (d) InSe. The Ge layers\nare above the substrate layers. The unit of charge density is e=bohr3. Charge densities in the regions pointed out by black\narrows show signi\fcant di\u000berences among di\u000berent systems. (e) DOS and (f) energy-solved spin-mixing parameter along zaxis\nb2\nz(\") of ML-Ge under Ez=-7 V/nm and on di\u000berent substrates. \"edgeis the band edge energy at the valence band maximum\nor conduction band minimum. The step or sudden jump in the DOS curve corresponds to the edge energy of the second\nconduction/valence band or the SOC-induced splitting energy at K. (g) The temperature-dependent e\u000bective spin-mixing\nparameter\nb2\nz\u000b\nof various ML-Ge systems.\nderstanding, we turn to examine the modulus square of\nthe spin-\rip matrix elements, and compare their qual-\nitative trend with our FPDM simulations. Note that\nmost matrix elements are irrelevant to spin relaxation\nand we need to pick the \\more relevant\" ones, by de\fning\na statistically-averaged function. Therefore, we propose\nan e\u000bective band-edge-averaged spin-\rip matrix element\njeg\"#j2(Eq. 8). Here the spin-\rip matrix element can be\nfor general scattering processes; in the following we focus\non e-ph process for simplicity. We also propose a so-called\nscattering density of states DSin Eq. 9, which measures\nthe density of spin-\rip transitions and can be roughly re-\ngarded as a weighted-averaged value of the usual DOS.\nBased on the generalized Fermi's golden rule, we approx-\nimately have \u001c\u00001\ns/jeg\"#j2DSfor EY spin relaxation (see\nthe discussions above Eq. 11 in \\Methods\" section).\nAs shown in Fig. 5a, \u001c\u00001\nsis almost linearly propor-\ntional tojeg\"#j2DSat 20 K. As the variation of DSamong\nML-Ge on di\u000berent substrates is at most three times (see\nFig. 3e and Fig. S6), which is much weaker than the\nlarge variation of \u001c\u00001\ns, this indicates that the substrate-\ninduced change of \u001csis mostly due to the substrate-\ninduced change of spin-\rip matrix elements. Although\njeg\"#j2was often considered approximately proportionalto\nb2\u000b\n, resulting in \u001c\u00001\ns/\nb2\u000b\n, our results in Fig. 3\nin the earlier section indicate that such simple approx-\nimation is not applicable here, especially inadequate of\nexplaining substrate dependence of \u001csat lowT.\nTo \fnd out the reason why jeg\"#j2for di\u000berent sub-\nstrates are so di\u000berent, we \frst examine the averaged\nspin-\rip wavefunction overlap jo\"#j2(with the reciprocal\nlattice vector G= 0), closely related to jeg\"#j2(Eq. 18\nand Eq. 17). From Fig. 5b, \u001c\u00001\nsandjo\"#j2have the same\ntrend, which implies jeg\"#j2andjo\"#j2may have the same\ntrend. However, in general, the G6=0elements ofjo\"#j2\nmay be important as well, which can not be unambigu-\nously evaluated here. (See detailed discussions in the\nsubsection \\Spin-\rip e-ph and overlap matrix element\"\nin the \\Methods\" section).\nTo have deeper intuitive understanding, we then pro-\npose an important electronic quantity for intervalley\nspin-\rip scattering - the spin-\rip angle \u0012\"#between two\nelectronic states. For two states ( k1;n1) and (k2;n2) with\nopposite spin directions, \u0012\"#is the angle between \u0000Sexp\nk1n1\nandSexp\nk2n2or equivalently the angle between \u0000Bin\nk1and\nBin\nk2.\nThe motivation of examining \u0012\"#is that: Suppose two\nwavevectors k1andk2=\u0000k1are in two opposite valleys7\n(a)\nSubstrate !K(meV) Contribution\nGe@-7V/nm 7.7 78%\nGe-GeH 6.9 70%\nGe-SiH 7.1 64%\nGe-GaTe 6.4 90%\nGe-InSe 7.2 99%\nFIG. 4. (a) The phonon energy at wavevector Kof the mode\nthat contributes the most to spin relaxation, and the per-\ncentage of its contribution for various systems at 20 K. We\nconsider momentum transfer K, as spin relaxation is fully de-\ntermined by intervalley processes between KandK0valleys.\n(b), (c) and (d) Typical vibrations of atoms in 3 \u00023 supercells\nof (b) ML-Ge@-7 V/nm, (c) ML-Ge-SiH, and (d) ML-Ge-\nInSe of the most important phonon mode at Karound 7 meV\n(shown in (a)). The red arrows represent displacement. The\natomic displacements smaller than 10% of the strongest are\nnot shown. (e) The layer-projected phonon dispersion of ML-\nGe-InSe within 12 meV. The red and blue colors correspond\nto the phonon displacements mostly contributed from the ma-\nterial (red) and substrate layer (blue) respectively. The green\ncolor means the contribution to the phonon displacements\nfrom the material and substrate layers are similar. The pur-\nple boxes highlight the two most important phonon modes\naroundKfor spin relaxation.(f) Phonon dispersion of ML-\nGe-InSe within 12 meV with substrate atoms (InSe) being\n\fxed at equilibrium structure and only Ge atoms are allowed\nto vibrate.\nQand -Qrespectively and there is a pair of bands, which\nare originally Kramers degenerate but splitted by Bin.\nDue to time-reversal symmetry, we have Bin\nk1=\u0000Bin\nk2,\nwhich means the two states at the same band natk1\nandk2have opposite spins and \u0012\"#between them is\nzero. Therefore, the matrix element of operator bAbe-\ntween states ( k1;n) and (k2;n) -Ak1n;k2nis a spin-\ripone and we name it as A\"#\nk1k2. According to Ref. 26,\nwith time-reversal symmetry, A\"#\nk1k2is exactly zero. In\ngeneral, for another wavevector k3within valley - Qbut\nnot\u0000k1,A\"#\nk1k3is usually non-zero. One critical quan-\ntity that determines the intervalley spin-\rip matrix ele-\nmentA\"#\nk1k3for a band within the pair introduced above\nis\u0012\"#\nk1k3. Based on time-independent perturbation theory,\nwe can prove that\f\fA\"#\f\fbetween two states is approxi-\nmately proportional to\f\fsin\u0000\n\u0012\"#=2\u0001\f\f. The derivation is\ngiven in subsection \\Spin-\rip angle \u0012\"#for intervalley\nspin relaxation\" in \\Methods\" section.\nAs shown in Fig. 5c, \u001c\u00001\nsof ML-Ge on di\u000berent\nsubstrates at 20 K is almost linearly proportional to\nsin2(\u0012\"#=2)DS, where sin2(\u0012\"#=2) is the statistically-\naveraged modulus square of sin\u0000\n\u0012\"#=2\u0001\n. This indicates\nthat the relation jeg\"#j2/sin2(\u0012\"#=2) is nearly perfectly\nsatis\fed at low T, where intervalley processes dominate\nspin relaxation. We additionally show the relations be-\ntween\u001c\u00001\nsandjeg\"#j2DS,jo\"#j2DSandsin2(\u0012\"#=2)DSat\n300 K in Fig. S7. Here the trend of \u001c\u00001\nsis still approxi-\nmately captured by the trends of jeg\"#j2DS,jo\"#j2DSand\nsin2(\u0012\"#=2)DS, although not perfectly linear as at low T.\nSince\u0012\"#is de\fned by Sexpat di\u000berent states, \u001csis\nhighly correlated with Sexpand more speci\fcally with\nthe anisotropy of Sexp(equivalent to the anisotropy of\nBin). Qualitatively, the larger anisotropy of Sexpleads to\nsmaller\u0012\"#and longer\u001csalong the high-spin-polarization\ndirection. This \fnding may be applicable to spin re-\nlaxation in other materials whenever intervalley spin-\rip\nscattering dominates or spin-valley locking exists, e.g., in\nTMDs[5], Stanene[27], 2D hybrid perovskites with persis-\ntent spin helix[7], etc.\nAt the end, we brie\ry discuss the substrate e\u000bects\non in-plane spin relaxation ( \u001cs;x), whereas only out-of-\nplane spin relaxation was discussed earlier. From Table\nSI, we \fnd that \u001cs;xof ML-Ge@-7V/nm and supported\nML-Ge are signi\fcantly (e.g., two orders of magnitude)\nshorter than free-standing ML-Ge, but the di\u000berences\nbetween\u001cs;xof ML-Ge on di\u000berent substrates are rela-\ntively small (within 50%). This is because: With a non-\nzeroEzor a substrate, the inversion symmetry broken\ninduces strong out-of-plane internal magnetic \feld Bin\nz\n(>100 Tesla), so that the excited in-plane spins will pre-\ncess rapidly about Bin\nz. The spin precession signi\fcantly\na\u000bects spin decay and the main spin decay mechanism\nbecomes DP or free induction decay mechanism[28] in-\nstead of EY mechanism. For both DP and free induc-\ntion decay mechanisms[20, 28], \u001cs;xdecreases with the\n\ructuation amplitude (among di\u000berent k-points) of the\nBincomponents perpendicular to the xdirection. As\nthe \ructuation amplitude of Bin\nzof ML-Ge@-7V/nm and\nsupported ML-Ge is large (Table SI; much greater than\nthe one ofBin\ny), their\u001cs;xcan be much shorter than the\nvalue of ML-Ge at zero electric \feld when EY mechanism\ndominates. Moreover, since the \ructuation amplitude of\nBin\nzof ML-Ge on di\u000berent substrates has the same or-8\nFIG. 5. The relation between \u001c\u00001\nsand the averaged modulus square of spin-\rip e-ph matrix elements jeg\"#j2, of spin-\rip overlap\nmatrix elementsjo\"#j2andsin2(\u0012\"#=2) multiplied by the scattering density of states DSat 20 K. See the de\fnition of jeg\"#j2,\njo\"#j2andDSin Eq. 8, 19 and 9 respectively. \u0012\"#is the spin-\rip angle between two electronic states. For two states ( k;n) and\n(k0;n0) with opposite spin directions, \u0012\"#is the angle between \u0000Sexp\nknandSexp\nk0n0.sin2(\u0012\"#=2) is de\fned in Eq. 24. The variation\nofDSamong di\u000berent substrates is at most three times, much weaker than the variations of \u001c\u00001\nsand other quantities shown\nhere.\nder of magnitude (Table SI), \u001cs;xof ML-Ge on di\u000berent\nsubstrates are similar.\nCONCLUSIONS\nIn this paper, we systematically investigate how spin\nrelaxation of strong SOC Dirac materials is a\u000bected by\ndi\u000berent insulating substrates, using germanene as a pro-\ntotypical example. Through FPDM simulations of \u001csof\nfree-standing and substrate supported ML-Ge, we show\nthat substrate e\u000bects on \u001cscan di\u000ber orders of magni-\ntude among di\u000berent substrates. Speci\fcally, \u001csof ML-\nGe-GeH and ML-Ge-SiH have the same order of mag-\nnitude as free-standing ML-Ge, but \u001csof ML-Ge-GaTe\nand ML-Ge-InSe are signi\fcantly shortened by 1-2 orders\nwith temperature increasing from 20 K to 300 K.\nAlthough simple electronic quantities including charge\ndensities, DOS and spin mixing\nb2\nz\u000b\nqualitatively ex-\nplain the much shorter lifetime of ML-Ge-GaTe/InSe\ncompared to ML-Ge-GeH/SiH in the relatively high T\nrange, we \fnd they cannot explain the large variations\nof\u001csamong substrates at low T(i.e. tens of K). We\npoint out that spin relaxation in ML-Ge and its inter-\nfaces at low Tis dominated by intervalley scattering pro-\ncesses. However, the substrate-induced modi\fcations of\nphonons and thermal vibrations of substrates seem to be\nnot important. Instead, the substrate-induced changes\nof the anisotropy of Sexpor the spin-\rip angles \u0012\"#which\nchanges the spin-\rip matrix elements, are much more cru-\ncial.\u0012\"#is at the \frst time proposed in this article to the\nbest of our knowledge, and is found to be a useful elec-\ntronic quantity for predicting trends of spin relaxation\nwhen intervalley spin-\rip scattering dominates.Our theoretical study showcases the systematic inves-\ntigations of the critical factors determining the spin re-\nlaxation in 2D Dirac materials. More importantly we\npointed out the sharp distinction of substrate e\u000bects on\nstrong SOC materials to the e\u000bects on weak SOC ones,\nproviding valuable insights and guidelines for optimizing\nspin relaxation in materials synthesis and control.\nMETHODS\nFirst-Principles Density-Matrix Dynamics for Spin\nRelaxation\nWe solve the quantum master equation of density ma-\ntrix\u001a(t) as the following:[19]\nd\u001a12(t)\ndt= [He;\u001a(t)]12+\n0\nBBB@1\n2P\n3458\n<\n:[I\u0000\u001a(t)]13P32;45\u001a45(t)\n\u0000[I\u0000\u001a(t)]45P\u0003\n45;13\u001a32(t)9\n=\n;\n+H:C:1\nCCCA;\n(1)\nEq. 1 is expressed in the Schr odinger picture, where the\n\frst and second terms on the right side of the equa-\ntion relate to the coherent dynamics, which can lead\nto Larmor precession, and scattering processes respec-\ntively. The \frst term is unimportant for out-of-plane\nspin relaxation in ML-Ge systems, since Larmor preces-\nsion is highly suppressed for the excited spins along the\nout-of-plane or zdirection due to high spin polarization\nalongzdirection. The scattering processes induce spin9\nrelaxation via the SOC. Heis the electronic Hamiltonian.\n[H;\u001a] =H\u001a\u0000\u001aH. H.C. is Hermitian conjugate. The\nsubindex, e.g., \\1\" is the combined index of k-point and\nband.P=Pe\u0000ph+Pe\u0000iis the generalized scattering-\nrate matrix considering e-ph and e-i scattering processes.\nFor the e-ph scattering[19],\nPe\u0000ph\n1234 =X\nq\u0015\u0006Aq\u0015\u0006\n13Aq\u0015\u0006;\u0003\n24; (2)\nAq\u0015\u0006\n13=r\n2\u0019\n~gq\u0015\u0006\n12q\n\u000eG\u001b(\u000f1\u0000\u000f2\u0006!q\u0015)q\nn\u0006\nq\u0015;(3)\nwhereqand\u0015are phonon wavevector and mode, gq\u0015\u0006\nis the e-ph matrix element, resulting from the absorp-\ntion (\u0000) or emission (+) of a phonon, computed with\nself-consistent SOC from \frst-principles,[29] n\u0006\nq\u0015=nq\u0015+\n0:5\u00060:5 in terms of phonon Bose factors nq\u0015, and\u000eG\n\u001b\nrepresents an energy conserving \u000e-function broadened to\na Gaussian of width \u001b.\nFor electron-impurity scattering[19],\nPe\u0000i\n1234=Ai\n13Ai;\u0003\n24; (4)\nAi\n13=r\n2\u0019\n~gi\n13q\n\u000eG\u001b(\u000f1\u0000\u000f3)p\nniVcell; (5)\nwhereniandVcellare impurity density and unit cell vol-\nume, respectively. giis the e-i matrix element computed\nby the supercell method and is discussed in the next sub-\nsection.\nStarting from an initial density matrix \u001a(t0) prepared\nwith a net spin, we evolve \u001a(t) through Eq. 1 for a long\nenough time, typically from hundreds of ps to a few \u0016s.\nWe then obtain spin observable S(t) from\u001a(t) (Eq. S1)\nand extract spin lifetime \u001csfromS(t) using Eq. S2.\nComputational details\nThe ground-state electronic structure, phonons, as well\nas electron-phonon and electron-impurity (e-i) matrix\nelements are \frstly calculated using density functional\ntheory (DFT) with relatively coarse kandqmeshes in\nthe DFT plane-wave code JDFTx[30]. Since all sub-\nstrates have hexagonal structures and their lattice con-\nstants are close to germanene's, the heterostructures\nare built simply from unit cells of two systems. The\nlattice mismatch values are within 1% for GeH, GaTe\nand InSe substrates but about 3.5% for the SiH sub-\nstrate. All heterostructures use the lattice constant 4.025\n\u0017A of free-standing ML-Ge relaxed with Perdew-Burke-\nErnzerhof exchange-correlation functional[31]. The in-\nternal geometries are fully relaxed using the DFT+D3\nmethod for van der Waals dispersion corrections[32].\nWe use Optimized Norm-Conserving Vanderbilt (ONCV)\npseudopotentials[33] with self-consistent spin-orbit cou-\npling throughout, which we \fnd converged at a ki-\nnetic energy cuto\u000b of 44, 64, 64, 72 and 66 Ry forfree-standing ML-Ge, ML-Ge-GeH, ML-Ge-SiH, ML-Ge-\nGaTe and ML-Ge-InSe respectively. The DFT calcula-\ntions use 24\u000224kmeshes. The phonon calculations em-\nploy 3\u00023 supercells through \fnite di\u000berence calculations.\nWe have checked the supercell size convergence and found\nthat using 6\u00026 supercells lead to very similar results of\nphonon dispersions and spin lifetimes. For all systems,\nthe Coulomb truncation technique[34] is employed to ac-\ncelerate convergence with vacuum sizes. The vacuum\nsizes are 20 bohr (additional to the thickness of the het-\nerostructures) for all heterostructures and are found large\nenough to converge the \fnal results of spin lifetimes. The\nelectric \feld along the non-periodic direction is applied\nas a ramp potential.\nFor the e-i scattering, we assume impurity density is\nsu\u000eciently low and the average distance between neigh-\nboring impurities is su\u000eciently long so that the interac-\ntions between impurities are negligible, i.e. at the dilute\nlimit. The e-i matrix gibetween state ( k;n) and (k0;n0)\nisgi\nkn;k0n0=hknjVi\u0000V0jk0n0i, whereViis the poten-\ntial of the impurity system and V0is the potential of the\npristine system. Viis computed with SOC using a large\nsupercell including a neutral impurity that simulates the\ndilute limit where impurity and its periodic replica do\nnot interact. To speed up the supercell convergence, we\nused the potential alignment method developed in Ref.\n35. We use 5\u00025 supercells, which have shown reasonable\nconvergence (a few percent error of the spin lifetime).\nWe then transform all quantities from plane wave basis\nto maximally localized Wannier function basis[36], and\ninterpolate them[29, 37{41] to substantially \fner k and\nq meshes. The \fne kandqmeshes are 384\u0002384 and\n576\u0002576 for simulations at 300 K and 100 K respectively\nand are \fner at lower temperature, e.g., 1440 \u00021440 and\n2400\u00022400 for simulations at 50 K and 20 K respectively.\nThe real-time dynamics simulations are done with our\nown developed DMD code interfaced to JDFTx. The\nenergy-conservation smearing parameter \u001bis chosen to\nbe comparable or smaller than kBTfor each calculation,\ne.g., 10 meV, 5 meV, 3.3 meV and 1.3 meV at 300 K,\n100 K, 50 K and 20 K respectively.\nAnalysis of Elliot-Yafet spin lifetime\nIn order to analyze the results from real-time \frst-\nprinciples density-matrix dynamics (FPDM), we com-\npare them with simpli\fed mechanistic models as dis-\ncussed below. According to Ref. [18], if a solid-state\nsystem is close to equilibrium (but not at equilibrium)\nand its spin relaxation is dominated by EY mechanism,\nits spin lifetime \u001csdue to the e-ph scattering satis\fes (for\nsimplicity the band indices are dropped)10\n\u001c\u00001\ns/N\u00002\nk\n\u001fX\nkq\u00158\n<\n:jg\"#;q\u0015\nk;k\u0000qj2nq\u0015fk\u0000q(1\u0000fk)\n\u000e(\u000fk\u0000\u000fk\u0000q\u0000!q\u0015)9\n=\n;;(6)\n\u001f=N\u00001\nkX\nkfk(1\u0000fk); (7)\nwherefis Fermi-Dirac function. !q\u0015andnq\u0015are\nphonon energy and occupation of phonon mode \u0015at\nwavevector q.g\"#is the spin-\rip e-ph matrix element\nbetween two electronic states of opposite spins. We will\nfurther discuss g\"#in the next subsection.\nAccording to Eq. 6 and 7, \u001c\u00001\nsis proportional to jg\"#\nqj2\nand also the density of the spin-\rip transitions. Therefore\nwe propose a temperature ( T) and chemical potential\n(\u0016F;c) dependent e\u000bective modulus square of the spin-\n\rip e-ph matrix element jeg\"#j2and a scattering density\nof statesDSas\njeg\"#j2=P\nkqwk;k\u0000qP\n\u0015jg\"#;q\u0015\nk;k\u0000qj2nq\u0015P\nkqwk;k\u0000q; (8)\nDS=N\u00002\nkP\nkqwk;k\u0000q\nN\u00001\nkP\nkfk(1\u0000fk); (9)\nwk;k\u0000q=fk\u0000q(1\u0000fk)\u000e(\u000fk\u0000\u000fk\u0000q\u0000!c); (10)\nwhere!cis the characteristic phonon energy speci\fed\nbelow, and w k;k\u0000qis the weight function. The matrix\nelement modulus square is weighted by nq\u0015according to\nEq. 6 and 7. This rules out high-frequency phonons at\nlowTwhich are not excited. !cis chosen as 7 meV\nat 20 K based on our analysis of phonon-mode-resolved\ncontribution to spin relaxation. w k;k\u0000qselects transitions\nbetween states separated by !cand around the band edge\nor\u0016F;c, which are \\more relevant\" transitions to spin\nrelaxation.\nDScan be regarded as an e\u000bective density of spin-\n\rip e-ph transitions satisfying energy conservation be-\ntween one state and its pairs. When !c= 0, we\nhaveDS=R\nd\u000f\u0010\n\u0000df\nd\u000f\u0011\nD2(\u000f)=R\nd\u000f\u0010\n\u0000df\nd\u000f\u0011\nD(\u000f) with\nD(\u000f) density of electronic states (DOS). So DScan\nbe roughly regarded as a weighted-averaged DOS with\nweight\u0010\n\u0000df\nd\u000f\u0011\nD(\u000f).\nWithjeg\"#j2andDS, we have the approximate relation\nfor spin relaxation rate,\n\u001c\u00001\ns/jeg\"#j2DS: (11)\nSpin-\rip e-ph and overlap matrix element\nIn the mechanistic model of Eq. 6 in the last section,\nthe spin-\rip e-ph matrix element between two electronic\nstates of opposite spins at wavevectors kandk\u0000qof\nphonon mode \u0015reads[29]g\"#;q\u0015\nkk\u0000q=D\nu\"(#)\nk\f\f\f\u0001q\u0015vKS\f\f\fu#(\")\nk\u0000qE\n; (12)\n\u0001q\u0015vKS=s\n~\n2!q\u0015X\n\u0014\u000be\u0014\u000b;q\u0015@\u0014\u000bqvKS\npm\u0014; (13)\n@\u0014\u000bqvKS=X\nleiq\u0001Rl@VKS\n@\u001c\u0014\u000bjr\u0000Rl; (14)\nVKS=V+~\n4m2c2rrV\u0002p\u0001\u001b; (15)\nwhereu\"(#)\nkis the periodic part of the Bloch wavefunc-\ntion of a spin-up (spin-down) state at wavevector k.\u0014is\nthe index of ion in the unit cell. \u000bis the index of a di-\nrection. Rlis a lattice vector. Vis the spin-independent\npart of the potential. pis the momentum operator. \u001bis\nthe Pauli operator.\nFrom Eqs. 12-15, g\"#can be separated into two parts,\ng\"#=gE+gY; (16)\nwheregEandgYcorrespond to the spin-independent\nand spin-dependent parts of VKSrespectively, called El-\nliot and Yafet terms of the spin-\rip scattering matrix\nelements respectively.[28]\nGenerally speaking, both the Elliot and Yafet terms\nare important; for the current systems \u001cswith and with-\nout Yafet term have the same order of magnitude. For\nexample,\u001csof ML-Ge-GeH and ML-Ge-SiH without the\nYafet term are about 100% and 70% of \u001cswith the Yafet\nterm at 20 K. Therefore, for qualitative discussion of \u001cs\nof ML-Ge on di\u000berent substrates (the quantitative calcu-\nlations of\u001csare performed by FPDM introduced earlier),\nit is reasonable to focus on the Elliot term gEand avoid\nthe more complicated Yafet term gY.\nDe\fneVE\nq\u0015as the spin-independent part of \u0001 q\u0015vKS,\nso thatgE=D\nu\"(#)\nk\f\f\fVE\nq\u0015\f\f\fu#(\")\nk\u0000qE\n. Expanding VE\nq\u0015as\nP\nGeVE\nq\u0015(G)eiG\u0001r, we have\ngE=X\nGeVE\nq\u0015(G)o\"#\nkk\u0000q(G); (17)\no\"#\nkk\u0000q(G) =D\nu\"(#)\nk\f\f\feiG\u0001r\f\f\fu#(\")\nk\u0000qE\n; (18)\nwhereo\"#\nkk\u0000q(G) isG-dependent spin-\rip overlap func-\ntion. Without loss of generality, we suppose the \frst\nBrillouin zone is centered at \u0000.\nTherefore,gEis not only determined by the long-range\ncomponent of o\"#\nkk\u0000q(G), i.e.,o\"#\nkk\u0000q(G= 0) but also the\nG6= 0 components. But nevertheless, it is helpful to\ninvestigate o\"#\nkk\u0000q(G= 0) and similar to Eq. 8, we pro-\npose an e\u000bective modulus square of the spin-\rip overlap\nmatrix elementjo\"#j2,11\njo\"#j2=P\nkqwk;k\u0000qP\n\u0015jo\"#\nk;k\u0000q(G= 0)j2\nP\nkqwk;k\u0000q: (19)\nInternal magnetic \feld\nSuppose originally a system has time-reversal and in-\nversion symmetries, so that every two bands form a\nKramers degenerate pair. Suppose the k-dependent spin\nmatrix vectors in Bloch basis of the Kramers degenerate\npairs are s0\nkwiths\u0011(sx;sy;sz). The inversion symme-\ntry broken, possibly due to applying an electric \feld or\na substrate, induces k-dependent Hamiltonian terms\nHISB\nk=\u0016BgeBin\nk\u0001s0\nk; (20)\nwhere\u0016Bgeis the electron spin gyromagnetic ratio.\nBin\nkis the SOC \feld and called internal magnetic \felds.\nBinsplits the degenerate pair and polarizes the spin along\nits direction. The de\fnition of Bin\nkis\nBin\nk\u00112\u0001SOC\nkSexp\nk=(\u0016Bge); (21)\nwhere Sexp\u0011\u0000\nSexp\nx;Sexp\ny;Sexp\nz\u0001\nwithSexp\nibeing spin\nexpectation value along direction iand is the diagonal\nelement ofsi. \u0001SOCis the band splitting energy by SOC.\nSpin-\rip angle \u0012\"#for intervalley spin relaxation\nSuppose (i) the inversion symmetry broken induces Bin\nk\n(Eq. 21) for a Kramers degenerate pair; (i) there are two\nvalleys centered at wavevectors Qand\u0000Qand (iii) there\nare two wavevectors k1andk2nearQand\u0000Qrespec-\ntively. Due to time-reversal symmetry, the directions of\nBin\nk1andBin\nk2are almost opposite.\nDe\fne the spin-\rip angle \u0012\"#\nk1k2as the angle between\n\u0000Bin\nk1andBin\nk2, which is also the angle between \u0000Sexp\nk1\nandSexp\nk2. We will prove that for a general operator bA,\n\f\f\fA\"#\nk1k2\f\f\f2\n\u0019sin2\u0010\n\u0012\"#\nk1k2=2\u0011\f\f\fA##\nk1k2\f\f\f2\n; (22)\nwhereA\"#\nk1k2andA##\nk1k2are the spin-\rip and spin-\nconserving matrix elements between k1andk2respec-\ntively.\nThe derivation uses the \frst-order perturbation theory\nand has three steps:\nStep 1: The 2\u00022 matrix of operator bAbetween k1and\nk2of two Kramers degenerate bands is A0\nk1k2. According\nto Ref. 26, with time-reversal symmetry, the spin-\rip\nmatrix element of the same band between kand\u0000kis\nexactly zero, therefore, the spin-\rip matrix elements ofA0\nk1k2are zero at lowest order as k1+k2\u00190, i.e.,A0;\"#\nk1k2\u0019\nA0;#\"\nk1k2\u00190.\nStep 2: The inversion symmetry broken induces Bin\nk\nand the perturbed Hamiltonian HISB\nk(Eq. 20). The\nnew eigenvectors Ukare obtained based on the \frst-order\nperturbation theory.\nStep 3: The new matrix is Ak1k2=Uy\nk1A0\nk1k2Uk2. Thus\nthe spin-\rip matrix elements A\"#\nk1k2with the inversion\nsymmetry broken are obtained.\nWe present the detailed derivation in SI Sec. III.\nFrom Eq. 22, for the intervalley e-ph matrix elements\nof ML-Ge systems, we have\n\f\f\fg\"#\nk1k2\f\f\f2\n\u0019sin2\u0010\n\u0012\"#\nk1k2=2\u0011\f\f\fg##\nk1k2\f\f\f2\n: (23)\nAs\f\f\fg\"#\nk1k2\f\f\f2\nlargely determines \u001csof ML-Ge systems,\nthe di\u000berences of \u001csof ML-Ge on di\u000berent substrates\nshould be mainly due to the di\u000berence of sin2\u0010\n\u0012\"#\nk1k2=2\u0011\n.\nFor the intervalley overlap matrix elements, we should\nhave\f\f\fo\"#\nk1k2\f\f\f2\n\u0019sin2\u0010\n\u0012\"#\nk1k2=2\u0011\f\f\fo##\nk1k2\f\f\f2\n. Since\f\f\fo##\nk1k2\f\f\f2\nis of order 1,\f\f\fo\"#\nk1k2\f\f\f2\nis expected proportional to\nsin2\u0010\n\u0012\"#\nk1k2=2\u0011\nand have the same order of magnitude as\nsin2\u0010\n\u0012\"#\nk1k2=2\u0011\n.\nFinally, similar to Eq. 8, we propose an e\u000bective mod-\nulus square of sin2\u0010\n\u0012\"#\nk1k2=2\u0011\n,\nsin2(\u0012\"#=2) =P\nkqwk;k\u0000qsin2\u0010\n\u0012\"#\nk;k\u0000q=2\u0011\nP\nkqwk;k\u0000q: (24)\nDATA AVAILABILITY\nThe data that support the \fndings of this study are\navailable upon request to the corresponding author.\nCODE AVAILABILITY\nThe codes that were used in this study are available\nupon request to the corresponding author.\nACKNOWLEDGEMENTS\nWe thank Ravishankar Sundararaman for helpful dis-\ncussions. This work is supported by the Air Force Of-\n\fce of Scienti\fc Research under AFOSR Award No.\nFA9550-YR-1-XYZQ and National Science Foundation\nunder grant No. DMR-1956015. This research used\nresources of the Center for Functional Nanomaterials,12\nwhich is a US DOE O\u000ece of Science Facility, and the\nScienti\fc Data and Computing center, a component of\nthe Computational Science Initiative, at Brookhaven Na-\ntional Laboratory under Contract No. DE-SC0012704,\nthe lux supercomputer at UC Santa Cruz, funded by NSF\nMRI grant AST 1828315, the National Energy Research\nScienti\fc Computing Center (NERSC) a U.S. Depart-\nment of Energy O\u000ece of Science User Facility operated\nunder Contract No. DE-AC02-05CH11231, and the Ex-\ntreme Science and Engineering Discovery Environment\n(XSEDE) which is supported by National Science Foun-\ndation Grant No. ACI-1548562 [42].AUTHOR CONTRIBUTIONS\nJ.X. performed the \frst-principles calculations. J.X.\nand Y.P. analyzed the results. J.X. and Y.P. designed all\naspects of the study. J.X. and Y.P. wrote the manuscript.\nADDITIONAL INFORMATION\nSupplementary Information accompanies the pa-\nper on the npj Computational Materials website.\nCompeting interests: The authors declare no com-\npeting interests.\nREFERENCES\n[1] N. Tombros, C. Jozsa, M. Popinciuc, H. T. Jonkman,\nand B. J. Van Wees, Nature 448, 571 (2007).\n[2] A. Avsar, H. Ochoa, F. Guinea, B. Ozyilmaz, B. J. van\nWees, and I. J. Vera-Marun, Rev. Mod. Phys. 92, 021003\n(2020).\n[3] M. Drogeler, C. Franzen, F. Volmer, T. Pohlmann,\nL. Banszerus, M. Wolter, K. Watanabe, T. Taniguchi,\nC. Stampfer, and B. Beschoten, Nano Lett. 16, 3533\n(2016).\n[4] A. Avsar, J. Y. Tan, M. Kurpas, M. Gmitra, K. Watan-\nabe, T. Taniguchi, J. Fabian, and B. Ozyilmaz, Nat.\nPhys. 13, 888 (2017).\n[5] P. Dey, L. Yang, C. Robert, G. Wang, B. Urbaszek,\nX. Marie, and S. A. Crooker, Phys. Rev. Lett. 119,\n137401 (2017).\n[6] J. Xu, H. Takenaka, A. Habib, R. Sundararaman, and\nY. Ping, Nano Lett. 21, 9594 (2021).\n[7] L. Zhang, J. Jiang, C. Multunas, C. Ming, Z. Chen,\nY. Hu, Z. Lu, S. Pendse, R. Jia, M. Chandra, et al. ,\nNat. Photon. 16, 529 (2022).\n[8] A. Habib, J. Xu, Y. Ping, and R. Sundararaman, Phys.\nRev. B 105, 115122 (2022).\n[9] C. Ertler, S. Konschuh, M. Gmitra, and J. Fabian, Phys.\nRev. B 80, 041405 (2009).\n[10] A. W. Cummings, J. H. Garcia, J. Fabian, and S. Roche,\nPhys. Rev. Lett. 119, 206601 (2017).\n[11] D. Van Tuan, F. Ortmann, A. W. Cummings, D. Soriano,\nand S. Roche, Sci. Rep. 6, 1 (2016).\n[12] P. Zhang and M. Wu, New J. Phys. 14, 033015 (2012).\n[13] J. Li, M. Goryca, K. Yumigeta, H. Li, S. Tongay, and\nS. Crooker, Phys. Rev. Mater. 5, 044001 (2021).\n[14] Z. Ni, E. Minamitani, Y. Ando, and S. Watanabe, Phys.\nChem. Chem. Phys. 17, 19039 (2015).\n[15] T. Amlaki, M. Bokdam, and P. J. Kelly, Phys. Rev. Lett.\n116, 256805 (2016).\n[16] K. Zollner, A. W. Cummings, S. Roche, and J. Fabian,\nPhys. Rev. B 103, 075129 (2021).\n[17] I. \u0014Zuti\u0013 c, J. Fabian, and S. D. Sarma, Rev. Mod. Phys.\n76, 323 (2004).\n[18] J. Xu, A. Habib, S. Kumar, F. Wu, R. Sundararaman,and Y. Ping, Nat. Commun. 11, 2780 (2020).\n[19] J. Xu, A. Habib, R. Sundararaman, and Y. Ping, Phys.\nRev. B 104, 184418 (2021).\n[20] J. Xu, K. Li, U. N. Huynh, J. Huang, V. Vardeny, R. Sun-\ndararaman, and Y. Ping, arXiv:2210.17074 (2022).\n[21] T. Giousis, G. Potsi, A. Kouloumpis, K. Spyrou, Y. Geor-\ngantas, N. Chalmpes, K. Dimos, M.-K. Antoniou, G. Pa-\npavassiliou, A. B. Bourlinos, et al. , Angew. Chem. 133,\n364 (2021).\n[22] S. Lei, L. Ge, S. Najmaei, A. George, R. Kappera, J. Lou,\nM. Chhowalla, H. Yamaguchi, G. Gupta, R. Vajtai, et al. ,\nACS Nano 8, 1263 (2014).\n[23] J. E. Padilha and R. B. Pontes, Solid State Commun.\n225, 38 (2016).\n[24] M. Ali, X. Pi, Y. Liu, and D. Yang, AIP Advances 7,\n045308 (2017).\n[25] D. Kochan, S. Irmer, and J. Fabian, Phys. Rev. B 95,\n165415 (2017).\n[26] Y. Yafet, in Solid state physics , Vol. 14 (Elsevier, 1963)\npp. 1{98.\n[27] L. Tao and E. Y. Tsymbal, Phys. Rev. B 100, 161110\n(2019).\n[28] M. Wu, J. Jiang, and M. Weng, Phys. Rep. 493, 61\n(2010).\n[29] F. Giustino, Rev. Mod. Phys. 89, 015003 (2017).\n[30] R. Sundararaman, K. Letchworth-Weaver, K. A.\nSchwarz, D. Gunceler, Y. Ozhabes, and T. A. Arias,\nSoftwareX 6, 278 (2017).\n[31] J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev.\nLett. 77, 3865 (1996).\n[32] S. Grimme, J. Antony, S. Ehrlich, and H. Krieg, J. Chem.\nPhys. 132, 154104 (2010).\n[33] D. R. Hamann, Phys. Rev. B 88, 085117 (2013).\n[34] S. Ismail-Beigi, Phys. Rev. B 73, 233103 (2006).\n[35] R. Sundararaman and Y. Ping, J. Chem. Phys. 146,\n104109 (2017).\n[36] N. Marzari and D. Vanderbilt, Phys. Rev. B 56, 12847\n(1997).\n[37] A. M. Brown, R. Sundararaman, P. Narang, W. A. God-\ndard, and H. A. Atwater, ACS Nano 10, 957 (2016).13\n[38] P. Narang, L. Zhao, S. Claybrook, and R. Sundarara-\nman, Adv. Opt. Mater. 5, 1600914 (2017).\n[39] A. M. Brown, R. Sundararaman, P. Narang, A. M.\nSchwartzberg, W. A. Goddard III, and H. A. Atwater,\nPhys. Rev. Lett. 118, 087401 (2017).\n[40] A. Habib, R. Florio, and R. Sundararaman, J. Opt. 20,\n064001 (2018).[41] A. M. Brown, R. Sundararaman, P. Narang, W. A. God-\ndard III, and H. A. Atwater, Phys. Rev. B 94, 075120\n(2016).\n[42] J. Towns, T. Cockerill, M. Dahan, I. Foster, K. Gaither,\nA. Grimshaw, V. Hazlewood, S. Lathrop, D. Lifka, G. D.\nPeterson, R. Roskies, J. R. Scott, and N. Wilkins-Diehr,\nComput. Sci. Eng. 16, 62 (2014)."    },    {        "title": "1908.00927v2.Two_dimensional_orbital_Hall_insulators.pdf",        "content": "Two-dimensional orbital Hall insulators\nLuis M. Canonico,1Tarik P. Cysne,2,3Tatiana G. Rappoport,2,4and R. B. Muniz1\n1Instituto de Física, Universidade Federal Fluminense, 24210-346 Niterói RJ, Brazil\n2Instituto de Física, Universidade Federal do Rio de Janeiro,\nCaixa Postal 68528, 21941-972 Rio de Janeiro RJ, Brazil\n3Departamento de Física, Universidade Federal de São Carlos,\nRod. Washington Luís, km 235 - SP-310, 13565-905 São Carlos, SP, Brazil\n4Department of Physics and Center of Physics, University of Minho, 4710-057, Braga, Portugal\n(Dated: March 4, 2020)\nDetailed analyses of the spin and orbital conductivities are performed for different topological\nphases of certain classes of two-dimensional (2D) multiorbital materials. Our calculations show\nthe existence of orbital-Hall effect (OHE) in topological insulators, with values that exceed those\nobtained for the spin-Hall effect (SHE). Notably, we have found non-topological insulating phases\nthat exhibit OHE in the absence of SHE. We demonstrate that the OHE in these systems is deeply\nlinked to exotic momentum-space orbital textures that are triggered by an intrinsic Dresselhaus-\ntype of interaction that arises from a combination of orbital attributes and lattice symmetry. Our\nresultsstronglyindicatesthatotherclassesofsystemswithnon-trivialorbitaltexturesand/ororbital\nmagnetism may also exhibit large OHE even in their normal insulating phases.\nThe OHE, similarly to the SHE, refers to the creation\nof a transverse flow of orbital angular momentum that is\ninduced by a longitudinally applied electric field [1]. It\nhas been explored mostly in three dimensional metallic\nsystems, where it can be quite strong [2–5]. For systems\nin which the spin-orbit coupling (SOC) is sizeable, the\norbital and spin angular momentum degrees of freedom\nare coupled, establishing an interrelationship between\ncharge, spin, and orbital angular momentum excitations.\nHowever, the OHE does not necessarily require SOC, it\ncan be associated to the presence of orbital textures [5]\nand be especially significant in various materials.\nChiral orbital textures in the reciprocal space have\nbeen discussed in connection with orbital magnetism at\nthe surface of spmetals [6], photonic graphene [7] and\nalso in topological insulators with strong SOC. More re-\ncently they were observed in chiral borophene [8], single-\nlayer transition metal dichalcogenides [9] and tin tel-\nluride monolayers for photocurrent generation [10]. Or-\nbital magnetism is enhanced in surfaces [11], indicating\nthat orbital effects can be crucial in 2D materials, which\ncan also be evidenced by the observation of orbital tex-\ntures in van der Waals materials. Still, OHE remains\nmostly unexplored in 2D materials [12, 13].\nHere, we investigate the role of orbital textures for the\nOHE displayed by multi-orbital 2D materials. We pre-\ndict the appearance of rather large OHE in these systems\nboth in their metallic and insulating phases. The orbital\nHall currents can be considerably larger than the spin\nHall ones, and be present even in the absence of SHE.\nTheir use as information carriers widens the development\npossibilities of novel spin-orbitronic devices.\nIn our analyses, we consider a minimal tight-binding\n(TB) model Hamiltonian that involves only two orbitals\n(pxandpy) per atom in a honeycomb lattice [14, 15]:H=X\nhijiX\n\u0016\u0017st\u0016\u0017\nijpy\ni\u0016spj\u0017s+X\ni\u0016s\u0000\n\u000fi+\u0015I`z\n\u0016\u0016\u001bz\nss\u0001\npy\ni\u0016spi\u0016s;\n(1)\nwhereiandjdenote the honeycomb lattice sites posi-\ntioned at~Riand~Rj, respectively. The symbol hijiindi-\ncates that the sum is restricted to the nearest neighbour\n(n.n) sites only. The operator py\ni\u0016screates an electron of\nspinsin the atomic orbitals p\u0016=p\u0006=1p\n2(px\u0006ipy)\ncentred at ~Ri. Here,s=\";#labels the two electronic\nspin states, and \u000fiis the atomic energy at site i, which\nmay symbolise a staggered on-site potential that takes\nvalues\u000fi=\u0006VAB, when site i belongs to the A and B\nsub-lattices of the honeycomb arrangement, respectively.\nThe transfer integrals t\u0016\u0017\nijbetween the p\u0016orbitals centred\non n.n atoms are parametrised according to the standard\nSlater-Koster TB formalism [16]. They depend on the\ndirection cosines of the n.n. interatomic directions, and\nmay be approximately expressed as linear combinations\nof two other integrals ( Vpp\u001bandVpp\u0019) involving the p\u001b\nandp\u0019orbitals, where \u001band\u0019refer to the usual compo-\nnents of the angular momentum around these axes.\nSince our model does not include the orbital pz, it is\nrestricted to a sector of the `= 1angular momentum\nvector space spanned only by the eigenstates of `z\f\fp\u0006\u000b\nassociated with m`=\u00061. Within this sector it is useful\nto introduce a pseudo angular momentum SU(2)-algebra\nwhere the Pauli matrices act on\f\fp\u0006\u000b\n. In this case, there\nis a one-to-one correspondence between the representa-\ntions of the Cartesian components of the orbital angular\nmomentum operators in this basis and the usual Pauli\nmatrices, and `zis not conserved (details are given Sec.\nI of the supplementary material - SM). The last term in\nEq.1 describes the intrinsic atomic SOC.\nThis simple model describes relatively well the low-arXiv:1908.00927v2  [cond-mat.mes-hall]  2 Mar 20202\nFigure 1: (a) Band structure calculations along some symme-\ntry lines in the 2D BZ for Vpp\u0019= 0,Vpp\u001b=1 eV, and\u0015I= 0.\nThe blue line represents the results for VAB= 0:0, and the\nred line for VAB= 0:8. (b) Orbital Hall conductivities cal-\nculated for the same sets of parameters. The insets show the\nin-plane contribution to the orbital angular momentum tex-\ntures calculated in the neighbourhoods the \u0000(left inset) and\nK(right inset) symmetry points of the 2D Brillouin zone, for\nVAB= 0:0. The left and right inset textures are associated\nwith the lower flat and dispersive bands, respectively.\nenergy electronic properties of novel group V based 2D\nmaterials [15, 17, 18]. Its topological characteristics were\npreviously investigated in the context of optical lattices,\nand it has been verified that it exhibits a rich topological\nphase diagram, which includes quantum spin-Hall insu-\nlator (QSHI) phases [17, 19–22].\nFollowing Ref. 21 we shall assume, for simplicity, that\nVpp\u0019= 0andVpp\u001b= 1eV. Our focus is on three dis-\ntinct phases that manifest themselves depending on the\nparameters specified in Eq. (1). In the absence of SOC\nand sub-lattice resolved potentials, the electronic band\nstructure consists of four gapless bulk energy bands, two\nof which form Dirac cones at the KandK0symmetry\npoints of the 2D first Brillouin zone (BZ), whereas the\nother two are flat. Each flat band is tangent to one of\nthe dispersive bands at the \u0000point, as Fig. 1 (a) illus-\ntrates.\nOur results for the orbital Hall conductivities ( \u001bz\nOH),\ncalculated as functions of energy by means of the Kubo\nformula [23], with the orbital current defined as J`z\ny=\n1\n2f`z;vyg, are shown in Fig. 1 (b) for VAB= 0:0(blue\nline), and for VAB= 0:8(red line). Details of these\ncalculations are given in Sec. II of SM. Here we notice a\nstrong orbital Hall conductivity, which peaks at energies\nclose to where the flat bands touch the dispersive bands\nat\u0000. ForVAB6= 0, the electronic structure develops\nan energy gap around E= 0that eliminates the original\nDirac cones in the vicinities of KandK0. The flat bands,\nhowever, remain tangent to the dispersive bands at \u0000, as\nshown in Fig. 1 (a), and the large OHE in this case also\noccurs for energies close to where they touch each other.\nThe insets of Fig. 1 (b) depict the in-plane contribu-\ntion to the orbital angular momentum textures, calcu-\nlated on a circle around the \u0000(left inset) and K(right\ninset) symmetry points of the 2D first BZ. They are both\ncomputed for VAB= 0. The colours of the arrows em-\nphasise their in-plane azimuthal angles. At the \u0000point,the texture displays a dipole-field like structure, whereas\nin the vicinity of the Kpoints it is identical to the spin-\ntexture produced by the Dresselhaus SOC in zinc blende\nlattice systems [24]. Here, the texture is not caused by\nSOC, but results only from the orbital features and crys-\ntalline symmetry, as we shall subsequently show.\nIn the presence of SOC, three energy gaps open: one\noriginating from the K(K0)points, and the other two at\n\u0000, while the flat bands acquire a slight energy disper-\nsion - see Fig. SI of the SM. When the relative values\nof\u0015IandVABvary, this model exhibits a rich topolog-\nical phase diagram [21]. We shall focus on three phases\nthat display distinct topological gap features. They are\nclassified by sets of spin Chern numbers (i,j,k,l) associ-\nated with the four \"-spin bands, namely A1 (1,-1,1,-1),\nB1 (1,0,0,-1), and B2 (0,1,-1,0), according to the nota-\ntion of Ref. 21. We remind that the spin Chern numbers\nfor the#-spin sector have opposite signs. This codifica-\ntion clearly indicates that when the system is in the A1\nphase the two lateral energy gaps are topological, but the\ncentral one is not. The reverse occurs in the B2 phase,\nwhere only the central energy gap is topological. Last\nbut not least, all the three energy gaps are topological\nin the B1 phase. This is explicitly verified in the left\npanels of Fig. 2, which show the spin Hall conductivi-\nties (\u001bz\nSH) (red curves) calculated as functions of energy\nfor three different sets of parameters that simulate sys-\ntems in each of these phases. In the absence of sublattice\nasymmetry ( VAB= 0) and for\u0015I= 0:2, the system as-\nsumes the B1 phase and becomes a QSHI within all the\nthree energy gaps, as the quantised plateaux of \u001bz\nSHin\nFig. 2 (a) show. For \u0015I= 1:1andVAB= 0:8, the system\nis in the B2 phase, which exhibits a quantised spin Hall\nconductivity plateau in the central energy gap, and two\nnon topological side gaps within which it behaves as an\nordinary insulator, displaying no QSHE, as Fig. 2 (c) il-\nlustrates. For \u0015I= 0:2andVAB= 0:8, the system takes\non the A1 phase, where it becomes a QSHI for energies\nwithin the lateral energy gaps, but behaves as a conven-\ntional insulator inside the central gap, as portrayed in\nFig. 2 (e).\nThe corresponding orbital Hall conductivities ( \u001bz\nOH)\ncalculated for the three phases (blue curves) are also de-\npicted in the left panels of Fig. 2, together with the\nrespective densities of states (grey lines) represented in\narbitrary units. We notice that within the lateral gaps,\n\u001bz\nOHexhibits plateaux with much higher intensities than\nthose of the SHE. However, in contrast with the latter,\nthe OHE is not quantised. Its plateaux heights depend\nupon\u0015IandVAB, increasinginmodulusasthegapwidth\nreduces, thoughlimitedbytheOHEvaluefor \u0015I!0(see\nSec. IV of SM). A remarkable result illustrated in Fig.\n2 (c) is the existence of finite OHE within the two (non-\ntopological) side energy gaps of phase B2, where the sys-\ntem becomes an ordinary insulator with no QSHE. This\nis particularly interesting because there are no electronic3\nedge states crossing these energy gaps (see Sec. V of the\nSM), and raises the question on how the orbital Hall cur-\nrent propagates through the system in this case. It is\nalso noticeable that the OHE is an odd function of the\nFermi energy ( EF)and vanishes in the central energy\ngaps for all three phases. This is due to symmetry lim-\nitations of this simplified model, which we shall address\nsubsequently.\nFigure 2: Spin Hall conductivity (red), and orbital Hall con-\nductivity (blue), together with the density of states (grey),\ncalculated as functions of energy for: (a) \u0015I= 0:2and\nVAB= 0- B1 phase; (c) \u0015I= 1:1, andVAB= 0:8- B2\nphase, and (e) \u0015I= 0:2, andVAB= 0:8- A1 phase. The\ndensities of states are depicted in arbitrary units. Panels (b),\n(d) and (f) show the associated orbital textures, calculated\nfor the lower\"-spin band, with the same sets of parameters,\nrespectively. The density plots illustrate their corresponding\nh`zipolarisations.\nIt is also important to examine how disorder affects\nthe transport properties of these systems. To simulate\nit we consider on-site potentials \u000fiwith values randomly\ndistributed within [-W/2, W/2], where W is the disorder\nstrength. We calculate the spin- and orbital-Hall conduc-\ntivities for different values of W using Chebyshev poly-\nnomial expansions and the Kubo-Bastin formula, which\nare efficiently implemented in the open-source software\nKITE. Similarly to what we have previously found forthe SHE [22], the orbital Hall plateaux are robust to rel-\natively strong Anderson disorder. Details of these calcu-\nlations are described in sections VI, VII and VIII of the\nSM.\nWe know that OHE is linked to orbital textures in re-\nciprocal space [5], and to establish this relationship we\nhave calculated these textures for the \"-spin lowest en-\nergybandintheentire2DfirstBZ.Theresultsareshown\nin panels (b), (d), and (f) for systems in the B1, B2, and\nA1 phases, respectively. The orbital characters for all \"-\nspin eigenstates are depicted in Fig. SV of the SM. It is\nworth noticing that when either \u0015IorVABare different\nfrom zero, the orbital textures display finite out-of-plane\ncomponents for each spin direction. However, due to\ntime reversal symmetry the `zorbital polarisations for\ninverted spin directions are opposite, and consequently\nthe total`zpolarisation vanishes. The structure of the\nin-plane texture, nevertheless, remains the same for both\nspin components, which means that the in-plane orbital\ntexture survives. It is also noteworthy that both the low-\nest two energy bands as well as the upper ones display\nopposite in-plane orbital textures for this simple model.\nConsequently, the OHE vanishes at the onset of the cen-\ntral energy gap, where the accumulated in-plane orbital\ntexture of the occupied states becomes zero. The absence\nof electronic states within an energy gap leads to a con-\nstant value for \u001bz\nOH[2, 25, 26] in its range, which justifies\nthe lack of OHE in the central energy gap found for the\nthree phases.\nContour curves are also shown for certain values of\nEFranging from the bottom of the energy band to the\nbeginningofthelowestenergygap. Inallphases, wenote\nthat close to the \u0000point, where the lowest energy band\nvalue is minimum, there is virtually no in-plane orbital\nangular momentum texture, and the OHE is very small.\nAsEFincreases the in-plane orbital texture builds up,\nassuming a dipole-field like configuration. Eventually,\nwhenEFapproaches the onset of the first energy gap, it\ndevelops a Dresselhaus-like arrangement near the K and\nK’ points, with opposite signs in each valley.\nIn order to uncover the raison d’etre of these exotic or-\nbital textures that promote OHE in this systems we de-\nrive an effective theory near the Dirac points KandK0.\nAround them, the orbital angular momentum texture is\nperfectly captured by a linear approximation in the crys-\ntalline momentum, whereas it requires a fourth-order ex-\npansion near the \u0000point. Our effective Hamiltonian Heff\ncan be expressed in terms of SU(2)\nSU(2)orbital and\nsub-lattice algebras, and written as: Heff=H0+HAB+\nHSOC+H`. HereH0=\u0000~vF(kx\u001bx+\u001cky\u001by)istheusual\nDirac Hamiltonian, with Fermi velocity vF=ap\n3\n2~,ade-\nnotes the lattice constant, and \u001c=\u00061for theKandK0\nvalleys, respectively. HSOC=s\u0015I`zrepresents the SOC,\nwheres=\u00061for\"and#spin electrons, respectively.\nHAB=VAB\u001bzis the sub-lattice resolved potential. H`4\nbreaksthedegeneracybetween `zeigenstatesandisgiven\nby:\nH`=\u0000~vF\n4\u001c(k+`+\u001b\u001c+k\u0000`\u0000\u001b\u0016\u001c)\u0000p\n3~vF\n2a(`x\u001bx+\u001c`y\u001by);\n(2)\nwhere\u001b\u001c=\u001bx+i\u001c\u001by,\u0016\u001c=\u0000\u001c,`\u000b(\u000b=x;y) are the\norbital angular momentum matrices in the corresponding\nHilbert space, k\u0006=kx\u0006iky, and`\u0006=`x\u0006i`y.\nAs shown in the section X of the SM, in the absence\nofH`each valley presents two degenerated Dirac cones.\nThe first term in the right hand side of Eq. (2) alters\nthe Fermi velocity of the Dirac cones and leads to an in-\nplane orbital texture profile similar to the one portrayed\naround the \u0000point. The second term, however, pro-\nduces a Dresselhaus-like splitting in the Dirac cones and\nis primarily responsible for the orbital angular momen-\ntum texture found in our TB calculations. Our effective\ntheory confirms that the exotic in-plane texture exhib-\nited by these 2D systems is an intrinsic property that\narises solely from the px-pyorbital characteristics and\ncrystalline symmetries.\nFigure 3: (a) Energy band spectrum calculated for the simple\nmodel with second n.n. hopping integrals V(2)\npp\u001b=\u00000:2. Here\nwe keepVpp\u0019= 0,Vpp\u001b= 1,\u0015I= 0:2andVAB= 0:8. (b)\nSH (red line) and OH (blue line) conductivities calculated as\nfunctions of energy. The grey line depicts the density of states\n(DOS) in arbitrary units. The inset shows a closeup of the\ncentral energygaphighlightingthenon-zero valueoftheOHE\nwithin this energy range.\nWe shall now address the absence of OHE in the cen-\ntral energy gap as results from our calculations. This\nlimitation actually comes from a combination of electron-\nhole and parity symmetries, which lead to energy levels\nthat are symmetric with respect to the zero energy for\nthis simple model[21]. One way of breaking it is by in-\ntroducing second n.n. hopping integrals, as Fig. 3 (a)\nillustrates. Here, just as a proof of concept, we kept\nVpp\u0019= 0, and choose the second n.n. hopping integrals\nVpp\u001b2=\u00000:2. In this case only the central energy gap\nsurvives, and within it the system assumes an ordinary\ninsulating phase. Fig. 3 (b) clearly shows that the SHE\nvanishes in this energy range, whereas the OHE is fi-\nnite. Here, the in-plane orbital texture associated with\nFigure 4: SH (red line) and OH (blue line) conductivities\ncalculated as functions of energies for flat bismuthene: (a)\nwithout sublattice asymmetry ( VAB= 0) and (b) with VAB=\n0:87. The insets highlight the non-zero values of the OHE\nwithin the corresponding central energy-gap ranges.The grey\nline depicts the DOS in arbitrary units.\nthe second lowest energy band no longer cancels the con-\ntribution from the first band. Thus, the OHE does not\nvanish at the onset of the central energy gap and keeps\nits non-zero value constant within it. This result, al-\nthough relatively small in this particular case, unequiv-\nocally shows that it is possible to obtain a finite OHE\nfor a non-topological insulating phase, as we previously\nfound for the lateral energy gaps of the B2 phase. Hav-\ning shown that this effect happens for our simple-model\nsystem, it is instructive to inquire into the possibility of\nobserving it in a real system. A candidate is the recently\nsynthesised flat bismuthene grown on SiC, whose low en-\nergy electronic properties are reasonably well described\nby an effective TB model Hamiltonian that includes only\ntwo orbitals ( pxandpy) per atom [14, 15, 17, 18]. It\nis a real solid state system, typical of a promising class\nof 2D materials based on the group group VA elements\nthat exhibit relatively large energy gaps. In fact, a very\ngood TB fit of both the valence and conduction bands\nof flat bismuthene can be obtained with the inclusion of\nsecond n.n. hopping integrals, as Fig. SX of the SM\nillustrates. Results for the associated SHE and OHE cal-\nculated as functions of EFfor planar bismuthene em-\nploying a Chebyshev polynomial expansion method are\nshown in Fig.4. We clearly see in this case that the spec-\ntraarenotsymmetricwithrespecttothezeroenergyand\nthe right-hand side gap disappear. Results for VAB= 0,\ndepicted in Fig.4 (a), show that the remaining gaps are\ntopological, displaying a quantised SHE, and significant\nOHE. For sufficiently large sublattice asymmetry, how-\never, the central gap ceases to be topological, exhibiting\nnoSHE,asFig.4(b)illustrates. Notwithstanding, theor-\nbital Hall conductivity is appreciable within this energy\nrange. This validates our original prediction that pure\norbital angular momentum currents can be triggered by\na longitudinally applied electric field in some normal in-\nsulators.\nIn summary, we have performed detailed analyses of\nthe spin and orbital Hall conductivities for a class of 2D5\nsystems, relating the corresponding OHE, SHE and or-\nbital textures. Our calculations show the existence of\nOHE in topological insulators, with values that exceed\nthose obtained for the SHE. Remarkably, we also obtain\nOHE for normal insulating phases where the SHE is ab-\nsent and no edge states cross their energy gaps. We show\nthat the OHE in these systems is associated with exotic\nmomentum-space orbital textures that are caused by an\nintrinsic Dresselhaus-type of interaction. This is rather\ngeneralandshowthatcertain2Dinsulatingmaterialscan\ngenerateorbitalangularmomentumcurrentsthatmaybe\nuseful for developing novel spin-orbitronic devices.\nWe acknowledge CNPq/Brazil, FAPERJ/Brazil\nand INCT Nanocarbono for financial support, and\nNACAD/UFRJ for providing high-performance com-\nputing facilities. TGR acknowledges COMPETE2020,\nPORTUGAL2020, FEDER and the Portuguese Foun-\ndation for Science and Technology (FCT) through\nproject POCI-01- 0145-FEDER-028114. TPC acknowl-\nedges São Paulo Research Foundation (FAPESP) grant\n2019/17345-7.\n[1] B. A. Bernevig, T. L. Hughes, and S.-C. Zhang, Phys.\nRev. Lett. 95, 066601 (2005), URL https://link.aps.\norg/doi/10.1103/PhysRevLett.95.066601 .\n[2] T. Tanaka, H. Kontani, M. Naito, T. Naito, D. S.\nHirashima, K. Yamada, and J. Inoue, Phys. Rev. B\n77, 165117 (2008), URL https://link.aps.org/doi/\n10.1103/PhysRevB.77.165117 .\n[3] H. Kontani, T. Tanaka, D. S. Hirashima, K. Yamada,\nand J. Inoue, Phys. Rev. Lett. 100, 096601 (2008), URL\nhttps://link.aps.org/doi/10.1103/PhysRevLett.\n100.096601 .\n[4] H. Kontani, T. Tanaka, D. S. Hirashima, K. Yamada,\nand J. Inoue, Phys. Rev. Lett. 102, 016601 (2009), URL\nhttps://link.aps.org/doi/10.1103/PhysRevLett.\n102.016601 .\n[5] D. Go, D. Jo, C. Kim, and H.-W. Lee, Phys. Rev. Lett.\n121, 086602 (2018), URL https://link.aps.org/doi/\n10.1103/PhysRevLett.121.086602 .\n[6] D. Go, J.-P. Hanke, P. M. Buhl, F. Freimuth,\nG. Bihlmayer, H.-W. Lee, Y. Mokrousov, and S. Blügel,\nScientific Reports 7(2017), URL https://doi.org/10.\n1038/srep46742 .\n[7] A. Nalitov, G. Malpuech, H. Terças, and D. Solnyshkov,\nPhysical Review Letters 114(2015), URL https://doi.\norg/10.1103/physrevlett.114.026803 .\n[8] F. C. de Lima, G. J. Ferreira, and R. H. Miwa, Nano\nLetters (2019), URL https://doi.org/10.1021/acs.\nnanolett.9b02802 .\n[9] Y. Chen, W. Ruan, M. Wu, S. Tang, H. Ryu, H.-Z. Tsai,\nR. Lee, S. Kahn, F. Liou, C. Jia, et al., Visualizing exotic\norbital texture in the single-layer mott insulator 1t-tase2\n(2019), arXiv:1904.11010.\n[10] J. Kim, K.-W. Kim, D. Shin, S.-H. Lee, J. Sinova,\nN. Park, and H. Jin, Nature Communications 10(2019),\nURL https://doi.org/10.1038/s41467-019-11964-6 .[11] M. Tischer, O. Hjortstam, D. Arvanitis, J. Hunter Dunn,\nF. May, K. Baberschke, J. Trygg, J. M. Wills,\nB. Johansson, and O. Eriksson, Phys. Rev. Lett.\n75, 1602 (1995), URL https://link.aps.org/doi/10.\n1103/PhysRevLett.75.1602 .\n[12] I. V. Tokatly, Phys. Rev. B 82, 161404 (2010),\nURL https://link.aps.org/doi/10.1103/PhysRevB.\n82.161404 .\n[13] V. o. T. Phong, Z. Addison, S. Ahn, H. Min,\nR. Agarwal, and E. J. Mele, Phys. Rev. Lett. 123,\n236403 (2019), URL https://link.aps.org/doi/10.\n1103/PhysRevLett.123.236403 .\n[14] F. Reis, G. Li, L. Dudy, M. Bauernfeind, S. Glass,\nW. Hanke, R. Thomale, J. Schäfer, and R. Claessen,\nScience 357, 287 (2017), ISSN 0036-8075, URL http:\n//science.sciencemag.org/content/357/6348/287 .\n[15] Y. Shao, Z.-L. Liu, C. Cheng, X. Wu, H. Liu, C. Liu,\nJ.-O. Wang, S.-Y. Zhu, Y.-Q. Wang, D.-X. Shi, et al.,\nNano Letters 18, 2133 (2018), pMID: 29457727,\nhttps://doi.org/10.1021/acs.nanolett.8b00429, URL\nhttps://doi.org/10.1021/acs.nanolett.8b00429 .\n[16] J. C. Slater and G. F. Koster, Phys. Rev. 94,\n1498 (1954), URL https://link.aps.org/doi/10.\n1103/PhysRev.94.1498 .\n[17] G. Li, W. Hanke, E. M. Hankiewicz, F. Reis, J. Schäfer,\nR. Claessen, C. Wu, and R. Thomale, Phys. Rev. B\n98, 165146 (2018), URL https://link.aps.org/doi/\n10.1103/PhysRevB.98.165146 .\n[18] T. Zhou, J. Zhang, H. Jiang, I. Žutić, and Z. Yang, npj\nQuantum Materials 3, 39 (2018).\n[19] C. Wu, D. Bergman, L. Balents, and S. Das Sarma, Phys.\nRev. Lett. 99, 070401 (2007), URL https://link.aps.\norg/doi/10.1103/PhysRevLett.99.070401 .\n[20] C. Wu and S. Das Sarma, Phys. Rev. B 77,\n235107 (2008), URL https://link.aps.org/doi/10.\n1103/PhysRevB.77.235107 .\n[21] G.-F. Zhang, Y. Li, and C. Wu, Phys. Rev. B 90,\n075114 (2014), URL https://link.aps.org/doi/10.\n1103/PhysRevB.90.075114 .\n[22] L. M. Canonico, T. G. Rappoport, and R. B. Muniz,\nPhys. Rev. Lett. 122, 196601 (2019), URL https://\nlink.aps.org/doi/10.1103/PhysRevLett.122.196601 .\n[23] G. D. Mahan, Many-particle physics (Springer Science &\nBusiness Media, 2013).\n[24] G.Dresselhaus, Phys.Rev. 100, 580(1955), URL https:\n//link.aps.org/doi/10.1103/PhysRev.100.580 .\n[25] P. Streda, Journal of Physics C: Solid State Physics\n15, L1299 (1982), URL https://doi.org/10.1088%\n2F0022-3719%2F15%2F36%2F006 .\n[26] M. Milletarì, M. Offidani, A. Ferreira, and R. Raimondi,\nPhys. Rev. Lett. 119, 246801 (2017), URL https://\nlink.aps.org/doi/10.1103/PhysRevLett.119.246801 .\n[27] See Supplemental Material, which includes Refs. [28–41]\nfor additional analyses, and details on the numerical\nsimulations.\n[28] H. Röder, R. N. Silver, D. A. Drabold, and J. J. Dong,\nPhys.Rev.B 55, 15382(1997), URL https://link.aps.\norg/doi/10.1103/PhysRevB.55.15382 .\n[29] A. Weiße, G. Wellein, A. Alvermann, and H. Fehske,\nRev. Mod. Phys. 78, 275 (2006), URL https://link.\naps.org/doi/10.1103/RevModPhys.78.275 .\n[30] A. Ferreira, J. Viana-Gomes, J. Nilsson, E. R. Mucci-\nolo, N. M. R. Peres, and A. H. Castro Neto, Phys. Rev.6\nB83, 165402 (2011), URL https://link.aps.org/doi/\n10.1103/PhysRevB.83.165402 .\n[31] Z. Fan, A. Uppstu, and A. Harju, Phys. Rev. B 89,\n245422 (2014), URL https://link.aps.org/doi/10.\n1103/PhysRevB.89.245422 .\n[32] N. Leconte, A. Ferreira, and J. Jung, in 2D Materials ,\nedited by F. Iacopi, J. J. Boeckl, and C. Jagadish (El-\nsevier, 2016), vol. 95 of Semiconductors and Semimet-\nals, pp. 35 – 99, URL http://www.sciencedirect.com/\nscience/article/pii/S0080878416300047 .\n[33] J. H. Garcia, A. W. Cummings, and S. Roche,\nNano Letters 17, 5078 (2017), pMID: 28715194,\nhttps://doi.org/10.1021/acs.nanolett.7b02364, URL\nhttps://doi.org/10.1021/acs.nanolett.7b02364 .\n[34] L. M. Canonico, J. H. García, T. G. Rappoport,\nA. Ferreira, and R. B. Muniz, Phys. Rev. B 98,\n085409 (2018), URL https://link.aps.org/doi/10.\n1103/PhysRevB.98.085409 .\n[35] M.Anđelković, S.Milovanović, L.Covaci, andF.Peeters,\narXiv preprint arXiv:1910.00345 (2019).\n[36] J.H.García, L.Covaci, andT.G.Rappoport, Phys.Rev.\nLett.114, 116602 (2015), URL https://link.aps.org/\ndoi/10.1103/PhysRevLett.114.116602 .\n[37] J. H. Garcia and T. G. Rappoport, 2D Materi-\nals3, 024007 (2016), URL http://stacks.iop.org/\n2053-1583/3/i=2/a=024007 .\n[38] A. Bastin, C. Lewiner, O. Betbeder-matibet, and\nP. Nozieres, Journal of Physics and Chemistry of\nSolids 32, 1811 (1971), ISSN 0022-3697, URL\nhttp://www.sciencedirect.com/science/article/\npii/S0022369771801476 .\n[39] M. Anđelković, S. M. João, L. Covaci, T. G. Rappoport,\nJ. M. V. P. Lopes, and A. Ferreira (In Preparation), URL\nhttp://quantum-kite.com .\n[40] J. Sinova, D. Culcer, Q. Niu, N. A. Sinitsyn, T. Jung-\nwirth, and A. H. MacDonald, Phys. Rev. Lett. 92,\n126603 (2004), URL https://link.aps.org/doi/10.\n1103/PhysRevLett.92.126603 .\n[41] F. Dominguez, B. Scharf, G. Li, J. Schäfer, R. Claessen,\nW. Hanke, R. Thomale, and E. M. Hankiewicz, Phys.\nRev. B 98, 161407 (2018), URL https://link.aps.org/\ndoi/10.1103/PhysRevB.98.161407 .1\nSupplementary material for “Spin and Charge Transport of Multi-Orbital Quantum\nSpin Hall Insulators”\nHAMILTONIAN IN RECIPROCAL SPACE AND ORBITAL ANGULAR MOMENTUM OPERATORS\nDEFINITION\nThe Hamiltonian given by Eq. (1) of the main text can be rewritten in the reciprocal space. Using the basis\nf\f\fp+;A\u000b\n;\f\fp\u0000;A\u000b\n;\f\fp+;B\u000b\n;\f\fp\u0000;B\u000b\ng, the hopping term reads,\n~H0(~k) =0\nBBB@0 0 ~A(~k)~D(~k)\n0 0 ~C(~k)~B(~k)\n~A\u0003(~k)~C\u0003(~k) 0 0\n~D\u0003(~k)~B\u0003(~k) 0 01\nCCCA; (S1)\nwhere\n2~A(~k) =txx(~k) +tyy(~k) +i\u0010\ntxy(~k)\u0000tyx(~k)\u0011\n; (S2)\n2~B(~k) =txx(~k) +tyy(~k) +i\u0010\ntyx(~k)\u0000txy(~k)\u0011\n; (S3)\n2~C(~k) =txx(~k)\u0000tyy(~k) +i\u0010\ntxy(~k) +tyx(~k)\u0011\n; (S4)\n2~D(~k) =txx(~k)\u0000tyy(~k)\u0000i\u0010\ntxy(~k) +tyx(~k)\u0011\n; (S5)\nThe transfer integrals t\u000b\f(~k)are Fourier transforms of Slater-Koster coefficients in the honeycomb lattice,\nt\u000b;\f(~k) =2X\nm=0\np\u000b;A\f\f^V\f\fp\f;B\u000b\n(m)e\u0000i~k\u0001~ am; (S6)\nwhere,mruns over the three nearest-neighbours of a site in sublattice A, that are located in sublattice B, and ~ amis\nthe vector connecting the atom at sublattice A with its mth neighbor at sublattice B. The Slater-Koster integrals are\ngiven by\nt\u000b\u000b(~k) =n2\n\u000b(m)Vpp\u001b+ (1\u0000n2\n\u000b(m))Vpp\u0019; (S7)\nt\u000b\f=\u0000n\u000b(m)n\f(m)\u0000\nVpp\u0019\u0000Vpp\u001b\u0001\n; (S8)\nwithn\u000b;\f(m)being the direction cosine connecting the site of sublattice Awithm-th first neighbor on sublat-\nticeB. In all results presented in this work, we set Vpp\u0019= 0, unless it is mentioned. This condition can be\nrelaxed without changing any of the main conclusions of our work. In this basis, the SOC term is diagonal,\nHs\nsoc=s\u0015Idiag(1;\u00001;1;\u00001)and the sublattice potential is given by HAB=VABdiag(1;1;\u00001;\u00001).\nAs mentioned on the main text, due the absence of the pzorbital, the electronic states are restricted to the subspace\nassociated with m`=\u00061only, hence the angular momentum operators can be redefined in terms of a SU(2)\u0000algebra\nas:\nlz=\f\fp+\u000b\np+\f\f\u0000\f\fp\u0000\u000b\np\u0000\f\f;\nlx=\f\fp+\u000b\np\u0000\f\f+\f\fp\u0000\u000b\np+\f\f;\nly=i\u0010\f\fp\u0000\u000b\np+\f\f\u0000\f\fp+\u000b\np\u0000\f\f\u0011\n(S9)\nKUBO FORMULA FOR LINEAR RESPONSE CONDUCTIVITY\nInthemaintext,wecomputethespinandorbitalHallconductivityfordifferenttopologicalphasesotheHamiltonian\nof Eq. (1) in the main text. For the cases of the pristine system, we used Kubo formalism to compute both OHE and2\nSHE. In this formalism, the spin Hall (SH) and orbital Hall (OH) \u0011-polarized response, in ^ydirection, to an electric\nfield applied in the ^xdirection is given by,\n\u001b\u0011\nOH(SH)=e\n~X\nn6=mX\ns=\";#Z\nB:Z:d2k\n(2\u0019)2(fm~k\u0000fn~k)\nX\u0011\nn;m;~k;s; (S10)\n\nX\u0011\nn;m;~k;s=~2Im\"\n s\nn;~k\f\fjX\u0011y(~k)\f\f s\nm;~k\u000b\n s\nm;~k\f\fvx(~k)\f\f s\nn;~k\u000b\n(Es\nn;~k\u0000Es\nm;~k+i0+)2#\n(S11)\nWere\u001b\u0011\nOH(SH)istheorbitalHall(SpinHall)DCconductivitywithpolarizationin \u0011-direction, \nX\u0011\nn;m;~k;sistherelated\ngauge-invariant Berry curvature. In Eq. S11, Es\nn(m);~kandj s\nn(m);~k\u000b\nare eigenvalues and eigenvectors of Hamiltonian\nof Eq. (S1), for n(m)Bloch band, with n;m = 1;::;4(in crescent order of energy), and s=\";#spin-sector. Velocity\noperators are defined by, vx(y)(~k) =@H(~k)=@kx(y), whereH(~k)is the tight-binding Hamiltonian in reciprocal space.\nThe current operator in ^ydirection is defined by jX\u0011y(~k) =\u0000\nX\u0011vy(~k) +vy(~k)X\u0011\u0001\n=2, whereX\u0011=^`\u0011(^s\u0011)for OH (SH)\nconductivities polarized in \u0011direction.\nAs it was discussed in the main text, this model presents a non-vanishing \u001bz\nOH, even in absence of SOC, in contrast\nto the spin Hall ( \u001bz\nSH) response which depends on the presence of SOC or exchange interaction. Added to this, in the\npresence of an exchange term, it was shown that the model presents a non-vanishing \u001bx\nOHassociated with in-plane\npolarized orbital Hall effect. It is important to mention that equations S10 and S11 are valid only in the clean limit\nand do not take into account the effect of disorder. However, as was briefly pointed in the main text, the effect of\ndisorder should not affect our results for insulating phases (Fermi energy inside an electronic gap) due to the absence\nof the Fermi surface, responsible to generates the leading-order contribution in the computation of vertex corrections\n[26]. We confirm the robustness of our results against Anderson disorder using a real-space computation method\nwhich is discussed in the next sections of the SM.\nANALYSIS OF THE BAND STRUCTURE\nWe have examined the orbital Hall conductivity properties of three distinct topological phases displayed by the\nHamiltonian Hdefined by Eq. (1) in the main text. They are labelled as B1, A1, and B2 phases, according to the\nclassifications used in Ref. 21. Figure SI shows the \"-spin electron energy bands for the system in these three phases.\nThe spin-#bands can be deduced by applying a time-reversal symmetry operation on H. Panel (a) illustrates the band\nstructure of the B1 phase, calculated for \u0015I= 0:2Vpp\u001b, andVAB= 0. We notice that the SOC causes three energy\ngaps to open, one originating from the K(K0) points, and the other two at \u0000, while the flat bands acquire a slight\nenergy dispersion. Panel (b) shows the energy bands for the system in the A1 phase, calculated with \u0015I= 0:2Vpp\u001b, and\nVAB= 0:8Vpp\u001b. The sub-lattice potential affects each valley differently, as expected, because it breaks the degeneracy\nbetween eigenvalues at the KandK0symmetry points. By examining the opposite spin polarisation one finds that\nthis phase exhibits a strong spin-valley locking, as discussed in Refs. 18, 21, 22. Panel (c) displays the energy bands\nfor the system in the B2 phase, calculated with \u0015I= 1:1Vpp\u001bandVAB= 0:8Vpp\u001b. In this case, \u0015Iis comparable but\nslightly larger than VAB, and we note that they lead to effects that are similar to those exhibited panel (b), including\na strong spin-valley locking with valley polarisation stronger than in the previous case due to the relatively large\nvalues of\u0015IandVAB.\nEVOLUTION OF THE ORBITAL HALL EFFECT PLATEAUX\nIn the main text, it was mentioned that the height of the orbital Hall plateaux within the lateral gaps depends upon\nthe SOC coupling constant and the sub-lattice resolved potential. To demonstrate this, we show in Figure SII results\nfor the spin and orbital Hall conductivities calculated for different sets of parameters for the B1, A1, and B2 phases.\nThe results depicted in each panel of Figure SII are obtained for a fixed value of VABand two different values of \u0015I\nthat are represented in the left and right columns, respectively. In panel (a) we show the conductivities calculated\nforVAB=0;\u0015I= 0:2Vpp\u001band\u0015I= 1:0Vpp\u001b, which correspond to situations in which the system is in the B1 phase. It\nis clear that the height of the OHE plateau decreases as the SOC increases. In fact, the height of the plateau scales3\nFigure SI:\"-spin electron energy bands calculated as functions of wave vectors along some symmetry directions in the 2D\nBrillouin zone for three distinct topological phases: (a) B1 with \u0015I= 0:2Vpp\u001bandVAB= 0. (b) A1 with \u0015I= 0:2Vpp\u001band\nVAB= 0:8Vpp\u001b(c) B2 with \u0015I= 1:1Vpp\u001bandVAB= 0:8Vpp\u001b.\nwith the size of the lateral gap, being close to the maximum value of the metallic limit for very small gaps. The same\ntrend is observed in the other two phases, in contrast with the heights of the spin Hall plateaux that remain the same\nin all cases .\nZIGZAG NANO-RIBBONS SPECTRA\nThorough the main text we analysed the spin and orbital Hall effects for the three distinct topological phases\nB1;A1andB2. Tofurthersubstantiateourfindingsofthenon-zeroorbitalHallconductivityinthetriviallyinsulating\nphases, we analysed the energy spectrum of a zigzag nano-ribbon in our system for the three distinct phases. Figure\nSIII shows the spectra for each of the phases studied in the main text. As expected, the number of pairs of edge\nstates corresponds with the index Z2of each of these phases. Panel (a) shows the energy bands corresponding to the\nphaseB1, here the most interesting features are the pairs of edge channels that cross the gap and the fully symmetric\nspectrum for both spin polarizations. Panel (b) displays the spectrum of the A1phase, here we can see the strong\nspin-valley locking that results from the inversion symmetry breaking produced by staggered sub-lattice potential.\nInterestingly here we can observe the absence of edge states traversing the central gap. Finally panel (c) shows the\nband structure of a ribbon in the phase B2. Here we can see how due the strong spin-orbit coupling and staggered\nsub-lattice potential the edge states in both the lateral gaps do not cross the gap while the edge estates of the central\ngap are crossing again. The results are fully consistent with the spin Cher number characterization of these phases\ndone in Ref [21]. The results of panel (c) are the most striking ones, because they indicate that differently from\nthe spin Hall conductivity, the orbital Hall effect plateau does not require electronic conducting channels to have a\nconstant non-quantized value.\nlatex onecolumn undefined\nCHEBYSHEV POLYNOMIAL EXPANSION\nTo study the transport and spectral properties of the honeycomb lattice with px\u0000pyorbitals we used the\nChebyshev polynomial expansion. In this numerical method, the Green and spectral functions are accurately\nexpanded in terms of Chebyshev polynomial of first kind of the Hamiltonian matrices[28, 29]. This set of polynomials\nare commonly chosen due their unique convergence properties, their relation with the Fourier transform and their\nconvenient recurrence relations that allows the iterative construction of higher order polynomials[28, 29]. In recent\nyears this method has gained much attention in the study of the transport properties of 2D systems[30–34]. Because\nof its high scalability, It was used to study of topological phase transitions induced by disorder[22], and more recently,\nin the analysis of the electronic properties of graphene encapsulated between two twisted hBN structures[35]. The\nmethod requires a rescaling of the Hamiltonian and it spectrum to make them fit into the interval (\u00001;1)where4\nFigure SII: Spin Hall conductivity \u001bz\nSH(red) and orbital Hall conductivity \u001bz\nOH(blue)calculated for: (a) VAB= 0and\n\u0015I= 0:2Vpp\u001b(left) and\u0015I= 1:0Vpp\u001b(right). (b) VAB= 0:8Vpp\u001band\u0015I= 0:2Vpp\u001b(left) and\u0015I= 0:5Vpp\u001b(right). (c)\nVAB= 0:8Vpp\u001b,\u0015I= 1:1Vpp\u001b(left) and\u0015I= 1:5Vpp\u001b(right)\nthe Chebyshev polynomials are defined and consequently the convergence of the method is assured. This scaling is\nachieved by means of the transformations ~H= (H\u0000b)=aand ~E= (E\u0000b)=awherea\u0011(ET\u0000EB)=(2\u0000\u000f)and\nb\u0011(ET+EB)=2. In the later ETandEBrepresents the top and bottom limits of the spectrum, respectively, and \u000f\nis a small cut-off parameter introduced to avoid numerical instabilities.\nWith this later conditions fulfilled, the Chebyshev polynomial expansion of the density operator considering N\npolynomials can be written as:5\nFigure SIII: Zigzag Nano-ribbons spectra for phases (a) B1with\u0015I= 0:2Vpp\u001bandVAB= 0, (b) A1 with \u0015I= 0:2Vpp\u001band\nVAB= 0:8Vpp\u001b, and (c) B2 with \u0015I= 1:1Vpp\u001bandVAB= 0:8Vpp\u001b.\n\u001a\u0010\n~E\u0011\n=1\n\u0019p\n1\u0000~E2N\u00001X\nm=0gm\u0016mTm\u0010\n~E\u0011\n; (S12)\nwheregmis a kernel introduced to control the Gibbs oscillations produced by the sudden truncation of the series\nexpansion[28, 29]. The coefficients are calculated with \u0016m=hTrTm\u0010\n~H\u0011\ni, in whichh:::irepresents the average\nover different disorder configurations. The calculation of the density operator of a given system is reduced to the\ncomputation of the trace of a matrix. To further decrease the computational cost of the calculation of quantities such\nas the density operator, instead of calculating the full trace of the polynomial matrices[29], we simply approximate\nthe expansion coefficient \u0016mas\n\u0016m\u00191\nRhRX\nr=1h\u001erjTm\u0010~~H\u0011\nj\u001erii (S13)\nIn the laterj\u001erirepresent a set of random vectors which are defined as j\u001eri=D\u00001=2PD\ni=1ei\u001eijii. Herefjiigi=1;:::;D\ndenotes the original basis set, in which orbitals and spins on the lattice sites are treated equivalently, Drepresents\nthe dimension of the Hamiltonian matrix, and \u001eiis the phase of each of the state vectors that comprise each of the\nrandom vectors. Ris the number of random vectors used in the trace estimation and the convergence of the later\ngoes as 1=p\nDR.\nCHEBYSHEV POLYNOMIAL EXPANSION OF KUBO FORMULA\nTo compute the spin and orbital conductivities of disordered systems, we employed the efficient algorithm developed\nby J. García et. al.[36, 37], which is based in the Chebyshev expansion of the Kubo-Bastin formula[38]:\n\u001b\u000b\f(\u0016;T) =i~\n\nZ+1\n\u00001dEf(E;\u0016;T)\n\u0002Trhj\u000b\u000e(E\u0000H)j\fdG+\ndE\u0000j\u000bdG\u0000\ndEj\f\u000e(E\u0000H)i; (S14)6\nin which \nrepresents the area of the 2Dsample,f(E;\u0016;T)is the Fermi-Dirac distribution for the energy E,\nchemical potential \u0016and temperature T.G+(G\u0000)symbolise the advanced(retarded) one electron Green function.\nAs it can be seen from (S14) the Kubo-Bastin formula is expressed as a current-current correlation function. Then,\nto adapt this formula to calculate the spin hall conductivity \u001bz\nSH, we define j\u000bas the current-density operator like\nj\u000b\u0011jx=ie\n~[x;H]andj\fas the spin current-density as j\f\u0011js\ny=1\n2f\u001bz;vygwhere\u001bzis the usual Pauli’s matrix and\nvyis they-Component of the velocity operator. For the computation of the orbital Hall conductivity, again \u001bz\nOHwe\ndefine the current operator j\u000basj\u000b\u0011jx=ie\n~[x;H]and we write j\fas the orbital current density operator, which\nis defined like j\f\u0011js\ny=1\n2f`z;vygwhere`zis thez. It is noteworthy to mention that for the spin hall conductivity\ncalculations we used the open-source code from the KITE project[39].\nNUMERICAL SIMULATION OF THE DISORDERED CASE\nIt is instructive to investigate how disorder affects the OHE in these two-dimensional systems and more specifically,\nhow it modifies the plateaux in the orbital Hall conductivity that, as discussed before, is not dominated by conducting\nedge states. For this purpose we include in our Hamiltonian an on-site Anderson disorder term \u000fiwhose values are\nrandomly picked from an uniform distribution that goes from\u0002\n\u0000W\n2;W\n2\u0003\n, in whichWrepresents the Anderson disorder\nstrength and then proceeded with the aforementioned Chebyshev polynomial expansions to compute the density of\nstates (DOS), and the transverse components of the spin and orbital conductivity tensors. In these calculations we\nhave considered systems of 8\u0002256\u0002256orbitals, Chebyshev polynomials up to the order M= 1280and we averaged\noverR= 150random vectors. It is noteworthy to mention that due the large number lattice sites that we are\nconsidering, we restricted ourselves to only one disorder realization, this is based on the assumption that almost every\npossible configuration is contained on our system due it large size.7\nFigure SIV: Spin (red) and orbital (blue) Hall conductivities calculated as functions of energy for: (a) \u0015I= 0:2Vpp\u001bVAB= 0,\n(b)\u0015I= 0:2Vpp\u001bVAB= 0:8Vpp\u001b, and (c)\u0015I= 1:1Vpp\u001bVAB= 0:8Vpp\u001bin the presence of disorder. The left, central and right\npanels show the results obtained in the relatively weak ( W= 0:05Vpp\u001b), intermediate ( W= 0:2Vpp\u001b) and strong ( W= 0:4Vpp\u001b)\ndisorder regimes, respectively. The grey lines represent the density of states calculated for the same set of parameters.\nFigure SIV shows the spin and orbital Hall conductivities calculated for both weak ( W= 0:05Vpp\u001b), intermediate\n(W= 0:2Vpp\u001b) and strong ( W= 0:4Vpp\u001b) disorder. Similarly to what was previously observed for the SHE [34], the\norbital Hall plateau remains present, even for a relatively strong disorder that closes the lateral gaps. Our preliminary\nresults indicate that the orbital Hall effect in two-dimensional insulators is robust against Anderson disorder.\nORBITAL TEXTURE ANALYSIS\nIn contrast with the SHE, our calculations show that the OHE is not quantised, and occurs even in the absence of\nmetallic edge states. In order to explore the origin of the OHE in this model system, we investigated the characteristics\nofitsorbitalangularmomentumin reciprocalspace withinthe2DfirstBZ.Tothisend, wecomputetheorbitaltexture\nin reciprocal space defined as,\n~Ls\nn;~k=\n`x\u000bs\nn;~k^x+\n`y\u000bs\nn;~k^y+\n`z\u000bs\nn;~k^z; (S15)8\nWhere,\n`x;y;z\u000bs\nn;~k=\n s\nn;~k\f\f`x;y;z\f\f s\nn;~k\u000b\nis the expected value of angular-momentum operator in reciprocal space for\nstates of Bloch band nand spin sector s. To study the orbital texture and how it affects the OHE, we separate the\nin-plane textures (\n`x;y\u000bs\nn;~k), which are represented by arrows, see Fig. SV and panels of Fig.2 of main text, and\nout-of-plane textures (\n`z\u000bs\nn;~k), which we represent as a color plot (dark blue color for\n`z\u000b\n\u00191and dark red color for\n`z\u000b\n\u0019\u00001). Following the semi-classical argument of Ref. [40], it is possible to show that \u001bz\nOHis a consequence of the\nexistence of non-trivial in-plane orbital texture (\n`x;y\u000bs\nn;~k). Some features of the function \u001bz\nOH(Ef)can be understood\nfrom these textures, as we briefly mentioned in the main text, and now we detailed here.\nFigure SV: Orbital character of the \"-spin eigenstates of H[Eq. S1 or Eq. (1) of the main text] calculated for: (a) \u0015I= 0:2Vpp\u001b,\nandVAB= 0; (b)\u0015I= 0:2Vpp\u001b, andVAB= 0:8Vpp\u001b; (c)\u0015I= 1:1Vpp\u001b, andVAB= 0:8Vpp\u001b.\nFigure 2 of the main text displays both the in-plane and the out-of-plane orbital polarisations of the lowest \"-spin\nenergy band for the B1, A1 and B2 phases. Results for the #-spin bands can be easily obtained by time-reversal\nsymmetry operation. In Figure SV we complement our analysis by showing the orbital textures of the four \"-spin\nenergy bands for each one of the three phases. The orbital projections depicted in panel (a) were calculated for\n\u0015I= 0:2Vpp\u001bandVAB= 0, and correspond to the case in which the system assumes the B1 phase. Clearly, the\nin-plane orbital textures of the first and second energy bands are opposite to each other, and the same happens to the\nthird and fourth bands, which leads \u001bz\nOH(Ef)to be an odd function of Fermi energy, and consequently, the absence\nof OHE in the central gap. As it was shown in Fig. 3 of the main text, if we include second neighbors hopping\nin the tight-binding Hamiltonian, the particle-hole symmetry around the central gap is broken, and the cancelation\nof in-plane orbital texture is lost, leading to the appearance of a central plateau in the orbital Hall conductivity.\nIt is also noteworthy that h`zi\"\nn;~kfor the second and third bands are opposite, as well as around the K(K0) and \u0000\nsymmetry points. Conversely, the first and fourth bands respectively exhibit h`zi\"\nn;~k\u0019\u00071in the vicinities of the \u00009\npoint, but virtually vanishing values around KandK0. Panel (b) displays the orbital projections of the eigenstates\ncorresponding to the A1 phase, calculated for \u0015I= 0:2Vpp\u001bandVAB= 0:8Vpp\u001b. One of the main eye-catching\ncharacteristics of this phase is the opposed out-of-plane orbital polarisations around the K0andKpoints, which is\na manifestation of the orbital-valley locking produced by VAB. Similarly to phase B1, the out-of-plane polarisations\nof the first and second \"-spin energy bands are opposed to the fourth and third ones, respectively. In addition, the\nin-plane orbital angular momentum polarisations for this phase exhibit the same configuration as those obtained for\nthe B1 phase,which means that, also in this phase, sigma is an odd function of Fermi energy, with no central plateau.\nHowever, due to the orbital-valley locking, the corresponding absolute values are smaller, which explains the different\ncurve derivative of the OHE in the phase A1 when compared with the OHE of the phase B1. Finally, panel (c) shows\nthe orbital character of the system, calculated for \u0015I= 1:1Vpp\u001bandVAB= 0:8Vpp\u001b, when it is in the B2 phase.\nIn this case we find that h`zi\"\nn;~k\u0019\u00001for the lowest energy band, which goes along with a substantial reduction of\nthe in-plane texture. Similarly to the previous cases, h`zi\"\nn;~kfor the lowest and highest energy bands are inverted.\nHowever, there a noticeable change in h`zi\"\nn;~kin comparison with the results obtained for the A1 phase, which is\naccompanied by a relatively strong orbital-valley locking produced by the combined action of the large values of \u0015I\nandVAB.\nLOW-ENERGY APPROXIMATION\nAs discussed in the main text, our effective Hamiltonian in the vicinity of the K=K0point can be expressed in\nterms ofSU(2)\nSU(2)orbital and sub-lattice algebras. Expanding the matrix of Eq. S1 near valleys K= 4\u0019=3a\nandK0=\u00004\u0019=3a, we obtain, up to first order in electronic momentum, the effective theory\nHeff=\u0000~vF(kx\u001bx+\u001cky\u001by) +s\u0015I`z+VAB\u001bz+H`: (S16)\nHere,vF=ap\n3\n2~Vpp\u001brepresents the Fermi velocity, and ais the lattice constant; \u001c=\u00061for theKandK0valleys,\nrespectively, and s=\u00061for\"and#spin electrons, respectively. The last term H`breaks the degeneracy between `z\neigenstates and can be separated in two contributions:\nH`=H`k+HD;whereH`k=\u0000~vF\n4\u001c(k+`+\u001b\u001c+k\u0000`\u0000\u001b\u0016\u001c)andHD=\u0000p\n3~vF\n2a(`x\u001bx+\u001c`y\u001by):(S17)\n\u001b\u001c=\u001bx+i\u001c\u001by,\u0016\u001c=\u0000\u001c,`\u000b(\u000b=x;y) represent the orbital angular momentum matrices in the corresponding Hilbert\nspace,k\u0006=kx\u0006iky, and`\u0006=`x\u0006i`y.\nFigure SVI shows a comparison between the energy band spectra obtained by our tight-binding (blue dashed lines)\nand effective models (red solid lines) calculations in the vicinities of KandK0. In the left column we notice for the\nthree phases that our effective linear model describes rather well the two inner energy bands, but fails to properly\ndo so for the two outer ones. This can be corrected with the inclusion of quadratic terms in our approximation, as\nillustrated in the right column of Figure SVI. It is noteworthy that the orbital texture near KandK0are very well\ndescribed by our effective model. Nevertheless, to reproduce the orbital texture in the vicinity of \u0000, it is necessary to\nperform an even higher-order expansion up to 4th order.\nTo provide insight on how H`affects the energy spectrum and orbital textures of this model, we examine the\ncorresponding contributions of each term in Eq. S17. For simplicity, we consider only one spin sector. In this case,\nthe energy spectrum of H0=\u0000~vF(kx\u001bx+\u001cky\u001by)consists of two degenerate Dirac cones that are associated with\nthe two eigenstates of the angular momentum pseudo-spinor. Similarly to what occurs in graphene, the inclusion of\nHAB=VAB\u001bzopens an energy gap in the spectrum, while HSOC=s\u0015I`zacts as an orbital exchange interaction,\nshifting upwards (downwards) the Dirac cone associated with the `zeigenvalue +1(-1). To understand how H`\nmodifies the energy spectrum, we introduce a multiplicative factor that regulates its overall intensity and inspect the\nenergy band structure of H0+\u0011H`for two different values of \u0011in the following situations: (i) H`K6= 0;HD= 0, (ii)\nH`K= 0;HD6= 0, and (iii)H`K6= 0;HD6= 0. The results for the energy bands calculated as functions of kxfor\nky= 0are exhibited in Figure SVII. In panels (a) and (d) we note that H`Klifts the orbital degeneracy of the two\nDirac cones for kx6= 0, by differently renormalising their corresponding Fermi velocities. Panels (b) and (d) show\nhowHDaffects the energy bands. HDdoes not depend upon the wave vector ~k, and has the same functional form10\nFigure SVI: Comparison between the tight-binding energy band calculations (blue dashed lines) with the eigenvalues of our\neffective Hamiltonians in the vicinities of KandK0(red solid lines). The eigenvalues obtained with the linear and quadratic\norder expansions are depicted in the left and right panels, respectively. The results are for: (a) \u0015I= 0:2Vpp\u001bVAB= 0, (b)\n\u0015I= 0:2Vpp\u001bVAB= 0:8Vpp\u001b, and (c)\u0015I= 1:1Vpp\u001bVAB= 0:8Vpp\u001b.\nof a Dresselhaus SOC for Dirac Fermions. It may be regarded as equivalent to a Dresselhaus SOC for orbital states.\nAs expected, HDleads to a Dresselhaus-like band splitting, without opening a gap at E= 0. In panels (c) and (f)\nof Figure SVII, we clearly see the formation of a single Dirac cone and the two outer bands when both H`Kand\nHDare present. It is worth recalling that to reproduce the flat-bands, it is necessary to consider high-order terms in\nk. Similarly to what is observed in quantum anomalous Hall insulators, the gap opening at E= 0is a consequence\nof the interplay between the orbital equivalent of a SOC and an exchange interaction. There is, however, a rich\nphenomenology involving the contributions of the distinct terms in Eq. S16 that arises when \u0011is varied, but this goes\nbeyond the scope of the present discussion.\nFinally, we examine the role of H`andHDin the orbital texture of this model system. Figure SVIII shows the\norbital textures calculated for: (a) HD6= 0andH`K= 0; (b) forHD= 0andH`K6=, and (c) for the effective complete\nHamiltonian without SOC and VAB. By comparing the three panels, it is clear that the orbital texture of our effective\nmodel is basically governed by the Dresselhaus-like coupling associated with the orbital angular momentum spinor,\nwhich reproduces rather well the in-plane texture of our tight-binding calculations near K.11\nFigure SVII: Energy bands calculated as functions of kx(forky= 0) by means of our effective theory around the Ksymmetry\npoint. Panels (a), (b) and (c) depict the results obtained for \u0011= 0:3in the cases: H`K6= 0;HD= 0;H`K= 0;HD6= 0and\nH`K6= 0;HD6= 0, respectively. Panels (d), (e), and (f) show the results calculated for the same cases, but with \u0011= 1:0.\nFigure SVIII: Comparison between of the in-plane texture profile for: (a) HD6= 0andH`K= 0; (b)HD= 0andH`K6= 0,\nand (c)HD6= 0andH`K6= 0.\nSECOND NEAREST NEIGHBOURS AND ORBITAL TEXTURE ANALYSIS\nAs mentioned in the main text, the absence of OHE plateau in the central electronic spectrum gap of px-py-model of\nEq. (1) is a consequence of the combination of the particle-hole and parity symmetries of spectrum which translates in\ncancellation of in-plane orbital texture at half-filling. To understand better the consequences of the breaking of these\nsymmetries, we introduce a toy-model of the px-pyHamiltonian, where we have included second nearest-neighbours\nhopping. This model is described by,\nH=X\nhijiX\n\u0016\u0017st\u0016\u0017\nijpy\ni\u0016spj\u0017s+X\nhhijiiX\n\u0016\u0017st\u0016\u0017\nijpy\ni\u0016spj\u0017s+X\ni\u0016s\u000fipy\ni\u0016spi\u0016s+X\ni\u0016shz\n\u0016spy\ni\u0016spi\u0016s; (S18)\nhere as before, iandjrepresents the honeycomb lattice sites whose position is given by ~Riand~Rj, respectively. The12\nsymbolshijiandhhijiiindicates that the summations are restricted to the nearest and second nearest neighbour\nsites respectively. The operator py\ni\u0016screates an electron of spin sin the atomic orbitals p\u0016=p\u0006=1p\n2(px\u0006ipy)\ncentred at ~Ri. Here,s=\";#labels the two electronic spin states, and, now, \u000fiis the atomic energy at site i, which\nencodes the effect of the combination of a sublattice potential VAB, and the on-site energy of porbitals\"p. This terms\ntake values \u000fi=\"p\u0006VAB, when site i belongs to the A and B sub-lattices of the honeycomb arrangement, respectively.\nFigure SIX: Comparison between the orbital (spin- \") texture profiles of the px-pywith only nearest neighbours (a), and the\norbital texture of the same model when second nearest neighbours are considered (b). Left: Orbital Texture profile for the\ndeepest energy band. Center: Orbital Texture profile for the second lowest energy band. Right: addition of the orbital textures\nin Left and Right panels with the in-plane component scaled by a factor 5.\nAs it was shown in figure 3 of the main text, the principal effect of the particle-hole and parity symmetries breaking,\na consequence of the inclusion of the second nearest neighbours, is the appearance of an orbital Hall conductivity\nplateau in the central gap. In order to uncover the connection between the appearance of this plateau and the orbital\ntextures, we analyse the texture profiles of the two deepest energy bands for two different cases of this model. In\npanel (a) of the figure SIX are shown the orbital textures of the two deepest energy bands(left and central panels) of\nthe simple model that does not include second nearest neighbours [Eq. (1) in the main text] and their summation\n(right panel) in which the in-plane components of the texture are in a larger scale to make easier the analysis of\ntheir details. The in-plane component of orbital textures in left and central panels present the dipole configuration\naround the \u0000point and the anti-vortices in the KandK0points, and the out-of-plane component appears due to the\ninversion symmetry breaking produced by the inter-lattice potential. At the right panel of the figure (a), we show that\nthe addition of the orbital textures of the left and central panels results in a zero net in-plane orbital texture. Once\nthat orbital Hall conductivity ( \u001bz\nOH) appears as a consequence of dynamics of in-plane orbital texture, in presence of\nan external electric field, this explains the absence of OHE in the central gap of the simplified px-pymodel of Eq. (1)\nin the main text. Now, in panel (b) of the figure SIX, we consider the orbital texture of the Hamiltonian with the\ninclusion of second nearest-neighbours hopping (see. Eq. S18). Again, the left and the central panels of the figure\nshow the orbital textures of two deepest bands and the right panel shows the sum of these two textures, with the\nin-plane component multiplied by a scaling factor to facilitate its visualization. To maintain the resemblance between13\nthe aforementioned case and this new case, we set the same Slater-Koster parameters that we used for the phase A1\nof the simplified model with the addition of \"p=\u00000:3andVpp\u001b2=\u00000:2. With these parameters, as it was shown\nin figure 3 (a) of the main text the energy bands of this modified model are not particle-hole symmetric, an effect\ncaused exclusively by the inclusion of second nearest-neighbours. We note in Fig. SIX (b) that the overall features of\nin-plane orbital-texture of two deepest bands (left and central panels) are not qualitatively modified, i.e., still present\na dipole-like texture near \u0000-point and anti-vortices textures at valleys. However, as can be seen in Fig. SIX (b), right\npanel, the exact cancellation of the in-plane texture of two deepest bands is lost, causing the existence of a net in-\nplane orbital texture which produces an OHE in the central gap of the spectrum shown in figure 3 (b) of the main text.\nOnce shown by means of the simply px\u0000pymodel that the orbital Hall effect is present in systems where the\nparticle-hole and parity symmetries are absent, we now focus on a real material. For this purpose, we have chosen\nthe flat bismuthene grown on SiC as a candidate for the observation of OHE in the central plateau. The observation\nof orbital-insulator phase in the central gap of bismuthene should be easier in the experimental point of view, once it\ncorresponds to neutrality situation. In the past, this system has been studied by means of the aforementioned minimal\npx-pytight-binding Hamiltonian [14, 22, 41]. However, we have noticed that by including second nearest-neighbours\nin the tight-binding Hamiltonian used in Ref. 22 the electronic structure is better reproduced. In the bismuthene/SiC\nheterostructure, the break of inversion symmetry induces a small Rashba SOC,\nHR= 2i\u0015RX\nhi;jiX\n\u0016\u0017spy\ni\u0016\u0016s[^z\u0001(~ \u001b\u0002^eij)]\u0016sspj\u0017s+H:c: (S19)\nwhere~ \u001bsymbolises the Pauli vector, ^eijdenotes the unit vector along the n.n. inter-site direction of ~Rj\u0000~Ri,\u0015Ris the\nRashba SOC constant, and \u0016sdesignates the opposite spin direction specified by s. We will consider this term only in\nthe fitting of tight-binding Hamiltonian to DFT spectrum and, we neglect it in transport calculations presented here.\nThe reasons is that the non-conserving spin character of this coupling complicates the analysis of orbital texture and\nits typical small value does not alter the main conclusions of our discussion, as it was checked by us. In figure SX is\nshown a direct comparison of the DFT energy band structure obtained in Ref. 14, with and without the inclusion of\nRashba spin-orbit coupling. From this figure, it is rapidly noticeable the agreement between the DFT energy bands\nand the tight-binding model in describing the top of the valence band and the bottom of the conduction bands and\nthe indirect gap in the \u0000point. In the table SI are shown the two-centre integral parameters used in the description\nof this model. Once shown the agreement of energy band of the complete model, we are going to restrict ourselves to\nthe situation in which the Rashba SOC is neglected and the system is subject to a staggered potential VAB= 0:87.\nThe first of these constraints is to avoid complications in the analysis due to possible contributions to the orbital\ntexture by the Rashba SOC, which does not conserve spin as a good quantum number, and the second one is to leave\nthe system in a topological phase similar to the phase A1of thepx\u0000pymodel with only nearest neighbours. This\nwill allow us to focus on the analysis of the Orbital texture and its connection with the orbital Hall conductivity.\nTable SI: Second Nearest-neighbour two-centre energy integrals, and spin orbit coupling constants (all in eV) for the Bi/SiC.\nTwo-centre integrals Intrinsic SOC Rashba SOC On-site energy\nVpp\u001b= +1:51522\u0015I=\u00000:435\u0015R= 0:032\u000fp=\u00000:279865\nVpp\u0019=\u00000:575788\nVpp\u001b2=\u00000:18\nVpp\u00192=\u00000:00658\nIn figure SXI is displayed the orbital textures of the two deepest energy bands of bismuthene grown over SiC in\nthe phaseA1and without Rashba SOC. The principal difference that is noticeable when one looks at this figure is\nthe change of the out-of-plane orbital texture of the spin- \"sector, with respect to textures of A1-phase in previous\ncases, produced by the change of sign of the spin-orbit coupling. The in-plane components of orbital texture for the\nmodel with parameters of bismuthene shown in Fig. SXI left and centre panels, do not present noticeable differences\nfrom those of Fig. SIX (b) of the model with second nearest neighbours. Again, as can be seen from the right panel\nof Fig. SXI, there is a non-zero total in-plane orbital texture when we add up the textures of two deepest bands\nof left and central panels what again explain the existence of OHE in the central gap, as shown in Figure 4 of the\nmain text. This suggests the recent synthesized flat bismuthene as a realistic platform to observe the orbital Hall\ninsulator phase. The central plateau will persist by the inclusion of the Rashba term [Eq. (S19)] on Hamiltonian,14\nFigure SX: Comparison between the DFT energy bands (blue doted line) and the tight-binding model with second nearest\nneighbours bands (red solid line) for: (a) \u0015R= 0:032eV and (b) \u0015R= 0.\nonce the spectrum keeps the particle-hole asymmetry. When the Rashba term is included, we cannot separate the\ntextures by spin sectors because it breaks the sz-symmetry. So the previous analysis of sum of orbital texture must\nbe done summing the four lowest energy bands to obtain total texture related to the central plateau. But the main\nconclusions are the same and we do not present this analysis here.\nFigure SXI: Orbital Textures profile of spin- \"sector for the two lowest energy bands of bismuthene over SiC in which the\nRashba SOC is neglected and VAB= 0:87eV. Left: Orbital texture profile of the lowest energy band. Center: Orbital texture\nprofile of the second lowest energy band. Right: Addition of the later texture profiles. The resultant in-plane orbital texture\nare scaled by a factor 5to facilitate the visualization"    }]